Hydrated Electron Dynamics Explored with 5-fs …Hydrated Electron Dynamics Explored with 5-fs...

RIJKSUNIVERSITEIT GRONINGEN

Hydrated Electron DynamicsExplored with

5-fs Optical Pulses

PROEFSCHRIFT

ter verkrijging van het doctoraat in de

Wiskunde en Natuurwetenschappen

aan de Rijksuniversiteit Groningen

op gezag van de

Rector Magnificus, dr. D.F.J. Bosscher,

in het openbaar te verdedigen op

maandag 13 maart 2000

om 16.00 uur

door

Andrius Baltuška

geboren op 26 november 1971

te Leningrad (Sovjetunie)

Promotor: Prof. dr. D.A. Wiersma

Referent: Dr. M.S. Pshenichnikov

Beoordelingscommissie:

Prof. dr. K. Duppen

Prof. dr. J. Knoester

Prof. habil. dr. A.P. Piskarskas

ISBN 90-367-1209-2

Contents

Foreword

Chapter 1

General Introduction...................................................................................................................11.1 Why use femtosecond spectroscopy in condensed phase? ..............................................21.2 The hydrated electron .....................................................................................................31.3 Basic principles of ultrashort pulse generation...............................................................81.4 Spectroscopic utility of existing 5-fs laser systems ......................................................131.5 Few-cycle pulse characterization .................................................................................151.6 Techniques of nonlinear spectroscopy..........................................................................171.7 Scope of this Thesis......................................................................................................20References................................................................................................................................22

Chapter 2

All-Solid-State Cavity-Dumped Sub-5-Fs Laser......................................................................252.1 Introduction...................................................................................................................262.2 Cavity-dumped Ti:sapphire laser .................................................................................282.3 White-light continuum generation..................................................................................302.4 Measurement of spectral phase.....................................................................................322.5 Temporal analysis of the white light pulse....................................................................362.6 Fiber output: experiment vs. numerical simulations ......................................................382.7 Compressor design .......................................................................................................392.8 Pulse duration measurement..........................................................................................432.9 Reconstruction of 5-fs pulse from the IAC and spectrum..............................................452.10 Pitfalls of IAC...............................................................................................................472.11 Summary and outlook....................................................................................................49References................................................................................................................................51

Chapter 3

SHG FROG in the Single-Cycle Regime ..................................................................................553.1 Introduction...................................................................................................................563.2 Amplitude and phase characterization of the pulse .......................................................593.3 Propagation and focusing of single-cycle pulses...........................................................603.4 The SHG FROG signal in the single-cycle regime........................................................623.5 Ultimate temporal resolution of the SHG FROG...........................................................683.6 Approximate expression for the SHG FROG signal .....................................................693.7 Numerical simulations ..................................................................................................713.8 Type II phase matching .................................................................................................773.9 Spatial filtering of the second-harmonic beam..............................................................803.10 Conclusions ..................................................................................................................82References................................................................................................................................84

Chapter 4

FROG Characterization of Fiber-Compressed Pulses ..............................................................874.1 Introduction...................................................................................................................884.2 The choice of the SHG crystal ......................................................................................894.3 Case study: Two contradicting recipes for an optimal crystal ......................................924.4 SHG FROG apparatus ..................................................................................................964.5 SHG FROG of white-light continuum...........................................................................974.6 SHG FROG of compressed pulses..............................................................................1014.7 Conclusions and Outlook............................................................................................105Appendix I: Wigner representation and Wigner trace error...................................................106References..............................................................................................................................109

Chapter 5

Four-Wave Mixing with Broadband Laser Pulses..................................................................1115.1 Introduction.................................................................................................................1125.2 The formalism for ultrafast spectroscopy with 5-fs pulses .........................................1135.3 Case study: Blue pulse characterization by third-order FROG...................................1195.4 Ultimate temporal resolution of SD and TG experiments............................................1215.5 Heterodyned detection and frequency-resolved pump–probe .....................................1235.6 Conclusions ................................................................................................................124References..............................................................................................................................126

Chapter 6

Early-Time Dynamics of the Photo-Excited Hydrated Electron..............................................1276.1 Introduction.................................................................................................................1286.2 Experimental...............................................................................................................130

6.2.1 Femtosecond laser system.....................................................................................1306.2.2 Transient grating and photon echo experiments ..................................................1326.2.3 Generation of hydrated electrons .........................................................................133

6.3 Results and Discussion...............................................................................................1366.3.1 Intensity-dependence measurements ....................................................................1366.3.2 Pure dephasing time of hydrated electrons ..........................................................1376.3.3 Transient grating spectroscopy ............................................................................1416.3.4 Early-time dynamics: the microscopic picture.....................................................1466.3.5 Theoretical model .................................................................................................148

6.4 Conclusions ................................................................................................................153References..............................................................................................................................155

Chapter 7

Ground State Recovery of the Photo-Excited Hydrated Electron............................................1587.1 Introduction.................................................................................................................1597.2 Short-lived vs. long-lived p-state: Manifestation in pump–probe...............................1637.3 Experimental...............................................................................................................1717.4 Results and discussion................................................................................................173

7.4.1 The measured traces .............................................................................................1737.4.2 The fit of transient spectra....................................................................................1787.4.3 The inversion of temporal data to potential surfaces at long times ....................1827.4.4 The multidimensional relaxation model ...............................................................186

7.5 Conclusions ................................................................................................................189Appendix I: Modulation of pump–probe spectra...................................................................191References..............................................................................................................................194

Samenvatting.........................................................................................................................196

List of Publications...............................................................................................................199

Chapter 1

General Introduction

Abstract

In this Chapter we introduce the reader to the technique and advantages of employing

ultrashort laser pulses for time-resolved spectroscopy in the condensed phase. We will then

turn your attention to the hydrated electron, – the main goal of this study. The hydrated

electron is one of the simplest conceivable physical systems, yet the grasp of its dynamics at

a fundamental level is extremely important. It is unique in the sense that it provides an

opportunity to confront results of state-of-the-art nonlinear optical experiments with quantum

molecular dynamics simulations. In order to obtain a comprehensive insight into the

dynamics of the hydrated electron, unprecedented time resolution is required. Consequently,

a laser delivering extremely short pulses should be designed. We outline two principle ways

of few-cycle-pulse generation: first, directly from a laser oscillator and, second, through

external-to-the-cavity spectral broadening and subsequent pulse recompression. Further, we

summarize past achievements in this field, current status, and future perspectives of ultrashort

pulse generation in order to place this work in perspective with current technology. Also, the

issue of femtosecond pulse characterization, the prerequisite for any practical use of the few-

cycle optical waves, is addressed. Finally, we provide a concise overview of spectroscopic

techniques used in the experiments on the hydrated electron and we define the scope and

outline of the Thesis.

Chapter 1

2

1.1 Why use femtosecond spectroscopy in condensed phase?

The femtoseconds (1 fs = 10-15 s) is the fundamental time scale on which many molecular

processes occur [1]. The use of laser pulses that contain only a few oscillations of the

electromagnetic field allows us to capture a “snapshot” of the spectral dynamics, as the nuclei

remain “frozen” at a given internuclear separation for the duration of the pulse [2]. In other

words, time-domain spectroscopic techniques open the possibility of creating a time-window

through which molecular motions can be explored [3]. Impressive rapid development of

femtosecond pulse generation [4,5] has provided experimentalists with state-of-the-art tools

for time-domain nonlinear optical spectroscopy [3]. In fact, many recent breakthroughs in

photochemistry, photobiology, and physics [6-8] were made possible due to the ability of

researchers to time-resolve the primary processes by using ultrashort laser pulses [9,10].

Recently, Prof. Ahmed H. Zewail, (Linus Pauling professor of Chemical Physics, California

Institute of Technology) was awarded the 1999 Nobel prize in Chemistry for his work in

studying chemical processes on the femtosecond time-scale, thus establishing the science of

femtochemistry.

Femtosecond spectroscopy provides the unique option to study ultrafast chemical

processes in the condensed phase. Indeed, rapid molecular events such as bond dissociation

[2,11] or bond twisting [12] can be observed “live” only when resolved in time. Much like

conventional stroboscopic photography [13], which is widely used to capture moving image

on a millisecond time scale, the use of a “femtosecond stroboscope” enables us to take

glimpses of nuclear motions [14], bond-twisting [12], molecular dissociation/recombination

[2,11]. In many cases, such as the study of ultrafast liquid phase dynamics, the time mapping

with femtosecond pulses of the spectro-temporal dynamics [15] is the simplest and most

informative experimental route. Usually, in such systems optical spectra of the solute consist

of a number of individually unresolved lines that are tremendously broadened due to the

strong coupling with the solvent. Consecutive femtosecond-resolution snapshots of electronic

relaxation and dephasing processes in the system [3] frequently allow unraveling of the

information encrypted in the absorption spectrum. However, to attain an adequate temporal

resolution on the femtosecond time-scale is only possible by employing ultrashort laser

pulses.

In this Thesis the methods of femtosecond spectroscopy are applied to study the process

of energy relaxation in photo-excited hydrated electrons, – a ubiquitous species in irradiated

aqueous systems. In order to outline the scope of this research, the reader is first introduced to

the paradigm of the hydrated electron. The following Section (1.2) describes the hydrated

electron as an experimental and theoretical test ground that covers a broad variety of

problems in physics ranging from the behavior of hot electrons in semiconductors to the

mechanisms of chemical reactions in the liquid phase.


3

1.2 The hydrated electron

Since the first observation of solvated electrons in liquid ammonia in 1864 [16], the study of

excess electrons in liquids has been an area of vast interest for both chemists and physicists

[17,18]. The existence of such electrons in aqueous solutions, known as hydrated electrons,

was first postulated in 1952 independently by Stein [19] and Platzman [20] as a necessary

species to explain the details of some chemical reactions in the liquid phase. After a decade

of accumulating indirect evidence, the hydrated electron was finally discovered in 1962 by

Boag and Hart [21,22]. For the first time, scientists were able to measure its visible–near

infrared (IR) absorption spectrum in a pulse radiolysis experiment on water.

Excess electrons in condensed-phase media play a crucial role in the dynamics of

important chemical processes. Among those are solution photochemistry, non-radiative

electronic transitions and charge transfer reactions. Unlike free electrons that are delocalized,

electrons in polar solvents become self-trapped because of their interactions with the solvent

environment. Owing to the strong solute–solvent coupling, the evolution of the electronic

structure is completely determined by the rearrangement of the solvent molecules.

The study of hydrated electrons is particularly interesting from the point of view of the

solvent involved. Of all solvents in chemistry, water is undoubtedly the most important one,

owing to its outstanding role in nature. Because of its large dipole moment and strong

hydrogen bonding, water crucially influences the outcome of many chemical reactions. For a

number of chemical transformations in aqueous systems, the fluctuations of water molecules

couple to the reaction coordinates and determine free energies of a reaction, thus ultimately

controlling the reaction dynamics [23]. Elucidation of the nature of the coupling of these

fluctuations to the electronic states of solutes is all-important for creating a complete picture

of aqueous chemical reactivity.

Solvent and solvation dynamics in water has been the subject of many theoretical and

experimental studies [24,25], and the investigation of the structural and dynamical properties

of water is a long-standing tradition in science. The detailed understanding of solute–solvent

interactions also has a number of direct practical implications, one of which is the dynamics

of chemical reactions. This process is critically affected by the motions of surrounding

solvent molecules, which are coupled to the reactant energy levels [26]. Because all chemical

reactions involve the rearrangement of electrons, the time-scale over which the solvent acts to

stabilize the new charge distribution of the reacting species can determine how rapidly, if at

all, a particular reaction can cross into its transition state. Consequently, obtaining a better

grasp of the solvent and solvation dynamics has been high on the physical-chemistry agenda

for a long time [3]. In the past decade molecular dynamics simulation studies and ultrafast

experiments on dye solutions, have unearthed the basic picture of the solvation process as

well as the relevant time scales [15,26-30]. For instance, molecular dynamics simulation

studies [31,32] and time-dependent Stokes-shift experiments on a coumarin dye in water [33]

both showed the initial solvation process to be exceptionally fast. However, because of the

lack of time resolution, the first 50 fs, during which most of these dynamics are thought to

Chapter 1

4

occur, remained unexplored [33]. Also, many important aspects of early time dynamics

remained unresolved because it was unclear how the intramolecular vibrational dynamics

could be separated from the solvation-dynamical process itself [15,30]. Recently, it was

shown that comparative studies on the same dye in different solvents could be used to

distinguish the two processes [15]. However, it is still of paramount importance to employ a

probe for solvation dynamics that had no internal degrees of freedom such that all dynamics

observed in the solvation process can be attributed to the solute–solvent coupling. Localized

electrons are ideally suited for this purpose since no internal energy redistribution, as a

consequence of solute–solvent interaction, is possible for a bare particle, such as an electron.

The energies of its bound electronic states, and the potential energy surfaces associated with

them are very sensitive to solvent configurations. Thus, the localized electron can be viewed

as an exceptional instrument for extracting information about the solvation process in a polar

liquid.

Another motivation for a detailed study of the hydrated electron is the fact that this

species is ideally suited for quantum molecular dynamics simulations in the liquid phase. The

unique possibility to directly confront the results of such computer studies with the results of

femtosecond spectroscopy allows verification of the basic a priori assumptions and

calculation methods put into the computer modeling. For instance, it is important to

understand to which extent one should employ quantum-mechanical character of electron

interactions with its nearest neighboring water molecules and when the switch to a simpler,

classical description of the molecular motion is justified. Also, because there are no internal

degrees of freedom in the electron itself, the hydrated electron is ideal for verifying the

correctness of the model potentials that describe interactions between the molecules of liquid

water.

Fig.1.1: The structure of the nearest solvation shell of hydrated electron in glassy water (adapted fromRef. [34]). Note that the six water molecules are oriented with their OH bonds towards the center ofthe electron charge distribution.


5

As outlined above, the hydrated electron is a unique probe of aqueous dynamics and an

excellent test field for computer simulations. Below we briefly describe the views, formed to

the present day, on the actual structure and dynamics of this species. This will serve as a

background to identify the issues that will be later addressed in this Thesis.

Upon laser– or radiation–induced ionization of a liquid-phase chemical species, the

excess electron, which is initially generated in the delocalized conduction band, rapidly

becomes localized, or “trapped”, in a micro-cavity existing among solvent molecules [31].

The localized electron subsequently undergoes electronic relaxation and becomes what is

known as an equilibrated hydrated electron. The structure of this species in crystalline water

was revealed in an electron-spin-echo study [34]. It was shown that each electron is

surrounded on average by six water molecules with their OH bonds directed toward the

electron (Fig.1.1). Recent numerical computation studies on the hydrated electron in liquid

water [35,36] confirm the idea that the first solvation shell is composed of approximately six

water molecules. However, the details of the exact structure are still under discussion. One

hypothesis suggests that the electron might be attached closer to one of the “dangling

protons” that are not involved in the hydrogen bonding of the molecules forming the solvent

cage [37].

Fig.1.2: Overview of the lowest electronic transition in the hydrated electron. (a) Absorptionspectrum. The smooth solid curve shows experimentally measured absorption (Ref. [42]) at roomtemperature. Squares depict the result of quantum molecular simulations (Adapted from Ref. [35]).The dashed curves correspond to the individual absorption components originating from three non-degenerate s–p transitions. (b) Electronic wavefunction plots for typical ground, s-like, state andlowest three excited, p-like, states. (Reproduced from Ref. [39].) The lateral side of each plotcorresponds to a distance of 12.3 Å.

The localization of a hydrated electron in the solvent cavity gives rise to bound

eigenstates, which are modulated by the coupling to the fluctuations of surrounding water

molecules. The high sensitivity of the electronic states of the hydrated electron to the aqueous

environment results in an intense broad electronic absorption spectrum that peaks at 720 nm

(Fig.1.2a, smooth line). The breadth of this spectrum (>350 nm) is a direct manifestation of

the strong underlying coupling with the solvent. Molecular dynamic simulation [38] have

Chapter 1

6

shown that the lowest energy eigenstate of the hydrated electron is nearly spherical and

corresponds to an s-like state. The first excited state was found to consist of three non-

degenerate p-like orbitals. The wavefunctions of the ground and excited states that were

generated from computer simulations [38] are depicted in Fig. 1.2b. The absorption spectrum,

produced in these computational studies, (Fig.1.2a, squares) consists of a superposition of

three non-degenerate s–p transitions (Fig.1.2a, dashed curves) with a small contribution of

the transitions to higher delocalized states. The fluctuation broadening by ~0.4 eV of the

individual s–p transitions accounts for roughly half of the total spectral width; the remaining

width being attributed to a splitting of these transitions by a comparable amount [39]. While

correctly predicting the width of the experimentally observed absorption band (Fig.1.2a,

smooth solid curve), these simulations, however, failed to reproduce the actual transition

frequency.

Interestingly, computer studies [35,40] revealed that upon the promotion to the p-state,

the size of the charge distribution of the electron grows nearly by a factor of two along the

axial lobes of the p-wavefunction (Fig.1.2b) but remains unchanged in the other two

directions. To accommodate this change, the surrounding water cavity takes on a peanut

shape. At the same time the energy of the unoccupied ground state is raised, while the energy

of the occupied excited state remains roughly the same. Figure 1.3 shows a typical dynamic

history of the s- and p-states of one hydrated electron, which emerged from a non-adiabatic

quantum simulation procedure [41].

Fig.1.3: Adiabatic eigenstates of the hydrated electron for a typical trajectory. Solid and dashed linesdenote the ground and first excited states, respectively. Diamonds mark occupied states. Excitationtakes place at t=0. (Reproduced from Ref. [41].)

For the depicted trajectory, at times before the excitation (t=0) the electron occupies the

lower, s-state. The electron eventually crosses back from the excited p-state (at t=200 fs for

the shown trajectory) and the equilibration of the s-state, the energy of which has been

substantially raised, takes place. The p–s transition is accompanied by a collapse of the

spatial extent of the electron, thus creating a void in the solvent. As is evident from this

simulation, the fluctuations of the surrounding solvent shell modulate the energies of the


7

electron eigenstates by nearly an eV on the time scale of tens of femtoseconds, which is a

manifestation of the strong coupling to the solvent. Although the s–p energy gap in these

simulations is more than two times larger than the actual transition frequency, this picture,

nonetheless, gives an excellent insight into the time scales and the dynamics of the energy

relaxation processes.

Pioneering femtosecond optical studies on the hydrated electron were conducted over a

decade ago by Migus et al. [23,43] when lasers became available, which generated pulses of

about 100-fs duration in the wavelength region where the equilibrated aqueous electron

absorbs (-800 nm). In these experiments, the electrons were generated by multiphoton

ionization of neat water and their transient absorption was studied using a super-continuum

probe. These measurements traced the initial process of electron localization in the solvent

cavity, which was found to occur in 110 fs [43] to 180 fs [23]. Following the localization of

the quasi-free electron on a pre-existing trap, further relaxation to a deeper well takes place in

~250 [43] to 500 fs [23].

A different approach to the femtosecond spectroscopy of hydrated electrons was

undertaken by Barbara and coworkers [44,45]. In this scheme, the electron, which already

resides in the thermalized s-state, was photo-excited to the p-state. The dynamics of the

relaxation back onto the equilibrated s-state yielded time constants of 1.1 ps and ~300 fs, the

latter being close to the time resolution of the laser spectrometer.

Both femtosecond time-resolved experiments [23,43,45,46] and numerous

computational studies [35,40,41,47-49] provide a strong stimulus for new experiments. Many

questions surrounding the hydrated electron still remain unanswered. This Thesis will address

some of the important issues. First, according to the theoretical work of Schwartz and Rossky

[50], the expected initial solvent response occurs on a time scale below 25 fs. Is this

prediction correct? Second, what kind of molecular motion, i.e. libration, free rotation,

vibration, translation or a combination of these is behind the initial ultrafast process of

excitation relaxation? Third, there is no clear answer as to how long the excited state of the

hydrated electron remains occupied. Within both the theoretical simulations and the

interpretation of the femtosecond data arise conflicting values. These figures for the lifetime

of the excited state cluster around two points, i.e. at ~200 fs [48,51] and at ~1 ps [50,52].

Finally, a self-consistent model, describing the whole process of energy relaxation and which

also can explain the experimental observations remains to be elucidated.

It is important to realize, however, that a substantially improved time resolution and an

adequately broad spectral range are required to resolve most of these issues and, in particular,

to catch a glimpse of the earliest dynamics of the solvent response to the photo-excitation of

the hydrated electron. In this Thesis we will present the results of state-of-the-art nonlinear

optical experiments on this vastly important and intriguing chemical species. The

unprecedented time-resolution of these experiments was achieved owing to the use of 5-fs

laser pulses. The ability to produce such record-short pulses, as well as to precisely control

and measure their properties owes its existence to several major breakthroughs that propelled

Chapter 1

8

the ultrafast laser technology to new heights. The following Section is devoted to the issues

surrounding generation of these pulses.

1.3 Basic principles of ultrashort pulse generation

Because of the Fourier transform relation between the time and the frequency domains, the

ultimate temporal resolution of a nonlinear spectroscopic experiment cannot be better than

the inverse bandwidth of the applied laser pulse(s). Therefore, in an attempt to sharpen the

time resolution of ultrafast spectroscopy, the laser pulse has to be supplied with an adequately

broad spectrum. Unlike the conventional incoherent light, the relative phases of different

frequency modes comprising an ultrashort pulse must be locked together, or modelocked

[4,53]. In order to produce a short intense burst of laser radiation, the individual cavity mode

frequencies must cooperate so that they are all in phase at one instance in time. The

illustration of this concept is given in Fig.1.4.

(c)

(b)

(a)

MirrorMirror

Lasermedium

Lasermedium

Fig.1.4: Principle of mode-locked laser operation. (a) A laser medium is sandwiched between twomirrors, one of them partly transmissive. (b) Different laser modes exist in a cavity under a conditionthat an integer number of half-periods of the wavelength equals the cavity length. (c) A constructivesuperposition of different modes at one point creates a high-intensity burst. (Adapted from Ref. [54].)

A straightforward way to generate wide spectra of laser radiation is to employ broad

bandwidth gain media. Among different materials that can be used for this purpose, to date,

Ti:Al2O3, (titanium doped sapphire) [55] has the widest known gain spectrum. This medium


9

has excellent optical and thermo-mechanical properties and can be optically pumped in the

green into the absorption band. The gain band of Ti:sapphire, which peaks near 800 nm,

supports hundreds of nanometers of oscillator frequencies. To generate ultrashort pulses,

however, the laser must be mode-locked. The discovery of the so-called Kerr-lens

modelocking in Ti:sapphire [56] in 1991 truly revolutionized ultrashort pulse technology.

Because the gain medium and the modelocking device are one in the same, Ti:sapphire

oscillators can be made very uncomplicated and robust. Rapid development of the lasers

based on this medium resulted in routine generation of pulses in order of 10 fs in duration

around central wavelength of 800 nm [57-59]. These new self-modelocking solid-state lasers

became the workhorse of the nonlinear optics laboratories of the nineties replacing the mode-

locked dye lasers [60,61], which dominated throughout the eighties and offered, at the time,

the best available time resolution (left broken line in Fig.1.5). In the last few years, a dramatic

decrease in the duration of the pulses, obtained directly from Ti:sapphire oscillators, has been

achieved (right part of Fig.1.5). This came from the continual improvement in mirror designs,

which were able to support ever-wider bandwidths. With now available broadband cavity

optics, present-day state-of-the-art oscillators deliver pulses shorter than 6 fs [62-64], which

is a remarkable technological achievement.

20001995199019851980197519701965

Year

10 fs

100 fs

1 ps

1 fs

10 ps

Ti:sapphire laser

dye laser

compressed

Fig.1.5: Evolution of the shortest pulse duration. (Courtesy of Günter Steinmeyer, ETH Zürich).Hollow symbols indicate pulses obtained by the technique of fiber-chirping and compression (seetext).

Another way to generate ultrashort pulses relies on the techniques of spectral

broadening outside the laser. All methods in this class are similar in spirit, since they employ

nonlinear optical frequency-mixing [65] (or wave-mixing) to generate new spectral

Chapter 1

10

components, thus producing a richer frequency content than that of the initial pulse. The

difference among external-to-the-cavity methods of spectral broadening lies in the order of

the nonlinearity (how many quanta are involved in the light–matter interaction that produces

a new photon) and in its physical origin. The efficiency of any nonlinear optical process

depends on the magnitude of nonlinear susceptibility, interaction length, and intensity of the

laser pulses. The higher the order of nonlinearity, the higher the laser intensity required. The

basic concept of spectral broadening via frequency mixing can be understood from Fig.1.6,

demonstrating the consecutive increase of the spectral width with the harmonic number. For a

Gaussian pulse and an ideal frequency conversion process, the minimal achievable duration

of the pulse supported by the spectrum of nth harmonic is proportional to n1 of the input

pulse duration.

While the bandwidth of the pulse sets the lower attainable limit on the pulse duration,

the actual pulse duration also depends on the spectral phase of the complex electric field of

the pulse. The phase determines how different frequency components of the pulse are delayed

with respect to each other. Synchronization of all these spectral modes, referred to as pulse

compression [1,4], is as vital in obtaining ultrashort durations as is the generation of large

bandwidth.

nω0

3ω02ω

0

I(ω

)/I m

ax

Frequency ω

___

√n

∆τ∆τ

ω0

I(t)

/Im

ax

Time t

Fig.1.6: Bandwidth growth and reduction of corresponding minimal pulse duration in the process ofharmonic generation for an Gaussian input pulse and ideal frequency conversion conditions. The toppanel depicts normalized spectral intensity, while the corresponding pulse duration (assuming flatphase) is shown in the bottom panel.

The most commonly used method to widen the spectrum of intense laser pulses without

shifting its central frequency is self-phase-modulation (SPM) [4,66]. SPM, in fact, is a four-

wave mixing process originating from a nearly instantaneous third order nonlinear

susceptibility in transparent media [67]. It is based on the modulation of the refractive indexthat has a nonlinear part depending on the intensity )(tI of the light wave propagating in the

medium, i.e. )()( 20 tInntn += . The concept of bandwidth widening due to pure SPM action

is illustrated by Fig.1.7.


11

The use of mode-guiding structures such as quartz optical fibers [67] and gas-filled

capillaries [68,69] provides the ability to maintain efficient spectral broadening over a long

distance. Additionally, unlike the optical filament in bulk materials [66] the spectrally

broadened output of single-mode fibers is spatially uniform [67]. The pure SPM produces

pulses with highly modulated spectra (Fig.1.7) and very unmanageable phases, which is a

great obstacle in pulse compression. The situation is remedied by combining the action of

SPM and material dispersion [67]. In general, the pulse leaving the fiber is many times longer

than its bandwidth-limited duration, and carries a change of the oscillation period of the

electric field from the leading to the trailing edge of the pulse, called chirp. Therefore, the

technique described above is usually referred to as fiber chirping.

∆ω

I(ω

)/I m

ax

Frequency ω

2π/∆ωI(t)

Time t

Fig.1.7: Bandwidth growth due to pure SPM action. The top panel shows the input intensity of aGaussian pulse. The corresponding normalized spectra are presented in the bottom panel.

The development of this technology culminated in 1987, setting a pulse duration record

for almost a decade. By compressing mode-locked dye laser pulses chirped in a single mode

glass fiber a pulse of ~6 fs was generated [9]. A laser system, the design and applications of

which are described in this Thesis, essentially draws on the same quartz fiber technology.

However, the wavelength region, the oscillator, and the capabilities are substantially

different. The technological advancements implemented in this work and presented in this

Thesis only became available in recent years. In particular, the most important breakthroughs

are: 1) the use of sophisticated dielectric “chirped” optics, on which our pulse compressor is

based, and 2) the reliable amplitude–phase measurement of the produced white light

continuum and of the compressed pulses. Thanks to the progress in developing new phase

correction methods, the pulse duration also became shorter (see Fig.1.8 and hollow triangles

in Fig. 1.5). Last but not least, to achieve spectral broadening by SPM, which requires high

input intensities, the standard Ti:sapphire oscillator was cavity-dumped [70] thus increasing

by an order of magnitude the energy of the output pulses and providing a flexible control over

Chapter 1

12

the repetition rate. The latter, which can be as high as 1 MHz, is a particularly valuable asset

for efficient data collection.

Fig.1.8: Shortest measured pulse duration world record registered by the Guinness world record bookand held by the Ultrafast Laser and Spectroscopy Laboratory, Groningen University. (Detail ofGuinness Diploma is shown).

The pulse chirping in glass waveguides, however, has a fundamental limitation because

these fibers cannot withstand greater intensities that are required for further bandwidth

growth without suffering optical damage. Furthermore, the effective SPM length constitutes

another limitation, because the nonlinear interaction rapidly becomes inefficient with a drop

in the pulse intensity. The latter is lowered due to the increase in duration, which the pulse

experiences as a result of chirping. Consequently, sub-5-fs pulses with energies of few tens of

nano-Joules are probably the limit attainable with single-mode glass fibers. A breakthrough

was achieved with the demonstration of pulse compression using a hollow fiber (capillary)

filled with noble gas [68,69], which can produce sub-5-fs pulses with energies exceeding

0.5 mJ. It should also be mentioned that, contrary to this high-intensity approach, the use of a

specialized fiber [71] with a zero-dispersion wavelength at 780 nm allowed generation of an

extremely broadband white light spectrum by chirping the output of a low-power oscillator.

However, no results on the pulse compression of this unprecedented continuum have been

reported yet.

Another frequency-mixing technique, parametric chirped-pulse amplification [72,73] is

also an efficient tool for the generation of few-cycle pulses. In this scheme energy is

transferred from a strong, not necessarily ultrashort, pulse to a weak wide bandwidth seed


13

pulse. Tunable sub-5-fs 7-µJ pulses in the visible range have been obtained in such

parametric amplifiers [74,75].

Generally, there is no fundamental limitation that would prevent achieving pulse

durations down to a single optical cycle. The traditional notions of pulse envelope and phase

remain fully applicable for a single-cycle pulse, (i.e. a pulse the electric field of which carries

merely one full oscillation of the light wave) [76]. Although sub-cycle pulses are in principle

possible, the decrease of duration below one oscillation period in the visible–near-IR optical

range seems to be very difficult from the standpoint of propagation in space of such pulses.

Namely, the problem is caused by the appreciably high amplitude of spectral components that

are close to the zero frequency and, consequently, have infinite divergence. Therefore, one

optical cycle of the light wave at 800 nm, which is about 2.6 fs, is the lowest practical limit

for a pulse with this carrier wavelength. The shortest modern optical pulses in this spectral

region carry only a couple of such cycles [68,69,77,78]. The use of higher carrier frequencies

in principle allows producing even shorter pulses, which would still carry many more optical

cycles. Different possibilities to reach attosecond (1 as = 10-18 s) pulse duration at high carrier

frequencies are now the topic of intensive discussion [79-83]. It is not unlikely, that the trains

of attosecond pulses have already been produced by high-harmonic generation [84,85] but

have not yet been measured. The applications of attosecond waveforms, however, are beyond

the scope of optical nonlinear spectroscopy in the visible and near-infrared spectral regions,

to which the attention of this Thesis is confined.

1.4 Spectroscopic utility of existing 5-fs laser systems

The laser source for ultrafast spectroscopy must meet several specific requirements. In this

Section we provide a comparison of the existing ultrashort lasers with respect to their ability

to cope with the demands of a typical third-order spectroscopic experiment. Optical pump-

probe, transient grating, and photon echo [1,3] are the examples of such techniques. Our aim

is to study electronic relaxation of molecules in a fluid environment and more specifically to

the subject of this thesis, to investigate the hydrated electron. A survey of different laser

schemes producing few-cycle optical pulses, which were discussed above, is summarized in

Table 1.

First, the laser radiation has to be coupled to extremely broad absorption spectra that in

the case of the hydrated electron exceeds 5000-1 cm in breadth. Therefore, the laser frequency

spectrum must support pulses as short as 5 fs. Time-resolving of the fastest relaxation

processes in this and similar systems, predicted to proceed on a 20-fs time scale, requires the

very best resolution the ultrashort pulses can offer.

Second, to stay in the low-perturbation regime, only a small amount of the ground-state

population must be transferred to the excited state. For an electronic transition with the dipole

moment strength in order of 1 Debye and the 5-fs pulses focused into a spot with ~20-µm

diameter, a 10% level of the change of electronic state population is achieved at pulse

energies as low as 5 nJ. Therefore, a cavity-dumped oscillator is the equipment of choice

Chapter 1

14

because of its sufficient but, compared with the amplified systems, quite modest output

energy.

Table 1. Brief summary of characteristics of different few-cycle laser sources

Technique Shortestpulse

FWHM,fs

Pulseenergy,

mJ

Repetionrate, kHz

Advantages Disadvantages Referen-ces

Ti:sapphireoscillators <6 ~2·10-6 105

Simplicity, lowcost, reliability

Fixed wavelength‡.Difficult rep. ratereduction

[62-64]

CavitydumpedTi:sapphirelaser +quartz fiber

<5 ~10-5 1–1000Flexible rep. ratecontrol. Higherenergy and broaderspectrum thanoscillators.

Fixed wavelength‡

[77,78]

Hollow-fiberpulsecompression

<5 0.5 ~1Very highintensities suitablefor variety of strongfield applications

Requires laseramplifiers. Rep.rate determined bypump lasers.

[68,69]

Noncollinearopticalparametricamplifiers†

<5 10-3 ~1Tunable pulses Requires laser

amplifiers. Rep.rate determined bypump lasers. Highcomplexity

[73-75]

Another important point of concern is the repetition rate of the laser. Because of low-

energy requirement (as indicated above) and the low typical nonlinear susceptibility, the

resulting signal that should be experimentally measured is very weak. Wishing to reduce the

time of needed statistical data averaging and to avoid problems with the drift in the

parameters of the laser output, the highest possible repetition rate in such a situation is, of

course, ideal. The amplifier-based systems [68,69,73-75] at the present time operate at the

repetition rate of one or several kilo-Hertz and depend in this respect on the pump sources.

Contrary to these, simple laser oscillators produce trains of pulses with up to 100 MHz

repetition rates [62-64]. This, however, becomes rather detrimental for spectroscopic

purposes since a fresh sample volume must be exposed to each individual laser shot to

prevent the heating effects and the pile-up of long-lived electronic states. Taking into account

the reasonable speed of sample replacement, ~10 m/s for a liquid jet, and a ∅=20 µm of the

irradiated region we arrive at the upper limit of the laser repetition rate. This is ~0.5 MHz,

which still ensures that a total sample volume is replaced between the laser shots.

Regrettably, for the wide-bandwidth emitting oscillators the task of repetition rate reduction

becomes an impassable obstacle. For the cavity-dumped laser, however, the adjustment of the

inter-pulse separation by any integer number of cavity roundtrips can be naturally achieved. ‡ Retaining the shortest pulse duration† Data for the signal wave


15

In fact, this laser breeches the gap between the “howitzers” – amplified lasers and their 5-fs

producing extensions– and the “peanut shooters” – plain oscillators in terms of both the

ammunition size and the rate of fire. This makes the Ti:sapphire cavity-dumped oscillator,

equipped with the 5-fs option, an invaluable asset in the armory of an ultrafast laser

spectroscopist.

Despite the fact, that such a cavity-dumped laser system, which will be presented in

detail in Chapter 2, probably represents the best solution for the research described in this

Thesis, one should not forget its principal limitations. The most significant of them is the lack

of broad wavelength tunability. A limited tunability can be achieved, however, by employing

a wavelength-selecting element. This, however, comes at the price of sacrificing the pulse

duration. To this end, the performance of ultrashort-pulse non-collinear parametric amplifiers

[73,74] remains unparalleled. Despite the staggering set-up complexity, these systems were

able to provide a quick spectroscopic turnout [86,87] justifying the efforts and expenses put

into their construction.

1.5 Few-cycle pulse characterization

The pulses that were described in the previous Section are the shortest man-made waveforms

produced to date. For the duration of such pulses light travels merely a distance of a couple of

microns. No direct methods of measuring these pulses seem to be possible. Therefore,

indirect or correlation techniques must be used. The simplest is the autocorrelation [88] –

effectively a time-gating of the pulse using its own delayed replica and optical instantaneous

nonlinearity as a shutter. Such autocorrelation traces, obtained as a function of time-delay,

give a fair assessment of the pulse duration, however only limited information on the pulse

shape and practically no information on its phase are available.

Fig.1.9: Ultrashort laser pulse measurement by FROG. In the depicted version of FROG technique(SHG FROG), one measures the dispersed signal of intensity autocorrelation in a second-harmoniccrystal.

The rigorous solution to the problem of the exact pulse shape and phase measurement

of an arbitrary ultrashort laser pulse was found six years ago and resulted in the development

of the technique of frequency-resolved optical gating (FROG) [89-91]. Essentially, FROG

consists of the measurement of any type of autocorrelation signal as a function of both delay

and frequency, and the inversion algorithm that is capable of extracting the precise amplitude

and phase of the electric field (Fig.1.9). The introduction of FROG heralded another

Chapter 1

16

revolution related to the ultrashort pulses, which took off shortly after the advent of the self-

mode-locked Ti:sapphire lasers. Indeed, the search for better ways of diagnostics was sparked

and remained motivated by the rapid progress in the generation of ultrashort pulses.

Unlike FROG, which essentially employs time-gating, another group of pulse

measuring techniques is based on spectral interferometry [4,92,93]. A frequency-domain

interferogram reflects the relative phase difference between the two arms of the

interferometer, as is well known from the conventional white-light interferometry [4]. If the

phase of the pulse travelling in one arm is already known, the task of extracting the relative

phase of the pulse propagating in another arm becomes straightforward [94]. A great

breakthrough in this group of techniques occurred with the invention of SPIDER [92], an

acronym for spectral interferometry for direct electric field reconstruction. This method

utilizes two replicas of an unknown pulse, which are frequency-shifted with respect to each

other. The phase is then reconstructed by a non-iterative algorithm that is applied to the

spectral interferogram of the two up-converted replicas. The spectral shear between them is

obtained through the mixing with two local oscillator fields, each of which has its own

characteristic frequency. In the practical implementation of this method [92], each replica of

the test pulse is up-converted in a nonlinear crystal with another, strongly chirped pulse,

which is derived from the same laser. Because the two replicas of the ultrashort pulse are

delayed with respect to each other, they overlap in time with different portions of the third,

chirped pulse. Therefore, the resulting up-converted spectra become shifted in frequency to a

different extent.

Both FROG and SPIDER have shown their capability in measuring pulses shorter than

6 fs [95-97]. For a nonlinear spectroscpist, wishing to characterize pulses directly at the

position of the sample, FROG, however, presents a more natural choice, since in itself it is an

exactly the same spectroscopic experiment, only performed in a material with instantaneous

nonlinearity. The use of SPIDER in this case would require a separate set-up. Since the

pulses in question easily become broadened even as they travel through air, the pulse being

measured and the one further used in the spectroscopic experiment may no longer be the

same, which is not acceptable. Additional limitation in the pure frequency-domain technique,

such as SPIDER, originates from the fact that any time-domain picture of the pulse is

obtained indirectly, i.e. through the use of Fourier transform. In this situation, even if the

spectral phase is measured correctly, an error in recording the laser spectrum can easily

produce a significantly different, from the real one, pulse duration. The techniques

performing a direct time-domain gating, such as FROG, are free from this limitation as either

the over- or underestimation of the true autocorrelation width, in not too-pathological cases

[91], cannot be larger that ~1.5 times that of the measured one.

The mathematical description of FROG (and SPIDER as well) data is based on the

assumption of ideal nonlinearities, where the implications of using finite-thickness real media

and finite-diameter beams are ignored. Therefore, for our very short and extremely broadband

pulses it becomes imperative to study these effects and their possible impact on the outcome


17

of the pulse characterization experiment. This analysis is performed in Chapters 3 and 5 of

this Thesis for the second- and third-order nonlinearities, respectively.

1.6 Techniques of nonlinear spectroscopy

In the experimental study of hydrated electrons, presented in this thesis, we employ several

techniques of time-resolved third-order nonlinear spectroscopy. All these methods aim at

measuring third-order polarization, 3

P , which is induced by an excitation laser field(s) and

is subsequently read-out by a delayed probe pulse field. The scan of the time delay between

the excitation pulse(s) and the probe pulse, measures, in one form or another, the temporal

decay of 3P . This provides key information on the lifetimes and dephasing times of

electronic states [3]. The spectro-temporal evolution of nonlinear polarization additionally

reflects the change in transition energies between occupied and unoccupied electronic states.

In the case of hydrated electrons, such changes of transition frequencies indicate the on-going

modification of the potential well containing the electron, which is a direct consequence of

positional readjustment of the water molecules. The ultimate, albeit not a straightforward

task, is to translate the collection of spectro-temporal snapshots of the nonlinear polarization

into a sequence of time- and space-resolved images. These images represent the motion of

individual water molecules in time as the latter act as energy dissipation channels for the

photo-excitation energy deposited on the hydrated electron.

Fig.1.10: Concept of the optical pump-probe experiment. The solid balls represent population ofelectronic states. The pump pulse creates photo-excitation, whereas the probe pulse monitors thehistory of population decay as a function of time elapsed since the excitation.

Optical pump-probe and two- and three-pulse photon echo techniques will be applied in

this work. The first two methods use identical simple geometry (i.e. two beams intersecting in

the sample) where one pulse serves for excitation and another one as a probe. The variation

used in our experiments, of the three-pulse echo, called transient grating, employs two pulses

for excitation, which are coincident in time but carried in two separate beams. Despite the

identical source of nonlinear response that is measured by these different techniques, some

are better suited to probe one aspect of the problem and some to tackle another. A detailed

mathematical description of these nonlinear spectroscopic techniques, and their respective

comparison is given in Chapter 5.

Chapter 1

18

Pump-probe is by far the simplest and most popular nonlinear spectroscopic technique.

As schematically shown in Figure 1.10, the first (excitation) pulse creates a population of

electrons in a higher (first excited) electronic state leaving a “hole” in the ground-state

population. At the wavelength of this electronic transition, the sample becomes temporarily

more transparent for the light travelling through it. On the other hand, absorption from the

now occupied excited state to higher states (not shown) makes the sample temporarily more

opaque at the respective transition wavelengths. These induced transparency and opaqueness

(i.e., absorption) of the sample are recorded as a function of the delay and change in

frequency of the probe pulse by measuring the change in its transmission through the sample.

Fig.1.11: Concept of transient-grating scattering experiment. A pair of pulses, coincident in time,creates a refractive index grating in the sample across the beam intersection area. A fraction of thedelayed (probe) pulse intensity diffracts off this grating that decays in time. It is the intensity of thediffracted beams (shown by sideways arrows), which is detected in this experiment.


19

In the transient grating experiment, (schematically presented in Figure 1.11), the

interference between the pair of excitation pulses imprints a spatial grating in the electron

population in the excited state, N∆ , and, consequently, in the spatial profile of the refractive

index, n∆ , across the sample. One then measures the intensity of scattered light of the

delayed probe pulse that interrogates the decaying population grating (Fig.1.11, bottom

right). A similar concept is employed in the two-photon echo experiment with the difference

being that the second pulse, which participates in the formation of the grating, and the pulse

scattered from it are one and the same. For this reason, this technique is also known as self-

diffraction. While transient grating is better suited to measure electronic population relaxation

time, self-diffraction is preferable to study the time of electronic dephasing.

From the viewpoint of detection, these spectroscopic experiments can be categorized as

homodyne and heterodyne methods. The scattering techniques measure a background-free

signal, which means that the scattered beam does not overlap spatially with any of the

incoming laser beams. Therefore, this is an intrinsically homodyne detection and the

magnitude of the measured signal is proportional to 23P . Thus, only the amplitude of the

induced polarization becomes available in these experiments while its phase remains

unknown. Besides, the background-free scattered signal is proportional to the product of

intensities of all three incoming pulses. Consequently, high laser intensity is required in this

type of experiment.

In the pump-probe experiment, the signal of interest propagates collinearly with theprobe field, prE . This field performs the familiar function of the local oscillator in a

heterodyne detection technique [98] and, therefore, the magnitude of the registered signal is

proportional to [ ]*3Im prEP >< . Therefore, in case prE is a purely real function, one can extract

the imaginary part of the induced polarization. Additionally, heterodyning in the pump-probe

measurement provides a way to amplify a weak signal, which becomes particularly valuable

for studying the longer time-scales of energy relaxation. Unlike the signal in grating

scattering, the one in the pump-probe scheme linearly depends on the excitation intensity and

on the optical density of the sample. Therefore, even with very low-energy laser pulses, one

can easily resolve transient spectra instead of using wavelength-integrated detection.

The methods applied in this Thesis are only several of many possible techniques.

Examples of modified third-order experiments that provide access to the real part of the

induced polarization, i.e. [ ]3Re P , as well as to imaginary one, [ ]3Im P , can be found in Ref.

[99]. Yet another efficient approach, the one based on the study of non-resonant fifth-order

response in liquid media, has been recently developed [100]. New, emerging spectroscopic

techniques could help clarify many questions surrounding the interesting and challenging

system of the hydrated electron. We foresee great perspectives for the application of the

femtosecond infrared spectroscopy [101]. By temporally and spectrally resolving of the

transient dynamics of the OH bond that has absorption in the infrared, one would obtain an

invaluable direct insight into the motions of the solvent molecules with respect to how they

Chapter 1

20

respond to the photo-excitation of the solvated electron. Another promising technique that

has been gaining its strength in the last years due to the enormous progress in femtosecond

technology is combined femtosecond visible – X-ray spectroscopy [102-104]. Recent

experiments on GaAs lattice dynamics studied by picosecond x-ray diffraction [105] clearly

demonstrated the feasibility of this approach toward physical and chemical processes.

1.7 Scope of this Thesis

In this Thesis a unique set-up is employed in nonlinear spectroscopic experiments on the

hydrated electron. The demand for time resolution, imposed by this spectroscopic system is

the basis for our efforts to construct a suitable ultrashort laser. We describe a versatile laser

system, based on a Ti:saphhire cavity-dumped oscillator which is able to produce pulses

below 5 fs at up to 1 MHz repetition rate.

The unprecedented short pulse duration and, moreover, the tremendous spectral width

attained require a careful approach to the description of the nonlinear signals obtained with

such pulses. Many theoretical aspects of nonlinear optics have to be scrutinized before they

can be applied to such pulses. A possible breakdown of some basic concepts, such as the

slowly-varying amplitude approximation [106] and rotating wave approximation, or RWA,

[3] have to be considered. Other concepts, such as the definition of the pulse carrier

frequency [107] become rather awkward. The choice of either the time- or frequency-domain

approach to describing nonlinear optical signals is also important. While these two

descriptions are generally connected via Fourier transform relations, the time-domain

language is generally considered to be more appropriate for the ultrashort pulses [3,67]. This

language is, however, totally inadequate to deal with such effects as spectral filtering [108],

spectral mode-size variation, precise inclusion of dispersion, spectral variation of nonlinear

susceptibilities, spectral sensitivities of the light detectors, etc. For these reasons, the

frequency-domain description is adopted, whenever possible, throughout this Thesis. The

appropriate and inappropriate conditions for switching from the frequency- to the time-

domain language are also demonstrated.

Since the precise knowledge of the amplitude and phase is crucial for the successful

compression of the pulse resulting from fiber-chirping, and for the spectroscopic applications

of the resulting compressed pulse, a substantial portion of this thesis is devoted to the

problem of pulse characterization. Second harmonic generation (SHG) FROG is chosen

because of its high sensitivity, simplicity, and low intensity requirements.

We next turn to a study of the hydrated electron. However, before even conceiving any

experiments on the femtosecond time scale, yet another, so far unmentioned, technical

problem has to be solved – production of hydrated electrons. Therefore, we provide a detailed

account on the generation of the hydrated electrons by cation photo-ionization with

nanosecond UV pulses. Further, measurement of different specific properties of the hydrated

electron species produced in this way are presented.


21

Subsequently, the results of femtosecond experiments are reported and interpreted.

First, the combined analysis of the absorption spectrum and two-pulse photon echo, or self-

diffraction, signals indicated a homogeneous nature of spectral broadening and a very short

electronic dephasing time, T2, which equals ~1.6 fs. A perfect fit of the absorption band was

obtained by using a Lorentzian contour, modified to account for the breakdown of the RWA,

and the deduced value of T2. Such a short dephasing time once more underscores the

importance of having as short as possible the duration of the pulses available for the

measurements. Next, transient grating experiments with 5-fs pulses were performed on the

hydrated electrons in water and heavy water to capture the initial step of the photo-excitation

relaxation. The dependence obtained points strongly towards the librational motion of the

water molecules as a primary channel of excess energy dissipation. The transient grating and

further measurements of transient absorption on the femtosecond and picosecond time-scales

form conclusive evidence for a short-lived excited state and a slower, picosecond, hot-

ground-state relaxation. Finally, these results are explained in a self-consistent model of

electronic-state potentials, in which the energy potential of the p-state has a significantly

steeper curvature and is strongly displaced with respect to the ground state potential well.

The reader will find this Thesis organized as follows. Chapter 2 presents a thorough

account on the working of the cavity-dumped laser and the design of the pulse compression

scheme. Chapter 3 examines the problem of pulse characterization by SHG FROG, with

durations down to one cycle considered. Chapter 4 describes experimental necessities and

results of the FROG measurement. Characterization of, first, a chirped and, next, 4.5-fs,

pulses compressed from it, are described in detail. Chapter 5 develops and presents the

general formalism for the third-order nonlinear spectroscopy using the frequency-domain

approach and makes contact with the conventionally used time-domain description. Chapters

6 and 7 gives the account of femtosecond spectroscopic experiments performed on the

solvated equilibrated electron in water. Specifically, Chapter 6 deals with photon-echo

measurements, whereas in Chapter 7 we study the pump-probe signals obtained from the

hydrated electron. Based on our experimental findings, in these two chapters we formulate a

new insight into the mechanics of the molecular response of liquid water surrounding the

electron.

