Analysis of the equations of discrete electron–wave interaction and electron beam bunching in periodic and pseudoperiodic slow-wave structures

700
ISSN 1064-2269, Journal of Communications Technology and Electronics, 2008, Vol. 53, No. 6, pp. 700–713. © Pleiades Publishing, Inc., 2008.
Original Russian Text © V.A. Solntsev, R.P. Koltunov, 2008, published in Radiotekhnika i Elektronika, 2008, Vol. 53, No. 6 pp. 738–751.
INTRODUCTION
Various approaches to calculation of the field
excited by an electron beam are used in the analysis of
interaction of the electron beam with the electromag-
netic field in microwave vacuum devices. In a large
number of studies, an approximate representation of
resonator slow-wave structures (SWSs) by equivalent
RLC circuits or chains of two-ports is used for solution
of the problem (see, for example, [1–6]). Another
approach is based on solution of the inhomogeneous
Maxwell equations in the electromagnetic theory of
excitation. In this case, the excited field is expressed in
the form of a series of SWS eigenmodes either without
separation [7] of the quasistatic field of the spatial
charge or with such a separation [8].
Various physical laws of linear and nonlinear dis-
crete interaction of electron beam with electromagnetic
waves in a traveling-wave tube (TWT), backward-wave
tube, and similar devices have been studied through the
use of these methods and application of 1- to
2.5-dimensional models of the electron beam. Software
systems for the analysis, optimization, and design of
these devices have been developed.
Today, much attention is given to the development
of methods and software for simulation of 2D and 3D
nonlinear processes in microwave electron devices. The
methods are based on direct numerical solution of the
Maxwell equations through the use of grid methods and
the particle-in-cell method (for example, KARAT,
MAFIA, MAGIC codes, etc.). However, the use of such
methods in the design of devices is complicated
because they require significant computer resources.
Therefore, methods of the electrodynamic theory of
excitation of slow-wave structures are still important.
These methods can be used for the analysis of the struc-
ture of the excited fields; for isolation of the waves that
are synchronous with the electron beam; and for isola-
tion of the field of the spatial charge, a process that
includes the dynamic corrections to this field.
In this study, analysis of the equations of discrete
interaction of rectilinear electron beams with the elec-
tromagnetic field of periodic SWSs is performed
through the use of the difference theory of excitation
[9]. These equations are generalized to pseudoperiodic
SWSs with nonuniformly spaced interaction gaps and
the phase distribution of the electromagnetic field in the
gaps that corresponds to this arrangement. These sys-
tems differ from usual periodic SWSs in that they com-
bine the properties of selection of spatial harmonics and
the properties of microwave filters, i.e. frequency selec-
tion [10, 11]. The possibility of unified description of
interaction in both the passbands and the rejection
bands of the structures is clarified. The description uses
the electrodynamic parameters determined from the
characteristics of the electromagnetic field. These
parameters can be calculated either via processing of
the results of 3D simulation of the SWS field through
the use of HFSS, ISFEL-3D, or similar software or by
means of equivalent circuits and chains of two-ports
simulating the SWS. The processes of electron bunch-
ing in the given field of a periodic SWS are compared
(in the linear approximation) to the processes occurring
in a pseudoperiodic SWS. This comparison confirms
the selective properties of pseudoperiodic SWSs.
Analysis of the Equations of Discrete Electron–Wave Interaction
and Electron-Beam Bunching in Periodic and Pseudoperiodic
Slow-Wave Structures
V. A. Solntsev and R. P. Koltunov
Received October 18, 2007
Abstract—A comparative analysis of the equations of discrete interaction of linear electron beams with the
electromagnetic field of periodic and pseudoperiodic slow-wave structures (SWSs) is given. The analysis is
based on the difference theory of excitation of SWSs. The possibility of unified description of interaction in
both passbands and rejection bands of the systems is clarified. The description uses the electrodynamic param-
eters determined from the characteristics of the electromagnetic field. Comparison of the processes of electron
bunching in a given field of periodic and pseudoperiodic SWSs is made in the linear approximation. The com-
parison verifies the selective properties of pseudoperiodic systems.
PACS numbers: 41.75.Fr, 52.35.-g
DOI: 10.1134/S1064226908060120
MICROWAVE ELECTRONICS

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 6 2008
ANALYSIS OF THE EQUATIONS OF DISCRETE ELECTRON–WAVE INTERACTION 701
1. SOURCE EQUATIONS OF INTERACTION
OF AN ELECTRON BEAM
WITH THE ELECTROMAGNETIC FIELD
Let us consider a linear electron beam in an electro-
dynamic structure with Q interaction gaps, which are
nonuniformly spaced with step Lq (Fig. 1). Let us con-
sider only the 1D model and take the nonlinear interac-
tion equations from [12] as the source equations. The
equations of motion of electrons (the cross-sections of
an electron beam) can be expressed as follows:
(1)
(2)
where e and m are the charge and the mass of an elec-
tron; v(z, t0) is the velocity of the electron in cross sec-
tion z of the beam, the electron entering the slow-wave
structure at moment t0; and t(z, t0) is the moment the
electron arrives in cross section z.
The sum on the right-hand side of Eq. (1) deter-
mines the longitudinal quasistatic field of the spatial
charge. This field has complex amplitudes
(3)
of harmonics nω of the main signal, where
(4)
∂
∂z
-----
mv
2
2
----------
⎝ ⎠
⎛ ⎞
= eRe E z( ) iωt–( )exp Eˆ
n z( ) inωt–( )exp
n 1=
∞
∑+
⎝ ⎠
⎜ ⎟
⎛ ⎞
,
∂t
∂z
-----
1
v z t0,( )
------------------,=
Eˆ
n
Γn
inωε0
--------------
Jn z( )
S
------------,=
Jn z( )
J0
π
----- inωt z t0,( )[ ]exp ωt0( )d
0
2π
∫=
are the complex amplitudes of the harmonics of cur-
rent, Γn are the depression coefficients of the longitu-
dinal field of the spatial charge at frequency nω, and
J0 is the effective dc current of the beam. In formula (3),
we do not take into account the terms with the coordi-
nate derivative of the current. These terms were stud-
ied in [12] and, as a rule, they make small contribu-
tions.
It should be noted that the longitudinal quasistatic
field corresponding to the sum on the right-hand side of
Eq. (1) can be expressed in terms of the Green’s func-
tion determining the interaction of two cross sections of
the electron beam. The corresponding transformations
have been given in [12].
Parameter (z) is the amplitude, averaged over the
beam’s cross section, of the field of synchronous waves
excited in the SWS at frequency ω of the basic signal.
This amplitude is considered below.
2. ANALYSIS OF THE STRUCTURE
OF THE EXCITED FIELDS
AND THE EXCITATION EQUATION
IN DIFFERENCE FORM
The initial point is the theory of excitation of elec-
trodynamic structures, which is based on the series
expansion of the total field (x, y, z, t) = Re (x, y,
z)exp(–iωt) excited at frequency ω in terms of forward
(+s) and backward (–s) eigenmodes (x, y, z) of the
structure and separation of quasistatic field (x, y, z) of
the spatial charge.
E
E E
E s±
E
ˆ
E+s
Gap
q – 1
Gap
q
Gap
q + 1 E–s
Lq – 1 Lq
zq 1–
–
zq 1–
+
zq
–
zq
+
zq 1+
–
zq 1+
+
zq 1– zq zq 1+
e z( )
dq 1– dq dq 1+
z
z
Electron
beam
1
Fig. 1. Layout of an SWS with nonuniformly spaced interaction gaps and the field distribution in the gaps: q = 1, 2, …, Q.

