PSFC/JA-16-71

Novel linear analysis for a gyrotron oscillator based on a spectral approach

J. Genoud, 1 T. M. Tran,1 S. Alberti,1 F. Braunmueller,1 J-Ph. Hogge,1 M. Q. Tran,1

D.S. Tax,2 W.C. Guss,2 and R.J. Temkin2

1 Swiss Plasma Center, Ecole Polytechnique F´ed´erale de Lausanne, Station 13, CH-1015 Lausanne, Switzerland. 2 Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139 USA.

March 2016

Plasma Science and Fusion Center Massachusetts Institute of Technology

Cambridge MA 02139 USA Work at MIT was supported by the U.S. DOE Office of Fusion Energy Sciences under Grant No. DE-FC02-93ER54186. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.

Novel linear analysis for a gyrotron oscillator based on a spectral approachJ. Genoud,1, a) T. M. Tran,1 S. Alberti,1 F. Braunmueller,1 J-Ph. Hogge,1 M. Q. Tran,1 D.S. Tax,2 W.C. Guss,2

and R.J. Temkin21)Swiss Plasma Center, Ecole Polytechnique Federale de Lausanne, Station 13, CH-1015 Lausanne,Switzerland.2)Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge,MA 02139 USA.

(Dated: December 22, 2015)

With the aim of getting a better physical insight in linear regimes in gyrotron, a new linear model wasdeveloped. This model is based on a spectral approach for solving the self-consistent system of equationsdescribing the wave-particle interaction in the cavity of a gyrotron oscillator. Taking into account the wall-losses self-consistently and including the main system inhomogeneities in the cavity geometry and in themagnetic field, the model is appropriate to consider real system parameters. The main advantage of thespectral approach, compared to a time-dependent approach, is the possibility to describe all the stable andunstable modes, respectively with negative and positive growth rate. This permits to reveal the existence ofa new set of eigenmodes, in addition to the usual eigenmodes issued from cold-cavity modes. The proposedmodel can be used for studying other instabilities such as for instance backward waves potentially excited ingyrotron beam tunnels.

PACS numbers: ?

I. INTRODUCTION

Gyrotrons are microwave sources exploiting the cy-clotron resonance maser instability to emit radio-frequency waves in the GHz up to the THz frequencyrange. These devices are mainly used as high power de-vices for fusion application but also as low power devices,as for instance for DNP-NMR (Dynamic Nuclear Po-larization enhanced Nuclear Magnetic Resonance) spec-troscopy. The full description of the interaction be-tween the slightly relativistic electron beam and the reso-nant mode of the cavity requires self-consistent nonlinearmodels1–3. On the other hand, to describe the excitationlimits, linear models are appropriate and have alreadybeen extensively studied2–6. Most of the models thathave been used in the past to calculate the self-excitationcondition, expressed in terms of a starting current, arenot self-consistent2,5,7–9, in the sense that the empty-cavity longitudinal eigenmode is used. However, the ne-cessity of using a self-consistent model has been discussedin several publications6,10–13, especially for backward-wave interactions where self-consistent effects are impor-tant. The majority of these models are time-dependent,describing the time evolution of the most unstable mode.The model presented in this paper is based on a spectralapproach, permitting to study not only the most unsta-ble mode, but also all the stable and unstable modes. Inthat sense, this approach is a novelty and allows to gaina deeper physical insight on the linear dynamics takingplace inside the cavity of a gyrotron oscillator. Moreover,the spectral approach is more convenient than a time-dependent approach for starting-current calculation, re-

a)jeremy.genoud@epfl.ch

quiring less computational resources, as discussed herein.In section II, the model and its derivation is described.In section III, the numerical implementation is presented.The model validation against different models as well asexperimental results is made in section IV. The applica-tion of the novel model to two distinct cases for whichdetailed experimental results exist: a), low power high-quality factor gyrotron and, b) high-power low-qualityfactor gyrotron, is presented and discussed in section V.Section VI concludes the paper.

