Dynamic Airy imaging through high-efficiencybroadband phase microelements by femtosecondlaser direct writingZE CAI,1 XINBO QI,1 DENG PAN,1 SHENGYUN JI,1 JINCHENG NI,1 ZHAOXIN LAO,1 CHEN XIN,1 JIAWEN LI,1,2

YANLEI HU,1 DONG WU,1,3 AND JIARU CHU1

1CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Precision Machinery and Precision Instrumentation,University of Science and Technology of China, Hefei 230027, China2e-mail: jwl@ustc.edu.cn3e-mail: dongwu@ustc.edu.cn

Received 6 January 2020; revised 20 March 2020; accepted 30 March 2020; posted 30 March 2020 (Doc. ID 387495); published 14 May 2020

Manipulating Airy beams to symmetric Airy beams (SABs) with abruptly autofocusing and self-acceleratingproperties has attracted much attention. With such a particular propagation dynamic, SABs may provide greatpotential in dynamic signal imaging. On the other hand, the generation of SABs by spatial light modulatorssuffers from the limitations of phase gradient accuracy, low optical efficiency (<40%), and a bulky footprint.Therefore, exploring imaging applications and optimal generation methods of these Airy-type beams deservesfurther research. Here, based on the coordinate transformation of SAB, an asymmetric Airy beam (AAB) is real-ized. Symmetric/asymmetric cubic phase microplates (SCPPs/ACPPs) are designed and fabricated for generatingSAB/AAB. The SCPP/ACPP demonstrates superior performance: compact construction (60 μm × 60 μm ×1.1 μm), continuous variation of phase, high efficiency (∼81% at 532 nm), and broadband operation from405 to 780 nm. Dynamic imaging under monochromatic and polychromatic lights is realized by the SAB/AAB, indicating various results at different propagation distances with a certain initial signal. Further investi-gation reveals that the SCPP on a soft substrate maintains its physical dimensions and optical propertiesunchanged during stretching. Our work enables wide potential applications in integrated optics, beam manipu-lation, and imaging. © 2020 Chinese Laser Press

https://doi.org/10.1364/PRJ.387495

1. INTRODUCTION

Since the experimental demonstration of Airy beams in 2007[1,2], self-accelerating beams have attracted great attention dueto wide applications in the generation of curved plasma chan-nels in air [3], accelerating electrons [4], optical manipulationof particles [5], laser microfabrication [6,7], optical signal trans-mission [8], and microscopy [9,10]. By extending the Airyfunction to the cylindrical coordinate, circular Airy beams(CABs) were obtained [11,12]. The CAB is a circularly sym-metric beam with an abruptly autofocusing property, which hastriggered a great deal of research on optical tweezers [13], lightbullets [14,15], and femtosecond laser polymerization [16].Inspired by the CAB, the symmetric Airy beam (SAB) arose[17–20], which first auto-focalizes with a single central lobeand then accelerates into four Airy lobes in quadrate symmetry.The SAB is one of the caustics, which are the loci where raysare locally focused [21,22], so it can be inferred that the objectcan be imaged along the four caustic curves. The opticalimaging is widely employed in both scientific research and

industrial communities. Up to now, great efforts have beendevoted to improving imaging quality, such as breaking thediffraction limit [23,24] and correcting geometrical and chro-matic aberrations [25,26]. Nevertheless, research on dynamicimaging is rare, and it has potential in optical encryption.Furthermore, additional transformation of SABs to other Airy-type beams may encourage more promising applications.

For the generation of SABs, the most common methodis based on symmetric cubic phase modulation by spatiallight modulators (SLMs). However, the pixels of commercialSLMs are in micrometer size, which limits the accuracy ofthe phase gradient. The efficiency of SLMs is still a challenge(usually < 50%) [27]. Furthermore, the bulky footprint ofSLMs imposes restrictions on the application in integrated op-tics. Attempts like photorefractive crystals [28], liquid crystals[29,30], digital micromirror devices (DMD) [20], nanoanten-nas [31], and metasurfaces [32,33] can be adopted to avoidsome of these problems. However, the above choices cannotsimultaneously satisfy the conditions of continuous phase

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variation, high efficiency, and compact construction. Recently,femtosecond laser micro/nanofabrication [34–38] has beendemonstrated to be a promising technique to process three-dimensional (3D) optical microdevices like photonic crystals[39], waveguides [40], and microlens groups integrated onthe fiber facet [36,41] in high precision and flexibility, whichcan be a great choice for precise fabrication of micro-nano op-tical elements with high performance and flexible integration.

