A generalized linear theory of the discrete electron–wave interaction in slow wave structures

ISSN 1064�2269, Journal of Communications Technology and Electronics, 2010, Vol. 55, No. 11, pp. 1271–1284. © Pleiades Publishing, Inc., 2010.Original Russian Text © V.A. Solntsev, R.P. Koltunov, 2010, published in Radiotekhnika i Elektronika, 2010, Vol. 55, No. 11, pp. 1362–1374.

1271

INTRODUCTION

The analysis of the equations of the discrete electron–wave interaction in slow�wave structures (SWSs) showsthat this interaction developing in SWS passbands andstopbands can uniformly be described with the use of thesecond�order finite�difference excitation equation andthe local coupling impedance entering this equation [1].The application of the theory of waveguide excitationenables us to represent the equations in a general formwithout using partial equivalent circuits or other modelSWSs.

At the same time, in contrast to Pierce’s theory [2],where the coupling impedance characterizes the interac�tion of electrons with a single SWS eigenmode via a singlespatial harmonic of this wave that is synchronous with thebeam, the local coupling impedance characterizes theinteraction of electrons with the total field of two—for�ward and counterpropagating—SWS waves in interactiongaps and does not become infinite at cutoff frequencies[1, 3, 4]. Various versions of the description of such inter�action have repeatedly be considered in the literature [5–9]. The approach based on the simultaneous use of (i) thedifference excitation equation derived in [10] from thegeneral theory of waveguide excitation and (ii) the localcoupling impedance entering this equation is the mostrigorous. Therefore, it is reasonable to apply the local cou�pling impedance rather than Pierce’s coupling impedancein the theory of the discrete electron–wave interaction.This technique eliminates the difficulties related with theinfinite value of Pierce’s coupling impedance at cutoff fre�quencies and makes it possible to consider the processesdeveloping during the passage from a passband to a stop�band, including the passage from the Cerenkov mecha�nism of interaction in a traveling�wave tube (TWT) to theklystron mechanism.

The local impedance, as well as the usual couplingimpedance, can be calculated and analyzed during pro�cessing of the results of 3D simulation of electromagneticfields in cold SWSs with the use of the well�known HFSS,ISFEL�3D, etc., codes; simplified model SWSs in theform of equivalent circuits and systems, including multi�port chains; or other methods [11].

In this study, a linear theory of the discrete electron–wave interaction occurring in SWS passbands and stop�bands is developed on the basis of the difference excitationequation and the local coupling impedance. The featuresof the equations for pseudoperiodic SWSs where the stepand parameters of a structure’s elements consistentlychange are considered. This analysis makes it possible toseparate one spatial harmonic of the operating mode, theremaining harmonics being suppressed [12].

1. THE ORIGINAL EQUATIONS OF THE LINEAR ELECTRON–WAVE

INTERACTION AND AN ALGORITHMOF THEIR SOLUTION

Consider a rectilinear electron flow in an SWS with Qinteraction gaps that, generally, are nonequidistantlyspaced with step Lq (Fig. 1). We restrict the considerationto a 1D model and use the known linear equation [1, 13]for high�frequency (HF) current J of the beam

(1)

where is the electron wave number; is the plasma wave number; z is the longi�

tudinal coordinate; ω is the circular frequency; S is theeffective area of the beam’s cross section; Г is thedepression coefficient (determined from known rela�

( )p p

22 2 2

02 2 ,e ed J dJih h h J ih S E

dzdz− + Γ − = − ωε

ve eh = ω

p p eh = ω v

MICROWAVE ELECTRONICS

A Generalized Linear Theory of the Discrete Electron–Wave Interaction in Slow�Wave Structures

V. A. Solntsev and R. P. KoltunovReceived June 8, 2010

Abstract—A linear theory of the discrete interaction of electron flows and electromagnetic waves in slow�wave structures (SWSs) is developed. The theory is based on the finite�difference equations of SWS excitation.The local coupling impedance entering these equations characterizes the field intensity excited by the elec�tron flow in interaction gaps and has a finite value at SWS cutoff frequencies. The theory uniformly describesthe electron–wave interaction in SWS passbands and stopbands without using equivalent circuits, a circum�stance that allows considering the processes in the vicinity of cutoff frequencies and switching from the Cer�enkov mechanism of interaction in a traveling�wave tube to the klystron mechanism when passing to SWSstopbands. The features of the equations of the discrete electron–wave interaction in pseudoperiodic SWSsare analyzed.

DOI: 10.1134/S1064226910110100

1272

JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 55 No. 11 2010

SOLNTSEV, KOLTUNOV

tionships [13]) of the longitudinal field induced by the

spatial charge at frequency is the plasma

frequency determined with disregard of the depressionof the longitudinal field of the spatial charge, i.e., atΓ = 1; the effective area of the cross section of the elec�tron beam is determined by the relationship

is the beam’s effective direct current; ε0 is

the permittivity of free space; is the sum of thelongitudinal components of the forward� and counter�propagating�wave electric fields that is averaged overthe section of the beam [1]

(2)

In expression (2), are the dimensionlessexcitation coefficients (wave amplitudes) determinedby the equations [7, 13]

(3)

where

are the distribution functions (averaged over the sec�tion of the beam) of the longitudinal electric field ofthe forward and counterpropagating waves,

is the specific coupling impedance at point (x0, y0, z0),ψ(х, у) is the distribution function of the HF currentdensity in the cross section of the beam, Ns is the wave

norm, is the amplitude of the chosen field compo�

p0

0 e

Jem S

ω =

ε v

2

1 ,( , )

eS

Sx y dS

=

ψ∫

0 ee

SJ JS

=

( )E z

0 0

( )

( ) ( )exp( ) ( ) ( )exp( ).s s s s s s

E z

C z e z ih z C z e z ih z− −

= + −

0 ( )sC z±

0 0

( ) ( )exp( ),2

s ss s

dC R J z e z ih zdz

±= ∓∓ ∓

,( ) ( , ) ( , , )

e

s s z

S

e z x y e x y z dS± ±

= ψ∫

0 00 2 s ss

s

E ERN

−

= −

0sE

±

nent at point (x0, y0, z0) where the distribution func�tion of this component is unity.

