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Volume 77, number 1 OPTICS COM MUN ICATIONS 1 June 1990

O P T I C A L K LY S T R O N C O N F I G U R AT I O NF O R A H I G H G A I N X -R AY F R E E - E L E C T R O N L A S E R

J u a n C. G A L L A R D O a n d C l a ud i o P E L L E G R I N IBrookhaven National Laboratory, Upton, NYl1973, USA

Received 5 December 1989

We present theory and numerical simulations of the performance of an x-ray free-electron aser in the amplified spontaneoemission mode with an optical klystron undulator configuration. This d evice can produce picosecond pulses of high brightnwith a significantly shorter und ulator than a con ventional FEL.

1 . In t roduc t ion

T h e a m p l i f i e d s p o n t a n e o u s e m i s s i o n ( A S E ) r e-g i me o f a f r e e - e lec t r on l as e r (F EL ) can b e u sed t op r od uce coh e r en t so f t x - r ay r ad i a t i on i n a l ong un -du lato r ( 5 ~

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an optical field which, in combination with the un-dulator magnetic field, acts back over the electrons

to introduce an energy modulation in the beam. Thisenergy modulation is transformed into bunching ofthe electrons as they go through the dispersive sec-tion. The optimal posi tion of the dispersive magnetoccurs at the point where the laser power starts togrow exponentially which correspond t o a gain lengthLcj, defined as the distance for e-folding of the laserpower. This forced bunching results in a discontin-uous jump of the gain as the electrons enter the ra-diator undulator.

To describe this system below saturation, we use

the linearized, 1D FEL theory for a helical undul atorgiven in refs. [5,6]. The FEL dynami cs are then de-scribed by defining three collective variables [ 5 ] forthe laser field X, the bunching function Y, and theenergy spread Z,

X = a , ( l a )

Y= (exp( -i~uo)0) , (lb )

Z = p - ' (e xp( - i~uo)rl> , (l c )

where ~'0 is the initial electron phase, O = ~ ' - ~ ' o t , 7 0

is the initial electron energy and ~/= (7-70)/7o.

P = (K~o/87r~C)2/3(4~reC2 ~e/~)1/3

is the FEL parameter with 20 the wiggler period, Kthe undulator parameter; re is the classical electronradius and ~ is the particle density of the electronbeam. In the limit of small p the FEL equations ofmotion [6] with initial conditions X(0)=Z(0)=0and Y(0) = Yo are

J ( = i d X - i Y , (2a)

)>=Z, (2b)

2= -x. (2c)

where d is the detuning para meter d= ( r 2 -7 ? ) / 2 y R pand 72 = (20/22) ( 1 + K 2 ) and J ( = d X / d rwith r = 4 g p N , Nbeing the number of undulatorperiods.

Nontrivial solutions of the form exp(i/t r) exist if~t satisfy the characterist ic equ ati on/ t 2 (/~- 6) + 1 = 0.In the high-gain self-spontane ous emission, the max-imum gain rate is obtained with 6=0; keeping the

fastest growing root we write

X= - ~ Yo/z~" exp (i/ll r) . (3a)

Y= -~ Yo exp (i/~ r ) , ( 3b )

Z = i l Yo/q exp(i/ll r) , (3c)

where ~t ~= (1 - i x/ 3) / 2 and /1~' is the complexconjugate.

The dispersive section can be taken as an instan-taneous interaction at rD with the following con-straints on the dynamical variables (X< and X> de-note the values right before and after the dispersioninteraction, respectively),

X> =X < , (4a)Y> = Y< - ~ Z < , (4b)

Z> =Z < , (4c)

where ~ = p k D with

D - { e ~ a i d z , ( i d z , , B t ) ( z , , ,)2 .

