A principle of selection of modes and their spatial harmonics in periodic waveguides and, in particular, in spatially developed slowing systems for multibeam traveling-wave tubes (TWTs) is elaborated. The essence of the principle is in the following: varying along the length of the system its period and at least one more parameter that determines the phase shift per period, one can provide constant phase velocity of one spatial harmonic and destroy other spatial harmonics, i.e., reduce their amplitudes substantially. In this case, variations of the period may be significant, and the slowing system becomes nonuniform, or pseudoperiodic; namely, one of the spatial harmonics remains the same as in the initial periodic structure. Relationships are derived for the amplitudes of the spatial-wave harmonics, interaction coefficient, and coupling impedance of the pseudoperiodic system. The possibility of the mode selection in pseudoperiodic slowing systems when the synchronism condition is satisfied for the spatial harmonic of one mode is investigated. The efficiency of suppressing spurious spatial harmonics and modes for linear and abrupt variation of spacing is estimated. The elaborated principle of selection of spatial harmonics and modes is illustrated by an example of a two-section helical-waveguide slowing system.

1. Journal of Communications Technology and Electronics, Vol. 43, No. 11,1998, pp. 1193-1198.
Translated from Radiotekhnika i Elektronika, Vol. 43, No. 11, 1998, pp. 1285-1290.
Original Russian Text Copyright © 1998 by Solntsev.
English Translation Copyright © 1998 by МАИК Наука/Interperiodica Publishing (Russia).
ELECTRODYNAMICS AND WAVE PROPAGATION
Pseudoperiodic Waveguides with Selection of Spatial
Harmonics and Modes
V. A. Solntsev
Received June 9, 1998
Abstract — A principle of selection of modes and their spatial harmonics in
periodic waveguides and, in particular, in spatially developed slowing systems for
multibeam traveling-wave tubes (TWTs) is elaborated. The essence of the principle
is in the following: varying along the length of the system its period and at least
one more parameter that determines the phase shift per period, one can provide
constant phase velocity of one spatial harmonic and destroy other spatial
harmonics, i.e., reduce their amplitudes substantially. In this case, variations of the
period may be significant, and the slowing system becomes nonuniform, or
pseudoperiodic; namely, one of the spatial harmonics remains the same as in the
initial periodic structure. Relationships are derived for the amplitudes of the
spatial-wave harmonics, interaction coefficient, and coupling impedance of the
pseudoperiodic system. The possibility of the mode selection in pseudoperiodic
slowing systems when the synchronism condition is satisfied for the spatial
harmonic of one mode is investigated. The efficiency of suppressing spurious
spatial harmonics and modes for linear and abrupt variation of spacing is
estimated. The elaborated principle of selection of spatial harmonics and modes is
illustrated by an example of a two-section helical-waveguide slowing system.
INTRODUCTION
The selection of modes and their spatial harmonics is important for high-power
amplifiers and microwave oscillators with electron beams: traveling-wave tubes
(TWTs), backward-wave tubes, gyrotrons, etc. The output microwave power of a
device P JUη= is determined by the electron-beam current J, accelerating
voltage U, and efficiency η .

