Nuclear
Instruments and Methods in Physics Research A318 (1992) 568-575
North-Holland
Chaotic-electron
orbits in a linearly-polarized wiggler free
electron
laser
i. .
Michel, A
.
B,)urdier ' and J
.M.
Buzzi
Laboratoire
de Pliysique des Milieux loirisés, Ecole Polytechnique, Centre
Cede-z,
Frairce
The
trajectory of an electron in a linearly-polarized wiggler with an axial guide field is found to be nonintegrable
.
There is
evidence
for chaos from numerical calculations of Poincaré maps and nonzero Liapunov exponents
.
Resonances can be predicted
from
a one-dimensional Hamiltonian perturbed by a small "time-dependent" quantity
.
1.Introduction
Stochastic
electron orbits are found when consider-
ing
a field configuration consisting of a linearly-
polarized
wiggler magnetic field and a uniform axial
magnetic
field
.
This situation is interesting in the case
of
a weak wiggler and a very small radius beam [1-3]
.We
simplify the problem using a canonical transforma-
tion
and perform Poincaré sections
.
The results are
confirmed
by calculating Liapunov exponents with two
methods.
Finally, the equations of motion are derived
from
a one dimensional "time-dependent" Hamilto-
nian .
Resonances can then easily be predicted
.
2.
Theoretical formulation of the problem
2.1.
Guiding center system
The
motion of one electron in a FEL with a lin-
early-polarized
wiggler Bw and a guide field B
�
is
considered .
The self-fields produced by the electron
beam
are neglected
.
The motion of the electron takes
place
in the following magnetic field
B=ezBO+exBw
sin k,,,z,
(1)the
rorresponding Hamiltonian is
eB
`
H=
c2
PX + (PJ,+eBOx+
w
cos kwz
k
+P~
w
Also
in Centre d'Etudes de Limeil-Valenton, 94195 Vil-
leneuve-Saint-Georges
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.
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© 1992 - Elsevier Science Publishers B
.V .
All rights reserved
NUCLEARINSTRUMENTS&METHODSIN
PHYSICS
RESEARCHSection
A
National
de la Recherche Scientifique, 91128 Palaiseau
As
the Hamiltonian is not an explicit function of
time,
H is a constant of motion
.We
have plotted the trajectory of an electron in the
(x,
y) plane
.
The motion looks chaotic for some initial
conditions
(fig
.
1), as will be confirmed by performing
Poincaré
sections and calculating nonzero Liapunov
exponents.Two
additional constants of motion are obtained
simply
by integrating the two first equations of Hamil-
ton
We
find a canonical transformation such that two of
the
conjugated variables Q,, P, are proportional to C1
and
C,
.
This transformation is given by the generating
function
F,(x,
y, z, P,,P,, P3)
=(Pt-eBoy)x+P,(y-
P
� n
) +P,z
.CL �
1
In
the new variables, one obtains
H
=
c
2 e`BO Q
;
+
P2
+ keBw cos k w Q3
,
+P
;w
)
+m2c4
As
expected, H is independent of the parameters
Q,
and P,, which determine the guiding center trajec-
tories .
C,
= P, + eB
�
y,
(3)C.,
= P,
.
Unfortunately,
they are not in involution
IC,,
C,} = eB
� . (4)
Having failed in finding a third constant of motion,Poincaré maps have been plotted to demonstrate thenonintegrability of the motion . To do so, the followingnormalized equations of motion, derived from eq . (6),have been solved numerically:Â I A
A
Q2 = y(P2+a,, cos Q3),
A
;, P:3Q3 =- "
P,
yA
fleQ2,
yÂ
Î,
AP, =a,,Q2 sin Q; .
L. Michel et al. / Chaotic-electron orbits in
The dimensionless variables defined by r=ckwt, Qi =k wQi, Pi = Pi/inc, aw = eBw/mckw, have been intro-duced, as well as the Lorentz factory = H/tnc 2, Hgiven by eq . (6) .
A
a FEL
569
The motion occurs in a three dimensional space(Q2, Q;, 150. The plane (Q2, 150 with Q; = 0 (mod2-rr) is chosen to be Poincaré surface of section. Thenumerical method used is a fourth-order Runge-Kutta.Figs . 2 and 3 show nonintegrable surface-of-sectionplots.
