IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. PS-13, NO. 6, DECEMBER 1985
Self-Consistent Kinetic Description of the FreeElectron Laser Instability in a Planar Magnetic
WigglerRONALD C. DAVIDSON AND JONATHAN S. WURTELE
Abstract-The linearized Vlasov-Maxwell equations are used to in-vestigate detailed free electron laser (FEL) stability properties for atenuous relativistic electron beam propagating in the z direction throughthe planar wiggler magnetic field B°(x) = -BK, cos koze.. Here, B,,=constant is the wiggler amplitude, and X0 = 2w/ko = constant is thewiggler wavelength. The theoretical model neglects longitudinal pertur-bations (6/ = 0) and transverse spatial variations (a/ax = 0 = alay).Moreover, the model is based on the Vlasov-Maxwell equations for theclass of self-consistent beam distribution functions of the formfb(Z, p, t) = 1ib6(px) b(PY) G(z, pz, t), where p = -ym v is the mechanicalmomentum, and Py is the canonical momentum in the y direction. Forlow or moderate electron energy, there can be a sizable modulation ofbeam equilibrium properties by the wiggler field and a concomitantcoupling of the kth Fourier component of the wave to the componentsk + 2ko, k + 4ko, * * in the matrix dispersion equation. In the diagonalapproximation, investigations of detailed stability behavior range fromthe regime of strong instability (monoenergetic electrons) to weak res-onant growth (sufficiently large energy spread). In the limit of ultra-relativistic electrons and very low beam density, the kinetic dispersionrelation is compared with the dispersion relation obtained from a linearanalysis of the conventional Compton-regime FEL equations. Finally,assuming ultrarelativistic electrons and a sufficiently broad spectrumof amplifying waves, the quasi-linear kinetic equations appropriate tothe planar wiggler configuration are presented.
I. INTRODUCTION1REE ELECTRON LASERS (FEL's) [1]-[4], as evi-l'denced by the growing theoretical [5]-[33] and exper-imental [34]-[41], [42, refs. therein], [43]-[47] literatureon this subject, can be effective sources for the generationof coherent radiation by intense electron beams. Recentexperimental investigations [44]-[47] have been very suc-cessful over a wide range of beam energy and currentranging from experiments at low energy (150-250 keV)and low current (5-45 A) [45], to moderate energy (3.4MeV) and high current (0.5 kA) [46], [47], to high energy(20 MeV) and low current (40 A) [44]. Theoretical stud-ies have included investigations of nonlinear effects [5]-[15] and saturation mechanisms, the influence of finite ge-ometry on linear stability properties [16]-[21], novel mag-netic-field geometries for radiation generation [21]-[26],
Manuscript received April 3, 1985. This work was supported in part bythe Office of Naval Research and in part by Los Alamos National Labora-tory.
R. C. Davidson is with Science Applications International Corporation,Boulder, CO 80302 and permanently with the Plasma Fusion Center, Mas-sachusetts Institute of Technology, Cambridge, MA 02139.
J. S. Wurtele is with the Plasma Fusion Center, Massachusetts Instituteof Technology, Cambridge, MA 02139.
Bo+ BB
SEe
/Y L s I N S N
Fig. 4. Planar wiggler configuration and coordinate system.
and fundamental studies of stability behavior [27]-[33].Because of the increased experimental emphasis on planarwiggler geometry [44], [46], [47], in the present analysiswe make use of the linearized Vlasov-Maxwell equationsto investigate detailed FEL stability properties for a ten-uous relativistic electron beam propagating in the z direc-tion (Fig. 1) through the constant-amplitude planar wig-gler magnetic field (1):
B0(x) = -Bu cos kozX.Here, Bw = constant is the wiggler amplitude, and Xo =2wr/ko = constant is the wiggler wavelength. The theoret-ical model neglects longitudinal perturbations (60 = 0)and transverse spatial variations (a/ax = 0 = alay), andthe beam density and current are assumed to be suffi-ciently low that equilibrium self-fields have a negligibleeffect (E° = 0 = BO). As illustrated in Fig. 1, the radia-tion field is assumed to be plane polarized with electric-and magnetic-field components 6E = 3EY(z, t) eY and6B = 6BO(z, t) ex. Moreover, the theoretical model isbased on the Vlasov-Maxwell equations for the class ofself-consistent beam distribution functions of the form [30,31, eq. (9)]
fb(Z , P , t) =nb6(Px) 6(Py) G(z9,Pz, t)
where , = -ym v is the mechanical momentum, and P,, =py- (eBwlcko) sin koz - (elc) 6Ay (z, t) is the canonicalmomentum in the y direction, which is exactly conserved.Note from (9) that the transverse motion of the beam elec-trons in the x and y directions is assumed to be "cold."The kinetic stability analysis in Sections II-V is based ona detailed investigation of the linearized Vlasov-Maxwell
DAVIDSON AND WURTELE: FEL INSTABILITY IN PLANAR MAGNETIC WIGGLER
equations for the perturbed distribution function 3G(z, pZ, t)= G(z, pz, t) - GO(z, pz) and the perturbed vector poten-tial 6A,(z, t). Although the principal emphasis is on tem-poral growth (FEL oscillator), extension of the analysis tospatial growth (FEL amplifier) is relatively straightfor-ward.As motivation for this article, we remind the reader that
conventional treatments [8], [9] of the Compton-regimefree electron laser instability for a planar magnetic wigglerare based on an analysis of the single-particle orbit equa-tions assuming a monochromatic waveform. The self-con-sistent evolution of the wave amplitude and phase are thencalculated [8], [9] from Maxwell's equations, where thewiggler-induced current is determined by a statistical av-erage over the single-particle orbits. While such an ap-proach [8], [9] has appealing features (e.g., the model isnonlinear and incorporates trapped-electron dynamics),there are also some shortcomings. For example, the anal-yses in [8] and [9] assume a monochromatic waveform forthe radiation field, ultrarelativistic electrons, and an ei-konal approximation to the wave field. Moreover, the sta-tistical averaging procedure is partially based on an intu-itive superposition of particle orbits. The present kineticanalysis, based on the Vlasov-Maxwell equations, is in-tended to investigate linear stability properties for a planarwiggler FEL from a different perspective. The outline andobjectives of the article can be summarized as follows.
a) We make use of the linearized Vlasov-Maxwellequations (Sections II and III) to provide a thorough ex-amination of FEL stability properties for perturbationsabout the general class of self-consistent beam equilibriaGO(z, pz) = U(pz) Go (yo) (see (16)). Here, U(pz) is theHeaviside step function defined by U(pz) = + 1 for pz >0, and U(p7) = 0 for pz < 0. Moreover, -yomc2 = [m2c4 +c p2 + (e2B2 Ik2) sin2koz] 1/2 is the electron energy inthe equilibrium wiggler field. The basis for performingstatistical averages is well established in the Vlasov-Max-well formalism.
b) In Section III, to evaluate the perturbed distributionfunction 6G(z, pz, t), use is made of the exact particle tra-jectories in the equilibrium wiggler field -Bk, cos koz ex.The analysis makes no a priori restriction to ultrarelativ-istic electrons. Indeed, for low or moderate electron en-ergy, there can be a sizable modulation of beam equilib-rium properties by the wiggler field and a concomitantcoupling of the kth Fourier component of the wave field tothe components k + 2ko, k + 4ko, *. v . This is evidentfrom the formal matrix dispersion equation (58) and thedefinition of electron susceptibility x(k, w, koz) in (63).
c) In the diagonal approximation, (58) reduces to thedispersion relation (77). In Section IV, we make use of(77) to investigate the detailed dependence of the FELgrowth rate on the choice of distribution functionGo (y0). Investigations of stability behavior range from theregime of strong instability (monoenergetic electrons) toweak resonant growth (sufficiently large energy spread).For the case of weak resonant growth, the growth ratesare calculated numerically for parameter regimes charac-
teristic of the Los Alamos experiment [44], and the Liv-ermore experiments planned on the Advanced Test Accel-erator (ATA) [47].
d) The limiting case of ultrarelativistic electrons andvery low beam density is considered in Section V. Wecompare the resulting kinetic dispersion relation (106) withthe dispersion relation (127) obtained from a linear anal-ysis of the conventional Compton-regime FEL equations[8], [9]. This comparison is made for a general beam equi-librium Go+(yo). Differences between the two dispersionrelations can be traced to the eikonal approximation andthe assumption of very narrow energy spread made in [8]and [9].
e) Finally, assuming ultrarelativistic electrons and a suf-ficiently broad spectrum of amplifying waves, in SectionV we summarize the quasi-linear kinetic equations appro-priate to the planar wiggler configuration considered inthe present analysis. This represents a straightforward ex-tension of the quasi-linear theory developed for the case ofa helical magnetic wiggler field [15]. The quasi-linear dis-persion relation (128), the kinetic equation (129) for thedistribution of beam electrons Go(-y0, t), and the kineticequation (131) for the wave spectral energy density Sk(t)describe the self-consistent nonlinear evolution of the beamelectrons and radiation field in circumstances where thewave autocorrelation time is short in comparison with thecharacteristic growth time (92).
