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Volume 50, numb er 6 OPTICS COMM UNICATIONS 15 July 1984

C O L L E C T I V E I N S T A B I L IT I E S A N D H I G H- G AI N R E G I M E I N A F R E E E L E C T R O N L A S E R

R . B O N I F A C I O * , C . P E L L E G R I N INational Syn chrotron L ight Source, Brookhaven Nat ional Laboratory , Upton, N Y 119 73, USAa n dL .M . N A R D U C C IPhy sics Dep artm en t , Dr exe l University, Philadelphia, PA 19104, USAReceived 5 April 1984

We study the behavior of a free electron laser in the high gain regime, and the condit ions for the emergence o f a col-lective instability in the electron beam-undulator-field system. Our equations, in the appropriate limit, yield the traditionalsmall gain formula. In the nonlinear regime, numerical solutions of the coupled equations of mo tion support the correct-ness of our proposed em pirical estimator for the build-up t ime o f the pulses, and indicate the existence of optimum parame-ters for the produ ction of high peak-power radiation.

S t u d i e s o f t h e f r e e e l e c t r o n l a s e r ( F E L ) i n t h e h i g hg a i n r e g i m e h a v e s h o w n t h a t w i t h a n a p p r o p r i a t e s e -l e c t io n o f th e e l e c t r o n d e n s i t y , d e t u n i n g a n d u n d u -l a t o r l e n g t h , t h e r a d i a t i o n f i e l d a n d t h e e l e c t r o nb u n c h i n g c a n u n d e r g o e x p o n e n t i a l g r o w t h a s a r es u l to f a c o l le c t i v e i n s ta b i l i ty o f t h e e l e c t r o n b e a m -u n d u l a t o r - r a d ia t i o n f i e ld s y s t e m [ 1 - 8 ] . I n th i s p a -p e r , w e s t u d y t h e c o n d i t i o n s f o r t h e o n s e t o f t h is i n -s t a b il i ty u s i n g a n e w s e c u la r e q u a t i o n f o r t h e c h a r a c -t e ri s ti c c o m p l e x f r e q u e n c i e s o f t h e F E L s y s t e m . O nt h e b a s i s o f th e s e r e s u l t s , w e s h o w h o w o n e c a n r e -de r i ve t he sma l l - s i gna l ga i n f o r mul a and e s t ab l i sh t hec o n d i t i o n s f o r i ts v a l id i t y . W e a l s o c o n s i d e r t h e p r o b -l e m o f t h e i n i ti a ti o n o f la s e r a c t i o n a n d o f t h e g r o w t ho f t h e r a d i a t i o n f i e ld f r o m n o i s e , a n d p r o p o s e a f o r -m u l a t o e v a l u a te t h e l e t h a r g y ( b u il d - u p ) t i m e o f t h ef i r s t p u l s e . F i n a l l y , w e s t u d y t h e n o n l i n e a r r e g i m e o ft h e F E L b y n u m e r i c a l m e t h o d s a n d o b t a i n r e m i tst h a t s u g ge s t t h e e x i s te n c e o f a n o p t i m u m e f f ic i e n c yo f t h e d e v i c e .

I n t h e d e r i v a t io n o f o u r w o r k i n g e q u a t i o n s w e s e -l e c t t h e p h a s e a n d t h e e n e r g y a s t h e b a s i c e l e c t r o nv a r ia b l e s, a n d a s s u m e t h e s l o w l y v a r y i n g p h a s e a n d* On leave from the University of M ilano, via Celoria 16,Milano, I taly.0 0 3 0 - 4 0 1 8 / 8 4 / $ 0 3 . 0 0 E ls e v ie r S c i e n c e P u 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 c s P u b l i s h i n g D i v i si o n )

