508 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 3, JUNE 1998

Gyro-TWT with a Helical OperatingWaveguide: New Possibilities to Enhance

Efficiency and Frequency BandwidthGregory G. Denisov, Vladimir L. Bratman, Alan D. R. Phelps, and Sergei V. Samsonov

Abstract—A helical corrugation of the inner surface of anoversized cylindrical waveguide provides, for certain parameters,an almost constant value of group velocity and close to zerolongitudinal wavenumber of an eigenwave for a very broadfrequency band. The use of such a helical waveguide as anoperating section of a gyrotron traveling wave tube (gyro-TWT)allows significant widening of its bandwidth and an increase inthe efficiency at very large particle velocity spreads. In this paper,the new concept is confirmed by theoretical analysis and “cold”measurements of the helical waveguide dispersion. Results of alinear and nonlinear theory of the helical gyro-TWT as well astwo designs for subrelativistic (80 keV, 20 A) and relativistic (300keV, 80 A) electron beams are also presented. For both designs,parameters providing a very broad frequency band (about 20%)and high efficiency (above 30%) have been found. When thetransverse velocity spread is increased from zero up to a very highvalue of 40%, simulations showed only a 20%–30% narrowing inthe frequency band and a 20% decrease in electron efficiency. Thetheoretical analysis demonstrates important advantages of thehelical gyro-TWT over the “smooth” one in frequency bandwidth,sensitivity to electron velocity spread, and stability to parasiticself-excitation.

Index Terms—Broadband, electron cyclotron amplifier, freeelectron amplifier, gyro-amplifier tube, gyro-TWT, gyrotron am-plifier, gyrotron traveling wave tube, helical waveguide, high-efficiency gyrodevice, high-power microwave TWT, microwaveamplifier tube.

I. INTRODUCTION

UP TO NOW, among the many known gyrodevices[1]–[3], only the gyrotron [4], [5] and gyroklystron

[6]–[8] are sufficiently developed and have high efficiency.At the same time, such an attractive broad-band amplifieras the gyro-TWT [9], [10], in spite of long and intensiveefforts (see, e.g., [11]–[14]), is actually still in the researchstage. This is linked with inherent difficulties of realizationof gyro-TWT’s with a conventional microwave system inthe form of a cylindrical waveguide. The most attractiveoperating regime for such devices is that of grazing incidenceof the wave and beam dispersion characteristics when theaxial electron velocity is close to the group velocity of the

Manuscript received September 16, 1997; revised March 26, 1998. Thiswork was supported by the United Kingdom DERA and Gycom, Ltd., NizhnyNovgorod, Russia.

G. G. Denisov, V. L. Bratman, and S. V. Samsonov are with the Institute ofApplied Physics, Russian Academy of Sciences, Nizhny Novgorod, 603600,Russia.

A. D. R. Phelps is with the Department of Physics and Applied Physics,University of Strathclyde, Glasgow, G4 0NG, U.K.

Publisher Item Identifier S 0093-3813(98)04271-4.

wave. Correspondingly, gyro-TWT’s with weakly relativisticelectron beams operate at frequencies near cutoff which limitstheir frequency band and reduces their stability to gyrotronoscillation. At the same time, gyro-TWT’s driven by electronbeams with relativistic particle axial velocity may have amuch broader frequency band, but due to operation withrelatively high Doppler upshift (CARM-TWT regime), theyare very sensitive to particle velocity spread. In this paper,which is based on the idea first published in [15], a newmicrowave system for the gyro-TWT is developed. Thissystem represents an oversized waveguide with a specialhelical corrugation of the inner wall. The main idea of usingthe helical corrugation consists in the radical change of thesmooth waveguide dispersion in the region of close-to-zerolongitudinal wavenumbers which allows the group velocityof the eigenwave and the band of the resonant electron-waveinteraction to be significantly increased. This is achievedby means of coupling on the corrugation two partial wavesof a regular waveguide with cutoff frequencies near andfar below the operating frequency respectively. The firstpartial wave resonantly interacts with electrons while theadmixture of the second wave makes the group velocity of theeigenwave nonzero. Changing the geometrical parameters ofthe corrugation, one can control the eigenwave group velocityand adjust it to the longitudinal velocity of an electron beamfor a rather broad frequency band around the point wherethe longitudinal wavenumber is equal to zero. These featuresallow very attractive regimes of gyro-TWT operation to berealized.