- \ ? m c 2 ) o o

A simple example of a dispersive section is given bya three-section magnet of total length s and field

B D ( z ) = B o, 0~

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x > ( o = ' * ~ )3#, Y< (ZD) (1 - i /x i

e x p [i/t1 ( 3 - r D ) ] ( 7 )

T h i s l a s t e x p r e s s io n s h o w s t h a t t h e e n h a n c e m e n t f a c-t o r i n t r od uced by an op t i c a l k ly s t ron i s g iven by t hef ac to r ~ ~ ~ wh ich , f o r t he pa r am e te r s cons id e r ed i no u r e x a m p l e i s o f t h e o r d e r o f 1 0 3 . To o b t a i n a neq u i v a l e n t g rowth i n t he f i e l d i n t h e conve n t i ona lF E L c o n f i g u ra t i o n , th e l e n g t h o f th e u n d u l a t o r c a nb e e s t i m a t e d f r o m e x p ( i / q A r ) . ~10 3 which , f o r t hev a l ue o f p u s ed , co r r e spon ds t o ~ 400 m agne t pe -r i o d s . T h i s s imp l i f i e d mod e l i s on ly v a l i d i n t he l i n -ea r r eg ime a n d fo r a pe r f ec t e l e c t ron beam ; i t is no t

i n t end ed t o d e sc r i be s a tu r a t i on .To u n d e r s t a n d t h i s i n c r e a s e i n t h e o u t p u t p o w e r106 we have u sed a ID s imu la t i on cod e t o so l ve

th e fu l l s e t o f FEL equa t i on s i nc lud i ng ene rgy sp r e adan d e m i t t ance . T o s imu la t e a re a l i st i c e l e c t ron beamwi t h ene rgy sp r ea d and emi t t ance , we u se a un i fo rmd i s t r i b u t i on i n ~ 'o and a p ro d uc t o f a ga us s i an d i s-t r i bu t i o n i n ene rgy s p r ead , t r an s ve r se pos i t i o n a ndin j ec t i on a n g l e . I n t ab l e l we l i s t t h e pa r ame te r s o ft he e l e c t r on b ea m and undu l a to r ; we no t i c e t ha t t oach i eve t h i s r a t he r la rg e va lue o f p w e mus t r e so r t t o

ex t e rna l qu ad r u po l e f ocus ing [2 ] a l on g the un du -la tor wi th a typ ica l e ffec t ive f l - func t ion f l* ~ 1 . Th isad d i t i o n a l f o cus ing i s e s s en t i a l to r e a ch t he e l e c t ronb e a m b r i g h tn e s s r e q u i r ed f o r a n x - ra y F E L [ 1 0 ] . A d -d i t i o n a l c o m m e n t s a r e p e r t i n e n t h e r e r e g a r d i n g t h eb r i l li a n ce o f t he e l e c t r on beam p re sen t ed i n t ab l e 1 .T h e r e a r e a n u m b e r o f e x p e r i m e n t a l g r o u p s s t u d y i n gl a se r-d r i ven pho toca thode e l ec t ron guns . Los A la -m o s [ 11 ] ha s p ro duc e d an e l ec t ron b eam o f no r-m a l i ze d e m i t ta n c e o f 4 0n m m m r a d a n d s i m u l a t io n ss t u d i e s a t B r o o k h a v e n N a t i o n a l L a b o r a t o r y ( B N L )[ 1 2 ] h a v e s u g g e s te d a n e m i t t a n c e o f 6 n m m m r a d .

Table 1Electron and undulator parameters used in the simulations

wavelength (n m) 2.5energy (Ge V) 1.02p 103 1.2L(m) 8.1undulator period (cm) 1.0gain length L G (m) 0.75peak current (A) 400normalized emittance (ram mrad ) 1.0external focusing f' (m) 1.0

