PSFC/JA-17-32

Theory of Linear and Nonlinear Gain in a Gyroamplifier Using a Confocal Waveguide

Alexander V. Soane, Michael A. Shapiro, Jacob C. Stephens, Richard J. Temkin

July 2017

Plasma Science and Fusion Center Massachusetts Institute of Technology

Cambridge MA 02139 USA This work was supported by the National Institutes of Health (NIH), National Institute for Biomedical Imaging and Bioengineering (NIBIB) under Contract EB004866 and EB001965. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.

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Theory of Linear and Nonlinear Gain in aGyroamplifier using a Confocal WaveguideAlexander V. Soane, Student Member, IEEE, Michael A. Shapiro, Member, IEEE,

Jacob C. Stephens, Member, IEEE and Richard J. Temkin, Life Fellow, IEEE

Abstract—The linear and nonlinear theory of a gyroampli-fier using a confocal waveguide is presented. A quasi-opticalapproach to describing the modes of a confocal waveguideis derived. Both the equations of motion and the mode ex-citation equation are derived in detail. The confocal waveg-uide circuit has the advantage of reducing mode competitionbut the lack of azimuthal symmetry presents challenges incalculating the gain. In the linear regime, the gain calcu-lated using the exact form factor for the confocal waveguideagrees with an azimuthally averaged form factor. A beamletcode including velocity spread effects has been written tocalculate the linear and nonlinear (saturated) gain. It hasbeen successfully benchmarked against the MAGY code forazimuthally symmetric cases. For the confocal waveguide,the beamlet code shows that the saturated gain is reducedwhen compared with results obtained using an azimuthallyaveraged form factor. The beamlet code derived here ex-tends the capabilities of nonlinear gyroamplifier theory toconfigurations that lack azimuthal symmetry.

I. Introduction

A. Gyrotron Physics

The physics of the interaction between an electromag-netic wave and an electron beam for application to gy-rotrons has been studied extensively [1]. One of the im-portant challenges in gyrotron research is the developmentof powerful gyrotron amplifiers, especially at high frequen-cies [2–10]. As the gyrotron frequency increases, it is ad-vantageous for the gyrotron amplifier to operate in a higherorder mode of the interaction circuit to minimize spacecharge effects and ohmic loss. One possible method of re-ducing the mode competition in overmoded waveguides isthe use of a confocal waveguide structure [6,11]. However,this configuration results in a coupling between the elec-tron beam and the electromagnetic wave that is nonuni-form in the azimuthal direction. Such nonuniformity isalso found in other gyrotron configurations, such as splitresonators [12,13] and quasioptical gyrotrons [14–16]. Thepurpose of this paper is to present a detailed linear andnonlinear gyrotron interaction theory that accounts forthe nonuniform azimuthal variation of the electromagneticmode in confocal waveguides.

The generalized linear and nonlinear theory of gyrotrontraveling wave amplifiers has been previously derived [1,2].The nonlinear equations take into account the geometry ofthe interaction, namely the coupling factor that describes

A.V. Soane, M. A. Shapiro, J. C. Stephens, and R. J. Temkin arewith the Plasma Science and Fusion Center, Massachusetts Instituteof Technology, Cambridge, MA. 02139. e-mail: sasha08@mit.edu

Manuscript received May XX, 2017. This work was supportedby the National Institutes of Health (NIH), National Institute forBiomedical Imaging and Bioengineering (NIBIB) under ContractEB004866 and EB001965.

the overlap between the electromagnetic mode and theelectron beam. Gyrotron amplifier theory, as developedin [2], has focused on analytical solutions for TE modes ofa circular pipe. In this paper, we derive the quasi-opticalapproximation for the open modes of a confocal resonator,and use this to develop the gyrotron equations in paral-lel to [2]. The open geometry of the confocal resonator isinherently lossy, and we show the analytical solution forthese losses. We subsequently derive both the equationsof motion and field excitation equations, presented in theappendix, in the context of beamlets.

B. DNP/NMR at MIT

The Francis Bitter Magnet Lab at MIT currentlyhas gyrotron oscillators for dynamic-nuclear-polarization-enhanced nuclear magnetic resonance (DNP/NMR) re-search at 140, 250, 330, and 460 GHz [17–23]. Previously,pulsed DNP has been achieved using an IMPATT diodedriver [24, 25]. The pulse length of 50 ns at 35 mW wasable to excite 1% of the sample’s linewidth. In order tocapture the entire linewidth, a shorter and more powerfulpulse is needed, on the order of 100 W to 1kW at 1 to 10ns. Gyro-amplifiers are a good candidate for generationof the pulses needed for pulsed DNP/NMR. Additionally,the frequency scaling of gyro-amplifiers is a useful featurefor accessing various frequencies. To date, amplification ofshort pulses has been demonstrated at 140 GHz [26], andat 250 GHz [10].

Contemporary gyro-amplifiers take advantage of a vari-ety of design approaches. Lossy-wall gyro-amplifiers havebeen designed and operated at 35 GHz [5, 27] and at 95GHz [28]. An alternative design feature is a helically-corrugated interaction circuit [8,29]. We present a confocalinteraction circuit as an alternative to lossy-wall designs.Gyrotrons with confocal circuits continue to be studied in-tensively [30–34].

II. Quasi-optical Approximation of a ConfocalResonator

A. Membrane function

We begin with a description of the confocal geometry.As seen in Fig. 1, the confocal geometry consists of twomirrors positioned such that their radius of curvature Rc, isequal to that of their separation distance, L⊥ (Rc = L⊥).The aperture, or total width of each mirror, is 2a. Asthis geometry is not closed, the width is adjusted in orderto either increase or decrease the diffractive loss of thesupported HEmn modes. These supported spatial modeshave m variations along x and n variations along y. Fig.

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1 shows the HE06 mode, in which there are six variationsalong the y direction.

Fig. 1. The confocal geometry including the field distribution of theHE06 mode. The dashed line represents the annular electron beam.

For a closed waveguide, supported modes are describedby the membrane function Ψ, which obeys the wave equa-tion:

∇2⊥Ψ + k2

⊥Ψ = 0 (1)

in which the transverse wavenumber k⊥ is real (k⊥ =k⊥r). In consideration of the fact that the confocal waveg-uide is not a closed geometry, we introduce a quasi-opticalapproach to finding an approximate membrane function[35,36]. The formulation for a closed geometry [1] relies onthe exact solution to the wave equation in Eq. 1. Transla-tional symmetry in the z component reduces the problemto two dimensions. Additionally, we assume a solution thatis based on a modified plane wave in the y direction:

U(x, y) = B(x, y)e−ik⊥y (2)

As indicated in Eq. 2, we assume a wave propagating inthe y direction. Variation in x is absorbed into the func-tion B(x,y), an appropriate step if assuming the paraxial

approximation of ky =√k2⊥− k2

x ' k⊥ − k2x2k⊥

. Now, thewave equation Eq. 1 becomes

∂2B

∂x2+∂2B

∂y2− 2ik⊥

∂B

∂y= 0 (3)

Since the field distribution is a modified plane wave prop-

agating in the y direction, we can neglect ∂2B∂y2 as small

compared to the term 2ik⊥∂B∂y . With this simplification,

Eq. 3 becomes[ ∂2

∂x2− 2ik⊥

∂

∂y

]B(x, y) = 0 (4)

Using the form of Eq. 2 for U , we provide a gen-eral solution to the membrane function Ψ as two counter-propagating waves:

Ψ = −iU − iU∗ (5)

The confocal geometry, when examined with a quasi-optical approximation, lends itself readily to a Gaussianbeam solution for B(x,y). Thus, U may be expressed as

U(x,y) =( 2

π

) 14

√w0

w(y)exp

(− x2

w2(y)

)×

exp(− ik⊥ry− i

k⊥rx2

2R(y)+i

2arctan

y

b

) (6)

where b = k⊥rw20/2, following the derivations in [35].

