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H A M I L T O N I A N M O D E L O F A F R E E E L E C T R O N L A S E R

R o d o lf o B O N I F A C I O 1 , F e de ri co C A S A G R A N D EDipart irnento di Fis ica del l 'Univers itb, Via Celor ia 16, 20133 M ilan, I ta ly

a n dC l a u d i o P E L L E G R I N IBr ookhave n N at iona l Labora tor y , Upton , N Y 11973 , USAR e c e i v e d 3 S e p t e m b e r 1 9 8 6

B o t h t h e C o m p t o n a n d t h e R a m a n r e g i m e s o f a f r ee e l e c t r o n l a se r a r e d e s c r i b ed b y a re l a ti v is t ic h a m i l t o n i a n w h i c h o r i g-i n a t e s t h e e v o l u t i o n e q u a t i o n s f o r 2 N + 2 c a n o n i c a l l y c o n j u g a t e e l e c t r o n a n d f i e ld v a r ia b l e s , w i t h t h e s p a c e c o o r d i n a t e a st h e i n d e p e n d e n t v a r i a b l e . S p a c e c h a r g e a n d f i e ld c o n t r i b u t i o n t o e l e c t r o n t r a n s v e r s e v e l o c i t y a r e i n c l u d e d . S c a l e d v a r i a b l e sa r e i n t r o d u c e d w h i c h a l lo w f o r a d e s c r i p t i o n o f t h e b e h a v i o u r o f t h e s y s t e m in t e r m s o f a s in g le e l e c t r o n - b e a m p a r a m e t e r .

T h e f r ee e l ec t r o n l a se r ( FE L ) i s p o t en t i a l l y an i d ea l t o o l f o r b a s i c an d ap p l i ed r e s ea r ch a s a p o w er f u l s o u r ce o ft u n ab l e co h e r en t r ad i a t i o n , g en e r a t ed v i a t h e i n j ec t i o n o f r e la t i vi s ti c e l ec t r o n s i n an u n d u l a t o r [1 ] . T h e co u p l ed d y -n am i cs o f t h e e l ec t r o n s an d t h e r ad i a t i o n f i e ld is s u ch t h a t u n d e r p r o p e r co n d i t i o n s t h e r ad i a t i o n em i t t ed b y t h e ac -ce lera ted par t ic les can g row exponen t ia l ly a long the undu la to r . Th is h igh-gain r eg ime i s due to a co l lec t ive ins tab i l -i t y o f t h e s y s t em [ 2 - 5 ] an d h as b een r ecen t l y d em o n s t r a t ed b o t h i n t h e s i ng le -p ass am p l i f ie r co n f i g u r a t i o n [ 6 ]w h i ch w e s h a l l co n s i d e r h e r e , an d ev en i n t h e o s c i l la t o r m o d e o f o p e r a t i o n [ 7 ] .

T h e c l a ss i ca l h am i l t o n i an ap p r o ach h as b een s u cces s f u l ly u s ed t o d e s c r i b e t h e F E L p r o ces s s in ce t h e ea r l y d ay so f s i n g l e - e lec t r o n , sm a l l -s i gn a l t r e a t m en t s [ 8 ] . We h av e ap p l i ed t h i s ap p r o ach t o t h e i n v es t ig a t i o n o f co l l ec t i v e e f-f ec t s in t h e h i g h - g a in r eg i m e . I n p a r t i cu l a r, a m an y - e l ec t r o n h am i l t o n i an m o d e l f o r FE L am p l i f ie r s w as di s cu s s ed i nref . [ 3 ] in the l imi t o f low dens i ty and neg l ig ib le space-charge . Then in r ef s. [ 4 ,5 ] F EL d yna m ics was d iscussedd r o p p i n g t h i s r e s t ri c t i o n . I n t h e s e p ap e r s s o m e t e r m s d u e t o t h e r ad i a t i o n f i e ld co n t r i b u t i o n t o e l ec t r o n t r an s v e r s ev e l o c i t y w e r e n eg l ec t ed ; t h o u g h s m a l l i n m a n y ca s es , t h ey can b eco m e i m p o r t an t f o r v e r y l o n g o r t ap e r ed u n d u l a t o r s .I n t h i s p ap e r w e p r e s en t a s e t o f ev o l u t i o n eq u a t i o n s , v a li d b o t h i n t h e C o m p t o n an d i n th e R am an r eg i m es , w h i chi s m o r e g en e r a l t h an t h a t o f r e fs . [ 4 ,5 ] an d p r e s e r v es t h e h am i l t o n i an s t r u c t u r e o f th e s y s t em , s o t h a t en e r g y co n -servat ion and L iouv iUe theorem in a (2N+ 2) -d imens ional phase space ar e va l id . A su i tab le sca l ing o f var iab les a l -l o w s f o r a d e s c r i p t io n o f FE L d y n a m i cs o n l y in t e r m s o f o n e p a r am e t e r .

