S y n c h r o t r o n L a b o r a t o r y. P a s a d e n a, C a l i f o r n i a. J. H. M u l l i n s. A u g u s t 7,
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- Juliana Morton
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3 R e p o r t C T S L C A L I F O R N I A I N S T I T U T E O F T E C H N O L O G Y S y n c h r o t r o n L a b o r a t o r y P a s a d e n a, C a l i f o r n i a A N A C H R O M A T I C I N F L E C T O R F O R T H E C A L T E C H S Y N C H R O T R O N J. H. M u l l i n s A u g u s t 7, S u p p o r t e d i n p a r t b y t h e U. S. A t o m i c E n e r g y C o m m i s s i o n C o n t r a c t N o. A T ( ) - 6 8
4 -1- C o n t e n t s I. I n t r o d u c t i o n p. 2 II. A p p r o x i m a t i o n s a n d N o t a t i o n p. 2 III. M a t r i x R e p r e s e n t a t i o n o f a S i n g l e E l e m e n t p. 4 IV. S o m e G e n e r a l T h e o r e m s p. 7 V. D e s i g n C o n s i d e r a t i o n s f o r C a l t e c h M a c h i n e p. 1 3 V I. S p e c i f i c C h o i c e o f S y s t e m p. 1 6 V I I. D e r i v a t i o n o f t h e T r a n s m i s s i o n M a t r i x p. 1 7 V I I I. E x a m p l e o f A c t u a l D e s i g n p. 2 3 IX. S o m e G e n e r a l O b s e r v a t i o n s p. 2 6 X. S y n c h r o t r o n A c c e p t a n c e a n d E m i t t a n c e M a t c h i n g p. 2 7 X I. S a m p l e T r a n s f e r R e q u i r e m e n t s p. 2 9 X I I. Q u a d r u p o l e L e n s e s p. 3 0 Χ I I Ι. S u g g e s t e d S y s t e m s p. 3 4 X I V. D o u b l e t p. 3 5 X V. T r i p l e t p. 3 6 X V I. Q u a d r u p l e t p. 4 0 X V I I. S a m p l e Q u a d r u p l e t S y s t e m p. 4 6 X V I I I. S u m m a r y p. 4 8
5 -2- I. I n t r o d u c t i o n T h e f o l l o w i n g is a n a n a l y t i c i n v e s t i g a t i o n, u s i n g f i r s t o r d e r t h e o r y o n l y, o f a p r o p o s e d a c h r o m a t i c i n f l e c t o r f o r t h e C a l t e c h S y n - c h r o t r o n. N o t h i n g n e w i n p r i n c i p l e is i n v o l v e d i n a n y o f t h e t h e o r y p r e s e n t e d b e l o w, a n d s i m i l a r t y p e s o f a n a l y s e s h a v e b e e n c a r r i e d o u t b y o t h e r s 1 ). T h e i n v e s t i g a t i o n w a s p r o m p t e d b y t h e n e e d t o f i n d a s p e c i f i c a c h r o m a t i c i n f l e c t o r t o c o u p l e a 1 0 M e V e l e c t r o n l i n a c t o t h e C a l t e c h s y n c h r o t r o n. I n p a r t i c u l a r, it w a s d e s i r e d to f i n d a s y s t e m s u c h t h a t t h e i n j e c t o r w a s p l a c e d i n a c o n v e n i e n t l o c a t i o n, a n d, i f a n y e l e c t r o s t a t i c e l e m e n t s w e r e u s e d, t h a t t h e y h a v e r e a s o n a b l y l o w r e q u i r e d f i e l d s t r e n g t h s. II. A p p r o x i m a t i o n s a n d N o t a t i o n I n t h e f o l l o w i n g w e w i l l e m p l o y x a s t h e r a d i a l d i s p l a c e m e n t o f a p a r t i c l e f r o m t h e c e n t r a l o r b i t o f t h e s y s t e m, w h e r e w e w i l l m a i n t a i n t h e s e n s e o f x t h r o u g h o u t t h e s y s t e m b y r e f e r e n c e t o t h e c e n t r a l p a t h. θ w i l l b e d e f i n e d a s dx/d s, ( f o r s m a l l v a l u e s, t h e a n g l e o f t h e p a r t i c l e p a t h w i t h r e s p e c t t o t h e c e n t r a l p a t h ), w h e r e s is t h e d i s t a n c e a l o n g 1 ) S e e, f o r e x a m p l e : L e e T e n g, " A c h r o m a t i c B e n d i n g S y s t e m s ", A N L A D ; F. C. M i l l s, " A c h r o m a t i c B e a m B e n d i n g a n d P o s i t i o n S h i f t i n g S y s t e m s ", M U R A T e c h n i c a l N o t e, A C C P r o p o s a l, F i l e N o. P ; S. P e n n e r, R. S. I. 32, ( ) ; R a p h a e l L i t t a u e r, " S o m e D e s i g n C o n s i d e r a t i o n s f o r I n j e c t i o n O p t i c a l S y s t e m s ", u n p u b l i s h e d C o r n e l l r e p o r t.
6 -3- t h e c e n t r a l p a t h, a g a i n m a i n t a i n i n g a c o n s i s t e n t d e f i n i t i o n o f t h e s i g n o f t h e q u a n t i t y b y r e f e r e n c e t o t h e c e n t r a l p a t h. T h e q u a n t i t y k is d e f i n e d a s ΔΡ/P, t h e d e v i a t i o n o f t h e p a r t i c l e m o m e n t u m f r o m t h e c e n t r a l m o m e n t u m. T h e d e g r e e o f a p p r o x i m a t i o n u s e d h e r e is t h e u s u a l f i r s t o r d e r o n e. T h a t is, t h e p a r t i c l e s a r e p r e s u m e d t o o b e y s t r i c t l y t h e d i f f e r e n t i a l e q u a t i o n i n t h e h o r i z o n t a l p l a n e, w h e r e s is t h e c e n t r a l p a r t i c l e p a t h d i s t a n c e p a r a m e t e r, R is t h e r a d i u s o f c u r v a t u r e o f t h e c e n t r a l p a t h, a n d Q is d e f i n e d b y Q 2 = 1 - n m f o r m a g n e t i c e l e m e n t s w h e r e a n d B o is t h e m a g n e t i c f i e l d a t t h e c e n t r a l p a t h, (x + R = r) Q 2 = 2 - ne f o r e l e c t r o s t a t i c e l e m e n t s, w h e r e a n d E o is t h e r a d i a l e l e c t r i c f i e l d a t t h e c e n t r a l p a t h. ( T h e l a t t e r e x p r e s s i o n Q 2 = 2 - n e a p p l i e s o n l y i n t h e r e l a t i v i s t i c l i m i t ; i n g e n e r a l,
7 -4- it is 3 - (v/c)2 - n e. L i k e w i s e t h e c o e f f i c i e n t o f k g i v e n a b o v e is t r u e o n l y in the r e l a t i v i s t i c l i m i t f o r the e l e c t r i c case.) O f c o u r s e, it is t a c i t l y a s s u m e d t h a t x/r, θ, a n d k a r e a l l s m a l l q u a n t i t i e s. III. M a t r i x R e p r e s e n t a t i o n o f a S i n g l e E l e m e n t T o s i m p l i f y t h e a l g e b r a, w e e m p l o y h e r e 3 x 3 o p t i c a l t r a n s m i s s i o n m a t r i c e s f o r t h e system. F u r t h e r m o r e, w e h e n c e f o r t h r e s t r i c t o u r atten- t i o n o n l y to c o n s t a n t g r a d i e n t b e n d i n g e l e m e n t s w i t h n o r m a l i n c i d e n c e. T h a t is, the c e n t r a l p a t h i m p i n g e s n o r m a l l y u p o n t h e f ace o f t h e b e n d i n g e l e m e n t, s uch as in a w e d g e m a g n e t. I f X o, θ o, a n d k a r e t h e c o o r d i n a t e s o f a p a r t i c l e e n t e r i n g o n e side o f a b e n d i n g e l e m e n t, the c o o r d i n a t e s o f the p a r t i c l e e m e r g i n g f r o m the o t h e r side a r e g i v e n b y s o l u t i o n of the d i f f e r e n t i a l e q u a t i o n as kfinal = k (since, r i g o r o u s l y I n a m a g n e t i c e l e m e n t, a n d t o f i r s t o r d e r i n a n e l e c t r o s t a t i c o n e 2 ), k is cons t a n t.) 2)A c t u a l l y, w e m a y say t h a t k = c o n s t a n t r i g o r o u s l y i n t h e e l e c t r o s t a t i c c a s e a s w e l l, if w e a r e i n t e r e s t e d i n the p a r t i c l e p a r a m e t e r s o n l y w h e n t h e p a r t i c l e is o u t s i d e t h e b e n d i n g e l e m e n t, i.e., h a s r e t u r n e d t o the o r i g i n a l e l e c t r o s t a t i c p o t e n t i a l. T o f i r s t order, h o w e v e r, the r e s u l t s are e x a c t l y t h e same f o r a n e m e r g i n g p a r t i c l e (except f o r f r i n g i n g f i e l d e f f ects) a s t h o u g h w e h a d u s e d t h e f o r e g o i n g d i f f e r e n t i a l e q u a t i o n a n d k e p t k c o n s t a n t t h r o u g h the system. T h e a c t u a l c h a n g e i n p a r t i c l e e n e r g y is t a k e n u p in t h e d e f i n i t i o n o f Q.