II. THEORETICAL MODEL AND ASSUMPTIONSA. Theoretical Model
In the present analysis, we consider a relativistic elec-tron beam propagating in the z direction through the planarwiggler magnetic field (Fig. 1)
B°(X) = -BIv3 cos k(zx. (1)
Here, BW = constant is the wiggler amplitude, and X0 =
2w/ko is the wiggler wavelength. The electron beam is as-sumed to have uniform cross section, and the beam den-sity and current are assumed to be sufficiently small thatthe effects of equilibrium self-electric and self-magneticfields can be neglected. Moreover, for a tenuous electronbeam, the analysis is carried out in the Compton regime;thus longitudinal perturbations are neglected (6/ = 0).We consider transverse electromagnetic fields with one-
dimensional spatial variations, where a/ax = 0 = alay,and aIaz is generally nonzero. Introducing the perturbedvector potential
6A(Lx, t) = 6Av(Z, t) e ,1 (2)the electromagnetic field perturbations 6&(x, t) and6B(x, t) can be expressed in the Coulomb gauge as
,T t)- -c dt A(x, t) -c-6A (z, t) yc at c at
aB(,t) =V x 6A(xc, t) = - 6AN(z, t) ~. (3)
465
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. PS-13 NO. 6, DECEMBER 19QS
There are two exact single-particle constants of the motionin the combined equilibrium and perturbed field configu-ration described by (1) and (3). These are the mechanicalmomentum p, and the canonical momentum PY transverseto the beam propagation direction. Here, P1. is defined by
=pe A z) _- 6A,, (z, t)P,= C C ,(z (4)
where -e is the electron charge, c is the speed of light invacuo, and A O,W(z) is the vector potential for the equilib-rium wiggler field in (1), i.e.,
A °0 (z) - sin koz. (5)-ko
In general, the beam distribution function fb(z, p, t)evolves according to the nonlinear Vlasov equation [30],[31]
+ u - e bE++ X(B°+6B)
aP fb(z, p, t) =O (6ar
The particle velocity v and momentum p are related by
In (10), 'YT(Z, pz, t) Mc2 is the particle energy evaluatedforp -= 0 and P, = p1 - (elc) (A°.,,, + 6A,) = 0:
YT(Z, pz t) I + 2 + 2 [A (z)+ 6A1 (z,m2c2 m2C4 t]
(11)
and vz is the axial velocity defined by vz = a(YTmC2)/ap,i.e.,
v7 = pz'YTM
(12)
Moreover, substituting (9) into (8), the Maxwell equationfor 6A y (z, t) becomes [30], [31 ]
i2 a2\ r12(c2at2adz2)b6Ay(z, t) = - 2 (Ay° + 6A1)
- A~ K dp-Goy1" 'Yo
K dpz'YT
(13)
A12,)where G(z, pz, t) evolves according to (10), @2 =
) 47rnbe2/m is the nonrelativistic plasma frequency squared,and yo(z, pz) is defined by (see (11) with 6A, = 0)
p=(1 + p2/M2C2)1/2 (7)
where m is the electron rest mass. In (6), the field polar-ization is prescribed by (3) and bAy(z, t) is determinedself-consistently from the Maxwell equation
(c :j2) bAy (z, t) = - e K d3pvy [fb(z, p, t)
fOb(Z, P)] (8)
Here, f2(z, p) is the equilibrium distribution function(a/at = 0) in the absence of perturbed fields (6A5 = 0).
B. Nonlinear Vlasov-Maxwell DescriptionIn the present analysis, we consider the class of exact
solutions to the nonlinear Vlasov equation (6) of the form
fb(Z, p, t) = nbb(Px) 6(Py) G(Z, pz v t) (9)
where P1, is defined in (4), nb = constant is the ambientelectron density, and G(z, pz, t) is the one-dimensional dis-tribution function in the phase space (z, pz). In (9), theeffective transverse motion of the beam electrons is"cold." Making use of the fact that Px and Py are exactconstants of the motion in the combined equilibrium andperturbed fields (see (1) and' (3)), we substitute (9) into(6) and integrate overpx and py. This readily gives for thenonlinear evolution of the one-dimensional distributionfunction G(z, pz, t) [30], [31]
{ + vz - Mc2 (8- YT) a j G(z, pz, t) = 0 (10)
ayo(z, pZ) = 1 + 22 +e B sin2 koz) (14)
In (13), Go(z, pz) is the beam equilibrium distribution thatsatisfies the steady-state Vlasov equation (10) with a/at=0 and bAy = 0. That is, GO(z, pz) solves
|vz a -Mc (+ Yo) a Go(z, pz) = 0 (15)
where -yo(z, pz) is defined in (14), and vz is defined byvz = a(,yOmc2)Iapz = pz/yOm.
C. Beam Equilibrium PropertiesAny distribution function GO(z, pz) that is a function of
the single-particle constants of the motion in the equilib-rium field configuration described by (1) is a solution tothe steady-state Vlasov equation (15). Unlike the case ofa helical wiggler [31], the axial momentum pz is not anexact invariant in the planar wiggler described by (1).However, the particle energy -yomc2 defined in (14) is anexact invariant in the equilibrium field configuration.Therefore, in the present analysis, we consider the classof equilibrium solutions to (15) where the particles aremoving in the positive z direction (pz > 0) and GO(z, p.)has the general form
GO(z, pz) = U(pz) Go(yo). (16)
Here, U(pz) is the Heaviside step function defined by
(+1I PZ > OU(pt) >0
0 PZ< 0.
(17)
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DAVIDSON AND WURTELE: FEL INSTABILITY IN PLANAR MAGNETIC WIGGLER
It is assumed that none of the electrons is "trapped" bythe equilibrium wiggler field. That is, the form ofGo(Qyo) in (16) is such that
2 > 1 + a2 (18)
where aw is defined by
eB,a = (19)
mc2koThe choice of Go ('y0) in (16) and in the stability formalismdeveloped in Sections II and III is quite general.To illustrate the spatial modulation (in z) of beam equi-
librium properties by the wiggler field, we consider theexample of a monoenergetic electron beam where
G (2 1)I2
GoA() ^ b(yo - "Y),ymc(20)
and the constant 5 satisfies A2 > 1 + a2. Making use ofdp7 = (Yomi2c2Ip:) d-yo, the equilibrium electron densitynb(z) =nb i dpzGO(z, P) can be expressed as
n°(z) = nb (52 ^1)/2 y d -5Pz6(_A)
= nbiC (2 7 ) djyo 6eyo- 5) (21)
where use has been made of (16) and (20). From (14) wesubstitute pz = +mc[y- 1 - a2 sin2 koZ] 2 in the in-tegrand in (21) and obtain
nb(Z) = 2 sin2 koZ) 12 (22)^2 < ~ 1 k 0z"
where K2 < 1 is defined by
2 a2K _ (23)-y - 1
Depending on the size of K2 we note from (22) that therecan be a substantial modulation of the equilibrium beamdensity by the wiggler field. For example, if ko = 1 cm 15y = 3, and Bw = 1.7 kG, then a, = 1 and K2 = 8, andthe peak-to-minimum density modulation in (22) is about6 percent.