In the case of the discrete interaction between elec�trons and the field of a periodic or pseudoperiodicstructure, as a rule, we can neglect the field phase vari�ation in the interaction space within one period (step)of a structure along the SWS axis, i.e., within the inter�action gap. This assumption is evident when the inter�action gap has a sufficiently small width and when theinteraction develops at the frequencies beyond theSWS passband, a situation where the field phase iseither the same over the entire volume of the structureor stepwise changes by π from gap to gap.

The analysis performed in [1] shows that field (2) at

the qth step ( ) can be represented in the form

(4)

where = is the real functionof the field distribution averaged over the beam’s sec�tion. This function is the same for forward and coun�terpropagating waves.

For plane interaction gaps of width dq, we have within a gap and beyond it. For con�

tinuous functions , we can introduce an equiva�lent plane gap of width dq. We can determine this widthassuming that the field intensity at the center of theequivalent plane gap is Eq and introducing voltage Uq

across the equivalent gap:

(5)

The induced current within the qth step is specified bythe expression

(6)

q qz z z− +

≤ ≤

( ) ( ) ,q q qE z e z E=

( )qe z ( ) ( )[ ]exps s qe z ih z z± ±

−

( ) 1qe z ≡ ( ) 0qe z ≡

( )qe z

( )

( )

q

q

q

q

z

q q z

zqq q

q qz

E e z dzU

d e z dzE E

+

+

−

−

= − = − =

∫∫

,

.

( ) ( )1 .q

q

z

q qq

z

J J z e z dzd

+

−

= ∫

Cap

Electronflow

E+S E–S

zzq–1

Lq–1 Lq

Cap Capq–1 q+1q

zq+1zq

Fig. 1. Schematic of an SWS with the discrete electron–wave interaction.

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 55 No. 11 2010

A GENERALIZED LINEAR THEORY OF THE DISCRETE ELECTRON–WAVE 1273

For periodic structures with period L and identicalequivalent interaction gaps of width d, we have the finite�difference excitation equation [1, 10]

(7)

where = is a second�orderfinite difference and This equation can berepresented as an equation for the voltages across equivalent gaps:

(8)

where is the local coupling imped�ance simultaneously taking into account the forwardand counterpropagating waves. It is important that thisimpedance is not infinite at the boundaries of thetransparency band of the SWS and is continuous inpassing from one band to another [1, 4].

According to (3) and (5), equivalent width d of theinteraction gap depends on the distributions of thefield and HF current density over the section of thebeam. Therefore, it is reasonable to introduce the

averaged specific coupling impedance in the considered 1D interaction model and representZs in the form

(9)

This averaging is similar to the averaging of the cou�pling impedance over the section of the electron beamfor the case of continuous interaction in TWTs withsmooth, for example, helical, SWSs [13].

A finite�difference excitation equation of form (8) canalso be obtained when a periodic SWS is represented as achain of identical two�ports, each describing one periodof the SWS. In this case, the local coupling impedance fora folded SWS coincides with one element of the transmis�sion matrix of a two�port accurate to the imaginary unity[4, 14]:

(10)

( )2 02 1 cos sin ,q q s s s qE E iR J dΔ + − ϕ = − ϕ

2qEΔ 1 12q q qE E E

+ −− +

.s sh Lϕ =

q qU E d= −

2 2 (1 cos ) ,q q s s qU U iZ JΔ + − ϕ =

0 2 sins s sZ R d= ϕ

( )20

s sR R d L=

2 sin .s s sZ R L= ϕ

12.sZ iA=

We represent pseudoperiodic structures with variablestep (spacing interval) Lq between interaction gaps andcell parameters changing consistently with the step as achain of nonidentical two�ports. The general finite�differ�ence relationships describing excitation with a preas�signed current of chains of nonidentical three�ports aregiven in [15]. Here, we follow [14] and present a simplederivation of the second�order finite�difference excitationequation for a chain of nonidentical two�ports that isshown in Fig. 2.

When the exciting current is Jk, the currents and volt�ages at the (k – 1)th and kth steps are coupled as follows:

(11)

Eliminating from these relationships currents Ik

and Ik+1 and taking into account the reciprocity con�

dition for a two�port we obtain thedifference equation of excitation of a chain of non�identical two�ports that describe a pseudoperiodicSWS

(12)

For a periodic SWS, the elements of the transmissionmatrix are identical for all of the two�ports, and Eq. (12)

coincides with (8), because

Equations (1), (6), (7), and (8) or (12) form a self�con�sistent system of equations describing the linear discreteelectron–wave interaction in the passbands and stop�bands of an SWS. This system should be completed withthe boundary conditions at the ends of a chosen SWS seg�ment. The boundary conditions are determined by thematching conditions of the cold structure and the modu�

( )

( )

11 1 12 1

1 11 11 12

21 1 22 1

1 11 21 22

,

,

,

.

k kk k k

k kk k k k

k kk k k k

k kk k k k

U A U A I

U A U A I J

I J A U A I

I A U A I J

+ +

− − −

−

+

+ +

− − −

−

= +

= + −

+ = +

= + −

11 22 12 21 1,k k k kA A A A− =

112 121 1 22 11 121 1

12 12

.k k

k k kk k k kk k

A AU U A A U A JA A

−

+ −− −

⎛ ⎞+ − + = −⎜ ⎟

⎝ ⎠

( )11 221cos .2

s A Aϕ = +

A1 Ak – 1 AQAk + 1Z1 ZQ

Ik + 1Ik – 1

Uk–1 Uk + 1

Ik

Uk

JkGe

q = 1 q = k – 1

Ak

q = Qq = k q = k + 1

J+kJ–

k

Fig. 2. Chain of nonidentical two�ports excited by induced currents Jk; Z1 and ZQ are the load impedances at the ends of the chain.

1274


SOLNTSEV, KOLTUNOV

lation of the electron beam at the beginning of the struc�ture.