II. MODEL DESCRIPTION

The new model is based on the nonlinear model of thecode TWANG14, describing self-consistently the wave-particle interaction for a single transverse TEm,p mode.Together with its linearization it is described in detailsin references6,14,15. An important point to be stressed isthat the amplitude and phase of the rf-field profile wereconsidered to vary on a time-scale much longer than theelectron transit-time in previous models6,14 (in the modelpresented in this paper and in a new reduced 1D Particle-In-Cell (PIC) model15, this assumption is relaxed). Theequations describing respectively the perpendicular andparallel electron motion for each electrons are

dPdz

=

(−iγ − sΩc

ω0

pz+s

2

d(lnB0)

dz

)P

+ s(−γF ∗ + ipzF′∗)C0

p2s−2⊥pz

, (1)

and

dpzdz

= −C0

pzIm(F ′P)− 1

2

p2⊥pz

d(lnB0)

dz. (2)

2

Whereas the wave-equation for the wave electric field en-velope F (z, τ) is(

2i∂

∂τ+

∂2

∂z2+ κ2‖0

)F = iI

⟨C0P∗

pz

⟩, (3)

with the boundary conditions at the beginning (zin) andend (zout) of the interaction space:

∂F

∂z

∣∣∣∣zin/out

= ∓ik‖0

k0F (zin/out, t). (4)

More general frequency-independent non-reflectingboundary conditions have been recently implemented inthe model16, permitting to extend the validity domainof the model to non-stationary regimes17,18.

The two normalized variables are the normalized ax-ial position z = ω0

c z and the normalized time τ = ω0t.ω0 is a reference frequency herein chosen as the cutofffrequency in the constant radius cavity section and c isthe speed of light. P = ps⊥ exp(−iΨ) is the complex per-pendicular momentum, p⊥ and pz are the perpendicularand parallel electron momentum. Ψ is the slow-timescaleelectron phase. Other quantities are: γ the electron rel-ativistic factor, Ωc the non-relativistic cyclotron angularfrequency and s the harmonic number. B0 is the exter-nal magnetic field amplitude and F = e

mec2ss

2ss!Ek⊥

is the

normalized electric field envelope with F ′ = ∂F/∂z itsspatial derivative, e and me are the electron charge andmass, respectively. E is the complex electric field and k⊥is the perpendicular wave number. k0 = ω0/c and k‖0are respectively the reference and parallel reference wavenumber. In the wave equation, the average of the source

term is defined as: 〈V 〉 = 1N

∑Nj=1 Vj for N electrons at

a given z and τ , where V stands for any electron beamdynamical quantity.

The three dimensionless constants C0, κ‖0 and I aregiven by:

C0 ≡(k⊥k0

)s(sΩcω0

)1−s

Jm−s (k⊥Rg) , (5)

κ2‖0 ≡ 1− k2

⊥k2

0

[1− (1 + i)

(1 +

m2

ν2mp −m2

)δskRw

], (6)

I ≡ eZ0

mec2IbCmp

(ss

2ss!

)2

, (7)

with

Cmp ≡π

2(ν2mp −m2)J2

m(νmp). (8)

In these expressions, Rw and Rg are the cavity wall andguiding center radius, Jm is the Bessel function of thefirst kind of orderm and νmp the pth root of its derivative.Z0 is the vacuum impedance. Ib is the beam current. Thewall losses are taken into account self-consistently in the

wave-equation via a correction factor in the normalized

parallel number κ‖0. δsk =√

2ω0µ0σ

is the skindepth, µ0

the vacuum permeability and σ the wall conductivity.In TWANGlin6, the electron equations of motion have

been linearized, considering the electric field as a first or-der term. The model has been simplified, the rf-magneticfield (proportional to dF/dz) has been neglected whichimplies that the perturbed parallel electron momentumpz1 = 0. Moreover no velocity spread for the electronbeam have been considered. The system of 2N + 1 equa-tions for the N electrons and the wave equation can be re-duced to three equations by defining two moments of theelectron distribution function π1(z, τ) = 〈P∗1 〉, π2(z, τ) =⟨P1e

2iψ0⟩

and the electric field envelope F (z, τ). Con-sidering an electron distribution function F(P1,Ψ0, z, τ)depending explicitly on the three dynamic variables P1,Ψ0, z and the evolution variable τ , the assumption ofa constant electric field during the electron transit-timecan be relaxed. In this case, the average of the spatialderivative of a function G(P1,Ψ0) can be expressed as⟨

dGdτ

⟩=

∂

∂τ〈G〉+

∂

∂z〈Gvz〉 . (9)

This expression can be derived starting from the phase-space continuity equation:

∂F∂t

+∂

∂P1

(F dP1

dt

)+

∂

∂Ψ0

(F dΨ0

dt

)+

∂

∂z(Fvz) = 0.