2. To raise the output of relativistic devices, electron beams are applied that are
accelerated by the voltage U> 100 kV, as well as explosive-emission cathodes that
produce a large current. In such devices, one has to use large-current electron
accelerators to obtain high-power electron beams; therefore, they are manufactured
as large stationary installations.
Another method of raising the power output consists in increasing the electron-
beam current with the help of hot cathodes when the voltage U < 100 kV is
limited. The beam-current density is restricted by the capacity of the electron-
optical system; therefore, the beam cross section must be increased in order to
increase the current. In this case, cross-sectional dimensions may be much greater
than the wavelength in the slowing system. There may be a large number of modes
and spatial harmonics in such overdimensioned (or spatially developed) systems
that cause electron-field multimode interaction, as well as amplification and
oscillation instability. Thus, the selection becomes one of the major problems when
the electron-beam current and the device power output are to be increased.
At present, different techniques are used to suppress spurious oscillations and
modes, from the utilization of selective absorbers in TWTs and up to the
applicationof open resonators and open waveguides in orotrons, gyrotrons, and
free-electron lasers.
This paper addresses the principle of selection of spatial harmonics and modes
in periodic waveguides and slowing systems; in a general form, this principle was
formulated in [1, 2], and developed then in [3, 4]. The essence of the principle
consists in employing periodic electrodynamic systems with nonuniformly spaced
electron-field interaction gaps and a specified relation between the gap spacing and
the gap-field phase, which makes it possible to select one spatial harmonic or mode
and suppress the others.
The considered technique of selection in slowing systems is similar to the method
of suppressing the sidelobe maxima of nonuniform antenna arrays. Such
nonuniform systems may be considered as cryptoperi-odic or pseudoperiodic,
wherein the amplitudes of one or several harmonics remain the same as in the
initial periodic system, and the amplitudes of other spatial harmonics decrease. A
planar logarithmic spiral or the synchronous spirals considered in [2] represent
examples of pseudoperiodic systems. To some extent, one can assign to this class
two- or three-section systems with different spacing in the sections but with the
same phase velocity of one of the spatial harmonics in all the sections [5].
Generally, one can apply the considered principle of selection to slowing systems
of any type introducing the nonuniformity of both the spacing and the respective
phase shift over the spacing by varying the dimensions or configuration of the
system elements from one space to another (for instance, the dimensions of slots in
a comb-type structure, cross section of a helical waveguide, etc.). Here, we will
consider the influence of the distributions of spacings and the field phase over the
spacings on the amplitudes of spatial harmonics that determine the efficiency of
the electron-field interaction.

3. 1. AMPLITUDES OF SPATIAL HARMONICS AND THE
SYNCHRONISM CONDITION
To describe the electron-field interaction in pseudo-periodic systems, one can
use amplitudes of spatial harmonics and the interaction coefficients and coupling
impedance for the system as a whole or the local interaction coefficients for
individual gaps. Consider a general method of calculating these quantities.
Assume that it is given the longitudinal electric field distribution along the
system comprising Q spacings of different length , 1,2...qL q Q=
0
( ) ( )exp[ ( )]zE z E f z i zψ= (1)
Distribution of the real amplitude ( )f z and phase ( )zψ is determined by the
type of the system (uniform periodic or nonuniform).
Applying the Fourier transformation, we define the amplitudes ( )E h of spatial
harmonics by the relations
1
( ) ( )exp( )
2
zE z E h ihz dh
π
∞
−∞
= ∫
(2)
1
( ) ( )exp( )zE h E z ihz dz
l
∞
−∞
= −∫
In the general case, amplitudes ( )E h are continuous functions of the
wavenumber h and differ from the spectral density only by the factor 1/l, where
l is the length of the system. Let us represent them as a sum over the Q spacings of
the system:
1
1
( ) exp[ ( )]
Q
q q q q
q
E h U M i hz
l
ψ
=
= −∑ (3)
where ( )q qzψ ψ= is the average field phase at the qth spacing;
( )
/2
/2
1
( ) ( )exp ( )
q q
q q
z L
q q q
q q z L
M h f z z h z z dz
f d
ψ ψ
+
−
⎡ ⎤= − + −⎣ ⎦∫ (4)
is the local electron-field interaction coefficient;
/ 2
/ 2
1
( )
q q
q q
z L
q
q z L
f f z dz
d
+
−
= ∫
is the average value of the field amplitude at the qth spacing; 0
q q qU E f d= is the
rf voltage at the qth spacing; and q qz and d are the mean coordinate and
effective width of the qth gap. Note that for the gridless gaps, the choice of qd and