a m p l i t u d e a p p r o x i m a t i o n f o r t h e r a d i a t i o n f ie l d a sd o n e a l s o i n e a r l i e r d e v e l o p m e n t s [ 9 , 1 0 ] . I n t h e r e -m a i n d e r o f t h e p a p e r w e s h al l a d o p t t h e f o l l o w i n gn o t a t i o n s : z r e p r e s e n ts t h e d i r e c t i o n o f p r o p a g a t i o no f t h e e l e c t r o n b e a m a n d o f t h e e l e c t r o m a g n e t icw a v e ; it a ls o re p r e se n t s t h e u n d u l a t o r a x i s ; x a n d ya r e t h e t r a n sv e r s e c o o r d i n a t e s ; B 0 d e n o t e s t h e s t r e n g t ho f t h e h e l i ca l m a g n e t i c f ie l d a n d ~ '0 a n d N O t h e p e r i o dl e n g t h a n d t h e n u m b e r o f p e ri o d s o f t h e u n d u l a t o r ,r e s p e c t i v e l y ; t h e u n d u l a t o r p a r a m e t e r i s K = e B o k o /( 21r mc2) , wher e m c 2 i s t h e e l e c t r o n r e s t e n e r g y ; k i st h e w a v e l e n g t h o f t h e r a d i a t i o n f i e ld , 3 ' is t h e e l e c -t r o n e n e r g y i n u n i ts o f m c 2, /3z ~ 1 i s t he l o ng i t ud i na le l e c t r o n v e l o c i t y a n d /~ p = K /T t h e a m p l i t u d e o f t h et r a n s v e rs e v e l o c i t y ; th e e l e c t r o n p h a s e , 4 , r e l a t iv e t ot h a t o f t h e e l e c t r o m a g n e t i c w a v e , is c o n n e c t e d t o za n d t b y t h e r e l a t i o n ~ = 27rz /X 0 + 2n ( z - cO ~ k; t h er e s o n a n t e n e r g y 3 tR is r e l a te d t o k 0 , k a n d K b y T2 =~ ,0 ( 1 + r 2 ) / 2 ; k , and , f i na l l y , t he undu l a t o r f r equency6o0 i s g i ven by 600 = 27rC13z/XO.W i t h th e s e n o t a t i o n s , t h e F E L w o r k i n g e q u a t i o n sc a n b e w r i t t e n a s [ 9 , 1 0 ]~bi = ~ 0 ( 1 - T 2 / 7 2 ) , ( 1 )

ec K [a exp(i~b/) + c .c . ] , (2)T i - 2 m c 2 7 /3 7 3

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V o l u m e 5 0 , n u m b e r 6 O P T I C S C O M M U N I C A T IO N S 15 July 1984

( o 1 a ( e - i+ / 3 ' > ( 3 )+ c ~ 7 !w h e r e j la b e l s t h e / t h e l e c t r o n i n t h e b e a m (J" = 1 , 2 , . . .,A re , w i t h N e t h e t o t a l n u m b e r o f e l e c t r o n s ) ; t h e a v e r -a g e C ) i s c a r r i e d o u t o v e r a ll e l e c t r o n s i n a b e a ms li ce o f l e n g t h X a t t h e p o s i t i o n z - ( l ~ z ) C t , w h e r eq~ z i s t h e a v e r a g e l o n g i t u d i n a l v e l o c i t y . T h e r e m a i n -i n g p a r a m e t e r s h a v e t h e f o l l o w i n g m e a n i n g : n e i s t h ee l e c t r o n b e a m l o n g i t u d i n a l d e n s i t y a t p o s i t i o n z -( # z ) C t , F , i s a n e f f e c t i v e b e a m t r a n sv e r s e c r o s s s e c t i o nd e s c r ib i n g t h e o v e r l a p o f t h e b e a m w i t h t h e r a d i a t i o nf i el d w h o s e a m p l i t u d e E 0 a n d p h a s e 0 0 h a v e b e e nc o m b i n e d i n th e c o m p l e x a m p l i tu d e a = i E 0 e x p ( i0 0 ) .I t i s i m p o r t a n t t o s t r e s s t h a t i n t h i s d i s c u s s i o n 3 ' i s n o tr e s t ri c t e d t o b e a p p r o x i m a t e l y e q u a l t o th e r e s o n a n tv a l u e 3' R , u n l i k e e a r li e r t r e a t m e n t s o f t h i s p r o b l e m .F o r t h e p u r p o s e o f o u r s u b s e q u e n t a n a l ys i s , i t i s c o n -v e n i e n t t o r e w r i t e e q s . ( 1 . 3 ) u s i n g t h e v a r i a b l e sz ' = z - ( [ 3 z ) C t , t ' = t , ( 4 )w i t h t h e r e s u l t :( a l a t ' ) O j = 6 o 0 ( I - 3 ' ~ I ~ - ) , ( 5 )

e c K [t~ e x p ( - i / ) + c . c . ] , ( 6 )( a / a t ' ) 3 ' j - 2 m c 2 3 " . ic a a -[ ( 1 - - ( / 3 z ) ) ~ z , + l ~ 7 ] - 2 7 r n e ( Z ' ) ~ ( e - i / 3 ' ) z , ( 7 )