702
SOLNTSEV, KOLTUNOV
It was rigorously determined [8] that, for periodic
waveguides,
(5)
where C±s(z) are the excitation coefficients, which are
determined by the density of the RF current of the beam
and the characteristics of eigenmodes via the following
relationship:
(6)
where Ns is the norm of the wave and S(z) is the cross
section of the structure. Parameters and are
taken at the frequency 0.
In the 1D model of interaction, the density of the RF
current is expressed in terms of distribution function
ψ(ı, Û)in the beam’s cross section:
(7)
The distribution function is normalized by the relation-
ship
so that J(z) is the RF current of the beam and
is the effective area of the beam’s cross section.
In accordance with the Floquet theorem, the field of
an eigenmode in a periodic waveguide is expressed as
(8)
where (x, y, z) are the distribution functions, which
are periodic along the z axis; hs are the wave numbers;
and is the amplitude of the chosen field component
at point (x0, y0, z0), where the distribution function of
this component is equal to unity. Let us introduce the
electric field averaged over the beam’s cross section:
(9)
Then, with consideration for formulas (7) and (8),
Eqs. (6) for dimensionless excitation coefficients C±s(z)
E x y z, ,( ) Cs z( )Es C s– z( )E s– -+
s
∑=
–
ω˜
ω
----
ω˜ 0→
lim C˜ sE˜ s C˜ s– E˜ s–+( ) E
ˆ
,+
dC s±
dz
-----------
1
Ns
------ j x y z, ,( )E s+− x y z, ,( ) S,d
S z( )
∫±=
C˜ s± E˜ s±
ω˜
j x y z, ,( ) J z( )ψ x y,( )z0.=
ψ x y,( ) Sd
Se
∫ 1,=
S ψ
2
x y,( ) Sd
Se
∫⎝ ⎠
⎜ ⎟
⎛ ⎞
1–
=
E s± x y z, ,( ) E s±
0
e s± x y z, ,( ) ihsz±( ),exp=
e s±
E s±
0
Ez z( ) ψ x y,( )Ez x y z, ,( ) S.d
Se
∫=
can be transformed into the equations for dimensional
values (z) = C±s(z)
(10)
where
are the distribution functions, averaged over the beam’s
cross section, of the longitudinal electric field of the
forward and backward waves and
is the specific coupling impedance at point (x0, y0, z0).
General expression (5) for the excited field includes
one or several series terms with spatial harmonics that
are synchronous with the electron beam. The remaining
terms of the series determine the dynamic corrections
to quasistatic field (x, y, z) of the spatial charge. As a
rule, these corrections can be disregarded owing to
rapid convergence of the series [8]. It was mentioned in
[13, 14] that the beam is synchronous with the spatial
harmonics of the forward and backward waves in peri-
odic SWSs at the frequencies close to the edges of the
passband. Therefore, let us take into account the quasi-
static field of the spatial charge and the two terms of
series (5) that correspond to these waves. Averaging in
accordance with (9) yields the value of longitudinal
field (z),
and, after separation of the fields of forward and back-
ward waves in sum (z), we obtain
(11)
This expression takes into account all spatial har-
monics of the excited field because distribution func-
tions (z) of the field of eigenmodes are not expanded
into a series in terms of spatial harmonics. At the same
time, owing to separation of the forward (+s) and back-
ward (–s) waves, the usual difficulty remains: The cou-
pling impedance of these waves becomes infinitely
large at the edges of the passbands. It has been shown
in a number of studies published earlier, for example in
[15–17], that this difficulty can be eliminated if the
excited field of forward and the backward waves is con-
sidered as a whole. The approach based on the use of
the difference equation of excitation of electrodynamic
structures [9] is the most general.
C s±
0
E s±
0
dC s±
0
dz
-----------
Rs
0
2
-----J z( )e s+− z( ) ihsz+−( ),exp+−=
e s± z( ) ψ x y,( )e s± z, x y z, ,( ) Sd
Se
∫=
Rs
0 2Es
0
E s–
0
Ns
-----------------–=
Eˆ
Ez
Ez z( ) E z( ) Eˆ z( ),+=
E
E z( ) = Cs
0
z( )es z( ) ihsz( ) C s–
0
z( )e s– z( ) ihsz–( ).exp+exp
e s±

