PFC/JA-90-36

CARM Amplifier Theory

and Simulation

Chen, C.; Danly, B.G.; Davies, J.A.t;

Menninger, W.L.; Wurtele, J.S.; Zhang, G.t

Plasma Fusion Center

Massachusetts Institute of Technology

- -Cambridge, MA 02139

September 1990

t Permanent Address: Department of Physics, Clark University, Worcester, MA

Submitted to: Nuclear Instruments & Methods (special issue).

This work was supported by U.S. D.O.E. Contract DE-FG02-89ER-14052.

ABSTRACT

The theory and simulation of cyclotron autoresonance maser (CARM) amplifiers arepresented, including studies of amplifier phase stability, multimode phenomena, and sus-ceptibility to absolute instabilities. Recent results include particle-in-cell simulations ofthe onset of absolute instabilities and numerical modeling of multimode effects. Phasestability studies indicate that the output phase of CARM amplifiers may be relativelyinsensitive to fluctuations in beam energy, pitch, and current, for optimized designs;simulations show a phase sensitivity of ~ 2* per percent beam energy variation. Anexperimental design for a long-pulse 17 GHz CARM amplifier is presented.

1

1. INTRODUCTION

Future linear colliders will require high frequency rf sources together with high gradi-ent accelerating structures in order to be economically feasible. The cyclotron autores-onance maser (CARM) is a promising source for application as an rf accelerator driver,and a program to develop CARM amplifiers at 17 GHz for this application is presentlyunderway at the MIT Plasma Fusion Center [1].

Experiments at a 17 GHz will be performed using two different technologies for gener-ation of the high voltage electron beam required by the CARM. A long-pulse (2.5 ps, 1 psflattop), 700 kV pulse modulator and a short-pulse (50 ns), 1.2 MeV induction acceleratorwill be employed for generation of the electron beam. This will allow a comparison oftwo alternate methods for producing the ~ 50 ns rf pulses required by the high gradientstructures. A long-pulse, modulator-driven CARM together with pulse compression tech-niques, or an induction-linac-driven CARM are both capable in principle of deliveringthe required rf pulses to the structure.

In this paper, several theoretical issues are discussed, including the behavior of theCARM amplifier with the presence of multiple modes, particle-in-cell simulations of am-plifier stability, and an analysis of phase stability in the CARM amplifier. Finally, anexperimental design for a long-pulse CARM amplifier is presented.

2. MULTIMODE CARM AMPLIFIER THEORY

Many planned CARM amplifier experiments will operate in overmoded waveguide.Until now, CARM amplifiers have been analyzed under a priori assumption that onlya single waveguide mode interacts with the electron beam. The goal of this sectionto develop a formalism which can treat overmoded systems with an arbitrary numberof transverse-electric (TE) and transverse-magnetic (TM) modes. The present analysisconsists of two approaches: (kinetic) linear theory and three-dimensional, self-consistentcomputer simulations. For simplicity, we present here the analysis of TE1j modes cou-

pling to an azimuthally symmetric electron beam at the fundamental cyclotron frequency,while maintaining the general features of multimode CARM interactions.

It can be shown that a complete set of nonlinear equations describing an over-moded CARM amplifier with multiple TEIn modes can be expressed in the dimensionlessform [2,3]

=- - 1 X.(rL,)AcosPn , (1)d! pZ

dp A 1 d6 dAn s- -Xn(rA,r,) -+-) A cosn+- si , (2)

di fix n n di IdI

2

d4,' 1 + + + V

di TO 6 AA.d 2 1+ -A- W,(rL, r,) V 2- + -')A., sini~ dA p- cs1 (3)

PP.,Pi #ii, d+PzC~~i

+ Tn )An (i) exp { i_ 2in + \n (id I

=ig X (rL, r,) e-ln- expf i -z+ n(i) , (4)

where the normalized coupling constant gn is defined by

= 4(026 - 1) .(5)

- 1)[J1 (Vn)] 2 \A (5)

