paul l. csonka- double beat-wave mechanism to keep particle in phase with accelerating plasma wave

8/3/2019 Paul L. Csonka- Double Beat-Wave Mechanism to Keep Particle in Phase with Accelerating Plasma Wave

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P r e s e n t e d a t t h e 2 n d Workshop o nt fcssar A c c e l e r a t i o n o P a r t i c l e sUCLA, L os A n g e i e s , C A. J a n . 7 - 1 7 , 1 9 83

B N L 3 6 0 9 3DOUBLE BEAT-WAVE MECHANISM TO KEEP PARTICLE DE 85 0 0 8 2 8 8IN PRASE WITH ACCELEKATING PLASMA WAVE

P a u l L . C s o n k a *B r o ok h av e n N a t i o n a l L a b o r a t o r y , U p t o n , NT 11973ABSTRACT

Two mechanisms are described by which the phase matching betweenaccelerated particle and accelerating plasma wave in a plasma beat-waveaccelerator can be maintained indefinitely. 1) Interference betweentwo beat-waves to cancel a) the fields in the interfering electromag-netic waves, b) electron oscillations which would otherwise be set upby those waves. 2) Alternating plasma density used in conjunction withone or two beat-waves.INTRODUCTION

The potential well which accelerates particles in a plasma beat-wave accelerator/ 1' moves with a speedv ci 1 - (u /w) 1 , (1) Pwhere v g wil!eventually overtake the potential well, and that puts on upper limit onthe energy they can acquire. This limitation can be overcome by theproposed surfatron mechanisnr " {n which the particle trajectorymakes an angle 9 / 0 with the line along which the potential welltravels. Two difficulties arise in connection with the surfatron mecha-

nism. First, for 9 ^ 0 the potential well will tend to curve the par-ticle trajectory. To compensate for that, one has to impose an exter-nal magnetic field. Second, sisce the particle velocity has a compo-nent parallel to the plasma wave front, the transverse size of the ac-celerator has to be larger than if 9 0 were to hold, and in addition,some of the electromagnetic wave energy will be wasted. The second dif-ficulty can be significantly alleviated through the ingenious device of"transverse optical mixing."^*'Two alternatives to the surfatroa mechanism will be described inthe present note. Both insure continuing phase matching between-theaccelerated particle and the accelerating plasma wave. In both mecha-nisms phase matching is maintained even if the particle and plasma wavetravel parallel to each other (9 * 0 ) , and in neither mechanism is an

external magnetic field required.DOUBLE BEAT-WAVE MECHANISM

To insure continued phase matching through this mechanism, oneuses not one but two beat-waves. These beat-waves are arranged so thatthey interfere constructively whenever the particle is in phase withthe accelerating potential well created by both of them. On the otherhand, when the particle gets out of phase with the potential well

1 3 W H I 1 M T F D


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created by, say, the first beat-wave, the second beat-wave will tend tocancel the effect of the first one, so that the effect of decelerationwill be correspondingly reduced. The particle will then be allowed toessentially drift through these regions until it again gets into phasewith the accelerating potential, where it will be accelerated further,etc. Choose four electromagnetic plane waves of equal amplitudes, A,and equal polarization stste, all traveling parallel to the z axis,which is also the direction of motion of the accelerated particle, Letall four waves be in phase, with the electric field maximum at t - 0and z 0. Denote the circular frequency of the ith wave by u^ Ci 1,2, 3, 4 ) , the electron density in the plasma by p e, and the plasma fre-quency by to_ Choose

to til ID ( 2 a )to, - to. to ( 2 b )u>3 - ujj - Aw , ( 2 c )

The Aid is to be specified later. With this choice the first two waveswill set up a beat-wave with beat frequency u ) - , and so will the thirdand fourth wave.Assume, as usual for plasma beat wave accelerators, that tOp/u 1,for i 1, 2, 3, 4 . Then the phase velocity of the plasp" wave set upby an electromagnetic wave pulse will equal the group velocity of thatpulse.* 1' Denote the group velocity in the plasma of the first and sec-ond beat-wave by vgi and v.g respectively. The wave vector of theplasma wave set up by the first and second beat-waves, respectively,will be, m to , wp l * i cg l, (U to

