Nuclear Instruments
North-Holland
293
2-
ster to the Editor
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Received 15 December 1989
Properly taking into account the coupling between different
that the exponential gain of the fundamental radiation wa
that of the fundamental. This l
to almost
saturation
wavelength . This is the underlying mechanism of the resonant-frequency
e self-amplified spontaneous emission (SASE) of a
harmonic radiation field in a high-gain fre -electron
laser (FEL) is a well-known phenomenon, bo?n theoreti
cally predicted (see refs. [1,2]) and experimentally observed in the ELF experiment in Livermore [3].
In the simulations and in the experiments there is the
evidence of a nonpurely exponential growth of the
harmonic radiation . This behavior cannot be explained
by the usual linearization of the FEL equations (see ref.
[1]), which leads to a set of decoupled. equations for the
harmonic components of the radiation field. In particular, the growth rate of the harmonics predicted by the
decoupled model is always smaller than the one in the
fundamental (see ref. [1]). We present here a simple
analytical model for the coupling between the fundamental and its third harmonic, via the strong secondharmonic bunching parameter induced by the exponential growth of the fundamental field. In this analysis we
find that the usual cubic equation for the third harmonic
has a driving term, due to the coupling with the fundamental. As a consequence, this inhomogeneous term
gives rise to a growth rate three times larger than that of
the fundamental.
A one-dimensional numerical analysis confirms these
results, and shows that each harmonic bunching parameter exhibits a similar behavior and the saturation values are of the same order of magnitude, i.e. of the order
of unity. This is intuitively explained : the strong bunching occurring during the SASE regime can be approximated by a delta-function electron phase distribution,
which has a flat spectral composition, leading to equal
bunching parameters on each harmonic component . A
strong bunching, combined with the resonant condition,
leads to a strong burst of coherent spontaneous emission. This is the physical basis for the method of produring XUV radiation by resonant-frequency tripling in
a two- i er
amplifier
elsewhere (see ref. 141).
Let us
u
form (
refs. [1,2,5]) :
(a. exp[ih (kz -
) l - cc.),
where
is an d index d
e c--tosta
'
(wi
l7e.
harmonic, and for the
=
k,,z,
expression
-Y . sin
period. e c
and A., is the length of the wi
observing
the radiation o
only odd-order harmonics,
ref.
[5]).
The
upi
the propagation axis (
a
slowly
v
"
6
radiaMaxwell-Lorentz equations for
gths. using the
tion field, averaged over many wavelengths.
.
[2],
dimensionless units of ref
read:
where j runs over the electron and over the
components of the field. Fh is the difference of
functions defined as
F(x) =
(-1)(h-1)/2
J(a-1'/2(hx)
-J(a + »/2
and J = a2,/(1 + aw )2, where aw = e .,/
the wiggler parameter.
The term bh = (ex - ih®), is the bun
o ic
eter of the electrons in the hth-h
nent.
0168-9002/9G`/$03.50 0 1990 - Elsevier Science Publishers B.V . (North-Holland)
k
2
M a-
10
5 .0 :
e
b
l1 2 (X
A3 1 2 (
to almost identical val
dwe plot the
~--_,ich e tibits a growth
;~s shown in fist. 1. Mus is
the str g
igle comy exnit
.
a (e + d 1 "~_j I ()
al cmg tho i,
.~ litude and bunching
d its third harmonic
see
3). VVe can
,e
.0
f t bunch-
s system,
ction with the
brmlng an
to that of
®
10.0
shown as functions of the dimensionless distance
ref. [IL we can introduce the electron collective variable
--- < p
- ihf)). With the approximations
JA, I ,
3
2
ik@>
1
<P ewe have the following system of equations:
d._ .
= ''b"
d2
(5)
di = F3b3,
dab'
= i 2 ba ,
dz3
2
dz2
3
d z3
-
F'
(6)
(
aba,
= 3iF3b3 + 3i FI
d dab2 .
10 .0
nie, 1 b2 I, as a function
ess, distance -F.
(8)
equations describe the linear regime of the two
coupled wavelengths system. In ref. [11 a similar analysis held a system of decoupl
equations for the
harmonics, which cannot describe the physical situation
in which the high-order harmonics experience gain
without any harmonic initial field, with an unbunched
beam. From eqs. () and (6) we have the usual asymptotic exponential growth for the fundamental field Al,
namely Aa = A® ex iXli), where A® is a complex constant and A1 = F2®3(1/2 /2). Hence A1 is proportional to ex F2®3Fz/2). Given this expression of Al,
we can evaluate a similar expression for bl. Inserting
these functions in eq. (7) for b2, we can evaluate the
source term of . (8) for b3.
e asymptotic expression
for the driving term appearing at the
S of
. (8) is
iv
5.0 z
( )
d iv~
from Alb, - (3i/2)
form of
() becomes
3
22
d1ó 3 ®
3i 112
3 = - gFA®
3 ex
3iAI I).
e3®A,a.
e LHS of eq. (9) gives the usual cubic
tion for b3 (and hence for A3), in
limit. In fact, neglecting the term appearing in the RHS
(we can do ° only if 1: 1, i.e. in the low-gain case)
and looking for exponential solutions of e form b3 4x
ex ik 3i), we Obtain the equation A3 + 3F3 _ , which
ves the asymptotic solution b3 proportional to
exp( F2"3 1/2), i.e. with a growth rate smaller than
the fundamental, c
since I F3 I < I Fy 0. But, in the high' cam, considering the inhomogeneous term for
(9), the asymptotic solution for the bunching (and hence
for the harmonic field) is proportional to e 3iA g z).
From this result it follows that At diverges, in the linear
regime, as ex 3F2/3jF3 1/2), i.e. it has a growth rate
three times larger than the one for the fundamental
wavelength. More generally, the solution for the thirdharmonic bunching (and for the third-harmonic field
amplitude), is a linear combination of two ex nentials,
one given by the deeoupled linear analysis, with a
growth rate proportional to A 3 (which is smaller than
the one for the fundamental, A,), the other, driven by
the exponential instability of the fundamental wavelength, with a growth rate of Al. In rig. 3 we show both
the terms b3 and bi in a logarithmic plot as functions of
the dimensionless distance along the wider, 1. A definite proportionality in the exponential growth regime is
seen, confirming our asymptotic analytical results. This
is even more clearly seen in fig. 4, where we plot
log I b3I as a function of log I bt I . We can clearly see in
this plot a linear-type dependence, with a slope of about
3
3, due to the proportionality
between I b3 l and I bt 1 .
,
With a simple analytical model, we have described
the mechanism of coupled-harmonic coherent sponta
neous emission in a high-gain FEL. The exponential
instability experienced by the fundamental wavelength
e would like to
.C u
rec ° g our minds on the problem of s
-vi z
ti
F.T.
le
radiation
(many) lp
r
fact
t the nonlinear
e would like to s
bunching,
here
analyticaHy
harmonic driving of the
in
the
case
of
a
very
predicted, works also
parameter, a w -4r. 1, w
drastically reduces, due to the very low value of
1
lions coupling factors. I
a s
wi er parameter,
coupling factor to almost 1, shortens
the growth of the bunching in ea,
driven by the fundamental one.
if io, OpL Commun.
[11 J. Murphy, C. Fel ini and R.
.
53 (1985) 197
=ini and L. ard
[21 R. nifacio, C.
50 (1984) 373.
private communication .
[31 E.i. Scharl
d E.T.
Bonifacio,
L.
De Salvo Souaa, F. i
[41 R.
.,
1989;
Scharl
.11
QE-17 (19,81)
t
[51 .B. Colson, IEEE J.
1417.