The difference equation of excitation for the total
field (z) ≡ (z) of forward and backward waves in
the section ∆zq = – can be obtained from Eqs. (10)
and (11) in the same way as was done in [9], i.e., via
setting down expressions for the finite differences of
field Eq(z) and excitation coefficients (z). Here, we
derive the difference equation for the 1D interaction
model on the basis of the approach developed in [18]
for periodic structures. The use of this approach permits
generalization of the obtained results to pseudoperiodic
systems.
Let us separate the total field of forward and back-
ward waves in (11) into three parts:
(12)
The first term corresponds to the field of the forward
wave entering the selected section from the left; the
second term, the field of the counter-propagating wave
traveling from the right; and ∆Eq(z), the field excited by
the current of this segment of the beam. In accordance
with (10)–(12), we have
(13)
(14)
(15)
If the interaction between the electrons and the field
of a periodic or a pseudoperiodic structure is discrete,
then, as a rule, the change of the field phase in the inter-
action space over one period of the system can be dis-
regarded. This circumstance is evident when the length
of the interaction gap is short enough and when the
interaction takes place at the frequencies lying outside
the SWS passband. In this case, the field phase either
remains uniform throughout the volume of the system
or changes stepwise by π [19]. Let us suppose that the
field phase is constant inside the section ≤ z ≤
selected above. Then, we have
(16)
where (z) is the real distribution function that is the
same for the fields of forward and backward waves (for
example, owing to the symmetry considerations) and
ψ±q are the phases of the fields of forward and backward
Eq E
zq
+
zq
–
C s±
0
Eq z( ) Cs
0
zq
–
( )es z( ) ihsz( )exp=
+ C s–
0
zq
+
( )e s– z( ) ih s– z( )exp ∆Eq z( ).+
∆Eq z( ) Cs
0
z( ) Cs
0
zq
–
( )–[ ]es z( ) ihsz( )exp=
+ C s–
0
z( ) C s–
0
zq
+
( )–[ ]e s– z( ) ih s– z( ),exp
Cs
0
z( ) Cs
0
zq
–
( )
Rs
0
2
----- J z˜( )e s– z˜( ) ihsz˜–( )exp z˜,d
zq
–
z
∫–=
C s–
0
z( ) C s–
0
zq
+
( )
Rs
0
2
----- J z˜( )es z˜( ) ihsz˜( )exp z˜.d
z
zq
+
∫–=
zq
–
zq
+
e s± z( ) ih±sz( )exp eq z( ) iψ q±( ),exp=
eq
waves inside the selected section, which are constant
within the section.
From (13)–(15) and (16), we find that increment
∆Eq(z) of the field can be expressed as
(17)
The first integral of the bracketed expression deter-
mines increment ∆E+q(z) of the field of the forward
wave from to z, and the second integral determines
increment ∆E–q(z) from z to . These increments are
constant everywhere inside the qth section ∆zq = –
, and at ( ) = ( ) = , we have
(18)
This expression corresponds to the so-called induction
theorem [20], which was used even in the first studies
on the theory of TWTs [21]. Here, this expression was
obtained with consideration for the longitudinal distri-
bution of field (z) over selected section ∆zq. In the
case of flat interaction gaps with width d, we have
(z) ≡ 1 inside the gap and (z) ≡ 0 outside the gap.
For continuous functions (z), the choice of the length
of this section depends on the parameters of the struc-
ture: In the case of a periodic structure, the section
length may be equal to the structure period ∆zq ≡ L, and
in the case of an aperiodic structure, the section length
may be equal to the step ∆zq = Lq between the electron–
field interaction gaps. Hence, an equivalent flat gap
with width dq can be introduced. On the assumption
that the field intensity in the middle of the equivalent
flat gap is Eq and with the introduction of gap voltage
Uq, the width can be determined from the following for-
mula:
(19)
∆Eq z( )
Rs
0
2
-----eq z( ) i ψq ψ q–+( )[ ]exp–=
× J z˜( )eq z˜( ) z˜d
zq
–
z
∫ J z˜( )eq z˜( ) z˜d
z
zq
+
∫+ .
zq
–
zq
+
zq
+
zq
–
e zq
–
e zq
+
eq
∆Eq zq
+
( ) ∆E q– zq
–
( )=
=
Rs
0
2
-----eq z( ) i ψq ψ–q+( )[ ] J z˜( )eq z( ) z˜.d
zq
–
zq
+
∫exp–
eq
eq eq
eq
dq
Uq
Eq
------– Eq eq z( ) zd
zq
–
zq
+
∫
⎝ ⎠
⎜ ⎟
⎜ ⎟
⎛ ⎞
Eq– eq z( ) zd
zq
–
zq
+
∫ .= = =

704
SOLNTSEV, KOLTUNOV
The integral in (18) determines the current induced
in the qth gap. This current is determined from the
expression
(20)
Now, the total field of forward and backward waves
that is excited by the electron beam in the qth section of
the system can be expressed as
(21)
In the case of reciprocal systems, the absolute value
of phase shift ϕq in any section is independent of the
direction of propagation of the wave: Only the sign of
ϕq changes with a change in the direction of propaga-
tion. Therefore,
Let us set const = 0 for the eigenmodes of the system.
Then, in the case of periodic systems, where ϕq ≡ ϕs, we
have
and in the case of aperiodic systems with a variable
phase progression at step ϕq, the following formula
holds:
(22)
Excitation coefficients (z) of the forward and
backward waves are determined from (14) and (15),
and when condition (16) is met, these coefficients can
be expressed as a sum of excitations over individual
steps:
(23)
Let us consider excitation of a current with slowly
varying dimensionless amplitude I(z) and wave number
he = ω/ve by the wave. In this case, we have
J(z) = J0I(z)
Jq
1
dq
----- J z( )eq z( ) zd( ).
zq
–
zq
+
∫=
Eq z( ) eq z( ) Eq,⋅=
Eq Cs
0
zq
–
( ) iψq( )exp C s–
0
zq
+
( ) iψ q–( ) ---exp+=
–
Rs
0
2
----- i ψq ψ q–+( )[ ]Jqdqexp .
ψq 1+ ψ q 1+( )–+ ψq ϕq ψ q– ϕ q––+ +=
= ψq ψ q–+ const.=
ψq q 1–( )ϕs, ψ q– Q q–( )ϕs,–= =
ψq ϕq', ψ q– – ϕq'.
q' q=
Q 1–
∑=
q' 1=
q 1–
∑=
C s±
0
Cs
0
zq( ) Cs
0
z1( )
Rs
0
2
----- Jq'dq' iψ q'–( ),exp
q' 1=
q 1–
∑–=
C s–
0
zq 1+( ) Cs
0
zQ( )
Rs
0
2
----- Jq'dq' iψq'( ).exp
q' q=
Q 1–
∑–=
ihez( )exp
and, disregarding the amplitude variation in a step, we
obtain from (20)
(24)
where
Parameter Mq is the coefficient of interaction between
the electron beam and the field in the step. Substitution
of (22) and (24) into (23) yields the phase factors under
the sum sign, which characterize the summation of
radiation from individual steps of the system. Let us
consider different variants.
Smooth Slow-Wave Structures
In this case, subinterval ∆z of the SWS can be taken
small enough:
Summation of the radiations from individual steps and
the spatial resonance are possible if only one wave is
synchronized with the electron beam. This wave may
be the forward wave, when he – hs ≅ 0 (if he > 0), or the
backward wave, when he + hs ≅ 0 (if he < 0).
Periodic Slow-Wave Structures
In this case, let us take the period of the system as a
subinterval:
Radiations from individual steps are summed in phase
when
(25)
where m = 0, ±1, ±2… is the index of the spatial har-
monic of the structure wave and the value of corre-
sponding parameter he determines the velocity of the
mth spatial harmonic, vm = ve = ω/he. The condition of
summation of radiation can be rearranged in the follow-
ing form:
In a periodic system near the edges of the passbands,
where ϕs = hsL = pπ (p = 0, 1, 2, …), these conditions
are simultaneously met for the spatial harmonics of the
Jq J0I zq( )Mq ihezq( ),exp=
Mq
1
dq
----- eq z( ) ihe z zq–( )[ ]exp z.d
zq
–
zq
+
∫=
∆z Ӷ λe 2π/he, zq q∆z,= =
ϕq ϕs≡ hs∆z, hezq ϕq'
q'
∑+− he hs.+−∼=
∆z L, zq qL, ϕq ϕs≡ hsL,= = =
hezq ϕq'
q' 1=
q 1–
∑+− heL hsL.+−∼
heL hsL+− 2πm,=
c
vm
-------
c
vs
----- m
λ
L
---.±=

