Wave coupling in sheet- and multiple-beam traveling-wave tubesGregory S. Nusinovich, Simon J. Cooke, Moti Botton, and Baruch Levush Citation: Phys. Plasmas 16, 063102 (2009); doi: 10.1063/1.3143123 View online: http://dx.doi.org/10.1063/1.3143123 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v16/i6 Published by the American Institute of Physics. Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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Wave coupling in sheet- and multiple-beam traveling-wave tubesGregory S. Nusinovich,1,a� Simon J. Cooke,2 Moti Botton,3,b� and Baruch Levush2

1Science Application International Corporation, 1710 SAIC Dr., McLean, Virginia 22102, USA2Naval Research Laboratory, Washington, DC 20375, USA3IREAP, University of Maryland, College Park, Maryland 20742-3511, USA

�Received 31 March 2009; accepted 6 May 2009; published online 4 June 2009�

To increase the power level of the sources of coherent electromagnetic radiation at frequencies from100 GHz up to the terahertz range it makes sense to develop devices with a spatially extendedinteraction space. Sheet-beam and multiple-beam devices belong to the category. In the presentpaper the small-signal theory of traveling-wave tubes with sheet-beam and multiple sheet-beamconfigurations is developed. It is shown that in such tubes the wave coupling on electron beams mayoccur even in small-signal regimes. The wave coupling and its role for amplification of forward andexcitation of backward waves in such amplifiers is studied. Also the effect of transversenonuniformity of the electromagnetic field on the device operation is analyzed and illustrated byseveral examples. © 2009 American Institute of Physics. �DOI: 10.1063/1.3143123�

I. INTRODUCTION

At present there is a strong interest in increasing thepower level of linear-beam devices at upper millimeterwaves �W-band and above�.1 To master this short wavelengthregion at a reasonably high-power level, it is necessary todevelop sources of coherent electromagnetic radiation with aspatially extended interaction space. Transverse dimensionsof the interaction space in such devices should be muchlarger than a wavelength, so they should operate at high-order modes and utilize electron beams with large cross sec-tions �i.e., high perveance beams�. This fact motivated stronginterest to such concepts as sheet-beam and multiple-beamconfigurations of traveling-wave tubes �TWTs� and klystrons�see, e.g., Ref. 2 and references therein� and sheet-beam free-electron lasers.3 Note that, as a rule, the users of such sourceswould like to operate at low voltages �see, e.g., Refs. 4–6and references therein�, in contrast to similar concepts ofhigh-power microwave sources for accelerator applications7

which are typically driven by high voltage modulators.A number of three-dimensional codes have been devel-

oped for accurate designing of such devices �see, e.g., Chap.10 in Refs. 2 and 8�. However, before starting designing anyspecific device with the use of these codes, it is always ex-pedient to obtain some insight into the physics of basic pro-cesses by using an analytical theory. The small-signal theoryfor sheet-beam klystrons has been developed in a rather gen-eral form in Ref. 9. Then, it was shown10 that, in contrast totraditional microwave tubes with axis symmetric cylindricalinteraction space, in such configurations as sheet-beam andmultiple-beam devices the coupling between competingmodes exists even in a small-signal regime. Such coupling insheet-beam klystrons had been analyzed in Ref. 10. Thebeam coupling to competing modes in multiple-beamklystrons was studied in Ref. 11. In Ref. 12, the linear theoryof a sheet-beam free-electron laser was developed and the

excitation of modes with different numbers of field variationsin a wide transverse direction was studied. In Ref. 13, thesmall-signal theory for a sheet-beam TWT with a ridgedwaveguide slow-wave structure had been developed and itwas shown that the dispersion diagram calculated with theuse of such simple theory agrees very well with results ob-tained with the use of SUPERFISH code. The wave dispersionand growth analysis of a low-voltage sheet-beam TWT witha planar grating slow-wave structure had been performed inRef. 14.

Below, a simple analytical small-signal theory of sheet-beam and multiple-beam TWTs is presented. This theory al-lows one to study effects of the geometry of the interactionspace on the wave coupling and competition in such tubes.This paper is organized as follows. In Sec. II, the basic equa-tions of the small-signal theory of a multiwave sheet-beamtraveling-wave tube and the conditions for self-excitation ofcoupled backward waves are formulated. In Sec. III we ana-lyze the effect of wave cross coupling in sheet-beamtraveling-wave amplifiers on amplification of forward andexcitation of backward waves. In Sec. IV, the effect of trans-verse nonuniformity on device operation is studied. First, weanalyze a single sheet-beam device with isolated waves.Then, the effect of splitting one sheet electron beam intoseveral beamlets on the beam coupling to waves with differ-ent spatial structures is analyzed. It is shown how such split-ting can improve the selectivity of excitation of a specificmode. Lastly, we study the effect of beam geometry on thewave coupling in such configurations. In Sec. V we discussthe results obtained and the applicability of the developedtheory to sheet-beam traveling-wave amplifiers with a highaspect ratio. Section VI contains a brief summary of results.