(10)For a function G(P1,Ψ0), the average of the time deriva-tive is :⟨

dGdt

⟩=

∫∫dP1dΨ0

dGdtF

=

∫∫dP1dΨ0

[∂G∂P1

dP1

dtF +

∂G∂Ψ0

dΨ0

dtF]

= −∫∫

dP1dΨ0G[

∂

∂P1

(F dP1

dt

)+

∂

∂Ψ0

(F dΨ0

dt

)]=

∫∫dP1dΨ0G

∂

∂tF +

∫∫dP1dΨ0G

∂

∂z(Fvz)

=∂

∂t

∫∫dP1dΨ0GF +

∂

∂z

∫∫dP1dΨ0GFvz

=∂

∂t〈G〉+

∂

∂z〈Gvz〉 .

(11)

For the first three steps respectively, the chain rule, anintegration by part and the continuity equation 10 havebeen used. With this relation, the system is reducedto a closed system of 3 equations for the two complexmoments π1(z, τ), π2(z, τ) and the electric field envelopeF (z, τ):

∂π1

∂τ+

∂

∂z(π1βz) = A1π1 + iC1π2 − C2C0F

∂π2

∂τ+

∂

∂z(π2βz) = A2π2 − iC1π1(

2i∂

∂τ+

∂2

∂z2+ κ2‖0

)F = iC3C0π1,

(12)

3

where A1 = i∆0 + iC1 + βzδ and A2 = i∆0 − iC1 + βzδ.The boundary conditions for a gyrotron cavity are:

π1(zin, τ) = 0 π2(zin, τ) = 0 F (zin, τ) = 0 (13)

dF

dz

∣∣∣∣zout

= ik‖

k0F (zout, τ). (14)

The dimensionless constants are

∆0 = 1− sΩcγ0ω0

, (15)

C1 =p2⊥0

2sγ20

, (16)

C2 = sp2s−2⊥0 , (17)

C3 =eZ0

mec2pz0

IbCmp

(ss

2ss!

)2

. (18)

Finally, by Fourier transforming equations (12) in time,the partial differential equations (PDEs) can be rewrittenas ordinary differential equations (ODEs). The Fouriertransformation is done assuming F ∼ e−iΩτ and π1,2 ∼e−iΩτ , with Ω = (ω − ω0)/ω0 being the normalized dif-ference between the rf-frequency ω and the reference fre-quency ω0. The system of equations in its spectral formis given by:

iΩπ1 =d

dz(π1βz)−A1π1 − iC1π2 + C2C0F

iΩπ2 =d

dz(π2βz)−A2π2 + iC1π1

−2ΩF =d2F

dz2+ κ2‖0F − iC3C0π1.

(19)

This same set of equations could be used for studyingbeam-wave parasitic oscillations in smooth-wall beam-ducts with appropriate boundary conditions.

III. NUMERICAL IMPLEMENTATION

The equations are solved together as a general eigen-value problem

AX = ΩBX. (20)

This is not a linear system, since the matrix A dependson the eigenvalue Ω, via the boundary conditions 14, aswill be shown below. The real and imaginary parts ofthe eigenvalue Ω are respectively the frequency and thegrowth rate of the wave with Im(Ω) > 0 for an unstablemode. The eigenvectors X are composed by the threecomplex fields π1(z), π2(z) and F (z). The discretizationin space is done using a finite difference scheme for thefirst and second order derivatives. It is expressed herefor an equidistant discretization for more clarity (∆z =zi − zi−1, ∀i ∈ [2, Nz]).:

dπ

dz

∣∣∣∣i− 1

2

' πi − πi−1

∆z∀i = 1, ..., Nz. (21)

Here, i and i − 1 are indices denoting points on the dis-cretization grid and i − 1

2 denotes a point centered be-tween the two former points. Nz is the number of points.The second order spatial derivative is:

d2F

dz2

∣∣∣∣i

' Fi+1 − 2Fi + Fi−1

(∆z)2∀i = 1, ..., Nz. (22)