In this paper, we experimentally realize miniature symmetricand asymmetric cubic phase plates (SCPPs and ACPPs) by fem-tosecond laser nanofabrication based on spectral coordinatetransformation of cubic phase, and we generate symmetricand asymmetric Airy beams (SABs and AABs) for dynamic im-aging. The fabricated SCPP and ACPP demonstrate the bene-fits of compact construction, continuous variation of phase,high efficiency (∼81% at 532 nm), and broadband light modu-lation (405–708 nm), which can be helpful in integrated optics.All these advantages enable the generation and application ofSABs and AABs to be more efficient, ensuring the achievementof dynamic imaging, which means a certain input signal butvarious imaging results during propagation. On the other hand,broadband operation makes the imaging enigmatic. Only byextracting the element of exact wavelength at the certain propa-gation distance can we get the precise signal, which mayprovide a potential method for message extraction.Moreover, flexible fabrication of femtosecond laser capacitatesthe SCPP on a soft substrate, enabling the SCPP to performsmartly in multifarious conditions. The SCPP fabricated ona polydimethylsiloxane (PDMS) substrate exhibits highmechanical flexibility and robustness. The SCPP can maintainits shape and size, even optical properties unchanged duringsevere stretching.

2. THEORETICAL CALCULATIONS

Solving the normalized paraxial equation of diffraction underthe condition of introducing an exponential aperture functionresults in the envelope of a finite Airy beam [1,2]:

ϕ�ξ, s� � Ai�s − �ξ∕2�2 � iaξ� exp�as − �aξ2∕2� − i�ξ3∕12�� i�a2ξ∕2� � i�sξ∕2��: (1)

Here Ai�s − �ξ∕2�2 � iaξ� represents the Airy function, a is thedecaying factor to truncate the tail of the infinite Airy beam,and a ≪ 1. s � x∕x0 represents a dimensionless transverse co-ordinate, where x0 is an arbitrary transverse scale. ξ � z∕kx20 isa normalized propagation distance, k � 2πn∕λ is the wavenumber, n is the refractive index of the medium, and λ isthe free-space wavelength. The Fourier spectrum of the Airybeam is given by

Φ0�k� � exp�−ak2� exp�i�k3 − 3a2k − ia3�∕3�: (2)

This implies that the finite Airy beam can be generated throughthe Fourier transform of a Gaussian beam with a cubic phase.The cubic phase spectrum [Fig. 1(a1)] for a two-dimensional(2D) Airy beam is in the form of φAB�kx , ky� � k3x � k3y ,where kx and ky are Fourier spectrum coordinates, and the si-mulated intensity distribution and propagation dynamic of thecorresponding Airy beam are shown in Figs. 1(a2) and 1(a3),respectively.

By converting the spectral coordinates with the absolute valuein the spectral cubic phase term expressed as φSAB�kx , ky� �jkx j3 � jkyj3 [Fig. 1(b1), kx , ky: −12π to 12π], the SAB canbe generated, and the complex amplitude of the SAB can beexpressed as [19]

u�s� � exp�−as�fAi�−s� � i�Gi�−s��g∕2� exp�as�fAi�s� � i�Gi�s��g∕2, (3)

where Gi�s� � 1π

R∞0 sin�13 t3 � st�dt. To carry out the simula-

tion with the phase mask, the diffraction theory is adopted:

U �x, y, z� � exp�ikz�iλz

ZZ �∞

−∞E int�kx , ky� exp�iφSAB�kx , ky��

× exp�ik2z

��x − kx�2 � �y − ky�2��dkxdky : (4)

E in � A exp�−k2∕ω2� is the incident Gaussian light,t�kx , ky� � exp�−ik�k2x � k2y �∕2f � is the phase transfer functionof the lens, f is the focal length, and the dimensionless coordi-nate is normalized by sx � x∕x0, sy � y∕y0,

ξξf� z∕ξf , and

Fig. 1. Symmetric and asymmetric Airy beams and their dynamicimaging of a letter “T”. (a1) Cubic phase mask. (a2) Intensity dis-tribution of the corresponding 2D Airy beam at the Fourierplane. (a3) Propagation dynamic of the Airy beam versus �sx , ξ�.(b1) Symmetric cubic phase mask. (b2) The side-view profiles of thecorresponding SAB. (b3) Intensity distribution of the SAB at the dis-tances ξ∕ξf � 0, 1, 2, 3 from the initial plane. (c1) Asymmetric cubicphase mask. (c2), (c3) The corresponding AAB (c2) during propaga-tion and (c3) at the certain planes. (d) Schematic illustration of dy-namic imaging of a letter “T” by the SAB at different distances.Insets: simulation. In the gray scale pattern, black corresponds to 0and white to 2π radians. In the intensity scale, [0,1] for each patterncorresponds to [0, Imax].

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ξf � f ∕kx20. As shown in Figs. 1(b2) and 1(b3), the SAB firstauto-focalizes with a single central lobe, and then it splits andaccelerates into four Airy lobes in quadrate symmetry. Sinceall four Airy lobes present equal acceleration, the propagationdynamic of the SAB versus �sx , ξ� is consistent with the one ver-sus �sy, ξ� as illustrated in Fig. 1(b2). The main lobe of SABdemonstrates absolute symmetry and shows the same shapein both lateral and vertical directions at arbitrary propagationplanes [Fig. 1(b3)].

