1372 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 33, NO. 4, AUGUST 2005

Design of High-Power Millimeter-WaveTM01–TE11Mode Converters by the

Differential Evolution AlgorithmShiwen Yang, Senior Member, IEEE, and Anyong Qing, Member, IEEE

Abstract—In this paper, a novel approach based on the differ-ential evolution algorithm is proposed for the optimal design ofhigh-power millimeter-wave TM01–TE11 circular waveguidemode converters. The axes of the waveguide mode convertersbent in one plane are represented by cubic B-splines, and thecoordinates of the B-spline nodes are parameters to be optimized.Numerical optimization results show that the proposed approachcan be used to design highly efficient mode converters with veryshort converter lengths. Furthermore, the input and output of theoptimized mode converters can be in the same direction.

Index Terms—Differential evolution, high-power microwave,mode converter.

I. INTRODUCTION

H IGH-POWER microwave (HPM) circular waveguidemode converters have found widespread applications

over the years, such as the mode conversion between a HPMsource and a HPM transmission waveguide, or the modeconversion between a HPM source and a HPM antenna. Onetype of HPM waveguide mode converters, the –circular waveguide mode converters, are of importance to theapplications of HPM sources, such as virtual cathode oscilla-tors (VCOs) or relativistic backward wave oscillators (BWOs).These types of HPM sources usually launch one or mixtures of

modes in an overmoded circular waveguide, which areapparently not suitable for their succeeding applications. On theother hand, the mode in a circular waveguide is usuallypreferred as the input mode of a HPM antenna. Consequently,high performance – mode converters are usuallynecessary. The – conversion is much easier tobe performed, by using the conventional sinusoidal converterwhose wall radius varies in multiple periods of beat wavelength[1], [2]. However, the – conversion is not as simple,due to their long beat wavelength in an overmoded circularwaveguide operating at millimeter-waves. Conventional ser-pentine converters can be used to perform direct –conversion, but the converter length is generally quite long[3]–[8]. The first author has proposed an optimization approach

Manuscript received November 26, 2004; revised March 10, 2005. This workwas supported in part by the Innovative Research Team Program of the Univer-sity of Electronic Science and Technology of China.

S. Yang is with the Department of Microwave Engineering, School of Elec-tronic Engineering, University of Electronic Science and Technology of China,Chengdu 610054, China (e-mail: swnyang@uestc.edu.cn).

A. Qing is with the Temasek Laboratories, National University of Singapore,Singapore 117508.

Digital Object Identifier 10.1109/TPS.2005.852426

for the design of relatively short mode converters, by selectingthe converters’ axes to be in the shape of one section of somespecially selected geometry such as cubic parabola, Gaussiancurve, or combinations of two arcs [9]. Nevertheless, the tan-gential directions at the input and output of those waveguideaxes are generally not in the same direction, which meansthat the preceding and succeeding waveguides are also notin the same direction and are apparently not convenient forsome applications. In this case, additional waveguide bendsare needed to change the output directions, which may causethe deterioration of the output mode. Moreover, the converterlength in [9] still seems to be lengthy for millimeter-waveapplications.

In this paper, a novel approach is proposed to overcomethe aforementioned drawbacks in the design of very highefficient – circular waveguide mode converters atmillimeter-wave band. The approach is based on the combina-tions of cubic B-spline technique and the differential evolution(DE) algorithm [10]–[14]. The converter’s axis is representedby a cubic B-spline, and the node coordinates of the B-splineare parameters to be optimized by DE algorithm. The DE algo-rithm has recently been gaining in acceptance and has alreadybeen applied in many areas, such as magnetic bearing design[10], electromagnetic inverse scattering [11], [12], and antennaarray pattern synthesis [13], [14]. However, DE has rarely beenapplied to waveguide problems, especially for mode converterdesign. Similar to the genetic algorithm (GA) [15], DE alsobelongs to the broad class of evolutionary algorithms, but DEhas many advantages such as the strong search ability andthe fast convergence ability over GA or any other traditionaloptimization approach, especially for real valued problems[12]. Since the optimization of a mode converter design usuallybelongs to real valued problems, it is reasonable to believethat DE will also be advantageous for the optimal design ofwaveguide mode converters.