Equations (1)-(3) describe the dynamics of each individual particle, and Eq. (4) describes

the slowly varying wave amplitude An(i) and phase shift 6(i) for the TE1j mode. InEqs. (1)-(5), i = wz/c is the normalized interaction length; w = 27rf is the angular

frequency of the input signal; n = /w = eBo/mct is the normalized nonrelativistic

cyclotron frequency; P. = p,/mc = -yo,, Pj = pj/mc = -yfLj, and -y = (1+P+3I)1/2 are,respectively, the normalized axial and transverse momentum components, and relativistic

mass factor of the beam electron; Ib is the beam current; IA = mc3/e 2 17 kA is the

Alfv~n current; Xn(rL,r,) = Jo(knrg)JI(knrL) and Wn(rL,rg) = Jo(knrg)JI(knrL)/knrLare geometric factors; Jo(x) is the zeroth-order Bessel function; JI(x) = dJ1 (x)/dx is the

derivative of the first-order Bessel function; vn is the nth zero of Jj(x); kn = vl'/r, is the

transverse wavenumber associated with the T E1 , mode; 0, = w/ck,, =k -is the normalized phase velocity of the vacuum TEIn waveguide mode; rTL = p±/mil, isthe electron Larmor radius; r, is the electron guiding-center radius which is assumed

to be constant. In Eq. (4), < ... > denotes the ensemble average over the particle

distribution, and typically more than 1024 particles are used in the simulations. The rf

power flow over the cross section of the waveguide for the TEin mode, Pn(z), is related

to the normalized wave amplitude An by the expression

Pn(i)= - + - ) (6)2gn e2 IA di n

where m 2 c5/e 2 c 8.7 GW.

3

By performing the Laplace transform of the linearized Maxwell-Vlasov equations, a

dispersion relation and amplitude equations can be derived for the cyclotron resonance

maser interaction with multiple TEIn modes coupling to a cold, thin (knr, < 1), az-

imuthally symmetric electron beam. To leading order in c2k'/(w - 1c/y - kv,)2 , it

can be shown that the Laplace transform of the equations for the amplitudes En(z) ~An(z) exp[ik2 ,z + 6b(z)] can be expressed approximately in the matrix form [3]

2 N ,(W2 + c 2S2 )52 _ k2 + - E( + nn kn a'

n(2 + , (L - l/_y + i_ n)

= sEn(O) + E W ,kIvt Ej,(0) (7),= (w - £c/i + i~s)2

where use has been made of initial condition dEn(O)/dz = 0, and the dimensionless

coupling constants Inn, are defined by

40 2 I1 Xn(rL , r,)X.,(rL , r,)Inn' _L ,(8)

(J)[(V2 - 1)(V2, - 1)]1/2j,(Vn)j,(Vn,).()

In Eqs. (7) and (8), s = ik, is the argument in the Laplace transform; 3, = vIc

and #_ = v./c are, respectively, the normalized axial and transverse velocities of the

equilibrium beam electrons; and ckN is the largest cut-off frequency below the operating

frequency w. Therefore, the amplitudes and dispersion relation can be obtained by solving

Eq. (7) and performing the inverse Laplace transform of En(s).

For two coupled modes, TE1, and TEIn,, it is readily shown from Eq. (7) that the

dispersion relation is

k + kn - (kZ + kn,- k,

=[Innk (k,2 + k + -n+ + k - ](w2 -c2k) . (9)

When the two modes are well separated, and Ennki (k + k2, - w2 /c 2) > ,(k2 +

k - w 2 /c2), corresponding to the electron beam mode, w = kvz + Oc/y, in resonance

with the TE1 n mode, w = c(kj + k2) 1/2, the coupled-mode dispersion relation in Eq. (9)

becomes the usual single-mode dispersion relation [2,4]

kZ+ k2_),esiw 2 (10)S c2 (w - Oc7- k..v)2

4

for the TEI, mode.