T h e h am i l t o n i an eq u a t i o n s w i t h t i m e a s t h e i n d ep en d en t v a r i ab l e can b e d e r i v ed f ro m t h e m o d i f i ed H am i l t o npr inc ip le [9 ]

t25 f (PxdX/dt +py dy/dt +Pz dz/dt - H ) d t = O . (1 )t 1

Si n ce i n o u r p r o b l em o n e f o l l o w s t h e ev o l u t i o n o f t h e s y s t em a l o n g th e u n d u l a t o r ax i s z , w e ch an g e t h e i n d ep en -d en t v a r i ab l e f r o m t t o z , an d u s i n g H = E w e o b t a i n [ 1 0 ]

i A l s o I s t i t u t o N a z i o n a l e d i F i s i c a N u c l e a r e , S e z i o n e d i M i l a n o , M i l a n , I t a l y .

0 0 3 0 - 4 0 1 / 8 6 / $ 0 3 .5 0 E l s ev i e r Sc i en ce Pu b l i s h e r s B .V .( N o r t h - H o l l a n d P h y s i cs P u b l i sh i n g D i v i s i o n )

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V o l u m e 6 1, n u m b e r 1 O P T I C S C O M M U N I C A T I O N S 1 J a n u a ry 1 98 7

z 26 f ( P x d X / d z + p y d y / d z - E d t / d z + P z ) d z = O. ( 2)

21In eq . ( 2 ) ( x , P x ) , C v , p y ) a n d ( t , - E ) a p p e a r a s c a n o n i c a l v a ri a b l e s w i t h r e s p e c t t o a n e w h a m i l t o n i a n H 1 = - p z .H e n c e w e c a n w r i t e

d x / d z = - ~ p z / ~ p x , d y / d z = - ~ p z / ~ p ) , , d P x / d z = ~ p z / ~ X , d P . v / d z = ~ p z /O ) ' , (3 )d t / d z = O p z /O E , d E / d z = - O p z/ O t. 14 )

E q s . ( 4 ) ar e o u r w o r k i n g e q u a t i o n s . N o w l e t H b e t h e r e l a t i v is t i c h a m i l t o n i a n f o r o n e e l e c t r o n i n t e r a c t i n g w i t h e l e c -t r o m a g n e t i c f ie l ds

H = c { [ p - ( e / e ) A ] 2 + m 2 c 2 } 1 / 2 + e V = 7 m e 2 + e V - K 1 StI n e q . ( 5 ) w e a s s u m e t h a t t h e v e c t o r p o t e n t i a l A = A ( z ) is t ra n s v e r s e a n d V = V ( z ) r e p r e s e n t s s p a c e - c h a r g e e f f e c t sd u e t o d e n s i t y f l u c t u a t i o n s i n a n e l e c t r o n b e a m . H e n c e e q s . (4 ) b e c o m e

d t / d z = ( l / m c 2 ) ~ p z / ~ 7 , ~ ~ )d T / d z = - ( 1 / m c 2 ) ( O p z / O t - e E z ), E z = - d V / d z , ( 6 ' )

w h i l e f r o m e q . ( 3 ) i t f o l lo w s t h a t p x = p y -- c o n s t . , s o t h a t o n e c a n s e t P x = p y = 0 t h u s o b t a i n i n g f r o m e q . ~ 5)P z = m c ( 7 2 - l - a 2 ) 1 /2 , a = - e A / m c 2 " (7 )