forward (upper sign) and backward (lower sign) waves
whose indexes mfwd and mbkwd differ by p: mbkwd =
mfwd + p. Therefore, at the frequencies lying near the
edges of the passbands, it is necessary to take into
account excitation of two waves by the electron beam:
the forward wave and the backward wave. For the total
field of these waves, a second-order finite-difference
equation can be formulated. Let us derive this equation
for a 1D model of interaction from expression (21) with
consideration for formulas (14) and (15) and under the
assumption that, for a periodic SWS, dq ≡ d.
Let us form a second-order finite difference via the
shift of index q in (21) by ±1
Taking into account (16) and (20), we find from (14)
and (15) that
Since ψq ± 1 = ψq ± ϕs and ψ–(q ± 1) = ψ–q ϕs, substitu-
tion of these expressions into ∆2Eq and reduction of
similar terms yields a second-order finite-difference
equation for the total excited field:
(26)
This equation is the same as the equation derived in [9]
in a different way. In addition, it can be rearranged into
an equation for the gap voltages Uq = –Eqd:
(27)
It is important that impedance d2sinϕs on the
right-hand side of this equation does not tend toward
infinity at the edges of the SWS passband. Let us prove
this statement for periodic lossless structures as done in
[22].
∆
2
Eq Eq 1+ 2Eq– Eq 1–+ Cs
0
zq 1+( ) iψq 1+( )exp= =
+ C s–
0
zq 2+( ) iψ q 1+( )–[ ]exp
Rs
0
2
-----Jq 1+ d– 2Eq–
+ Cs
0
zq 1–( ) iψq 1–( )exp
+ C s–
0
zq( ) iψ q 1–( )–( )
Rs
0
2
-----Jq 1– d.–exp
Cs
0
zq 1+( ) Cs
0
zq( )
Rs
0
2
-----dJq iψ q–( ),exp–=
C s–
0
zq 2+( ) C s–
0
zq 1+( )
Rs
0
2
-----dJq 1+ iψq 1+( ),exp+=
Cs
0
zq 1–( ) Cs
0
zq( )
Rs
0
2
-----dJq 1– iψ q 1–( )–[ ],exp+=
C s–
0
zq 1+( ) C s–
0
zq( )
Rs
0
2
-----dJq iψq( ).exp+=
+−
∆
2
Eq 2Eq 1 ϕscos–( )+ iRs
0
ϕsJqd.sin–=
∆
2
Uq 2Ud 1 ϕscos–( )+ iRs
0
Jqd
2
ϕs.sin=
Rs
0
In the passband, we have E–s = , Ns = ±4Ps,
where Ps is the power flow and Ws is the stored energy
per unit length of the system. Group velocity vgr
becomes zero at the cutoff frequencies. Therefore,
near cutoff frequency ω0 corresponding to the 0 type
and
near cutoff frequency ωπ corresponding to the π type. In
this case,
in the passbands with normal dispersion of the funda-
mental spatial harmonic n = 0 and
when the dispersion of the fundamental spatial har-
monic n = 0 is anomalous, facts that are evident from
Fig. 2.
In addition, expansion of sinϕs = sinhsL in the neigh-
borhood of the cutoff points ϕs = 2πn and ϕs = π(2n + 1)
at any of cutoff frequencies ω0 or ωπ yields
Es*+−
Rs
0 Es
0 2
2Ps
----------, Ps Wsvgr Ws
dω
dhs
--------,= = =
vgr
dvgr
dhs
---------- ω0( ) hs
2π
L
------n–
⎝ ⎠
⎛ ⎞ , n 0 1 2 …, , ,= =
vgr
dvgr
dhs
---------- ωπ( ) hs
π
L
--- 2n 1+( )–
⎝ ⎠
⎛ ⎞ , n 0 1 2 …, , ,= =
dvgr
dhs
---------- ω0( ) 0,
dvgr
dhs
---------- ωπ( )> 0<
dvgr
dhs
---------- ω0( ) 0,
dvgr
dhs
---------- ωπ( ) 0><
ω
ω0a
ωπa
ωπn
ω0n
0 π
L
---
2π
L
------
hs
2
1
Fig. 2. Dispersion characteristics for the SWSs with (1) nor-
mal and (2) anomalous dispersion of the fundamental spa-
tial harmonic.