8 -5- In the a b o v e, so is the p a t h l e n g t h o f the c e n t r a l orbit, a n d thus so/r = φ, t h e g e o m e t r i c a l a n g l e o f t h e b e n d i n g e l e m e n t. If w e t h e n r e p r e s e n t the p a r t i c l e c o o r d i n a t e s as a v e c t o r (xθk ), w e m a y t h e n d e s c r i b e the t r a n s m i s s i o n t h r o u g h t h e e l e m e n t b y the o p t i c a l m a t r i x w h e r e, a s usual, S i m i l a r l y, the t r a n s m i s s i o n o f a p a r t i c l e t h r o u g h a f i e l d - f r e e s e c t i o n o f l e n g t h l is g i v e n b y t h e m a t r i x a n d t h a t t h r o u g h a simple, a c h r o m a t i c l ens o f f o c a l l e n g t h f b y
9 -6- The m a t r i x o f t h e b e n d i n g m a g n e t M e g i v e n a b o v e m a y b e s i m p l i f i e d e n o r m o u s l y b y t a k i n g a s t h e r e f e r e n c e p l a n e s the p r i n c i p a l p l a n e s o f the system. T o i l l u s t r a t e, if w e m u l t i p l y Me o n t h e r i g h t a n d o n the left b y the m a t r i x w h e r e d = R/Q t a n ( Q φ/2), w h i c h is e q u i v a l e n t t o m o v i n g the r e f e r e n c e p l a n e s a d i s t a n c e d t o w a r d the c e n t e r o f t h e b e n d i n g e l e m e n t a l o n g the l i n e s d e f i n e d b y the d i r e c t i o n o f the i n c i d e n t and e x i t i n g c e n t r a l rays, w e f i n d T h e b e n d i n g e l e m e n t a c t s t hus as a lens o f f o c a l l e n g t h R/q s i n Q φ, e x c e p t t h a t d i s p e r s i o n is i n d i c a t e d b y t h e n o n - v a n i s h i n g t e r m < M 2 3 > = 1/Q s i n Q φ. I n the case o f t h e u n i f o r m w e d g e m a g n e t o r c o a x i a l c y l i n d r i c a l e l e c t r o s t a t i c e l e m e n t, Q = 1, a n d d = R t a n ( φ / 2 ). T h e two p l a n e s t hus b o t h p a s s t h r o u g h the i n t e r s e c t i o n o f t h e i n c i d e n t a n d e x i t i n g c e n t r a l r ays. F o r simplicity, w e w i l l w r i t e t h e m a t r i x as
10 -7- ρ = Q/R s i n Q φ w h e r e s = 1/Q s i n Q φ a n d a l l r e f e r e n c e w i l l b e t o the p r i n c i p a l p l a n e s in t h e s u b s e q u e n t a n a l y s i s. IV. S o m e G e n e r a l T h e o r e m s B e l o w a r e g i v e n some g e n e r a l r e s u l t s, some o f w h i c h w i l l b e f o u n d u s e f u l in w h a t f o l l o w s. T h e f i r s t t h e o r e m is t h a t the d e t e r m i n a n t o f a t r a n s m i s s i o n m a t r i x m u s t b e unity. In t h e f o r m w i t h w h i c h w e w i l l b e c o n c e r n e d, t h i s t h e o r e m implies o n l y t h a t A D - C B = 1, a n d n o s t a t e m e n t s m a y be m a d e a b o u t E a n d F. H o w e v e r, t h e r e s u l t is u s e f u l in d e t e r m i n i n g the f i n a l f o r m of the m a t r i x (and f o r c h e c k i n g a l g e b r a i c a n d n u m e r i c a l r e s u l t s ). N e x t, g e n e r a l a c h r o m a t i c i t y f o r a t r a n s m i s s i o n m a t r i x r e q u i r e s t h a t b o t h E a n d F v a n i s h, a n d t h u s r e p r e s e n t s t w o r e l a t i o n s t h a t m u s t b e s a t i s f i e d. T h i s, in g e n e r a l, is t a n t a m o u n t to the r e m o v a l o f two d e g r e e s of f r e e d o m a v a i l a b l e in t h e d e s i g n o f a n a c h r o m a t i c system. If one f u r t h e r w i s h e s to s p e c i f y t h e e n t i r e m a t r i x, o n e r e q u i r e s o n l y t h r e e
11 -8- m o r e r e l a t i o n s i n t h e f o r m w e e m p l o y, s i n c e t h e u n i t y d e t e r m i n a n t c o n d i t i o n t a k e s c a r e o f t h e f o u r t h e l e m e n t. T h u s a m i n i m u m o f f i v e p a r a m e t e r s m u s t b e a v a i l a b l e i n o r d e r f o r o n e t o s p e c i f y a s y s t e m o f t h i s f o r m c o m p l e t e l y. A l t h o u g h t h e s y s t e m f i n a l l y p r o p o s e d i n t h i s r e p o r t h a s n o i n h e r e n t s y m m e t r y p r o p e r t i e s, m a n y a c h r o m a t i c s y s t e m s c o n s t r u c t e d h a v e, a n d t h e r e a r e s o m e g e n e r a l r e s u l t s w h i c h a r e u s e f u l f o r t h e s e s i t u a t i o n s. G i v e n t h a t a s y s t e m c o n s i s t s o f t w o s i m i l a r p a r t s, w e t r e a t t w o c a s e s. F i r s t, l e t u s a s s u m e t h a t t h e s e c o n d p a r t c o n s i s t s o f t h e m i r r o r r e f l e c t i o n o f t h e f i r s t p a r t a b o u t a r e f e r e n c e p l a n e n o r m a l t o t h e c e n t r a l p a r t i c l e p a t h. W e c a l l t h i s c a s e I. W e c a n a l s o c o n s i d e r t h e s a m e s i t u a t i o n e x c e p t t h a t t h e c u r v a t u r e o f t h e e l e m e n t s i n t h e s e c o n d p a r t h a v e b e e n a l l r e v e r s e d. T h i s w e w i l l c a l l c a s e II. C a s e I m i g h t b e c a l l e d a b e a m b e n d i n g s y s t e m, C a s e I I a b e a m d i s p l a c e m e n t s y s t e m, s i n c e i n t h e l a t t e r c a s e t h e c e n t r a l r a y e m e r g e s n e c e s s a r i l y p a r a l l e l t o t h e e n t e r i n g c e n t r a l r a y. C A S E I C A S E I I
12 -9- T h e t r a n s m i s s i o n m a t r i x o f a p a r t i c l e t r a v e r s i n g t h e s e c o n d p a r t o f t h e s y s t e m c a n b e r e l a t e d t o t h a t o f t h e f i r s t, in t h e f o l l o w i n g w a y. F o r C a s e I t h e f o r m c a n b e f o u n d b y r e q u i r i n g t h a t i f a p a r t i c l e t r a v e r s e s t h e f i r s t h a l f o f t h e s y s t e m, h a s i t s d i r e c t i o n r e v e r s e d a t t h e c e n t r a l p l a n e, t r a v e l s b a c k w a r d t h r o u g h t h e f i r s t h a l f, a n d is r e v e r s e d a g a i n a t t h e b e g i n n i n g, t h e t r a n s m i s s i o n m a t r i x m u s t b e u n i t y. ( O f c o u r s e, w e h a v e t o i m a g i n e a l s o r e v e r s i n g t h e s i g n o f t h e m a g n e t i c f i e l d s in t h e m a g n e t i c e l e m e n t s, b u t t h i s d o e s n o t e n t e r t h e a l g e b r a. ) R e v e r s a l o f a p a r t i c l e d i r e c t i o n is m a t h e m a t i c a l l y e q u i v a l e n t, i n t h i s e a s e, t o m u l - t i p l i c a t i o n b y ( i. e., c h a n g i n g t h e s i g n o f θ) T h u s if t h e t r a n s m i s s i o n m a t r i x o f a p a r t i c l e g o i n g b a c k w a r d s t h r o u g h a s y s t e m T is T *, w e h a v e, s i n c e Ν θ = Νθ-1 F o r C a s e II, w e p e r f o r m t h e s a m e o p e r a t i o n e x c e p t w e m u s t r e v e r s e t h e s i g n o f k a t t h e s a m e p o i n t w e r e v e r s e t h e s i g n o f θ e a c h t i m e, s i n c e t h e s e c o n d h a l f o f t h e s y s t e m is t h e f i r s t h a l f r e v e r s e d i n d i r e c t i o n, a s i n C a s e I, e x c e p t t h a t t h e m o m e n t u m d i s p e r s i o n is a l s o r e v e r s e d in s i g n. T h u s, w h e r e
13 w e h a v e, s i n c e o r * w h e r e T f * t h e t r a n s m i s s i o n m a t r i x o f t h e s e c o n d h a l f o f t h e s y s t e m. A l g e b r a i c a l l y, t h e r e s u l t s a r e g i v e n b y t h e f o l l o w i n g : i f t h e n
14 A n d the f i n a l t r a n s m i s s i o n m a t r i c e s o f the s y s t e m s a r e g i v e n b y C a s e I C ase II T h e r e f o r e, b o t h s y s t e m s are n e c e s s a r i l y o f the f o r m N o w, if w e r e q u i r e a c h r o m a t i c i t y, w e r e q u i r e U 1 3 = U 2 3 = 0. F o r C a s e I, t h i s i m p l i e s B F = A F = 0, a n d since A D - B C = 1, the g e n e r a l r e q u i r e m e n t is t h a t F = 0 (Case I ). F o r C a s e II, w e r e q u i r e D E = C E = 0, a nd, a g a i n u s i n g A D - BC = 1, w e f i n d E = 0 (Case II). So t h e r e a re d e f i n i t e re- q u i r e m e n t s on the f o r m o f the h a l f s y s t e m i n o r d e r to a c h i e v e a c h r o m a t i - city.