Other equilibrium properties calculated from (16) and(20) are also modulated as a function of z. For example,the average beam velocity in the z direction is defined byVzob(z) = [l dpz (pzlyom) Go]/(S dpzGo). Following the pro-cedure used in the previous paragraph, it is readily shownfrom (20) that
Vb(z) = Vb(l - K2 sin2 k0z)112 (24)
where Vb = c(52-1)l/2/5. Combining (22) and (24), itfollows that no(z) V%(Z)) = lbVb = constant (independentof z). This corresponds to a constant flux of electrons,which is expected from the continuity equation understeady-state conditions.
D. Linearized Vlasov-Maxwell EquationsWe now make use of (10) and (13) to derive the linear-
ized Vlasov-Maxwell equations for small-amplitude per-turbations 6G(z, pz, t) and 6A, (z, t) about the beam equi-librium described by (16) for general choice of Go (y0). Inthis regard, it is useful to introduce the normalized per-turbed vector potential ay (z, t) defined by
ay(z, t) = in2 bAy(z, t)
and to express (11) in the equivalent form
r(Z, pz, t) = L1 + 2PZ + a , sin2 koz+ 1/2
+asin kozay(z, t) + a2(Z, t)
(25)
(26)where aw = eB/imc2ko.For small-amplitude perturba-tions, (26) can be expanded to give the approximate re-sults
aw sin kozYT= + a,(z, t)
Yo
1 1 a, sin koz- =-- av(z, t)
'YT 70 y0where y0(z pz) is defined by
yo-(1 +P<2 + a2 sin2 koz)
(27)
(28)
Equation (27) is valid to leading order in the perturbedvector potential ay(z, t), assuming 2|a,,,ay I<< y 2We now express the distribution function G(z, pz, t) as
its equilibrium value plus a perturbationG(z, pz, t) = Go(z, p) + 6G(z, pz, t) (29)
and make use of (27) and (29) to simplify (10) and (13).Retaining only the linear terms proportional to 6G(z, pz, t)and ay(z, t), the Vlasov equation (10) gives
+ _PZ a - Mc2 aO a} 3G(z, pZ, t)-YOM azapPz a- PZ a,,sin koza(z,t)- GO(z pz)
+ mc2aw 9 (s a (z, t)) a GO(z, pz) (30)
which describes the evolution of the perturbed distributionfunction. Making use of (15) to simplify the right-handside of (30), the linearized Vlasov equation can be ex-pressed as
IEEE TRANSACTIONS ON PLASMA SCIENCE. VOL. PS-13, NO. 6, DECEMBER 1985
where 'yo(z, pz) is defined in (28), and GO(z, pz) has thegeneral form in (16). Finally, the perturbed vector poten-tial a, (z, t) evolves according to (13). Linearizing (13) andmaking use of (27) and (29), we obtain
1 a2 a2 ~2 dpz-I+ Go(z, pz)
at az C\nv -P G0(z, Pz))3 ay(Z, t)-2ataZ si2 k0 'dYo
2a sin dpz=- 2a sin koz J-G(z, pz,t) (32)
where W2 = 4wrnbe2/m, and 6G(z, pz, t) evolves accordingto (31).
Equations (31) and (32) are the final versions of the lin-earized Vlasov-Maxwell equations used in the formal sta-bility analysis in Section III. Keep in mind that (31) and(32) are valid for small-amplitude perturbations about thegeneral class of spatially modulated beam equilibria GO(z,pz) = U(p) G +(,yo) (see (16)). No a priori restriction hasbeen made to a specific choice of Go+(yo), nor has K2a2/(_y2 - 1) << 1 been assumed.
III. DERIVATION OF THE GENERAL EIGENVALUEEQUATION
In this section, we make use of the linearized Vlasov-Maxwell equations (31) and (32) to investigate FEL sta-bility properties. First a formal solution for 6G(z, pz, t) isobtained from (31) using the method of characteristics(Section III-A), and then the particle orbits are calculatedin the equilibrium field configuration (Section III-B). Fol-lowing a derivation of the exact eigenvalue equation foray(z, t) (Section III-C), we then simplify the eigenvalueequation in the diagonal approximation (Section III-D),where the coupling of the kth Fourier component of ay tothe k + 2ko, k + 4ko, * - * components is neglected.
A. Solution for bG by the Method of CharacteristicsThe formal solution for 6G(z, pz, t)can be obtained from
the linearized Vlasov equation (31) by using the methodof characteristics. For the case of temporal growth (single-pass FEL oscillator), the solution to (31) is given by
where the initial value (at t' = -oo) has been neglected.Here, z'(t') and p'(t') = yO'mdz'/dt' are the phase-space trajectories in the equilibrium wiggler field-B., cos koz'e .. Since 'y6 = 'yo = constant (independentof t '), the axial orbit z '(t') satisfies
dz = +c(Qy - 1 - a2 sin2 koZ)"12 (34)
where p' > 0 is assumed, and use has been made of (28).In (33) and (34), the boundary conditions on z'(t') andp'(t') are z'(t' = t) = z and p'(t' = t) = pz = oyomv-.That is, the phase-space trajectory (z', p') passes through(z, pz) at time t' = t.We examine (33) forpz > 0 and make use of GO(z, pz) =
U(pz) Go y0) (see (16)). It readily follows that
-, Go (^y) = mc (35
in the integrand in (33). Since zy = yo = constant alongan equilibrium trajectory, the factor aG +/ayo can be takenoutside of the t' integral in (33). This gives
G(z, pz, t)= wYo dtY v''Yo deyo -XOC
a(36)
forp > 0. We further simplify (36) by making use of
d- [sin k0z'ay (Z', t'1)]
=-(t' + vz ,I, [sin k0z' ay(z', t')] (37)
where dldt' is the time derivative along an equilibrium or-bit. Substituting (37) into (36) and integrating by partswith respect to t', we find (forp > 0)
6G(z, p~, t) = aW sin koz ay(z, t) adGoYo aYo
aw aGo v
'Yo aYoa
dt' sin koz ' ~- ay(z'f, t'1)
(38)where use has been made of z'(t' = t) = z, anday(z', t' -+ -oo) = 0 is assumed.
In Section III-C, the formal solution for aG(z, pz, t) in(38) is substituted into the linearized Maxwell equation(32), and properties of the resulting eigenvalue equationfor ay(z, t) are investigated. Although the principal em-phasis in the present analysis is on temporal growth (FELoscillator), for future reference we conclude this sectionby stating the generalization of (38) to the case of spatialgrowth (FEL amplifier). Some straightforward algebragives (forpz > 0)
aw sin koz ay(z, t) adGo'Yo a-Yo
_aw(Go (,Z dz' sin koz' ay(z t
(39)where ay(z' -oo, t') = 0 is assumed, and t'(z') is theinverse solution of (34) with boundary conditionst = z) = t and v'(z' = z) = vz.
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DAVIDSON AND WURTELE: FEL INSTABILITY IN PLANAR MAGNETIC WIGGLER
B. Particle Orbits in the Equilibrium Wiggler FieldThe orbit integral on the right-hand side of (38) requires
a determination of the particle trajectory z '(t') in the equi-librium wiggler field B°( x) = -B cos k0z e Defining
X (1 -~~ 1/2=0 - )12(40)
anda2K2 = (41)
the equation of motion (34) can be expressed as
d (koz') = ko3oc[1 -K 2 sin2 k0z ]i 1/2.dt I
The solution to (42) can be expressed in terms of thliptic integral of the first kind
r d7q'F(7, K) =J [1 - K2 sin2 l]I/2In this regard, we introduce the shorthand notation
F = F(7rI2, K)
F' = F[irI2, (1 - K2)1/2]
F, = F(koz, K).
(42)
e el-
Integrating (42) from t ' = t to time t' gives
F(koz ', K) - F(koz, K) = f30Cko(t' - t) (45)where z'(t' = t) = z. Moreover, (45) can be inverted togive the explicit solution for z '(t'). We find [48]
Z'(tW) = Z + /FC(t - t)oo
+ Zzjsin 2n(4z + OF3ckoTr) - sin 2noz]nI
(46)Here, T = t' - t, and the phase kz and average speed Fare defined by [48]
Z= - Fz
7=
Moreover, the oscillation amplitude z, in (46) is definedby [48]
assumption has been made that K2 = a2JI(y 0 - 1) << 1in deriving (46) from (42). That is, depending on the sizeof K2, the oscillatory modulation of the axial orbit in (46)can be strong.
