PFC/JA-81-23 INTENSE FREE ELECTRON LASER HARMONIC GENERATION IN A LONGITUDINAL MAGNETIC WIGGLER Ronald C. Davidson Wayne A. McMullin 10/81 INTENSE FREE ELECTRON LASER HARMONIC GENERATION IN A LONGITUDINAL MAGNETIC WIGGLER Ronald C. Davidson t Plasma Research Institute Science Applications, Inc. Boulder, Colorado 80302 and Wayne A. McMullin Plasma Fusion Center Massachusetts Institute of Technology Cambridge, Massachusetts 02139 ABSTRACT The free electron laser instability is investigated theoretically for a tenuous relativistic electron beam propagating parallel to a longitudinal wiggler field (Bo + 6Bainkoz)%. The stability analysis is based on the linearized Vlasov-Maxwell equations for perturbations about the beam equilibrium f4 = (nb/2w5)6(p_-mV)6(p,-mmV). For (W D/b)(eB/BO) of order unity or larger, it is found that the transverse electron orbits experience a stmng temporal modulation at harmonics ofkov = koVb. In the stability analysis, this orbit modulation is manifest by radiation amplification occurring near the simultaneous zeroes of w = kc and w - klb - Web = nAeuVb for general harmonic number n, which corresponds to an upshifted (for n > 1) wavenumber I = (I - Vb/c)' (web/c + nkoVb/c). For (wbA/OVb)(B/Bo) > 1, the instability is typically broadband with several harmonics cxcited. Moreover, depending on beam and field parameters,'the characteristic maximum growth rate can be substantial with Itnm/cko = ( '/2)[(V2 /4c2)(W 'c2k)J2(L-,6B/koVBa)j'/3. Iermianent Address: PIlarna Fusion Center. NhIssachuscits Insitute of Technulogy, Cambridge. Massachusetts 02139 I I. INTRODUCTION In recent years there has been a growing interest in the use of intense relativistic electron beams to generate coherent radiation in the centimeter, millimeter, and submillimeter wavelength regions of the electromagnetic spectrum. Two principal classes of radiation generation mechanisms have received considerable attention. First, there is the cyclotron maser instability 1-10 in which the electron beam propagates along a solenoidal magnetic field Bok, and the basic mechanism is the electromagnetic Weibel instability driven by an excess of beam thermal energy in the perpendicular direction relative to the parallel direction (Tn> T11). Second, there is the class of free electron laser (FEL) instabilities 11-16 in which the electron beam propagates in combined axial and wiggler magnetic fields Bo +6B, and the basic instability mechanism is associated with the enhanced axial bunching produced by the beam interaction with the (spatially periodic) wiggler field. For the most part, analyses of the FEL instability have been restricted to circumstances where the wiggler magnetic field 6B is primarily transverse to the axial field Bok. Recently, however, McMullin and Bekefi'7,18 have proposed the Lowbitron (acronym for longitudinalwiggler beam interaction)as an attractive field configuration for intense FEL radiation generation. In the Lowbitron, the oscillating wiggler field 6B is primarily in the axial direction and the instability mechanism is a hybrid of the Weibel and axial bunching mechanisms for the cyclotron maser 1-10 and standard FEL"-- instabilities. To be more specific, in Refs. 17 and 18, and in the present analysis, a relativistic electron beam with average axial velocity V and sufficiently large transverse thermal velocity VL (before entering the interaction region ) propagates along the axis of a multiple-mirror (undulator) magnetic field with axial periodicity length L = 21r/ko, and axial and radial magnetic fields. B2 and B, given by 19 6B B" = A) 1 + -ij(kqr )sinkz], B" = -bB(k-)r)coskoz, %here I,,(kpr) is the modified Bescl function of the first kind of order