Typical results from the computer simulations and kinetic theory are summarizedin Figs. 1-3. Figure 1 shows the dependence of rf power in the TE11 and TE12 modeson the interaction length, for (a) single-mode CARM interactions and (b) the CARMinteraction with the two modes coupling to the beam. The system parameters in Fig. 1correspond to the TE11 mode in resonance, and the TE12 mode off resonance, withthe electron beam. The solid curves are the simulation results obtained by integratingnumerically Eqs. (1)-(4), and the dashed curves are obtained analytically from Eq. (7).The inclusion of the coupling of the TE11 and TE12 modes results in instability for theTE12 mode in Fig. 1(b), while the single-mode theory predicts complete stability for theTE12 mode in Fig. 1(a). In fact, in Fig. 1(b), the TE12 mode grows parasitically withthe dominant unstable TE11 mode and the two coupled modes have the same spatialgrowth rate -ImAk, > 0, corresponding to the most unstable solution of the dispersionrelation in Eq. (9). Because the TE11 mode is in resonance with the beam mode and theTE12 mode is detuned from the corresponding resonance, the TE12 mode suffers greaterlaunching losses than the TE11 mode in the interaction. This is one of the general featuresof the multimode CARM interaction, namely, all of the coupled modes have the samesmall-signal growth rate but suffer different launching losses depending upon detuningcharacteristics.

Another interesting feature of the multimode CARM interaction is that the rf powerdistribution among the coupled modes at saturation is insensitive to the power distribu-tion at z = 0 but is sensitive to detuning. Figure 2 shows the results of the simulationsfor the coupling of the TE11 and TE12 modes with two different distributions of input rfpower. Here, only the TE12 mode is plotted because the TE11 mode remains virtuallyunchanged for the two cases. Figure 3 depicts the detuning characteristics of saturationrf power distribution over four coupled TEln modes (n = 1,2,3, 4), as obtained from thesimulations with an input power of 100 W per mode. By increasing the axial magneticfield B0 in Fig. 3, the beam mode is tuned through the resonances with the TE11 , TE1 2 ,

TE13 , and TE14 modes at B0 = 3.74, 4.29, 5.33, and 6.98 kG, respectively. The fractionalrf power for a given mode reaches a maximum at its resonant magnetic field, while therf power decreases rapidly for off-resonance modes. In the transition from one resonanceto another, however, two adjacent competing modes close to the resonance can havecomparable rf power levels at saturation.

3. SIMULATIONS OF CARM ABSOLUTE INSTABILITY

Most designs of CARM amplifiers, including our design in Sec. 5, rely on linear theoryto determine parameter regimes where the absolute instability [5,6] is below threshold.The theoretcial models used in the calculation of the threshold current assume that

5

there is no input signal at the drive frequency and that the system in infinite in the axialdirection. Both these conditions are, obviously, violated in any amplifier experiment. Inorder to investigate the possibility of suppressing the absolute instability by injecting alarge input signal, the MAGIC particle-in-cell code (7] is being used simulate a CARMamplifier. Parameters have been choosen so that, in the absence of an injected signal, theabsolute instability will experience strong growth. At present, the cylindrically symmetricversion of the program is being employed, limiting this preliminary analysis to TEonwaveguide modes.

The growth of the TEO, mode downshifted convective instability and absolute in-stability from numerical noise (with no driver) is illustrated in Figs. 4(a) and 4(b).System parameters are waveguide radius = 1.5 cm, hollow beam radius = 0.75 cm,gamma = 2.967, beam current = 50A, applied longitudinal magnetic field = 13.75 kG,and 01 = 1.0. Figure 4(a) shows E9 vs. z at t = 3.6 nsec. The long-wavelengthoscillation of Eq corresponds to the convective downshifted guide wavelength of 28 cmpredicted by linear theory. Figure 4(b) shows the same field at t = 8.1 nsec. The longwavelength is now consistent with the absolute instability guide wavelength of 18cm pre-dicted by linear theory. Apparently, the growth of the downshifted convective instabilityhas been suppressed by that of the slower growing absolute instability.

