competing instabilities in the circular free-electron laserraman.physics.berkeley.edu › papers ›...

Competing instabilities in the circular free-electron laser Yasushi Kawai and Hirobumi Saito The Institute of Space and Astronautical Science, Yoshinodai 3-I-1, Sagamihara, Kanagawa 229, Japan

Jonathan S. Wurtele Department of Physics and Plasma Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Received 2 November 1990; accepted 31 December 1990)

A small signal theory of the circular free-electron laser (FEL) is developed. A matrix dispersion relation, which includes coupling between the transverse magnetic (TM) and transverse electric (TE) waveguide modes, is derived from a Eulerian fluid model. The full dispersion equation is then expanded around the TM and TE mode resonant frequencies of the circular coaxial waveguide. The growth rate for frequencies near the TM mode resonance agrees with previous results obtained from a nonlinear pendulum model of the circular FEL, and becomes the negative mass growth rate as the wiggler field strength approaches to zero. It is shown that the dispersion relation expanded near the TE mode resonance has a coupling with the wiggler field that is different from the usual FEL mechanism. In the limit of a weak wiggler field, the dispersion relation for frequencies near a TE resonance reduces to that of the cyclotron maser. Numerical calculations of the growth rate and the ratio of the amplitudes of TE and TM modes are presented.

I. INTRODUCTION

The free-electron laser (FEL) has the potential to be a useful source of coherent radiation over a broad range of the electromagnetic spectrum. One substantial drawback to the FEL is its size and cost. As a consequence, much research effort has gone into developing compact free-electron lasers. Among the novel schemes under investigation is the circular FEL.

In the circular free-electron laser, electrons circulate azimuthally in the gap of a concentric coaxial waveguide, wiggling in the axial direction due to a radially directed azimuthally periodic wiggler magnetic field (see Fig. 1). The analysis of the circular FEL is complicated by the presence of the axial and wiggler fields and the coaxial waveguide geometry. In particular, the circulating beam is subject to a broadband negative mass instability as well as an FEL instability, and will couple to transverse electric (TE) and transverse magnetic (TM) waveguide modes.

Initial investigations of this device, first proposed by Be- kefi,’ assumed no drift velocity in the axial direction, a cold, thin electron layer, and coupling of the beam to a TM waveguide mode. Linear analysis’ included the FEL instability, but, since beam space-charge forces were neglected,‘missed the negative mass instability. Saito and Wurtele developed3 a nonlinear model from the single-particle pendulum equations in the applied wiggler and axial magnetic fields and self-consistent coupling to a TM waveguide mode at cutoff. This analysis3 included beam space charge and showed that the circular FEL instability is coupled with the negative mass instability of a hollow, rotating electron layer. Subse- quently, linear analysis was extended to allow for axial drift,4 and, in the limit of low gain, distributions of electron gyro- centers.” Experimental observations, which confirm the basic concept, have also been reported.6

In general, the radiation field in the coaxial waveguide of the circular FEL, even at cutoff, is a combination of TE

and TM modes, which are coupled by the particle dynamics in the applied and perturbed fields. The main purpose of this paper is to provide a simple theoretical framework to under- stand the relations between the competing instabilities that emerge in the circular FEL: ( 1) the FEL instability (a resonant instability near the TM mode frequency); (2) the negative mass instability without the wiggler;‘-” (3) a resonant instability with the wiggler near the TE mode frequency, which is discussed for the first time in this paper [see Eq. (63)]; and (4) the cyclotron maser instability without the wiggler near the TE mode frequency.’

A macroscopic Eulerian fluid formalism, similar to that used to study the cyclotron resonance maser,’ is employed to analyze the circular FEL. The dispersion relation thereby obtained contains the negative mass instability and the cyclotron maser instability in the limit of weak wiggler fields,

ROTATING ELECTRON BEAM

CONDUCTING i

WALL PERMANENT MAGNETS

FIG. 1. Schematic configuration of the circular free electron laser.

1485 Phys. Fluids B 3 (6), June 1991 0899-8221 I91 I061 485-l 1$02.00 @ 1991 American Institute of Physics 1485

Downloaded 04 Mar 2003 to 128.32.210.150. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

and the circular FEL instability for strong wiggler fields. It allows for the calculation of the relative excitation of the TE and TM modes as a function of the detuning.

In Figs. 2(a) and 2(b), we plot vacuum TM (TE) dispersion relations [ w/2rr = o(p) /277-( GHz), solid lines] and the corresponding beam modes (dashed lines). Here p(q) are the azimuthal (radial) mode numbers, and w. is the cyclotron frequency. The resonance of the electron beam and the radiation occurs near the crossing point of the beam mode and the electromagnetic wave, and, since we are con- sidering waves at cutoff in a coaxial waveguide, the mode numbers are integers. For TM waves, the resonant beam mode is upshifted by the wiggler field azimuthal mode number, iV.