T h e p r o p a g a t i o n t e r m ( 1 - < # z > ) a / a z ' i n e q . ( 7 ) i s i m -p o r t a n t t o d e s c r i b e t h e e v o l u t i o n o f t h e p u l s e i n t h eF E L , e sp e c i a l ly w h e n t h e a c c u m u l a t e d p a t h d i f f er -e n c e A L = L p h - L e l = ( c - o ) t i n t b e t w e e n t h e p h o -t o n s a n d t h e e l e c t r o n s d u r i n g a n i n t e r a c t i o n t i m e i sc o m p a r a b l e t o t h e l e n g t h o f t h e e l e c t r o n b u n c h i ts e l f.N o t e t h a t t h e p a t h d i f f e r e n c e A L c a n a l s o b e e x -p r e s s e d i n t h e f o r m C t i n t (1 - ( /~ z) ) = X 0 N 0 ( 1 - ( / 3 z ) ) =N O X . I n t h i s p a p e r , w e o n l y c o n s i d e r s i t u a t i o n s w h e r et h e l e n g t h o f t h e e l e c t r o n b u n c h i s s u f f i c i e n t ly l a r g ert h a n N O A ; t h u s , w e n e g l e c t t h e p r o p a g a t i o n t e r m a n da s s u m e t h e l o c a l e l e c t r o n d e n s i t y n e ( g ' ) t o b e c o n -s t a n t .

T h e l i n e a r s t a b i l i t y a n a l y s i s o f e q s . ( 5 ) - - ( 7 ) i s g r e a t -l y a i d e d b y t h e i n t r o d u c t i o n o f a s u it a b le s e t o f c o l -l e c t i v e v a r i a b l e s [ 8 ] . F o r t h i s p u r p o s e , w e f i r s t i n t r o -d u c e t h e r e la t iv i s ti c p l a s m a f r e q u e n c y~ p = ( 4 r r re n o C 2 / 3 " 3 0 ) l/ 2 , ( 8 )

