PFC/JA-86-54
INFLUENCE OF UNTRAPPED ELECTRONS ON THE
SIDEBAND INSTABILITY IN A
HELICAL WIGGLER FREE ELECTRON LASER
Ronald C. Davidson
and
Jonathan S. Wurtele
September, 1986
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
INFLUENCE OF UNTRAPPED ELECTRONS ON THE
SIDEBAND INSTABILITY IN A HELICAL WIGGLER FREE ELECTRON LASER
Ronald C. Davidsont
Science Applications International Corporation, Boulder, CO 80302
Jonathan S. Wurtele
Plasma Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139
ABSTRACT
The detailed influence of an untrapped-electron population on the sideband
instability in a helical wiggler free electron laser is investigated for smallamplitude perturbations about a constant-amplitude (as = const.) primary electromagnetic wave with slowly varying equilibrium phase 6 . A simple model is
adopted in which all of the trapped electrons are deeply trapped, and the equilibrium motion of the untrapped electrons (assumed monoenergetic) is only weakly
modulated by the ponderomotive potential. The theoretical model is based on the
single-particle orbit equations together with Maxwell's equations and appropriate statistical averages. Moreover, the stability analysis is carried out in
the ponderomotive frame, which leads to a substantial simplification in deriving
the dispersion relation. Detailed stability properties are investigated over a
wide range of dimensionless pump strength B/rbck0 and fraction of untrapped
electrons f = Au b. When both trapped and untrapped electrons are present,
there are generally two types of unstable modes, referred to as the sideband
mode, and the untrapped-electron mode. For fu = 0, only the sideband instabil-
ity is present. As fu is increased, the growth rate of the sideband instability
decreases, whereas the growth rate of the untrapped-electron mode increases
until only the untrapped-electron mode is unstable for fu = 1. It is found that
the characteristic maximum growth rate of the most unstable mode varies by only
a small amount over the entire range of f u from fu = 0 (no untrapped electrons)
to fu = 1 (no trapped electrons).
This suggests that it is a serious oversight
to neglect an untrapped-electron component when calculating the detailed linear
and nonlinear evolution of the beam electrons and the radiation field.
Permanent address: Plasma Fusion Center, Massachusetts Institute of
Technology, Cambridge, MA 02139.
2
I. INTRODUCTION AND SUMMARY
Free electron lasers (FELs),1
4
as evidenced by the growing experi-
mental5- 22 and theoretical 23 -70 literature on this subject, can be
effective sources for coherent radiation generation by intense relativistic
electron beams.
Recent theoretical studies have included investigations
of nonlinear effects 23 -4 7 and saturation mechanisms, the influence of
finite geometry on linear stability properties,48-53 novel magnetic field
geometries for radiation generation, 48 ,54 -58 and fundamental studies of
stability behavior. 59-70
One topic of considerable practical interest is
the sideband instability36 which results from the bounce motion of electrons
trapped in the (finite-amplitude) ponderomotive potential.
Both kinetic
23 -25
and single-particle36-47 models of the sideband instability have been
developed, and numerical simulations 39 -47 have been carried out.
However,
with the exception of the recent kinetic formalism developed by Davidson
et al.,23-25 the analytical treatments have consistently neglected the effects of
any untrapped-electron population.
The purpose of the present analysis is to investigate the detailed
influence of untrapped electrons on the sideband instability.
Small-
amplitude perturbations are assumed about a constant-amplitude (a = const.)
5
primary electromagnetic wave with slowly varying equilibrium phase 6 .
Moreover, we adopt a simple model in which all of the trapped electrons
are deeply trapped, and the equilibrium motion of the untrapped electrons
(assumed monoenergetic) is only weakly modulated by the ponderomotive
potential.
The theoretical model (Sec. II) is based on the single-particle
orbit equations together with Maxwell's equations and appropriate statistical averages. 36 ,37
Like our recent treatment 37 of the sideband instability
3
(which neglects the effects of untrapped electrons), the present analysis
is carried out in the ponderomotive frame, which leads to a substantial
simplification in the analysis.
The theoretical model and assumptions are described in Sec. II.
A
tenuous, relativistic electron beam propagates through a constant-amplitude
helical wiggler magnetic field with wavelength X0 = 21T/k
malized amplitude aw
=
0
= const., nor-
2
eBw/mc k0 = const., and vector potential specified
by [Eq.(1)]
mc2
A(x)
=
--
a (coskozix + sinkoz
The model neglects longitudinal perturbations (Compton-regime approximation
with 60 ~ 0) and transverse spatial variations (a/ax = 0 = a/3y).
Moreover,
the analysis is carried out for the case of finite-amplitude primary
electromagnetic wave (ws ,ks) with right-circular polarization and vector
potential specified by [Eq.(2)]
2
in
A (xt) =
-
a (z,t) {cos[ksz
-
st + 6s(Zt)]
sin[ksz - wst + 6s(z t)]W
,
where the normalized amplitude S(z,t) and wave phase 6s(z,t) are treated
as slowly varying (Eikonal approximation).
A detailed investigation of the
sideband instability simplifies considerably if the analysis is carried out
in the ponderomotive frame 37,7 1 ,72 moving with velocity [Eq.(3)]
v
=
"s
ks + k0
4
In the ponderomotive frame ("primed" variables), the nonlinear
evolution of as(z',t') and 6'(z',t') is described by Eqs.(5) and (6).
Here, correct to lowest order in (w's )
served in the ponderomotive frame (dy/dt'
3
s/3t')
=
<< 1, energy is con-
0), and the axial orbit
(ks + k0 )/ p is the wave2 2 -2
number of the ponderomotive potential, and yp = (1 - vp/c ) 2. Moreover,
e
(t') = k'z (t') solves Eq.(13), where k
are related by the
the real oscillation frequency w'5 and wavenumber k'
5
dispersion relation (7).
In obtaining Eqs.(5), (6), (7), and (13), it
is assumed that all electrons have zero transverse canonical momentum,
i.e., P'. = 0 = P'..
xj
In Sec. III, Eqs.(5), (6) and (13) are used to investigate the influence of untrapped electrons on the sideband instability for smallamplitude perturbations about a primary electromagnetic wave with constant
amplitude a5 = const. (independent of z' and t').
electrons are treated as distinct components.
The trapped and untrapped
Moreover, the principal
assumptions in the present analysis are the following (Sec. III.A).
(a) All of the trapped electrons are deeply trapped with a sharply
defined energy y
=
=
[1 + (aw
-
)2I.
This implies that the
trapped electrons are spatially localized ("bunched") near the bottom of the
ponderomotive potential (Fig. 2).
The average density of the trapped electrons
in the ponderomotive frame is nT = n
yp
(b) All of the untrapped electrons have a sharply defined energy
y' = j'
>
j' = [1 + (aw + a)21, where j'' is sufficiently large that
the motion of the untrapped electrons is only weakly modulated by the
ponderomotive potential (Fig. 2).
The average density of the untrapped
electrons in the ponderomotive frame is n'
= 6/
(c) Consistent with (a) and (b), we assume that the perturbations
0
are about a quasi-steady equilibrium state characterized by as = const.