In Section II, the principle of gyro-TWT operation in com-parison with the gyrotron and the gyrotron-backward-waveoscillator (gyro-BWO) is discussed. In Section III, results oftheoretical and experimental study of wave dispersion in thehelically corrugated waveguide are reported. In Section IV, atheory of electron-wave interaction for the helical gyro-TWTincluding amplification of the operating wave in both linearand nonlinear regimes, as well as self-excitation of spuriousmodes, is developed. In Section V, results of the optimizationfor new designs of subrelativistic and relativistic amplifiersare given.

II. PRINCIPLE OF GYRO-TWT OPERATION

Operation of all gyrodevices is based on the stimulated cy-clotron radiation of electrons moving along helical trajectoriesin a homogeneous magnetostatic field . In the

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DENISOV et al.: GYRO-TWT WITH A HELICAL OPERATING WAVEGUIDE 509

interaction region of a gyrodevice, the cyclotron resonancebetween an operating electromagnetic wave and electronsoccurs

(1)

Here, and are the frequency and the axial wavenumberof the wave, and are the axial velocityand the cyclotron frequency of the electrons, andis thecyclotron harmonic number. According to (1), gyrodeviceshave an important peculiarity, namely, the possibility of theresonant interaction of electrons with fast electromagneticwaves. Correspondingly, in most gyrodevices, cavities andwaveguides with smooth metal walls are used as microwavesystems.

The main varieties of gyrodevices can be classified bythe magnitude of the axial wavenumber. In the gyrotron,the operating cavity mode is formed with waves propagatingalmost across the magnetostatic field and having very smallaxial wavenumbers

(2)

where . This provides very weak sensitivity to thevelocity spread and is one of the main merits of the gyrotron.In the gyro-TWT, electrons interact with a cotraveling wave,

, the feedback is absent, and convective amplification ofthe wave occurs. In the gyro-BWO, particles excite a counter-traveling wave, , and internal distributed feedback andan absolute instability are realized.

As a microwave system for the gyro-TWT, a cylindricalmetal waveguide with a circular cross section is most fre-quently used. The “Achilles’ heel” of such a system is aparasitic gyrotron self-excitation which forces one to choosethe operating wave with a relatively large axial wavenumberin order to provide the stable amplification regime, and,correspondingly, results in significantly higher sensitivity tothe velocity spread in comparison with the gyrotron. It wouldbe very attractive to avoid the drawback of the gyro-TWTmentioned above and enhance its efficiency. A possible wayfor such an improvement is a radical change in the waveguidedispersion [16], provided, in particular, by using a circularcylindrical waveguide with a helical corrugation of its innerwall [15].

III. D ISPERSION OF THEWAVES IN A CIRCULAR

CYLINDRICAL WAVEGUIDE WITH HELICAL CORRUGATION

The helical corrugation which resonantly scatters the op-erating near-cutoff mode into a traveling wave and providesone-way energy output from the cylindrical cavity is usedsometimes for gyrotrons [17], [26]. In [15], an analogousmethod was suggested for increasing the wave group veloc-ity and starting currents in the gyro-TWT in the region ofclose-to-zero axial wavenumbers. Let us discuss this methodconsidering an oversized waveguide with the helical profile ofits inner surface represented in cylindrical coordinatesas follows:

(3)

Here, is mean radius of the waveguide, andare amplitude, azimuthal, and axial numbers of the corrugation,

Fig. 1. Diagram showing parameters of partial waves to be coupled by thecorrugation.

respectively, and is the corrugation period. In order toprovide coupling, the near-cutoff mode

(4)

formed by waves having small axial wavenumbers ,with a far-from-cutoff traveling wave

(5)

parameters of the corrugation have to obey the Bragg reso-nance conditions

(6)

Here, are are azimuthal numbers of the waves. Theresonant coupling of the waves corresponds to the intersectionof their dispersion curves on the dispersion diagram (Fig. 1).More exactly, the intersection occurs between one of the wavesand a spatial harmonic of the second wave. For the desireddispersion, they are mode and the first spatial harmonicof wave , namely, , where