15 It is i n t e r e s t i n g to note, a lso, t h a t in e i t h e r C ase I o r C a s e II, if w e f u r t h e r r e q u i r e t h a t e i t h e r o f the o f f d i a g o n a l e l e m e n t s U 12 o r U 2 1 b e zero, t h e n the d e t e r m i n e n t a l r e q u i r e m e n t a 2 = 1 f i x e s a = ±1. T h i s r e q u i r e m e n t a l s o i m p l i e s t h a t one o f t h e o r i g i n a l f o u r e l e m e n t s A, B, C, or D is a l s o zero. F o r o u r p u r p o s e s, h o w e v e r, the r e l e v a n t t h i n g is that, since U 1 1 = U 22 in the s y m m e t r i c o r a n t i s y m m e t r i c s y s t e m s, as w e l l as the re- q u i r e m e n t o n the d e t e r m i n a n t, t h e n t h r e e r e l a t i o n s o n l y a r e n e ce s s a r y o n the h a l f s y s t e m in o r d e r t o c o m p l e t e l y s p e c i f y the f i n a l a c h r o m a t i c system. F o r e x a m p l e, if t h e f i n a l m a t r i x is to b e o f the f o r m t h e n w e r e q u i r e f o r C ase I, u p p e r sign B = C = 0, F = 0 Case I, l o w e r sign A = D = 0, F = 0 Case II, u p p e r sign B = C = 0, E = 0 Case II, l o w e r sign A = D = 0, E = 0
16 V. D e s i g n C o n s i d e r a t i o n s for C a l t e c h M a c h i n e F o r a s a t i s f a c t o r y i n j e c t i o n s y s t e m, t h e f o l l o w i n g g o a l s w e r e set. T h e s e g o a l s a r e n o t n e c e s s a r i l y t h e o n l y p o s s i b l e o n e s, b u t w e r e r a t h e r c h o s e n o n a s o m e w h a t p e r s o n a l b a s i s a s b e i n g c o n v e n i e n t f o r c o n s t r u c t i o n a n d p l a c e m e n t o f e l e m e n t s. W e r e q u i r e, f i r s t, t h a t t h e s y s t e m b e a c h r o m a t i c ( t o f i r s t o r d e r ). S e c o n d, t h e r e a r e d i f f i c u l t i e s i n b r i n g i n g t h e b e a m i n t o t h e m a c h i n e, so t h a t o n l y c e r t a i n p a t h s a r e s u i t a b l e. I n p a r t i c u l a r, it w a s d e s i r e d t o e m p l o y a n e l e c t r o s t a t i c d e f l e c t o r f o r t h e f i n a l b e n d i n g e l e m e n t in o r d e r t o a v o i d p r o b l e m s w i t h s t r a y f i e l d s n e a r t h e s e p t u m. H o w e v e r, a t 1 0 M e V, t h e p r o p o s e d i n j e c t i o n e n e r g y, q u i t e h i g h f i e l d s a r e r e q u i r e d i n o r d e r t o a c h i e v e r e a s o n a b l e r a d i i o f c u r v a t u r e. S i n c e t h e a p e r t u r e w i l l h a v e t o b e o f t h e o r d e r o f a c e n t i m e t e r, l a r g e v o l t a g e s a r e a l s o c a l l e d f o r. I n o r d e r t o a v o i d s p a r k i n g p r o b l e m s, w e h a v e s e t a n u p p e r l i m i t o f 6 0 k v / c m o n t h e e l e c t r i c f i e l d, i m p l y i n g a l o w e r l i m i t o f ~ 1. 8 m e t e r s o n t h e r a d i u s o f c u r v a t u r e. S u c h a b e n d i n g e l e m e n t w i l l n o t a l l o w t h e b e a m to c l e a r t h e m a g n e t c o i l s, so it is p r o p o s e d to f o l l o w a s h o r t e l e c t r o s t a t i c e l e m e n t w i t h a n " H " m a g n e t p l a c e d i n t h e t a n g e n t t a n k a n d c l o s e t o t h e b e a m l i n e t o c l e a r t h e p a r t i c l e p a t h o f the m a g n e t c o i l s, a s s h o w n i n t h e s k e t c h. I t w a s f u r t h e r d e c i d e d to r e s t r i c t t h e i n d i v i d u a l e l e m e n t s to u n i f o r m f i e l d w e d g e m a g n e t s o r c o a x i a l c y l i n d r i c a l e l e c t r o s t a t i c d e f l e c t o r s (q = 1 )
17 for simplicity in construction, and to separate the vertical and horizontal focussing problems. A further goal was the design of the system such that it places the injector in a convenient, somewhat predetermined position. Having assigned these goals, we may now analyze, in general, the number of parameters required. Each element has two; a radius of curvature R and a geometrical angle φ. Further, there is a drift section between each element. Then a symmetric or antisymmetric system has 3J parameters for the half system, where J is the number of elements in the half system. A non-symmetric system has 3J - 1 parameters, where J is the number of elements. In the system we desire, 5 parameters have already been determined; i.e., we have specified the radii of curvature and geometrical angles of the first two elements, and the
18 d i s t a n c e b e t w e e n t h e m 3 ). N o w, t h e a c h r o m a t i c r e q u i r e m e n t p r o v i d e s two r e l a t i o n s h i p s f o r t h e n o n - s y m m e t r i c system, o r one r e l a t i o n s h i p f or the h a l f s y s t e m in t h e s y m m e t r i c o r a n t i s y m m e t r i c c ases. I n o r d e r t o s p e c i f y c o m p l e t e l y t h e t r a n s m i s s i o n m a t r i x, w e r e q u i r e t h r e e m o r e r e l a t i o n s h i p s in the n o n - s y m m e t r i c case, or t w o o n t h e h a l f s y s t e m o f the symmetric o nes. F o r o u r p u r p o s e s, h o w e v e r, it w a s d e c i d e d t h a t it is u n n e c e s s a r y to s p e c i f y c o m p l e t e l y the t r a n s m i s s i o n m a t r i x. I n s t e a d, it w a s d e c i d e d t h a t the m a t r i x s h o u l d b e o f the f o r m a n d n o t t o s p e c i f y the v a l u e o f Λ, s ince t h e e l e m e n t -Λm c a n b e m ade zero b y s i m p l y t r a n s l a t i n g the i n p u t r e f e r e n c e p l a n e a d i s t a n c e Λ u p s t r e a m f r o m the p o i n t o r i g i n a l l y c h o s e n f o r t h e a n a l y s i s. W i t h the s o m e w h a t less s t r i n g e n t r e q u i r e m e n t g i v e n a bove, the t o t a l n u m b e r o f d e t e r m i n e d p a r a m e t e r s o r r e l a t i o n s h i p s is g i v e n below: S y m m e t r i c o r A n t i s y m m e t r i c System: 1. N u m b e r o f r e q u i r e m e n t s o n h a l f s y s t e m = 7 2. M i n i m u m n u m b e r o f e l e m e n t s in h a l f s y s t e m s u c h that 3J > 7; J = 3 3)T h e r e is, to b e sure, some l a t i t u d e in a s s i g n i n g t h e s e p a r a m e t e r s, b u t t o b e safe, w e a s s u m e t h e y are fixed.