5
(independent of z' and t') and 36 /at' = 0. However, a slow spatial
0
variation of the equilibrium phase 6
is required [Eq.(41)]
Following a discussion of the quasi-steady equilibrium state (Sec.
III.B), we analyse the linearized wave and particle orbit equations
(Sec. III.C), and derive the dispersion relation (70) for small-amplitude
perturbations in the ponderomotive frame (Sec. III.D).
Here, it is
assumed that the perturbed amplitude 6 s(z',t'), the perturbed phase
6'(z',t'), etc., vary as
exp(-i(Aw')t' + i(tk')z'
where Im(Aw') > 0 corresponds to instability (temporal growth).
The dis-
persion relation (70) relates Aw' to Ak' and other system parameters
such as ao, k , P , K', etc.
Finally, in Sec. IV, the dispersion relation (70) is used to investigate
detailed properties of the sideband instability including the effects of
the untrapped electrons.
First, we transform Eq.(70) back to the laboratory-
frame frequency w = ws + Aw and wavenumber k = ks + Ak making use of the
transformation in Eq.(71) relating (Aw,Ak)to (Aw',Ak').
In this regard,
it is convenient to introduce the shorthand notation [Eq.(73)]
A= AW - v
Ak
v Ak
AK= k0
where vp= Ws/(ks + ko),
-
c ks
and (ws,ks) are the frequency and wavenumber of
the primary electromagnetic wave in the laboratory frame.
After some
algebraic manipulation (Sec. IV.A), it is straightforward to show that the
dispersion relation (70) can be expressed in the equivalent form [Eq.(80)]
6
1
B TCO0.B)
(AQ - cAK)
(A-)
(rc03
(r ck
O)2
(AO)(AQ2
B6
40 2(rck
Q2
-B
-
22 - 21
c 2 2
2~n ~ck
cAK) 2
au (rTcko) 3 + 4(AQ)(AQ - cK)
'cko
x
2
+
2 (rTcko)
(2
Tck
2 A 2
)+
+
k
2
(+2 2
B
In Eq.(80), au
4
[Eq.(75)] is a measure of the ratio of
=
the untrapped electron density to the trapped electron density, and
s' = (1 + vp/c)s' [Eq.(78)] is proportional to the speed of the untrapped
electrons in the ponderomotive frame.
Moreover, OB is the bounce fre-
quency of deeply trapped electrons defined by [Eq(24)]
1 + vc )
B=
+ aw
k
in the laboratory frame, and rT is the (small) dimensionless gain parameter
defined by [Eq.(81)]
3
rT
1
4
a
(4 T2e /m) (1 + v /c)
(1 + a2)3/
where use has been made of
w
YpC
= YTyp.
K0
<<
vp/c
Consistent with Assumption (b) at
the beginning of Sec. III, we require that .ck0 be sufficiently large
in comparison with QB in Eq.(80) in order that the untrapped-electron
motion be only weakly modulated by the ponderomotive potential (Fig. 2).
7
Equation (86), which is equivalent to Eq.(80), constitutes the final
dispersion relation which is analysed numerically in Sec. IV.B.
For fu =
Eq.(86) is
Pidb = 0, which corresponds to no untrapped electrons ( u = 0),
the familiar dispersion relation for the sideband instability 37 ,73 in
circumstances where the equilibrium wave phase is slowly varying [Eq.(41)].
For f u
0, however, it is found that the untrapped electrons can significantly
modify stability behavior (Sec. IV.B).
Detailed stability properties are
investigated over a wide range of dimensionless pump strength
B /Fbck0
(where rb = n rT/nT) and fraction of untrapped electrons fu =
u b. When
both trapped and untrapped electrons are present, there are generally
two types of unstable modes, referred to as the sideband mode, and the
untrapped-electron mode.
present.
For fu = 0, only the sideband instability is
As fu is increased, the growth rate of the sideband instability
decreases, whereas the growth rate of the untrapped-electron mode increases
until only the untrapped-electron mode is unstable for fu
=
1.
The present analysis indicates that the detailed properties of
Im(A2)/rbck0 versus AK/rbk0 are quite different for the two unstable modes.
Equally important, however, it is found that the characteristic maximum
growth rate of the most unstable mode varies by only a small amount over
the entire range of f u from fu = 0 (no untrapped electrons) to fu = 1 (no
trapped electrons).
8
THEORETICAL MODEL AND ASSUMPTIONS
II.
A. Basic Equations and Assumptions
A tenuous, relativistic electron beam propagates in the z-direction
through a constant-amplitude helical wiggler magnetic field with wavelength X0 = 27r/k 0 = const., normalized amplitude aw = eBw/mc 2 k0 = const.,
and vector potential specified by
mc2
Aw(x) =
-
e
aw(cosk 0 zsx + sink 0 z)
.
(1)
The model neglects longitudinal perturbations (Compton-regime approximation,
6p 2 0) and transverse spatial variations (a/ax = 0 = a/ny).
Moreover, the analysis is carried out for the case of a finite-amplitude
primary electromagnetic wave ( s,k s) with right-circular polarization and
vector potential specified by37
mc2
A (x,t) = e
s(z,t) cos[ksz - wst + 6s(z,t)] x
(2)
-
sin[ksz - wst + 6s (z,t)]
where the normalized amplitude
,
s(z,t) and wave phase 6 (z,t) are treated
as slowly varying (Eikonal approximation).
Here, -e is the electron
charge, m is the electron rest mass, and c is the speed of light in vacuo.
In Eqs.(l) and (2), the wiggler magnetic field is determined from B =
v
x
A ,and
the electromagnetic wave field is determined from Bs = v x As
and Es= -c~ 1 As/at.
A detailed investigation of the sideband instability
simplifies considerably if the analysis is carried out in the ponderomo37 7 1 72
tive frame moving with velocity , ,
9
v
=
P
(3)
s
ks + k0
Therefore, the present analysis is carried out in ponderomotive-frame
variables (z',t',y') defined by the Lorentz transformation
= y (z - vPt) ,
z'
(4)
t' = y p(t - vpz/c)2
= yp(y - vppz /mc
Y
1 ,ymc2 _
where yp = (1 - v2 2()/c
2c4 + c2
p
2
2 + c2
2 + c2
2)
is the
mechanical energy, and the components of momentum (p',p',p') are related to the velocity v' = dx'/dt' by p'
=
y'mv'.
In the ponderomotive frame, the slow nonlinear evolution of as(z',t')
and 6'(z',t') is described by
2w'
a
-
k'c 2 a4)T
+ ss
W'
at'
+
-
2'as
+
at' +
2
_w
s
1
e2a
m
az'
k'c 2
37
az Im
w
'=
L'
,
(5)
y
L
4Te2aw 1
a
sin(e' + 6')
is
cos(e'
js,
+ 6')
Y
()
6
are related
where the real oscillation frequency w'5 and wavenumber k'
5
by the dispersion relation 37
c2 k 2
In Eqs.(5)-(7), (
'''>
+
2-1
denotes statistical average, and the axial
and energy y (t') of the j'th electron solve 37
orbit e6 (t') = k'z
3
pj (t')
iJs
10
c2 k'2a
d2
d-'
Im [es xp(i
2
s +
+
i 6 ')
(8)
c2 k'a
Re exp(ie )
is
Y
1 dz
a
+
c dt' at
az
.exp(i')
and
d
d
a
w Re et'
at'
_
y
dt'
sexp(i 6s')
.