(Fig. 1). For negligibly small corrugation depth,these modes satisfy the following unperturbed dispersion equa-tions:

(7)

where and are wavenumbers corresponding to thecutoff frequencies of modes and , respectively. In theinteresting region of parameters which is located around theBragg resonance for axial wavenumbers, , andnear the cutoff frequency of the mode , where

and are the “geometrical” andfrequency mismatches, respectively, (Fig. 1),the dispersion curves (7) can be approximated as a parabolaand a straight line

(8)

If the corrugation amplitude is small compared with thewavelength, , then the method of perturbation canbe used for calculation of the field structure and dispersioncharacteristics [18]–[20]. Following this method, we considerthe amplitudes of the partial waves and as functions ofthe axial coordinate, which are described by a set of coupledwave linear differential equations. In accordance with (8), in

510 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 3, JUNE 1998

(a)

(b)

Fig. 2. Dispersion of partial (thin lines) and eigen (thick lines) wavesof “cold” helical waveguide with different parameters of “geometrical”mismatch.

the region of parameters mentioned above, this set consistsof one equation of the second order and another one of thefirst order

(9)

where and is the coupling coefficient ofthe waves which is proportional to the relative amplitude ofcorrugation and depends also on the azimuthal and radialindices of the partial waves [18]–[20].

Searching for the eigenwaves of the set (9) in the form of, one obtains the dispersion equation (see also [26])

(10)

where the variables and parameters andare normalized to . The left part of (10) is, obviously,proportional to the product of the left parts of (8), describingdispersion of the uncoupled partial waves. Equation (10) can

Fig. 3. Photograph ofX-band waveguide with helical corrugation.

be simplified to the form containing one less number ofindependent parameters

(10 )

where.

Equation (10) has three roots: one of them,, is alwaysreal and two other roots, and , are real atand complex conjugates at , whereis a mismatch corresponding to the cutoff frequency foreigenwaves (Fig. 2). Eigenwave

at has positive group velocitywhich can be controlled near the point where its phasevelocity is infinite (at zero geometrical mismatch, ,it takes place when the frequency mismatch is linked with thecoupling coefficient by the relation ) by changing theparameters of the corrugation (Fig. 2). These features makethis eigenwave attractive for the use as an operating mode forthe gyro-TWT.

The approach described above was used to design severalwaveguide sections with helical corrugations. For relativelysmall corrugation depth corresponding to the coupling co-efficient , this method leads to a good agreementwith the experiment. In the “cold” experiment a 40-cm-longwaveguide of 14.5-mm mean radius with a three-fold helicalcorrugation of 1.5-mm amplitude and 37.5-mm period whichcoupled mode TE near cutoff and the TE traveling waveof this waveguide (for this case, and

) was tested (Fig. 3). Experimental points for twodifferent dispersion curves (Fig. 4) were found by means oftwo different methods.

In order to obtain curve 1, which is the main interest, theangle of rotation was measured for the linearly polarized TEwave when passed through the waveguide. As a matter offact, the dispersion of rotating waves in a waveguide withhelical corrugation strongly depends on the direction of theirrotation: the dispersion of the waves under Bragg conditions

DENISOV et al.: GYRO-TWT WITH A HELICAL OPERATING WAVEGUIDE 511

Fig. 4. Dispersion of partial waves, i.e., modes of a smooth waveguide (- - -) and eigenmodes of the spiral waveguide (—);—results of measurement.

(6) is significantly modified, while the dispersion of the wavewith the opposite rotation is practically unperturbed (close tothe TE wave dispersion in the smooth circular waveguide)and can be used as the reference one (for larger corrugationdepth, the dispersion characteristics of the reference waveshould be known more accurately). The difference in theaxial wavenumbers of the rotating waves results in thepolarization rotation of the initial linearly polarized wave(similar to the Faraday effect in gyrotropic media). The pointswhere the phase difference , where is the waveguidelength, amounts to and, correspondingly,the polarization vector at the output of the waveguide isperpendicular or parallel to the initial one are shown in Fig. 4.

In order to find experimental points for the dispersioncurve 2, resonant frequencies of different axial modes

of the cavity, formed by the waveguide section,were measured. The phase difference , corresponding tothese modes, and their measured frequencies determined thesegments parallel to the axis in the dispersion diagram.Upon matching of one of the ends of each segment with thecalculated curve, good agreement for the positions of the otherends were obtained (Fig. 4).