Consider an SWS section containing Q steps. Wesolve the boundary�value problem for the system ofequations of interaction with the help of the shootingmethod and select the boundary conditions at thebeginning of the section such that the boundary con�ditions at the end of the section are fulfilled. For thelinear theory of interaction, it suffices to apply threeiterations: two test iterations and one iteration that ischosen on the basis of test ones and yields a solutionsatisfying the boundary conditions at the ends of thesection. At each iteration, we use the solution to theequations of the linear theory of interaction obtainedbelow and represented in the form of recurrence for�mulas. These formulas make it possible to calculatethe current and field in the qth gap from their values inthe previous interaction gaps. As is seen from excita�tion equations (7) and (8), the field in a given gap isdetermined by the field value in the two previous gaps.As the boundary conditions for the field, we specify thecoefficients of reflection in the cold structure from thefirst gap, Γ1, and from a load connected to the Qth gap,ΓL. For a periodic SWS, the field of the structure gapscan be represented as the sum of the forward andcounterpropagating waves with amplitudes changing along the structure due to induced current Jq

(13)

In the last, Qth, gap, the counterpropagating wave isformed only due to the reflection (with reflection coeffi�cient ΓL) of the forward wave from the load connected tothis gap, because the external signal is not applied here.Therefore,

(14)

and we obtain for q = Q

(15)

Now, we assume that in the next to last, (Q – 1)th, gap,the amplitudes of the forward and counterpropagatingwaves are the same as in the last, Qth, gap:

. (16)

This assumption is strictly fulfilled in the cold structurewithout an electron beam. In the presence of an electronbeam, the field changes only slightly from gap to gap;therefore, relationship (16) is fulfilled approximately. As isseen from (7) and (8), (16) is fulfilled strictly if there are noinduced currents in the last two gaps; i.e.,

(17)

and the electron flow does not excite these gaps.Assuming that relationships (16) are fulfilled approxi�

, ,s qC±

( )[ ] ( )[ ], ,exp 1 exp 1 ,

1,2, .q s q s s q sE C i q C i q

q Q−

= − ϕ + − − ϕ

= …

( )( ) ( )( )L, ,exp 1 exp 1 ,s Q s s Q sC i Q C i Q−

− − ϕ = Γ − ϕ

( ) ( )( )L ,1 exp 1 .Q s Q sE C i Q= + Γ − ϕ

, 1 , , 1 ,,s Q s Q s Q s QC C C C− − − −

= =

1 0,Q QJ J−

= =

mately or strictly, we obtain from (13) with allowanceof (14) that, for q = Q − 1,

(18)

For ΓL, we find

(19)

A feed line is connected to the input of gap q = 1.This line excites forward�wave field Ein in the first gap;in addition, a forward wave is formed due to the reflec�tion (with reflection coefficient Γ1) of the counter�propagating wave from the line. Therefore, in the firstgap, and (13) yields the total field inthis gap in the form

(20)

Assume that, in the second gap,

(21)

As at the output, these relationships are exact if thereare no exciting currents in the first two gaps:

(22)

Otherwise, (21) is fulfilled approximately. With allow�ance for (21), (13) yields for the field in the second gap

(23)

We look for solutions with the help of iterations whenEin, Γ1, ϕs are specified identical at all of the iterations andonly amplitude C–s of the counterpropagating wave at thebeginning of the section is unknown. An algorithm fordetermination of this amplitude is presented below.

2. DIMENSIONLESS VARIABLESAND RECURRENCE RELATIONSHIPS

FOR THE CURRENT AND FIELD

We introduce dimensionless variables according to[13]: is a dimensionless coordinate; ε is aparameter that, generally, can arbitrarily be chosenlike, for example, the amplification parameter in aTWT or the ratio of the plasma frequency to the oper�

ating frequency; and is the spatial�charge parameter.

Dimensionless field F is determined from the relation�ship

(24)

where e and m are the electron charge and mass,respectively, and vе is the electron velocity.

( ) ( )L

L1

exp exp.

1s s

Q Qi i

E E−

− ϕ + Γ ϕ

=

+ Γ

( )

( )

( )( ) ( )

( ) ( )

L1

1

1

1

exp

exp

1 expexp 2 .

1 exp

s Q Q

s Q Q

Q Q ss

Q Q s

i E E

i E E

E E ii

E E i

−

−

−

−

− ϕ −Γ = −

ϕ −

− ϕ= − − ϕ

− − ϕ

in 1 ,1,s sC E C−

= + Γ

( )in1 ,1 11 .sE E C−

= + + Γ

,2 ,1 ,2 ,1, .s s s sC C C C− −

= =

1 2 0.J J= =

( ) ( ) ( )[ ]in2 ,1 1exp exp exp .s s s sE E i C i i−

= ϕ + − ϕ + Γ ϕ

eh zζ = ε

( )p22

σ = Γ ω εω

( ) ( )2 2exp exp ,2

ee e e

e EF E i ih zm hU

ζ= − = − −

εω ε εv



High�frequency current J(z) of the beam is repre�sented through current dimensionless amplitude I(z) as

(25)

As a result, we obtain from (1), (13), and (14) theequation for the dimensionless HF current

(26)

Specifying values and of the currentand its derivative, respectively, at the beginning of the

qth step and applying the method of variation ofconstants for solving Eq. (26), we find at the qth step

(27)

Following [1], we consider equivalent plane gaps;

then, we have for withina gap. In this case, the integrals from expressions (27)are calculated analytically. Within the qth gap, we

obtain for

(28)

where is the amplitude of the beam velocitymodulation and

(29)

is the normalized field amplitude in the gap.Beyond gaps, in the drift space of the electron

beam, we only take into account the spatial�charge

forces, use (28) at , and choose as the originat the qth step.