w h e r e 3 '0 i s t h e i n i t ia l e n e r g y , a n d r e t h e c l a s s ic a le l e c t r o n r a d iu s , a n d t h e s o - c a ll e d P ie r c e p a r a m e t e rp = ( ~ K ( 3 ' 0 / 3 ' R ) 2 ~ 2 p / O 0 ) 2 / 3 . ( 9 )F u r t h e r m o r e , w e i n tr o d u c e t h e q u a n t i t y~ 0 = c 0 ( 1 - 7 2 1 3 ' 2 ) , ( 1 0 )a n d r e s c a l e t h e t i m e v a r i a b l e a s f o l l o w s :r = 2 W O P ( T R / 3 " o ) 2 t . ( 1 1 )I n t e r m s o f t h e n e w s c a le d v a r ia b l e s4 i = i - b 0 t ' r j - 3 ' i / ( p 3 ' 0 ) , ( 1 2 )A - a e x p ( i ~ o t ) [ ( 4 , m c 2 3 " O n O p 2 ) 1 / 2 ,t h e n o n li n e a r eq u a t io n s o f m o t i o n ( 5 ) - ( 7 ) t a k e t h ef o r m( d / d r ) 4 ] = ( 1 1 2p ) (1 - 1 ] 0 2 r 2 ) , ( 1 3 )( d / d z ) r j = - ( l l p ) [ ( A / r i ) e x p ( i 4 / - ) + c . c . ] , ( 1 4 )d A / d r = i a A + ( ] / p ) < e - i* I v > . ( 1 5 )N o t e t h a t i n t e r m s o f e qs . ( 1 3 ) - - ( 1 5 ) , t h e d y n a m i c so f th e F E L i s c o n t r o ll e d b y o n l y t w o p a r a m e t e r s , t h eP i e r c e p a r a m e t e r p ( e q . ( 9 ) ) a n d 6 = A / p , w h e r e A i st h e u s u a l d e t u n i n g ( 3'0 2 - 3 '2 R ) /( 2 3' 2R ) . B e c a u s e w e n e -g l e c t s p a c e - c h a r g e f o r c e s , w e s h a l l a s s u m e i n t h e f o l -l o w i n g t h a t p i s s u f f i c i e n t ly s m a l l e r t h a n u n i t y . I t isa l so w o r t h n o t i n g t h a t e q s. ( 1 3 ) - ( 1 5 ) a r e c o n s i s t e n tw i t h t h e c o n s e r v a t i o n l a wL = I A [2 + ( F ) = c o n s t a n t , ( 1 6 )o r a l s oL = m c 2 n o ( 3 " ) + E 2 / 4 r = c o n s t a n t , ( 1 6 ' )w h i c h c a n b e r e a d i l y r e c o g n i z e d a s t h e c o n s e r v a t i o no f e n e r g y f o r t h e e l e c t r o n b e a m - r a d i a t i o n f i el d s y s-t e m . T h e m e t h o d d e v i s e d t o a n a l y z e t h e s t a b i l i t y o ft h e s y s t e m i s b a s e d o n t h e p r o c e d u r e s u g g e s t e d in r e f.[ 8 ] . T h e e q u a t i o n s a r e l i n e a ri z e d a r o u n d t h e e q u i l i b -r i u m s t a t e A 0 = 0 , F 0 j = l / p , ( e x p ( - i n 4 0 ) ) = 0 a n dp e r t u r b e d b y l e t t i n g A = a , F i = ( l / p ) ( 1 + 7/1-) a n d4 i = 4 0 / + 6 4 i . T h e l i n ea r l iz e d e q u a t i o n s f o r m t h eb a s is f o r a c l o s e d f o r m l i n e a r s y s t e m o f e q u a t i o n s f o rt h e c o l l e c t i v e v a r i a b l e sx = ( 6 4 e x p ( - i 4 0 ) ) , ( 1 7)

3 7 4

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y = ( 1 /p ) (W e x p ( - i ~ k 0 ) ) , ( 1 8 )a n d f o r t h e f i e l d p e r t u r b a t i o n a . T h e s e t a k e t h e f o r md x l d r - - y ( 1 9 )d y / d r = - a , ( 2 0 )d a / d T " = - i S a - i x - p y . ( 2 1 )N o n t r i v i a l s o l u t i o n s w i t h a t i m e d e p e n d e n c e o f t y p ee x p ( i X r ) e x i s t i f a n d o n l y i f X i s a s o l u t i o n o f t h ec h a r a c t e ri s t ic e q u a t i o nX3 - 8 X 2 + p X + 1 = 0 . ( 2 2 )T h e r e s u l t s o f e a r li e r a n a l y s e s [ 1 - 8 ] c a n b e r e c o v e r e db y s e t t i n g f o r m a l l y p = 0 i n e q . ( 2 2 ) . C l e a r l y , e x p o -n e n t i a l g r o w t h s , a n d t h u s , u n s t a b l e b e h a v i o r , r e s u l tsi f t h e c u b i c e q u a t i o n ( 2 2 ) h a s o n e r e al a n d t w o c o m -p l e x c o n j u g a t e r o o t s . I n t h i s c a s e, t h e i m a g i n a r y p a r to f t h e e i g e n v al u e m e a s u r e d t h e r a t e o f g r o w t h o f t h eu n s t a b l e s o l u t i o n . T h e i n s t a b i l i t y c o n d i t i o n c a n b ee a s i ly d e r iv e d f r o m e q . ( 2 2 ) : i n t e r m s o f t h e p a r a m -e t e r s p a n d 8 i t t a k e s t h e f o r m ( f i g . 1 )O3 - - ~ p 2 6 2 * 9 p 6 - 6 + ~ - ~ > 0 . ( 2 3 )T h e t y p i c a l b e h a v i o r o f t h e e i g e n v a lu e s o f e q . ( 2 2 ) a sa f u n c t i o n o f d e t u n i n g i s s h o w n i n f ig . 2 . T h e e i g e n -v a l u e s a r e r e a l w h e n 6 e x c e e d s a c e r t a i n t h r e s h o l dv a lu e t h a t d e p e n d s o n p a c c o r d i n g t o e q . ( 2 3 ) , w h i let w o o f t h e e i g e n v a lu e s f o r m a c o m p l e x c o n j u g a t ep a i r w h e n 6 < 8 t h r .