n, and 6B/,i1 < I is related to the iwrror ratio It by It = (1+ 5/L6)/( I - L/Do). In circumstances %here the beam radius Rb is sufficiently small with koRb < 1, then the leading order oscillation (wiggle) in the applied field is primarily in the axial direction, and BO and B? can be approximated by [Eq. (3)] B,= Bo 1 + 6Bsinkoz), BP, =0, in the beam interior, where r < Rh. Assuming a tenuous electron beam, McMullin and Bekefi 17,18 consider the amplification of right-circularly polarized electromagnetic waves with frequency w and wavenumber k propagating parallel to the axial magnetic field (BO + OBsinkoz), for the case of a small-amplitude longitudinal wiggler with ~Wcb 6BCI where w) = eo/-ymc is the relativistic electron cyclotron frequency, and -mc Vl/c2 _ 1i/c 2 )-1/2 mc2 is the electron energy. For 6B 30 0, it is found 2 = 17,18 (1 - that there can be intense FEL amplification (Imw > 0) for w and k near the simultaneous zeroes of w = kc and w - kVb - Wcb = koVb, which corresponds to an upshifted wavenumberki, Vb1/c)-d(wCb/c - koW/C) ygVj/c 2 ) (1+ where use has been made of I + V/C2 = y 2 1/2. -+ For 2Vc 2 < i, it is clear from the above expression for Ai that Lowbitron radiation generation can be at even shorter wavelengths than for the cyclotron maser or for a standard free electron laser with transverse wiggler field 6B. In the present analysis, we make use of the Vlasov-Maxwell equations to investigate intense FEL radiation generation in the longitudinal wiggler configuration, removing the restriction of weak wiggler amplitude, and allowing (wab/kVb)(6B/BO) to be of order unity or larger. Equilibrium and stability properties (Secs. Il and IIl, respectively) are investigated for the choice of bcan equilibrium distribution finction Job [Fq. (6)] b= *(p.- )j( p 21r p_ 2 --V- jmh) where n6 = const is the beam density, pi = (pl + pl)'/ 2 is the transverse momentum, and p, is the axial momentum. The beam density is assumed to be sufficiently low that the equilibrium electric and magnetic self fields have a negligibly small effect on the particle trajectories and stability behavior. Moreover, the influence of finite radial geometry is neglected in the present analysis, and perturbed quantities are allowed to vary only in the z-direction. Treating (web/ko,)(5B/Bo) as finite, it is found (Sec. II) that the S1 x 6Bsinkoz', force on an electron can lead to a large temporal modulation of the transverse orbit at all harmonics of kovt = koVb. The detailed orbit analysis (Sec. II) shows that the nth harmonic modulation of the transverse motion is strongest whenever [Eq. (10)] wcb 6B kD0% Bo'' where v, = Vb and -y = -1 are assumed, an,1 is the first zero of J(z) = 0, and J(z) is the Bessel function of the first kind of order n. In the formal stability analysis for a tenuous electron beam (Sec. III), the transverse orbit modulation is manifest by radiation amplification (Imw > 0) occurring near the simultaneous zeroes of w = kc and w - kVb web = nkoVb for general harmonic number n [Eq. (41)]. This corresponds to an upshifted (for n > 1) wavenumber kn [Eq. (40)) in (I+ Vb/c)-(Wcb/c + nkoVb/c) (I + _y2 /C2) ' Moreover, the characteristic maximum growth rate Imw for the nth harmonic is found to be [Eq. (45)] I I [= V 2 4 y2I Wyb 6B C.PJVBO "n 2 C2P The striking feature of Eqs. (40) and (45) is that by choosing the appropriate value of (wb/k4)(bB/Bo) s a,,,, FEL radiation generation can be made intense for moderate values of harmonic number n, corresponding to a significant upshift in wavenumber A and frequency a, = kIc. Moreover, the inmtability is inherently broadband in that when (wi,/ko1)(WB/LD) is chosen to maximize In"w for a particular value of n, many adjacent hannonics still exhibit substantial growth. 