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Electromagnetic Properties of a TrapezoidallyCorrugated Slow Wave Structure for

Backward Wave OscillatorsMd. Ruhul Amin, Member, IEEE, Kazuo Ogura, Member, IEEE, Junpei Kojima, and Rakibul Hasan Sagor

Abstract— In this paper, the electromagnetic properties ofa beam free trapezoidally corrugated slow wave structure(TCSWS) are numerically and experimentally investigated.The dimensions of the TCSWS are selected so that it canbe deployed in X-band backward wave oscillators (BWOs).The theoretical dispersion characteristics for the axisymmetricTM01–TM03 modes are obtained by solving the dispersion equa-tion numerically. In order to verify the numerical calculation,a cold test is conducted using a 6-period TCSWS cavity anda vector network analyzer (VNA). The cavity is excited by asmall needle-type antenna and frequency of the VNA is sweptover the passbands. The dispersion characteristics are obtainedfrom frequency values of the discrete resonant peaks of reflectionmeasurement of each TM mode. Experimental results show anexcellent agreement with numerical ones. The numerical and coldtest results reveal that TCSWS can be used as an alternative tosinusoidally corrugated slow wave structure in BWO experimentswith an advantage of ease of fabrication.

Index Terms— Backward wave oscillator (BWO), cold test,dispersion relation, slow wave structure (SWS), vector networkanalyzer (VNA).

I. INTRODUCTION

BACKWARD wave oscillators (BWOs) and traveling wavetubes (TWTs) are the typical examples of slow wave-

type high-power microwave (HPM) devices where the spacecharge waves of an axially streaming electron beam interactwith the electromagnetic (EM) wave of the resonating structurethat results in microwave generation. In order for the energyexchange to occur, the phase velocity of the EM wave mustbe slowed down by properly designed slow wave structures(SWSs). The BWOs operate in the negative group velocityregime. In order to obtain negative group velocity in thedispersion diagram, periodic-type SWSs are needed [1], [2].In the earlier stage of development, the spatial periodicity wasintroduced by inserting dielectric disks or irises periodically

Manuscript received November 14, 2013; revised January 14, 2014 andFebruary 18, 2014; accepted February 18, 2014.

M. R. Amin and R. H. Sagor are with the Department of Electrical andElectronic Engineering, Islamic University of Technology, Gazipur 1704,Bangladesh (e-mail: aminr_bd@yahoo.com).

K. Ogura and J. Kojima are with the Graduate School of Sci-ence and Technology, Niigata University, Niigata 950-21, Japan (e-mail:teogura@eng.niigata-u.ac.jp).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2014.2308434

inside a metallic waveguide. For high power operation ofthe devices, RF breakdown was an alarming consequenceof using dielectric loaded SWS. In an attempt to avoidRF breakdown and increase the power level, SWSs were madeby modifying the inner wall of the metallic waveguide with-out inserting any dielectric material. Sinusoidally corrugatedSWS (SCSWS) is the result of such efforts [3]–[19]. TheSCSWSs are being used in BWOs successfully to generatemicrowave in the order of GW in centimeter wavelength range[15]–[20]. In [19], a peak microwave power of 4.3–5.3 GW(pulsewidth is ∼20 ns) with 31%–22% efficiency at 9.4 GHzis experimentally predicted for an X-band relativistic BWOusing SCSWS. The possibility of increasing output power ofBWO by using pulse compression technique has been reportedin [20]. The deployment of a compressor at the output of theBWO with SCSWS resulted in a peak output power of 3.2 GW(FWHM pulsewidth is ∼2.2 ns) compared with 0.7 GW(FWHM pulsewidth is ∼15 ns) without compressor. However,making a precise sinusoidal radial profile on the inner wallof a cylindrical metallic structure requires high precisionnumerically controlled machine. As a viable alternative of theSCSWS, a trapezoidally corrugated SWS (TCSWS) has beenstudied in [21] in presence of an annular IREB. The Rayleigh–Fourier (R–F) method was used to derive dispersion relationof the TCSWS under the constraint of Rayleigh hypothesis[12], [13] and temporal growth rate for the fundamentalTM01 mode was extensively studied and compared with thatof SCSWS. In this paper, beam free dispersion characteristicsof the TCSWS and SCSWS for TM02 and TM03 are calcu-lated. In order to verify the dispersion characteristics of theSWSs experimentally, modular SWSs are constructed and coldtest is carried out using SWS cavity having 6-period and aVNA [8], [9].

