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 .

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.

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( )2zE z E h ihz dhπ

∞

−∞

= ∫

(2) 1( ) ( )exp( )zE h E z ihz dzl

∞

−∞

= −∫

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 qq

E h U M i hzl

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 qq q z L

M h f z z h z z dzf d

ψ ψ+

−

⎡ ⎤= − + −⎣ ⎦∫ (4)

is the local electron-field interaction coefficient; / 2

/ 2

1 ( )q q

q q

z L

qq z L

f f z dzd

+

−

= ∫

is the average value of the field amplitude at the qth spacing; 0q 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

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 2q q

q

d dM 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:

( ) ( ) ( )( )1

1 expQ

q q m qq

E h U M h i h h zl =

= −∑ (8)

The amplitude of the selected mth harmonic that meets conditions (5) or (6) will be maximal:

( ) ( )1

1 Q

m q q mq

E h U M hl =

= ∑ (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

( ) ( )UQE h M hl

=

where

( ) ( )( )1

1( ) expQ

q m qq

M h M h i h h zQ =

= −∑ (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( )2E h M h

K h Zh 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

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 qqm

E h i h h zE 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 = = .

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 jj

q qz L qL L=

−= = + Δ∑

and expression (12) takes the form

( ) ( )1

( ) 1 exp 1 1( ) 2

Q

mqm

E h Li h h L qE 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

11

21 1 1

1 1

( ) 1 exp( )

1 exp

Q

mqm

Q

mq Q

E h i h h qLE h Q

Li h h Q q Q LQ L

=

= +

= − − +

⎛ ⎞⎡ ⎤+ − + −⎜ ⎟⎢ ⎥

⎣ ⎦⎝ ⎠

∑

∑ (14)

The diagrams in Figs. 4a and 4b show that even a two-section system enables one to suppress some spatial harmonics.

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 dxv xω

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

01 020

2, 1кр

c Rv vv L

π λλ

⎛ ⎞= ≈ − ⎜ ⎟⎜ ⎟

⎝ ⎠

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:

1 2,Z Z=

( )lim1,2

1,2 1,2

1,21 / c

ZZ 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,

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.

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