B y a s s u m i n g t h a t 7 2 >> 1 + a 2 w e h a v eP z = m c [ 7 - (1 + a 2 ) / 2 7 ] . ( 7 ' )

N o w w e s p e c i f y t h e d i m e n s i o n l e ss v e c t o r p o t e n t i a l a ( z , t ) a s t h e s u m o f a m a g n e t o s t a t i c , s p a t i a l ly p e r i o d i c u n d u l a -t o t p o t e n t i a l a 0 ( z ) a n d a r a d i a t i o n f i e ld p o t e n t i a l a L ( z , t ) :

a = a 0 + a L , a 0 = ( a o /X / - 2 ) [0 e x p ( - i k o z ) + c . c .] ,a L = - ( i / % / ' 2 ) {Oa L ex p [ i (k LZ - C OLt)] - - c . c . } , ( 8 )

wh ere k 0 = 27r/X0 = C O o/ e i s t h e w a v e n u m b e r a s s o c i a t e d w i t h t h e u n d u l a t o r p e r i o d i c i t y , k L = 2 7r /k L = COL/C is ther a d i a t i o n w a v e n u m b e r , a n d c i r c u l a r p o l a r i z a t i o n i s t a k e n f o r a h e l i c a l u n d u l a t o r . T h u s e q . ( 7 ' ) b e c o m e s

P z = m e { 7 - ( 1/ 2" )') [ 1 + a o + i a 0 ( a [ e x p ( - i 0 ) - c . c . ) + l a L I 2 ] } , ( 9 )w h e r e 0 i s t h e e l e c t r o n - f i e l d p h a s e

0 = (k L + k o ) z - C O L t. ( 9 ' )A s d O / d z = k L + k 0 - c o L d t / d z , t h e e v o l u t i o n e q u a t i o n s f o r t h e v a r ia b l e s 0 a n d 7 c a n b e o b t a i n e d a t o n c e f r o m( 6 ) , ( 6 ' ) a n d ( 9 ) ; o r , a lt e r n a t i v e l y , v i a a c a n o n i c a l t r a n s f o r m a t i o n f r o m ( t , - E ) t o ( 0 , 7 ) , a s w e l l k n o w n i n a c c e le r a -t o r p h y s i c s [ 1 0 ] .

W i t h o u t a n y f u r t h e r a p p r o x i m a t i o n w e g e td O f / d z = k o ( 1 - 7 2 / 7 2 ) + ( k L / 2 T ? ) [ i a o ( a L e x p ( i O / ) - - c . c .) - - [ a L l 2 ] , ( 1 0 )d T j / d z = - k L ['} a0 ( a L e x p ( i O j ) / T j + c . c . ) + i ( c o p / c o L ) 2 ( ( e x p ( - - i 0 ) ) e x p ( i 0 / ) - - c . c . ) ] , ( 11 )

wh ere COp = ( 4 7 r e Z n / m ) 1 /2 i s t h e p l a s m a f r e q u e n c y , n = N / V t he e l e c t r o n n u m b e r d e n s i t y , 7 R = [ CO L( 1 + a ~ ) / 2 COO]1/t h e r es o n a n c e e n e r g y ( a t z e r o i n it ia l f ie l d) i n r e st e n e r g y u n i ts , ( e x p ( i 0 ) ) = N - 1 E N 1 e x p ( - i 0 / ) t h e e l e c t r o n5 6

