Pseudoperiodic waveguides with selection of spatial harmonics and modes


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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.

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Pseudoperiodic waveguides with selection of spatial harmonics and modes

  1. 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 PROPAGATIONPseudoperiodic Waveguides with Selection of SpatialHarmonics and ModesV. A. SolntsevReceived June 9, 1998Abstract — A principle of selection of modes and their spatial harmonics inperiodic waveguides and, in particular, in spatially developed slowing systems formultibeam traveling-wave tubes (TWTs) is elaborated. The essence of the principleis in the following: varying along the length of the system its period and at leastone more parameter that determines the phase shift per period, one can provideconstant phase velocity of one spatial harmonic and destroy other spatialharmonics, i.e., reduce their amplitudes substantially. In this case, variations of theperiod may be significant, and the slowing system becomes nonuniform, orpseudoperiodic; namely, one of the spatial harmonics remains the same as in theinitial periodic structure. Relationships are derived for the amplitudes of thespatial-wave harmonics, interaction coefficient, and coupling impedance of thepseudoperiodic system. The possibility of the mode selection in pseudoperiodicslowing systems when the synchronism condition is satisfied for the spatialharmonic of one mode is investigated. The efficiency of suppressing spuriousspatial harmonics and modes for linear and abrupt variation of spacing isestimated. The elaborated principle of selection of spatial harmonics and modes isillustrated by an example of a two-section helical-waveguide slowing system.INTRODUCTIONThe selection of modes and their spatial harmonics is important for high-poweramplifiers and microwave oscillators with electron beams: traveling-wave tubes(TWTs), backward-wave tubes, gyrotrons, etc. The output microwave power of adevice P JUη= is determined by the electron-beam current J, acceleratingvoltage U, and efficiency η .
  2. 2. To raise the output of relativistic devices, electron beams are applied that areaccelerated by the voltage U> 100 kV, as well as explosive-emission cathodes thatproduce a large current. In such devices, one has to use large-current electronaccelerators to obtain high-power electron beams; therefore, they are manufacturedas 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 islimited. The beam-current density is restricted by the capacity of the electron-optical system; therefore, the beam cross section must be increased in order toincrease the current. In this case, cross-sectional dimensions may be much greaterthan the wavelength in the slowing system. There may be a large number of modesand spatial harmonics in such overdimensioned (or spatially developed) systemsthat cause electron-field multimode interaction, as well as amplification andoscillation instability. Thus, the selection becomes one of the major problems whenthe electron-beam current and the device power output are to be increased.At present, different techniques are used to suppress spurious oscillations andmodes, from the utilization of selective absorbers in TWTs and up to theapplicationof open resonators and open waveguides in orotrons, gyrotrons, andfree-electron lasers.This paper addresses the principle of selection of spatial harmonics and modesin periodic waveguides and slowing systems; in a general form, this principle wasformulated in [1, 2], and developed then in [3, 4]. The essence of the principleconsists in employing periodic electrodynamic systems with nonuniformly spacedelectron-field interaction gaps and a specified relation between the gap spacing andthe gap-field phase, which makes it possible to select one spatial harmonic or modeand suppress the others.The considered technique of selection in slowing systems is similar to the methodof suppressing the sidelobe maxima of nonuniform antenna arrays. Suchnonuniform systems may be considered as cryptoperi-odic or pseudoperiodic,wherein the amplitudes of one or several harmonics remain the same as in theinitial periodic system, and the amplitudes of other spatial harmonics decrease. Aplanar logarithmic spiral or the synchronous spirals considered in [2] representexamples of pseudoperiodic systems. To some extent, one can assign to this classtwo- or three-section systems with different spacing in the sections but with thesame phase velocity of one of the spatial harmonics in all the sections [5].Generally, one can apply the considered principle of selection to slowing systemsof any type introducing the nonuniformity of both the spacing and the respectivephase shift over the spacing by varying the dimensions or configuration of thesystem elements from one space to another (for instance, the dimensions of slots ina comb-type structure, cross section of a helical waveguide, etc.). Here, we willconsider the influence of the distributions of spacings and the field phase over thespacings on the amplitudes of spatial harmonics that determine the efficiency ofthe electron-field interaction.