II. BASIC EQUATIONS

A. Dispersion equations for coupled forward waves

We will use a self-consistent set of equations describingthe stationary operation of a TWT. Such set is formed byequations for electron motion and equations describing the

a�Permanent address: IREAP, University of Maryland, College Park, MD20742-3511, USA.

b�Permanent address: Hebrew University, Jerusalem, Israel.

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wave excitation. Since in amplifiers with large transversecross section of the interaction space an electron beam cansimultaneously interact with several waves having close cut-off frequencies, we will consider the beam interaction withsuperposition of such waves. In equations for electron mo-tion the only important is the space harmonic of the waves’electric field, which is synchronous with electrons. The axialcomponent of this electric field of all waves can be repre-sented as

Ez = Re�ei�t�s

As�z��s�r���e−ikzsz� ,

where kzs is the axial wavenumber of the synchronous spaceharmonic of the sth wave.

Equations for electron motion can be written as twoequations, viz., the equation for the electron energynormalized to the rest energy, �, and the equation for aslowly variable phase of electron motion which describesperturbations in the electron phase due to the modulationof the electron axial velocity v by the rf field,�=�t0+���0

zdz� /v−z /v0�:

��

�z= − Re�ei��

s

As�s� , �1�

��

�z=

�

�2 − 1−

�0

�02 − 1

. �2�

Equations �1� and �2� should be supplemented by the bound-ary conditions at the entrance ��0�=�0, ��0�=�0� �0,2��.In Eqs. �1� and �2� we introduced normalized axial co-ordinate z�=�z /c and normalized wave amplitudeA�= �eA /mc��exp−i�kzs−� /v0�z� which are given inEqs. �1� and �2� without primes. In Eq. �2� we expressed theelectron velocity normalized to the speed of light �=v /c viathe normalized energy using �=1 /1−�2; initial value ofthe normalized energy is determined by the beam voltage�0=1+eVb /mc2. Note that our definition of a slowly varyingelectron phase differs from the standard one which definessuch phase as the phase of electron motion with respect tothe phase of the rf field �st=�t−kzz=��t−z /vph� wherevph=� /kz is the phase velocity of the wave. We had chosensuch definition because we consider electron interaction withseveral waves having different phase velocities.

The wave excitation in the case of operation far enoughfrom cutoff can be described by the first-order equationwhich in our notations can be given as

dAs

dz+ i�ksAs = Is�

S�

��s�� 1

��

0

2�

e−i�d�0�ds�. �3�

The function ��r��� in Eq. �3� describes the electron beamcurrent distribution over the cross section, jz0=−Ib��r���;its normalization condition is �Sb

�ds�=1. We also intro-duced the normalized detuning �ks=�ph,s

−1 −�0−1 between the

unperturbed electron velocity �0 and the phase velocity of agiven wave �ph,s=vph /c and the normalized beam currentIs= �eIb /mc3��c3 /�2Ns� where Ns is the wave norm differentfor different waves.

The small-signal theory is based on the assumptionabout smallness of the wave amplitude and perturbations inelectron motion caused by interaction with the wave. Linear-izing Eqs. �1� and �2� with respect to these perturbations onecan readily integrate the equations. Then, Eq. �3� with thelinearized perturbation in electron phase in the case whenthere are S waves excited by an electron beam can be re-duced to a set of linear uniform differential equations forwave amplitudes

d3As

dz3 + i�ksd2As

dz2 = − i�s�

Dss�3 As�

� . �4�

Here the terms in the right hand side are equal to

Dss�3 =

Is

��02 − 1�3/2�

S�

��s��s�ds�. �5�

Assuming exponential dependence of the wave amplitudeson the axial coordinate Ase−ihz reduces Eq. �4� to the set ofcubic equations for the propagation constant h

�s=1

S �h2�h − �ks�As + �s�

Dss�3 As�

� � = 0, �6�

whose solution can be found from a corresponding Sth order�1sS� determinant. When nondiagonal terms in this de-terminant are equal zero �as in the case of conventional de-vices with axis symmetric interaction space� one readily getsfrom Eq. �6� the dispersion equations for isolated waves. Inthis case, Eq. �6� for each wave becomes equivalent to thecubic dispersion equation derived by Pierce;15 the diagonalterms Dss are equal to the Pierce gain parameter which wewill denote by CP. In the case of S coupled waves, Eq. �6�has 3S solutions.

To analyze the effect of the beam geometry on the cou-pling to various modes it makes sense to use the Pierce gainparameter C0 for a beam with a very small cross sectionlocated in a small spot on axis. Then, the Pierce gain param-eter for a beam with a large cross section can be given asCP=C0B where

B3 =1

��0� 2�S�

� � 2ds�. �7a�

When the geometry of a waveguide and a beam allows oneto separate transverse variables in Eq. �7a�, the parameter Bcan be represented as B=BxBy where the x and y denote thetransverse vertical and horizontal coordinates, respectively.When there is a set of beamlets with center coordinates yc,j

and the width in the horizontal direction �� j the parameterBy determined by Eq. �7a� can be rewritten as