To illustrate the effects of an input signal, we apply a driver of 25 kW to a highlyunstable system with the same waveguide dimensions and beam energy as in Fig. 4, butwith 0L/011 = 1.6, beam current = 500 A, and an applied longitudinal magnetic fieldof 19.43 kG. The frequency of the driver is 34.272 GHz (equal to that of the upshiftedintersection of the uncoupled beam and TEO, waveguide dispersion relations). At earlytimes growth is predominately in the upshifted TEO, mode. It is evident from Fig. 5,where E9 vs. z is plotted at a time of t = 1.4 nsec, that an instability of a longerwavelength, equal to that of either the TEO, downshifted (A. = 3.9 cm) or absoluteinstabilities (Ag = 3.8 cm), has developed in the region z < 0.16 m. If the power of thedriver is increased to 2.5 MW, then, as is evident from Fig. 6, no significant longer-wavelength growth has appeared within 3.6 nsec. Further theoretical and numericalstudies will be needed to quantitatively understand this phenomena and to determine ifit is advantageous for CARM amplifiers to operate in this regime.

4. CARM AMPLIFIER PHASE STABILITY

An analysis of the phase stability of CARM amplifiers to perturbations in electronbeam parameters has been performed [8]. For a high peak power rf source to be attractivefor use in powering the next generation linear collider, the source must produce rf with astable, non-varying phase. Both the phase stability during the single 50 ns - 1 As rf pulse,and the shot-to-shot phase stability must meet certain requirements which depend on the

6

particular design of the linear collider. Failure of the source to deliver phase stable rfoutput usually results in an unacceptable variation of the energy of the electron beam inthe linear collider. The rf phase in a high gain amplifier will be sensitive to the variationsof the electron beam energy and current during the pulse. The typical phase stability ofconventional klystrons is approximately 8* per percent beam energy variation.

In the cyclotron autoresonance maser, the question of phase stability is complicatedby the fact that the beam transverse momentum pi. (or, correspondingly, the transversevelocity fj_) is independently variable from the beam energy. For a free-electron laser, the

beam transverse motion is set by the wiggler, such that O. = a./ y. The phase stabilityof the CARM amplifier must therefore be investigated with respect to potential variationin at least three beam parameters, energy 7, transverse momentum pj, and current I.These parameters are interrelated. For example, the current is correlated to the voltagefluctuations for space-charge limited emission from the cathode, and the beam pi will becorrelated with fluctuations in the beam energy through the wiggler which is employedto produce the pi (assuming that a wiggler is used to spin up the beam).

Phase stability studies for several CARM amplifier designs have been carried out. Forthe parameters outlined in Table 1, the net rf phase variation with fluctuations in bothbeam voltage and .# is shown in Fig. 7 for energy E = mC2 (t - 1) and 13 L, when Yand O_ are assumed to be independently variable parameters. The contours are straightlines. In these figures, the phase shift between the amplifier input phase and the amplifieroutput phase has been taken to be zero for the mean values of the design parameters.

In a real CARM amplifier which utilizes a Pierce-Wiggler for the helical beam for-mation, the wiggler produces a correlation between the beam 0j. and the instantaneousbeam energy in the wiggler. This correlation depends on how close the device is op-erating to resonance between the guide field and the wiggler field, exact resonance atthe entrance of the wiggler occurring when -y Ok .c = eBo/m. By a careful choice of thewiggler and guide field parameters, this wiggler induced correlation between fluctuationsin y and fluctuations in # can be chosen in such a way as to significantly reduce the rfphase fluctuation for variations in the electron beam energy. An example follows. Forthe parameters in Table 1, for a wiggler designed with A, = 15 cm, Bo, = 1.964kG, awiggler field strength of B,, = 35.8 G, and a length of L, = 45 cm, the wiggler inducedcorrelation between changes in beam voltage and the resulting change in O1 is shown inFig. 7 as the arc intersecting the mean value zero phase shift point of the contour plot. Asis apparent from this figure, by careful choice of the wiggler induced correlation between- and #., the rf phase stability of the CARM amplifier can be greatly enhanced. Fig. 8presents a summary of the phase variability for independent voltage and O.# variation,and for beam voltage when the beam is intentionally correlated (corrected) by properdesign of the wiggler system. Uncorrellated variations are ±8* per percent energy vari-ation, ±15* per percent variation in 0j, but only ~ ±2* per percent variation in beam

7

energy, over a narrow range of energy change (- 2.5%), for the properly designed cor-

relation between & and y. For large variations in the beam energy, the wiggler inducedcorrelation should result in the expected large variations in amplifier output phase; thewiggler induced correlation will only correct the phase for small changes about the designenergy.