  3. 3. 1. AMPLITUDES OF SPATIAL HARMONICS AND THESYNCHRONISM CONDITIONTo describe the electron-field interaction in pseudo-periodic systems, one canuse amplitudes of spatial harmonics and the interaction coefficients and couplingimpedance for the system as a whole or the local interaction coefficients forindividual gaps. Consider a general method of calculating these quantities.Assume that it is given the longitudinal electric field distribution along thesystem 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 thetype of the system (uniform periodic or nonuniform).Applying the Fourier transformation, we define the amplitudes ( )E h of spatialharmonics by the relations1( ) ( )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 thewavenumber h and differ from the spectral density only by the factor 1/l, wherel is the length of the system. Let us represent them as a sum over the Q spacings ofthe system:11( ) exp[ ( )]Qq q q qqE h U M i hzlψ== −∑ (3)where ( )q qzψ ψ= is the average field phase at the qth spacing;( )/2/21( ) ( )exp ( )q qq qz Lq q qq q z LM h f z z h z z dzf dψ ψ+−⎡ ⎤= − + −⎣ ⎦∫ (4)is the local electron-field interaction coefficient;/ 2/ 21( )q qq qz Lqq z Lf f z dzd+−= ∫is the average value of the field amplitude at the qth spacing; 0q q qU E f d= is therf voltage at the qth spacing; and q qz and d are the mean coordinate andeffective width of the qth gap. Note that for the gridless gaps, the choice of qd and
  4. 4. qf is to a certain extent arbitrary, because only their product is defined. Variationof the voltage from one gap to another is determined, for the chosen form (1) of thefield representation, both by the distribution function ( )f z and losses in thesystem. Formally, one may not separate these factors and take into account lossesfrom 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 interactioncoefficient (4), one may assume, as a rule, that within the qth gap, ( ) qzψ ψ≈ thefield is constant in the gap, ( ) qf z f= we obtain the familiar expressionsin /2 2q qqd dM h h⎛ ⎞= ⎜ ⎟⎝ ⎠.In the general case, the written relationships enable one to take into account thedistribution of the field amplitude and phase within one spacing. Usually, variationof the field phase within a spacing can be ignored for slowing systems withdiscrete electron-field interaction; then, the interaction coefficients are real andpositive, 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 harmonicwith the maximal amplitude. Physically, conditions (5) mean the in-phase additionof the electron radiation from individual gaps where interaction takes place whenelectrons 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 takinginto account that 1q q qL z z+= − , we can write the equivalent conditions ofsynchronism 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 alsopossible 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 thsynchronous spatial harmonic is taken at each spacing.Taking into account the conditions of synchronism (5) and (6), one can rewriteexpression (3) for the ampli tudes of spatial harmonics:
  5. 5. ( ) ( ) ( )( )11expQq q m qqE h U M h i h h zl == −∑ (8)The amplitude of the selected mth harmonic that meets conditions (5) or (6) willbe maximal:( ) ( )11 Qm q q mqE 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 determinedonly by ( )q mM h .In a nonuniform waveguide with different spacings qL , condition (2) can besatisfied 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, whichis less dense than in a periodic waveguide. Thus, selection of spatial harmonicstakes place.Such a mechanism can also be used for mode selection. Separating one spatialharmonic of the operating mode, one can suppress other spatial harmonics of, notonly this mode, but of other modes as well.2. THE COUPLING IMPEDANCEAND INTERACTION COEFFICIENTFOR PSEUDOPERIODIC SLOWING SYSTEMSThe relations derived above enable us to calculate the amplitudes of spatialharmonics 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 parametercharacterizing the interaction efficiency. For a TWT, the coupling impedance ofthe slowing system is usually chosen as such a parameter; however, in the case ofstructures with pronounced non-uniformity and a small number of gaps (forexample, pseudoperiodic systems), it is expedient to apply also the interactioncoefficient which is similar to the quantity used in the theory of klystrons. Let usdetermine these values according to the rules that are applied to the definition ofthe known quantities.According to (8), for a lossless structure ( qU U≡ ), we have( ) ( )UQE h M hl=where
  6. 6. ( ) ( )( )11( ) expQq m qqM h M h i h h zQ == −∑ (10)Fig. 1. Section of the pseudoperiodic waveguide comprising Q spaces; q denotes the the electron-field interaction coefficient averaged over the total length of thesystem that depends on the wavenumber h.To determine the average coupling impedance ( )K h for a bilaterally matchedpseudoperiodic system with a limited length, one can use the relationship( ) ( )2 22 2( )2E h M hK h Zh P ϕ= = (11)where 2/2Z U P= is the gap characteristic impedance, and /hl Qϕ = is theaverage phase shift per spacing.The interaction coefficient considered here is generally a complex quantity;however, when calculating the coupling impedance and investigating suppressionof spatial harmonics, only its modulus is of importance.The obtained relationships allow us to calculate the efficiency of interaction of theelectrons with the field of spatial harmonics corresponding to different modes fordifferent wavenumbers and arbitrary number Q of the interaction gaps that havedifferent interaction coefficients qM , voltages qU , and arbitrary phase distributionqΨ over the gaps. This method makes it possible to optimize pseudoperiodicstructures from the viewpoint of suppressing spurious modes and spatialharmonics.3. ANALYSIS OF THE SPATIAL HARMONIC SELECTIONAssuming, 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. 7. characterize suppression of spatial harmonics with respect to the mth harmonic,which satisfies conditions (5) and (6), by the quantity1( ) 1exp( ( ) )( )Qm qqmE hi 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 theperiodic system; / 0, 10L L QΔ = = .Fig. 3. Suppression of spatial harmonics in a section of the pseudoperiodic system with linearvariation of the spacing for (a) / 0.1, 10L L QΔ = = and (b) / 0.05, 20L L QΔ = = .