For an index i ∈]1, Nz[, the two first equations of (19)become:

π1i−1

(− βz

∆z+β′z2− A1

2

)+ π1i

(βz∆z

+β′z2− A1

2

)+

+ π2i−1

(− iC1

2

)+ π2i

(− iC1

2

)+

+ Fi−1

(C2C0

2

)+ Fi

(C2C0

2

)=

= π1i−1

(iΩ

2

)+ π1i

(iΩ

2

), (23)

and

π2i−1

(− βz

∆z+β′z2− A2

2

)+ π2i

(βz∆z

+β′z2− A2

2

)+

+ π1i−1

(iC1

2

)+ π1i

(iC1

2

)=

= π2i−1

(iΩ

2

)+ π2i

(iΩ

2

), (24)

where all the indices for the elements βz, β′z = ∂βz

∂z , A1,

A2, C0, C1 and C2 have been omitted. Those elementsare evaluated at zi− 1

2. The third equation of 19 becomes:

Fi−1 − Fi(

2−∆z2κ2‖0

)+ Fi+1+

− π1i

(i∆z2C3C0

)= −2∆z2ΩFi. (25)

Here, the elements κ‖0, C0 and C3 are evaluated at zi.The value of FNz+1 is needed in the last wave-equation(25). It is expressed as function of FNz

and FNz−1, usingthe radiation boundary condition (14):

FNz+1 = 2i∆zk‖(Ω)

k0FNz

+ FNz−1. (26)

An important point to be noted is that, in this expression,k‖ depends on the eigenvalue Ω, and thus the operatorA in (20) depends on the eigenvalue Ω. The boundaryconditions at the entry (13) impose:

π11= 0, π21

= 0, F11= 0. (27)

With this discretization, the eigenvalue problem (20)takes the formA11 A12 A13

A21 A22 0A31 0 A33(Ω)

Π1

Π2

F

= Ω

B11 0 00 B22 00 0 B33

Π1

Π2

F

,

(28)

4

where Π1 = (π11, π12

, ..., π1Nz), Π2 = (π21

, π22, ..., π2Nz

)and F = (F1, F2, ..., FNz

) are the three complex fields ona discretized vector form. The matrices Aij contain atmost three non zeros terms by row. Hence, sparse ma-trices are used to optimize numeric performance. Twomethods have been tested to solve the eigenvalue prob-lem. The first method, which will be retained, solves theproblem using the Matlab19 function eigs, which is basedon routines from the Arpack20 library. Since the verylast term of the matrix A33 depends on the eigenvalueΩ, the eigenvalue problem is solved iteratively, startingfor the first iteration with Ω = 0, i.e. from the arbi-trary reference frequency ω0. The second method to solvethe eigenvalue problem consists in searching the roots ofdet(A(Ω) − ΩB) using the Muller’s method21. As thismethod uses quadratic interpolation, three initial guessesare needed. One iteration of the first method gives forexample those three guesses. On figure 1 are reported thetime needed by the two methods to perform 10 iterations,solving only the wave equation and for an equidistantdiscretization in the longitudinal direction z. The firstmethod is faster for large Nz (Nz > 600), and thus forlarger matrices. Moreover, by adding the time neededto compute the eigenvector in the second method, thefirst method is faster for Nz > 400. For this reason, thefirst method, calculating the eigenvalues with the Matlabscript eigs has been retained.

0 200 400 600 800 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Nz

Tim

e f

or

10

ite

rati

on

s [

s]

1st

method : "eigs"

2nd

method : "det"

Figure 1. Time needed by the two methods to solve the eigen-value problem with 10 iterations (average on 10 runs).

IV. CODE DEVELOPMENT AND BENCHMARKING

A. Validation against other codes

The cavity longitudinal profile used in the simulationsis shown in figure 2 along with the external magneticfield and the electron beam position. This is the cavity of

a 260 GHz gyrotron17 used for DNP-NMR spectroscopyresearch.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

2

4

6

x 10−3

z [m]

RW

[m

]

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0.9

0.95

1

Bm

ax [

T]

Cavity

Electron beam

Magnetic field

Figure 2. Longitudinal profile of the DNP-cavity (in blue),position of the electron-beam (in red, dashed) and normalizedmagnetic field profile (in green, dashed-dotted).