Further coordinate transformation can occur by introducinga ratio factor in the kx or ky coordinate that enables the spec-tral phase turn into the form φAAB�kx , ky� � jkxj3 � tjkyj3,where t is a transformation factor, and t > 0, t ≠ 1, resultingin the SAB converting into another Airy-type beam. It canalso be realized by giving different values of transverse scalesx0 and y0 [42], but the introduction of the transformation fac-tor normalizes the relationship between two transverse scales.As shown in Fig. 1(c1), with the transformation factor t � 1.5,the phase mask exhibits a denser phase range in ky coordinate(kx : −12π to 12π, ky: −18π to 18π) compared to the sym-metric cubic phase mask [Fig. 1(b1)], and thus we call it anasymmetric cubic phase mask. Thanks to the transformationfactor in the ky coordinate of the spectral phase, the correspond-ing beam shows different autofocusing and accelerating profiles[Fig. 1(c2)]. The beam auto-focalizes longer and acceleratesslower in the sy−ξ plane than that in the sx−ξ plane,and the main lobe is first elongated in the vertical directionduring autofocusing and then elongated in the lateral directionwhen splitting and accelerating [Fig. 1(c3)]. Since the beam isnot consistent in both directions, we call it an asymmetric Airybeam (AAB). Although SABs and AABs both displaysimilar propagation features like autofocusing and accelerating,they show different accelerating frames (SAB: quadrate, AAB:rectangular) and disparate main lobe shapes (SAB: circle,AAB: ellipse).

These Airy-type caustics (SAB and AAB) are a series of raysforming local foci, and each ray is tangent to the main lobeat one point. Thanks to the prominent propagation featuresof SABs/AABs, they have potential in dynamic imaging.Corresponding to the propagation dynamic of the SAB, we in-fer that a letter “T” imaged by the SAB first auto-focalizes andbecomes smaller, and then it splits into four of the same “T”sand accelerates in quadrate symmetry, forming diverse imagingresults at different planes. The schematic illustration of dy-namic imaging of a letter “T” by an SAB is shown in Fig. 1(d).To verify this inference, the simulation of “T” imaged by anSAB/AAB was carried out. As displayed in the inset ofFig. 1(d), the result by an SAB presents a letter “T” duringthe autofocusing term, and then it turns into a Chinese word“开” at ξ∕ξf � 1.97, which means “open” in English.Therefore, dynamic imaging by an SAB can convey differentinformation during propagation. As for dynamic imaging car-ried by an AAB, the initial result of a letter “T” is the same asthe one by an SAB, except for a little elongation in the verticaldirection, which corresponds to the main lobe of AAB.However, dynamic imaging during propagation is quite dispa-rate owing to different accelerating frames, such as the Greeksymbol “π” formed at ξ∕ξf � 1.37.

3. EXPERIMENTAL SECTION

A. Sample Preparation and FabricationThe femtosecond laser source (Coherent Chameleon) hasan 80 MHz repetition rate, 800 nm center wavelength, and75 fs pulse width. The photoresist [SZ2080 mixed with1 wt.% 4,4-bis(diethylamino)benzophenone (BIS) as photoini-tiator, provided by IESL-FORTH, Greece] [43] dropped on aglass was prebaked at 100°C for half an hour to evaporate thesolvent in the SZ2080. The SCPP/ACPP was fabricated byfemtosecond laser layer-by-layer writing, and the fabricationprocess was controlled by an XY scanning unit while the stepbetween two layers was realized by a nano-positioning stage[Fig. 2(a)]. The optimal fabrication parameter was 7.8 mWlaser power at 1 ms exposure time. After polymerization, thesample was developed in 1-propanol for 30 min until all un-polymerized photoresist was washed off. The fabrication pro-cess on a PDMS substrate is the same as above, but we shouldpretreat the PDMS substrate before fabrication. First, PDMSwas mixed with the crosslinker (Sylgard 184, Dow Corning) ata mass ratio of 10:1. After that, the mixture was spin-coated onglass at 300 r/min for 200 s, followed by curing at 80°Cfor 6 h, which resulted in 150 μm thickness of PDMS.Then the PDMS was peeled off from the glass, cut into slices,and covered on a glass slide. Afterwards, it was treated by oxy-gen plasma at 75 W for 50 s (Mingheng PDC-MG). Later, thetreated PDMS was soaked in a solution mixed with poly(vinylalcohol) (PVA), oxalic acid, and water in a 1:10:500 mass ratioto increase the adhesion between the photoresist and PDMSfor 12 h. Finally, it was taken out from the solution and blow-dried. After photoresist drop, prebake, fabrication, develop-ment, and peeling off the PDMS from the glass slide, theSCPP/ACPP was fabricated on a PDMS substrate.