IV. I NTERACTION OF ELECTRON BEAM WITH

THE WAVES OF THE HELICAL WAVEGUIDE

A theory of the helical gyro-TWT can be developed byinclusion of effects of the operating mode scattering on thecorrugation into known gyrotron equations [21]. For a small-signal regime, such an approach results in a quite complicatedfifth-order system of linear differential equations; the mostserious difficulties arise when solving a boundary problem forthe excitation of spurious modes. Fortunately, for the mostattractive regimes of the operating mode amplification, the setof equations can be reduced to a third-order system whichformally coincides with that for the usual “smooth” gyro-

TWT, differing from the latter by the “cold” dispersion of theoperating wave and the value of its coupling with an electronbeam. Analogous reduction is especially fruitful for a nonlinearanalysis of the helical gyro-TWT operating regimes.

A. Equations for Gyro-TWT with the Helical Waveguide

We will describe the motion of electrons in a homogeneousmagnetostatic field and monochromatic RF fieldof the helical waveguide by the Lorentz equations usingLagrange’s variables

(11)

where is the particle momentum. Assuming that theRF field frequency is close both to the cutoff frequency of thepartial wave , being a TE-mode, and to a harmonic of thecyclotron frequency, we will neglect transverse componentsof the field and the nonresonant interaction of electronswith wave . In this case, the axial component of the electronmomentum is not changed, const. Neglecting also the RFspace charge as well as the perturbation in transverse modestructure caused by the corrugation and electron beam, the RFfield in (11) can be represented in the following form:

(12)

where functions anddescribe the radial structure of the resonant mode andand

are the Bessel function and its derivative.

512 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 3, JUNE 1998

The axial structures of both partial modes in the presenceof the electron beam are described by a set of equations

(13)

where is the beam current, is thedimensionless norm of mode andis the entrance time of each electron. A self-consistent set of(11)–(13) transforms into the well-known gyrotron equations[21] at zero coupling coefficient, and into (9) at zeroelectron beam current .

Averaging over fast cyclotron oscillations, taking into ac-count conservation of axial electron momentum, changingfrom the momentum to the energy and phase withrespect to the resonant modeand normalizing (11)–(13), one obtains the reduced nonlinearequations of the gyro-TWT

(14)

where is the normalized guiding center of theelectrons, and

. The set of self-consistent nonlinear steady-stateequations (14) with the corresponding boundary conditions forthe waves and the electrons allows the analysis of differentregimes of amplification and parasitic excitation of the gyro-TWT.

B. Linear Amplification

We will perform the analysis of small-signal gain and stabil-ity of the gyro-TWT using the linearized averaged equations

(15)

which have been obtained from (14) assuming that the am-plitudes of the modes and the normalized amplitudes

of the perturbations in electron energies,

, and phases, ,

are small. Here,

is the cyclotron resonance mismatch, and

is the Pierce parameter. Searching for a solution

of system (15) of the form , we obtain a dispersionequation for the gyro-TWT as follows:

(16)

where is the magnetic field mismatch.Similarly to the way the coupling parametercontrols the

coupling of partial modes of a “cold” waveguide, the Pierceparameter controls the coupling of the “cold” eigenwavesof the helical waveguide with the electron beam. In a generalcase, (16) has five different complex roots .If some of them have a positive imaginary part, ,then amplification of the corresponding “hot” eigenwavestakes place. At fixed parameters of the microwave system andelectron beam, functions determine the dispersion ofeigenwaves of the “hot” system. In the case when an electronbeam perturbs the partial waves stronger than a corrugation,

, we obtain very complicated behavior of the dispersioncurves, but in order to realize the advantages of the helicalwaveguide, we need to couple an electron beam with theparticular eigenwave having the attractive properties (curve 1in Fig. 2). Thus, the coupling provided by a corrugation shouldbe stronger than that caused by an electron beam .In addition, the magnetic field should be tuned such thatthe synchronism with the desirable wave having dispersion

is fulfilled

For this case, we can significantly simplify the dispersionequation (16) and reduce it to the third-order equation

(17)

which is identical to that for a usual gyro-TWT with a smoothcylindrical waveguide under the assumption of interaction ofelectrons with the only traveling wave [9], [10] (see also e.g.,[27]–[29]). In (17), is a function of which satisfies (10)and describes the “cold” dispersion of the operating eigenwaveof the helical waveguide (curve 1 in Fig. 2),is the modified gain parameter, and coefficient

(18)

is responsible for the content of the resonant partial modein the operating eigenwave 1. In the case of the usual gyro-TWT for the chosen normalization of parameters, one has

DENISOV et al.: GYRO-TWT WITH A HELICAL OPERATING WAVEGUIDE 513

, where is the normalized group velocityof the synchronous wave.