Code Model Num. implementation

Cavity (cold-cavity) spectral FEM

TWANGlin temporal FEM + RK

TWANGlinspec spectral FDM

Table I. Model and numerical implementation for the cold-cavity code “Cavity” and the time-evolution linear code“TWANGlin”, both used to benchmark the new spectral code“TWANGlinspec”. FEM, FDM and RK stand for Finite El-ement Method, Finite Difference Method and Runge-Kutta,respectively.

First of all, TWANGlinspec was tested without cur-rent (Ib = 0), i.e., without beam-wave interaction. Itwas benchmarked against the code Cavity, a cold-cavitycode based on finite element method (see table I). Forthe comparison Nz = 4225 points for the discretizationwere chosen as well as 10 iterations in order to have anumerical convergence. The reference frequency ω0 wasset to the cutoff-frequency of the flat section of the cav-ity. The relative discrepancy between the first eigenvaluegiven by the two codes never exceeds 0.09% for both thereal and imaginary parts. The discrepancies for highermodes are slightly larger, as expected, as a higher num-ber of points is needed for the sampling (according to theNyquist-Shannon sampling theorem).

A second validation step consisted in adding the self-consistent interaction with the electron beam. For thiscase, the comparison was done with the results of thecode TWANGlin6 (see table I). To solve the equations,the code TWANGlin uses a fourth order Runge-Kuttamethod for the two beam equations related to the mo-ments π1 and π2 and a finite element method for the waveequation. In figure 3 are shown the real and imaginary

5

parts of the eigenvalues computed with the two codes,corresponding respectively to the frequency and growthrate of the wave, for different average step size < dz > ofthe non-equidistant spatial discretization. The number ofpoints Nz used for the discretization varies between 423(far-right point) and 16897 (far-left point). As expected,the convergence is faster with the code TWANGlin, us-ing a third order finite element method for the wave-equation, whereas a first order finite difference method isused in TWANGlinspec. In the following, at least 2000points are considered for the discretization.

260.5

260.501

260.502

260.503

260.504

freq

uen

cy [

GH

z]

0.5 1 1.5 2 2.5

x 10−4

0

0.5

1

1.5

2

2.5x 10

7

<dz> [m]

Γ [

s−

1]

TWANGlinspec

TWANGlin

a)

b)

Figure 3. Frequency (a) and growth rate (b) versus theaverage step size ||dz|| computed with the spectral codeTWANGlinspec (in blue) and the time-dependent codeTWANGlin.

B. Validation against experiments

The model in TWANGlinspec allows to consider realexperimental system parameters with all the spatial inho-mogeneities such as magnetic field or cavity radius inho-mogeneities. An additional validation is done comparingthe starting-current calculated with TWANGlinspec withexperimental measurements. In gyrotrons, the startingcurrent is the self excitation condition, i.e. a thresholdcurrent above which a mode is excited. The results ofTWANGlinspec for the DNP-gyrotron17 are presented infigure 4 along with experimental results. The simulationparameters are listed in the first part of table II. Theagreement between TWANGlinspec and the experimen-tal results is remarkably good over the whole range ofmagnetic field. With a time-dependent code, the calcu-lation is also possible but requires long calculation, giventhe very small growth rate close to the starting current.With a spectral code, the calculation is faster, as it re-duces to finding the root of the eigenvalue. Indeed, the

imaginary part of the eigenvalue Ω in (20) corresponds tothe wave growth rate and the current, where this valuebecomes positive corresponds to the starting current.

Most of the start-up scenarios in gyrotron operationsare based on non-self-consistent starting current calcula-tion. The starting current of the analyzed DNP-gyrotronwas calculated with such a fixed-field model and addedin Fig. 4. A significant discrepancy is found betweenthe correct starting current and the non-self consistentcalculations, for example for magnetic field lower than9.52T, corresponding to the positive detuning regime inwhich high-power gyrotrons are operated. A fully self-consistent model is thus preferable for start-up scenar-ios studies, as it has already been mentioned in severalpublications6,8,10–13.

The TWANGlinspec code, based on a fully self-consistent model, is numerically efficient and allows tocompute the starting current for a given transverse mode.