19 C o n c l usion: a six e l e m e n t s y s t e m is r e q u i r e d. (Or 5 w i t h s y m m e t r i c p l a n e in c e n t e r o f No. 3.) N o n - S y m m e t r i c System: 1. N u m b e r o f r e q u i r e m e n t s o n s y s t e m = 9 2. M i n i m u m n u m b e r o f e l e m e n t s s u c h t h a t 3 J - 1>9; J = 4 3. Conclusion: a f o u r e l e m e n t s y s t e m is r e q u i r e d. T h u s it is s e e n t hat in e i t h e r a 6 e l e m e n t s y m m e t r i c s y s t e m o r a f o u r e l e m e n t n o n - s y m m e t r i c system, t h e r e r e m a i n two f r e e p a r a m e t e r s w i t h w h i c h w e are a l l o w e d t o d e t e r m i n e the f i n a l p l a c e m e n t o f the injector. VI. S p e c i f i c Choice o f S y s t e m A l t h o u g h the symmetric s y s t e m s u g g e s t e d a b o v e h a s m a n y p l e a s i n g f e a t u r e s, s u c h as r e q u i r i n g o n l y t h r e e d i s t i n c t t y p e s of e l e m e n t s, a n d t h e r e m a y b e some r e d u c t i o n in g e o m e t r i c a l a b e r r a t i o n s, it w a s d e c i d e d t h a t the r e d u c t i o n in t o t a l n u m b e r o f e l e m e n t s t o f our, p o s s i b l e w i t h the n o n - s y m m e t r i c system, w a s s u f f i c i e n t l y a t t r a c t i v e t o w a r r a n t its a d o p t i o n. T h e a l g e b r a i n v o l v e d is s o m e w h a t m o r e severe, since the 6 e l e m e n t s y s t e m r e q u i r e s the c o n s i d e r a t i o n o f o n l y the h a l f system, o r t h e m u l t i p l i c a t i o n o f 6 m a t r i c e s (3 b e n d i n g e l e m e n t s, 3 d r i f t sections) w h i l e the f o u r e l e m e n t s y s t e m r e q u i r e s the m u l t i p l i c a t i o n of 7 (4 b e n d - i n g e l e m e n t s, 3 d r i f t sections).
20 -17 - VII. D e r i v a t i o n o f t h e T r a n s m i s s i o n M a t r i x The c a l c u l a t i o n is g i v e n b e l o w in steps. W e r e t a i n t h e t e r m i n o - l o g y o f ρ a n d s as d e f i n e d e a r l i e r, so t h a t t h e f i n a l m a t r i x a l l o w s a f ree c h o i c e o f Q = 1 - n. H o w e v e r, w e w i l l a c t u a l l y u s e o n l y Q = 1. F u r t h e r m o r e, it is t o b e n o t e d t h a t e i t h e r r i g h t o r left b e n d i n g ele- m e n t s are a l l o w e d, as t h i s is s i m p l y d e t e r m i n e d b y the s i g n o f s; if s > 0 is t a k e n as the " n o r m a l " b e n d i n g d i r e c t i o n, a n s < 0 c o r r e s p o n d s t o a r e v e r s e m a g n e t c u r v a t u r e (i.e., to c h a n g i n g t h e s i gn of b o t h φ a n d R, so t h a t ρ is a l w a y s p o s i t i v e d e f i n i t e ). W e a s s u m e h e r e t hat a l w a y s -π < Q φ < π. The i n p u t a n d o u t p u t p l a n e s of the b e n d i n g ele- m e n t s are a l w a y s t a k e n as the p r i n c i p a l p l a n e s, a n d the d r i f t s e c t i o n l e n g t h b e t w e e n the e l e m e n t s is m e a s u r e d f r o m t h o s e p l a n e s. The s k e t c h b e l o w c l a r i f i e s the n o t a t i o n O U T P U T P L A N E I N P U T P L A N E T h a t is, ρ 1, s 1 are t h e p a r a m e t e r s o f the e l e c t r o s t a t i c e l e m e n t, a n d ρ 2, s2, etc., are the p a r a m e t e r s o f the m a g n e t s t a k e n in s e q u e n t i a l o r d e r f r o m the m a c h i n e to the injector. The d i s t a n c e l 2 is t h e d r i f t
21 space b e t w e e n the e l e c t r o s t a t i c e l e m e n t a n d the f i r s t m a g n e t, a n d l 2, l 3, are the o t h e r d r i f t spaces in t h e same order. (Taken b e t w e e n p r i n c i p a l p l a n e s o f the elements'.) T h e n if M i is t h e t r a n s m i s s i o n m a t r i x o f the it h b e n d i n g e l e m e n t, a n d Li t h a t o f the Lt h d r i f t region, the m a t r i x w e s e e k is g i v e n b y The r e s u l t s follow. L3M 3 is L 2M 2 g i v e n a b o v e w i t h s u b s c r i p t s c h a n g e d f r o m 2 t o 3. T h e n
22 t h e n L 4M 4 is g i v e n b y the e x p r e s s i o n L2M 2 w i t h the o b v i o u s s u b s c r i p t s changed, so f i n a l l y
23 -20 -
24 -21 - T h i s is n o w t h e m a t r i x u p o n w h i c h w e m u s t impose the t r a n s m i s s i o n c o n d i t i o n s. In o r d e r to s i m p l i f y the s u b s e q u e n t a l g e b r a, w e w i l l c o n s i d e r s1, ρ 1, l 2, s2, ρ 2 as pr e d e t e r m i n e d c o n s t a n t s. T h e n w e d e f i n e (1a) (1b) (1 c) T h e n the m a t r i x w e r e q u i r e is w h e r e w e w i l l not s p e c i f y Λ. T h i s l e a d s t o the f o l l o w i n g f o u r relations: A c h r o m a t i c C o n d i t i o n (ΤA )13 = ( Τ A )23 = 0: (2a) (2b)
25 R e q u i r e m e n t (TA )21 = 0: (2c) R e q u i r e m e n t ( T A )2 = 1/m : (2d) F o u r s o m e w h a t s i m p l e r r e l a t i o n s c a n b e d e r i v e d f r o m t h e f o r e g o i n g b y t a k i n g s u i t a b l e l i n e a r c o m b i n a t i o n s. If (2a) is m u l t i p l i e d b y β a n d (2b) b y α a n d the r e s u l t s a d d e d, w e f i n d (3a) T o o b t a i n t h i s r e s u l t, w e m u s t o b s e r v e t h a t T h e n if we m u l t i p l y (2a) b y ϒ a n d (2b) b y - l 2, add, a n d u s i n g α ϒ + β l 2 = 1, w e f i n d (3b) F u r t h e r, if w e m u l t i p l y (2c) b y - l 4, (2d) b y (1 - l 4 ρ 4 ), a dd, w e o b t a i n (3c)
26 -23 - F i n a l l y, m u l t i p l i c a t i o n o f (2d) b y ρ 4 a n d a d d i t i o n to (2c) y i e l d s (3d) T h e f i n a l e l e m e n t -Λm, is g i v e n b y (4) The r e l a t i o n s 3 a, b, c, d, are the o n e s w e s h a l l e m p l o y in a s p e c i f i c d e s i g n, a n d w e w i l l d i s c o v e r the v a l u e o f Λ o n l y a f t e r the d e s i g n is m a d e. V I I I. E x a m p l e o f A c t u a l D e s i g n F o r p u r e l y n u m e r i c a l r e a s o n s, it is c o n v e n i e n t to r e w r i t e the r e l a t i o n s g i v e n a b o v e i n a s o m e w h a t d i f f e r e n t w a y. S i n c e t h e r e are f o u r e q u a t i o n s a n d six u n k n o w n s ( l 3, ρ 3, s3, l 4, ρ 4, s4 ), t w o o f t h e m m u s t be s p e c i f i e d b e f o r e the s y s t e m is f u l l y d e t e r m i n e d. One o f t h e a r b i t r a r y c o n d i t i o n s w h i c h w e set is t h a t the o r i e n t a t i o n o f the i n j e c t o r is s u c h t h a t its l o n g d i m e n s i o n (the b e a m line) be m a d e p a r a l l e l to the w a l l o f the l a b o r a t o r y. T h i s is r e a l l y a c o n d i t i o n o n the s u m φ 3 + φ 4, since φ 1 a n d φ 2 are p r e d e t e r m i n e d, b u t it is m o r e c o n v e n i e n t to a c t u a l l y s p e c i f y s3 a n d s4 s e p a r a t e l y so as to s a t i s f y the a n g u l a r c o n d i t i o n, a n d solve f o r the o t h e r p a r a m e t e r s. T h e n b y c h o o s i n g s e v e r a l such sets o f s 3 a n d s4, w e m a y f i n d the s y s t e m m o s t s a t i s f a c t o r y in the o t h e r p a r a m e t e r s.
27 T h e r e f o r e, c o n s i d e r i n g s3 a n d s4 as k n o w n q u a n t i t i e s, w e w r i t e the f o l l o w i n g f o u r e x p r e s s i o n s d e r i v e d f r o m 3 a, b, c, d. (5a) (5b) (5c) (5d) D e t e r m i n a t i o n o f s3 a n d s4 t h e n a l l o w s t h e s e r e l a t i o n s t o b e e m p l o y e d in n u m e r i c a l o r d e r to y i e l d the o t h e r p a r a m e t e r s. A s p e c i f i c e x a m p l e is g i v e n b e l o w. A set o f p a r a m e t e r s w h i c h a l l o w the b e a m to be c a r r i e d to the s e p t u m are r 1 = m e t e r s s1 = ( φ ) l 2 = m e t e r s r 2 = m e t e r s s2 = ( φ 2 = 4 θ )
28 -25 - thus: ρ 2 = ρ 1 = α = β = ϒ = W e n o w t a k e m = +1, (for u n i t y t r a n s f e r m a t r i x e x c e p t f o r t he "drift" t e r m -mλ ). W e set φ 3 = φ 4 = 44.7, s3 = s4 = T h e n w e f i n d l 3 = m e t e r s l 4 = m e t e r s ρ 3 = o r r3 = m e t e r s ρ 4 = o r r4 = m e t e r s a n d a l s o -Λm = - Λ = m, t hat is, Λ is a p o s i t i v e n u m b e r. This implies t h a t w e m u s t a d d m e t e r s to the f l i g h t p a t h to a c h i e v e t rue u n i t y t r a n s f e r. T h i s is s a t i s f a c t o r y, since w e a l r e a d y h a v e to m o v e the e x i t (output) p l a n e a b o u t 14 c m t o w a r d the m a c h i n e (to t he e n d o f the e l e c t r o s t a t i c section) for p r o p e r i n j e c t i o n, a n d t he e n t r a n c e (input) p l a n e a b o u t 15 c m t o w a r d the i n j e c t o r to g e t it t o the f r o n t f ace o f the e n t r a n c e m a g n e t (M4 ). T h e n w e m e r e l y have to c o n s i d e r the i nput p l a n e t o e x i s t some 93 c m b e f o r e t h e e n t r a n c e m a g n e t (M4 ) p ole face.