This technique for local reduction of the CARM amplifier phase sensitivity to beam

energy variations could significantly improve the prospects for a high-gain CARM am-

plifier with a high degree of phase stability.

5. EXPERIMENTAL DESIGN

Experimental designs for both a long-pulse and short-pulse CARM amplifier have

been carried out. The long-pulse experiment will utilize a 0.27 pP, 700 kV electron gun,and the MIT 2.5ps (1 ps flat-top), 700 kV pulse modulator. For operation with pulse

lengths of 1 ps, the design of the CARM amplifier must optimize the device efficiencywithin the constraints imposed by the requirement that the amplifier not be susceptibleto absolute instabilities [5,6]. Because the 1 ps pulse length is significantly longer than

the typical e-folding times of the absolute instability, the design parameters of the device

must be well within the calculated stability limits.

The beam pitch angle a =_ po which corresponds to 50% and 80% of the critical

coupling for instability is shown in Fig. 9 as a function of the normalized detuning from

resonance A for three different beam voltages. This detuning is defined by

2 (1 - 01jo/,30) 01 :)2 _3; 2

The total efficiency as a function of detuning, with beam a = 6,o determined as a

function of A from Fig. 9, is shown in Fig. 10 for the case of a cold beam. The drive

power was Pd = 1 kW, and the guide radius r. = 1.11 cm for these simulations. Results

were obtained from a code which integrates a single mode version of Eqns. (1)-(4). The

device efficiency is seen to increase markedly for magnetic fields below resonance; this is

primarily because the allowable beam Opo increases substantially as the magnetic field islowered. For fixed Opo the device efficiency does not vary with field as dramatically as inFig. 10. Amplifier efficiency enhancement has been shown possible with magnetic fieldtapering [9,10]. The design parameters for a choice of 550 kV as the beam voltage are

shown in Table 2.

6. CONCLUSIONS

CARM theory and simulation have been used to study amplifier performance in over-moded systems. A multimode theory, which can predict the transverse mode excitation,

8

including launching losses, is in good agreement with multiparticle simulations. It was

shown analytically, and confirmed in the simulations, that all of the coupled modes growat the same growth rate as the dominant unstable mode, but suffer different launchinglosses, which depend upon detuning. The saturated rf power in each mode was found to

be insensitive to input rf power distribution, but sensitive to detuning.

The absolute instability can be suppressed with sufficiently large input rf power forbeam parameters which would otherwise be absolutely unstable. Further studies of this

phenomena are required to determine if high gain and suppression of the absolute insta-

bilty can be achieved simultaneously.

The sensitivity of the CARM induced phase shift in the rf signal to variations in thebeam volatage and O.L has been examined in detail. When a Pierce wiggler is used tospin-up the beam, the correlations between errors in energy and perpindicular velocitycan be made to cancel (to first order) near the operating point. For the design presentedin Sec. 4, sensitivities are, assuming no correlations between energy and 6.L, ±8* per

percent energy variation and ±15* per percent variation in #1., but only ±2* perpercent variation in beam energy, over a narrow range of energy change (~ ±2.5%),when the errors are correlated by the Pierce wiggler. Experimental design has beenpresented, and the experiments are underway.

ACKNOWLEDGEMENT

Supported by the U.S. Department of Energy, Office of Basic Energy Sciences, ContractDE-FG02-89ER14052.

9

References

[1] B.G. Danly, J.S. Wurtele, K.D. Pendergast, and R.J. Temkin. CARM driver forhigh frequency RF accelerators. In F. Bennett and J. Kopta, editors, Proceedings ofthe 1989 Particle Accelerator Conference, pages 223-225, I.E.E.E., 1989.

[2] A.W. Fliflet. Linear and nonlinear theory of the Doppler shifted cyclotron resonancemaser based in TE and TM waveguide modes. Int. J. Electron., 61:1049, 1986.

[3] C. Chen and J.S. Wurtele. Multimode interactions in cyclotron autoresonance maseramplifiers. Phys. Rev. Lett., 1990. Submitted for Publication.