By3 =

1

��0� 2�j=1

J1

�� j�

yc,j−��j/2

yc,j+��j/2

��y� 2dy . �7b�

Note that Eq. �7a� is essentially the same as the factor��M2d � / , where d is an element of the transverse areaand M is the gap factor, used by Pierce in his definition ofthe gain parameter �cf. �4.69� and �4.78� in Ref. 15�. As

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follows from Eq. �5�, the Pierce gain parameter for a beamleton axis in our notations can be defined as

C03 =

Is ��0� 2

��02 − 1�3/2 . �8�

In our case, in addition to diagonal terms, there are alsononzero nondiagonal terms which can be defined by the fol-lowing equation similar to Eq. �7a�:

Bss� =1

�s�0� 2�S�

��s��s�ds�. �9�

These terms are responsible for the wave cross couplingwhich will be analyzed below. Note that the wave couplingin sheet-beam free-electron lasers was studied in Ref. 16under assumption that a sheet beam occupies all space fromwall to wall in the wide horizontal direction. Our analysisbelow will be focused on studying sheet-beam configurationswith different geometries in this wide direction.

When the interaction length is long enough �so only thegrowing wave should be taken into account�, the small-signalgain, as follows from Eq. �6�, is equal to �cf. Ref. 15�

Gs-s�dB� = 47.263C0BN − 9.54, �10�

where N is the number of axial wavelengths in the interac-tion length.

Below we will consider several examples characterizingthe effect of the beam geometry on the excitation of variousmodes. However, prior to this consideration, we will formu-late the starting conditions for excitation of coupled back-ward waves.

B. Self-excitation conditions for coupledbackward waves

As known �see, e.g., Ref. 17�, the self-excitation condi-tions for backward waves follow from the dispersion equa-tion describing propagation of these waves in the directionopposite to streaming electrons and the boundary conditionscorresponding to the absence of electron energy and phasemodulation at the entrance and the absence of waves enteringthe interaction space from collector side in the case of acircuit with a well-matched output. The dispersion equationcan be written in the form of Eq. �6� with the only differencethat the last term in figure brackets should change the signbecause now we consider the waves propagating in the op-posite direction and, hence, the norms of these waves pro-portional to the power flow are negative �these norms arepresent in the definition of the normalized beam current pa-rameter, so, if we change the sign plus to minus before thelast term in Eq. �6�, then we assume in the definition of thebeam current parameter the absolute value of the norm�.

To determine the boundary conditions, let us representeach of S coupled waves as a sum of three partial waves thatcorresponds to the cubic dispersion equation for each iso-lated wave As=�l=1

3 Cl�s�eihlz and use similar representation

for small perturbations in electron energy and phase. Then,the absence of energy and phase modulations at the entrancecan be defined with the use of, respectively, Eqs. �1� and �2�as

�l=1

3 Cl�s�

hl= 0, �

l=1

3 Cl�s�

hl2 = 0, �11�

and condition for zero amplitude of the backward wave in acertain cross section defining the starting length can be givenfor each wave as

�l=1

3

Cl�s�e−ihlLst = 0. �12�

So, to obtain the self-excitation conditions for backwardwaves we should solve the set of Eq. �6� �with changed signbefore the last term� and Eqs. �11� and �12�. These solutionsshould determine the starting length and the oscillation fre-quency. The latter is contained in Eq. �6� in the normalizeddetuning �ks=�ph,s

−1 −�0−1 dependent of the wave frequency.

The number of such solutions increases with the number ofcoupled waves S. Of course, the most dangerous among themis the solution yielding the smallest starting length.

III. EFFECT OF WAVE CROSS COUPLING

As was explained above, the cross coupling betweenwaves appears when the determinant defined by Eq. �6� con-tains nonzero nondiagonal terms. Let us consider this effectfor forward and backward waves separately.

A. Cross coupling of forward waves

Consider as an example the case of two coupled forwardwaves. For simplicity, assume that self-coupling coefficientsof both waves are equal: Dss=CP. Then propagation con-stants in Eq. �6� can be normalized to CP. Let us representaxial wave numbers of both waves as kz1=kz0+�k andkz2=kz0−�k where kz0= �kz1+kz2� /2 is the mean value of theaxial wave numbers. Then, introducing b=� /CP �where�=1 /�ph−1 /�0 is the detuning between the electron initialvelocity and the phase velocity of the wave with the meanaxial wave number� and �=�k / �� /c�CP one can reduce Eq.�6� to the following form:

�h2�h − b� + 1�2 − �2h4 = � . �13�

In Eq. �13� the term in right hand side describes the ratio ofcross-coupling terms to the Pierce gain parameter in a corre-sponding power:

� =D12

3 D213

CP6 . �14�

When the self-coupling coefficients for two waves arenot equal, one can normalize all parameters in Eq. �6� to thePierce gain parameter of the first wave. Then, instead of Eq.�13�, one readily derives a little more complicated equation:

�h2�h − b − �� + 1��h2�h − b + �� + C� = � . �15�

Here C is the ratio of diagonal terms for two waves:C=D22

3 /D113 =CP�2� /CP�1�. When C=1 Eq. �15� reduces to

Eq. �13�.Note that when the functions �s describing transverse

distribution of the axial components of synchronous wavesare real D12

3 =D213 the cross-coupling parameter is also real

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and positive. When the cross coupling is absent, Eq. �13� and�15� split into two dispersion equations for two separatewaves similar to those analyzed by Pierce.15 Now the veloc-ity parameter b defined for the mean value of the axial wavenumber is shifted by �� for the waves with different axialwave numbers and the presence of coupling, as follows fromEq. �13�, should change the real and imaginary parts ofpropagation constants, i.e., affect the gain and the phase ve-locity of the waves.