T h e s m a l l si g n a l g a i n f o r m u l a e m e r g e s i n a n a t u r a lw a y f r o m o u r a n a l y s i s i n t h e l i m i t p ~ 0 , a n d f o r s u f -

a* 8Fig. 1 . Ins tabi l i ty bou ndary in the ~o, a) p lane. Fo r 8 < ~ * ,the solut ions of eqs. (1 3)- (15 ) are unstable for a l l va lues ofp . Fo r se lected values of p (e .g . , f i" in the f igure) uns table be-havior occurs for a < 6 thr"

0.5

n 5

J / J [

10 0 ( ~ t hr ~ 10Fig. 2 . The behavior of thre e e igenvalues of the secular equa-t i on a s a func t i on o f t he de tun ing pa ram e te r a and fo r p =0.1 . The ver t ica l axis labels both the real and imaginary par ts .The real par ts have been scaled by a fac tor of 10 to f i t thedisplay. For a suff ic ient ly posi t ive value of a ( i . e ., a > 6 thr" the e igenvalues are rea l (curves c , d , e) . A t thresh old , two ofthe real eigenvalues degenerate i nto o ne, while, for the samevalue of a , th e imaginary par ts (curves b , b ' ) b ecome di f fer -ent f rom ze ro. The real par t of the com plex conjugate e igen-values for 8 < a thr i s label led by a .

f i c i e n t l y l a r g e v a l u e s o f 16 I . I n t h i s l i m i t , t h e e i g e n -v a l ue s ta k e t h e a p p r o x i m a t e f o r mX 1 ~ 8 ( 1 - 1 / 6 ) , X 2 , 3 " ~+ 1 / 6 1 / 2 , 8 > 0 ,X 1 ~ 6 ( 1 - 1 / 6 ) , X 2 , 3 " + 1 / 1 6 1 1 / 2 , 8 < 0 , ( 24 )a s o n e c a n c o n FL rr n q u a l i t a t i v e l y f r o m f i g . 2 . T h e o u t -p u t f i e l d A ( r ) i n t h e l i n e a r r e g im e c a n b e c a l c u l a t e d a sa li n e a r s u p e r p o s i t i o n o f e l e m e n t a r y e x p o n e n t i a l f u n c -t i o n s w h o s e c o e f f i c i e n t s a r e t o b e F L xe d f r o m t h e i n i-t i al c o n d i t i o n s . A l e n g t h y , b u t s t r a i g h t f o rw a r d c a lc u -l a t i o n y i e l d s t h e f o l l o w i n g e x p r e s s i o n s f o r t h e s m a l ls i g n a l g a i n :G = [ l A ( r ) l 2 - - I A 0 1 2 ] / I A 0 1 2

= (4 / 8 3 ) ( 1 - c o s 6 r c o s r / x / r 6- - ~ 8 3 / 2 s i n S r s i n r / x / r 6 ) , 8 > 0 ,

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G = ( 4 / 6 3 ) ( 1 - c o s f i r c o s h z / V ~-~1613/2 s i n 6 r s i n h r / x / ~ ] ) , 6 < 0 . ( 25 )

I n o r d e r t o m a k e c o n t a c t w i t h t h e u s u a l s m a ll -s ig n a lga in f or m ula , i t i s no t en oug h to r equ i r e th a t 16 [ bes u f f i c ie n t l y la r g e r t h a n u n i t y , b u t o n e a l s o m u s t i m -p o s e t h e c o n d i t i o n r [ x / ~ < 1 . I n t h i s c a s e , e q . ( 2 5 )b e c o m e s

1G "~ ( 4 /63 ) ( 1 - cos 6 T - - ~ 6 "C f a n 6 z ) ( 2 6 )w hich , in f ac t , agr ees w i th th e s tand ar d exp r es s ionf o r G .

