The Group of 1D First-Order Optical Systems

7/30/2019 The Group of 1D First-Order Optical Systems

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3The Group o f 1DSystems Firs t -Order Opt ica l

3.1 Introduct ionI n t h e p r e v i o u s c h a p t e r w e h a v e t r a c e d a li n e o f a p p r o a c h f o r i n v e s t i g a t i n g t h er e l a ti o n b e t w e e n t h e r a y - o p t ic s d e s c r i p t io n o f p a r a x i a l p r o p a g a t i o n t h r o u g h1 D o p t i c a l s y s t e m s a n d t h e s y m p l e c t i c L i e g r o u p o f 2 x 2 r e a l m a t r i c e s , a s s oc i -a t e d w i t h t h e q u a d r a t i c p o l y n o m i a l s i n t h e c o n j u g a t e v a , r i a , b l e s q a n d p u n d e rP o i s s o n b r a c k e t o p e r a t i o n a n d m a t r i x e x p o n e n t i a t i o n .

W he n cons id e r i ng r ays wi t h sm a l l i nc l i na t i on an g l es t o t he ax i s (p2


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genera t ing func t ion g e n e r a t i n g m a t r i xf r e e - r e e d i u m 1 ' )sect ion ~t> = --+ K - ( I )) ~))th in lens 1q2 - - + K , - ( _ ( ~ (~ ))

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The Group of the 1 D First-Order Optical Systems 11 3

is e s t a b l i s h e d . S e c t i o n 3 .6 i l l u s t r a t e s t h e l i n k o f t h e a t t r a c t i v e a n d r e p u l s is eo s c i l l a t o r - l i k e H a m i l t o n i a n d y n a m i c s w i t h f r a c t i o n a l F o u r i e r t r a n s f o r m e r s a n db e a m e x p a n d e r s ( or r e d u c t o r s ) , w h i c h i n d i v i d u a l i z e i n t h e (q , p ) p l a n e s c a l e dr o t a t i o n s a n d s q u e e z e s . T w o i n t e r e s t i n g r e a l i z a t i o n s o f o p t i c a l m a t r i c e s a r e d e -s c r i b e d . O n e r e a l i z a t i o n , w h i c h a p p l i e s l o c a l l y n e a r t h e g r o u p i d e n t i t y , i n v o l v e sth e ~ 1 q 2 - , l q p _ a n d l p 2 - g e n e r a t e d o p t i c a l s y st e m s , a n d a c c o r d i n g l y is v i s ua l -i z e d in t he op t i c a l pha se p l a ne a s a q - she a r f o l low e d by a s c a l ing a nd the n bya p - s h e a r (w 3 .5 ) . T h e o t h e r r e a l i z a t i o n , w h i c h i n c o n t r a s t a p p l i e s t o t h e e n t i r e

i q 2 _ 1 1 q2gr oup , i nvo lve s t he ~ , -~qp-, a n d ~ (p 2 + ) - g e n e r a t e d o p t i c a l s y s t e m s ; i ti s p i c tu r e d in t he ( q, p ) p l a ne a s a r o t a t i on f o l low e d by a s c a l ing a n d a ve r t i -c a l s h e a r (w 3 .7 ) . C o n c l u s i v e c o n s i d e r a t i o n s a b o u t t h e l i n k b e t w e e n q u a d r a t i cp o l y n o m i a l s a n d p a r a x i a l r a y m a t r i c e s a r e g i v e n i n S e c t . 3 . 8 . F i n a l l y , S e c t .3 . 9 d e s c r i b e s t h e m e t h o d t o i n t e g r a t e H a m i l t o n ' s e q u a t i o n f o r t h e o p t i c a l r a ym a t r i x , d e t a i l i n g s e p a r a t e l y f o c u s i n g a n d d e f o c u s i n g q u a d r a t i c i n d e x m e d i a .

3 .2 Ray m atrix of composite optical systemsO p t i c a l s y s t e m s , c o n s i s t in g o f s e v er a l c o m p o n e n t s c o n n e c t e d t o g e t h e r i n c as -c a d e, c a n b e h a n d l e d s y m p l y b y m u l t i p l y i n g t h e r a y m a t r i c e s o f t h e i n d i v i d u a lo p t i c a l d e m e n t s , a r r a n g e d i n t h e r e v e r s e o r d e r [1 ]. I n f a c t, e a c h o p t i c a l d e m e n th a s i t s o w n i n p u t a n d o u t p u t t ) l a n e s a n d a c t s o n t h e r a y i n c i d e n t o n i t s i n p u tp l a n e i n a c e r t a i n w a y t o p r o d u c e t h e r a y a t i t s o u t p u t p l a n e . T h u s t h e r a yp r o p a g a t i o n t h r o u g h a s e q u e n c e o f N o p t i c al e le m e n t s w i t h r a y - m a t r i c e s M 1 ,M.~ , .. ., M N i s a c c ou n te d f o r by the f o l low ing c ha in o f ma t r i x r e l a t i on s

U 1U 2oo ~

U N

- - M l U 0 ,- M 2 u l , (3 .2 .1)- - M N U N _ I ,

w h e r e as u s u a l u j d e n o t e s t h e r a y - c o o r d i n a t e v e c t o r u j - (qj ,pj)T a t t h ef ixe d r e f e r e nc e p l a ne I I j , j = 0 , .., N ( F ig . 3 .2 ) . T h e in t e r m e d ia t e p l a ne s I I j ,j = 1, .., N - 1 p l a y t h e d o u b l e ro l e a s t h e o u t p u t a n d i n p u t p l a n e s r e s p e c t i v e l yf o r t h e j - t h a n d ( j + 1 ) - t h o p t i c a l e l e m e n t , w h i l s t H 0 a n d H N ar e t h e i n p u t a n do u t p u t p l a n e s f o r t h e f i r s t a n d l a s t c l e m e n t a s w e l l a s f o r t h e o v e r a l l s y s t e m .

C o m b i n i n g t h e i n p u t - o u t p u t r e l at i o n s ( 3 .2 .1 ), w e o b t a i n t h e t r a n s f e r la wf o r t h e r a y c o o r d i n a t e s f r o m H o t o H N i n t h e f o r m

u N -- M N M N _ 1 - . - M 2 M l r o = M u 0. ( 3.2 .2 )A c c o rd i n g ly , t h e r a y m a t r i x M o f t h e o v e ra ll s y s t e m c o m e s t o b e t h e p r o d u c t

M = M N M N _ I - - " M 2 M 1 (3 .2 .3 )


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Input p lane Output p lane

FI( l l~]I1E 3 .2 . Pro paga t io~l o f a ligl~t ray t ln'o~gl~ a seq~mncc of op t ica l c lc~ncnts . Th e op t ic a l~natrix ()f t im ()vcral l s ys tc ~ is t im I)ro(l~c t o f the i~t(livi(l~ml ray I~mtriccs f ro ~ l l ,, to H N .

w it h t lm ~ml, ri(:(;s M ~, . ., M x l)(; ing a rr a~ ge (l i~ t l le i llW;l 'S(', ()l '( l( ;r fr ()~ th at inwlfi(:l~ t,l~(; (~o rres l)() ~(li ~g (,h:l~(',~fl, ar e t)la.(~(;(l ()~ t |m ax is wit,t~ r ('st)( :ct Lo t, hc(lir('~('ti()~ ()f t,l~(; i~ (: () ~i ~g ray .

3 .2 .1 77~ , ic / , : a ' n , d t h , m l c , n s c , s 'A s al l (;xa lllI)h ', ()f a (:()llll)()llllr r ,";yst,clll, WC r162 a t h , i t ' k h:lls, wlli(:lll)asi( ~ally (:()llsi sts ()f tw() sl)ll(;ri(: al sllrfil(~(;s S(:l) arat(; (l t)y 1,1,11 at)I) r(:('ial )h; (tis-l, ml(:(; . ~ 1 ' 1 1 r ra y 1)r() l )a ,ga , t i () l l l , ] lr r a t h , i ~ : k lo l l s l l l a ,y l ) ( ; s ( ; ( ; l l a , s ( ' ( ) l lq )r i s ingl, l l (; f ()l l ()w i]lg s(;(l l l(; l l (: (; ()f ra y-l , l ' a ,~ls f()r lIlal , i ()l ls: 1) r(:fl 'a ,(: l ,i ()~l a l, l , ll ( ; fl' ()l l l, s l l r fa ,cew i t l l r ( ' f ra ( ' l , i lw4 1)()w(: r ~ , 2 ) f l ' cc t )r ( ) l )aga , l , i( ) l~ l h r() ~gl~ l ,hc lc ,~s ( ) f a x i a l t ,h i ( :k -~(;ss (t m~(l 3) r (;f ra( 'ti() ~ al, t, m t)a,(~k s~lrfa(~(; witt~ t)()w(;r 7)2 (F ig . 3.:l.a,) ). T h ethi( :k l(;~s ~m,t,rix is ti ta n ()l)tain(;(l D(m~ tim ()r(h;r(;(l t)r()(t~(:t, ()f r(;fl 'a('~tion-K(;(;i)l'()t)a,ga,l,i()]~-rt;fra('ti()~ l~m,tri('cs, wl~i(~l~ ti~m,lly g iv es

( 1 -0 7 ~ ' i) ) (3 .2 .4 )M th i ck l en s - - - - P 1 - - 07)2 'wt~(;r(; t,t~r fo(:a l t)()w (;r 7 ()f the, le ns is 7 - 7)~ + 7)2 - i)7)~ 7)2, a,n(t i) d e n o t e sth e axia,l ttfi(:k~(; ss ()f tl~(; l(;~s (tivi(Ie(t |)y th(; r(;la.tiv(; re, fra,(:tiv(; in(t(;x.As a fl~rther (~,xa,~q)h;, wc consider a, t h i n lens, for~m.lly define,(] a ,s consistingof t,w() re fr ac ti ng s~rfa.(:(;s wi th neg ligi l)le set)a.ra, t ion in bet, we en , so t, hat, a ra yi n c i d e n t a,t a, g iv e , n p o i n t o n o n e s u r f a c e e m e r g e s a , p t ) r o x i m a , t e l y a ,t t h e s a m eh e i g h t f r ( ) m t he , o t h e r s ~ r f a, c e . T h e r e f e r e n c e p l a n e s I I i a n ( t I io a,re, t a k e n t o(:()in('i(h~ with t,h(; t)la,nc ()f tt~e lens, a,lt,hough Hi re la , t e s to inc idenL rays whi l s t ,I io r e la t e s t o e m e r g i n g r a y s ( F i g . 3 . 3 .b ) ) . T h e r a y m a t r i x o f t h e t h i n l e n s i st h e n o b t a i n e ( t a s t h e p r o d u c t o f t w o r e f r a c t i o n m a t r i c e s r e l e v a n t t o t h e tw ol e n s i n t e r f a c e s . T h e r e s u l U n g m a t r i x l o o k s li k e a, r e f r a c t i o n m a t r i x , i .e .,

M th in lens - - _ ' ] '9 1 ' "


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F I G U R E 3 . 3 . (a) Prop agat ing th rough a thick lens, the light ray experiences the refractionat the front surface, the propagation through the lens and another refraction at the backsurface. (b) Being the refracting surfaces of a thin lens separated by a negligible distance,the light ray experiences only the refractions at the lens front and back surfaces.t i le f o c a l p o w e r 7 o f t i le l e n s b e i n g t i l e S U l il o f t h e r e f r a c t i n g p o w e r s o f t h etw o su r fa , ces : 7 = P l - 31 -P 2 . T h e l en s p o w e r 7 is c o m m o n l y e x p r e s s e d i n t e r m so f t h e r e d u c e d f o c a l l e n g t h f a,s 7 - 1 / f , w i t h f - f 1 / 7 ~ 0 1 - f 2 / ~ / ~ 0 2 , ~ , 0 1 a n dn % d e n o t i n g t h e r e f r a c t i v e i n d i c e s o f t h e m e d ia , i n t h e o b j e c t a ,n d i m a g e s p a c e s .

T h e f o r m a l a n a l o g y b e t w e e n t h e r e f r a c t i o n a n d t h i n l en s m a t r i c e s e s t a b -l is h e s t h e o p t i c a l e q u i v a l e n c e b e t w e e n a r e f r a c t i n g i n t e r fa c e w i t h a, g i v e n r e -f r a c t i n g p o w e r 7 a n d a t h i n l e ns w i t h t h e s a m e f o c u s i n g p o w e r .

D i s r e g a r d i n g t h e r e a l d e s i g n , i n t h e f o l l o w i n g w e w il l r e f e r t o a t h i n l e n sa,s t h e o p t i c a l e l e m e n t d e s c r i b e d b y t h e r a y m a t r i x ( 3 .2 . 5 ) . A l s o , a ,c c o r d i n gt o t h e u s u a l p r a c t i c e , w e w i l l r e f e r t o converging, o r positive, l e n s e s , w h e nf > 0 , a n d t o diverging, o r negative, l e n s e s , w h e n f < 0 . A s is w e l l k n o w n ,p o s i t i v e l e n s e s c a u s e c o n v e r g i n g s p h e r i c a l w a v e s t o c o n v e r g e m o r e r a p i d l y , a n dd i v e r g i n g s p h e r i c a l w a v e s t o d i v e r g e l e s s r a p i d l y o r t o c o n v e r g e ; i n contrast,n e ga , t i v e l e n s e s c a u s e c o n v e r g i n g s p h e r i c a l w a ,v es t o c o n v e r g e l es s r a p i d l y o r t od i v e r g e , a n d d i v e r g i n g w a v e s t o d i v e r g e m o r e r a p i d l y .

3.3 Th e subgroup o f f ree pro pag at ion and thin lens m atricesF r e e - m e d i u m s e c t i o n s a,n d t h i n l e n se s a re t h e b a s i c c o m p o n e n t s o f e v e r y f ir s t-o r d e r o p t i c a l s y s t e m . A s i t is p r o v e d , i n f a c t , e v e r y o p t i c a l t r a n s f o r m a t i o n c a nb e r e a l i z e d b y a f i ni t e s e q u e n c e o f f re e p r o p a g a t i o n s a n d l e ns o p e r a t i o n s , o re q u i v a l e n t l y e v e r y 2 x 2 r e al s y m p l e c t i c m a t r i x c a n o p t i c a l l y b e s y n t h e s i z e dp r o p e r l y a r r a n g i n g f r e e p r o p a g a t i o n s e c t i o n s a n d t h i n l e n s e s .