We will refer to the beat-wave between the two beat-waves as"super beat-wave." The amplitude (envelope) of the super beat-wavevaries as

U J - 4|ACOS J ((WJ.- 3>t " (kj - k )z]]# AlAcos i [ (^t - jB | U% 2 - (jJ-E^-)^,} (

1 1

ALJ | {0 . - 0 4|Aco -Z


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I f t h e a c c e l e r a t i n g p o t e n t i a l w e l l i s L + l o ng a n d t r a v e l s w i t ha s p e e d v - ( a l l q u a n t i t i e s i n t h i s p a p e r a re m e a s u r e d i n t h e l a b o r a t o r yf r a m e ) , tfiea t h e t i m e i t t a k e s f o r a p a r t i c l e w i t h s p e e d v t o m o v ea c r o s s t h e a c c e l e r a t i n g p o t e n t i a l w e l l , f r o m o n e e n d t o t h e o t h e r , i st L L + / ( v - v _ ) . T h e d i s t a n c e t he p a r t i c l e t r a v e l s i n t he p l a s a a d u r -i n g t h i s t i m e i s( 5 )

a s s u m i n g v i s e s s e n t i a l l y u n c h a n g e d d u r i n g t he p r o c e s s .W e d e s c r i b e t wo d i s t i n c t m e t h o d s t o r e a l i z e t h e d o u b l e b e a t -w a v e m e c h a n i s m .a ) I n t he f i r s t m e t h o d & d i s c h o s e n s o th a t w h i l e t h e p a r t i c l et r a v e l s a d i s t a n c e L , o n e h a l f o f a c o m p l e t e s u p e r b e a t - w a v e p e r i o dp a s s e s o v e r i t sif - ( t o 1 - f c ^ t j - ( k j -: k 3 ) t ) . ( 6 )

L e t Ajjj b e t h e w a v e l e n g t h o f t h e p l a s m a w a v e g e n e r a t e d b y t h e f i r s te l e c t r o m a g n e t i c b e a t - w a v e . A s s u m e L X ^ j . T h e n t o i n s u r e c o n d i t i o n( 6 ) , o n e n e e d s

T o i l l u s t r a t e , c o n s i d e r t h e c a s e w h e n U j - 1 . 88 1 0 1 5 s e c " * , W -- 5 . 6 4 1 0 1 3 s e ' ' , a n d v c . T h e n t L 2 . 5 1 0 ~ 1 0 s e c , L 7 . 8c m , a nd o n e n e e d s : d u - 1 . 2 7 * 1 0 1 0 s e e ' 1 , s o t h a t A ii/u ij 6 . 8 1 0 " * 6.b ) I n th s e c o nd m e t h o d o n e c h o o s e s & u i t o b e s u c h t h a t w h i l e t h ep a r t i c l e t r a v e l s a d i s t a n c e 2 L , t he s e c o n d b e a t - w a v e m o v e s a h e a d o f t h ef i r s t o n e b y a d i s t a n c e A ^ j :