Now, let us modify difference equation (8) of exci�tation of a periodic SWS through introducing dimen�

( ) ( ) ( )0 exp .eJ z J I z ih z=

22

2 .d I I iFd

+ σ = −ζ

( )qI −

ζ ( )' qI −

ζ

q−

ζ

( ) ( ) ( )

( ) ( ) ( ) ( )

' sin ' '

1cos ' sin .

q

q q q q

iI F d

I I

−

ζ

ζ

− − − −

ζ = − ζ σ ζ − ζ ζσ

+ ζ σ ζ − ζ + ζ σ ζ − ζσ

∫

q qE E U d≡ = − .q qz z z− +

< <

q q e qh d− − +

ζ ≤ ζ ≤ ζ + ε = ζ

( ) ( ) ( ) ( ) ( )

( )( )( ) ( )

( ) ( ) ( )

( ) ( )

( )( ) ( )

2

2 2

2 2

1cos sin

exp 1 exp1

sincos ,

sin

cos exp1

1 exp cos

q q q q

m qq q

qq

q q

m qq q q

q q

I I iV

iiF i

i

V iI

V F i

i

− − − −

−

−

−

−

− −

−

− −

− −

ζ = ζ σ ζ − ζ − ζ σ ζ − ζσ

⎧⎛ ⎞ζ − ζ ⎪ε+ − + ζ − ζ⎨⎜ ⎟ε ε− ε σ⎝ ⎠ ⎪⎩

⎡ ⎤⎫σ ζ − ζ ⎪× − σ ζ − ζ⎢ ⎥⎬

εσ⎢ ⎥⎪⎣ ⎦⎭

ζ = − ζ σ σ ζ − ζ

⎛ ⎞ζ − ζ ε+ ζ σ ζ − ζ + −⎜ ⎟ε − ε σ⎝ ⎠

× − ζ − ζ σ ζ − ζε

( ){ }sin ,qi −⎡ ⎤− εσ σ ζ − ζ⎣ ⎦

'V iI=

( )2

exp

2m q e q

qe e

E ih zF

h U

−

= −

ε

0mqF = q

+

ζ

sionless field amplitude in the gap (the amplitudebeing averaged over period L):

(30)

Taking into account that = and = we obtain from (8) and (30) the differ�

ence equation for the dimensionless gap field

(31)

where is the normalized

local coupling impedance; is the HF transit

angle for electrons covering SWS period L; and isthe dimensionless current induced in the qth gap, i.e.,

The expression for can be simplified by select�ing normalizing parameter ε. Consider a number ofvariants. First, following the technique applied in [13]for a TWT, we introduce parameter ε according to therelationship

(32)

where and −J0 > 0, because е < 0.

Then,

(33)

Introduced parameter ε is similar to the TWT amplifi�cation parameter but slightly differs from the latter. Forcomparison, let us switch from a periodic SWS to asmooth one and assume that L → 0. Then,

and we obtain

(34)

where is the specific coupling imped�ance averaged over the beam’s section, because,according to (3) and (6), equivalent width d of the gapdepends on the field and HF�current distributionsover the beam’s section. The difference from the con�ventional definition of ε is in the factor hs/he [13].

For a smooth structure, where L = dz → 0, excitationequation (31) takes the form

(35)

where

qF

( )21 exp .

2m q

q q e qee

UdF F ih zL Uh L

= = −

ε

2qUΔ 1 12q q qU U U

+ −− +

1qz±

,qz L±

( ) ( )1 1exp exp 2 cos ,sq e q e q s qF i F i F iZ I+ −

ϕ + − ϕ − ϕ = − �

002 22

ss s

e e e

Z JeJZ Zm L U

= =ω ε ϕ εv

e eh Lϕ =

qI�

( )0 exp .q q e qJ J I ih z=�

sZ

( )

( )

3 002 3 3,

2 4ss

e e e

Z JeJ Z

m h L U h L

−

ε = =

ω

2

0,2

ee

mUe

= − >

v

( )22 .s eZ h L= ε

( ) ( )2 2

0 2 0 2 0 3sin sins s s s s s sd dZ R d R L R h LL L

= ϕ = ϕ →

3 02 ,

2s

se

eJ hRhm

ε =

ω

( )20

s sR R d L=

2 22

2 22 2 ,s e

e

h hd F dFi F iIdd h

−ε + + = −

ζζ ε

.eh zζ = ε

1276


SOLNTSEV, KOLTUNOV

This equation describes the excitation of the total fieldby the forward and counterpropagating waves of a smoothstructure at arbitrary frequency ω.

For TWTs containing SWSs close to smooth struc�tures, for example helical SWSs, we have hs ≠ 0 at frequen�cies substantially differing from cutoff values, and, whenhs ≈ hе, the electron flow effectively excites only a singleforward wave that is synchronous with the flow. Dimen�sionless field F+ of this wave satisfies the known first�orderexcitation equation

(36)

This equation can be derived from (35) by means ofthe method of variation of the constant or immediatelyfrom expression (2) for the total field with the use of nor�malization (24) and definition (34) of the amplificationparameter. In expression (2), only the first term, whichcorresponds to the forward wave, should be taken intoaccount.

If, instead of (32), parameter ε is introduced accordingto the relationship

(37)

we obtain

(38)

and the dimensionless amplitude of the HF gap volt�age

(39)

The simplest form of the dimensionless excitationequation is obtained when ε is chosen as follows:

(40)

Then, we have

, (41)

(42)

The dimensionless current ,

which is induced in the qth gap and enters the excita�tion equations, is obtained via the substitution of beamconvection current (25) and linear�approximationexpressions (28) for this current into integral (6). Forsimplicity, we calculate the induced current with disre�gard of the effect of the spatial�charge forces within agap.