In the special limiting case where K2 < < 1, the oscil-latory modulation in (46) is weak, and the various ellipticintegral factors defined in (44) and (47)-(49) can be ap-proximated by [48]
2F/ r = 1 + K2/4F' = ln (4/K)F2= (1 + K2/4) koz
z = k0za = K2/16
nk0 (16) (50)
(43) when K2 << 1. Of course, (50) leads to a correspondingsimplification in the expression for z '(t') in (46). In par-ticular, when K2 << 1, (50) can be approximated by
z '(t') = z + 3FC(t - t)00
+ E zj[sin 2n(koz + fFckoT) - sin 2nk0z](44) n= I
/C 1 \(m1)where SF = (1 - K2/4) So and Zn = K218kon. With n = 1,(51) is the familiar approximate expression for longitudi-nal motion in a planar wiggler.
C. General Eigenvalue Equation for ay/z. t)We now examine the linearized Maxwell equation (32)
for the case of temporal growth (FEL oscillator). Substi-tuting (39) for 6G(z, pz, t) into (32) gives the equation foray(z, t):
( l 82 a2 2 N
-dtd2+ - S(koz)j ay(z, t)C2a2 aZ2 C2o , 2+ E a sin 2oz+
C o 2 a-o
X dt' sin koz ' ,- a, (z', t ) (52)
2 1 anZn = _
2nkon 1 + a
where
a = exp (-irF'IF).
(48)
(49)Equation (46) is a very useful representation of z '(t') forthe subsequent simplification of the orbit integral (38) inSection III-C and Section IV. In this regard, no a priori
where Ct)p = 4vnbe2Im Z'(t') is the axial orbit defined in(46) for K2 < 1, and S(koz) is the spatially modulated formfunction defined by
S(koz) = z0 G[G+(7o) - a2 sin2 koz0 'Yo
(53)
While the formal stability analysis in Sections III-C andIII-D is presented for general G +(yo), for future reference
469
(47)
G + (70) I aG + (yo)x 0 02
ly 0 -yo a-yo
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. PS-13, NO. 6, DECEMBER 1985
we state here the explicit functional form obtained forS(koz) for the special case where Go (yo) corresponds tomonoenergetic electrons. Substituting (20) into (53) andcarrying out the integration over pz gives
1 1S(kz) = (1 - k2 sin2 k0Z)3/2
x(k, , koz) =
(54)
where K2 = a2w(-y2 - 1), and use has been made ofdpz = (Yom2c2/pz) dyo (see Section 1I-C). As expected,the strength of the spatial modulation of S(koz) depends on
the size of K.Equation (52) is analyzed using a normal-mode ap-
proach, where ay (z, t) is assumed to be of the form [31]
ay(z, t) = aiy(z) exp (-iwt), Im w > 0. (55)
Substituting (55) into (52) gives the eigenvalue equationfor ay (z):
tj2 + C a2 2_uS(kOz)3 y (z)
=icoia2 sin koPz °d G dt'o PyO0V 00
x exp [ - ico(t' - t)] sin koz' ay (z') (56)
where z '(t') is defined in (46). In general, (56) should besolved numerically for the eigenfunctions aiy(z) and eigen-frequency w. For present purposes, it is useful to representay(z) as the Fourier series
(z)= E ayk exp (ikz)k
(57)
where k = 27rnIL, n is an integer, L is the periodicitylength in the z direction, and the summation extends fromn = -oo to n = +oo. Substituting (57) into (56), we
obtainZ {@2- c2k2 -2S(k0Z)k
-WPX(k, o, koz)} ayk exp (ikz) = 0. (58)
Here, X(k, w, koz) is the dimensionless wiggler-inducedsusceptibility defined by
where fF, z, and Zn are defined in (47) and (48). To sim-plify the exponential factors
00
exp { }n= J
in (60), we make use of the identity00
exp (ib sin ae) = E Jm(b) exp (imce) (61)m = -oo
where Jm(b) is the Bessel function of the first kind of orderm. Defining
k±+ko02 anb1+ = (k + ko) Zn kn n1+a2n (62)
where a is defined in (49), the expression for the suscep-tibility X(k, c, koz) in (60) can be expressed in the equiv-alent form
x(k, c, koz) =
(59)
where r = t' - t, and the axial orbit z'(t') is defined in
(46). Substituting (46) into (59) readily gives
1iwa
dpz aGo 0
- 4 J I2d Tx ([exp (2ikoz) - 1]
x exp {-i[co- (k + ko) OFC] T
00 00 00
. U E E Jm(b+) Jmn(bQ)n-= Lm--oo m=' 0-oo
x exp [im(2nkOFCcr)]
x exp [i(m - mi') 2nIz]j+ [exp (-2ikoz) - 1]
x exp {-i[k - (k - ko) ORC] 4
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00
x ln = I
00 00
Li Jm(bn) Jm'(bn)m=-00M=-00
x exp [im(2nkOOFlcr)]
x exp [i(m- i') 2n4oij). (63)
The expression for X(k, w, koz) in (63) can be further sim-plified, depending on the parameter regime and frequencyrange under investigation (Section IV).To summarize, (58) and (63) are the final results of this
section, and are fully equivalent to the eigenvalue equation(56) for ay(z). In this regard, several points are note-worthy. First, in the limit of zero wiggler amplitude, (58)and (63) give the familiar dispersion relation w 2 -
c2k2 + cacx for electromagnetic waves propagating in thez direction (Here, a, = o dpzG ('yO)/IyO follows from (53)for a2 = 0.) Second, the susceptibility x(k, c, koz) de-fined in (63) depends on kcz. This spatial modulation oc-curs through the factors exp (±2ikoz), through the depen-dence of fz on koz (see (47)), and through the integrationoverp, in (63) (see also Section Il-C). As a consequence,the kth Fourier component wave amplitude aiyk in (58) isgenerally coupled to the wave components ay,k+ 2ko'aY, k+4k0, etc. Third, in deriving (58) and (63), no a prioriassumption has been made that the spatial modulation (kozdependence) of S(koz) and X(k,w, koz) is weak or that theparameter K2 = a2I/(&yo - 1) is small. Finally, (58) and(63) have been derived for perturbations about the generalbeam equilibrium GO(z, pz) = U(pz) GoQ(0), and the for-malism can be used to investigate detailed FEL stabilityproperties over a wide range of system parameters con-sistent with the assumptions and theoretical model de-scribed in Section II.
D. Diagonal Approximation to the Dispersion RelationThe simplest approximation to (58) is where we retain
diagonal terms and neglect the coupling of ayk to the k +2ko, k + 4ko, * * * Fourier components. In this regard, thequantities S (koz) and x (k, co, koz) can formally be ex-pressed as average values plus terms that depend explicitlyon koz. That is
S(koz) = (S> + E SI exp (il2koz)1*0
x(k, c, koz) = <X> (k, c) + E Xl exp (il2koz) (64)1.0
where the average values < S > and < X > (k, c) are definedby
(2r d(koz)<S> = 3 2 S(koz)
<x> (k, c) = | 27r X(k, co koz). (65)
D(k, ) = 2 _ c2k2 _ A2<S> _ 2<X) (k, ) 0.
(66)
In (66), the average quantities <S> and <X> (k, w) arecalculated from (65), making use of the definitions ofS(koz) and x (k, c, koz) given in (53) and (63) for generalG (yo).The diagonal dispersion relation in (66) is used in Sec-
tion IV to investigate FEL stability properties over a widerange of system parameters. It is important to emphasizethat neglecting the coupling to off-diagonal terms in (58)is likely to be a good approximation insofar as the param-eter K2 = a2/(_y2 - 1) is sufficiently small.
IV. FREE ELECTRON LASER STABILITY PROPERTIESIn this section, we make use of (53), (63), (65), and the
diagonal dispersion relation (66) to investigate detailedFEL stability properties.