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1 1 6 L i n e a r R a y a n d W a v e O p t i c s n P h a s e S p a c e

W e d e n o t e b y T ( d ) a n d L ( f ) t h e o p t i c a l m a t r i c e s r e l e v a n t r e s p e c t i v e l y t op o r t i o n s o f a u n i fo rm re f r a c t i v e i n d e x m e d i u m ,

T (d ) - - c d K - - - ( 10 d )l ' ( 3 3 . 1 )a,n(1 ttli~l l(;::S(;S,

L ( . f ) - ( , ~ K , _ _ ( 1 0 ) (3 .3 .2)- 1 / . f 1 "

W(; r('(:all ttm.t et ml(t . f (l(;~ (,t(, th( ; s(,(.ti() ll l('llgt, ll a.ll(t ttl( ; l( 'lls f()(:a.1 l('llgt, ll, t)()t,hs(:a,l(',(l 1)y 1,11(; r(',l(wa,ld, I'(',fra,(:tiv(', ill( |(',x . Als() , th in l(',lls llla,l,l'i(:(~,s a,l'(', (' ,( tl liva le ll tly(tcn()t(;(l t)y L ( . f ) ( ) r L (7 ), 7 1)(;i~g tl~(~ f()(:a,1 t)()w(;r: 7 = l / f . Tl~(', lil~fit ('a,s(;s( ) f ( t --, () ( s e ( : t i ( ) ~ ( ) f ~ ( ; g l i g i t ) l ( ; l ( ; ~ g t l ~ ) m ~ ( t f - - ~ 4 - o o ( l ( ; ~ s ( ) f ~ ( ~ g l i g i t ) l ( ~ f ( ) ( : a lt)()w(',r) i(leld,ify tlm llllit,y nm tr ix I: T(( )) = I a,n(1 L( -t-o c) = I.Inl,(~r(~stillgly, T a.n(l L a r(~ lmi lll() (lll lar ml(l, r(;st)(;(:tively, lq)t )er - ml(l l()wer-tri a,l lgl lla ,r llm,tri(:(',s. 'l'll(~y ar(', tll(', r(',I)res(~ld, a,tiv(~s ()f l, ll(; tw () AI )(; lim i ()i~(~-t)a,r a.~( ;t(; r s~fi)gr()~q)s ()f tt~(', sy~q )l(;( :ti(: gr (n q) 5~p(2, IR) , g(;~(;ra,t(;(l r(;sI)(',(:tiv(;lyt)y t,l~(~ a lg(4)r a ('~l(;~(;~d,s K _ m~(l K + as ilh~s trat( ;(l il~ (ll~a.l)t(;r 2 (!i!i 2.3, 2.4.1an( t 2. 4.2 ) [2]. T lm s~fi)gr()~q) a,(l (liti vity l)r()t)(~rty (2 .3. 12 ) witl~ r(;sl)(;(:t t() th ei)a,l'a.lll(',l,(',r ()f l, II(; alg (:l) ra- t() -gr ()~ q) (;Xl)()~ (;~tiati(n~ wr it( ;s as

T(d~ )T (d ~ ) = T ( d ~ )T (d I ) = T (d ) , ( 3 .3 .3 )witl~

d - d ~ 4- d ~ , ( 3 . 3 . 4 )a .n ( 1 s i n ~ i la , r ly

L(.f, )L (.f ,) = L(. /~ )L(.fl ) = L( .f ) , (3.3.5)witl~ 1 1 1

+ . 7 . , ' ' "1 7 ~ , f lTh(~ a,t)()v(~ r(4a,ti(n~s ext) res s in f()rnm,1 tem ps tim well kn( )wn t)r() tmrt y tha, t,

th e se(ll lell(:(' , ()f two free st)a,ce sec tion s d 1 an(t d~ is equ iv ale nt t()()l ie sec tionof le ng th d = d I + d~ and , simil arly, two casca,de(t lenses f~ a,n(t .f~ (w ith nof~ fusepa, ra , ti ( )n i~ be tw ee n) a ,re equiva, lent to the lens f - f~ +f~.As a,n inu ne( t ia te con seq uen ce of (3 .3 .3) and (3 .3 .5) , we also ha,ve

T ( d ) T ( - d ) = T ( - d ) T ( d ) - I , ( 3. 3.7 )a n d

L ( f ) L ( - f ) = L ( - f ) L ( f ) = I , ( 3. 3. 8)whic h s ta, t e the ex i s ten ce and un ic i ty o f the inverse to every T and L- t ypem a t r i x a , c c o rd i n g t o

[T(d) ] -1 - W ( - d ) - e - d K - (3.3.9)


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The Group of the 1D First -Order Opt ical Systems 117

a n d [ L ( f ] -1 _ L ( - f ) - e - ~ K + . ( 3 .3 .1 0 )F o r m a ll y , s in c e b o t h T ( - d ) a n d L ( - f ) r e p r o d u c e t h e t y p o l o g y o f t h e m a -

t r i c e s (3.3 .1 ) an d (3 .3.2 ) , r e l a t i o n s (3 .3 .9 ) an d (3 .3.1 0 ) co n f i rm t h a t T an dL - t y p e m a t r i c e s d o fo r m t w o p r o p e r o n e - p a r a m e t e r s u b g r o u p s o f S p ( 2 , R ) ,w h e n r e g a r d e d a s r e a l u p p e r a n d l o w e r t r i a n g u l a r m a t r i c e s w i t h u n i t d i a g o n a len t r i e s . O p t i ca l l y , r e l a t i o n (3 .3 .1 0 ) e s t ab l i s h e s t t m t t h e i n v e r s e o f a len s s y s t ems h o u l d b e a l en s w i t h o p p o s i t e v a l u e o f t h e fo ca l l en g t h . Th i s i s a p h y s i ca l l yr e a l i z a b l e s y s t e m ; r e l a t i o n ( 3 . 3 . 8 ) c a n t h e r e f o r e b e i m p l e m e n t e d b y t h e s i m -p l e s eq u en ce o f t w o l en se s w i t h o p p o s i t e f o ca l l en g t h s . A s t h e l en s f a l t e r st h e d i r e c t i o n o f propagation of t h e ingoing r a y, t h e le n s - f a c t s o p p o s i t e l yo n t h e i n t e r m e d i a t e r a y, w h i c h t h e n r e g a i n s t h e o r i g in a l d i r e c t i o n o f p r o p a g a -t i o n . H e n c e m a t r i c e s ( 3 . 3 . 2 ) d o f o r m a s u b g r o u p e v e n w h e n i n t e r p r e t e d a s t h eo p t i c a l m a t r i c e s o f t h i n l en s es .

O n t h e c o n t r a r y , b y ( 3 . 3. 9) t h e i n v e rs e o f t h e f r e e p r o p a g a t i o n b y t h e d i s -t a n c e d s h o u l d b e t h e fr e e p r o p a g a t i o n b y t h e d i s t a n c e - d , w h i c h in p ri n c i p l em ay a ,p pea, r n o t p h y s i ca l l y r ea l i z ab l e s i n ce d i s t an ces o ccu r n a t u r a l l y a s p o s i t i v eq u a n t i t i e s . T h e r e f o r e i n a t r i c e s ( 3 . 3. 1 ), w h e n i n t e r p r e t e d a s o p t i c a l i n a t r ic e s f o rp r o p a g a t i o n t h r o u g h p o r t i o n s o f a h o m o g e n e o u s m e d i u m , d o n o t s t ri c t ly f o r ma, s u b g ro u p . I n t h e n ex t p a, r ag ra , p h w e w i ll see t h a t t h e p ro p a g a t i o n co r r e s p o n d -i n g to "n e g a t i v e" v a l u e s o f d can o p t i c a l l y b e r ea l i z ed b y a , s u i t ab l e s e q u en ce o ff r ee p ro p ag a , t i o n s e c t i o n s a n d t h i n l ens e s . Th e i n v e r s e o f t h e f r ee p ro p ag a , t i o nm a t r i x , i .e . t h e m a t r i x T ( - d ) , a c q u i re s t h e n a p h y s ic a l c o n c r e te n e s s , a l t h o u g hu n l i k e T ( d ) i t d o e s n o t c o r r e s p o n d t o a s i n g l e a n d u n i q u e o p t i c a l e l e m e n t . A c -co rd i n g l y , r e l a t i o n (3 .3 .7 ) b eco m es o p t i c a l l y s y n t h es i za b l e , t h e e f f ect o f t h e f r eep r o p a g a t i o n b y d b e i n g c o m p l e t e l y v a n i s h e d b y t h e p r o p a g a t i o n t h r o u g h t h es y s t e m T ( - d ) , a s t h e r a y e v e n t u a l l y r e c o v e rs t h e i n it i a l p o s i t i o n a n d d i r e c t i o n .

A p a r t f r o m t h e i r p h y s i ca l r e a , l i z a ,b i l i t y , "n eg a t i v e" f r ee p ro p ag a t i o n s , w h enf r a m e d w i t h i n t h e c o l n I n o n c o n v e n t i o n s r eg a, r d i n g t h e o p t i c a l s y s t e m s , c a n b eg i v e n a p r a c t i c a l i n t e r p r e t a t i o n w h i c h r e l a t e s t o t h e c h a r a c t e r , r e a l o r v i r t u a l ,o f t h e r a y s w e a r e d e a l i n g w i th . L e t u s w r i t e d in t e r m s o f t h e a x i a l c o o r d i n a t eas d = Z o - z i ( t h e r e f r a c t i v e i n d e x b e i n g o m i t t e d f o r s i m p l i c i t y ) , s o t h a t

T - l ( z o - z i ) = T ( - ( Z o - z i ) ) = T ( z i - Z o) . (3 .3 .11)S u p p o s e t h a t t h e r a y c o o r d i n a t e s ( q o , p o ) a r e k n o w n a t t h e p l a n e Z o . E v i d e n t l yt h e r a y c o o r d i n a t e s ( qi, p i ) a t t h e p r e c e d i n g p l a n e z i m a y b e c a l c u l a t e d b y t h ei n v e r s e ma t r i x T -1 (Zo - z i) = T (z i - Z o ) . In s y mb o l s , w e h av e

( q : ) . o ) ( q o ) , < 0 ( . . 1 . )I f t h e p o r t i o n o f s p a c e f r o m z i to Zo i s n o t f r ee , b u t o ccu p i ed b y s o me o p t i ca ls y s t e m , t h e r a y a t z i , w e m i g h t o b t a i n t h r o u g h T ( z i - Z o ) , i s n o t t h e a c t u a l r a y


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118 Linear Ray and Wave Op tics in Phase Space

a t z i , t )u t t h e v i r t u a l r a y ( s e e w 1 .8 .1 ), n a m e l y t h e r a y t h a t w o u l d p r o d u c e t h eg i v e n r a y a t Zo, i f t h e i n t e r v e n i n g s p a c e b e t w e e n zi a n d Zo w e r e f r e e . E x e m p l a r yt o t h i s v i e w a r e so m e o p t i c a l sy s t e m s , w e w i ll d e sc r i b e b e l o w a s , fo r i n s t a n c e ,t h e Fo u r i e r t r a . n s f o r m i n g c o n f i g u r a , t i o n r e a l i z e d b y ( t i v c r g i n g t h i n l e ~se s .Th e a . l)ove dis cu ss ion gives a t)hysi(:a .1 co nc re ten es s to f ree t ) rot ) aga t ion an dth i n l ens ma t r i ( : e s , T ( d) an d L ( f ) , w i th neg a t iv e va,lu('~s ()f d a,n (t .f . Th e pre sen tt r e a t m e n t a ,( :(p~ires in(lee( t a, g( ;n(; ral va l i( t i ty , ( :Ora l) r i sing w i th in th e s am e for-~m,lisl~ tl~(; (:a,s(;s ()f (liv(;rgil~g l(;~scs, v ir tu a l ()t)jc(:ts al~(1 virt ~m l i~m,g(;s.

3 . 3 . 1 f ~} 'ee p r v p a g a t i o n t h ' ,v u g h " n e g a t i v e " d i st a rt c esFr(;(; t)r()I)a,ga, ti()ll s (:()rr('~st)()ll(liIlg t() n (; ga tiv (; va,llu;s ()f l,ll(; (lisl,ml(:(; (:aal t)(; rea,1-ize(t t)y lts in g a tillit(; ln m d )c r ()f l, iill l(,ns('~s s(;I)a.ra, (;(l I)y tilfil,('~ (list alm (;s. T h islnm(tlfiv()(:a,lly (;sta,t)lisll(;s l,ll(: l)llysi(:a l r(;a,liza,t)ility ()f tl m ilw(;rs(; ()f fre e t)ro t)-aga.ti()ll l)y a. ('.(nlstru(:til)h '. ()l)ti(:al sy st (n ll, ml(t ll(;ll(:(; (:()llq)()S(;S tlu ; (:()lltra.stt)ctwe('. ll tll(; "ll m tll eI lm ti( :a l" ml(l "()I)ti(:al" t)etm,vi()ln" ()f tll( ; s(;t, ()f Iim,tri(:cs(3.3 .1) in fa.v(nlr ()f tim gr(nlI) s trl l(: tllr e ()f tim se t ()f lill( 'ar ()l)ti(:a.l sy st(; nls .