Results of numerical study of the sixtth order Eq. �13�are shown in Fig. 1. Here Fig. 1�a� shows old results15 for thecase when �=0. Solid and dashed lines depict imaginary andreal parts of solutions, respectively. In Fig. 1�b� results aregiven for �=0.1 and �=0.1. The presence of cross-couplingcauses splitting of solutions: one of the waves has a larger

increment than in the case of an isolated wave, while anotherhas a smaller one. The effect of increasing the detuning be-tween wave numbers is illustrated by Figs. 1�c� and 1�d�which are plotted for �=0.2 and �=0.3, respectively. As thedetuning � increases, in the range of � from 0.2 to 0.25 thereis an intersection of two solutions for real parts of complexwaves �see Fig. 1�c� where such intersection takes place atb�1.1�. However, at larger �’s, starting from ��0.26, thereis no intersection of real parts, but the intersection of imagi-nary parts appears, as shown in Fig. 1�d� for b�0.7. As thecross coupling described by the parameter � increases, sodoes the splitting of propagation constants of two waves, asshown in Fig. 1�e� where �=0.2 and �=0.1. �Results shownin this figure should be compared with those in Fig. 1�b�.�

FIG. 1. Real and imaginary parts of wave propagation constants as functions of the detuning between electron and wave phase velocity �velocity parameter�for different values of the cross-coupling parameter � and the mismatch between axial wavenumbers of two waves �: �a� �=0, �b� �=�=0.1, �c� �=0.1,�=0.2, �d� �=0.1, �=0.3, and �e� �=0.2, �=0.1. Both waves have the same coupling coefficient B to the beam.

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B. Cross coupling of backward waves

Let us now consider in the same fashion the effect ofwave coupling on excitation of backward waves. For sim-plicity, assume again that there are only two coupled wavesand that self-coupling coefficients of both waves are equal.Then, the dispersion equation equivalent to Eq. �13� for for-ward waves can be written in the following form:

�h2�h − b� − 1�2 − �2h4 = � . �16�

In a more general case when two coupled waves havedifferent self-coupling coefficients, the dispersion equationcan be given as

�h2�h − b� − 1��h2�h − b − �� − C� = � . �17�

This equation is equivalent to Eq. �15�.Results of calculations for starting conditions of coupled

backward waves are shown below. Figures 2–4 illustrate thecase of equal Pierce gain parameters for both waves, i.e.,solution of Eq. �16� with corresponding boundary conditionsgiven above. In Figs. 2 and 3 real and imaginary parts of thesolutions of Eq. �16� are shown in the absence of cross cou-pling ��=0—Fig. 2� and in its presence ��=0.2—Fig. 3�.The number of solutions shown in the figures is smaller thansix because some of them are degenerate. In the absence ofcross-coupling �Fig. 2� solutions for two waves are displaced

along the horizontal axis because as one can easily find fromEq. �16� the role of velocity parameter for each of the un-coupled waves is played by either b+� �for one wave� orb−� �for another wave�. Each of these uncoupled normalwaves consists of three partial waves, two of them are com-plex conjugate while the third one has only real part. In thepresence of cross coupling, as shown in Fig. 3, when thevelocity parameter b is smaller than 1.04 all partial waveshave nonzero imaginary parts but all real parts are degener-ate �there are only three solutions shown in Fig. 3�a��, whileat larger velocity parameters two partial waves have zeroimaginary parts but the degeneracy of their real parts isbroken.

Figure 4 illustrates the method of solving the transcen-dental Eq. �12�, which is the boundary condition formulatingthe absence of incoming waves from the collector side. Heresolid and dashed lines show solutions of real and imaginaryparts of this equation, respectively. So, the intersection ofthese lines determines solutions of Eq. �12�, i.e., eigen-numbers for the starting length and the velocity parameter.As one can see, in addition to the primary solution yieldingthe smallest starting length �Lst=1.9917, which because of aweak wave coupling is slightly different from the solutionfound in Ref. 17 for a single-wave backward-wave oscillator�BWO�� there is also the next solution at larger values of the

FIG. 2. Real and imaginary parts of propagation constants of backward waves having the same coupling to the beam in the absence of cross coupling.

FIG. 3. Real and imaginary parts of propagation constants of the same backward waves in the presence of cross coupling.