4. qf is to a certain extent arbitrary, because only their product is defined. Variation
of the voltage from one gap to another is determined, for the chosen form (1) of the
field representation, both by the distribution function ( )f z and losses in the
system. Formally, one may not separate these factors and take into account losses
from the very beginning using function ( )f z and assuming that ( )zψ is real.
This method is convenient in the presence of reflections in the system, when
( )zψ may be a complicated function. When calculating the interaction
coefficient (4), one may assume, as a rule, that within the qth gap, ( ) qzψ ψ≈ the
field is constant in the gap, ( ) qf z f= we obtain the familiar expression
sin /
2 2
q q
q
d d
M h h
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
.
In the general case, the written relationships enable one to take into account the
distribution of the field amplitude and phase within one spacing. Usually, variation
of the field phase within a spacing can be ignored for slowing systems with
discrete electron-field interaction; then, the interaction coefficients are real and
positive, 0qM > . In this case, the maximal values of ( )E h can be obtained,
according to (3), for the wave-numbers mh h= that satisfy Q conditions:
2 ,m q qh z qmψ π= + 1,2....q Q= (5)
where the integer 0, 1...m = ± determines the number of the field spatial harmonic
with the maximal amplitude. Physically, conditions (5) mean the in-phase addition
of the electron radiation from individual gaps where interaction takes place when
electrons move synchronously with the mth spatial harmonic to the velocity
/e m mv v hω= = .
Introducing the field-phase shift 1q q qϕ ψ ψ+= − at the qth space and taking
into account that 1q q qL z z+= − , we can write the equivalent conditions of
synchronism for every spacing:
2 ,m q qh L mϕ π= + 1,2...q Q= (6)
The synchronism of electrons and field in nonuniform slowing systems is also
possible under more general conditions:
2 ,m q q qh L mϕ π= + 1,2...q Q= (7)
where 0, 1, 2...qm = ± ± varies from spacing to spacing, i.e., as if a particular qm th
synchronous spatial harmonic is taken at each spacing.
Taking into account the conditions of synchronism (5) and (6), one can rewrite
expression (3) for the ampli tudes of spatial harmonics:

5. ( ) ( ) ( )( )
1
1
exp
Q
q q m q
q
E h U M h i h h z
l =
= −∑ (8)
The amplitude of the selected mth harmonic that meets conditions (5) or (6) will
be maximal:
( ) ( )
1
1 Q
m q q m
q
E h U M h
l =
= ∑ (9)
In a periodic waveguide, ,q aL L ϕ ϕ≡ ≡ , and q qψ ϕ= ; therefore, conditions
(5) and (6) are met for an infinite series of spatial harmonics 'm m= when
( )' 2 ' /m mh h m m Lπ= + − , the difference in their amplitudes being determined
only by ( )q mM h .
In a nonuniform waveguide with different spacings qL , condition (2) can be
satisfied for one harmonic by choosing the appropriate phases qψ . For mh h≠ ,
this condition is either not satisfied or holds for the wave-number spectrum, which
is less dense than in a periodic waveguide. Thus, selection of spatial harmonics
takes place.
Such a mechanism can also be used for mode selection. Separating one spatial
harmonic of the operating mode, one can suppress other spatial harmonics of, not
only this mode, but of other modes as well.
2. THE COUPLING IMPEDANCE
AND INTERACTION COEFFICIENT
FOR PSEUDOPERIODIC SLOWING SYSTEMS
The relations derived above enable us to calculate the amplitudes of spatial
harmonics of a pseudoperi-odic slowing system. To analyze interaction of the elec-
tron beam with the field, it is also necessary to know the value of the parameter
characterizing the interaction efficiency. For a TWT, the coupling impedance of
the slowing system is usually chosen as such a parameter; however, in the case of
structures with pronounced non-uniformity and a small number of gaps (for
example, pseudoperiodic systems), it is expedient to apply also the interaction
coefficient which is similar to the quantity used in the theory of klystrons. Let us
determine these values according to the rules that are applied to the definition of
the known quantities.
According to (8), for a lossless structure ( qU U≡ ), we have
( ) ( )
UQ
E h M h
l
=
where

6. ( ) ( )( )
1
1
( ) exp
Q
q m q
q
M h M h i h h z
Q =
= −∑ (10)
Fig. 1. Section of the pseudoperiodic waveguide comprising Q spaces; q denotes the spacing
number.
is the electron-field interaction coefficient averaged over the total length of the
system that depends on the wavenumber h.
To determine the average coupling impedance ( )K h for a bilaterally matched
pseudoperiodic system with a limited length, one can use the relationship
( ) ( )
2 2
2 2
( )
2
E h M h
K h Z
h P ϕ
= = (11)
where 2
/2Z U P= is the gap characteristic impedance, and /hl Qϕ = is the
average phase shift per spacing.
The interaction coefficient considered here is generally a complex quantity;
however, when calculating the coupling impedance and investigating suppression
of spatial harmonics, only its modulus is of importance.
The obtained relationships allow us to calculate the efficiency of interaction of the
electrons with the field of spatial harmonics corresponding to different modes for
different wavenumbers and arbitrary number Q of the interaction gaps that have
different interaction coefficients qM , voltages qU , and arbitrary phase distribution
qΨ over the gaps. This method makes it possible to optimize pseudoperiodic
structures from the viewpoint of suppressing spurious modes and spatial
harmonics.
3. ANALYSIS OF THE SPATIAL HARMONIC SELECTION
Assuming, for the sake of simplicity, that all the gaps are equal, so that
( ) ( ), 1,2...q lM h M h q Q= = we obtain ( ) ( )q m l mM h M h= . We shall