Fig. 2. Femtosecond laser two-photon polymerization of symmetricand asymmetric cubic phase microplates. (a) Schematic illustrationof the fabrication of SCPP and ACPP. (b) Image of the designedSCPP according to the phase mask by converting the phase to thecorresponding height and discretization with a step of 100 nm.(c) SEM and AFM micrographs (half of the image) of the fabricatedSCPP. (d) Height profiles of the desired, designed, and fabricatedSCPP along the dashed lines in (b) and (c). (e), (f ) Top-view SEMand three-dimensional AFM images of the fabricated ACPP.

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B. Characterizations of SCPP/ACPPThe scanning electron microscope (SEM) images were takenwith a secondary electron SEM (ZEISS EVO18) operated atan accelerating voltage of 10 keV with a working distance of8 mm after depositing ∼10 nm gold. The topography of theSCPP/ACPP was scanned at a rate of 0.5 Hz in air in tappingmode with a commercial atomic force microscope (AFM,MFP3D-origin OXFORD) using a Tap300Al-G tapping modetip. The optical images were taken by a commercial opticalmicroscope (Leica DMi8-M, Germany).

C. Generation and Observation of SAB/AABA Gaussian beam from a 532 nm laser propagated through a 40×objective lens and the SCPP/ACPP, and after magnification byanother 40× objective lens, the generated SAB/AAB was capturedby a CCD camera at and after the focal plane of the first objectivelens [Fig. 3(a)]. A series of images was taken at various propaga-tion planes. The position and full width at half-maximum(FWHM) of the main lobe in each image at a certain propagationdistance were measured and marked in Figs. 3(b)–3(d). To gen-erate the chromatic SAB/AAB, we replaced the laser source with ahalogen lamp, and when signal masks were placed after the lightsource, dynamic imaging of signals was realized.

D. Mechanical Stretch of SCPP on a PDMSSubstrateFour guide screws coupled with clamping pieces controlled bylinear motors constitute our homemade stretch device. ThePDMSs were fixed on two clamping pieces in the x directionfor a single-direction stretch test. The stretched length wasprecisely given by linear motors driven by their controllers.The homemade stretch device was installed on an inverted

microscope (Leica DMi8-M, Germany) to observe the SCPPduring the stretch process, and it was introduced into theSAB generation setup to verify the consistency of intensity dis-tribution before, during, and after stretch.

4. RESULTS AND DISCUSSION

A. Design and Fabrication of SCPP/ACPPCompared to phase modulation by an SLM, generation ofSABs and AABs through the specially designed phase deviceswith continuous phase variation shows space saving as well ashigher optical efficiency. Therefore, we decided to transformthe phase profile to the geometric height, which can be fabri-cated by femtosecond laser direct writing precisely [Fig. 2(a)].As shown in Fig. 2(b), a symmetric cubic phase mask is firstconverted through the thickness-accumulated phase equationh�x, y� � λφ�x, y�∕2π�nm − 1� and then discretized. λ is thefree space wavelength of incident light and is set to550 nm, and nm represents the refractive index of the photo-resist (∼1.5). Therefore, the corresponding height range isabout 0–1100 nm. The beam quality can be evaluated bythe mean square error (MSE) between the continuous phaseand discrete phase with different step N and normalized byusing the value for N � 1 written as MSE�N � �P

i,j�I continuous�i, j�−IN �i, j��2P

i,j�I continuous�i, j�−I 1�i, j��2

[44]. For the case N � 11, the MSE

tends to be 0.002, demonstrating a good quality.Furthermore, the step tends to disappear after fabrication inthe course of unpolymerized resin rinsing (self-smoothing ef-fect) [45], forming a continuous phase. However, the MSEincreases with the decrease of N [MSE�5� � 0.0216]; mean-while, a greater step can cause invalidation of the self-smooth-ing effect. Therefore, the step of discretization is set to 100 nm,and the number of layers is N � 11 [Fig. 2(b)].

The fabricated SCPP with a size of 60 μm × 60 μm ×1.1 μm presents high quality, which is verified by the SEM mi-crograph and AFM image [Fig. 2(c)]. Topography of the fab-ricated SCPP extracted from the AFM result along the dashedline in Fig. 2(c) reveals the great phase continuity and agreeswell with the desired profile [Fig. 2(d)]. Some tiny fluctuationscaused by the fabrication and measurement errors are unavoid-able. On the other hand, the high aspect ratio of local areasmakes it difficult to dip the AFM probe into the bottom, whichresults in a mismatch between the extracted height and thedesired one near the corner of the SCPP. An ACPP can be de-signed and fabricated similarly to the SCPP, and the top-viewSEM and three-dimensional AFM images are depicted inFigs. 2(e) and 2(f ), respectively, demonstrating great fabricationquality.