Arr a,n gi~ lg l, llre(; l(uls(;s al)l )r( )I)r iat( 4y (:h()s(ul ml(l Sl)a~:(;(t ()ira fl'()l~ tim () th er s()11(; lll ay sy~tl~('.siz(', a "~ wg ativ (:" fr(;('. I)r() I)ag ati()n . I~ fa(:t, tl~('~ f()r ~m l i(t(;l~tity

(A ~ ) _ L( . f a ) T ( d . ~ ) L ( f . e ) T ( d ~ ) L ( . f , ) , (3 .3 .13)(:a.~ |)(; s()lv(;(t fi)r tll(: l)a ra,~l(:t( ;rs of tl m ()I)ti(:a,1 a,rra,~g(uim~d, ()~ tll(: ri gh t, i.e.tl~e f()('al l(n@d,s . f ~ , f . ~ , f a a~(t t,l~e (lista~( '(:s d, , d., , in ()r (h'r t() I)r()(tu(:c a"n(;ga, t ive" free t)r()I)a.gati()~ nmtrix, f i)r whi(:h in(tee(l we nn~st l~a,ve

A = D = 1 , ( : = 0 , B < 0 . ( 3.3 .1 4 )S ~ l) st i t ut in g l ,h(; va,, 'i ()~s T an(l L ~na.tri( :cs en t(; rin g (3.3 .13) , w(; ()l) ta , in for

th e e~d, i( ;s of t im nm, rix (n~ t lu; left, t im exp res si()~ s

A - 1 - ~ B B - ( t ~ + ( t ~ ( 1 - ~ )f2 f l ' f2 '1 1 d _ _ ~ ). 1 1 d 2 D D - - 1 + d l ( f 2 f ' * q t . . f 2 f a - - f ' ~ 'C : f 2 f 3 + f 2 f a f l ' . , .w h i c h b y ( 3 . 3 . 1 4 ) p r o v i d e t h e r e l a t i o n s

(3.3.15)

I t is i m m e d i a t e l y e v i d e n t t h a, t f2 m u s t b e p o s i ti v e . A l so , it c a n ea s i l y b e c h e c k e dt h a t f~ a n d f a a s w e l l m u s t b e p o s i ti v e . T h e r e f o r e , i d e n t i t y ( 3 . 3. 1 3 ) s a t i s f y i n gr e q u i r e m e n t s ( 3 .3 . 1 4 ) is c o n c e v a i b l e w i t h a l l t h e t h r e e l e n se s p o s i t i v e :

L , s > o . (3 .3 .17 )

1 1 1 d 2 d I d 2 1 1 1f l - - f 2 f 3 + f2 f3 ' f l -- f 3 ' f 2 > ~ + d--~" (3 .3 .16)


9/56

The Group of the 1D Fi rs t -Order Opt ical Systems 1 1 9

0 1 < >

d > 0

d< O

.... < .......................................................< ............................................

__2, n ~ 9! o o ], i ]! n, i !

1.d IFI, 13oi.f .f .1~ !!A A A! Ii l l i i I f ! i

" ' 3 1 3 1 '1 . . . . . . . . . . . . . . . . . . . . . . . : . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . j

FI G U RE 3.4. A possible three-lens configuration imp lemen ting the free propa gation by a"negative" distance.

C l e a r l y t h e c h o i c e o f t h e p a r a m e t e r s , i .e ., fo c a l l e n g h t s a n d d i s t a n c e s , s a t is -f y i n g t h e a b o v e r e l a t i o n s is n o t u n i q u e . H o w e v e r , i n o r d e r t o p r o v e t h e e x i s t e n c eo f s o m e o p t ic a l c o n f i g u r a t i o n b e i n g i n a c c o r d w i t h ( 3 . 3 . 1 6 ) , w e c h o o s e

f2 - fa - f . (3 .3 .1 8)C o n s e q u e n t l y , w e h a v e

1 2 d 2f l = f + Y > 0 , (3 .3 .1 9)w h i c h c a n b e s a t is f i ed , f or in s t a n c e , b y s e t t i n g

d 2 - 3 f . (3 .3 .2 0 )I t f o l l o w s t h a t

f l - f , d l - 3 f - d 2 , ( 3 . 3 . 2 1 )t h u s y i e l d i n g a s y m m e t r i c c o n f i g u r a t i o n w i t h t h r e e i d e n t i c a l l en s es e q u a l lys p a c e d f r o m e a c h o t h e r ( F ig . 3 .4 ). T h e B - e n t r y o f t h e r a y m a t r i x c o m e s t o be

B - - 3 f < O. ( 3 . 3 . 2 2 )T h e v a l u e o f f w i ll b e f ix e d a c c o r d i n g t o s o m e s p e ci fi c r e q u e s t o n t h e v a l u e

a T h e r e a d e r m a y w o r k o u tf B . T h u s , f r o m B - - d , d > 0 , w e o b t a i n f - 5 "o t h e r c o n f i g u r a t i o n s s a t i s f y i n g ( 3 . 3 . 1 6 ) .


10/56

1 20 L i n e ar R a y an d W a v e O p t i c s n P h a s e S p a c e

3.4 Optical matrices factorized in terms of free propagationsections and thin lenses

E v e r y 2 x 2 r e a l s y m t ) h ; c t i c m a t r i x c a n b e u n d e r s t o o d a,s a r a y m a t r i x r e p -rcs(',~d,i~lg a. fir st -o r( le r ot)ti(:a l syst,( ',nl. Ih',n(:e, first-()r( t(;r ()t)tica.1 s y s I , e l I I S C a l l|)e reg a.r( te( t a,s ot)tit:a,1 mm,l() gs ()f I)hysit:a,1 t)ro(:ess(',s (t(',s(:ril)e(t |)y 2 x 2 re alsynlt)lc(:(,i(: maI , ri(:(',s. I(, (:a,n 1)(; l)r()vc(t , in fa ct , tlm, t (',very 2 x 2 re al sy mt) le( :-ti(: l l m tr ix is exI)r( ',ssit)l(; as tl le I)r()(lll( 't ()f a fill i t(: l l luld)( 'r ()f T ml(l L- lik emal , ii(',(:s, (,tHis a.ls() I)l'()vi~ lg (,Ira,l, e v e r y ()I)tic:a.1 sys(,(~,lll (:a.ll 1 )(; r(',a,lize,(1 a s a,na.i)t)r()i )ria(,(', ([i~fit(',) s(',(t~t(;~(:(', ()f fl'(;(', t)r()t)a,ga.l,i(n~ s(', (:ti (ms a,II(] (,lfin l(',n ses [3 ].

th' ,re w(', (;xmlli11(', tll('~ r(' a,li zal )ili ty ()f (',v(',ry Hp (2 , N ) syst(' ,lll |)y ml ()t)tic:ali l , l ' l ' i l , l l g ( ~ , l l l ( ~ , l l I , ()f tl li ll l(',llS(',S ml (l fr(',(',-~ll(',(liln~l S(',('t,i(nlS. W(', slla, ll 11()1, (t wel l ()ntll(', (tll(; sti( nl r(; ga ,r( tin g tll(: lll ill i] ln ni l ]nu~fl)(',r ()f h',~s(',s ~(;(',(1(',(1 1,() r(;aliz(~ ai)a,rt i(:~l lar sy ste I~ l. Tl~ is (lll(',s(,i()li is a~ aly z(; (1 i~ (l(;I,ail i~ [a.n]; tl~(;l'(:, as a, res ~fl t()f a.II a,(:(:lll 'a,(,(: i~w(:stiga,(,i()~ ()f tl m va.ri(n~s r(:g i()~s ()f Hp (2 ,1R ), (,lm.(, (:a.lt 1)(:r(:a.(:h(:(l l)y (n~(:, (,w() an (l (,l~r(:(: ](:~s (:()nligura,(,i (nls, it is sl~()w~ t lm t (:v ery(:I(:~(:l~(, i~ (,I~(: gr ()~ I) (:a,~ l)(: r(:ali z(:(l l)y a n ()I)ti (:al (:()l~lig~m~(,i()l~ i ~w ()Iv ing n()l~i()r(: t]m.ll tlu'(:(: h:llS(:s.

W(: firs(,ly (:()~si(l(:r ()l)(,i('al (:()l~Iig~tra,ti()~s, (l(:~m.l~(li~g f()r (,In'(:(: l)a,rani(:(,crs(,() I)(: (l(:li~(:(l, m~(l l)r()v(: tl m, t t,l~(:y yi(;l(l sld(,a.l)l(: ()l)(,i('al sy~(,l~(:s(:s f()r (:l(:~(:I~tsin Sl)(:(:ili(: r(:gi(n~s ()f (,I~(: Hp (2 , lR) gr( nq ). ~l'h(:l~, ~ ()r (: (:(n~l)l(:x a.rrm~g (:n~(:ntswi(,l~ il~(:r(:a,si~g ~nl~l)(:rs ()f l(:~s(:s m~(l fr(:(: l)r(>l)a,ga,(,i()~ s(:(:(,i()~s ar(; (:()nfigurc(li~ ()r( l( ' r t () s l~()w ( l int ~n~i~ ()( l~lar l )~r( ' ly ( l iag() iml ~mt, r i ( '( 's at ( ' ( ) l ) ( i ( 'al ly r eal i z-a,l)l(:. T lf is is l)a.si(: t() (,I~(: ('()~(:h~s iv(: l)r()()f a,s (:la,l)()I'a.t(:(l i~ ,~ 3. 4. 4, (:()~ (:(; rnin g(,I~(: I)()ss il)ili(,y ()f r(:I)r( :s(:~( ,i~g (:v( :ry r(:a,l sy~q )l(:(:(,i(: ~m,(,l'ix wi(,l~i~ th e c()n -~(:(:(,(:(I i(l(:~(,i(,y (:()~l)( )~(: n(, ()f (,l~(: gr () ~l ) in tl m f()rn~ a,s (,h(: fa.(:t()r(:(l l)r o( lu (:t ofT , L a~(l I)~r(:ly (liag()~m,1 ~mt, ri('( ;s, h~ th e f()rtl~( '()~ d~g (tis(:~ssi()~ tl~e ma ,tr ixT ( d ) is ~n~(t(;rs(,()()(l a,s (t(; s(:r il)i ~g (,r~fly fl'(;(; t)r()l)a,ga,(,i(n~ s(;(:(,i()l~S, a,~(t he n c ethe t ) a , ram( ; t e r d i s l i mi t e ( t t( ) n ( ) mm ga t ive va lues . Th i s i s i n o r ( l e r t () ev i den ceth e rcleva,n(:e o f t lm in h(:r( :nt , scmig r()ul) ( inst( :a.( l ( )f gr()~H)) s( , r~u: t,~rc of these t of th e free,-t)r()t)aga,ti()~ nm,ti'i(:( '~s to I) re ve nt (,1~(', sin q) h; thr e(; t ) a r a m e t e rc o n f i g u r a , t io n s , c ( ) m t ) ri s i n g o n e o r t w o l e n s e s, f r o m e x h a u s t i n g a ll t h e t h r e e -p a r a m e t e r S p ( 2 , IR) s y s t e m s . T h e, r e s t r i c t i o n o n t h e m a t r i x T(d) i s r e l a xed i nw 3 . 4 .4 , w h e r e " n e g a t i v e " ( t is t a n( : es a r e a l l o w e d .

3 . 4 .1 T L T a n d L T L c on f i g u r a t i o n sT h e s i m p l e s t r e a l i z a t i o n o f a 2 x 2 s y m p l e c t i c m a t r i x b y m e a n s o f t h i n le n s e sa n d f re e p r o p a g a t i o n s e c t i o n s , o n e m a y e n v i s a g e , s h o u l d i n v ol ve a n ar ra n g e -m e n t a l l o w in g fo r t h r e e p a r a m e t e r s , w h i c h t h e n m i g h t b e t w o d is t a n c e s a n do n e f o c al le n g t h o r c o n v e r s e l y t w o f oc a l l e n g t h s a n d o n e d i s t a n c e . T h e r e s u l t in gc o n f ig u r a t io n s w i l l b e d e n o t e d a s T L T a n d L T L , r e s p e c t i v e l y .


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Th e Group o f the 1D Firs t -Order Opt ica l Sys tem s 121/ AT h e n , le t M - ( ~ ~ )) b e a s y I n p l e c t ic i n a t r i x . S u p p o s i n g M h a s n o n n u l l

e n t r i es , t h e f o ll o w i n g s e q u e n c e o f o p t i c a l e l e m e n t s c a n b e h y p o t h e s i z e d\ /

( ~ BD) - - T ( d 2 ) L ( f ) T ( d i )h a v i n g M a s t il e re l e v a n t r a y m a t r i x . T h e l e n g t h s d l , d 2 a n d fa c c o r d i n g t o t h e r e l a t i o n s

(8 .4 .1)a re d e t e r m i n e d

1 - d -z A 1 - ~ D , 1 C , d I + d 2 d l d2s , s s - s - B . ( 3 . 4 , 2 )A s f m a y ra n g e fr o m - o o t o + o c , w h i l s t d I a n d d 2 a r e t h o u g h t a s n o n n e g -

a t i v e q u a n t i t i e s , n o t a l l t h e p o s s i b l e v a lu e s o f t h e m a t r i x e n t r i e s A , B , C , Dc a n b e r e a c h e d . I n f a ct , o n l y t h e t w o f a m i l ie s o f s y m p l e c t i c m a t r i c e s"

/ 'A


12/56

12 2 L i n e ar R ay a n d W a v e O p t ic s n P h a s e 5 p a c e

T e l e sc o p i c a n d i m a g i n g s y st e m sT h e L T L c o n f i g u r a t i o n ( 3. 4. 4) is a p p r o p r i a t e t o s y n t h e s i z e s y m p l e c t i c m a t r i c e sh a v in g C = 0 . I n f a c t , t h e i d e n t i t y

((a 2 > O) _ L ( L ) T ( d ) L ( f ~ ) (3 .4 .7)is a,(:t~lal)le with d = .f, + f2 = B a,n(t A = - . f 2 / f l .W e re(:a,ll f i ' o~ !i 1.8.4 tl~a,t () t)ti(:al sys tel ~S (les(:riber t)y ray-~ m,tr i( :es w it h(7 = () ar e lla,Ille(l a foca l ( )r te le ,scop ' i c sys t, e~ s , as | ,l~e r ()f the c ~ er g en tra y ( lel)c~(ls () lily ()~l t ,l~(; (] ire(:l , i(m of | , t~e in(:i( lent ray. T l ~ s an y ra y ( :om ingin to t,l~(' sysl,ex~ l)a,ra,lh'l 1,() th e ()t)t,i(:al ax is (;~( ;rg(; s t)a.rallel to th e ax is a.s wel l.