Chapter 1

22

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Tilsch, T. Tschudi, and U. Keller, IEEE J.Select. Topics Quantum. Electron. 4, 169 (1998).65. Y. R. Shen, The principles of nonlinear optics (Wiley, New York, 1984).66. R. R. Alfano and S. L. Shapiro, Phys. Rev. Lett. 24, 592 (1970).67. G. P. Agrawal, Nonlinear fiber optics, 2nd ed. (Academic Press, Inc., San Diego, CA, 1995).68. M. Nisoli, S. D. Silvestri, R. Szipöcs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz,

Opt. Lett. 22, 522 (1997).69. M. Nisoli, S. Stagira, S. D. Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, C.

Spielmann, and F. Krausz, Appl. Phys. B 65, 189 (1997).70. M. S. Pshenichnikov, W. P. de Boej, and D. A. Wiersma, Opt. Lett. 19, 572 (1994).71. J.K. Ranka, R.S. Windler, and A.J. Stentz, Opt. Lett. (to be published) (2000)72. A. Dubietis, G. Jonušauskas, and A. Piskarskas, Opt. Commun. 88, 437 (1992).73. G. Cerullo, M. Nisoli, S. Stagira, and S. De-Silvestri, Opt. Lett. 23, 1283 (1998).74. A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, Appl. Phys. Lett. 74, 2268 (1999).75. A. Shirakawa, I. Sakane, and T. Kobayashi, Opt. Lett. 23, 1292 (1998).76. T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).

Chapter 1

24

77. A. Baltuška, Z. Wei, M. S. Pshenichnikov, and D. A. Wiersma, Opt. Lett. 22, 102 (1997).78. A. Baltuška, Z. Wei, M. S. Pshenichnikov, D.A.Wiersma, and R. Szipöcs, Appl. Phys. B 65,

175 (1997).79. P. Corkum and M. Ivanov, Phys. Rev. Lett. 71, 1995 (1993).80. I. P. Christov, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. A 57, 2285 (1998).81. A. E. Kaplan, Phys. Rev. Lett. 73, 1243 (1994).82. A. E. Kaplan, S. E. Straub, and P. L. Shkolnikov, J. Opt. Soc. Am. B 14, 3013 (1997).83. M. Ivanov, P. B. Corkum, T. Zuo, and A. Bandrauk, Phys. Rev. Lett 74, 2933 (1995).84. Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 79,

2967 (1997).85. Z. Chang, A. Rundquist, H. Wang, I. Christov, M. M. Murnane, and H. C. Kapteyn, IEEE J.

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(1999).96. L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C.Iaconis, and I. A.

Walmsey, Ultrafast Optics 24, 1314-1316 (1999).97. A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, Opt. Lett. 23, 1474 (1998).98. M. D. Levenson and S. S. Kano, Introduction to nonlinear laser spectroscopy (Academic Press,

New York, 1988).99. W. P. de Boeij, PhD thesis, Univ. Groningen (1997).100. T. Steffen, PhD thesis, Univ. Groningen (1998).101. C. Chudoba, E. T. J. Nibbering, and T. Elsaesser, Phys. Rev. Lett. 81, 3010 (1998).102. F. Raksi and K. Wilson, J. Chem. Phys. 104, 6066 (1996).103. M. Ben-Nun, J. Cao, and K. R. Wilson, J. Chem. Phys. A 101, 8743 (1997).104. J. Cao and K. R. Wilson, J. Chem. Phys. A 102, 9523 (1998).105. C. Rose-Petruck, R. Jimenez, T. Guo, A. Cavalleri, C. W. Siders, F. Raksi, J. A. Squier, B. C.

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Chapter 2

26

2.1 Introduction

Ever since pulsed lasers were invented there has been a race toward shorter optical pulses [1].

Next to the fact that the breaking of any record is a challenge, a major scientific driving force came

from dynamical studies showing that ultrashort pulses were essential to the exploration of

elementary processes in chemistry, photobiology and physics. For instance, the primary step in

bond-breaking reactions (femtochemistry) [2], the rate of electron-transfer in photosynthetic

reaction centra [3,4], and the time scales of relaxation processes in condensed phase [5,6] could

only be time-resolved with femtosecond excitation pulses. On the road toward femtosecond pulse

generation a better grasp of the underlying physics proved to be essential. A milestone here was

the development of the colliding pulse modelocked (CPM) laser [7]. When the importance of a

careful balance between the group delay and dispersion on pulse formation was recognized [8,9],

sub-100 fs optical pulses became feasible [10,11]. Further development ultimately led to the

prism-controlled CPM laser [12,13] which delivered pulses of -30 fs. It was this CPM laser that

laid the foundation for many groundbreaking experiments in the past decade, from the observation

of wavepacket motion in chemical reactions [14] to the exploration of carrier dynamics in

semiconductors [15,16].

Another crucial invention for ultrashort optical pulse generation was the technique of fiber

pulse compression [17]. In this method, a relatively long pulse is injected into a single mode fiber.

Via the combined action of self-phase modulation [18] and dispersion it becomes spectrally

broadened, carrying an almost linear chirp [19]. This spectrally and temporally broadened pulse

is subsequently compressed by a pair of gratings [20-22], prisms [10,23] or their combination

[24,25] to a much shorter pulse. The compressor’s action is to retard in a well-defined manner the

frequencies of the pulse that have advanced. Pulse compression of the amplified output of the CPM

laser culminated in the generation of optical pulses of 6 fs (assuming a hyperbolic secant pulse

profile) in 1987 [25]. The electric field of such a pulse exhibits only 4.5 oscillations at its

FWHM\1. With these ultrashort pulses photon echoes in solution could be studied for the first time

[28-30], while their large spectral width turned out to be very useful for pump-probe experiments

in photobiology [31,32].

A new era in ultrafast laser technology began with the development of the fs modelocked

Ti:sapphire laser [33] which routinely generates pulses of about 10-15 fs [34-39]. In addition, this

laser exhibits low amplitude noise and is extremely reliable. It is not surprising that in the past

five–seven years Ti:sapphire based lasers have replaced the CPM lasers in many laboratories as

new ultrafast light sources. Sub-10-fs pulse formation from a Ti:sapphire laser also looked

promising since the fluorescence bandwidth of the lasing material [40] supports pulses as short

as 4 fs. The efforts to construct an oscillator matching the whole bandwidth continue. Just in a

\1 The frequently cited number of 3 oscillations [6,25] refers to the duration of the intensity envelope, which,in contrast to the electric field, contains no oscillation at the optical frequency. This misleading notation hasalso been applied to very recent results [26,27].

All-Solid-State Cavity-Dumped sub-5-fs Laser

27

couple of years the duration of pulses obtained directly from a laser dropped from -7-8 fs [41-48],

what seemed to be a practical limit at the time [44,49,50], to about 5.5 fs in 1999 [26,27].

Despite the fact that a relatively simple oscillator producing very large-bandwidth sub-10-fs

pulses seems to be an attractive option, its employment for nonlinear spectroscopy is rather

impossible. Here the problem of repetition rate reduction becomes detrimental, since no shot-to-

shot sample refreshment is feasible for the pulse trains generated at the typical 80-100 MHz

repetition rates. Pulse picking the output of such a laser outside the cavity is very problematic

owing to the large bandwidth. For example, the spectral content of such short pulses would be

appreciably dispersed in space in case an acousto-optic pulse picker is employed. Alternatively,

an electro-optical switch would modify the output spectrum because the polarization of different

wavelength components could not be turned for the whole spectral interval simultaneously to the

same degree. Additionally, a combination of a Pockels cell plus polarizers introduces a large

amount of bulk dispersion, which is difficult to compensate. The repetition rate reduction by up-

scaling the cavity length is also unfeasible as it crucially affects the laser stability and makes

mode-locking operation more difficult. Clearly, other alternatives should be sought.

With the development of a 13-fs cavity-dumped laser, pulse compression was shown to be

a viable route toward pulses of less than 6 fs [51]. Another very promising development was the

use of a hollow fiber for spectral broadening of ultrashort pulses [52]. A distinct advantage of a

hollow fiber is that it can stand high intensities, allowing pulses of millijoule energy to be

compressed.

Another spectacular development took place in the ultrashort pulse generation from non-

collinear optical parametric amplifiers pumped by the second harmonic of Ti:sapphire. Utilizing

the uniquely broad phase-matching bandwidth of the Type II BBO crystal [53], tunable sub-10-fs

pulses have been produced in the visible and infrared [54,55]. The shortest pulses obtained by this

technique in the visible measure only 4.5 fs. (See Table 1 in Chapter 1.)

In our early attempts to compress the fiber-chirped output of a cavity-dumped Ti:sapphire

laser, we succeeded in the production of about 5-fs pulses at repetition rates of up to 1 MHz

[56,57]. While the shortest pulses were attained using a prism–grating compressor, slightly longer

pulses were obtained from the higher throughput prism–chirped mirror compressor. It was also

suggested that with custom-designed chirped mirrors, shorter pulses with higher pulse energies

should be possible. Nisoli et al. recently showed that by using a hollow fiber -20-µJ, sub-5-fs

pulses can be generated at a 1-kHz repetition rate [58].

In this Chapter, we report the generation of sub-5-fs pulses from a cavity-dumped

Ti:sapphire laser using a prism/chirped mirror/Gires-Tournois interferometer compressor. Group

delay measurements of the generated continuum, which served as input for the design of this novel

compressor, are discussed. It is shown that the pulse shape and spectral phase can be determined

from the collinear autocorrelation function in combination with the optical spectrum. The similar

pulse shape is calculated when the optical spectrum and phase difference between the pulse

compressor and continuum is used as input.

Precise knowledge of the amplitude and phase of ultrashort pulses is extremely important

in many experiments, especially when dynamics occur on the time scale of the pulse width. An

Chapter 2

28

example is femtosecond photon echo in solution, where explicit use of the pulse shape in

calculations of the echo relaxation has shown to be essential [59,60]. Another example is coherent

control of wave packet motion and bond-breaking reactions [61-63]. More generally, a detailed

description of a molecule–light field interaction requires full knowledge of the electric field. The

applicability of the so-called slowly varying envelope approximation in experiments with

ultrashort pulses becomes questionable [64]. In fact this approximation may break down and in that

case new effects are to be expected.

Chapter 2 is organized as follows. In Section 2.2 we discuss the cavity-dumped laser.

Generation of the continuum to be compressed is described in Section 2.3. In Section 2.4 the

spectral phase of the white light is discussed, while in Section 2.5 the temporal shape of the

continuum is dealt with and compared to calculations based on spectral phase measurements. In

Section 2.6 the spectral and temporal shape of the continuum are commented on. Section 2.7 deals

with the compressor. In Section 2.8 the pulse duration is measured by fringe-resolved

autocorrelation. In Section 2.9 we demonstrate how the amplitude and phase of the compressed

pulse can be reconstructed from measurement of the interferometric autocorrelation and the

spectrum of the pulse. Section 2.10 outlines experimental uncertainties of the interferometric

autocorrelation. Section 2.11 provides a summary and suggests some applications of this compact

sub-5-fs 2-MW laser.

2.2 Cavity-dumped Ti:sapphire laser

Figure 2.1 displays the schematic of the self-mode-locked cavity-dumped Ti:sapphire laser used

for continuum generation. It represents the next version of an earlier reported design [51].

Compared to the conventional Ti:sapphire oscillator [34,38,65], its cavity dumped counterpart

incorporates an additional mirror fold around an acousto-optic modulator [66]. In this way the

intracavity pulse energy is stored in a relatively high-Q cavity, which can be switched out of the

resonator at any desired repetition rate. The maximal pulse energy of a cavity-dumped Ti:sapphire

laser is typically a factor of 10 higher than that from its non-cavity-dumped counterpart. A careful

cavity design ensures the Kerr-lens self-mode-locking action is not disturbed by the extra fold and

by the added dispersion due to the Bragg cell. The best performance of the system is achieved

when the fold mirrors of the cavity dumper are separated by nearly a confocal distance and the

mirror fold around the Ti:sapphire crystal is set to the inner edge of the second stability zone [51].

This configuration allows the system to operate under soft-aperture Kerr-lens mode-locking

conditions, thus making the oscillator less sensitive to perturbations caused by the cavity-dumping

process and mechanical instabilities [51,67-69] than in the first stability zone. Besides, in the latter

case the hard aperture needed to initiate the mode-locking reduces the intracavity power.

Compared to the earlier reported design [51] the current version of cavity-dumped

Ti:sapphire laser has been significantly improved and presents a more versatile and compact

master oscillator. First, the argon-ion pump laser has been replaced by an intracavity doubled

Nd:YVO4 laser (Spectra Physics “Millennia”). The superior beam pointing stability and noise

characteristics of this diode-pumped solid-state laser allow the pump power to be reduced to


29

~4 W. Second, by introducing a high reflector (HR1) in the prism arm, the cavity has been folded,

which led to a more compact laser. Third, the output coupler OC has been placed at the non-

dispersive end of the cavity, providing an additional output at 82 MHz. Finally, we have saved

space by replacing the output (cavity-dumped) pulse pre-compressor, consisting of four prisms

[57], with two chirped mirrors [70].

Fig.2.1: Schematic of an all-solid-state sub-5-fs laser. Ti:Sa: 4-mm long Ti:Sapphire crystal (Union Carbide);L1: f = 12.5 cm lens; M1-M4: r = -10 cm cavity mirrors; HR1, HR2: high reflectors (CVI); OC: T = 2%output coupler (CVI); M5: pick-off mirror (Newport BD2); IP1, IP2: intracavity 69° fused silica prisms;CM1, CM2: chirped mirrors for pre-compression (R&D Lezer Optika); WLG - white-light generator; OAP:30° off-axis parabolic mirror (Kugler) with low dispersion overcoated silver coating; GTI1, GTI2: Gires-Tournois interferometers (R&D Lezer Optika); CM3, CM4: chirped mirrors for pulse compression (R&DLezer Optika); P1, P2: 45° fused silica prisms; RM: low dispersion overcoated silver roof mirror (R&DLezer Optika). The cavity-dump beam that in reality is ejected in the vertical plane is depicted here as beingdisplaced in the horizontal plane. The compressor output beam passes just above GT2. The solid arrowthrough the OC shows the 82-MHz output used in cross-correlation experiments. The whole set-up occupiesa work space of 1×1.5 m2 on an optical table.

With a 3-mm thick Bragg cell (Harris), driven by a 5-W electronic driver (CAMAC

Systems), the laser dumps 13–15-fs, 40-nJ pulses at a 1-MHz repetition rate. Pulses in excess of

45 nJ are generated when the RF signal is amplified to 16 W using a RF power amplifier

(CAMAC Systems). Even higher pulse energies are available at lower repetition rates. Figure 2.2a

presents the dynamics of the pulse train inside the cavity seen through the output coupler by a high-

speed photodiode. In this illustration, every 82nd pulse is cavity-dumped (rapid drops), which

leads to an interval of intracavity energy recovery lasting a few tens of cavity roundtrips. Notably,

the oscillations on the intracavity pulse train go on even after the intracavity pulse energy has been

fully recovered. This, however, does not affect the pulse-to-pulse stability of the cavity-dumped

Chapter 2

30

output, since the modulation of the intracavity train automatically adjusts itself to the periodicity

of RF bursts on the Bragg cell. (Compare the trace levels at the points immediately preceding the

two successive RF bursts in Fig.2.2a.) Pulses with energies below 40 nJ can be stably generated

with repetition rates reaching to 1.5-1.7 MHz. At even higher repetition rates the period between

successive RF bursts becomes insufficient for the full recovery of the intracavity pulse energy, and,

therefore, the RF power must be lowered to sustain stable cavity-dumping operation.

The use of the CAMAC RF driver provides an excellent contrast ratio between the

preceding/trailing and dumped pulses. The contrast of the pulse switching between the cavity-

dumped and the preceding by ~13 ns pulse has been found to be better than 1:1000 (Fig.2.2b).

Unfortunately, it is impossible to measure the contrast with the trailing pulse due to electronic

noise on the photodiode signal (right-hand part of Fig.2.2b). The given above figure of 1:1000 is

similar to the contrast ratio achieved using an electro-optical cavity-dumper [71], be it that in the

latter case the repetition rate is limited by ~10 kHz.

Fig.2.2: Oscilloscope traces of the cavity-dumping dynamics. (a) Intracavity pulse train corresponding tocavity dumping of ~40-nJ pulses at the repetition rate of 1 MHz. (b) A cavity-dumped pulse. The verticalarrow indicates the location of the pulse preceding the cavity-dumping event. The sensitivity of the traceshown in the inset is enhanced 250 times. The numbers below the traces indicate the value of the verticaland horizontal grid, respectively.

2.3 White-light continuum generation

The pre-compressed pulses from the cavity dumped laser, with 75 nm spectral bandwidth around

790 nm, were launched into a single-mode quartz fiber (Newport, F-SV, 2.75 µm core diameter)

through a 18/0.35 microscope objective (Melles Griot). The optimal fiber length calculated

according to Ref. [19] was ~1 mm; however, for practical reasons connected to mounting of the

fiber, we used a piece of ~2–3 mm. Angular alignment of the fiber along the longitudinal axis

proved necessary to prevent polarization rotation of the light passing through the fiber as a result

of chromatic anisotropy. A 3D piezo-driven (Piezo-Jena) fiber positioning stage is used to

simplify the alignment procedure. To keep the fiber tip dust-free, a constant flow of dry nitrogen

was applied to the focusing area. No damage to the fiber was observed for up to 40-nJ pulses.


31

Other types of fibers from different manufacturers were also tested; however the ability to

withstand high input intensities (~10 TW/cm2) seems a unique property of Newport single-mode

fibers. As of now, we have no explanation for this phenomenon.

The fiber output is collected by an off-axis parabolic mirror (OAP, Fig.2.1), which ensures

achromatic and nearly dispersion-free beam recollimation. The focal length of our custom-

manufactured parabola (Kugler) is ~7 mm and the inclination angle to the parabola axis is 30°.

Note, that the production of such a mirror with a high optical surface quality is a great

technological challenge, since the required post polishing of a diamond-turned parabola is difficult

due to the high curvature of the aspherical profile. The OAP is made of aluminum and is coated

by a silver- and a low-dispersion protective dielectric coatings (R&D Lezer-Optika, Hungary).

500 600 700 800 900 1000 11000.0

0.5

1.0

Fiber output

Laser

Inte

nsity

[ar

b.]

Wavelength [nm]

-20 -10 0 10 200

1

3.7 fs

FT assuming flat

spectral phase

Inte

nsity

[ar

b.]

Time [fs]

Fig.2.3: Fiber output (solid line) and cavity-dumped laser (filled contour) spectra. The inset shows the pulseobtained by Fourier-transforming the fiber output spectrum assuming constant spectral phase.

The white light continuum, generated by self-phase modulation exhibits approximately a

fourfold spectral broadening compared to initial spectrum (Fig.2.3). The optimal pulse energy for

injection into the fiber was found to be ~35 nJ, as judged by the quality of the generated continuum.

In this case, the pulse energy measured after recollimation of the continuum is about 18 nJ. The

long-term stability of the continuum intensity measured at several wavelengths varies from ~0.7%

rms at the edges of the spectrum (below 500 nm and above 1100 nm) to less than 0.5% rms near

the central frequencies.

The blue-shifted wing of the continuum reaches into the UV, and the red-shifted part stretches

into the near infrared, even beyond the spectral cut-off of the silicon detector used for the spectral

measurements (Fig.2.3). The shortest pulse attainable by compression of this continuum is obtained

by Fourier transformation of the spectrum assuming a flat spectral phase. This yields pulse

duration of ~3.7 fs (Fig.2.3, inset). Note that despite the irregular spectrum of the continuum the

ideally compressed pulse looks very clean. It is also worth pointing out, that the low intensity

wings of the continuum – excluded in the compression scheme to be discussed later – carry enough

Chapter 2

32

intensity for a variety of spectroscopic applications. Moreover, the use of a long piece of fiber

enables delivery of the pulse to a remote point in applications where the spectral bandwidth rather

than the pulse width is important [72].

2.4 Measurement of spectral phase

Pulse compression aims at the removal of spectral phase distortions accumulated by self-phase

modulation and propagation through dispersive media. Precise knowledge of the phase

characteristics of a chirped pulse is therefore vitally important to the design of an appropriatepulse compressor. In order to fully characterize a pulse one needs to know its spectrum )(

~ωI and

spectral phase )(~ ωϕ or its time dependent intensity )(tI and temporal phase )(tϕ . The temporal

and spectral descriptions are complementary and follow from each other by Fourier

transformation. Since no detector is fast enough to resolve the temporal shape of a pulse on a fs

time scale, indirect methods have to be used to resolve the exact intensity profile of a fs pulse.

In recent years a number of techniques of indirect phase and pulse shape retrieval have been

proposed [73-84]. For instance, in various implementations of frequency resolved optical gating

(FROG, see for example [77-79] and Chapter 3 of this thesis) a spectrally dispersed signal of the

autocorrelation-type is recorded. When a well-known phase retrieval algorithm is used to analyzeFROG traces )(tI and )(tϕ can be recovered.

Another approach to phase retrieval is spectrally resolved up-conversion [25,74-

76,81,82,84,85] or down-conversion [86]. In this method, the analyzed pulse (further called probe

pulse) is mixed with a well-characterized reference pulse in a nonlinear crystal. The resulting

signal is a (spectrally-resolved) cross-correlation. The knowledge of the parameters of the

reference (gate) pulse in this case dismisses the fundamental problem of pulse retrieval from

autocorrelation or FROG, where the gate pulse is unknown since it is a replica of the pulse to be

characterized.

The resulting signal at the sum frequency is dispersed through a monochromator and can be

expressed as

2),(2

),(

1)(

~)(

~)(),( ω

ωωωτ

ωωτ d

Lk

eeEERS

Lkii

pr Ω∆−

−ΩΩΩ∝ΩΩ∆

∫ (2.1)

where )(ΩR stands for the spectral sensitivity of the detector, [ ])(~exp)(~

)(~ ωωω rrr iAE ϕ= and

[ ])(~exp)(~

)(~ ωωω ppp iAE ϕ= are the (complex) amplitudes of the reference and probe pulses,

respectively, τ is a delay between them, and L is the interaction length. Here we assume that the

nonlinearity is instantaneous. A non-instantaneous response will be considered in the Chapter 3.

The phase mismatch for Type I (oo–e) interaction [87] is given as

)()()(),( Ω−+−Ω=Ω∆ EOO kkkk ωωω . (2.2)


33

with k denoting the wavevectors for ordinary (O) or extraordinary (E) waves. Similarly to FROG,

the cross-correlation signal ),( τΩS contains explicit information about the complex electric field

of the probe pulse, provided that the reference pulse has been fully characterized.

In principle, the knowledge of the reference field permits direct recovery of the probe field

through a so-called Wigner deconvolution [76,84]. The implementation of a FROG-like iterative

inversion algorithm, however, is also well justified [85,86] given experimental uncertainties of

the measured cross-correlation spectrogram.

A valuable asset of the spectrally-resolved up-conversion technique is that in some special

cases a time-consuming algorithm of phase retrieval can be replaced by a straightforward analysis.

For instance, if the spectrum of the reference pulse is sufficiently narrow and its spectral phase is

constant, Eq.(2.1) simplifies significantly and becomes:

( )

Ωϕ−−Ω×

Ω−+−Ω

ΩΩ∝Ω

ωτω

ωωτ

d

dAA

LkkkRS

prrp

ErOrO

)(~)(

~

2

)()()(sinc)(),(

22

22

, (2.3)

where Ar(t) stands for the temporal amplitude of the reference pulse. Note that the magnitude of

the measured signal is proportional to Ω2 [88] (See Chapter 3). This often omitted factor gains

importance with increased spectral bandwidth of the probe pulse.

Eq.(2.3) shows that if the delay τ is scanned for a given setting of a monochromator Ω, the

maximum of the up-converted signal directly reflects the group delay of the probe field [81]:

ω

ωωωτ

d

d pprpp

)(~)(

ϕ=+=Ω . (2.4)

Another important trait of spectrally-resolved up-conversion is that all factors limiting the

acceptance bandwidth, like phase-matching or spectral response of a detector, do not influence the

position of the maxima. These factors only affect the signal intensity. Furthermore, the phase-

matching conditions are also relaxed for spectrally-resolved up-conversion compared to second-

harmonic FROG, because the necessary acceptance bandwidth of the crystal is smaller by

approximately a factor of two.The spectral phase of the chirped white light is readily obtained from )(Ωpτ by integration

of Eq.(2.4):

∫=ϕ ωωτω dpp )()(~ (2.5)

The aforementioned technique is valid for reference pulses whose spectral bandwidth are

Chapter 2

34

appreciably narrower than those of the probe pulse. However, by choosing a spectrally infinitely

narrow pulse the duration of the up-converted signal becomes infinitely long, limiting the time

resolution. In the other extreme limit, when an infinitely short reference pulse is used, the up-

converted signal would be detected with an infinitely broad spectrum, limiting the resolution in

the frequency domain. Therefore, there is an optimal reference pulse duration, which yields a

compromise between temporal and spectral resolutions.

Fig.2.4: Normalized probe-reference correlation signals at different wavelengths. The monochromatorsettings are indicated on the left side of each trace. The schematic of the crosscorrelation experiment isshown in the inset. Probe pulse stands for the white-light continuum while the reference pulse at 82-MHzrepetition rate is derived directly from the Ti:sapphire laser. PMT - photomultiplier tube.

In our experiment we cross-correlated the chirped white light pulse with a laser pulse from

the output coupler (shown as a solid arrow through the OC in Fig.2.1) in a 100-µm thick BBO

crystal. The importance of having an independent reference beam at 82 MHz from the laser now

becomes evident. This pulse has a suitable duration and spectral width for the reference pulse in


35

the cross-correlation experiment. In order to have this pulse chirp-free, it was passed through a

4-prism compressor. The interferometric autocorrelation of this pulse was found to be in excellent

agreement with the one calculated from the pulse spectrum assuming a constant spectral phase.

To measure the group delay across the continuum the up-converted signal was scanned as

a function of time delay between the white light and reference pulse at different wavelengths

selected by a monochromator. The layout of this experiment is shown in the inset to Fig.2.4. The

spectral resolution of the monochromator was ~1 nm. Due to the limited phase-matching bandwidth

of the crystal, small angular tuning was necessary to obtain reliable measurements of the infrared

and the visible components of the white-light spectrum. Typical normalized up-converted profiles

at different settings of the monochromator are depicted in Fig.2.4. The corresponding frequencies

of the white light can be obtained knowing the central wavelength of the reference pulse (800 nm).

The duration of the up-converted signals increases toward the blue-shifted wing of the continuum.

This is explained by a faster change of the spectral phase of the probe pulse within the spectral

width of the reference pulse, compared to the relatively slow changing phase in the infrared region,

where material dispersion is considerably lower. The modulation appearing in some profiles is

due to intensity variations in the spectrum around the central frequency of the continuum (Fig.2.3).

The up-converted signals cover the fundamental wavelengths of the white-light from 0.55 to 1.2

µm. Note that the bandwidth of the white-light that can be up-converted stretches much further into

the infrared region than can be reliably measured (Fig.2.3) using a silicon photodiode array. This

means that the real bandwidth of the white-light continuum as well as the shortest achievable pulse

duration (Fig.2.3, inset) are most probably underestimated.

10000 12500 15000 17500-200

-100

0

100

200

300

400

-1

Gro

up d

elay

τw

lc [

fs]

Energy [cm ]

1200 1000 800 600Wavelength [nm]

Fig.2.5: Group delay of the white-light continuum retrieved from the probe-reference cross-correlation.Solid circles denote the first momenta of the up-converted temporal profiles and the solid line is apolynomial fit to the experimental points.

To obtain the group delay across the white-light spectrum weighted averages of the time-

dependent up-converted traces were measured. There are two reasons why this approach is

superior to evaluation of τ(ω) from the peak positions as given by Eq.(2.4). First, the actual peak

positions might be additionally shifted due to the unevenly distributed spectral intensity in the

probe pulse. Second, by calculating weighed averages one uses the information from all

Chapter 2

36

experimental points and not only from the maxima [75,76,84]. The group delay of the white-light

continuum is shown in Fig.2.5 as solid points. The solid line represents a low-order polynomial

fit used in the further calculations of the pulse compressor. The estimated group delay dispersion

is ~380 fs2 at the wavelength of 600 nm and decreases to ~220 fs2 at 1 µm. We will return to the

discussion of the apparent nonlinearity in the group-delay in Section 2.6.

In closing to this Section we note that the measurements described in it were repeated

several times using slightly different fiber lengths. The results were found to be identical - within

experimental uncertainty - to those presented in Figs.2.4 and 2.5, which indicates a remarkable

long-term stability of the spectral phase.

2.5 Temporal analysis of the white light pulse

To verify the group delay measurements, we studied the properties of the white-light continuum

in the time domain. To this end, we compare the wavelength-integrated cross-correlation trace,

recorded with pulses that are considerably shorter than the duration of the white pulse, with the

calculated temporal profile of the continuum. The latter is obtained by Fourier transformation of

the electromagnetic field, taking into account the spectral phase calculated according to Eq.(2.5).

The amplitude of electromagnetic field is derived from the measured spectrum. The continuum is

mixed with the reference pulse in a 15-µm thick BBO crystal and the up-converted signal detected

using a photomultiplier tube (PMT) [17,89,90].

-200 0 200 4000.0

0.5

1.0

Inte

nsity

[ar

b. u

nits

]

Time [fs]

700 900 11000.0

0.5

1.0R'(λ

p)

Wavelength [nm]

Fig.2.6: Comparison of the experimental (open circles) and computed (solid line) cross-correlation betweenthe white-light continuum and the reference pulse (filled contour). The solid curve was obtained bynumerical correlation of the reference pulse with the white-light pulse and corrected for the spectralsensitivity R´(λp). The overall spectral response of the detector and up-conversion efficiency of a 15-µmBBO crystal is displayed in the inset.

The measured signal is displayed in Fig.2.6 (open circles). Negative times represent the

leading and positive times the trailing edge of the pulse. Note that the direction of time is known

unambiguously since no time reversal symmetry is present in a cross-correlation experiment. The

red-shifted components of the spectrum are concentrated in the leading edge of the pulse and the


37

blue-shifted ones are trailing behind.

To compare this experimental pulse shape with the calculated one, several factors should be

taken in consideration. First, the finite duration of the reference pulse needs to be taken into

account. Second, the spectral response of the detector and the relevant phase-match conditions

must be regarded. The up-converted signal can be calculated by integration of Eq.(2.3) over

frequency:

∫ ΩΩ= dSSCC ),()( ττ . (2.6)

Taking into consideration the fact that the probe pulse is spectrally narrower than the white-light

continuum, we may assume that each given instant corresponds to a single instantaneous frequency.

In this approximation Eqs.(2.6) and (2.3) yield:

( )∫ −∝ dttEtERS rppCC

22)()()(')( ττωτ , (2.7)

where )(τω p denotes the instantaneous frequency of the probe field and the overall spectral

sensitivity is:

( ) ( )[ ]( ) ( )( )

2

)()()(sinc

)()()('

2

2

Lkkk

RR

prErOpO

prprp

τωωωτω

τωωτωωτω

+−+×

++=(2.8)

The correction term ( ))(' τω pR , comprising a spectrally-varying conversion efficiency, the phase-

matching factor of the crystal and the spectral response of the PMT, is depicted in the inset to

Fig.2.6. The main spectral distortions occur at the high-frequency part of the white-light continuum

(i.e. in the trailing edge) where phase-mismatch in the nonlinear crystal increases due to the

increased dispersion. The resulting temporal shape of the continuum calculated according to

Eq.(2.7) is depicted in Fig.2.6 as a solid line, and agrees reasonably well with the experimentally

measured data, given all the assumptions made. This also indicates that the spectral phase was

measured correctly. The asymmetry of the pulse (Fig.2.6) and its spectrum (Fig.2.3) will be

addressed in more detail in the following Section.

In the previous Section we described the measurement of group delay by frequency-resolved

cross-correlation. The obtained spectral phase correctly describes the wavelength-integrated trace

presented in this Section. The question remains, however, whether the precision of group delay

estimation is satisfactory. In our approach we captured the gross features of the spectral phase

distortion of the white-light pulse. As was mentioned in Section 2.4, a more complete set of

parameters can be recovered if a FROG-like inversion is applied to up-converted traces. This

technique has been recently called XFROG [85]. It implies that the reference pulse is separately

fully characterized by FROG prior to the inversion of the cross-correlation trace. While this

Chapter 2

38

approach is feasible in our case, it is not entirely welcome since the final result of the continuum

characterization also depends on the error in the separate measurement of the reference pulse.

Instead, in Chapter 4 we will be able to recover refined information on the spectral phase of the

uncompressed, as well as the compressed pulse through a SHG FROG measurement. Lying ahead,

however, is the solution of the difficulties, which were rather unimportant in the case of a narrow

– compared to the probe pulse – reference bandwidth in the spectrally-resolved cross-correlation

experiment. The issues of the frequency mixing of two identical ultrabroad bandwidths will be

examined in detail in Chapter 3.

2.6 Fiber output: experiment vs. numerical simulations

In a single-mode fiber, spectral broadening occurs due to the self-phase modulation (SPM), while

a combination of SPM and normal (or positive) group velocity delay (GVD) acts to smoothen the

chirp [19]. The dynamic evolution of a pulse propagating in a single-mode fiber is described by

the nonlinear Schrödinger equation (NSE) [91]. When only SPM and group-delay dispersion are

considered, the solution of the NSE yields a symmetric power spectrum, which corresponds to a

symmetric rectangular-like pulse in the frequency domain and an almost linear chirp over most of

the pulse duration [19]. It has been shown that linear frequency chirp, corresponding to a parabolic

spectral phase, can be compensated by a quadratic compressor [19].

However, experiments [22] and numerical studies [92-96] have shown that higher-order

dispersion and nonlinearities become increasingly important for propagation of femtosecond

pulses, even for fibers shorter than 1 cm. In order to account for the intensity dependence of the

group velocity, the conventional NSE should be extended to include a nonlinear correction term

involving the time derivative of the pulse envelope, the so-called optical shock term [92]. This

means that the part of the pulse that has the highest peak intensity, moves at a lower speed than the

low-intensity wings. This effect, named self-steepening, causes pulse asymmetry and has been

widely discussed in the literature (see, for example Chapter 4 of ref. [91] and references therein).

In absence of mechanisms that stabilize this self-steepening process, the latter leads to an infinitely

sharp pulse edge that creates an optical shock, similar to the development of an acoustic shock on

the leading edge of a sound wave. Moreover, in this case the spectral phase of the pulse undergoes

fast fluctuations that are difficult to compensate in a compressor.

Significant progress in numerical modelling of pulse propagation in fibers was made by

taking into consideration both the optical shock term and higher-order dispersion [92-95]. It was

shown that these two effects acting together suppress severe oscillations in the chirp. The

predicted strongly asymmetric pulse shape and power spectrum agree reasonably well with the

measured properties of our white-light pulse\1. The nonlinearity of the chirp near the leading edge

\1 The results of simulations most relevant to our experiments, are presented in ref. [94] in Fig.2 (pulseshape and chirp) and Fig.12 (pulse spectrum and spectrum phase).


39

of the pulse fully agrees with our measurements. It is worth noticing that despite the fact that the

spectrum is asymmetric, the bandwidth introduced by the higher-order terms can effectively be

used to obtain pulses shorter than those from the purely SPM-broadened spectra [95].

2.7 Compressor design

A light pulse broadened by SPM action in a fiber and by propagation through bulk material, can

be compressed by passing it through a suitable optical element with anomalous (or negative)

dispersion [97]. The group delay (or the spectral phase) is conventionally expanded into a Taylor

series around a central frequency ω0 [25]:

K+−ϕ ′′′′+

−ϕ ′′′+−ϕ ′′≅ϕ

=

300

20000

))((6

1

))((2

1))((

)(~)(

0

ωωω

ωωωωωωωωωτ

ωd

d

(2.9)

where )(ωτ is the group delay, )( 0ωϕ ′′ is the group delay dispersion (GDD), )( 0ωϕ ′′′ is the

third-order dispersion (TOD), )( 0ωϕ ′′′′ is the fourth-order dispersion (FOD), etc. Note that a

constant (non-frequency dependent) group delay has been disregarded in Eq.(2.9). This equation

shows that in first order one should match the GDD of the compressor to the GDD of the pulse, in

second order the TOD’s of pulse and compressor should be matched, and so on.

The quest for optical pulse compression emerged soon after the invention of sub-nanosecond

lasers. The first report on extracavity pulse compression concerned a mode-locked He-Ne laser

[98]. Over the past three decades a number of compressors have been proposed and successfully

implemented: resonant Gires-Tournois interferometers (GTI’s) [99], resonant vapour delay lines

[17], diffraction gratings [20,100] and prism pairs [10,23]. In particular, a combination of gratings

and prisms [24,25] was triumphantly used to achieve 6-fs pulses [25]. This compressor can

compensate for both GDD and TOD over a very broad spectral range [101]. Recently chirped

mirrors [102] revolutionized the technology of ultrashort pulse generation.

Design of an appropriate high-throughput pulse compressor becomes increasingly difficult

for larger bandwidth of the chirped pulse. In addition, the spectral region over which any of the

aforementioned compressors provides adequate phase compensation, narrows rapidly with the

increase of the chirp rate. The requirements for compression of the white-light continuum arise

from the fact that both GDD and TOD are positive as is evident from Fig.2.5. Therefore, one

should aim for a compressor that exhibits both negative GDD and negative TOD. In our previous

experiments [57], the spectral range of the prism-grating compressor was broadened by careful

balancing the GDD against the FOD, such that nearly transform-limited 5-fs pulses were obtained.

This seems to represent the current limit of this technique for pulses chirped in a fused-silica fiber.

Moreover, an oscillatory residual phase remained – as a trade-off between phase corrections of

different orders – which led to sidelobes on the 5-fs pulse. Another inherent drawback of the

grating-prism compressor is its low throughput, typically ~25% [57]. Note that a 200-nm

Chapter 2

40

bandwidth of recently designed high-efficiency (~90%) gratings [103] is not sufficient for pulse

compression down to 5 fs.

0

100

200

3001000 800 600

0.0

0.5

1.0Prism cut-off

45° prism

compressor(a)

Wavelength [nm]

0

10

20

30

40

Gro

up d

elay

[fs]

(b)

Gires-Tournois Interferometer

0.0

0.5

1.0

Tra

nsm

issi

on /

Ref

lect

ion

0

5

10

15

(c)

Chirped

mirror

0.0

0.5

1.0

1000 800 600

0.0

0.5

1.0Overcoated silver mirror

Wavelength [nm]

0

1

2

3

(d)

Tra

nsm

issi

on /

Ref

lect

ion

Gro

up d

elay

[fs]

0.0

0.5

1.0

0

10

20

30

40

(e)

Beam splitter

0.0

0.5

1.0

0

10

20

30

40

(f)

1 m of air

0.0

0.5

1.0

10000 12500 150000

200

400

Gro

up d

elay

[fs]

(g)

Energy [cm-1

]

Total

0.0

0.5

1.0

Tra

nsm

issi

on

1000 800 600

0.0

0.5

1.0

Wavelength [nm]

10000 12500 15000

Energy [cm-1]

10000 12500 15000

Energy [cm-1

]

Fig.2.7: Overview of optical elements used in the compressor and autocorrelator: 45° fused silica prismcompressor (a), chirped mirror (b), Gires-Tournois interferometer (c), overcoated silver mirrors (d), 1 mof air (e), beam splitter in the autocorrelator (f) and total compressor (g). Group delays of various dispersivecomponents are indicated by solid lines (left axis) while dotted lines show transmittance or reflectance (rightaxis). The compressor itself comprises three parts: a prism pair, chirped mirrors and Gires-Tournoisinterferometers. Solid circles in (g) are experimentally measured group delay depicted with the reversed signand used as the desired group delay of the compressor. Reflection on the beam splitters is not taken intoaccount in the overall throughput. The interprism pathlength in air is included in the data for the prismcompressor.

A major advance in pulse compression technology was made by the introduction of a

compressor based on chirped mirrors and prisms [104]. In contrast to gratings, chirped mirrors

can be made that have a large acceptance bandwidth and a very high reflectivity at the same time.

With this prism-chirped mirror compressor, pulses of 20-fs were amplified to the millijoule level

[105]; more recently a similar compressor was used to generate 5.5-fs, 6-nJ pulses at 1-MHz

repetition rate [57] and sub-5-fs, 20-µJ pulses at a 1-kHz repetition rate [58].


41

To improve on our previous compression scheme [57], we designed a novel high throughput

compressor. To obtain the required negative GDD and FOD a fused-silica prism compressor was

used (Fig.2.7a), which, however, overcompensates the TOD when used alone. Recently it was

shown that ultra-broadband chirped mirrors can be made that exhibit negative GDD and positive

TOD (Fig.2.7b), while having a reflectivity exceeding 99% over a bandwidth of 600-1100 nm

[70,106]. This means that a combination of chirped mirrors and a prism compressor provides

flexible control over TOD across a large spectral range [70]. For higher-order phase corrections,

broadband dielectric GTI’s [107-109] have been shown to be suitable (Fig.2.7c). GTI’s counteract

the FOD of the prism pair, which becomes significant above 900 nm. These ideas lead us to a

compressor design that consists of a prism pair, ultra broadband chirped mirrors, and dielectric

GTI’s.

We employed dispersive ray-tracing analysis [101,110] to compute the group delay of the

three-stage compressor. The use of Eq. (2.9) to calculate the spectral phase by a Taylor expansion

becomes impractical since the compressor should span the region from 600 to 1100 nm.

Wavelength-dependent refractive indices were calculated from dispersion equations while

corresponding refraction angles in the prism compressor were obtained by using Snell’s law.

Subsequently, the total accumulated phase of the prism compressor and bulk material was

computed at each wavelength. By numerical differentiation of the phase, the group delay of the

prism part of the compressor was obtained. This group delay was added to the group delay of the

reflective optics to compute the overall group delay of the compressor. The resulting group delay

COMPRτ is than compared to the measured group delay of the white-light continuum WLCτ , but taken

with opposite sign. Subtracting the calculated group delay from the desired, we find the residual

group delay

)()()( ωτωτωτ WLCCOMPRRES −= , (2.10)

by integration of which we obtain the residual spectral phase )(ωRESϕ . To further characterize

the compressor performance, the input white-light spectrum (Fig.2.3) modified by the compressor

throughput (Fig.2.7g, dotted curves) is calculated. Taking into account the residual phase, the

temporal shape and phase of the compressed pulse is then computed via a Fourier transformation.

An overview of all optical elements used in the compressor and autocorrelator is presented

in Fig.2.7. The previously employed [57] unprotected gold-coated mirrors with 90% peak

reflectivity and rapidly growing absorption below 600 nm, were substituted by low dispersion and

higher reflectivity overcoated silver mirrors (Fig.2.7d). The dispersion due to propagation in air

[111] was also found to play an essential role for 5-fs pulses (Fig.2.7e).

The compressor performance is optimized by varying the number of reflections on the

dispersive mirrors, by changing the interprism spacing and by varying the prism apex angles.

Optimal performance is judged by looking for the shortest pulse through second harmonic

generation in the autocorrelator. Hence, the pathlength in air from the compressor output, the 0.5-

mm thick beam splitter at 45° incidence (Fig.2.7f) and reflections off the autocorrelator mirrors

should be included in calculations as well. Pulse broadening due to dispersion inside the

Chapter 2

42

autocorrelator nonlinear crystal was not considered because of its negligible effect. Reflectivity

curves (Fig.2.7, dotted lines) and group delays (Fig.2.7, solid lines) of the chirped mirrors, GTI’s,

the overcoated silver mirrors and beam splitters were provided by the manufacturer (R&D Lezer-

Optika, Hungary).

The angle of incidence onto the fused-silica [112] prism, being a sensitive parameter, was

chosen to correspond to the least deviation angle for the sake of experimental convenience.

Simulations show that the apex angles smaller than 45° are impractical because they call for

unreasonably large interprism separation. Moreover, with increased prism separation the positive

dispersion of air (Fig.2.7e) between the prisms becomes more important so that the whole

compressor would need to be put in a vacuum. Note, that the amount of the TOD could also be

reduced by employing doubled-prism pairs as has been demonstrated previously [58,65].

Optimal compression (Fig.2.7g) was obtained for 5 reflections of the chirped mirrors,

2 reflections of GTI mirrors, and use of a 45° prism compressor with the following settings:

~5.2 mm of prism material for the 800-nm wavelength ray and ~115 cm distance between apices.

The root mean square error of the residual group delay amounts to ~1.5 fs. The blue wavelength

cut-off of the prism compressor coincides with the abrupt reflectivity drop of the chirped mirrors,

thus no additional loss of the spectral content originates from the prism part of the compressor. The

compressor throughput is fairly flat between 600 and 1100 nm and amounted, at the beginning, to

~75%, mainly due to eight reflections from the non-Brewster-angle prisms. When a low-dispersive

anti-reflection coating is deposited on the surfaces of the prisms, the total compressor throughput

reached ~90%.

The Fourier transform of the compressor output spectrum assuming constant phase, yields

a pulse of ~4.2 fs in duration, i.e. somewhat longer than the Fourier transform of the input spectrum

(Fig.2.3, inset). This lengthening of the pulse occurs due to the loss of spectral components in the

near infrared and visible part of the continuum (Fig.2.7g, dotted curve). Residual phase correction

should be feasible by installation of a programmable phase mask [113] into the pulse compressor.

The applicability of this technique has recently been demonstrated for pulses as short as 10 fs

[114]. Spectral shaping would also allow manipulating of the spectrum leading to cleaner optical

pulses [115].

With the compressor being set up near the cavity-dumped laser and white-light generator

(Fig.2.1) the overall size of the system is 1×1.5 m2. This compactness makes our sub-5-fs laser

system extremely robust and ensures that the cavity alignment is retained for a long time. When

starting up the laser the only thing needed is to correct for the sub-micron drift of the fiber tip. Due

to the short warming-up time of the diode-pumped “Millennia”, the stable regime of sub-5-fs

operation is achieved within minutes. The compactness of the laser source presents a distinct

advantage in experiments because it allows building the experimental setup close to the laser

thereby limiting pulse propagation through air.

2.8 Pulse duration measurement

Accurate pulse-width measurement of pulses containing only a few oscillations is quite a


43

challenge. An easy and informative method to judge the compression quality is the second-order

interferometric autocorrelation (IAC) [97,116,117]. An additional benefit from this technique is

that it can be used as an on-line tool. Of course, the technical demands to be made for a 5-fs

autocorrelator are substantial. In our experiments we employ a Mach-Zehnder interferometer

[57,118-120], which has the advantage of being fully symmetric with respect to both arms. Note

that the “magic” 0 : 1 : 8 ratio between the minimum, the asymptotic level and maximum of the IAC

trace [116] is obtained only if the intensities of two interfering beams are strictly equal. If one

intensity exceeds the other one by a factor of β, the re-normalized ratio becomes

8:)1/(64:0 ββ +− with the asymptotic level being between 1 and 4. Imperfectness in

alignment of the interferometer leads to the same result.

Fig.2.8: Schematic of Mach-Zehnder interferometer for measurements of interferometric autocorrelation.BS1, BS2: 50% ultra-broadband beam splitters centred at 800 nm; M1-M4: flat low dispersion overcoatedsilver mirrors; M5: r = -10 cm, low dispersion overcoated silver mirror; BBO: 15-µm thick BBO crystal;M6: r = -10-cm protected aluminum coated mirror; PD: photodiode; PMT: photo multiplier tube; PZT:piezo transducer; HVA: high voltage amplifier; DAC: digital-analog converter; ADC: analog-digital converter.All optics were obtained from R&D Lezer Optika, Budapest.

The input beam is split and recombined in such a way that each of the beams travels once

through an identical beam splitter while both reflections occur on the same coating-air interfaces

(Fig.2.8). To match the beam splitters [46], the initial horizontal polarization of the compressed

pulse is rotated by a periscope. A 15-µm BBO crystal is used for second-harmonic generation.

Such a thin crystal is required to avoid dispersion-induced pulse broadening and to ensure a

sufficiently broad phase-matching bandwidth.

The moving arm of the interferometer is driven by a piezo transducer (PZT) which is

controlled by a computer via a digital-analog convertor (DAC) and a high voltage amplifier

(HVA). After having moved the M3-M4 arm to a new position, the measurement of the second

Chapter 2

44

harmonic intensity is performed by sampling and digitalization of the photomultiplier (PMT)

signal. The experimental points obtained in this way are depicted in Fig.10 by open circles. To

introduce an on-line calibration of the time axis, a He-Ne laser beam is aligned in a direction

opposite to the white light. The signal of the photodiode (PD) monitoring the interference fringes

at the wavelength of He-Ne laser [121] is used for precise time calibration (Fig.2.9, lower

panel).This allows autocorrelation measurements to be performed with ~0.2 fs accuracy

throughout the whole scanning region of ~100 fs. A typical time step is ~0.1 fs, or 23 points per

oscillation period at 800-nm wavelength at the rate of ~60 ms for a 100-fs scan.

The typical IAC shown in Fig.2.9 was obtained by setting the compressor according to the

calculated optimal settings, whereupon the amount of prism material was balanced so as to get the

shortest autocorrelation. Compared to our earlier result [57], the wing structure of the IAC is

substantially reduced, which demonstrates the superior characteristics of this compressor. We

verified the importance of the GTI’s for high-order dispersion correction by changing the angle

of incidence from the design angle of 45o to ~15o. In this case the group delay curve shifts toward

shorter wavelength (Fig.2.7c) resulting in broadening of the central part of the IAC function and

an appreciable increase of the amplitude in its wings.

-20 0 200

1

4.6 fs

Inte

nsit

y [a

rb. u

nits

]

Time [fs]

-30 -20 -10 0 10 20 30

0

2

4

6

8

Inte

nsity

[ar

b. u

nits

]

Delay [fs]

Fig.2.9: Interferometric autocorrelation (IAC) of the compressed pulse. Open circles: experimental points.solid line: calculated IAC of deduced pulse shape shown in the inset. Bottom panel depicts He-Ne laserinterference fringes used for on-line time calibration.