706
SOLNTSEV, KOLTUNOV
where “+” corresponds to the normal dispersion of the
fundamental spatial harmonic n = 0 and “–” corre-
sponds to the anomalous dispersion. Thus, in the case
of periodic waveguides, an everywhere-limited local
total coupling impedance that accounts for both the for-
ward and backward waves simultaneously can be intro-
duced. This impedance is expressed as
(28)
Coupling of this impedance with coupling imped-
ance Ks, m of the mth spatial harmonic is determined
from the following formulas:
where es, m and hs, m are the dimensionless amplitudes
and wave numbers of spatial harmonics. As a result, we
have
(29)
In the passband, where only one synchronous wave
is excited and is bounded, either difference equa-
tion (26) or source expression (11) may be used to find
the field. If the source expression is used, we should
take into account only the forward wave, which corre-
sponds to a first-order finite-difference equation. For
example, in the center of the passband, ϕs = π/2 and
sinϕs = 1 and, on the basis of (29), a known expression
for the coupling impedance can be obtained in terms of
Zs. In this case, Zs is the wave impedance. Formally, the
solution of difference equation (26) can be expressed as
a sum of forward and backward waves in the form of
(11) through the use of the method of variation of con-
stants, which is analogous to the method used for ordi-
nary differential equations.
At the edges of passbands ϕs = 0, π, 2π, …, the value
of Zs remains bounded, while Ks, m ∞. Outside the
passbands, a reactive attenuation depending on fre-
quency appears, while the real parts of the wave num-
ber and the phase shift per period remain constant.
Therefore,
Rs
0
ϕssin
Es
0 2
2Ws
----------
L
dvgr
dhs
----------
ω0 ωπ,
±
----------------------------,=
Zs Rs
0
d
2
ϕs.sin=
Ks m,
Es m,
2
2hs m,
2
Ps
------------------ Rs
0 es m,
2
hs m,
2
--------------,= =
es m,
1
L
--- e z( ) ihs m,–( )exp z, hs m,d
L
2
---–
+
L
2
---
∫ hs
2πm
L
-----------,+= =
Ks m, Zs
es m,
2
ϕs m,
2
ϕssin
-----------------------, ϕs m, hs m, L.= =
Rs
0
Let us use the following expression [8] for calcula-
tion of the local coupling impedance:
This expression is valid for both lossy systems and eva-
nescent waves. Disregarding ohmic losses in the SWS,
we have E–s = and, in the numerator, obtain qua-
druplicated total energy Ws of the sth wave per unit
length of the SWS on average. As a result, outside the
passbands, we obtain
Phase shift ϕs and local impedance Zs entering dif-
ference equation (27) are sufficient for description of
excitation of a periodic SWS inside, outside, and at the
edges of the passband. These parameters can be calcu-
lated either through processing of the results of com-
puter simulation of the SWS fields or through the use of
simplified SWS models, i.e., RLC circuits and
waveguide–resonator models.
Let us consider a simple example: a folded
waveguide, which can serve as a model of an SWS in
the form of a chain of coupled cavities (CCC) with
turned-around coupling slots or a long line without
reflections from the folds. Propagation of the wave
along the system folds is described by wave number hw
and wave impedance Zw, which are determined from the
following formulas:
where kc is the cutoff wave number and is the wave
impedance at k ӷ kc. In the case of the H10 mode of a
rectangular waveguide with sides a and b < a, we have
kc = , ≈ , and Z0 = = 377 Ω. In the case
of a long line, kc = 0 and is determined by the geom-
etry of this line. In this case, the wave impedance can
have a significant value, ≥ Z0, for slot lines or inter-
digital structures, which have been applied even in
backward-wave tubes with multirow SWSs [23].
hs p
π
L
--- ihs'' ω( ), p+ 0 1 2 …, , ,= =
Ns
2
L
---ε0 EsE s– Vd
hsd
dω
-------.
VL
∫–=
Es*+−
Rs
0
i
dhs''
dω
--------
Es
0 2
2Ws
----------, Zs
Es
0 2
d
2
2Ws
----------------
dhs''
dω
-------- hs''L.sinh= =
hw k
2
kc
2
– , Zw Zw
0 1
1 kc/k( )
2
–
-----------------------------,= =
Zw
0
π
a
--- Zw
0 b
a
---Z0
ε0
µ0
-----
Zw
0
Zw
0

In the direction of the SWS axis, the wave numbers
of the spatial harmonics take the values determined
from the formula
where S is the length of the system loop within system
period L, which includes two interaction gaps. When a
is sufficiently large (the CCC coupling slot is wide),
parameter kc determines the low-frequency edge of the
SWS passband, where the phase shift of the field
between adjacent gaps is π. As a rule, in TWTs with a
folded SWS (including CCCs), the operating mode is
the first spatial harmonic of the field in the region of the
cutoff frequency near the π type. For this harmonic, we
have
the phase shift between adjacent gaps is
and the magnitude of the local coupling impedance is
since it follows from the definition of wave impedance
that
The frequency dependence of local coupling imped-
ance Zs presented in Fig. 3 shows that its value contin-
uously changes during transition from the passband to
the rejection band and increases as we go further into
the rejection band. The last effect can be attributed to
the resonator properties of the SWS in the rejection
band; therefore, in this case, the mechanism of interac-
tion of the electron beam with the field has klystron fea-
tures.
The obtained general relations and the presented
example demonstrate that the difference equations of
excitation of a periodic SWS that include two parame-
ters, namely, the phase shift per period (that is complex
in the general case) and the local coupling impedance,
can be used for description of discrete electron–wave
interaction inside, at the edge, and outside the SWS
passband.
hs m,
hwS 2πm+
L
--------------------------,=
hs 1,
hwS 2π+
L
---------------------- k
2
kc
2
–
S
L
---
2π
L
------,+= =
ϕs 1, hs 1,
L
2
--- k
2
kc
2
–
S
2
--- π,+= =
Zs Zw ϕs 1,sin Zw
0
1
kc
k
----
⎝ ⎠
⎛ ⎞
2
–
kS
2
------
⎝ ⎠
⎛ ⎞sin
1
kc
k
----
⎝ ⎠
⎛ ⎞
2
–
--------------------------------------------- ,= =
Zw
Uq
2
2Ps
-----------
Es
0
d
2
2Ps
-------------- Rs
0
d
2
.= = =
Pseudoperiodic Slow-Wave Structures
In these structures, the interaction gaps are spaced
nonuniformly with variable step Lq, but phases ψq of the
field (phase shifts ϕq) are chosen so that the radiations
are summed for the given mth spatial harmonic. The
following condition corresponds to this structure:
(30)
or
(31)
because zq = . In this case, the amplitude of
the mth spatial harmonic of the forward or backward
wave is the same as in the initial periodic system, while
the remaining harmonics of the forward and backward
waves are suppressed, i.e., selected. Thus, in a pseudo-
periodic SWS, as in smooth systems, the main contri-
bution to interaction with the electron beam is made by
one (forward or backward) wave.
3. LINEAR THEORY OF ELECTRON-BEAM
BUNCHING IN PERIODIC
AND PSEUDOPERIODIC SYSTEMS
FOR THE CASE OF DISCRETE INTERACTION
In order to analyze and compare the processes of
electron bunching in periodic and pseudoperiodic
SWSs, let us consider a linear theory of bunching in
given field (z) of the system. Equations (1)–(4) are
the source expressions. Let us linearize these equations
via, as usual, the expression
(32)
where ϑ is the perturbation caused by the RF field. In
the linear theory, it is supposed that |ϑ| Ӷ 1, the field
is considered only at frequency ω, and the field is
assumed to be small (~|ϑ|). Substituting (32) into
Eqs. (1)–(4) and taking only linear terms of the Taylor
series expansions in ϑ, we obtain
(33)
(34)
(35)
where the index 1 of the current and the depression
coefficient is omitted.
hezq ϕq'
q' 1=
q 1–
∑– 2πm q 1–( )=
heLq ϕq+− 2πm, q 1 2 …Q 1–, ,= =
Lq'q' 1=
q 1–
∑
E
ωt z t0,( ) ωt0 hez ϑ z t0,( ),+ +=
∂
∂z
-----
mvev˜
e
---------------
⎝ ⎠
⎛ ⎞
= Re E z( )
Γ
iωε0
-----------
J z( )
S
----------+ i ωt0 hez+( )–[ ],exp
∂ϑ
∂z
-------
ω
ve
2
------v˜ ,–=
J z( )
J0
π
----- iωt0( )iϑexp ωt0( ) ihez( ),expd
0
2π
∫=