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starting length and velocity parameter. �These solutions aremarked by numbers “1” and “2” in the figure.� The secondsolution yields the starting length Lst

�2��3.7594 which is al-most two times larger than the first one. Since our variablesare normalized to the Pierce gain parameter proportional tothe cubic root of the beam current, this difference in startinglengths for a system of a given length means that the startcurrent of the second mode is about 6.7 times higher than thestart current of the primary mode. This agrees with resultsshown in Fig. 18 of Ref. 17 for the zero value of the space-charge parameter where it was shown that the ratio of startcurrents of spurious to fundamental mode in a single-modebackward wave oscillator is about 6.4.

Figures 5 and 6 illustrate the case when Pierce gain pa-rameters of two waves are different. In Fig. 5, axial profilesof the waves are shown; solid and dashed lines show theabsolute value of the wave amplitude and its phase, respec-tively. Figures 5�a� and 5�b� correspond to the fundamentaland second modes, respectively. The second mode has twoaxial variations, as expected. In both Figs. 5�a� and 5�b� twoprofiles are shown which correspond to two solutions of our

set of equations. Of course, the solution yielding a smallerstarting length �solution 1� is more dangerous.

Resulting dependences of the starting length and veloc-ity parameter on the cross-coupling coefficient � are shownin Fig. 6 by solid and dashed lines, respectively. Figure 6�a�corresponds to the case of equal Pierce gain parameter forboth waves, while Fig. 6�b� corresponds to different param-eters. In the case shown in Fig. 6�a� the values of the startinglength, Lst=1.973, and the velocity parameter, b=1.522, forzero cross coupling between the waves ��=0� are the sameas in Ref. 17. The velocity parameter of the second wave issmaller by 0.2 because for the second wave the role of ve-locity parameter is played by the sum b+� and in the caseshown �=0.2. In the case shown in Fig. 6�b�, when thecross coupling is absent, the starting length of the secondwave is 0.5−1/3 times larger than the starting length of thefirst wave that corresponds to the difference in the Piercegain parameters.

The most important conclusion which can be drawnfrom Fig. 6 is the fact that, as the wave cross coupling in-creases, the starting current of one wave decreases, i.e., thecross coupling between backward waves makes a devicemore prone to parasitic self-excitation. For instance, in thecase shown in Fig. 6�a�, when cross coupling increases fromzero to �=0.5, the starting current in a device of given in-teraction length decreases by about 43%. Also, as followsfrom Fig. 6�b�, when in a device with nonequal Pierce gainparameters of two waves their cross coupling increases fromzero to �=0.2, the starting current of the fundamental modedecreases by about 14%. Comparison of these two numbersallows one to conclude that the effect of cross coupling is thestrongest when the waves are equally coupled to the beam.This conclusion agrees with the general theory of coupledoscillatory systems, stating that the coupling between twooscillatory systems is strongest when these systems haveidentical parameters.

IV. EFFECT OF TRANSVERSE NONUNIFORMITY

In this section we analyze the effect of the transversenonuniformity of the rf field on the beam-wave and wave-wave interactions. First, we consider the effect of transversenonuniformity �in the vertical and horizontal directions� on

FIG. 4. Solutions of the real �solid lines� and imaginary �dashed lines� of theboundary condition �12�. Eigenvalues of the velocity parameter and startinglength are determined by intersections of these curves.

FIG. 5. Axial profiles of the amplitude �solid lines� and phase �dashed lines� of the fundamental �a� and parasitic �b� modes.

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the coupling of a single sheet electron beam to an isolatedwave. Then, we analyze the effect of splitting this beam intoseveral parallel beamlets on the beam coupling to an isolatedwave. Lastly, we study the effect of transverse nonuniformityon the multibeam coupling to two coupled waves

A. Coupling of a single beam to an isolated wave

1. Effect of nonuniformity in the vertical direction

Let us start from analyzing the effect of rf field nonuni-formity in the vertical direction. Assume that a waveguide isuniform in a wide horizontal y-direction and a beam has aconstant thickness in the vertical direction, so the beam andwaveguide parameters do not depend on the horizontaly-coordinate. Such configuration is shown schematically inFig. 7 and is used here for illustrative purposes only.

In this case the parameter B characterizing the role oftransverse nonuniformity of the rf field on the small-signalgain is equal to �cf. Ref. 11�

B3 = Bx3 =

1

2�1 +

sin w

w� , �18�

where w=m��Wb /W� is proportional to the ratio of the beamthickness Wb to the distance between plates W, m is a num-ber of field variations in the vertical direction. So, as thebeam thickness increases, this parameter decreases from 1�for a very thin beam� to 0.5. The latter corresponds to abeam filling all space between plates when the vertical indexof mode equals one or to a beam filling the space corre-sponding to one vertical variation when this index is largerthan one. Equation �18� is valid for the case when the wavenumber in the vertical direction is real. When plates containa periodic structure �like a grating� this number can beimaginary. In such case �see, e.g., Ref. 15�

B3 = Bx3 =

1

2 cosh��W/2��1 +sinh��Wb�

�Wb� . �19�

Here � is the absolute value of the vertical wavenumber.