(9)
(ks + k0 )/yp is the wavenumber of the pondero-
In Eqs.(8) and (9), kp
motive potential, and y! is defined by
3
,2
Y 2 =
+
in the ponderomotive frame.
-mc 2y /az
Eq.(10).
2+a - 2awRe
zj + a
s1 + i6')
sa
(10)
In obtaining Eqs.(8) and (9) from dp ./dt'
ZJ
and dy/dt' = ay /at'
we have neglected
=
2 << 1 + a2 in
Moreover, it is assumed that all electrons have zero transverse
canonical momentum, i.e., P
=
0 = P .
There is some latitude in specifying the precise operational meaning36
of the statistical averages
(
---
)
occurring in Eqs.(5)-(7).
For
present purposes, let us assume that the orbits z (t') and y (t') have
been calculated from Eqs.(8) and (9) in terms of the initial values z (0)
and y (0).
Then the simplest definition of the statistical average
(
over some phase function
>
(e'5 (0)
,Y(0)) is given by
e(0),y(0)))
1 KZ
7de6
...- 2
=n
J
20
0
1
dy6G(e6,y6)P(e6,y6)
.
11
Here, h is the average density of the beam electrons in the ponderomotive frame, and G(e6,y6) is the (probability) distribution of electrons
in initial phase e6 and energy y6.
Moreover, L' = 27r/k' is the basic
periodicity length in the ponderomotive frame.
Equations (5)-(9) constitute a closed description of the nonlinear
evolution of the system.
(8) and (9)
In this regard, further simplification of Eqs.
is possible by virtue of the assumption of slowly varying
wave amplitude and phase (Eikonal approximation), i.e.,
-1 a
- [asexp(i'.)
IWS' >> lasexp(16'.)
,
(12)
-1 3
Ik
>>
[asexp(i')]
-
.
isexp(id')
In particular, to lowest order, it is valid to neglect the local time
and spatial derivatives on the right-hand sides of Eqs.(8) and (9).
This gives the approximate dynamical equations 37
c2k 2a
d2
6
dt'-2
is
+
W Im
sexp(i
Y2
s + id')
s
=
0
,
(13)
d
-
y' = 0
dt'
(14)
(
The major benefit of carrying out the analysis in the ponderomotive frame
is evident from Eqs.(13) and (14).
To lowest order, the particle energy
y can be treated as constant in Eqs.(5)-(7) and (13).
In the subsequent analysis, we make use of the closed description
of the nonlinear evolution of the system provided by Eqs.(5)-(7) and
Eqs.(13) and (14).
12
B. Definitions and Notation
For future reference, in this section we establish the basic definitions and notation to be used in the stability analysis in Secs. III and IV.
The wave frequency and wavenumber (w',k') in the ponderomotive frame
are related to the wave frequency and wavenumber (w,k) in the laboratory
frame by
W' = y (w - kv ) ,
k ' = 79(k - wv /C 2
2/c2
. As a special case, we
/(k + ko)
where vp =W
p ss
0P and yp= (1 - v
obtain w'= y(W - k v ) from Eq.(15), which gives
'=
(16)
yk 0v
In the present analysis, it is also assumed that the electron beam
is sufficiently tenuous that beam dielectric effects can be neglected in
the dispersion relation (7) (and its laboratory-frame analogue).
gives the vacuum dispersion relation w 2
c k ,for
=
This
c2 k'2, or equivalently w
the primary electromagnetic wave.
=
Assuming a forward-moving
electromagnetic wave, we solve the simultaneous resonance conditions
Ws = +cks
(17)
Ws = (ks + k )v
0 p
for ws and ks.
This readily gives the familiar results 37
Ws = Y (1 + vp/c)k0
v
'
(18)
2
k= Yp (1 + vP/c)(v p c)ko
13
where y
p
=
(1-v /c2 -1, ana vp =
pP
s/(ks + ko) is (nearly) synchronous
k)
with the average axial velocity Vb of the beam electrons.
Moreover,
from Eq.(18), the ponderomotive wavenumber k= (ks + k0 )/yp can be
expressed as
k' = yp(1 + vp/c)k 0
(19)
.
In circumstances where perturbations are about a primary electromag-
netic wave with amplitude a
=
const. (independent of z' and t'), it is
useful in analysing the orbit equation (13) to introduce the bounce fre-
23
quency wB(YJ) defined by
=
(c2 k 2aa
>
0 and a
>
2 )1
.
(20)
p ws j
B(YP
Here, aw
0/
0 are assumed without loss of generality, and
is the effective bounce frequency of deeply trapped electrons
A detailed analysis 23 of Eqs.(10) and (13) shows
with energy y'.
(y)
3
that the zero-order electron motion is untrapped for energies y
satisfying (Figs. 1 and 2)
y
[1 +(aw +)2
.
(21)
That is, when Eq.(21) is satisfied, the particle motion is modulated by
s/dt' does
the ponderomotive potential, but the normalized velocity de is
, the elecnot change polarity (Fig. 1). On the other hand, for y <
trons are trapped, and the zero-order motion described by Eq.(13) is
cyclic, corresponding to periodic motion in the ponderomotive potential.
From Eqs.(10) and (13),
it is readily shown that the minimum allowable
energy of a trapped electron is 23
'
=
+ (aw
-
a)2
.
(22)
14
Because ^ <<s aw in the regimes of practical interest, we note from Eqs.
(21) and (22) that the characteristic energy of a trapped electron is
approximately
'
(1 + a2w).
The stability analysis in Secs. III and IV specializes to the case
untrapped electrons with
where there are two classes of electrons:
energy y
= j
>
i', and deeply trapped electrons with energy Y=
For the deeply trapped electrons, the effective bounce frequency in the
laboratory frame is defined by QB =
GB
Because
s <<
=
(c2k 2awO
aw we estimate
use of Eq.(19) to express k
~ 9
j'
= yp
' i.e.,
B
(23)
.
2
2
(1 + a w
(1 + v /c)k
in Eq.(23), and make
Equation (23) can then
be expressed in the equivalent (and more familiar) form
awas
(1 + a2
1 +
QB =
c
ck0
(24)
.