The remainder of this paper is organized as follows. In Sec. II, the basic assumptions and model are presented. In Sec. III, the perturbed fluid motion is solved in terms of perturbed electromagnetic fields. In Sec. IV, the perturbed fluid motion is substituted into the perturbed Maxwell equations and the full matrix dispersion equation of the radiation fields is obtained. In Sec. V, the cubic dispersion relations near the TM and TE resonances are developed by an expansion of the full dispersion relation. In Sec. VI, numerical solutions of the dispersion relation are presented.

Il. BASIC MODEL AND ASSUMPTIONS We consider, as shown in Fig. 1, the interaction of an

electron fluid annular layer, which rotates between the concentric walls of a coaxial waveguide and wiggles axially, with the modes of the coaxial guide. In this analysis, we use cylindrical coordinates (r,B,z). The radii of the outer and the inner conducting walls are a and 6, respectively, The electron layer is assumed to be infinitely long in the axial direction, narrow in the radial direction, of uniform density, and to have no streaming velocity to thez direction. The electron beam is assumed to be sufficiently tenuous so that its self- fields in the equilibrium state can be neglected.

The beam rotation is maintained by a uniform magnetic field

B, =%,, (1) where i and B, are the unit vector in the axial direction and

t-a)

0.0 , / W=(p+N)tio

-50 -N 0 50

P

the magnitude ofthe uniform magnetic field, respectively. A periodic wiggler magnetic field, created by an assembly of magnets behind two concentric metal cylinders (see Fig. 1) I generates the wiggling motion of the beam. In free space, the wiggler field B, must have V+i, = VxB, = 0. These conditions can be exactly satisfied,’ when the wiggler is indepen- dent of z, if the field has the form

B, = - +’ COS(NB)

* 40 +8 2 sin(N@)

q;)“-‘-(+)“+‘] (.$)‘“*-“‘*” (2)

Here, N is the azimuthal &wiggler periodicity, B, is the magnitude of the wiggler, ?( 8) are the unit vector in the radial (azimuthal) direction, and ii( 6) are the outer (inner) radius of the wiggler magnets. Note that both a and ii, and b and &, need not to be equal. At r = FO= (iiN- “6 N + ‘) ‘lzN, the azimuthal component of the field vanishes. The approximate wiggler field used in this paper is

B, = - PBB,(F(Jr)sin(Ni9), (3) which is valid as long as r is close to FO. Furthermore, since the beam has no streaming velocity in the z direction, all the electromagnetic modes are assumed to be at cutoff, k, = 0.

The fields for a uacuum TM mode at cutoff are given by

SEr,, = 0, SE$$ = 0,

S&, = C,X, (k Fr)exp(icuL”t - I@@),

SB,, = (C,p/ck;f,Mr)X,(k,T,Mr)exp(iu,T,t - ipO), (4)

SB, = (C,/ic)X; (k;$“r)exp(imTrt - ip6),

m,p = 0,

where mTM = ck TM P9 , and the fields for a vacuum TE mode at cutoff arTgiven by

(b)

-50 -N 0 E i0

P

FIG. 2. Vacuum dispersion curves for (a) the vacuum TM mode (solid line) and the beam mode (dashed line) and (b) the vacuum TE mode (solid line) and the beam mode (dashed line). Here Q = 0.07 m, 6 = 0.055 m, and N = 12.

1486 Phys. Fluids B, Vol. 3, No. 6, June 1991 Kawai, Saito, and Wurtele 1486


SE,, = - (C&k~“r)ZP(k,T,Er)exp(iW,T,Et - ipf?), SE,, = iC,cZ; (k Er) exp( ioFt - ipe), SEz,, = 0, SB,,, = 0, SB,, = 0, (5)

SB,, = C2ZP (k Er) exp (itiz’,“t - ip@ ,

where 02 = ck g. Here p(q) are the azimuthal (radial) mode numbers and C, and C, are constants. The functions X,, and Z,, are defined by

Xp (k,,r) - Y, (k,&J, (k,,& - J,, (kpq4 Yp (k,,r), (6)

Z,, (k,r) z Y; (k,,~)J, (k,r) - J; (k,& Y, (k,,d, (7)

which automatically satisfy the boundary conditions at r=u

X&,,p) = 0, (8) Z;(k,,a) = 0. (9)

The wave numbers k Ty ( k zt) satisfy

XP(kLMb) = 0, (10) Z;(k;,Eb) = 0, (11)

which are equivalent to the boundary conditions at r = b,

s& = 0, (12)

(13)

wiggler fields, y = [ 1 - (v,/c)‘] - “* is 3/O if we neglect the second- and the higher-order terms if a,/y,.