I f I ~ - ( ) , 1 ,1 ~ (; s i ~ t zh ; h ;~ s ( ' ( ) ~ f iN~ ra ,t ,i ( ) ~ ( 3 .4 . 1 ) ca , ~ 1 ) (', (lesigl~e(1 witl~ 1 / . f -- ( 7 , d . ~ / d , = - A a.~,(l l / f -= l / d I + 1 /d .~ ( f o c a l r i()l,)i~ , ()r(ler 1,()give

A < 0 ( )c: < () A ,) - T ( d ' 2 ) L ( f ) T ( d , )" (3 .4 .8)limite(1, ~f ( ' (~r ,sc, t,r ne ga ti w ; wd lle s of t , t~c eil tr i es A a n(l (~. A(:(:or~ling toti le f~r rela,t,i~ll, tll~ i11terllw,~tia,te lellS is I)~siti w~ a,ll~l r t~e llll~h; rst~o~ l as(:Olllt)Ose


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The Group of the 1D Fi rs t -Order Opt ical Systems 123

f = _ l d = A - I ( ,

H i I - I o I - I i 1 7 0! f ! ! f !i i i ii i i ii i i i, , ( o - V _ ~ , ,i , -" [ c ~ c l Q - - - > ; ii i a ,i t f = I d = D - ! i ii i - ~ - - U i i~ -- f d - - i i--~- d f

(a) (b)

FIG UR E 3.5 . Opt ica l rea liza tion of a symplectic mat r ix having ( a ) D - - 0 ( c o l l i m a t i n gsys tem) and ( b ) A - 0 ( i m p e r f e c t F o u r i e r t ransforming sys tem) .

w i t h A - 0 a c c o u n t f or t h e p r o p a g a t i o n f r o m t h e p l a n e l o c a t e d b y t h e a x ia lsu 1 -D t o t h e s e c o n d a r y f o c a lo s i t i o n o f t h e p r i m a r y p r i n c i p a l p l a n e a s cI t 0 1p lane o f t he sys t em (w 1 .8.4 ).

E v i d e n t l y , a D - 0 m a t r i x c a n b e r e al iz e d b y t h e o n e le n s s e t u p( ~ - C - 1o ) - T ( d ) L ( f ) T ( f ) (3 .4 .9 )

w i t h t a k i n g1 d A-1 (3 .4 .10)f c , c ,

w h i c h t h e n d e m a n d f or C < 0 a n d A < 1 in o r d e r t h a t b o t h T ( d ) a n d T ( f )r e p r e s e n t p o s i ti v e f r ee p r o p a g a t i o n s e c ti o n s.

L i k ew i s e, to p r o d u c e a n A = 0 m a t r i x , t h e r e v e r s e c o n f i g u r a t i o n c a n b ea r r a n g e d a c c o r d i n g t o

( ~ - C - 1D ) -- T ( f ) L ( f ) T ( d ) (3 .4 .1 1)w i t h

1 d D - 1 ( 3 . 4 12)f c , - c ,t h u s r e q u i r i n g t h a t C < 0 a n d D < 1 .

Th e s i ng le l ens dec om pos i t i ons (3 .4 .9) and (3 .4 .11) e s t ab l i sh , w i th in t hes p e ci fi e d r e s t r ic t i o n s , t h e e q u i v a l e n c e o f c o l l i m a t i n g a n d f o c u s i n g s y s t e m s t ot h i n l e n s e s , w h o s e i n p u t a n d o u t p u t p l a n e s b e p r o p e r l y p l a c e d . I n ( 3 . 4 . 9 ) t h ei n p u t i s t a k e n a t t h e p r i m a r y f o c a l p l a n e o f t h e l e ns a s t h e o u t p u t a t d i s t a n c ed A-1 f rom the s econ dar y p r i nc ip a l p lane , wh i ch fo r a t h i n l ens is t he lens- - Cp lan e (F ig . 3 .5 . a ) ) . R ever se ly , in (3 .4 .11) t h e i np u t i s t ak en a t t he d i s t anc ed - o - 1 f r o m t h e l e ns p l a n e a n d t h e o u t p u t a t t h e s e c o n d a r y f o ca l p l a n eC( F i g . 3 . 5 . b ) ) . F o c u s i n g s y s t e m s p e r f o r m t h e imper fec t F o u r ie r t r a n s f o r m o f t h e


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124 L i n ea r R a y a n d W a v e O p t i c s in P h a s e S p a c e

V I i V l o I - I i ~ I o

i ' I : " - Y o C l i ii i i ii , _ ~ ' ' x ) i i, i T i ~

i : , , i ,

. t . f . - - . . i ~ . t - - . - - - - - . t ~

( w ( b )

FI(]I~ I~E 3.6. S i~glc le~s rcali zati o~ of a~ m~ti(liag()~ml sy~I)l( :ctic ~natrix (pc)fi:ct F o u r i e rtransf i)r~i~ g sys te~ ). The "2f " sysl,c~n witl~ ( a ) a p()sitivc l(:~s m~(l (b ) a ~mgativc lens.

i~I)~(, sig ~m l, (,1~(: a lX~l)li(,~(l(: (lis(,ril)~(,i()~x ( )f ti le ()~1,t)~(, sig~m,l t)( :i~ g i~(,(:rt)r(:t(:( ta.s (,1~(: l: ()~r i( :r (,rm~sfi)r~xl ()f 1,1~(: inl )~( , ~n~l(,il)li(:(1 1)y a (l~m.(lra.(,i(: t)lxa,s(: t( :r m .

Tll( : fi~r(,l~('r r(:(t~(:s(, A = () i~ (3.4 ..9), ()r D - () i1~ (3 .4 .1 1) , yi(:l(ls d -- .f, (,ll~st)r()vi(li~xg fi)r (,Ix(: pe ' ~ : f ec t F()~n'i(:r t, ra~|sfi)rlxx il~g sys(,(:l~l t,l~(: si~tzl(: l(:~xs r( :a li za ,ti on

() - 1 / ( ' ) _ T ( f ) L ( f ) T ( f ) ( 3 .4 . 13 )F ( f ) - - (7 ()l Th (: a.ll(,i(liag ()xml llm .(,rix a.l)()v(: (l(:s(a'il)(:s ill g(:n(:ra.l (,h(: ()l) ti-i(,h f - c "(:a,l l)r()Im.ga,(,i()ii fl'()lll (,If(: l) ri n m ry fl)(:a,l l) lan(: ()f th(; syst,(:Ixl (,() (,If(: s(:( :(m (lar y

(m(:, ()ll wlli(~ll th(: l)(:rf(:(:(, F()In'i(:r ( ,rai lsf( )rn l ()f ill(: illg ()illg sign a,l is f()I'mc(l (,~1.8 .4 ). A('(~()r(lillg t() (3. 4. 13 ), tlx(: silxxl)l(:st r(:a.liza,ti()Ix ()f s11(:ll a sy st(: lll (:( resi sts()f a l)()sitiv(; tllill l(:lls, wit, If tlx(: ligllt ra ys |)(:illg l)r()l) agat(:(l fr()i11 tll(: fr(m t t()th(: ])a(~k ft)t~al I)lml(: (s(:(: tll(: s(:ll(:xxx(: in Fi g. 3. 6 .a ) ft)r t, x(: (: x( :n q) lar y (:a.s(: ofa bi-( ' ()l lv(:x l(:ns). Il l t)ri lwiI)l(: , th(: TLT (l(:s ign (3.4.13) sh()lfl(t s(,ri(: t ly admiton ly I)()sitiv(: l(:ns(:s, ( '()rr(:sI)()ll(|ing t() ll(:ga,tiv(: va,hms ()f C in th(: g iv en ma, tri x.H()w(:v(:r, lx(:gativ(: wfll ms ()f f ar c a,lh)w(:(t as w(:ll. Tlx(: r(: slfl tin g ()t)(,i(:al con fig -ura, ti()ll is (l(:I)i(:t(:(t ixx Fi g. 3.6 .t)) f()r ttl(: il lu st ra ti v( : (:a,s(: ()f a, t)i-(:()~l(:a,ve len s.Ta k i ng i Il t( ) a ,( :( :o llnt the m( :a n ing o f the s ign o f th ( : fo (:al l eng t h f , which , ina . ccor( t wi th th e ( :om mo n ( :onv en t ions (w167.8 .1 and 3 .3 ) , e s ta ,b l i shes the cha r -a c t e r , r (: al ()r v i r t u a l , o f t h e o b j e c t a n d i m a g e r a y s , t h e r e a d e r m a y e a s i l y v e ri f yt h a t t h e f or ma .1 r e p r e s e n t a t i o n f o r t h e o p t i ( ' al s e t u p i n t h e f i g u r e t u r n s t o b e.illst t h e T L T p r o d u c t ( 3 .4 . 13 ) , w i t h f r e t a i n i n g i ts o w n s i gn in t h e T - m a t r i c e s .T h u s , l l n i m o ( t u l a r a n t i d i a g o n a l m a t r i c e s c a n b e r e a l i ze d b y u se o f o n e le n s o f

1 p l a c e d m i d w a y b e t w e e n t w o r e f er e n c e pl a n e s s e p a r a t e d( )ca l l en g th f c ,b y 2 I fl . T h i s c o n f i g u r a t i o n is r e p o r t e d a s t h e 2 f s y s t e m .

F i n a l l y w e n o t e t h a t t h e d o u b l e l e n s c o n f i g u r a t i o n ( 3 .4 . 4) c a n a ls o b e v a l u e dt o c o r r e s p o n d t o a r a y m a t r i x w i t h D = 0 , A = 0 , o r A = D = 0 . I n p a r t i c u l a r ,


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T h e G r o up o f t h e 1 D F i rs t -O r d er O p t i c a l S y s t e m s 12 5

[ - [ if

1 7 o, i o

f - ~ ,f ~! ~= f

( a) ( b )

[1 1-1o7 7

X \

FIG UR E 3 .7 . Doub le l ens r ea li za t ion o f an an t id i agona l sym plec t i c m a t r ix ( p e r f e c t F o u r i e rt r ans fo rm ing sys t em ) . The F o u r i e r t u b e wi th ( a ) posit ive lenses and ( b ) negative lenses.

i n F i g . 3 .7 w e s h o w t h e d o u b l e l en s s e t u p s ( a l s o k n o w n a s Four i er t ub es ) , s u i t e dt o p e r f o r m t h e p e r f e c t F o u r i e r t r a n s f o r m o f t h e o b j e c t d i s t r i b u t i o n .

3 . 4 . 3 P u re m a g n i f i er sTh e b as i c o n e an d t w o l en s co n f i g u ra , t i o n s (3 .4 .1 ) an d (3 .4 .4 ) a r e n o t s u i t ed t or e a l i z e u n i m o d u l a r p u r e l y d i a g o n a l m a t r i c e s , w h i c h w i t h i n t h e o p t i c a l c o n t e x ta r e i n t e r p r e t e d a s t h e r a y t r a n s f e r m a t r i c e s o f p u r e m a g n i f i e r s ( w 1 .8 .4 ) . E i t h e ra d d i t i o n a l t h i n l e n s e s o r a d d i t i o n a l f r e e p r o p a g a t i o n s e c t i o n s o r b o t h m u s t b ea d d e d t o t h e o ptic a,1 a r r a n g e m e n t s (3.4.1) a n d (3.4.4), thus i n t r o d u c i n g f u r t h e rp a r a m e t e r s , w h i c h c an b e a d j u s t e d in o r d e r t h a t b o t h t h e o f f -d i ag o n a l e l e m e n t si n t h e r e s u l t i n g r a y m a t r i x m a y v a n i s h .

W e c o n s i d e r t h e o p t i c a l i d e n t i t y (3.4.8) f o r t h e i m a g i n g m a t r i x , w h i c h f o rc o n v e n i e n c e i s r e w r i t t e n a s

A < 0 0a - l ) - T ( d , ) L ( L ) T ( d l ) , (3.4.14)C < 0w i t h

1 1 1 d ,,= + - - C , A - - - - . ( 3 .4 . 15 )f l d l d2 d lC o n c a t e n a t i n g t h e T L T s e q u e n c e ( 3 . 4 . 1 4 ) w i t h a t h i n l e n s o n t h e l e f t o r r i g h t ,t h e l o w e r -l e ft e n t r y o f t h e r e s u l t i n g o p t i c a l m a t r i x m a y v a n i s h . A c c o r d i n g l y ,w e m a y r e a l i z e t h e p u r e m a g n i f i e r m a t r i x b y t h e o p t i c a l a r r a n g e m e n t

0 M -1 - L ( f 2 ) T ( d 2 ) L ( I I ) T ( d I ) , ( 3 . 4 . 1 6 )t h e m a g n i f ic a t io n M b e i n g d e t e r m i n e d b y t h e p a r a m e t e r s f a n d d a s

M d2 f2 1 1 1 (3.4 .17 )d l - - f l ' f l = d l + d2


16/56

1 26 L i n e a r R a y a n d W a v e O p t i c s n P h a s e S p a c eV i i F lo

, f , f 2 i

/ ' 2 1 < > 'i

& f , IM - - - - - - - - - - - - - - I(I ~ h ; - " * ( I :

FI G U RE 3.8. 'I'wo-lc~s rc alizatio~ of a I)~rc ~nag~filier.

whi , :h a lso yM(t d~ = . f , + f2 (Fig . 3 .8) .Th( ; nmgnif i( :a , ti ( )n .hd is (: lea ,r ly nega. t ivc, t tn ls s ign ify i l lg an inv( ;rs i(nl ( )f th era y t)()sitio ll w itt l r(;st)('~(:t t() tlm ()t)ti(:al a,xis. W (; (:ml ()])taill a l)()sitiv(; ina.g-nifi(:ati()n t)y (:as(:a~lillg tw() llla,gllifi(;rs like (3.4.16). Ill fa,(:t, l)(;illg a strikingt)r()l)(;rty ()f (tiag()lm.1 llml, ri(:( 's t ll a t tl m y llnflt, it)ly ill|,() (lia g()lm l llml,ri(:(;s a,s

-~ () M ~ - ' - () M ~ (3 .4 .18 )() M lw it h 3d - ,hd ~3 d 2, w(; ( 'ml lmv(; 3d > () wi tll 1)()ill 3d~ < () ml(1 3/[ 2 < ().

E(t~m.ti()~s (3 .4 .1 6 ) a l~(l (3 .4 .1 8) ar c t)a.si(: t() ()~ r t)~rt)()s(;s, 1)(;( 'm~s(', th(; y rei )-re se nt tlw, l)r()()f tlla,t Inlr(; ly (lia,g()na,l synq )h;(: ti(: nm, tri(:(;s (:ml t)e syld, hesi z(;dt)y (',()I~(:a,te~m, i~ g tl~i~ l(;~ ses m~(1 fre e t)r()l)ag a,ti()~ se(: ti()n s.