3 The numerical examples discussed in Sec. III (Figs. 1 and 2) assume a moderate value of VL with Vt/c = 0.5. In this regard, it is important to note from Eqs. (40) and (45) that the amplification can also be substantial for considerably smaller values of VL/c. As an example, consider the choice of beam and field parameters given by V/c = 0.1, 'b = 2, V/c = 0.86, W./c 2k4 = 0.05, and 6B/Bo = 1/3. Then, if we choose we/cko = 10.84, it follows that (Wrb/koV)(6B/Bo) = 4.2 = a3,1, which maximizes the growth rate for n = 3. From Eq. (45), the maximum n = 3 growth rate is given by 0.036, = and from Eq. (40) the amplified wavenumber is 96. = Note that this corresponds to a substantial upshift in wavenumber relative to the standard estimate for an FEL with transverse wiggler, i.e., relative to (1+ Vb/c)Ybko k = (I + ggV2j/c2) = for V/c = 0.86, VL/c = 0.1 and -a 7.15k0 , = 2. In order to achieve moderate values of the parameter (web/koVb)(6B/Bo), it is necessary to operate at relatively high values of magnetic field. For example, if L = 27r/ko = 8cm, -yb = 2, and wes/cko = 10.84 (the value chosen in the above example), then BO = 28.34kG is required. Finally, we note that the influence of finite radial geometry on FEL emission in a longitudinal wiggler field has been investigated by Uhm and Davidson 19 for the case of a thin annular electron beam. Their analysis, 19 however, was restricted to small transverse energies (thin beam), relatively weak applied fields with (wb/ko14)(6B/1) fundamental (n = 1) mode. 4 < 1, and to the Il. EQUILIBRIUM CONFIGURATION AND PARTICLE TRAJECTORIES A. Equilibrium Configuration We consider an intense relativistic electron beam propagating along the axis of a multiple-mirror (undulator) magnetic field with axial periodicity length L = 21r/ko. It is assumed that the electron beam radius R6 is sufficiently small that kDR&<1, (1) and that kor < I is a good approximation over the radial cross-section of the electron beam. Here, cylindrical polar coordinates (r, 6, z) are introduced, where r is the radial distance from the axis of symmetry, and z is the axial coordinate. For kor < 1, the axial and radial magnetic fields, BO(r, z) and B0(r, z), can be approximated near the axis of the multiple-mirror system by' 9 BO = Bo I+ + I6Bkor2sinkoz, sin BO = - 6Bkorcoskoz, (2) where 6B = const and 6B/Bo < I is related to the mirror ratio R by R = (1 + 8B/Bo)/(1 - SB/Bo). For present purposes, it is assumed that koRb is sufficiently small that field contributions of order kor6B (and smaller) have a negligibly small influence on the particle trajectories and FEL stability properties. That is, we approximate the axial and radial magnetic fields in Eq. (2) by BO = Bo I + 6Bsinkoz], BO = 0, (3) in the subsequent analysis. Assuming a tenuous electron beam with negligibly simall equilibrium self fields, then the electron imotion in the C(Uilibritim longitudinal wiggler field describcd by Fq. (3) is 5 I characterized by the four single-particle invariants Pz, p2= P- (p2 + p8) = (m 2c4 + c2 p2L+ Pe = r[Pe C2p)1/2 A (rz) - (4) where p, is the axial momentum, pi = (p2 + pe)1/ 2 is the perpendicular momentum, ymc 2 is the electron energy, Pe is the canonical angular momentum, and A = (rBo/2)[1+ (6B/Bo)sinkozj is the equilibrium vector potential for the axial magnetic field Bz in Eq. (3). Here, m is the electron rest mass, -e is the electron charge, and c is the speed of light in vacuo. Note that ymc2 = const can be constructed from the invariants p, and p!, which are independently conserved. Any distribution function f4(x, p) that is a function only of the single-particle invariants in Eq. (4), is a solution to the