2. Effect of rf field nonuniformityin the wide horizontal direction

Consider two systems: waveguide with open ends anddielectric loaded waveguide.

a. Waveguide with open ends. Consider the space be-tween two plates unbounded from both sides. In such openwaveguide, the waves have transverse structure in the widehorizontal direction, which can be described by Hermitepolynomials Hn. The functions describing the distribution ofthe axial electric field of the wave in the y-direction can begiven as18,9

Ez,n = fn = e−i�n+1/2� 1n!

Hn���e−�2/4. �20�

In Eq. �20�, n=0,1 ,2 , . . . where �n+1� is the number of fieldvariations along the y-axis, is the phase factor unimportantfor our consideration, the variable � is introduced as�=4y /Ly where the distance Ly characterizes the width offield peaks in the y-direction. Hermite polynomials in Eq.�20� are determined, in accordance with Ref. 18, as

Hn = �− 1�ne�2/2 dn

d�n �e−�2/2� . �21�

Note that such distributions corresponding to real wavenum-bers in the wide horizontal direction can be realized in thecase of slow waves when the axial wavenumbers in the nar-row vertical direction are imaginary. Since now we are inter-ested in the effect of the horizontal nonuniformity, we will

FIG. 6. The starting length and the velocity parameter as functions of the cross coupling between two waves with equal �a� and nonequal �b� Pierce gainparameters.

FIG. 7. Cross section of an electron slab positioned between two conductingplates.

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calculate the coupling factors By3 which for the case of sev-

eral beamlets was given by Eq. �7b�.Substituting the functions given by Eqs. �20� and �21� in

Eq. �7b� one can study the effect of the beam width on thecoupling to various modes and effect of beamlet location on

the coupling to a given mode. Here we will study the modeswith one, three, and five variations interacting with a singlesheet beam. We consider only the modes with odd number ofvariations because there is a cross coupling between them,while the cross coupling between modes with odd and evennumber of variations for a single sheet beam located sym-metrically with respect to the midplane of this circuit is ab-sent. For such a beam �7b� yields �cf. Ref. 10�

By3 =

�

2b��b� , �22�

where ��b� is the error function tabulated elsewhere19 andits argument is proportional to the ratio of the beam width2Lb to Ly: b=22�Lb /Ly�.

The dependence of the coupling parameter �22� onthe normalized beam width expressed via the parameter�b=4Lb /Ly is shown in Fig. 8�a�. �Figures 8�b� and 8�c� il-lustrate the effect of splitting one sheet beam into severalbeamlets and will be discussed later.� As follows from Fig.8�a�, when Lb is less than or equal to Ly, the beam couplingto the mode with one variation is much stronger than tomodes with three and five variations. In the region of Lb

�1.5Ly the values of this parameter for modes with one andthree variations are almost the same and, when Lb�2Ly, theyare rather close for all modes considered.

b. Dielectric loaded waveguide. Next we consider a di-electric loaded waveguide where two dielectric layers areattached to the top and bottom walls of a rectangular wave-guide �see Fig. 9�. The beam is passing between two dielec-tric layers, interacting with the slow propagating wave. Simi-lar waveguide geometry was studied in Ref. 4 �see alsoreferences therein�, but there a solid cylindrical beam wasconsidered, while we study the device with a sheet electronbeam, which can be split into several parallel beamlets.

The axial electric field is a combination of trigonometricfunctions for the y direction and exponential functions forthe x direction in between the dielectric layers. The functionsdescribing the distribution of the axial electric field of thewave in transverse direction for the lowest symmetric modeare

FIG. 8. Coupling coefficients as functions of the normalized width for adevice with one �a�, three �b�, and five �c� beamlets. Solid, dashed, anddotted lines the coupling coefficients for modes with one, three and fivevariations, respectively.

FIG. 9. Cross section of a sheet electron beam positioned between twodielectric layers in a rectangular waveguide.

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Ez�x,y� = cos�kyy�

��cos�k̄xh�

sin�k̂xd�sin�k̂x�a − x�� h x a

cos�k̄xx� − h x h

cos�k̄xh�

sin�k̂xd�sin�k̂x�a + x�� − a x − h .� �23�

In Eq. �23�, ky =� /b, k̄x, and k̂x are the transverse wavenum-bers in the center and dielectric sections, respectively, satis-fying the following dispersion equation:

cos�k̄xh�

sin�k̂xd�=

k̂x

k̄x

sin�k̄xh�

� cos�k̂xd�. �24�

Above, 2h is the width between the dielectric layers, d is thewidth of the dielectric layer, and a=h+d. The dependence ofthe coupling factors on parameters of a sheet beam in thismodel is illustrated by Fig. 10 which is similar to Fig. 8.

B. Coupling of a multiple sheet beamto an isolated wave

Beam coupling to the waves in open structures was de-termined in Ref. 10 for sheet electron beams having rectan-gular and elliptical cross sections. Below we analyze the ef-fect of splitting sheet beams of rectangular cross section intoseveral parallel beamlets on the beam coupling to themodes with different indices n. When we have a set ofbeamlets with center coordinates yc,j and the width in thehorizontal direction �� j the coupling parameter is deter-mined by Eq. �7b�.