T h e rea,(t (; r ( :an eas i l y ver ify th a t t lm sa,~n(; o t) t i( :a l ( : ( )~f ig~ra, t i( )n as in(3 .4 .1 6 ) in a y t)e ()l)tai~(;(1 a,(l(li~g a, fr(;(; t)r()t)a,gati()I~ s(;(:ti()~ t() tlm ri gh t ofth e (h)ub le lens rca l iza t i ( )n (3 .4 .7 ) ( )f a ,n a,f i)ca,1 nm, r ix .P u r e m a g n i f i e r s a n d Fb u'ri~ ;'r t ' i n n s f o r m i n g s ys tt ~ m sR c m a , r k a b ly , a, p u r e ly m a g n i f y in g c o n f ig u r a t i o n c a n a l s o b( ; r( ;a ,( :h ed t )y e n l a r g -ing the two h ;ns ( :onf igura, t ion (3 .4 .16) to ( : ( )n lpr ise a ,n a , (h t i t iona,1 f l ' ee propa,-g a t i o n s e c t i o n , t h u s y M d i n g

D ( . A 4 ) - T ( d 2 ) L ( . f 2 ) T ( d ) L ( . f l ) T ( d l ) , (3 .4 .19 )w h e r e t h e i nv o l ve d l e n g t h s a r e r e l a t e d t h r o u g h

AA - _1 2 1 d d + dl .A d -1- d-2- -- 0 (3 .4 .20)f l - - f l ~ .A / I "P r e s e r v i n g t h e r e s t r i c t i o n o f 0 _< dx , d~ d 2 ~ C


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T h e G r o up o f t h e 1 D F i rs t -O rd e r O p t ic a l S y s t e m s 127r l i

i . t i i . I S .i ii ii !i

ii= ,u= ?_ _ v _' . t l . f i ' . t i

1 - ] o P l iI a . / I i l : *

. ~ ii :

_ ! I/ 2 - ' ' [ i . f l ' . 1 2.(a) (h)

r i oi

i

i!

ii

F I G U R E 3 .9 . O p t i c a l r e a li z a t io n o f a p u r e m a g n i f i e r w i t h . h a l < 0 b y c a s c a d i n g t w o F o u r i e rt r a n s f o r m i n g s y s t e m s . ( a ) I .h d < 1 ( b ) . h d I > 1 .

We obse rve t ha t t he s econd o f (3 .4 .20 ) m ay be s a t i s f i ed by s e t t i ng d I --f l , d 2 - f2 a n d a c c o r d i n g l y d - f l + f2 - T h e re s u l t i n g t w o l en s c o n f i g u r a t i o nt u r n s t o b e t h e s e q u e n c e o f t w o 2 f s y s t e m s , d e s c r i b e d i n w 3 .4 .2 a n d p i c t u r e din F ig . 3 .6 . a ) . There fo re wc wr i t e

D ( A // ) - F ( f 2 ) F ( f l ) .A /[ - - [ z f l " (3.4.21)O n a c c o u n t o f t h e c o n s i d e r a t i o n s o f w 3 .4 .2 c o n c e r n i n g t h e i n t e r p r e t a t i o n o f t h es ig n o f t h e f o c al l e n g t h s in t h e i m p l e m e n t a t i o n o f t h e F o u r i e r t r a n s f o r m b y t h i nlenses , the focal l engths f l a ,nd f2 above ca .n be le t to be a , rb i t rar i ly pos i t iveo r n e g a t i v e , t h u s a l l o w i n g f o r b o t h n e g a t i v e a n d p o s i t i v e m a g n i f i c a t i o n s .

Accord ing t o (3 .4 .21 ) , a pu re m agn i f i e r can be r ea l i zed by us ing t wo l enses ,s p a c e d s o a s t o h a v e a c o m m o n f o c a l p o i n t , a n d e m b r a c e d b y t w o f r e e - m e d i u ms e c ti o n s, w h o s e l e n g t h s a r e a d j u s t e d a c c o r d i n g t o t h e f o c a l l e n g t h o f t h e c o r-r e spond ing a ,d j acen t l ens . Thus , i n t he who l e sys t em bo t h t he e f f ec t i ve l eng t hand t he op t i ca l power van i sh . As a l l exam ple , F ig . 3 .9 shows two pos s ib l er e a l i z a t io n s o f a n M < 0 c o n f i g u r a t i o n b y c o n c a t e n a t i n g t w o F o u r ie r t r a n s -f o r m e r s , b o t h h a v i n g p o s i t i v e l e n s e s ; t h e r e s p e c t i v e p e r f o r m a n c e s a r e p i c t u r e df or a n i n c o m i n g b u n d l e o f p a r a l l e l r a y s. T h e p r o p a g a t i o n t h r o u g h b o t h s y s t e m sm a n i f e s t s a n i n v e r s io n o f t h e r a y - c o o r d i n a t e w i t h r e s p e c t t o t h e o p t i c a l a x i sa l o n g w i t h a m a g n i f y i n g o r d c m a g n i f y i n g e ff ec t a c c o r d i n g t o w h e t h e r ~ > 1

< 1. I n p a r t i c u l a r , w e e m p h a s i z e t h a t t w o id e n t i c a l F o u r ie r t r a n s f o r m e r sri n s e q u e n c e p r o d u c e a p u r e i n v e r s io n o f t h e r a y c o o r d i n a t e s , i n a c c o r d w i t ht h e g e n e r a l p r o p e r t y o f t h e F o u r i e r t r a n s f o r m . T h e n , a r a y e m e r g i n g f r o m a s o-q u e n c e o f f o u r i d e n t ic a l F o u r i e r t r a n s f o r m i n g s y s t e m s h a s t h e s a m e c o o r d i n a t e s(pos i t i on a nd i nc l i na t i on ) a s wh en en t e r i ng t o ( see a lso w 3 .6 .1 ) .

In con t ra s t , f14 > 0 con f ig u ra t i ons de m an d fo r l enses of opp os i t e s i gns .F i g u r e 3 .1 0 s h o w s t w o p o s s ib l e r e a l i z a ti o n s . T h e p a t h o f a n i n g o i n g b u n d l e o f


18/56

128 Linear Ray and Wave Opt ics in Phase Space

F l i l - l o

: 1iI

a0

l 00

V ii

0,)

E l oi

iii )i

.-4.

FIGURE 3.10. Optical rcalizatioI~ of a p u re m a g n i f i e r with M > () by casca(li~g two Fo~ricrtrm~sfor~ni~g systelnS. ( a ) M < l (b ) M > I.

t)a,ra,lM ra y s is t, ra(:(~(t, sll()w illg tim (:()llt, a,sl, illg (~tt'(~(:ts ()f tim tw() sy st cn ls , timfirst t)r()( llu:illg a, re(llu:ti() ll ( .A4 < 1) ()f th e tra xls ve rsc size ()f t, le tnnl ( lle, th eS O , ( : ( ) l l ( l , ( ) l l t, l(} (:()ld, a,ry , a,ll eXl)a,llsi()ll (.A 4 > 1).

T im ot) ti( :al ( :()~lf igurati()I ls t)i( :tl lrc(l i l l Fi gs. 3.9 ml(t 3.10 nm y tlm ll 1)e lnl( lcr-st()()(t a s tll(~ ()t)ti(:al m la l() gs ()f I)llysi(:a,1 I)r()('ess es, (les(:ril)e(1 t)y 2 x 2 sylllI)l(~(:t,i(:t)l n'e ly (tiag()na,1 llm .tri(:e s w i t h ll(~ga,tive all( l I)()sitiv(~ (~ t,r ie s, r(~sI)(~(:tiv(~ly. A ls o,a s a, t)() siti v(; ~a ,g ~i fi (w ('i.1,11 1)(~, a,rra,~g'(~(t })y (:a,s(:a~li~g tw () n(;gat,iv(~ l ~ m g ni fy in gsy st e m s, f(n~r I)()sitive F (n~rier traa M '()rnf ing ( :()nfig~mtti()ns ( :a,n |)e ( :()n(:a,t(;na,t(;(tto sy ~ th e s iz e th e ()I)ti(:a,1 mm, l()g ()f m~y I)l~ysi(:a,l I)r()('ess (tes(:ri|)e (l t)y a, 2 x 2synq)le( ' t i( : ( tiag(nm,1 I)()sitive ~na,tr ix.

Fi na l ly , we n ( ) te tha, t th e ( : ()nfig~mtti( )n (3 .4 .1 6) ( :a ,n a ls() t)e (mla,rge(t to(-o m pr is c a,n a,(t(titi()~m,1 th in le~ s t h u s lca,(li~g t() tim thr(~(~-leI~S sc t~ t)

D( .A4) - L ( . f , ~ ) T ( d , ~ ) L ( f ) T ( d I )L( . f l ) . (3 .4 .22 )W e i n v i t e t h e r e a d e r t o f i n d o u t t h e e x p l i c i t e x p r e s s i o n s l i n k i n g t h e i n v o l v e dt ) a r a m e te r s d I , d 2 , f l , f 2 a n d f . A l s o , w c s u g g e s t t o p r o v e t h e r c a l i z a b i l i t y

I _ 1 1 a ,n(t ac co rd in gl yf th e ( :onf igura, t ioI l w it h f l - d i , f2 - d2 , 7 - ~ + ~A /I _ 1 2 w h i c h is i n t e r p r e t e d a s t h e s e q u e n c e o f t w o F o u r i e r t u b e s .T h e s ub g r o up o f p u r e m a g n i f i e r sT i l e i de a l m a g n i f i e r s f o r m a p r o p e r s u b g r o u p o f t h e s e t o f 1 D l i n e a r o p t i c a ls y s t e m s . T h i s s t r a i g h t f o r w a r d l y f ol lo w s f r o m t h e p r o p e r t i e s o f t h e u n i m o d u l a rd i a g o n a l m a t r i c e s , w e h a v e d e n o t e d b y D ( 3 / ) . I n f a ct , a s a l r e a d y s h o w n , t h ep r o d u c t o f t w o m a t r i c e s i n t h e s e t { D ( 3 d ) , A d r e a l} b e l o n g s t o t h e s e t a s w e li ,a,n d h e n c e j u s t r e w r i t i n g ( 3 .4 . 1 8) w e h a v e

D ( . A 4 1 ) D ( . M 2 ) - D ( . A / [ 2 ) D ( . M 1 ) - D ( A d ) , ~ 4 - . A / ~ I M 2 . ( 3 . 4 . 2 3 )


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T h e i n v e rs e o f a p u r e m a g n i f i e r is th e m a g n i f i e r w i t h r e c i p r o c a l m a g n i f i c a t io n :[ D ( .s - 1 - D ( . / ~ - I ) , ( 3 . 4 .2 4 )

a n d t h e u n i t m a t r i x I c a n b e i n t e r p r e t e d a s t h e o p t i c a l m a t r i x o f t h e m a g n i fi e rw ith A/l = 1:

D (1 ) = I . (3 .4 .25 )W e n o t e t h a t , a c c o r d i n g t o ( 3 . 3 . 7 ) a n d ( 3 . 3 . 8 ) , t h e u n i t m a t r i x c a n b e o p t i -

c a l l y s y n t h e s i z e d b y c a s c a d i n g t w o , p o s i t i v e a n d " n e g a t i v e " , f r e e p r o p a g a t i o nsec t i ons o r two t h in l enses hav ing oppos i t e foca l l engh t s . Due t o (3 .4 .25 ) , t hei d e n t i t y m a t r i x c a n a l s o b e i m p l e m e n t e d b y c a s c a d i n g t w o p o s i t i v e o r t w o

1 and henc e i t can be r ea l i zed ase g a t i v e p u r e m a g n i f i e r s w i t h A 42 = M I 't w o , p o s i t i v e a n d n e g a t i v e , F o u r i e r t u b e s n e s t e d o n e i n t o t h e o t h e r o r a s f o u rp o s i t i v e F o u r i e r t r a n s f o r m i n g s y s t e m s c o n c a t e n a t e d o n e t o t h e o t h e r w i t h t h ere l a t i ve foca l d i s t an ces be ing r e l a t ed by f l f 3 = f~ f4 .P u r e m a g n i f i e r s w i t h o u t i n v e r s i o nP o s i t iv e i d e a l m a g n i f i e r s c o n s t i t u t e a p r o p e r s u b g r o u p o f t h e g r o u p o f u n i m o d -u l a r p u r e l y d i a g o n a l m a t r i c e s { D ( A J ) , ,k 4 r e a l} , b e c a u s e c a s c a d i n g t w o p o si -t i ve m agn i f i e r s p rod uce s a pos i t ive m agn i f i e r . In w 2 .4 .3 we have e l abo ra t e d fo rt h e p u r e m a g n i f i e rs w i t h o u t i n v e r si o n t h e e x p o n e n t i a l r e p r e s e n t a t i o n , r e w r i t -t e n i n ( 3. 1. 2 ), i n v o l v i n g t h e t r a c e l e s s m a t r i x K 3 , r e p r e s e n t a t i v e o f t h e m i x e dq u a d r a t i c m o n o m i a l l p q . As we wi l l s ee l a t e r , under spec i f i c r e s t r i c t i ons t hesca l e m a t r i x S (m ) p l ays a c ruc i a l ro l e a s a bu i l d ing "b lock" fo r t he r ep re -s e n t a t i o n o f o p t i c a l m a t r i c e s i n a fa c t o r e d p r o d u c t f o r m i n v o l v in g a s w el l t h eT a n d L m a t r i c e s . T h e r e f o r e , w e f u r t h e r c o i n I n e n t h e re o n t h e s y s t e m S ( m ) ,s u g g e s t i n g i n p a r t i c u l a r s o m e p o s s i b l e o p t i c a l r e a l i z a t i o n s o f i t.

W e m a y r e p h r a s e r e l a t i v e l y t o t h e s c a l i n g m a t r i x_ ) _S ( m ) m - 1 ~ ? It r ( 3 . 4 . 2 6 )

t h e f o r m a l c o n s i d e r a t i o n s d e v e l o p e d i n w 3 .3 i n r e s p e c t t o t h e f re e p r o p a g a t i o ns e c ti o n a n d t h i n l e ns in a t ri c e s, T ( d ) a n d L ( f ) , t h u s c o m p l e t in g t h e v i e w o ft h e t h r e e o n e p a r a m e t e r s u b g r o u p s o f @ ( 2 , I R), g e n e r a t e d b y t h e a l g e b r a b a s i s{ K _ , K + , K 3 } b y e x p o n e n t i a ti o n t h r o u g h a r ea l p a r a m e t e r .