B. Perturbed motion of the electron layer

In this subsection a relation between the perturbed motion of the electron layer and the perturbed electromagnetic fields will be obtained.

The equation of motion for the perturbation is (see Ref. 7 for details of the formalism)

r’, 2 (y-v) = (yo + aye + vq- (v+) = - (e/m) (E + VXB), (19)

where the overdot denotes the substantial derivative d /dt = a /at + (v. + 6v) l V and where quantities with sub- script o are unperturbed, quantities with S are perturbed, and y, v, E, and B denote the sum of perturbed plus unperturbed parts.

Inserting Eq. ( 17) for the unperturbed electron motion into Eq. ( 19)) neglecting second- and higher-order terms in the perturbed variables and in addition, second and higher terms in ufi (since a,/yg l), and with dV,,/& = w,/d and a /& = 0, yields

( &+wo-$ sv,-w,y2,sv*-rc:-----sv, > do vzo c2 r

Hereafter, when the meaning is clear, superscripts TM and TE are omitted.

III. PARTICLE MOTION A. Equilibrium particle motion

In the absence of the wiggler field, the equilibrium motion, Vo, of the concentric electron layer is a pure rotation around the z axis,

f, = (O,r~o,O), (14) where o. is the cyclotron frequency

a0 E eB,/m y,, (15) e( > 0) and m are the absolute value of the electron charge and the rest mass of electrons, respectively, and y. is defined by

yoo” [ 1 - (a,,/c)‘] -1’2, (16) where ijeo is the azimuthal component of fW

If the wiggler field is sufficiently weak, so that the ratio of the wiggling velocity to the pure rotational velocity is much smaller than unity, then the equilibrium velocity, vo, of the electron layer with a uniform axial field and a radial wiggler field, to the first order in a,/~, is (see Appendix A),

vg = (0,rw,,2”2c(u,/yo)cos(NB)), (17) where

a,. z er$,/2’/*mcN (18) is the normalized vector potential of the wiggler field evaluated at the center, r = r,, of the thin electron layer. Note that the Lorentz factor of the equilibrium motion with the

=-- m;o (SE, + veOSBz - u,SB,), (20)

= -?- (SE, + v,SB,), my0

Veo~zo -w,sv,+y2,=

C2 C2

do eBm +“/2o-p -su,+

my0

= -e (6E, - v,,SB,). my0

(21)

(22)

Terms with a /dz can be dropped since we assumed that the system is invariant in the axial direction. Note also, that, in the absence of a streaming axial velocity component, the beam should not couple to modes with k, # 0, and we do not consider this coupling here. Terms proportional to fields and the product of the unperturbed velocities and fields with the perturbed velocities have been kept. The higher-order corrections, in a,/y,, to the unperturbed electron motion yield,’ after appropriate averaging, Bessel function corrections to the FEL interaction and higher harmonic corrections. These are small corrections for small a,/~,.

Next we expand all the perturbed variables with a Four- ier decomposition in the azimuthal directions,

SF(r$,t) = 2 SF,, (r)exp(iwt - i&l). n

(23)

1487 Phys. Fluids B, Vol. 3, No. 6, June 1991 Kawai, Saito, and Wurtele 1487 Downloaded 04 Mar 2003 to 128.32.210.150. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

The equations of motion can then be simplified. Equations (20)-(22) can be solved when the mode number of S&, SE,, and SB,, which are the components of TE fields with k, = 0, isp f N, and the mode number ofSB,, &Be, and SE,, the components of TM fields with k, = 0, is p.

The perturbed velocities and the perturbed fields can be related by the matrix equation,

is1 - @Or’

- @OfM

WO iW5 ih,d - iNqfo

OOAl iflA,d + iNA,w,d is

(24)

where

cd, ziT&/r, AOEv~0/c2,

A, Ev~0iJo/C2, i& =2-‘4(a,/y,),

iI=: - (p + Nb,, f&a -pwo,

8,~ - (dmyo)(W,p+N + v80W.p+N -%&b,,,), I

I ! _ - 30 _

=

i(ti’ -@;:I W-C-0;

X2= - tdmyo)(SE,,+, +‘&.S&,),

X3= - (e/my01 (SE,,p - v~~SB, 1. (25)

In deriving the matrix Eq. (24)) we have restricted our analysis to those terms that have azimuthal mode numberp or p + N, and neglected terms with mode numbers p - N andp + 2N. The perturbed fields in Eq. (25) do not need to satisfy the vacuum waveguide Eqs. (4)-( 7) when an electron layer is present.