  8. 8. Linear variation of the spacing, ( 1)qL L q L= + − Δ , where LΔ is the spacingincrement. In this case, we have1( 1)2qq jjq qz L qL L=−= = + Δ∑and expression (12) takes the form( ) ( )1( ) 1exp 1 1( ) 2QmqmE h Li h h L qE h Q L=⎛ Δ ⎞⎡ ⎤= − − + −⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠∑ (13)which determines the ratio of this field to the synchronous field depending on thedifference of wavenumbers and the nonuniformity parameter /L LΔ .Figure 2 shows this ratio for a section of the periodic waveguide with 10Q = andLΔ = 0. In the periodic waveguide, the main maxima correspond to the spatialharmonics, and their values are equal, because the interaction coefficients for allgaps were assumed to be the same when formula (13) was derived. A finite width ofthe lobes close to the main maxima and the presence of sidelobes are caused by thefinite 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 spacingvariation and number of gaps. As seen from Figs. 3a and 3b, the amplitudes ofspurious 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 ofsuppression becomes more pronounced.Abrupt variation of the spacing. A two-section system with one abruptvariation of the spacing is the simplest version of a pseudoperiodic structure. Inthis case, the section parameters are chosen so that the selected spatial harmonicretains its value, and other harmonics are to a certain extent suppressed. Let thefirst and second sections comprise, respectively, 1Q gaps with the spacing 1L and2Q gaps with the spacing 2L . In this case, considering one mode in both sections,we obtain from (12)( )( )( ) ( )11121 1 11 1( ) 1exp( )1expQmqmQmq QE hi h h qLE h QLi h h Q q Q LQ L== += − − +⎛ ⎞⎡ ⎤+ − + −⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠∑∑(14)The diagrams in Figs. 4a and 4b show that even a two-section system enables oneto suppress some spatial harmonics.
  9. 9. 4. EXAMPLES OF PSEUDOPERIODIC SYSTEMSIn spiral slowing systems, the wave slowing factor essentially depends on thepitch, 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 thelaw of winding, these spirals selected either the fundamental spatial harmonic m = 0(logarithmic spiral) or higher harmonics m = ±1, ±2... (synchronous spirals). Thewave can travel along these systems without reflections due to continuous variationof the pitch.In other systems, an abrupt variation of the spacing (or of the pitch) can beimplemented in the simplest manner. In this case, the problem of matching separatesections of the system arises, so that one has to choose their parameters not only inorder 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 inhelical 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 thereexists a dense spectrum of spatial harmonics propagating in such waveguides, whichrequires their selection. A cross-sectional view of a helical П-waveguide is shownin Fig. 5. Consider the possibility of separating the fundamental harmonic using atwo-section waveguide. Electron beams interact with the waveguide field in the gapsof a width d at a distance R from the axis. Therefore, radius R is the same in bothwaveguide sections, and the pitch L, ridge width A, height B, and gap width d maybe 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 integral0( )qxq dxv xωΦΨ = ∫which is easily calculated for a sectional waveguide.Choosing 2 1(0.7 0.8) ,L L= − , we obtain the coefficient of the harmonicsuppression 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 bethe same, i.e.,201 0202, 1крc Rv vv Lπ 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 thewaveguide sections:
  10. 10. 1 2,Z Z=( )lim1,21,2 1,21,21 / cZZ ξλ λ=−where limZ is determined mainly by the ridge parameters (capacitance), and ξcharacterizes the effect of the bend and the transit channel.Fig. 4. Suppression of spatial harmonics in a two-section pseudoperiodic system with an abruptvariation 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 Aand 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. 11. suppress spurious harmonics and preserve the amplitude of the fundamentalharmonic.CONCLUSIONThe possibility of selecting spatial harmonics and modes in pseudoperiodicsystems has been studied. The method of selection is based on the coordinatedvariation of the spacing (pitch) and phase distribution along the system, whichprovides the constant phase velocity of one spatial harmonic and destroys otherspatial harmonics. The efficiency of suppressing spurious spatial harmonics andmodes is evaluated. Using a sectional helical П-waveguide as an example, we haveshown that it is possible to match simultaneously the phase velocity of theoperating spatial harmonic and characteristic impedances of the sections.In order to apply the developed principle of selection of spatial harmonics andmodes in slowing systems of various types (helical and resonator systems) in thegeneral case, it is necessary to study and select discontinuities and their distributionin the system that would provide simultaneously the efficient electron-beaminteraction and mode selection, as well as obtaining the filtering properties whichgovern the matching with external circuits.ACKNOWLEDGMENTSThe work was supported by the Russian Foundation for Basic Research, project no.97-02-16577.REFERENCES1. 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, posvyashchennayadnyu 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 RegionSymposium 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 RadioPhysics), Saratov, 1996, Book 1, part II, p. 157.