7. characterize suppression of spatial harmonics with respect to the mth harmonic,
which satisfies conditions (5) and (6), by the quantity
1
( ) 1
exp( ( ) )
( )
Q
m q
qm
E h
i h h z
E h Q =
= −∑ (12)
Let us consider various cases.
Fig. 2. Distribution of spatial harmonics with respect to wavenumbers for a section of the
periodic system; / 0, 10L L QΔ = = .
Fig. 3. Suppression of spatial harmonics in a section of the pseudoperiodic system with linear
variation of the spacing for (a) / 0.1, 10L L QΔ = = and (b) / 0.05, 20L L QΔ = = .

8. Linear variation of the spacing, ( 1)qL L q L= + − Δ , where LΔ is the spacing
increment. In this case, we have
1
( 1)
2
q
q j
j
q q
z L qL L
=
−
= = + Δ∑
and expression (12) takes the form
( ) ( )
1
( ) 1
exp 1 1
( ) 2
Q
m
qm
E h L
i h h L q
E h Q L=
⎛ Δ ⎞⎡ ⎤
= − − + −⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠
∑ (13)
which determines the ratio of this field to the synchronous field depending on the
difference of wavenumbers and the nonuniformity parameter /L LΔ .
Figure 2 shows this ratio for a section of the periodic waveguide with 10Q = and
LΔ = 0. In the periodic waveguide, the main maxima correspond to the spatial
harmonics, and their values are equal, because the interaction coefficients for all
gaps were assumed to be the same when formula (13) was derived. A finite width of
the lobes close to the main maxima and the presence of sidelobes are caused by the
finite length of the waveguide section under consideration.
In the pseudoperiodic waveguide, certain main maxima, i.e., spatial harmonics,
are suppressed, and the degree of suppression depends on the rate of the spacing
variation and number of gaps. As seen from Figs. 3a and 3b, the amplitudes of
spurious spatial harmonics can be reduced up to 0.4-0.5 of their value in the peri-
odic structure. As the number of spacings increases from 10 to 20, the efficiency of
suppression becomes more pronounced.
Abrupt variation of the spacing. A two-section system with one abrupt
variation of the spacing is the simplest version of a pseudoperiodic structure. In
this case, the section parameters are chosen so that the selected spatial harmonic
retains its value, and other harmonics are to a certain extent suppressed. Let the
first and second sections comprise, respectively, 1Q gaps with the spacing 1L and
2Q gaps with the spacing 2L . In this case, considering one mode in both sections,
we obtain from (12)
( )( )
( ) ( )
1
1
1
2
1 1 1
1 1
( ) 1
exp
( )
1
exp
Q
m
qm
Q
m
q Q
E h
i h h qL
E h Q
L
i h h Q q Q L
Q L
=
= +
= − − +
⎛ ⎞⎡ ⎤
+ − + −⎜ ⎟⎢ ⎥
⎣ ⎦⎝ ⎠
∑
∑
(14)
The diagrams in Figs. 4a and 4b show that even a two-section system enables one
to suppress some spatial harmonics.