T h e n i f t h e t w o b e a t - w a v e s a r e i n p l a c e v h s n t h e p a r t i c l e i s i n s i d e a na c c e l e r a t i n g p o t e n t i a l w e l l , thctt t h e ? w i l l a g a i n b e ifi p h a s e w h e n t h ep a r t i c l e i s r e a c h e s t h e n e x t p o t e n t i a l w e l l . H a l f w a y b e t w e e n t h o s e t w oe v e n t s , h o w e v e r , t he e f f e c t o f t h t w o b e s t - w a v e * c a n c e l , w h i c h t e n dst o e l i m i n a t e a n y d e c e l e r a t i o n w h i c h w o u l d o t h e r w i s e o c c u r .A s a n i l l u s t r a t i o n a g a i n c h o o s e u> , u>_ a n d v a s i n t h e e x a m p l eg i v e n u n d e r a . _ T h e n f r om E q . 8 ) o n e f i n d s i u * 0.779 * 1 0 1 5 s e c " 1 ,i . e . i u / u j / 2 U | .T h i s m e t h o d g e n e r a l l y r e q u i r e s h i g h e r A u / u ) | v a l u e s t h a n th e m e t h o dd e s c r i b e d u n d e r a ) , t h at f e a t u r e i s s o m e t i m e s d e s i r a b l e .W h e n m e t h o d b ) i s u s e d , t h e co m m o n p o l a r i z a t i o n o f t h e w a v e s w i t hc i r c u l a r f r e q u e n c y u>j a n d u) m a y d i f f e r f r o m t he c o m m o n p o l a r i z a t i o n o ft he o t h e r t w o w a v e s .

A L T E R N A T I N G D E N S I T Y ( D O U B L E B E A T W A V E ) M E C H A N I S MI n t h i s m e c h a n i s m t he p l a s m a i s s u b d i v i d e d i n t o r e g i o n s w i t h d i f -f e r e n t e l e c t r o n d e n s i t i e s . I n r e g i o n s o f th e f i r st t y p e t h e e l e c t r o n


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d e n s i t y i f P ^ , w h i l e i n r e g i o n s o f t he se c o n d t y p e , t he d e n s i t y o fe l e c t r o n s i s p e g . W e v i l l r e f e r t o r e g i o n s o f t he f i r s t an d s e c o n dt y pe A - r e g i o n s a n d B - r e g i o n s r e s p e c t i v e l y . T h e p l a s m a f r e q u e n c y i nt h e A - r e g i o n s a n d B - r e g i o n s w i l l b e d e n o t e d b y u ) p^ a n d w p r e s p e c -t i v e ly. C h o o s e t w o e l e c t r o m a g n e t i c p l a n e w a v e s w i t h e q u a l a m p l i t u d e s ,a n d e q u a l p o l a r i z a t i o n , t r a v e ll i n g a l o n g t h e z a x i s . L e t t h e i rf r e q u e n c i e s , d) a n d ^ s a t i s f y

* x - 2 - M p 4 . ( 9 . )T h e s e t w o w a v e s w i l l s e t u p a b e a t - w a v e w h i c h w i l l g e n e r a t e r e s o n a n te l e c t r o n o s c i l l a t i o n s i n a n y r e g i o n - A . O n t h e o t h e r h a n d , i f P ^ and.p e A a r e s u f f i c i e n t l y d i f f e r e n t , t h e n t h i s s a n e b e a t - w a v e c a n n o t s e t u pr e s o n a n t e l e c t r o n o s c i l l a t i o n s i n a n y B - r e g i o n . C h o o s e t h e l e n g t h o fa n y A - r e g i o n t o b e L , c o r r e s p o n d i n g t o t h e v a l u e o f v a t t h e t i m e w h e nt h e p a r t i c l e p a s s e s t h r o u g h t h a t r e g i o n . T h e n i f t h e b e a t - w a v e i sc o r r e c t l y p h a s e d r e l a t i v e t o t h e p a r t i c l e , i t w i l l c o n t i n u o u s l y a c c e l e r -a t e i t w h i l e t he p a r t i c l e m o v e s a c r o s s t h e A - r e g i o n . A s t h e p a r t i c l el e a v e s t h e a c c e l e r a t i n g p o t e n t i a l w e l l , a n d w o u l d e n t e r c d e c e l e r a t i n gw e l l , t he p a r t i c l e e n t e r s a B - r e g i o n w h e r e i t e x p e r i e n c e s n o s i g n i f i -c a n t a c c e l e r a t i o n , d r i f t s t h r o u g h t h i s r e g i o n , e n t e r s t h e n e x t A - r e g i o nw h e r e it i s a c c l e r a t ed f u r t h e r , e t c .O n e c a n i m p r o v e t h e a b o v e s c h e m e b y u t i l i z i n g t w o m o r e e l e c t r o m a g -n e t i c p l a n e w a v e s w i t h e q u a l a m p l i t u d e s a nd p o l a r i z a t i o n , b o t ht r a v e l l i n g a l o n g t h e z a x i s . C h o o s e t h e i r f r e q u e n c i e s a c c o r d i n g t o