When fitting the IAC to a hyperbolic secant envelope, we get a pulse of ~3.7 fs, a Gaussian

a pulse of ~4.4 fs is obtained. The former value clearly violates the earlier derived spectral-

limited pulse duration of ~4.2 fs (Section 2.7). Furthermore, neither of these pulse shapes

reproduces the wing structure on the experimental IAC. This clearly indicates that one should be

extremely cautious about fitting the IAC of a short pulse to an a priori assumed pulse profile,

especially when the pulse spectrum is not smooth. It should also be noted that the standard

deviation, conventionally used in fitting routines to judge the fit quality [122], can hardly serve as

a criterion in favor of any particular pulse shape. Most of the experimental points in the IAC are


45

located at the slopes of the fringes where the gradient is too high to recognize any anomaly. As a

matter of fact, only 8-10 points at the extrema of the IAC are meaningful which clearly is not

sufficient to discriminate between the different pulse profiles. We will address the problem of

retrieving the pulse shape from the IAC in the next Section.

2.9 Reconstruction of 5-fs pulse from the IAC and spectrum

In the previous Section we showed that a fit of the interferometric autocorrelation (IAC) to an a

priori analytical pulse intensity profile is not warranted. Clearly, it would be a major step forward

if the pulse shape could be retrieved from the IAC without having to rely on any assumption

concerning the temporal profile of the electric field. It was pointed out by Naganuma et al. [73]

that information on the phase and amplitude of the pulse is, in principle, contained in the IAC and

pulse spectrum. Several algorithms have been applied over the years to treat the problem of pulse

reconstruction from its autocorrelation [123-128]

The normalized interferometric autocorrelation signal can be expressed as [73]:

( ) [ ][ ])2exp()(Re

)exp()(Re421)(

02

01

τωττωτττ

iF

iFGIAC

−+

++=, (2.11)

where, for the sake of clarity, in the expression of the complex electric field in the time domain

we separated the term oscillating at the carrier frequency ω0. The constituent terms of the sum in

Eq.(2.11) are:

∫ −= dttItIG )()()( ττ , (2.12)

∫ −−+

= dttEtEtItI

F )()(2

)()()( *

1 ττ

τ , (2.13)

∫ −= dttEtEF )()()( 2*22 ττ . (2.14)

Here )(τG stands for the (background-free) intensity autocorrelation, and )(2 τF represents the

second harmonic field autocorrelation. Note that when the temporal phase is a constant thefunctions )(τG and )(2 τF become identical [73]. This property can be exploited to determine

whether the compressed pulse carries any residual chirp. Since the carrier frequencies of )(τG

and )(2 τF are different, the simplest way to extract this information from the IAC (Fig.2.9) is to

Fourier transform the IAC, with the constant background (unity level in Eq.2.11) subtracted. One

then obtains a spectrum composed of )(~

ωG at zero frequency, )(~

1 ωF at the fundamental frequency

ω0, and )(~

2 ωF at the second harmonic frequency 2ω0 (Fig.2.10). Note that since the latter two

components are projected at both positive and negative frequencies, their magnitudes are reduced

Chapter 2

46

by a factor of two compared to the ratios given by Eq.(2.11). As can be seen from Fig.2.10, the

functions )(~

ωG and )(~

2 ωF are quite similar, which confirms our earlier conclusion that the

compressor has removed most of the chirp in the white-light continuum. Nonetheless, the smallasymmetry of )(

~2 ωF indicates that there is some residual chirp in the compressed pulse\1.

Peatross et al. recently demonstrated temporal decorrelation of intensity autocorrelationfunction )(τG [125,126] which yields the modulus of the pulse electric field in the time domain.

Combining this data with the modulus of the electric field in the frequency domain (square foot of

spectral intensity), in the second stage of their two-stage algorithm they extract phase information

that corresponds to these two moduli.

-30000 0 12500 25000

Am

plitu

de

×4

F2(ω)

F1(ω)G(ω)

Energy [cm-1]

Fig.2.10: Fourier-transform of the experimental interferometric autocorrelation function. The mirror imageof the spectrum at the negative frequencies is not shown. The close similarity between the zero and double-frequency peaks indicates that the compressed pulse is almost chirp-free.

Concisely, the problem of phase retrieval from a collinear (IAC) or non-collinear (intensity)

autocorrelation and the fundamental spectrum is summarized in the following integral equations

written for the Fourier transforms of )(τG and )(2 τF , which are denoted here as )(~

ωG and

)(~

2 ωF :

[ ]2

)(~)'(~exp)()()(~

ωωωωωωωω ′−′ϕ−ϕ−′′= ∫ dSSG , (2.15)

\1 Note that there is a distinct difference between transform-limited (or spectral-limited) and chirp-freepulses. For instance, in the case of an asymmetric spectrum, a transform-limited pulse does carry somechirp, even if its spectral phase is constant.


47

[ ]2

2 )(~)'(~exp)()()(~

ωωωωωωωω ′′−ϕ+ϕ′−′= ∫ dSSF , (2.16)

where 2

)(~

)( ωω ES = is a fundamental spectrum measured by a spectrometer or, alternatively,

Fourier-transformed from an interferogram [73], and )(~ ωϕ is the unknown spectral phase. As can

be seen from the structure of these equations, Eq.(2.15) contains a modulus square of the

autocorrelation of the frequency-domain electric field, while Eq.(2.16) contains a modulus square

of its autoconvolution. Regretfully, the knowledge of only a modulus of the autoconvolution

prevents the possibility of a straightforward deconvolution of the complex electric field. In fact,)(

~2 ωF is equivalent to a directly measured second-harmonic spectrum of the pulse. While the

fundamental and the second harmonic spectra in principle uniquely define the amplitude and phase

of a pulse (with time-direction ambiguity) [124], the influence of different spectral phases on theshape of )(

~2 ωF is frequently only very minute. On the other hand, the profile of )(

~2 ωF is

critically affected by the frequency-dependent conversion of the fundamental field into the second

harmonic radiation (See Section 3.6 for the detailed explanation of the spectral filtering effect).

Like )(~

2 ωF , the term )(~

ωG is also susceptible to spectral filtering. However, the recognition of

different pulse shapes from )(~

ωG is much more reliable, compared to )(~

2 ωF .

To deduce the parameters of the compressed pulse, we applied a two-stage phase retrieval

algorithm [126,129] to the Fourier transform of the experimental data from Fig.2.9. The resulting

temporal intensity profile is depicted in the inset to Fig.2.9. The IAC corresponding the retrieved

complex electric field is shown as a solid curve alongside the experimental trace. Since our

procedure relied only on the treatment of the IAC part corresponding to the intensityautocorrelation, )(τG , it is interesting to see how the calculated terms )(1 τF and )(2 τF comply

with the overall measured IAC trace. Clearly, certain discrepancies appear around ±10-fs delay

(Fig.2.9). Their most likely explanation is in the mentioned above spectral filtering effect, which

was not considered in the employed phase retrieval routine.

2.10 Pitfalls of IAC

IAC, or collinear autocorrelation, measurement has a number of advantages and disadvantages.

For example, the background-free autocorrelation or FROG measurement of sub-10-fs pulses in

the non-collinear beam arrangement becomes difficult because of the need to overlap the

intersecting beams at the point where the beam waist is the smallest. This is required to minimize

the geometrical smearing effect, or delay blurring, (See Section 3.5). In this respect, the collinear

alignment is much easier and IAC does not suffer from delay blurring. Unlike the intensity

autocorrelation that is determined solely by the temporal intensity shape, whatever the phase is,

IAC is phase-sensitive [116]. IAC also possesses a certain degree of intuitiveness in the form of

fringes oscillating at the carrier frequency. This can almost directly be related to the number of

optical cycles contained by the electric field of the pulse.

However, the very merit of IAC – its fringes – may turn against it. Unlike the background-

Chapter 2

48

free autocorrelation, in which a very high dynamic range may be achieved, the IAC asymptotically

approaches its background (ideally 1/8 of its peak value) in oscillatory fashion. Importantly, these

oscillations are on both sides of the background level. As can be seen from Eq.(2.11), the terms

comprising an IAC, have different periods of oscillations. Therefore, some IAC fringes may cancel

giving a pleasing aesthetic appearance of the trace that in reality does not correspond to a nice

pulse. The “lucky” interplay of the fringes is especially easy to achieve when there are merely 3-5

significant fringes on the IAC trace.

By its nature, a collinear fringe-resolved autocorrelation cannot be single-shot. Thus, to

avoid the “mop-up” of the IAC wings by the statistical averaging of the fringes, the mechanics of

the autcorrelator should be engineered in the way ensuring reliable interferometric stability. This,

however, cannot prevent such fringe averaging if the phase and/or the spectrum of the laser output

fluctuate in time. Obviously, the statistical sum of slightly different IAC’s may produce an

unpredictable result.

As has been pointed out by R. Trebino [130], the inhomogeneity in the spatial distribution

amplitude-phase characteristics across the beam also may result in the fringe averaging effect.

Figure 2.11 gives an example of an IAC of a 4.6-fs pulse that is spectrum-limited and has a top-hat

spectrum (Fig.2.11a, inset). The solid lines in Fig.2.11a and 2.11b represent the ideal IAC and

intensity autocorrelation, respectively. The laser beam is assumed to be Gaussian, and all

frequency components have identical sizes. We now model spatial chirp by assuming that at the

position of the SHG crystal the beam is dispersed linearly with frequency in the horizontal

direction to ~1.7 of its vertical size. Therefore, the spectrum is somewhat red-shifted on one side

of the beam, and blue-shifted on the other. The spectrum at each point across the beam in horizontal

direction now corresponds to a slightly longer pulse than the one obtained from the total spectrum,

as reflected by a slight broadening of the background-free autocorrelation (Fig.2.11b, dotted line).

However, the period of oscillation of IAC traces corresponding to different points across the beam

now vary by about a third of the period length. This efficiently damps oscillations in the overall

resulting IAC depicted as dotted line in Fig.2.11a. beginning with the third fringe. Clearly, the

“new” IAC gives the impression of a wing-free pulse.

The situation discussed above is not unrealistic. For instance, it can easily take place in

presence of chromatic aberrations in the focusing into the SHG crystal. The described situation is

also likely to occur following a small angular “adjustment” of a prism or grating compressor while

judging the perfectness of compression by the quality of IAC. Unbalancing the arm-length in a

prism or grating compressor would lead to the same effect. Even if the identical spectra are

measured across the beam profile, there is no guarantee that the spectral phase is identical in all

points. This compromises the use of IAC unless it is backed up by other pulse measuring methods.


49

10000 150000

1

Inte

nsity

[ar

b. u

nits

]

Energy [cm-1]

-30 -20 -10 0 10 20 300

2

4

6

8

(a)In

tens

ity

[arb

. uni

ts]

Delay [fs]-30 -20 -10 0 10 20 30

0

1(b)

Inte

nsit

y [a

rb. u

nits

]

Delay [fs]

Fig.2.11: Simulation of spatial-chirp effect on the autocorrelation measurement via second-harmonicgeneration. (a) collinear autocorrelation. (b) background-free autocorrelation. Solid curves represent idealautocorrelation traces of the pulse, the spectrum of which is shown in the inset. Dotted curves showcomputed autocorrelations in presence of spatial chirp.

In the previous Section we also hinted at the importance of the spectral filtering effect which

we so far disregarded in our measurement of the compressed pulse. In combination with the factors

described in this Section, this might have caused under- or over-estimation of the pulse

characteristics. To solve all these problems and obtain a rigorous amplitude-phase

characterization, in Chapter 4 we return to the measurement of the compressed pulse by means of

frequency-resolved non-collinear autocorrelation (FROG).

Summarizing our ideas about IAC, we suggest that great caution should be exercised in

dealing with it and that circumstances potentially jeopardizing its validity should carefully

examined. In any case, the quality of the pulse reconstruction should be judged by the ability to

reproduce the wing structure of the measured IAC trace.

2.11 Summary and outlook

In this Chapter we have discussed a compact and robust light source that generates sub-5-fs, 2-MW

pulses at variable repetition rates of up to 1 MHz, using a novel three-stage compressor. The phase

characteristics of the compressor have been analyzed using dispersive ray tracing and mapped unto

the measured group delay of the continuum. The fidelity of this approach has been confirmed by

the fact that the pulse shape derived from the optical spectrum and the calculated residual phase,

fit the measured autocorrelation function very well. It has also been shown that the interferometric

autocorrelation and optical spectrum of the compressed pulse comprise sufficient information to

derive the temporal pulse intensity and its phase.

We foresee several applications of this ultrafast laser. First, it is an almost ideal tool for

ultrafast spectroscopy, if not for the short pulse then for the white light continuum that can be used

as a probe for spectral events from the blue-green part of the spectrum to the near infrared region

(500 nm to 1.3 µm). The large bandwidth of this laser may also be of use in optical coherence

tomography measurements [72]. For the near future we aim for an all chirped-mirror compressor

Chapter 2

50

which would enable an even more compact design of this laser. With smaller diode pumped light

sources coming on the market there is every reason to believe that soon it will be possible to built

a sub-5-fs cavity-dumped laser that fits onto an a breadboard of only 1 m by 0.5 m. This may be

an important asset for many applications.

Another replica of the 5-fs set-up that is described in this chapter has been built in our

laboratory to adapt to the needs to carry out several parallel projects in nonlinear spectroscopy

in solutions. Both laser systems displayed a very high performance in terms of both long- and

short-term stability over more than two years of their intensive use. Typical cycles of experiments

lasted longer than 24 hours of non-stop data collection. Several interesting systems have been

studied using the white-light pulses, for example, solvation dynamics of small rigid ions [131],

solvent-controlled electron-transfer reaction [132], and photon-echo and pump-probe studies on

equilibrated solvated electron. The latter study is included in Chapters 6 and 7 of this Thesis.

Meeting the demands of particular experiments [132], tunable excitation pulses of about 20-fs

duration and 10-fs probe pulses were tailored from the white light to facilitate two-color

experiments. Considering the overall simplicity of our set-up, favorably distinguishing it from very

complex amplified laser systems with parametric generators for wavelength tuning, this

remarkable versatility in our case was achieved a at relatively low expense.


51

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Chapter 2

All-Solid-State Cavity-Dumped Sub-5-Fs Laser

Abstract

We discuss in detail a compact all-solid-state laser delivering sub-5-fs, 2-MW pulses at repetition

rates up to 1 MHz. The laser system employed is based on a cavity-dumped Ti:sapphire oscillator

the output of which is chirped in a single-mode fiber. The resulting white-light continuum is

compressed in a high-throughput prism/chirped mirror/Gires-Tournois-interferometer pulse

compressor. The preliminary pulse duration measurement is carried out by a collinear fringe-

resolved autocorrelation. The temporal and spectral phase of the sub-5-fs pulses are deduced from

the measured autocorrelation trace and optical spectrum.

Chapter 2

54

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(Rochester, October 20-24, 1996), p. 163.126. J. Peatross and A. Rundquist, J. Opt. Soc. Am. B 15, 216 (1998).127. A. Baltuška, A. Pugžlys, M. S. Pshenichnikov, and D. A. Wiersma, in CLEO'99 (OSA technical

digest, Baltimore, May 23-28, 1999).128. T. W. Yan, Y.-Y. Jau, C.-H. Lee, and J. Wang, in CLEO'99 (OSA technical digest, Baltimore,

May 23-28, 1999).129. A. Baltuška, Z. Wei, M. S. Pshenichnikov, D.A.Wiersma, and R. Szipöcs, Appl. Phys. B 65,

175-188 (1997).130. R. Trebino, private communication (1997).131. M. S. Pshenichnikov, A. Baltuška, R. Szipöcs, and D. A. Wiersma, in Ultrafast Phenomena XI,

edited by T. Elsaesser, J. G. Fujimoto, D. A. Wiersma, and W. Zinth (Springer, Berlin, 1998).132. H. P. den Hartog, A. Baltuška, A. Pugžlys, S. Umapathy, M. S. Pshenichnikov, and D. A.

Wiersma, J. Chem. Phys. (submitted) (1999).

Chapter 3

Second Harmonic Generation Frequency-Resolved

Optical Gating in the Single-Cycle Regime

Abstract

The problem of measuring broadband femtosecond pulses by the technique of second-

harmonic generation frequency-resolved optical gating (SHG FROG) is addressed. We derive

the full equation for the FROG signal, which is valid even for single-optical-cycle pulses. The

effect of the phase-mismatch in the second-harmonic crystal, the implications of the beam

geometry and the frequency-dependent variation of the nonlinearity are discussed in detail.

Our numerical simulations show that under carefully chosen experimental conditions and

with a proper spectral correction of the data the traditional FROG inversion routines work

well even in the single-cycle regime.

Chapter 3

56

3.1 Introduction

Recent progress in complete characterization of ultrashort pulses reflects the growing demand

for detailed information on pulse structure and phase distortion. This knowledge plays a

decisive role in the outcome of many applications. For instance, it has been recognized that

pulses with identical spectra but different spectral phases can strongly enhance efficiency of

high-harmonic generation [1], affect wavepacket motion in organic molecules [2,3], enhance

population inversion in liquid [4] and gas [5] phases, and even steer a chemical reaction in a

predetermined direction [6]. Moreover, a totally automated search for the best pulse was

recently demonstrated to optimize a pre-selected reaction channel [7]. Then, by measuring the

phase and amplitude of the excitation pulses, one can perform a back-reconstruction of

potential surfaces of the parent molecule.

The complete determination of the electric field of femtosecond pulses also uncovers

the physics behind their generation as has been demonstrated in the case of fs Ti:sapphire

lasers [8,9]. Such information is invaluable to determine the ways of and ultimate limits for

further pulse shortening. Last, owing to the great complexity of broadband phase correction

required to produce spectrum-limited pulses with duration shorter than 5 fs [10-13], the

characterization of the white-light continuum as well as compressed pulses becomes

mandatory.

A breakthrough in the full characterization of ultrashort pulses occurred six years ago

with the introduction of frequency-resolved optical gating (FROG) [14,15]. FROG measures

a two-dimensional spectrogram in which the signal of any autocorrelation-type experiment is

resolved as a function of both time delay and frequency [16]. The full pulse intensity and

phase may be subsequently retrieved from such a spectrogram (called FROG trace) via an

iterative retrieval algorithm. Notably, no a priori information about the pulse shape, as it is

always the case for conventional autocorrelation measurements, is necessary to reconstruct

the pulse from the experimental FROG trace.

In general, FROG is quite accurate and rigorous [17]. Because a FROG trace is a plot

of both frequency and delay, the likelihood of the same FROG trace corresponding to

different pulses is very low. Additionally, the great number of data points in the two-

dimensional FROG trace makes it under equivalent conditions much less sensitive to noise

than the pulse diagnostics based on one-dimensional measurements, such as the ordinary

autocorrelation. Last but not least, FROG offers data self-consistency checks that are

unavailable in other pulse measuring techniques. This feedback mechanism involves

computing the temporal and spectral marginals that are the integrals of the FROG trace along

the delay and frequency axes. The comparison of the marginals with the independently

measured fundamental spectrum and autocorrelation verifies the validity of the measured

FROG trace [9,18,19]. To date, FROG methods have been applied to measure a vast variety

of pulses with different duration, wavelength and complexity [20].

SHG FROG in the single-cycle regime

57

A number of outstanding features make FROG especially valuable for the measurement

of extremely short pulses in the range of 10 fs and below.

First, since FROG utilizes the excite-probe geometry, common for most nonlinear

optical experiments, it is ideally suited to characterize pulses that are used in many

spectroscopic laboratories. Unlike other pulse diagnostics [21-25], FROG does not require

splitting of auxiliary laser beams and pre-fabrication of reference pulses. This fact is of great

practical relevance, since the set-up complexity in many spectroscopic experiments is already

quite high [26-32]. Therefore, it is desirable to minimize the additional effort and set-up

modifications that are necessary for proper pulse diagnostics. FROG directly offers this

possibility. Pulse characterization is performed precisely at the position of the sample by

simply interchanging the sample with a nonlinear medium for optical gating. The last point

becomes especially essential for the pulses consisting of only several optical cycles [10-

13,33] currently available for spectroscopy. The dispersive lengthening that such pulses

experience even due to propagation through air precludes the use of a separate diagnostics

device. Thus, FROG is the ideal way to measure and optimize pulses on target prior to

carrying out a spectroscopic experiment.

Second, it is still possible to correctly measure such short pulses by FROG even in

presence of systematic errors. Several types of such errors will inevitably appear in the

measurement of pulses whose spectra span over a hundred nanometers or more. For example,

a FROG trace affected by wavelength-dependent detector sensitivity and frequency

conversion efficiency can be validated via the consistency checks [9]. In contrast, an

autocorrelation trace measured under identical conditions may be corrupted irreparably.

Third, the temporal resolution of the FROG measurement is not limited by the sampling

increment in the time domain, provided the whole time-frequency spectrogram of the pulse is

properly contained within the measured FROG trace. The broadest feature in the frequency

domain determines in this case the shortest feature in the time domain. Therefore, no fine

pulse structure can be overlooked [20], even if the delay increment used to collect the FROG

trace is larger that the duration of such structure. Thus, reliability of the FROG data relies

more on the proper delay axis calibration rather than on the very fine sampling in time, which

might be troublesome considering that the pulse itself measures only a couple of micrometers

in space.

Choosing the appropriate type of autocorrelation that can be used in FROG (so-called

FROG geometries [18,20]), one must carefully consider possible distortions that are due to

the beam arrangement and the nonlinear medium. Consequently, not every FROG geometry

can be straightforwardly applied to measure extremely short pulses, i.e. 10 fs and below. Inparticular, it has been shown that in some ><χ 3 -based techniques (for instance, polarization-

gating, transient grating etc.) the finite response time due to the Raman contribution to

nonlinearity played a significant role even in the measurement of 20-fs pulses [34].

Therefore, the FROG with the use of the second harmonic generation in transparent crystals

Chapter 3

58

[35-37] and surface third-harmonic generation [38], that have instantaneous nonlinearity,

presents the best choice for the measurement of the shortest pulses available to date.

Another important experimental concern is the level of the signal to be detected in the

FROG measurement. Among different FROG variations, its version based on second

harmonic generation (SHG) is the most appropriate technique for low-energy pulses.

Obviously, SHG FROG [35] potentially has a higher sensitivity than the FROG geometries

based on third order nonlinearities that under similar circumstances are much weaker.

Different spectral ranges and polarizations of the SHG FROG signal and the fundamental

radiation allow the effective suppression of the background, adding to the suppression

provided by the geometry. The low-order nonlinearity involved, combined with the

background elimination, results in the higher dynamic range in SHG FROG than in any other

FROG geometry.

In general, the FROG pulse reconstruction does not depend on pulse duration since the

FROG traces simply scale in the time-frequency domain. However, with the decrease of the

pulse duration that is accompanied by the growth of the bandwidth, the experimentally

collected data begin to deviate significantly from the mathematically defined ideal FROG

trace. Previous studies [8,9] have addressed the effect of the limited phase-matching

bandwidth of the nonlinear medium [39] and time smearing due to non-collinear geometry on

SHG FROG measurement which become increasingly important for 10-fs pulses. The

possible breakdown of the slowly-varying envelope approximation and frequency

dependence of the nonlinearity are the other points of concern for the pulses that consist of a

few optical cycles. Some of these issues have been briefly considered in our recent Letter

[40].

In this Chapter we provide a detailed description of SHG FROG performance for

ultrabroadband pulses the bandwidths of which correspond to 3-fs spectral-transformed

duration. Starting from the Maxwell equations, we derive a complete expression for the SHG

FROG signal that is valid even in a single-cycle pulse regime and includes phase-matching in

the crystal, beam geometry, dispersive pulse-broadening inside the crystal and dispersion of

the second-order nonlinearity. Subsequently, we obtain a simplified expression that

decomposes the SHG FROG signal to a product of the ideal SHG FROG and a spectral filter

applied to the second harmonic radiation. Numerical simulations, further presented in this

Chapter, convincingly show that the approximations made upon the derivation of the

simplified expression, are well justified.

The outline of the Chapter is the following: in Section 3.2 we define the pulse intensity

and phase in time and frequency domains. In Section 3.3 the spatial profile of ultrabroadband

pulses is addressed. The complete expression for SHG FROG signal for single-cycled pulses

is derived in Section 3.4. We discuss the ultimate time resolution of the SHG FROG in

Section 3.5. The approximate expression for the SHG FROG signal, obtained in Section 3.6,

is verified by numerical simulations in Section 3.7. In Section 3.8 we briefly comment on

Type II phase-matching in SHG FROG measurements. Possible distortions of the


59

experimental data resulting from spatial filtering, are considered in Section 3.9. Finally, in

Section 3.10 we summarize our findings.

3.2 Amplitude and phase characterization of the pulse

The objective of a FROG experiment lies in finding the pulse intensity and phase in time, that

is )(tI , )(tϕ or, equivalently, in frequency )(~

ωI , )(~ ωϕ . The laser pulse is conventionally

defined by its electric field:

))(exp()()( titAtE ϕ= , (3.1)

where )(tA is the modulus of the time-dependent amplitude, and )(tϕ is the time-dependent

phase. The temporal pulse intensity )(tI is determined as )()( 2 tAtI ∝ . The time-dependent

phase contains information about the change of instantaneous frequency as a function of time

(the so-called chirp) that is given by [41,42]:

t

tt

∂ϕ∂

=)(

)(ω . (3.2)

The chirped pulse, therefore, experiences a frequency sweep in time, i.e. changes frequency

within the pulse length.

The frequency-domain equivalent of pulse field description is:

))(~exp()(~

)exp()()(~ ωϕω≡ω=ω ∫ iAdttitEE , (3.3)

where )(~

ωE is the Fourier transform of )(tE , and )(~ ωϕ is the frequency-dependent (or

spectral) phase. Analogously to the time domain, the spectral intensity, or the pulse spectrum,

is defined as )(~

)(~ 2 ω∝ω AI . The relative time separation among various frequency

components of the pulse, or group delay, can be determined by [42]

ω∂ωϕ∂

=ωτ)(~

)( . (3.4)

Hence, the pulse with a flat spectral phase is completely “focused” in time and has the

shortest duration attainable for its bandwidth.

It is important to notice that none of the presently existing pulse measuring techniquesretrieves the absolute phase of the pulse, i.e. pulses with phases )(tϕ and 0)( ϕ+ϕ t appear to

be totally identical [43]. Indeed, all nonlinear processes employed in FROG are not sensitive

to the absolute phase. However, the knowledge of this phase becomes essential in the strong-

Chapter 3

60

field optics of nearly single-cycled pulses [44,45]. It has been suggested [46], that the

absolute phase may be assessable via photoemission in the optical tunneling regime [47].

In fact, the full pulse characterization remains incomplete without the analysis of

spatio-temporal or spatio-spectral distribution of the pulse intensity. In this Chapter we

assume that the light field is linearly polarized and that each spectral component of it has a

Gaussian spatial profile. The Gaussian beam approximation is discussed in detail in the next

Section.

3.3 Propagation and focusing of single-cycle pulses

The spatial representation of a pulse which spectral width is close to its carrier frequency is a

non-trivial problem. Because of diffraction, lower-frequency components have stronger

divergence compared with high-frequency ones. As a consequence, such pulse parameters as

the spectrum and duration are no longer constants and may change appreciably as the beam

propagates even in free space [48].

We represent a Gaussian beam field in the focal plane as:

ω

+−

ωπω=ω

)(2ln2exp

)(

12ln2)(

~),,(

~2

22

d

yx

dEyxE , (3.5)

where )(ωd is the beam diameter (FWHM) of the spectral component with the frequency ω

and x and y are transverse coordinates. The normalization factors are chosen to provide the

correct spectrum integrated over the beam as measured by a spectrometer:

dxdyyxI ∫ ∫∝2

),,(~

)(~ ωω E (3.6)

We now calculate the beam diameter after propagating a distance z:

2

2 )0,(

21)0,(),(

ω=ω+=ω=ω

zd

czzdzd , (3.7)

where c is the speed of light in vacuum. To avoid the aforementioned problems, we require

diameters of different spectral components to scale proportionally as the Gaussian beam

propagates in free space, i.e.

constzd =ω=ω )0,(2 (3.8)


61

The constant in Eq.(3.8) can be defined by introducing the FWHM beam diameter d0 at the

central frequency ω0 . Therefore, the electric field of the Gaussian beam given by Eq.(3.5)

becomes

ωω+

−ωω

πω=ω

020

22

00

2ln2exp12ln2

)(~

),,(~

d

yx

dEyxE (3.9)

At this point, the question can be raised about the low-frequency components the size of

which, according to Eq.(3.9), becomes infinitely large. However, the spectral amplitude of

these components decreases rapidly with frequency. For instance, the spectral amplitude of a

single-cycle Gaussian pulse with a central frequency ω0 is given by

ωω

−π

−=ω2

0

2

12ln2

exp)(~A (3.10)

Consequently, the amplitude of the electric field at zero frequency amounts to only 0.1% of

its peak value.

The spatial frequency distribution was observed experimentally with focused terahertz

beams [49] and was discussed recently by S. Feng et al. [50]. Note that our definition of

transversal spectral distribution in the beam implies that confocal parameters of all spectral

components are identical:

2ln42ln4

)0,( 020

2 ω=

ω=ω=

dzdb (3.11)

This is totally consistent with the beam size in laser resonators where longer wavelength

components have a larger beam size. The spatial distribution of radiation produced due to

self-phase modulation in single-mode fibers is more complicated. First, the transverse mode

is described by the zero-order Bessel function [51]. Second, near the cut-off frequency the

mode diameter experiences strong changes [52]. However, for short pieces of fiber

conventionally used for pulse compression and reasonable values of a normalized frequency

V [51] it can be shown that a Gaussian distribution given by Eq.(3.9) is an acceptable

approximation. The situation with hollow waveguides [53] is quite different since all spectral

components have identical radii [54].

Another important issue concerns beam focusing which should not change the

distribution of spectral components. Since the equations for mode-matching contain only

confocal parameters [55], the validity of Eq.(3.9) at the new focal point is automatically

fulfilled provided, of course, the focusing remains achromatic.

Chapter 3

62

-2 -1 0 1 2

0

0

(a)

ω -∆ω/2

ω +∆ω/2

Inte

nsity

x [mm]

500 750 1000 1500

(b) x = 0

x = 0.5 mm

x = 1.0 mm

Inte

nsity

Wavelength [nm]0.0 0.5 1.0

0.0

0.5

1.0

(c)

Nor

mal

ized

cen

tral

freq

uenc

y

x [mm]

0.0

0.5

1.0

Norm

alized pulse duration

Fig.3.1: Spatial parameters of an ideal single-cycle Gaussian pulse centered at 800 nm. (a) Spatialintensity profiles of two spectral components that are separated by FWHM ∆ω from the centralfrequency ω0. (b) Intensity spectra as a function of transverse coordinate x. (c) Dependence of pulsecentral frequency (solid curve) and pulse duration (dashed curve) on transverse coordinate x. Thebeam axis corresponds to x=0.

Although Eq.(3.9) ensures that different spectral components scale identically during

beam propagation and focusing, it also implies that the pulse spectrum changes along the

transversal coordinates. Fortunately, this effect is negligibly small even in the single-cycle

regime. Figure 3.1a shows the spatial intensity distribution of several spectral components of

a Gaussian single-cycle pulse with a central wavelength 800 nm. As one moves away from

the beam axis, a red shift is clearly observed (Fig.3.1b), since the higher frequency spectral

components are contained in tighter spatial modes. However, the change of the carrier

frequency does not exceed 10% (Fig.3.1c, solid line), while the variation of the pulse-width is

virtually undetectable (Fig.3.1c, dotted line). Therefore, this kind of spatial chirp can be

disregarded even for the shortest optical pulses.

3.4 The SHG FROG signal in the single-cycle regime

In this Section, the complete equation is derived which describes the SHG FROG signal for

pulses as short as one optical cycle. We consistently include such effects as phase-matching

conditions in a nonlinear crystal, time-smearing effects due to non-collinear geometry,


63

spectral filtering of the second harmonic radiation, and dispersion of the second-order

nonlinearity.

We consider the case of non-collinear geometry in which the fundamental beams

intersect at a small angle (Fig.3.2). As it has been pointed out [39] the pulse broadening due

to the crystal bulk dispersion is negligibly small compared to the group-velocity mismatch.

This means that the appropriate crystal thickness should mostly be determined from the

phase-matching conditions. For instance, in a 10-µm BBO crystal the bulk dispersion

broadens a single-cycle pulse by only by ~0.1 fs while the group-velocity mismatch between

the fundamental and second-harmonic pulses is as much as 0.9 fs.

Fig.3.2: Non-collinear phase matching for three-wave interaction. )(ωk and )( ω−Ωk are the wave-

vectors of the fundamental fields that form an angle α with z axis. )(ΩSHk is the wave-vector of the

second-harmonic that intersects z axis at an angle β.

We assume such focusing conditions of the fundamental beams that the confocal

parameter and the longitudinal beam overlap of the fundamental beams are considerably

longer than the crystal length. For instance, for an ideal Gaussian beam of ~2-mm diameter

focused by a 10-cm achromatic lens the confocal parameter, that is, the longitudinal extent of

the focal region, is ~1.2 mm. This is considerably longer than the practical length of the

nonlinear crystal. Under such conditions wavefronts of the fundamental waves inside the

crystal are practically flat. Therefore, we treat the second harmonic generation as a function

of the longitudinal coordinate only and include the transversal coordinates at the last step to

account for the spatial beam profile (Eq.(3.9)). Note that the constraint on the focusing is not

always automatically fulfilled. For example, the use of a 1-cm lens in the situation described

above reduces the length of the focal region to only 12 µm, and, in this case, it is impossible

to disregard the dependence on transverse coordinates.

We assume that the second-harmonic field is not absorbed in the nonlinear crystal. This

is well justified even for single-cycle pulses. Absorption bands of the crystals that are

transparent in the visible, start at ~200 nm. Consequently, at these frequencies the fieldamplitude decreases by a factor ( ) 001.02ln2/exp 2 ≈π− (Eq.(3.10)) compared to its

maximum at 400 nm. We also require the efficiency of SHG to be low enough to avoid

depletion of the fundamental beams. Hence, the system of two coupled equations describing

nonlinear interaction [56] is reduced to one. The equation that governs propagation of the

Chapter 3

64

second harmonic wave in the +z direction inside the crystal can be obtained directly from

Maxwell’s equations [57]:

),(')',()'(),( 22

2

02

2

002

2

tzPt

dttzEttt

tzEz

t

SHSH><

∂∂

µ=−ε∂∂

µε−∂∂ ∫

∞−

, (3.12)

where ),( tzESH is the second harmonic field, µ0ε0=1/c2, ε is the relative permittivity, and

),(2 tzP >< is the induced second-order dielectric polarization. By writing both ESH(z,t) and

P<2>(z,t) as a Fourier superposition of monochromatic waves, one obtains a simple equivalent

of Eq.(3.12) in the frequency domain:

),(~

),(~

)(),(~ 22

02

2

2

ΩΩµ−=ΩΩ+Ω∂

∂zPzEkzE

zSHSHSH

>< , (3.13)

where ),(~

ΩzESH and ),(~ 2 Ω>< zP are Fourier transforms of ),( tzESH and ),(2 tzP >< ,

respectively, Ω is the frequency and )(ΩSHk is the wave-vector of the second harmonic field:

)(~)( 0022 ΩεµεΩ=ΩSHk , with )(~ Ωε being the Fourier-transform of the relative

permittivity )(tε .

In order to simplify the left part of Eq.(3.13), we write the second harmonic field as a

plane wave propagating along z axis:

))(exp(),(~

),(~

zikzzE SHSHSH ΩΩ=Ω E , (3.14)

whence Eq.(3.13) becomes:

( )zikzPzz

zz

ik SHSHSHSH )(exp),(~

),(~

),(~

)(2 2202

2

Ω−ΩΩµ−=Ω∂∂

+Ω∂∂

Ω ><EE (3.15)

So far we have made no simplifications concerning the pulse duration. Now we apply

the slowly-varying amplitude approximation [57], i.e.

),(~

)(2),(~

ΩΩ<<Ω∂∂

zkzz SHSHSH EE , (3.16)

in order to omit the term ),(~

2

2

Ω∂∂

zz

SHE .

Note, that the use of the time-domain description of the signal wave propagation results

in a second-order differential equation, similar in its structure to Eq.(3.15). Unlike Eq.(3.15),

though, simplification of the time-domain expression requires a rejection of the second-order


65

temporal derivative of the envelope, i.e. )(4

)(2

2

tEtT

tEt per ∂

∂π∂∂

<< , where perT is the

characteristic period of light oscillation. Such a move implies the assumption of the slow

envelope variation as a function of time. This condition is not fulfilled for the pulses that

carry only a few cycles, since the change of the envelope within one optical period is

comparable to the magnitude of the envelope itself. Brabec and Krausz [58], who explored

the time-domain approach for the propagation of nearly monocycle pulses, found out that the

rejection of the second-order derivative term is warranted in the case when the phase and the

group velocities of light are close to each other. To this point we notice that the application of

non-equality (3.16) to the frequency-domain Eq.(3.15) does not require any assumptions on

the change of the temporal envelope altogether. Therefore, non-equality (3.16) is safe to

apply even to monocyclic pulses, provided there is no appreciable linear absorption at lengths

comparable to the wavelength. The only point of concern is related to the lowest frequencies

for which kSH becomes close to zero. However, as we have already mentioned in Section 3.3,

the amplitude of such components does not exceed 0.1% of the maximum and therefore can

be disregarded. Consequently, Eq.(3.15) can be readily solved by integration over the crystal

length L:

∫ Ω−ΩΩ

Ωµ=Ω ><

L

SHSH

dzzkzPn

ciL

0

20 ))(exp(),(~

)(2),(

~E (3.17)

where )(~)( Ωε=ΩSHn is the refractive index for the second harmonic wave. Now we should

calculate the second-order polarization ),(~ 2 ΩzP >< . We assume that two fundamental fields

cross in the xz plane at a small angle 2α0 (Fig.3.2). The inclination with the z axis of eachbeam inside the crystal is then [ ] )(sin)(arcsin)( 00 ωα≈αω=ωα nn . We denote the relative

delay between the pulses as τ. An additional delay for off-axis components of the beam due

to the geometry can be expressed for a plane wave ascxcxcxnx //sin/)(sin)()( 00 α≈α=ωαω=τ′ for the beam propagating in +α direction,

and cxx /)( 0α−≈τ′ for the beam in -α direction. The electric fields in the frequency domain

can be found via Fourier transforms:

( )( ))/(exp)(

~)(

~)/(exp)(~)(~

02

01

ταωωω

αωωω

−−=

=

cxiEE

cxiEE(3.18)

In order to calculate the second-order dielectric polarization induced at frequency Ω by the

two fundamental fields, we should sum over all possible permutations of fundamental

frequencies:

Chapter 3

66

( )

( ) ( )

( )[ ] ,)/2()()(exp

)(~

)(~

,,~)/(exp

)(~

)(~

,,~),(~

0

20

2122

ωατωωω

ωωωωχατ

ωωωωωχ

dcxzkzki

cxi

dEEzP

zz ++−Ω+

×−Ω−ΩΩ+Ω=

−Ω−ΩΩ=Ω

∫∫

><

><

EE

><

(3.19)

In Eq.(3.19) we included frequency-dependence of the nonlinear susceptibility( )ω−ΩωΩχ >< ,,~ 2 and represent the fundamental field analogously to Eq.(3.14). The electric

field of the second harmonic therefore becomes

( ) ( )

ω

ω−Ωω∆

α

+τω+ω−Ωω∆

ωω−Ω

×ω−ΩωΩχα+τΩΩΩµ

=Ω ∫ ><

dLk

c

xi

Lki

cxin

LciLSH

2

),(sinc

2

2

),(exp)(

~)(

~

,,~)/(exp)(2

),(~

0

20

0

EE

E

(3.20)

where ∆ Ωk( , )ω ω− is the phase mismatch given by the equation:

( ) ( ) ),(cos)()(cos)()(cos)(),( 2010 ω−ΩωβΩ−ω−Ωαω−Ω+ωαω=ω−Ωω∆ SHknknkk ,

(3.21)

with n1 and n2 being the refractive indices of the fundamental waves, and ),( ω−Ωωβ being

the angle between )(ΩSHk and the z axis inside the crystal. The appearance of this angle can

be easily understood from Fig.3.2. The momentum conservation law determines the direction

of emitted second harmonic field:

)()()( Ω=ω−Ω+ω SHkkk , (3.22)

where k(ω) and k(Ω−ω) are the wave-vectors of the incident fundamental waves. In the case)()( ω−Ω≠ω kk , β is non-zero and it can be found from the following equation#:

)(

)()()()(sin),(sin 21

0 Ωω−Ωω−Ω−ωω

α=ω−ΩωβSHk

nknk(3.23)

# In fact, if the second harmonic is an extraordinary wave, the magnitude of )(ΩSHk in Eq.(23) is a

function of ),( ω−Ωωβ . The problem of finding the exact values of both )(ΩSHk and ),( ω−Ωωβcould be easily solved by employing the relations of crystaloptics and Eq.(3.23). However, Eq.(3.23)

alone gives an excellent approximation for ),( ω−Ωωβ if one chooses 0

)(=β

ΩSHk .


67

Since β is of the same order of magnitude as the intersection angle, the correction),(cos ω−Ωωβ is required only in the ∆k expression (Eq.(3.21)). Elsewhere this correction

can be dropped.

The values of the wave-vectors and refractive indices in Eqs.(3.21) and (3.23) depend

on the actual polarization of the three interacting waves. Thus, for Type I we obtain:

( ) ( ) ),(cos)()(cos)()(cos)(),( 00 ω−ΩωβΩ−ω−Ωαω−Ω+ωαω=ω−Ωω∆ EOOOO knknkk

(3.24)

and for Type II:

( ) ( ) ),(cos)()(cos)()(cos)(),( 00 ω−ΩωβΩ−ω−Ωαω−Ω+ωαω=ω−Ωω∆ EOOEE knknkk

(3.25)

Here indices O and E correspond to the ordinary and extraordinary waves, respectively.

To calculate the total FROG signal, one should integrate the signal intensity

2

0 ),(~)(

),(~

ΩΩ

ε=Ω Lc

nLI SH

SHSH E (3.26)

over the transverse coordinates x and y. Hence, for the second-harmonic signal detected in

FROG we obtain:

( )

dxdLk

c

xi

Lki

d

x

nc

QL

LS

SH

2

0

0

2

0

2

0

2/3

003

22

2

),(csin

2

2

),(exp)(

~)(

~

1,,~2ln4exp2ln

)(2

)(

),,(

ωωωα

τωωω

ωω

ωωωωχωπωε

τ

−Ω∆

++

−Ω∆−Ω

×

Ω−−ΩΩ

Ω

−

ΩΩΩΩ

=Ω

∫∫Ω

><

EE

(3.27)

In Eq.(3.27), )(ΩQ is the spectral sensitivity of the photodetector. We also took into

consideration transverse profiles of the fundamental beams as given in Section 3.3.

Thus far we have limited our discussion to the case of low-efficiency second-harmonic

generation, i.e. when the depletion of the fundamental waves can be disregarded. In the high

conversion efficiency regime, however, additional effects play an important role. While the

second-harmonic intensity depends quadratically on the crystal length L in the case of

undepleted pump [59], in the high-efficiency regime, conversion efficiency “saturates” for

more intense spectral modes but remains proportional to L2 for the weaker ones.

Consequently, the FROG traces measured in a Type I SHG crystal in presence of significant

pump depletion typically have both spectral and temporal marginals broader compared with

Chapter 3

68

the low conversion efficiency case. Hence, despite seemingly increased bandwidth in the

high-efficiency regime, the FROG trace is intrinsically incorrect. The case of the high-

efficiency SHG in a Type II crystal [60,61] is more complex than in the Type I and can result

in both shortening and widening of the temporal width of the FROG trace. Another important

example of the second-harmonic spectral shaping in the high-conversion-efficiency regime is

the nonlinear absorption of the frequency-doubled radiation inside the SHG crystal [62].

Therefore, the high-efficiency second-harmonic conversion is a potential source of systematic

errors in a FROG experiment and should be avoided.

To conclude this Section, we would like to make a remark on the frequency – as

opposed to time – domain approach to the wave equation Eq.(3.12) in the single optical cycle

regime. Clearly, the former has a number of advantages. The spectral amplitude of a

femtosecond pulse is observable directly while the temporal amplitude is not. The frequency

representation allowed us to include automatically dispersive broadening of both fundamental

and second-harmonic pulses as well as their group mismatch, frequency-dependence of the

nonlinear susceptibility, frequency-dependent spatial profiles of the beams, and the blue shift

of the second-harmonic spectrum (analog of self-steepening in fibers [51]). Furthermore, we

have made a single approximation given by Eq.(3.16), which is easily avoidable in computer

simulations. Eq.(3.20) can also be used to describe the process of SH generation in the low

pump-depletion regime to optimize a compressor needed to compensate phase distortions in

the SH pulse. Extension of the theory to the high conversion efficiency by including the

second equation for the fundamental beam is also straightforward. Note that a similar

frequency-domain approach to ultrashort-pulse propagation in optical fibers [63] helped solve

a long-standing question of the magnitude of the shock-term [51,64].

3.5 Ultimate temporal resolution of the SHG FROG

In the general case of arbitrary pulses, the complete expression for the SHG FROG signal

given by Eq.(3.27) must be computed numerically. However, for the limited class of pulses,

such as linearly-chirped Gaussian pulses Eq.(3.27) can be evaluated analytically. Such

analysis is valuable to estimate the temporal resolution of the SHG FROG experiment.

The geometrical smearing of the delay due to the crossing angle is an important

experimental issue of the non-collinear multishot FROG measurement of ultrashort pulses.

As can be seen from Eq.(3.27) the dependence on the transverse coordinate x yields a range

of delays across the beam simultaneously which “blurs” the fixed delay between the pulses

and broadens the FROG trace along the delay axis. Analogously to Taft et al. [9], we assume

Gaussian-intensity pulses and, under perfect phase-matching conditions, obtain the measured

pulse duration τmeas that corresponds to a longer pulse as given by

222 tpmeas δ+τ=τ , (3.28)

where τp is the true pulse duration, and tδ is the effective delay smearing:


69

cdt f /0α=δ , (3.29)

with fd being the beam diameter in the focal plane, and 2α0 the intersection angle of the

fundamental beams.

We consider the best scenario of the two Gaussian beams separated by their diameter don the focusing optic. In this case the intersection angle fd /2 0 =α , and the beam diameter

in the focal plane d f df = λ π/ , where f is the focal length of the focusing optic. Therefore,

the resultant time smearing amounts only to 4.02/ ≈πλ=δ ct fs at λ=800 nm. This value

presents the ultimate resolution of the pulse measurement in the non-collinear geometry.

Interestingly, this figure does not depend on the chosen focusing optic or the beam diameter

d, since the beam waist is proportional whereas the intersection angle is inversely

proportional to the focal distance f. It should be noted that the temporal resolution

deteriorates if the beams are other than Gaussian. For instance, if the beams of the same

diameter with a rectangular spatial intensity profile replace the Gaussian beams in the

situation described above, the resultant temporal resolution becomes 0.7 fs.

Additional enhancement of the temporal resolution could be achieved either by placing

a narrow slit behind the nonlinear medium [65], as will be discussed in Section 3.9, or by

employing a collinear geometry [66,67].

3.6 Approximate expression for the SHG FROG signal

In this Section, our goal is to obtain a simplified expression for SHG FROG that can be used

even for single-cycle optical pulses. We start from the complete expression given by

Eq.(3.27) and show that the measured signal can be described by an ideal, i.e. perfectly

phase-matched SHG FROG and a spectral filter applied to the second-harmonic field.

Throughout this Section we consider Type I phase-matching.

In order to simplify Eq.(3.27), we make several approximations. First, as was shown in

the previous Section, under carefully chosen beam geometry the effect of geometrical

smearing is negligibly small. For instance, it causes only a 10% error in the duration

measurement of a 3-fs pulse, and can be safely neglected. With such approximation, the

integral along x in Eq.(3.27) can be performed analytically. Second, we assume that 2/Ω≈ωand apply this to modify the factor that is proportional to the overlap area between different

fundamental frequency modes: ( ) 2//1 Ω≈Ωω−ω . Third, we expand )(ωOk and

)( ω−ΩOk into a Taylor series around 2/Ω=ω and keep the terms that are linear with

frequency#. Hence, for Type I phase-matching we write: # Alternatively, one can perform Taylor expansion around the central frequency of the fundamentalpulse 0ω=ω [22,39,43]. However, in this case the first derivative terms do not cancel each other and

must be retained. Our simulations also prove that the expansion around 2/Ω=ω provides a betterapproximation when broadband pulses are concerned. The practical implications of bothapproximations are also addressed in Section 4.3

Chapter 3

70

( ) ( ) ( ) ( )2/,2/)()2/(cos2/2, 0 ΩΩ∆=Ω−ΩαΩ≈ω−Ωω∆ kknkk EOO (3.30)

Fourth, we estimate dispersion of the second treat the second-order susceptibility),,(~ 2 ω−ΩωΩχ >< from the dispersion of the refractive index. For a classical anharmonic

oscillator model [56], )(~)(~)(~),,(~ 1112 ω−ΩχωχΩχ∝ω−ΩωΩχ ><><><>< , where

1)()(~ 21 −Ω=Ωχ >< n . Equation (3.27) can now be decomposed to a product of the spectral

filter )(ΩR , which originates from the finite conversion bandwidth of the second harmonic

crystal and varying detector sensitivity, and an ideal FROG signal ),( τΩSHGFROGS :

),()(),,( τΩΩ∝τΩ SHGFROGSRLS , (3.31)

where

( )2

exp)(~

)(~

),( ωωτωω−Ω=τΩ ∫ diS SHGFROG EE , (3.32)

and

( )( )[ ] ( )

ΩΩ∆

−Ω−ΩΩ

ΩΩ=Ω

22/,2/

sinc1)2/(1)()(

)()( 22222

3 Lknn

nQR OE

E

. (3.33)

In Eqs.(3.31-33) we retained only the terms that are Ω-dependent.