For a conventional linear FEL, no resonance can be achieved with the wave number (k, - k, ), where k, and k, are the wave number of the FEL radiation and the wiggler field, respectively, because the velocity of a resonant electron, up = w/(k, - k, ), exceeds the velocity of light. For the circular EEL, where (p - N) terms are somewhat analogous to the (k, - k, ) terms in a linear FEL, the velocity of a resonant electron does not exceed light velocity, as noted by previous authors.5

Here, however, terms that have mode number (p - N) are neglected because the p + N resonance corresponds to a higher frequency, and thus is of greater interest for developing compact radiation sources.

Equation (24) can be inverted to yield the perturbed velocities in terms of the unperturbed motion and the rf electromagnetic fields:

Nmotw, - A,w,M iiw-Wi) _ /T \

00 R Nib L

yo2tQ2 -&I iyo2(LP - 0; 1 iS(f?? (25)

-4)

\ N@oOo Nw,RA, A, - -

i?(fL’-a$) -7

iG(Cl”--wg) itl

Here we retained only first-order terms in u,/y,. Note that SU, and 6v, have mode number p f N while SV, has mode number p.

IV. FIELD EQUATIONS In order to obtain the dispersion relation, the Maxwell

equations must be used to find the self-consistent fields gen- erated by the perturbed currents.

Since k, = 0, the six Maxwell equations break into two uncoupled (except through the particle motion) sets of three equations for the three nonzero field components of the TM and TE waves. Equations (27) correspond to the TM mode, which has the azimuthal mode number p, and Eqs. (28) describe the TE wave, which has the azimuthal mode numberp + N:

SEzs, = WCYB,,~, -%f!- dr

= iwSB,, I

(27)

--f-dr6B r dr

rp,/, + * SBr,, = ,uoSJz., + 5 SE,, r

I

p+N W,, + N r = iio~Jr,, ,- N - w S-f%., + N, C2

1 arsEeP + N - (7v +

i(P + N) W,, + N = _ iwsB z,p + NY (28) r r

-$.W,+N = ~&Jog + N + 5 %,, + .v,

where SJ and ,u~ are the perturbed current density and the magnetic permeability in the vacuum. When coupled to the electron beam, the TM mode primarily interacts through the FEL like coupling of the electric field with the periodic u, motion from the wiggler. The resulting beam energy change (EY) has the ponderomotive mode number p -i- N. The TM mode is driven by the SJ, current. The TE mode is driven by the SJ, current, which has mode number p + N. These modes are coupled through the electron dynamics in the combined applied magnetic fields and rf waves as seen in Eq. (26).

In order to solve the coupled fluid and field equations the ratio between the geld components at the inner edge (r, )



and at the outer edge ( r2) of the beam need to be determined. These are given by the wave admittances”

d+E - P( a /a-) SE*,, 1 &, + N

PSE,, r = r, = 7 SB,,,, + N ,r ’

d-= + r(a /WSE,, 1 f%, + N =--

POE,, r = I, i SB,, + N r, and

(29)

b = -(P+N)~B,~+N . SE,,, + N +

r(a/dr)SBz,,+, ,=r, =’ @&,,N ,

b-E + (P+N)SBz,,+N Ir2 (30)

. SE,, + N

r(a/Jr)SB,+N r=r, = - ’ %,p + N I, *

In general, the wave admittances are determined by the boundary conditions at the inner and outer conducting walls. For coaxial cylindrical guides they can be solved for explicitly in terms of the Bessel function solutions for the vacuum modes. The functional form of the wave admittances is not required until the dispersion relation is solved for a particular set of system parameters. In vacuum, the dispersion relation for the TM mode is just d, + d- = 0, and the dispersion relation for the TE mode is 6, + b- = 0. Using the results of Appendix B, it is clear that the fields reduce to the vacuum fields given by Eqs. (4)-(7) in the absence of the electron layer. For our geometry, the admittances are well known; lo for completeness we sketch a derivation in Appendix B.

The boundary condition at the inner (r, ) and the outer ( r2) edges of the electron beam is that the radiation field be continuous.

The field equation can now be simplified. Integration of the third equation of Eqs. (28) from the inner boundary of the electron layer r = rr to the outer boundary of the electron layer r = rZ yields

s

II zpo (31)

r, SJ@,, + N dr i- T 2Ar6Eo,, + N (ro),

where 2Ar = r, - r,. Since SJ,,, N = 0 (see Sec. IV), the first equation of Eqs. (28) reduces

[(P+N)/rl~Bz,,+~= - CW/C’)SE~,,+N- (32) Substituting this equation into Eq. (3 1) and performing some algebra yields

s$(ro) = % j-“ cjJo,p+ N &, r,

where

b++b- + Ar(p+N) 2 r

(33)

Here

S&r) = - irf!&,p+ N(r)/@ + N) (35)

is defined. Knowledge of S4 is sufficient to calculate all the components of the TE mode.