A. Simplified Dispersion RelationFor present purposes, two main approximations are
made in evaluating (X> (k, o) from (63) and (65). First,it is assumed that K2 = a I(y2- 1) is sufficiently smallthat q5 can be approximated by
z = koz (67)
in the expression for x (k, c, koz) in (63). Referringto Section Ill-B and (47) and (50), it is evident thatK2/4 << 1 is the appropriate small parameter for validityof (67). Second, the r dependence in the integrand in (63)is generally of the form
exp {-i [ -(k + ko) IFC- m(2nko) OFFC] 4 (68)
and
exp {-i[co- (k - ko) /FC- m(2nko) OFC] 74. (69)
In the subsequent stability analysis, we retain contribu-tions to the r integral in (63) that exhibits resonant behav-ior at the simple upshifted FEL resonance [8], [9]:
C = (k + kO) [3FC. (70)
That is, in contributions to (63) associated with the factorin (68), we retain only the m = 0 term, and in contribu-tions to (63) associated with the factor in (69), we retainonly the m = 1, n = 1 term.Making use of (63) and (65) and the assumptions in the
preceding paragraph, the susceptibility (x> (k, co) can beexpressed as
(X> (k, v) - - 1 icoa X d(koz) o dpz aG+4 o2e r o 0 ( 0
x i d7- exp {-[o(k + ko) O3FC 7-1-00
Substituting (64) and (65) into (58) and retaining only thediagonal terms gives the dispersion relation
x ([exp (2ikoz) - 1]
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00 00
x U E JO(b+) Jm, (b )nf= m =-00
x exp [-im'(2n) k0z] + [exp (-2ikoz) - 1]00
XJ,(b-) E Jm,(b-)
x exp [i(1 - i') 2koz])
(71)
We carry out the T integration in (71) and average over the(fast) koz oscillations in the integrand. For example, thefirst term in the factor [exp (2ikoz) - 1] combines withthe m' = 1, n = 1 term to give a nonzero average value,whereas the -1 term in the factor [exp (2ikoz) - 1] com-bines with the m' = 0 term to give a nonzero averagevalue. After some straightforward algebra, we obtain
-1 2a d(koz) , dpzKX4> (k, o) 22 o 24 Jo 2r Jo Yo
x G0 _Y K(y)co- (k + kO) I3FC K~
where K(Qyo) is defined by00
K(yo) = IT J2(b+) - Jo(bt) J1(bt)n = 1
-JO(b- ) J1(b-) + J'(b-).
(72)
(73)Here, for small values of a= K2/16, it follows from (50)and (62) that b+ can be approximated by
n k±k 2 K1K2n ko ~n 1 (74)
0 = D(k, w) = - c2k2 - C?, KS) + COPa t!
2x d (koz) i dpz aGo /ayo Ko 2 7r o 70 w - (k + ko) OSFC (77)
which is the final form of the dispersion relation used inthe remainder of this paper. Here, K(Q0) is defined in (75)with bt [(k + ko)!ko] K2/8, and SF is defined by F7r30I2F (1 K2/4) $0 where 00(1 - 1/y2)'/2 (see(47) and (50)).
B. Resonant Free Electron Laser InstabilityThe dispersion relation (77) can be used to investigate
detailed FEL stability properties over a wide range of sys-tem parameters when K2 = aWIQ(yg - 1) is sufficientlysmall. In this section, we calculate the growth rate'Yk = Im w in circumstances corresponding to weak reso-nant instability. In particular, it is assumed that the growthrate is sufficiently small and the energy spread of the beamelectrons is sufficiently large that the inequality
(k + ko) Auvz(78)
is satisfied. Here, A vz is the axial velocity spread char-acteristic of G (-y0) over the range of unstable phase ve-locities. Of course, A vz is also related to the beam emit-tance. In (77) we express co = Wk + i'Yk and expand forsmall growth rate Wk. This gives
0 = D(k, C0k + iYk) = Dr(k, Wk)
+ iL Di(k, ak)+ Dr(k, ck) +CO+Ck , k +k (79)
where
For the range of k values of interest for the FEL instability,(k + kO)1kO >> 1, b- is typically of order unity, andb+ << I for n 2 2. Therefore an excellent approximationto (73) is given by
Moreover, ifwe approximate bj b bI(klko) (K218)for k >> ko, then (75) reduces to the familiar factor
K(yo) = [Jo(b1) - J1(b1)]2, which occurs in standard sin-
gle-particle analyses [8] of the FEL instability in theCompton regime. As a further point, it should be notedthat we have retained the spatial integral I 2 d(koz)/27r ...
in the expression for (X > (k, c) in (72). This averages over
the (weak) dependence on koz of the momentum integral00 dpz00 dyoMd00P deyo mc
oY 1 yOQY2 - 1)1/2 (1 - K2 sin2 koz)12
(76)that occurs in (72).
Substituting (72) into (66) gives
Dr(k, Ok) - lim Re D(k, Wk + iTk)0k° +
and
Di(k, Wk) = lim Im D(k, Uk + iyk)~q--.0 +
Making use of
1 Plimlikm+ k (k + ko)tFC + iYk Wk (k + kO)fFC
- i76[ik - (k + ko) OFC]where P denotes the Cauchy principal value, we set thereal and imaginary parts of (79) separately equal to zero.This readily gives
0 = Dr(k, Wk) = COk - Ck -Up + 4 wpae,2ak
X 2 d(koz) 00 dpz PaGo/aKyoo 2ir o o k - (k + ko) OFC
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andA2 2Di(k, w) _ U)paw4k
X, = - -aDrIaWk 4aDrIaWk
'2 d(koz) dp z aG__Jo 2 0K(oyo)
X 6[(k - (k + ko) OFdC (81)
where aDrlawk denotes aDr(k, Wk)IaWk. Equation (80) isthe dispersion relation that determines the real oscillationfrequency Re X = ok, whereas (81) determines the growthrate Im w = 'Yk for specified beam distribution functionG (yo).Note from (81) that the instability is driven by resonant
electrons with velocity
OFC = (k + k0) (82)
Here, for small K2, /3F is defined in terms of yo by /3F =2(l - K2/2), where 2 = 1 - 1I'y2. That is, f3 can be
expressed as
/321 I+ a 2/2F =_2 1+ aW2 (83)'o
We denote by yo = 'Yr the resonant energy where O3F('Yr)C = Wkl(k + ko). Making use of (83) then gives for 'r
r= 21c2(k+ ko)2) (84)
To simplify (81), we convert the p, integral to an integralover yo as in (75) and make use of the identity
where use has been made of (83) and (84). Moreover, intypical parameter regimes of interest, the principal-valueterm in (80) makes a negligibly small contribution toaDr/aCk, and it is valid to approximate aDrIacok = 2Wk.Carrying out the integration over koz in (81) for small K2,and making use of (85), we obtain after some straightfor-ward algebra
^2 a2-k =Im @ = 8 1k + kol c (1 + a2w2)
FaG1I+]X K(Yr) 'Yrmc 0 ° (86)
_a'y° -oY =Yr'
Here, the resonant energy 'Yr is defined in (84), and K('yo)is defined in (74).The expression for the growth rate in (86) has a wide
range of validity, subject to the inequality in (78). Notefrom (86) that instability exists (Yk > 0) over the entire
INSTABILITYFOR aG+l/d o
I G+(y)
G O0
|~~~~~~r
I ~~~~~~~~~Ay
Fig. 2. Schematic of Go(-yo) versus -y0. The region of positive slope with[aGo /dyej0=,Y, > 0 corresponds to instability in (86).
range of 'y for which aGo laYO0 =Yr > 0 (Fig. 2). Thecorresponding real oscillation frequency Wk of course isdetermined self-consistently from (80).We now make use of (78) to determine the range of
validity of (86). From (83), the characteristic velocityspread A v, is related to the characteristic energy spreadA'y by /3FAvZIC = (1 + a242) A'y/y'y. For O3F / and'y= , where
A mc2 and jc are the mean energy and meanaxial velocity, respectively, of the beam electrons, we ob-tain the estimate for A vz:
(1 + a' /2) A'Avz C A2 A (87)
Moreover, for C2 << c2k2, the characteristic wavenumberof the instability (denote by k) can be estimated from thesimultaneous solution to ck = kc (see (80)) and Wk =(k + ko) fc. This gives the familiar result
A A AA 2
A 0(I + 0 0(1 + 0)k= (1_ )-ko = + ki
'2)22 (88)
where use has been made of (83). Finally, if we furtherestimate aG1+7a'yjO= Y lImc((A'y)2 in (86), then the in-equality 'ykl(k + ko) A v2j << 1 in (78) can be expressedin the equivalent form
Equation (89) is equivalent to (78) and can be satisfied byrelatively modest values of fractional energy spread A'ybjK
It should also be noted that the instability bandwidth A kis readily estimated from the simultaneous resonance con-ditions kc = Uk = (k + ko) OF. This gives
(Ak) (1 - /) = (k + ko) Avzlc (90)where we have approximated F /3 and k = k (see (88)).Making use of (87), (88), and (90), the normalized band-width A k/k can be expressed as
Ak I A'yk 0 eY (91)
Equation (91) gives a simple estimate of A k/k in terms ofthe fractional energy spread A'/A.