Furthermore, w and R are

w2(y) = w20

[1 +

( 2y

k⊥rw20

)2](7)

1

R(y)=

y

y2 + (k⊥rw20/2)2

(8)

The perpendicular wavevector component, k⊥r, may befound by considering the boundary conditions of Eq. 6 un-der the confocal geometry. In particular, at a coordinate(x,y) = (0,L⊥/2) we know that the phase front curvatureR(y) needs to match the confocal mirror radius, Rc. Con-sequently, k⊥r is found to be

k⊥r =π

L⊥

(n+

2m+ 1

πarcsin

√L⊥2Rc

)(9)

Since L⊥ =Rc for the confocal geometry,

k⊥r =π

L⊥

(n+

m

2+

1

4

)(10)

These results are in agreement with those derived byNakahara [37], following Goubau [38]. In this section wehave derived the quasi-optical approximation of the mem-brane function that describes the field distribution of aconfocal resonator. Because the confocal geometry is open,RF power may leak out of the sides of the waveguide. Thus,a mode HEmn will, whilst propagating axially, lose powerthrough transverse diffraction. This leads to a loss per dis-tance that is useful for suppressing undesired (lower-order)modes that do not support the microwave field intended foramplification.

B. Diffractive losses in a confocal resonator

We follow the derivation in [11, 36] in our discussion oflosses incurred by waveguide modes HEmn in the confocalgeometry. The wave vector k⊥ is decomposed into both areal and an imaginary component, k⊥r + ik⊥i. Eq. 10 isthe real component; a more general expression is needed:

k⊥ =π

L⊥

(n+

m

2+

1

4+δ

π

)− i Λ

2L⊥(11)

In Eq. 11 we have introduced a small phase shift δ as wellas an imaginary component for ik⊥i. Λ is directly relatedto the diffraction losses. As discussed in Section A, thequasi-optical solution to the modes in a confocal geometryis found by superimposing propagating Gaussian waves toform a standing wave between the curved mirrors. Diffrac-tion occurs because the transverse width of the confocalmirrors is insufficient to capture the transverse extent of

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the waveguide modes HEmn. With each subsequent re-flection, the Gaussian wave loses a fraction of its power.Schematically, lower-order modes (with fewer variations inthe y-direction) have a broader transverse “footprint”, somore readily diffract for a given confocal aperture width.This effect is shown in Fig. 2.

Fig. 2. The confocal geometry including the field distribution ofthe HE06 mode at 140 GHz (a) and HE04 mode at 94 GHz (b).This field distribution is found for an aperture of 2a = 4 [mm] andRc = 6.83 [mm].

Λ is written in terms of the Fresnel diffraction parameterCF and the radial spheroidal wavefunction expressed in

prolate spheroidal coordinates [36,39] R(1)0,m(ξ1, ξ2):

Λ = 2 ln[√ π

2CF

1

R(1)0,m(CF , 1)

](12)

The Fresnel diffraction parameter defined as CF =k⊥ra

2/L⊥, where a is half the width of the confocal mir-

rors. In Eq. 12, R(1)0,m(CF ,1) is a function of CF as well

as m, which specifies the transverse mode content of theHEmn waveguide modes. For the first three values of m

(0, 1, 2) the values of R(1)0,m(CF ,1) are shown in Fig. 3 and

log(Λ) are shown in Fig 4, both as a function of CF .

Fig. 3. The exact values of R(1)0,m(CF , 1) plotted against CF for

various values of m; the lines connect calculated points.

In the wave vector component kz, losses incurred are dueto the imaginary component, kzi. If we consider a loss ratein terms of decades per axial distance, then the loss rate isexpressed as

Fig. 4. The exact values of log(Λ) plotted against CF for variousvalues of m; the lines connect calculated points.

Loss Rate =20

ln (10)kzi (13)

in which

kzi = Im

√(ωc

)2

− k2⊥ = −k⊥rk⊥i

kzr(14)

where kzr =√

ω2

c2 − k2⊥r is assumed to be not close to

0 (i.e. the mode is not close to cutoff; k2zr 2k⊥rk⊥i).

We calculate the attenuation in decades per unit lengthas a function of CF (or aperture a). Figure 5 shows acomparison between the theoretical loss rate and numericalsimulation results from the commercial numerical softwareprogram CST Microwave Studio. For the theoretical lossrate, Eq. 13 was used. These results are for a frequency of140 GHz in a confocal geometry with Rc = 6.83 mm.

Fig. 5. Comparison of loss rate due to diffraction using Eq. 13and numerical simulation software CST for an HE06 mode. Here,Rc = 6.83 mm and the frequency is 140 GHz.

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C. Field equations and RF Lorentz force

With the general solution of Eq. 5, we may write thevector fields of ~E and ~H as

Ex =k

k2⊥r

∂Ψ

∂y=

k

k⊥r(U∗−U) +

k

2k3⊥r

∂2

∂x2(U∗−U) (15)

Ey =− k

k2⊥r

∂Ψ

∂x= i

k

k2⊥r

∂

∂x(U +U∗) (16)

Hx =kzrk2⊥r

∂Ψ

∂x=−i kzr

k2⊥r

∂

∂x(U +U∗) (17)

Hy =kzrk2⊥r

∂Ψ

∂y=kzrk⊥r

(U∗−U) +kzr

2k3⊥r

∂2

∂x2(U∗−U) (18)

along with the general solution that Hz = U +U∗ and,because of the transverse electric nature of this waveguidemode, Ez = 0, with k = ω/c.

Equations 15-18 describe the transverse components un-der the quasi-optical approximation. Using these compo-nents, the RF Lorentz (~G) force may be found. The com-ponents of this force are necessary in order to analyze theinteraction in this waveguide mode. A thin, annular elec-tron beam of radius Rg (the “guiding center”) is injectedinto the confocal cavity, as seen in Fig. 6. At each positionaround the guiding center exist populations of electronswith gyroradius rc. The coordinate transform between theguiding center (X,Y ) and the beam center (x,y) is givenby

x=X + rc cosθ (19)

y = Y + rc sinθ (20)

It is particularly useful to find the radial and azimuthalcomponents of the Lorentz force (Gr & Gθ) at every guid-ing center due to the azimuthally-symmetrical electronbeam [1].

Fig. 6. An exploded view of the geometry at every guiding center.(X,Y ) are the coordinates of the location of the guiding center.