The equations for the TM mode can be combined in a similar fashion. Substitute the first equation into the third

equation of Eqs. (27), multiply by r, and integrate from r, to r, to find

r2s&,, - r,SBo,P + ip2Ar[ 1 - ( rw/cp)2]SBr,P

s

r2 =+O rsJ, dr. (36)

rl

Since the electron layer is sufficiently tenuous so that w,, I r, =W,;p I,, zl?B, Ir,,, Eq. (36) reduces to

Sa(r,) = -7 rsJ,, dr, (37)

where

(38)

Here

Sa(r)~(c/iW)SE,,p(r), (39) is defined. All the components of the TM mode can be calculated from Sa.

Next, the right-hand side of Eqs. (33) and (37) must be evaluated in terms of the velocity perturbations, which can, in turn, be expressed in terms of the perturbed fields.

The perturbed current density within the electron layer is given by

SJ = - en,& - e&v,. (40) Combining Eq. (17) for the unperturbed velocity, and the perturbed equation of continuity,

&(n,+Sn) +V~(n,Sv+Snv,) =o, (41)

yields the perturbed density,

6n = - (d /dr) (n,rSv, ) + in,lSv,

ir(w - Iwo) 7 (42)

where the azimuthal harmonic number I is integer. The unperturbed electron number density distribution

is assumed to be a step function,

no = i ii,, rI <r<r,, 0, otherwise.

(43)

Since the density distribution is very steep, the most impor- tant term in Eq. (42) is the term proportional to dn,,/dr and the perturbed density is just

Snz - au, dn, i(w - Iwo) dr ’

(44)

The leading term in the axial and azimuthal perturbed currents is thus, after Fourier decomposition, from Eqs. (40)-(44),

sJtQ + N z (ev,,/iQ) [Ti,G(r - r,) - ii,S(r - r2)]

(45) 6Jz.p - -(eijfi/ifl)[TioS(r-rr,) -Ti,S(r- r2)]

=k,ps N’

Therefore, only the r component of the perturbed velocity is required to obtain the perturbed current in the Eulerian variable fluid model.7 Restricting to the synchronous mode,



I@- (p+J=f)wol< wo, and using Eq. (26)) SU,, + N reduces to

svr,, + N = - 1 e *o my0 (f%,, + N + &o SB,, >

(@I - APO)74 - 4

XL my0

(SE,, - %o%~ + N ) - (46)

V. DISPERSION RELATIONS A. Full matrix dispersion equation

We obtain the dispersion equation, in matrix form, by substituting Eqs. (35), (39), (45), and (46) into Eqs. (33) and (37). This yields

i -- 1 1*6&B;j+ ( - a, > 0:8&- 2L12 iI*

1 J- “; eJ=ch YO

- 21/2 - 1 - 1 ( ( - a, 2 YO a > > 42Gfi * 2 Ya

w3@12 1 -iF

(47)

where Sa = 0 (51)

f,=(p+MG/Z and

h, = (p + N) 2H /2p, (48)

0; = Eoe2 ( r, - rl 1 /~o~omro. (49)

Here Be, Q, and w, are uBo/c, the electric permittivity in the vacuum and an effective plasma frequency. The quan-

[14&~fi(1/i-P)]s#=0. (52) Equation (5 1) expresses the idea that there is no TM

wave when, in the absence of the wiggler field, and Eq. (52) is just the dispersion relation for the negative mass instability.

tity (w,/wo)‘fi~ is the ratio of the self-magnetic field to the axial uniform magnetic field.’ For the parameters of interest here, this ratio is much less than unity.

Note that this matrix has a nontrivial solution when

[w - (p + Nbo12 = ~32e[;(aw/~oYo)2~2 -f,], (50) i.e., the determinant of the dispersion matrix vanishes.

From the matrix form of the dispersion relation (47), the ratio of the TE and TM wave can, for the first time, be calculated.

Although the Eq. (47) is valid as long as 10 - (p + N)w,l (<o,, is satisfied, it is useful to expand the disper-

sion relation around mTM, the resonant frequency of the vacuum TM mode, and tirE, the resonant frequency of the vacuum TE mode. Note that these modes do not necessarily have the same value ofp.

B. The dispersion equation near the TM mode resonance

If a,. = 0, then the matrix dispersion equation (47) de- couples into two equations,

I

With w - w,F, and assuming that the electron layer is infinitely thin, the full dispersion relation can be expanded near a TM,, mode. A new matrix dispersion relation

t

i -- 1 1+w:p:fi& - 0, - 21/2 1 ( - a, > 4B;fi -L YO

1 w ck2 (3=(3~ 2x/2 ( > &tf2--

YO fl2 r 1

1-L a, ‘&f*-!& ( > 2 Yo

(53)

is obtained, where

J%= 12x,, (kpqro)* a2X;(k,,,a)’ - b2X;(kppb)2

and

hckpp - w.