Let us consider the case when a sheet electron beam issplit into several beams and analyze the effect of the width ofsuch beamlets on the coupling to the modes under study.Results of this analysis are shown in Figs. 8�b� and 8�c�

which correspond to the cases of three and five beamlets,respectively; solid, dashed and dotted lines show the cou-pling coefficients By

3 to modes with one, three and five varia-tions. In the case shown in Fig. 8�a�, which was briefly dis-cussed above, a center of a single beam is located at �0=0,the coupling coefficients are shown as functions of the beamwidth ��.In the case shown in Fig. 8�b� the centers of threebeamlets are located at �01=−5, �02=0, and �03=5. Thesecoordinates correspond to peaks of the mode with threevariations. Correspondingly, the maximum width of beamletsis equal to ��max=5; when ��=��max the beamlets merge.Finally, in Fig. 8�c� the case of five beamlets is shown. Inthis case, coordinates of the beamlet centers are chosen to beequal to �01=−3.2, �02=−1.6, �03=0, �04=1.6, and �05=3.2.Such positions are close to the peaks of the mode with fivevariations. Correspondingly, the maximal normalized widthof such beamlets is equal to 1.6.

As one can see in Fig. 8�b�, when there are three narrowbeams positioned in the peaks of the mode with three varia-tions, these beams are coupled to the desired mode by morethan 50% stronger than to the mode with one variation and4.5 times stronger than to the mode with five variations.However, when the width of these beamlets approaches itsmaximum value, the coupling to the mode with three varia-tions becomes even a little smaller than to the mode with onevariation and remains larger than that to the mode with fivevariations only by about 25%, so the concept looses its se-lectivity practically completely. In the case of five beamletsshown in Fig. 8�c�, the situation is quite similar: when beam-lets are narrow, the mode with five variations has an obviousadvantage over other modes, but it loses this advantage whenthe beam width approaches its maximum. Similar conclu-sions can be drawn from Fig. 10 illustrating the effect ofbeam coupling to the waves in dielectric loaded waveguides.

C. Cross coupling of waves in multiple sheet-beamconfigurations

In this case, there are too many parameters for doinggeneral treatment of the problem. Therefore to illustrate theapplicability of this formalism consider the following ex-ample. Assume that a wave with three variations in they-direction �m=3� is chosen as the operating wave; corre-spondingly, an electron beam is split into three beamlets andthe normalized width of these beamlets is equal to 1.0. Thecenters of these beamlets are located at y0=0 and y0= �5as was assumed above, in Sec. IV B. Thus, the filling factorfor these beamlets, i.e., the ratio of beamlet width to thespacing between them is equal to 1 /5�0.447. As followsfrom Fig. 8�b�, the beam coupling parameter defined byEq. �7b� for this choice of a mode and a beam is equal toBm=3

3 =1.6. Consider now two cases of competing waves. Inthe first case, assume that the parasitic wave is the mode withone variation �m=1�. For this mode the beam coupling pa-rameter is equal to Bm=1

3 =1.15, i.e., it is large enough; how-ever, the cross-coupling parameter is rather small. As followsfrom Fig. 4�b�, this parameter is equal to �31

3 =−0.17. Thesenumbers mean that in Eq. �15� the parameter in the LHS isequal to C=0.72 and the cross-coupling parameter in its right

FIG. 10. Coupling coefficients as functions of the normalized width for adielectric loaded device with one �a�, three �b�, and five �c� beamlets. Solid,dashed, and dotted lines the coupling coefficients for modes with one, three,and five variations, respectively.

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hand side is equal to �=0.0113. So, the effect of cross cou-pling in this case should be negligibly small. Consider nowanother case: assume that the parasitic wave is the wave withfive variations in the y-direction �m=5�. As follows fromFigs. 8�b� and 11�b�, now we have B=0.3 and �=0.09. So, inthis case, in line with results shown in Fig. 11, the effect ofcross coupling on the wave growth should be more signifi-cant. To give here a more accurate numbers, it is necessary tospecify other dimensions of a waveguide, which would char-acterize the difference in cutoff frequencies, and beam pa-rameters. Then, one will be able to determine a correspond-ing value of the parameter �, for which solutions of Eq. �15�should be found.

Further demonstration of the presented formulation isthe cross-coupling coefficients for waves in the dielectricloaded waveguide considered in the example above. Thesecross-coupling coefficients are shown in Fig. 12 for the samebeam parameters used in Fig. 8 �namely, one, three, and fivebeamlets positioned accordingly�. It is interesting to note thatthe three to five coupling term is extremely small in thiscase. However, choosing a different positioning for thebeamlets we can find different values for the cross-couplingcoefficients. Of course, changing the position of the beamletsaffects not only the cross-coupling terms but also the lineargain term. It can therefore be concluded that in choosing thenumber and position of beamlets in a multiple sheet-beam

device care must be taken to minimize the cross couplingbetween high-order modes and maximize the gain of thelower order modes.