We start with the radial component of the RF Lorentzforce.

Gr = (Ex− βzHy) cos θ+ (Ey + βzHx) sin θ+ β⊥Hz (21)

where βz and β⊥ are the axial and perpendicular electronvelocities, vz and v⊥, normalized to the speed of light c,respectfully. We expand Eq. 21 in azimuthal harmonics:

Gr =∑l

Glr exp(−ilθ) (22)

As we are concerned with the fundamental harmonic, thefirst coefficient of the expansion (l = 1) should be found.With a coordinate transformation to the guiding center(X,Y ), we may use Eqs. 15-18, 21 to express G1r in termsof the function U .

G1r =−1

2

k−βzkzrk⊥r

(U −U∗)− 1

2

k−βzkzr2k3⊥r

( ∂2U

∂X2−

∂2U∗

∂X2

)− 1

2

k−βzkzrk2⊥r

( ∂U∂X

+∂U∗

∂X

)+β⊥k⊥rrc

1

2(U −U∗)

(23)

In the limit of k⊥rrc → 0, and using Eqs. 5 and 18 wecan further reduce Eq. 23 with reference to Ψ as

G1r = −ik − βzkzr2k2⊥r

( ∂Ψ

∂X+ i

∂Ψ

∂Y

)(24)

Equation 24 relates the fundamental harmonic’s radialRF Lorentz force to the membrane function, Ψ, the latterof which we have found via the quasi-optical approxima-tion of the confocal field distribution. An analogous calcu-lation may be computed for the azimuthal component ofthe Lorentz force, Gθ, which relates to the Cartesian fieldcomponents as

Gθ = (Ey + βzHx) cos θ − (Ex − βzHy) sin θ (25)

Using a similar expansion Gθ =∑lGlθe

−ilθ, and re-stricting ourselves to the fundamental harmonic (l = 1),a similar treatment leads to an expression for G1θ in termsof Ψ:

G1θ = −k − βzkzr2k2⊥r

( ∂Ψ

∂X+ i

∂Ψ

∂Y

)(26)

Equations 24 and 26 comprise force terms that will beused in deriving the self-consistent set of gyro-TWT equa-tions.

We have shown that within the quasioptical approxima-tion of the membrane function Ψ (Ex. 5) we reduce thetheory of the open waveguide gyro-TWT to the theory ofthe closed waveguide gyro-TWT [1]. The derivations inthe Appendices help to apply the theory [1] to this partic-ular geometry under the simplifications introduced by theGaussian beam approximation.

III. Beamlets

The self-consistent, nonlinear equations that describethe interaction physics in a gyro-amplifier were developedby Yulpatov [40] and later generalized by Nusinovich and

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Fig. 7. Schematic of a HE06 mode supported by a confocal geom-etry. The annular electron beam is shown as a dashed circle, andis discretized into distinct beamlets, shown as red dots. The cou-pling factor at different beamlets depends on the (asymmetric) fielddistribution of the HE06 mode.

Li [2]. The derivation of the gain equations is summarizedin Appendix A. We introduce the concept of beamlets as atool to be used for numerically solving the gain equationsfor a gyro-amplifier that is not cylindrically symmetric.

The spatial geometry of the interaction between the elec-tron beam and the electromagnetic wave is embodied bythe form factor, Ls (for the fundamental harmonic, s= 1),of the waveguide mode. The form factor is described fullyin Appendix A. In the case of the confocal cavity, the formfactor takes on different values at different guiding centers.This is seen in Fig. 7, which schematically features the an-nular electron beam discretized, overlaid on to the HE06

spatial mode of the confocal waveguide.

The field pattern is clearly different at individual pointsaround the guiding radius of the electron beam, and thismust be taken into account for the simulation of the gainfrom this interaction. In fact, the local coupling factorL1 can be significantly different at various locations aboutthe annular electron beam’s interaction with the confocalmode. In Appendix A, we numerically find this variationin L1, shown in Fig. 8. The value of Rg = 1.6 mm waschosen to optimize coupling to the second and fifth peaksalong the y-direction. A smaller beam diameter to coupleto the third and fourth peaks was not considered practicaldue to the required cathode size and space charge effects.

IV. Linear Gain

In this section we show that the asymmetry of a modeand electron beam is irrelevant in the computation of lineargain. The linear gain derived assumes that the magneticfield is uniform and constant over the interaction length.It suffices to simply take the average of the form factor,〈|L1|2〉 in the computation of linear gain.

We begin with the differential equations describing theevolution of electron phase, electron energy, and field am-plitude, denoted respectively by [1] as θ, w, and C. Theequations are derived in Appendix A. That derivationpresents a stationary theory for the waveguide excitation

no more

Fig. 8. Coupling factor L1 around an electron beam interacting withthe HE06 mode of a confocal geometry with mirror radius Rc = 6.83mm and beam radius Rg = 1.6 mm. Numerically, the coupling factorvaries significantly with guiding center location.

due to an electron beam. In this approach, the field equa-tion is of the first order with respect to axial distance, z,and is appropriate to describe the excitation of a waveg-uide mode at a single frequency. When operating close tocutoff, a different approach based on a second order fieldequation is taken, as discussed in [41].

A fundamental approximation taken in this derivation isthat open modes of the confocal waveguide are treated withthe excitation theory assuming closed waveguides. Thisassumption is appropriate when the confocal modes areconsidered to be confined, and therefore excitation of leakywaves by the electron beam is not included.

For N guiding centers, these differential equations be-come

dθndz

=1

1− bwnµwn−∆+

1

2√

1−wnIm[CL1ne

−iθn ] (27)

dwndz

=−√

1−wn1− bwn

Re[CL1ne−iθn ] (28)

dC

dz=− 1

NIo

N∑n=1

1

4π

∫ 2π

0

√1−wn

1− bwnL1n

∗e−iθndθ0n (29)

where the subscript n denotes the nth guiding center ofan annular electron beam. Effectively, the net field ampli-tude is the sum of contributions of electrons at all guidingcenters, which in a numerical code will be discretized. Theparameters b, µ, and ∆ are related to the initial conditionsof the electron beam-RF interaction at the beginning of thecircuit (z = 0) and are defined in Appendix A. Addition-ally, the values of θn, wn, and C at z = 0 are, respectfully, auniform distribution of the electrons over the entire Larmorradius in phase (0 to 2π), wn|z=0 = 0, and C(z = 0) nu-merically determined by the power in the waveguide mode.Furthermore, the integral in Eq. 29 is taken over the initialphase θ0n at z = 0. In general, the terms L1n are differentfor different guiding centers. Using the substitution

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C = Ce−i∆z (30)

as well as a first order expansion

θn = θ0n−∆z+ θ(1)n (31)

wn = w(1)n (32)

we can rewrite Eqs. 27-29 in the following form:

dθ(1)n

dz= (µ−∆b)w(1)

n +1

2Im[CL1ne

−iθ0n ] (33)

dw(1)n

dz=−Re[CL1ne

−iθ0n ] (34)

dC

dz− i∆C =− 1

NI0

N∑n=1

1

2π

∫ 2π

0

[(b− 1

2

)w(1)n +

iθ(1)n

]1

2L∗1ne

iθ0ndθ0n (35)