steady-state (8/st = 0) Vlasov equation. In general, to construct radially confined self-consistent beam equilibria with nb(r -+ co) = 0, it is necessary that f,(x p) depend explicitly on the canonical angular momentuum Ps. For present purposes, however, we assume that fb has no explicit dependence on P, and consider the class of self-consistent beam equilibria of the form That is, although Eq. (5) does satisfy the steady-state Vlasov equation, the influence of finite radial geometry is neglected in the present equilibrium and stability analysis. In addition, although the FEL stability analysis will be formulated for general choice of Jf(pL, p,) in Section Ill. detailed stability properties are investigated for the specific choice of equilibrium distribution function -6(p- = 2 !7rpL - /bmVb)6(pz - 1',mV), (6) where n6 = f d3pfo = const is the beanm density, the constants Y, and VL are related to the relativistic miass factor b by Vb = jf d'p(p./vn)fD/(f dOpfp) = (1 _ - 2 /CI)-1/2. and the constant can he identified with the aver.igc beam velocity in 6 the axial direction. Note that Eq. (6) corresponds to an electron beam equilibrium that <p, >)(,-- < v, >)1(f d3pf2) = 0, where < p >= [f d3 p(p, T11 = is cold in the axial direction with effective axial temperature (f dap b')/(f - d pf ). On the other hand, the effective perpendicular temperature TL is non-zero, with T_ = (1/2)(f d 3 ppvjf )/(f d 3pfi) = -ybmV2 /2. It is precisely the thermal anisotropy (T. > T11) that provides the free energy source to drive the (Weibel-ike) cyclotron maser and FEL instabilities discussed in Section III. B. Particle Trajectories The orbit equations for an electron moving in the axial magnetic field B.O(z) = B0 + SBsinkoz described by Eq. (3) are given by dp',/dt' = -(e/c)v'AB?(z'), dp,/dt' = (e/c)v',B(z') and dp',/dt' = 0, where p'(t') = 'ymv'(t') and - = (1+ p2/m2c2)/ 2 = const. The axial trajectory (', p,) that passes through the phase space point (z, p.) at time t'= t is given by p= P, = z + vir, (7) where r = t' - t and , = p,/ym = const is the axial velocity. Defining t'+(t) t'(t')+ ivY(t') and making use of Eq. (7), it is straightforward to show that v'+(t') satisfies V'+= iWe 1+ isin(koz + kovr)lv'+, (8) where w, = eBo/-ymc is the relativistic cyclotron frequency for electron motion in the average axial field Bo. Integrating Eq. (8) with respect to t' and enforcing v'+(t' = t) = v, + ivy = verp(iO) where (vi, v,) = (vcosO, vsinO) is the perpendicular velocity at time t' = t, gives V, = VeXP. v'+(t') VWeZP[?9 + , iwcTr + + *S .io8B coskoz - cos(kez ovz~vrj+ kov.-r)(9 From Fq. (9). %%e note that p', (t') = niv'I(t')I = const, as expected. Flowever, the inditiduat trinse rc V'(l') and v'('). can be strungly modulated as a funLctin oF 'lecit COmpis1C1. and t' by the ]y iiudinal %ig'lcr field 6Bsinkoj:. Dkpending on the siYc of 7 I SB/Bo, this can lead to a significant enhancement of radiation emission relative to the case where 5B = 0. For future reference, it is convenient to Fourier decompose the koz dependence in Eq. (9), making use of the Bessel function identity Jm(b)ezp(-ima+ ezp(ibcosa) = E' im7r/2), where Jm(b) is the Bessel function of the first kind of order m. This gives v'(t')= Vtexp(i)E Jm ().n mn (kov.- Bo (kov. Bo wc6B)(,)n X ezp[i(Wcr + mkovr)]ezp[i(m - n)koz], where F,,,n denotes E0 _, (10) E"n_,. From Eq. (10), for (wc/kov,)(B/Bo) of order unity, we note that the temporal modulation of the perpendicular velocity can be strong at harmonics of kov, .= koV. Fally, when Eq. (10) is integrated with respect to t' to determine the transverse particle orbit z'(t') + iy(t'), we note that there are resonant contributions proportional to (w. + mkovY)-I.