bonifacio 1987 0054

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Text of bonifacio 1987 0054

  • 8/14/2019 bonifacio 1987 0054

    1/6

    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

    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 )

    55

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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