2b 1d first-order optical systems the ray-transfer matrix

7/30/2019 2B 1D First-Order Optical Systems the Ray-Transfer Matrix

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21D F i r s t - O r der O pt ica lRay - Tr ans f e r M a t r ix Systems: T h e

2 .1 In troduct ionW e a p p l y t h e s t r a t e g y d e v i s e d in C h a p t e r 1 t o c o n s t r u c t t h e o p t i c a l m a t r i c e s ,d e s c r i b i n g t h e p r o p a g a t i o n o f l ig h t ra y s t h r o u g h l i n e a r s t i g m a t i c a n d s i m p l ea s t i g m a t i c o p t i c a l s y s t e m s . T h e b a s i c to o l s of t h e s t r a t e g y a r c t h e o p t i c a lH a m i l t o n i a n H ( q.~ , % , p x , p y , z ) a n d t h e a s s o c i a t e d H a m i l t o n ' s e q u a t i o n s f o rt h e e v o l u t i o n o f t h e l ig h t r a y a s p r o p a g a t i n g t h r o u g h t h e s y s t e m a ,lo ng t h e o p t i -ca,1 z ax i s . T h e s y s t e m i s d e s c r i b ed b y t h e r e fr a, c t i v e i n d ex fu n c t i o n n ( q , , q y , z ) ,a s t h e r a y is d e s c r i b e d b y t h e c a n o n i c a l l y c o n j u g a t e va, i a b l e s (q x, % ) a n d( p: ~,p y ), s p e c i f y i n g t h e c a r t e s i a n c o o r d i n a t e s a n d t h e d i r e c t i o n c o s i n e s o f t h er a y a t i t s c r o s s i n g p o i n t w i t h t h e r e f e r e n c e s c r e e n t r a n s v e r s a l t o t h e o p t i c a la x i s . M o r e p r e c i s e l y , t h e o p t i c a l m o m e n t a , ( p x , P y ) a r e t h e d i r e c t i o n c o s i n e s o ft h e r a y s c a l e d b y t h e l o c a l va .lu e o f t h e r e f r a c t i v e i n d e x . T h e C a r t e s i a n s p a c es p a n n e d b y t h e r a y p o s i ti o n a n d m o m e n t u m c o o r d i n a t e s i s t h e g e o m e t r i c a lo p t i c a l p h a s e s p a c e . T h e r a y i s r e p r e s e n t e d b y p o i n t s i n t h e 4 D o p t i c a l p h a s es p ace a ,n d a c c o r d i n g l y t h e r a y p a t h i n rea,1 s p a c e c o r r e s p o n d s t o a t r a j e c t o r yo f t h e r a y r e p r e s e n t a t i v e p o i n t in p h a s e s p a c e [1].

T h e o p t i c a l H a m i l t o n i a n h a s th e p e c u l i a r f o r m o f t h e s q u a r e - r o o t f l m c t i o nH ( q , p , z ) = - v / n 2 ( q , z ) - p 2 , h a v i n g s o m e r e s e m b l a n c e t o t h a t o f a r e l a t i v is t i cp a r t ic l e . T h e a s s o c i a t e d L i e o p e r a t o r k , ( q , p , z ) , o b t a i n e d f r o m H b y P o i s s o nb r a c k e t : L /~ - { -, H } , r u l e s t h e e v o l u t i o n o f t h e r a y v a r i a b l e s q ( z ) a n d p ( z ) , asf u n c ti o n s o f t h e p r o p a g a t i o n p a r a m e t e r z , a c c o r d i n g to H a m i l t o n ' s e q u a t i o n s

d ( ; ( ( z ) ) _ - ~ u ( q , p , z ) ( q ( z ) ' ~ ( 2 . 1 1 )- z i d e n o t i n g t h e a x i a l p o s i t i o n w h e r e t h e r a y c o o r d i n a t e s a r e k n o w n ; q a n d ph e r e a r e h a n d y a b b r e v i a t i o n s o f ( q , , q y ) a n d ( p x , p y ) , r e s p e c t i v e l y .