This paper is organized as follows. A summary of thederivation of dispersion relation is presented in Section II.Section III presents the numerically obtained dispersion char-acteristics of TM02 and TM03 modes. Experimental (cold test)results and their comparison with numerical ones are presentedin Section IV. Finally, conclusions are drawn in Section V.

II. DISPERSION RELATION OF THE TCSWS

Longitudinal section of the proposed TCSWS is shown inFig. 1(a). The SWS is considered cylindrical, infinitely long,

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Fig. 1. (a) Longitudinal section of the cylindrical TCSWS. (b) Inner radialprofile of the SWS for Fourier expansion.

and axisymmetric. In order to utilize the R–F method used toderive the dispersion relation of an axisymmetric SCSWS, theradial profile of the proposed TCSWS is expressed by Fourierseries in the axial direction. Using the notations of Fig. 1(b),the Fourier series of the radial function R(z) of the TCSWSis derived as [21]–[23]

R(z) = R0 + 6h

π2

∞∑

υ=1

(−1)υ−1

(2υ−1)2

×[sin

((2υ−1)π

3

)+ sin

(2(2υ−1)π

3

)]cos(2υ−1)K0z

(1)

where R0 is the average radius, h is the corrugation amplitude,z0 is the period of corrugation, K0 = 2π/z0, and the integerυ is the harmonic number. Although (1) is a sum of infinitenumber of terms, an exact trapezoidal profile is obtainedfor υ ≥ 8. Detail derivation of the dispersion relation ispresented in [21]. For the beam free case, (17) of [21]becomes

∞∑

m,n=−∞An [1 + (m − n)Qn]P J

mn =∞∑

m,n=−∞Dmn · An =0 (2)

where Qn = K0kzn/β2n , Dmn = [1+(m−n)Qn](P J

mn), P Jmn =∫ π

0 cos [(n − m)u] J0 (βn R(z))du, u = K0z, kzn = kz + nK0and n = 0,±1,±2, . . . is the Floquet harmonic number, kz isthe axial wavenumber, β2

n = (ω/c)2 −k2zn , βn is the transverse

wavenumber, and c is the velocity of light in vacuum.In (2), An is a column matrix represents the amplitude

of Floquet harmonics. The dispersion relation results fromthe fact that in the solution of (2), at least some An �= 0.This implies that

det

[ ∞∑

m.n=−∞Dmn

]= 0 (3)

TABLE I

SIZE PARAMETERS OF THE SCSWS AND TCSWS

is the required generalized beam free dispersion relation for thecorrugated SWS. Specific cases can be analyzed by selectingthe appropriate expression for R(z) in the integration. Thedeterminant Dmn is a function of ω and k. In general, therank of the determinant Dmn is infinite. However, in numericalcomputation, we must truncate the rank to a finite value. Ifwe consider that the number of Floquet harmonics n(= N) isequal to the number of Fourier harmonics m(= N), then n andm in Dmn are limited to −(N − 1)/2 ≤ n, m ≤ (N − 1)/2.The rank of the determinant is then 2N . A tradeoff is made onthe value of N to obtain converged dispersion characteristics.The convergence is tested by the periodicity f (kz + nK0) =f (kz) of the dispersion diagram in the f − k plane [21]. Thevalue of N that is suitable for TM01 mode calculation may notyield converged dispersion characteristics for the higher ordermodes.

III. DISPERSION CHARACTERISTICS OF TM MODES

Table I shows the size parameters of an X-band SCSWScomparable TCSWS, which were determined in [21].Using these parameters and the same numerical procedure,the dispersion curves for TM02 and TM03 modes are obtainedin the first Brillouin zone (−K0/2 ≤ kz ≤ K0/2), whichare shown in Fig. 2(a) and (b), respectively. It can bevisualized from the figures that dispersion curves for TM02mode of both the structures almost coincide as TM01 mode[21], however, a significant difference is observed in thedispersion characteristics of TM03 mode for the parameters ofTable I.

