JAERI-Conf 95-021

PB41

Nonlinear Resonances in a Multi -Stage
Free-Electron Laser Amplifier
Satoshi HASHIMOTO, Ken TAKAYAMA*
Department of Accelerator Science, The Graduate University for Advanced Studies
1-1 Oho, Tsukuba, Ibaraki, 305 Japan*National Laboratory for High Energy Physics (KEK)
1-1 Oho, Tsukuba, lbaraki, 305 Japan

Abstract
Nonlinear resonances in the longitudinal phase-spaceof a multi-stage Free-Electron Laser for the Two-Beam
Accelerator have been studied. We have developed a
new analytic theory based on the macroparticle modeland the perturbation method. A resonance-structureobserved in simulations is found to be modeled by thenonlinear pendulum equation and to depend on awaveguide dimension.

1. Introduction
A Two-Beam Accelerator (TBA) is a possible candidate
for future linear colliders, in which rf power required fora high-gradient linac is provided from a Multi-stage FreeElectron Laser (MFEL) [1,2] as shown in Fig. 1. The
MFEL has some unique features. First a bunched beam
drives each FEL. Second it has a periodicity: after
amplification of input seed-power in each FEL, the driving
beam is re-accelerated with an induction unit for energy
replenishment to go to the succeeding stage. This factmeans that rf and beam characters vary periodically anda bucket evolves rapidly in each PEL.

longitudinal phase-space, and degrade the performanceof the MFEL as an rf source. We suppose that theresonances should be serious when the power-density of
the amplified rf is strong.
This paper presents the nonlinear resonance in theMFEL for the recent version of TBA [4,5], not for theearly one [3]. Section 2 briefly shows the simulation
results which show the existence of the resonance. We
show in section 3 that by use of the macroparticle model[6-8], the motion of electron in the phase space of theMFEL can be described by the nonlinear pendulum
equation with periodically and rapidly time-varying"mass" and "length". Section 4 shows how the resonancecan be theoretically analyzed with the perturbative
calculation.

2. Simulation
The well-known one dimensional FEL simulations[9,10] have been performed. We have assumed arectangular waveguide TE01 mode as a signal wave.
Typical parameters of the MFEL of interest are listed inTable 1.

The high rf power-density with a rapidly

increasing ponderomotive force would give rise to strong
resonances. In order to evaluate effects of the rf powerdensity on the resonances, only the waveguide width a·
is varied while the other parameters are fixed. For a
relatively wide waveguide (a' ;::::10 em), our previous
simulation shows that the beam propagates from the firstto the 300-th stage without detrapping, and maintains theoriginal bunch shape [7]. For cases of smaller waveguideFig. 1

Schematic picture of TBAIFEL

a'= 8, 4 em, meanwhile, the fourth and third-integer

One of the interesting issues in the MFEL is theresonances between the synchrotron motion in a bucket
and the periodicity of the MFEL [3]. The resonances
lead to a formation of islands within a rf bucket inresonances are observed as shown in Fig. 2. These
resonances are considered to be caused by the strong rf
power-density resulting from the reduced waveguide
width. When these resonances occur, the beam continues
to lose its population, the rf power decreases, and the-365-

JAERI-Conf 95-021

signal phase changes [4]. These are significant problem
for the multi-stage rf-source where transport of a highcurrent beam over a long distance is indispensable tomaintain a constant amount of amplified rf power.

amplitude, oks = wsfc- ks the shift of longitudinalwavenumber from its value in vacuum, ws rf angularfrequency, ks = ~(wJcY- (n/a'Y the wavenumber

for TE01 mode, J, beam current density, e, normalized

Table 1.

signal-field, kw the wavenumber of wiggler, (/)s the signal

FEL parameters for TBA/FEL

beam current I ebeam energy yenergy gain per period ll.y
wiggler wave length A.w

2 kA
23
126 emwiggler length per period L w

52 em

wiggler peak field Bw
signal frequency

H(e,5;z)

10 MW
20-4 emwaveguide height b •number of FEL stage

3 em
300

26,..,.,.,.,...,.,.,..,.,.....,.,.,...........,..-.....,.........,
24

24~ 22~ 22~ 20

18

16-3-2u_,~1

16 -3-2

Ponderomotive phase (rad.)

-1·0

12

Ponderomotive phase (rad.)

(a)

(b)

FIG. 2Longitudinal phase-space structure by FEL
simulations. (a) the fourth-integer resonance (waveguide*.

*width a = 8 em). (b) the third-integer resonance (a = 4em).

whereG(z)=ws(1+a:)/cy;,F(z)=aweJy•.
Neglecting friction terms which are proportional to 5'
because of their smallness, we obtain a nonlinear
pendulum equation
5'' + GF{sin vr.(coss -1) + cosvr. sins} ... 0, (3)from the Eq. (2), where primes denote differentiation
with respect to z. The macroparticle model assumes
Y. oc aw; hence GF in Eq. (3) is determined mainly byEq. (3) for a· = 8, 4 em. We observe that Eq. (3) canwell reproduce the results of the FEL simulations seen inFig. 2. Thus, we consider the Hamiltonian (1) withe:,E oc X= -qJs determined by the macroparticle model
as a theoretical base to analytically assess the nonlinear
resonances in the MFEL.