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6 0 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

I n p a r t i c u l a r , i f p r o p a g a t i o n i n t h e c l o s e p r o x i m i t y o f t h e a ,x is is c o n c e r n e d ,t h e o p t i c a l H a m i l t o n | a n s i m p l if i e s in t o t h e f o r m y-~ - n ( q z ) r e s e m b l i n g t h e2 n ' 'H a m i l t o n i a n o f a n o n r e l a ti v i s ti c p a r t i c l e m o v i n g u n d e r t h e p o t e n t i a l - n ( q , z )[2 ]. T h e o p t | c a, 1 m o m e n t a , p :r a n d p y c o m e t o b e d i r e c t l y l i n k e d t o t h e i n c l i n a t i o na n g l e s o f t h e r a y i n x a n d y d i r e c t i o n s . I n a d d i t i o n , a s t h e r e f r a c t i v e i n d e x i s asm()ot,}~ f lm( : t io n of st )a ,( :e , i t m a y be des ( : r i l ) e ( t by a, qu a( I ra | i ( : f l lnc t , i (m of ther a y ('()()r(lina.t(;s (q :r ,% ) ()v(;r t,l~(; t)(;rt, i~(;~lt, sIlm,ll i '(;gi()ii a.(:ros s t,l~(; ax is . T h eH am il t ( )n ia n f im(:t , i( )n l )c( : ()mcs th e n ( t~m,(t ra . |, i( : in | ) () t ,h q a , t~( l p ( : ( )or( t inat , cs ,t ,h~ls yi( 'l( l i~lg t , w liIwa ,riz( '( l f()r~11 ()f Ha,~ filI,o~ e(l~m ti()~s (2 .1 .1 ) as

w h e r e 1,11e ray-t) a,I-a ,lllel,er i~l(let)(;Ii(h;ld, llm,l,ri x H ( z ) is ()t)l,a,iIm(1 fl'()111 t, ll( ' Ita ,nf il-t,()nia.n H ( q , p , z ) I)y I)()iss()ll I)rat:k( 'l, i l lg wil, l l t im


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1D Fi rs t -Order Op t ica l Sys tems: The R ay- Trans fe r Mat r ix 61

Classical M e c h a n i c s

S i n g l e - p a r t i c l e d e s c r i p t i o nconveyed by the canonica|tyconjusate variabtes q(c) and p(c)through Ham i[ton 's equations

G e o m e t r i c a l O p ti c s

E n s e m b l e s t a t i s t i c a l d e s c r i p t i o nconveyed by the phase-spacedensity distribution p(q,p;z) through

Liouvitte's equation

F I G U R E 2 .1 . E x e m p l i f i c at i v e sc h e m e o f th e s i n g l e - p a r ti c le a n d e n s e m b l e d e s c r ip t i o n s o fa c l a s s i c a l d y n a m i c a l s y s t e m , e . g . a p o i n t p a r t i c l e , w h i c h a r e m i r r o r e d i n t o t h e g e o m e t r i -c a l- o p t ic s a p p r o a c h t o l ig h t p r o p a g a t i o n i n t e r m s o f t h e s i n g l e - r ay v a r i a b le s q ( z ) , p ( z ) a n dt h e r a y - b u n d l e d e n s i t y p ( q , p ; z ) i n p h a s e s p a c e .

r a y p a r a m e t e r s , f o ll o w in g t il e r a y p a t h i n t h a t p l a n e , w h e r e t h e p r o p a g a t i n gr a y k e e p s o n ly i ng . T h e p r o b l e m b e c o m e s o n e - d i m e n s i o n a l , a s t h e m o t i o n o f t h er a y r e p r e s e n t a t i v e p o i n t t a k e s p l a c e in a 2 D s u b s p a c e o f t h e 4 D o p t i c a l p h a s es p a c e , a n d a c c o r d i n g l y H a n d M c o m e t o b e 2 x 2 m a t r i c e s . L i k e w i s e , t i le p r o p -a g a t i o n t h r o u g h s i m p l e a s t i g m a t i c s y s t e m s , w h i c h a r c s y m m e t r i c a b o u t t w oo r t h o g o n a l p l a n e s i n t e r s e c t i n g t h e o p t i c a l a x is , c a n b e a c c o u n t e d f or b y 2 x 2m a t r i c e s , b e i n g t h e m o t i o n i n t h e o p t i c a l p h a s e s p a c e d e c o m p o s a b l e i n t o t w oi n d e p e n d e n t m o t i o n s i n t i l e t w o 2 D s u b s p a c e s . W e w i l l l i I n i t o u r d i s c u s s i o nt o f i r s t - o r d e r s t i g m a t i c a n d s i m p l e a s t i g m a t i c o p t i c a l s y s t e m s , w h i c h , b e i n gd e s c r i b a b l e w i t h i n t h e 1 D m o d e l , w i ll s h o r t l y b e r e f e r re d t o a s 1 D s y s t e m s .