E v i d e n t l y , S - li ke m a t r i c e s c o n s t i t u t e a p r o p e r s u b g r o u p o f S p ( 2 , I R) ; m o r ep rec is e ly , t he s e t {S (m ) , m > 0 } i s i nc l ud ed i n to t he sub gro up {D(A / I ), A /I r ea l}o f u n i m o d u l a r d i a g o n a l m a t r i c e s D ( A / I ). T h e n , a s p a r t i c u l a r c a s e s o f ( 3 .4 . 23 )a n d ( 3 .4 .2 4 ), w e c a n s t a t e t h e c l o s u r e p r o p e r t y

S ( T ~ I ) S ( ~ 2 ) = 8 ( ~ 2 ) 8 ( ~ 1 ) = 8(77-~), m - - m lm 2 > 0 (3 .4 .27 )


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fo r ma , t r i ces ( 3 . 4 . 2 6 ) , a n d h e n c e t h e e x i s t e n c e a n d u n i c i t y o f t h e i n v e r s e b e i n g[ S ( I ~ ,. ) ] - 1 - - S ( l r t - 1 ) . ( 3 . 4 . 2 8 )

I n a , d d i t i o n , t h e p r e v i o u s l y d e v e l o p e d c o n s i d e r a t i o n s c o n c e r n i n g t h e o p t i c a lr e a l iz a , ti o n o f p u r e -m a g n i f i c a t i o n d e v i c e s w i t h , A/I > 0 a p t ) l y to t h e s y s t e m( 3 . 4 . 2 6 ) a s we l l . Th u s , t i m s y m p l e c t i c d i a g o n a l m a , t r i c ( ' , s , g e n e ra t e d b y t h ea,lg('])ra, nmt, rix K a, (:an ()t)ti(:a,lly t)e r ealiz(;(t |)y (:()n(:a,t(',~m,ting tw() s y st e m sl ike ( 3 . 4 . 16 )( )r tw o Fouri( ' ,r t ra ,~sf orm ers (3 .4 .13 ) , the relewm l, fo(:a ,1 l( ;ngt , hs|)eing related t() th(; nm.gnifi( 'a, t i()~ '~, a,s

. f ~ = - , , , , , . f , . ( 3 . 4 . 2 9 )A n (;xa,~q)le is giv(;~ i~ F ig. 3.10 . Al so, we ('m~ (:as(:a,(l(; tw() ~mgat, v(; ~m ,gnif ierst() sy~tl~(',siz(', tim ~m,t,rix (3. 4.2 6 ), tim resu ll, i~g (:(mfig~n'a, i()~ n m y t)e reg ar( te(ta.s ()t)ti(:a.1 a,I~al()gs ()f tl~(', ~a.trix ~ , , s K a , tim, i~v()lv(;(t f()(:al l(;~gt, ~s f l , f2, .f:~, .f ~1)(; ing s~ita|)ly ( : l ~ ( ) s ( 'a ~ t ( ) sat is fy t im r( ; la , t i (m

f . e . f 4 - ( , , ~ / 2 . f , f , a . (3 .4 .30)3 . 4 . 4 S y m p l e r : t i c m a t ' s ' i c e s a s o p t i c a l m a t ' s ' i c e sAt)t)r{ )t)ria.te (:()ll.jllr {)f T , L a.ll{t D - li k e ilm.l, ir {:a,Ii 1)e a.r rm lg ed to beeql liva ,lent t() ally giv(',ll sy lllI )le ctic ll m tr ix , (;x(:llutiIlg tim a.ld, clia,g()lm,1 nm,t, i-(:es, f or w hi( :h, a,s (lis(:llsse(1 ill .~ 3.4 .2, th e 2.f sysl, ell ls a.ll(t t, lm F( nn 'ie r tu t) est) ro vi de t h e slfit, a,t)le ()l)l,i(:al rea ,liza ,ti(m s. In tif fs (:()llll('~(:l,i()ll, il, is w ()rt,h em-ph as izi ng onc e a ,ga. ii l t im ( l it f 'erence, re la t ive ly t{) t im gr{nlI) I )r()perty , of t, hema t r ic es T ' s a .( :( :o r( ting to w he th er th ey a re rega, r{ te (l as In~r(' ly ~m, he m at i ca len t i t i es or i n te rI) ret ed a .s f i)rnm.1 l,oo ls for n~o{telling ot)tir syst, e~ns. M a , t h e -m a t i c a , l l y , the 2 x 2 ~n in~o{lu la r rea l up pe r t r i a ng u la r ma , tr ir (3 .3 .1 ) do fo rma, su bg ro up o f tl~c syn q) lc c t i c g rou t ) S p(2 , I~) . In co~ t ras t , { )p t ica l ly t tmy rep re-sen t ra ,y-t )roI)aga, ti (m th rougl~ I)ort io ns of a, l~(m~ogeimous nm(t i mn , a ,~(t h e n c ed o n o t fo rm a . s u b g ro u p b e c a u s e t h e i n v e r s e o f a, f r ( ; e -p ro p a g a t i o n s e c t i o n c a n -n o t b e s y n t h e s i z e d a,s a f r e e - t ) ro p a g a t i o n s e c t i o n a s we ll ; i t s h o u l d d e m a n d i nfa c t for s e c t i o n s o f "n e g a t iw~ " l e n g t h a n d t h e re b y is n o t p h y s i c a l l y r e a l i z a b l e .I n t h e f o l l o w i n g w e r e g a , r d T ( d ) j u s t a s t h e u p p e r t r i a n g u l a , r m a t r i xo t )t i ca l ly s y n t h c s i z a b h ; b y p o r t i o n s o f a h o m o g e n e o u s m e d i u m o r b y m o r e c o m -p le x a r r a n g e m e n t s o f l en s es s e p a r a t e d b y u n i f o r m m e d i u m s e c ti o n s a c c o rd i n gt o w h e t h e r t h e p a r a m e t e r d t a k e s o n a p o s i t i v e o r n e g a t i v e v a l u e .

I t is e a s il y p ro v e d t h a t e v e r y s y m p l e c t i c m a t r i x w i t h n o n v a n i s h i n g u p p e r -l ef t e n t r y c a n b e r e a li z e d a s t h e r a y m a t r i x o f a n o p t i c a l s y s t e m , c o m p o s e d b yt h e s e q u e n c e o f a f re e p r o p a g a t i o n , a p u r e m a g n i f i e r a n d a t h i n l en s, n a m e l y

( A c # 0 D ) - - L ( f ) D ( . M ) T ( d ) ( 3 . 4 . 3 1 )


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The Group of the 1D Fi rs t -Orde r Opt ical Systems 131

w i t h t h e p a r a m e t e r s d , 3 /[ a n d f b e i n g r e l a te d t o t h e e n t r i e s o f t h e m a t r i x a sd _B M - A f = A ( 3 .4 . 3 2)A ' ' C"

Need les s t o s ay , t he op t i ca l a r r angem en t on t he r i gh t o f (3 .4 .31 ) i s bas i ca l l yc o m p o s e d b y t h i n l e n s e s a n d f r e e p r o p a g a t i o n s e c t i o n s .

I n t e r e s t i n g l y , s i n c e D r e p r e s e n t s a n o p t i c a l s y s t e m w i t h v a n i s h i n g e f f e c t i v el e n g t h a n d f o c a l p o w e r , t h e p r e s e n c e of t h e m a t r i c e s T a n d L i n t h e s y n t h e s i s(3 .4 .31 ) d i r ec t l y r e l a t e s t o t he en t r i e s B a nd C , r e spec t i ve ly . T hus , i f B = 0o r C = 0, co r resp ond ing ly T o r L d i s a pp ea r s f rom (3 .4 .31 ) .

I f A = 0 , and henc e D ~ 0 , t he r e ade r m a ,y eas i l y ve r i fy t ha t t he r ev e r seo r d e r i n g T D L m a y b e e f f e c t i v e l y a p p l i e d :

, 1. (3.4. 33)How ever , fo l lowing t he sugg es t i on p u t fo rw ard i n w 3 .4 .2 we can ve r i fy t hefeas ib i l i t y o f t he op t i ca l equ iva l ence

(~ --D -1 ) -- F (- ~ )T(~), (3.4.34)wh ich unl ike (3 .4 .33) coInp r i ses t il e case D = 0 as wel l. Oi l ac co un t of t i leo p t i c a l r e a l i z a t i o n o f F ( f ) b y a 2 f s y s t e m , it is e v i d e n t t h a t t h e p r o d u c t i n( 3 . 4 . 3 4 ) d e s c r i b e s t i l e p r o p a g a t i o n f r o m a n i n p u t p l a n e , p l a c e d a t d i s t a n c e df r o m t h e p r i m a r y f o c al p la n e o f t h e l en s , to t h e s e c o n d a r y f o c al p la n e . T h ep r e s e n c e of T ( d ) r e v e a ls in f a c t t h a t s y s t e m ( 3 .4 . 34 ) p e r f o r m s t h e i m p e r f e c tF our i e r t r a ns fo rm o f t i le i npu t s i gna l , and so m a . rks t i le d i ff e rence wi t h r e spec tt o a pe r fec t F ou r i e r t r ans fo r m ing sys t em , co r r esp ond ing t o d = 0 , i .e . D = 0 .

E q u a t i o n ( 3 .4 .3 1 ) a n d a ll o t h e r s , o b t a i n a b l e c h a n g i n g t h e o r d e r o f t h e o p -t i c a l c o m p o n e n t s i n t h e p r o d u c t c o n f i g u r a t i o n o n t h e r i g h t , r e p r e s e n t t h e c o n -c l us iv e s te p o f t h e i n v e s t i g a t i o n p r e s e n t e d i n th i s s e c t io n . W e m a y a c c o r d i n g l ys t a t e t h a t e v e r y 2 x 2 s y m p l e c t i c m a t r i x c a n a r i se a s a n o p t i c a l m a t r i x ; i n o t h e rw o r d s , e v e r y S p ( 2 , R ) s y s t e m c a n b e r e a li z e d a ,s a n o p t i c a l s y s t e m c o m p o s e db y a s u i t a b l y a r r a n g e d a n d c h a r a c t e r i z e d s e t of t h i n l e n s e s s e p a r a t e d b y ( p o s -i t i v e ) f r e e p r o p a g a t i o n s e c t i o n s . T h i s e s t a b l i s h e s t h e h o m o m o r p h i s m b e t w e e nt h e s y m p l e c t i c g r o u p Sp(2 , R ) a n d t h e g r o u p o f t h e 1 D lin ea ,r o p t i c a l s y s t e m s .3.5 W ei-N orm an representation of optical elem ents: L S Tsynthes isI n t h e p r e v i o u s s e c t i o n i t h a s b e e n s h o w n t h a t e v e r y r e a l s y m p l e c t i c m a t r i xw i t h n o n v a n i s h i n g d i a g o n a l i s e x p r e s s i b l e i n a p r o d u c t f o r m i n v o l v i n g t h e m a -t r ic e s T , L a n d D . I n g e n e ra l , t h e m a t r i x T is t o b e c o n s i d e r e d f r o m a p u r e l y


22/56

1 32 L i n e a r R a y an d W a v e O p t ic s n P h a s e 5 p a c e

m a t h e m a t i c a l v i e w p o i n t a s a r e a l u n i m o d u l a r u p p e r - t r i a n g u l a r m a t r i x , a d m i t -t i n g t h e e x p o n e n t i a l r e p r c s e nt a , t i o n i n t e r m s o f t h e a l g e b ra , m a t r i x K , a n dh e n( :e d e s c r i b i n g f r o m t h e o p t i c a l v i e w p o i n t a s i n gl e f r e e - p r o p a g a t i o n s e c t i o no r a s u i t a b l e a , r r a n g e m e n t o f f r ee - in e (t i lm ~ s e c t i o n s a n d t h in l e n s e s.H e r e w e w il l s ee t h a t , r e s o r t i n g t o t h e we ll k n o w n W e i - N o r m a n t h e o r e mo f L i e g r o u p t h e o r y [ 4 . 1 ] , s y ~ t ) l e ( : t i ( : n m t r i c e s h a v in g a p o s i t i v e A o r D e n t r ya,d~f it a,n or de re d t ) ro( tu( : t forn~ rel) res e~t at i ( )~ in te n~ s of T , L a ,n(t S- l ike~lm,tI'i('(;s; Im ,~ll(;ly i~l I,('r~ls ()f a l~t)t)(;r trimit L~ la,r ~lm,trix w it li ~m it (t ia,g( ma lentl ' i( ;s , a, ~lninl()( l~fla,r t)()sitiv ( ' -(h 'f initc ( lia,g()nal m a tr ix , ml(l a h)we r tr i an gu la rIIm,ti'ix w i t h ul ii t (lia,g()lial (;~ltI'ies. In fa,('t, tll('~ ()t)ti(:a,1 a , r I ' a , l l g e l l l e l l t

( ~ B ) _ L ( 7 ) ) S ( . ,, ,. ) T ( d ) ( 3 . 5. 1 )(h;~xm,~(ls for tll(; I)ara ,~( ;t( ;rs P , ' t~, m~(1 d 1)(; rela, te(l t() tl~(; ~m,tl' ix eh ;l ~e nt s t)y

7 ' - _ c . , ' , , , - A , , t - B_.4 ( ;3.5 .2).j~st a,s (3 .4 .3 2) witl~ M rel)la(:(;(l t)y ' t t t . A s 't~t > (), i(h;~d,ity ( 3. 5 .1 ) is a,(l~fis-sit)h; ()nly f()r llmtri( ' .es w itll t)()sitive ( liag oila l ( ;I lI , y A, w llih; D is il l ge ne ra la,lh~w('~(l t() rmlg~'~ fr(~lll - -C~ t,(~ OO. Tl l(; rea,(l (;r i im y w ;r if y t lm t th (' fa,('.t()riza,-t,i(~ils w ll er e t, l('~ limt, rix L is (~ll tll(; l(;ft ()f tll('~ xl m tri x T (i. e., t, lc I~ST, SL Ta,n(1 LT S (:(mfigllra,ti(ms) ar e slli tal) le t ,() sy nt ll es iz e sYnll)h;( :ti( : xlm,tri(:(;s hav in gA > ( ) a ,~ ( t - ~ < I ) < c~ . I~ (:~)ld,ra,st, t, tle fa,(:t()riza,ti~)~ls wll('xo, tl m nm ,t ri x Lis (m th(; ri gh t (~f T (wtli( ' ll lea,(l t,(~ t tlo TS L , STI~ a~l(t TI~S ( '(~Ilfigl~ra ti(~s) (:a,na,(:(:()~nd, fo r D > () a,x~(l A a,rl fitr a,ri ly ra ,~ gi ~g t,l~r(~gl~ t,l~(; re al lin e.