The analysis of (16) and (17) shows that for a large regionof electron beam parameters, including subrelativistic andrelativistic energies, a very broad frequency-band gain canbe obtained by optimizing the parameters of the corrugationand the value of the magnetostatic field. A good agreementbetween solutions of (16) and the simplified (17) up todemonstrates the correctness of the approach of the dominantinteraction of electrons with the only eigenwave of the “cold”system. This approach allows us also to simplify significantlya nonlinear analysis of the helical gyro-TWT (see Section III-D), which can be performed using the equations of the usualgyro-TWT with changed dispersion and coefficient .

For sufficiently small -beam current , zero axialwavenumber of the “cold” eigenwave and geometrical mis-match and , and the magnetic field mismatchcorresponding to exact cyclotron resonance between electronsand this wave , (17) is reduced to the simplest form

(19)

which gives for the small-signal gain .

C. Stability to Self-Excitation

Let us study now a problem of the helical gyro-TWTstability with respect to parasitic self-excitation of the eigen-modes 1 and 2. The topology of the dispersion curves (Fig. 2)shows that mode 2, having a region with very small groupvelocity, can be easily excited like an operating mode of ausual gyrotron-oscillator. But fortunately, for the attractiveregimes of the amplifier operation, when the corrugation issufficiently deep, large separation of dispersion characteristics1 and 2 occurs, and, correspondingly, one can expect thatparasitic mode 2 is excited at magnetic fields significantlyhigher than the operating ones. In order to analyze this problemwe have to use the whole fifth-order set of linear equations(15) unlike the case of the operating regime when third-orderequations are sufficient. For the sake of determinacy, among alot of possibilities we consider a scheme of the system havinga certain practical interest. Let a beam of electrons at theentrance of the interaction region be nonmodulated

(20)

A helical waveguide is bounded at this point by such a narrowcutoff section that the simplified boundary conditions

(21)

for partial waves can be used. We will assume also that theelectron-wave interaction region is limited at by asharp drop in the magnetic field while the helical waveguideis infinite from the collector end which corresponds to anideal matching for the eigenwaves (electromagnetic energyflow from infinity to the interaction region is absent)

(22)

(a)

(b)

Fig. 5. (a) Frequency and (b) starting current of self-excitation forgyro-TWT’s (V = 80 kV, � = 1:2) with smooth (thin lines) and helical(thick lines) waveguides(k0L = 100; � = 0:17;�g = 0:18).

Here, the eigenwave should be represented as a linearsuperposition of amplitudes describing partial waves ofa “cold” waveguide. Equations (15) with boundary conditions(20)–(22) represent a boundary problem. At fixed energy andpitch-angle of the electrons as well as parameters of thehelical waveguide and magnetostatic field, eigenvalues of thisproblem are the frequency and starting electron beam currentof the self-oscillations, or normalized parametersand .A numerical analysis of the problem confirms the predictiondeclared above that at the optimal corrugation, when thedispersion curves are far apart from each other, the regimeof the self-excitation with minimum starting current and theoperating one are separated in the value of the magneticfield by a significantly larger amount than those for theusual gyro-TWT. Numerical simulations with some concreteparameters show that the starting current of the self-excitationfor helical gyro-TWT’s at optimal magnetic field is sufficientlylarger than the operating current, whereas usual gyro-TWT’sneed some additional measures to prevent the self-excitation(Fig. 5).