The superscript (1) denotes the first-order terms for wnand θn. To handle the integral over θ0n in Eq. 35, weintroduce a change of variables:

wn(1) =

1

2π

∫ 2π

0

w(1)n L∗1ne

iθ0ndθ0n (36)

θn(1)

=1

2π

∫ 2π

0

θ(1)n L∗1ne

iθ0ndθ0n (37)

With this change of variables, Eqs. 33-35 become

dθn(1)

dz= (µ−∆b)wn

(1)− i

4

1

2π

∫ 2π

0

CL1nL∗1ndθ0n (38)

dwn(1)

dz=−1

2CL1nL

∗1n (39)

dC

dz− i∆C =− 1

NI0

1

2

N∑n=1

[(b− 1

2

)wn

(1) + iθn(1)]

(40)

As a final step, we note that we can sum over the Nguiding centers index, n, by taking the express averageover the guiding centers:

¯θ(1) =1

N

N∑n=1

θn(1)

(41)

¯w(1) =1

N

N∑n=1

wn(1) (42)

Finally, we can simplify Eqs. 38-40 to:

d ¯θ(1)

dz= (µ−∆b) ¯w(1)− i

4C

1

N

N∑n=1

|L1n|2 (43)

d ¯w(1)

dz=−1

2C

1

N

N∑n=1

|L1n|2 (44)

dC

dz− i∆C =−1

2I0

(b− 1

2

)¯w(1)− i

2I0

¯θ(1) (45)

In considering linear gain, we assume that ddz = iΓ. Con-

sequently, Eqs. 43-45 can be reduced to

(µ−∆b) ¯w(1)− iΓ¯θ(1)− i

4〈|L1|2〉C = 0 (46)

iΓ ¯w(1) +1

2〈|L1|2〉C = 0 (47)

1

2I0

(b− 1

2

)¯w(1) +

i

2I0

¯θ(1) + (iΓ− i∆)C = 0 (48)

in which 〈|L1|2〉 ≡ 1N

∑Nn=1 |L1n|2 is the averaged form

factor over the guiding centers. We note that Eqs. 46-48show that the reduced first-order expansion of consideringmultiple beamlets has simplified to the standard expres-sion for linear growth as derived in [2], provided that theform factor in question is computed from an average of allguiding radii. Indeed, with the variable I ′0 = 1

4I0〈|L1|2〉,we can find that the polynomial dictating the gain term Γis

Γ2(Γ−∆) + Γ(b− 1)I ′0 + (µ−∆b)I ′0 = 0 (49)

Thus, the linear gain when considering form factors thatdepend on the guiding center is identical to that of anapproach that averages over all of the guiding centers.This fact reflects the superposition principle in the linearregime. This is useful computationally, because it showsthat if the form factor L1 is calculated from

√〈|L1|2〉, the

resulting value can be used in Eq. 49 to compute the lineargain directly, but not the nonlinear gain.

V. Nonlinear Code Results

The nonlinear beamlet code was developed for appli-cation to interaction circuits which lack azimuthal sym-metry (e.g. a confocal waveguide operating in the HE06

mode). For the beamlet method, the annular beam is di-vided into discrete beamlets, each with a coupling factorL1n, calculated according to their relative location in thefield distribution (c.f. Fig. 7 and 8). A second methodusing a single beamlet and beam averaged coupling coef-ficient 〈L1〉 is presented for comparison. For the beamletmethod in all instances, 100 total beamlets are utilized,which well-discretizes the spatial variance of the fields. Asa means of verifying the beamlet code, the gain for a cir-cular waveguide operating in the azimuthally symmetricTE03 mode was calculated and compared with the predic-tions of MAGY [42]. Figure 9 shows that the gain predictedby MAGY and the beamlet code are almost identical.

7

Fig. 9. The gain for a TE03 mode as predicted by the beamlet andMAGY codes. The operating condition simulated is for a 141 GHzsignal at 5.085 T, 37 kV, a pitch factor of 0.9, 3 A beam current,input power 50 mW, and a pipe radius of 3.54 mm.

The gain of a confocal waveguide operating in the HE06

mode is shown in Fig. 10 for both the beamlet method andthe beam averaged coupling coefficient approach. Thesesimulations were performed for 140 GHz with 50 mW in-put power, 5.085 T, 37 kV, 3 A beam current, confocalrail spacing Rc = 6.83 mm, a pitch factor α of 0.9, and aloss per unit length of 3 dB/cm, which represent typicaloperating values.

Fig. 10. Confocal circuit gain versus interaction distance calculatedusing a single beam averaged coupling factor 〈L1〉, and 100 beamlets.

A difference of ∼ 1.7 dB in peak gain is computed be-tween the code using an averaged L1 and that of the beam-let simulation as seen in Fig. 10. This 1.7 dB differencein peak gain is observed to be invariable over a range ofinput power, as seen in Fig. 11 for the same operatingconditions. Although the saturated gain difference of 1.7dB appears small, it is important when considering power.For the case shown in Fig. 10, the estimated saturated

output power is 500 W for the averaged case, but is only360 W when accurately calculated by the beamlet theory.

Fig. 11. The peak confocal circuit gain versus input power as cal-culated using the beam averaged coupling factor method and thebeamlet method.

Electron beam velocity spread is an important factorthat can impact the predicted gain of a gyrotron amplifierand has been studied analytically as well as numerically[43,44]. We take the approach used in [44] to model the ef-fect of a non-cold electron beam (a distribution of electronvelocities is introduced). The velocity spread model, whichconserves total energy, assumes a Gaussian distribution inthe perpendicular velocity component:

f(β⊥) =1√πσ

exp(− (β⊥ − β⊥0)2

σ2

)(50)

where β⊥0 is the mean value of β⊥ and σ is the width.The developed code was used to study the effect of ve-

locity spread on the saturated gain in the confocal mode.The calculated gain in Fig. 12 is shown for an operatingpoint of 5.085 T, 37 kV, 3 A beam current, a pitch factorof 0.8, confocal radius 6.83 mm, 3 dB/cm waveguide loss,and 1 W input power at 140 GHz.

The difference between saturated gain as calculated withan averaged coupling factor and the beamlet approach isshown in Fig. 12.

The results in Fig. 12 emphasize the importance of prop-erly accounting for the effect of velocity spread on circuitgain. Again, as in Fig. 12, the use of a beam averagedcoupling coefficient is found to overpredict the saturatedcircuit gain.