The existence of chaotic trajectories is confirmed bycalculating nonzero Liapunov exponents by two ap-proaches . The first consists of considering two nearbytrajectories with an initial tangential vector with normd�. The distance d� between those trajectories is calcu-lated numerically, and as soon as d�/d� is greater thana quantity between 2 and 3, we renormalize d,, to d,,.The Liapunov exponent [4] (fig . 4) is given by
1 nmrx d= lim
log ,,(8)~~~mra x t�mrx
(dj)d� --0
The second approach consists of integrating thedifferential equation on the tangent vector [4,5] wi =
Fig. 1 . Projection of an electron trajectory on the (x, d ) plane, for B�^ 1 .98 T, B,, =1 .5 T, !1, = 2, H = 3, and for the followinginitial conditions : x =0.17. j = 0, -^ = 0, Pr = 0, P,. =0 and P_ = 2.13 .
VIL FELTHEORY
N cc~
-0.16
-1.52
-2.87 -0
.64
-0.32
0.01
0.33
0.65
n Q2Fig
.2 .Surfaceof
sectio
nplot
sfor
chaoti
ctraj
ectori
eswit
h0),
=tl(
mod
2-,,),
for
H
=3,B
�=1,2
1T
.B �=1 .5
Tand
!1,=
1 .25.
N (a
09E°--
-0.36
-1.67
-2.98 -1
.01
-0.42
0.18
0.78
1.37
n Q2
Fig.3 .No
nint
egra
ble
surfac
eofs
ection
plots
with
~)3=
tl(mod
2r,) .for
f9=3
.B �
=1 .9l
îT,Bw
=1 .5
Tan
d!1,
'=? .
1 V r n F 4 4 w y ti
(x,, - xz,), where x,, and x;, are the coordinates oftwo neighboring trajectories at the same time, t. Thetime evolution for w is found by linearizingdx,dt = VAX)
to obtaindw
Integrating eq . (10) numerically, d(t) = II w(t) Il isderived . The corresponding Liapunov exponent is givenby
1a" = lim -(log d(t) - log d(0)) .
(12)r--.m
tRenormalizations may be necessary when d(t) be-
comes too large. For a sufficiently long time, the expo-nent converges as shown in fig . 5. This figure and fig . 4show that with two different approaches we have ob-tained almost the same result . This confirms the nonin-tegrability of the system .
?.2. Reduction of the problem to a one-dimensional"time-dependent" system
Let us return to the Hamiltonian (eq. (2)) . Theequations of motion are
px = -je&
(P, +eB,x+eL"cosk,,z~ ,
y "B ,
(P = e" sin k",z[ P,.+eB,x+ek" cos k",z~,
my t
"
my my
L. Alrclrel et al. / Chaotic-electron orbits in a FEL
( 13 )
Let us divide two of ills
dx
x
P,dz i P.'
dPr
Ps
eB, rd z
i
P,
z can then be consideredEq . (2) gives
H=P - -, - P,= - m-'C-
T
r-
- P,,+eB�x+eB&"
We note that (-P-) cdimensional Hamiltonian itime . Eqs. (14) can indeeding Hamilton's equations
dx
a(-P)d z
_
aP,
'dP, _
a(_P~ )a-Z ax
For a weak pump, thelinearized by setting
P-=P_{,+P_t+P-,+x=x�+x,+x,+ ---,Pt =Pf+PO+Pr,+ . . .
Fig. 4. Liapunov exponent obtained by numerically integrating two trajectories: one correspondiinitial conditions: Q3 T Qz = Q, P, = 0.35, P3 = 2.13 and the other one with very close initial cor
-=M(x(t))w. (10)dtwhere
aVM= ax' (11)
572
The different variables are evaluated under thecondition that the frequency is far from the magnc-toresonance [1]. The amplitude of the electron motionis assumed to be small . We consider a weak pump anda homogeneous solution with an amplitude of the sameorder of magnitude as the one of the particular solu-tion .