29 T h e s y s t e m s e e m s a n a c c e p t a b l e one, as t h e i n j e c t o r e n d s u p p a r a l l e l w i t h t h e l a b o r a t o r y w a l l, a n d w e l l r e m o v e d f r o m the m a c h i n e. A r o u g h s k e t c h is s h o w n b e l o w. It is a l s o t o b e n o t e d t h a t a r a d i u s o f c u r v a t u r e o f 1 / 3 m e t e r f o r 10 M e v e l e c t r o n s r e q u i r e s a m a g n e t i c f i e l d o f a b o u t gauss, a c o n v e n i e n t f i g u r e f o r m a g n e t d e s i g n. O t h e r v a l u e s can, of course, b e f o u n d u s i n g d i f f e r e n t v a l u e s of s3 a n d s4, (and b y c h a n g i n g s1, s2, ρ 1, ρ 2, l 2 slightly), as the a b o v e is o n l y one s p e c i f i c e x a m p l e. One can a l s o a d j u s t m. IX. S o m e G e n e r a l O b s e r v a t i o n s It m i g h t a l s o b e c o n v e n i e n t to f i n d a set o f p a r a m e t e r s s u c h t hat the i n j e c t o r e n d s u p p o i n t e d in the o p p o s i t e d i r e c t i o n, i.e., it w o u l d b e w e s t o f the m a c h i n e, p o i n t e d north, a n d i n j e c t i n g i nto the N. W. t a n g e n t t a n k. T h i s t u r n s o u t to b e h a r d t o d o w i t h t h e s t i p u l a t i o n s w e
30 h a v e m a d e, a s c a n b e s een b y e x a m i n a t i o n o f (5b). A l l q u a n t i t i e s i n the n u m e r a t o r o n t h e r i g h t a r e n e c e s s a r i l y p o s i t i v e e x c e p t α, a n d s1 is r e q u i r e d to b e a s m a l l n u m b e r. T h u s it is d i f f i c u l t to m a k e s4 a n e g a t i v e q u a n t i t y w i t h o u t a l s o m a k i n g l4n e g a t i v e. S u c h a s y s t e m w i t h o u t a n e g a t i v e s4 w o u l d b e v e r y clumsy, as a l l the r e v e r s a l in b e a m d i r e c t i o n w o u l d h a v e t o b e d o n e b y M 3. A b e r r a t i o n s O n l y r o u g h e s t i m a t e s h a v e b e e n m a d e o f t h e a b e r r a t i o n s o f the system. T h e s e e s t i m a t e s i n d i c a t e that, o w i n g t o t h e s m a l l b e a m a n d b e a m d i v e r g e n c e, the d e t r i m e n t a l e f f e c t o f the g e o m e t r i c a l a n d c h r o m a t i c a b e r r a t i o n s s h o u l d b e n e g l i g i b l e. T h i s d o e s not, h o w e v e r, r e l i e v e t h e r e q u i r e m e n t s o n the g o o d n e s s o f the e l e m e n t s t h e m s e l v e s, or t h e c o n t r o l o f t h e m a g n e t i c fields. X. S y n c h r o t r o n A c c e p t a n c e a n d E m i t t a n c e M a t c h i n g It w o u l d a p p e a r that the o u t p u t o f t h e p r o p o s e d i n j e c t o r (~ 0. 6 c m d i a m e t e r spot, 1 m r spread) is r o u g h l y in t h e p r o p e r r a n g e f o r t h e s y n c h r o t r o n a c c e p t a n c e. If, h o w e v e r, w e d e c i d e to choose, f o r e x a m p l e, a s m a l l e r s pot a n d l a r g e r a n g u l a r s p r e a d in t h e m e d i a n p l a n e f o r the i n j e c t i o n p h a s e space, t h i s c a n be d o n e in two w a y s. F i r s t, w e c o u l d a d j u s t the v a l u e o f m in t h e f o r m u l a s ( 5a,b,c,d), t o t h e p r o p e r value, o r w e c o u l d s i m p l y f o c u s the p r o p e r e m i t t a n c e o n the i nput p l a n e o f a n m = 1 i n f l e c t o r w i t h a u x i l i a r y lenses. T h e l a t t e r c o u r s e w o u l d involve some i n c r e a s e in t h e s p a c i n g b e t w e e n t h e i n j e c t o r a n d the i n f l e c t o r
31 i n p u t p l a n e t o a c c o m o d a t e t h e l e n s e s, b u t p r o b a b l y n o t m u c h. W e w i l l a d o p t t h e l a t t e r c o u r s e, o w i n g t o i t s m u c h g r e a t e r f l e x i b i l i t y. P r e f e r a b l y, t h e e m i t t a n c e m a t c h i n g l e n s s y s t e m s h o u l d b e a d j u s t a b l e o v e r a m o d e r a t e r a n g e, a s t h e o p t i m u m m a t c h i n g m u s t e v e n t u a l l y be d e t e r m i n e d e x p e r i m e n t a l l y. H o w e v e r, a s a n i n i t i a l g u i d e w e w i l l a s s u m e t h e s y n c h r o t r o n a c c e p t a n c e a s c a l c u l a t e d b y P e c k *, a n d w e w i l l a s s u m e t h e w o r s t c a s e f o r t h e o u t p u t o f t h e l i n a c, w h i c h is t h a t o f a d i f f u s e s o u r c e w i t h t h e a n g u l a r l i m i t s ± 1 m r a n d s p o t s i z e 6 m m d i a m e t e r. I n d e s i g n i n g t h e s y s t e m, h o w e v e r, w e m u s t k e e p i n m i n d t h a t t h e l i n a c o u t p u t m a y, i n f a c t, b e a m o r e c o r r e l a t e d s o u r c e, so t h a t i m p r o v e m e n t m a y b e a c h i e v e d i n c a p t u r e b y t a k i n g a d v a n t a g e o f t h e c o r r e l a t i o n o f d i v e r g e n c e a n g l e w i t h r a d i a l p o s i t i o n. T h i s is t h e p r i m a r y r e a s o n f o r r e q u i r i n g t h a t t h e m a t c h i n g s y s t e m b e f l e x i b l e. F o r t h e p u r p o s e o f t h e f o l l o w i n g d i s c u s s i o n, w e w i l l a s s u m e t h a t a l l e m i t t a n c e m a t c h i n g e l e m e n t s m u s t b e p l a c e d b e t w e e n t h e i n j e c t o r a n d t h e i n p u t o f t h e i n f l e c t o r. T h i s s e p a r a t e s t h e v e r t i c a l f r o m t h e h o r i z o n t a l r e q u i r e m e n t s, w h i c h a r e q u i t e d i f f e r e n t, a n d a l s o s e p a r a t e s, in f i r s t o r d e r, t h e e m i t t a n c e m a t c h i n g p r o b l e m f r o m t h e r e q u i r e m e n t o f f i r s t o r d e r a c h r o m a t i c i t y i n t h e b e a m b e n d i n g s y s t e m. T h u s w e c o n s i d e r t h a t w e m u s t c a r r y t h e o u t p u t o f t h e l i n a c i n t h e h o r i z o n t a l p l a n e t o t h e h o r i z o n t a l i n p u t p l a n e o f t h e i n f l e c t o r w i t h t h e p r o p e r o p t i c a l * C h a r l e s P e c k, "C a p t u r e E f f i c i e n c y o f a C o n s t a n t G r a d i e n t S y n c h r o t r o n ", C a l i f o r n i a I n s t i t u t e o f T e c h n o l o g y S y n c h r o t r o n L a b o r a t o r y R e p o r t C T S L
32 a d j u s t m e n t s. F o r t h i s p u r p o s e, w e w i l l a s s u m e t h a t the i n f l e c t o r h a s u n i t t r a n s f e r to the s y n c h r o t r o n septum. T h e v e r t i c a l e m i t t a n c e, o n the o t h e r h a n d, m u s t b e t r a n s f e r r e d, w i t h t h e p r o p e r a d j u s t m e n t s, f r o m the i n j e c t o r t o the s y n c h r o t r o n septum. W e w i l l c o n s i d e r the b e a m b e n d i n g s y s t e m t o h a v e no v e r t i c a l f o c u s s i n g p r o p e r t i e s, ( w h i c h is t r u e t o f i r s t o r d e r ). T h e d i s t a n c e t h r o u g h the i n f l e c t o r is a b o u t 5 m e t e r s i n the p r o p o s e d system. XI. S a m p l e T r a n s f e r R e q u i r e m e n t s F r o m the c a l c u l a t i o n s o f Peck, it c a n b e s e e n t h at, f o r 10 M e v i n j e c t i o n, a spot size o f 6 m m is s o m e w h a t t o o l a r g e in t h e h o r i z o n t a l plane, w h i l e d i v e r g e n c e a n g l e s s o m e w h a t g r e a t e r t h a n ± 1 m r c o u l d b e t o l e r a t e d. A l s o, the C o r n e l l linac a p p e a r e d to p e r f o r m s o m e w h a t b e t t e r w i t h r e s p e c t to d i v e r g e n c e a n g l e t h a n the ± 1 m r q u o t e d. T h u s f o r pur- p o s e s o f i n i t i a l d e s i g n, w e w i l l a s s u m e t h a t a m a g n i f i c a t i o n o f 0. 5 is r e q u i r e d in t h e h o r i z o n t a l plane. T h a t is, w e n o m i n a l l y r e q u i r e a t r a n s f e r m a t r i x o f (or its negative) in the h o r i z o n t a l p l a n e b e t w e e n t he i n j e c t o r o u t p u t a n d the i n f l e c t o r input. In the v e r t i c a l plane, the a c c e p t a n c e r e q u i r e - m e n t s a r e m u c h less severe. A t r a n s f e r m a t r i x o f