VI. Discussion

The equations of motion for electrons and the mode ex-citation equations for the microwave fields have been de-rived. Using the quasi-optical approximation for the spa-tial modes of a confocal waveguide, we see that there isa difference in the peak gain as calculated for the case ofan averaged coupling factor, L1, or using a beamlet codeto sample the interaction at different guiding centers. Due

8

Fig. 12. Confocal circuit gain versus interaction distance for variousspecified RMS velocity spread values. Solid curves are calculatedusing a beam averaged coupling factor 〈L1〉 and dashed curves arecalculated using the beamlet method.

to nulls in the confocal modes, portions of the annularelectron beam are not interacting strongly with the mode.Consequently, electrons at these guiding centers may notfully bunch, and the energy that they carry will remainin the transverse velocity component. Qualitatively, thisphenomenon of imperfect bunching due to local nulls inthe electromagnetic spatial modes of an interaction circuitwill occur in waveguide designs that support such spatialmodes, and may be accounted for in numerical simulationsby using beamlets. In the case of the confocal waveguidemodes, it is seen that the interaction between electrons ina given guiding center with the spatial mode depends onthe location of the guiding center relative to the confo-cal geometry. Therefore, calculating an averaged interac-tion value may over-sample the net coupling between theelectron beam and the waveguide mode. This will resultnot only in a higher predicted gain but also in a dimin-ished effect from increased waveguide loss, as sections ofthe electron beam that in reality are not interacting withthe waveguide mode are included via the averaging calcu-lation. In the beamlet case, which accurately accounts forthe asymmetry of the waveguide mode, certain regions ofthe electron beam do not factor prominently into the gy-rotron interaction. Consequently, those beamlets that areinteracting with a strong field region amount to the entiretyof the gyrotron interaction. Increased waveguide loss low-ers the field strength, which adversely affects the bunchingmechanism, leading to a greater difference in the predictedgain when comparing the averaged coupling against thebeamlet code. We note that the numerical results pre-sented do not distinguish between diffractive and ohmicloss, and therefore the conclusions apply to both physi-cal situations. Finally, the inclusion of velocity spread wasdemonstrated. At moderate velocity spread values, the dif-ference in calculated gain has only a minor dependence onthe absolute velocity distribution. This is consistent with

the argument that it is the spatial distribution of asymmet-ric modes, and the poor coupling between beam and modeat certain spatial locations, that contributes to a differencein calculated saturated gain.

The work presented has up to this point not consideredthe effect of guiding center drift. The ~E cross ~B force mayintroduce an azimuthal rotation of the guiding center po-sitions. For a purely transverse electric mode, such as theHE06 mode of the confocal waveguide, the axial compo-nent of the electric field is zero. Thus, the possible sourcesfor this azimuthal guiding center drift are due to the radialelectric components from the DC space charge depressionand the microwave field. For the operating point presentedin Fig. 9 (5.085 T, 37 kV, α = 0.9, and a pipe radius of3.54 mm) the DC space charge depression is about 0.5 kV.This results in a total accumulated azimuthal drift over a20 cm interaction length of about 7 degrees, which is smallenough to be negligible in the analysis. The transverse mi-crowave E-field also can contribute to the azimuthal drift.For 500 W of power, this E-field is an order of magnitudesmaller than the DC E-Field and it also is only present inthe last centimeter of the gain section, so that it can beignored.

This article has focused solely on an annular beam con-figuration. A sheet beam, as demonstrated in [45], mightrepresent an alternate approach to enhance the couplingif the sheet beam is aligned to interact with the middlepeak of an HE0m mode of a confocal waveguide with modd. Additionally, a large orbit gyrotron could also beused [46].

acknowledgment

The authors would like to thank Dr. Emilio Nanni forhis advice in the initial discussions that led to this work.

Appendices

A. Derivation of governing equations for agyro-twt

A. Motion equations

The motion equations that govern the dynamics of anelectron in a waveguide field begin with the fundamentals[1]:

dp⊥dz

=− e

vzRe(CGθe

i(ωt−kzrz))

(A.1)

p⊥dψ

dz=

e

vzRe(CGre

i(ωt−kzrz))

(A.2)

where C is the waveguide mode amplitude, p⊥ is theperpendicular electron momentum, Gθ and Gr are the RFLorentz force components, and ψ is the electron phase inits gyro-orbit. In order to keep track of the phase ad-vance due to electrons gyrating in the magnetic field, usingthe longitudinal momentum pz we introduce the variablehH = eBo/cpz (which conveniently allows us to write theLarmor radius as rc = p⊥

hHpz) and Bo is the magnetic field.

9

The longitudinal momentum is pz = γmcβz. Then, thegyrophase can be defined as

θ = hHz + ψ (A.3)

Equation A.3 is useful to quantify the phase of thewaveguide mode with respect to the gyrophase:

ωt− kzrz − θ = −ϑ (A.4)

Furthermore, as follows from Eqs. 24, 26

dp⊥dz

=e

vzRe(Ck − βzkzr

2k⊥rL1(X,Y )e−iϑ

)(A.5)

where the coupling factor L1 is

L1(X,Y ) =1

k⊥r

( ∂Ψ

∂X+ i

∂Ψ

∂Y

)(A.6)

We can then take Eqs. A.3 and A.4 and find that

dϑ

dz=dψ

dz+

Ω + kzrvz − ωβzc

(A.7)

where Ω = eBo/γmc is the cyclotron frequency. Now,we rewrite Eq. A.2 as

p⊥dϑ

dz+ω− kzrvz −Ω

vzp⊥ =

e

vzRe(C(−i)k−βzkzr

2k⊥rL1(X,Y )e−iϑ

) (A.8)

For γ0,βz0,β⊥0, and Ω0 defined at the beginning of theinteraction (z = 0), we use normalized electron energy, w,and the parameter b:

w = 21− kzr

k βz0

β2⊥0

γ0− γγ0

(A.9)

b=kzrkβ2⊥0

1

2βz0(1− kzrk βz0)

(A.10)

(It can be seen that the product bw = kzrk

12βz0

γ0−γγ0

). Thecomponents of momentum may be rewritten to incorporatenormalized electron energy as follows:

pz = pz0(1− bw) (A.11)

p⊥ = p⊥0

√1−w (A.12)

We introduce the parameter µ,

µ =β2⊥0

(1− k2zr

k2

)2(

1− kzrk βz0

) (A.13)

The term ω−kzrvz−Ωβzc

in Eq. A.7 may be expressed withb and µ as

ω − kzrvz − Ω

βzc=

ω

cβz0(1− bw)

(ω − kzrvz0 − Ω0

ω− µw

)(A.14)

Therefore, Eq. A.7 is conveniently expressed as

dϑ

dz=− ω

cβz0(1− bw)

(ω− kzrvz0−Ω0

ω−µw

)+

emγ0

pz0(1− bw)

1

p⊥0

√1−w

×

Re(− iC

1−βz0 kzrk2k⊥r

kL1(X,Y )e−iϑ) (A.15)

For further convenience, we may define the detuning pa-rameter ∆:

∆ =ω − kzrvz0 − Ω0

ω(A.16)

and normalize length as z′ = kz. Then, the relative gy-rophase ϑ, in normalized coordinates z′, varies as

dϑ

dz′=

1

βz0(1− bw)

(µw−∆+

1− kzrk βz0

γ0β⊥0

√1−w

Im[ eC

2mc2k⊥rL1e

−iϑ]) (A.17)

To find the differential of the normalized electron energy,w, we may take the derivative of Eq. A.12:

dw

dz′=−2

√1−w

1− bw

(1−βz0

kzrk

) 1

γ0βz0β⊥0

×Re( eC

2mc2k2⊥rL1e

−iϑ)

(A.18)

We denote additional normalized variables with a prime:

µ′ =µ

βz0(A.19)

∆′ =∆

βz0(A.20)

C ′ =eC

mc2γ0βz0β⊥0

(1−βz0

kzrk

) 1

k⊥r(A.21)

The above simplifications allow us to summarize thevariation of both the relative gyrophase, ϑ, and normalizedelectron energy, w, with respect to normalized coordinatesz′:

dϑ

dz′=

1

1− bw

(µ′w−∆′+

1√1−w

Im[C ′

1

2L1e

−iϑ])(A.22)

dw

dz′=−2

√1−w

1− bwRe(C ′

1

2L1e

−iϑ)

(A.23)

Equations A.22 and A.23 fully characterize the dynam-ical properties of an electron beam interacting with awaveguide mode at any guiding center (X,Y ).