The eq~fiva,h;~t:e (;xt)I't,,ss(:(t i~l (3.5.1) a,s w(;ll a,s in all tl~e ()tl~er fa,(:t()I'e(tform s, obta , ina, |)h ; chm ~gi~ g t l~e ( ) r tter r th e ( :om t)on ent ~m,t , i ( :es in the t )r () (l -~ lc t o n t h e r i g h t , s t a ,t e s t h e I ) os s il )i l it y o f r e p r e s e n t in g e v e r y S p ( 2 , I~) e l e m e n tw it h t )os i t i ve u t) t)er -h ;f l , o r lo we r- r i gh t en tr y a ,s the or (lere(1 1) ro( tuct of e le-m e n t s ( t r a w n f r o m th e o n( ~t )a ,r a, ~n e te r s u b g r o u t ) s g e n e r a t e d | )y t h e c o r r e s p o n d -ing a, lgel )r a, ma , t r i ( ' e s K , K + , an d K 3. Th us , on ac( ' ou~ t o f the e x t )on en t ia lr e t ) r esen ta t io I~ s o f the ma , t ri ( :es invo lv ed in the t ) ro (h lc t , we (:a,n wr i te

w i t h 7 , s a n d d d e f i n e d t )y ( 3 . 5 . 2 ) ; n a m e l y , 7 c , s - - 2 1 n A , d - ~ .A c c o r d i n g l y , e v e r y 2 x 2 s y m p l e c t i c m a t r i x w i t h p o s i t i v e u p p e r - l e f t o r lo w e r -

r i g h t e n t r y c a n b e i n t e r p r e t e d a s a n o p t i c a l m a t r i x a r i s i n g f r o m t h e a p p r o p r i a t ec o n j u c t i o n s o f ( p o s i t i v e o r n e g a t i v e ) f r e e - p r o p a g a t i o n s , t h i n l e n s e s a n d r e ci p r o-(: al s ( :a l in g s y s t e m s . W i th in t i le (: o n t e x t o f 1 D l i n e a r o p t i c s t i l e d e c o I n p o s i t i o n( 3 .5 .3 ) a c q u ir e s a n u t m o s t r e l e v a n c e w h e n a p p r o a c h i n g p r o p a g a t i o n p r o b l e m s .I n t h a t c a s e t h e i n it i a l v a lu e s o f t h e r a y m a t r i x e n t r i e s a r e s e t i n o r d e r t or e p r o d u c e t h e i d e n t i t y m a t r i x . T h e c o n t i n u i t y o f t h e p r o c e s s a s s u r e s t h a t i n a


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H iir !

0 A D - B ( ' = I iQ . i0 ""

. m>

df r e e - p r o p a g a t i o n s e c t i o n

( p o s i t iv e o r n e g a t i v e )

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mpos i t i ve magn i f ie r

n oIi

i

fth in lens

O A D - B C = IJE lower - t r iang u la r un i t -d iagon a l mat r ix u p p e r - t r i a n g u l a rEl .:~ uni t - d iag ona l matr ix ( ,, ;p(2,R)hyperbol ic subgroup) u ni t -d iago nal ma tr ixL ( 5 ' p ( 2 , R ) p a r a b o l i c s u b g r o u p ) ( 5 ; p ( 2, R ) p a r a b o l i c s u b g r o u p )O~

- y ,

9 ( - . .. .. .. . / -~ (, q q q$ A ~ - . ~ ' - - I ]t~t " q -she ar overa l l sca l ing p - s h e a r

F I G U R E 3 .1 1. W e i - N o r m a n s y n t h e s i s o f a n o p t ic a l m a t r ix .

n e i g h b o u r h o o d o f t h e i n i ti a l p o s it i o n , i .e . in a n e i g h b o u r h o o d o f t h e i d e n t i t y ,t h e e n t r ie s A a n d D r e m a i n g r e a t e r t h a n z e r o . T h e r e p r e s e n t a t i o n ( 3 .5 . 3 ) ist h e r e f o r e " l o c a ll y " a l lo w e d in t h a t n e i g h b o u r h o o d , w h e r e w e c a n w r i t e

M ( z , z i ) - c p ( z )K + c s ( z) K a c d ( z ) K - , M ( z i , z i ) - I , (3 .5 .4)t h u s d e c o m p o s i n g t h e o v e ra ll p r o b l e m o f d e t e r m i n i n g t h e t r a n s f e r m a t r i xM ( z , z i ) i n to t h e " p a r t i a l " p r o b l e m s o f d e t e r m i n i n g t h e p a r a m e t e r s 7 ) (z ) , s ( z )a n d d ( z ) . N o t a b l y t h e f a c t o r s i n ( 3 . 5 . 4 ) d i r e c t l y r e l a t e t o t h e " d y n a m i c s " a s s o -, l p 2c i a t e d r e s p e c t i v e l y w i t h t h e q u a d r a t i c m o n o m i a l s l q 2 l q p a n d ~ , w h i c h c a ne a s il y b e i n v e s t i g a t e d a n d p o s s i b l y e x p r e s s e d t h r o u g h d e f i ni te f u n c t i o n a l f o r m so f t h e r e l e v a n t c h a r a c t e r i s t i c p a r a m e t e r s P ( z ) , s ( z ) a n d d ( z ) (see w167.5 .2 and2 . 4 ) . A s a n e x a m p l e , w h e n c o n s i d e r i n g t h e p r o p a g a t i o n i n a p a r a b o l i c i n d e xpro f i l e m ed ium , we wi l l s ee t ha t t he d i f f e ren t i a l equa t i ons fo r t he m a t r i x e l e -


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1 34 L i n e a r R a y an d W a v e O p t i c s n P h a s e S p a c e

m e n t s A , B , C , D t u r n i n to d if fe r e n t ia l e q u a t i o n s f or t h e p a r a m e t e r s 7 , s , do f t h e d e c o m p o s i t i o n ( 3 . 5 . 4 ) , w h i c h , o f fe r i n g a n a l t e r n a t i v e p a r a m e t r i z a t i o n o ft h e p r o b l e m , m a y p r o v i d e a d e e p e r f e e l i n g fo r t h e b e h a v i o u r o f t h e s y s t e m .

F i g u r e 3 .1 1 su m n m , r i z e s t h e v a r i( m s i n t e r p r e t a t i o n s o f t h e L S T d e ( : o m p os i -t ion (3 .5 .1 ) w i th in the g rm l t ) theorc ti ( :a ,1 co n t ex t as wel l as wi th in the op tica ,1c o n te x t i n t e r m s o f b o th o p t i c a l s y s t e m a , r r a n g e m e ~ l t s a,Ii(t t )h a se -t )l a, Im t r a n s -f()rnm,ti()n sequen(:es.

M()re over , as in t i le l i lm ar a t ) I ) r ( )x inmtio n t im ray tA,r a ,n s fe r m a t r i x M i sth e Iim,i,rix r( '4)n'st' lltativ('~ ()f t, tl('~ ra y tr a n sf e r ()t)cra,t()r 9)l, rel at, i(m (3. 5.4 ) a.lsosta , tes t] l at t ] le r ay tI'a, llS[(W ()i)( ; ra,t t )r () f a,] ly ()]I t t i rst- t)I ' (h 'x 1D ( ) t) t i ( :a l sys tenlis fa,(:tt)rizal)h'~ in a ll('~iglfl)t)llrh()()(t ()f th e (;nt ra,ll( 'c t)la,nc illl,() tlw, t)r()(tu(:t oft i le Lie t r an sf or nm ti (m s g('a l('xa.te(1 | )y t lm l l l ( )n(n l l ia ls ~ ml( l a ,s2

~ ( z , z~ ) - ( ,P ( ~ )K + r 9 ~ l ( Z ~ ,Z i) - I , (3 .5 .5 )w he re K , K ~ ml(l K :~ ar e tile Lic () l)cra, t()rs ass()( :ia, te( t wi tll t lm (tll()tc( l mon o-,,r ials (see .~ 1.5.2 an(l 2.3) [4] .

Rt'~stri( :tly to tl~(: l i~mar al) t)r()xi~m, tiol~ t he ~m .tr ix an(l ot)eI'a.t()r i( le ntit ies,( : ] . 5 . 4 ) /- I,1 1 (| ( : ] . 5 . 5 ) , tn '(wi(h: t l m a , ~ s w e r t,(, l , l~( ; I,,(,],1( ' ,~ (,f z-,w(h,ri,~g, w h i ( . hlma,v()i( lal) ly ari se s wl~(;~ al)l)I' (m(:l~i~g tim i~t, e gr a ti ,m ()f ttm ~ilt ,()~ 's e( l~ at ion sin tl~e (:a,se of a, z-(h; t)el~ (li~ g IIa ~i lt, ()~ im ~, (t~(; t,() 1,1~(' i~ gel~(;ra.1 ~()~(:()~mn~-tatiw'~ nat,~tre ()f ~m ,tri( 'es a,n(t ()I)erat()rs . In w 1.5 .3 w(; n~(;~iti()I~('(t tha,t th efa(~l,()rizati()~i ()f 9/1 ca ~ 1)(, (~()l~t,i~nt(;(t t,() i~(:l~ (h, h i g h e r or (h 'r t(,rn~s, a(:(',()unt-i~ g fl)r t, ic a.1) crra, ti()~ s. Ea~ 'l~ ,~f 1,1~(' a,(htit, i,)~m,1 fi~ :t( )rs ta k( ;s tl~(; fi )n ~ of a, L ietra,nsfi)n~m ,tio~, a,ltl~()~lgl~ it, ( 'a ,~ ()t t)e giwu~ a 2 x 2 ~m ,tr ix ret) rcs e~ ta,t ion .

3 . 6 R otations and sq ueezes in the pha se planeI n w 1 .5 . 2, w h e n d i s c u s s i n g t h e t ) h a s e- s t) a c e d y n a m i c s g e n e r a t e d b y th e L i et I ' a ns f o rn mt ion s ass ( ) (' i a te (t w i th ( tynam i( :a l f lm( : t ions qua(h 'a t i ( : in t i l e t )o s i t iona n ( t m ( ) m e n tu m r a y v a r i a | ) l e s , w e h a v e ( : on s i( te r e( t t h e t ) o ly n o m ia l s 1 ( p2 + q2 )1 (p2 q2) . W e hav(~ sh()we( t th at th ey g ive r ise to ( tef in i te ly d i f f e r e n tnd f f -m o t i o n s i n t h e p h a s e p l a n e , t h e c o r r e s p o n d i n g t r a j e c t o r i e s l y i n g r e s p e c t i v e l ya l o n g a c i rc le a n d a b r a n c h o f h y t ) e r b o la . H e r e w e r e c o n s i d e r i n d e t a i l t h ea b o v e b y n o m i a l s i n o r d e r t o i d e n t i f y t h e o p t i c a l s y s t e m s t h e y m a y d e s c r i b e .

3 . 6 . 1 A t t r a c t i v e o s c i l l a t o r , phase- p l ane ro t a t i on and f r a c t i ona lF o u r i e r t r a n s f o r mW e d e n o t e a s H a .o. t h e q u a d r a ti c h o m o g e n e o u s p o l y n o m i a l

H a o 1 2 ~_~12 2. . + , ( 3 . 6 . 1 )" J s


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T h e Group o f the 1D First-Order Optical Systems 135w h i c h t y p i c a l l y m o d e l s t h e a t t r a c t i v e o s c i l la t o r -l i k e d y n a m i c s . T h e s p a c e c o-o r d i n a t e q h a s b e e n e x p l i c i t l y sc a l e d b y a,n a p p r o p r i a t e l e n g t h f a c t o r f s ' w h o s eo p t i c a l i n t e r p r e t a t i o n is c l a ri fi e d b e lo w .

AT h e c o r r e s p o n d i n g L i e o p e r a t o r k a.o . is~- 0 q 0L~.o. - p Oq f 2 O p ' ( 3 . 6 . 2 )

t . . ~ . .w h i c h w e r e w r i t e i n t e r m s o f t h e o p e r a t o r s K _ a n d K + , d e f in e d b y ( 2 .3 . 5 ), i nt h e c o n v e n i e n t f o r m

1 " _ _ 1 _ 1 K + ) ( 3 . 6 . 3 )L ~.o . - f s ( L K + ~ 9A c c o r d i n g l y t i l e m a t r i x r e p r e s e n t a t i v e H a .o . l i n e a r l y re l a t e s t o t h e s p (2 , ]R )b a si s m a t r i c e s K _ a n d K + a s.ao ( ~. - 1 / f s 2 1 _ _ ~ s_ s l ( f s ) , ( 3 . 6 . 4 )w h e r e w e h a ve d e n o t e d b y K ~ ( f s ) t h e a l g e b r a e l e m e n t

1 ( 0 f s )K ( f s ) - l ( f s K + ~ K + ) - 1 . ( 3. 6. 5 )1 - - - 1 / f ~ 0 "T i l e c o n v e n i e n c e o f s u c h a, s e e m i n g l y o d d d e f i n i t i o n w i l l become a p p a r e n t l a t e r .