33 in the v e r t i c a l p l a n e b e t w e e n o u t p u t o f the i n j e c t o r a n d the i n f l e c t o r s e p t u m w o u l d p r o b a b l y b e b e s t, b u t w e p r o b a b l y do n o t r e q u i r e s u c h a s t r i n g e n t c o n d i t i o n. T h e p r i m a r y t h i n g is a c t u a l l y t o k e e p the v e r t i c a l spot size s m a l l e n o u g h to c l e a r t h e 1.5 c m g a p p l a n n e d f o r the t a n g e n t t a n k m a g n e t. O n the o t h e r h and, it is p r o b a b l y d e s i r a b l e to a l l o w the b e a m to b e c o m e as l a r g e as p o s s i b l e v e r t i c a l l y in o r d e r to r e d u c e space charge e f f e c t s. XII. Q u a d r u p o l e Lenses B e c a u s e o f t h e i r f l e x i b i l i t y a n d r e l a t i v e l y s m a l l a b e r r a t i o n s, w e p r o p o s e to e m p l o y q u a d r u p o l e m a g n e t i c lenses f o r t h e e m i t t a n c e m a t c h i n g. F o r r e a s o n s o f c o m p l e t e n e s s, w e w i l l d e v e l o p h e r e t h e r e q u i r e d e q u a t i o n s to f i r s t order. W e c o n s i d e r a q u a d r u p o l e lens to h a v e t h e f o l l o w i n g properties: (1 ) t h e m a g n e t i c f i e l d v a n i s h e s a l o n g the axis, (2) the m a g n e t i c f i e l d g r a d i e n t in the x d i r e c t i o n a n d in t h e y d i r e c t i o n is c o n s t a n t, a n d (3) t h e r e is no a x i a l c o m p o n e n t o f the m a g n e t i c field. A s s u m p t i o n (3) is, o f course, n o t v a l i d in the f r i n g i n g r e g i o n, b u t is a p p r o x i m a t e l y true i n the i n t e r i o r o f a lens. T h e n, if t h e m a g n e t i c f i e l d g r a d i e n t is d e f i n e d b y
34 -31 - w h e r e w e a r e c o n s i d e r i n g x i n the h o r i z o n t a l p lane a n d y i n the v e r t i c a l plane, (and in a r i g h t - h a n d e d c o o r d i n a t e system, z is t h e p a r t i c l e d i r e c tion) the p a r a x i a l e q u a t i o n s o f m o t i o n o f a p a r t i c l e in the l ens are g i v e n b y w h e r e a n d the s i g n refers to the s ign o f c h a r g e o f t h e p a r t i c l e, ε is the t o t a l e n e r g y o f the p a r t i c l e, (e.g., in M K S u n i t s ε is i n e l e c t r o n volts) r e f e r t o the v a l u e s at z = 0. T h e n if the q u a d r u p o l e h a s a l e n g t h L, the o p t i c a l t r a n s m i s s i o n m a t r i c e s are g i v e n b y H orizontal: Vertical:
35 O f course, t h e d e s i g n a t i o n " h o r i z o n t a l " a n d " v e r t i c a l " are m e r e l y c o n v e n i e n t, a n d h e n c e f o r t h w e s h a l l c a l l the m a t r i x h a v i n g the f o r m g i v e n f o r the h o r i z o n t a l the " f o c u s s i n g p l a n e ", a n d t h a t g i v e n f o r t h e v e r t i c a l the " d e f o c u s s i n g p l a n e ", a n d a s s u m e t h a t the m a g n e t i c f i e l d s are so o r i e n t e d t h a t κ is real. It can b e e a s i l y s h o w n that f o r a n y o p t i c a l m a t r i x o f t h e f orm, if C 0, t h e n w e m a y r e d u c e it to a g e n e r a l t h i c k lens f o r m i f w e m o v e the r e f e r e n c e p l a n e s to the p r i n c i p a l p l a n e s o f t h e system. The f i r s t p r i n c i p a l p lane is l o c a t e d a d i s t a n c e P 1 = 1-D/C a h e a d o f the o r i g i n a l f i r s t r e f e r e n c e plane, and the s e c o n d p r i n c i p a l p l a n e a dis- t a n c e P2 = 1-A/C b e y o n d the o r i g i n a l s e c o n d r e f e r e n c e plane. T h e n in t h i s c a se the f o c a l l e n g t h f = -(1/C). T hus for the q u a d r u p o l e lens, the f o c a l l e n g t h in t h e f o c u s s i n g p l a n e is g i v e n b y f+ = 1/(κ sin κl), a n d in t h e d e f o c u s s i n g plane f- = - 1/(κ sinh κl), a n d the p r i n c i p a l p l a n e s are l o c a t e d a d i s t a n c e
36 -33 - in the f o c u s s i n g p l a n e and in the d e f o c u s s i n g p l a n e t o w a r d the c e n t e r o f t h e lens ( f r o m the edges). I n e s s e n t i a l l y a l l p r a c t i c a l cases, h o w e v e r, a s i m p l i f y i n g a p p r o x i m a t i o n m a y b e m a d e. I f it h a p p e n s t h a t κ L is small, t h e n s i n κ L s i n h κ L, a n d 1/f+ κ2l - 1/f-. M o r e e x p l i c i t l y, if w e l o o k m o r e c a r e f u l l y a t the e x p a n s i o n f o r sin a n d sinh, w e f i n d the c o n d i t i o n that, if t h e n a n d or the two p r i n c i p a l p l a n e s c o i n c i d e in the c e n t e r o f the q u a d r u p o l e. Thus w e m a y t r e a t the q u a d r u p o l e lens as a t h i n lens, w h o s e f o c a l l e n g t h s in the h o r i z o n t a l a n d v e r t i c a l p l a n e s are e q u a l in m a g n i t u d e and o p p o s i t e in sign. T h i s is the a p p r o x i m a t i o n w e s h a l l e m p l o y in a l l f u r t h e r c o n s i d e r a t i o n s.
37 X I I I. S u g g e s t e d S y s t e m s W e w i l l n o w d e r i v e t h e a p p l i c a b l e r e l a t i o n s f o r some s y s t e m s w h i c h s a t i s f y o u r r e q u i r e m e n t s. It w i l l b e s e e n t h a t a l t h o u g h o n e q u a d r u p o l e t r i p l e t s y s t e m e x i s t s, it is v e r y c l u m s y, a n d the m o s t s a t i s f a c t o r y one is a q u a d r u p l e t. F o r t h e p u r p o s e o f t h e d e r i v a t i o n, w e a s s u m e t h a t w e r e q u i r e a m a t r i x in t h e h o r i z o n t a l p l a n e, a n d in the v e r t i c a l plane, w h e r e M = ± 1 /2 a n d N = ± 2. S y s t e m s o f t h i s n a t u r e, w i t h t h e v a n i s h i n g M 2 1 e l e m e n t w e s h a l l t e r m " t e l e s c o p i c ", w h e r e a s t h o s e in w h i c h t h e e l e m e n t M 12 v a n i s h e s, w e s h a l l t e r m " i m a g i n g ". W h a t w e w o u l d a c t u a l l y p r e f e r is a n i m a g i n g, t e l e s c o p i c s y s t e m in the h o r i z o n t a l p l a n e b e t w e e n i n j e c t o r o u t p u t a n d i n f l e c t o r input, a n d a n i m a ging, t e l e s c o p i c s y s t e m i n t h e v e r t i c a l p l a n e b e t w e e n i n j e c t o r o u t p u t a n d i n f l e c t o r s e p t u m. H o w e v e r, it is e a s i l y s e e n t h a t a t e l e s c o p i c s y s t e m c a n b e c o n v e r t e d t o a n i m a ging, t e l e s c o p i c o n e b y a s i m p l e t r a n s - l a t i o n o f r e f e r e n c e p l a n e s, w h e r e a s a n o n - t e l e s c o p i c s y s t e m c a n n o t b e c o n v e r t e d t o t e l e s c o p i c b y a n y s u c h t r a n s l a t i o n. It w i l l a l s o b e c o m e
38 a p p a r e n t t hat, if w e c h o o s e M = + 1 /2 a n d N = -2, it is in g e n e r a l c o m p a r a t i v e l y e a s y to a r r a n g e t h a t t h e r e q u i r e d t r a n s l a t i o n s axe c o m m e n surate w i t h t h e a c t u a l p l a c e m e n t o f i n j e c t o r, i n f l e c t o r, a n d s e p t u m in o u r c ase. T h u s f o r now, w e r e q u i r e o n l y d o u b l e t e l e s c o p y w i t h the c o r r e c t v e r t i c a l a n d h o r i z o n t a l m a g n i f i c a t i o n s. X I V. D o u b l e t It is t r i v i a l to s h o w t h a t a d o u b l e t c a n n o t s a t i s f y t h e r e q u i r e m e n t s. T h e t r a n s m i s s i o n m a t r i x f o r a d o u b l e t is g i v e n b y w h e r e g 1 = 1 / f 1, g 2 = 1 / f 2, the f o c a l l e n g t h r e c i p r o c a l s o f l e n s o n e a n d lens two, r e s p e c t i v e l y. T h e t r a n s m i s s i o n in the o r t h o g o n a l p l a n e is o b t a i n e d b y c h a n g i n g the s i g n s o f g 1 a n d g2. T h u s if w e r e q u i r e d o u b l e t e l e s c o p y, w e r e q u i r e T h i s is n o n p h y s i c a l.