&(d, fd-1

(54) k 1 = -- P q,(k~o)2

[a’X;(ka)‘- b’X;(kb)*]. (56)

(55) This matrix dispersion relation has a nontrivial solution only when

Here - Im I is the growth rate. In deriving Eq. (53)) the wave admittances d, and d- have been expanded about the

r3 + 2AoP + (Aw’ + w;B;f,)I- = #~,/~~)*u~f~w

TM resonant frequency using d, + d- = 0 for k = k $” (57) and or, equivalently, when



(w2--c2k~)~~o- (P+N)WolZ+ dPkf-,l = - (a,/y,) W .fh2 (58)

is satisfied. Here Aw= (p + N)w, - ckP7 ) the detuning parameter between the beam mode and the vacuum waveguide mode.

The left-hand side of the dispersion relation (58) is the product of the electromagnetic and the beam modes, which are coupled by the FEL interaction term on the right-hand.

side of Eq. ( 58). For fi > 0, the circular FEL has no collec- tive (Raman) regime.3 At high currents, the system is domi- nated by the negative mass instability; the latter instability remains when a,/y,, = 0.

C. The dispersion equation near the TE mode resonance

In a similar fashion, an expansion around the TE mode resonance yields

1-@2gh,&+

1 dB 2,h, s12

where

J,Z h,s

;+&w~o)2

a211 - [(p + N)/kp,a]21Zp+N (kpqa)2 - b2{l - [(p + N)/kp,b]2}Zp+N(kwb)2 ’

(59)

(60)

The derivation of Eq. (59) requires the expansion of b, and b- about the TE resonant frequency, using b, + b- = 0 for k=krnand w

X(a2 [l -(Ty]Zp+,(ka)’

-b2[l -(%)‘I Z,+,(kb)‘).

(61)

The dispersion relation becomes, for w - tiTE,

r3 + 2Aw12 + [Aa2 - ~(a,/yo)2w~~2,h2] l? = ozp2,h,o

(62) or, equivalently,

(a2 - c2k&){[u - (p + N)w,]~ - $(a,/yo)2tiiP2,h2) =-- 2w33;h,~~. (63)

Equation (63) is reduced to cyclotron maser instability when a, = 0, but can exhibit an instability due to the wiggler when a2, > 0 and h, < 0. Indeed, comparison with the TM dispersion equation (58) shows that the wiggler field term in Eq. (63) is (mathematically) analogous to the beam space- charge term in Eq. (58). Thus, the wiggler field will tend to reduce the cyclotron resonance growth rate when h, < 0.

D. Stable and unstable regions of parameter space

The stabe and unstable regions of parameter space can be quickly evaluated at zero detuning, Aw = 0. The dispersion relations (57) and (62) reduce to

r3 + w32ef,r = J(a,/yo)2df2u (64)

1491 Phys. Fluids B, Vol. 3, No. 6, June 1991

I and

r3 - ~(a,/yo)2&?~h2T = W$‘2,hp. (65)

The necessary and sufficient conditions for the growth of the instability near the TE or TM mode resonances are

D2=4U3 + 27V2>0,

with

(66)

hf -4P”e fi9 VT, =J(a,/yo)2wi f2w > 0, (67)

UT, = - t(a,/yo)2wi/? ih2, VT, =wipsh,u>O.

We plot the stable and unstable regions in Figs. 3 (a) and 3(b). In the TM mode resonant case (a,/y,,)2 CC VT, and hence the line that satisfies

&.I > 0, (68)

VT, = 0,

corresponds to the negative mass instability. In the TE mode resonant case, ( a,/yo)2 a UT, and hence the line that satisfies

VT, > 0, u,, = 0

(69)

corresponds to the cyclotron maser instability. Note from Fig. 3(b) that the cyclotron maser instability is not posi- tioned on the boundary between the stable region and the unstable regions.

VI. NUMERICAL EXAMPLES

A planned experiment ’ ’ will investigate the circular FEL with the parameters of Table I. Using the expansions of the full dispersion relation about the TM and TE resonances,

Kawai, Saito, and Wurtele 1491


is) (b)

UTM

FIG. 3. Stable and unstable regions of parameter space at (a) the TM mode resonance frequency and (b) the TE mode resonant frequency. Shaded aeas indicate unstable region. Broad lines indicate (a) the negative mass instability (NMI) and (b) cyclotron maser instability (CMI). The parameters (iand y are defined in the text.

the growth rates and composition of the unstable mode have been numerically evaluated. In the numerical work, all the growth rates have been normalized by wP4.