Continuing with definitions, we denote the average density of the
trapped electrons in the ponderomotive frame by
average density of the untrapped electrons by n'
= YTyp, and the
i
=
u/y p
It is con-
venient to introduce the corresponding plasma frequencies defined by
2
WpT
4 Tn
e2
4 Te2
m
(25)
pu
A
47rr'e
m
2
4, ue
dgn
A detailed investigation of Eqs.(5) and (6) (Secs. III and IV) shows
15
that the appropriate small parameters, E
<< 1 and e'
U
<< 1, used in
analysing the wave equations are defined by
aw
Ejck
S2
=
a
(26)
a w
w pu
e u'ck'
s'"'s
Here, w' = Y k0 p is defined in Eq.(16), 9' is the energy of the untrapped
are defined in Eqs.(22) and (25).
and
2',
electrons, and
from Eqs.(25) and (26) that ej and e'
are related by Ej
Note
=e
In Sec. III, we will find that Ef is related to the (slow) variation of
by 36a0/az' = efck, [Eq.(41)].
the equilibrium wave phase 6
Finally, for future reference, we introduce the small dimensionless
parameter r
37
defined by23 ,
T)
1 a2
- 71
4
2
T
_
(1 + v /c)
1 << 1.(27)
k0
vp/c
In the absence of untrapped electrons (Pi'
= 0), the quantity (3)irTck0/2
can be identified with the linear gain (temporal growth rate) in the
weak-pump regime (B /rTck0
<
1).37
Moreover, from Eqs.(19), (23)
(26) and (27), it can be shown that rT and Ej are related by
E
From Eq.(28), we note that e
=
2rT
(rT ck 0
.
<< 1 necessarily requires that the zero-
order wave amplitude a be sufficiently large that (B /rTcko)2 >> 2rT
(a small parameter).
(28)
16
III.
STABILITY ANALYSIS FOR SMALL-AMPLITUDE PERTURBATIONS
A. Assumptions and Model
We now make use of Eqs.(5), (6) and (13) with y
= const. [Eq.(14)]
to investigate detailed stability properties for small-amplitude perturbations about a quasi-steady equilibrium state.
The principal assumptions in
the present analysis are the following:
(a) All of the trapped electrons are deeply trapped with a sharply
defined energy y=
T ~
=
[1 + (aw
-
) 2.
From Eq.(13), this
implies that the trapped electrons are spatially localized ("bunched") near
the bottom of the ponderomotive potential with e' + 6' ~ 2n7, where n = 0,
±1, ±2,'-" is an integer.
The average density of the trapped electrons in
the ponderomotive frame is n
= YTp
(b) All of the untrapped electrons have a sharply defined energy
y
=
)21, where j' is sufficiently large that the motion
' > y'= [1 + (aw +w ^0su
of the untrapped electrons is only weakly modulated by the ponderomotive
potential.
Strictly speaking, this requires that j 2 _ 42 be large in
comparison with the total well depth 4a w s '
(c) Consistent with (a) and (b), we assume that the perturbations are
about a quasi-steady equilibrium state characterized by a5 = const. (independent of z' and t') and 36 5 /at' = 0. However, a slow spatial variation
of the equilibrium phase
60
is required [Eq.(41)]. 37 ,71
In the subsequent analysis, we denote the axial coordinate of the
(deeply) trapped electrons with energy 9j- ~'
by e8
= k'zj(t'), and the
axial coordinate of the untrapped electrons with energy jK' is denoted by
e' = k'z'(t').
The corresponding bounce frequencies are defined by
17
WB(Y)
WBT
2)
(c2k'2 a a
=
(29)
WBu
WB(Y')
(c2 k 2 a
=
2)
a/
where ao = const. is the equilibrium amplitude of the primary electromagnetic
wave.
Note from Eq.(29) that WBu
(Y-/Yu)wBT = BT because j' typically
=
exceeds j'' by only a small amount for awa
<< 1. Making use of Assumptions
(a) - (c), it readily follows from Eqs.(5), (6) and (13) that the nonlinear
wave equations and the equations of motion for the trapped and untrapped
electrons can be expressed as
2
ck
+
at'
.2
s
W
a
a
T sin(ej+6) +
as
az'
at'
2w , 'u
2
2
N + 6 s'+ aw pu cos (e ' + 6')> u
T cos
a
2ws'i'
ks' a
W' az'
C
(<sin(e'+ 6s) >u
2-' '
2 ,
s
2
(30)
(31)
2w ' i'
and
2
e
+ ^BT
dt
sin(ej + 6') = 0
(32)
+ 6') = 0
(33)
as
2
e' + 2Bu
dt'
In Eqs.(30) and (31),
sin(e'
aSO
2 and
2 are defined in Eq.(25), and the statistical
~pT
anpu
average <.''>u denotes an average over initial phases of the untrapped
electrons, i.e.,
27
fde'(0)
-u
u..
0
'
(34)
18
defined in Eq.(26), the nonlinear
In terms of the small parameters eT and E'
wave equations (30) and (31) can be expressed in the equivalent form
a
-
i- a
c~k'
+ --
s
at,
-
as = sjck'ssin(ej + 6') +
Pck's
<sin(e' + 6 )>u
,
(35)
az'
and
a
s-
c 2k
+ -
,
ck'a cos(e
+
6') + £ ck a<cos(eu + 6')>
.
(36)
The coupled equations (32), (33), (35) and (36) constitute a closed description of the nonlinear evolution of the system within the context of
Assumptions (a) - (c).
We now make use of Eqs.(32), (33), (35) and (36) to investigate detailed
properties of the sideband instability (including the influence of both
trapped and untrapped electrons) for small-amplitude perturbations about
a primary electromagnetic wave with constant amplitude a and slowly varying
phase 6 . Each quantity is expressed as its equilibrium value plus a perturbation, i.e.,
a=
s
0
= 0 +
el
e'
+
=
6 as'
(37)
el
0 + 6e'.
For the deeply trapped electrons with ej + 6'
loss of generality in Eqs.(32), (35) and (36).
2nn, we take n
=
0 without
19
B. Equilibrium Model
We first consider equilibrium solutions to Eqs.(32), (33), (35) and
(36) in the absence of perturbations, i.e., 6as = 0, 6' = 0, 6ej = 0 and
68' = 0. Consistent with the assumption that j is sufficiently large in
comparison with
', the zero-order orbit of an untrapped electron calculated
from Eq.(33) can be approximated by
6 0 = 80(0) +
'ck't'
(38)
.
Here, a'c = const. is the average velocity of an untrapped electron in the
ponderomotive frame.
In Eq.(38), note that the modulation of the electron
orbit by the ponderomotive potential has been neglected.
Making use of
Eq.(38) and the definition of the phase average in Eq.(34) it follows
trivially that
(39)
<sin(e u + 6 )>u = 0 = <cos(e u + 6 )>
That is, in Eqs.(35) and (36), the untrapped electrons do not contribute to
any change in the equilibrium amplitude a and phase 60.
Therefore, an
appropriate quasi-steady equilibrium state consistent with Eqs.(32), (35)
and (36)
is described by 37
(40)
aa0
at,
= 0 =
s
a
80
az'
s
and
at,
s
(41)
Cck'
az' 6s =
20
Note from Eq.(41) that c << 1 is required in the present analysis in order
that the change in 6
is small over the scale length of the ponderomotive
potential (X' = 2'rk'
). Making use of Eq.(28), the inequality
j
<
1
is equivalent to (B /rTck0)2 >> 2r T, where rT is the small parameter defined in Eq.(27).
To summarize, the equilibrium state is characterized by free-streaming
untrapped electrons [Eq.(38)], trapped electrons with e0 + 6
and a primary electromagnetic wave with constant amplitude
0s
[Eq.(41)].
slowly varying phase with 36 s/az' = c'ck'
T
a0
= 0 [Eq.(40)],
[Eq.(40)] and
C. Linearized Equations
We now linearize Eqs.(32), (33), (35) and (37) for small-amplitude
perturbations about the equilibrium state described by Eqs.(38)-(41).