As a rule, width d of interaction gaps is much smallerthan plasma wavelength λp in an electron beam when this

.s e

e

h hdF i F Id h

+

+

−− = −

ζ ε

2 1 ,eh Lε =

0

2s s

e

JZ Z

U=

( )exp .qq e q

e

UF ih z

U= −

2 00 .2

ss

e e e

Z JZ eJm L U h L

ε = =

ωv

1sZ =

( )0

exp .qq e q

s

UF ih z

Z J= −

( )0

expqq e q

JI ih z

J= −

�

wavelength is determined with allowance for the finitenessof the beam’s cross section from the relationship

Hence,

and . Therefore, expressions (28) can

be simplified with disregard of the effect of the spatial�charge forces within gaps. Passing to the limit in (28)as σ → 0, we obtain within the gap

These expressions take into account electron

grouping caused by gap field and by the initialvelocity modulation at the gap input. This modulationdetermines the derivative of the current. Using theabove expressions and calculating the integrals from(6), we obtain

(43)

where is the angle of electron transit in anequivalent plane interaction gap of width d;

; and

(44)

are the dimensionless active and reactive, respectively,components of the conductance of the interaction gapthat are normalized by . In (43), the first termdetermines the contribution of the current modulationof the electron beam at the beginning of the gap to theinduced current, the second term determines the con�tribution of the electron�beam velocity modulationaccounting for additional grouping within the gap, andthe third term determines the contribution of the

beam grouping in the gap caused by field to theinduced current.

pp

p p

i.e., 22 , .e

e

dω ππ

= Γ λ =λ Γω

v

v

�

( )p

2 1,pq e

e

dh d d−

ω ωσ ζ − ζ ≤ σε = Γ ε = π

ωε λv

�

p 1ω

εσ = Γω

�

( ) ( ) ( )( )

( )2

exp

1 exp 1 ,

m qq q q q

qq

I I iV iF i

ii

−

− − −

−

−

⎛ ⎞ζ − ζζ = ζ − ζ ζ − ζ + −⎜ ⎟

ε⎝ ⎠⎧ ⎡ ⎤⎫ζ − ζ⎛ ⎞ζ − ζ⎪ ⎪× ε + −⎢ ⎥⎨ ⎬⎜ ⎟

ε ε⎢ ⎥⎝ ⎠⎪ ⎪⎩ ⎣ ⎦⎭

( ) ( ) exp 1 exp .m q qq qV V iF i i

− −

−

⎛ ⎞ ⎧ ⎛ ⎞⎫ζ − ζ ζ − ζζ = ζ + − ε −⎨ ⎬⎜ ⎟ ⎜ ⎟

ε ε⎝ ⎠ ⎩ ⎝ ⎠⎭

mqF

( ) ( ) ( )( ) ( )[ ]

21 2

exp2

,

q q q

mq

I I M V i M

F Y iY

− − θ⎡ ⎤= ζ − ζ ε −⎢ ⎥⎣ ⎦

+ ε θ θ − θ

�

eh dθ =

sin2 2

M θ θ=

( )

( )

1 2

2 2

2 1 cos sin( ) ,

2sin 1 cos( )

Y

Y

− θ − θ θθ =

θ

θ − θ + θθ =

θ

0 2 eJ U

mqF



With allowance for (30) and (43), relationships (28)and (31) make it possible to calculate the current,velocity, and field at the beginning of the (q + 1)th step

of structure from their values at the beginning of

the qth step of as well as fields Fq–1 in the (q – 1)thgap. Assuming that the qth gap is associated with

and the qth drift space is asso�

ciated with we should apply completerelationships (28) for recalculating the current andvelocity from the beginning to the end of the qth gap

and use relationships (28) at in the qth driftspace. This recurrence calculation scheme allowsdetermination of the distributions of the field, current,and velocity of electrons along the structure withoutseparating individual waves and solving the corre�sponding characteristic equations, a circumstancethat makes it possible to analyze the amplificationcharacteristics of a TWT with the discrete interactionbetween electrons and the field of a pseudoperiodicSWS.

With allowance for the forward and counterpropagat�ing waves, excitation equation (31) yields for a periodicSWS

(45)

When only the forward wave of the structure, i.e., onlythe first term in (2), is taken into account, we obtain,instead of excitation equation (8), the elementary rela�tionship for voltages

. (46)

This relationship can also be obtained by applying themethod of variation of the constant to (8). Passing tonormalized field (30), we obtain in this case

(47)

instead of (45).For a pseudoperiodic structure, the voltages in excita�

tion equation (12) can be normalized with the help ofrelationship (30) where the value of a certain step of thestructure or the average value of this step is used as L.Then, we obtain

(48)

where and is the value that ischosen for normalization and that is identical for allsteps. If parameter ε is defined, as in the case of a

periodic structure, by relationship (32), then, hasform (33).

1q−

+ζ

,q−

ζ

,q q q eh d− + −

ζ ≤ ζ ≤ ζ = ζ + ε

1,q q+ −

+ζ ≤ ζ ≤ ζ

0mqF =

( ) ( )1 1exp 2 cos exp .q e q s q e s qF i F F i iZ I+ −

ϕ = ϕ − − ϕ − �

( )2

1 exp2s

q q q sR LU U J i

+

⎛ ⎞= + ϕ⎜ ⎟⎝ ⎠

( )1 exp2sin

sq q q s e

s

ZF F I i+

⎡ ⎤= − ϕ − ϕ⎢ ⎥ϕ⎣ ⎦

�

( ) ( )1121 1 1 11

12

1 12 1222 11 1

1212

exp exp

,

qq

q q q qqq

q qq q

q s qq

LAF i F iLA

A AA A F iZ IAA

−

+ + − −−

−

−

ϕ + − ϕ

⎛ ⎞− + = −⎜ ⎟⎝ ⎠

�

,q e qh zϕ = 12 sA iZ= −

sZ

As a result, according to (28) and (45), we have the fol�lowing recurrence relationships:

(49)

(50)

(51)

where current induced in the qth gap is determinedby expression (43) and the quantities

and

are the transit angles (measured in plasma wave�

lengths) for the qth step and gap, respectively, and

is the analogous transit angle for the qthdrift space.