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. PS-13, NO. 6, DECEMBER 1985
To summarize, the expression for weak resonant growthrate in (86) is valid within the context of (89). In SectionIV-C, we make use of (86) to investigate numerically thelinear growth properties in parameter regimes character-istic of the Los Alamos FEL experiment [11], [44], andthe Livermore FEL experiment planned on the AdvancedTest Accelerator (ATA) [47].
In conclusion, the analysis in this section also has fun-damental implications for the range of validity of differentnonlinear models for describing the evolution of the FELinstability. For a very narrow wave spectrum, the nonlin-ear development is coherent, and the dynamics of elec-trons trapped in the pondermotive potential play a criticalole in determining the evolution of the system [8], [9].On the other hand, if the instability is sufficiently broadband that the wave autocorrelation time Tac is short incomparison with the characteristic growth time yk-1, thena multiwave quasi-linear model [15] is appropriate, andparticle trapping is unimportant. The basic condition forvalidity of the quasi-linear description is that the wavespectrum be sufficiently broad that [15], [49], [50]
cTac IA[k - (k + ko) vz] << 7- 1
10-3 k/ko-Fig. 3. Plot of normalized growth rate Y/klkoc versus klko obtained from
(86) and (94) for parameters characteristic of the Livermore FEL exper-iments planned on ATA [47]. Here the dimensionless parameters ^y =100, c2c2kO2 = 3.6, and a, = 1.7 correspond to beam current lb = 1.9kA, beam radius rb = 0.45 cm, wiggler amplitude B,. = 2.3 kG, wigglerwavelength X0 = 8 cm, and beam density hb = 6.3 x 10" cm-'. Thefigure illustrates the dependence of the growth rate on fractional energyspread for A-y/r = 1, 2, and 3 percent.
(92)where A[ck - (k + ko) v,] = (Ak) c(1 - ,B) is the char-acteristic spread of [Wk - (k + ko) v,] over the extent ofthe amplifying wave spectrum. Equation (92) can then beexpressed in the equivalent form
'Yk -<<1.c(Ak) (1 - 03) (93)
Making use of c(Ak) (1 - j3) =(k + ko) Av,, (93) re-
duces to the inequality 'Ykl(k + ko) A v << 1, which isidentical to (78). That is, the condition (see (78) or (89))for weak resonant instability and validity of the expressionfor 'Yk in (86) is identical to the condition (see (92) or (93))that the unstable wave spectrum be sufficiently broad thatquasi-linear theory gives a valid description of the nonlin-ear evolution of the system. This of course assumes thatthe bandwidth of the initial (input) signal is comparable toA k defined in (90) or (91).
C. Stability Properties for Weak Resonant GrowthIn this section, we make use of (86) and (89) to inves-
tigate numerically the stability properties for weak reso-
nant growth. As one example, which corresponds to theparameter range planned for the Livermore FEL experi-ments [47] on the ATA, we consider the case where thebeam energy is y = 100, the beam current is Ib = |-elx fb7rrbfc = 1.9 kA, the beam radius is rb = 0.45 cm,the wiggler amplitude is B, = 2.3 kG, and the wigglerwavelength is X0 = 27r/ko = 8.0 cm. This gives nb = 6.3x 10ll cm-3, a. = eB /mc2ko = 1.7, ^4/C2ki2 = 47rnbe2/mcW0 = 3.6, bt = 0.23 (from (73) and (88)), and K(' =
100) = 0.78 (from (74)). The inequality in (89) then re-
duces to (Ay-/j)3 >> 6 x 10-7, which requires a frac-tional energy spread in excess of 0.9 percent for the growthrate expression in (86) to be valid. The total effective valueof Ay/-'y for the Livermore FEL experiment on ATA may
2.3 2.4 2510-3 k/ko0
Fig. 4. Plot of normalized growth rate 'Yklko c versus klko obtained from(86) and (94) for parameters characteristic of the Los Alamos FEL ex-periment [11], [44]. Here, the dimensionless parameters e - 41, O2/2 2 an o m
c ko = 0.21, and a, = 0.76 correspond to Ib = 40 A, rb 0.09 cm,Bw,. = 3 kG, -yo = 2.73 cm, and nb = 3.3 x lol cm-3. The figure illus-trates the dependence of the growth rate on fractional energy spread for&Ay/j = 1, 2, and 3 percent.
be in the range of 1-2 percent. As a second example, whichcorresponds to typical operating parameters for the LosAlamos FEL experiment [11], [44], we consider the casewhere e = 41, Ib = 40 A, rb = 0.09 cm, B, = 3 kG, ando= 2.73 cm. This gives nb = 3.3 x 10ll cm-3 , a,w
0.76, 6.,/c2k0 = 0.21, bt = 0.113, and K(' = 41) = 0.89.The inequality in (89) then reduces to (A'y/-)3 >> 2.0 x10-6, which requires a fractional energy spread in excessof 1.3 percent. The effective value of Ay/A in the Los Ala-mos FEL experiments is typically 1-2 percent.
Typical numerical results obtained from (86) are pre-sented in Figs. 3 and 4, where 'yk/kOc is plotted versus
k/ko for the two choices of beam and wiggler parametersgiven in the previous paragraph [11], [44], [47]. Here,(80) has been approximated by 02k = kc, and the beamdistribution function Go(-yo) is assumed to be Gaussianwith
G~~(y0)= 1 ~ (.,Y2 1)/2 2
G 0 =(r 1/2mcA'y Yo exp 2(A)
(94)
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where 5 >> 1 and Ay/y << 1 are assumed. In both Figs.3 and 4, the growth rate has been plotted for values of A'y15corresponding to fractional energy spreads of 1, 2, and 3percent. Note that as the energy spread is increased, thedecrease in maximum growth rate is proportional to(Ay)-2, and the increase in instability bandwidth Ak isproportional to A-y (see (91)). Furthermore, in Figs. 3 and4, we have chosen the energy spread to be consistent withthe validity criterion in (89), with Ay/I = 1 percent cor-responding to the limit of the range of validity.Some further comments are appropriate with regard to
the FEL experiments planned on ATA [47], which willoperate in both the amplifier and (single-pass) oscillatormodes. Since the input signal in the amplifier configura-tion will be provided by a laser with very narrow band-width, (91) is not the appropriate estimate of Aklk for theamplifying wave spectrum, nor will the criterion in (93)(required for validity of quasi-linear theory) be satisfied.Application of the present analysis should therefore be re-stricted to the oscillator configuration in which the signalgrows from low-level broad-band noise. Furthermore, theenergy spread applicable to ATA should be estimated byincluding transverse beam emittance, which has been as-sumed to vanish in the present analysis. Therefore thepresent model, which assumes weak resonant instability,should be applied only if the total effective energy spreadexceeds 1 percent. From Fig. 3, for Ay/5 = 1 percent, wenote that the maximum growth rate corresponds to an e-folding distance of CI/[Yk]MAX = 3 m.