In
this regard, it is convenient to introduce the normalized amplitude perturbation 6A
defined by
6a
7A
as
(42)
For ej + 6' ~ 0, it is straightforward to show that the small-amplitude
6'ea, 6As and 6'
perturbations 6se,
d2
evolve according to
2
66T + WBT(Se
+ 6') = 0
(43)
,
dt'
' +
Bucos(e0 + 6 )6e'
dt'
(44)
=
-B Im (6As + i6')exp(ieu + i5 )
,
21
(
+
cc2k '
-
3
-)6AS
= ejck (6ej + Z')
T
W ' az'p
at'
s
(45)
e'ck < (6e' + 6')cos(e 0 + 60 )
+
u
up
a
-
+
c2kc' a
-2 s
at,
az'
u
U
6_' + e ck 6As
(46)
=
-e'ck' <(66' + 6')sin(e
+ 6
In analysing Eqs.(43)-(46) it is useful to express
6e' = 6i'exp(ie 0 + i6 ) + 6 '*exp(-ie0 - i6 )
where the complex amplitude 6*' = 6*P + i6*i
denotes the complex conjugate of 6*'.
(47)
,
is slowly varying, and 6p'*
Making use of Eqs.(44) and (47)
it is straightforward to show that 6p' evolves
and 6u = 0u (0) + auck't',
u
according to
2
d2+
d
+
'ck d
dt'
iu
2i$'ck'
+
'2c2 k2] 1
0
0
2
ucos(e
+
6
auck
j 6i'
s
u
Bu
(48)
Bu(6As + i6')
2i For untrapped electron energy ju' sufficiently large in comparison with
'
=
2 cos(e 0 + 60 ) in comparison
[1 + (aw + 0s)2]1, it is valid to neglect Bu
s
u
2c2 k 2 in Eq.(48). Therefore, Eq.(48) can be approximated by
with up
d2
\d
d
+
2i 'ck
d
1
2c2k 2 (6 R
+
I )=
-
-
Bu
+ i ')
(49)
22
In Eq.(49), 6P, 6P , 6As and 6' are all real quantities, and it follows
that 6p'*= 6ip - i6Pj evolves according to
d2
-
2iW'ck'
dt
d
dt'
2 2k2
c
1
(6
p
u
(6As - i6') .
- i^
(50)
2i
Evidently, Eq.(49) [or Eq.(50)] describes the slow evolution of 6*p and 6
in response to the amplifying wave perturbations 6As and 6'.
Substituting Eq.(47) into Eqs.(45) and (46), and making use of
u
-0
'ck't' and the definition of the phase average in Eq.(34), it is
e (0)
u + up
straightforward to simplify the untrapped-electron contributions in the
We readily obtain
linearized wave equations.
<
(6e'
+ 6')cos(e0 + 60))u = <6eu'cos(eo + 60)>u = 6qp
<
(6e'
+ 6')sin(e0 + 60 )u
= <u6e'sin(e
+ 6 )>u = -&p{
(51)
,
.
(52)
10 + 60) or sin(e 0 + 60
In Eqs.(51) and (52), the average of 6' times cos(e
vanishes because 6' is assumed to be slowly varying.
Substituting Eqs.(51)
and (52) into Eqs.(45) and (46), we obtain for the evolution of 6As and 6'
-
a
c 2k '
+
-
at,
-
a
A = ejck'(6e
+ 6')
+ E'ck'6
,
(53)
s az'
a
c2 k'
+ at'
s
-
a
6' = -esck'As + E'ck'6
.
(54)
a zI)
To summarize, in the present analysis the final set of coupled, linearized equations for 66e,
(50), (53) and (54).
64),
64',
^As and 6' is given by Eqs.(43), (49),
23
D. Dispersion Relation in Ponderomotive Frame
We now assume that the t'- and z'-dependence of the perturbations
in Eqs.(43), (49), (50), (53) and (54) is proportional to
exp[-i(Aw')t' + i(Ak')z'] ,
where Im(Aw')
>
(55)
0 corresponds to instability (temporal growth).
Consistent
with neglecting beam dielectric effects in the dispersion relation (7), we
also approximate c2 k'/w' = c in Eqs.(53) and (54).
The linearized equations
(43), (49), (50), (53) and (54) then become
6)
(Aw' - Vck+
-
u
-
(Aw' + S'ck')2(6*
uip
' 2+
=
2
2
-
= -
idj)
=
2i
w(u(6^As + iS')
1
2
2i WBu(6As
-
(56)
,
-
(57)
(58)
16')
+ _') + e'ck'6+,
(59)
- i(Aw' - cAk')6' = -E'ck'6As + E'ck'6*'.
(60)
-i(Aw'
- cAk')6As = ejck'(de
In Eqs.(57) and (58), it is useful to introduce the untrapped-electron susceptibilities X+ and X~ defined by
2
-2
+ Ak',A ')=+
-
(Aw'
+
'ck')
2
X~(Ak',Aw') =
(Aw'
Bu
+ a'ck')
(Aw'
-
2
Bu
a'ck')
2
Bu(62)
(Aw' - 'ck')
'
(61)
24
From Eqs.(57) and (58) we readily obtain
1
6P
=
- (ix-6As + x +s)
(63)
4
= 1 (-x+A 5 s + ix')
which express 6t
and phase 6'.
(64)
,
4
and 64j directly in terms of the perturbed amplitude 6As
Solving Eq.(56) for 6ej + 6' in terms of 6', and substituting
Eqs.(63) and (64) into Eqs.(59) and (60) give two coupled homogeneous equations for 6As and 6'.
Setting the resulting two-by-two determinant equal
to zero, we obtain, after some straightforward algebraic manipulation,
(Aw'
- cAk') + 1
e'ckx-
2
(65)
(Aw)2
11
2 77 + - 'ck x
(Awl) - BT
u kj
=Eck'
ck' + - Ek
p
]-
Equation (65) is the desired dispersion relation which relates the (complex)
oscillation frequency Aw' to the wavenumber Ak' and the system parameters
Ej,
C ', ck', etc.
Here, Ej., e', x+ and x~ are defined in Eqs.(26), (61)
and (62).
Before investigating detailed stability properties (Sec. IV), we show
that the dispersion relation (65) reduces to familiar results in two
limiting cases:
electrons (Pj
=
(a) no untrapped electrons (p' u = 0), and (b) no trapped
0).
= 0):
No Untrapped Electrons (i'
u
that F-'= 0, and Eq. (65) reduces to
For
i'
u
=
0, it follows from Eq.(26)
25
(A
'2c2k 2
2
k')
-
(Al
2c k 2
c
(66)
2
(Awl )
wBT
-
Equation (66) can be expressed in the equivalent form
0 = 1
2c2k2
c p
cAk')
(Aw'
2
WBT
(Aw')
(67)
,
which is the familiar dispersion relation 37,73 for the sideband instability
assuming slowly varying equilibrium phase S and no untrapped electrons.
The detailed stability properties predicted by Eq.(67) are investigated
in Ref. 37.