9. 4. EXAMPLES OF PSEUDOPERIODIC SYSTEMS
In spiral slowing systems, the wave slowing factor essentially depends on the
pitch, radius of winding, and velocity of the wave propagation along the coiled line.
In [2], a family of planar pseudoperiodic spirals was considered; depending on the
law of winding, these spirals selected either the fundamental spatial harmonic m = 0
(logarithmic spiral) or higher harmonics m = ±1, ±2... (synchronous spirals). The
wave can travel along these systems without reflections due to continuous variation
of the pitch.
In other systems, an abrupt variation of the spacing (or of the pitch) can be
implemented in the simplest manner. In this case, the problem of matching separate
sections of the system arises, so that one has to choose their parameters not only in
order to obtain equal phase velocities of the operating spatial harmonic in the sec-
tions, but also to provide minimal reflections. Let us consider how this is done in
helical H- or П-waveguides, which were proposed in [6,7] for application in high-
power wide-band TWTs. In particular, it is demonstrated in these papers that there
exists a dense spectrum of spatial harmonics propagating in such waveguides, which
requires their selection. A cross-sectional view of a helical П-waveguide is shown
in Fig. 5. Consider the possibility of separating the fundamental harmonic using a
two-section waveguide. Electron beams interact with the waveguide field in the gaps
of a width d at a distance R from the axis. Therefore, radius R is the same in both
waveguide sections, and the pitch L, ridge width A, height B, and gap width d may
be varied. As a result of variation of the waveguide cross section, the phase velocity
( )v v xΦ Φ= of the wave traveling along the curvilinear waveguide axis x changes,
and the phase at the qth spacing is determined by the integral
0
( )
qx
q dx
v x
ω
Φ
Ψ = ∫
which is easily calculated for a sectional waveguide.
Choosing 2 1(0.7 0.8) ,L L= − , we obtain the coefficient of the harmonic
suppression for these values, which is presented in Figs. 4a and 4b.
In this case, the slowing factor of the fundamental wave in both sections must be
the same, i.e.,
2
01 02
0
2
, 1
кр
c R
v v
v L
π λ
λ
⎛ ⎞
= ≈ −⎜ ⎟⎜ ⎟
⎝ ⎠
where, for the cutoff wavelength, we have an approximate relationship
'/avB Al dλ π≈ .
Another condition consists in matching the charac teristic impedances of the
waveguide sections:

10. 1 2,Z Z=
( )
lim1,2
1,2 1,2
1,21 / c
Z
Z ξ
λ λ
=
−
where limZ is determined mainly by the ridge parameters (capacitance), and ξ
characterizes the effect of the bend and the transit channel.
Fig. 4. Suppression of spatial harmonics in a two-section pseudoperiodic system with an abrupt
variation of the spacing; 1 210, 5.Q Q Q= = = , 2 1/ 0.8L L = (a) and 2 1/ 0.7L L = (b).
Fig. 5. Helical П-waveguide.
The two conditions written above can be satisfied by changing the ridge width A
and height В together with pitch L and gap width d. Thus, one can match the char-
acteristic impedances of separate waveguide sections and, at the same time,

11. suppress spurious harmonics and preserve the amplitude of the fundamental
harmonic.
CONCLUSION
The possibility of selecting spatial harmonics and modes in pseudoperiodic
systems has been studied. The method of selection is based on the coordinated
variation of the spacing (pitch) and phase distribution along the system, which
provides the constant phase velocity of one spatial harmonic and destroys other
spatial harmonics. The efficiency of suppressing spurious spatial harmonics and
modes is evaluated. Using a sectional helical П-waveguide as an example, we have
shown that it is possible to match simultaneously the phase velocity of the
operating spatial harmonic and characteristic impedances of the sections.
In order to apply the developed principle of selection of spatial harmonics and
modes in slowing systems of various types (helical and resonator systems) in the
general case, it is necessary to study and select discontinuities and their distribution
in the system that would provide simultaneously the efficient electron-beam
interaction and mode selection, as well as obtaining the filtering properties which
govern the matching with external circuits.
ACKNOWLEDGMENTS
The work was supported by the Russian Foundation for Basic Research, project no.
97-02-16577.
REFERENCES
1. Solntsev, V. A., Proc. SPIE Int. Soc. Opt. Eng., 1994, vol. 2250, p. 399.
2. Solntsev, V. A., Radiotekh. Elektron. (Moscow), 1994, vol. 39, no. 4, p. 552.
3. Solntsev, V. A., Abstracts of Papers, 50 nauchnaya sessiya, posvyashchennaya
dnyu radio (50th Scientific Session Devoted to the Day of Radio), Moscow,
1995, part II, p. 136.
4. Solntsev, V. A. and Solntseva, K. P., Abstracts of Papers, Black Sea Region
Symposium on Applied Electromagne-tism, Athens, 1996, p. 13.
5. Silin, R. A., Elektron. Tekh., Sen 1: Elektronika SVCh, 1976,no. 11, p. 3.
6. Mukhin, S. V. and Solntsev, V. A., Izv. Vyssh. Uchebn. Zaved., Radioelektron.,
1990, vol. 33, no. 10, p. 35.
7. Amirov, V. A., Kalinin, Yu. A., Kolobaeva, Т.Е., et al., in Lektsii po SVCh-
elektronike i radiofizike (Lectures on Microwave Electronics and Radio
Physics), Saratov, 1996, Book 1, part II, p. 157.