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V o l um e 6 1, n u m b e r 1 O P T I C S C O M M U N I C A T I O N S 1 J a n u ar y 1 98 7b u n c h i n g p a r a m e t e r . I n e q s . ( 1 0 ) , ( 1 1 ) w e h a v e a d d e d a n i n d e x 4 t o d i s t in g u i s h d i f f e r e n t e l e c t r o n s i n a b e a m . F u r -t h e r m o r e , t h e s p a c e - c h a r g e c o n t r i b u t i o n t o d T / / d z i s o b t a i n e d f r o m e q . ( 6 ' ) u s i n g t h e r e s u l t o f r ef . [ 5 ] f o r t h e f i r s th a r m o n i c c o n t r i b u t i o n t o E z ,

( E z ) = - ( i a z r e n / k L ) [ ( e x p ( - - i 0 ) ) e x p ( i 0 j ) - - c .c . ].T h e f i e ld e q u a t i o n c a n b e d e r i v e d a s i n r e f. [ 4 ] f r o m M a x w e l l e q u a t i o n s b y n e g l e c t i n g s li p p ag e e f f e c t s , a n d r e a d s

d a L / d Z = ( ~ 2 / 2 C ~ L ) ( a 0 ( e x p ( - - i 0 ) / 7 ) - - i ( 1 / 7 ) a L ) , ( 1 2 )w h e r e a g a in a n y b r a c k e t ( ) m e a n s a n a v e ra g e N - 1 G N 1 T h e l a s t te r m i n e q . ( 1 2 ) , w h i c h is u s u a l l y n e g l e c t e d ,c o m e s f r o m t h e r a d i a t i o n f i e l d c o n t r i b u t i o n t o e l e c t r o n t r a n s v e rs e v e l o c i t y , j u s t l i k e t h e t e r m s d e p e n d i n g o n t h ef i e l d i n e q . ( 1 0 ) . H o w e v e r , w e s t r e s s t h a t i t i s n o t n e c e s s a r y t o d e r i v e e q . ( 1 2 ) a s u s u al f r o m M a x w e l l e q u a t i o n ss i n c e, as w e d e m o n s t r a t e h e r e , e q s . ( 1 0 ) , ( 1 1 ) a n d ( 1 2 ) c a n b e o b t a i n e d a s h a m i l t o n e q u a t i o n s f r o m a u n i q u e h a m i l -t o n i a n f o r b o t h e l e c t r o n a n d f i e ld v ar i ab l e s.

A c t u a l l y , b y i n t r o d u c i n g t h e s c a li n g a s i n r e f . [ 4 ] :t~ j = O / - ~ , P / = ( 1 / p ) (~ [ j / ( ~ f) o ) , A = ~ L a L e x p ( i 6 Y . ) / O a p ( p ( ~ f ) O ) t ] 2 , = 2k0(72R/(7)02) O Z ,

( 1 3 )e qs . ( 1 0 ) - ( 1 2 ) b e c om e

d ~ j / d ~ . = ( 1 / 2 p ) ( 1 - 1 / 0 2 1 ' 2 ) + i ( 1 / p ) ( A e x p ( i ~ j ) / P 2 - c . c . ) - ~ a [ A t Z / P2 ( j = 1 . . . . N ) ( l a a )d F j /d ~ ? = - ( l / p ) ( A e x p ( i ~ j ) / P ! + c . c . ) - i o ( ( e x p ( - i~ b )) e x p ( i ~ i ) - c . c . ) ( j = 1 . . . . N ) ( 1 4 b )