B. Generation of SAB/AABThe SAB/AAB can be realized through the phase modulationof SCPP/ACPP [Fig. 3(a)]. To characterize the propagationfeature of the generated SAB/AAB, we measured a series of in-tensity patterns [Fig. 3(b1)] in the 0–200 μm range, wherez � 0 coincides with the focal plane position of the firstobjective lens. The experimentally generated SAB first auto-focalizes during 60 μm propagation and then splits to four mainlobes that accelerate in quadrate symmetry. As for the AAB

Fig. 3. Generation of symmetric and asymmetric Airy beams.(a) Schematic of the experimental setup for generating and observingSAB and AAB. (b1)–(b2) Intensity distribution of the (b1) SAB and(b2) AAB generated by the fabricated phase plates at the propagationplanes z � 0, 60, 120, and 180 μm. (c) Transverse acceleration of themain lobes of the SAB and AAB as a function of beam propagationdistance. (d) Lateral and vertical FWHMs of the main lobes of theSAB and AAB varying with propagation distance. Error bars in (c),(d) indicate standard deviation (s.d.). The ellipses with arrows indicatethe corresponding coordinate axis according to the color.

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[Fig. 3(b2)], it also has an autofocusing property during thefirst 60 μm propagation but is elongated in the vertical direc-tion. After splitting, it exhibits inconsistent acceleration thatleads to a discrepancy in the lateral and vertical directions.These experimental results [Figs. 3(b1) and 3(b2)] are ingood agreement with the numerical simulations [Figs. 1(b3)and 1(c3)].

We also quantitatively analyzed the distinction of the propa-gation features between the SAB and AAB. Figure 3(c) depictsthe transverse acceleration of the SAB/AAB as a function ofbeam propagation distance. It is obvious that the SAB showsidentical propagation dynamics in both the x − z and y − zplanes, which means the same autofocusing distance and equalacceleration. However, the measured result reveals that the AABowns a longer autofocusing distance but accelerates muchslower in the y − z plane than in the x − z plane. The denserphase range in the ky coordinate caused by the transformationfactor t � 1.5 can account for these results. Moreover, ourexperimental measurements [pink (x − z) and purple (y − z)points toward the right (SAB) and left (AAB)] are consistentwith the theoretical trajectories [solid (x − z) and dashed (y − z)lines with black color towards the right (SAB) and red colortowards the left (AAB)] extracted from the simulated results[Figs. 1(b2) and 1(c2)].

Although the SAB and AAB display similar propagation fea-tures, the shapes of their main lobes are clearly distinct. Afterobtaining the normalized intensity along the lateral and verticaldirections of SAB/AAB at z � 0, 60, 120, and 180 μm, wemeasured the FWHMs of their main lobes, and the variationtendency is recorded in Fig. 3(d). The FWHMof the main lobeof the SAB drops rapidly during the first 60 μm (autofocusingduration); after splitting, it still decreases but much less (60–120 μm), and finally it increases little. The results in both thelateral and vertical directions are consistent. For AAB, theFWHM of the main lobe is quite different in the two direc-tions. In the lateral direction, the FWHM first decreases slowlyduring the autofocusing phase (0–60 μm), then it increasesduring the next 60 μm acceleration, and finally it becomes al-most constant. However, the vertical result reveals that theFWHM drops fast during 120 μm propagation, and thenthe decrease slows down quickly and tends to fade away.These can be explained by the different autofocusing distancesand shapes of the main lobes in the lateral and vertical direc-tions, which have been discussed above. All these experimentalresults are in accordance with the simulated distributions[Figs. 1(b3) and 1(c3)].

C. Broadband OperationDefined as the rate of laser power of the transformed SAB/AABdivided by the laser power before the sample, the optical effi-ciency of the SCPP was measured. Figure 4(a) depicts therelationship between the optical efficiency of SCPP and thewavelength of incident light. Notice that the efficiency differsslightly at the wavelengths of 405 nm (∼59%), 590 nm(∼66%), 633 nm (∼65%), and 780 nm (∼60%) but exhibitsa large gap between 488 nm (∼29%) and 532 nm (∼81%).As we described in the former section, the height profile cor-responding to the phase variation is related to the wavelength ofincident light. Therefore, the incident light with a wavelength

away from 550 nm suffers from incomplete phase modulationand low efficiency. Nevertheless, an optical efficiency around60% is still considerable. It is satisfactory that high and stableoptical efficiency (81%� 4%) can be realized at the wave-length of 532 nm due to the integrity and continuity of thephase. The transmission loss from the glass substrate and sometiny flaws on the SCPP result in the loss of laser power aftertransmission, which is unavoidable. Another special point isthat the SCPP shows a poor 29% efficiency at 488 nm causedby the absorption of the photoresist in the spectral range of450–500 nm [43].