To a first approximation, eq . (16) leads to the fol-lowing equations of motion
_eooP~_ a
P~ u r -B(, ( P',
with
As a consequence, we have
x�_ -1P,, .eB,?
x, +w2x, _ -wi BW
cos k,vz,
t7
B�k W
L. 141ichel et al / Chaotic-eléctron orbits in u
18
H,
1/2,
P~n = , -pYu -m'c' - (P~, +eBuxo)y
(19)
(20)
where to,, = eB,,/P~_t, .Our hypothesis regarding the amplitude of the mo-
tion leads to the following solution for x�
(21)
Calculating P,, and x,, neglecting second order terms,leads to
(22)
s
FEL
This equation predicts one resonance for k, = ±w� ,which is not considered because of our hypothesis tobe far from the magnetoresonancc .
Other resonances were predicted when taking intoaccount third order terms. We have
eBx - + u~ �x
=w�P_ z m�x t +kwpO
_P"IP.-2
Prep2
pôwith
P.
S+ ;wt,X
2Pci~Pz0
+EX, P, �tù 2 cos k�,z + E-wi,2 P,,, cos- k,, z
,
where E=BW/kWB{ , is supposed to be a small quan-tity.
There is a resonance whenever
k,, -- nw{ ,,
(25)
with n = ±
, ±3.The intersections between the trajectories and a
surface of section were determined . To do so, dimen-sionless variables were introduced : Pi =Pi/mc, z =k,,z, x = k�,x, A, = .R,/ckW, ar =eAW/mc, H=H/mc4, z --- ck,,t. The plane (x, P,) with z=0 (mod270 is crosen to be the Poincaré surface of section .
In fig;. 6 and 7 a period-three and
island appearcorrespaading to then = 3 and tr =
resonance condi-tion, respectively .
(23)
(24)
Fig . 5. Liapunov exponent obtained with the linearized equations, for the same initial conditions as Fig . 4 (curve 1) and with BW = 0(curve 1I).
L. Abcltel e1 crL / Cltaritir-c~lc>c'trn~t urbilà in u FEI
(â
4a
1MM
-4,41h-̂
'
-130
\;,
Vic
N"
-0.8
7
-0.3
6
V64
2.3
6
1.1
3
-103
-2.566
-1I
1
1-0
.54
0.02
0.59
Fig.
8.Surfaceof
sectionplotsfor%tocha%tic
traj
ecto
ries
with
z=0(mod
21r),for
Fig.9.
Surfaceof
section
plotsforst
ocha
stic
traj
ecto
ries
with
i=0(mod
2r),
for
H=
3,B,, =
2 .56
T,B,
,=
0 .65TandP
.=2.59
.
F1=
3,BO
=2.
2T,B,,=
0.8T
and
fi,=
2.25
.
2 .S5-
L. Afichel et al. / C7raoric-elecrron orhirti irr a FEL
By performing Poincarc sections and calculatingLiapunov exponents, it has been shown that the motionof one electron in a uniform magnetic field and in theheld of a linear wiggler is nonintegrable . This problemcan be reduced to a one dimensional "time-dependent"motion . In the case of a weak pump, resonances havebeen easily predicted .
3. Discussion
References
575
-1 .78
-1 .13
-0 .48 -
0.17-0 .82
xFig. 10 . Surface of section plots for stochastic trajectories with Z^ = 0 (mod 21r). for N = 3, B, = 1 .98 T, B� = 0.65 T and 12, = 2.
Figs . S-1(1 show that for other conditions, we have
Acknowledgmentsstochastic trajectories which confirm that the system is
The authors wish to thank Prof . G. Laval and Dr. S.nonintegrablc. Bouquet for their useful suggestions.
[1] Y.Z . Yin and G . Bekefi. J . Appl . Phys. 5 5 (198-1) 33.[2j K.D . Jacobs. Ph.D. Thesis, MIT (1956).[3j T.C . Marshall. Free-Electron Lasers (Macmillan . New-
York, London . 19tî5).[-1j S .N . Rashand . Chaotic Dynamics of Nonlinear Systems
(Wiley, 1990).G . Benettin . L . Galgani and J.M . Strelcyn . Phys. Rev . Al4(1976) 2338.
[51 A .J . Lichtenberg and M.A . Lieberman, Regular andStochastic Motion (Springer Veriag, New York . 1983) .
V1I. FEL THEORY