No Trapped Electrons (nj = 0):
For
=
0, it follows from Eq.(26)
that Ej = 0, and Eq.(65) can be expressed as
&2
1
0
(Aw' - cAk')
-
-
2
u
-1
Bu
eck'
(Aw' -
'ck ')(
6
(68)
x
where e'
u and
(Aw'
1
- cAk') + - uck'
22IA 9
Bu
Bu
a'+Sck') u
+
72
p
2 are defined in Eqs.(26) and (29), and use has been made
Bu
of Eqs.(61) and (62).
Apart from a sign, the two factors in Eq.(68) are
identical under the (simultaneous) reflections Aw'
-+
-Aw' and Ak'
-+--Ak'.
Setting the first factor in Eq.(68) equal to zero gives the dispersion
relation
(Aw' - cAk')(Aw' -
upck')
a2 2 c k'2
= w Pu3
where use has been made of Eqs.(26) and (29).
(69)
Consistent with Assumption (b),
we note that Eq.(69) is independent of the equilibrium wave amplitude a .
26
When Aw' and Ak' are transformed back to the laboratory frame, it is
straightforward to show that Eq.(69) is similar to the Compton-regime dispersion relation66
obtained in the small-signal limit in the absence of
trapped electrons.
We now return to the full dispersion relation in Eq.(65).
Alternate Form of the Full Dispersion Relation:
It is useful to re-
write Eq.(65) in an alternate form which clearly delineates the trappedand untrapped-electron contributions.
Making use of Eqs.(61) and (62),
rearranging terms in Eq.(65), and multiplying Eq.(65) by [(Aw')2 _
]/(Aw) 2(A
2BT
- cAk) 2, it is straightforward to show that the dispersion
relation can be expressed in the equivalent form
2
1
(Al)2
2
1
=-
ck 'Bu (
, 2 c2k'2
BT _
(Aw
- CAk')
BT
[(Aw)2
2c k
- cAk')2 [2
,w1)2(Aw'
2
(70)
x
1
2 Eu'ck
+ ejck' [(Aw') 2 +
Bu + 4Aw'(Aw' - cAk')u'ck'
'2 c2 k 22]
Her,
e,
',
2' 2
1 +
(Awl)
- eBT
BTand Euare defined in Eqs.(26) and (29).
of untrapped electrons (c'
In the absence
0), we note that Eq.(70) reduces directly to
the familiar dispersion relation (67) for the sideband instability.
That
is, the effects of the untrapped electrons (the terms proportional to E')
are incorporated on the right-hand side of Eq.(70).
27
IV.
ANALYSIS OF DISPERSION RELATION
A. Dispersion Relation in Laboratory Frame
We now transform the full dispersion relation (70) back to the
laboratory frame.
From Eq.(15), it follows that
Aw'= yP (A - vp Ak)
(71)
Ak'
=
yp [Ak - (vp/c2
where Aw and Ak are the frequency and wavenumber of the perturbations in
the laboratory frame.
Making use of Eq.(71) and y
= (1 - V /C2 -1,
it is
straightforward to show that 37
Aw' - cAk' =
where ks
vAk
p(1 + vp/c)[ (Aw - vp Ak) - ck0 -2 -
y (1 + V /c)(V
/c)k0 is defined in Eq.(18).
,
(72)
We further intro-
duce the shorthand notation
AQ =Aw - vp Ak
(73)
v
AK = k0
Ak
c ks
Then, from Eqs.(71)-(73), Aw' and Aw' - cAk' can be expressed in the
equivalent form
Aw'
y AQ
(74)
Aw' - cAk'
=
yp(1 + vp/c)(A2 - cAK)
28
Equation (74) expresses Aw' and Aw' - cAk' directly in terms of AS and AK,
which are related to Aw and Ak in the laboratory frame by Eq.(73).
To simplify the dispersion relation (70), it is convenient to introduce the dimensionless parameter
au
--
'/
' 3
_- ) ,
(75)
which is a measure of the ratio of the untrapped electron density to the
=
n'/fj
trapped electron density,
nBuT.
Making use of the definitions of
2 [Eq.(25)],
T and pu
= yp(1 + v /c)ko [Eq.(19)],
' = ypkovp [Eq.(16)], k'
pi
p u
p
p
pu
C and e' [Eqs.(26) and (28)], B and B [Eq.(29)], and au [Eq.(75)],
2 can be expressed
some straightforward algebra shows that E ck' and E'ck
in the equivalent forms
ejck' =
2rTck
k0
Yp(1 + vP/c)
(76)
,
B
and
E'ck'u 2
Tck
= 2a
3 Y (i + vp/c)
(77)
.
Here, rT is the (small) dimensionless gain parameter defined in Eq.(27),
and the bounce frequency &B = (c2 kp2aw05 - 2 p )i of the deeply trapped
electrons is defined in the laboratory frame in Eqs.(23) and (24).
Finally, making use of k' = y (1 + v /c)ko [Eq.(19)], we note that
y (1 + v p/c)'ck.
It is useful to define
=
so that
'ck' =
(1 + vp/c)
'ck' can be expressed in the compact form
(78)
29
'ck
= yp u'ck
Pu
(79)
0
BT/p and
After some algebraic manipulation that makes use of QB
Eqs.(74), (76), (77) and (79), it is straightforward to show that the dispersion relation (70) can be expressed in the equivalent form
2
1
B
0 B)
(tn - cAK)
(AS)
(r cko) 3 U
u
6
Q
42 2(rck
2
(A ) (A2 - cAK)
)2
a2
2
- O
22
222
c k 22
(80)
u (Tcko
x
rTck0
+ 2(rTck)
T
3
+ 4(n0)(AE
.c2 k]
2 [()2 +
0
- cAK) 'cko
2
+
1 +(AS)
a c ko
Here, au = ( uT/)(j,/%) 3 [Eq.(75)], AQ = A-
2-
v Ak [Eq.(73)], AK
=
k0 (vp/c)Ak/ks [Eq.(73)], and rT is the (small) dimensionless gain parameter
defined in Eq.(27).
For awa << 1, we estimate
w s
'
(1 + a 2)
in Eq.(27),
-w
and rT can also be expressed in the more familiar form
1
r
-
(4,,
a2
w
T
4 (1 + a )
where use has been made of
e2 /m) (1 + v /c)
ypc k0
=
<
1
,
(81)
vp/c
p* Consistent with Assumption (b) at
pT
at the beginning of Sec. III, we require that a'cko be sufficiently large
in comparison with 2B in Eq.(80) in order that the untrapped-electron
motion be only weakly modulated by the ponderomotive potential.
30
Equation (80) constitutes the final dispersion relation which is
For au = 0, which corresponds to no
analysed numerically in Sec. IV.B.
37 7 1
untrapped electrons (^n' U = 0), Eq.(80) is the familiar dispersion relation ,
for the sideband instability in circumstances where the equilibrium wave
f 0, however, it is found that
For a
phase is slowly varying [Eq.(41)].
the untrapped electrons can significantly modify stability behavior
(Sec. IV.B).
B. Numerical Results
In analysing the dispersion relation (80) it is sensible to introduce
the total density of beam electrons nb =T + nu.