Intensity distributions of SAB/AAB over a visible light band(405–780 nm) are recorded as well. As displayed in Fig. 4(a),the intensity distributions of the SAB/AAB at the same propa-gation distance (z � 180 μm) exhibit different frame sizes withsimilar features. We surmised that the acceleration is relevantto the wavelength of incident light, which has been confirmedin Airy beams. To verify this assumption, we measured thelength between two main lobes of the SAB and AAB in the

Fig. 4. Broadband generation of symmetric and asymmetric Airybeams. (a) Experimentally measured intensity profiles and opticalefficiency of the generated SAB and AAB over a broadband illumina-tion from 405 to 780 nm. All the intensity profiles are captured atz � 180 μm. Error bars in efficiency represent s.d. (b) Distance be-tween main lobes in lateral and vertical directions of the generated SAB(upper) and AAB (lower) as a function of beam propagation distance at488, 532, and 633 nm. (c1) and (d1) Simulated (top row) and exper-imental (bottom row) results of the chromatic (c1) SAB and (d1) AABat z � 0, 60, 120, 180 μm. (c2) Simulated (left) and experimental(right) results of the chromatic SAB in the x − z plane; (d2) experi-mental profiles of the chromatic AAB in the x − z plane (left) andin the y − z plane (right).

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lateral and vertical directions at different propagation planes atthe wavelengths of 488, 532, and 633 nm. It can be seen inFig. 4(b) that the generated SAB at 488 nm shows the shortestautofocusing distance (d � 0 means autofocus) and acceleratesthe fastest, while the SAB at 633 nm auto-focalizes the longestdistance and accelerates much slower. As for AAB, though thedistances in the lateral and vertical directions are unequal(dx > d y), the results in both directions are consistent, whichis coincident with the conclusion of SAB. The autofocusingdistance shortens and the acceleration increases with the de-crease of wavelength. All the points in Fig. 4(b) represent themeasured results, and solid and dashed lines are fitted curves.

After replacing the monochromatic light with a polychro-matic light (halogen lamp), the generated SAB and AAB exhibitchromatic characteristics. Here we name them chromatic SABand chromatic AAB. Figure 4(c1) depicts the simulated (toprow) and experimental (bottom row) results at z � 0, 60, 120,and 180 μm. Chromatic SAB is the same as the SAB but color.Though the multicolor presentation during propagation hasto be avoided in most imaging applications, it may be usefulin multicolor imaging display. Chromatic SAB is a beam com-prising a series of SABs with various wavelengths. Thanks tothe disparate autofocusing distances and acceleration of SABsat different wavelengths, contemporaneous SABs from a poly-chromatic light shift differently during propagation, resultingin the formation of chromatic SAB. As shown in Fig. 4(c1),with the increase of propagation distance, the multicolor be-comes more and more obvious, which can be obtained fromthe side-view result as well [Fig. 4(c2)]. All the experimentaldistributions agree well with the simulated results. Similar tochromatic SAB, the chromatic AAB also displays multicolorduring propagation with distinct acceleration in the lateraland vertical directions as shown in Figs. 4(d1) and 4(d2).

D. Dynamic Airy ImagingBecause the main lobe of the SAB/AAB is formed by a seriesof rays locally focused during propagation, every point of themain lobe of the SAB/AAB is a focus, which makes it suitablefor imaging. Dynamic imaging was experimentally realized bythe SAB/AAB when we imposed a signal mask after the lightsource [Fig. 5(a)]. As shown in Fig. 5(b1), an English letter “T”imaged by the SAB at different propagation distances displaysdiverse results. It becomes smaller during the first 60 μm propa-gation, which is caused by the autofocusing effect, and then itsplits into four of the same “T”s that accelerate in quadratesymmetry. These four “T”s partly overlap with each other[Fig. 5(b1), z � 118 μm] in the initial period of accelerationbut finally completely separate [Fig. 5(b1), z � 180 μm]. Afterelaborately observing the process of overlap and separation, wefound that four overlapping “T”s form a Chinese character“开” which means “open” in English at the distance of118 μm. It is significant that a special signal can be transformedfrom one language to another language, expressing entirely dif-ferent meaning by the SAB at a certain propagation distance.Similarly, the imaging of an English letter “T” by the AABalso auto-focalizes from 0 to 60 μm, zooming out the sizeof letter, then bursts into four “T”s, and accelerates in the rec-tangular frame [Fig. 5(b2)]. Therefore, the overlapping result isfairly different from the one by the SAB. Due to the slower

acceleration in the vertical direction, the “T”s in the verticaldirection completely overlap together, while a part of them sep-arates in the lateral direction, which constitutes a Greek symbol“π” at the unique distance of 82 μm [Fig. 5(b2), z � 82 μm].All the experimental results [Figs. 5(b1) and 5(b2)] show agood agreement with the simulations [Fig. 1(d), inset]. Notethat if we want to pass a “π” signal to the receiver, we can senda “T” message to the receiver, who can obtain the true infor-mation via an AAB at a certain propagation distance, and thechoice of “开” signal can be realized by an SAB. This providesus with an opportunity for information transformationduring propagation.