In Fig. 4(a), the peak of growth rate near the TM resonance is plotted as a function of rotating beam current (solid line), and the ratio

Y/r /Sal/( ISal + [SqSf2)‘i2 (70)

is given by the dashed line. The parameters are those in Table I, w/27r = 10.25 GHz, and the radial location ofthe electron beam is r. = 0.0659 m. For this experiment, the growth near the TM resonance should result primarily in a TM mode and scale like the FEL instability (Im I? -Pb3),

In Fig. 4(b), the growth rate (solid line) and ratio Y (dashed line) near the TE resonance are graphed as function of the rotating beam current (A/m) for the parameters of Fig. 4(a), except that r, = 0.0595 m and w/2n = 11.34 GHz. Near the TE resonance, the cyclotron resonance instability is enhanced by the wiggler at high current, where the resulting unstable mode has a substantial TM component.

In Fig. 5, the peak growth rate and ratio Y are plotted as a function of wiggler strength for the TM [Fig. 5(a) ] and TE mode resonance [Fig. 5 (b) 1.

As a, increases, the peak growth rate near the TM eigenmode resonance increases nearly linearly, and, except

TABLE I. Parameters of the computed models, We calculated the growth rate and the ratio Y at the frequency near the TM resonance (TMR) and theTE resonance (TER). Mode numbers in theazimuthal and radial direc- tionsare denoted byp and q, respectively. The thickness ofthe electron layer Aris supposed to be0 in our calculation. Other parameters, namely a, b, N, y,,, X, and a, are the outer and the inner radii of the coaxial waveguide, the azimuthal wiggler periodicity, the Lorentz factor ofelectrons, current in the unit axial length, and the normalized vector potential of the wiggler field.

4 Mode a(m) b(m) N p q v, (Ah) a, Ar(m)

TMR 0.070 0.055 12 3 1 3.0 200. 0.3 0.0 TER 0.070 0.055 12 3 0 3.0 zoo. 0.3 0.0


for very small values of a,, the unstable mode is TM. Near the TE resonance, increasing the wiggler field has only a small impact on the growth rate, but results a substantial TM component to the unstable mode. Note also that the magnitude of the growth is larger for the TE mode than for the TM mode.

VII. CONCLUSIONS We have used a fluid formalism to derive, for the first

time, the combined excitation of TE and TM in a circular FEL. Our main assumptions were that the fields are at cutoff and the electron layer is thin in the radial direction and has no free-streaming axial velocity component. The formalism has been used to evaluate the expected growth rates and unstable mode composition for a planned experiment.

ACKNOWLEDGMENTS This work was supported partly by the Scientific Re-

search Fund of the Japanese Ministry of Education, Science and Culture (Grant No. 01790568) (U.K.) and partly by the Institute of Space and Astronautical Science where one of us (J,W.) enjoyed a visiting research fellowship. One of the authors (Y.K.) is grateful to the Japan Society for the Promotion of Science, for financial support.

APPENDIX A: MOTION IN THE WIGGLER AND AXIAL FIELDS

A full treatment of the unperturbed motion in the wiggler and axial fields would be quite complicated, and, fortunately is not needed. For the parameters of interest here a,/y4 1, so that the change in the fluid motion when the wiggler field is added to the axial field will be small.

The correction to the pure rotating motion from the wiggler field can be found from the linearized fluid equations,



hB) Im(&)

2xlo-2 ___-___-__------ r------

TM !r

I

?r (b) I&$

I .o 2x10-*-

TE

0.5

0’ O*O 0. 0 5x10* I X103 0 5x10’ I x103

A +I

A m +I m

FIG. 4. The peak growth rate (solid line) and ratio Y [dashed line, Eel. (70) ] areplottedasafunctionofrotatingbeamcurrent~(A/m) = eii,r,(r, - rr)oo for (a) o near the TM resonance and (b) o near the TE resonance. Other parameters in (a) and (b) are those given in Table I; also, in (a) w/2s = 10.25 GHz and r,, = 0.0659 m and in (b) w/2n = 11.34 GHz and r, = 0.0595 m.