39 X V. T r i p l e t T h e t r a n s m i s s i o n m a t r i x f o r a t r i p l e t is g i v e n b e l o w. I t c a n b e o b t a i n e d f r o m t h e t r a n s m i s s i o n m a t r i x g i v e n a b o v e f o r t h e f o u r m a g n e t s y s t e m b y l e t t i n g s 1, s2, s 3, s4 0ρ4 g 1 ρ 3 g 2 ρ 2 g 3 ρ 1 0 l 4 l 1 l 3 l 2 l 2,l1 0 T h e i n p u t a n d o u t p u t p l a n e s o f t h e s y s t e m a r e t a k e n a t t h e p r i n c i p a l p l a n e s o f l e n s o n e a n d l e n s t h r e e r e s p e c t i v e l y. g 1 = 1 / f 1, e t c.
40 -37 - W e n o w f o r c e the m a t r i x c o n d i t i o n s w e w i s h to impose, i.e., for the m a t r i x g i v e n a b o v e, a n d f o r t his m a t r i x w i t h the signs o f g 1, g 2, a n d g 3 r e v e r s e d. S i n c e w e are n o t s p e c i f y i n g LH a n d LV the u n i t y d e t e r m i n a n t c o n d i t i o n m a k e s it n e c e s s ary to s p e c i f y o n l y N, M, a n d the t w o zero e l e m e n t s. T h u s w e h a v e f o u r e q u a t i o n s. T h e s e f o u r e q u a t i o n s c a n b e c o m b i n e d l i n e a r l y t o y i e l d f o u r s i m p l e r o nes, w h i c h are g i v e n below: (6a) (6b) (6c) (6d)
41 S i n c e t h e r e a r e f o u r e q u a t i o n s a n d five u n k n o w n s, w e e m p l o y o n e (g2 ) as a s cale f a c t o r, a n d w r i t e t h e d i m e n s i o n l e s s q u a n t i t i e s S o m e e x t e n s i v e a l g e b r a i c m a n i p u l a t i o n s y i e l d t h e f o l l o w i n g set o f r e l a t i o n s (7a) (7b) (7c) (7d) The f i n a l e q u a t i o n is a q u a d r a t i c in λ 1, w h i c h c a n b e s o l v e d d i r e c t l y. T h e s o lution, w h e r e w e a l s o n o t e t h a t α12- α 22 = M N, is g i v e n b y
42 (8 ) W e m a y n o w i n v e s t i g a t e t h e s o l u t i o n s. F o r N = 2, M = 1 /2, w e f i n d w h i c h is, o f c o u r s e, n o n p h y s i c a l. F o r N = -2, M = - 1 /2, w e f i n d w h i c h is a g a i n n o n p h y s i c a l. T h e l a s t c a s e t o i n v e s t i g a t e is N = -2, M = 1 /2. ( T h e c a s e o f N = 2, M = - 1 / 2 is p h y s i c a l l y t h e s a m e, a s it is j u s t t h e N = -2, M = + 1 /2 s y s t e m r e v e r s e d i n d i r e c t i o n, a n d w i t h t h e h o r i z o n t a l a n d v e r t i c a l p l a n e s i n t e r c h a n g e d. ) T h i s c a s e y i e l d s t h e t w o r e a l s o l u t i o n s H o w e v e r, o n l y t h e f i r s t is p h y s i c a l, s i n c e l1 a n d l2 m u s t b o t h b e p o s i t i v e. T h i s i n f e r s t h a t λ 1 a n d λ2 m u s t h a v e t h e s a m e s i g n. I t c a n b e s e e n f r o m t h e p r e v i o u s e q u a t i o n s that t h e v a l u e s M = 1 /2, N = -2, λ 1 = 3 5 / 2 8 y i e l d a n e g a t i v e v a l u e f o r λ 2. T h e f i r s t s o l u t i o n, h o w e v e r,
43 -40 - g i v e s t h e p h y s i c a l v a l u e s, e x p r e s s e d i n t e r m s o f t h e f o c a l l e n g t h (f2 = 1 / g 2 ) o f l e n s t w o i n t h e h o r i z o n t a l p l a n e T h i s s y s t e m is t h e o n l y o n e o f t h e c o m b i n a t i o n s o f m a g n i f i c a t i o n w h i c h i s p h y s i c a l l y r e a l i z a b l e, a n d i t is v e r y c l u m s y. T h e v e r y l a r g e d i f f e r e n c e b e t w e e n l 1 a n d l 2 ( l 2 / l 1 = 49) i m p l i e s t h a t t h e s y s t e m w i l l t a k e u p a l a r g e s p a c e w i t h r e a l l e n s e s o f s i g n i f i c a n t t h i c k n e s s. W e t h u s g o o n t o t h e q u a d r u p l e t. X V I. Q u a d r u p l e t T o p r o c e e d w i t h t h e q u a d r u p l e t c a s e i n t h e s t r a i g h t f o r w a r d w a y a s w a s d o n e w i t h t h e t r i p l e t l e a d s t o i m p r e s s i v e a l g e b r a i c d i f f i c u l t i e s. I n s t e a d, w e h a v e f o u n d s a t i s f a c t o r y s o l u t i o n s b y p r e - g r o u p i n g t h e q u a d r u p l e t i n t o t w o d o u b l e t s w i t h s p e c i a l p r o p e r t i e s o f t h e i r o w n. T h i s s y s t e m is a m e n a b l e t o f a i r l y s i m p l e a l g e b r a i c t r e a t m e n t.