D. Stability Properties for Monoenergetic ElectronsWe now consider FEL stability properties in circum-
stances where the beam energy spread A'y is sufficientlysmall that the inequality in (89) is not satisfied. In partic-ular, we make use of the diagonal dispersion relation (77)to investigate stability properties for monoenergetic beamelectrons where Go ('yo) = (Smc) -t (5y2 _ 1)1/2 6(Yo 5)(see (20)). In this regard, it is assumed that 5 is suffi-ciently large that K2 = a /l(^2 - 1) can be treated as asmall parameter. Therefore OF can be approximated byOF = (1 - K2/4) So in (76), where /0 = (1_-ly2)1/2.Moreover, from (54), ( S > can be approximated by
< S>) I-( + -K2 (95)
for k2 « 1 and Go'(yo) specified by (20). We substitute(20) into (77), convert the integral over p, to an integralover oyO (see (75)), and integrate by parts with respect to-yo. This gives
/2 ,2 22 2k2 _- P 3 2 1-)w4
CK() (1+ K /4)aRx +K(z) (1 + 2 (k + ko) c L-Ai
w (k+k_1 rc] L +yoim-i1
where K(yo) is defined in (74), SF is defined in (83), and3 = 3FQ(YO = 5) is given by
(97)= (1 - 7
Making use of (83), we obtain
[aF
daeo zo, = a
(1 + a'j2)A3 (98)
The dispersion relation (96) can then be expressed in thecompact form
CO cN2+32)
1 WP 2 C_Nk+y)4 ' aw [CO) - (k+ k0(c]
A 2 C(k + ko) CN2(5)+I
w[co(k +k)]2 (99)
where N1 (5) and N2(^) are defined by
NL(5) = - 1)1/2 O (KQ(yo) (1+ K214)
(100)
and
N2(y) = K(a) (1 + K2/4) (1 + aw/2). (101)
Note that both N1 (5 ) and N2(5 ) are typically of orderunity.
Equation (99), supplemented by the definitions in (74),(97), (100), and (101), constitutes the final dispersion re-lation for monoenergetic electrons. Equation (99) is afourth-order algebraic equation for the complex eigenfre-quency ', and can be used to investigate detailed stabilityproperties over a wide range of the dimensionless param-eters c%p/c2ko, a , and 5. For purposes of obtaining a sim-ple estimate of the characteristic growth rate, we examine(99) for ($2/c2k0) (a2 /A3) << 1 and (a, k) closely tuned to(w, k) satisfying the simultaneous resonance conditions
= +c2k+-= (k + ko) /c.
3 -124
(102)
Note that (w-, k) determined from (102) differs slightlyfrom (88) because of the inclusion of the A2 contributionin (102). We now examine (96) for (co, k) close to (co, k).Expressing Cw = X + 6co and k = k + bk, then for bk =0 and Ibwl1/-I << 1, (96) gives
[w - (k + ko) fc] a-yo 0 -,Y- 1)1/
(96)
(6c)3 = 8 5 N2(5) c(k + ko). (103)
Equation (103) can be used to examine the characteristic(maximum) growth rate for 6k = 0. This gives
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. PS-13, NO. 6, DECEMBER 1985
Im(w) = (3)12 lo N2() c(k + ko) (104)
Keep in mind that (104) is valid only for negligibly smallenergy spread, and the range of validity of (99) does notoverlap with the range of validity of (86). In circum-stances where the approximations K2 « 1, klkof » 1,and I= 1 are valid, we find N2(') K(e) (1 + a I2).Equation (104) becomes, after use of (88),
Im (6w) _ (3) 2KpawK(Qy) ckoI (105)
which corresponds to the familiar expression for the cold-beam Compton-regime growth rate.
V. STABILITY PROPERTIES FOR ULITRARELATIVISTICELECTRONS
In this section, we consider (77) in the limit of an ul-trarelativistic, tenuous electron beam and compare the re-sulting dispersion relation (Section V-A) with the disper-sion relation obtained from a linear analysis [51] of thestandard Compton-regime FEL equations [8], [9] basedupon a superposition of single-particle orbits (SectionV-B). Finally, in Section V-C, we extend the quasi-linearkinetic equations derived by Dimos and Davidson [15] fora helical wiggler magnetic field to the case of an ultra-relativistic electron beam propagating through a planarmagnetic wiggler.
A. Kinetic Dispersion Relation for UltrarelativisticElectrons
For an ultrarelativistic, tenuous electron beam with»y>> l and 2 << C2k2, we approximate K2 « and
00 dpr02
4... =Mc dyo ...
0 To
in (77). In this case, the dispersion relation (77) can beapproximated by
0 = D(k, w) 2- c2k2 + &2pa2 wmc4p
co dyo aG laTyo2 w - (k + ko) FCK(Co) (106)
where K(yo) and fF(Tyo) are defined in (75) and (83). Here,G;(Tyo) is centered about T0 = e >> 1 with characteristicenergy spread AT << j'. For OF = 1 and To » 1, theaxial velocity 3FC occurring in (106) can be approximatedby (see (83))
I + a'/22-yo
(107)
Integrating by parts with respect to Tyo in (106) and makinguse of (107), the dispersion relation (106) can be expressedin the equivalent form
co c2 = -cpawomc l dyoG1o(yo)
x | (k + ko) c K(yo) 1 (1X 2~~~~~~~~51([ -(k + kO) f3FC] To
In (108), for temporal growth (FEL oscillator case), thewavenumber k is real and the oscillation frequency w iscomplex. It is convenient to express
W = kc + bw (109)
where 6w is complex and corresponds to the wiggler-in-duced modification to the vacuum dispersion relation X =kc (see (108) with a, = 0). We also introduce the quantityAw(yo) defined by
1 + a-/2Aw=-k c +(k +ko) c 2T (110)
Making use of (107), (109), and (110), it is readily shownthat
w - (k + kO) IFC = 6w + Aw (111)
and the dispersion relation (108) can be expressed as
2kc6w + (6w)2 = j2a2 (kc + bw) mc dyo Go (yo)
(k + ko) cK('yo) 1 ( +aQ (6w+ AC)2 To
+ w
[2yo 3K(yo) - To aK/aT0]y(6w + AW)
(112)
Here, Aw(yo) is defined in (110), and the dispersion rela-tion (112) is fully equivalent to (108) with SF approximatedby (107).
B. Linearized Compton-Regime FEL EquationsFor purposes of comparison, we now investigate linear
stability properties within the context of the standardCompton-regime FEL equations [8], [9] which describethe interaction of the beam electrons with a monochro-matic electromagnetic wave with wavenumber k and fre-quency kc. For the jth electron, with energy Tyj, the phasefunction Oj and frequency shift Awj are defined by
Oj = (k + ko) zj - kct
axj = -ko c +c (k + ko) I1=-k0c+ ~~~2Tyj (113)
In the notation of this paper, assuming ultrarelativisticelectrons, the Compton-regime equations [81, [9] are givenby
476
DAVIDSON AND WURTELE: FEL INSTABILITY IN PLANAR MAGNETIC WIGGLER
d =j _k2awg Im [ay exp (iOj)]
d ck- 0 = -,AW + - a ,g Re [a~ x i) 15dt1Oi 2,yj2 epiO] (15
d iupawg-ay =
exp
Equations (114)-(116) describe the coupled nonlinear ev-olution of the electrons and the radiation field (assumedmonochromatic). In (116), (<j> denotes the ensemble av-erage over NT electrons,
I NT
NTj=l (17
and the amplitude factor g is defined by
g = Jo(b) - J1(b)= [K(A)]1/2. (118)
Here, b is defined by b = b+(,yo = A) a /(4 + 2awhich is a valid approximation for e » 1, K2 « 1, andk/ko = 2 A2/(l + a2/2) (see (74) and (89)). Moreover, theidentification g = [K(Aj)]1/2 has been made in the ultra-relativistic limit (see (75) and (118)).