Assume that each two�port of the chain represent�ing the SWS is reciprocal. Then, in (48) and (51), we

can use the phase shift by cell instead of ele�

ments of the transmission matrix.For periodic SWSs, solution (45), (49), (50) can be

sought in the form of the eigenwaves of an SWS with anelectron beam, and, for the current, electron velocity,and field, the complex phase shift corresponding tostep in each ith wave can be determined from theconditions

Then, we obtain a fourth�order characteristic equa�tion, because excitation equation (45) couples Fq + 1 notonly with Fq but also with Fq – 1. This can readily be seen ifwe formally introduce the variable Gq = Fq – 1, which leads

( )( )( )( )

2

1 2 21cos sin

1sin 'exp cos cos

1 'exp cos sin sin ,

q q q q q q

dd q

d d q

L iI I iV Fd

i i

i i i

+

ε= θ − θ +σ − ε σθ⎡× − θ + − θ θ

⎢⎣ εσ

⎤− − θ − θ + εσ θ θ⎥⎦εσ

( )( )( )( )

1 2 2sin cos1

sin 'exp cos sin

'exp cos sin cos ,

q q q q q q

dd q

d d q

LV I V Fd

i i

i i i

+

−ε= − σ θ + θ +− ε σ

θ⎡× − θ + − θ εσ θ⎢⎣ εσ

⎤− − θ − θ + εσ θ θ⎦

( )

( )

( )( )

121 1

12

1 1222 11 11

12

1121 1 11

12

exp

exp

exp ,

q

q s q q

qq q

q qq

qq

q q qqq

AF iZ I iA

AA A F iA

LA F iLA

+ +

−

+−

−

− − +−

= − − ϕ

⎛ ⎞+ + − ϕ⎜ ⎟⎝ ⎠

− − ϕ + ϕ

�

qI�

2q qq q q

e q

LL

ωθ = εσϕ = = π

λv

qd

e

dω

θ = εσθ = =v

2 qL

dπ

'q q dθ = θ − θ

qsϕ

qijA

iγ

( ) ( )

( )

1, , 1, ,

1, ,

exp , exp ,

exp .q i q i i q i q i i

q i q i i

I I i V V i

F F i+ +

+

= γ = γ

= γ

1278


SOLNTSEV, KOLTUNOV

to the relationship Gq + 1 = Fq completing (45), (49), and(50). Physically, this result corresponds to the presence offour electron waves in periodic SWSs with an electronbeam that have been investigated, for example, in [16, 17]with the help of the method of equivalent circuits. Here,we consider the direct solution of the problem based onrecurrence relationships (45) and (49)–(51). This solu�tion makes it possible to analyze the processes of the dis�crete electron–wave interaction in both periodic andpseudoperiodic SWSs.

It is convenient to represent relationships (49)–(51) ina matrix form as

(52)

where coefficients aij generally depend on q. Thesecoefficients can readily be obtained from (45) and(49)–(51).

We apply the following algorithm to solve theboundary�value problem. Let us fix two pairs of lin�

early independent values ( ) and

calculate two pairs of values ( )

using a program that realizes (52). Since the equationsare linear, the relationship coupling the output andinput values is linear:

(53)

We obtain matrix В and the inverse matrix A = B–1:

(54)

Specifying output reflection coefficient ΓL, we find from expression (19):

(55)

and calculate and from expression (54):

(56)

(57)

as well as the field amplification .

When Γ1 is given, we find from the above formulas

and i.e., the input signal and the backward

(relative to output field ) radiation. We find from(56) and (57) that

(58)

(59)

Strictly speaking, the amplification in a tube should beconsidered as the ratio of the forward�wave amplitude inthe Qth gap to the forward�wave field that arrives from theinput line and is observed in the first gap. According to(15), we have

(60)

or

(61)

From the obtained expressions, we can readily findthe ratio of the wave amplitudes in input and output

lines connected to the first and Qth gaps, respec�tively.

1 11 12 13

1 21 22 23

1 31 32 33 34 1

,

,

,

q q q q

q q q q

q q q q q

I a I a V a F

V a I a V a F

F a I a V a F a F

+

+

+ −

= + +

= + +

= + + +

( ) ( )(1) (1) (2) (2)1 2 1 2, , ,F F F F

( ) ( )(1) (1) (2) (2)1 1, , ,Q Q Q QF F F F

− −

(1,2) (1,2) (1,2)1 11 1 12 2

(1,2) (1,2) (1,2)21 1 22 2

,

.

Q

Q

F B F B F

F B F B F

−

= +

= +

1 11 1 12

2 21 1 22

,

.Q Q

Q Q

F A F A F

F A F A F−

−

= +

= +

1Q QF F−

( )( ) ( )( )L

L

1 exp exp,

1Q s e s e

Q

F i i

F−

− ϕ − ϕ + Γ ϕ + ϕ

=

+ Γ

1 ,Q

FF

2

Q

FF

( )in ,1111 ,s

Q Q Q

FF FF F F

−

= + + Γ

( )( ) ( )( )in ,121exp exp 2s

s e sQ Q Q

FF Fi iF F F

−

− ϕ − ϕ = + − ϕ + Γ

1

QF

F

in

Q

FF

,1,s

Q

F

F−

.QF

( )( )

( )

2 1

,1

exp

,exp 2 1

s es Q Q

Q s

F FiF F F

F i−

− ϕ − ϕ −

=

− ϕ −

( )( ) ( )( )( )

( )

in

1 21 1exp 2 exp 1

.exp 2 1

Q

s s eQ Q

s

FF

F Fi iF F

i

− ϕ + Γ − − ϕ − ϕ + Γ

=

− ϕ −

( )( )( )

in in L

, exp 1,

1s Q Q s e

UC F i Q

KE F

− ϕ − ϕ −= =

+ Γ

1 21 1 1

exp( 2 ) 1 1 .1 (exp( 2 ) ) exp( ( ))(1 )

sU

s s eQ Q

iKF Fi iF F

− ϕ −=

+ Γ− ϕ + Γ − − ϕ − ϕ + Γ



3. THE FEATURES OF THE DISCRETE ELECTRON–WAVE INTERACTION NEAR

THE CUTOFF FREQUENCIES OF FOLDED PERIODIC SLOW�WAVE STRUCTURES

Folded SWSs include a folded waveguide, an interdig�ital line, a coaxial–radial line, a chain of coupled resona�tors with coupling slots turned through 180°, and otherstructures where the wave energy is mainly transferredalong structure loops and multiply crosses the SWS axis.The first spatial harmonic of the wave traveling along sucha structure is an operating one, because the couplingimpedance (and the local coupling impedance) of thefundamental forward spatial harmonic is zero due to thegeometric rotation of the field phase through 180° in adja�cent interaction gaps. In order to reveal the features ofapplication of the derived system of equations of the elec�tron–wave interaction, let us consider an SWS that is auniform folded waveguide with disregard of reflectionsfrom its bends (Fig. 3a). In this case, we can pass to a rec�tilinear waveguide that is penetrated by an arbitrarilyfolded electron flow moving in adjacent interaction gapsin the opposite directions (Fig. 3b). This circumstanceshould be taken into account in excitation equations(49)–(51) with the coefficient (–1)q of the induced cur�rent. In these equations, ϕs is assumed to be the field phaseshift along the waveguide over a rectified loop of length lcorresponding to the step D = L/2:

where is the wave number in free space and is the critical wave number. The phase shift

of the nth spatial harmonic along the z axis of the SWSis on period L

and on the step D = L/2 with allowance for the geo�metric turn of the waveguide through 180°