10

B. Mode excitation equations

Just as an electron is affected by the presence of a waveg-uide mode’s EM field, the waveguide mode itself is alsochanged by the electron beam. Following the derivationin [1], the mode amplitude Cs (of the s-th mode of thewaveguide) interacts with the electron beam via the mode’selectric field as integrated over the waveguide cross sectionS⊥:

dCsdz

= − 1

Ns

∫S⊥

~jω · ~E∗sds⊥ (A.24)

in which ~jω is the current density component at the an-gular frequency ω. The normalization factor Ns of the s-thmode is

Ns =c

4π

∫S⊥

( ~Es× ~H∗s + ~E∗s × ~Hs)~z0ds⊥ =

c

2πRe

∫S⊥

( ~Es× ~H∗s )~z0ds⊥ = 4Ps

(A.25)

In Eq. A.25, Ps is the Poynting vector. The normaliza-tion factor Ns is explicitly calculated in Appendix SectionB. Since the electric field is purely transverse, the inte-grand in Eq. A.24 is ~jω · ~E∗s = ~jω⊥ · ~E∗s . Additionally, weknow that by charge conservation in the electron beam,at any cross sectional slice the charge entering and ex-iting is conserved: jzdt = jz0dt0. So, in considering thetransverse current, we can relate it to the axial compo-nent by ~j⊥d(ωt) = ~j⊥

jz0jzd(ωt0). Transforming to frequency

space [1]:

~jω⊥ · ~E∗s =1

π

∫ 2π

0

~j⊥ · ~E∗s e−iωtd(ωt) (A.26)

Using the formulation to relate d(ωt) to d(ωt0), Eq. A.26may be rewritten as

1

π

∫ 2π

0

~j⊥jz0jze−iωt ~E∗sd(ωt0) =

1

πjz0

∫ 2π

0

~p⊥pz

~E∗s e−iωtd(ωt0)

(A.27)Using the relations for momentum in Eqs. A.11 and

A.12 as well as recognizing that the area integral in Eq.A.24 reduces to (jz0S⊥), the differential equation for thegrowth of the amplitude C (dropping the mode number s)becomes

dC

dz=

1

N

1

π(jz0S⊥)

∫ 2π

0

p⊥0

√1− w

pz0(1− bw)E∗1ϑ(X,Y )eiϑdϑ0

(A.28)in which E1θ is the azimuthal component of the elec-

tric field (which interacts with the gyrating current com-ponent). The interaction is at the fundamental harmonic,so in the usual expansion Eϑ =

∑Elϑe

−ilϑ we are inter-ested in l = 1.

Using the field Eqs. 15 and 16 and the fact that Eϑ =Ey cosϑ−Ex sinϑ, the fundamental harmonic E1ϑ is

E1ϑ = − k

2k2⊥r

( ∂Ψ

∂X+ i

∂Ψ

∂Y

)= − k

2k⊥rL1 (A.29)

This is a result that follows the derivation for G1r in Eq.21. Thus, the amplitude differential dC/dz is rewritten as

dC

dz= − 1

N

1

πIb

∫ 2π

0

p⊥0

√1− w

pz0(1− bw)

k

2k⊥rL∗1e

iϑdϑ0 (A.30)

where Ib is the beam current. It is convenient to rewriteEq. A.30 in normalized form:

dC ′

dz′=− 1

N

1

πIb

e

mc2γ0β⊥0βz0k⊥r

(1− kz0

kβz0

)×

β⊥0

βz0k⊥r

∫ 2π

0

√1−w

1− bwL∗12eiϑdϑ0

(A.31)

We make one final simplification and introduce the nor-malized current, I0, defined as

I0 = 2( eIbmc3

) 1− kz0k βz0

γ0β2z0(k⊥r

k )2

c3

ω2N(A.32)

where

eIbmc3

=Ib(A)

17000(A)(A.33)

In Eq. A.33 the fraction mc3

e reduces to 17000 amps in SIunits. Finally, the differential of the normalized amplitudeC ′ in normalized units z′ is expressed as

dC ′

dz′= −I0

1

2π

∫ 2π

0

√1− w

1− bwL∗12eiϑdϑ0 (A.34)

Equations A.22, A.23, and A.34 comprise the (normal-ized) self-consistent set of equations that describe gyro-TWT operation. In Eq. A.34, a loss term may be intro-duced in the system if we consider an imaginary compo-nent to the axial wavevector, kzi. Properly normalized,loss in the system is represented by dC′

dz′ ∼ −k′ziC′ where

k′zi = kzi/k.

In the absence of loss, it is convenient to derive an energyconservation relation. Since energy is transferred betweenthe electrons and the field (power ∼ |C|2), we may use Eqs.A.34 and A.23 to find that

|C ′|2 − |C ′z=0|2 = I01

2π

∫ 2π

0

wdϑ0 (A.35)

The form of Eq. A.35 is an expression of the conservationof energy in the system and relates the change of microwaveenergy to that of the electrons.

11

B. Normalization factor

The normalization factor first introduced in Eq. A.25has an analytical expression dependent on the field dis-tribution of the operating mode. In our case, the quasi-optical approximation of the confocal resonator is used tofind an analytical solution for the normalization factor. Tobegin, we may relate the normalization factor Ns to thefield distribution U by taking the explicit cross product inEq. A.25. Expanding the cross product, we find that

~Es × ~H∗s · ~z0 = ExH∗y − EyH∗x (B.1)

Referring to Eqs. 15 - 18, Eq. B.1 is equivalently

ExH∗y−EyH∗x =

kkzrk2⊥r

(U −U∗)2 +kkzrk4⊥r

( ∂U∂X

+∂U∗

∂X

)2 (B.2)

Equation B.2 is a general result; we may consider thequasi-optical approximation of the field distribution in aconfocal resonator by using the expression for U given inEq. 6 and write the integral in Eq. A.25:

Ns =c

2π

kkzrk2⊥r

∫∫ [− 4

√2

π

w0

wexp

(− 2x2

w2

)sin2

(k⊥ry+

k⊥rx2

2R− 1

2arctan

y

b

)+

1

k2⊥r

4

√2

π

( 2x

w2

√w0

wexp

(− x2

w2

)cos(k⊥ry+

k⊥rx2

2R

− 1

2arctan

y

b

)+k⊥rx

R

√w0

wexp

(− x2

w2

)sin(k⊥ry

+k⊥rx

2

2R− 1

2arctan

y

b

))2]dxdy

(B.3)