9.5 9.55 9.6 9.65 9.70

10

20

30

40

50

60

70

80

90

100

B [T]

I sta

rt [

mA

]

TWANGlinspec

fixed−field

experiments

q = 3 q = 4q = 2q = 1

Figure 4. Starting current calculated with TWANGlinspec (inblue), with a fixed-field code (in black, dashed) and measured(red points).

V. RESULTS

In this section the model is applied to two different gy-rotrons for which detailed experimental results have beenobtained: a) DNP-gyrotron17, already discussed earlier,characterized by a low-order high quality factor operat-ing mode (low-power gyrotron) and b) a high-order lowquality factor (high-power gyrotron) for which the ex-perimental results are presented in the paper by Tax etal.22.

6

Parameter Value

Transverse mode TE7,2

Beam radius 1.394 mm

Pitch angle 1.9

Acceleration voltage 15.5 kV

Wall conductivity 2.9 · 107 S/m

Magnetic field 9.55 T

Beam current 10 mA

Table II. Simulation parameters. The magnetic field, beamradius and pitch angle values are set in the middle of theconstant radius section of the cavity.

A. DNP-gyrotron

As already mentioned, unlike time-dependent mod-els, which describe the most unstable mode evolution,the spectral approach gives information on both stable,Im(Ω) < 0, and unstable modes, Im(Ω) > 0. This is il-lustrated in figure 5, for the simulation parameters givenin table II. The horizontal axis represents the real partof the eigenvalues, corresponding to the wave frequency,whereas the vertical axis represents the imaginary part ofthe eigenvalues, corresponding to the wave growth rate.In this case, only one mode is unstable (Γ > 0). Thefield profiles corresponding to these eigenmodes are ob-tained via the eigenvector X in Eq.(20). The amplitudeand phase of the electric field profile for the four modeshighlighted in figure 5 are shown in figures 6 and 7. Forthe unstable mode, the electric field profile correspondsto the mode with one axial variation. The closest modein the complex plane in figure 5 is the mode with two ax-ial variations. By following the diagonal formed by thesemodes in the complex plane, all “usual” modes with in-creasing axial variations are found. On the other hand,all the modes situated in the bottom part of the com-plex plane, with Γ < −2 ·109 s−1, have their electric fieldprofile mainly situated in the uptaper following the con-stant radius section of the cavity, as can be seen in figure7. For the following, a distinction is made between the“usual” eigenmodes and these “uptaper” eigenmodes.

Analysis of the novel “uptaper” modes

With the above-described DNP gyrotron, non-stationary oscillations have been observed17,18. In acertain range of operating parameters, equidistant side-bands appear in the frequency spectrum. The “uptaper”modes show several similarities with these sidebands.First the “uptaper” modes, unlike the “usual” modes,are almost equidistant in frequency. Secondly, their elec-tric field profiles are peaked in the uptaper. This hasalso been observed with the nonlinear code TWANG forsidebands with frequency lower than the cutoff frequency.The third similarity is the dependency on the pitch angle

258.5 259 259.5 260 260.5 261 261.5 262 262.5−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

x 109

frequency [GHz]

Γ [

s−

1]

eigenvalues

q = 1

q = 2

"singular" mode 1

"singular" mode 2

Figure 5. Real (frequency) and imaginary (growth rate) partsof the first 14 eigenvalues computed with TWANGlinspec.The dashed vertical red line indicates the cutoff frequency inthe constant radius cavity part. The horizontal continuousblack line indicates the limit between stable (Γ < 0) andunstable (Γ > 0) modes. The four symbols indicate whichmodes are considered in figures 6 and 7. (cf. table II forparameters).

.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.2

0.4

0.6

0.8

1

|E|

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

100

200

300

Ph

ase o

f E

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

5

x 10−3

z [m]

RW

[m

]

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.18

9

10B

[T

]

q = 1

q = 2

b)

a)

c)

Figure 6. (a) and (b) Amplitude and phase of the electricfield profile along the longitudinal direction, calculated withTWANGlinspec for two of the modes highlighted in figure 5.(c) Cavity and magnetic field profile. The dashed black lineindicates the end of the cavity constant radius part. (cf. tableII for parameters).