-In this regard, the present analysis assumes that v, = V is sufficiently far removed from cyclotron resonance (with w, + mkoVb orbits do not exhibit large (secular) transverse excursions. 8 # 0) that the particle III. STABILITY PROPERTIES A. Linearized Vlasov-Maxwell Equations In this section, we make use of the linearized Vlasov-Maxwell equations to investigate stability properties for perturbations about the equilibrium configuration described by Eqs. (3) and (5). All perturbed quantities 60 are expressed in the form (11) 60(z, t) = 6t(z)ezp(-iwt), where Imw > 0, and perpendicular spatial variations are assumed to be negligibly small (a/9ax = 0). In addition, it is assumed that the beam is sufficiently tenuous that lon- gitudinal field perturbations can be neglected in the present analysis, i.e., 6# = 0 ~ (ComptonRegime). (12) The transverse electromagnetic field perturbations, 6E = (iw/c)5A and 6B = V X 84 can be expressed as 6E = 6E.(z, t)e, + 6Ey(z, t)e, = i w[6A(z)e. + SAy(z)eyexp(-iwt), (13) and 6B= 6B,(z, t)e, + By(z, t)e = [ 86A,(z) 8*A3 (z) - .~z~w1(4 ey eXp(-it). e, + (14) Moreover, the vector potential 6A = 6A,(z)e, + 6AY(z)ey is determined self-consistently in terms of the perturbed distribution function 6fb(z, p, t) = 6fb(z, p)exp(-iwt) from the Maxwell equation V X 6B = -(47re/c) f d'prifS + (1/c)(6/Ot)6E,i.e., (2 + T2) 6A(z) = fd3pt4Sfb(z, p). (15) Making use of the inethod of characteristics, the linearized Vlasov equation for 6fA(z, p, t) can be integratcd to give ,= 11dt' [--iw(1' - 1)1(6 /:(') +42 ~ (p cc~ 9 t(1p',- 2 O ,?" ()16) I where the particle trajectories (z', p') solve dp'/dt' = -(e/c)S X B,(z')e. and dx'/d'= d, with "initial" conditions z'(t' = t) = z and p'(t' = t) = p. After some straightforward algebra that makes use of Eqs. (7), (13) and (14), and the fact that p2 and p, are exact single-particle invariants (independent of t') in the equilibrium magnetic field BO = BO + 6Bsinkoz, the perturbed distribution function 65A(z, p) in Eq. (16) can be expressed in the equivalent form ) - - = 6f(z p)= d x e p(-iwr)[v'(r) 6A,(z + vr) + v' (r) 6A,(z+vzr) C P±L 8CP± x dr exp(-iwr) V,(r)6A,(z + var) + v' (r)6Ay(z + tr) f (17) where t',(r)and 'y(r)are determined from Eq. (9)[or Eq. (10)], and f(pi,pz) is independent of t'. In the subsequent analysis, it is convenient to Fourier decompose 6A.(z) and 6A,(z) according to 6A,(z) = 6A,(k)ezp(ikz), (18) [v'6A_(k) + vl6A+(k)], (19) k and to express v,6A,(k) + v'6A y(k)= where V (r) - V(r)ii/'(r)and 6A.(k) 6A,(k)± i6A,(k). Making use of Equations (17) - (19), the perturbed distribution function 5A(z, p) is given by I 64(z, p) = i C Xf iiezp(ikz) +k Pi a '-L \ 8 PZ- PjL 89Pw JJ dr exp[-i(w - k )I)r[v'(r)6A-(k) Nlaking use of Eqs. (15) and (20) and f d p = f' do f that JJ! dqefxp(+20i) + v'(r)6A+(k)J. (20) dp; f dpip1 . and noting = 0. it is straight fOiv-ard to show that the ei-ce'malue equations ror 10 6A±(z) completely decouple to give + 6A(z)= 1 41re2 exp(ikz) k x Jdf 3Pi-L2 P±L x where v2 = 2 /J2m 4.L an P, (apz Z+ dr ezp[-i(w - kv.)r]O'(r)6At(k), (21) ':/vjezp(Fi4).Here, t'y = 2 , and O (r)is defined by O' = t',~F iV is defined in Eq. (9) [or Eq. (10)1. Without loss of generality, we consider the lower sign in Eq. (21), which corresponds to the branch with right-hand circular polarization. Substituting Eq. (10) into Eq. (21) and carrying out the r integration gives ( 02 + w2flA-(z) = k -2 Xm,n(k, w)eSA_..(k)zp[i(k + ink 0 nkb)z], - (22) k m,n \I where the susceptibility Xm,n(k, w) is defined by Xm,n(kw) = X 2 02 1 2 4rC d kovBo]\ v2 (w -kv 3 -mkov.-w) YWc~ 0 p Changing the k- summation variable from k j Op±. + koviBo-) 1 (O4 P. 9f + k mk - nkD p, -+ (23) pL k, Eq. (22) can also be expressed as (a2 + !6A-(z).