44 F i r s t, l e t u s c o n s i d e r t w o s y s t e m s o f a n y k i n d w i t h a d r i f t r e g i o n L b e t w e e n t h e m. W e c o n s i d e r the t w o s y s t e m s M 1 a n d M 2 b e t w e e n t h e i r r e f e r e n c e p l a n e s. T h e n the t r a n s m i s s i o n b e t w e e n the i n p u t p l a n e o f M1and the o u t p u t p l a n e o f M 2 is g i v e n b y N o w t h e s y s t e m w e are s e e k i n g is d e f i n e d b y s e t t i n g the p r o p e r e l e m e n t s to the r e q u i r e d v a l u e s. W e d e f i n e A 1 *, B 1 *, e tc., as the a b o v e elem e n t s w i t h the s i g n s o f a l l lens f o c a l l e n g t h s r e v e r s e d, t h e n M 1 *, M 2*, are the t r a n s m i s s i o n s i n t h e o p p o s i t e (vertical) p l a n e i f M 1, M 2 are s y s t e m s o f q u a d r u p o l e s. T h e n the d e f i n i n g r e l a t i o n s a r e (9a)
45 (9b) (9c) (9d) W e n o w a s s u m e t h a t M 1 a n d M 2 a r e q u a d r u p o l e d o u b l e t s, w i t h t h e r e f e r e n c e p l a n e s t a k e n a t the c e n t e r s o f t h e l e n s e s. W e f u r t h e r n o w a s s u m e t h a t the t w o d o u b l e t s a r e t h e m s e l v e s t e l e s c o p i c i n t h e h o r i z o n t a l p l a n e, i.e., C 1 = C 2 = 0. F o r a d o u b l e t, t h i s c o n d i t i o n, a l o n g w i t h the a b o v e d e f i n i n g r e l a t i o n s, y i e l d s w h e n m a n d n are t h e m a g n i f i c a t i o n s o f t h e t e l e s c o p i c d o u b l e t s M 1 a n d M 2 r e s p e c t i v e l y. T h e n, a g a i n r e f e r r i n g to the d o u b l e t m a t r i x p r e v i o u s l y g i v e n, w e m a y f i n d (10a)
46 (10b) (10c) (1 0 d) (1 0 e) (1 0 f) w h e r e a = s p a c i n g b e t w e e n l e n s e s i n f i r s t d o u b l e t ; b = s p a c i n g b e t w e e n l e n s e s i n s e c o n d d o u b l e t ; g, h a r e r e c i p r o c a l s o f f o c a l l e n g t h i n t h e h o r i z o n t a l p l a n e ; a n d g 1, g 2 r e f e r t o f i r s t a n d s e c o n d l e n s e s i n f i r s t d o u b l e t ; h 1, h 2 r e f e r t o f i r s t a n d s e c o n d l e n s e s i n s e c o n d d o u b l e t. ( 1 1a) (11b) A l s o, s i n c e w e s e e t h a t S i m i l a r l y,
47 N o w, u s i n g m = 1 - a g 1, n = 1 - b h 1, w e m a y f i n d t h a t (12a) (12b) W e n o w d e v e l o p t h e e q u a t i o n f o r t h e v e r t i c a l m a g n i f i c a t i o n N. B y e l i m i n a t i n g L b e t w e e n (9c) a n d ( 9d), w e f i n d T h e m a t r i x d e t e r m i n e n t a l c o n d i t i o n r e q u i r e s so (13) T h e e x p r e s s i o n f o r L is o b t a i n e d b y s o l v i n g (9d) f o r L a n d s u b s t i t u t i n g i n t h e e x p r e s s i o n s (10) a n d ( 1 2 ). T h e r e s u l t is (1 4 ) O r, i f w e e l i m i n a t e b w i t h t h e a i d o f ( 1 3 ), w e h a v e (15a)
48 T h e a b o v e e q u a t i o n, a l o n g w i t h M = mn, p l u s ( 1 3 ), a n d t h e s e t ( 1 0), c o m p l e t e l y d e f i n e s t h e s y s t e m. W e s t i l l h a v e a t o u r d i s p o s a l, h o w e v e r, t h e i n d i v i d u a l d o u b l e t m a g n i f i c a t i o n s, s u b j e c t t o t h e c o n d i t i o n t h a t m n = M, t h e s c a l i n g f a c t o r " a ", a n d t h e a c t u a l p l a c e m e n t o f t h e s y s t e m w i t h r e s p e c t t o t h e b e n d i n g s y s t e m arid t h e i n j e c t o r. I n o r d e r t o e x a m i n e t h e s e e f f e c t s, w e r e q u i r e a n e x p r e s s i o n f o r t h e p o s i t i o n o f t h e i n p u t a n d o u t p u t p l a n e s f o r w h i c h t h e s y s t e m is b o t h i m a g i n g a n d t e l e s c o p i c. O f c o u r s e, t h e v e r t i c a l a n d h o r i z o n t a l p l a n e s a r e n o t n e c e s s a r i l y t h e s a m e. T o o b t a i n t h i s e x p r e s s i o n, w e m u l t i p l y t h e t r a n s m i s s i o n m a t r i x o n t h e r i g h t a n d l e f t b y d r i f t d i s t a n c e s P 1 a n d P 2 i n t h e h o r i z o n t a l p l a n e, a n d P 1 * a n d P 2* i n t h e v e r t i c a l p l a n e, a n d s e t e q u a l t o a n i m a g i n g m a t r i x. T h u s T h u s w e r e q u i r e ( h o r i z o n t a l ) (16a)
49 (v e r t i c a l ) ( 1 6 b) a n d (17a) (17b) X V I I. S a m p l e Q u a d r u p l e t S y s t e m W e c h o o s e N = -2, M = + 1 /2. W e a l s o s e t, s o m e w h a t a r b i t r a r i l y, m = n = 1 / 2. T h e n a n d
50 w h e r e t h e f o c a l l e n g t h s o f t h e l e n s e s i n t h e s e c o n d d o u b l e t. S u c h a s y s t e m s e e m s a c c e p t a b l e. F o r e x a m p l e, i f a = 0. 1 m e t e r, t h e n L 1 m e t e r a n d t h e s t r o n g e s t l e n s r e q u i r e d h a s a f o c a l l e n g t h o f cm, w h i c h s e e m s r e a d i l y o b t a i n a b l e f o r 1 0 M e v e l e c t r o n s. T o f i n d t h e r e f e r e n c e p l a n e s f o r w h i c h t h i s s y s t e m b e c o m e s f o c u s s i n g, w e r e f e r t o e q u a t i o n s (1 6) a n d ( 17). W e f i n d E a E * a F o r p u r p o s e s o f c o m p a r i s o n, w e s e t P 1 = P 1* = 0 ; i. e., t h e i n j e c t o r o u t p u t is f o c u s s e d o n t h e i n p u t l e n s. T h e n Ρ 1 = a P 2* = a S i n c e t h e h o r i z o n t a l i n p u t p l a n e f o r t h e p r o p o s e d i n f l e c t o r a p p e a r s a p p r o x i m a t e l y 9 0 c m i n f r o n t o f t h e i n p u t m a g n e t, it is q u i t e e a s y t o m a k e P1 n e g a t i v e. I f w e m a k e a = 0. 1, t h i s a l l o w s ~ 2 5 c m c l e a r a n c e b e t w e e n t h e e n d o f t h e m a g n e t a n d t h e f i r s t q u a d r u p o l e. I n t h i s c a s e P2 * m. T h i s d o e s n o t q u i t e m e e t t h e r e q u i r e d c o n d i t i o n, a s t h e t o t a l d i s t a n c e b e t w e e n t h e f i n a l q u a d r u p o l e a n d t h e s e p t u m w o u l d b e a b o u t 4. 3 m e t e r s.
51 H o w e v e r, t h e 2. 3 m e t e r d i s c r e p a n c y is n o t a t a l l s e r i o u s. F o r the w o r s t c a s e, b e a m d i v e r g e n c e a f t e r t h e m a g n i f i c a t i o n o f 2 i n the v e r t i c a l p l a n e is ± 0. 5 mr, a n d it m a y b e c o n s i d e r a b l y b e t t e r. T h i s m e a n s a m a x i m u m (worst case) spot s p r e a d o f a n a d d i t i o n a l ~ 2. 3 ram o u t o f a s p o t size in the v e r t i c a l o f 12 m m. E v e n t h i s c a n b e c o m p e n s a t e d s o m e w h a t b y c h a n g i n g t h e a d j u s t m e n t o f t h e q u a d r u p o l e s s l i g h t l y t o p r o d u c e a slight f o c u s s i n g a c t i o n i n t h e v e r t i c a l p l a n e. T h i s s o l u t i o n seems q u i t e s a t i s f a c t o r y. X V I I I. S u m m a r y B e l o w are g i v e n t h e r e l e v a n t d i m e n s i o n s a n d p a r a m e t e r s f o r the c o m p l e t e s u g g e s t e d i n j e c t i o n system. T h e s y s t e m is s h o w n " s t r a i g h t e n e d o ut", as it w e r e, f o r c l a r i t y. M ' s are m a g n e t s (except t h a t M 1 is a n e l e c t r o s t a t i c b e n d i n g element); Q 's are q u a d r u p o l e s. L e n g t h s (meters):
52 l1 = l 2 = l3 = l4 = d = b = 0. 2 L = a = 0. 1 M a g n e t s : u n i f o r m f i e l d we d g e : d e f l e c t i o n a n g l e s a n d r a d i i o f c u r v a t u r e ( d e g r e e s, m e t e r s ) : M 1 ; M 2 ; M 3 ; M 4 ; Q u a d r u p o l e s : f o c a l l e n g t h s i n h o r i z o n t a l p l a n e s : ( m e t e r s ) : Q 1 : Q 2 : Q 3 : Q 4 :
53 A C K N O W LE D G M E N T S T h a n k s are e x p r e s s e d to P r o f e s s o r R a p h a e l L i t t a u e r o f C o r n e l l f o r m a n y o f t h e i d e a s u s e d in t h i s r e p o r t. M a n y h e l p f u l s u g g e s t i o n s f r o m P r o f e s s o r s R o b e r t L. W a l k e r a n d M a t t h e w Sands, a n d Dr. R o b e r t F. D e e r y are a c k n o w l e d g e d. P r o f e s s o r W a l k e r a n d Dr. D e e r y w e r e a l s o o f g r e a t a s s i s t a n c e in c h e c k i n g t h e i d e a s a n d the s o m e w h a t i n v o l v e d a l g e b r a. The a u t h o r is a l s o g i v e n t o u n d e r s t a n d t h a t t h e 3 x 3 m a t r i x t e c h n i q u e f o r the m a g n e t s w a s s u g g e s t e d to P r o f e s s o r L i t t a u e r ( from w h o m t h e a u t h o r l e a r n e d o f them) b y Dr. K a r l B r o w n, o f t h e M i c r o w a v e L a b o r a t o r y at S t a n f o r d,
54
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A2 Assignment lambda Cover Sheet. Ready. Done BP. Question. Aa C4 Integration 1 1. C4 Integration 3
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More informationPower Series. Part 1. J. Gonzalez-Zugasti, University of Massachusetts - Lowell
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