The FROG signal given by Eq.(3.32) is the well-known classic definition of SHG

FROG [14,18,35] written in the frequency domain. The same description is also employed in

the existing FROG retrieval algorithms. Note that the complete Eq.(3.27) can be readily

implemented in the algorithm based on the method of generalized projections [68]. However,

relation (3.31) is more advantageous numerically, since the integral Eq.(3.32) takes form of

autoconvolution in the time domain and can be rapidly computed via fast Fourier transforms

[69]. It is also important that the use of Eq.(3.31) permits a direct check of FROG marginals

to validate experimental data.The spectral filter )(ΩR , as given by Eq.(3.33), is a product of several factors

(Fig.3.3). The 3Ω -term (dotted line) results from Ω -dependence of the second-harmonic

intensity on the spatial overlap of the different fundamental frequency modes♦, and from the2Ω dependence that follows from Maxwell’s equations. The meaning of the latter factor is

that the generation of the higher-frequency components is more efficient than of the lower-

frequency ones. The combined 3Ω dependence leads to a substantial distortion of the second-

♦This dependence should be disregarded for the output of a Kerr-lens mode-locked laser [70] and fora hollow fiber [43,54]


71

harmonic spectrum of ultrabroadband pulses. For instance, due to this factor alone, the up-

conversion efficiency of a spectral component at 600 nm is 4.5 times higher than of a 1000-

nm one.

300 400 500 600 7000

1

|χ<2>|2

sinc2

Ω3

R(Ω)SHE

ffic

ienc

y [a

rb. u

nits

]

SH wavelength [nm]

Fig.3.3: Constituent terms of spectral filter R(Ω) given by Eq.(3.33): the Ω3 dependence (dotted line),estimated squared magnitude of second-order susceptibility χ<2> (dash-dotted line), the crystal phase-matching curve for a Type I 10-µm BBO crystal cut at θ=29° (dashed line), and their product (solidcurve). The second-harmonic spectrum of a 3-fs Gaussian pulse is shown for comparison (shadedcontour).

The variation of the second-order susceptibility with frequency, expressed in Eq.(3.33)

as the dependence on the refractive indices, plays a much less significant role than the 3Ωfactor (dotted line). According to our estimations for BBO crystal, the squared magnitude of

><χ 2~ for the 600-nm component of the fundamental wave is only 1.3 times larger than for the

1000-nm component. Such a virtually flat second-order response over the immense

bandwidth is a good illustration of the almost instantaneous nature of ><χ 2~ in transparent

crystals. Nonetheless, the estimation the contribution of the ><χ 2~ dispersion is required for

the measurement of the optical pulses with the spectra that are hundreds of nanometers wide.

The last factor contributing to )(ΩR is the phase-matching curve of the SHG crystal

(Fig.3.3, dashed line). The shape and the bandwidth of this curve depend on the thickness,

orientation and type of the crystal. Some practical comments on this issue will be provided in

Section 4.2.

3.7 Numerical simulations

In this Section we verify the approximations that were applied to derive Eqs. (3.31–33). In

order to do so, we numerically generate FROG traces of various pulses using the complete

expression Eq.(3.27) and compare them with the ideal FROG traces calculated according to

Chapter 3

72

Eq.(3.32). To examine contributions of different factors to pulse reconstruction, we compare

FROG inversion results with the input pulses.

Fig.3.4: Simulation of SHG FROG signal for an ideal 3-fs Gaussian pulse for Type I phase-matching.(a) ideal FROG trace, as given by Eq.(3.32). (b) complete FROG trace as given by Eq.(3.27).(c) spectral filter curve R(Ω) computed according to Eq.(3.33) (shaded contour) and the ratio ofFROG traces given in (b) and (a) at several delays (broken curves). (d) spectral marginal of the tracesshown in (b) (solid curve) and autoconvolution of the fundamental spectrum (dashed curve). TheFROG traces here and further on are shown as density plots with overlaid contour lines at the values0.01, 0.02, 0.05, 0.1, 0.2, 0.4, and 0.8 of the peak second harmonic intensity.

Two types of pulses with the central wavelength at 800 nm are considered: 1) a

bandwidth-limited 3-fs Gaussian pulse, and 2) a pulse with the same bandwidth that is

linearly chirped to 26 fs. We assume that the fundamental beam diameter in the focus is

fd =20 µm and the beams intersect at 02α =2°. Therefore, the geometrical delay smearing

that was defined in Section 3.5 [Eq.(3.29)] amounts to =δt 1.2 fs. The thickness of the Type I

BBO is L=10 µm. As we pointed out in Section 3.4, such a short crystal lengthens the pulse

less than 0.1 fs, and, therefore, dispersive pulse broadening inside the crystal can be


73

disregarded. The crystal is oriented for the peak conversion efficiency at 700 nm#. The

spectral sensitivity of the light detector Q(Ω) is set to unity.

Fig.3.5: Simulation of SHG FROG signal for a linearly-chirped 26-fs Gaussian pulse. The conditionsare the same as in Fig.4. (a) ideal FROG trace, as given by Eq.(3.32). (b) complete FROG trace asgiven by Eq.(3.27). (c) spectral filter curve R(Ω) computed according to Eq.(3.33) (shaded contour)and the ratio of FROG traces given in (b) and (a) at several delays (broken curves). (d) spectralmarginal of the traces shown in (b) (solid curve) and autoconvolution of the fundamental spectrum(dashed curve).

The results of FROG simulations for each type of pulses are presented in Figs.3.4 and 3.5.

The ideal traces calculated according to Eq.(3.32) are shown in Figs.3.4a and 3.5a, while the

traces computed using Eq.(3.27) are displayed in Fig.3.4b and 3.5b. The FROG trace of the

# The phase-matching angle is slightly affected by the non-collinear geometry. Due to the fact that thefundamental beams intersect at an angle 2αo, the equivalent phase-matching angle is different fromthat in the case of collinear SHG: ncollinear /0α+θ=θ , where n is the refractive index of the

fundamental wave at the phase-matching wavelength. For instance, the 800-nm phase-matched cut ofa BBO crystal for 2αo=2° becomes θ=29.6° instead of θcollinear=29° for collinear SHG. This fact shouldbe kept in mind since the phase-matching curve is quite sensitive to the precise orientation of thecrystal.

Chapter 3

74

3-fs pulse is also noticeably extended along the delay axis as the consequence of the

geometrical smearing. For the 26-fs pulse, as should be expected, this effect is negligible. The

spectral filtering occurring in the crystal becomes apparent from the comparison of the

spectral marginals that are depicted in Figs.3.4d and 3.5d. Calculated marginals are

asymmetric and substantially shifted toward shorter wavelengths.

By computing a ratio of the FROG signals given by Eq.(3.32) and Eq.(3.27) we obtain

delay-dependent conversion efficiency, as shown in Figs.3.4c and 3.5c. The spectral filter

R(Ω) calculated according to Eq.(3.33), is shown as shaded contours. Clearly, at the small

delays τ the conversion efficiency is almost exactly described by R(Ω). With the increase of

pulse separation, the approximation given by Eq.(3.33) worsens, as both the conversion peak

position and the magnitude change. The rapid ratio scaling at non-zero delays for the 3-fs

pulse (broken curves in Fig.3.4c) is mostly determined by the geometrical smearing rather

than by the phase matching, as in the case of the chirped pulse (Fig.3.5c). On the other hand,

the deviations from R(Ω) at longer delays become unimportant because of the decreasing

signals at large pulse separations.

To estimate the significance of the spectral correction of the distorted FROG traces and

feasibility of performing it in the case of extreme bandwidths, we examined FROG inversion

results of the numerically generated traces using the commercially available program from

Femtosoft Technologies. Four different cases were considered for each type of pulses: a) an

ideal phase-matching (zero-thickness crystal); b) a 10-µm BBO crystal with the parameters

defined above; c) the trace generated in the case (b) is corrected by R(Ω); and, last, in d)

geometrical smearing is included as well. In its essence, the case (d) is similar to (c), but in

(d) the FROG trace was additionally distorted by the geometrical smearing. The results of the

FROG inversion of the cases (a) - (d) are presented in Fig.3.6.

In the case (a), the 3Ω dependence is exclusively responsible for the spectral filtering

that substantially shifts the whole FROG trace along the frequency axis. Both the bandwidth-

limited and the chirped Gaussian pulses converged excellently to their input fields, but

around a blue-shifted central frequency. In (b), where the phase-matching of a 10-µm BBO

crystal is taken into account as well, the central wavelength is even more blue-shifted due to

spectral filtering in the crystal. A small phase distortion is obtained for both types of pulses.

The retrieved 3-fs pulse is also artificially lengthened to ~3.4 fs to match the bandwidth

narrowed by the spectral filtering in the crystal. The results of FROG retrieval of the same

trace upon the correction by R(Ω) (case (c)) indicate an excellent recovery of both the

bandwidth-limited and the chirped pulses.

Finally, in the case (d) the geometrical smearing had a negligible effect on the 26-fs

pulse. However, the FROG of the shorter pulse converged to a linearly chirped 3.3-fs

Gaussian pulse. This should be expected, since the FROG trace broadens in time and remains

Gaussian, while the spectral bandwidth is not affected. In principle, like the spectral

correction R(Ω), the correction for the temporal smearing should also be feasible. It can be


75

implemented directly in the FROG inversion algorithm by temporal averaging of the guess

trace, produced in every iteration, prior to computing the FROG error.

-3 0 30

1

Inte

nsity

Time [fs]-3 0 3

Time [fs]-3 0 3

Time [fs]-3 0 3

Time [fs]

500

750

1000

1500

Chirp [nm

]

500 750 1000 15000

1

Inte

nsity

Wavelength [nm]500 750 1000 1500

Wavelength [nm]500 750 1000 1500

Wavelength [nm]500 750 1000 1500

Wavelength [nm]

-2

0

2

Line

arly

chi

rped

Gau

ssia

n pu

lse

Ban

dwid

th-li

mite

d 3-

fs G

auss

ian

puls

e

Corrected by R(Ω)+

Geometrical smearing

Corrected by R(Ω)BBO θ=33.4°

L=10µm

Zero crystal

thickness

Group delay [fs]

-25 0 250

1

Inte

nsity

Time [fs]-25 0 25

Time [fs]-25 0 25

Time [fs]-25 0 25

Time [fs]

500

750

1000

1500

Chirp [nm

]

500 750 1000 15000

1

Inte

nsity

Wavelength [nm]500 750 1000 1500

Wavelength [nm]500 750 1000 1500

Wavelength [nm]500 750 1000 1500

Wavelength [nm]

-40

-20

0

20

40

(d)(c)(b)(a)

Group delay [fs]

Fig 3.6: Retrieved pulse parameters in the time and frequency domains for various simulated FROGtraces. (a) perfectly phase-matched crystal, no geometrical smearing. (b) Type I 10-µm BBO crystalcut at θ=33.4°, no geometrical smearing. (c) same as in (b), the FROG trace is corrected according toEq.(3.33). (d) same as in (c) but with the geometrical smearing included. Dashed curves correspond toinitial fields, while solid curves are obtained by FROG retrieval.

Chapter 3

76

Several important conclusions can be drawn from these simulations. First, they confirm

the correctness of approximations used to obtain Eq.(3.31-33). Therefore, the spectral

correction given by R(Ω) is satisfactory even in the case of single-cycle pulses, provided the

crystal length and orientation permits to maintain a certain, though not necessarily high, level

of conversion over the entire bandwidth of the pulse. Second, a time-smearing effect does not

greatly affect the retrieved pulses if the experimental geometry is carefully chosen. Third, the

unmodified version of the FROG algorithm can be readily applied even to the shortest pulses.

Forth, it is often possible to closely reproduce the pulse parameters by FROG-inversion of a

spectrally filtered trace without any spectral correction [43]. However, such traces rather

correspond to similar pulses shifted in frequency than to the original pulses for which they

were obtained.

5 10 15 20 25 301E-5

1E-4

1E-3

Sys

tem

atic

err

or

Duration of bandwidth-limited pulse [fs]

Fig.3.7: Dependence of the systematic FROG trace error on the pulse duration. FROG matrix size is128×128. The dotted curve corresponds the trace after the spectral correction given by Eq.(33). Theerror due to geometrical smearing of a perfectly phase-matched trace is shown as a dashed curve,while the error of a spectrally corrected and geometrically smeared FROG is given by the solid curve.The parameters of the crystal and of the geometrical smearing are the same as above. The centralwavelength of the pulse is kept at 800 nm.

In order to quantify the distortions that are introduced into the SHG FROG traces by the

phase-matching and the non-collinear geometry and that cannot be removed by the

R(Ω)-correction, we compute the systematic error as rms average of the difference between

the actual corrected FROG trace and the ideal trace. Given the form of the FROG error [19],

the systematic error can be defined as follows:

2

1,)(

),,(),(

1 ∑=

Ω

τΩ−τΩ=

N

ji

jiji

SHGFROG R

LSaS

NG , (3.34)


77

where ),( τΩSHGFROGS and )(ΩR are given by Eq.(3.32) and Eq.(3.33), and ),,( LS τΩ is

computed according to Eq.(3.27). The parameter a is a scaling factor necessary to obtain the

lowest value of G. The dependence of G on the duration of a bandwidth-limited pulse for the

128×128 FROG matrix that has optimal sampling along the time and frequency axes is

presented in Fig.3.7. As can be seen, the systematic error for ~5-fs pulses becomes

comparable with the typical achievable experimental SHG FROG error. It also should be

noted, that the contribution of geometrical smearing is about equal or higher than that due to

the spectral distortions remaining after the spectral correction.

The systematic error should not be confused with the ultimate error achievable by the

FROG inversion algorithm. Frequently, as, for instance, in the case of linearly-chirped

Gaussian pulses measured in the presence of geometrical smearing, it means that the FROG

trace continues to exactly correspond to a pulse, but to a different one. However, for an

arbitrary pulse of ~3 fs in duration it is likely that the FROG retrieval error will increase due

to the systematic error.

3.8 Type II phase matching

So far, we limited our consideration to Type I phase-matching. In this Section we briefly

discuss the application of Type II phase-matching to the measurement of ultrashort laser

pulses.

In Type II the two fundamental waves are non-identical, i.e. one ordinary and one

extraordinary. This allows the implementation of the collinear SHG FROG geometry free of

geometrical smearing [67]. The FROG traces generated in this arrangement in principle does

not contain optical fringes associated with the interferometric collinear autocorrelation and,

therefore, can be processed using the existing SHG FROG algorithms. However, the fact that

the group velocities of the fundamental pulses in a Type II crystal become quite different, has

several important implications. First, the second-harmonic signal is no longer a symmetric

function of the time delay [39]. Second, because the faster traveling fundamental pulse can

catch up and pass the slower one, some broadening of the second-harmonic signal along the

delay axis should be expected [39].

In order to check the applicability of the collinear Type II SHG FROG for the

conditions comparable to the discussed above in the case of Type I phase matching, we

performed numerical simulations identical to those in the previous Section. The same pulses

were used, i.e., the bandwidth-limited 3-fs pulse at 800 nm and the pulse with the same

bandwidth stretched to 26 fs. The thickness of the Type II BBO is L=10 µm, and the crystal

oriented for the peak conversion efficiency at 700 nm (θ=45°). The expression for the

spectral filter, adapted for Type II, is given by:

( )( )( )[ ] ( )

ΩΩ∆

−Ω−Ω−ΩΩ

ΩΩ=Ω

22/,2/

sinc1)2/(1)2/(1)()(

)()( 222223 Lk

nnnn

QR EOEE

, (3.35)

Chapter 3

78

where the phase mismatch# is

( ) ( ) )()2/(2/2/,2/ Ω−Ω+Ω=ΩΩ∆ EEO kkkk (3.36)

The results of FROG simulations are presented in Figs.3.8 and 3.9. The FROG trace of

the 3-fs pulse (Fig.3.8b) is practically symmetrical along the delay axis. However, despite the

fact that no geometrical smearing has occurred, this trace is evidently broadened along the

delay axis. Consequently, the FROG inversion of this trace after the spectral correction yields

a longer ~3.3-fs pulse. The elongation of the trace is due to the temporal walk-off of the

fundamental waves, which in this case is about 1 fs.

Fig.3.8: Simulation of SHG FROG signal for an ideal 3-fs Gaussian pulse for Type II phase-matching. (a) ideal FROG trace, as given by Eq.(3.32). (b) complete FROG trace as given byEq.(3.27). (c) spectral filter curve R(Ω) computed according to Eq.(3.33) (shaded contour) and theratio of FROG traces given in (b) and (a) at several delays (broken curves). (d) spectral marginal ofthe traces shown in (b) (solid curve) and autoconvolution of the fundamental spectrum (dashed curve).

# Unlike in the case of Type I phase-matching, the first derivative terms do not cancel each other butthey have been disregarded anyway.


79

The magnitude of this temporal distortion is approximately equal to the geometrical smearing

discussed in the previous Section. The trace of the chirped pulse, produced under the same

conditions (Fig.3.9b), is much more severely distorted than in the case of the bandwidth-

limited pulse. The straightforward use of this trace is virtually impossible because of its

strong asymmetry.

Fig.3.9: Simulation of SHG FROG signal for a linearly-chirped 26-fs Gaussian pulse. The conditionsare the same as in Fig.3.8. (a) ideal FROG trace, as given by Eq.(3.32). (b) complete FROG trace asgiven by Eq.(3.27). (c) spectral filter curve R(Ω) computed according to Eq.(3.33) (shaded contour)and the ratio of FROG traces given in (b) and (a) at several delays (broken curves). (d) spectralmarginal of the traces shown in (b) (solid curve) and autoconvolution of the fundamental spectrum(dashed curve). Note the skewness of the FROG trace in (b).

As in the Type I case, the conversion efficiency, obtained as a ratio of the ideal and

simulated FROG traces, continues to correspond nicely the spectral filter R(Ω) (Figs.3.8c and

3.9c, shaded contours) at near-zero delays. Conversion efficiency at other delays, however,

sharply depends on the sign of the delay τ. Similar to Type I phase-matching, the frequency

marginals (Figs.3.8d and 3.9d) are substantially blue-shifted. It is also apparent from

Figs.3.8c and 3.9c, that the phase-matching bandwidth in this case is somewhat broader than

in the analogous Type I crystal.

Chapter 3

80

We can conclude from our simulations that Type II SHG FROG offers no enhancement

of the temporal resolution and is less versatile compared to the non-collinear Type I

arrangement. Additionally, the collinear Type II SHG FROG requires a greater experimental

involvement than in the Type I SHG FROG. However, for some applications such as

confocal microscopy, where the implementation of the non-collinear geometry is hardly

possible due to the high numerical aperture of the focusing optics, the use of Type-II-based

FROG appears quite promising [67].

3.9 Spatial filtering of the second-harmonic beam

In this Section, we show how spatial filtering of the second-harmonic beam can corrupt an

autocorrelation or FROG trace. Unfortunately, this type of distortion can pass undetected

since the FROG trace may still correspond to a valid pulse, but not the one that is being

measured.

Fig.3.10: Delay-dependent change of the second-harmonic direction in the case of a chirped pulse.

As it was already mentioned in Section 3.4 [Eq.(3.23)], the direction in which a second

harmonic frequency is emitted varies because of the non-collinear geometry. Even though the

intersection angle of the fundamental beams is small, this effect becomes rather important for


81

the measurement of broadband pulses due to the substantial variation of the wave-vector

magnitude across the bandwidth.

Let us consider a certain component of the second-harmonic signal that has a frequencyof 02ω (Fig.3.10). This component can be generated for several combinations of fundamental

frequencies, for example, such as the pairs of 0ω and 0ω , and of δω+ω0 and δω−ω0 . The

direction in which the 02ω component is emitted for each pair can vary, as determined by the

non-collinear phase-matching. Therefore, as can be seen from Fig.3.10, the direction of the

second-harmonic beam changes as a function of delay between the fundamental pulses. This

phenomenon is utilized in the chirp measurement by angle-resolved autocorrelation [71,72].

To illustrate the effect of spatial filtering of the second-harmonic beam, we examine the

same Gaussian pulses linearly chirped to 26 fs, which were used in the numerical simulations

described above. We keep the same geometrical parameters as in the previous sections of thisChapter, i.e. fd =20 µm and 2α0=2°. The resulting dependence of the autocorrelation

intensity as a function of the second-harmonic angle in the far field is depicted in Fig.3.11a.

The tilt of the trace clearly indicates the sweep of the second-harmonic beam direction. The

signal beam traverses approximately half the angle between the fundamental beams, and the

magnitude of this sweep scales linearly with the intersection angle. The autocorrelation trace

obtained by integration over all spatial components of the second-harmonic beam is depicted

in Fig.3.11b (solid curve). The FROG trace corresponding to this autocorrelation, i.e.

measured by detecting of the whole beam, is entirely correct and allows recovery of the true

pulse parameters.

-1.0

-0.5

0.0

0.5

1.0

-60 -40 -20 0 20 40 60

(a)

SH a

ngle

[de

g]

Delay [fs]-60 -40 -20 0 20 40 60

0

1(b)

Inte

nsity

Delay [fs]

Fig.3.11: Angular dependence of the non-collinear second-harmonic signal for a linearly-chirpedGaussian pulse in the far field. (a) autocorrelation intensity as a density plot of delay between thefundamental pulses and the second-harmonic angle. (b) autocorrelation intensity trace obtained byintegration over all spatial components of the second-harmonic beam (solid curve) and the tracesdetected through a narrow slit at the second-harmonic angle of 0° (dashed curve) and 0.4° (dottedcurve). The pulse is stretched to ~5 times the bandwidth-limited pulse duration. The intersection angleof the fundamental beams is 2°

Chapter 3

82

The situation, however, becomes different if only a portion of the second harmonic

beam is selected. In the considered example, the autocorrelation or FROG, measured through

a narrow slit placed on the axis of the second harmonic beam, would “shrink” along the delay

axis, as shown in Fig.13b (dashed curve). The width of this trace is ~10% narrower than the

true autocorrelation width. Positioning of the slit off the beam axis (Fig.3.11b, dotted curve)

leads to the shift of the whole trace along the delay axis, and, for some pulses, to asymmetry

in the autocorrelation wings. In the case of Gaussian pulses examined here, the FROG traces

measured with such spatial selection remain self-consistent, disregarding the delay shift. The

spectral marginal of such FROG traces is exactly the same as in the case of the whole-beam

detection. Consequently, the FROG retrieval of the spatially filtered traces yields pulses of

correct bandwidth but less chirped than in reality.

The described effect should not be identified alone with the pulses that are much longer

than the bandwidth limit, since even the bandwidth-limited pulses with asymmetric spectra

carry a chirp in time. Therefore, careful collecting of all spatial components of the second

harmonic field is extremely essential. We also underline importance of measuring an

independent autocorrelation trace in front of spectrometer, since its comparison with the

temporal marginal of the FROG trace might signal improper spatial filtration occurring in the

FROG detection.

In Section 3.5 we have already mentioned the desirability to enhance the temporal

resolution of a non-collinear measurement by placing a slit behind the nonlinear medium.

This reduces the effective spot of the second harmonic beyond the size of the diffraction-

limited focus. However, placing a slit into the collimated beam would cause the spatial

selection considered above. To avoid such undesirable distortion, one should position the slit

behind the crystal within the Rayleigh range, or, alternatively, into the scaled image of the

crystal plane projected by an achromatic objective lens. The realization of both these options

is rather difficult and becomes really necessary only if the beams are poorly focusable.

3.10 Conclusions

In this Chapter, we have developed the SHG FROG description that includes the phase-

matching in the SHG crystal, non-collinear beam geometry, and dispersion of the second-

order nonlinearity. The derived master equation is valid down to single-cycle pulses.

Subsequently, thorough numerical simulations have been performed to estimate the

separate roles of the crystal phase-matching, geometrical smearing and spatial filtering of the

SHG signal. These simulations have shown that the conventional description of FROG in the

case of Type I phase-matching can be readily used even for the single-cycle regime upon

spectral correction of the FROG traces, provided the beam geometry, the finite crystal

thickness and phase-matching bandwidth are chosen correctly.

The SHG FROG of very short pulses with Type II phase-matching in a BBO crystal is

shown to be rather impractical, since the group velocity mismatch between the two


83

fundamental waves of different polarization causes a delay smearing similar to the one

originating from geometrical blurring in the non-collinear measurement.

We also show that, while the spectral correction of the FROG traces helps the recovery

of true pulse characteristics, the systematic error of the FROG trace, nonetheless, increases

with the increase of the spectral breadth of the pulse. This is due to the fact that the effective

spectral filter applied by the second harmonic generation process on the FROG trace varies

for different delay values. Consequently, higher FROG trace retrieval errors should be

expected from the inversion algorithm for the bandwidths supporting durations shorter than

5-6 fs.

Chapter 3

84

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Chapter 4

FROG Characterization of Fiber-Compressed Pulses

Abstract

The technique of SHG FROG is applied to measure the white-light continuum pulses in the

spectral region of 500-1100 nm. The obtained spectral phase of these pulses served as a target

function for the pulse compressor design. The pulses around 800 nm produced by

compression were also characterized by SHG FROG. The resulting pulse duration measures

4.5 fs which corresponds to ~2.5 optical cycles.

Chapter 4

88

4.1 Introduction

This Chapter puts to work the ideas about the SHG FROG pulse characterization, which have

been developed in Chapter 3. We demonstrate practical application of this pulse measuring

technique to strongly chirped ultrabroadband pulses and compressed 4.5-fs pulses from the

cavity-dumped-laser based white-light generator that has been presented in Chapter 2. The

use of SHG FROG allows us to finally obtain rigorous electric field characterization in terms

of amplitude and phase, which Chapter 2 clearly lacked.

Prefacing the account on the experimental results, several practical implications of

dealing with the extraordinarily broad bandwidths will be considered here. The basic

experimental requirements raised to the FROG apparatus are the adequate bandwidth of

phase matching of the SHG crystal, the low overall dispersion of the optical elements. Next

to it, the device should be able to yield exact replicas of the pulse that do not differ from each

other in their spectral content or phase.

In particular, the choice of the frequency-doubling to be employed for the FROG

measurement is a very delicate issue. This is partly due to the fact that the demand of a high

signal to noise ratio and, therefore, the need to employ thicker, more efficient doublers finds

itself in conflict with the necessity to keep the phase-matching bandwidth broad and,

consequently, the crystal thickness low. Another riddle to solve is the right angular

orientation of the crystal or the wavelength it is cut for. The difficulty here comes from the

fact that the “red” crystal orientation typically provides nearly flat frequency conversion

efficiency over the most of the bandwidth of an ultrabroadband pulse. Such an orientation,

however, dispenses with the blue-shifted wing of the spectrum where the conversion

efficiency dramatically falls. On the other hand, the use of the “blue” phase-matching

significantly lowers frequency-conversion efficiency in the red wing of the spectrum. To

address these issues and to find a reasonable balance that satisfies the demands of various

pulses, a useful criterion on crystal selection is developed in this Chapter. Further, the merits

of the two most commonly used SHG crystals, BBO and KDP, are compared among

themselves. We also present a case study of two contradicting recipes concerning the most

suitable, for measuring sub-5-fs pulses, cut angle of the crystal.

We next focus our attention on the working of the FROG apparatus and the

peculiarities of the measurement of strongly chirped and nearly fully compressed laser pulses

by this technique. The spectral phase of the white light pulse measured before and after the

pulse compressor permits a good verification of the ray-tracing routine employed to design it.

We subsequently present valuable observations on how extra information about the level of

pulse compression can be gained from a simple examination of the SHG FROG trace that is

normally considered quite unintuitive.

This Chapter is organized as follows: Section 4.2 advises on the choice of the SHG

crystal. Section 4.3 provides a quantitative study of the effect of phase-matching wavelength

FROG-characterization of fiber-compressed pulses

89

on the outcome of SHG FROG. Section 4.4 we describe our SHG FROG apparatus. SHG

FROG characterization of the white-light continuum and 4.5-fs pulses is demonstrated in

Sections 4.5 and 4.6, respectively. Finally, in Section 4.7 we summarize our findings.

4.2 The choice of the SHG crystal

In this Section, we provide several guidelines for the right SHG crystal selection in the FROG

measurement. On the one hand, the crystal should be thick enough to generate an appropriate

level of the second-harmonic signal for a high dynamic range measurement. One the other

hand, the thickness of the crystal should be sufficiently small to provide an appropriate phase

matching bandwidth and minimize pulse broadening in the crystal.

Obviously, when choosing the crystal one must consider the bandwidth of the pulse that

has to be characterized. We employ a simple criterion to estimate the required crystal

thickness: the conversion efficiency calculated according to Eq.(3.33) must be higher than

50% of the peak conversion efficiency everywhere over the double FWHM of the FROG

spectral marginal. For the pulses that are Gaussian in frequency, the ideal spectral marginal,

or the autoconvolution of the fundamental spectrum, is 2 times broader than the pulse

bandwidth. Using this criterion, we evaluated BBO and KDP crystals, which are typically

employed for the ultrashort pulse measurement. Both considered crystals are cut for Type I

phase-matching at the wavelength of 800 nm and 600 nm. Figure 4.1 depicts the appropriate

crystal thickness of the BBO (solid curve) and KDP (dashed curve) as a function of duration

of a bandwidth-limited Gaussian pulse.

5 10 15 20 250

20

40

60

80

100

800 nm(a)

BBO

KDP

Cry

stal

thic

knes

s [µ

m]

Duration of bandwidth-limited pulse [fs]

5 10 15 20 250

20

40

60

80

100

600 nm(b)

BBO

KDP

Cry

stal

thic

knes

s [µ

m]

Fig.4.1: Crystal thickness required for SHG FROG measurement as a function of the pulse duration atthe central wavelength of 800 nm (a) and 600 nm (b). The crystals are cut for Type I phase-matching,which corresponds to θ=29° for BBO (solid line) and θ=44° for KDP (dashed line).

As can be noticed from Fig.4.1, an approximately 10-µm BBO should be employed to

measure 5-fs pulses at 800 nm. The adequate thickness of the KDP crystal is approximately

Chapter 4

90

2.5 times larger due to its lower dispersion. However, while clearly providing an advantage in

thickness, the KDP crystal has a disadvantage in the SHG efficiency. The signal level that

can be obtained with a 2.5 times thinner BBO crystal is still approximately by a factor of 6

larger than in KDP because of the higher nonlinear coefficients and lower phase-matching

angle in the BBO crystal [1]. Therefore, BBO is a more suitable choice for characterization of

weak-intensity pulses. For high-intensity pulses, where the low level of the second-harmonic

signal is not really the issue, KDP presents a better choice [2].

With the growth of the phase-matching bandwidth of the crystal, the 3Ω dependence

(Eq.(3.33)) begins to dominate the conversion efficiency. As it was shown in Section 3.6, this

dependence blue-shifts the second-harmonic spectrum. In case the phase-matching bandwidth

of the SHG crystal is wider than required by the pulse bandwidth, angular tuning of the

crystal can effectively counteract such blue shift [2]. To illustrate the point, we consider a 10-

µm BBO crystal applied to measure 8-fs Gaussian pulses at 800 nm. Figure 4.2a shows the

blue-shift of the FROG spectral marginal (filled circles) with respect to the autoconvolution

(solid curve) if the crystal is perfectly phase-matched at 800 nm, i.e. θ=29°. However, after

adjusting the phase-matching angle to θ=24.4° that now corresponds to the central

wavelength of 970 nm (Fig.4.2b) the phase-matching curve of the crystal (dashed curve)

nearly perfectly balances the 3Ω dependence (dotted curve). The overall conversion

efficiency becomes almost flat and no spectral correction of the FROG trace is required.

Experimentally, Taft et al. [3] demonstrated the effectiveness of the angular adjustment that

enabled them to yield correct FROG data.

The mutual compensation of the 3Ω and phase-matching terms is only possible for relatively

long (~10 fs) pulses. As a thinner crystal is chosen to measure shorter pulses, the high-

frequency slope of the phase-matching curve becomes relatively steeper than the low-

frequency one (Fig.4.2c, d). This is to be expected, since crystal dispersion is low in the

infrared and increases approaching the UV absorption band. Tuning the central wavelength of

the crystal from 800 nm (Fig.4.2c) to 970 nm (Fig.4.2d) substantially narrows the SH

spectrum in the blue due to the crystal phase-matching. Even worse, the FROG trace can

hardly be corrected for the imposed spectral filter since the conversion efficiency becomes

extremely low in the blue wing (Fig.4.2d). This should be contrasted to the 800-nm-cut case

when the correction is still possible (see Fig.3.6). Therefore, in order to extend the phase

matching-bandwidth in the blue, one should consider using a crystal with the phase-matching

wavelength blue-shifted with respect to the central frequency of the pulse. For example, a

L=10 µm BBO crystal oriented for peak conversion efficiency at 700 nm is more suitable for

the measurement of sub-5-fs pulses centered at 800 nm than the same crystal tuned to

970 nm. Although the 700-nm-cut crystal has poorer conversion efficiency in the infrared it,

nonetheless, allows the extension of the phase matching below 600 nm. Consequently, this

crystal has an appreciable efficiency of frequency conversion all over the spectrum of a 5-fs

pulse and, therefore, the FROG traces can be validated upon spectral correction. In contrast,

information about the blue spectral wing is completely filtered out if the crystal oriented for


91

970 nm is used. The quantitative analysis of how the poor choice of the SHG crystal can

affect the FROG recovery of a sub-5-fs pulse will be given in the next Section.

360 380 400 420 440 4600

1

SH wavelength [nm]

(a)

Inte

nsity

360 380 400 420 440 460

SH wavelength [nm]

(b)

300 400 500 6000

1

SH wavelength [nm]

(c)

Inte

nsity

300 400 500 600

SH wavelength [nm]

(d)

Fig.4.2: Correction of frequency conversion efficiency by crystal orientation for 8-fs (a,b) and 3-fs(c,d) bandwidth-limited Gaussian pulses. A Type I 10-µm BBO crystal is oriented for the phase-matched wavelength of 800 nm (a,c) and 970 nm (b,d). The phase-matching curve and the Ω3

dependence are shown as the dashed and dotted lines, respectively. The solid curves depict theautoconvolution of fundamental spectra, while spectral marginals of FROG traces are given by filledcircles. In (b), no spectral correction of the FROG trace is required for an 8-fs pulse because of thered-shifted phase-matched wavelength. In contrast, the use of the 970-nm phase-matched crystalirreparably corrupts the second-harmonic spectrum in the case of a shorter 3-fs pulse (d). Note thedifference in horizontal scales in (a), (b) and (c), (d).

In closing to this Section, we mention an interesting property of very thin crystals, i.e.

those that have a thickness in order of a few microns. In such thin crystals the differentiation

between the Type I (oo-e interaction) and Type II (eo-e interaction) becomes less strict. For

instance, if we speak about a Type I 800-nm-cut crystal this means that the phase mismatch,

k∆ , for this wavelength is zero. However, if the crystal thickness, L , is very small then theproduct Lk EEO−∆ , albeit never reaching a zero value, becomes comparable to the magnitude

of Lk EOO−∆ for the wavelengths detuned from the phase-matching frequency. Therefore, we

can no longer neglect the contribution to the SH signal produced by Type II interaction.

Additionally, even for the fundamental waves that have a perfect linear polarization the

Chapter 4

92

second harmonic beam, obtained in this case, becomes somewhat depolarized. This situation

reciprocates for thin Type II crystals where the mixture of the oo-e contribution adds up to

the total signal. This has far-reaching consequences. For instance, this means that in collinear

Type II FROG experiments some fringes that are due to the interference of the SH waves,

produced by each interacting fundamental wave, will always be present, no matter how

perfectly orthogonal the polarizations of fundamental beams are kept. The mentioned here

property is considerably stronger for BBO than for KDP for which, for the same thickness, no

such effect takes place. Finally, we point out that the necessity to account for both Type I and

Type II contributions applies only to sub-10-µm BBO crystals.

4.3 Case study: Two contradicting recipes for an optimal crystal

In the previous Section we outlined the issue of a proper crystal choice and offered a solution.

However, no particular quantitative assessment of the damaging role of a poorly chosen

crystal was given there. In this Section, we ascertain the possible distortions of the amplitude-

phase measurement, which arise from the orientation of a 10-µm BBO crystal applied to

measure sub-5-fs at 790 nm. Two distinctly different recommendations had been given on

that account in this Thesis and in the recent publication by Chen et al. [2], respectively.

The first suggestion claims that, in order to extend the phase-matching bandwidth to

cover the wavelengths below 600 nm, a 700-nm 10-µm BBO crystal should be used and the

FROG trace must be necessarily corrected to the frequency-doubling efficiency, that

significantly drops in the NIR region.

The second recipe, on the contrary, recommends the use of a 970-nm-centered 10-µm

BBO crystal. The red-shifted cut-wavelength balances off the 2Ω term in the conversion

efficiency, which gives rise to a very broad and nearly symmetric phase-matching band

around the carrier wavelength of the pulse, 790 nm. Because of the large width of the latter

band, no additional spectral correction of the FROG trace is necessary, according to Ref. [2].

The apparent contradiction between the two recipes (subsequently in this Section called

recipes I and II) arises from the difference in the calculation of the approximate expressionfor the spectral filter, )(ΩR , imposed by the SHG process on the FROG trace (Eq.(3.33)). To

comply with the line of reasoning in Ref. [2], in the expressions throughout this Section we

drop the dispersion of the second-order nonlinearity and the extra Ω -dependence responsible

for the variation in the mode sizes of different frequency components. The resulting

expression for the spectral filter is then given by:

∆

Ω∝Ω2

sinc)( 22 kLR . (4.1)

The two recipes stated above are based on two different Taylor expansions of the effective

phase mismatch, k∆ : around 2/Ω (see Eq.(3.30)) in the first case, and around the carrierfrequency of the pulse, 0ω [2,4,5], in the second. The resulting respective approximations are:


93

)(2

2)()( Ω−

Ω

≅Ω∆ EOI kkk , (4.2)

and

( )

∂∂

−∂∂

−Ω+−≅Ω∆00 2

000)( )2()(2)(

ωω ωωωωω EO

EOII kk

kkk . (4.3)

The spectral filters calculated using these approximate expressions for 970-nm and 700-nm

cut BBO crystals are plotted in Fig.4.3. Indeed, the dashed line (970-nm-cut crystal) in

Fig.4.3b in the approximation given by Eq.(4.3) seems to provide a much better choice than

the 700-nm one.

300 400 500 6000

1(a)

Eff

icie

ncy

[arb

. uni

ts]

SH wavelength [nm]300 400 500 600

0

1 (b)E

ffic

ienc

y [a

rb. u

nits

]

SH wavelength [nm]

Fig.4.3: Spectral filter for 10-µm BBO crystal for two different approximations of the phasemismatch. (a) and (b) represent the approximation given by Eq.(4.2) and Eq.(4.3), respectively. Solidand dashed curves are calculated for phase-matching wavelength of 700 nm and 970 nm, respectively.The shaded contour shows autoconvolution of super-Gaussian intensity spectrum supporting 4-fspulses.

To test the implications of these two recipes and to verify the better approximation of the

phase mismatch, we simulated SHG FROG measurements of a 4.5-fs pulse at 790 nm. In

order to follow a realistic scenario, we assume that the spectrum of the laser pulses is super-

Gaussian and the bandwidth supports pulses as short as 4 fs. The autoconvolution (SHG

FROG spectral marginal) of the spectrum is shown in Fig.4.3a,b alongside with the spectral

filters.

We next assume that the pulse is not perfectly compressed, and a small amount of a

quartic spectral phase (cubic group delay) broadens the pulse to ~4.5 fs. The chosen fourth-

order phase distortion roughly describes the residual phase of a hypothetical pulse

compressor such as, for instance, a combination of prisms and diffraction grating or chirped

mirrors. We now compute the FROG traces of this pulse according to Eq.(3.27) for a 700-nm

and 970-nm-cut crystal. Note that since Eq.(3.27) is exact, no approximations about the phase

mismatch are made in this calculation. Following the two recipes given above, the FROG

Chapter 4

94

inversion algorithm is applied to the resulting traces: with the spectral correction given by

Eq.(4.1,2) for the 700-nm crystal and directly to the other trace.

The pulse reconstructions corresponding to the two recipes are given in Fig.4.4b and

Fig.4.4c. (dashed curves) along with the input pulse parameters (thin lines in Fig.4.4a-c). The

respective FROG errors [6] for the matrix size of 128×128 are 0.0008 for recipe I, and 0.0042

for recipe II. The main source of error in the first case originates from the fact that, as has

been shown in Section 3.6, the spectral filter is somewhat delay-dependent, which is not

reflected in Eq.(4.1,2). The FROG error in the second case is much more substantial.

However, the pulse duration is almost accurately recovered in both these cases. There is only

a minor asymmetry on the recovered pulse in Fig.4.4c and the deviation from the input phase

is not significant for the portions of the pulse carrying any appreciable intensity. Clearly,

better criteria should be employed to judge if the pulse reconstruction in one ore both these

cases were faulty. A possible candidate for such a criterion can be the intensity-weighed

phase error [6]. Another recently demonstrated approach uses a comparison of Wigner

representations of the input and reconstructed pulses [7]. A particular merit of Wigner

representation is that it constitutes a very intuitive two-dimensional “fingerprint” of a laser

pulse field. The properties of such Wigner traces are summarized in Appendix I. Particularly,

such frequency-time-domain plots, also known as chronocyclic representation [8],

asymptotically show the sequencing of frequency components in time according to the group

delay.

The respective Wigner traces and the group delays of the input pulse and the two

recovered pulses are depicted in Fig.4.4d–f. The Wigner trace error, defined in Appendix I,

amounts to 0.035 for recipe I and to 0.238# for recipe II. According to Ref. [7], the error in

excess of 0.15 represents an unacceptably poor pulse reconstruction. Indeed, while the

Wigner trace of the recovered following recipe I pulse (Fig.4.4e) is nearly identical to the

input one (Fig.4.4d), the result produced by recipe II exhibits a clearly different behavior. The

inspection of Wigner traces in Fig.4.4d and Fig.4.4f conspicuously shows difference in

instantaneous frequency spectra of the two pulses, especially at times below and above the

half-width of the pulse. Obviously, the intensity of frequency components belonging to the

spectral wings, which are displaced in time (Fig.4.4f), is very small to affect the overall

correctness of the pulse duration measurement. On the other hand, such intensities are still

usable for a variety of nonlinear optical experiments, for instance, frequency-resolved optical

pump–probe. Consider the example, where the pulse discussed here serves as a probe in such

a spectroscopic experiment. The false phase reconstruction, using recipe II, makes us believe

that the sample interacts simultaneously with the blue and red frequencies of the probe pulse.

Therefore, a possible increase of the nonlinear response of the matter at positive pump-probe

delays, when the blue frequencies arrive in reality, can be mistakenly interpreted as having to

do with the dynamics of the sample. Consequently, misjudged phase distortions of the pulse

# Since time-direction ambiguity is present in SHG FROG reversal, the orientation of traces inFig.4.4c and Fig.4.4f was chosen accordingly to the smallest Wigner trace error.


95

can lead to an erroneous interpretation of the data obtained in a nonlinear spectroscopic

experiment.

Fig.4.4: Comparison of SHG FROG recovery of a 4.5-fs pulse. (a) spectrum and phase of the inputpulse. (b) and (c) time-domain intensity and phase of retrieved pulses (dotted curves) and input pulse(solid curves) for two different crystal choices (see the text for details). (d),(e), and (f) Wigner tracesof the input and retrieved pulses. The thick curves show group delay. The convention on contour linesin (d)-(f) is the same as the one adopted in Chapter 3, dotted contour lines represent negative values.

In summary to this Section, we performed the quantitative assessment of the effect of

using differently phase-matched crystals for the measurement of sub-5-fs pulses. Our

simulations confirm that, in the calculation of the spectral SHG filter, the approximation of

the phase mismatch given by Eq.(4.2) is superior to the one specified by Eq.(4.3). We finally

comment that the approximation by Eq.(4.2) has a clear physical sense since, due to the

increase of the crystal material dispersion approaching UV resonances, the phase mismatch

must grow sharply toward the high frequencies. This produces a very distorted sinc2 shape, –

Chapter 4

96

the feature, which the approximation calculated from Eqs.(4.1,3) entirely fails to reproduce.

We also point out that the amount of spectral filtering introduced by a 10-µm BBO crystal in

the measurement of 5-fs pulses around 800 nm does not significantly affect the figure of the

pulse duration. Therefore, in the given situation, it is unlikely that the error in labeling the

pulse by its intensity FWHM would rise above 10% as a consequence of varying the phase-

matching angle.

4.4 SHG FROG apparatus

In our experiments, we used pulses from a self-mode-locked cavity-dumped Ti:sapphire

oscillator compressed upon chirping in a single-mode fused silica fiber. We measured the

white-light continuum (WLC) pulses directly at the fiber output and, again, upon the their

compression performed as described in Chapter 2.

The SHG FROG apparatus (Fig.4.5) is based on a phase and amplitude balanced multi-

shot autocorrelator designed for sub-5-fs short pulses [9]. The input beam was split and

recombined in such a way that each of the beams travels once through an identical 50% beam

splitter while both reflections occurring on the same coating-air interfaces#. To match the

beam splitters, the initial horizontal polarization of the laser beam was rotated by a periscope.

The moving arm of the autocorrelator was driven by a piezo transducer (Physik Instrumente)

which is controlled by a computer via a digital-analog converter and a high voltage amplifier.

The precise time calibration was provided by an auxiliary Michelson interferometer. The

photodiode monitored the interference fringes that serve as time calibration marks.

Fundamental pulses were focussed in the nonlinear crystal with an r=-25-cm spherical

mirror at near normal incidence to minimize astigmatism. Due to the low curvature of the

mirrors, delay variations within each beam are less than 0.1 fs. To achieve simultaneous up-

conversion of the entire fundamental bandwidth, we employed a 10-µm-thick BBO crystal

cut for a central wavelength of 700 nm (EKSMA Inc.). Dispersive lengthening of a 5-fs pulse

by such crystal does not exceed 0.02 fs. The blue-shifted central wavelength permits one to

extend the phase-matching bandwidth below 600 nm as shown in Fig.3.4c. The cut angle of

the crystal was verified with a tunable 100-fs laser. Retro-reflection of the beams from the

crystal surface provided exact reference for crystal orientation. This enables us to accurately

calculate R(Ω) required for data correction according to Eq.(3.33). A visible-IR PC1000

(Ocean Optics) spectrometer was used to detect the fundamental spectra.

Two different second harmonic detection systems were employed in the measurements

of the compressed and the chirped pulses. In the case of compressed pulses, a well-

characterized UV- Vis PC1000 (Ocean Optics) spectrometer was used. Therefore, the FROG

traces could be readily corrected by R(Ω), as described above.

# For shorter pulses, one should use lower-reflectivity beam splitters that have a broader reflectivityrange and flatter spectral phase.


97

Fig.4.5: Schematic of the SHG FROG apparatus. Spectrometer and its coupling optics are not shown.

In the case of the strongly chirped pulses a combination of a scanning monochromator

and a photo-multiplier tube provided the dynamic range necessary to measure the spectral

wings (see next Section). The reason for this was the following: The dynamic range of the

measurement in a CCD-based spectrometer is determined not only by the spectral sensitivity,

which is adequately high, but by the charge spreading all over the array due to overload of

some channels. To further extend the dynamic range, a lock-in amplifier was used to detect

the second-harmonic signal. Because of the unknown spectral sensitivity Q(Ω), the spectral

correction of the FROG traces in this case was performed according to the method suggested

in Taft at al. [3], i.e., by using the ratio of the autoconvoluted fundamental spectrum and the

spectral marginal.

4.5 SHG FROG of white-light continuum

As has been shown in Chapter 2, the study of the group delay of the chirped WLC, is the

corner stone of the pulse compression. The phase measurement of the pulses leaving the fiber

permits to assess the feasibility of the pulse compression in general. Understandably, the

spectral phase must be sufficiently smooth to allow compensation by the existing dispersion

control means. A measurement of the spectral intensity, on the other hand, provides only a

limited insight and reveals the minimum duration of the would-be compressed pulse. As an

example of virtually uncompressible pulses, one might consider the case of spectral

broadening due to a pure self-phase modulation. Furthermore, the task of building an

appropriate pulse compressor is substantially eased if the phase distortion on the pulse is

measured beforehand. This becomes increasingly important with the growth of the pulse

spectral bandwidth that puts severe limitations on dispersion tunability of the pulse

compressor. Therefore, it is desirable to replace a great deal of “trial and error” work by

measuring the phase distortion and computing the settings of the pulse compressor.

Somewhat counter-intuitively, the FROG measurement of a strongly chirped pulse is

considerably more complicated compared with the case a bandwidth-limited pulse with the

Chapter 4

98

identical spectrum. First, the up-conversion signals are weaker due to the lower peak power.

This is evident, since the second harmonic intensity of a pulse that is stretched to the ten

times its initial duration drops down 100 times.

Second, a higher dynamic range is required because of the uneven temporal spread of

spectral wings. This occurs due to the high-order material dispersion. To explain this, we

consider two spectral components with frequencies separated by 1000 cm-1. The group delay

accumulated between them after passing 1 mm of quartz amounts to 4 fs if these components

are situated around 1000 nm and exceeds 11 fs in the case of 600 nm. Evaluating roughly, the

corresponding elements of the FROG trace scale ~7 times in intensity. In our experiments,

the bandwidth of the WLC that needs to be captured in the FROG trace is broader than

10000 cm-1, and, therefore, the signal intensity varies very strongly across the resultant

FROG traces.