We define the fraction
of beam electrons that are untrapped (f ) and the fraction of beam electrons
that are trapped (fT)
by
n
fu
fu
^A
nb
(82)
nT
fT
u'
nb
nb
,
in the ponderomotive frame.
Therefore, fu and fT are also given by fu
u
ub'/6
and fT =
6f/F.)
nT
p and
(Keep in mind that n' = n/y
up9
=
YTp
nb
=
p
are the densities
We further define the gain factor rb associated
with the total beam density by
r
r =
(83)
,
nT
where r
is defined in Eq.(81).
follows from Eq.(75) that au = 6/
Because (_'/j
= 6u
-
)3 ~ 1 for awa
<< 1,
it
T is an excellent approximation.
31
can be expressed in terms
and r Tcabeepesditrm
Therefore, from Eqs.(82) and (83), aur3
U T an
of r b and fu by
u b'
u T
(84)
= (1 - f
r
where fu
u
'
b
u b'
T
b is the fraction of beam electrons that are untrapped.
In the numerical analysis of Eq.(80), we normalize all frequencies
to rbck0 and introduce the dimensionless parameters
-
-A Q
A=2
cAK
,
AK = -
,
rbck0
rbck0
(85)
B
ack0
rbck0
rbck0
=
B b/rbck0 is a dimensionless measure of the pump
In Eq.(85), note that 2B
Substituting
strength (amplitude of the primary electromagnetic wave).
Eqs.(84) and (85) into Eq.(80), we find that the dispersion relation can
be expressed in the equivalent form
1-
(AQ
-
+ 4()(Z
-
2
-
8))
(
u
uf
UB
AK)
-2
-f
K )Fu + 2(1
B
u
(AQ)
2(
f )2-4
(1
, 2
B
~ 2+
]
1 +
2
2
B
32
The dispersion relation (86) has been solved numerically for the
normalized growth rate Im(AQ)
quency Re(AS)
=
Im(A2)/rbck0 and the normalized real fre-
= Re(a)/rbckO versus the normalized wavenumber AK
AK/rbk0 over a wide range of system parameters
and au =
f=u n
b'
Typical results are illustrated in Figs. 3 - 8 for a fixed
b.
value of Z6y = 3
B = QB/rbck
x
21/3 = 4.3267, and normalized pump strength ranging from
B /rbck0 = 21/3 = 1.2599 (Figs. 3 - 5), to QB /rbck0 = 0.5 (Fig. 6), to
QB rbck0 = 0.2 (Figs. 7 and 8).
In Fig. 3, we illustrate typical numerical results and establish the
sign conventions inherent in the dispersion relation (86).
for OB
In particular,
bck0 = 21/3 and fu = nui b = 0.5, Fig. 3 shows plots of the
normalized growth rate Im(AQ)/rbck0 and real oscillation frequency
Re(Ao)/rbck0 versus normalized wavenumber AK/rbk0 obtained from Eq.(86)
for the two classes of unstable solutions.
The results in Figs. 3(a) and
3(b) pertain to the unstable mode driven by the untrapped electrons, whereas
the results in Figs. 3(c) and 3(d) pertain to the unstable mode driven by
the trapped electrons.
For fu =u
b = 0.5, of course both classes of
unstable modes are affected by the other population of electrons.
With
regard to the symmetries inherent in Eq.(86) and evident in Fig. 3, we note
that
ReAQ(-AK)
=
-ReAQ(AK)
ImAQ(-AK)
=
ImAO(AK)
(87)
are (necessarily) satisfied by both classes of unstable modes.
Equation (87)
assures that the Fourier transform functions for the perturbed quantities
^As,6',6e',
Ts etc., correspond to transforms of real-valued functions.
s
33
For simplicity of notation, keeping in mind the symmetries in Eq.(87)
and Fig. 3, throughout the remainder of this paper we display only the
stability results corresponding to the right-most growth curves in Figs. 3(a)
and 3(c).
That is, in Figs. 4 - 8, the stability results are presented only
for the right-most lobes of the growth rate curves.
Figure 4 shows plots of the normalized growth rate Im(AQ)/rbck0 versus
AK/rbk0 obtained from Eq.(86) for
B /rbck0
=
21/3 and fraction of untrapped
electrons ranging from fu = 0 [Fig. 4(a)] to fu = 1 [Fig. 4(e)].
For fu = 0,
Fig. 4(a) corresponds to the familiar growth rate curve 37,73 for the sideband instability assuming slowly varying equilibrium wave phase and that all
of the electrons are deeply trapped.
[Indeed, for fu = 0 and
B = 21/3
Eq.(68) can be solved analytically,37 which gives a useful calibration of
the numerical results.]
Adding an untrapped electron component, it is
evident from Figs. 4(b) - 4(e) that a new unstable mode (driven by the untrapped electrons) is introduced.
The untrapped-electron mode is repre-
sented by the dotted curves in Figs. 4(b) - 4(e), whereas the sideband
mode is represented by the solid curves.
As expected for zero energy spread,
the untrapped-electron mode in Figs. 4(b) - 4(e) has a relatively broad
bandwidth in AK-space.
Moreover, as fu is increased (thereby decreasing the
fraction of trapped electrons), the growth rate and bandwidth of the sideband
instability continue to decrease as fu is increased from fu = 0.2 [Fig.4(b)],
to fu = 0.5 [Fig. 4(c)], to fu = 0.8 [Fig. 4(d)].
Indeed, for fu = 1 (no
trapped electrons), the sideband instability is completely absent (as expected),
and only the instability driven by the untrapped electrons is present
[Fig. 4(e)].
It is evident from Figs. 4(a) - 4(e) that the properties
of Im(LQ)/rbk0
versus AK/rbk0 differ in detail for the two unstable modes. However, an
34
equally striking feature of Fig. 4 is that the characteristic maximum growth
rate of the most unstable mode varies by only a small amount (less that 25%)
between the case where there are no untrapped electrons [fu = 0 in Fig. 4(a)]
to the case where there are no trapped electrons [fu = 1 in Fig. 4(e)].
This suggests that it is a serious oversight to neglect the role of an
untrapped-electron component when calculating the detailed linear and nonlinear evolution of the beam electrons and the radiation field.
Figure 5 shows plots of the normalized real frequency Re(AQ)/rbck0
versus AK/rbk0 obtained from Eq.(86) for
B /rbck0 = 21/3, and fu = 0
[Fig. 5(a)], fu = 0.5 [Fig. 5(b)] and fu = 1 [Fig. 5(c)].
The system param-
eters in Figs. 5(a), 5(b) and 5(c) are identical to Figs. 4(a), 5(c) and 5(e),
respectively.
Moreover, Re(AQ) is plotted only over the unstable range of
AK, and the solid curves in Fig. 5 correspond to the sideband mode whereas
the dotted curves correspond to the untrapped-electron mode.
Evidently,
Re(Ao) increases monotonically with AK for the sideband mode [Figs. 5(a) and
5(b)].
Furthermore, the magnitude of Re(AQ) is somewhat larger for the
untrapped-electron mode [Figs. 5(b) and 5(c)].
Moreover, Re(An) is approxi-
mately constant for the untrapped-electron mode for AK in the range LK/Tbko > 5.