The fundamental spatial harmonic is a backward wave,because for ϕ0 = ϕs – π, 0 ≤ ϕs < π For the first spatialharmonic, ϕ1 = ϕs + π. The slowing factor of the oper�ating harmonic is determined from the relationship

(62)

Here, is the low�frequency (LF) boundary of themain passband associated with ϕ1 = π. This boundaryis determined by the critical frequency of the foldedwaveguide:

(63)

2 2, ,s s s sh l h k kϕ = = −

2k = π λ

2s sk = π λ

, 2 2 , 0, 1, 2, ...,s n s n nϕ = ϕ + π = ± ±

( ), 2 2 1 .n s n s nϕ = ϕ − π = ϕ + π −

2

1

1

2

1,

12

2 1 .

c lkD D D

c l

π

π π π π

⎛ ⎞ϕ λ λ= = − +⎜ ⎟λ⎝ ⎠⎛ ⎞⎛ ⎞λ λ⎜ ⎟= − +⎜ ⎟⎜ ⎟λ λ λ⎝ ⎠⎝ ⎠

v

v

πλ

cr 1,

, 0, .2

chDπ

π

π

λλ = λ = =

v

It is seen from (62) that the form of the dispersioncharacteristic is governed by the only parameter, High�frequency boundary λ2π of the main passband isassociated with ϕs,1 = 2π (the Bragg resonance). Thisboundary is absent in an ideal waveguide withoutreflections; however, in a real structure, this boundaryexists because of reflections from the waveguide bends.

For the local coupling impedance of a uniform foldedwaveguide, we have the elementary relationship

(64)

which is obtained in [1].

As is shown in [1, 3, 4], the local coupling impedancechanges continuously in passing from a transparencyband to an opacity band, and the maximum of Zs mayoccur in either band at variable dimensions of thewaveguide. The absolute value of Zs depends on the shapeof the section of a folded waveguide.

Figure 4 shows the calculated dispersion of the plusfirst spatial harmonic of the forward and backward (theminus first spatial harmonic of the counterpropagatingwave) waves, which have equal phase velocities π at thecutoff frequency v

π. The calculation is performed for the

three typical SWS variants considered in [4] for the slow�ing factor с/v

π=4. The wave characteristics are presented

not only in the SWS passband but also in the LF stopband

.lπ

λ

s s ss

kZ Z Z lhh

= ϕ =1 01sin sin

Electron flow

q + 1D

q

l

Electron beam

Slow�wave

D

l

(a)

(b)

Fig. 3. (a) Slow�wave structure that is a uniform foldedwaveguide with an electron flow and (b) its representation inthe form of a rectilinear waveguide.

structure

1280


SOLNTSEV, KOLTUNOV

λ > λπ, a circumstance necessary for the analysis of the

interaction in this band, where the phase shift is complex:

(65)

The linear discrete interaction of waves in a periodicSWS with an electron flow is described by Eqs. (45), (49),and (50), which are represented in form (52). In the com�putations, we use normalizations (32) and (33) and fixcomplex phase shift ϕs in a chosen frequency band, the

( )1

2

"' ' ', 0, ,

2" " 1 .

s s s s

s s

i

lh l π

±

π

ϕ = ϕ + ϕ ϕ = ϕ = π

π λϕ = ± = ± −

λ λ

parameter of the mismatch of electron velocities relativeto the first spatial harmonic

(66)

spatial�charge parametet σ2, the angle (or )

of the electron transit in a plane interaction gap,plasma transit angle θq for a period, total number Q ofinteraction gaps in an SWS section, reflection coeffi�cients Г1 and ГL on the section boundaries, and ampli�fication parameter ε. The Mathcad system is appliedto solve the equations.

1 1

1

',e e

e

ϕ − ϕ −ξ = =

εϕ ε

v v

v

edL

θ = ϕdL

8

6

4

2

1.61.20.80.40 1.41.00.60.2 λ λπ

⁄

c vs,n,ϕ''⁄

1

2

3

= π

1

2

3

321

3

2

1

02.01.61.20.80.40.2 1.81.41.00.6

(а)

(b)Zs Zin

0⁄

3

2

1

ϕ''1

ϕ''s

λ λπ

⁄

Fig. 4. (a) Dispersion and (b) the local coupling impedance of the forward and backward waves in a folded waveguide without reflections:l/λ

π = (1) 0.29, (2) 0.5, and (3) 0.75.



Figure 5 shows an example of the calculated distribu�tions of the field and HF current of the beam along theSWS at the cutoff frequency (ϕ1 = π) and at the center ofthe passband (ϕ1 = 3π/2). The computations are per�formed at constant values of the dimensionless parame�ters: the velocity parameter ξ = 0 (i.e., under the condi�tion of the exact synchronism of electrons and the firstspatial harmonic of the field) and the amplificationparameter ε = 0.05. Note that, according to Fig. 3b, thelocal coupling impedance entering amplification param�eter (32) depends on frequency weaker than the standardcoupling impedance. Therefore, the second condition isapproximately fulfilled even without adjusting the currentof the electron beam. At the inputs of the first and secondgaps, the field values corresponding to the forward travel�ing wave are specified. Here, the input counterpropagat�ing radiation is assumed to be zero (C–s,1 = 0) owing tofixed output reflection coefficient ΓL, which is calculatedduring the computations. As is seen from Fig. 5, a solutionto the obtained system of equations has no singularities onthe boundary of the passband and the field and currentdistributions retain their characters as the cutoff fre�quency is approached.