The paper is organized as follows. Section II describes thetheoretical construction of the proposed approach. The coupledmode theory is used to analyze the – bent waveguidemode converters, and the DE coupled with B-spline is used tooptimize the geometry of the converters. Section III then demon-strates the effectiveness of the proposed approach by presentingnumerical results from several optimizations and by comparingthe performance of the designed mode converters to that of pre-vious ones. Finally, Section IV presents the conclusion of thisstudy.

0093-3813/$20.00 © 2005 IEEE

YANG AND QING: DESIGN OF HIGH-POWER MILLIMETER-WAVE – MODE CONVERTERS 1373

Fig. 1. Scheme of the TM –TE bent waveguide mode converter.

II. DESIGN APPROACH

For a circular waveguide with a constant radius and theaxis bent in the x–y plane, the mode conversion from a(TE or TM) mode to mode is governed by the fol-lowing first-order coupled mode differential equations, given by[16]–[18]:

(1)

(2)

where represents the complex amplitude of the forward(backward) waves for different modes, is the com-plex propagation constant, is the attenuation constant due toohmic loss, and is the real propagation constant along the axis.In (1) and (2), or represents the coupling coefficient be-tween two corresponding modes propagating in the same direc-tion or in opposite direction, respectively. The general expres-sions for the coupling coefficients between different types ofmodes are available in [16]. Based on the coupling coefficientsfor curvature, a mode couples to mode only incondition that . Therefore, critical or dangerousspurious modes can be roughly spotted and taken into accountin the simulation.

The scheme of such a bent waveguide mode converter isshown in Fig. 1. For bent waveguide mode converters withlarge curvature, the integral element in (1) and (2) has to bealong the waveguide axis. Suppose that the coordinates of thewaveguide axis can be described by a one-dimensional function

in the plane, which is also the polarizationplane of the output mode, then is given by

(3)

Therefore, the differentials about in (1) and (2) can be substi-tuted by differentials about . Consequently, the coupled dif-ferential equations (1) and (2) become

(4)

(5)

For a – converter of length in the direction, iftotally modes are taken into account, then the boundary con-ditions can be written as [9]

(6)

(7)

where the first element of the -dimensional vector in (6) or(7) represents the complex amplitude of the mode, thesecond element represents that of the mode, etc.. Note that(6) implies that there is only the mode of amplitude 1.0 atthe input, while (7) implies that there is no reflection for all themodes at the output. Due to the boundary conditions’ simplicity,the boundary value problem, which is composed of (4), (5), (6)and (7), can be solved using a transmission matrix method [3].

We further suppose that the waveguide axis starts from theorigin (0, 0), and can be approximated by a cubicB-spline with nodes , namely

(8)

where , , and .The node coordinates can be selected as uniformly dis-tributed from to , whileand are unknowns to be determined. The constraint for theconverter’s input and output to be in the same direction can beguaranteed by selecting the first type of boundary conditionsfor the cubic B-spline, as given by

(9)

Based on the boundary condition (9) and the continuity of thefirst order derivatives of the defined cubic spline function (8),the second order derivatives can be determined recursively.

In order to obtain the unknown parametersand for a – bent waveguide mode converter withhighest possible conversion efficiency and shortest possibleconverter length (along the waveguide axis), DE algorithmis used as the optimization approach. The aforementionedunknowns are taken as the optimization parameter vector

. The cost function can be expressedas follows:

(10)

where is the number of evolution generations, is the con-version efficiency to mode at the output, is the totalconverter length along the waveguide axis, is the maximumpower level of spurious modes within the converter, which isalso used to allow a tradeoff between the converter length and

1374 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 33, NO. 4, AUGUST 2005

Fig. 2. Power distributions for the DE optimized mode converter with N = 7

and P = 0:1.

the bandwidth to be achieved for a given optimization, andto are the corresponding weighting factors for each term.