= - Xrn~n(k + nko - ikow) /Ak n X 6A_(k + kkn nkuj - (24) in)ezp(ikz), where Xm,(k + nflk - mk0 , w) = I :re x- :2 d apJ,,, -_-_ (w - kv- - nk4ov: - w,.) p + It. JB 1, 1 (k + nko - nk)I) - in ROl!_2 5) p- C,)pL) Fourier decomposing the left-hanx side of Eq. (24) gives the final dispersion equation E XN(k + NkO, w)6A-(k + Nko)= 0, Db(k, w)5A-(k)+ (26) N960 £ where EN#O0 - = + N1* Db(k, w) C - k2 + Xo(k, w), (27) and XN(k + Nko,w) = Xn -N,n(k + Nk,w). (28) n=-cc The dispersion equation (26), and the supporting definitions in Eqs. (25), (27) and (28), can be used to investigate detailed stability properties for a broad range of beam equilibria I(p2, p.). B. Stability Analysis For present purposes, we specialize to the choice of distribution function in Eq. (6). After some straightforward algebra, the susceptibility Xn,-N,n(k + Nko, w) defined in Eq. (25) reduces to Xn-N,n(k + I_ Nko, w) 2 /i) _________ _ -kV ? V -W) V2 (k x I-21w -(k + Nko)V) - jWeb + Nko)c 2 Vb + fw2- (k + Nk4)(k + nko)c2(2 C2 (w - kVb - nkoVAb-wb ) 6B k j, WcA, kOVb B) n-N,n (29) for the choice of beam equilibrium in Eq. (6). Hcre, web = e)/Ymc. W2 = 4xnhe2 /ryYm, and e is defined by . n . 12 I Moreover, the quantity Do(k, w) [Eq. (27)] can be expressed as Db(k, w)= W - k2 2 2 +2bE j2 W2 j -(wk - _ 2 V2L k2 x-2( -kW) b - _2 V n + ' C2 Vb nkVb -wcb) -3, koV Bo (w 2 _k 2 c3-nkokc2 (W-k2C-fl kok-2) C2 (W- kWb - nkoV6 - caes) V2 _- By shifting the k variable successively from k to k + 1k 0 , where -1, -2, B0 web 1B e= (31) 1, 2, 3... and f = --, the dispersion equation (26) gives an infinite-order matrix dispersion equation coupling the amplitudes 5A-(k), 5A_(k ± ko), 5A_(k ± 2kD), ---. For the case of a tenuous electron beam (small wpb), the coupling to the off-diagonal elements (proportional to Xv, N = ±1, ±2,. --) can be neglected to lowest order. Setting the determinant equal to zero then yields the approximate dispersion relation +cc 1 Db(k + iko, w) = 0, (32) or equivalently, Do(k, w) = 0, (33) where k is treated as a continuous variable. Making use of Eq. (31), the approximate dispersion relation (33) can be cxpressed as y2L i1 -W2 A2 C2 2 B n(kol%, Bo (w - kb - nkoV - wc_6)2 2W 2 c2 E _ .i 2 72 2 c 2 + 2--- 2 A _~6 1+12 _k.2 _wb 8B kouV B) j cb 2, ~k eb _____6B)_____ ( - kVb - pb nkVb - W )2' 6B 2(w - kVb) + (kV2/b)e V6 n. Bo ( ko-nkr - we),b (34) We now make use of Eq. (34) to investigate FEL and cyclotron mascr stability properties for a tCnuOuS electron beam. In this regard. for 5B = 0, only the n = 0 terms survive in Fq. (34) and the di'pen.ion relacion exihits the standard cyclotron maser instability for wave peittrhations clo,.c to cclotroii resonance with w - k1 - we. z= 0. On the other 13 hand, for 6B 3 0, we find that the system exhibits instability near each of the resonances w - kV - w6 ~ nkoVb, for n = 0, ±1, ±2, ---. Moreover, for a particular choice of harmonic number n, the instability growth rate Imw depends sensitively on the amplitude 5B of the axial wiggler field as measured by the dimensionless parameter (Wb/koVb)(6B/BO). Cyclotron Maser Instability for 6B = 0: As a simple reference case, it is useful to consider Eq. (34) for a tenuous beam with 6B = 0. Making use of J2 (0) = 1 and J'(0) = 0 for n 3 0, Eq. (34) reduces to - k2)I+ T22w (_ - kVb - Wb)2 w= - c2 (w (35) kVbw- W)'(3 which is similar in form to the dispersion relation for the standard cyclotron maser instabilityl(with 6B = 0), neglecting the influence of finite radial geometry. For the forward-travelling : +kc, we approximate w 2 - c2k 2 ~ 2kc(w - kc) and the electromagnetic wave with w dispersion relation (35) reduces to the approximate form (w - kc)W - = kVb - wcb) 2 + - kwVb)(w 1 y21 kVb - - 