B,, as6, 2&,&&j ----= r de - e &,B,,

r my0 6fi ahO 6,, asc,

r ar -+7x= - ?- &,B,, my0

fi,, a66, IF= + d- (Sfi,B, + tie& 1, my0

(AlI

where Sf is the perturbed velocity from the pure rotating motion and &,,/& = w. under the constraint of constant energy. The equilibrium velocity with wiggler fields is obtained as

(a) In+&) 1zr

2x10-2 I-----

__---- I-

___------- L

I .o

7 ?I-

TM

vo = To + 6f. (AZ)

Using SC, Q COO in Eq. (A 1 ), we obtain the z component of the velocity

SC, = 2”%(a,/yo)cos(Ne). (A31

From the definition of the cyclotron frequency, Eq. ( 15 ) , the first equation of Eqs. (A 1) yields

as -J=SB,. de (A4)

Substitution of Eqs. (3), ( 15), and (A4) into the second

(b)

1 TE

I x lo- *\ Irn(7 ,/,,,/ /=- io.5

FIG. 5. The peak growth rate (solid line) and ratio Y [dashed line, Eq. (70) ] are plotted as a function of a,/y, for (a) w near the TM resonance and (b) o near the TE resonance. Other parameters in (a) and (b) are those given in Table I; also, in (a) w/2n = 10.25 GHz and r, = 0.0659 m and in (b) w/2n = 11.34 GHz and r,, = 0.0595 m.



equation of Eqs. (Al ) yields a second-order differential APPENDIX 6: CALCULATlONS OF THE WAVE equation for SC,, ADMITTANCE

z + &fir = (%)’ (2) cNsin(2NB). (A5)

From the boundary condition, we obtain

SC, = - 4Gy 1 ($)’ (6) sin@N@. L46)

Substitution of this equation into Eq. (A4) results in

Si& = - 4;;; 1 (2,’ (?--) cosC2NR. (A7)

Furthermore, if i?) 1 (this condition is satisfied in al- most all cases of interest here) Eqs. (A6) and (A7) become

SC,= ---$(~~(&)sin(2fVO)

and

s6,= -$-(~~(&)c0scwei.

(A81

(A9)

From Eqs. (A2), (A8), and (A9) we find that SC, is first order in (a,/y,), while Sii, and SC, are second order. The motion of the fluid must, of course, be relativistic because SU, and SE, scale as ( c/uBo ).

The assumption that &,4tr,, yields the relation that ij,/c% 2”‘a,/y, and so a,/~, must be very small compared to unity. Hence, when a,/y,& 1, second- and higher-order terms can be neglected. The unperturbed velocity is then given by Eq. (17).

d ~ _ r(a/arm,p 1 m,, f N c

P=& r = r, = -i- SB, + N r,

For the reader’s convenience we sketch the derivation of wave admittance b,, b-, d,, and d-. This derivation is similar to that given in an earlier investigation” of beam instabilities in coaxial waveguides.

Equations (27) and (28) can be reduced to two Bessel equations for E,, and Bz,p + N outside of the E layer,

I a -- ( rgW.p+N r dr ar >

-I- =,p f N = 0

and

~~(r~IsE,,)+(~-~)6E,,~=O. (B2)

In general, the solution of Bessel’s equation is the sum of the first kind of Bessel function Jl(rw/c) and the second kind of Bessel function Y, (rw/c) . Boundary conditions re- quire that

asB,, + iv =o (B3) dr

atr=aandr=b,and

SEz*, = 0 (B4) atr=aandr=b.

MakinguseoftheEqs. (Bl)-(B4),onecanobtainthe magnetic wave admittance d,, d- and the electric wave admittance b,, b- as

r u J6 [ Wc)r21 Y, [ (w/c)al - J,[ (w/c)a] Y;[ (w/c)r2] E-2 PC 6[(~/chlYpb/c)4 -J,[(w/c)a]Y,[(w/c)r,] ’

d-E + r(6’/dr)SEz,p 1 = -- mv t N

paEz,p r= I) i m,, * N I+, I,

t-0 J~[(w/c)rIIYp[(w/c)bl -J,[(w/c)b]Y;,[(W/c)r,] =--1 PC J,[(W/c)rllYp[(O/c)bl -J,[(w/c)b]Y,[(w/c)r,] ’

(B5)

b + ~ _ (p + N)SB,+N . 6E,, + N

‘(a/arm,p+N ‘Err =l sJ%,,+. r)

= _ (P+N)C J,+,[(w/c)r21Y,:+,I(w/c)al -J~+,[(U/C)a]Y,+,[(o/c)r,]

r2w J~+N[(~/~)r2]y~+~[(o/c)a] -J ~+N[(W/c)QIY~+N[(W/c)r,] ’

b- ~ + (p + N)SBZ,P + N . @-L f N r(a/ar)m,,+. rEr, = --I SE~,~+~ r,

(B6)

= + (P+N)c JP+N[(W/C)r,IY~+N[(w/c)bl -J~+N[(w/c)b]Yp+NI(W/c)rl] row J;,.,[(W/c)rllY~+N[(W/c)bl -J 6+N[(W/c)blY~+N[(O/c)rt] ’

1494 PM. Fluids B, Vol. 3, No. 6, June 1991 Kawai, Saito, and Wurtele 1494


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