The third complication is purely numerical, since FROG inversion demands greater

matrix sizes to provide the adequate sampling in both time and frequency domain. For the

sake of speed, the FROG inversion algorithms employ fast Fourier transform (FFT) [10]. To

avoid the loss of information in the change from the time to the frequency domain and vice

versa, FFT requires an equal number of points N in both these domains. Therefore, if the

FROG matrix covers the total delay of Nτ∆ in the time domain, where ∆τ is the time step,

the spectral width represented in this trace is τ∆/N . Compared with bandwidth-limited

pulses, the pulses stretched in time require larger ∆τ to contain the whole time information of

the FROG trace in the matrix used in the FROG inversion algorithm. This narrows the

spectral window covered by the matrix. Consequently, the number of points N, that in FFT is

an integer power of two, must be increased to fully represent the FROG trace in the matrix

used by the algorithm. This has an appreciable effect on the calculation speed. The change of

N from 2n to 2n+1, where n is an integer number, slows the FROG retrieval by a factor of)1(4 1−+ n . In other words, by changing a 128x128 matrix with a 256×256 one increases the

calculation time by a factor of ~4.5.

Lastly, we point out the experimental inconvenience. In the case of strongly chirped

pulses the crystal alignment and the detected FROG trace become very sensitive to the delay-

dependent change in the direction of the second harmonic beam, as has already been

discussed in Section 3.9.

The SHG FROG traces of the chirped WLC in our experiments were recorded in 2.5-fs

delay steps and converted into 256×256 matrices for processing. To reveal the conditions best

suited for the compression of the WLC we varied the parameters of the pulses entering the

fiber, by changing of the settings of the prism precompressor. The intensity and the chirp of

the input pulses, derived by SHG FROG, are shown in Fig.4.6a. The measured and retrieved

FROG traces of the WLC are depicted in Figs.4.6c and d, and the retrieved WLC spectra and

the group delay are shown in Fig.4.6b. The combined action of self-phase modulation and

dispersion leads to a nearly linear group delay over most of the spectrum (Fig.4.6b, solid

curves).


99

Fig.4.6: Experimental results of FROG measurement of the strongly chirped white-light continuum(WLC). (a) temporal intensity (shaded contours) and chirp (solid curves) of the pulses entering asingle-mode fused-silica fiber. (b) measured and (c) retrieved SHG FROG traces of the WLC. (d)retrieved spectral intensity (shaded contours) and the group delay of the WLC (solid curves). Theamount of bulk material (fused silica) used to pre-chirp the input pulses is indicated in right topcorners of (a). Note that the input pulse energy is kept constant, while the respective scaling of theWLC spectra in (d) is preserved.

The departure of the overall group delay from a linear asymptotic can be partly explained by

the bulk dispersion of the fiber, air, and the beamsplitters in the FROG apparatus. For

Chapter 4

100

instance, while the optimal fiber length was estimated to be 1 mm [11], we employed a 2-mm

piece for the practical convenience and in order to clean the exiting mode structure.

The WLC spectrum changes dramatically with the change of the input pulses (Fig.4.6b,

shaded contours). The widest and least modulated spectrum corresponds to the almost chirp-

free input pulse (Fig.4.6b, the third from the top panel). Positive as well as negative chirping

leads to a substantial narrowing of the WLC spectrum. In contrast, the overall behavior of the

group delays shown as solid lines in Fig.4.6b, remains virtually unaffected. This ensures

efficient pulse compression under different experimental conditions.

500 600 700 800 1000

-100

0

100

200

Gro

up d

elay

[fs

]

Wavelength [nm]

Fig.4.7: Group delay of the designed pulse compressor. Solid curve is calculated by dispersive ray-tracing and is depicted reversed in time. The broken curves are the measured group delay of the WLCreproduced from all panels in Fig.4.6b.

Group delay measurements of the generated continuum served as a target function for

the design of the three-stage, high throughput compressor, consisting of a quartz 45°-prism

pair, broadband chirped mirrors and thin-film Gires-Tournois dielectric interferometers [9].

The spectral bandwidth of the compressor is 590–1100 nm and is limited by the reflectivity

of the employed chirped mirrors [12]. (See Fig.2.7b.) The phase characteristics of the

compressor have been analyzed using dispersive ray tracing and mapped onto the measured

group delay of the continuum. Figure 4.7 depicts the measured group delay for different

pulses, entering the fiber (shown as broken curves) which are reproduced from Fig.4.6b and

the calculated group delay of the pulse compressor (solid line). As one can see, our design

compensates the group delay of the white light everywhere across the compressor bandwidth.

The adjustment of the material of the prism-pair allows final fine optimization of the

compressor dispersion, as judged form the FROG trace of the compressed pulses.


101

4.6 SHG FROG of compressed pulses

The FROG traces of the compressed pulses were recorded by incrementing the time delay

between the arms in steps of 0.5 fs. The acquired two-dimensional arrays of points were

converted into a 128×128 FROG matrix. The experimental and retrieved FROG traces of

compressed pulses are depicted in Figs.4.8a and b. The FROG error amounted to 0.004 and is

mainly caused by the noise in the spectral wings which scaled up after the spectral correction

of the FROG trace. The temporal marginal of the FROG trace has a nice correspondence with

the independently measured intensity autocorrelation (Fig.4.8c) obtained by detecting the

whole second-harmonic beam. This suggests that no spatial filtering of the second-harmonic

beam has taken place. Comparison of the FROG frequency marginal and the autoconvolution

of the fundamental spectrum (Fig.4.8d) indicates that no loss of spectral information has

occurred.

Fig.4.8: The results of SHG FROG characterization of compressed pulses. (a) experimental and (b)retrieved traces. (c) temporal marginal (filled circles) and independently measured autocorrelation of4.5-fs pulses (solid curve). (d) frequency marginal (filled circles) and autoconvolution of thefundamental spectrum (solid curve).

Chapter 4

102

Figure 4.9 shows the retrieved intensity and phase in the time and frequency domains.

To remove the time direction ambiguity in the measurement of the compressed pulses, we

performed an additional FROG measurement introducing a known amount of dispersion (a

thin fused silica plate) in front of the FROG apparatus. The obtained pulse duration is 4.5 fs

while variations of the spectral phase (dashed line in Fig.4.9b) is less than ±π/4 across the

whole bandwidth. These results fully confirm our previous analysis based on the

interferometric autocorrelation [9].

-40 -20 0 20 400

1 (a)

Inte

nsity 4.5 fs

Time [fs]600 800 1000

(b)

-π

0

π

Pha

se

Wavelength [nm]

Fig.4.9: Retrieved parameters of 4.5-fs pulses in the time (a) and frequency (b) domains. The FROG-retrieved intensity and phase are shown as shaded contours and dashed curves, respectively.Independently measured spectrum (filled circles) and computed residual phase of the pulsecompressor (dash-dotted curve) are given in (b) for comparison.

To additionally verify both the self-consistency of our compressor calculations and the

accuracy of the FROG retrieval, we compare the obtained spectral phase of the 4.5-fs pulse

(Fig.4.9b, dashed curve) with the predicted residual phase of the pulse compressor (Fig.4.9b,

dash-dotted curve). The close similarity of the two reassures us of the correctness all used

procedures, including the measurement of the chirped WLC, the knowledge of the dispersion

of compressor constituent parts, the numerical routines employed for the ray tracing analysis,

and, finally, the characterization of the compressed pulses.

The electric field of the compressed pulses is shown in Fig.4.10. Approximately 2.5

optical cycles comprise its half-width. The heavy oscillations in the wings, however, indicate

the imperfection of spectral phase correction and the toll that the modulation of the spectrum

takes on the temporal structure of the pulse. The importance of the pulse energy carried in the

wings of the pulse is only moderately critical for some experiments, for instance, such as><3χ - or higher-nonlinearity-order spectroscopies, where the signal is proportional to a

certain power of intensity. This leads to a “clean-up” of the signal. Indeed, the prominent


103

wing structure of the electric field (Fig.4.10) is effectively “suppressed” in the intensity

profile (Fig.4.9a) because of the square dependence between the field and the intensity.

-20 0 20

Continuous

monochromatic

light-wave

Optical cycle

@790 nm (2.63 fs)

Real part of E(t)

|E(t)|E

lect

ric

fiel

d E

(t)

Time [fs]

Fig.4.10: Reconstructed electric field of 4.5-fs pulses. The electric field of continuous light-wave at790 nm is drawn for reference.

The SHG FROG traces are generally considered unintuitive due to their symmetry

along the delay axis [13-15]. We found out that in the case of nearly bandwidth-limited

pulses, one can significantly increase the amount of information available from the simple

visual inspection of the trace. In order to do so, every trace in the time domain at its

corresponding second-harmonic wavelength should be normalized to unity. Effectively, this

represents the FROG trace as a series of normalized autocorrelations. In the case of the pulse

with an arbitrary spectrum and the flat spectral phase, such representation of the SHG FROG

trace would give a streak of uniform thickness around zero delay. The result of such

operation applied to the FROG trace of the 4.5-fs pulse is presented in Fig.4.11a. The

variation of the thickness, that is, the width of autocorrelation at a given second-harmonic

wavelength♣, which can be seen in Fig.4.11a indicates the non-perfect pulse compression

without the necessity to run the FROG inversion algorithm.

Figure 4.11b shows two autocorrelation traces derived from the spectrogram in

Fig.4.11a at two separate wavelengths. The FWHM of the autocorrelation at 350 nm is

merely 6 fs which is indicative of an ~4-fs pulse duration. However, the autocorrelation at

470 nm is three times broader. Such a difference clearly illustrates the effect of the spectral

filtering in nonlinear crystal as well as second harmonic detection on the autocorrelation

width. This also underscores the importance of pulse characterization by frequency-resolved ♣ Here we apply the term “autocorrelation” to a slice of a frequency-resolved autocorrelation of thepulse intensity purely for the sake of convenience. In an arbitrary case, such a slice in itself is notnecessarily an autocorrelation function of any real non-negative distribution.

Chapter 4

104

(e.g., FROG) rather than non-frequency-resolved (e.g., intensity autocorrelation) methods if

one deals with such broadband pulses.

Fig.4.11: Normalized FROG data of the 4.5-fs pulses. (a) SHG FROG trace of compressed pulsesnormalized along the delay axis as described in the text. (b) autocorrelation traces derived from theFROG trace at the second-harmonic wavelength of 350 nm (solid curve) and 470 (dashed curve).Note that because of spectral selection the pulse duration estimated from the autocorrelation width canbe both lower and higher than the real one and differ by as much as a factor of 3.

Finally, we note that the width of the autocorrelation traces, such as the ones shown in

Fig.4.11a, can be directly related to the instrument response of a spectroscopic experiment.

For instance, the temporal resolution of a kinetic trace in a frequency-resolved pump-probe

experiment [16,17] detected at 950 nm will be ~12 fs, albeit the weighted average pulse

duration is 4.5 fs [18,19]. Therefore, the frequency-resolved measurement (as FROG) brings

invaluable information even if the correct estimation of the pulse width could be achieved by

other, simpler means, such as the autocorrelation measurement.

In closing to this Section, we summarize the accumulated here knowledge about the

amplitude-phase properties of the compressed pulse by constructing a “fingerprint” form of a

Wigner spectrogram (see Appendix I). As has been shown above in the example discussed in

Section 4.3, direct information about the time sequence of light–matter interaction with the

frequencies throughout the pulse spectrum is readily available from the Wigner plot. For

instance, a simple examination of the spectrum and spectral phase (or group delay) (Fig.4.9b)

gives us the idea about the peak moment of time when the heaviest presence of a certain

frequency component is observed. Such an examination, however, is unable to show a

relative measure of how much, at any moment, and for how long, in total, this frequency

component will be felt by matter on its passage through the latter. The Wigner plot

(Fig.4.12), on the other hand, allows such assessment by a simple visual inspection.


105

-40 -20 0 20 40

600

700

800

900

1000

Time [fs]

Wav

elen

gth

[n

m]

Fig.4.12: Wigner (chronocyclic) representation of the compressed pulses. Negative values are markedby dotted contour lines

Several useful observations can be made on the basis of the trace depicted in Fig.4.12.

First, we notice that the IR wing of the pulse precedes the arrival of the main body. This

feature is inherited from the chirped pulse where the red-shifted frequencies are advanced

while the blue-shifted ones are delayed. The failure to properly retard the IR wing is mostly

explained by the dominating role in the infrared of the reversed (above 850 nm) third-order

dispersion of the prism compressor. Second, we notice that the frequency components,

corresponding to the sharp peaks on the spectrum (Fig.4.6d), occupy much broader time

intervals than the rest of the “well-behaved” spectrum. These peaking frequencies dominate

instantaneous intensity spectra seen at times ±20 fs around the main pulse. To some extent,

the behavior of such sharp spectral irregularities may be viewed as a superposition of

different pulses that are distinguished by a narrower spectral and broader temporal width. The

implications of this on the interpretation of pump–probe data will be addressed in Chapter 7.

4.7 Conclusions and Outlook

SHG FROG is a powerful and accurate pulse diagnostics technique that is ideally suited for

the measurement of a vast variety of pulses. In particular, the instantaneous nonlinearity, high

sensitivity, and broadband response allow measuring the shortest pulses available to the date.

The FROG measurement of the pulses that are shorter than 5 fs is nowadays probably the

Chapter 4

106

only available means to evaluate the pulse parameters and the temporal resolution of a

nonlinear spectroscopic experiment.

We have applied the developed theory to the SHG FROG measurement of 2.5-optical-

cycle pulses with a central wavelength around 800 nm. To the best of our knowledge, these

are the shortest pulses that have been completely characterized to date. We have also

successfully measured strongly non-spectral-limited weak-intensity pulses generated at the

fiber output. These two key experiments that are required to design, test and optimize the

pulse compressor have both been performed without a single change in the SHG FROG

apparatus. Under the given conditions, no other pulse measuring technique known to the

present day allows similar versatility.

FROG characterization of the chirped spectrally broadened pulses offers an important

shortcut in the generation of the ever-shorter pulses via external compression. The direct

phase measurement of the output of glass fibers, as demonstrated in this Chapter, hollow

waveguides [20] and parametric amplification [21-23] provides a rigorous target function for

the pulse compressor design. In particular, we foresee clear benefits for two direct methods of

pulse compression: adaptive dispersion control and all-mirror compression.

In the first case, the whole pulse compressor or one stage of it consists of the computer-

controlled intensity and phase masks [24] or an acousto-optical modulator [25]. The required

phase pattern can be calculated and set to match the target function measured by FROG. Such

straightforward finding of the optimal conditions allows eliminating the time-consuming

iterative search based on the feedback [26] and guarantees the correctness of the phase

corrections.

In the second case, in which no flexible control over the resulting dispersion of the

pulse compressor is permitted, the precise knowledge of the target function is even more

important. The well-developed theory of the chirped mirrors [27] makes it possible to design

the adequate dielectric layer structure that in many cases almost perfectly follows the

required dispersion curve, measured by FROG. In general, the phase distortion of nearly any

complexity can be compensated for by a mirror that is based on the gradient change of the

refractive index instead of the discrete dielectric layers, as is the case in the currently

available chirped mirrors [12]. No doubt that with the growing interest in the intracavity

[28,29] and extra-cavity broadband dispersion control [9,21,22,30], the possibility of

manufacturing the gradient-index structures will shortly become available. Therefore, the

phase measurement of chirped pulses gains paramount importance.

Appendix I: Wigner representation and Wigner trace error

A Wigner representation of a pulse [8], ),( ωtW , which is a two-dimensional trace in the

time–frequency space, is straightforwardly calculated from the complex electric field infrequency, )(

~ ωE :


107

ΩΩ−

Ω

+

Ω

−= ∫ dtiEEt )exp(2

~2

~),( * ωωωW (4A.1)

Alternatively, in a similar fashion, one can produce a Wigner trace starting from the complexelectric field in the time domain, )(tE .

τωτττ

ω ditEtEt )exp(22

),( *∫

+

−=W , (4A.2)

where )(~

ωE and )(tE are a Fourier pair.

Because ),( ωtW is a function of both time and frequency, it can be conveniently

plotted as a two-dimensional spectrogram in the time-frequency domain. To reflect this fact,

Wigner representation of the light pulse is also called chronocyclic [8].Integration of ),( ωtW along ω or t produces pulse intensity in time or pulse

spectrum, respectively.

∫=≡ ωω dttEtI ),()()( 2 W , (4A.3)

∫=≡ dttEI ),()(~)(~ 2ωωω W , (4A.4)

),( ωtW is a real function that can be both positive and negative. The marginals of ),( ωtW

given by Eqs.(4A.3,4) are non-negative.

The Wigner representation is very intuitive since the shape of the contour basically

reflects the group delay. In fact, for each time value it gives the instantaneous spectrum of

frequencies [31]. For some classes of pulses, such as double pulses, however, this

intuitiveness is lost [32]. Next to the intuitive properties, the Wigner trace contains a delicate

balance between the amount of phase- and amplitude-information. While an element of a

Wigner trace scales accordingly to the intensity, phase information remains responsible for

the precise location in the time-frequency domain of the Wigner trace element corresponding

to this intensity. Because Wigner traces give a linear distribution of a field, their comparison

is significantly more sensitive than a comparison of corresponding FROG traces. Forinstance, like Wigner traces, ><3χ -based FROG traces also provide a quite intuitive delay-

versus frequency distribution of the FROG signal. However, due to the fact that FROG is

based on a nonlinear frequency-mixing, the response from weaker spectral components can

be hidden under the pile-up of the signal at a given frequency.

The Wigner trace error, proposed in Ref. [7] computes the error between two Wignermatrices, ),(0 ωtW and ),( ωtW , in the following form:

Chapter 4

108

[ ] [ ]2

,

0,

2

,

,0

, ),(),(),( ∑∑ −=N

ji

jiji

N

ji

jijijiji ttt ωωαωε WWW , (4A.5)

where α is a scaling factor that minimizes ε , and N is the size of the matrix. The error εtakes values from 0 to 1, the upper limit being the worst case scenario in which the

discrepancy between the two matrices equals the value of the initial matrix itself. A valuable

property of ε is that it is insensitive to the matrix size N and to the sampling along the time

and frequency axes. The precise lateral overlap of the two Wigner traces in the time-

frequency space is required to correctly compute ε . This can be easily arranged by

optimizing the respective overlap of their marginals (i.e. temporal and spectral intensities).

According to Ref. [7], the error level below ε =0.15 corresponds to a reasonableamplitude-phase reconstruction of the target pulse represented by ),(0 ωtW , and the error

below ε =0.03 is considered excellent.


109

References

1. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of non-linear opticalcrystals (Springer-Verlag, Berlin, 1991).

2. Z. Cheng, A. Fürbach, S. Sartania, M. Lenzner, C. Spielmann, and F. Krausz, Opt. Lett. 24, 247(1999).

3. G. Taft, A. Rundquist, M. M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N.Fittinghoff, M. A. Krumbügel, J. Sweetser, and R. Trebino, IEEE J. Select. Topics in QuantumElectron. 2, 575 (1996).

4. J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, IEEE J. Quantum. Electron. 28, 2285(1992).

5. A. M. Weiner, IEEE J. Quantum Electron. 19, 1276 (1983).6. K. W. DeLong, D. N. Fittinghoff, and R. Trebino, IEEE J. Quantum Electron 32, 1253

(1996).7. S. Yeremenko, A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, Appl. Phys. B

(submitted) (1999).8. J. Paye, IEEE J. Quantum. Electron. 28, 2262 (1992).9. A. Baltuška, Z. Wei, M. S. Pshenichnikov, D.A.Wiersma, and R. Szipöcs, Appl. Phys. B 65,

175 (1997).10. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C

(Cambridge University Press, New York, 1996).11. A. Baltuška, Z. Wei, M. S. Pshenichnikov, and D. A. Wiersma, Opt. Lett. 22, 102 (1997).12. E. J. Mayer, J. Möbius, A. Euteneuer, W. W. Rühle, and R. Szipöcs, Opt. Lett. 22, 528

(1997).13. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. Sweetser, M. A. Krumbügel, B. Richman, and

D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).14. K. W. DeLong, R. Trebino, J. Hunter, and W.E.White, J. Opt. Soc. Am. B 11, 2206

(1994).15. K. W. DeLong, R. Trebino, and D. J. Kane, J. Opt. Soc. Am. B 11, 1595 (1994).16. Femtosecond laser pulses, edited by C. Rullère (Springer-Verlag, Berlin, 1998).17. C. H. B. Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, Chem. Phys. Lett. 132, 341 (1986).18. A. Kummrow, M. F. Emde, A. Baltuška, D. A. Wiersma, and M. S. Pshenichnikov, Zeit. Phys.

Chem. 212, 153 (1999).19. M. F. Emde, A. Baltuška, A. Kummrow, M. S. Pshenichnikov, and D. A. Wiersma, in Ultrafast

Phenomena XI, edited by T. Elsaesser, J. G. Fujimoto, D. A. Wiersma, and W. Zinth (Springer,Berlin, 1998), pp. 586.

20. M. Nisoli, S. D. Silvestri, and O. Svelto, Appl. Phys. Lett. 68, 2793 (1996).21. A. Shirakawa, I. Sakane, and T. Kobayashi, in XIth International Conference on Ultrafast


22. G. Cerullo, M. Nisoli, S. Stagira, and S. D. Silvestri, Opt. Lett. 23, 1283 (1998).23. T. Wilhelm, J. Piel, and E. Riedle, Opt. Lett. 22, 1494 (1997).24. A. M. Weiner and A. M. Kan'an, IEEE J. Select. Topics in Quantum Electron. 4, 317 (1998).25. C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, Opt. Lett. 19, 737

(1994).26. D. Yelin, D. Meshulach, and Y. Silberberg, Opt. Lett. 22, 1793 (1997).27. R. Szipöcs and A. Köházi-Kis, Appl. Phys. B 65, 115 (1997).28. L. Xu, C. Spielmann, F. Krausz, and R. Szipöcs, Opt. Lett. 21, 1259 (1996).

Chapter 4

110

29. I. D. Jung, F. X. Kärtner, N. Matuschek, D. H. Sutter, F. Morier-Genoud, G. Zhang, U. Keller,V. Scheuer, and M. Tilsch, Opt. Lett. 22, 1009 (1997).

30. S. Sartania, Z. Cheng, M. Lenzner, G. Tempea, C. Spielmann, F. Krausz, and K. Ferencz, Opt.Lett. 22, 1562 (1997).

31. Y. Meyer, in Wavelets:Algorithms and Applications (Society for Industrial and AppliedMathematics, Philadelphia, 1993).

32. M. Joffre, private communiacation (1997).

Chapter 5

Four-Wave Mixing with Broadband Laser Pulses

Abstract

In this Chapter, we derive a complete equation that describes the signal measured in third-

order nonlinear spectroscopy. This equation is applicable to laser pulses down to one optical

cycle in duration. We show that even for extremely short pulses the signals obtained in

photon echo spectroscopy can be described in the conventional way, provided care is taken of

the spectral filtering effect and experimental beam arrangement. The typical four-wave

mixing experiments are considered: transient grating, self-diffraction, and pump–probe.

Chapter 5

112

5.1 Introduction

The use of extremely short 5-fs pulses, described in the preceding three chapters, provides

obvious advantages to a spectroscopic experiment. Next to the very high temporal resolution,

the broad bandwidth associated with short pulses allows covering an impressive spectral

window at once. On the other hand, any experiment with 5-fs pulses is a daunting task.

Besides the trivial experimental nuisances such as the pulse lengthening during its

propagation before and inside the sample because of group velocity dispersion, there are also

more fundamental problems to be addressed.

The conventional description of nonlinear signals applicable to multi-cycle pulses

becomes questionable for the pulses that consist merely of a couple of optical fringes.

Clearly, in the latter case the conventionally employed slowly varying envelope

approximation [1-3] implying that the temporal variation of the pulse amplitude is negligible

on the duration of an optical cycle, can no longer be maintained. Furthermore, the phase-

matching bandwidth [4,5], which is limited due to dispersion in the nonlinear medium,

rapidly gains importance with the broadening of the pulse spectrum. Another point of serious

concern is the frequency-dependent variation in the sensitivity of the photodetector employed

to register the signal generated in the nonlinear process. In combination, the above listed

features of an experiment with broadband pulses result in what is known as a spectral-filter

effect [6-8]. On top of that, artificial lengthening of the observed time dependencies is a

direct consequence of the noncollinear geometry employed in spectroscopic experiments.

Evidently, if a portion of the signal field is filtered out in frequency and/or the signal is

artificially “blurred” in time this might crucially influence the measured data [9] and

subsequently lead to its erroneous interpretation.

To address these issues, we present a comprehensive theoretical analysis in which the

frequency– and time–domain formalism of ultrafast nonlinear spectroscopy is thoroughly

reexamined. The complete expressions valid even for single-cycle-pulse applications are

derived for the nonlinear signal in the frequency and time domains. Among others, we show

that one does not need to invoke the slowly varying envelope approximation in its

aforementioned meaning, i.e. rejecting derivatives of the time-domain electric field. We also

assert that the influence of geometrical delay smearing does not introduce a significant

distortion of the observed traces provided that the geometry is carefully optimized.

This Chapter is organized as follows. In Section 5.2, we discuss the formalism for

optical four-wave mixing spectroscopy with extremely short laser pulses that consist of only

a few optical cycles. We summarize the experimental conditions required to link the time–

and frequency– domain observables. Section 5.3 investigates a particular case in which the

spectral-filter effect is shown to jeopardize the outcome of the self-diffraction experiment. In

Section 5.4 we discuss the impact of beam geometry on the outcome of the spectroscopic

measurements with 5-fs pulses. The formalism for frequency-resolved pump-probe

experiment is outlined in Section 5.5. Finally, in Section 5.6, we present our conclusions.


113

5.2 The formalism for ultrafast spectroscopy with 5-fs pulses

In this Section, we derive the master equation that describes spectroscopic observables and is

valid even for single-cycle optical pulses. Using the frequency–domain framework, we

consistently include the effects of phase-matching, dispersive pulse broadening, dispersion of

the third-order nonlinearity, and frequency dependence of the resulting nonlinear signal. The

frequency–domain formalism is then recast in the time–domain, which is conventionally used

in the description of transient spectroscopy with short light pulses. We subsequently show

that despite the ultrabroad bandwidth associated with 5-fs pulses, the effect of spectral

filtering can be disregarded under proper experimental conditions. This allows a

straightforward transition from the frequency–domain representation to the time–domain one,

with the latter offering a simpler formalism. Most importantly, this simplifies the

experimental task by lifting the otherwise unavoidable necessity to frequency-resolve the

signals generated by the ultrabroadband pulses.

We consider the case of non-collinear geometry in which three beams Ei(z,t) (i=1-3)

intersect at small angles in a nonlinear medium (Fig.5.1.). The corresponding configurations

for two types of non-collinear third-order experiments are depicted in Fig.5.1b and 5.1c. The

self-diffraction (SD, Fig.5.1b) and transient grating (TG, Fig.5.1c) signals are equivalent to

the two- and three-pulse stimulated photon-echo signals originating from the systems with

phase memory.

Fig.5.1: (a) Schematic representation of the pulse sequence in a three-pulse nonlinear spectroscopicexperiment. E1,2,3 are the input fields, and E4 is the signal due to the third-order nonlinear process. t12

and t23 are the delay between pulses E1–E2 and E2–E3, respectively. (b) Self-diffraction (two-pulse

photon echo) configuration. Two conjugated signal are emitted in the directions k4 and 4k′ . (c)

Transient grating in a “box” geometry.

Chapter 5

114

Focusing conditions of the beams are chosen such that the confocal parameter [5] and

the longitudinal beam overlap of the fundamental beams are considerably longer than the

interaction length. For simplicity, we assume that neither of the fields is absorbed in the

nonlinear medium and that the nonlinear response is purely third order. The input beams then

induce a third-order nonlinear polarization P<3>(z,t) that serves as a source for the signal field

E4(z,t). The approach used here is similar to the treatment of second-order nonlinear

polarization in Section 3.4 (see Eq.3.13). By writing both P<3>(z,t) and E4(z,t) as a Fourier

superposition of monochromatic waves, one obtains an equation that governs propagation of

the signal wave in the +z direction inside the nonlinear medium [10]:

),(~

),(~

)(),(~ 32

042442

2

ΩΩ−=ΩΩ+Ω >< zPzEkzEz

z µ∂∂

, (5.1)

where ),(~

4 ΩzE and ),(~ 3 Ω>< zP are Fourier transforms of ),(4 tzE and ),(3 tzP >< ,

respectively, Ω is the frequency and )(4 Ωzk is projection of the wave-vector of signal field

)(~)( 0022

4 ΩΩ=Ω εµεk onto the z-axis, with )(~ Ωε being the Fourier-transform of the

complex relative permittivity )(tε .

To simplify the left part of Eq.(5.1), we write the signal field as a plane wave

propagating along z axis:

[ ]zikzzE z )(exp),(~

),(~

444 ΩΩ=Ω E , (5.2)

and substitute it into Eq.(5.1):

[ ]zikzPzz

zz

ik zz )(exp),(~

),(~

),(~

)(2 432

042

2

44 Ω−ΩΩ−=Ω∂∂

+Ω∂∂

Ω ><µEE (5.3)

Identically to the application of non-equality (3.16), we now neglect the second-order

derivative over the signal electric field [5,10]:

),(~

)(2),(~

444 ΩΩ<<Ω zkzz z EE

∂∂

(5.4)

on the grounds that were discussed in Section 3.4. Equation 5.3 then has a simple solution by

integration:


115

∫ Ω−ΩΩΩ

=Ω ><L

z dzzikzPn

ciL

0

43

4

0 ))(exp(),(~

)(2),(

~ µ4E (5.5)

where )(~)(4 Ω=Ω εn is the refractive index for the signal wave and L is the thickness of the

nonlinear medium.

In order to calculate the third-order dielectric polarization induced at frequency Ω by

the fundamental fields, we should sum over all possible permutations of fundamental

frequencies weighted according to the third-order susceptibility [11]:

( )[ ]

[ ]))('''(''exp

))'''()''()'((exp)''',(~

)'',(~

)',(~

,''','~'''),(~

231212

32132

33

ttiti

zkkkizz

zddzP

zzz

egeg

++−Ω−−

×+−Ω++−+−Ω

×Ω+−−−=Ω ∫∫ ><><

ωωωωωωωωωω

ωωωωωωχωω

EE

E *1

(5.6)

where )'',(~

ωziE is a Fourier transform of ),( tzEi . Analogously to Eq.(5.2), the phase

accumulated as the result of linear propagation, ,z)(k , is styled into a separate oscillating

term. In Eq.(5.6), t12 and t23 are the delays between pulses E1–E2 and E2–E3 , respectively. In

the SD case (Fig.5.1b) t23 is set to zero and t12 is scanned while in the TG experiment (Fig.1c)

t12=0 and t23 is scanned. Representation of the frequency-dependent third-order nonlinear

susceptibility, ( )Ω+−−−><egeg ωωωωωχ ,''','~ 3 , is based on the interaction of the input fields

with an electronic transition with the frequency egω . The inclusion of the third-order

susceptibility due to Raman and two-photon processes is also straightforward. The particular

expression for ><3~χ will be discussed below.

To calculate the signal field, one should integrate the signal intensity over the

longitudinal coordinate z according to Eq.(5.5). This can be performed analytically for a low-

efficient nonlinear process (E1,2,3= const), as it is usually the case in spectroscopic

applications:

( )

++−Ω−−Ω∆

×

Ω∆+−Ω

×Ω+−−−Ω

Ω=Ω ∫∫ ><

))('''(''2

)'',',(exp

2)'',',(sinc)'''(

~)''(

~)'(

~

,''','~''')(2

),,(~

231212

32

3

4

023124

ttitiL

ki

Lk

ddn

Lcitt

z

z

egeg

ωωωωω

ωωωωωω

ωωωωωχωωµ

EEE

E

*1 (5.7)

The phase mismatch

)()'''()''()'()'',',( 4321 Ω−+−Ω++−=Ω∆ zzzzz kkkkk ωωωωωω (5.8)

Chapter 5

116

should be calculated for each particular geometry, given in Fig.5.1b,c.

Equation 5.7, which will be extensively used in this Chapter, is valid even for single-

cycle optical pulses. The frequency representation allows us to include in a self-consistent

way dispersive broadening of interacting pulses and frequency-dependence of the nonlinear

susceptibility. Besides, we avoid the introduction of the carrier frequency [12] the definition

of which becomes confusing for a few-cycle pulses. We also draw attention to the Ω term in

front of the integral that follows directly from the Maxwell equations and reflects the fact that

higher frequencies are generated more efficiently. It is this term that is responsible for the

effect of self-steepening of the pulses propagating in optical fibers [13].

The total spectrally-resolved signal registered by a quadratic detector is written as

2

231244

023124 ),,(~)()(

),,(~

ttc

QnttI Ω

ΩΩ=Ω Eε (5.9)

with )(ΩQ being the spectral sensitivity of a monochromator-detector combination.

From the point of view of practical application of 5-fs pulses, we now quantify the

differences between the complete frequency-resolved signals of TG and SD computed

according to Eqs.(5.7–9) with const=><3~χ and the ideal frequency-resolved TG and SD

signals for an instantaneous nonlinear response [14]:

( ) 2

231212

322312

))('''(''exp

)''',(~

)'',(~

)',(~

'''),,(

ttiti

zzzddttI

+−+−×

+−Ω=Ω ∫∫ωωω

ωωωωωω EEE *1

ideal4

(5.10)

The comparison of the respective complete and ideal signals provides us with

information on the spectral filter effect, that is, a combined influence of the spectral

variations in the generation efficiency of the signal field and in its detection. To simulate the

conditions of our experiments on hydrated electrons (see Chapter 6), in the calculation of the

complete SD and TG traces we included dispersive properties of a 100-µm layer of water

[15,16] and the impact of the non-collinear beam geometry on the phase-mismatch given by

Eq.(5.8). The small thickness of the medium is crucial to prevent dispersive broadening of the

pulse inside the jet. The lengthening of a 5-fs 800-nm pulse caused by a 100-µm layer of

water is less than 0.1 fs and, therefore, is negligible. The ideal frequency-resolved traces were

calculated according to Eq.(5.10). The spectral filters for the SD and TG cases, obtained as

the ratios of the complete [Eqs.(5.7–9)] vs. ideal [Eq.(5.10)] signals, are presented in Fig.5.2.

The dashed and dotted curves correspond to TG and SD, respectively, for the case of a flatspectral response of the detector ( constQ =Ω)( ). Apparently, both filters are dominated by

the 2Ω -dependence that originates from the Ω -term in Eq.(5.5). The curve representing the

SD filter is somewhat steeper compared with the one in the TG case. This reflects the fact


117

that the phase mismatch for SD is greater since SD is intrinsically a non-phase-matched

geometry [14].

600 700 800 900 10000.0

0.5

1.0

TG

SDDetector

Total

Inte

nsity

[ar

b. u

nits

]

Wavelength [nm]

Fig.5.2: Spectral filters for two configurations of photon-echo experiment in water. Shaded contourrepresents the spectrum of ideal 5-fs pulses. The spectral filter calculated for self-diffraction is shownby a dotted line, and the filter for transient grating is presented by a dashed line. The dash-dotted linedepicts the typical spectral sensitivity of a silicon light detector, Q(λ). The spectral filter for transientgrating corrected by Q(λ) is given by a solid curve. The thickness of the water layer is taken 100 µmand the intersection angles of the beams are 4°. Note that the solid curve (the overall spectral filter inthe TG case) is nearly flat in the wavelength region up to 900 nm because the photo-detectorsensitivity balances off the more efficient generation of the nonlinear signal at higher frequencies.

The taking into account of a typical real spectral sensitivity of a silicon photodiode,)(ΩQ (dash-dotted curve in Fig.5.2) results in the overall spectral filter for TG depicted by

the solid curve. Noteworthy, the overall spectral filtering effect is nearly frequency-

independent throughout most of the spectrum of a 5-fs pulse (shaded contour in Fig.5.2)

because the photo-detector sensitivity balances off the 2Ω -dependence. Therefore, we can

disregard the effect of spectral filtering in case it is counterweighed by the proper choice of

the spectral response of the detector. This is an important conclusion for the practical purpose

of nonlinear spectroscopy with 5-fs pulses since it justifies the use of less cumbersome

spectrally unresolved detection of TG and SD signals.

Now we demonstrate how to arrive to the conventionally used time–domain description

of ultrafast spectroscopy [4]. As we already pointed out, to match the information obtained in

a SD or TG experiment, Eq.(5.9) should be integrated over all frequency components in order

to obtain the total energy of the signal field detected by a quadratic detector. According to

Parseval’s theorem [17], the amount of energy carried by the signal is the same whether we

compute it in the time domain or in the frequency domain. Therefore, the following formula

is a time-domain expression for the same signal:

Chapter 5

118

∫∞

∞−

=2

231244

02312 ),,(2

)(),( tttEdt

c

nttS

πω

ε (5.11)

where

[ ])()()()(exp

),,()(

)()()(2

),,(

2312312211212312323

321312233

23122

0 0 0

123*

132144

023124

ttitititititti

tttRtttt

ttttttttdtdtdtn

Lcttt

eg +++−−+−++−−×

−−−×

−−−−−−= ∫∫∫∞ ∞ ∞

ωωωωωωωωω

ωµ

E

EEE

(5.12)

and the so-called nonlinear response function is introduced as a Fourier transform of the

nonlinear susceptibility [4]:

[ ]∫∫∫ −−−= ><321332211321

3321 exp),,(~),,( ωωωωωωωωωχ dddtitititttR (5.13)

In Eq.(5.12) we also extracted the oscillations of electrical fields at the optical frequency ω i:

( )tittE iii ω−= exp)()( E (5.14)

Note, that in the case of ultrabroadband optical pulses the transition between the

frequency–domain description formulated by Eqs.(5.7–9) and the time–domain representation

summarized by Eq.(5.11–13) becomes valid only in the case of a flat spectral filter. In other

situations when the spectral filtering of the SG or TG signals does occur (regardless of its

reason), the correctness of Eq.(5.12) is not warranted and one must use more general

Eqs.(5.7–9).Equation 5.13 provides the link between the nonlinear response function ),,( 321 tttR

and the third-order susceptibility ><3~χ . For the former, extensive formalism of non-

Markovian dynamics, based on the pathway propagation in the Liouville space [4] has been

developed. Here we restrict ourselves to a simple model of a homogeneously broadened two-

level system. In this case, the nonlinear response function is given as

−

+−=

1

2

2

314

4

321 exp),,(T

t

T

ttNtttR eg

h

µ(5.15)

where egµ is the transition dipole moment, N is concentration, T1 and T2 are the population-

relaxation and dephasing times, respectively, and

11

1*2

12 )2()( −−− += TTT (5.16)


119

with *2T being the pure dephasing time. Fourier-transformation of Eq.(5.15) yields a well-

known result [18]:

( )

)(

1

)''(

1

)'(

1

)'''(

1

,''','~

12

12

12

11

4

4

3

Ω−+

−−+

−−−−−

=Ω+−−−

−−−−

><

egegeg

eg

egeg

iTiTiTiT

Ni

ωωωωωωω

µ

ωωωωωχ

h

(5.17)

The second sum term in square brackets in Eq.5.17 is included to account for the fact that><3~χ possesses symmetry with respect to 'ω and ''ω , and the total expression of ><3~χ is a

sum of all frequency permutations [5,10]. The situation addressed here is of direct relevance

to the experiments described in Chapter 6 and 7. Third-order susceptibilities for different

four-photon processes like Raman scattering or two-photon absorption can be calculated in a

similar fashion. The two-level system can also be dressed in a vibrational manifold to account

for coherent excitation of several Frank-Condon transitions [4].

5.3 Case study: Blue pulse characterization by third-order FROG

In the previous Section we demonstrated a fortunate combination of the beam geometry,

medium properties, and detector sensitivity. The spectral filter resulting from it is benign and,

consequently, it does not seriously affect the correctness of the wavelength-integrated

detection of SD and TG traces. Obviously, under less fortunate circumstances the spectral

filtering can play a significantly more damaging role.

Here we address such a situation by exploring the problem of SD and TG FROG

measurement of a blue pulse around 400 nm with an ~10-fs duration. Exactly this problem

has been recently confronted experimentally in the attempts to characterize tunable pulses

around this wavelength generated in gas-filled hollow fibers [19,20]. The severity of the

spectral filtering in this wavelength region is aggravated by the steeply rising bulk dispersion

in both crystals and glasses because of the proximity of the resonance absorption lying in the

UV. The spectral filter calculated for SD and TG measurement configurations in a BBO

crystal and quartz (fused silica) is depicted in Fig.5.3.

Here the frequency-dependant conversion efficiency is shown against the spectral

content of the pulse. Compared with the SD FROG, the TG FROG (dotted line) provides

much wider spectral window that is determined by the self-steepening effect, i.e. more

effective generation of blue spectral components. The broadening of the spectral window is a

direct consequence of the “box” geometry used in TG FROG [14]. In the SD FROG case, the

central frequency components are substantially suppressed while the wings are enhanced. The

resulting broader spectrum corresponds to a shorter pulse. To illustrate the latter statement,

Chapter 5

120

Fig.5.4 depicts an ideal SD FROG trace of an 11-fs pulse, calculated using conventional

expression [14], and the full SD FROG trace calculated according to Eq.(5.7–9).

360 380 400 420 440 4600.0

0.5

1.0

Inte

nsity

[ar

b. u

nits

]

Wavelength [nm]

Fig.5.3: Spectral filtering effect in SD and TG FROG techniques. As a nonlinear medium, a 100-µmthick slab of BBO (solid curve, SD) or fused silica (dashed curve, SD, and dotted curve, TG) is used.Angles between interacting beams are set at 4o. A spectrum of a 10-fs spectral-limited pulse is shownas a shaded contour for a comparison.

Fig.5.4: Ideal SD FROG trace of a slightly-chirped 11-fs pulse centered around 400 nm. (b): SDFROG trace of the same pulse calculated according to Eq.(5.7–9). (c): Temporal pulse intensitiesretrieved from ideal (dotted curve) and calculated (solid curve) SD FROG data, i.e. (a) and (b),respectively. A 100-µm BBO crystal is employed as a nonlinear medium. Angles between interactingbeams are set at 4o. Note that the trace on (b) appears to belong to a chirp-free pulse.


121

As is apparent from the ideal trace, the pulse is slightly chirped. However, the full SD FROG

trace looks as if the pulse were chirp-free. Moreover, the pulse retrieved from the full trace

(Fig.5.4c, solid curve) is noticeably shorter than its counterpart (Fig.5.4c, dashed curve)

recovered from the ideal trace. The same applies to the SD autocorrelation traces (not

shown), i.e. temporal marginals of the SD traces that would be measured in the experiment

using wavelength-integrated detection. This artificial temporal width shortening is mostly the

result of phase-matching: signal spectral components with the same frequency but generated

from different frequency combinations of fundamental waves have different phase shifts and

therefore can interfere constructively or destructively.

In conclusion, we have shown the case of strong spectral filtering in a SD experiment,

which severely compromises the correctness of the measured characteristics, – in this case,

pulse shape and duration.

5.4 Ultimate temporal resolution of SD and TG experiments

In this Section, we address geometrical smearing – the effect deteriorating the temporal

resolution of a nonlinear spectroscopic experiment as a direct consequence of employing non-

collinear beam geometry. This type of distortion originates from the fact that in a beam,

inclined at an angle to a plane, different transverse components of a pulse travel different

distances before reaching the plane. This means that a fixed delay between two pulses

propagating in two intersecting beams changes into a range of delays across the waist of the

beams in the intersection region. The very same idea of yielding a range of delays

simultaneously is utilized in single-shot pulse autocorrelation techniques [21].

The described above “delay blurring” can be of a serious concern dealing with the laser

pulses that have duration shorter than 10 fs. This issue has been addressed previously in

connection with the temporal resolution of a non-collinear pulse duration measurement via

second-harmonic generation [7,8]. Analogously to Section 3.5, here we evaluate the influence

of the geometrical smearing on the width of self-diffraction and transient grating traces.

For arbitrary pulses and beam profiles, the shape of the resulting traces should be

computed numerically by integrating Eq.(5.9) over each transverse component of the beam.

For linearly chirped Gaussian pulses with Gaussian spatial profile, however, these traces can

be calculated analytically. Assuming that the nonlinear response of the medium is

instantaneous, one can calculate from Eq.(5.10) that the ideal SD or TG trace has a Gaussian

intensity profile in time. Its width, 0τ , is by a factor of 2/3 broader than the pulse duration.

The width of the actual signal, measτ , which has been stretched by geometrical smearing, can

be expressed by

220

2 tmeas βδττ += , (5.18)

Chapter 5

122

where β is a scaling constant dependent on the employed beam geometry, and δ is the

effective delay smearing given by

c

dt f

2

αδ = (5.19)

Here fd is the beam diameter in the focal plane and α is a small intersection angle between

the interacting beams (Fig.5.1b,c). As has been stated in Section 3.5, the lowest value of tδfor Gaussian pulses and beams amounts to 0.4 fs if the central wavelength of the pulse is 800

nm. For the beam profiles other than Gaussian the value of tδ becomes larger.

2 4 6 80

2

4

6

8

Self-Diffraction

Transient grating

No delay

smearing

Sig

nal F

WH

M [

fs]

Beam intersection angle α [deg]

Fig.5.5: Geometrical smearing of transient grating and self-diffraction traces as a function of beamintersection angle. The temporal widths of the observed signals are shown by solid and dashed curvesfor transient grating and self-diffraction, respectively. The duration of ideal Gaussian pulses is 5 fsand the nonlinear response is assumed instantaneous. The focal length of the focusing optics is 125mm and the FWHM of the collimated Gaussian beams is 2 mm.

For self-diffraction the constant β equals 4/3, while for transient grating in the “Box”

beam arrangement β takes the value of ≈5/3. The influence of geometrical smearing on the

width of the trace observed in these two measurement configurations is illustrated in Fig.5.5.

As can be seen from Fig.5.5, the temporal resolution of the self-diffraction experiment is

somewhat higher compared to transient grating. This is explained by the fact that the

smearing in the case of transient grating takes place in xz and yz planes (Fig.5.1c)

simultaneously. In any case, for intersection angles smaller than 10° the lengthening of the

detected signal does not exceed 10%.

Therefore, the effect of geometrical smearing on the generated signals is insignificant

even for experiments with pulses as short as 5 fs, provided the intersection angle is kept

sufficiently small and the beams are properly focused.


123

5.5 Heterodyned detection and frequency-resolved pump–probe

In the previous sections of this Chapter, we have discussed the implications of third-order

nonlinear optical experiments in which the direction of the signal beam differs from that of

the input field(s). Therefore, the signal in SD and TG experiments is essentially background-

free and proportional to the modulus squared of the nonlinear polarization (see Eqs.(5.6,9)).

Despite clear advantages provided by the absence of the background, there are also some

inconveniences. These are a typically weak nonlinear polarization; no information on the

phase of it; a faster, by a factor of two, decay of the TG traces than the actual decay of

induced nonlinear polarization, and square dependence of the signal intensity on the medium

length. Therefore, it may be desirable to combine information from such an experiment with

the measurement in which the signal can by enhanced by hederodyne detection and is linearly

proportional to the nonlinear polarization at the same time.

We now turn our attention to the case of optical pump–probe experiment where the

signal wavevector shares its direction with one of the input fields. Thus, the latter field can be

viewed as a local oscillator that heterodynes the signal field. A third-order pump–probe

experiment involves a double interaction with the pump pulse and a single interaction with a

probe pulse. To comply with the notation in Section 5.2, we assume that the field of the pumppulse is 21 EE ≡ ; 23t is the delay between pump and probe; and 3E is the field of the probe

pulse. The total spectrally-resolved signal registered by a quadratic detector in the direction

of the probe pulse wavevector is then [4,11,22-25]:

[ ] 2

234*3234

2

30

2

2343023

),(~)(~),(~Re2)(~)(

),(~

)(~)(

),(~

tEEtEEc

n

tEEc

ntITOTAL

Ω+Ω⋅Ω+ΩΩ

=

Ω+ΩΩ

=Ω

ε

ε(5.20)

The first term of the sum in Eq.(5.20) is delay-independent and, therefore, it acts as a constant

background that can be readily subtracted (e.g., employing a lock-in or synchronous

detection). The second term is a heterodyned signal. The last (homodyne) term in Eq.(5.20) is

negligibly small compared to the second one provided the conversion efficiency was low

enough. After consulting Eqs.(5.6,7), for the heterodyned signal we obtain:

[ ][ ])(~),(~Im)(

)(~

),(~

Re)(2

),(~

*323

3

*3234023

Ω⋅ΩΩ∝

Ω⋅ΩΩ

=Ω

>< EtPL

EtEc

ntI HET ε

(5.21)

In order to construct transient absorption spectra, in assumption that actual optical density

change, which is due to nonlinear response, is very small, one computes a ratio

Chapter 5

124

2

3

230

2323

)(~

),(~

)(

),(),(

Ω

Ω≅

ΩΩ∆

−=ΩE

tI

T

tTtS HET

FRPP , (5.22)

where ),( 23tT Ω∆ denotes the induced change in the sample transmission, )(0 ΩT is a steady-

state transmission. The spectral sensitivity of the detector, omitted in Eq.(5.20), cancels out in

Eq.(5.22). Unlike the steady-state absorption spectrum that is invariant to the light source

which was used to measure it, the pump-probe spectrum, in general, depends on the actual

properties of both the pump and probe pulse. The dependence on the spectral width and

frequency of the pump is very well known from the transient hole burning spectroscopy [11]

where the first laser pulse creates a spectral “hole“ in the absorption spectrum of an

inhomogeneously broadened transition. The width of the hole in this case is mostly

determined by the homogeneous line-width [4,11]. One cannot, however, neglect the relationbetween the specific shape of ),( 23tSFRPP Ω and the amplitude and phase of the pump and

probe pulses even for completely homogeneously broadened absorption lines, especially ifthe population (longitudinal) relaxation time, 1T , is not significantly longer than the duration

of the pump pulse, τ∆ . Indeed, if τ∆>>1T in a two-level homogeneously broadened

transition, then the memory about specific details of the excitation will be lost upon thecomplete thermalisation of the excited state. In the opposite case, ),( 23tSFRPP Ω can still bear

modulation imprinted on it by the laser pulse even at pump-probe delay times that are

substantially longer than the duration of the pulse(s) and electronic dephasing, given by thevalue of 2T . We will return to the problem of modulation on the pump–probe spectra in the

discussion on our experimental results on the hydrated electron in Chapter 7.