T h e r e f o r e , f u r t h e r i m p r o v e m e n t s a r e n e e d e d t o s a t -i sf y t h e e m i t t a n c e r e q u i r e m e n t s . O n t h e o t h e r h a n d ,

t h e p e a k c u r re n t I = 4 0 0 A f o r a 1 p s e l e ct r o n p u ls eand e ne rgy sp r ead aE~ 10 -3 c o r r e spon d s t o a l on -g i tud ina l b r i l l i anceB L = l/trEy= 200 A which i s ra therm o d e s t c o m p a r e d w i t h t h e e x p e r i m e n t a l r e s u l t s a tLos A lamos and t he expec t ed va lue s a t B N L. As i l -l u s t r a t ed i n f i g . l , t h e s a tu r a t i on power a c h i eved byan op t i c a l k ly s t ron i s l a rge r t han t he one ob t a inedw i th a s t and a rd w igg l e r and , m ore imp or t a n t , we ob -se rve s i gn i f ic an t pow er l eve l s a t l e s s t han ha l f t he un -du l a to r l eng th . As expec t ed , i n t roduc in g t he beame n e rg y s p r e a d a n d e m i t t a n c e r e d u c e s t h e o u t p u t

pow er l eve l o f t he l a s e r; how eve r, i n t he h igh ga inr eg ime , l arge r ga in t han i n t he ca se o f th e s t anda rdun du l a to r c a se i s o bse rved even fo r e ne rg y sp r eadAy/y a s l a rg e a s p . A l tho ugh ou r r e su l ts we re ob -t a ined i n t he con t ex t o f a ID t heo ry, f o r the s e t o fpa r am e te r s u sed , t he ga in l eng th o f t h e dev i ce i ssho r t e r t han t he Ray l e igh range o f t he l a se r and , con -sequent ly, 3D effec ts wi l l no t s ign i f ican t ly mod i fy ourconc lus ion s [ 13 ] .

I n s u m m a r y , w e h a v e s h o w n t h a t t h e o p t i c a l k ly s -t r o n c o n f i g u r a t io n o f a n F E L c a n b e u s e d t o a d v a n -

t age i n t he h igh ga i n co l l e c t i ve r e g ime , m a k ing a so f tx - r ay F E L i n t h e A S E m o d e m o r e f e a s ib l e b y r e d u c -

.,6

1 0 8

1 06

10 4

10 2

I 0 0

, o - Z

, 0 4

I I I l I l

i o - 60 1.4

/ f /

, \/ . S . o g o O , , TO ,p - / . . . - - 1# /. /i l / l / 0 /

J , , " / 0 0 0 -i~//~//0" 0 0 0

0 / 0 " 0 0l I I 1 t

0 . 2 0 . 4 0 . 6 0 , 8 1 . 0 1 . 2

z / L E N G T H - U N D U L A T O R

Fig. 1. Laser power versus z/L from the simulations for bothstandard undulator and optical klystron, a t= 0.0 (solid);

aE=0.1% (d ash) and aE=0.2% (circle).

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i n g in h a l f t h e u n d u l a t o r l e n g th a n d m a i n t a i n i n g t h el a se r ou tpu t power.

Acknowledgements

We wi sh t o t h ank J . Go lds t e in f o r adv i ce on t hen u m e r i c a l a l g o r i t h m . T h i s w o r k w a s s u p p o r t e d b yt h e U .S . D e p a r t m e n t o f E n e rg y u n d e r C o n t r a c t n u m -b e r D E - A C 0 2 - 7 6 -C H 0 0 1 6 .

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[ 5 ] R. Bonifacio, L. Narducci and C. Pel legrini, Optics C om m.50 (1984) 373.

[6 ] J . Murphy, C . Pe l l egr in i and R . Boni fac io , Opt ics Comm.53 (1985) 197.

[ 7] P.B. W ilson and J . Griff in , in: High energy electron l inacs;appl icat ions to s torage r ings rf system and l inear col l iders ,Physics of high energy part ic les accelerators , eds. R.A.Car r igan , F.R . Huson and M. Mon th , AlP C onfe rence Proc .87 (1982) p. 450.

[8] W.B. Colso n and I . Boscolo, Phys. Rev. A31 (1985) 2353.[9 ] A .S . Ar tam anov e t a l. , Nuc l. Ins t rum. Me thods 177 (1980)

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