III. OPTIMIZATION RESULTS

To illustrate the approach described above, – bentwaveguide mode converters with a radius of mm anda center operating frequency of are considered.The target is to synthesize the shapes of the converters’ axes de-scribed by corresponding cubic B-splines, with highest conver-sion efficiency to the mode and shortest converter length.The ohmic losses for a copper waveguide wall are included inthe simulation. According to the coupling principle, six propa-gating waveguide modes are included in the simulation, whichare as follows: , , , , , and .Based on the simulation results, other spurious modes seemto be negligible. Since the converter has a constant radius, thecutoff modes cannot become propagating modes, their effectson the mode conversion process are negligible. It is found thatdifferent numbers of node points of the cubic B-spline, aswell as the maximum allowable power level of spurious modes

within the converter will affect the design.First, and are selected as and , re-

spectively. Typical DE simulation parameters, such as the popu-lation size , the mutation probability , and the crossoverprobability are set as follows: ,

, . Fig. 2 shows the DE optimized relative powerdistributions of forward waves as functions of the axial dis-tance . The axial converter length cm, whichis at least 3 cm shorter than those of the converters with theshape of cubic parabola or Gaussian curve in [9]. While using aconventional serpentine – mode converter, the totalconverter length is about 2.48 m [3], which is five times theconverter length in Fig. 2. The conversion efficiency in Fig. 2is , which is very close to the power transmissionefficiency at the output and actually means a completeconversion to the mode. Moreover, it can be seen fromFig. 2 that the mode is the strongest spurious mode nearthe input, while the mode is the strongest spurious modenear the output. The highest spurious mode level within the con-verter is about 0.0936. According to the simulation, the powerlevels for backward waves are generally below 80 dB at the

Fig. 3. Output power levels versus frequency for the mode converter withN = 5 and P = 0:1.

Fig. 4. DE optimized shapes of the converters’ axes.

center frequency. Furthermore, neglecting the backward waveschanges the boundary value problem to an initial value problem,which allows forward integration and increases the optimizationspeed significantly. Thus, for simplicity, backward waves arenot included in the DE optimization. Fig. 3 plots the converter’soutput power levels versus frequency for different modes and

. If 90% mode conversion to the mode is accept-able, the bandwidth of the optimized converter is about 25.1%,ranging from 30.5 to 39.3 GHz. Fig. 4 shows the shape of theconverter’s axis in the x–y plane (solid line). As can be seen, theinput and output directions are indeed the same.

Next, and are selected as and ,respectively. Generally, greater will increase the robustnessin the design. However, the DE population size will alsoincrease rapidly, which costs longer CPU time for convergence.The relative power distributions as functions of the axial dis-tance for this converter are plotted in Fig. 5. By comparingto Fig. 2, it is observed that although the – conver-sion is no longer in a monotonous fashion; the conversion effi-ciency is still quite high . The spurious mode levelwithin the converter is also successfully suppressed below 0.1.Fig. 6 demonstrates the converter’s output power levels versusfrequency for different modes and . The band-width for this case is about 24.0%. The shape of the converter’saxis in the x–y plane is plotted in Fig. 4 (dashed line). Again, theinput and output are in the same direction. The axial converter

YANG AND QING: DESIGN OF HIGH-POWER MILLIMETER-WAVE – MODE CONVERTERS 1375

Fig. 5. Power distributions for the DE optimized mode converter withN = 11

and P = 0:1.

Fig. 6. Output power levels versus frequency for the mode converter withN = 11 and P = 0:1.

Fig. 7. Power distributions for the DE optimized mode converter withN = 11

and P = 0:2.

length is about 41.20 cm, which is a 2.72 cm reduction as com-pared to the previous case.

As a last example, consider and . In thiscase, the highest spurious mode level within the converter is al-lowed to be relatively higher, thus, the converter length is ex-pected to be shorter. Fig. 7 presents the relative power distribu-tions as functions of the axial distance for this converter. Ascompared to Fig. 5, the power levels of the and

Fig. 8. Output power levels versus frequency for the mode converter withN = 11 and P = 0:2.

