The lith Symposium on Accelerator Science and Technology, Harima Science Garden City, 1?97

A Superrnode Analysis of Michelson Resonator FELs
Ryoichi HAJIMA and Shunsuke KONDO
Departwent of Quantum Engineering and Systems Science, University of Tokyo,
Hongo 7-3-1, Bunkyo-ku, Tokyo 113, JapanAbstract

2 Lasing Dynamics in Michelson FEL

Evolution of laser field in a Michelson resonator FEL
is studied via supermode analysis. It is found that thelasing dynamics in a Michelson FEL can be describedby three dimensionless variables: cavity loss, equivalentcavity detuning and the interference parameter. Smallsignal gain and temporal profile of laser pulse can beobtained as eigenvalue and eigenstates of the supermodeequation.

. We start with one-dimensional FEL equations described in terms of scaled dimensionless variables anduse the same notation as re£.[1].
Let normalized cavity detuning for primary and secondary arms of a Michelson resonator be 81 and 82 respectively, and we define positive sign of 8 for cavity
shortening from perfect synchronism of electron bunchinterval. The laser pulse is reflected by mirrors of thecavity and evolves· through successive interaction withelectron bunches every round trip, where the laser pulse
pushed forward or backward relative to the electron
bunch according to the cavity detuning from the synchronism.
The input field for the (n+1) pass is given as

1 Introduction
Evolution of laser filed in a short-pulse free-electron
laser (FEL) oscillator shows various behavior, stable,
limit cycle and chaotic lasing, according to resonatorparameters: loss and detuning [1]. Controlling the lasing behavior is, therefore, a key issue to realize wide
application of FELs in scientific and industrial fields.
Michelson FEL was first developed to increase spectral purity of FEL by establishing phase coherence between successive optical pulses in the resonator, wherethe secondary arm provides extra delay of one rf period[2].
Numerical analyses recently showed that a Michelson resonator also has ability to suppress the emergenceof chaos even with parameters which gives chaotic lasing in a conventional two-mirror resonator FEL [3]. The
path difference between two arms of the Michelson resonator for the chaos suppression is chosen so as to provide small delay shorter than the electron bunch. Themechanism of chaos suppression might be considered asa result offrequency filtering or suppressed superradiant,the detail has not been cleared however.
In the present paper lasing dynamics of a Michelson FEL is studied via supermode analysis which gives
intuitive understanding of lasing dynamics in a Michelson FEL and allows further investigation of the chaossuppresswn.
In section 2, we introduce lasing dynamics in aMichelson FEL. An eigenmode equation which gives supermodes in a Michelson FEL is derived in section 3.

A~n+ 1 )(~- 81- 82)= rt;A~n)(~- 82) + rr;A~n)(~- 81) ,where A is the slowly varying complex amplitude of theradiation in dimensionless form, A 0 and At are field at
the entrance and the end of und ulator, respectively, ~ =(ct- z) /b. is longitudinal coordinate in the moving frame
andnormalized by slippage distance b. = >.Nw,
are transmittance and reflectance of the beam splitterin the Michelson resonator and r represents energy loss
at mirrors.
An ideal beam splitter has no energy loss : +=1. The beam splitter, however, still allows leakage ofoptical power due to phase mismatch between radiations
reflected from two arms, that is frequency filtering, theintrinsic effect of a interferometer.
We limit the analysis to small single-pass gain andassume both cavity detuning and loss are small, andhence Eq.(1) can be expanded in Taylor series andrewritten with the FEL equations:r;t;t; r;8A(~,T) _ *8A(~,T)~A(' )
8TV
8~+ 2 <,,T= 17 ( exp[-iO(~, T)]) - p* :~ 17 ( exp[-iO(~, T)])(2)

8 2 0(~,T))l}ae = - {. A (~' T ) exp ["-(z(} ~' T + c.c.

(3)•leakageIFig. 1 Michelson resonator FEL(1)where short bunch limit is assumed and parameters arenormalized by small-signal gain"{ following ref. [1]: T ="{n, a = a 0 j"{, a 0 /2 = 1- r, v* = (81 + 82 )/'Y- p*,
p* = r (t;82 + r;81) h·Equation (2) shows that lasing dynamics in a Michelson FEL is determined by three dimensionless parameters: scaled cavity loss a, equivalent cavity detuning v*
and the interference parameter p*.

-329-The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

The second term in the right-hand side of Eq.(2) isproportional to the derivative of bunching term along (-axis, which will vanish, if the magnitude of bunching is
uniform along ( -axis, that is the case no phase variation exists in driving term. The second term, therefore,
represents the interferometric property of a Michelson
resonator and never appears in FELs with conventional
two-mirror resonators.
The interference parameter in the definition of theequivalent cavity detuning reduces the mirror detuning
51 by factor (1- rr;). This relaxation of effective detuning results in the broadening of cavity detuning curve,
which is one of the most important effects in a Michelson
FEL [2].
It is also found that the values of v* and p* arerestricted by 51 and 52 . Since the cavity mirror andthe beam splitter have reflectance of 0 ::; r ::; 1 and0 ::;+1, the interference parameter may havethe valuer; t; ::;Solutions of Eq.(12) depends on the boundary condition for <I>(() which is divided into three cases as follows:( 1) 51 < 0 and 52 < 0The laser pulse is pushed backward every round trip and~(0) = 0 is given as the boundary condition. No nonzero
solutions for A((, T) exist in this case a:; well as cavity
lengthening in conventional two-mirror resonator FELs.
(2) 51 > 0 and 52 > 0The laser pulse is pushed forward every round trip and~(1) = 0 is given as the boundary condition. Nonzero
solutions for A((, T), that is supermodes having positivegain, can be obtained with the algorithm described inref. [1]. Figure 2 shows small-signal gain of the first twosupermodes as a function of equivalent cavity detuning,where the interference parameter is chosen as p* = 0.01
and p* = 0.1. It can be seen that the supermode gain
curves of Michelson FELs part from the gain curve of
conventional FELs with increasing of p*.