The O-type HPM devices like TWT and BWO need SWSthat can support EM wave with phase velocity less than thevelocity of light in vacuum. However, TWT works in theslow wave regime with positive group velocity and BWO inthe negative group velocity regime that corresponds to either−K0/2 ≤ kz ≤ 0 or equivalently K0/2 ≤ kz ≤ K0. Groupvelocity can be calculated by taking the derivative of thedispersion equation as

vg = dω

dk. (4)

Group velocity for the axisymmetric TM modes for theTCSWS is calculated and the results are shown in Fig. 3. Lightline (ω = kc) and beam line with β = 0.9 (that correspondsto 660 keV of beam energy) (ω = βkc) are also shown inthe upper part of the figure that contains the dispersion curvesof the first three TM modes. Group velocity of these modesis shown in the lower part of the figure. The intersection ofthe beam line occurs with the TM01 mode in the negativegroup velocity region that corresponds to BWO operation,

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AMIN et al.: EM PROPERTIES OF A TCSWS 3

Fig. 2. Dispersion characteristics of the TCSWS (solid) and SCSWS (dotted).(a) TM02 mode. (b) TM03 mode. Structure parameters are shown in Table I.

on the other hand, for the TM02 mode, the intersection occursin the positive group velocity region where TWT operates.The possible operation points are indicated by black dots inthe figure.

IV. EXPERIMENTAL RESULTS OF DETERMINATION OF THE

DISCRETE POINTS ON THE DISPERSION CURVE

Experimental approach is considerably important not onlyfor the verification of the dispersion characteristics of theSWS, but also for the validation of computational results[8], [24]. The continuous dispersion characteristics (Fig. 2)are obtained under the assumption of infinite length structure.If the length of SWS is finite, reflections from the endof the structure produces standing EM wave. The standingwave quantizes the axial wavenumber that consequently resultsin generation of axial resonant modes corresponding to a

Fig. 3. Dispersion curves and the corresponding group velocities for the firstthree TM modes of the TCSWS with size parameters shown in Table I. Thelight line and beam line with β = 0.9 are also shown in the upper part of thefigure. The possible operating points for BWO and TWT are also indicatedby the black dots.

particular TM baseband. In general, any periodic structurewith end reflections having N number of periods exhibits(N + 1) resonant axial modes. Each axial mode is character-ized by a discrete resonant frequency ( fr ) and a wavenumber(kr ) [8]. The discrete resonant point (kr , fr ) must lie on thecorresponding continuous dispersion curve. If the dispersioncurve of a particular mode is monotonic within the firstBrillouin zone, the axial modes will be equally spaced in thenormalized wavenumber space 0 ≤ kz0 ≤ π . In order toverify the dispersion characteristics, frequency of the axialresonant modes is determined by performing cold test on6-period TCSWS and SCSWS cavities using a VNA. Shortingplates are placed at both ends of the 6-period SWS with a small

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Fig. 4. Cold test setup for the reflection measurement of a 6-period SWS.Axial modes are excited by a needle-type antenna.

hole at one plate to admit a needle-type antenna for the excita-tion of the axial modes. The antenna is connected to S11 portof the VNA by an SMA coaxial cable for reflection measure-ment. The 6-period SWS cavity resonates at seven distinct fre-quencies, which are equispaced in the normalized wavenumberplane between 0 and π . Fig. 4 shows the experimental setup forthe cold test, and Fig. 5(a)–(c) shows the results of reflectionmeasurement and their comparison with the numerical ones,respectively, for TM01, TM02, and TM03 modes. The left por-tion of Fig. 5 shows the reflection profile and the comparisonof the numerical and experimental results of frequency of theaxial modes with normalized wavenumber (kz0 = i · π/6;i = 0 · · · 6) are depicted in the right. As can be seen from thefigures that the experimental and numerical results agree excel-lently except for the π axial mode of TM02 passband. It is tobe noted that the amplitude of corrugation h of the constructedTCSWS did not match exactly to 0.38 cm. So, the numericalcomputation is repeated for measured h = 0.377 cm and theresults are compared with their experimental counterparts. Thereflection profile gives the resonance peaks that correspond tothe axial modes of the SWS. From the discrete wavenumbersand frequencies (kr , fr ) of the reflection measurement, thedispersion characteristics of the SWS can be determinedsynthetically using the mathematical approach reported in [24].The theoretical and cold test values of the resonant frequencyof the axial modes of TM01 through TM03 passbands for theTCSWS and SCSWS are compared in Table II.

The closeness of the theoretical and cold test frequenciesof the axial modes of SCSWS and TCSWS is evident fromTable II. However, the π mode of the TM02 passband did notappear in the reflection measurement using the needle-typeantenna.

Fig. 5. Results of the reflection measurement and their comparisonwith numerical ones (h = 0.377 cm). (a) TM01 mode. (b) TM02 mode.(c) TM03 mode.

