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).
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H. S. Uhm and R. C. Davidson, Phys. Fluids 24, 1541 (1981).
20
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