T h e c o n c e p t u a l p a t h , d e v e l o p e d i n C h a p t e r 1 a n d b r i ef ly s y n t h e s i z e d h e r e ,p e r t a i n s t o ti le s in g l e - r a y p i c t u r e o f l ig h t p r o p a g a t i o n . E v i d e n t l y w i t h i n t il er a y - p i c t u r e o f g e o m e t r i c a l o p t ic s a l ig h t d i s t r i b u t i o n m a y b e s e e n a s a n e n s e m -b l e of u n c o r r e l a t e d r a y s. A s n o t e d i n w 1 .2 , a c c o u n t o f t h e p r o p a g a t i o n o f t h el ig h t d i s t r i b u t i o n b y i n t e g r a t i o n o f H a m i l t o n ' s e q u a t i o n s f o r e a c h r a y in t h e e n -s e m b l c is, o f c o u r se , i m p r a c t i c a b l e . A n a l t e r n a t i v e s t r a t e g y m u s t b e e n v is a g e d .W e m a y p r e s u m a b l y g u e s s t h a t , a s t h e s i n g l e -r a y d e s c r i p t i o n o f l ig h t p r o p a -g a t i o n p a r a l l el s t h e H a m i l t o n i a n f o r m u l a t i o n o f t h e s i ng l e p a r t i c l e d y n a m i c s ,t h u s a r a y - e n s e m b l e d e s c r i p t i o n m i g h t b e e l a b o r a t e d p a r a l l e l i n g t h e s t a t i s t i c a lf o r m u l a t i o n o f t h e d y n a m i c s o f a n e n s e m b l e o f p a r t i c l e s [3, 4]. I n o t h e r w o r d s ,r e f e r r i n g t o F i g . 2 . 1 , w e m a y s a y t h a t , a s w e h a v e d e l i n e a t e d t h e c o r r e s p o n -d e n c e b e t w e e n c l a s s i c a l m e c h a n i c s a n d g e o m e t r i c a l o p t i c s t h r o u g h t h e u p p e rp a t h , s o w e d e l i n e a t e h e r e s u ch a c o r r e s p o n d e n c e t h r o u g h t h e l o w e r p a t h . I n -d e e d , L i o u v i l l e ' s e q u a t i o n f o r t h e p h a s e - s p a c e d e n s i t y d i s t r i b u t i o n p ( q , p ; z ) is


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6 2 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

d e r i v e d i n S e c t . 2. 2, w h e r e a r e a l so e x a m i n e d s o m e b a s ic p r o p e r t i e s o f t h ehL ie t r a n s f o r m a t i o n s . T h e L i e o p e r a t o r I _, a n d a c t o r ( t in g l y t h e ra y m a t r i x M

c o n f i r m a s t h e b a s i c to o l s t o d e s c r i b e t h e l ig h t b e a m t ) r o p a g a t i o n w i t h i n t h er a y -m ( ) ( te l o f g c o m e t r i ( :a l o p t i c s. S o m e c o n s i d e r a t i o n s a i m e d a t r e l a t i n g t h eg e o m e t r i c a l a n d t ) h y s ic a l i n t e r p r e t a t i o n s o f p ( :lo se t h e s e c t i o n . T h (u ~, i n s t) i re dt )y th ( : f lm( : t i (ma l f ( )n n ( )f th ( : f i r st - ( )r ( t (: r ( )t )t i( :s Ha m i l t ( )n ian , be in g ( t~a ( l ra t i ci~l th(: ray varia])h:s , w(: s tar(, t() (:()~lstr~l(:t the ~m,(,rix ()t)t i(:s , gen(:ra,t(:(t | )y the

l ~ 1 l(pm,(h'a,ti(' n~()n()nfia,ls ~ t? }q~ a,n( I ~ q p , (,I~(: (:()rr( :sl)(m (ling ()l)ti('a,l s y s te m s l)e-in g th(u~ i(te ntifi( :(t a.s a, s(:(:ti()n ()f l~()~()g(:n(:()~s n~(:(ti~m~, a r efr a.(: ti~g su rf ac ean(1 a,1~ i(teal ~a ,gj~iii(:r , r(:s t)e(:t iv(:ly (S e(:t . 2 .4) . Fiscally, S e(:t . 2.5 i lh ~s tra te st h e t)l~as(:-st)a,(:e t)i(:(,~r(: ()f tl~(: tra,i~sf()r~m.t,i()~s (:fi'(:('t(:(t t)y tt~(: a,t)()v(: (t~( )te( tsyst,(:~s ()~ t)()(,l~ a, s il~gl( : ra y a ,~(1 s ()~(: e x(: I~I ) lary I ) ~( l l ( :s ( )f ray s .