1 o ( 1 / F ) ) A , ( 1 4 c )t A /d ~ = ( l / p ) ( e x p ( - i ) / r ' ~ + i ( 6 - ~w h e r e t h e g e n e r a l i z e d P ie r c e p a r a m e t e r p , t h e d e t u n i n g p a r a m e t e r ( a t z e r o i n i t ia l f i e ld ) 6 a n d t h e s p a c e - ch a r g e p a -r a m e t e r o a r e d e f i n e d a s

p = (1 / ( 'r~0) a~ a 0 ( ( 7 ) 2 / " y 2 R ) ~ p / ~ 0 ) 2 /3 , 6 --- ( 1 / 2 p ) ( ( 7 ) 2 - - 7 2 ) / 7 2 , ~ = 4 p ( 1 + a 2 ) / a 2 . ( 1 5 )N o t i c e t h a t t h e la s t t e r m s o f e q s . ( 1 4 a ) a n d ( 1 4 c ) , w h i c h a r e d u e t o t h e t r a n s v e r s e v e l o c i ty i n d u c e d b y t h e r a d i a t i o nf i el d , h a v e t h e s a m e c o e f f i c i e n t o o f t h e s p a c e -c h a r g e t e rm , . S i n c e a d e p e n d s e s s e n t ia l l y o n O , th e w h o l e s y s t e m o fe v o l u t i o n e q u a t i o n s ( 1 4 ) i s r u l e d o n l y b y t h e p a r a m e t e r p .

E q s . ( 1 4 a - c ) c a n b e o b t ai n e d i m m e d i a t e ly f r o m t h e f o l lo w i n g h a m i l t o n i a nN ( l " j + ~ - N e x p ( - i ~ 0 j ) c . c . )~ = 1 2 0 j~ l'= I / p ' 2 p J ) + v ( .4 * j = l F j - ( 6 - ~ o ( 1 / P ) ) N I A [ 2 + a N l ( e x p ( - i ~) )1 2

N ( r / + ~ . c o s / C 0 N s in ~ /

' (6 ' o ( 1 / P ) ) ( ~ b 2 + r 2 ) + a N l ( e x p ( - i ~ ) ) [ 2 ( 1 6 )~ - ~w h e r e

A - (~ b 0 + i P o ) / ( Z N ) 1 / 2 , ( 1 7 )a s th e c a n o n i c a l e q u a t i o n s

d ~ /j/ d~ , = 3 H / a P j , d P j / d ~ = - 3 H / O ~ j ( j = O , 1 . . . . U ) . ( 1 8 )N o t e t h a t w h e n j = 0 i n e q . ( 1 8 ) , t h e t w o r e a l e q u a t i o n s c a n be c o m b i n e d i n t o d A / d Y , = - ( i /N)(aR/aA*) ,w h i c hl e a d s d i r e c t l y t o e q . ( 1 4 c ) .

5 7

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Vol um e 61 , num b e r 1 OPTI CS COM M UN I CATI ONS 1 J a nua ~ ' 1987E q s . ( 1 4 ) a d m i t a n o t h e r c o n s t a n t o f m o t i o n b e s i d e s ( 1 6 ) , n a m e l y

( U ) + I A I 2 = c o n s t . , ( 1 9 )w h i c h r o l e s t h e g l o b a l e n e r g y e x c h a n g e b e t w e e n t h e e l e c t r o n s a n d t h e f i e ld .

E q s . ( 1 4 ) g e n e r a l iz e t h o s e o f r e f . [ 5 ] , w h e r e a l s o h a r m o n i c s w e r e c o n s i d e r e d f o r a p l a n a r u n d u l a t o r , s i nc e t h e yi n c l u d e a l l c o n t r i b u t i o n s d u e t o t h e e f f e c t o f t h e r a d i a t i o n f ie l d o n t h e e l e c t r o n t r a n s v e rs e v e l o c i t y . D r o p p i n g t h e s et e r m s a n d t h e s p a c e - c h a r g e t e r m s a s w e l l o n e o b t a i n s t h e e q u a t i o n s o f r e f. [ 4 ] . I n t h e li m i t