In Fig. 6, the normalized pump strength is reduced to 2B /rbck0 = 0.5.
In particular, Fig. 6 shows plots of the normalized growth rate Im(Ao)/rbck0
versus AK/rbk0 obtained from Eq.(86) for 2Bb/rbck0 = 0.5 and fraction of
untrapped electrons ranging from fu = 0 [Fig. 6(a)] to fu = 1 [Fig. 6(e)].
In Fig. 6, the general features of the growth rate curves for the sideband
mode (solid curves) and the untrapped-electron mode (dotted curves) are
qualitatively similar to Fig. 4, although the bandwidth of the sideband
instability is considerably larger for the smaller value of QB/Pbck0 chosen
in Fig. 6 [compare Figs. 4(a) and 6(a)].
Moreover, the maximum growth rate
35
of the untrapped-electron mode shifts from negative values of AK for
fu
0.5 [Figs. 6(b) and 6(c)] to positive values of AK for fu
[Figs. 6(d) and 6(e)].
>
0.5
As in Fig. 4, it is evident from Fig. 6 that the
characteristic maximum growth rate of the most unstable mode varies by
only a small amount over the entire range from fu = 0 [Fig. 6(a)] to fu = 1
[Fig. 6(e)].
However, the detailed properties of Im(AQ)/rbk0 versus
AK/r bk0 differ considerably for the two modes.
Finally, in Figs. 7 and 8, the normalized pump strength is reduced
further to
B /rbck0 = 0.2.
Shown are plots of Im(AQ)/rbck0 (Fig. 7) and
Re(AQ)/rbck 0 (Fig. 8) versus AK/rbko obtained from Eq.(86) for "B/rbcko = 0.2
and values of fu ranging from fu = 0 to f = 1. As in Figs. 4 and 6, only
the sideband mode is unstable for fu = 0 [Fig. 7(a)], whereas only the
untrapped-electron mode is unstable for fu = 1 [Fig. 7(e)].
Finally, as
in Figs. 4 and 6, the characteristic maximum growth rate of the most unstable mode varies by only a small amount over the entire range of fu
considered in Fig. 7.
36
V. CONCLUSIONS
This paper has investigated the detailed influence of untrapped
electrons on the sideband instability in a helical wiggler free electron
laser.
Small-amplitude peturbations are assumed about a constant-amplitude
(as = const.) primary electromagnetic wave with slowly varying equilibrium
phase 6 [Eqs.(40) and (41)].
A simple model is adopted in whch all of the
trapped electrons are deeply trapped, and the equilibrium motion of the
untrapped electrons (assumed monoenergetic) is only weakly modulated by
the ponderomotive potential.
The theoretical model is based on the single-
particle orbit equations together with Maxwell's equations and appropriate
statistical averages (Sec. II).
Like our recent treatment 37 of the side-
band instability (which neglects the effects of untrapped electrons), the
present analysis is carried out in the ponderomotive frame, which leads
to a substantial simplification in deriving the dispersion relation (70)
(Sec. III).
Transforming Eq.(70) back to the laboratory-frame frequency
W = Ws + Aw and wavenumber k = ks + Ak, detailed properties of the sideband
instability are investigated, including the effects of the untrapped
electrons (Sec. IV).
The resulting dispersion relation (86) has been analysed numerically
over a wide range of dimensionless pump strength
of untrapped electrons fu = nu /ib
B /Fbck0 and fraction
To briefly summarize, when both trapped
electrons and untrapped electrons are present, there are generally two
types of unstable modes, which we refer to as the sideband mode, and the
untrapped-electron mode.
present (as expected).
For fu = 0, only the sideband instability is
As fu is increased, the growth rate of the
sideband instability decreases, whereas the growth rate of the untrappedelectron mode increases until only the untrapped-electron mode is unstable
for fu = 1 (Figs. 4, 6 and 7).
37
It is evident from the present analysis that the detailed growth
properties are quite different for the two unstable modes.
However,
a very important feature of the stability results is that the characteristic
maximum growth rate of the most unstable mode varies by only a small amount
over the entire range of fu from fu = 0 (no untrapped electrons) to fu=
(no trapped electrons).
This suggests that it is a serious oversight to
neglect the role of an untrapped-electron component when calculating the
detailed linear and nonlinear evolution of the beam electrons and the
radiation field.
ACKNOWLEDGMENTS
This research was supported in part by the Office of Naval Research.
38
V1.
1.
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43
FIGURE CAPTIONS
Fig. 1.
In the ponderomotive frame, electron motion in the phase space
(z',p') occurs on surfaces with y' = const.
Fig. 2.
Plot of the equilibrium ponderomotive potential W(z') =
i 2 + 4a asin 2 [(1/2)(k z' + o)
versus k Z1.
W(z')
is
the envelope of turning points with pj = 0 and (as,6')
(0,60) in Eq.(10).
y! ::'
=
Deeply trapped electrons have energy
[1 + (a -
o
)2
s
-w
have energy y! =
that j 2 _
Fig. 3.
5'
>
5'
[Eq.(22)].
=
2 = 4a w
ws
Untrapped electrons
[1 + (aw + as)2] [Eq.(21)].
Note
1.
Plots of the normalized growth rate Im(AE)/rbck0 and
real frequency Re(AQ)/rbck0 versus AK/rbk0 obtained from
Eq.(86) for the untrapped-electron mode [Figs. 3(a) and 3(b)]
and the trapped-electron mode [Figs. 3(c) and -3(d)].
Results
= 0.5.
are presented for s2 B/rb cko= 21/3 1 'U
u = 3 x 21/3 , and fu =05
Fig. 4.
Plots of the normalized growth rate Im(AQ)/rbck0 versus AK/rbk0
obtained from Eq.(86) for 0B/r bck0 = 21/3,
u = 3
x
21/3, and
(a) fu = 0, (b) fu = 0.2, (c) fu = 0.5, (d) fu = 0.8, and
(e) fu = 1.
Fig. 5.
Plots of the normalized real frequency Re(AL)/rbck 0 versus AK/rbk0
obtained from Eq.(86) for QBb/rbck0 = 21/3
u = 3
x
21/3, and
(a) fu = 0, (b) fu = 0.5, and (c) fu = 1.
Fig. 6.
Plots of Im(A)/rbck0 versus AK/rbk0 obtained from Eq.(86) for
0B/Fbck0 = 0.5,
=
3
x
21/3, and (a) fu = 0, (b) fu = 0.2,
(c) fu = 0.5, (d) fu = 0.8, and (e) fu = 1.
44
Fig. 7.
Plots of Im(AQ)/rbck0 versus AK/rbk0 obtained from Eq.(86) for
9B/rbck0 = 0.2, a= 3 x 21/3, and (a) fu = 0, (b) fu = 0.2,
(c) fu
Fig. 8.
=
0.5, (d) f
= 0.8, and (e) fu
=
1.
Plots of Re(LsI)/rbcko versus AK/rbko obtained from Eq.(86) for
II
=
1/3
QB/rbck0 = 0.2, B= 3 x 2/, and (a) fu = 0, (b) fu = 0.5, and
(c) fu
=1
45
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