5.6 Conclusions

To solve the non-trivial fundamental issues related to nonlinear spectroscopy with the optical

pulses that consist of 2.5 optical cycles, we developed a general formalism describing the

generated signal field in both the time– and frequency–domain. The frequency–domain

representation is found to be more powerful since it allows a consistent account of a variety

of effects, such as phase-mismatch, self-steepening, dispersive pulse broadening, etc.

Additionally, the use of the frequency–domain formalism removed the necessity to invoke a

number of approximations such as, for example, the slowly varying envelope approximation.

The derived formulation also avoids the use of parameters that are ill–defined for broadband

optical pulses such as, for instance, the carrier frequency of the pulse. Equations 5.7–10

constitute the backbone of the general description of a third-order nonlinear experiments.

Importantly, these equations remain valid and could be directly applied even for single-cycle

pulses.

We have developed a general procedure for calculating the spectral-filter effect. Such a

routine should be employed to optimize the experimental configuration for any third-order

spectroscopic experiment that utilizes laser pulses shorter than 10 fs. In particularly, one can


125

design a compensating filter to account for spectral-filtering effects, and place it in front of

the light detector. Notably, a careful choice of the beam geometry and selection of a

photodetector with the suitable spectral sensitivity, as has been done in our experiments, can

illuminate the need for a separate compensating filter. We next have demonstrated that the

ability to defeat the damaging role of the spectral-filter effect legitimizes a transition to the

typically employed for the multi-cycle pulses time–domain formulation. Importantly for the

weak-signal applications, the absence of spectral filtering eliminates otherwise unavoidable

requirement to frequency-resolve the signals.

Chapter 5

126

References

1. K. Shimoda, Introduction to laser physics, 2nd ed. (Springer-Verlag, Berlin, 1991).2. L. Allen and J. H. Eberly, Optical resonance and two-level atoms (Dover publications, Inc,

New York, 1987).3. P. N. Butcher and D. Cotter, The elements of nonlinear optics (Cambridge University Press,

Cambridge, 1990).4. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New

York, 1995).5. R. W. Boyd, Nonlinear optics (Academic Press, San Diego, 1992).6. A. M. Weiner, IEEE J. Quantum Electron. 19, 1276 (1983).7. G. Taft, A. Rundquist, M. M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. F.

Fittinghoff, M. A. Krumbuguel, J. N. Sweetser, and R. Trebino, IEEE J. Select. TopicsQuantum Electr. 2, 575 (1996).

8. A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, IEEE J. Quantum Electron. 35, 459(1999).

9. M. S. Pshenichnikov, A. Baltuška, R. Szipöcs, and D. A. Wiersma, in Ultrafast Phenomena XI,edited by T. Elsaesser, J. G. Fujimoto, D. A. Wiersma, and W.Zinth (Springer, Berlin, 1998).

10. Y. R. Shen, The principles of nonlinear optics (Wiley, New York, 1984).11. M. Schubert and B. Wilhelmi, Nonlinear optics and quantum electronics (John Wiley, New

York, 1986).12. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of femtosecond laser pulses

(American Institute of Physics, New York, 1992).13. G. P. Agrawal, Nonlinear fiber optics, 2nd ed. (Academic press, San Diego, 1995).14. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. Sweetser, M. A. Krumbügel, B. Richman, and

D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).15. Release on the Refractive Index of Ordinary Water Substance as a Function of Wavelength,

Temperature and Pressure (The International Association for the Properties of Water andSteam, Erlangen, Germany, 1997).

16. A. H. Harvey, J. S. Gallagher, and J. M. L. Sengers, J. Phys. Chem. Ref. Data 27, 761 (1998).17. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New

York, 1986).18. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).19. O. Duhr, E. T. H. Nibbering, G. Korn, G. Tempea, and F. Krausz, Opt. Lett. 24, 34 (1999).20. E. T. H. Nibbering, O. Duhr, and G. Korn, Opt. Lett. 22, 1335 (1997).21. G. R. Fleming, Chemical Applications of Ultrafast Spectroscopy (Oxford University Press,

New York, 1986).22. P. Cong, Y. J. Yan, H. P. Deuel, and J. D. Simon, J. Chem. Phys. 100, 7855 (1994).23. M. Chachisvilis, PhD thesis, Univ. Lund (1996).24. B. Wolfseder, L. Seider, G. Stock, and W. Domke, Chem. Phys. 217, 275 (1996).25. L. Seidner, G. Stock, and W. Domke, J. Chem. Phys. 103, 3998 (1995).

Chapter 6

Early-Time Dynamics of the Photo-Excited Hydrated

Electron

Abstract

Employing photon-echo techniques, we investigate the early relaxation dynamics of the

equilibrated hydrated electron within the first 200 fs upon photo-excitation. The use of 5-fs

laser pulses provided unprecedented temporal resolution of our measurements. We conclude

that the absorption spectrum of the hydrated electrons is primarily homogeneously

broadened. The comparison of two pulse photon echo experiments on pure water and

hydrated electrons allows us to measure the pure dephasing time of ~1.6 fs. The shape of the

absorption spectrum is described excellently by an extended Lorentzian contour with a

spectral width fully determined by the pure dephasing time. From the polarization-dependent

transient grating experiments we establish that the polarization anisotropy of the hydrated

electron falls to the zero value within 5 fs after initial excitation. A prominent role of a non-

Condon effect due to strong coupling of the electron to neighboring water molecules is

suggested. Based on the observed isotopic effect we conclude that the initial relaxation

dynamics are determined by the inertial response of the water molecules which is librational

in its origin. A microscopic picture of the early dynamics of the hydrated electron based on

the experimental results is presented. Finally, we develop a theoretical model based on

wavepacket dynamics, which is capable of capturing the subtle features of the experimental

data.

Chapter 6

128

6.1 Introduction

Upon injection into a fluid, an electron can be captured in a potential energy well formed by

neighboring molecules of the liquid. The first observation of such an electron, known

nowadays as the solvated electron, dates back to 1864, when Weyl reported a blue color of

solutions of metals in ammonia [1]. However, it took more than half a century until Kraus [2]

suggested that the blue color was caused by bound species: electrons trapped among the

ammonia molecules. The solvated electron has been the subject of numerous investigations

ever since. In the sixties the electron solvated in water, called the hydrated electron, was

discovered through the observation of the extraordinarily broad absorption spectrum spanning

from 500 to 1000 nm which appeared upon injection of electrons [3,4].

The vast interest in the hydrated electron from both theoretical and experimental points

of view is by no means accidental. The hydrated electron is a transient species in

charge-transfer reactions in biology, non-silicon solar-cell technology [5-8], and chemistry.

Important examples are photosynthesis [9], charge transport through biomembranes [10], and

long-distance charge transport in nerves [11]. The hydrated electron is also a key intermediate

in radiation chemistry [4] and electrochemistry [5-8]. Next to this, the hydrated electron is a

perfect test-ground for various theories of complex quantum–mechanical systems. The

three-dimensional confinement of the single electron by the surrounding water molecules

gives rise to discrete quantum states. This closely resembles a quantum dot [12], an entity

that has raised immense attention in recent years as a model system for atoms [13-15] and

molecules [16], and holds a great promise for the use in optical devices [17]. However, the

important difference between the quantum dot and the hydrated electron is that in the former

case the potential giving rise to the confinement is static whereas in the latter case it changes

rapidly in time due to dynamical fluctuations of the liquid surrounding.

Numerous computational studies have been performed to investigate the

quantum-mechanical status of the hydrated electron and the microscopic structure of its

surroundings. It became clear that the first shell of water around the electron is composed of

approximately six molecules with their OH-bonds oriented toward the electron [18,19]. (See

Fig.1.1.) A similar result was also found in electron-spin echo measurement on the electron

solvated in glassy water [20,21]. When the molecular dynamics simulations matured, they

succeeded in reproducing the general features of the absorption band shape, such as its

breadth and asymmetry [18,22-25], although the transition energies were somewhat

overestimated. After being shifted toward lower frequencies, the simulated spectra closely

resembled the experimental data. According to the extensive computational modeling

performed by the group of Rossky [18,22,23], the absorption spectrum of solvated electrons

is primarily caused by a strongly allowed transition from a roughly spherical localized s-like

ground state to a triple of p-like states that are also bound and localized. The existence of the

s-state indicates that on average the potential energy well, created by the molecules

surrounding the electron, is close to spherical. However, because of the asymmetries caused

Early-Time Dynamics of the Photo-Excited Hydrated Electron

129

by the dynamical nature of liquid water, the potential energy surface does not have a perfect

spherical shape, which results in three non-degenerate p-states.

The question about homogeneous vs. inhomogeneous broadening in the optical

absorption spectrum of the excess electron in water and other fluids, as well as the

explanation of its asymmetry and extraordinary spectral width, has remained a standing

problem for over three decades. Numerous attempts have been made to fit the experimental

data by various line shapes [26-30] and superposition of lines [31-33]. In the computer

simulations [22], by imposing an ordering of the p-states according to their energy, the

resulting absorption contour was decomposed into a superposition of the contributions from

separate s–p transitions. Each of the obtained three bands is claimed to be substantially

inhomogeneously broadened by different structures of the solvent surroundings. The energy

splitting between two adjacent s–p transitions found in these simulations is ~0.4 eV, which

nearly constitutes the width of the bands associated with each separate transition.

The dynamical behavior of the hydrated electron, i.e., the energy relaxation after an

instantaneous s–p excitation has been thoroughly modeled as well [18,19,25,34-41]. All

computer simulations predict that the solvation dynamics are essentially bimodal. The initial

decay is responsible for ~50% of the total energy relaxation and occurs at a 10–25 fs time

scale which is followed by a slower 130-250 fs decay. It is generally agreed that the latter

time scale results from the diffusional motion of water molecules into and out of the first

solvation shell.

However, the microscopic nature of the fast initial decay is still under considerable

debate. By comparing the frequencies from the power spectrum of the correlation function of

the s–p energy gap with the Raman spectrum of water, Staib et al [25] concluded that the

accelerated decay was determined by coupling to hindered rotations of water molecules,

generally called librations. Based on the dependence of the most rapid time scale in ordinary

and deuterated water, Barnett et al. found that the initial dynamics of the hydrated electron is

caused by free rotational diffusion of water molecules [34]. Conversely, Berg [41] and the

group of Rossky [36,37,40] revealed no isotope effect in the first 25 fs of the solvation

dynamics of the hydrated electron. They accordingly inferred that the origin of the initial

dynamics is translational. Park et al. concluded that the motions of the water molecules in the

first solvation shell of the hydrated electron are dominantly rotational through repulsion of

hydrogen-bonded hydrogen atoms and attraction of dangling hydrogen atoms [19].

The past decade has witnessed numerous studies of the hydrated electron with an

~200-fs time resolution [32,42-50]. The femtosecond time-resolved studies of hydrated

electrons were pioneered by Migus et al. [43]. The electrons were generated by multiphoton

ionization of neat water and studied by transient absorption of a super-continuum probe.

Later similar experiments were carried out by several groups evidencing the importance of

geminate recombination and pump-probe cross phase modulation in the recorded transients

[32,42,44-48].

Chapter 6

130

In another approach, the already-equilibrated hydrated electron is excited from the

ground s-state to the p-state using a short pulse, and the resulting solvation dynamics is

probed as a function of time with another, delayed, pulse. Following this route, the group of

Barbara found decays of ~300 fs and ~1.1 ps in a pump-probe experiment [51]. The results

were explained using a three-state model, where the fastest decay is caused by the relaxation

down to the ground state, giving rise to a not yet-equilibrated “hot” ground state. Recently

Assel et al. refined this model by including excited state solvation that took place before

relaxation back to the ground state [32].

More recently, experiments carried out with a substantially improved time resolution

revealed that the early solvation dynamics occur on a much shorter time scale [52-55]. In

these experiments, it was found that the ~300 fs decay is preceded by the dynamics on a

timescale of less than 50 fs. Interestingly, both the experiments with 35-fs-pulses [52,53] and

with 13-fs pulses [54,55] pointed to the librational nature of this initial solvation dynamics. It

became evident, however, that even shorter pulses are required to match the large spectral

width of the hydrated electron absorption. The rapid progress of state-of-the-art laser

technology in the last years has made such pulses available for spectroscopic experiments

[56-60].

In this Chapter, we report the latest results on the early dynamics of the hydrated

electron obtained with an unprecedented time resolution of 5-fs pulses. Based on the

comparison of photon echo signals from hydrated electrons and from water alone, we derive

a 1.6-fs pure dephasing time of the hydrated electrons. This value is fully consistent with the

line-shape of the absorption spectrum, which is shown to be overwhelmingly homogeneously

broadened. We demonstrate that the optical response of the hydrated electron is substantially

delayed with respect to the excitation pulses. This unexpected effect is explained using a

model in which the transition dipole moment of the electron increases after the excitation due

the strong coupling to the solvent molecules. Furthermore, it is shown that the effect is

librational in nature. Finally, we demonstrate that a simple model based on wavepacket

dynamics can account for the experimental results.

This paper is organized as follows. In Section 6.2, we describe the experimental set-up

for the ultrafast photon-echo spectroscopy and give a detailed account on the preparation of

equilibrated hydrated electrons. In Section 6.3, the results of two– and three–pulse transient

nonlinear spectroscopy on electrons in ordinary and deuterated water are presented.

Subsequently, a model is presented that satisfactorily explains the observed behavior. Finally,

in Section 6.4, we summarize our findings.

6.2 Experimental

6.2.1 Femtosecond laser system

The femtosecond spectrometer is based on a home-built cavity-dumped Ti:sapphire laser, that

has been described in detail in Chapter 2. The schematics of the set-up are presented in

Fig.6.1. Briefly, the cavity-dumped laser is a standard Ti:sapphire oscillator that incorporates


131

a Bragg cell inside the cavity. This allows us to produce 15-fs pulses at the central

wavelength of 790 nm at a desired repetition rate. The energy of the dumped pulses reaches

up to 35 nJ and is adjustable by setting a level of RF power applied to the Bragg cell.

Fig.6.1: Schematics of the femtosecond photon-echo spectrometer. BS1 is an R=30% beamsplitter,BS2 is a 50% beamsplitter. CP’s are compensating plates, and PD1-4 are silicon light detectors. E1,E2,and E3 are femtosecond excitation pulses. t12 and t23 are the delays between pulses E1-E2, and E2-E3,respectively. The bottom-left inset shows the image of the signal and excitation beam arrangement onthe recollimating mirror. The bottom-right inset shows the profiles of two-pulse photon echoesmeasured in two conjugate directions by PD1 and PD2.

To provide adequate time resolution for the study of the ultrafast dynamics of the

hydrated electron and broaden the spectral window of our measurements, the output of the

cavity-dumped laser is externally compressed to the pulse duration below 5 fs. The 15-fs

laser pulses, precompressed by a pair of fused silica prisms, are injected into a single-mode

quartz fiber through a microscope objective lens. The white-light continuum resulting from

the combined action of the self-phase modulation and dispersion in the fiber core is

collimated by an off-axis parabolic mirror to avoid chromatic aberrations and bulk dispersion

of a collimating lens. A portion of the white-light in the spectral range of 580-1060 nm

Chapter 6

132

(shaded contour in Fig.6.2) is then compressed in a state-of-the-art three-stage pulse

compressor that includes a pair of 45° quartz prisms, specially designed chirped dielectric

mirrors, and thin-film dielectric Gires-Tournois interferometers.

Immediately before performing photon echo spectroscopy on hydrated electrons, the

compressor is adjusted to yield the shortest duration of the pulses, which are characterized by

second-harmonic frequency-resolved gating (SHG FROG), as has been described in Chapter

4. Because dispersive broadening easily affects the 5-fs pulses even as they propagate

through air, the FROG characterization is carried out directly at the location of the

spectroscopic sample by replacing it with a very thin (10 µm) second-harmonic BBO crystal

(EKSMA).

6.2.2 Transient grating and photon echo experiments

The experimental arrangement used for the photon echo spectroscopy is depicted in Fig.6.1.

The beam carrying 6-nJ, 5-fs pulses is split into three channels of approximately equal

intensity by the 0.5-mm-thick beamsplitters BS1 and BS2 (pulses E1, E2, and E3 in Figure

6.1). Compensating plates (CP) of the same thickness are inserted into the beams to equalize

dispersion in all three channels. To minimize distortions in the pulse duration and preserve its

spectral content, silver mirrors overcoated by a thin protective layer are used throughout. To

match the reflectivity of the beamsplitters, the initial horizontal polarization of the laser beam

is turned 90° by a mirror periscope. The optical polarization of one of the pulses, E3 is further

turned by 45° with respect to the polarization of the pulses E1 and E2 by another periscope to

facilitate photo echo measurements in the parallel and orthogonal polarization directions.

Two independent optical delay lines, t12 and t23 are employed to fabricate desired sequences

of the three pulses.

The beams are focused into a 100-µm water jet, in which the hydrated electrons are

generated, and recollimated behind it by spherical mirrors with the radii of curvature R=-250-

mm. A small incidence angle on the spherical mirror is chosen to prevent astigmatism of the

beams within the intersection region. The intersection angles between the incident beams are

kept at ~4°. The waist of the focused beam is ~ 30 µm in diameter. The phase-matching

geometry of the laser beams is explained in the bottom-left inset to Fig.6.1, presenting the

enlarged image of the beam configuration on the recollimating mirror. The advantages of

employing such a beam arrangement have been discussed by de Boeij et al. [61].

The two- and three-pulse stimulated echo signals are detected simultaneously behind

the sample by silicon photodiodes PD1–4 equipped with built-in amplifiers. The photodiode

signals are processed by lock-in amplifiers, digitized, and stored in the computer memory.

The ability to determine the exact overlap of the pulses E1–E3 in time with a high

degree of precision presents a considerable experimental challenge and is vital for the photon

echo measurements with 5-fs pulses that occupy merely 1.5 µm in space. Fortunately, the

overall symmetry of the employed configuration makes it possible to accurately find t12=0

and t23=0 without employing additional means.


133

The two-pulse photon echo traces are measured as a function of delay between the

pulses E1 and E2. A typical result of a t12 scan is depicted in the bottom-right inset to Fig.6.1

Here the solid and the dashed curves represent the contours obtained in the two conjugate

directions, monitored by the photodiodes PD1 and PD2, respectively. Since these two-pulse

photon echo traces are intrinsically symmetric around t12=0, such a scan allows finding the

precise overlap of the pulses E1 and E2. Similarly, the precise location of t23=0, i.e. the

overlap between the pulses E2 and E3, can be verified by a t12 scan of the three-pulse echo

signals, which in the case of coinciding pulses E2 and E3 correspond to the signals depicted in

the inset to Fig.6.1.

The transient grating scans, a variety of three-pulse photon echo spectroscopy, are

performed by scanning the delay t23 between the time-coincident pair of excitation pulses

E1-E2.and the third (probe) pulse E3. The symmetry of the incident beam arrangement impliesthat in this case two exactly identical signals are emitted in k3+k2-k1 and k3-k2+k1 directions.

This also serves as a sensitive indicator to continuously monitor the perfect time overlap of

the excitation pulses. By rotating polarizing cubes in front of PD3-4, we record the

components of the transient grating signal parallel and perpendicular to the polarization of the

pulses E1 and E2.

6.2.3 Generation of hydrated electrons

The technique of hydrated electron generation through electron photo-detachment from

various types of anions [62] has been introduced almost immediately following the

observation of hydrated electron formation by the action of intense electron beams on water

[3,4]. The former production method offers a clear advantage, since it typically requires

merely one-photon ionization and, therefore, is employed in our experiments. Among

different complex ions, studied for the electron photo-detachment [62-65], ferrocyanide

(hexacyanoferrate(II), Fe(CN)64-) was found to have the highest quantum yield reaching the

value of 0.9 for 228-nm irradiation [66]. In the case when a ferrocyanide ion is photolyzed to

yield a hydrated electron, a Fe(CN)63- ion, hexacyanoferrate(III), forms in the solution which

gives rise to an absorption band at 415 nm [63,67]. The important fact that the absorption of

this photoproduct clearly lies outside the spectral range of our femtosecond experiment, is an

additional favorable aspect of employing ferrocyanide rather than another negative ion.

Hydrated electrons are generated by photo-ionizing a small amount of potassium

ferrocyanide [52,62,66,68] added to water, with the quadrupled output of a Nd:YLF laser

(263 nm). Potassium ferrocyanide was obtained from Merck. Water and heavy water of

HPLC grade were purchased from Aldrich and used without further purification. The

polished sapphire nozzle (Kiburtz) ensured good quality of the jet and, therefore, no addition

to the solution of the chemical substances stabilizing the jet surfaces [52] was needed. All

measurements were carried out at the room temperature.

The repetition rate of the YLF laser and, consequently, the repetition rate of the entire

spectrometer was set at 4 kHz. The UV pulses are focused directly into the intersection region

Chapter 6

134

of the pulses E1, E2, and E3 in the water jet. To avoid noticeable variation of electron

concentration within the interaction area of the femtosecond pulses, the spot size of the UV

beam on the water jet is approximately two times larger than the waist of the three other

beams. The triggering of the Nd:YLF laser is synchronized with the cavity-dumping of the

Ti:sapphire laser so that the UV pulse precedes the femtosecond pulse by ~200 ns. The

absorption spectrum of hydrated electrons has been measured in the spectral region 480–1100

nm with uncompressed white-light pulses and obtained as a difference in optical density of

the water jet in presence of, and without UV radiation. The typical absorption spectrum

(Fig.6.2, open circles) has the peak value of O.D.≈0.2 around 720 nm. The data of our

absorption measurement coincide very well with the known from the literature [33]

absorption spectrum of hydrated electrons (Fig.6.2, solid line) that have been directly injected

into a volume of water.

500 600 800 10000.0

0.1

0.2

5-fs pulse

e-

aq

O.D

.

Wavelength [nm]

Fig.6.2: Absorption spectrum of electrons in water. The solid line is the absorption of hydratedelectrons produced by electron beams in bulk water (adopted from Ref. [33]). The solid dots aremeasured with the femtosecond white-light continuum upon photo-ionization of potassiumferrocyanide with 263-nm pulses. The shaded contour shows the spectrum of the 5-fs pulses.

Unlike the injected electrons which reportedly in water have a lifetime of ~10 µs [3],

the electrons released through photo-ionization generally have a shorter recombination time.

The mechanism responsible for this shortening is a so-called scavenging process, whereby an

electron recombines with one of the ions of the donor molecules or other scavengers

introduced to the solution [64]. Thus, the variation of the concentration of ferrocyanide in the

solution has a two-side effect. On one hand, the increase of the concentration is directly

proportional to the amount of the electrons generated by photo-ionization. On the other hand,

it increases the rate of electron recombination with the ferrocyanide ions because of the

scavenging. To measure the recombination time, we recorded the change in absorption of the

hydrated electrons as a function of delay between the photo-ionizing UV pulse and the white-

light pulse. The normalized kinetics at 720 nm obtained at two different concentrations of

ferrocyanide are depicted in Fig.6.3 as circles. The intensity profile of the 120-ns UV pulse


135

shown alongside (shaded contour) was detected with a fast solar-blind photo-multiplier tube

(Hamamatsu) and recorded with a 1-GHz sampling oscilloscope (Hewlett Packard). The

delay between the two pulses was set electronically by varying the triggering time of the

Ti:sapphire and the Nd:YLF lasers in steps of 24 ns.

The photo-ionization process, countered by the electron scavenging, was modeled by a

simple balance equation similar to the one derived in Ref. [64] in which we included the

scavenging term and explicit pulse shape:

)()()(

tItN

dt

tdNUVα

τ+−= , (6.1)

here N(t) is a concentration of equilibrated hydrated electrons, τ is the time constant of

electron-ion scavenging, IUV(t) is the intensity of the UV pulse, and α is a constant reflecting

the quantum efficiency of photo-ionized electron generation. In Eq.(6.1) we assumed that the

hydrated electrons are formed instantaneously. Since the spectrum of the hydrated electrons

is formed with a time constant of 0.3–0.5 ps [67,69], this assumption is fair on the much

slower time scale of our experiment.

-200 0 200 400 600 800

Femtosecond

spectroscopy

Abs

orpt

ion

UV - IR Pulse Delay [ns]

Fig.6.3: Recombination dynamics of hydrated electrons generated by photo-ionization. The shadedcontour shows the intensity of the UV pulse used for photo-ionization. The dots represent measuredand normalized absorption changes at the wavelength of 720 nm for potassium ferrocyanideconcentrations of 0.4 g/l (solid circles, peak optical density ~0.2) and 4.0 g/l (hollow circles, peakoptical density ~0.05). The solid curves depict the fits obtained according to the procedure describedin the text.

Equation 6.1 was solved numerically to fit the data presented by circles in Fig.6.3

Digitized pulse shape (Fig.6.3, shaded contour) was used as the parameter IUV(t), and τ was a

fitting parameter to match the experimental dependence. The respective fits of the two data

sets are depicted in Fig.6.3 by solid lines. The decay time τ obtained for the ferrocyanide

concentration C=0.4 g/l is 115 ns, while for C=4.0 g/l τ=45 ns. Therefore, a tenfold increase

Chapter 6

136

of the ferrocyanide concentration results in an about twofold acceleration of the rate of

electron–ion recombination, while the absorption of hydrated electrons at its peak increases

by a factor of 5.

The femtosecond photon-echo spectroscopy on the hydrated electrons was performed in

the time window indicated on the falling edge of the absorption change traces in Fig.6.3 as a

vertical bar. While the concentration of the electrons is still near its peak at this delay, the UV

pulse is already largely over and, therefore, predominantly equilibrated hydrated electrons are

present in the solution at this time.

6.3 Results and Discussion

6.3.1 Intensity-dependence measurements

To verify the order of the nonlinearity contributing to the transient grating signal from the

hydrated electrons, we measured the dependence of this signal on the intensity of

femtosecond pulses. Since tuning the 5-fs pulse intensity is not feasible without destroying

the pulse duration or its spectral content, we employed for this purpose 15-fs pulses directly

from the cavity-dumped laser. The energy of the 15-fs pulses was changed by varying the RF

power of the cavity-dumper driver in the interval 5—30 nJ. This corresponds to the combined

intensity of the three pulses in the sample ranging from 0.6×1011 to 4.0×1011 W/cm2. The

power dependencies of the TG signal (i.e., t12=0) from electrons in water measured at two

different values of t23 delay are depicted in Fig.6.4 (dots). Solid lines shown alongside the

experimental data represent the cubic power dependence that is expected from nonlinearity

based on the third-order response. Clearly, no noticeable deviation from the third-power law

is reached with the intensities used.

5 10 30

102

103

104

20

P3

t2 3

=0 fs t

2 3=50 fs

TG

Sig

nal [

arb.

uni

ts]

Pulse energy [nJ]

Fig.6.4: Intensity dependence of the transient grating signal from hydrated electrons. Close and opencircles show data measured at delays between the excitation pulse-pair and the probe pulse of t23=0 fsand 50 fs, respectively. The solid lines depict third-power dependencies expected for the third-ordernonlinearity.


137

Previous intensity-dependence measurements of the pump–probe signal from hydrated

electrons revealed departure from a purely third-order nonlinear response for the intensities of

the excitation pulse in excess of ~3.0×1011 W/cm2 [52]. For higher intensities, leveling-off of

the signal was reported. Since the pulse intensities applied in our measurement did not exceed

4.0×1011 W/cm2, the saturation regime has not been reached yet.

The total intensity of the excitation pulses in the 5-fs experiments amounted to

2×1011 W/cm2. Therefore, the contribution of higher-order nonlinearities or saturation effects

is not expected in the experiments reported below.

The contribution of the pure water to the total nonlinear optical response of the sample

was checked by recording the photon echo traces in absence of UV pulses. This

contamination of the signal did not exceed ~5% of the signal peak value obtained in the

presence of the hydrated electrons and was confined to the region of delays within the

overlap of the 5-fs excitation pulses.

6.3.2 Pure dephasing time of hydrated electrons

The employed method for the generation of hydrated electrons provides us with a unique

opportunity to compare the photon echo signal from the hydrated electrons with the one

measured in pure water, that is in absence of photo-ionizing radiation. Since electronic hyper-

polarizability [70-72] heavily dominates the overall water response [73,74], it is well justified

to treat the nonlinearity as nearly instantaneous on the time scale of our pulses. Therefore, the

signals obtained from pure water correspond to the ultimate instrument response of the

spectrometer. This instrument function, among its other merits, automatically accounts for the

pulse duration, mode size, and spatial as well as spectral filtering in the detection. Therefore,

the differences in the shape of photon echo traces recorded in the presence and without the

UV radiation provide us with direct information on electronic dephasing of the hydrated

electrons.

The two-pulse photon echo signals from the water and hydrated electrons are shown as

solid circles in Fig.6.5a and 6.5b, respectively. A minute difference in the widths of these two

traces suggests that the electronic dephasing of the photo-excited hydrated electrons is

extremely fast. To fit to the experimental data (solid curves in Fig.6.5a and 6.5b), we used the

formalism developed in Chapter 5. The precise pulse parameters were obtained from

independent FROG characterization as described in Section 6.2.1. For water (Fig.6.5a), weassumed an instantaneous response function (Eq.(5.15)) )()()(),,( 321321 ttttttR δδδ∝ or, in

other words, the frequency-independent third-order susceptibility. In the case of the hydrated

electron (Fig.6.5a), complete Eq.(5.15) was used with the dephasing time T2 being the fitting

parameter. The population lifetime T1 was considered to be much longer than any relevant

experimental time scale, including pulse duration. Experimental evidence [52,53] supports

the idea that the excited state lifetime can be as large as hundreds of femtoseconds.

Therefore, the difference between the full electronic dephasing time, 2T , and the pure

dephasing time, *2T , which are connected by Eq.5.16, is negligible.

Chapter 6

138

-20 -10 0 10 20

0

1(b)

Inte

nsity

Delay t12

[fs]-20 -10 0 10 20

0

1(a)

Inte

nsity

Delay t12

[fs]

Fig.6.5: Results of two-pulse photon echo experiments on water alone (a) and hydrated electrons (b).Circles represent experimental data points and solid curves show fits obtained according to theprocedure described in the text.

The finite population lifetime of the electrons in the excited state causes the delay of the

echo trace in Fig.6.5b (its shift of further away from t12=0 compared to the data in Fig.6.5a).

The best fit to the experimental data yields the dephasing time of T2 = 1.6 fs. Note that this

value is reasonably close to theoretical estimations derived from a model based on the

Gaussian wave packet approximation for the bath [39,75,76]. The addition of any appreciable

amount of inhomogeneity immediately results in pulling the echo maximum away from zero

and appearance of noticeable asymmetry of the trace.

Evidently, such a 1.6-fs dephasing time should manifest itself in the absorption line

shape, which has to be substantially homogeneously broadened [77]. Here we stress that the

use of a standard Lorentzian line shape is not warranted for the spectra with the widths

comparable to the central frequencies. Instead, a more general relation should be used [78-

80]:

[ ]

22

2222

22

2

12

12

1

4)(

4

)(

1

)(

1Im)(Im)(

−

−

−−><

+−∝

+++

−−∝∝

T

T

iTiT

eg

egegA

ωωωω

ωωωωωωχωωσ

(6.2)

(For example, see Eq.(3.5.25) in Ref. [78]). In Eq.(6.2) Aσ is the absorption cross-section

and egω is the transition frequency. In the case of a narrow absorption band, i.e. egT ω<<−12 ,

one can make use of the approximation egωω ≅ , which immediately gives the conventional

Lorentzian contour:

22

2

22

)()(

−

−

+−∝

T

T

egA

ωωωσ (6.3)


139

In fact, the rejection of the second term of the sum in Eq.(6.2), resulting in Eq.(6.3), amounts

to application of the so-called rotating-wave approximation (RWA) [77,78,81].

The difference between the two line-shapes, given by Eq.(6.2) and Eq.(6.3), mostly

affects the asymptotic behavior in the spectral wings. Unlike the pure Lorentz function

Eq.(6.3), which produces a centro-symmetric contour, the line shape described by Eq.(6.2) is

essentially asymmetric. The latter contour has a more abrupt red wing and a prolonged blue

wing. The difference between the two line-shapes is plain to see in Fig.6.6. The need to

account for the experimentally observed asymmetry precluded the use of a single Lorentzial

line-shape in the past attempts to model the absorption spectrum of hydrated electrons.

Consequently, a collection of spectral lines [26,31,32] or a combined line-shape [27-30,33]

was required for a reasonable fit.

0 2 4 6 8 10 12 140

1

A/A

max

Frequency in units of T2

-1

Fig.6.6: Difference in line-shapes. Dashed curve is calculated according to Eq.(6.3) (SymmetricLorentzian contour), and solid curve is given by Eq.(6.2)

Employing the more general relation for a homogeneously broadened line-shape given

by Eq.(6.2), we obtained fits (solid curves in Fig.6.7) to the absorption spectra of electrons

injected into bulk water and heavy water at various temperatures. Solid circles in Fig.6.7a and

Fig.6.7b represent experimental data from Ref. [33] for water and heavy water, respectively,

at different temperatures. Evidently, the whole absorption spectrum can be excellently

reproduced by a homogeneously broadened line-shape. The dephasing time T2=1.7 fs,

deduced from the fit of the spectrum of solvated electrons in water at 298 K, perfectly agrees

with the one obtained from the photon-echo experiment. This leaves no room for doubts

about the homogeneous nature of spectral broadening of the hydrated electron absorption

band.

Having established that the absorption spectrum of the hydrated electron is

homogeneously-broadened, we address the results of MD simulations from which the

conclusion of inhomogeneous broadening of each s–p bands was made [18,22,23]. It is clear

Chapter 6

140

that the stipulated non-degeneracy of the p-states is a direct and natural consequence of the

not entirely spherically symmetric solvent cavity. However, the decomposition of the

absorption spectrum into three bands seems to be somewhat artificial. Indeed, as the energy

of a given p-state fluctuates in time as a result of the rearrangement of the surrounding

solvent molecules, this state might well become the highest or the lowest among the three p-

states.

10000 15000 20000 25000 30000 350000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6(a)

380 K

340 K

298 K

274 K

A/A

max

Wavenumbers [cm-1]

300 3502750

3000

3250

3500

1/T

2 [cm

-1]

T [K]

10000 15000 20000 25000 30000 350000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6(b)

380 K

340 K

298 K

274 K

A/A

max

Wavenumbers [cm-1]

300 3502750

3000

3250

3500

1/T

2 [cm

-1]

T [K]

Fig.6.7: Fit of absorption spectrum of electrons in H2O (a) and D2O (b). The circles show measuredabsorption of hydrated electrons produced by electron beams in bulk water (adopted from Ref. [33]).The solid curve shows the fit by homogeneously broadened line-shape calculated according toEq.(6.2). The insets present the value of 1/T2 resulting from the fit. Temperatures are indicated withthe respective absorption spectra. The artificial offset of 0.2 along the vertical axis between each twoadjacent curves is applied for better viewing.

According to the proposed ordering of the excited states by energy, one should then reassign

the transition to this p-state as belonging to a different absorption sub-band. As a

consequence of this artificial reassignment, the width of the Gaussian-like sub-bands revealed

by these simulations becomes nearly equal to the separation of the band centers (vide supra).

If the energy-dependent order were dropped, the absorption band associated with each s–p

transition eventually would have the same width as the whole absorption spectrum of the

solvated electron. This means that, depending on the current precise solvent surroundings, the

optical transition to a designated p-state can take place anywhere across the whole absorption

spectrum. The dynamic fluctuations of the solvent cavity cause constant “migration” of the

respective s–p absorption bands within the common envelope.

We state that the seeming confusion with the assignment of the three absorption sub-

bands arose from the averaging over an ensemble of solvent configurations (Ref. [18,22,23])

rather than averaging of the same configuration evolving in time. The same averaging of

multiple static cavity “snapshots” led to the conclusion of the inhomogeneous broadening of

each absorption line [18,22,23].


141

To the best of our knowledge, there has been no experimental evidence supporting the

hypothesis of the inhomogeneous nature of the line broadening in the absorption of an

electron in a fluid. Both picosecond [82] and femtosecond [32] transient pump–probe spectra

revealed no presence of hole-burning behavior. Our recent attempts to imprint a spectral hole

in the hydrated-electron absorption line employing a 15-fs excitation pulse centered around

800 nm and a broadband 5-fs readout pulse likewise resulted in a uniform and immediate

bleaching of the whole absorption contour, suggesting that the latter is overwhelmingly

homogeneously broadened. The lack of asymmetry in the two-pulse photon-echo signal

(Fig.6.5b) is another strong evidence in favor of overwhelming homogeneous broadening.

Besides, in the TG experiments we observed no quantum beats which are usually associated

with the coherent excitation of several transitions (vide infra).

Therefore, based on the current and previously accumulated experimental data we

conclude that the absorption spectrum of the hydrated electron is predominantly

homogeneously broadened. An important question now should be raised about the physical

origin of a very rapid dephasing associated with a very broad absorption band. Indeed, hardly

any nuclear motion on such a short time-scale should be expected. However, the electronic

dephasing is not necessarily the consequence of the rapid fluctuation of local structures, nor

does it mean that a large amount of energy has to be dissipated by the bath within a few

femtoseconds. It is well known that the difference in the frequencies of (harmonic) ground

and excited state potentials influences electronic dephasing through the so-called quadratic

electron–phonon coupling [83-87]. This mechanism also has been shown to lead to

prevalently homogeneous broadening of absorption lines [88]. Since multiphonon

interactions are involved in nonlinear electron–phonon coupling, the latter explains large

breadths of absorption spectra in the case of a relatively modest width of the spectral density

of the solute–solvent fluctuations [77].

To conclude this Section, we suggest that the absorption band of equilibrated hydrated

electrons is primarily homogeneously broadened with the corresponding dephasing time

T2≅1.6 fs. The asymmetry of the spectrum is explained by the frequently neglected

dependence of the optical absorption cross-section on frequency. A mechanism responsible

for the extraordinary width of the absorption spectrum is most probably the quadratic

electron–phonon coupling due to appreciable difference in the steepness of the ground– and

6.3.3 Transient grating spectroscopy

The early part of TG transients of the hydrated electron is shown in Fig.6.8a for parallel

polarizations of the excitation pulse pair and the probe pulse. The signal has a sharp peak

around zero which is followed by a prominent recurrence at ~40 fs. Subsequently, the signal

decays on an ~200 fs timescale.

Chapter 6

142

-50 0 50 100 150 200

0.0

0.5

1.0

||⊥

(a)

Inte

nsity

Delay t23 [fs]

-50 0 50 100 150 200

0.0

0.5

1.0(b)

Inte

nsity

Delay [fs]

Fig.6.8: Transient grating signals obtained from the hydrated electron in H2O (a) and their difference(b). Solid dots and open circles in (a) represent experimental data measured with the parallel andperpendicular polarization of excitation pulses, respectively. Solid curves in (a) depict the fitscalculated as described in Section 6.3.5. The difference between the two signals is shown in (b) byfilled diamonds while solid curve gives the two-pulse echo signal from Fig.6.5b for comparison. Notethat the non-zero difference is confined in the region where the pulses overlap in time.

Although the 5-fs pulses have enough spectral bandwidth to excite more than one of the

p-states at once, there is no indication of quantum beats that are associated with the presence

of several transitions [77]. From the calculated splitting between the different s-p subbands

[18,22,23] one expects the quantum beats with an ~10-fs period. This raises the question

about the exact meaning of the three p-states found in the quantum molecular dynamics

simulations. The separate bands arise from the following procedure: in each snapshot

corresponding to one time step in the simulation (1 fs), the three p-states were ordered by

energy, and the spectrum is subsequently decomposed into contributions from each of the

three transitions. This procedure does not incorporate the important factor of the time scale at

which the energy of each of the p-states changes in time. As was shown above, the electronic

dephasing for the s–p transition is exceptionally fast, giving rise to an extremely broad

homogeneous absorption line-shape. This means that every s–p transition rapidly samples all

possible energy differences within the absorption spectrum. Therefore, we cannot assign to

the latter three separate bands with each of them corresponding to a separate s–p transition.


143

As a matter of fact, the total spectrum consists of the sum of three bands with approximately

the same width as the total spectrum and the nearly identical central frequencies. As a result,

no quantum beats can be observed under these circumstances.

Next we focus on the origin of the narrow peak around zero and the subsequent

recurrence in the TG signal. In the earlier papers on experiments with a 15-fs time resolution

[54,55], we suggested that a similar shape of TG transients were caused by librational

motions of water molecules in the first solvation shell. Upon excitation a coherent

wavepacket is created, that undergoes underdamped oscillatory motion on the excited-state

potential surface. The return of the wavepacket to the inner turning point gives rise to the

recurrence in time-resolved optical signals. The signal amplitude near t23=0 is indeed higher

than that of the recurrence due to two main reasons: the damping of the wavepacket and

additional coupling between interaction pulses near zero-delay known as coherent artifact

[89-91]. The latter is a manifestation of extra contributions to the TG signal originating from

irregular time ordering of the excitation pulses. For example, around zero delays the second

pulse E2 is scattered from the grating imprinted by pulses E1 and E3 thus leading to the

increased signal.

However, if the polarization of the excitation pulse pair and the probe pulse is

orthogonal (Fig.6.8a, solid dots), the peak around zero vanishes. The second intriguing

feature of the transient is that its amplitude is precisely equal to the one obtained with the

collinear polarizations. This is highlighted in Fig.6.8b, where the difference between the two

signals is shown. For comparison, we also depicted the two-pulse photon echo signal from

Fig.6.5b. Obviously, the non-zero part of the difference between the TG signals with parallel

and orthogonal polarizations is confined to the region of the pulse overlap. Note that this is

highly uncommon because usually the signal obtained with parallel polarizations is stronger

than the one obtained with orthogonal polarizations because of polarization anisotropy [92].

The amplitudes of the two signals become equal only after some time as a consequence of

rotational diffusion processes, which randomize the preferential orientation of dipole

moments induced by the first pulse(s).

Clearly, the situation in the case of the hydrated electron is entirely different. As we

have shown in Section 6.3.2, the absorption spectrum of the hydrated electron is primarily

homogeneously broadened. The underlying mechanism is that the charge distribution in the

ground state rapidly samples many configurations with different orientations. The time scale

at which the sampling takes place is shorter than the duration of 5-fs excitation pulses. The

hydrated electron experiences a variety of possible orientations of the neighboring water

molecules during the excitation process. Therefore, the orientations of the excited p-state are

completely randomized by the time the excitation is completed. As a result, no anisotropy can

be observed after the excitation pulse pair ends.

We conclude that, in the case of the aqueous electron, the transient anisotropy

disappears in less than 5 fs. The important consequence is that when the polarizations of the

pump pulses and the probe pulse are perpendicular, the coherent coupling between them is

Chapter 6

144

extremely ineffective. Therefore, if the contributions with irregular time ordering (as, for

instance, E1-E3-E2) are omitted, the total number of permutations is decreased by a factor of

two [61]. Hence, the TG signal near t23=0 for perpendicular polarizations is expected to be

approximately one fourth of the signal with parallel polarizations. This is value is very close

to the one observed in the experiment (Fig.6.8a).

-40 -20 0 20 40

Perpendicular

Parallel

In

tens

ity

Delay t12 [fs]

Fig.6.9: Two pulse photon echo signals from hydrated electrons in H2O. Solid dots and open circlesshow the signals obtained, respectively, with parallel and perpendicular polarizations of the laserpulses. In order to ensure identical pulse intensities and focusing conditions, these measurements wereperformed with 15-fs laser pulses. Note that virtually no signal is detected in the case of perpendicularpolarizations.

To verify this finding, we performed a two-photon echo spectroscopy with parallel and

perpendicular polarizations of two pulses. In this experiment, the time ordering is essentially

identical to the irregular time ordering around zero delay in the TG scan. The use of 15-fs

pulses directly from the cavity-dumped laser allowed us to easily control the polarizations

while maintaining constant energy of the pulses. For this purpose, a zero-order λ/2-plate

(Karl Lambrecht) was introduced in one of the spectrometer arms. Note that this is hardly

possible in the case of 5-fs pulses since the λ/2-plate cannot equally rotate the polarization of

spectral components in the 600–1100 nm range. The results of the experiments are presented

in Fig.6.9. The signal in the case of perpendicular polarizations (solid circles) is suppressed

by a factor of ~30 compared to the signal with parallel polarizations (open circles). This

proves our conclusion that the efficiency of the excitation of the hydrated electron with two

pulses having orthogonal polarizations, is extremely low.

Based on the data obtained in polarization-dependent photon echo and TG

spectroscopy, we reach a conclusion that the signal at short delays is substantially weaker

than that at ~40 fs. Therefore, there is no recurrence but rather a delayed response in the TG

signal. The TG data for the orthogonal polarizations clearly indicate that the delayed response


145

cannot be explained by underdamped wave packet dynamics. In that case, the signal would

always be the largest at t23=0 due to inevitable damping of coherent motions.

The mechanism we suggest to explain the delayed response is that the transition dipole

moment of the hydrated electron increases as a function of time after the initial excitation.

Since the amplitude of the TG signal depends on the transition dipole moment (consult

Eqs.(5.9) and (5.17)), the signal increases before falling due to irreversible energy relaxation.

The rise in the transition dipole moment should be directly connected to the strong coupling

between the hydrated electron and water molecules that begin to readjust their positions after

the excitation to minimize the total free energy of the system. The dependence of the

transition dipole moment on nuclear degrees of freedom is known in molecular spectroscopy

as the non-Condon effect [93].

-50 0 50 100 150 200

0.0

0.5

1.0

||⊥

Inte

nsit

y

Delay t23

[fs]

Fig.6.10: Transient grating signals obtained from the hydrated electron in D2O. Solid dots and opencircles show TG signals measured with the parallel and perpendicular polarization, respectively. Solidcurves depict the fits calculated as described in Section 6.3.5.

We next determine the microscopic motions of water molecules underlying the

non-Condon effect. For this, we performed the TG experiment on the electron solvated in

heavy water (Fig.6.10). The TG signals are quite similar to those for normal water (Fig.6.8a)

but the maximum is delayed to longer times (~60 fs). However, if the time scale of the D2O

TG data is compressed by 2 , the TG signal becomes virtually indistinguishable from that

for H2O (Fig.6.11).

The difference in time scale can be explained by the fact that the moments of inertia of

H2O and D2O differ precisely by a factor of 2. This immediately leads us to the conclusion

that the maximum in the TG signals is caused by an overdamped librational motion of the

water molecules. The specific dependence on deuteration rules out translations because the

total mass of a water molecule changes only by a factor of 18/20. Another mechanism for the

2 dependence proposed by Barnett et al. [34] is free rotational diffusion of water molecules

surrounding the electron. However, this explanation is not very likely. Rotations of molecules

in a liquid can hardly be called free, especially in the case of a strongly hydrogen-bonding

Chapter 6

146

liquid like water. Furthermore, there is no signature of free rotations in the Raman spectrum

of water [94].

-50 0 50 100 150 200

0.0

0.5 H2O

D2O t

23/√2

Inte

nsit

y

Delay t23

[fs]

Fig.6.11: Comparison of the transient grating signals obtained from electrons hydrated in H2O (soliddots) and D2O (open circles). All data are given for perpendicular polarizations of excitation pulses.

The delay-axis for D2O has been compressed by a factor of 2 to highlight the isotopic effect.

6.3.4 Early-time dynamics: the microscopic picture

The microscopic picture underlying the early-time dynamics of the hydrated electron that has

emerged from our experiments is schematically presented in Fig.6.12. In this cartoon, only

four of the approximately six water molecules in the first solvation shell around the electron

are depicted for the sake of simplicity. The dark contour in the left panel shows the charge

distribution (i.e., squared modulus of the wavefunction) of the hydrated electron. We

assumed that the latter is confined in a harmonic potential formed by the neighboring water

molecules (right panel). The use of, for example, square-shaped potential does not change the

main conclusions. The relevant mean size of the electron cloud in the ground |g⟩ state was

matched to value reported by the group of Rossky (1.9 Å) [23] while the distances between

the electron and the water molecules were taken from Ref. [20].

Before excitation, the electron finds itself in the s-like ground state in equilibrium with

surrounding water molecules (Fig.6.12a, left panel). Upon excitation, the electron makes a

transition to the excited p-state wavefunction that is elongated in a particular direction

(Fig.6.12b, left panel). Since the excitation pulses are very short, the water molecules have no

time to react and, therefore, they still preserve their compact geometry. However, as time

progresses, the water molecules in the first solvation shell are being pushed away by the

expanded charge distribution of the electron (Fig.6.12c, left panel). We have already

concluded from the TG experiments that the underlying microscopic processes at this stage

are the librations of water molecules schematically shown by curved arrows in Fig.6.12c.

This reorientational motion makes the potential energy well in which the electron is confined,


147

shallower as shown in Fig.6.12c, right panel. Subsequently, the charge distribution of the

electron expands even further (Fig.6.12d), and so on.

Fig.6.12: Artist’s impression of the early-time dynamics of the hydrated electron. Left panel:configuration of the charge distribution of the electron (dark contour plot) and surrounding watermolecules. Right panel: the potential-energy well in which the electron is confined (red curve) and thewavefunctions in the ground <g| and excited <e| states. The wavefunction of the currently occupiedlevel is shaded. The delay time after the initial excitation is shown in the right bottom corner of theright panel. The thickness of the arrow corresponds to the magnitude of the transition dipole moment.

Chapter 6

148

Expansion of the cavity occupied by the electron has a profound effect on the

wavefunction of the unoccupied ground-state as well. As the water molecules in the first

solvation shell are readjusting their orientations, the ground-state wavefunction is elongating

in the same direction as the excited-state wavefunction (Fig.6.12, right panel). Therefore, the

dipole moment of the electronic p–s transition, µeg, becomes larger due to the increased

overlap between the probability densities of the ground and excited states. For instance, in the

case of a harmonic potential the transition dipole moment increases inversely proportional to

the square root of the oscillation frequency 01 ωµ ∝eg . The amplitude of the TG signal

depends on the fourth power of the transition dipole moment that changes in time. Hence, a

very moderate extension of the electron cloud leads to a substantial increase of the TG signal

as was observed experimentally (Fig.6.8 and 6.10).