In the small-signal regime, we linearize (114)-(116) andexpress [51]
Here, subscript "zero" labels unperturbed values in theabsence of the radiation field (bay = 0). Substituting (119)into (114)-(116) and retaining terms which are linear in theperturbation amplitudes, we obtain
d & = kcawg Im [bay exp (iOjo -iAwjo t)]dt 2yjd = c(k + ko)
where' Go(yo) is the energy distribution, and Qjo is the ini-tial phase in (119). In (124), we have converted the sum-mation over discrete particles in (117) to a continuum in-tegral over the distribution Go (yo). In obtaining (123), usehas been made of ('yjo exp (-iOo + iAw1jot)> = 0.
In (121)-(123), the vector potential bay(t) is expressedas ba = ba exp (-i6wt), where Im bw > 0 correspondsto instability (temporal growth). Integrating (121) from t= - oo to time t, and neglecting "initial" values (fort -oo), we obtain for 6yj (t)
kcawg mibay exp (iOjo - iAwjot - i6wt)j6, = Im
2,yjo (6w + Awojo) I
(125)
where Im(bc) > 0 has been assumed. Similarly, makinguse of (125), we obtain for 60j(t) from (122)
60j 27 aWg Re i6ay exp (iOjo - iAojot - i6wt)2 2=24Y gRe ex
x 1l(bw + Awj)
12Ia c(k +ko) )2 / (6++A)21
(126)
Substituting (125) and (126) into (123), and making use of(124), we find after some straightforward algebra
and the factor My exp (- i6wt) has been canceled fromboth sides of the equation.We now compare the kinetic dispersion relation (112)
with the dispersion relation (127) obtained from a linear(122) analysis of the standard Compton-regime FEL equations
[8], [9]. First, comparing (107), (109)-(111), and (120), itis evident that 6w + Aw = w - (k + ko) 3FC in bothdispersion relations. Moreover, the kinetic dispersion re-lation (112) reduces directly to (127) provided we makethe following approximations in (112):
(123)
In (123), the ensemble average < > denotes
a) 2kcbw + (bw)2 = 2kcbw on the left-hand side of(112);
b) kc + bw = kc on the right-hand side of (112); and
477
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. PS-13, NO. 6, DECEMBER 1985
c) K(Qyo) = K(7) and aK(y0)Iay0 0 on the right-handside of (112).
Approximations a) and b) are associated with the factthat the eikonal approximation has been made in derivingthe Compton-regime equations (114)-(116). These approx-imations are indeed justified because ILw1 << IkccI in theultrarelativistic tenuous beam limit. Moreover, Approxi-mation c) is also a reasonably good approximation be-cause G+(-y0) is strongly peaked around yo = j, and thevariation of K(Qyo) with energy yo is relatively weak.
This completes the proof of equivalence of the two dis-persion relations in the limit of an ultrarelativistic tenuouselectron beam.
C. Quasi-Linear Kinetic Equations for a Planar WigglerFor completeness, making use of the ultrarelativistic
tenuous electron beam assumptions enumerated at the be-ginning of Section V-A., we conclude this paper with asummary of the appropriate quasi-linear kinetic equationsfor the planar wiggler configuration considered in thepresent analysis. This represents a straightforward exten-sion of the quasi-linear kinetic equations developed byDimos and Davidson for the case of a helical wiggler mag-netic field [15], [50]. In this regard, for the quasi-linearanalysis to be valid, it is important to recognize that theamplifying wave spectrum must be sufficiently broad thatTac « -Y I where i- is the wave autocorrelation timedefined in (92).
In quasi-linear theory, the average background distri-bution function G+(?yo, t) is allowed to vary slowly withtime in response to the amplifying wave perturbations. Thecomplex oscillation frequency Ck(t) + ijYk(t) is then deter-mined adiabatically in time from the linear dispersion re-lation (see (106)):
where y » 1,I£ « c2k2, and K2 < 1 have been as-sumed, and OF is defined in (107). For ultrarelativisticelectrons with pm = -yomc, the appropriate extension of theparticle kinetic equation [15, eq. (30)] (or [49, eq. (12)])to the case of a planar magnetic wiggler is
Go ('yo, t) =- D(-o, t) Go+(o t)j (129)At daow7where the quasi-linear diffusion coefficient D(Qyo, t) is de-fined by
1Wp 2 K(y0)D(yo, t) = 2a 2
4 hbmc Yoco 1Fk(t)OOk(kk)/F±iY (130)
k-=-°oo k - (k + ko) OFC + iYkHere, for k2 >> ko, lk(t) = k2 6 Ay (k, t)12!8r is the effec-tive spectral energy density of the magnetic-field pertur-
bations, and £k(t) evolves according to the wave kineticequation
at £k(t) = 2Yk(t) Sk(t) (131)
where 'yk(t) is determined from (128).Equations (128)-(131) constitute a closed description of
the nonlinear evolution of the system in circumstanceswhere the amplifying wave spectrum is sufficiently broad-band that the inequality in (92) is satisfied. To summarize,as the wave spectrum amplifies, there is a correspondingredistribution of electrons in 'yo space, and a concomitantmodification of the growth rate 'yk(t). The details of thetime evolution and the stabilization process of course de-pend on the specific parameter regime, the initial distri-bution function G{(-yo, t = 0), and the input spectrum£k(t = 0). It is sufficient for present purposes simply tonote that (128) and (130) can be simplified considerably incircumstances corresponding to weak resonant instability[15] (see also Section IV-B and IV-C), and have been in-tegrated numerically [50] for certain simple functionalforms of GoQ(yo, t).
VI. CONCLUSIONS
In this paper, we have made use of the linearized Vla-sov-Maxwell equations (Sections II and III) to investigatedetailed FEL stability properties for a tenuous relativisticelectron beam propagating through the constant-ampli-tude helical wiggler magnetic field (1). The analysis wascarried out for perturbations about the general class of self-consistent beam equilibria GO(z, pz) = U(pz) Go+(yo) (see(16)). To evaluate the perturbed distribution function6G(z, pZ, t), use was made of the exact particle trajecto-ries in the equilibrium wiggler field, and there was no apriori restriction to ultrarelativistic electrons. Indeed, forlow or moderate electron energy, it was shown that therecan be a sizable modulation of beam equilibrium proper-ties by the wiggler field and a concomitant coupling of thekth Fourier component of the wave field to the componentsk + 2ko, k + 4ko, * . . . This is evident from the formalmatrix dispersion equation (58) and the definition of elec-tron susceptibility X(k, co, koz) in (63). In the diagonalapproximation, it was shown that (58) reduces to the dis-persion relation (77). In Section IV, we made use of (77)to investigate the detailed dependence of FEL growth rateon the choice of distribution function Go(-yo). Investiga-tions of stability behavior ranged from the regime of stronginstability (monoenergetic electrons) to weak resonantgrowth (sufficiently large energy spread). For the case ofweak resonant growth, the growth rates were calculatednumerically for parameter regimes characteristic of theLos Alamos experiment [44] and the Livermore experi-ments planned on the ATA [47].The limiting case of ultrarelativistic electrons and very
low beam density was considered in Section V. We com-pared the resulting kinetic dispersion relation (106) withthe dispersion relation (127) obtained from a linear anal-
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DAVIDSON AND WURTELE: FEL INSTABILITY IN PLANAR MAGNETIC WIGGLER
ysis of the conventional Compton-regime FEL equations[8], [9]. This comparison was made for general beamequilibrium Go (y0). Differences between the two disper-sion relations were traced to the eikonal approximation andthe assumption of very narrow energy spread in [8] and[9]. Finally, assuming ultrarelativistic electrons and a suf-ficiently broad spectrum of amplifying waves, in SectionV we presented the quasi-linear kinetic equations appro-priate to the planar wiggler configuration considered inthe present analysis. This represented a straightforwardextension of the quasi-linear theory developed for the case
of a helical magnetic wiggler field [15], [50]. The quasi-linear dispersion relation (128), the kinetic equation (129)for the distribution of beam electrons Go (Qyo, t), and thekinetic equation (131) for the wave spectral energy densitySk(t) describe the self-consistent nonlinear evolution of thebeam electrons and radiation field in circumstances wherethe wave autocorrelation time is short in comparison withthe characteristic growth time (92).
ACKNOWLEDGMENT
It is a pleasure to acknowledge the benefit of useful dis-cussions with A. Dimos in relation to the quasi-linear for-malism in Section V-C.
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