Equation B.3 is an exact solution; we may take twoapproximations that will result in an analytical expres-sion for the integrals. First, the argument of the trigono-

metric functions, k⊥ry + k⊥rx2

2R − 12 arctan y

b , describes aphase front. Under a paraxial approximation, the largeterm k⊥ry dominates, and therefore the argument of thetrigonometric functions reduces to that single term. Sec-ond, we notice that in Eq. B.3 the first term comes fromthe cross product component ExH

∗y whereas the remaining

two are from EyH∗x . As has been seen in the field distribu-

tion of the confocal resonator modes, the transverse termEx Ey. It follows that the first term of Eq. B.3 dom-inates, and we may drop the remaining two. With theseapproximations, the normalization reduces to

Ns 'c

2π

kkzrk2⊥r

∫∫ [− 4

√2

π

w0

w(y)

× exp(− 2x2

w2(y)

)sin2(k⊥ry)

]dxdy

(B.4)

The limits of integration are −∞< x <∞ and −L⊥/2<y < L⊥/2. We find that the normalization may be approx-imated as

Ns '−c

π

kkzrk2⊥L⊥w0

c3

ω2Ns=− c3πk2

⊥rω2ckkzrL⊥w0

=− π

k3kzr

k2⊥r

L⊥√L⊥

√k⊥r

(B.5)

in which the final step is found from evaluating the ra-dius R(y = L⊥

2 ) = L⊥ in Eq. 8 and therefore

w0 =

√L⊥k⊥r

(B.6)

References

[1] G. S. Nusinovich, Introduction to the Physics of Gyrotrons. TheJohns Hopkins University Press, 2004.

[2] G. S. Nusinovich and H. Li, “Theory of gyro-travelling-wavetubes at cyclotron harmonics,” International Journal of Elec-tronics, vol. 72, no. 5-6, pp. 895–907, 1992.

[3] K. L. Felch, B. G. Danly, H. R. Jory, et al., “Characteristicsand applications of fast-wave gyrodevices,” Proceedings of theIEEE, vol. 87, no. 5, pp. 752–781, May 1999.

[4] K. R. Chu, “Overview of research on the gyrotron traveling-wave amplifier,” IEEE Transactions on Plasma Science, vol. 30,no. 3, pp. 903–908, June 2002.

[5] K. R. Chu, H. Y. Chen, C. L. Hung, et al, “Theory and ex-periment of ultrahigh-gain gyrotron traveling wave amplifier,”IEEE Transactions on Plasma Science, vol. 27, no. 2, pp. 391–404, April 1999.

[6] J. R. Sirigiri, M. A. Shapiro, and R. J. Temkin, “High-power140-GHz quasioptical gyrotron traveling-wave amplifier,” Phys.Rev. Lett., vol. 90, 258302, June 2003.

[7] B. G. Danly, M. Blank, J. P. Calame, et al., “Development andtesting of a high-average power, 94 GHz gyroklystron,” IEEETransactions on Plasma Science, vol. 28, pp. 713–726, June2000.

[8] G. G. Denisov, V. L. Bratman, A. D. R. Phelps, et al., “Gyro-TWT with a helical operating waveguide: new possibilities toenhance efficiency and frequency bandwidth,” IEEE Transac-tions on Plasma Science, vol. 26, no. 3, pp. 508–518, June 1998.

[9] M. Blank, K. Felch, B. G. James, et al., “Development anddemonstration of high-average power W-band gyro-amplifiersfor radar applications,” IEEE Transactions on Plasma Science,vol. 30, no. 3, pp. 865–875, June 2002.

[10] E. A. Nanni, S. M. Lewis, M. A. Shapiro, R. G. Griffin, and R.J. Temkin, “Photonic-band-gap traveling-wave gyrotron ampli-fier,” Phys. Rev. Lett., vol. 111, 235101, Dec 2013.

[11] C. D. Joye, M. A. Shapiro, J. R. Sirigiri, and R. J. Temkin,“Demonstration of a 140-GHz 1-kW confocal gyro-traveling-wave amplifier,” IEEE Transactions on Electron Devices,vol. 56, no. 5, pp. 818–827, May 2009.

[12] I. I. Antakov, S. N. Vlasov, V. A. Gintsburg, A. L Goldenberg,M. I. Petelin, and V. K. Yulpatov, “CRM-oscillators with me-chanical tuning of frequency,” Elektron. Tekhn., ser. I, no. 8,pp. 20 – 25, 1975.

[13] G. F. Brand, Z. Chen, N. G. Douglas, et al., “A tunablemillimetre-submillimetre gyrotron,” International Journal ofElectronics, vol. 57, no. 6, pp. 863–870, 1984.

[14] A. W. Fliflet, T. A. Hargreaves, W. M. Manheimer, R. P. Fis-cher, and M. L. Barsanti, “Operation of a quasioptical gyrotronwith variable mirror separation,” Phys. Rev. Lett., vol. 62, pp.2664–2667, June 1989.

[15] S. Alberti, M. Q. Tran, J. P. Hogge, T. M. Tran, A. Bonde-son, P. Muggli, A. Perrenoud, B. Jodicke, and H. G. Mathews,“Experimental measurements on a 100 GHz frequency tunablequasioptical gyrotron,” Physics of Fluids B: Plasma Physics,vol. 2, no. 7, pp. 1654–1661, 1990.

[16] A. Bondeson et al., “Multimode analysis of quasi-optical gy-rotrons and gyroklystrons,” in Infrared and Millimeter Waves,K. J. Button, Ed. vol. 9, chap. 7, Academic Press, New York(1983).

12

[17] L. R. Becerra, G. J. Gerfen, B. F. Bellew, et al., “A spectrome-ter for dynamic nuclear polarization and electron paramagneticresonance at high frequencies,” Journal of Magnetic Resonance,Series A, vol. 117, no. 1, pp. 28 – 40, 1995.

[18] S. Han, R. Griffin, K. Hu, C. Joo, C. Joye, J. Sirigiri, R. Temkin,A. Torrezan, and P. Woskov, “Spectral characteristics of a 140-GHz long-pulsed gyrotron,” IEEE Transactions on Plasma Sci-ence, vol. 35, no. 3, pp. 559–564, June 2007.

[19] A. C. Torrezan, M. A. Shapiro, J. R. Sirigiri, R. J. Temkin, andR. G. Griffin, “Operation of a continuously frequency-tunablesecond-harmonic CW 330-GHz gyrotron for dynamic nuclearpolarization,” IEEE Transactions on Electron Devices, vol. 58,no. 8, pp. 2777–2783, Aug 2011.

[20] V. S. Bajaj, M. K. Hornstein, K. E. Kreischer, J. R. Sirigiri, P. P.Woskov, M. L. Mak-Jurkauskas, J. Herzfeld, R. J. Temkin, andR. G. Griffin, “250 GHz CW gyrotron oscillator for dynamicnuclear polarization in biological solid state NMR,” J. Magn.Reson., vol. 189, no. 2, pp. 251–279, Dec 2007.