The position of the maximum in the TG signal gives us an estimate for the time scale of

the expansion of the first solvation shell: ~50 fs. The frequencies that correspond to this time

coincide reasonably well with librational band of water which span the range ~300-900 cm-1,

and which also scales with 2 upon deuteration [94]. The decay following the maximum in

the signal is most probably caused by relaxation down to the ground state at the 125-fs time

scale [32]. The stimulated emission is rapidly diminished due to the decreasing population on

the excited state while at the same time the equilibrating ground state gives rise to induced

absorption.

6.3.5 Theoretical model

In order to put the microscopic picture of the solvation of the hydrated electron developed in

the previous Section on a more quantitative ground, we next introduce a simple model based

on wavepacket dynamics on the ground and the excited states (Fig.6.13). In this picture, we

assume that prior to the photo-excitation the ground state of the hydrated electron is

equilibrated. The excitation pulse pair creates a hole in the ground state distribution

(Fig.6.13a) and a wavepacket in the excited state (Fig.6.13b). This gives rise to a bleach of

the ground–state absorption and to a stimulated emission from the excited state. Assuming

that no population relaxation takes place yet on the duration of the applied laser pulses and

the optical transition is primarily homogeneously broadened, both the bleach and the

stimulated emission contours have approximately the same spectral shape as the steady-state

absorption band.

Immediately after the excitation, the contours of the bleach and the stimulated emission

are exactly identical. Subsequently, the downhill movement of the wavepacket on the excited

state potential takes place, which reflects the gradual expansion of the water cavity. Because

the energy gap between the two potentials decreases as the wavepacket moves on the excited

state surface, the stimulated emission band becomes increasingly red-shifted with respect to

the ground state bleach whose spectral position remains fixed (Fig.6.13a,b).


149

Fig.6.13: Schematic representation of the wavepacket dynamics on the ground– and excited–statepotentials. (a) Ground state bleaching. (b) Excited-state wavepacket movement, corresponding to thered-shifting in time of the stimulated emission spectrum. (c) Wavepacket propagation on the hotground state, causing the blue-shifting in time of the induced absorption spectrum. Solid verticalarrows indicate the excitation process.

The expression for the observed TG signal can be easily calculated in the slow

diffusion limit [77,95]. Based on the finding of predominantly homogeneous dephasing inSection 4.1, we assume the nonlinear response function of the ground state, ),,( 321 tttRgr , in

the form given by Eq.(5.15). The nonlinear response function associated with the populated

excited state is then given by [95]:

[ ] )(1exp),,(),,( 233321321 tMtitttRtttR grex −∆= ω , (6.4)

where ω∆ is the Stokes shift, i.e. the spectral shift of absorption (fluorescence) due to energyreorganization, and )(tM is the electronic bandgap correlation function [77]. Here we

assumed that )(tM changes slowly on the time of electronic dephasing characterized by 2T .

Adapting the model of an overdamped vibration [77], )(tM then can be expressed as:

2,exp)( Tttt

tM exex

>>

−= (6.5)

To obtain the spectral shape of the ground- and excited-state contributions we nowFourier-transform ),,( 321 tttRgr and ),,( 321 tttRex . The resulting third-order susceptibility for

the ground-state contribution to the nonlinear signal is given by Eq.(5.17) while the ><3~χ

contribution of the populated excited state is expressed by:

Chapter 6

150

( )

.

exp1

1

)''(

1

)'(

1

)'''(

1,''',',~

2312

12

12

11

4

4

233

Ω−

−−∆−+

×

−−+

−−×

−−−=Ω+−−−

−

−−

−><

exeg

egeg

egegegex

t

tiT

iTiT

iT

Nit

ωω

ωωωω

ωω

µωωωωωχ

h

(6.6)

Analogously to the wavepacket in the excited state, one can include into consideration the

wavepacket sliding down the (hot) ground state.

As described in the previous Section, the expansion of the solvent cavity forced by the

photo-excitation leads to the increase of the transient dipole moment magnitude as a function

of time, which gives rise to the non-Condon effect. Therefore, it is assumed that the change

of the transition dipole moment strength occurs on the same time scale as the motion of the

excited state wavepacket. This change does not influence the magnitude of the absorption

bleach band since the latter is caused by a static hole in the ground state. On the other hand,

the magnitude of the spectral contribution caused by the traveling wavepacket is dependent

on the changing strength of the transition dipole moment. Therefore, egµ in Eq.(6.6) should be

considered as a function of time, i.e.:

)()( 0 tFt NCeg µµ = , (6.7)

where 0µ is the initial dipole moment, and the non-Condon parameter, )(tFNC , is a slowly

changing function compared with the pulse duration.

After about 200 fs, the excited state wavepacket reaches the region where the potentials

of the ground and the excited states cross [32]. In this region, the wavepacket "leaks" from

one potential surface to another. However, the crossing occurs to a modified (hot) ground

state that has to “cool” before reaching the equilibrated steady state. Therefore, the hot

ground state wavepacket gives rise to an induced absorption band that shifts from the red to

the blue and eventually cancels the ground state bleach (Fig.6.13c). In the model, this

crossing process is taken into account by assuming that the excited state stimulated emission

vanishes with the same rate as the hot ground state absorption builds up.

We next calculate the TG signal according to the formalism developed in Section 2.1. If

the population relaxation time T1 is long compared to the pulse duration, in Eq.(5.17) and

Eq.(6.6) one can assume )'''(~ 3 ωωδχ −∝>< . Then, integrating Eq.(5.9) over frequency Ω and

making use of Eqs.(5.7), (5.17) and (6.6), and accounting for the hot-ground state

contribution, we arrive at:


151

( ) ( ) ( ) ( ) ( )∫ ΩΩ−Ω+ΩΩ∝= − dIttS grhotexgrTG

2802312 ,0 σσσµ (6.8)

Here, ( )ΩI is the spectrum of excitation pulses and the contributions from the ground,

excited, and hot ground state potentials are respectively written as

( ) ( )Ω=Ω Agr σσ (6.9)

( ) ( ) ( ) ( )[ ]( )2323234 exp1exp tttF exAcNCex γωσγσ −−∆+Ω−=Ω (6.10)

( ) ( ) ( )[ ] ( )( )2323234 expexp1 tttF grAcNCgrhot γωσγσ −∆+Ω−−=Ω− (6.11)

where ( )ΩAσ is the steady-state absorption spectrum, ∆ω is the Stokes shift, γex and γgr are

the rates at which the wavepacket moves on the excited and the ground states, respectively,

and γc is the crossing rate from the excited state to the ground state.

The expressions given by Eqs.(6.9–11) represent Lorentzian line shapes described by

Eq.(6.3). These constituent terms of the TG signal are schematically depicted in Fig.6.13a-c

in their respective order. The first term is static and stands for the hole in the ground state

(Fig.6.13a). The second and the third ones account for the moving wavepacket in the excited

(Fig.6.13b) and hot ground (Fig.6.13c) states, respectively. The use of RWA [78] [77] in

derivation of Eqs.(5.17) and (6.6) leads to symmetric Lorentzian contours, rather than the

asymmetric extended line-shape given by Eq.(6.2) employed to explain the absorption

spectrum in Section 6.3.2 The spectrally-integrated TG signal, however, is insensitive to such

a minute difference in the spectral contours, which justifies the use of centro-symmetric line-

shape in our calculations.

The physical meaning of the movement of the Lorentzian wavepacket can be explained

as follows: The difference in curvature of the ground and excited state potentials implies a

strong quadratic electron-vibration coupling that is responsible for the extraordinary amount

of homogeneous spectral broadening. On the other hand, linear electron-vibration coupling

determines, at large, the presence of the wavepacket dynamics. This is due to predominantly

linear change of the energy gap with nuclear coordinate between the ground and excited

states as a result of the displacement of their potentials with respect to each other.

The time-dependent non-Condon parameter is assumed to be equal to

( ) ( )[ ]2323 exp11 tftF exNCNC γ−−+= (6.12)

where fNC is a measure of the strength of the non-Condon effect or, in other words, the ratio

of the magnitudes of the initial and maximal dipole moments. The influence of the non-

Condon effect on the TG signal is illustrated in Fig.6.14. With a constant transition dipole

moment (fNC = 0), the TG signal peaks close to zero (Fig.6.14a). If the non-Condon strength

is chosen to be 0.5, the dipole moment grows as a function of time, and reach a maximum

Chapter 6

152

after ~100 fs (Fig.6.14b). Subsequently, it balances the drop in the TG signal due to

population relaxation, which results in a maximum that is significantly shifted away from

zero (Fig.6.14c).

TG

Sig

nal

(c)

(b)

(a)

1.0

1.2

1.4

FN

C

0 100 200

TG

Sig

nal

Delay t23 [fs]

Fig.6.14: Simulated transient grating signals for perpendicular polarization of excitation pulses. (a)Signal without non-Condon effect (fNC = 0). (b) Transition dipole moment as a function of time with anon-Condon effect present. (c) The transient grating signal from (a) when the non-Condon effect istaken into account.

In order to incorporate the fact that we are dealing with the pulses of finite duration, we

convoluted the polarization of Eq.(6.5) with the instrument response function. The following

parameters are taken for water: 1/γex = 33 fs; 1/γgr = 300 fs; 1/γc = 125 fs; ∆ω = 0.50 PHz; fNC

= 0.5. For heavy water, all rates are decreased by a factor 2 in accord to the established fact

that underlying microscopic dynamics are determined by librations. The calculated TG

signals for H2O and D2O are depicted in Figs.6.8a and 6.10, respectively. Evidently, by

inclusion of the non-Condon effect we are well able to reproduce the delayed maximum in

the TG signals, even though the rate of increase of the transition dipole moment is not an

independent variable. As can be seen from Eqs.(6.10) and (6.12), we ascribed a single

common rate, γex, to the evolution of both the transition dipole moment strength and the


153

dynamic Stokes shift. The fact that we can fit the experimentally observed dependencies

reflects the common microscopic origin of the dynamic Stokes shift and the non-Condon

effect: the unidirectional expansion of the solvent cavity caused by librational reorientation of

the water molecules.

We predict that the time constant of populating the hot ground state is faster than 100 fs

for H2O. Careful inspection of the measured TG traces (Figs.6.8a) shows that at this point

there is a slight bend in the curvature. Apparently, this is a signature of the interplay between

stimulated emission from the excited state that is being diminished and the induced hot

ground state absorption that is being increased simultaneously. After this, the induced ground

state absorption shift to the blue on a time scale of 300 fs. This shows up as the slower decay

in the transient grating. These points will be further addressed in Chapter 7.

6.4 Conclusions

Photon echo and transient grating spectroscopy on the hydrated electron performed with the

best time resolution available to date has provided a powerful insight in the microscopic

processes that underlie solvation dynamics.

Both two-pulse photon echo and TG experiments, involving two and three femtosecond

pulses, respectively, have been performed on equilibrated hydrated electrons. By comparing

two-pulse echo signals from hydrated electrons and from water alone, we have derived the

pure dephasing time of ~1.6 fs. We have shown that the absorption band of equilibrated

solvated electrons in water is predominantly homogeneously broadened and succeeded in

modeling the whole absorption spectrum of the hydrated electrons by a single homogeneous

line shape. The typically employed symmetric Lorentzian line shape has been abandoned in

favor of a more general expression for a homogeneously broadened line. Importantly, in the

line shape used in our fit, the long-puzzling issue of the asymmetry of the absorption

spectrum found a natural explanation. Further proof for the homogeneous nature of the

absorption spectrum comes from the absence of quantum beats on the 10-fs time scale in the

TG signal, pointing to the fact that the absorption spectrum does not consist of three separate

absorption bands.

The TG experiments on the hydrated electron with perpendicular polarziations of

excitation pulses have revealed a delayed response never observed before due to limited time

resolution. We have suggested that the delayed response is due to a non-Condon effect,

caused by librational motions of the water molecules surrounding the electron. The large

magnitude of the non-Condon effect is a direct consequence of the fact that the Hamiltonian

of the hydrated electron is fully determined by the configuration of the neighboring water

molecules. This renders the hydrated electron a convenient probe for the local structure of

water, the most-important-for-life but still mysterious liquid [21]. We have also shown that a

numerical model including the non-Condon effect which is due to initial librational motion of

water molecules, and subsequent population relaxation and ground state cooling can

reproduce the essential features of the observed signals, putting our ideas on solid grounds.

Chapter 6

154

Electron in polar liquids continues to be a vast experimental and theoretical field in

which many intriguing questions remain to be answered. There are still many conflicting and

not well understood issues concerning electron equilibration [47,96], energy relaxation of the

photo-excited electrons in fluids [32,52], the nature and the number of the bound localized

and unbound states of the electron trapped in the solvent cavity [22,50], etc. In the current

paper, we have addressed only the earliest part of the energy dissipation that directly follows

photo-excitations. The subsequent relaxation processes, albeit taking place on a slower time

scale, are no less interesting or less controversial. In particular, the questions about the

lifetime of the bound excited state and the involvement of the quasi-continuum states,

predicted by quantum molecular dynamics simulations [23], have to be addressed.


155

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Chapter 7

Ground State Recovery of the Photo-Excited

Hydrated Electron

Abstract

In this Chapter, we present a detailed frequency-resolved pump–probe study of the hydrated

electron dynamics, which was performed with 5-fs pulses. We analyze the difference in thebehavior of the pump–probe signal obtained from a two- and a three-electronic-level system.

This provides the guidelines to resolve a long-standing dilemma of the two contradicting

models that are used to describe the dynamics of the hydrated electron. The first modelpredicts a rapid, whereas the second one implies a slow electronic relaxation of the excited

state. The dynamics of the blue shift in the measured transient spectra is shown to correspond

perfectly to the behavior of the transients at various detection wavelengths. This providesstrong support for the short-lived excited state model. Next, the pump–probe spectra at all

delays are successfully fitted on the basis of this model, yielding the excited state lifetime of

~50 fs. Also, the equilibration of the ground state is shown to proceed with a predominanttime constant of ~1 ps. In accordance with our previous findings, the shortest decay time is

dominated by the librations of water molecules. Other decay components that exhibit no

isotopic effect are assigned to the translational motions of water molecules. To clarify thestructure of the potential surfaces at the long pump–probe decays, the time-domain

dependence of the energy gap is converted into the function of a generalized solvent

coordinate. Finally, a multidimensional model of the hydrated electron solvation is proposed.

Ground State Recovery of the Photo-Excited Hydrated Electron

191

equilibrium state at times longer than 300 fs can be described almost purely by a propagation

along the translational coordinate. The model consistently accounts for all time scales andtransition frequency changes observed in the pump–probe (Chapter 7) and transient-grating

(Chapter 6) experiments.

Appendix I: Modulation of pump–probe spectra

The pump–probe spectra in Fig.7.7 at early delays (up to 120 fs) exhibit a certain amount of

modulation, which subsequently disappears. Here, to explain this phenomenon, we perform acalculation of the transient spectra according to the complete expressions given by Eqs.(5.6)

and (5.21, 5.22). To avoid the problem with the evolution of the excited state contribution

after excitation, we now consider the transient spectrum at zero pump–probe delay (Fig.7.16,solid points). In this particular case, the contribution of the ground state bleach and stimulated

emission from the excited state are exactly identical. Thereupon, we utilize the expression of

the third-order susceptibility, given by Eq.(5.17) to describe both the “hole” in of the groundstate and the “particle” in the excited state. We further employ the value of T2=1.6 fs, derived

in Section 6.3.2, and T1=50 fs, which follows from the fit of transient spectra in Section 7.3.2.

The difference absorption contour computed, using the actual spectrum and the phase of the5-fs pulses, is depicted in Fig.7.16 by solid curve.

A good overlap with the measured data (solid points) is achieved upon shifting the

computed spectrum by the amount shown by the dashed line. The reason for this discrepancylies, most probably, in the excited state absorption to the continuum band (or another higher-

lying excited state). A similar deficit of the positive absorption component that rapidly shifts

to the blue causes apparent deviation of the measured transient spectra from their fits,reformed in the two-electronic-state model (Fig.7.7, top panel). Despite the need to account

for an additional transient absorption to a higher excited state, our calculations, which

employed the 3χ of a homogeneously broadened transition, successfully reproduce the fine

features of the measured signal.

A modulation of the pump–probe spectrum in the region of pump and probe spectral

overlap potentially may, in general, have an entirely different origin, e.g. the one associatedwith hole-burning of inhomogeneously broadened absorption spectra. Our present simulation

of the pump–probe signal in a homogeneously broadened system, however, dismisses this

idea. In fact, as has been discussed in Section 5.5, the spectral modulation pattern in ourexperiments is caused by a very short population lifetime, i.e. T1.


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7.1 Introduction

In Chapter 6 we have studied the earliest steps of the energy relaxation that take place within

the first 100 fs following the optical excitation of the s–p transition in the hydrated electron.

In this Chapter, we broaden the studied time window to include the picosecond delay range.Here we attempt to account for the whole complex scheme of photo-excitation and relaxation,

which includes different stages. Next to examining the initial rapid relaxation components,

we now turn our attention to the subsequent, slower processes. The exploration of thesecontributing pathways of the photo-excitation energy dissipation is important in two ways.

First, it should provide the answers about the mechanisms of the solvent response dominating

each particular stage of the relaxation. Second, by reconstructing all subsequent steps, we willbe able to verify our ideas about the initial energy relaxation process that occurs within just a

few tens of femtoseconds.

Based on the observation of the isotopic effect in the transient grating signal, weconcluded that inertial motion of water molecules is responsible for the initial step of energy

dissipation. According to the dynamical model proposed in Section 6.3.4, librations of water

molecules in the first solvation shell dominate the overall solvent response withinapproximately the first 50-100 fs, following the excitation. This is a reasonable conclusion,

considering the instantaneous character of the expansion that the spatial extent of the electron

charge distribution undergoes upon photo-excitation. Obviously, in that case, librations of theO-H (O-D) bonds of H2O (or D2O) molecules can proceed much faster than the translation

motion of the entire molecules away from the electron to accommodate the change in the size

of the latter. It is evident, therefore, that another, slower time-scale of energy relaxation mustset in, reflecting the change from initially predominantly librational to later predominantly

translational movement. Because of the almost unidirectional expansion of the side lobes of

the electron excited-state p-wavefunction [1,2], uneven forces propel the moleculessurrounding the electron from different sides. It is justified, therefore, to expect different time

constants to be responsible for the positional adjustment of the “on-axis” and “off-axis” water

molecules, which has been recently predicted in computer simulations [3]. Furthermore, theslowest time scale of energy relaxation must be identified with the typical diffusional motion

of the solvent molecules in and out of the shell surrounding the electron. The indication of the

radical change in the relaxation rate can be also found in the computer simulations [2], wherea transition from a rapid 25-fs component to an ~300-fs one has been predicted.

The extraction of the excess energy from the photo-excited electron to the surrounding

liquid water results in an electron transition back to the ground state that is now altered,compared with the pre-excitation situation [2,4]. Numerical calculations [2] predict that the

return to the modified s-state, which is more compact than the previously occupied p-state,

creates a void in the solvent, which corresponds to the sudden “implosion” of the electron.Unlike in the case of the instantaneous expansion of the photo-excited electron, which

forcibly drives the water molecules to rapidly reorient and give room, the closing of the

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solvent void in the wake of the “collapsed” electron is supposed to proceed at a much slower,

diffusion-governed pace.Summarizing the ideas presented above, we expect to see several time-scales

determining the later stages of energy relaxation, which take over the initial, libration-

dominated part. Accordingly, the goal that we pursue in this Chapter is to create a self-consistent model explaining all stages of the photo-excitation relaxation cycle. To this end we

have to find the energies released at each particular step of the energy relaxation, as well as to

determine the time scales and molecular dynamics behind these events. The capital questionsthat must be answered concern 1) the thermalisation time of the excited p-state, 2) its

lifetime, and 3) the time of the modified ground state evolution from the configuration

directly after the p–s transition to that of the equilibrated ground state.So far, the experimental evidence provided by the time-resolved transient absorption

spectroscopy on the equilibrated hydrated electron [5-11] remains, at large, inconclusive. For

reasons, which will be addressed later in this Chapter, the pump–probe data can beinterpreted using two contradicting models. According to one possible scenario [5,7,10,11]

the lifetime of the excited state is very short, in the order of 200 fs, and the “cooling off”

process of the modified (hot) ground state back to the equilibrium proceeds with a ps timeconstant. In the opposite interpretation [2,6,9,12], after the thermalisation that takes place on

a 200-300 fs time scale, the excited state (nonadiabatically) decays with a predominant time-

constant in the range from 700 fs to 1.2 ps. As it followed from molecular dynamicsimulations for this model, upon the p–s transition, the initial equilibrated ground state is

rapidly recovered in <100 fs [2,9]. Because the latter process is superimposed over a slower,

picosecond decay component, it cannot be experimentally resolved.The schematic qualitative representation of the above-described controversy is given in

Fig.7.1. Based on the librational nature of the initial solvent response, established in

Chapter 6, the first relaxation step is identical to the one in Fig.6.12. Despite the initialsimilarity of the two scenarios, they correspond to substantially different dynamical behavior

at the later energy relaxation stages. In the short-lifetime model, the energy involved in the

water cavity readjustment, accompanying the sudden increase in the size of the electroncharge distribution, is sufficient to cover the modified s–p energy gap. The remaining excess

energy that corresponds to the picosecond cooling of the ground state is then dissipated once

the water molecules close the void in the solvent, which has formed after the “burst” of theexcited state. On the contrary, the long–lifetime model implies that the energy involved in the

rearrangement of the first solvation shell is considerably smaller than the frequency of the s–p

gap. Therefore, the p-state remains conserved for a substantially longer (picosecond) time. Inthis model, as opposed to the other one, the energy transfer from the electron to the solvent

proceeds in a reversed fashion. This means that the main part of the excess energy,

corresponding to the transition from the p- to the modified s-state is transported on the slow(picosecond) time-scale, while the remainder is quickly dissipated within 100 fs or so. A

possible explanation of why this last step is so fast was put forward by Schwartz and Rossky


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[4] who suggested that the free rotation of the water molecules, which becomes possible due

to the formation of the void in the solvent, constitutes a rapid and efficient solvation channel.

Fig.7.1: Schematics of short-lived (left) and long-lived (right) photo-excited p-state models. See textfor details.

The resolution of the conflict outlined above, i.e. a short-lived p-state vs. a long-lived p-

state model, forms the core of this Chapter.It is essential to realize the great importance of determining the lifetime of the excited

state. Indeed, a short-lived p-state means that a very substantial part of the excitation photon

energy is rapidly transferred to the surrounding water molecules and, therefore, this great

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amount of energy is further absorbed by different-level solvation shells. To provide a channel

for such a fast and efficient initial dissipation of the excess energy, there must be a verystrong coupling of the hydrated electron to water molecules. In the opposite, long-lived p-

state scenario, a longer survival of the p-state signifies a more “autonomous” status (i.e. a

higher level of adiabaticity) of the electron whose ties with the surrounding water moleculesappear to be weaker. Therefore, the probability of the excited state depopulation is

substantially smaller. The eventual p–s transition in this case is triggered by much slower

events such as, for example, the rearrangement of the first solvation shell as a consequence ofthe diffusional relocation of the water molecules from the outer shells.

The experimental differentiation between these two controversial models should have a

profound impact on choosing the right approach to the molecular dynamics simulations of thehydrated electron in particular, and of the aqueous environment in general. The main

“sticking point” of the computer studies performed to this date [1,2,4,12-24] is the use of a

quantum mechanical description for the electron and a classical one for the water molecules.While an oscillator in a classical flexible molecule can accept any quantity of energy, in a

quantum-mechanical one it is capable of receiving only a limited amount [15,23,25]. The

quantized disposal of a large excess energy from the relaxing electron to real molecules must,therefore, be slower than in the case classical flexible molecules. In the (semi-)classical

approach, the use of which is justified because of the high computational efficiency, the

results of any calculation are predetermined by the built-in assumption of how much energy awater molecule is allowed to accept. Thus, a widespread range of predictions for the lifetime

of the excited state has emerged from the computer simulations of different authors.

For instance, the initial use of “rigid” water molecules in the calculation of Rossky andcoworkers resulted in the lifetime of the p-state in the order of 1 ps [22]. The implementation

of the “flexible” water potential [26] and the electron–water pseudopotential [27] in the work

of Neria et al. produced an ~120-fs and ~220-fs radiationless transition times for H2O andD2O, respectively, in a mixed classical–quantum-dynamical treatment [28]. These figures

subsequently rose to ~220 fs for H2O and ~800 fs for D2O in a semi-classical simulation [24].

The electronic relaxation time of 230 – 250 fs has also been found in other simulations[25,29]. A different, also “flexible” potential [30] was assumed in the work of the Rossky

group. Initially, the lifetime of the excited state of ~160 fs was inferred [23]. The following

attempts of Rossky et al. to restrict the over-flexibility of the water molecules yielded asteady increase of the lifetime. This resulted, first, in the lifetime figure of ~250 fs [15],

which subsequently was nearly doubled, to produce an average nonadiabatic p–s passage rate

of ~450 fs [2]. The latter value has been refined later [17], claiming a 240-fs and a 300-fstime of the excited state solvation for H2O and D2O, respectively, and a 1.1-ps lifetime of the

equilibrated p-state.

It should be expected, therefore, that with such a great reliance on a priori assumptions,the above cited results of numerical simulations present, at best, the attempts to reconcile the

times seen in the femtosecond experiments [5,7,8,11,31-33] with the computed results. This


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but once more signifies the importance of resolving the issue of the p–s relaxation

experimentally.In this Chapter, we explore the transient absorption dynamics of the hydrated electron,

photo-excited from its equilibrated ground state. This type of experiment, which involves

hydrated electron preparation by a UV pulse preceding the femtosecond pump–probe pulsesequence, was first demonstrated by the group of Barbara [5,7,8,11]. Their measurements had

a relatively poor, 300-fs time resolution. The latter was substantially improved in subsequent

studies of the hydrated electron by the same group [6,9], in which much shorter, ~25-fspulses were used. Similar pump–probe experiments were also reported by Laubereau and

coworkers [10] who, however, employed considerably longer, 170-fs pulses. Also, in our

previous pump–probe measurements of the hydrated electrons [34], we used 15-fs pulses,concentrating primarily on the measurements of the initial, librational dynamics.

Compared with the experimental work of other researchers, this study presents several

important advancements. First, the use of 5-fs excitation and probe pulses provides anunprecedented time resolution. Second, the adequately short duration of the excitation pulse

prevents distortion of the pump–probe signal as a consequence of the rapid energy loss in the

excited state. Third, the spectral dynamics in the range of 600 - 1050 nm is covered at onceby the spectrum of the 5-fs pulses. This removes the need to tune the wavelength of the probe

pulses and to synchronize them each time with the pump pulse. Therefore, it became possible

to record accurate transient spectra with a high density of data points. Fourth, a considerablyenhanced dynamic range of our measurements, up to 4 decades, allowed a highly precise

study of the kinetic as well as the spectral behavior in the pump–probe delay interval up to

several picoseconds. Finally, we successfully employed a modeling procedure that enabled usto unravel the contributions of the excited and (hot) ground states in the overall shape of the

transient spectra. Consequently, based on our fit of the experimental results we prove the

validity of the short-lived p-state model and conclude that the electronic depopulation of theexcited state proceeds on a 50-fs time scale.

This Chapter is organized as follows. In Section 7.2, we discuss the difference in the

pump–probe signals obtained if the system is modeled 1) in a three-electronic-level modelwith a long-lived first excited state, and 2) in a two-electronic-level model with a rapid

electronic relaxation and a subsequent ground state solvation. We produce practical

guidelines for the experimental discrimination between the two models. Section 7.3 providesthe details on the experimental procedure. In Section 7.4, the results of the 5-fs pump–probe

measurements are presented and fitted according to the short-lived p-state model. Finally, in

Section 7.5 we summarize the findings of our investigation.

7.2 Short-lived vs. long-lived p-state: Manifestation in pump–probe

In the previous Section, we outlined the basic controversy related to the rate of electronicrelaxation from the excited p-like state back to the s-like state. Another aspect of this problem

lies in the necessity to explain how many electronic states contribute to the transient

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absorption picture. Indeed, it is well known that, before its localization in a polar solvent, a

quasi-free electron is initially released into the conduction band, or continuum. Subsequently,in the process of localization, the electron occupies bound states of the liquid cavity, until it

finally solvates to the bottom of the lowest electronic state. Obviously, the transient dynamics

seen in a optical pump–probe experiment will depend on the number of bound electronicstates involved, on their position in frequency with respect to each other and to the

continuum, and on the dipole moment strength of each transition. Below we address

separately the views on the structure of the electronic energy levels, which emerged fromnumerical simulations, experiments probing electron trapping and solvation in water, and

from interpretation of the earlier pump–probe measurements on the equilibrated hydrated

electron.Molecular dynamics simulations [1] predict the existence of several bound states with

the strongly allowed transitions to the lowest three of them, i.e. the p-states (see Fig.1.2),

from the ground state. Individual bands corresponding to weaker-bound electronic statescomprise the blue tail of the absorption spectrum of the hydrated electron. With the

occasional exception of the fourth excited state, all higher excited states (nine in total) [1] lie

in the positive energy (continuum) region, i.e. they are delocalized, or unbound. In the samesimulation [1], the energy gap, separating the p-states from the edge of the continuum, was

estimated to be less than 1 eV.

Several important remarks can be made about such level structure. First, as it has beenargued in Chapter 6, the difference of energy levels associated with the three p-states in the

discussed simulation [1] was created artificially in these simulations due to the applied

energy sorting. Without this sorting, the variation of the frequency gap between the s- andeach of the p-states covers the whole absorption band. Therefore, because the different p-

levels overlap in energy, it would be impossible to discriminate among them in a pump–

probe experiment. Second, the general overestimation of the transition energies by ~1 eV inthis calculation [1] misplaced the position of the excited states to a higher energy value.

Consequently, practically all excited states above the p-states reach into the continuum band.

However, if the correct size of the s–p energy gap is taken into account, it leaves room forspeculation that some higher than p excited states may still be bound. Hence, in a pump–

probe spectrum, this would manifest itself as an excited state absorption contour centered at

the corresponding transition frequency. In short, from the cited molecular dynamicssimulations it is impossible to conclude to which extent, if at all, the continuum and high

excited-state absorption will contribute to the pump–probe signals. Consequently,

experimental answers should be sought.The first indirect probing of the electronic level evolution of the hydrated electron was

realized in the experiments on the trapping of a photo-ionized electron and its subsequent

equilibration (solvation) to the bottom of the potential well formed by the solvent cavity. Inthe measurements of this kind [31,32,35-40] a UV electron photo-detachment- and an IR

probe- pulses are applied. These experiments monitor the onset and evolution of the


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absorption band of the electrons as it changes into the steady-state absorption spectrum, such

as in Fig.6.7. Some of the transient spectra recorded in these studies revealed the presence ofan isosbestic point [31,32,35,36]. This was interpreted as a presence of a distinct step-wise

electronic transition between the two fixed states, the so-called “weakly-bound” and the

“strongly-bound” states, which the electron occupies during different stages of the trappingand solvation process. Recent, most elaborate and exhaustive studies (e.g. Ref. [41]) of the

equilibration of the photo-detached electrons in heavy water, however, have convincingly

demonstrated a continuous spectral blue shift that accompanies the evolution of the electronabsorption band associated with the electron in the lower, “strongly-bound” state. The

characteristic time of ~0.52 ps attributed to this shift must, therefore, describe the solvation

rate of the final (ground) state of the electron hydration. In brief, these experiments generallypoint at the existence of two distinct bound or semi-bound electronic levels, at least one of

which (the bottom one) undergoes a complex evolution.

The just mentioned UV-pump–NIR-probe studies of electron photo-detachment providea valuable insight into the electronic level structure and the time scales of electron solvation.

An advantage of this method lies in the relative experimental simplicity (only two laser

pulses are involved) and in the fact that the equilibrated ground state of the hydrated electronis not populated in the beginning. As a result, there are no contributions of the ground state

bleach in the measured evolution. In contrast to the UV-pump–NIR-probe method, the

presence of the latter contributions in the pump–probe signal from equilibrated hydratedelectrons (vide infra) significantly complicates the data interpretation.

Despite the mentioned advantages, the photo-ionization–probe experiments have severe

limitations. One of these, namely geminate recombination [39,42-45] makes the studies in thepicosecond delay-time range particularly difficult. In absence of electron recombination,

kinetics measured in this experiment would reach a plateau that could be used as a

background line. Because of the geminate recombination, which is the fastest processreducing the numbers of surviving hydrated electrons, the background of such kinetics shifts

in time. This strongly affects the precision with which the hydrated electron solvation can be

measured.Most crucially, however, such experiments have an intrinsically low time resolution,

because of the large phase mismatch between the UV and NIR pulses in the water sample.

For instance, despite the use of 90-fs pulses at 267 nm, the obtained time resolution of theexperiments described in Ref. [40] was about 150 fs due to the finite thickness of the sample.

This is the best time resolution of this type of experiments to the date. However, even this

value is not entirely sufficient to study the initial event of electron localization. The times oftrapping in the “weakly-bound” state reported in the literature on the issue are in the range of

100 to 300 fs [31,32,35,46], which is in all cases very close to the time resolution used in the

experiments. In fact, on the basis of one UV-pump–NIR-probe study, it has been suggestedrecently [40] that the electronic level structure of the hydrated electron may consist of a

single weakly-bound state. Obviously, the insufficient time resolution, limited by the duration

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of the UV pulse and the phase mismatch with the probe pulse, can affect the interpretation of

the measured results. It is important to note that possibly some intermediate short-livedelectronic states may be overlooked in this type of experiments. The problem of the low time

resolution can be more easily solved in an all-NIR-pulse pump–probe on equilibrated

hydrated electrons. Therefore, the latter variety of pump–probe, which is also employed inthe present work, is a powerful method to disclose both the structure of the electronic levels

as well as the rates of passage to the bottom of the ground state.

Unlike the UV-pump–NIR-probe method, which investigates the formation of thehydrated electron and its absorption band, the pump–probe measurements on the already

equilibrated species [5-7,9-11,34,47] study the changes in the absorption spectrum upon

photo-excitation of the s–p transition. Technically, this experiment involves a sequence ofthree laser pulses. First, a UV preparation pulse produces hydrated electrons. Subsequently,

upon equilibration of the electrons, the NIR pump and probe pulses are applied. Therefore,

the time resolution is not affected by the phase mismatch between the UV and the NIRpulses, nor does it depend on the duration of the preparation pulse. The fundamental features

of the transient absorption dynamics of the equilibrated hydrated electron became clear after

the pioneering experiments of the Barbara group [5,7,8,11], despite a relatively poor, 300-fstime resolution. The kinetic data showed bleach recovery for the wavelength region below the

peak of the steady state absorption (720 nm) and appearance of a red-shifted positive

transient absorption signal superceding the initial bleach on a sub-ps time scale. The decay ofthe pump–probe kinetics was found to be essentially bimodal. The average time constant at

different probe wavelength lay within the 0.3 – 0.8 ps interval. The variation in the time

scales of the pump–probe dynamics and the absence of an isosbestic point evidenced stronglyagainst a simple two-state model, where the energy levels of each state remain fixed.

Despite the clear advantages of this experimental route, the data interpretation,

however, is not straightforward. While in the UV-pump and NIR-probe spectroscopy, for thedelays outside the pulse overlap, the measured data consists of a pure absorption contribution,

in the equilibrated electron pump–probe one should discriminate between different

components of the transient spectra. As has been mentioned above, transient absorptionbleach and stimulated emission dynamics add to, and can be superimposed on transient

absorption from previously unpopulated states. In the case of the hydrated electron, the

excitation of the s–p transition creates a “hole” in the population of the s-state. The “hole”gives rise to the bleach component in the transient spectrum, which initially has the same

spectral position as the stimulated emission from the p-state that is now occupied. This

complicates the assignment of the contributions from different electronic states to the overallobserved pump–probe spectrum.

The latter difficulty is illustrated in Fig.7.2. The solid contour in Fig.7.2 schematically

shows a typical pump–probe spectrum of equilibrated hydrated electrons, which is wellknown from the experiments [6,7,10,11]. This spectrum can be decomposed into the negative

contributions of the ground-state bleach and stimulated excited emission and a positive


167

component due to excited state absorption. In principle, the resulting “butterfly” shape, can

have any wanted ratio of the negative vs. positive peak amplitude and can cross zero at anarbitrary wavelength, depending on the strengths of the transition dipole moment, the energy

gap between the two excited states, and their mutual curvature.

-∆O.D.

+∆O.D.

0

ωxing

Pump-probe signal

ωeg

Ground state bleach

Transient absorption

Frequency

Fig.7.2: Schematic shape of pump–probe spectrum of hydrated electron in presence of the occupied p-or, alternatively, hot s-state.

The complication outlined above brings us to the fundamental issue of this Section, –

finding the criteria that will make it possible to differentiate between the short-lived p-statemodel vs. the long-lived one in pump–probe on the equilibrated hydrated electrons. First, we

notice that the decay of the induced transparency might be, in general, explained by both the

recovery of the hole in the ground state and by the depopulation of the excited state. Second,the positive transient absorption changes correspond to a transition from a now occupied but

previously unpopulated state to another electronic state. As has been already mentioned, these

positive changes above the central wavelength of the steady-state absorption have beenobserved in all pump–probe experiments on the equilibrated hydrated electrons [5-8,10,11].

Two explanations of these pump–probe features correspond to different involvement of

various electronic states, which is outlined in two models depicted in Fig.7.3. The first one(Fig.7.3.a) complies with the short-lived p-state scenario, while the second one (Fig.7.3.b)

becomes necessary to explain a picosecond lifetime of the excited state.

In the short-lived p-state scenario, the transient component of the positive sign arisesdue to induced absorption from the hot ground state (Fig.7.3a). Consequently, the essential

evolution of the pump–probe dynamics at long pump–probe delay times takes place only on

two potential energy surfaces, that of the s- and of the p-state. Contrary to it, in the long-livedp-state model (Fig.7.3b) another, higher lying excited state must be involved to explain the

presence of the positive transient absorption.

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Fig.7.3: Two models of photo-excitation relaxation in hydrated electron. (a) short-lived p-state (Refs.[5,7,10]). (b) long-lived p-state (Refs. [2,6,9,12]).

Both these models have been applied to interpret experimental pump–probe results. The

initial explanation of the kinetic behavior observed in the early studies by the group of

Barbara [5,7,8] was given in accordance with the above-presented short-lived excited statemodel. However, in view of a short, <100 fs ground state equilibration time and a long ~1 ps

p-state lifetime predicted in the Rossky et al. simulations, the interpretation of the

experimental data was modified [17] in favor of the second, long-lived p-state model. Thefollowing pump–probe measurements conducted in the Barbara group with a much-improved

resolution were also interpreted along these lines [6,9]. Conversely, the model of a rapid

nonadiabatic electronic relaxation and a subsequent long vibrational cooling of the groundstate was upheld for solvated electrons in alcohols [17,48]. The scheme that involves a rapid

p–s relaxation and a ps equilibration of the s-state in the hydrated electron has re-appeared

recently in the study by the group of Laubereau [10]. Despite the use of inadequately long,~170-fs pulses, these researches inferred from the fit routine an ~190-fs lifetime of the

excited state and an ~1.2 ps time constant of the ground state recovery. Moreover, sub-100-fs

dynamics in the excited state was concluded, which is significantly faster than the limitimposed by the temporal resolution of the experiments.

Evaluating the properties of one relaxation model against another, one can easily see

that the long-lifetime scheme, which involves three different electronic states with arbitraryparameters, can describe virtually any dynamics within a limited spectral window of the

experiment. Therefore, the success of such a three-level model (Fig.7.3b) in explaining the

numerous pump–probe data is to be expected.Exactly the same pump–probe features, depicted in Fig.7.2 can be observed upon the

excited state relaxation in the two-electronic-state model that includes the ground state

solvation (Fig.7.2a). Consequently, neither the transient spectrum shape, nor the decay of thekinetic traces can directly betray the underlying system of electronic potentials. Therefore, a


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challenge remains to differentiate between the two models on the basis of more subtle

features of the transient spectral dynamics.To this end, we notice that there are two ways, in which the evolution of the transient

absorption spectrum proceeds in the three-level (i.e. long p-state lifetime) model. The first

way is the excited state solvation that causes the modification of the transition frequencybetween the first and the second excited states and also decreased the p-s energy gap. As a

result, the contour of the stimulated excited state emission moves to the red and the spectrum

of the induced absorption shifts as well (typically, to the blue). The second way lies in thenonadiabatic character of the excited state depopulation [17,18]. This basically means that the

probability of relaxation is related to the actual size of the s–p gap, i.e. the rate of

depopulation is higher for lower transition frequencies. While the first way (solvation)introduces the overall shift, the second one (nonadiabatic relaxation) changes the shape of the

spectral contour of each excited state contribution. Upon completion of the solvation process,

which is faster than the electronic relaxation, the spectral position of both the excited stateabsorption and stimulated emission further remains fixed. Subsequently, the overall transient

absorption spectrum decays with the given rate of the nonadiabatic relaxation for the given

wavelength. A useful quantity here becomes the frequency, at which the “butterfly” shape ofthe pump–probe spectrum (Fig.7.2) crosses zero. In the three-level system, this frequency

changes during the excited state solvation and remains “frozen” afterwards. Schematically,

this is shown in Fig.7.4. Therefore, tracing the position of this frequency from theexperimental data allows discriminating between the faster rate of solvation and a much

slower rate of electronic relaxation (Fig.7.4, right panel).

Fig.7.4: Evolution of zero-crossing frequency (top row) and kinetic traces (bottom row) in twodifferent models. Horizontal dashed line indicates the frequency of the s–p transition. See text fordetails.

Chapter 7

170

Despite the external similarity of the pump–probe spectra predicted by a three-level and

a two-level (or short-lived p-state) models, the behavior of the zero-crossing frequency in thelatter is entirely different. The reason for that is a continuous blue shift of the hot ground-

state absorption contour, which moves towards the initial size of s–p transition frequency

corresponding to an equilibrated hydrated electron. The decay of the pump–probe kinetictraces in this model is a direct consequence of the shifting population of the hot ground state

as it overlaps and “fills in“ the hole in the population created by the excitation pulse.

Consequently, the continuous spectral blue shift (e.g. monitored by the evolution of the zero-crossing frequency) and the decay of the kinetic traces must have identical rates (Fig.7.4, left

panel).

An additional virtue of the zero-crossing as a model “sensor” lies in its asymptoticfrequency at very long pump–probe delays. While the short-lived p-state model automatically

predicts the arrival of zero-crossing to the center of the steady-state absorption spectrum, the

asymptotic frequency in the long-lived p-state model depends on the interplay of theindividual state parameters. Therefore, in the latter case, a sheer coincidence would result in

the absorption peak being the asymptotic value.

Last but not least, the sliding of the hot ground state population downhill to recombinewith the “hole” has one more profound impact on the pump–probe spectra. Based on the

considerations of electronic population conservation, we evaluate yet another helpful

difference between the two models. In the three-electronic-level model, at least three signalcontributions have to be considered at any pump–probe delay, namely, 1) the ground state

bleach (“hole”), 2) excited state absorption, and 3) stimulated emission. One of them, the

transient absorption, can, in principle, have considerably different transition strength and,consequently, a much stronger or weaker contribution to the transient spectrum. Therefore,

the integral of the pump–probe spectrum along the frequency axis can produce a finite value.

A value of zero in this case signifies that, by coincidence, the excited state absorptioncontribution to the overall signal exactly cancels the combined weight of the ground state

bleach and the stimulated emission put together.

The situation in the two-level model is completely different. Here, upon the transfer ofthe entire displaced population to the hot ground state, only two contributions, the “hole” and

the hot ground state absorption, form the pump–probe spectrum. Close to the bottom of the s-

state potential well, both the transition dipole moment and the shape of the transientabsorption must automatically match those of the “hole”. Therefore, the integral over the

whole frequencies of the pump–probe spectrum should produce exactly zero, which signifies

a perfect balance between the “hole” and transient absorption. The straightforwardimplementation of this last criterion on experimental data maybe not easy, considering the

limited window of the laser spectrum and the breadth of the hydrated electron spectral

features. We will, however, put this stringent criterion to use in our fit procedure.In closing to this Section, we have identified several useful features that allow

discriminating in practice between a short-lived and a long-lived p-state model. These are: 1)


171

the comparison of the zero-crossing dynamics with the decay of kinetic traces, 2) the

asymptotic value of zero-crossing, and 3) the ratio of the positive vs. negative contributions tothe transient spectra. These recipes for model identification will be subsequently applied in

this Chapter on the measured results.

7.3 Experimental

In this Section we briefly outline the practical aspects of measuring pump–probe kinetics and

spectra of the hydrated electron, using 5-fs optical pulses. The basic elements of thefemtosecond spectrometer and the sample preparation have been discussed already in

Section 6.2. Therefore, here we only address the necessary changes in the set-up.

Fig.7.5: Schematic of set-up for pump–probe on hydrated electron. (Also see Fig.6.1).

The schematic of the pump–probe spectrometer is presented in Fig.7.5. The difference

with the self-diffraction geometry implemented here is in the orthogonal polarizations of

pump and probe beams. The reason for this is to avoid unwanted heterodyning of the weaknonlinear polarization by the scattering from the pump pulse, which takes place in the

sample. Because of the ultrafast dephasing in the hydrated electron, the use of orthogonal

polarizations for the pump and probe does not decrease the amount of useful signal. Thismeasure is necessary, since it is impossible to screen off the scattered pump light, which

spectrally exactly overlaps with the probe radiation, by any means of spectral selection. To

further protect the measured signal from the pump scatter contamination, a polarizer cubewas installed in front of the detection scheme. The probe beam was subsequently focussed

into a 1/8-m monochromator (CVI) and split behind it into two channels by a metal beam-

splitter with a calibrated reflection/transmission ratio over the spectral range of interest. Thetwo channels served for the measurement of the pump–probe signal (i.e. modulation of theprobe beam intensity, prI∆ ) and the reference (i.e. the probe beam intensity, prI ). Two

separate optical choppers set at different chopping frequencies served to modulate the

excitation and the reference beams. Accordingly, the pump–probe signal and the reference

intensity were registered by two separate digital lock-in amplifiers (SRS) and fed to thecomputer. The difference absorption signal was then computed as the prpr II /∆ ratio. The

Chapter 7

172

measurements consisted of recording sets of transient spectra and sets of kinetic traces at

different probe wavelengths.The spatial overlap of the 5-fs pump- and probe- beams, and of the UV-preparation beam

from the YLF laser was verified through an aperture ∅25 µm. The optical delay was

subsequently aligned so that no noticeable deterioration of the beam overlap was present forthe large scans of the pump–probe delay in excess of 200 ps.

0 5 10 15 80 90 100

-1

0

1×10-3

O.D. (a)

λprobe

=650 nm

Delay T [ps]

0 5 10 15 80 90 100-0.02

-0.01

0.00

(b)

Nor

mal

ized

Pum

p-Pr

obe

Sign

al

Delay T [ps]

Fig.7.6: A long delay-range pump–probe trace of hydrated electron in water at the detectionwavelength of 650 nm.

The contribution of pure solvent to the pump–probe signal was measured by blocking the

UV preparation pulses from the sample. The amplitude of this contribution comprised about10-15% of the peak signal from the hydrated electron. The response from water quickly

disappears outside the pump–probe overlap. Transient spectra in absence of UV radiation

were taken for each pump–probe delay in the delay region of ±50 fs and subtracted from thecorresponding spectra of hydrated electrons.

A typical long-range kinetic trace is presented in Fig.7.6. Identical large-delay scans (not

shown) at different wavelengths all resulted in a slight remaining bleach component, thedecay time of which is in the order of 100 ps or longer. The amplitude of this component is

very small, only about 0.002 of the peak amplitude, to have any effect on the results of the

ultrafast part of the pump–probe studied here. Therefore, it has been subtracted from allrelevant data. The presence of this scanty component is an interesting fact in itself, signifying

that the system has not returned to the original state even after 100 ps. One of the plausible

explanations of this effect might be direct photo-ionization of a small percentage of thehydrated electrons back to a quasi-free state. These electrons have higher chances of


173

recombination and may never return to a bound state. Hence, the “escaped” particles can no

longer be accounted for in the pump–probe measurement. Also, similar “non-conservation”of the pump–probe traces has been reported in the recent study of Laubereau and coworkers

[10], although they did not comment on this fact.

7.4 Results and discussion

7.4.1 The measured tracesThe representative transient absorption spectra of hydrated electrons in water at differentpump–probe delays are presented in Fig.7.7 (solid dots). The large spectral width of the 5-fs

pulses allowed reliable acquisition of the pump–probe data in the range of 600 – 1050 nm.

The depicted dependencies are the averages of multiple scans. To provide an adequate signal-to-noise ratio at all pump–probe delays, the number of scans was gradually increased in

accordance with the drop in the amplitude of the signal. Consequently, while only 5 scans

needed to be accumulated for the spectra in the delay range below 100 fs, more than 100spectral scans were required for the delays above 5 ps.