7

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.2

0.4

0.6

0.8

1

|E|

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−500

0

500

Ph

ase o

f E

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

5

x 10−3

z [m]

RW

[m

]

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.18

9

10

B [

T]

"singular" mode 1

"singular" mode 2

b)

a)

c)

Figure 7. (a) and (b) Amplitude and phase of the electricfield profile along the longitudinal direction, calculated withTWANGlinspec for two of the modes highlighted in figure 5.(c) Cavity and magnetic field profile. The dashed black lineindicates the end of the cavity constant radius part. (cf. tableII for parameters).

α = v⊥/v‖. In the sideband regime, the frequencies inexperiment are observed to fluctuate in time, while re-maining equidistant at each moment, as can be seen infigure 9, which is a spectrogram acquired with an hetero-dyne system. It has been shown18,23 that these frequencyfluctuations are related to an anode voltage fluctuation,with the same periodicity and causing a modulation ofthe pitch angle. The sidebands are much more affectedby this fluctuation than the main mode. The calculatedeigenvalue spectrum for three values of the pitch angle,with a maximum relative difference similar to the ex-perimental fluctuation (∆α

α ≈ 3%), is presented in fig-ure 8. One observes that the “uptaper” modes are alsomuch more affected than the “usual” modes, even thoughthe absolute frequency difference is more important forthe “uptaper” modes (∆f ≈ 150 MHz) than the maxi-mum sidebands frequency modulation in the experiment(∆f ≈ 50 MHz).

These few observed similarities would suggest thatthese “uptaper” equidistant stable eigenmodes could berelated to the experimentally observed equidistant side-bands. However, one should be careful while interpretingthe results coming from the linear theory, given the factthat the sidebands occur in strongly nonlinear regimes.The mechanism that would lead to the excitation of these“uptaper” modes cannot be studied with this model andwould require a nonlinear model adequate for studyingnon-stationary regimes such as TWANG-PIC15.

259 259.5 260 260.5 261 261.5 262−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

x 109

frequency [GHz]

Γ [

s−

1]

α = 1.87

α = 1.9

α = 1.93

Figure 8. Real (frequency) and imaginary (growth rate) partsof the first 15 eigenvalues calculated for three different pitchangle values. (cf. table II for parameters, B = 6.3 T).

Figure 9. Spectrogram of experimental signal during non-stationary oscillation regime. The side-bands are more af-fected by the pitch angle fluctuations than the main mode.

B. MIT 110-GHz gyrotron

Another study has been carried out based on the exper-imental results obtained with the 1.5MW, 110 GHz MIT-gyrotron described in the paper by Tax et al.22. In theexperimental results presented in this paper, it has beenshown that during the start-up phase, a TE21,6,4 modeis excited before the nominal TE22,6,1 mode. For thisstudy, the cavity profile and the parameters described intable III have been used. The system parameters of tableIII correspond to a specific time (t=1.45 microseconds)during the startup-phase described in Fig.3 of Tax’s pa-per. Before considering the real magnetic field profile,an uniform magnetic field has been used for the start-ing current calculation. As the results for the two caseswere significantly different, intermediate magnetic fieldprofiles have been considered to study their effect on thestarting current. Three of these different magnetic field

8

profile are presented in figure 10.The starting current of the first longitudinal mode

to be excited for the two transverse modes TE21,6 andTE22,6 are listed in table IV. For these parameters, allthe modes correspond to backward-wave modes. Witha constant magnetic field, the starting current of theTE21,6,4 mode is lower than for the TE22,6,3 mode, indi-cating that it is the mode which is excited first. This is ingood agreement with what is observed experimentally22.On the contrary, with the intermediate, Bint, or real mag-netic field profiles, Breal, shown in figure 10, the modeTE21,6,q has a larger starting current than the TE22,6,q

mode. Even though the relative difference in magneticfield amplitude is only around 2% at the end of the sim-ulated cavity region and lower than 0.2% until the endof the initial uptaper region, the starting current for theTE21,6,3 mode is almost two times larger for the real mag-netic field profile.

Another aspect to consider, in particular when such pa-rameter sensitivity is observed, is the voltage depressionassociated to electron beam space charge. To assess thiseffect, based on the system parameters of III, a space-charge depression of 4 kV has been calculated resultingin an electron beam kinetic energy of Eb = 46 kV. Theresults for the two transverse modes using the real mag-netic field profile are listed in table V. As it was the casefor the different magnetic field profile, the results dependstrongly on the accelerating voltage, especially for theTE21,6 mode.