D. Simplified Nonlinear Equations

When studying the saturated gain regimes, according tothe simplified linear approach of Section III-C we will use

514 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 3, JUNE 1998

Fig. 6. Scheme of proposed experiment with 300-keV 80-A helical gyro-TWT. 1: Explosive-emission cathode. 2: Anode with selecting aperture. 3: Electronbeam. 4: Kicker. 5: Wave launcher. 6: Helical waveguide. 7: Solenoid.

a first-order equation

(23)

for excitation of the operating eigenwave of the “cold” helicalwaveguide, with boundary condition

(24)

at the cathode end only, instead of a very complicated bound-ary nonlinear problem arising from (13) with the correspond-ing boundary conditions at both the cathode and collector ends.As for the particle motion (11), we have to put into themthe RF field of the operating eigenwave which coincides byits form with (12), but the amplitude is replaced by .After that, we can study (11) and (23) numerically with rathergeneral assumptions using both averaged and nonaveragedapproaches, taking into account particle velocity spread andtransverse components of the wave magnetic field (the latteris important when the axial wavenumber of the eigenwave

is relatively far from zero). Averaging and linearizing(11) and (23) leads to the simplified third-order dispersionequation (17). Numerical results obtained using the nonaver-aged equations are basically similar to those of the averagedapproach. The difference between them is most significantat the boundaries of the amplification band: the nonaveragedequations provide slightly (5%–15% for simulated parameters)narrower bands with the same maximum efficiencies. Thenumerical analysis shows that for parameters of the systemproviding the absence of the self-excitation, a high gain andelectron efficiency can be obtained for a very broad frequencyband and very large particle velocity spread.

V. DESIGNS OFSUBRELATIVISTIC AND RELATIVISTIC

SECOND-HARMONIC HELICAL GYRO-TWT’S

In order to check in more detail the concept of the gyro-TWT with a helical waveguide, a number of linear and

nonlinear simulations on the basis of equations discussedin Section IV-D were performed, and designs for subrel-ativistic and relativistic electron energies were developed.A few corresponding experiments are now in progress atthe Institute of Applied Physics, Nizhny Novgorod, Russia,and at the University of Strathclyde, Glasgow, U.K. For allthese investigations of gyro-TWT’s, we are going to usethin electron beams encircling the waveguide axis (Fig. 6).It is well known [22], [23] that such beams can excite incircular cylindrical waveguides only those modes for whichthe azimuthal indexes are equal to the cyclotron harmonicnumber , enhancing the mode selection in comparisonwith the case of hollow beams. Choosing the traveling TEwave as partial wave (far from cutoff) and the near-cutoffmode TE as the operating partial mode, we study theinteraction at the second cyclotron harmonic . In thiscase, a three-fold helical corrugation has to be usedfor resonant coupling of these modes and for providing thenecessary eigenwave dispersion. It is important that in thiscase the operating magnetic field is so low that a spuriousfundamental gyrotron excitation at TE mode is impossible,while the gyrotron excitation at higher cyclotron harmonics( , large orbit gyrotron regime) can lead to someproblems.

Taking into account the parameters of installations proposedfor the experiments, two different electron beams are assumedas drivers for the helical gyro-TWT’s: a subrelativistic beamhaving energy of 80 keV and current of 20 A and a moderatelyrelativistic one with 300-keV energy and 80-A current. Bothbeams are supposed to be initially rectilinear and sufficientlythin, acquiring large-enough transverse velocity (the operatingpitch-angle ) in short nonadiabatic magnetic systems(kickers [24], [25]) (Fig. 6). The driving RF signal for bothcases is assumed to be in the operating eigenwave of the “cold”helical waveguide which can be easily obtained in practiceby introducing to the input a properly rotating TE modeof the smooth circular waveguide and transforming it in asection with adiabatically increasing corrugation depth. Let usemphasize also that for operation at frequencies below cutoffof eigenmode 2, (Fig. 2), all the output power iscontained in eigenwave 1, which is easily transformed into

DENISOV et al.: GYRO-TWT WITH A HELICAL OPERATING WAVEGUIDE 515

(a)

(b)

Fig. 7. Simulated bandwidths of gyro-TWT’s with (a) smooth and (b) helical waveguides driven by a 80-keV 20-A electron beam with the velocity ratio� = 1:2 for different transverse velocity spreads: — — 0%, – - –18%, – – – 38%, - - - 53%.

a TE mode and radiated in the form of a nearly Gaussianwave beam (Fig. 6).