As demonstrated in the former section, the SCPP and ACPPshow a broadband operation from 405 to 780 nm.We also showthe dynamic imaging at the wavelengths of 633 nm and 488 nmwith different shapes of signals. As displayed in Fig. 5(c1), a“square” shows various imaging results at different propagationplanes via the SAB at the wavelength of 633 nm. At the initialautofocusing term, the picture shows a Chinese character “�?�,”which is the same shape as the square signal [Fig. 5(c1), z � 0],meaning “mouth.” Further acceleration enables the formation ofthe Chinese character “田” [Fig. 5(c1), z � 146 μm], meaning

Fig. 5. Dynamic imaging by symmetric and asymmetric Airybeams. (a) Schematic illustration for the characterization of dynamicAiry imaging. (b1), (b2) Imaging results of letter “T” at differentpropagation planes via the (b1) SAB and (b2) AAB at 532 nm.Different imaging results are exhibited at different propagation planesby the SAB and AAB. (c1) A square shows various imaging results atdifferent propagation planes via the SAB at 633 nm, the initial shape“square” (Chinese character “�?�”) at z � 0, a Chinese character “田”at z � 146 μm, and an ancient Chinese character “㗊” atz � 177 μm. (c2) A circle shows various imaging results at differentpropagation planes via the AAB at 488 nm, a number “0” at z � 0, ahorizontal number “8” at z � 67 μm, and a double-digit number“88” at z � 122 μm. (d) Imaging results of the letter “T” at differentpropagation planes via the ACPP under white light illumination.Different information can be extracted at a certain propagation planez � 90 μm with different wavelength elements, a Chinese word “开”in the blue ingredient (488 nm), and a Greek symbol “π” in the redingredient (633 nm).

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“cropland,” and the ancient Chinese character “㗊” [Fig. 5(c1),z � 177 μm], meaning “noisy.” On the other hand, a “circle”can realize various imaging results via the AAB during propaga-tion at the wavelength of 488 nm [Fig. 5(c2)]. Due to the ellip-tical main lobe of the AAB, the imaging result of a “circle” showsan ellipse like a number “0” at z � 0. After propagating 67 μm,it turns into a horizontal number “8” and then transforms into adouble-digit number “88” at z � 122 μm. Here we demon-strate some special symbols whose imaging results are commonsymbols or characters. Actually, any other symbols can be usedfor the application, because all symbols can be defined artificiallyas valuable information at some propagation planes, althoughthey look meaningless.

The dynamic variation process of the signal imaging underwhite light illumination is similar to that under a monochro-matic laser illumination. The difference is that the images donot show multicolor during the autofocus process but exhibitchromatic multiple letters during acceleration [the left columnin Fig. 5(d)] because of different accelerations according to vari-ous wavelength elements. It provides two choices that we canutilize the autofocusing term without chromatism in image ob-servation and take advantage of accelerating phase presentingrich color in a multicolor display. Furthermore, a mixture ofingredients with various colors blurs the picture and makesit hard to distinguish useful information at some propagationplanes, which should be avoided in multicolor display but maybe valuable in confidential information transmission and recep-tion. The second image of the left column in Fig. 5(d) displaysthe blurry image of a letter “T” at z � 90 μm via the AAB, inwhich we are unable to read some useful information. However,after extracting blue (488 nm), green (532 nm), and red(633 nm) elements in the picture, different informationemerges with a Chinese word “开” in the blue ingredientand a Greek symbol “π” in the red ingredient. Hence, withthe aid of polychromatic light, the valid result can only be ob-tained in a certain ingredient of wavelength at a unique propa-gation distance with an exact SCPP/ACPP.

E. Stretching Flexibility and RobustnessSince the SCPP/ACPP has demonstrated so much superior per-formance as mentioned above, given its strong mechanical flex-ibility and robustness, SCPP/ACPP has promising potential inreforming integrated optical systems by introducing miniatur-ized and fully flexible optical elements. To realize this aim, theSCPP is fabricated on a PDMS substrate, and then mechanicalstretch is imposed on the PDMS substrate [Fig. 6(a) inset]. Thestretch coefficient of the PDMS substrate with an SCPP de-fined as ε � Δd∕d 0, where Δd is the increment during thestretch and d 0 represents the original length between twoclamps, can be controlled flexibly. As shown in Fig. 6(a), giventhe stretch coefficient in the x direction varying in a wide rangefrom 0% to 85%, the SCPP can still maintain its shape.Meanwhile, the efficiency remains still as well, showing a minorfluctuation from 67% to 74% at 532 nm. Moreover, the effi-ciency presents a good consistency during a 100 cyclic stretchwith a stretch coefficient of 50% [Fig. 6(b)].