U 3 " \ " " p B 'a n d E q . ( 2 c ) . T h e s e t w o w a v e s w i l l s e t u p a s e c o n d b e a t - w a v e w h i c hw i l l g e n e r a t e r e s o n a n t e l e c t r o n o s c i l l a t i o n s i n a n y B - r e g i o n , b u t n o ti n th e A - r e g i o r s . I f th e s e c o n d b e a t - w a v e i s c o r r e c t l y p h a s e d r e l a t i v et o th e p a r t i c l e , i t w i l l a c c e l e r a t e i t a c r o s s B - r e g i o n s . T h u s t h e t w ob e a t - w a v e s t o g e t h e r w i t h t he a l t e r n a t i n g p l a s m a d e n s i t y * w i l l i n s u r ec o n t i n u o u s p a r t i c l e a c c e l e r a t i o n w h i l e t h e p a r t i c l e m o v e s a l o n g t he e n -t i r e l e n g t h o f t h e p l a s m a b e a t - w a v e a c c e l e r a t o r .T h e c o m m o n p o l a r i z a t i o n o f t h e f i rs t t w o e l e c t r o m a g n e t i c w a v e sn e e d n o t b e t h e s a m e A M t h e c o m m o n p o l a r i z a t i o n o f t h e o t h e r t w o .I n a l l t h e s c h e m e s d i s c u s s e d i n t h i s n o t e , c a r e a u s t b e t a k en t oc h o o s e Ato s o t h a t e x c e p t f o r t h e w a n t e d f r e q u e n c i e s , n o o t h e r b e a t f r e -q u e n c y b e t w e e n t h e v a r i o u s w a v e s s h o u l d o v e r l a p w i t h a n y r e s o n a n t f r e -q u e n c y r e g i o n s o f t h e p l a s m a .

A C K N O W L E D G M E N T SI a m g r a t e f u l t o J o h n M . D a w s o n , C h a n J o s h i a n d R o g e r G . E v a n s f o rd i s c u s s i o n o f t h i s p r o b l e m .

R E F E R E N C E S^ P e r m a n e n t a d d r e s s : I n s t i t u t e o f T h e o r e t i c a l S c i e n c e , U n i v e r s i t y o fO r e g o n , E u g e n e , O r e g o n 9 7 4 0 3 .( 1 ) T . T a j i a a a n d J . M . D a w s o n , K i y s . R e v . L e t t . 4 3 , 2 6 7 ( 1 9 7 9 ) .


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(2) T. Katsouleai and .M. Damon, IEEE Transaction! on Nuclear Sci-ence, Vol. HS-3O, 3241 ( 1 9 8 3 ) .

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(4) C. Joahi, W.B. Mori, T. Kataoulea*, J.M.Dawaon, J.M. indel,andD.W. Foralund, Nature 311., 525 ( 1 9 8 4 ) .

D I S C L A I M E RThis report w as prepared as an account of w ork sponsored by an agency of the United SlatesGovernm en!. Neithe r the United S tales Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or responsi-bility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represe nts that its use w ould not infringe privately ow ned rights, Refer-ence herein to aa y specific commercial p roduct, process, or service iby trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the Un ited States G overnment or any agency thereof. The view sand opinions of authors expressed herein do not necessarily state or reflect these cf theUnited States Government or any agency thereof.

paul l. csonka- double beat-wave mechanism to keep particle in phase with accelerating plasma wave

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