P n j = - p r / - 1 = ( 7 / - ( ' 7 ) 0 ) / ( ' ) ' ) 0 ~ i , 1 2 0 )w h i c h i s v a l i d u p t o p O , p l= - O 2 p , O l - - a ( 6 1 + p 2 ) . (2 5 )H o w e v e r , w e d o n o t f u r t h e r d i s c u ss t h e s t a b i li t y o f t h e f u ll d is p e r s i o n r e l a t i o n ( 2 4 ) b e c a u s e s i m i la r e q u a t i o n s w e r ee x t e n s i v e l y s t u d i e d s i n c e t h e l a te f o r t i e s i n c o n n e c t i o n w i t h m i c r o w a v e e l e c t r o n i c s [ 1 5 ] . O n t h e c o n t r a r y w e c o n -s i d er t w o l im i t c a se s , c o r r e s p o n d i n g t o t h e C o m p t o n a n d t h e R a m a n F E L r e g im e s [ 1 , 2 ,1 4 ] .5 8

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Volume 61, num ber 1 OPTICS COMM UNIC ATIO NS 1 January 1987

( i ) I n the l imi t p ~ 0 o f low d ens i ty an d neg l ig ib le space charge , eq . (24) r educes to the cub ic equa t ionX3 - 6 X 2 + 1 = 0 ( 26 )

a s s o c i a ted w i t h t h e r ed u ce d h am i l t o n i an ( 2 2 ) . I n t h i s ca s e th e s y s t em i s u n s t ab l e w h en 6 ~

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Volume 61, number 1 OPTICS COMMUNICATIONS 1 January 1987

I A i 2

1 . 5

1

0 . 5

0 2 4 6 8 1 0 1 2zFig. 2. Compton regime: Field intens ity IAI 2 versus longitu-dinal coordinate ~-calculated from eqs. (14), with initial condi-tions A0 = 0, (r].) 0 = I/p, ( ex p( - iV)) ~ 0.15. The parame tersare: p = 0.0l , 8 = 0.09, a o = 1.

2 --

L A I 2 i ~ - . . . . . . . . . .

I

o . ~ / II

o r j i _ _ _ i4 6 8 1 0 1 2

Fig. 3. Raman regime: Field intensi ty IAI 2 versus longitudinalcoord inate Y calculated as in fig. 2, but with: p = 0.1, ~ = 0.78.

Refe rences[ 1] See e.g.T.C. Marshall, Free-electron lasers (Macmillan, New York, 1985);

Free electron lasers, eds. J.M.J. Madey and A. Renieri (North -Holland, Amsterdam, 1985) (Nucl. Instr. and Meth. A 237 (1,2));Coherent and collective propertie s in the interaction of relativistic electrons and electromagnetic radiation, eds. R. Bonifacio,F. Casagrande and C. Pellegrini (North-Holland, Amsterdam, 1985) (Nucl. Instr. and Meth. A 239 (1)).

[2] N.M. Kroll and W.A. McMuUin, Phys. Rev. A 17 (1978) 300;I.B. Berns tein and J.L. Hirschfield, Phys. Rev. A 20 (1979) 1661;P. Sprangle, C.M. Tang and W.M. Manheimer, Phys. Rev. A21 (1980) 302;V.N. Baler and A.1. Mil'shtein , Sov. Phys. Dokl. 25 (1980) 112;A. Gover and P. Sprangle, 1EEE J. Quantum Electron. QE-17 (1981) 1196;G. Dattoli, A. Marino, A. Renieri and F. Romanel li, Electron. QE-17 (1981) 137 l;C.C. Shih and A. Yariv, Electron. QE-17 (1981) 1387.

[3] R. Bonifacio, F. Casagrande and G. Casati, Optics Comm. 40 (1982) 2 l 9.[4] R. Bonifacio, C. Pellegrini and L. Narducci, Optics Comm. 50 (1984) 373.[5 ] J.B. Murphy, C. Pellegrini and R. Bonifacio , Optics Comm. 53 (1985) 197.[6] D.B. McDermot t, T.C. Marshall, S.P. Sehles inger, R.K. Parker and V.L. Granatstein, Phys. Rev. Lett. 41 (1978) 1368;

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