These two parameter studies illustrate that for start-up scenario studies, where reliable information on thestarting current are needed, a precise knowledge on theexperimental parameters is required.

Parameters Value

Magnetic field maximum 4.38 T

Cathode voltage 50 kV

Pitch angle 0.7

Wall conductivity 1.45 ·107 S/m

Beam radius 10.1 mm

Modes considered TE21,6 TE22,6

Table III. Parameters used for the starting current calcula-tions.

ModeIstart[A]

Bconst Bint Breal

TE21,6,q 25.3 (q = 4) 27 (q = 4) 46.6 (q = 3)

TE22,6,q 27.1 (q = 3) 26.8 (q = 2) 29.5 (q = 1)

Table IV. Starting current calculated with TWANGlinspecfor the two transverse modes TE21,6 and TE22,6 and for thethree different magnetic field profiles shown in figure 10. Theassumed electron beam energy is Eb = 50 kV. q indicates thelongitudinal mode index of the mode.

0.019

0.02

0.021

z [m]

RW

[m

]

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.975

0.98

0.985

0.99

0.995

1

B/B

max

Breal

Bconst

Bint

Figure 10. Longitudinal profile of the MIT 110 GHz cavity(in blue) and normalized magnetic field profiles used for thecalculations (in green).

ModeIstart[A]

Eb = 50 kV Eb = 46 kV

TE21,6,q 46.6 (q = 3) 31.7 (q = 4)

TE22,6,q 29.5 (q = 1) 25.3 (q = 3)

Table V. Starting currents calculated with TWANGlinspecfor the two transverse modes TE21,6 and TE22,6 and for twodifferent electron beam energy. The real magnetic field profilehas been used for these calculations. q indicates the longitu-dinal mode index of the mode.

VI. CONCLUSION

A new self-consistent linear spectral codeTWANGlinspec has been developed and extendsthe existing TWANG6,14,15 series of codes. It has beenimplemented and validated against other codes. Themodel, taking into account the spatial inhomogeneitiesin the cavity geometry and in the magnetic field profile,and treating the ohmic wall losses self-consistently, iswell adapted to describe a real experimental system.The agreement with experiment from a low powerDNP-gyrotron is excellent and constitutes anotherbenchmark for the code. Unlike the time-dependentapproach, limited to the evolution of the most unstablemode, the spectral approach permits to get informationon both stable and unstable eigenmodes. Consequently,a new set of equidistant stable eigenmodes has beenidentified, revealing similar parametric dependencieswith respect to side-bands observed experimentally in anon-stationary regime. However, these non-stationaryoscillations occur in strongly nonlinear regimes, andthe mechanism that would lead to the excitation of the“uptaper” modes cannot be studied with this linearmodel and is still under investigation. The new code is

9

also convenient to calculate starting currents. Indeed,in the spectral formulation, the system takes the formof a general eigenvalue problem and the starting currentcalculation is done searching the current at which theimaginary part of the eigenvalue, corresponding to thegrowth rate, changes its sign. Compared to a timeevolution approach, this approach leads to considerablyshorter simulation time and is thus appropriate forparameter scans. Such a parameter study has beencarried out with a high power gyrotron and revealedthe importance of a precise knowledge on experimentalparameters for start-up scenario studies. In the future,the code TWANGlinspec will be adapted to includea higher order discretization method for the waveequation. It will also be used to study spurious exci-tation of instabilities in beam-ducts24–26, by adaptingthe boundary conditions and the wall losses with thecorresponding impedance boundary conditions.

ACKNOWLEDGMENTS

Work partially supported by by EFDA under GrantWP13-DAS-HCD-EC and by Fusion for Energy underGrant F4E-GRT-432 and within the European GyrotronConsortium (EGYC). The views and opinions expressedherein do not necessarily reflect those of the EuropeanCommission. EGYC is a collaboration of CRPP, Switzer-land; KIT, Germany; HELLAS, Greece; IFP-CNR, Italy.

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Work at MIT was supported by the U.S. DOE Office of Fusion Energy Sciences under Grant No. DE-FC02-93ER54186.