In order to compare the field characteristics of the axialmodes of SCSWS and TCSWS, field patterns of some of therepresentative axial modes of both structures are calculated.The results are shown in Fig. 6. It can be visualized in Fig. 6;

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AMIN et al.: EM PROPERTIES OF A TCSWS 5

TABLE II

COMPARISON OF RESONANT FREQUENCY OF THE AXIAL MODES OF TM01–TM03 PASSBANDS FOR SCSWS AND TCSWS

Fig. 6. Electric field patterns of TM01–TM03 modes at normalized wavenumber kz0 = π /2 of SCSWS and TCSWS. The color map depicts the normalizedvalue of the field strength. Blue: <20%, Green: 20%–50%, and Red: >50%.

there exists no significant difference in the field structure ofthe SWSs.

V. CONCLUSION

In this paper, the EM properties of a previously proposed[21] TCSWS are compared with those of an SCSWS to

establish a basis of its possible deployment in BWO exper-iments as a viable alternative of SCSWS. The theoreticalcalculation of the dispersion characteristics of the TCSWS isextended to higher order TM modes. Prior to the deploymentto a real BWO experiment, cold tests are usually performed onthe SWS to ascertain its frequency characteristics. To validate

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the theoretical dispersion characteristics, cold test is conductedon finite length SCSWS and TCSWS cavities having 6-periodsusing a VNA. Discrete frequencies of the resonant axial modesagree excellently with the theoretical values. The fields ofthe axial modes of both the structures also show similarpatterns. The growth rate study in [21] and the cold test resultspresented in this paper reveal that the TCSWS may performequivalently as the SCSWS in real BWO experiments. Hence,it can be concluded that the proposed TCSWS can be used asa viable alternative of the SCSWS.

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Md. Ruhul Amin (M’10) was born in Rampur,Bangladesh, on October 10, 1959. He received theB.Sc. degree in engineering from the University ofRajshahi, Rajshahi, Bangladesh, the M.Sc. degreein engineering from the Bangladesh University ofEngineering and Technology, Dhaka, Bangladesh,and the Ph.D. degree in electrical and electronicengineering from Niigata University, Niigata, Japan,in 1984, 1987, and 1996, respectively.

He is a Professor of electrical and electronicengineering with the Islamic University of Technol-

ogy, Gazipur, Bangladesh, a subsidiary organ of OIC in Bangladesh. Hiscurrent research interests include generation and application of high powermicrowaves, antenna theory, signal processing, and computational electromag-netics. He has authored and coauthored over 70 research papers in refereedjournals and conference proceedings including the IEEE TRANSACTIONS,IET, and Physical Review. He was a Visiting Academic Fellow with theGeorge Green Institute of Electromagnetic Research, University of Notting-ham, Nottingham, U.K. He served as the Dean of the Faculty of electricaland computer engineering with the Rajshahi University of Engineering andTechnology, Rajshahi.

Dr. Amin is a recipient of the Sir Thomas Ward Memorial Medal awardedby the Institution of Engineers, India. He is a Commonwealth Fellow and afellow of the Institution of Engineers, Bangladesh.

Kazuo Ogura (M’13) was born in Japan in 1957. Hereceived the B.S. degree in physics from OkayamaUniversity, Okayama, Japan, and the M.S. and Ph.D.degrees in physics from Kyoto University, Kyoto,Japan, in 1981, 1983, and 1989, respectively.

He is a Professor with the Electrical and Elec-tronic Engineering Department, Niigata University,Niigata, Japan. His current research interests includehigh-power microwave devices utilizing electronbeams and their applications to plasma heating andcontrol.

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AMIN et al.: EM PROPERTIES OF A TCSWS 7

Junpei Kojima was born in Japan in 1990. Hereceived the B.S. degree from Niigata University,Niigata, Japan, in 2013, where he is currently pur-suing the Degree with the Graduate School.

His current research interests include microwavedevices and propagation of the microwave.

Rakibul Hasan Sagor received the B.Sc. degreein electrical and electronic engineering from theIslamic University of Technology (IUT), Gazipur,Bangladesh, and the M.Sc. degree in electrical engi-neering from the King Fahd University of Petro-leum and Minerals, Dhahran, Saudi Arabia, in 2007and 2011, respectively.

He was a Lecturer with the Electrical and Elec-tronic Engineering Department, Stamford Univer-sity, Dhaka, Bangladesh, from 2008 to 2009. Heis currently an Assistant Professor of Electrical and

Electronic Engineering with IUT, a subsidiary organ of the Organization ofthe Islamic Conference in Bangladesh. His current research interests includethe generation and application of high-power microwaves, modeling andsimulation of high-frequency active devices optically controlled active devices,computational electromagnetics, and nonlinear integrated optics.