[4] K.R. Chu and A.T. Lin. Gain and bandwidth of the gyro-TWT and CARM ampli-fiers. IEEE Trans. Plasma Sci., PS-16:90-104, 1988.

[5] Y.Y. Lau, K.R. Chu, L.R. Barnett, and V.L. Granatstein. Gyrotron traveling waveamplifier: I. analysis of oscillations. Int. J. Infrared and Millimeter Waves, 2:373-393, 1981.

[6] J.A. Davies. Conditions for absolute instability in cyclotron resonance maser. Phys.Fluids, B1:663-669, 1989.

[7] B. Goplin and et al. MAGIC Particle-in-Cell Code Manual. Mission ResearchCorporation, Newington, VA.

[8] W.L. Menninger, et. al, Phase Stability of CARM Amplifiers, In Preparation, 1990.

[9] K.D. Pendergast, B.G. Danly, R.J. Temkin, and J.S. Wurtele. Self-consistent sim-ulation of cyclotron autoresonance maser amplifiers. IEEE Trans. Plasma Sci.,PS-16:122-128, 1988.

[10] C. Chen and J.S. Wurtele. Efficiency enhancement in cyclotron autoresonance maseramplifiers by magnetic field tapering. Phys. Rev. A, 40:489, 1989.

10

Table 1: Average Beam Parameters for Phase Stability Study

550 kV CARM Amplifier Design

Table 2: Long-Pulse CARM Amplifier Design Parameters

11

Parameter ValueEnergy, -y mc 2 1 MeV

Current, I 3.7 kA9PO= P.IPIl 0.6

Mode TE11Frequency 17.136 GHz

r, 1.4 cmDetuning, A 0

Parameter Design ValueFrequency 17.136 GHzMode TEI1Beam Energy 550 keVBeam Current 110 Arw, 1.111 cm

34 1.127OPO 0.59BZ0 3.91 kGDetuning (A) 0.35P;. 1 kWPet 18.2 MW77T, (oap = 0) 30 %Z.at 0.83 mSaturated Gain 43 dB

1 07

3106

0

CL10 4U-

102

1010.

10

W

a-W-

105

102

101

0

0.0

100.0

100.0

200.0

Z (cm)200.0

(a) I

30(

30C

Figure 1: The rf power in the TE11 and TE12 modes if plotted as a function of the inter-action length z for (a) single-mode CARM interactions and (b) the CARM interactionwith the two coupled modes. Note in (b) that the TE12 mode grows parasitically withthe dominant unstable TE11 mode at the same spatial growth rate due to mode coupling,despite the differences in launching losses.

12

SIMULATION--- LINEAR THEORY

7

TE11

TE 12

sr

- SIMULATION (b)r--- LINEAR THEORY

TEI

TE12

f a 18GHg Boa 3.92kGy2. 9 6 9= 0.61b 50 0 A r, a2.7cm

~ S

I ~I _

0.0

.0

107

10 5

104

03

102

10

100

10 1

0. 0 50.0 100.0

Z (cm)150.0 200.0

Figure 2: The TE12 rf power is plotted as a function of the interaction length for aCARM with the TE11 and TE12 modes. Here, the two solid curves depict the linear andnonlinear evolution of rf power for the TE12 mode obtained from the simulations withtwo input rf power distributions: (a) Po(TE11 ) = 1.0 kW and PO(TE12 ) = 1.0 kW, and(b) Po(TE11 ) = 1.0 kW and PO(TE12 ) = 1.0 W, while the two dashed curves are thecorresponding analytical results from Eq. (7).

13

W3:

0-

U-

--- SIMULATION--- LINEAR THEORY

~ -

f = 18GHZE12 rw = 2.7cm

B0 = 3.92kGTE 12 y= 2 .9 6

e- 0.6 -

Ib500A

i i i

I

1.00

0

LU.0.10

-J

z0

.10.01

3.0

I It ~ I

' I

----- S --.-- ..- -- -- =

- \I--T

- V- TE

- \ / /\ --- TE1 2- I j----T E

- / I\\ --TE- \

f = 18GHzI = 5 cm y=2.96

Ib=500A ePO--0.6. I I . I I

ii -12-

1314-

4.0 5.0 6.0 7.0 8.0BO(kG)

Figure 3: The (fractional) saturated rf power in four coupled TE, modes is plotted as afunction of detuning. Here, the values of the resonating magnetic field for the TEN, TE1 2 ,TE13 , and TE14 modes correspond to BO = 3.74, 4.29, 5.33, and 6.98 kG, respectively.