I n s p i t e o f t h e f a c t t h a t t h e e q u a t i o n s o f m o t i o no f t h e F E L a r e n o n l i n e a r , s o m e a s p e c t s o f t h i s p r o b -l e m c a n b e h a n d l e d a c c u r a t e l y b y a n a l y t i c m e a n s . T h ee v o l u t i o n b e l o w t h r e s h o l d ( 6 > 6 t h r ) i s g o v e m e d b yt h e l i n e a r a p p r o x i m a t i o n . I n t h i s r e g i m e , t h e e i ge n -va lues a r e r eal ( s ee f ig . 2 ) and the ou tpu t f i e ld di s-p l a y s sm a l l a m p l i t u d e o s c i l l a ti o n s w h e n p l o t t e d a s af u n c t i o n o f t i m e . O n v a r y i n g 6 , b e a t p a t t e r n s o r m o r ec o m p l i c a t ed - l o o k i n g m o d u l a t i o n e f f e ct s c an b e o b -s e rv e d , w h o s e o r ig i n c a n b e u n d e r s t o o d e n t i r e l y i nt e r m s o f th e e i g e n va l u e s o f t h e l i n e a r iz e d p r o b l e m .A r e p r e s e n t a t iv e e x a m p l e i s s h o w n i n f ig . 3 . I t m a y b ew o r t h m e n t i o n i n g t h a t w h i l e t h e t r a c e in f i g . 3 h a sb e e n o b t a i n e d b y t h e a p p r o p r i a te s u p e r p o s it i o n o fe x p o n e n t i a l f u n c t i o n s , t h e e x a c t s o l u t i o n o f e q s .( 1 3 ) - ( 1 5 ) i s i n d i s ti n g u i s h ab l e o n t h e s c a le o f t h isgr aph .

T h e s y s t e m e v o l u t i o n a b o v e t h r e s h o l d ( 6 < 6 t h r )

0 2 0 Z " 4 O

Fig. 3. Outpu t intensity IA12 for p = 0 .01 and 8 = 4.0. Theeigenvalues of the linearized equations are -0 .51 9, 0.628,3.066. The modulation is due to the b eat of the different ex-ponential terms in the solution.

=

I A

0 1 0 " C' 2 0

Fig. 4. O utput intensity 1,412 versus time above threshold.The parameters used in this simulation are p = 0.0021, 8 =1.86, no = 16.

i s en t i r e ly d i f f e r en t , a nd i s show n in f ig . 4 f o r thecase of ze r o in i t ia l f i e ld and an in i t i a l bunch ing pa-r a m e t e r I ( e x p ( -i ~ k )) l , s m a l l, b u t d i f f e r e n t f r o m z e r o .U n d e r u n s t a b l e c o n d i t i o n s , f l u c t u a t i o n s i n t h e e l e c -t r o n s i n j e c t i o n v e l o c i t ie s , o r t h e l a c k o f u n i f o r m i t yi n t h e i n it ia l d i s t r ~ u t i o n o f t h e e l e c t r o n p h a s e v a r i-ab les , o r the p r esence of an in i t i a l f i e ld w il l t riggerthe gr ow th o f a s igna l. The s igna l w i l l the n gr o w to ap e a k v a l u e a f t e r w h i c h i t o s c i l l at e s . T h i s b e h a v i o r i sve r y gener a l and i s independent o f the in i t i a l t r igger -i n g m e c h a n i s m a s lo n g a s th i s p e r t u r b a t i o n i s sm a l l .T h i s n o n l i n e a r r e g im e r e q u i re s n u m e r i c a l i n t e g r a t i o no f t h e f u l l e q u a t i o n s o f m o t i o n . T h i s w e h a v e d o n ef o r a n u m b e r o f v al u e s o f p a n d 6 .

