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

Cedex, France

.

0168-9002/92/$05 .00

© 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