[21] K. E. Kreischer, C. Farrar, R. Griffin, and R. Temkin, “The useof a 250 GHz gyrotron in a DNP/NMR spectrometer,” in 23rdInternational Conference on Infrared, Millimeter and TerahertzWaves, 1998, pp. 341–357.

[22] A. C. Torrezan, S. T. Han, M. A. Shapiro, J. R. Sirigiri, and R.J. Temkin, “CW operation of a tunable 330/460 GHz gyrotronfor enhanced nuclear magnetic resonance,” in 33rd InternationalConference on Infrared, Millimeter and Terahertz Waves, Sept.2008.

[23] M. K. Hornstein, V. S. Bajaj, R. G. Griffin, and R. J. Temkin,“Continuous-wave operation of a 460-GHz second harmonicgyrotron oscillator,” IEEE Transactions on Plasma Science,vol. 34, no. 3, pp. 524–533, June 2006.

[24] M. Bennati, C. T. Farrar, J. A. Bryant, S. J. Inati, V. Weis,G. J. Gerfen, P. Riggs-Gelasco, J. Stubbe, and R. G. Griffin,“Pulsed electron-nuclear double resonance (endor) at 140 GHz,”J. Magn. Reson., vol. 138, no. 2, pp. 232–243, June 1999.

[25] R. G. Griffin and V. Weis, “Electron-nuclear cross polarization,”Solid State Nucl. Magn. Reson. (USA), vol. 29, no. 1-3, pp. 66–78, Feb. 2006.

[26] H. J. Kim, E. A. Nanni, M. A. Shapiro, J. R. Sirigiri, P. P.Woskov, and R. J. Temkin, “Amplification of picosecond pulsesin a 140-GHz gyrotron-traveling wave tube,” Phys. Rev. Lett.,vol. 105, 135101, Sept. 2010.

[27] K. R. Chu, H. Y. Chen, C. L. Hung, T. H. Chang, L. R. Barnett,S. H. Chen, and T. T. Yang, “Ultrahigh gain gyrotron travelingwave amplifier,” Phys. Rev. Lett., vol. 81, pp. 4760–4763, Nov.1998.

[28] L. R. Barnett, W. C. Tsai, H. L. Hsu, N. C. Luhmann, C. C.Chiu, K. F. Pao, and K. R. Chu, “140 kW W-band TE01 ultrahigh gain gyro-TWT amplifier,” in IEEE International VacuumElectronics Conference held Jointly with 2006 IEEE Interna-tional Vacuum Electron Sources, 2006, pp. 461–462.

[29] V. L. Bratman, G. G. Denisov, S. V. Samsonov, A. W. Cross,A. D. R. Phelps, and W. Xe, “High-efficiency wideband gyro-TWTs and gyro-BWOs with helically corrugated waveguides,”Radiophysics and Quantum Electronics, vol. 50, no. 2, pp. 95–107, 2007.

[30] Y. Yang, S. Yu, T. Z. Zhang, Y. Zhang, and Q. Zhao,“Study of beam-wave interaction in a 170-GHz confocal gyrotrontraveling-wave tube,” IEEE Transactions on Plasma Science,vol. 43, no. 3, pp. 791–795, March 2015.

[31] Y. Yang, S. Yu, Y. Liu, T. Zhang, Y. Zhang, and Q. Zhao,“Efficiency enhancement of a 170 GHz confocal gyrotron trav-eling wave tube,” Journal of Fusion Energy, vol. 34, no. 4, pp.721–726, 2015.

[32] G. S. Nusinovich, S. Chainani, and V. L. Granatstein, “Effect ofthe transverse nonuniformity of the radiofrequency field on thestart current and efficiency of gyrodevices with confocal mir-rors,” Physics of Plasmas, vol. 15, 103106, 2008.

[33] W. Fu, X. Guan, and Y. Yan, “High harmonic terahertz confocalgyrotron with nonuniform electron beam,” Physics of Plasmas,vol. 23, 013301, 2016.

[34] W. Hu, M. A. Shapiro, K. E. Kreischer, and R. J. Temkin, “140-GHz gyrotron experiments based on a confocal cavity,” IEEETransactions on Plasma Science, vol. 26, no. 3, pp. 366–374,Jun 1998.

[35] H. A. Haus, Waves and Fields in Optoelectronics. PrenticeHall, 1984.

[36] L. A. Weinstein, Open Resonators and Open Waveguides. Boul-der, CO: The Golem Press, 1969.

[37] T. Nakahara and N. Kurauchi, “Guided beam waves betweenparallel concave reflectors,” IEEE Transactions on MicrowaveTheory and Techniques, vol. 15, no. 2, pp. 66–71, Feb. 1967.

[38] G. Goubau and F. Schwering, “On the guided propagation ofelectromagnetic wave beams,” IRE Transactions on Antennasand Propagation, vol. 9, no. 3, pp. 248–256, May 1961.

[39] J. A. Stratton, P. M. Morse, J. D. C. Little, and F. J. Cor-bato, Spheroidal Wave Functions, Including Tables of Separa-tion Constants and Coefficients. New York: Wiley, 1956.

[40] V. K. Yulpatov, “The nonlinear theory of interaction of periodicelectron beam with an electromagnetic wave,” Radiophysics andQuantum Electronics, vol. 10, pp. 471–476, 1967.

[41] N. S. Ginzburg, A. S. Sergeev, I. V. Zotova, and I. V. Zheleznov,“Time-domain theory of gyrotron traveling wave amplifiers op-erating at grazing incidence,” Physics of Plasmas, vol. 22, no. 1,p. 013112, 2015.

[42] M. Botton, T. M. Antonsen, B. Levush, K. T. Nguyen, andA. N. Vlasov, “MAGY: a time-dependent code for simulationof slow and fast microwave sources,” IEEE Transactions onPlasma Science, vol. 26, no. 3, pp. 882–892, June 1998.

[43] O. V. Sinitsyn, G. S. Nusinovich, K. T. Nguyen, and V. L.Granatstein, “Nonlinear theory of the gyro-TWT: Comparisonof analytical method and numerical code data for the NRL gyro-TWT,” IEEE Transactions on Plasma Science, vol. 30, no. 3,pp. 915–921, June 2002.

[44] G. S. Nusinovich and H. Li, “Large-signal theory of gyro travel-ing wave tubes at cyclotron harmonics,” IEEE Transactions onPlasma Science, vol. 20, no. 3, pp. 170–175, June 1992.

[45] M. E. Read, A. J. Dudas, J. J. Petillo, and M. Q. Tran, “De-sign and testing of an electron gun producing a segmented sheetbeam for a quasi-optical gyrotron,” IEEE Transactions on Elec-tron Devices, vol. 39, no. 3, pp. 720–726, Mar 1992.

[46] V. L. Bratman, Y. K. Kalynov, G. I. Kalynova, V. N. Manuilov,and P. B. Makhalov, “Frequency tuning in a subterahertz gy-rotron with a variable cavity,” IEEE Transactions on ElectronDevices, vol. 61, no. 10, pp. 3529–3533, Oct 2014.