10000 12000 14000 16000

-2

-1

0

1

-∆T

/T [a

rb. u

nits

] T=20 fs

Wavenumbers [cm-1]

10000 12000 14000 16000

T=40 fs

Wavenumbers [cm-1]

10000 12000 14000 16000

T=80 fs

Wavenumbers [cm

-1]

10000 12000 14000 16000

T=120 fs

Wavelength [nm]

Wavenumbers [cm-1]

10000 12000 14000 16000-1

0

1

-∆T

/T [a

rb. u

nits

] T=200 fs

Wavenumbers [cm-1]

10000 12000 14000 16000

T=300 fs

Wavenumbers [cm-1]

10000 12000 14000 16000

T=400 fs

Wavenumbers [cm-1]

10000 12000 14000 16000

T=500 fs

Wavenumbers [cm-1]

10000 12000 14000 16000-0.4

0.0

0.4

-∆T

/T [a

rb. u

nits

] T=1ps

Wavenumbers [cm-1]

10000 12000 14000 16000

×2

T=2 ps

Wavenumbers [cm-1]

10000 12000 14000 16000

×4

T=3 ps

Wavenumbers [cm-1]

10000 12000 14000 16000

×2 0

T=6.5 ps

Wavenumbers [cm-1]

1100 1000 900 800 700 600

Wavelength [nm]

1100 1000 900 800 700 600

Wavelength [nm]

11001000 900 800 700 600

Wavelength [nm]

1100 1000 900 800 700 600

Fig.7.7: Transient spectra of the hydrated electron obtained at different delays (solid dots). Solidcurves represent best fits to experimental data, which consist of the ground-state bleach and hot-ground state absorption contributions, as described in Section 7.3.2. Note different scaling of thevertical axis.

One measurement cycle of transient spectra in the delay range up to 7 ps took, on average,

~30 hrs, demonstrating extraordinary stability of the set-up. The pump–probe data showed

Chapter 7

174

excellent reproducibility, with the main source of experimental error attributed to the drop of

spectral amplitude in the wings of the probe pulse spectrum. Similar transient spectra havebeen recorded on the hydrated electron in heavy water (not shown).

0 1000 2000 3000 4000700

750

800

850

900

950

1000

1050

e-/H

2O

e-/D

2O

Zer

o-cr

ossi

ng w

avel

engt

h [n

m]

Delay T [fs]

0 100 200 300 400700

800

900

1000 Isotopic effect

e-/H 2O

e-/D 2O

e-/D 2O, T/ √2

Zer

o-cr

ossi

ng [n

m]

Delay T [fs]

Fig.7.8: Zero-crossing point as a function of pump and probe delay of hydrated electrons in water(solid circles) and heavy water (solid squares). Note that the asymptotic value in each case approachesthe position of the absorption maximum of the fully equilibrated electron (indicated by the dashedhorizontal lines). Solid curves show biexponential fits. The fit parameters are τ1=205±30 fs,A1=0.56±0.03 and τ2=1±0.1 ps, A2=0.44±0.03 for water, and τ1=215±20 fs, A1=0.64±0.03 andτ2=1±0.1 ps, A2=0.36±0.03 for heavy water. (The combined amplitude of A1 and A2 is normalized tounity). Inset shows the isotopic effect at early pump–probe delays. Vertical arrow indicates the end of

overlap between the data for H2O and, on the compressed by a factor of 2 time scale, data for D2O.

A dominating feature of the spectra in Fig.7.7 at initial pump–probe delays is an

absorption bleach contour, which stretches through most of the covered spectral window.

Another peculiarity is the rise and blue shift of the induced absorption. It corresponds to thecontour of positive amplitude, which appears in the infrared and progresses towards the

center of the steady-state absorption. This spectral evolution is easily traceable by the

position of zero-crossing that corresponds to the frequency where the strength of the inducedabsorption equals the amplitude of the ground-state bleach. Dashed vertical lines in Fig.7.7

denote the spectral evolution of the zero-crossing as the latter shifts toward the center of the

steady state absorption peak (720 nm). The values for the zero-crossing wavelength wereextracted from a linear fit of each measured contour in the adjacent to the crossing interval.

The size of the interval was set at 50 – 80 nm. The intersection of the resulting straight line

with the x-axis was subsequently taken as a zero-crossing point. Compared with the directuse of a single data point, in which the measured pump–probe spectrum intersects with the


175

horizontal axis, the outlined procedure is more trustworthy with respect to experimental

noise.To exploit the model-identification criteria, put forward in Section 7.2, we now plot the

zero-crossing wavelength, traced from each transient spectrum as described above, as a

function of the pump–probe delay. The corresponding dependence for hydrated electrons inwater and heavy water is depicted in Fig.7.8. It is clear that both cases exhibit a continuous

blue shift over the whole range of measured delays. The picosecond tails of both data sets

asymptotically arrive at the position of each respective peak of the steady-state absorption.The solid curves in Fig.7.8 depict biexponential fits starting from 100-fs delay.

0 2 4 6 8 10 12 1410

-3

10-2

10-1

100

λprobe

=900 nm

λprobe

=650 nm

Nor

mal

ized

|∆T

/T|

Delay T [ps]

-6

-4

-2

0

20 2 4 6 8 10 12 14

Delay T [fs]

λprobe

=900 nm

λprobe

=650 nm

-∆T

/T [

mil

li O

.D.]

Fig.7.9: Pump–probe kinetics of hydrated electron in water at detection wavelengths of 650 nm and900 nm. Circles depict measured data points, while curves represent biexponential fits (see text fordetails). The kinetic traces are scaled so the picosecond tails overlap. The parameters of the fits areτ1=1.04±0.08 ps, A1=-0.975±0.05 and τ2=4.9±0.3 fs, A2=-0.025±0.02 for 650 nm, and τ1=1.18±0.1 fs,A1=0.94±0.06 and τ2=5.1±0.4 fs, A2=0.06±0.02 for 900 nm. (The combined amplitude of A1 and A2

is normalized to unity. In the case of the 650-nm trace A1+A2 accounts for 1/17 of the total amplitudeof decay.) Inset shows the full extent of the measured transients.

We now turn our attention to the decay of pump–probe traces, which, according to themodel recognition recipe in Section 7.2, must provide the answer whether the time constants

of the spectral evolution and of pump probe decay are identical or not. The typical transients

at two different probe wavelengths, corresponding to the bleach recovery (λ=650 nm), and tothe build-up and subsequent recovery of induced positive absorption (λ=900 nm), are

Chapter 7

176

presented in Fig.7.9. The behavior of the kinetic tails in the delay range above 2 ps is clearly

bimodal. The biexponential fits for the detection wavelength of 650 nm and 900 nm, startingfrom a 1.7-ps value of delay, are shown by a dashed and by a solid curve, respectively. The

presence of the 1.1±0.1-ps component is thoroughly documented in the previous

experimental work of other groups [5-7,10], as has been discussed in the previous twoSections.

Two facts become apparent from the inspection of the dynamics of zero-crossing

(Fig.7.8) and the recovery of the transient kinetics (Fig.7.9). First, the dynamics of the blueshift clearly matches the picosecond rate in the tail of kinetic trace recovery. The presence of

the additional 3–5% 5-ps component in the observed transients does not affect this

conclusion, since the dynamic range of the kinetic measurement is significantly highercompared to the experimental precision, with which we can estimate the evolution of zero-

crossing. Therefore, in the available zero-crossing data, a 5-ps tail, if present at all, would

manifest itself merely as a raised background. This precludes the possibility to observe thisrate in the reported here experiment. Second, the asymptotic value of the blue shift in zero-

crossing invariably arrives at the peak of the equilibrated absorption spectrum for both water

and heavy water. In their combination, these two observations support, with a great certainty,the short-lived p-state model. A general fit of the pump–probe spectra, consistent with such a

scenario, will be presented in the following Section.

We now discuss several other issues surrounding the pump–probe measurementsdescribed in this Section.

Next to the well-known ~1-ps absorption recovery time, another, previously

unreported, 5-ps component is seen in our pump–probe data. As is apparent from the fitparameters, the relative contribution of this exponential tail to the overall pump–probe trace

is nearly 20 times smaller than that of the 1-ps component. Therefore, due to the sheer weight

of numbers, the 5-ps component cannot represent any substantial population relaxation of theexcited state. The fact that this decay rate has not been reported previously may be explained

in a number of ways.

As has been pointed out above, the dynamic range of our measurements presented hereis much higher than the one in the past experiments [5-7,10]. Therefore, this small component

could not have been resolved previously and the reported rates might have comprised the

weighed average of a faster and this slower component. A plausible explanation of thepresence of a slower rate in the pump–probe kinetics can be found in the recent visco-elastic

continuum simulations of the hydrated electron by Berg [3]. These simulations predicted a

small 5-ps relaxation component caused by the expansion and subsequent contraction of thewater cavity surrounding the electron in the direction perpendicular to the principal axis of its

p-wavefunction side-lobes.

Another explanation may be given considering the fact that the solvated electrons in ourexperiments are produced by photo-ionization of potassium ferrocyanide ions. It cannot be

excluded that for some photo-ionized ferrocyanide molecules the released electron stays in


177

the direct vicinity of its parent ion. Consequently, the electron can be trapped in a cavity that

has a ferrocyanide ion in the vicinity. This inevitably leads to an increase in the timeconstants of the transient absorption decay. For instance, for the electron in liquid methanol,

which has less mobile, and twice as heavy molecules compared with water, the relaxation

rates slow down nearly by an order of magnitude [48,49]. Therefore, it is possible that in ourpump–probe experiments we detect a statistical contribution from the electrons with different

environments.

Several important issues have to be addressed in connection with the spectral dynamicsat early delays. As is easy to notice in Fig.7.7 (top panel), the transient spectra in the delay

range up to 100 fs clearly show some modulation, which subsequently disappears (Fig.7.7,

middle panel). This phenomenon is addressed in Appendix I, where we relate it to thetemporal and spectral overlap of the pump and probe pulses and to a short lifetime of the

excited state.

Another aspect of the transient spectra in the sub-100-fs delay region is a large amount ofspectral shift (Fig.7.7, top panel). This suggests a great speed of population rearrangement,

either within the same electronic state due to rapid solvation or between two different

electronic states. Therefore, the use of extremely short excitation pulses is essential to“freeze” the nuclear motions of the solvent molecules [50] for the duration of pump. Indeed,

in our experiments with an 18-fs 800-nm excitation pulse and a 5-fs probe we observed some

deviations from the transient behavior shown in Fig.7.7, although the overall patternremained quite similar. With the longer excitation pulse duration, the negative absorption

changes were proportionally smaller, while the zero-crossing point was a few nanometers

offset to the red. A very logical explanation of this observation is the rapid populationmovement on the excited state potential, which covers a noticeable energy distance on the 18-

fs duration of the pump. This results in a red shift of the contour of absorption bleaching.

We now more closely examine the impact of deuteration on the spectro-temporalbehavior of the pump–probe signal. The impact of isotope substitution on the early delays is

illustrated in the inset to Fig.7.8, which presents the results of two identical measurements,

carried out on hydrated electrons in water and in heavy water. To ensure that the conditions,i.e. the pulse and beam characteristics, stay exactly identical, the experiments with these two

solvents in the 0 – 500 fs delay part were performed back-to-back. Like in the case of

transient grating (Fig.6.11), the scaling of the delay axis for D2O by a factor of 2 reveals aperfect overlap of the zero-crossing shift in water and heavy water in the initial part. The

subsequent rate of the blue shift, however, is nearly identical for both solvents, as is readily

seen from the parallel tails of the unscaled dependencies and nearly identical, 200-fs timeconstants obtained in the biexponential fit (Fig.7.8). Consistently with our previous

conclusions (Chapter 6), the observed results of the isotopic effect again show that the initial

solvent response within the first 100 fs in water and ~140 fs in heavy water is predominantlyof librational nature.

Chapter 7

178

As is clearly seen in Fig.7.8 (inset), the initial behavior of the zero-crossing shift is

highly non-exponential. Indeed, the measured dependence for water shows a “bump” around

a 50-fs delay. The fact that the same bump but on the time scales different by a factor of 2 ,is observed in both H2O and D2O rules out a possibility of the measurement artifact related to

the properties of the laser pulses. Observation of a similar bump in the transient gratingexperiments (see Chapter 6) led us to a hypothesis about a strong influence of a non-Condon

effect. The presence of a bump in the spectral position of the zero-crossing point indicates

that if the dipole-moment variation were caused by a non-Condon behavior, it should bewavelength-dependent. A possible alternative view on the origin of the bump can be given by

implying a short-lived excited-state absorption, which causes a delay in the transient signal

growth. We will return to the discussion about the role of the excited-state absorption in thefollowing Section.

7.4.2 The fit of transient spectraIn the previous Section, we have established that the observed pump–probe behavior is

consistent with a two-level model. We now perform a fit procedure of the transient spectra,

which allows us to decompose the overall response into separate contributions of eachinvolved state. The approach undertaken here relies on the same formalism and assumptions

that brought us to Eq.(6.8). The only difference is that the pump–probe signal is linearly

proportional to induced polarization (See Section 5.5), while in transient grating thisdependence is squared. Therefore, we are interested in the direct sum of the various states’contributions. Consequently, for each transient spectrum )(ΩPPS at the delay time T, we

obtain:

),,()(

),,(),,(

)(

exexexgrhotgr

grhotgrhotgrhotgrhoteggrgrgr

PP

aa

aa

S

ωσ

ωσωσ

ΓΩ−−

ΓΩ+ΓΩ−

=Ω

−

−−−− (7.1)

where gra and grhota − are the corresponding amplitudes of the contributions formed by the

population “hole” in the ground state and by the “hot” ground state, and σ ’s denote the

spectral shapes that are defined similarly to Eq.(6.2), i.e:

222220

22

04)(

4),,(

ΓΩ+Ω−

ΓΩ

Γ

Γ=ΓΩ

ωωσ gr

, (7.2)

where 0ω stands for the central frequency, and Γ is the characteristic half-width of each

contour. For the “hole” contribution )(),,( Ω≡ΓΩ Aeggrgr σωσ and 12−=Γ T , where )(ΩAσ

is the steady-state absorption spectrum given by Eq.(6.2). The contributions of the stimulatedemission and of the hot ground-state absorption are denoted by exσ and grhot−σ , respectively.


179

To fulfill the condition of population conservation, the amplitude of the excited state

contribution (which is proportional to p-state population) in Eq.(7.1) equals to the differenceof the total number of excited electrons and the amount that has already returned to the hot-ground state, i.e. grhotgr aa −− . Additionally, the scaling factor in the square brackets in

Eq.(7.2) ensures that each normalized σ -contour retains its area regardless of the size of Γ .The fit procedure runs as follows. The width grΓ is set at 3250 cm-1, which corresponds

to the best fit of the steady-state absorption at room temperature in Fig.6.7. Then the value of

gra , that is identical for all contours, is fixed. Subsequently, the widths and the central

frequencies of the stimulated emission contour, exσ , and the hot-ground-state induced

absorption, grhot−σ , and the amplitude grhota − , are globally fitted as free parameters. A

Levenberg–Marquardt routine was used for minimization [51]. The freedom in allowing thestimulated emission to have a somewhat different central transition frequency is provided

deliberately to account for the fact that a nonadiabatic electronic relaxation [17] is faster atnarrower energy gaps. The nonadiabatic relaxation shifts the position of grhot−σ to the red

with respect to exσ . The fits of individual pump–probe spectra, obtained in this model, are

depicted in Fig.7.7 as solid curves. As can be seen from the overlap with the experimental

points, the overall fit quality is very high.The summary of the fit parameters as a function of pump–probe delay is shown in

Fig.7.10. As is evident from Fig.7.10a (the initial delay part is shown enlarged in the inset),

the amplitude of the stimulated emission rapidly drops within the first 100 fs. The mono-exponential fit of this amplitude yields an ~55-fs time constant of the lifetime of the p-state.

The negligible amount of the excited state population, left after ~120 fs, allows taking awaythe exσ -contour from the fit of further spectra. Therefore, the solid curves presented in the

middle and the bottom rows of Fig.7.7 correspond to the three fit parameters, i.e. the width,amplitude, and position of grhot−σ . Keeping grhota − as a free parameter is necessary to

account for the gradual population build-up of this state. As can be seen from Fig.7.10a and c,at large pump–probe delays, when the population of the hot ground state approaches theequilibrium region, both the amplitude and the width of the grhot−σ converge into those of the

grσ -contour. Except for the initial 100 fs, corresponding to the build-up of the hot ground

state population, the ratio of grgrhot aa − stays reasonably close to unity, which is consistent

with the overall population conservation. The fact that this follows from the fit and is not

assumed a priori is a powerful check of the right model choice. Note that the small deviationsof the grgrhot aa − ratio from unity in the delay region of 0.5–4 ps are related to the

simultaneously occurring change in the spectral width grhot−Γ . The latter influences the

magnitude of the amplitude grhota − via the normalization factor scaling in Eq.(7.2)

The initial relatively small width of the grhot−σ (Fig.7.10b) is an indication of the rapid

funneling of the p-state population through a relatively narrow photochemical “sink” [52], i.e.a region of the excited state potential with the highest probability of crossing back to theground state potential. The width of grhot−σ , grhot−Γ , subsequently broadens during the

Chapter 7

180

downhill slide on the hot ground state. A more detailed explanation of this will be provided inthe following Section. The blue shift of the central position of the grhot−σ exhibits the

familiar, ~1.2-ps predominantly exponential evolution (Fig.7.10c). This rate is slightly

different from the one inferred from the evolution of the zero-frequency shift, which wasestimated to be 1±0.1 ps. The small discrepancy can be explained by the fact that the location

of the zero-frequency point is determined by the overlap of two asymmetric contours, i.e.

grhot−σ and grσ , which somewhat speeds up the rate of zero-frequency shift.

0

1

(b)

(a)

a hot-

gr/a

gr

0 100 200 300 400 5000

1

τ=55 fs

aho

t-gr/a

gr

Delay [ f s ]

0

1

Γ hot-

gr/Γ

gr

0 1 2 3 4 5 6

10

102

103

(c)

slope 1.2 ps

ν hot-

gr-

ν eg [

cm-1]

Delay T [ps]

Fig.7.10: Summary of fit parameters as a function of pump–probe delay. (a) Relative amplitudes ofthe hot ground state (solid circles) and excited state (open circles). Solid curve in the inset shows amono-exponential fit with a time constant of 55 fs. (b) Relative width of the hot ground stateabsorption spectrum. (c) Spectral shift of the peak of the hot ground state absorption with respect tothe initial transition frequency.

The frequency of the p–s gap, where the population of the excited state crosses back to

the ground state is around 9500 cm-1. This value corresponds to a substantial portion of the


181

initial transition frequency, i.e. ~14000 cm-1. Therefore, a large amount of energy deposited

on the hydrated electron is rapidly absorbed by the solvent with a characteristic transfer timeof ~50 fs.

Clearly, it was impossible to extract such a short, 50-fs lifetime of the excited state

from previous pump–probe studies [5-7,10], due to the lack of adequate temporal resolution.The values of the lifetime as short as 120 fs did emerge, however, from numerical

simulations [23,28]. These figures, nonetheless, are still higher by a factor of two than the

lifetime of the p-state that resulted from the fit of our pump–probe data. We suggest that thediscrepancy between the simulations and the interpretation of the experimental data maybe a

priori “built in” into the computed results. Set aside the fact that the electron–water

interaction is typically simulated by a “pseudopotential” [27,53], there is a distinct problemwith representing water molecules by a model potential [26,30] in general. For instance, it

has been realized for the simple point charge potential [30] that the effective stretch force

constant is different for an isolated water molecule, and for the one that has its OH bondaligned with the oxygen of the neighboring molecule. There is every reason to believe that,

unlike in the case of a partly screened oxygen atom, in the case of the hydrated electron, the

hydrogen bond formation might be even stronger. As an indication of this one should recallthe predominant OH bond alignment in the direction of the electron in the first solvation

cavity, revealed by the electron spin resonance study of glassy water [54] (see Fig.1.1).

Therefore, the description of the first solvation shell by a potential, which is identical to theone used to describe the bulk solvent, should result in an underestimation of the coupling

forces and the energy involved in the electron–water interaction. Accordingly, this must be

reflected in the lengthening of the computed electronic relaxation time. Even though theenergy deficit can be made up for by the addition of the interaction “pseudopotential”, the

fundamental properties, such as oscillation frequencies, of the neighboring to the electron

molecules will be accounted for incorrectly.A closer inspection of the fit quality at the 0–100 fs delays (Fig.7.7, top row) reveals a

deficit of positive transient absorption in the blue wing, which rapidly traverses the spectral

window of observation on the same time scale as the excited state depopulation. This,together with the calculations of the pump–probe spectra during the pulse overlap (see

Appendix I), and with the bump around 50-fs delay in the spectral shift of zero-crossing (see

previous Section), can be explained by excited state absorption to the continuum (c-state) [1](Fig.7.3a). The rapid blue shift and the subsequent total disappearance of this absorption

correspond to depopulation of the excited state. The inclusion of this transient absorption into

the overall fit was not attempted yet for two reasons. These are: 1) the shape of the p–cabsorption is unknown. Assuming, that the coupling strength between the p- and the various

states comprising the continuum manifold stays identical, the spectral shape of the

corresponding absorption would be described by a sigmoidal function, rising towards thehigh frequencies. 2) The lack of information on the strength of the p–c transition and on its

evolution in time extremely complicates any sensible fit with not-too-many parameters.

Chapter 7

182

Anyhow, because the population lifetime of the excited state is ~50 fs, the p–c contribution to

the pump–probe can be disregarded for the delays longer than 80 fs.The p–c transient absorption presents an alternative view to the hypothesis proposed in

the previous Chapter, which involved a non-Condon effect. To what extent the first or the

second, or both these processes together dominate the initial dynamics of the hydratedelectron energy relaxation remains to be clarified. Hopefully, measurements in an even

broader spectral window, than the one provided by the spectrum of 5-fs pulses, would give an

answer to this question.In summary to this Section, from the fit of the pump–probe spectra we have inferred a

very short, ~55-fs lifetime of the first excited state and obtained time dependencies of the

spectral width of the hot-ground state absorption and the corresponding size of the modified-s–p energy gap. In the following Section, we perform an inversion of the time-domain

picture, observed at the longer delays, into the generalized solvent coordinate.

7.4.3 The inversion of temporal data to potential surfaces at long timesThe desired goal of the pump–probe experiment, besides estimating the energy relaxation

rates, is to clarify the structure of the potential energy surfaces. Therefore, a transition shouldbe made from the measured time-dependence to the actual (spatial) solvent coordinate(s). It

has been shown by Zewail and coworkers [55-57] that such inversion is a relatively

straightforward task for the photo-dissociation experiments in molecular beams where quasi-free fragments reach, after some time, terminal velocity and retain their kinetic energy. In this

case, because of the linear dependency of the covered distance on the time, the inversion

involves a simple scaling of the coordinate space. The situation, however, is substantiallydifferent in condensed media. Here, because of the strong interactions with the surroundings,

the kinetic energy of a photo-excited particle is rapidly lost. Consequently, velocity along any

of the solvent coordinate undergoes a substantial and nonlinear decrease as a function oftime. Clearly, the inversion method of Zewail et al. is not applicable to the situation of the

hydrated electron.

Here we utilize a different approach to reconstruct the difference between the groundand excited state potentials of the hydrated electron. To do this, the temporal behavior of the

hot ground state absorption, obtained from the general fit of the transient spectra, is exploited.

The concept is illustrated in Fig.7.11. We apply a classical treatment to the downhillmovement of the hot-ground-state population. Our assumption relays the width of the grhot−σ

to the combined curvature of the ground and excited state potentials that are denoted,respectively, as grV and exV . According to this scheme, the recovered from the spectral fit

central frequency of grhot−σ as a function of delay, )(tgrhot−ω , provides the size of the current

gap energy gap, i.e. exV - grV . At the same time the spectral contour width, )(tgrhot−Γ is related

to the shape of )()( qVqV grex − as:


183

qtq

V

q

Vt grhot

t

gr

t

exgrhot

&

1)(

∂

∂∝

∂

∂−

∂∂

∝Γ −−

ω, (7.3)

where q is the generalized solvent coordinate, and q& is velocity. The expression (7.3) is

exact in the case of a Brownian oscillator model [58], which considers two identical

harmonic potentials that are displaced along the solvation coordinate. The spatial extent of

the wavepacket in this scenario remains constant [59]. Consequently, it is the curvature of thepotentials, which determines the width of the corresponding transient absorption. However, in

general Eq.7.3 presents only a first-order approximation that breaks down, for instance, in the

case of not-displaced potentials, with non-identical frequencies.

Fig.7.11: Concept of hydrated electron potential surface inversion from the width and spectralposition of the hot-ground-state absorption.

According to Eq.7.3, the tq → transition can be readily realized by computing the

following integral:

constdttt

tqt

grhot

grhot +Γ∂

∂∝ ∫

−

−

0)(

1)(

ω(7.4)

Chapter 7

184

Because the function q (t) is unambiguously defined for each delay instance, the inverted

function, t( q ), can be directly obtained.

The result of this procedure, applied to the fit parameters from Fig.7.10, is shown in

Fig.7.12. This picture corresponds to a strongly damped motion. Indeed, while about 2/3 of

the distance, separating the point of population return to the (hot) ground state and the Frank-Condon region, are covered within the first picosecond, the subsequent change of location is

very slow.

0 1 2 3 4 5 6-1.0

-0.5

0.0Frank-Condon region

Solv

ent c

oord

inat

e q [a

rb.u

nits

]

Time t [ps]

Fig.7.12: Time to solvent coordinate inversion of the data in Fig.7.10. The abscissa value of -1corresponds to the initial position of hot-ground state population after radiationless p–s transition.Zero stands for the coordinate of s–p transition of the equilibrated hydrated electron.

Having us obtained the t( q ) dependence, we are now able to plot the evolution of the

potential energy gap between the ground and excited state as a function of coordinate(Fig.7.13). Inspection of the graph shows that there is a distinct change of pattern that takes

place around the value of q≈-0.7. It is clear, that the electron trajectory after ~300 fs

following the photo-excitation corresponds to a nearly linear change of the s–p with thecoordinate q. Several important conclusions follow from this fact.

First, the linear dependence of the energy gap on the coordinate is consistent with the

picture of two identical harmonic potentials that are displaced. This a posteriori justifies theapplication of Eq.(7.3) for the long times in the case of the hydrated electron. Second, the

change of the behavior of the gap towards a linear coordinate dependence suggests that a

certain type of nuclear motion becomes dominant. Note, that that Eq.(7.3) has been obtainedfor the one-dimensional case. In the multidimensional scheme with involvement of several

solvent coordinates, which is typical in chemistry, the inversion problem does not have a

simple solution, such as the one given by Eq.(7.4). However, we can still rely on thisapproach if the energy relaxation has distinct phases, associated with different types of

molecular motions. One, nonetheless, should bear in mind that at different locations this

coordinate could, in principle, represent very different nuclear degrees of freedom. A


185

confirmation of the latter idea can be found in the reconstruction of the s–p energy gap as a

function of a single generalized solvent coordinate q presented in Fig.7.13. Indeed, theobserved picture can be readily understood by stepping from a single coordinate q to a

multidimensional space, which separates different degrees of freedom of the water molecules.

In the following Section, we further elaborate on the relaxation pathways on the basis of amulti-coordinate representation.

-1.0 -0.5 0.00

2000

4000

6000

8000

10000

12000

14000

2 ps

500 fs200 fs

p→s sink region

FC region

Wav

enum

bers

[cm

-1]

Solvent coordinate q [arb. units]

Fig.7.13: s–p gap as a function of generalized solvent coordinate. Solid and open circles show thevalues of grhot−ω and grhot−Γ , respectively, obtained from the fit (see Fig.7.10). Solid curve

computed by numerical integration of grhot−Γ over q (see text for details).

It is important to realize that it is impossible to assign the actual shape and curvature of

each potential, since the pump–probe experiment measures a gap between the excited states.

Therefore, an additional input from Raman or other-kind vibrational spectroscopy is requiredin order to obtain information on the curvature of the ground-state potential.

Before closing this Section, we point out an interesting aspect of the applied procedure.

It concerns the relation between the temporal and spatial evolution of the width and positionof grhot−σ , which are independent in the time domain but are connected in the q-space by

definition in Eq.(7.3). To verify that the numerical calculation of the temporal derivative ofthe discrete array of grhot−ω did not introduce any distortions in application of Eq.(7.4), we

can compute the function constdqqq

q

grhot +Γ∝ ∫ −

0

)()(ω , which follows from integration of

Eq.(7.3). The result of this validity check is plotted in Fig.7.13 as a solid curve. The good

Chapter 7

186

overlap of this dependency with the values grhot−ω indicates that there are no significant

inconsistencies in our calculations.

In summary of this Section, we have demonstrated a practical method of inversion of

the pump–probe data, which allows reconstructing the energy gap between the ground- andexcited-state potential surfaces of the hydrated electron.

7.4.4 The multidimensional relaxation modelIn Section 7.4.2 we have established the time scales and the sequence of the basic steps in the

energy relaxation process of the hydrated electron. According to the results of the global fit of

the pump–probe data, approximately 2/3 of the deposited by photo-excitation excess energyis dissipated with a time constant of 50 fs due to population relaxation from the excited state.

This process is deuteration-dependent and, therefore, is likely to be dominated by the

librational motion of the water molecules. The relaxation process following the excited statedepopulation is attributed to the cooling of the ground state. Contrary to the population decay,

no clear-cut dependence of this process on deuteration has been observed. Therefore, we

assume that collective translational motion of the water molecules takes over in this timerange. Subsequently, in Section 7.4.2 we have mapped the energy loss of the photo-excited

hydrated electron onto the generalized solvent coordinate. This provided further indication

that more than one characteristic motion of the water molecules participates in the process ofenergy dissipation. In this Section, we attempt to tentatively disentangle the contributions

associated with the evolution of different solvent coordinates from the overall response. In

other words, we aim at recreating a multidimensional trajectory of energy relaxation.Our idea is based on the clear separation of the time scales associated with a

predominantly librational and, subsequently, with a predominantly translational response of

the water molecules. Obviously, within 50 fs the translational motion cannot be substantial incomparison with the more rapid librations. This is also proved by the observed strong

dependence on the isotopic effect. On the other hand, Fig.7.13 suggests that after 300 fs

practically only translations determine the process of energy relaxation. This conclusion isbased on the linear dependence of the energy gap on the coordinate q. As has been pointed

out in Section 7.4.3, this is consistent with a picture of two displaced harmonic potentials

with identical frequency along this coordinate. As a result of the time scale separation, wecan consider that for the short (<50 fs) times the electron “propagates” mostly along the

librational generalized coordinate while for longer (>0.3 ps) times – along the translational

one. It is also clear that there should be an intermediate region where both coordinates changesimultaneously. This interval corresponds to the intermediate decay rates observed in all

transients. Although there are no conclusive data available on the actual behavior of each

coordinate in the intermediate time, we can, nonetheless, take a very good guess, since theamount of energy released to the solvent during this interval is known from the potential

inversion (Fig.7.13).


187

050

0010

000

Librational coordinate

50 fs

1 ps200 fs

300 fs

100 fs

- h ω

Translational coordinate

Exc

ess

ener

gy [

cm-1]

Fig.7.14: Schematic trajectory of hydrated electron energy relaxation as a function of two separatesolvent coordinates. At t=0 the hydrated electron is photo-excited from the origin of the referenceframe (photon energy is shown by a dotted line). Positions at different times following the excitationare indicated by hollow circles. See text for details.

The deduced trajectory of the relaxing hydrated electron is given in Fig.7.14. Uponoptical excitation the electron undergoes the s–p electronic transition. The initial excess

energy supplied by the photon is approximately 14000 cm-1. As we have established, the first

stage of the solvation process starts with an ultrafast energy loss to the librations of the watermolecules. The concept of librations should be understood in a broad sense, i.e. as hindered

rotations of water molecules, and bending and stretching of the O-H bonds. According to the

results of the global fit in Section 7.4.2, the change of the energy in the initial step is~9500 cm-1. This is remarkably close to the triple frequency of the O-H stretching mode in

liquid water (~3400 cm-1). However, since the electron in the excited state basically

undergoes a one-dimensional expansion, it is expected that only 2 out of 6 water molecules(Fig.1.1) can efficiently take up the energy of ~6800 cm-1. The rest is either absorbed by an

O-H stretch with a lower efficiency, or by one of the other four molecules in the first

solvation shell. Alternatively, it is distributed among several lower-frequency modes. Thededuced lifetime of ~50 fs compares very well with the time needed to transport energy to

multiple O-H bonds. Indeed, as determined by the inverse frequency, an excitation of a single

Chapter 7

188

O-H bond roughly takes place in 10 fs. Accordingly, the required time increases by a factor

of 2 in the case of D2O.As revealed by the fit results of the pump–probe data (Fig.7.10), by approximately

100 fs after the excitation most of the electrons have already returned to the (hot) ground

state. This corresponds to the “collapse” of the size of the hydrated electron, which nowoccupies a more compact hot s-state, leaving a void in the solvent. The more compact

configuration of the electron changes the character of the librational motion from forcibly

driven by the electron expansion to a more random one. Accordingly, the “distance” coveredby the electron along the librational coordinate stops increasing while the “propagation”

along the translational coordinate begins. During this phase, the water molecules in the first

solvation shell deploy the accumulated excess energy into a collective-type translationalmotion. It is likely that the existing hydrogen-bond network [60,61] helps to speed up the

energy transfer that occurs on an ~250-fs time scale.

The kinetic traces on the intermediate time scale do not exhibit any appreciable isotopiceffect. This can be explained by the fact that the excess energy received from the electron by

the librational modes of water molecules is subsequently released (passed on to further

solvation shells) through translational motion. Only after dissipating this extra energy, themolecules of the first solvation shell can close the void that has formed after the crossing

from the p-to the s-state.

0 1 2 3 4 5 6

300

350

400

450

500

550

Loc

al te

mpe

ratu

re [

K]

Time [ps]

275 300 325 350 37512500

13000

13500

14000

14500

15000 e

- in H2O

Peak

abs

orpt

ion

posi

tion

[cm

-1]

Temperature [K]

Fig.7.15: Local heating effect as a function of time since the photo-excitation. Solid circles show thetemperature calculated from the spectral position of the hot-ground state absorption. Solid curvedepicts a biexponential fit with the parameters A1=135 K, τ1=0.3 ps and A2=120 K, τ2=1.1 ps. Insetshows peak position of hydrated electron absorption from the data in Ref. [62] (hollow circles). Solid

line is given by expression Teg 75.2020730 −=ν , were T is the temperature in K.


189

Approximately after 300 fs, the librational coordinate returns to its initial value before

the excitation while the translational one reaches the maximum. This means that thelibrational energy has already been transferred into translational motions of the first and the

next solvation shells. From now on the energy is released in the form of local heating (i.e. the

local temperature is substantial raised) and spreads further away from the electron. Finally,by about 5 ps practically full equilibration of the hydrated electron is achieved.

The local temperature can be evaluated from the spectral position of the hot ground

state absorption. The change of the local temperature as a function of time is estimated inFig.7.15. Here we extrapolated the known linear dependence of the hydrated electron

absorption peak on temperature [62] and took the position of the hot-ground state absorption

in time from the fit results depicted in Fig.7.10c. Although the resulting value of the localtemperature exceeds 500 K, one should bear in mind that up to the times of ~300 fs the

excess energy is accumulated primarily in the (collective) librational modes of the water

molecules. Therefore, the change of the temperature, viewing the latter as a measure of themean kinetic energy of the water molecules, is unlikely to exceed ~100 K. Nevertheless, this

value is substantially higher than the one reported before [11].

In summary to this Section, we have proposed a complete multidimensional model ofthe energy relaxation in the hydrated electron. We distinguish three main phases in this

process. The first stage is a librational relaxation, which is responsible for an energy loss of

~9500 cm-1 with a time constant of ~50 fs. The second, an intermediate phase takes placeduring 100-300 fs following the excitation and accounts for the energy release in the order of

1000 cm-1. Collective translational modes are being excited during this time. The last step is a

purely translational relaxation, which proceeds with a time constant of ~1 ps and isresponsible for the further release of the remaining 3000 cm-1 of the excess energy. The local

heating of the hydrated electron environment is estimated to be around 100 K.

7.5 Conclusions

Frequency-resolved pump–probe measurements with 5-fs laser pulses rendered a

comprehensive picture of the energy relaxation pathways that a hydrated electron undergoesupon photo-excitation of the s–p transition. This work clarified several issues that have posed

a challenge for the ultrafast laser spectroscopy of this species, despite a decade of intense

efforts. For the first time, pulses of an adequately short duration and broad bandwidth wereapplied to uncover the initial spectral dynamics that rapidly evolves within a few tens of

femtoseconds. Additionally, a significant breakthrough in describing the energy relaxation

process on the picosecond time scale was achieved due to the unprecedented dynamic rangeof the measurements. Importantly, the correct understanding of the picosecond time scale also

appears to be vitally linked to the use of as short as possible excitation pulses, ensuring that

no substantial population relaxation occurs for the duration of the excitation pulse. In all, theuse of 5-fs pulses, employed is this study, has resulted in a considerable progress in the

research of the hydrated electron dynamics.

Chapter 7

190

The fundamental difficulty that has been solved in order to enable a correct pump–

probe data interpretation is the delineation of various contributions comprising the observedsignal. Namely, two different models of electronic relaxation can be used to account for the

transient spectral dynamics seen through the frequency window that is limited by the pulse

spectrum. To resolve the issue, in this Chapter we have developed the guidelines forrecognition of a “short-lived p-state + slow ground state solvation” vs. a “long-lived p-state +

fast ground solvation” models of the hydrated electron. The resulting recipes for the level

scheme identification are based on the combined analysis of the spectral as well as kineticdynamics and, in general, are very reliable in determining the right relaxation scenario.

The application of the derived criteria to the experimental data provided strong support

for the “short-lived p-state + slow ground state solvation” relaxation model. Subsequently, aglobal fit according to this model has been performed on the transient spectra and revealed an

~50-fs lifetime of the excited state. Indirect evidence of such a fast electronic relaxation has

also been provided by numerical simulations of the transient spectra in which the actualamplitude and phase of the pump and probe pulses have been used.

According to our current understanding, the relaxation dynamics of the hydrated

electron consists of three stages: 1) a rapid, 50-fs depopulation of the excited state, whichcorresponds to transfer of the energy of ~9500 cm-1 to the surrounding water molecules; 2) a

slow, 1-ps equilibration of the ground state; and 3) an intermediate phase between the first

two phases. Similarly to the behavior of the transient grating signals, reported in Chapter 6,the early delay region of up to about 100 fs is dominated by librational motions of the water

molecules, which is manifested by an ~ 2 difference in the time scales for H2O and D2O.The energy released to the solvent by the ground state thermalisation amounts to ~1/3 of theinitial s–p transition energy. The features of pump–probe spectra in the range of delays from

zero to 60 fs suggest the presence of a short-lived excited state absorption to a higher lying

state or to the continuum band. The disappearance of this contribution at delays longer than60 fs is fully consistent with the rapid depopulation of the first excited state.

A practical method for the inversion of the potential energy gap between the ground-

and the excited- state potential has been proposed and applied to the experimental data. Thisprocedure resulted in the reconstruction of the s–p energy gap as a function of the generalized

solvent coordinate. As has been suggested in the interpretation of our kinetic data, this

generalized coordinate represents a passage between, at first, a mostly librational to a mostlytranslational coordinate later on. This enabled us to produce a tentative multidimensional

picture of the energy relaxation that takes place upon a photo-excitation of the hydrated

electron. In the proposed model, a gradual transition between the two modes of the electron-to-solvent energy transfer has been put forward. The first mode determined by the collective

librations of the water molecules, accounts for the highest excess energy loss on a sub-100-fs

time scale. It is progressively superceded by the second mode, described by collectivetranslations of the water molecules, which is responsible for the dissipation of the

accumulated local heat. In the proposed explanation, the return of the hydrated electron to its

Chapter 7

192

10000 12000 14000 16000

1100 1000 900 800 700 600

-1

0

1

T=0

Inte

nsity

[ar

b. u

nits

]

Wavelength [nm]

10000 12000 14000 16000

-20

-10

0

10

20

-∆T

/T [m

illi O

.D.]

Wavenumbers [cm-1]

Fig.7.16: Simulation of transient spectrum at delay T=0 for hydrated electron in heavy water. Solidpoints show the measured spectrum, while the solid curve presents calculated results shifted up by thelevel of dashed line. Shaded contour depicts the spectral intensity of the laser pulses.

1100 1000 900 800 700 600

1

0

-A/A

max

Wavelength [nm]

10000 12000 14000 16000

-1

0

(b)(a)

T 1=500 fs

T1=50 fs

T1= 5 0 f s (×2.9)

-∆T

/T [

arb.

uni

ts.]

Wavenumbers [cm-1

]

T 1=500 fs

T1=50 fs

Fig.7.17: (a) Simulation of transient bleach at delay T=50 fs for a homogeneously broadened systemwith T2=50 fs (solid curve) and T2=500 fs (dashed curve). Shaded contour shows steady-stateabsorption, or, in other words, the transient bleach spectrum in the case of δ-pulse excitation. (b)Enlarged detail of pump–probe modulation.

To deepen our understanding of the role of T1 in the pump–probe spectra modulation

we next perform calculations at a delay where the main bodies of the two pulses are already

well separated in time. For the sake of demonstration, we only consider the contribution ofthe “hole” in the ground state and do not include the input of the excited state. This is


193

legitimate, since the transient spectra consist of a linear superposition of various

contributions. Therefore, the conclusions made on the basis of one selected contribution areapplicable to the whole combination as well. The results at the delay value of 50 fs, which

corresponds to a homogeneously broadened transition with T2=1.6 fs and with T2=50 fs and

T2=500 fs are depicted in Fig.7.17.As can be seen from this simulation, the relative size of the surviving modulation in

comparison with the magnitude of the overall signal is considerably larger for a short-lived

system. Consequently, the modulation seen in the pump–probe spectra of the hydratedelectron in Fig.7.7 at short delays is a direct outcome of the short value of T1.

Chapter 7

194

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Samenvatting

Als een enkel elektron geheel wordt omgeven door watermoleculen spreken we van een nat

of gehydrateerd elektron. Het gehydrateerd elektron is als het ware gevangen in een “kooi”

van watermoleculen. Dit voor de chemie zo belangrijke systeem werd in 1962 ontdekt. Sinds

die tijd trekt het de aandacht van experimentalisten en theoretici. Experimenteel is het

systeem van belang, omdat het elektron een cruciale rol speelt in radio- en elektrochemie en

in fotochemische reacties. Vanuit fysisch oogpunt is het nat elektron interessant als

modelsysteem voor kwantummoleculaire dynamica berekeningen en solvatatiedynamica.

Solvatatiedynamica is de vloeistofdynamica die optreedt bij verandering van een

elektronische toestand.

Omdat het elektron in een kooi van watermoleculen is opgesloten, heeft het systeem net

als een waterstofatoom discrete energieniveaus. Uit berekeningen blijkt dat een nat elektron

een s-achtige grondtoestand heeft en drie aangeslagen p-toestanden. De s–p overgang is

gecentreerd rond 720 nm en heeft een breedte van 350 nm. De eerste optische experimenten

aan het gehydrateerd elektron met behulp van femtoseconde lasers dateren uit het eind van de

jaren tachtig. Vanwege het destijds beperkte tijdoplossend vermogen (~200 fs) konden de

meest interessante dynamische processen niet worden ontrafeld. Zo voorspelden computer

simulaties dat de eerste stap in de solvatatiedynamica slechts 25 fs duurt. Pulsen veel korter

dan 25 fs zijn dus nodig om dit ultrasnelle proces te bestuderen.

Het doel van het onderzoek dat in dit proefschrift staat beschreven is het in kaart

brengen van de verschillende stappen die optreden bij de relaxatie van het geNxciteerde nat

elektron. Uit het onderzoek krijgen we een beeld van de rol die de watermoleculen spelen in

het relaxatieproces. Bewegingen van watermoleculen zorgen ervoor dat de energie van het

geNxciteerde elektron wordt overgedragen aan de omringende watermoleculen. De snelheid

van deze energieoverdracht is een maat voor de interactie tussen het elektron en water. Om

snelle optische experimenten aan het nat elektron te kunnen doen, werd in ons laboratorium

een laser gebouwd die bijzonder korte pulsen kan leveren. De kortste puls die gemaakt werd

duurt slechts 4,5 fs (1 fs = 10-15 s). In die tijd legt licht een afstand af van minder dan 2

micron. Het elektromagnetische veld van een dergelijke puls bestaat uit iets meer dan twee

oscillaties. Interessant is ook, dat dit de kortste lichtflits is die ooit is gemaakt. Dit record

heeft daarom een vermelding gekregen in het Guinness Book of Worldrecords.

In het promotieonderzoek zijn gehydrateerde elektronen bestudeerd met een aantal niet-

lineaire optische technieken die gebaseerd zijn op de derde-orde materierespons. Soortgelijke

experimenten, die gebruik maken van de tweede-orde respons van de materie, zijn nodig om

de laserpulsen zelf te karakteriseren. In al deze technieken wordt een aantal pulsen

gefocusseerd in het monster of in een niet-lineair optisch kristal. Het signaal wordt dan

gemeten als functie van de vertraging tussen de pulsen en/of van de golflengte van het licht.

Het gevormde signaal kan zowel in het tijd- als frequentiedomein worden beschreven. Voor

experimenten waarin pulsen worden gebruikt, wordt meestal voor een beschrijving in het

Samenvatting

197

tijddomein gekozen. Wij hebben voor een beschrijving het frequentiedomein gekozen,

vanwege het gemak waarmee een aantal effecten, samenhangend met de zeer grote spectrale

breedte van de pulsen, in de berekeningen verdisconteerd kunnen worden.

Om de ultrasnelle dynamica van het nat elektron betrouwbaar te kunnen meten, moeten

de amplitude en de fase van de femtoseconde pulsen vooraf bekend zijn. Deze eigenschappen

bepalen n.l. het intensiteitprofiel van de puls en de manier waarop de verschillende spectrale

componenten over de puls zijn verdeeld in de tijd. Om deze karakteristieken nauwkeurig vast

te leggen zijn verschillende meettechnieken gebruikt. De meest geavanceerde hiervan is de

zogenoemde FROG-methode, waar FROG staat voor Frequency-Resolved Optical Gating.

Deze techniek bestaat uit het meten van een autocorrelatie-type signaal, waarmee de spectrale

inhoud van de puls wordt vastgelegd. Via bekende algoritmes kan hieruit de amplitude en de

fase van de puls berekend worden. De aanpassing van deze techniek voor het meten van zeer

korte optische pulsen vormt een belangrijk deel van dit proefschrift.

Het in dit proefschrift beschreven onderzoek bestaat uit vier verschillende onderdelen.

Deze zijn: 1) het ontwerp van een geschikte fs laser, 2) het karakteriseren van de

gegenereerde fs pulsen, 3) de ontwikkeling van het formalisme voor het beschrijven van de

experimentele resultaten en tenslotte, 4) de studie van de dynamica van het gehydrateerd

elektron. Na een algemene introductie worden genoemde onderwerpen behandeld. In

hoofdstuk 2 wordt het ontwerp van de 4,5 fs laser beschreven. Als uitgangspunt wordt een

‘cavity-gedumpte’ Ti:saffier laser gebruikt. Vergeleken met een gewone laser worden door

het ‘cavity-dumpen’ pulsen met een hogere pulsenergie gegenereerd, wat een voordeel is bij

het verbreden van het spectrum. Deze spectrale verbreding is noodzakelijk om een korte puls

mogelijk te maken. Dit effect wordt gerealiseerd door de cavity-gedumpte puls in een fiber te

injecteren. De resulterende spectrale verbreding is het gevolg van niet-lineair optische

effecten, waarbij frequentiemenging optreedt. Om de ‘getjilpte’ puls te comprimeren tot 4,5

fs wordt deze door een compressor gestuurd, die bestaat uit twee prisma’s en een set getjilpte

spiegels.

De belangrijke karakteristiek van deze 4,5 fs puls voor onze experimenten is dat het

bijbehorend spectrum zich uitstrekt van 600 tot 1100 nanometer en dus een goede overlap

vertoont met het absorptiespectrum van het gehydrateerd elektron.

Hoofdstuk 3 beschrijft het onderzoek dat gedaan is naar de geschiktheid van de FROG-

methode voor het meten van de lichtpulsen, die uit een paar oscillaties van het elektrische

veld bestaan. Het complete formalisme om het FROG-signaal te berekenen wordt uitgelegd.

Vervolgens worden een aantal FROG-beelden berekend en via het FROG-inversie algoritme

geanalyseerd. De hiermee verkregen amplitude en fase worden vergeleken met de ingevoerde

pulskarateristieken om de betrouwbaarheid van de gebruikte techniek te bepalen. Op grond

van de gegeven analyse kan geconcludeerd worden dat het mogelijk is één-cyclus optische

pulsen volledig te karakteriseren, mits de juiste dikte en oriëntatie van het niet-lineair

optische kristal worden gebruikt.

Samenvatting

198

In hoofdstuk 4 worden de getjilpte en gecomprimeerde pulsen gekarakteriseerd, daarbij

gebruikmakend van de FROG-methode. Dit is niet alleen essentieel voor het maken van zeer

korte pulsen maar is ook noodzakelijk voor het praktisch gebruik van het fs lasersysteem.

Ook wordt in dit hoofdstuk de precieze duur van de gecomprimeerde puls bepaald.

Hoofdstuk 5 geeft een overzicht van de gebruikte niet-lineair optische technieken, zoals

twee- en drie-puls foton echo en ‘pump–probe’. Verschillende optische experimenten aan het

gehydrateerd elektron komen aan bod in de hoofdstukken 6 en 7. Foton echo metingen laten

bv. zien dat de faserelaxatietijd van het elektron korter is dan 2 fs, terwijl uit pump–probe

metingen blijkt dat de de aangeslagen toestand van het elektron vervalt op een tijdschaal van

50 fs. Hierbij komt het elektron in een hete grondtoestand terecht, die op een ps tijdschaal

afkoelt naar de oorspronkelijke grondtoestand. Door experimenten in gewoon en

gedeutereerd water te doen blijkt het mogelijk om de aard van de eerste stap in het

relaxatieproces te bepalen. Deze blijkt grotendeels bepaald te worden door libraties van

watermoleculen in de eerste waterschil. Na 100 fs nemen translatie bewegingen de

energieoverdracht over. In hoofdstuk 7 wordt op basis van deze metingen een

tweedimensionaal model ontwikkeld dat de overgang van een libratie- naar translatie-

gedreven respons van het gehydrateerd elektron verklaart.

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