2 .2 Ray-ensemble description of light propagationT h e ( : (> l ls i (l e ra . ti ( )l l sh ; w ' h q ) e < l i ll C l m l ) t e r I l l a , v e l e l) i r t i l e ( :< )l lC X; l) tl la ,ln ( |l ) ra (: t ic :a l ) r ( )( : e( ll l re i ) r t, l e s i l l g l e - r a y t r a < : i n g , i ll f i fl l l a , h ) g y w i t l l ( , l le a m i l -to11ia,ll (lyIm,Illit:s t)f a sillgl(: l)arl, it:le [ 1 ] . A s i l lgh : ra y is (h:s( 'r i t)er i l l rea l s t )a ce| )y t ,]i(: f()lu" (:a,~t)~it:a,lly ('.()Itjllga,t(: Z-(l(:I)(:ll(l(:1,t, (:()()r(Iix,a,t(:s ( q (z ), p ( z ) ) a , ~ ( I vi-s u a ll y r(:I)r(:s(:ll(,(:(I a,(, a,lly z ])y a, llm,(,lml~m,ti(:al I)()i~(, i~ tl~(: g( :() ~(: tri( :al- ol) ti(:a lI)ha.s(: sl)a,(:(:. TI~ (: r a y I)a,(,l~ i~ I)hy si(:a,l sl)a,(:(: (:()rr(:sl)()~(Is (,() a, (,ra,je('(,()ry o f t h ere la tiv (: rel)res(:~d,a,(,iv(: l)()i~d, in l)ha,s(: sl)a,(:e. T im I)()sit, (m a,~(I (lire(:(,i()i~ co o rd i-~m,t(:s ()f (,I~(: r a y l)r()v i(h: a,~ "i ~( liv i(l ~a ,l" (:Im,ra,(:(,(:riza,ti()~; (,Imir (:v()h~(,i()~, a n da,(:( :or( li~g ly th (: I)}m,s(.'-sl)a,(:(: t,ra , je( :t ( )ry ()f th( : ra y, is ( : ()nq)le t ( : ly m~(l ~n i( lu elydefine(t t )y I ta ,nf i l toI~ 'S eq~la . t ions in ( : ( )rres t )o~(te l~(: ( : wi th the s t )e( : i f ied inputd a t a . S ~ m h a (h : s ( 'r i l) t i o n i s n o l o n g e r v i a b l e w h (u ~ (h : a l i n g w i t h a ,n e n s e m b l e()f ra y s. W e ~(:(:(t ithu d, ify a,I)I)rt)I)ria,(,(: I)ara ,nm (,t:rs (,() I)r()vi(te a, "glt)l)a,l" c h ar -a ,c te r iza . t ion ( )f th e ray- ( :ns (: ln l )l e , m~( t th en e la, t) ( )ra t( : th ( : r (: leva ,~ t e vo lu t io neq u a t io n s to a ,( :(:( )u I~ t fo r th e t ) ropaga , t io I~ of the ray - | )u n( t l e .

E v i d e n t l y , w i t h i ~ t h e r a y t ) i c t u r e ( ) f g e o m ( : t r i c a l o t ) t ic s , a l i g h t t )(:a ,n ~ m a y b ed e s ( : r i b e d a s a, c ( )l h :( :t io n o f r a y s . I t c o v e r s a c e r t a i n a r e a o n t h e r ef er (: l~ ce s c r e e na n d ( :( re ta in s ra y s t h r o u g h o u t a c e r t a i n r a n g e o f i n c l i n a t i o n a n g l e s. H e n c e , itm a y t )e v i s~ m l ly r e t ) r e se n t e ( t i n t h e 4 D o p t i c a l p h a s e s p a c e b y a c e r t a i n f i n i tev o l u m e 12; e a c h p o i n t i n t h e v o l u m e V is r e p r e s e n t a t i v e o f a r a y i n t h e b u n d l e .I f t h e b u n d l e p r o p a g a t e s t h r o u g h a n o p ti c a l s y s te m , t h e p h a s e - s p a c e v o h l m e ,o r i g in a l l y o c c u p i e d b y t h e r a y s, c h a n g e s i ts s h a p e a n d m o v e s t h r o u g h p h a s es p a c e . T h u s , i n s t e a d o f f o l lo w i n g t h e m o t i o n o f e v e r y s i n g le r a y in t h e b u n d l e ,w e m a y o b s e rv e t h e m o t i o n o f t h e p e r t i n e n t p h a s e - s p a c e v o l u m e .