We also verified the consistency of the generated SAB dur-ing the stretch process. As illustrated in Figs. 6(c1)–6(c3), theSCPP (first row) almost remains the same, and the correspond-

ing SAB shows no difference both at the side view (second row)and at a certain propagation distance (third row) before[Fig. 6(c1)], during [Fig. 6(c2)], and after stretch [Fig. 6(c3)],as does the imaging result of the letter “T” (last row) atz � 118 μm. Further stretch with a higher stretch coefficient(>90%) makes the structure break along the vein. Surprisingly,the cracked parts join together perfectly, and the SCPP isrestored after releasing the stretch. The crack is too tiny to bedistinguished under an optical microscope, which can be veri-fied by the SEM image [Fig. 6(d)]. The unique architecturalfeature with dense rings, which enhances the tensile strength,may account for the robustness of the SCPP. Since the SCPPdisplays such a nice restoration, the corresponding intensityprofiles and imaging of the letter “T” are not influenced[Fig. 6(d)]. The robustness of the SCPP integrated on the softsubstrate makes it flexible to fit different scenarios.

5. CONCLUSION

Symmetrization of the spectral cubic phase of an Airy beamenriches the accelerating Airy-type beams, such as autofocusingCABs, which have circular symmetry with an Airy radial pro-file, autofocusing SAB with a quadrate accelerating frame. Ourintroduction of a ratio factor in a symmetric cubic phase results

Fig. 6. Flexibility and robustness of the symmetric cubic phase dur-ing mechanical stretch. (a) The length and width of the SCPP on aPDMS substrate almost remain unchanged during the stretch coeffi-cient in the x direction varying from 0% to 85%, and the optical ef-ficiency maintains stable as well. Error bars on Dx ,Dy , and efficiencyindicate s.d. (b) The optical efficiency of the SCPP on a PDMS sub-strate has a good consistency during 100 cyclic stretch with a stretchcoefficient of 50%. Error bars represent s.d. (c1)–(c3) Images of theSCPP on a PDMS substrate, the generated intensity profiles, and theimaging results (c1) before, (c2) during, and (c3) after stretch. TheSCPP can keep its structure and optical properties unchanged.(d) Higher stretch coefficient (>90%) makes the structure break alongthe vein. After releasing the stretch, the cracked parts can joint togetherperfectly. The SEM micrograph shows the tiny crack at the joint.Because of the nice restoration, the corresponding intensity profilesand imaging results are not influenced.

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in the autofocusing AAB with a rectangular accelerating frameand elliptical main lobes. The similar but different propertiesbetween SAB and AAB provide a choice for information extrac-tion on demand [“开” in Fig. 5(b1) and “π” in Fig. 5(b2)]. Inaddition, our effort on the coordinate transformation of spectralcubic phase provides a potential for manipulating the struc-tured light with diverse features to meet various requirementsin application. The design and fabrication of the SCPP/ACPPovercomes the shortcomings of the SLM, such as the limita-tions of phase gradient accuracy, low optical efficiency(<50%), and bulky footprint in optical setup. Further fabrica-tion on the soft substrate endows the SCPP/ACPP with thecapacities of flexibility and robustness, which may play a sig-nificant role in integrated optical systems. The dynamic imag-ing via the SAB/AAB, which presents the distinct informationfrom the input signal, broadens the application of Airy-typebeams, though the CAB and SAB have been widely used inoptical tweezers, light bullet, biomedical treatment, and femto-second laser polymerization. Broadband operation across thevisible region enhances the difficulty of identification by intro-ducing the variable of wavelength.

In summary, we demonstrate the design and fabrication ofan SCPP/ACPP with superior performance: compact construc-tion (60 μm × 60 μm × 1.1 μm), continuous variation ofphase, high optical efficiency (∼81% at 532 nm), and broad-band operation from 405 to 780 nm, through which the ex-perimental generation of an SAB/AAB with an autofocusingfeature and quadrate/rectangular self-accelerating frame is real-ized under both monochromatic light and polychromatic light.Dynamic imaging of different signals is realized via the SAB/AAB, showing different results during propagation with a cer-tain initial signal, which provides a potential for the encryptionand decryption of information during propagation.Furthermore, the SCPP fabricated on a soft substrate demon-strates mechanical robustness and flexibility, which can main-tain its size, optical efficiency, generated intensity distribution,and even imaging results unchanged before, during, and afterstretching. Our research on high-performance SCPPs/ACPPsconstructs a promising platform for the creation of compactoptical elements with high efficiency and broadband operationfor integrated optical systems, and the findings on dynamic im-aging provide potential applications in imaging science.

Funding. National Natural Science Foundation of China(51675503, 51805508, 51805509, 51875544, 61475149,61805230); Fundamental Research Funds for the CentralUniversities (WK2090090012, WK2090090021,WK2480000002, WK6030000103); Youth InnovationPromotion Association of the Chinese Academy of Sciences(2017495); National Key Research and Development Programof China (2018YFB1105400).

Disclosures. The authors declare no conflicts of interest.

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