2 wb). (36) For VL = 0, there are no unstable solutions to Eq. (36) with ImW > 0. On the other hand, for VL -7 0, Equation (36) exhibits the Weibel-like cyclotron maser instability driven by the thermal anisotropy (T_._ > T11) associated with the choice of beam equilibrium distribution function in Eq. (6). For short wavelengths satisfying (V2 /c 2)(k 2c 2/w2 ) > C2/V2 and (V2 /c 2 )(kc 2 /w ,) > 1, it is straightforward to show that the characteristic maximum growth rate obtained from Eq. (36) is (ImW)MAx In the subscquCEnt analysis it 'Aill - c /b be useiui to compare the FI. 613 _ 0 Aith the Weibcl grow%th rate in Eq. (37). 14 (37) growth rate obtained for 10 FEL Instability for 6B = 0: We now consider the dispersion relation (34) for 6B 7 0 and (w, k) near the simulataneous zeroes of w = kc, w = kVb + nkoVb + w, (38) for a particular value of harmonic number n. Retaining the nth term in each of the summations, Eq. (34) can be approximated by V2 W (w- kc) (w - kb - nkob - wcb)+ g 1 2 c2 + where J2 Wb = _ -~b pbb nkokc2 j2 2(w - kVb) + (koV2 /Vb)fn,n (W - koVb - fkb - Wb)J, (39) J2d(wc/koVb)(B/Bo)]. Apart from the additional eo,o contribution, it is evi- dent from Eq. (39) that the n = 0 results for 6B 3 0 are identical to Eqs. (36) and (37) with the replacement WP __+ Wp2J2. The present analysis of Eq. (39) is therefore limited to harmonic numbers n 3 0. Denoting the simultaneous solutions to Eq. (38) by (&i, kc), we find (1 + VbIc)-(bcb/C + nkoVb/c) S(+gyV21 /c 2) ( OGn = InCc, (40) = V /c 2 + l/-y, and positive n corresponds to an where use has been made of I - -/C2 upshift in frequency. Since n 3 0 and w - kVb - nkoVb - web = 0 are assumed, it is valid to neglect the final term on the right-hand side of Eq. (39), and the dispersion relation can be approximated by (w - kc) (w - kV 2 I C2 - nlkqVb - W)2 + nkokc 2 j2. n 2kc 6 (41) The d&persion relation (41) is a cubic equaition for dhe conplex omcillation frequency w. In this reqprd. it is strjivhafiOr%ard t) shov. ihat the w2J 15 icrin on the left-hiand side of Fq. r (41) can be neglected in comparison with the right-hand side of Eq. (41) whenever n2 >4 WJ_ 2k2 ,2j"(2 27 VC2 C (42) which is easily satisfied in regimes of experimental interest, for all harmonic numbers n > 1. Defining V 1%= 4kI,W2 (43) , k, where (Wn, kn) are defined in Eq. (40), the and expressing w = 'n + 6wand k = kn + dispersion relation (41) can be expressed in the equivalent form . (6w - c6k) (5w - Vb6k) 2 + (44) The dispersion relation (44) [or equivilently, Eq. (41)] has been solved numerically for a broad range of system parameters (Figs. 1 and 2). As a simple analytic estimate of the characteristic maximum growth rate, we consider Eq. (44) for k = 0. Making use of Eq. (42), the dispersion relation (44) reduces to (6w) 3 = r3, which gives the characteristic maximum growth rate and real frequency shift y2 Im 5w =-n=- n 2 2 Rew J2 Vnkocw2Pb4c) 1/3 , (45) =-jn, for 6k = 0. Several points are noteworthy from Eqs. (44) and (45). First, for given harmonic number n, the characteristic maximum growth rate In6w defined in Eq. (45) will be largest whenever the argument of J corresponds to the first zero of J'(z) = 0, i.e., when Wob 6B = an, kVb Bo (46) where a,, is the first zero of J',,(x) = 0. Second, comparing Eqs. (37) and (45), the FEL growth ratce [ hich scales as (VI/c) 2/3n ' 3 ] can be significantly larger th3n the Weibel gro% th rate [which sczles as (V1 /c)n/ inhcrentil (w,-I/kl 2 ]. Third, the FEH. instability described by Eq. (44) is bro ad band in the sense that naii% harmonics arc unstable (Iq. (45)] c, cn when )(il,) is chosen to niaxinize 2row th Iir a particular valuc of n [I 'q. (4i)j. From 16 Eqs. (40) and (45), we