TABLE IOPTIMIZED CHARACTERISTIC RESULTS FOR THE TM -TO-TE

BENT WAVEGUIDE MODE CONVERTERS

modes oscillate violently. Nevertheless, complete conversion isstill achieved at the output . Fig. 8 shows theoutput power levels versus frequency for different modes and

. The bandwidth for this case is only about15.7%, due to the higher spurious mode level within the con-verter. Nevertheless, the bandwidth is still sufficient for manyapplications. The shape of the converter’s axis in the x–y planeis plotted in Fig. 4 (dotted line). Again, the input and output arein the same direction. The axial converter length is only about38.1 cm; a 3.1 cm reduction as compared to the case forand .

Finally, for comparison purpose, the optimized characteristicresults for the aforementioned three types of – modeconverters are summarized in Table I. As can be seen, the spu-rious mode levels are essentially low at the output. Note that thesum of the output power levels equals to the power transmissionefficiency for each case, implying the energy conserva-tion. The shortest converter has an axial length of about 38.1cm, but it also has a narrowest bandwidth of about 15.7%. Ac-cording to the optimization, further increase of cannot reducethe converter length significantly. Moreover, this will increasethe simulation time rapidly. If the highest allowable spurious

1376 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 33, NO. 4, AUGUST 2005

mode level is increased, the mode converter design may notbe successful due to excessive power levels of spurious modes.

IV. CONCLUSION

A novel approach for the design of very short high powermillimeter-wave – bent waveguide mode convertersis documented in this paper. The shapes of the bent circularwaveguides’ axes are denoted by cubic B-splines local shapefunctions in a two-dimensional plane. Coupled mode theory isused to solve the wave amplitudes of various modes in the over-moded circular waveguides. The synthesis problem is convertedto an optimization problem and solved by the robust DE algo-rithm. Numerical optimization results show that the proposedapproach can be used to design high performance –mode converters with very short converter length, high conver-sion efficiency and suppressed spurious mode levels.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers,whose comments enhanced the quality of this paper consider-ably.

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Shiwen Yang (M’00–SM’04) received the B.Sc.degree in electronic science from East China NormalUniversity, Shanghai, China, and the M.Eng. degreein electromagnetics and microwave technology andthe Ph.D. degree in physical electronics from theUniversity of Electronic Science and Technology ofChina (UESTC), Chengdu, China, in 1989, 1992,and 1998, respectively.

From 1994 to 1998, he was a Lecturer at the Insti-tute of High Energy Electronics, UESTC. From 1998to 2002, he was a Research Fellow at the School of

Electrical and Electronic Engineering, Nanyang Technological University, Sin-gapore. He joined the Temasek Laboratories, National University of Singapore,as a Research Scientist in 2002. He is currently a Full Professor in the Depart-ment of Microwave Engineering, School of Electronic Engineering, Universityof Electronic Science and Technology of China, Chengdu, China. His researchinterests include high-power microwave components, antennas and arrays, andcomputational electromagnetics.

Anyong Qing (S’97–M’01) was born in Hu’nan Province, China, on May 27,1972. He received the B.E. degree from Tsinghua University, Beijing, China,the M.E. degree from Beijing Broadcasting Institute, Beijing, China, and thePh.D. degree from Southwest Jiaotong University, Chengdu, China, in 1993,1995, and 1997, respectively.

He is currently a Research Scientist in the Temasek Laboratories, NationalUniversity of Singapore, Singapore. From September 1997 to June 1998, hewas a Postdoctoral Fellow in the Department of Communication Engineering,Shanghai University, Shanghai, China. From June 1998 to June 2000, he wasa Research Fellow in the School of Electrical and Electronic Engineering,Nanyang Technological University, Singapore, and from July 2000 to June2001, he was a Member of Scientific Staff in the Electromagnetic Theory Divi-sion, Department of Electrical Engineering, University of Kassel, Germany. Hewas also with VS Electronics Pte Ltd., Singapore, as an RF Design Engineerand also with Sumitomo Electric Industries Ltd., Itami, Japan. His researchinterests include composite materials, acoustic and elastic waves propagation,computational electromagnetics, inverse problem, frequency selective surfaces,RFIC, and MMIC.