B e c a u se o f t h e n a t u r e o f t h e t r i g g er i n g m e c h a n i s m ,i n t u i t iv e l y , o n e w o u l d e x p e c t t h a t t h e t i m e r e q u i r e df o r t h e i n i ti a l p u ls e t o b u i l d u p ( l e t h a r g y t i m e ) s h o u l db e a f a i r l y se n s it i ve f u n c t i o n o f t h e m a g n i t u d e o f t h ei n i ti a l f lu c t u a t i o n . W e h a v e e x a m i n e d t h e d e p e n d e n c eo f t h e b u i l d u p t i m e o f t h e f i r s t p u l s e o n t h e i n i t i a lv a l u e o f th e b u n c h i n g p a r a m e t e r , a n d v e r i fi e d t h a t ( a )a s i g n if ic a n t f r a c t i o n o f t h e b u i l d u p p r o c e s s i s w e l ld e s c r ib e d b y t h e l i n e a r iz e d e q u a t i o n s ; a n d ( b ) t h e a r -r ival t ime of the f i r s t pe ak i s w e l l desc r ibed by thef o r m u l a :7"peak = - - (1 /I m X) In I (exp(- i~k0) l + 1 . (27 )A t e s t o f t h i s e q u a t i o n i s p r o v i d e d i n f i g . 5 , w h e r e w eh a v e p l o t t e d t h e a r ri v a l t i m e o f t h e f i r s t p u l s e c a lc u -l a te d f r o m t h e n o n l i n e a r eq u a t i o n s o f m o t i o n ( 1 3 ) -( 1 5 ) , a s a f u n c t i o n o f t h e i n i t ia l b u n c h i n g p a r a m e t e rI ( exp( - i~b0) ) l . O ne asp ec t o f cons ider ab le in te r es t f o r

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Volume 50, number 6 OF r lCS COMM UNICATIONS 15 July 1984

20

1 0

I I Io , o ~ , ,1 < , , o ) 1

Fig. 5. The arrival time of the first peak (lethargy time) isplotte d as a func tion of the logarithm of the ini t ial bunch-ing parameter (dots). The solid curve corresponds to eq.(26). The parameters used in this scan are no = 8, p = 0.4,= 1.25.

2P I A I m a x

0 . 4

0.2 I . - /

I I ] IO 0 1 0.1 p

Fig. 6. Dependence of the peak output intensi ty pPilmaxon p in the ne ighb orho od and just above the instabil ityboun dary l ine o f f ig. 1. The solid l ine is only a quali tat iveaverage of the point s .

t h e p u r p o s e o f o p t i m i z i n g t h e s y s t e m ' s p a r a m e t e r s i st h e e x is te n c e o f a m a x i m u m p e a k p o w e r o u t p u t a s af u n c t i o n o f p a n d 5 . W e h a v e v e r if ie d t h a t w h i l e t h em a x i m u m g r o w t h r at e is o b t a i n e d f o r ~ ~ 0 , t h em a x i m u m p e a k a m p l i tu d e o f t h e f ir st p u ls e o c c u r sf o r ~ ~ t ~t h r . T hus , w e have s can ned t he ( / 9 , 6 ) p l anei n th e n e i g h b o r h o o d o f , b u t a b o v e , t h e t h r e s h o l d l i nea n d f o r A 0 = 0 , a n d r e c o r d e d t h e p e a k o u t p u t i n t e n s i typ [ A [ 2 a x a s a f u n c t i o n o f p ( fi g. 6 ). N o t i c e t h a t i t fo l lo w sf r o m e q . ( 1 6 ) t h a t p ] A [ 2 = ( (T f - 7 0 ) / 7 0 ) , s o t h a tp [ A [ 2 g i ve s t h e e n e r g y t r a n s f e r f r o m t h e e l e c t r o n s t ot h e r a d i a t i o n . T h e s c a t t e r o f t h e p o i n t s is a l m o s t c e r -t a i n l y d u e t o t h e s l ig h t v a r i a t i o n s o f t h e c o n d i t i o n sf r o m r u n t o r u n . T h e s o l i d l in e , w h i c h i s o n l y a q u a l i-t a t iv e a v e r a g e t h r o u g h t h e p o i n t s , s u g g e s t s t h e e x i s-t e n c e o f a n o p t i m u m g a i n -d e t u n in g c o n d i t i o n s u c ht h a t t h e e f f i c i e n c y o f t h e s y s t e m is m a x i m u m f o ro p e r a t i o n ju s t a b o v e t h r e s h o l d . I t i s c l e a r t h a t i n t h ep r e s e n c e o f e f f i c i e n c i e s a s l a rg e a s , in p r i n c i p l e , 4 0 % ,t h e o l d a p p r o x i m a t e t r e a t m e n t s [ 1 - 8 ] i n w h i c h t h ee l e c t ro n m o m e n t u m is a s s u m e d t o v a r y o n l y b y s m a lla m o u n t s c a n n o t b e a d e q u a t e t o d e s c r ib e s i t u a ti o n sw h e r e s u c h la r ge e n e r g y e x c h a n g e s t a k e p l a c e b e t w e e nt h e e l e c t r o n b e a m a n d f i e ld . O n t h e o t h e r h a n d , i t i si n t u i t iv e l y o b v i o u s t h a t f o r s u f f i c i e n t l y s m a l l v a l u e so f t h e P i e r c e p a r a m e t e r , t h e e l e c t r o n e n e r g y w i ll s u f-f e r o n l y a l i m i t e d d e p l e t i o n s o t h a t e a r l ie r t r e a t m e n t ss h o u l d b e s u f f i c i e n t l y a c c u r a t e .