therefor. conclude that the Lowbitron field coniguration may be attractive for FEL radiation generation at very high frequencies. We have solved Eq. (44) [or Eq. (41)] numerically for the complex eigenfrequency w in the region 18k/kol < I (Figs. 1 and 2). The normalized growth rate Imw/ cko is plotted versus normalized wavenumber k/ko for a tenuous electron beam with 'a = 2, Vb/c = 0.71, Vjc = 0.5, 6B/Bo = 1/3 and Wpb/cko = 0.05. [If (for example) the beam radius satisfies k R2 - 1/3, then the beam current lb nb7rR 2 eVb (bc/4)(-ibmc 2 /e)(w2 /c 2 kg)(kjR ) = 5A for this choice of beam parameters. Moreover, 6B/Bo = 1/3 corresponds to a mirror ratio R = (1 + 5B/Bo)/(1 - 6B/BO) = 2.] In Fig. 1, the value of B0 (as measured by wcb/cko) is chosen to satisfy (web/cko)(6B/Bo)(c/V) = 1.8 which maximizes the growth rate Imw for n = I in Eq. (44). On the other hand, in Fig. 2, a larger value of B0 is chosen with (wcb/cko)(6B/Bo)(c/V) = 4.2, which maximizes the growth rate for n = 3. In both Figs. 1 and 2, the maximum growth rate at each hannonic occurs for 6k = k - k =- 0. Moreover, for 6k/kb < 0, the growth rate Imw exhibits a very abrupt threshold. On the other hand, for 6k/kb > 0, the growth rate falls off more slowly. In Fig. 1, onlythe first three harmonics exhibit a substantial growth rate with Imw/cko > 0.01, whereas in Fig. 2 the first five harmonics all exhibit large growth rates at much higher output frequencies than in Fig. 1. It is important to note from Eq. (44) and Fig. 2 that the applied magnetic field is required to be quite strong for intense radiation at higher harmonics. For example, if Xo = 2w/ko = 8cm., the the value we6/cko = 8.946 chosen in Fig. 2 corresponds to B0 = 23.9kG. 17 f IV. CONCLUSIONS In this paper, the free electron laser instability has been investigated theoretically for a tenuous relativistic electron beam propagating parallel to a longitudinal wiggler field (Bo + 6Bsinkoz)Z6. The stability analysis (Sec. III) is based on the linearized VlasovMaxwell equations for perturbations about the beam equilibrium fu = (nb/ 2 1rp_)(p_ $mVj6(p. - -mVb). For (w./koVb)(6B/Bo) of order unity or larger, it is found that the transverse electron orbits experience a strong temporal modulation at harmonics of kov. n2 koVb. In the stability analysis, this orbit modulation is manifested by radiation amplification occurring near the simultaneous zeroes of w = kc and w - kVb - we = nkoVb for general harmonic number n, which corresponds to an upshifted (for n > 1) wavenumber In = (1 - Vb/c)-'(web/c + nkoVb/c). For (wb/koVb)(B/Bl) > 1, the instability is typically broadband with several harmonics excited (Figs. 1 and 2). Moreover, depending on beam and field parameters, the characteristic maximum growth rate can be substantial with Imw/cko = (/J/2) [(V2/4c2 )(wo 6 /c2 kg)n/2,n(w, /k VABd)}1/3. An important extension of this theory will be to include the influence of finite radial geometry on equilibrium and stability properties. This research was supported in part by the Office of Naval Research and in part by the Air Force Aeronautical Systems Division. 8 FIGURE CAPTIONS 1 Plot of normalized growth rate Imw/cko versus k/ko [Eq. (44)] for = 2, 0.71, V/c = 0.5, Wpb/cko = 0.05, 6B/BO = 1/3 and wcb/cko = 3.83. 2. Plot of normalized growth rate Imw/cko versus k/kO [Eq. (44)] for % = 2, Vb/c = 0.71, Vjc = 0.5, wp/cko = 0.05, 6B/Bo = 1/3 and wel/cko = 8.946. 19 Vb/c i REFERENCES 1. R. Q. Twiss, Aust. J. Phys. 11, 564 (1958). 2. J. Schneider, Phys. Rev. Lett. 2, 504 (1959). 3. A. V. Gaponov, Izv. Vuz. Radiofizika2, 837 (1959). 4. J. L . Hirshfield and J. M. Wachtel, Phys. Rev. Lett. 12, 533 (1964). 5. J. 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