708
SOLNTSEV, KOLTUNOV
Multiplying (33) and (34) by exp(iωt0), integrating
the result over ωt0, and taking into account (35), we
obtain
(36)
(37)
where
(38)
(39)
and
(40)
are the dimensionless amplitudes of the RF current, the
electron velocity, and the field, respectively; ζ = εhez is
the dimensionless coordinate; ε is a small parameter
that, in a general case, may be chosen arbitrarily, for
example, as the amplification parameter of the TWT or
the ratio of the plasma frequency to the operating fre-
quency; σ2 = Γ(ωp/εω)2 is the parameter of the spatial
charge; and ωp = is the plasma frequency
without consideration for the depression.
On the basis of Eqs. (36) and (37), it is not difficult
to obtain the following linear equations for the dimen-
sionless RF current,
(41)
as well as an equation for dimensional RF current J,
(42)
These equations have repeatedly been obtained and
used in the theory of TWTs and klystrons. Here, these
equations will be used in order to study bunching of the
electron beam in a structure containing Q nonuniformly
spaced (with step Lq) identical interaction gaps.
In accordance with (21), at the qth step, we have
= (z) = (z)Eq and, correspondingly, F =
Fq (ζ)exp(–iζ/ε). Usage of the method of variation of
constants for solution of Eq. (41) yields at the qth step
dI/dζ iV,–=
dV/dζ iσ
2
I– F,+=
I
1
π
--- iϑ iωt0( )exp ωt0( ),d
0
2π
∫=
V
1
π
---
v˜
εve
-------- iωt0( )exp ωt0( ),d
0
2π
∫=
F
e
mωveε
2
--------------------Ez ihez–( )exp=
e
ε0m
---------
J0
veS
---------
d
2
I
dζ
2
-------- σ
2
I+ iF,–=
d
2
J
dz
2
-------- 2ihe
dJ
dz
------– Γhp
2
he
2
–( )J+ ihp
2
Sωε0E.–=
E Eq es
es
I ζ( )
iFq
σ
------- es ζ'( ) iζ'/ε–( ) σ ζ ζ'–( )sinexp ζ'd
ζq
–
ζ
∫–=
(43)
Constants Aq and Bq are determined by the values of
current I( ) and its derivative I'( ) at the beginning
of the qth step :
(44)
From (43) and (44), we obtain
(45)
Expressions (43)–(45) describe generation and
propagation of the waves of spatial charge in the elec-
tron beam that travels in a system of interaction gaps
with given fields Eq, (i.e., Fq).
Let us compare the bunching process in structures
with periodically and pseudoperiodically spaced gaps.
We will consider equivalent flat gaps with thickness d,
so that, within a gap, Eq = –Uq/d and (z) ≡ 1 at <
z < . In this case, the integrals in expressions (45) can
be calculated analytically. Let us obtain formulas
within the qth gap at ≤ ζ ≤ + εhed = :
+ Aq iσζ( )exp Bq iσζ–( ),exp+
I' ζ( ) iFq es ζ'( ) iζ'/ε–( ) σ ζ ζ'–( )cosexp ζ'd
ζq
–
ζ
∫–=
+ iσ Aq iσζ( ) Bq iσζ–( )exp–exp( ).
ζq
–
ζq
–
ζq
–
Aq I ζq
–
( )
1
iσ
-----I' ζq
–
( )+
iσζq
–
–( )exp
2
----------------------------,=
Bq I ζq
–
( )
1
iσ
-----I' ζq
–
( )–
iσζq
–
( )exp
2
-------------------------.=
I ζ( )
iFq
σ
------- es ζ'( ) iζ'/ε–( ) σ ζ ζ'–( )sinexp ζ'd
ζq
–
ζ
∫–=
+ I ζq
–
( ) σ ζ ζq
–
–( )cos I' ζq
–
( )
1
σ
--- σ ζ ζq
–
–( ),sin+
I' ζ( ) iFq es ζ'( ) iζ'/ε–( ) σ ζ ζ'–( )cosexp ζ'd
ζq
–
ζ
∫–=
– I ζq
–
( )σ σ ζ ζq
–
–( )sin I' ζq
–
( ) σ ζ ζq
–
–( ).cos+
es zq
–
zq
+
ζq
–
ζq
–
ζq
+
I ζ( ) I ζq
–
( ) σ ζ ζq
–
–( )cos V ζq
–
( )
1
σ
--- σ ζ ζq
–
–( )sin+=
+ iFq iζ/ε–( )
ε
2
1 ε
2
σ
2
–
------------------- 1
i
ε
-- ζ ζq
–
–( )exp+
⎩
⎨
⎧
exp