Figure 6 shows the calculated distributions of thefield and HF current of the beam along the structure inan LF stopband of the SWS. Here, andsynchronism is also possible but in the presence of

reactive attenuation that increases with the wave�

1 1"iϕ = π + ϕ

1",ϕ

length (Fig. 4a). The results are presented for the caseof the exact synchronism (ξ = 0) and constant ampli�fication parameter ε. At the inputs of the first two gaps,the field values corresponding to the decreasing waveamplitude are specified. It is seen that, in the evanes�cent SWS section, electron�beam grouping occurs. Asa result, the field decrease in the start section of the SWSsegment becomes less intense and, then, is replaced by thefield increase along gaps. Formally, the field is amplifiedalong the SWS. However, actually, the field amplificationsubstantially depends on the coupling between the evanes�cent SWS section and lead�in lines. This SWS section canbe considered as the multigap resonator of a klystron thatcan be coupled with these lines through butt�ends or inanother way. In a TWT, an evanescent SWS section can beapplied as an absorbing insert with current amplification.

Dependences of the field amplification in gaps on themismatch between the velocities of the electron flow andthe first spatial harmonic (ξ) in a passband at ϕ1 = 3π/2are depicted in Fig. 7. It is seen that, at the center of thepassband (at ϕ1 = π), where the interaction developsmainly with the forward wave, the amplification zone islocated approximately in the interval of the mismatchparameter ranging from –0.6 to 2, which corresponds toPierce’s theory. Near the band boundary at ϕ, amplifica�tion increases and the amplification band expands. Thisresult corresponds to the computations based on equiva�lent circuits of SWSs [16, 17].

1.2

0.9

0.6

0.3

1612840 18141062

|Fq|

0.04

0.03

0.02

0.01

1612840 18141062 q

|Iq|

1

2

1

2

(b)

(а)

Fig. 5. Distributions of the absolute values of the (a) field and (b) HF current of the beam along the SWS for ϕ1 = (1) π and (2) 3π/2. The

computation is performed for ε = 0.05, ξ = 0, θ = 0.718, and under the assumption that there is no reflected wave at the input.0.1,σ =

1282


SOLNTSEV, KOLTUNOV

The amplification in a passband is calculated for aTWT with a four�step double interdigital structureresembling the structure presented in [18, p. 312]. Forthis structure, the dispersion and coupling impedance

of the operating first spatial harmonic are foundwith the help of known procedures applied in the 3Dsimulation of systems [19] (Fig. 8). Local couplingimpedance Zs is calculated through the couplingimpedance according to the formula from [1]

,1sK

(67)

where es,m and ϕs,m are the dimensionless amplitudesand phases of spatial harmonics, respectively.

The amplification calculated with the help of the pro�gram developed in this study and the local couplingimpedance are presented in Fig. 9. The exact synchro�nism of the electron flow and the first spatial field har�monic is chosen at the wavelength λ = 1.475 cm, at which

2,

, 2, ,

,sins m

s m ss m s m

eK Z=

ϕ ϕ

0.2

0.1

840 1062

q

|Fq|

0.01

0.005

840 1062

|Iq|

1 2

3

2

1

3

(а)

(b)

Fig. 6. Distributions of the absolute values of the (a) field and (b) HF current of the beam over gaps at the reactive attenuation (1)0, (2) 0.2, and (3) 0.5.

1"ϕ =

25

20

15

10

5

0

–5

–100.4�0.2�0.4�0.6�1.0 0.8 1.0 1.41.2 1.6 1.8 2.00.20�0.8 0.6

K, dB

ξ

1

2

Fig. 7. Field amplification in a section consisting of 20 interaction gaps as a function of mismatch parameter ξ for ϕ1 = (1) π and (2) 3π/2.

The computations are performed for ε = 0.05, ξ = 0, θ = 0.718; (the parameter values coinciding with those from Fig. 5).0.1σ =



the mismatch parameter is ξ = 0 and the amplificationparameter is assumed to be ε = 0.052. The variation ofthese parameters in a frequency band is calculated fromthe data of Fig. 8 with the use of formulas (66) and (67).

CONCLUSIONS

The equations of a generalized linear theory of the dis�crete interaction between electron flows and electromag�netic waves of SWSs have been derived. The equationsuniformly describe the interaction in the frequency pass�bands and stopbands of SWSs. In contrast to known stud�ies (see, e.g., [16, 17], etc.) where equivalent circuits orother simplified model SWSs are applied, our approach isbased on the general theory of waveguide excitation, the

second�order finite�difference excitation equationderived from this theory, and the use of the local couplingimpedance instead of Pierce’s coupling impedance. Theequations have been obtained for periodic and pseudope�riodic SWSs that can select spatial harmonics and waves.Examples of calculated interaction in folded periodicSWSs have been presented, and the possibility of theeffective interaction in a stopband of a SWS has beendemonstrated.

ACKNOWLEDGMENTS

The authors are grateful to M.V. Nazarova for helpfuldiscussions of the study and D.S. Shabanov for the assis�tance in computations.

14

13

12

11

10

9

81.51.41.31.11.00.80.7 1.20.9

100

60

20

120

80

40

01.6 1.7

λ, cm

Ks,1, Ωc v1⁄

1

2

2π

π

Fig. 8. (1) Slowing factor с/v1 and (2) coupling impedance of the first spatial harmonic of a double interdigital SWS.,1sK

30

25

20

15

10

5

01.51.41.31.2

1200

800

200

1400

1000

600

01.6 1.7

λ, cm

Zs, Ω

400

G, dB

1

2

Fig. 9. (1) Local coupling impedance Zs and (2) amplification G in a frequency band for a TWT with an SWS section that is displayed inFig. 8 and contains 10 interaction gaps.

1284


SOLNTSEV, KOLTUNOV

This study was supported by the Russian Foundationfor Basic Research, project no. 10�02�00859�a.

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A generalized linear theory of the discrete electron–wave interaction in slow wave structures

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