A c k n o w l e d g e m e n t sO n e o f u s ( L M N ) w i s h es t o a c k n o w l e d g e t h e s u p -

p o r t o f t h e A r m y R e s e a r c h O f f ic e a n d t h e R e s e a r c hL a b o r a t o r i e s o f t h e M a r ti n -M a r i e tt a C o r p o r a t i o n . W ew i s h t o t h a n k D r . J . M u r p h y f o r m a n y u s e f u l d is c us -i o n s . T h e h e l p o f H . S a d i k y w i t h s o m e o f t h e n u m e r -i c a l c o m p u t a t i o n s i s a l s o g r a t e f u l l y a c k n o w l e d g e d .T h is w o r k h a s b e e n p a r t ia l ly s u p p o r t e d b y t h e U . S .D e p a r t m e n t o f E n e rg y .

R e f e r e n c e s

[ 1] N.M. Kroll and W .A. McM ullin, Phys. Re v. A17 (1978)300.[2] A. Gover and Z. Livni , Optics Comm. 26 (1978) 375.[3] I .B. Bemstein and J.L. Hirschfeld, Phys. Rev . A2 0(1979) 1661.3 7 7

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V o l u m e 5 0 , n u m b e r 6 O P T I C S C O M M U N I C A T I O N S 1 5 J u l y 1 9 8 4[4] C .C . Sh i l and A . Yar iv , IEEE J . Quan tum Elec t ronQ E 1 7 ( 1 9 8 1 ) 1 3 8 7 .[5 ] P . Sprang le , C .M. Tang and W.M. Ma nhe im er , Phys .R e v . A 2 0 ( 1 9 8 0 ) 3 0 2 .[ 6 ] G . D a t t o l i , A . M a r i n o , A . R e n i e r i a n d F . R o m a n e l l i ,

I E E E J . Q u a n t u m E l e c t r o n . Q E 1 7 ( 1 9 8 1 ) 1 3 7 1 .[ 7 ] A . G o v e r a n d P . S p r a n g le , I E E E J . Q u a n t u m E l e c t r o n .Q E 1 7 ( 1 9 8 1 ) 1 1 9 6 .

[ 8 ] R . Bo nifac io , F . Casagrande and G . Casca t i , Opt icsC o m m . 4 0 ( 1 9 8 2 ) 2 1 9 .[9 ] W.B . Colson and S .K . R ide , in : Phys ics o f quan tumelec t ron ics , Vol . 7 , eds . S .F . Jacob s e t a l . (Addison-Wesley , Read ing , MA 1980) p . 377 .

[ 10] C . Pe l legr in i , in : F re e e lec t ron lasers , eds . A .N . Ch es te rand S . Mar te l lucc i (P lenum Press, NY 1983) p . 91 .[ 1 1 ] M . A b r a m o w i t z a n d I . A . S t e g u n , H a n d b o o k o f m a t h e -m a t i c a l f u n c t i o n s ( D o v e r P u b l i c a t io n s , N Y , 1 9 7 0 ) e q .(3 .8 .2 ) , p . 17 .

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