(46)
As a rule, width d of interaction gaps is much
smaller than plasma wavelength λp in the electron
beam, which, with allowance for the limited dimen-
sions of the beam’s cross section, is determined from
the ratio = , i.e., λp = ӷ d. Hence,
σ(ζ – ) ≤ σεhed = = 2π Ӷ 1 and εσ =
Ӷ 1. Therefore, if the influence of the spatial-
charge forces within a gap is disregarded, expres-
sions (46) can be simplified. Proceeding to the limit
σ 0 and taking into consideration (36), we find
that, within a gap,
(47)
These expressions take into account the electron
bunching under the influence of gap field Fq and initial
velocity modulation at the input of the gap. The modu-
lation determines the derivative of the beam current in
accordance with (36). For comparison of the bunching
properties of periodic and pseudoperiodic structures
with gaps, it is convenient to write these expressions in
the dimensional form when the normalizing parameter
ε = 1 is assumed, so that ζ = εhez = hez, and when Fq is
replaced with the gap voltage Uq = –Eqd:
where Ue = – /2e is the accelerating voltage. After
these manipulations, we obtain
× i
σ ζ ζq
–
–( )sin
εσ
----------------------------- σ ζ ζq
–
–( )cos–
⎭
⎬
⎫
,
V ζ( ) I– ζq
–
( )σ σ ζ ζq
–
–( )sin V ζq
–
( ) σ ζ ζq
–
–( )cos+=
+ Fq iζ/ε–( )
ε
1 ε
2
σ
2
–
------------------- 1
i
ε
-- ζ ζq
–
–( )exp–
⎩
⎨
⎧
exp
---× σ ζ ζq
–
–( )cos iεσ σ ζ ζq
–
–( )sin–[ ]
⎭
⎬
⎫
.
2π
λp
------ Γ
ωp
ve
------
2πve
Γωp
--------------
ζq
–
Γ
ωp
ωε
-------ε
ω
ve
------d
d
λp
-----
Γ
ωp
ω
------
I ζ( ) I ζq
–
( ) iV ζq
–
( ) ζ ζq
–
–( ) iFq iζ/ε–( )ε
2
exp+–=
× 1
i
ε
-- ζ ζq
–
–( )
i ζ ζq
–
–( )
ε
-------------------- 1–exp+
⎩ ⎭
⎨ ⎬
⎧ ⎫
,
V ζ( ) V ζq
–
( )=
+ iFq iζ/ε–( )ε 1
i
ε
-- ζ ζq
–
–( )exp–
⎩ ⎭
⎨ ⎬
⎧ ⎫
.exp
Fq
e
mωve
---------------Eq
Uq
Ue
------
1
2hed
-----------,= =
mve
2
(48)
At the output of the qth gap at = + d, we have
(49)
where M is the interaction coefficient in the gap, which
is determined from the formula M = sin in the
case of a flat gap; θ = hed is the transit angle of electrons
in the gap; and zq = 1/2( + ) is the coordinate of the
gap center.
In the drift space between gaps, relations (46),
which describe kinematic bunching of electrons at
σ 0, are valid at Fq = 0. For the section between the
qth and (q + 1)th gaps, these relations can be expressed
as follows:
(50)
where θq = he(zq + 1 – zq).
Formulas (49) and (50) are the recurrence equations
describing the process of beam bunching in a system of
gaps with voltages Uq = |Uq|exp(iψq) at the gaps. In
order to compare the features of bunching in periodic
and pseudoperiodic systems of gaps, let us consider
thin gaps at d 0 and θ = 0. In the absence of mod-
ulation at the input of the first gap, the current in the qth
gap can be expressed as
(51)
I z( ) I zq
–
( ) iV zq
–
( )he z zq
–
–( )
Uq ihezq
–
–( )exp
Ue
------------------------------------+–=
×
z zq
–
–
2d
------------–
he z zq
–
–( )sin
hed
------------------------------ ihe
z zq
–
–
2
------------–
⎝ ⎠
⎛ ⎞exp+
⎝ ⎠
⎛ ⎞ ,
V z( ) V zq
–
( )
Uq ihezq
–
–( )exp
Ue
------------------------------------+=
×
he z zq
–
–( )/2[ ]sin
hed
---------------------------------------- ihe
z zq
–
–
2
------------–
⎝ ⎠
⎛ ⎞ .exp
zq
+
zq
–
I zq
+
( ) I zq
–
( ) iV zq
–
( )θ–
Uq ihezq–( )exp
2Ue
------------------------------------+=
× M iθ/2( )exp–( ),
V zq
+
( ) V zq
–
( )
Uq ihezq–( )exp
2Ue
------------------------------------M,+=
θ
2
---
θ
2
---
⎝ ⎠
⎛ ⎞
zq
–
zq
+
I zq 1+
–
( ) I zq
+
( ) iV zq
+
( )θq,–=
V zq 1+
–
( ) V zq
+
( ),=
Iq Iq 1– iVq 1– θq 1––=
= Iq 2– iVq 2– θq 2–– iVq 1– θq 1––
= Iq 3– iVq 3– θq 3–– iVq 2– θq 2–– iVq 1– θq 1––
= … i Vpθp,
p 1=
q 1–
∑–=
q 2 3 … Q., , ,=

710
SOLNTSEV, KOLTUNOV
In turn,
(52)
As usual, let us introduce bunching parameter Xr at
the rth step of the system. This step has the length Lr =
zr + 1 – zr and includes the gap and the drift space.
Assuming that the modulating voltages Ur =
|Ur|exp(iψr) have the same magnitude |Ur| = |U|, we find
(53)
Let us take into account that zr = at L0 = 0
and p = 1, 2, …, Q – 1. In this case, we obtain
(54)
Let us first consider the electron-beam bunching in
a periodic system with the constant step (period) Lp = L.
The change in the voltage phase over the period is
determined as
where vp is the phase velocity of the wave traveling
through the slow-wave structure, including the consid-
ered gaps. The voltage phase is determined from the
formula
where ψ1 = 0. Then, expression (54) for the current can
be rearranged as
(55)
Let us rearrange the exponent in expression (55) with
consideration for the following expression for the volt-
age phase over the period:
Vp Vp 1–
Up 1–
2Ue
------------M ihezp 1––( )exp+=
= Vp 2–
Up 2–
2Ue
------------M ihezp 2––( )exp+
+
Up 1–
2Ue
------------M ihezp 1––( )exp …=
=
Ur
2Ue
---------M ihezr–( ),exp
r 1=
p
∑
p 1 2 … Q 1–( )., , ,=
Xr
U
2Ue
---------Mθr, θr heLr.= =
Lmm 1=
r 1–
∑
Iq i Xp i ψr θm 1–
m 1=
r
∑–
⎝ ⎠
⎜ ⎟
⎛ ⎞
.exp
r 1=
p
∑
p 1=
q 1–
∑–=
ψ
ω
vp
------L,=
ψr r 1–( )ψ,=
Iq iX i r 1–( ) ψ heL–( )[ ].exp
r 1=
p
∑
p 1=
q 1–
∑–=
ψ
ω
ve
------L–
ω
vp
------L
ω
ve
------L–
ω
vp
------L 1
vp
ve
------–
⎝ ⎠
⎛ ⎞ ϕ0,= = =
where ϕ0 is the transit angle of electrons relative to the
wave.
As a result, we find
Let us reduce the sum in the last expression to
Use of the last expression yields the following expres-
sion for the first harmonic of the current:
(56)
Expression (56) coincides with the relations pre-
sented in studies that have been published earlier (see,
for example, [20, 24]).
On the basis of the value of the first harmonic of the
current, it is possible to obtain the electron power of
interaction of the electron beam and the traveling wave
in a sequence of Q thin RF gaps:
(57)
where is the complex conjugate value of the voltage.
The dependence of active power P' of interaction on
transit angle ϕ0 of electrons relative to the wave is pre-
sented in Fig. 4a for the case of a periodic system with
the total number of gaps Q = 10. This dependence was
obtained through the use of expressions (53) and (56)
for determination of the output power.
Iq iX i r 1–( )ϕ0( )exp
r 1=
p
∑
p 1=
q 1–
∑–=
= iX
ipϕ0( )exp 1–
iϕ0( )exp 1–
----------------------------------
p 1=
q 1–
∑–
=
iX–
iϕ0( )exp 1–
------------------------------- ipϕ0( )exp 1–( ).
p 1=
q 1–
∑
ipϕ0( )exp
p 1=
q 1–
∑ iϕ0( )exp i p 1–( )ϕ0( )exp
p 1=
q 1–
∑=
= iϕ0( )
i q 1–( )ϕ0( )exp 1–
iϕ0( )exp 1–
-----------------------------------------------exp
=
iqϕ0( )exp iϕ0( )exp–
iϕ0( )exp 1–
-----------------------------------------------------.
J zq( )
iJ0 X i
ω
ve
------zq⎝ ⎠
⎛ ⎞exp–
iϕ0( )exp 1–
-------------------------------------------=
×
iqϕ0( )exp iϕ0( )exp–
iϕ0( )exp 1–
----------------------------------------------------- q 1–( )–
= iJ0 X i
ω
ve
------zq⎝ ⎠
⎛ ⎞ iqϕ0( )exp q iϕ0( ) q 1–+exp–
iϕ0( )exp 1–( )
2
---------------------------------------------------------------------------.exp–
P Pq
q 1=
Q
∑
J zq( )Uq*
2
-------------------
q 1=
Q
∑ P' iP'',+= = =
Uq*