Taking into account possible reasons for limitation of theelectron transverse velocity (beam current losses during trans-port through the system, initial velocity spread), the meanelectron pitch angle in the simulations was chosen to be as highas . Correspondingly, electrons for the two designsmentioned above have significantly different (from the pointof view of the interaction) axial velocities: for 80-keV and for 300-keV energy. Optimization of thegyro-amplifiers’ parameters was directed to obtaining max-imum frequency bandwidth and minimum sensitivity to theelectron velocity spread. Correspondingly, at the preliminarystage of the simulations, the parameters of the “cold” helicalwaveguide, namely, coupling coefficient and geometrical

mismatch were varied to obtain a wide-band coincidence ofthe operating eigenwave group velocity with the axial electronvelocity near the point of zero axial wavenumber. At the nextstep, performed on the basis of the nonlinear nonaveragedequations (11) and (23), mainly magnetic field mismatch

and interaction length were optimized to provide asufficiently flat frequency characteristic of the amplification atan efficiency of about 30% and saturated gain above 30 dB(Figs. 7 and 8). The corrugation parameters were also slightlyvaried to adjust not only the synchronism mismatch

, but also the coupling coefficientbetweenthe operating wave and electron beam. Then, an optimizedversion was checked for stability by means of solution ofthe boundary problem (15), (20)–(22) (a starting current ofparasitic self-excitation should be higher than the operating

516 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 3, JUNE 1998

(a)

(b)

Fig. 8. Simulated bandwidths of gyro-TWT’s with (a) smooth and (b) helical waveguides driven by a 300-keV 80-A electron beam with the velocity ratio� = 1:2 for different transverse velocity spreads: — — 0%, – - –18%, – – –38%, - - -53%.

one). For the final variant, a dependence of the efficiency andgain on frequency was simulated for monoenergetic electronbeams with various velocity spreads (Figs. 7 and 8).The spread was determined as the distribution function widthat 10% of the maximum level.

In order to compare helical and conventional gyro-TWT’s,analogous simulations were carried out for the second-harmonic gyro-TWT’s with smooth waveguides driven bythe same electron beams. It is known that in this case themost wide-band regimes are realized at the grazing of theelectron beam and wave dispersion characteristics (in theoptimal case an uncoupled dispersion curve of the wave isplaced slightly above that of the beam).

The analysis performed shows important advantages of thegyro-TWT with the helical waveguide over the conventional

one. For the subrelativistic electron beam (80 keV, 20 A), thefrequency bandwidth of the “helical” gyro-TWT is simulatedto be significantly wider: 18% at zero velocity spread and13% with 29% peak efficiency at 38% spread (maximum gainamounts to 30 dB) in comparison with 10% at zero spread and4% with 25% peak efficiency at 38% spread for the “smooth”amplifier (Fig. 7). The starting current for the helical gyro-TWT at optimal magnetic field andlength, , is sufficiently above 20 Awhile for the smooth amplifier itamounts to 3.6 A (Fig. 5). For higher particle energy (300keV) the main advantage of the “helical” gyro-TWT is avery weak sensitivity to the particle velocity spread: at zerovelocity spread, the bandwidth of the “helical” gyro-TWTis only slightly wider than that for the “smooth” one with

DENISOV et al.: GYRO-TWT WITH A HELICAL OPERATING WAVEGUIDE 517

about the same level of efficiency, while at 38% spread, the“helical” amplifier has a bandwidth of 19% with 28% peakefficiency (38-dB gain), which is nearly twice as wide in thefrequency band and 2.3 times higher in the efficiency thanthose for the “smooth” version (Fig. 8). In addition, accordingto the simulations, the smooth gyro-TWT has a starting currentless than the operating one at the optimal magnetic fieldwhereas the amplifier with the helical waveguide is stable toself-excitation.