P a r a l l e l in g t h e m e t h o d s o f s t a t i s ti c a l m e c h a n i c s [4 ], w e i n t r o d u c e t h e p h a s es p a c e f l m c t i o n p ( q , p ; z ) , s p e c if y i n g t h e d e n s i t y o f r e p r e s e n t a t i v e p o i n t s t h r o u g hp h a s e s p a c e . P r e c i s e l y , p ( q , p ; z ) i s d e f i n e d i n s u c h a w a y t h a t , a t a n y a x i a l


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1D F irst -Order Op t ica l Systems: The R ay- Transfer M atr ix 63

1-[ iIII

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FIGURE 2.2. Naive view of tile phase-plane transformation of tile representative point ofa single ray and the representative region of a ray-bundle, effected by an optical system.( a ) A ray propagates to tile back focal point of a positive lens and ( b ) tile correspondingrepresentative point in the phase plane "moves" from Pi to Po. Likewise, ( c ) a beam ofparallel rays is focused to the lens back focus and ( d ) the corresponding representativesegment P1P2, lying on the q-axis, changes to the segment PAP4, lying on the p-axis.

p o s i t i o n z , t h e n u m b e r o f r e p r e s e n t a t i v e p o i n t s d N ' ( q , p ; z ) i n t h e v o l u m e e l -e m e n t d q d p a r o u n d t h e p o i n t ( q , p ) i n p h a s e s p a c e i s g i v e n b y t h e p r o d u c td N ' ( q , p ; z ) = p ( q , p ; z ) d q d p . C l ea r l y , p ( q , p ; z ) t a k e s o n v a l u e s t h r o u g h o u t t h eb e a m v o l u m e 12 i n p h a s e s p a c e , a n d v a n i s h e s o u t s i d e )2.

A s a n e x a m p l e , w e m a y r e f e r t o t h e s i m p l e s i t u a t i o n s i l l u s t r a t e d i n F i g .2 .2 , i n s p i r e d b y [ a ., ]. A p a r a x i a l r a y , i n c o m i n g p a r a l l e l t o t h e o p t i c a l a x i s o f af o c u s i n g l e n s , i s d e v i a t e d t o p a s s t h r o u g h t h e l e n s b a c k f o c u s F ' ( F i g . 2 . 2 . a ) ) .T h e i n p u t - r a y r e p r e s e n t a t i v e p o i n t P i , t h a t li es o n t h e q - a xi s o f t h e p h a s ep l a n e , " m o v e s " t o t h e o u t p u t - r a y r e p r e s e n t a t i v e p o i n t P o , t h a t i n c o n t r a s t l ie so n t h e p - a x i s ( F i g . 2 .2 . b ) ) . C o r r e s p o n d i n g l y , a b e a m o f p a r a l l e l r a y s h i n g i n go n t h e l e n s i s f o c u s s e d t o a s p o t i n t h e b a c k f o c u s . T h e i n c o m i n g b e a m i sr e p r e s e n t e d i n th e p h a s e p l a n e b y a s e g m e n t , l y i n g o n t h e q - a x is , o f e x t e n s i o n


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6 4 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

e q u a l t o t h e s p a t i a l w i d t h 2 w o f t h e b e a m ( F i g . 2 . 2 . c ) ) . L i k e w i s e t h e f a n o ft h e f o c u s s e d r a y s p a s s i n g t h r o u g h F ' i d en t i f ie s a s e g m e n t , l y i n g o n t h e p - a x i s ,o f e x t e n s i o n 2 ~ ( F i g. 2 . 2 . (t ) ). I t i s e v i d e n t t h a t t h e r e g i o n i n t h e o p t i( : al p h a s ep l a n e r e p r e s e n t i n g t h e b e a m i n o v e d a n d c h a n g e d i ts sh a t) e ; a c t u a l ly , i n t h es i m p l e ( :a s( '. () f t h e 1 D m o d e l t ) e r t a i n i n g t o F i g . 2 . 2 , o n l y t h e e x t e n s i ( m o f t h eze ro -a rea , t )ha , se -t ) l ane reg i on has be en a f fc ( : ted by t he l c il s f ( l( :l ls i ll g e f fec t . T i l e( te ns i ty ( i f rc I ) re se nt a t i ve t ) ( l il l t s i~ l t )ha ,se t ) lan( ' (' ,ha I lgc( t as wel l .