0.2

The equivalent detuning is also allowed to have value:

0.15

3 Supermode Analysis of Michelson FEL-t"c"iii

8A _ 11 *8A +::_A= B _ p*8B8T
8(28( '8B-=-iP8('8P
8( =-A '(8)where B = ( exp( -iB)) and P = (i.iexp( -iB)) are thecollective variables called as the bunching parameter andthe phase-momentum average, respectively.
Equations (6) - (8) have an eigenstate:A((, T)
B((,T)
P((, T)=exp [(tL- a/2)T] <I>(()
exp [(tL- a/2)T]IJ!(()
exp [(tJ- a/2)Tj r((}

(9)
(10)

(11)

Substituting Eq.(9)-(ll) into Eqs.(6)-(8), we findv*~ 111 - 11~" - ip*~' + i~ = 0 ,0.05.... ..

0(6)

(7)

(12)

which is similar to one derived in ref.[1] but includes thep* term resulting from the interference in a Michelson
resonator. Once an eigenvalue tL is found, single-passgain can be obtained as 9 = 2"(Re(tL) and linear phaseshift is given as ¢ = Im(tJ). The eigenfunction ~(()
represents a supermode which describes the evolution of
temporal profile of laser pulse.

0.1~

The FEL equations expressed in terms of collective
variables can be rewritten in a form including the term
representing the interferometric property of a Michelson
resonator:

00.02

0.04

0.06

v·0.080.1

0.12

Fig. 2 Small-signal gain of the first two supermodes inMichelson FEL (5 1 > 0, 52 > 0) vs equivalent
detuning: p* = 0.01 (dashed line), p* = 0.1(dotted line), conventional resonators ( continuous line).

(3) 51 52 < 0This is the case chaos suppression occurs [3]. A partof laser pulse is pushed forward and the rest is pushed
backward every round trip. The boundary condition is
not given in a form as <I>(() = 0 and Eq.(12) cannot besolved analytically. The profile of laser filed and smallsignal gain, therefore, are calculated numerically with apulse propagation code [3] instead of finding analyticalsolutions.
Calculated small-signal gain for Michelson FELs
with cavity detuning as 51 52 < 0 is plotted in Fig.3.
The leakage from beam splitter is also shown in Fig.4.
It shows that a Michelson FEL with cavity detuning as
81 52 < 0 has gain curve similar to the first supermode ofconventional FELs. The gain, however, does not reduce
as fast as the supermode with approaching the equivalent detuning to zero.
Figure 5 shows the profile of saturated laser filed inMichelson FEL with parameters 51 52 < 0, p* = 0.01,-330-The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

1.4/\

1.2
0.15
0~

c'iii

0000..

J\·

1.0
00.1

"'<(0•II \\

0.8,:

0.6

Q.

jf \\

""1\"...
"\\

0.4

0.05

1:0.2

1'.

\:\'·.

0, ....

L---~--~----~--~----~--~~

00.02

0.04 .

0.06
v"

0.08

0.1-30.12

Fig. 3 Small signal gain for Michelson FEL (8 1 82 < 0)vs equivalent cavity detuning: p* = 0.01 ( o ),p* = 0.1 (•), the first supermode for FEL withconventional resonators (solid line).

-2-1~

023Fig. 5 Laser field profile for Michelson FEL with parameters: p* = 0.01, a = 0.05, v* = 0.075(continuous line), v* = 0.025 (dashed line),
v* = 0.01 (dotted line).

1.4 .----..------,-- --,-----.----..- -----,----,

0.1 .-----.:-------.-- ---,----,-----..-- -----.----,

1.2
0.08

1.0"'<(0.80.6
0.4

0.02

0.2

0 L---~--~----~--~----~--~~
00.02

0.04

0.06
v"

0.08

0.1-3

0.12

Fig. 4 Normalized leakage power from the beam splitter of Michelson FEL: p* = 0.01 ( o ), p* = 0.1(•)< v* < 0.075 . Results for p* =0.1 are also shown in Fig.6. It shows that laser field
penetrates into the region f. > 1 where no laser filed
exists in case of FEL oscillators with conventional twomirror resonators. The penetration due to radiation field
pushed backward by cavity lengthening becomes more
remarkable with decreasing of v* and increasing of p*.

-2-1023~Fig. 6 Laser field profile for Michelson FEL with parameters: p* = 0.1, a = 0.05, v* = 0.075(continuous line), v* = 0.025 (dashed line),v* == 0.01 (dotted line).

a = 0.05 and 0.01

References[1] N. PioveU.a et al., Phys. Rev. E 52 (1995) 5470.
[2] E. B. Szarmes and J. M. J. Madey, IEEE QE 29(2)(1993) 465.
[3] R. Hajima et al., Proc. FEL'96 (to be published).

4 SummaryAnalytical study on Michelson FEL have been madeand it is found that lasing dynamics of Michelson FEL
can be described by three dimensionless parameters:
cavity loss, equivalent cavity detuning and the interference parameter.
We have derived the supermode equation for aMichelson FEL which gives small-signal gain and evolution of temporal profile of laser pulse. Laser field profileand small-signal gain in Michelson FEL is also obtained
numerically for various parameters.

-331:--