Now, let us consider a pseudoperiodic system with
the parameters
where hm = ω/vm is the wave number of the mth spatial
harmonic and vm is the phase velocity of this harmonic.
This condition can be presented in one of the following
forms:
(58)
where
or
(59)
In order to isolate the fundamental spatial harmonic
m = 0 with constant phase velocity v0, condition ϕq/Lq =
const should be met. In the general case, the distribu-
tion law of ϕq for extraction of the mth spatial harmonic
with constant phase velocity vm is expressed as
(60)
In this case, we can arrive at the expression
In the exponent of formula (54), we obtain
Thus,
Let us take a linear distribution of the step along the
system:
Then
hmzq ψq 2πqm,+=
hmLq ϕq 2πm,+=
Lq zq 1+ zq, ϕq– ψq 1+ ψq,–= =
c
vm
-------
ϕqλ
2πLq
------------ m
λ
Lq
-----.+=
ϕq
c
vm
-------
Lq
λ
----- m–
⎝ ⎠
⎛ ⎞ 2π.=
ψq 1+ ψq ϕq+ ψq 2π
c
vm
-------
Lq
λ
----- m–
⎝ ⎠
⎛ ⎞+= =
= ψq hmLq 2πm.–+
ψr
ω
ve
------zr– = hm he–( ) Lj – 2πrm
j 1=
r 1–
∑ = ϕ0
Lj
L1
-----
j 1=
r 1–
∑ 2πrm.–
i ψr ω
zr
ve
------–
⎝ ⎠
⎛ ⎞
⎝ ⎠
⎛ ⎞exp iϕ0
Lj
L1
-----
j 1=
r 1–
∑⎝ ⎠
⎜ ⎟
⎛ ⎞
.exp=
Lq L1 q 1–( )∆L.+=
zq Lj
j 1=
q 1–
∑ q 1–( )L1
q 1–( ) q 2–( )
2
---------------------------------∆L.+= =
Let us rearrange (53) into the form
In this case, expression (54) for the current will take the
form
(61)
If the step changes linearly, then
(62)
The dependence of active power P' on ϕ0 is pre-
sented in Fig. 4b. In comparison to Fig. 4a, Fig. 4b
clearly indicates that one maximum remains, while the
remaining maxima are suppressed. Transit angle ϕ0 of
electrons relative to the wave is determined by the
phase velocity of a spatial harmonic; therefore, this cir-
cumstance corresponds to the situation when the phase
velocity of one spatial harmonic is constant and the
remaining spatial harmonics are destroyed. If the num-
ber of gaps is increased, for example, to Q = 20, one
maximum increases and becomes narrower, while the
remaining maxima become more suppressed (Fig. 4c).
Thus, the linear theory of electron-beam bunching
in a given ﬁeld of a system of gaps conﬁrms the possi-
bility of effective suppression of interaction in a non-
Xp
1
2
---
U1
Ue
---------M1
ωLq
ve
---------- X1
Lq
L1
-----.= =
Iq iX1
Lp
L1
----- iϕ0
Lj
L1
-----
j 1=
r 1–
∑⎝ ⎠
⎜ ⎟
⎛ ⎞
.exp
r 1=
p
∑
p 1=
q 1–
∑–=
Iq iX1 1 p 1–( )
∆L
L1
-------+
⎝ ⎠
⎛ ⎞
p 1=
q 1–
∑–=
× iϕ0 r 1–( ) 1
∆L r 2–( )
2L1
-----------------------+
⎝ ⎠
⎛ ⎞ .exp
r 1=
p
∑
5
4
3
2
1
0 2.01.51.00.5
k/kc
Zs
Zw
0
--------
Fig. 3. Change of the local coupling impedance within (k >
kc) and beyond (k < kc) the SWS passband: kcS/2 = (solid
line) 2 and (dotted line) 3.

712
SOLNTSEV, KOLTUNOV
uniform pseudoperiodic structure at all spatial harmon-
ics except for one operating harmonic.
CONCLUSIONS
Analysis of the equations of discrete interaction of
electron beams with the electromagnetic ﬁeld of peri-
odic and pseudoperiodic SWSs has shown the possibil-
ity of a uniﬁed description in passbands and rejection
bands. This description can be realized through the use
of a difference equation of excitation and the local cou-
pling impedance, which, in contrast to the traditionally
used coupling impedance, does not contain discontinu-
ities at the edges of the bands. This description of the
interaction may serve as a basis for the software for 1D,
2D, and 3D simulation of microwave electron devices
with periodic and pseudoperiodic SWSs without the
use of equivalent circuits.
A linear theory of electron-beam bunching in peri-
odic and pseudoperiodic slow-wave structures with dis-
crete interaction has been presented. On the basis of
this theory, the processes of bunching in systems with
5
0
–5
9.426.283.140–3.14–6.28–9.42
P'
(‡)
5
0
–5
9.426.283.140–3.14–6.28–9.42
(b)
50
0
–50
9.426.283.140–3.14–6.28–9.42
(c)
ϕ0
Fig. 4. Dependence of active power P' on transit angle ϕ0 of electrons relative to the wave for different systems of gaps: (a) a periodic
system with ∆L = 0 and Q = 10, (b) a pseudoperiodic system with ∆L/L1 = 0.05 and Q = 10, and (c) a pseudoperiodic system with
∆L/L1 = 0.05 and Q = 20.

different arrangements of the interaction gaps have
been compared and the possibility of extraction of one
operating spatial harmonic in a pseudoperiodic system
has been conﬁrmed.
ACKNOWLEDGMENTS
This study was supported by the Russian Founda-
tion for Basic Research (project no. 07-02-00947).
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Analysis of the equations of discrete electron–wave interaction and electron beam bunching in periodic and pseudoperiodic slow-wave structures

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