14

II I'

I', ~

Li

0

0

N

£0>0

06-N

0..

00

(0

Li

0.O 0.2 0.4 0.6 0.8 1.0 1

Z(m)

0.0 0.2 0.4 0.6

Z (m)0.8 1.0

£0

2

1.2

Figure 4: (a) Plot of Eq vs. z for a CARM with parameters r, = 1.5 cm, Rb = 0.75 cm,i±/011l = 1, y = 2.967, Ib = 50 A, and an applied field of Bo = 1.375 T. This signal hasdeveloped from numerical noise and is shown at time 3.6 ns, (b) plot of Eq vs. z for thesame parameters as in (a), but at time 8.1 ns.

15

(a)

(b)

LLJ

0

0

>0

6-

0.00 0.08 0.16 0.24 0.32 0.40

Z(m)

Figure 5: Plot of Ea vs. z for a CARM with parameters r, = 1.5 cm, Rb = 0.75 cm,A11 = 1.6, y = 2.967, Ib = 500 A, and an applied field of Bo = 1.943 T. The driverpower is 25 kW. The wavelength to the right of z = 0.16 m is that of the driver, thewavelength in the launching loss region (to the left) is either that of the downshiftedconvective wave or of the absolute instability.

16

I

(O q

+0

6

0;-

7

I I A

0.16 0.24

z (m)

Figure 6: Plot of Eq vs. z for the same parameters as in Fig. 5, except that the driverpower is now 2.5 MW.

17

.0 I0.00 0.08 0.32 0.40

III

I

-3 -2

2-

E

0'i-

2 3

Figure 7: RF Phase Variation Contours in Energy - O. Plane. Straight line contoursare phase variations for uncorrelated .L and voltage V; the curved line represents thecorrelation between O. and V introduced by the wiggler.

-1 -0.5 0 0.5Percent Change

-U-Correlated-ArVotage varies

Beta-perp varies

1 1.5 2 2.5

Figure 8: CARM Amplifier Phase Stability for Uncorrelated Energy and #.3 Variationand for Wiggler-Correlated Beam

18

-1 0 1Percent Change in Beta-perp

4U,

30-

20-

10-

-10-

-20

-30-

-2.5

LLu.

-2 -1.5

40 deg

20 deg

0 deg-------------------------- ------------------------- .. -. --... -----. ...................... ....

-20 deg

-40 deg

I

TE1 1 17.136 GHz, Rw=1.111cm0.76

0-0500kV, I/IC=0. 8

*- *500kV, I/IC=0.50.70-- & -- &55kV, I/IC=0.8

A-A 550kV, I/IC=0.5

0.865 O-- 600kV, I/c=0.8 0S-U 600kV, I/Ic=0.5 0..A -.*

0.27 .. ...

0.6 - 0

0.45 -0-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

DETUNING A

Figure 9: Plot of amplifier beam pitch Opo, corresponding to 50% and 80% of criticalcoupling as a function of magnetic field, and parametrized by beam voltage, for fixedperveance.

19

T 17.136 GHz, Rw=1.111cm

50.0 O--O 500f/Ic=0.8

45.0 -.---o5001/Ic=0.540A.- 55CI/Ic=0.8

40.0 A..A55C/Ic=0.5

35.0 C.- 1 o60d/Ic=0.8. - 60[/1c=0.5

30.0 0.21

25.0

20.0

15.0

10.0

5.0

0.0-0.3 .-0.2 0 0.1

E-

rn

A

0.2 0.3 0.4 0.5 0.6 0.?

DETUNING A

Figure 10: Plot of effi; detuning for different beam voltages and for OPO values

corresponding to 504/ critical coupling.

20