•0.4

3. Nonlinear pendulum equation

,...

0.2.

W> 0

Following the Ref. 3 and using the definition of themacroparticle [6-8], the Hamiltonian for the MFEL canbe written asN{H(e,g,E,x;z)=L (kw-OkJ(r.+e)+
1JParticles,112

a = eB /.Jim ckwwew

the normalized wiggler..,.·f-o.4-3...,........,_2,.......-~1~o!:'-'-'-'""'"1~2~3( . )2c Y. + e

2'..

-0.2ws(1+a:)(E- NyJ' cos(vr. + 5) + dy. s}.
w
m,c2
N 112
Y. + ed.z(1)where r .. 11'. are the Lorentz factor and ponderomotive
phase of the macroparticle, e , 5 are deviations from theenergy and phase of the macroparticle, N the number of-a (eZ0 J,(2)es which is written in a term of the trigonometric function

C)

18

2

[8], es(z) =: (2K/albl)sin(lblz/2), where a, b, K, andK are constants defined in Refs. 6, 7.
Figure 3 shows the results of numerical integration of~ 20C)

=G(z)E /2- F(z)cosvr. cos5+F(z)sinvr.(sins-5),

17.1 GHz.

waveguide width aZ0 = 377Q,
E = Nm/e: /eZ0 J, and X= -qJs.
Expanding the Hamiltonian (1) in powers of efr. andretaining the dominant terms, we have

3.85-3.6 kGf.

input rf power l!nphase, e and m, are the charge and rest mass of electron,c the speed of light, z longitudinal coordinate,

FIG. 30.4

0.2

W> 0-o.2-o.4

Q-,.,
'~·

-3...,......,_2~-1,........l<'-0~1~2~3

1;

1;

(a)

(b)

Phase-space plots of the nonlinear-pendulum's

solution for (a) a' = 8 em and (b) a' = 4 em.

4. Isolated resonance theory
Using the isolated resonance theory [11] which has
been established in beam dynamics of circular-366-

JAERI-Conf 95-021

accelerators, we can calculate the size and position of
the primary resonance islands [4]. Expanding theHamiltonian (1) in powers of sand ejya:H(c,~;z) = Gc 2 /2 + F/i,2 /2 +(perturbations)(4)

where Fe = F cos 1/f a. The expression can be regarded
as the non-autonomous one degree of freedom
Hamiltonian for a pendulum with time-varying "mass"
G and "length" F;. affected by nonlinear perturbations.
Using the generating functions F,(£,£;z) =S£/ .,jG(z)and F;(~,O)=-(a+tan0)~ 2 /2f3, where a, f3satisfy

2/3/3" - /3' 2 + 4GF;/3 2 = 4,
a= -/3'/2,(Sa)(5b)

the Hamiltonian (4) is transformed to(6)H(J, O;z) = ( V 0 j f3(z) )1 +(perturbations).
Here f3 is referred to as a longitudinal amplitude functionin the MFEL, which is a quantity analogous to a transverse
amplitude function in circular accelerators. This function
represents the orbital evolution of the bunch envelope in
the MFEL. Instead of z, we use a new independent
variable a defined by a= (1/vJJ)/ f3(z')dz', whereV0 = (1/211:)jLw 1/ f3(z')dz' is referred to as thelongitudinal tune and 271:V0 is the phase advance per
FEL period. Retaining only the dominant terms aftersome straightforward mathematical manipulation andaccording to the isolated resonance theory [11], we havethe "time"-independent Hamiltonian for the isolated thirdinteger resonance:(7)
H,(J,O) = Sm 13 J + hJ 2 + h,} 312 sin(30 + E>J,
where 8 m 1 3 , ~ , h3 and 8 3 are all constants which dependon FEL parameters [4]. The Hamiltonian for the fourthinteger resonance also can be calculated in a similar way[4]. Eventually we arrive at the exact mathematicalformula necessary to theoretically asses the primaryresonance observed in the simulations. Fig. 4 showslines of the equi-Hamiltonian in the phase-space for a*=8, 4 em in the rectangular coordinates P = -fiJ sinO,W

cos 0 to compare with Figs. 2, 3. The calculatedQ=
position and size of the islands are in agreement withsimulations [4]. The transition of phase-space structuresin Figs. 2, 3 also can be explained with the above theory[4]. The longitudinal tune V0 increases with a decreaseof the waveguide width a'. When V0 is more than 1/4(1/3), the fourth (third)-integer resonance can occur.
This theory is able to give a crucial suggestion inchoosing practical MFEL parameters.

(b)

(a)
FIG. 4Phase-space plots of the equi-Hamiltonian for(a) the fourth-integer resonance (a'= 8 em) and (b) thethird-integer resonance (a'= 4 em).

Acknowledgments
The authors would like to thank Dr. H. P. Freund ofScience Applications International Corporation and Prof.
T. C. Marshall of Colombia University for helpful remarksand suggestions. One of the authors (S.H.) thanks Prof.
S. Hiramatsu of KEK for his hospitality. The numericalcalculations were performed on DEC 3000 AXP-500.

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