VI. CONCLUSION

A relatively shallow helical corrugation of the inner surfaceof an oversized circular cylindrical waveguide can radicallychange its dispersion characteristics. Such corrugation pro-vides, in particular, appearance of the wide-frequency bandregion where axial wavenumbers are small while wave-groupvelocity is high and nearly constant This makes the use ofhelical waveguides very attractive as microwave systems forgyro-TWT’s, where they allow the decrease of sensitivityto electron velocity spread, better matching of wave andelectron characteristics, and higher stability to spurious modeexcitation in comparison with the usual “smooth” gyro-TWT.In this paper, the above-mentioned idea is confirmed bymeans of theoretical simulations and measurements of the“cold” dispersion of helical waveguides. Linear and nonlineartheories of electron-wave interaction in a helical waveguide aredeveloped. Optimization of the parameters of the gyro-TWTis performed, and results of two designs for subrelativisticand relativistic electron energies are given. The correspondingexperiments are planned to be carried out at the Instituteof Applied Physics, Nizhny Novgorod, Russia, and at theUniversity of Strathclyde, Glasgow, U.K. Results of the paperallow us to hope for an important improvement of gyro-TWTparameters.1

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1Recently (in December 1997), we realized the helical gyro-TWT ex-perimentally. Using a 200-keV electron beam with a large spread in thelongitudinal velocity, we have measured single-frequency amplification in theX-band with a power of about 1 MW and an efficiency of more than 20%.(G. G. Denisov, V. L. Bratman, A. W. Cross, W. He, A. D. R. Phelps, K.Ronald, S. V. Samsonov, and C. Whyte, to be published.)

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[8] V. L. Granatstein and W. Lawson, “Gyro-amplifiers as candidate RFdrivers for TeV linear colliders,”IEEE Trans. Plasma Sci., vol. 24, no.3, pp. 648–665, 1996.

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[12] K. C. Leou, D. B. McDermott, and N. C. Luhmann, “Large-signalcharacteristics of a wide-band dielectric-loaded gyro-TWT amplifier,”IEEE Trans. Plasma Sci., vol. 24, no. 3, pp. 718–726, 1996.

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518 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 3, JUNE 1998

Gregory G. Denisov was born in Gorky, Russia,in 1956. He received the M.Sc. and Ph.D. degreesin physics from Gorky State University, Gorky, in1978 and 1985, respectively.

Since 1991, he has been Head of the Group ofMicrowave Systems for High Power Electronics atthe Institute of Applied Physics, Russian Academyof Science, Nizhny Novgorod. In 1991, he alsobecame an Associate Professor in the AdvancedSchool of General and Applied Physics at NizhnyNovgorod State University. His current research

interests include high-power radiation from cyclotron resonance masers (in-cluding gyrotrons and cyclotron autoresonance masers) and the developmentof powerful sources of millimeter- and submillimeter-wave-radiation forplasma diagnostics and ECRH and quasi-optical transmission lines andantennas.

Dr. Denisov received the Leninsky Komsomol Prize in 1987 and FusionPower Associates Award for the Excellence in Fusion Engineering in 1996.

Vladimir L. Bratman was born in Chirchik,Uzbekistan (former U.S.S.R.), in 1945. He receivedthe M.Sc. and Ph.D. degrees in physics from GorkyState University, Gorky, Russia, in 1967 and 1977,respectively, and the Doctor of Science degreefrom Tomsk Institute of High-Current Electronics,Russia, in 1992.

From 1970 to 1974, he was a Member of theTechnical Staff at the Gorky Research Institute“Salyut.” He joined the Gorky Radio PhysicalResearch Institute in 1974 and the Institute of

Applied Physics of the Russian Academy of Science in 1977. Since 1985,he is Head of the Short-Wavelength Relativistic Devices Group. In 1992, hebecame a Professor in the Advanced School of General and Applied Physics atthe Nizhny Novgorod (former Gorky) State University. His current researchinterests include high-power radiation from cyclotron resonance masers(including gyrotrons and cyclotron autoresonance masers) and from freeelectron lasers as well as the development of powerful sources of millimeter-and submillimeter-wave-radiation for plasma diagnostics and ECRH.

Alan D. R. Phelps, for a photograph and biography, see this issue, p. 381.

Sergei V. Samsonovwas born in Arzamas-16,Gorky region, Russia, in 1966. He received theM.Sc. degree from the Advanced School of Generaland Applied Physics of Nizhny Novgorod StateUniversity, Russia, in 1989 and the Ph.D. degreein physics in 1996 from the Institute of AppliedPhysics, Russian Academy of Science, Nizhny Nov-gorod, where he is currently working.

His current research interests include high-powerelectron devices, in particular, cyclotron resonancemasers.