I n o r d e r t o id e n t i f y t h e o p t i c a l s y s t e m d e s c r i b e d b y t h e h a r m o n i c o s c i l l a to r -l ik e H a m i l t o n i a n ( 3 .6 . 1 ), w e e x p o n e n t i a t e t h e t r a c e l e s s m a t r i x ( 3 .6 . 4) t h r o u g ht h e a x i a l d i s t a n c e A z - - z - z i. A s a r e s u l t , w e o b t a i n t h e " t r a n s f e r m a t r i x "f or t h e r a y p r o p a g a t i o n f r o m z i t o z i n t h e f o r m (a )

F ~ ( f s ) c AzH a _ c 2 ~ K l ( f s ) ( c ~ 1 6 2 . f s s i n r. . . . . ~ - - 1 (3 .6 .6 )- ~ s in r c o s r 'ZXz o f t h e c i r c u l a r f u n c t i o n s a sa v i n g i n t e r p r e t e d t h e d i m e n s i o n l e s s a r g u m e n t

t i le a n gl e 6 , c o m m o n l y m e a s u r e d i n u n i t y o f ~ t h r o u g h t h e t ) a r a I n e t e r (~:r ~ Z - - Z i 7 rf s - - c ~ ( z ) ~ . ( 3 . 6 . 7 )

N o t e t h a t , a s f s h a s t h e d i m e n s i o n o f l e n g t h , t h e o f f - d i a g o n a l e n tr i e s i n (3 .6 . 6 )h a v e t h e d i m e n s i o n s o f l e n g t h a n d ( l e n g t h ) - 1 r e s p e c t i v e l y , a s i t s h o u l d b e f o ra n o pt i ca l m a t r i x i n t h e r a y - c o o rd i n a t e a n d m o m e n t u m r e p r e se n t a t io n .

a As in the case of the m atr ices K _, K + and K a (see w 2 .4), we may com pute theexponential function of the m atrix K 1(fs) (as we ll of K 2(fs ) below) by simply applyingthe expo nential power series, since powers of g 1(fs) (and g 2 (fs)) are easily com puted. T heinterested reader m ay consult [14] in ch. 2 for a general account of the me thods of com putingexponential functions of matrices.


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136 Linear Ray and Wave Op tics in Phase 5pace

- - - " - ~ P ( z )

F I G U R E 3 .1 2 . I ) h as c - t) l a ne ( ly ~mmic g e n e r a t e ( l b y t h e a t t r a c t i v e o s c i l la to r H mn i l t o n i a n(3 .6 .1 ) . Th e ray r epr ese~ ta t iv e po i~ t s l ips a lo~ g th e e l l ipse ( le te r~ni~e(1 b y t im i~ f i tia l da taand the frost i ly parm~mter fs .

T h e tra~ sfl)r~ m.ti(n ~ a('t(;(l 1)y tim ~xm.trix F " ( f s ) ()~ th(; t)lm.s(,'-l)la,n(; wtr i-a | )h ;s ( q , p ) fr () ~ l, m i~itia .l (la.ta (q,s,Pi) is (l(;s(:ril)e(l 1)y

q ( z ) ..... qi (:()s q5 + Pi. fs sil~ (/),1p ( z ) - - ~ q i s i ~ 4 ) + p i ( :( )s 05,

wh icl l is ()f (:(Jllrs(; ill a(:(:(n'r wi ttl rc sll lt (1 .5 .23 ).T h e (:y(:li(: lm.t,llre (ff tim l)llas(; l)la,n(; lll()ti(m ( 3.6 .8) is al)l)ar (;llt. As tile ra y

I)r()t)agal,(;s (,llr()llgll (,ll(; "()l)(,i(:al" sys(,(;Ill (3 .6 .6 ), (,ll(; r(;I)r(;s(;~d,ativ(; I)()i~t Pin t im q-p t)la n(; l~()v(;s al()l~g (,l~e (;llit)s(;

q2 2q(fz2)2, + P ( z ) 2 - ~' e - - ~ + P i' (3.6 .9)T h e co nsta ,~t e is (t(;t(;rn~i~m(t t)y th e t)a ran ~e tcr fs a,n(l th e ini t ial Wd~leS; rspec ifies fi)r ea,(:t l va,hu', ()f t l lc a xia l t) ar am et er z t l lc a,xlgle of t lm r ot at io no f t h e ( : o rr c s t)o n (l i ~ g r e p r e s e n t a t i v e t )o in t P ( q ( z ) , p ( z ) ) with res t )co t to thein i t ia l t ) () in t P~(q~, p~) (F ig . 3 .12 ) . The ro ta t ion i s ( 'h ) ( :kwise o r ( :o~n te r ( : lockwiscaccor( t i I lg to the s ig l l of fs -E v i ( t c n t l y t h e n m t r i x F ' ~ ( f s ) r e l a t e s t o a r o t a t i o n b y r c t~ a l o n g a I l ing e n e ra l c l l i pt i (: a l c o n t o u r i n th e p h a s e -p l a n e . In f a c t, a s w il l b e c l a ri f ie d l a t e r ,i t c a n t ) a s i c a l l y b e i n t e r p r e t e d a s a p h a s e - p l a n e r o t a t o r . I n p a r t i c u l a r , w i t hf s = 1, F ' ~( 1 ) t a k e s t h e a p p e a r a n c e o f a p u r e r o t a t i o n m a t r i x b y t h e a n g l e r

- sin r cos rK= = 2 , t h e u n i t a r g u m e n t b e i n g o m i t t e d . A c c o r d i n g l yh e r e K 1 - - Z 1 (1 ) + K +

t h e r e p r e s e n t a t i v e p o i n t I n ov e s a l o n g a c i rc l e c e n t e r e d a r o u n d t h e o r i gi n w i t hra di us v(, - v /q] + p~.


27/56


In t e res t i ng ly , w i th r ~ ( i. e. c ~ - 1 ) we r ecover t he pe r fec t F our i e r t r an s -f o r m m a t r i x ( 3 .4 . 13 ) ,

0 f S ) - - ( ~ T r K l ( f S ) (3.6 11)F ( f s ) - - - 1 / f S 0 ' "( t h e u n i t o r d e r b e i n g o m i t t e d a s w e l l) , w h i c h t h e n is s e e n to r e l a t e t o t h e p h a s ep l a n e r o t a t i o n b y r r / 2 .

T h e m a t r i x F ~ ( f s ) is r e p o r t e d in t h e l i t e r a tu r e a s t h e f rac t iona l FouriertransfoTwz m a t r i x o f o r d e r c~ a n d f a m i l y p a r a m e t e r f s"T h e t e r m " f r a c t i o n a l " e v o ke s t h e c h a r m i n g v i si o n o f t h e " e x t e n s i o n t o c on -

t i nu um " o f a p roces s t ha t occu r s t h ro ug h f i n i t e -s i ze s t eps . I t is i n f ac t i nc o n f o r m i t y w i t h t h i s " i d e a o f f r a g m e n t a t i o n " t h a t t il e fr ac tio na ,1 F o u r i e r t r a il s -f o r m h a s b e e n i n t r o d u c e d w i t h i n d i ff e re n t c o n t e x t s o f b o t h m a t h e m a t i c s a n dp h y s ic s . C o n d o n , f o r i n s t a n c e , i n v e s t i g a t e d a b o u t a c o n t i n u o u s g r o u p o f f u n c -t io n a l t r a n s f o r m a t i o n s i s o m o r p h ic w i t h t h e g r o u p o f r o t a t i o n s o f a p la n e a b o u ta fi xe d p o i n t b y m u l t i p l e s o f a n a r b i t r a r y a n g le , t h u s g e n e r a l i z i n g th e p r o p e r t yo f t h e F o u r i e r t r a n s f o r m g r o u p , w h i c h in f a c t c o r r e s p o n d s t o r o t a t i o n s b y m u l -t ip les of t i l e r igh t a,ngle [ 5 . 1 ] . Likewi se , Nam ias e l a ,bo ra , t ed t i l e func t i ona l fo rmof F our i e r - l ike op e ra to r s , a,d ln i t t i ng , a s t he F our i e r t r a ,n s fo rm , t he Her ln i t e -G a u s s e i g e n f u n c t i o n s , b u t w i t h e i g e n v a l u e s e v e n l y s e p a r a , t e d b y a f r a c t i o n oft he im ag ina ry un i t [ 5 .2 ] .

T h e d e f i n it i o n o f t h e F o u r i e r t r a n s f o r m o f f r a c t i o n a l o r d e r w i t h i n t h e o p t i c a lc o n t e x t h a s b e e n i n s p i r e d b y t h e p e r f o r ma n c e o f a n o p t i c a l s y s t e m p e r c e i v e da s o b t a i n a b l e b y f r a g m e n t i n g t h e F o u r i e r t r a n s f o r m c o n f i g u r a t i o n i n t o a l a r g e ra n d l a r g e r n u m b e r o f s h o r t e r a n d s h o r t e r f r e e - s p a c e se c t i o n s i n te r l e a v e d w i t hwea ker and we aker l enses [5 .a ], o r equ i va l en t l y a s ca pab l e o f ex t e nd i ng t o a r -b i t r a r y a n g l e s t h e p r o p e r t y o f t h e F o u r i e r t r a , n s fo r m i n g s y s t e m t o r o t a t e b y ar i g h t a n g l e t h e W i g n e r c h a r t i n t h e i n h e r e n t W i g n e r p h a s e p l a n e [ 5 . 4 ] .

T h e r o t a t i o n b y a r i g h t a n g l e i n t h e o p t i c a l p h a s e p l a n e i s t h e u n d e r l y -i ng c a n o n i c a l t r a n s f o r m a t i o n a c t e d b y t h e F o u r i er o p e r a t i o n o n th e c o n j u g a t ev a r i a b l e s q a n d p . L e t t i n g t h e r o t a t i o n a n g l e v a r y c o n t i n u o u s l y l e a d s t o t h ef r a c t i o n a l F o u r i e r o p e r a t i o n . T h u s , t h e f i n i t e - s t e p t r a n s f o r m a t i o n s m a r k e d b ym u l t ip l e s o f t h e r i g h t a n g l e , c o r r e s p o n d i n g t o F o u r i e r t r a n s f o r m s y s t e m s p o s-s i b l y c o n c a t e n a t e d o n e t o t h e o t h e r , b e c o m e s p e c i f i c e v e n t s w i t h i n t h e s m o o t he v o l u t i o n o f t h e p r o c e s s g o v e r n e d b y t h e c o n t i n u o u s l y v a r y i n g a n g l e q S(z ) o v e rt h e 2 re r a n g e , a n d s o b y t h e c o n t i n u o u s l y v a r y i n g a x i a l p a r a m e t e r z . S u c h ap h a s e p l a n e r o t a t i o n b y a c o n t i n u o u s l y v a r y i n g a n g l e m a y b e a c h i e v e d b y s e c -t i o n i n g t h e b a s i c F o u r i e r t r a n s f o r m i n g c o n f i g u r a t i o n s i n t o a n i n c r e a s i n g n u m -b e r o f s i m i l a r c o n f i g u r a t i o n s a p p r o p r i a t e l y d e s i g n e d , a s w e w il l s ee la t e r . T h er e l a t i o n , a n d t h e r e l e v a n t v i s u a l i z a t i o n i n t h e o p t i c a l p h a s e p l a n e , b e t w e e n t h eF o u r i e r t r a n s f o r m a n d t h e W i g n e r d i s t r i b u t i o n f u n c t i o n is c l a ri fi e d in w 8 .4 .1 .


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A f t e r i t s i n t r o d u c t i o n i n t o t h e f ie ld o f o p t i c s , t h e ( : on c e pt o f f r a c t i o n a lF o u r i e r t r a n s f o r m a n d t h e r e l e v a n t f or m a, l i sm g a v e r i s e to a h u g e v a r i e t y o fa p p l i c a t i o n s, i n v e s t i g a t io n s a n d n e w f o r m u l a t i o n s i n a n i n c r e a s i n g l y e n r i c h e d( )t )t ic s s c e n a r io . I n t h e f o r t h c o m in g p a r a g r a t ) h s w e w i ll s im p ly c la ,r if y t h e c o n -( :cI) t of f ra ct io na l Fo ur ie r t ra n sf or m at a ver y bas ic l(;v( ;1 , a , (t ( lr ( ;ss ing the rea,(lcrt o t h e r e l a t e d l i t ,era , t~ re for a wi de r mid ( teet)er t, rcatn ie~l t [5] .F T " a c t i o n a l t : o w r i t ' . ' r t ' r a ' n . @~T w t m a t T " l e t ' s a s a s y ' m p l t ' c t i c s u b g T " o u pRc nm rka t) ly , f in" a f ixc(l wdlu ; ( )f th( ' fan l i ly Im,ra .n lcter f s , t l lc m a. tr i (:cs F ~ sp ana t ) ro t )c r Ab( ; l ia l l s l f l )g r r162 the symp lc( : t i c g r ( )l lt ) Sp (2 , R) wi th r cs t )c ( ' t toth e or de r (~, wtfi( 't l I lmy ra.llge ill t)r iI l( :it) lc tllr(n lgll all t i le r(;a,1 lilm. In fa ct,w it h (~ - 0 w(: (f i) ta,in the i( l cn tit y n m,tr ix; lla,llmly,

F ~ I . ( 3 . 6 . 1 2 )Also, l )y t h ( : a , ( l ( l i ( , i ( ) l l f ( ) n l n l l a s ( ) f i l l ( : ( : i r ( : l f l a . r f l u l t : i , i ( ) l l S , w ( : l m v ( :

< O S (4 ) A - r ) f s s i n ( 0 1 + 0 : ~ 1 )F ' ~ ' ( f s ) F ' ~ ( f s ) - ' . ~ i , , ( O , + ~ 2 ) ~ o . ~ ( r + % ) ( 3 . 6 13)/ - -~-= F " , + " ~ ( . f s ) - F , , ' ~ ( . f s ) F , , , ( f s )

wlfi (:h r(,fl c(:ts tll(', w('ll kll()wl~ I)r()I) ('rty tl~a,t (,l~(' ('()~I) ()siti (n~ ()f tw() r ()t at i() nsl)y th (: a,n gl(: s 4>, m ~( l ~ is (:(l~fiva,h:nt (,() th (: r()I,a,(,i()ll })y tl~ (: a,llgl(: (/5 -- ('/>, + ~ .T h e ad(l itivi( ,y t)r() l)( ' r ty (3.(i . 13) w it h r(;sI)( '~(:t ( ,() t im ()r( ler (~ lea(Is t () i( tc nti fytt~('~ i~ v( 'xs c t() F " ( f s ) a,s (,Ira ~m,t, ix (:()I'r(;s t)()~(l i~g t() (,lm ()I)l)()si(,('~ v a h m ()f (~,

( ( : ( )s < / ) - ' f s s i " 4 > ) ( 3 . 6 1 4)[ F " ( . f s ) ] - ' - F - ' ) ( f s ) - ~ s in r ( : i ) s 4 > "wlfi (:h t h e n r(:la,t(:s t() (,l~(: r()(,a(,i()n l)y tl~(: mlg l(: c/> i~i t,l~(: I'(~V(~l'S(~ l ir (: (: tion .

Also , t he n ,a , t r ix F~ ' ( / s ) , ' . a . , , t , e in t e

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