V. DISCUSSION

Let us estimate the frequency separation of possiblycompeting waves in a waveguide with closed walls. Considertwo waves at the same frequency and assume that both ofthem have only one variation in the vertical direction �thedistance between corresponding walls denote by a� and dif-ferent number of field variations m1,2 in the wide horizontaldirection whose width is A times larger than a �here A is theaspect ratio�. The frequency separation of these two wavescan be estimated with the use of the obvious relation be-tween the frequency and wave numbers

��

c�2

= kz1,22 + ��

a�2�1 + �m1,2

A�2� . �25�

Represent axial wave numbers as kz2=kz0+�k and kz1=kz0

−�k where the separation between axial wave numbers issmall enough: �k�kz0. In these notations the difference inaxial wave numbers normalized to � /c can be given as

FIG. 11. Cross-coupling coefficients as functions of the normalized width of beamlets in a waveguide with open ends.

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�k� =�k

�/c=

�

8

�

kz0a2

m12 − m2

2

A2 . �26�

In the small-signal theory developed above, we use the de-tuning of the axial wave numbers normalized to the Piercegain parameter �=�k� /CP. As follows from Eq. �26�, thisdetuning is small enough ���1� when

A2 ��

8

�

kz0a2

m12 − m2

2

CP. �27a�

For instance, for waves with one and three variations in widedirection this means

A2 � ��

kz0a2

1

CP. �27b�

This condition can be rewritten one more time as the ratio ofthe waveguide width W=aA to the wavelength

aA

�=

W

�� �ph

2CP, �27c�

where the wave phase velocity is defined for the mean valueof the axial wavenumber �ph=� /ckz0 and for the wave to bein synchronism with electrons this velocity is close to theelectron axial velocity. The last fact being taken togetherwith the known dependence of the Pierce gain parameter onthe voltage yields the following scaling law:

aA

�� �aA

��

min Vb

5/12, �28�

which shows that the effects studied above become essentialfor devices in which the normalized aspect ratio exceeds theminimal value proportional to the operating voltage in 5/12power.

In the design of a W-band sheet-beam broadband TWTdone at NRL �Ref. 20� a 20 kV, 4 A electron beam is usedand the impedance is approximately 4 �, with a beam tun-nel aspect ratio of 4:1. Correspondingly, the value of the

Pierce gain parameter is close to 0.06 and, hence, the beamtunnel width obeying Eqs. �27a� and �28� should exceed 1.5wavelength, while in the present design this width is severaltimes smaller. This fact is illustrated by Fig. 13 showing thedispersion diagram for a sheet-beam coupled-cavity TWThaving relatively small �a� and high �b� beam tunnel aspectratios �4 and 10, respectively�. The dispersion curves shownin the right figure have a large number of potentially com-peting modes. However, among them, close axial wavenum-bers have only pair of modes one with odd and another witheven number of variations in a wide horizontal direction.Such modes, in the case of a beam injected symmetricallywith respect to the waveguide axis, are not coupled. Themodes with odd number �one, three, and five� of variationsare still well separated.

For backward waves in the identical structure impedancevalues are typically significantly larger, of the order of 50 �.Correspondingly, the value of the Pierce gain parameter isclose to 0.14, and hence the beam tunnel width satisfyingEqs. �27a� and �28� would be of the order of 1.0 wavelengthin this device. While this is still larger than the beam widthfor the low aspect ratio structure, it is close to the beamwidth in the high aspect ratio structure �Fig. 13�b��. Hence,the effect studied above in a given design should be ratherweak, but could become important for higher aspect ratiodesigns, in which the starting current for backward waveoscillation could be reduced by this mechanism.

VI. SUMMARY

Results obtained above in Sec. II characterize a trade-offbetween the selectivity of a desired mode and the limitationson achievable current density. Indeed, to realize the sametotal current being unable to compress the beamlets in thewide horizontal direction, the beam current density in thecase of narrow beamlets should be higher than in the case ofwide beamlets. These results can help developers to estimatethe beamlet width and their positioning optimal for the caseof a limited current density. The formalism developed canalso be used for evaluating the same effects in waveguides ofarbitrary cross sections. This formalism can also be adoptedfor analyzing the effect of the transverse nonuniformity ofthe interaction space on excitation of parasitic backward

FIG. 12. Cross-coupling coefficients as functions of the normalized width ofbeamlets in a dielectric loaded waveguide.

FIG. 13. Dispersion diagrams for sheet-beam coupled-cavity TWTs havingbeam tunnel aspect ratios of �a� 4:1 and �b� 10:1. Odd-numbered curvescorrespond to modes having axial electric field distributions symmetricacross the beam width, while even-numbered curves correspond to antisym-metric modes.

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waves. Results obtained in Sec. IV can be important forevaluating the role of cross coupling between various modeson the performance of sheet-beam traveling-wave amplifierswith a high aspect ratio and different number of electronbeamlets. Results presented in Sec. V indicate that forpresent-day sheet-beam traveling-wave tubes the effectsstudied in this paper are still rather weak, but a further in-crease in the aspect ratio in sheet-beam TWTs of the nextgeneration can make the effect of wave coupling significant,especially for excitation of parasitic backward waves.

ACKNOWLEDGMENTS

This work was supported by the Office of NavalResearch.

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