I t is thcr(;fl ir~; na. t l l ral t () i ] l( tuire w h e th e r i t, is l ){issi tf le t~l cl lvisa,gc, a, lawfo r t,h~; ew lll lti ~m ~lf tll~', lflm.s~'~-sI)a,(:~; (l e n s it y f l ( q , p ; z ) , ( h i a.(:('tnnd, (if tha,t theev oh lti ~m ~f ev er y sin gl(; ra y ~/f t,ll(; CllS(;nfl/lt; is gt~W;nl('~l ]ly Ha ,llfilt~ nl's ( '(tlla,-t io ns . W~'~ wil l f imt th a t a . l l "ev(fl l lt , i~nf ' e(t lm ,t ion fi~r p ( q , p ; z) (:a .~l t)(' w ri t t e n(t()wll, ml(l zl(ita,llly ill a, [()1"111 st r i c t ly s imi la , r t ,( ) I lm l l i l t ( n l 'S C(l lm,t i( /l~s f li r th era,y (:()()r(lilmte s. M(ir(;(iV(;l ', w e wi ll s ec l,l~a.t l,lle t)lm.se sIla,(:e (le ~ si ty p ( q , p ; z )is ev (/lv(;(l lw tl~e r a y l, ral~s f(;r ()t)cr a,l,t/r ()r, wl~ e~ 1,l~(; li~l(;ar al)t)rt) xi~ m.l, itm isco nc en m (l , 1)y the ra y tra , l~sfer ~m .trix a ,~ld t ln~s w e will rega, il~ t l~e si l~gle-ra ,yt)i(:t~lre as l)asi(: t ,(/l,l~(; ray-e~ |s(;~fl)l(: I)i( '. t,~r(; a,s wel l. Tl~e (lis('~ssi(n~ l)r(;se~t,e(tin tt~(; ~(;xt, Im ragra ,tfl~s lm,si(:ally l )a ra lM s t, le (l(;riw~,ti(/l~ (if I ,i (l~ vi lh; 's tlle()l 'elllin st at is t i ( 'a l ~e( ' lm ,~i(:s [:~, 41. l)lm,se-Sl)a ,( ,.e t)a,se(t I~etl~o(ts ar e (ir(li~a. l 'y to ol s in~a .g~ (;ti( : (lliti(:s. I~(1(;(;(1, as is w (;ll k~()W l~, tl~(;


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1 D First-Order Op tical Systems: The Ray-Transfer M atrix 6 5

2 . 2 . 1 L i o u v i l l e 's eq u a t i o n f o r t h e r a y - e n s e m b l eW e c o n s i d e r t h e v a r i a t i o n o f t h e d e n s i t y f u n c t i o n p ( q , p ; z ) w i t h t h e p r o p a g a -t i o n v a r i a b l e z . I t i s m a t h e m a t i c a l l y g i v e n b y t h e t o t a l d e r i v a t i v e o f p ( q , p ; z )w i t h r e s p e c t t o z , n a m e l y

d 0 O p d q O p d pd z p ( q , p; z ) - - ~ z p ( q , p ; z ) + O q d z ~ O p d z ' (2 .2 .1 )w h i c h , o f c o u r se , d i s p la y s c o n t r i b u t i o n s f r o m b o t h t h e e x p l i c i t a n d i m p l i c i td e p e n d e n c e o n z , m a n i f e s te d r e s p e c ti v e ly t h r o u g h t h e p a r t i a l d e r i v a t iv e O p / O za n d t h e d e r i v a t i v e - c h a i n t e r m s , i n v o l v i n g t h e r a y c o o r d i n a t e s q a n d p ( a ). F o rb r e v i t y , w e u s e a g a i n t h e a b b r e v i a t i o n ( q, p ) f o r ( q x , q y , p x , p y ) .

R e p l a c i n g f o r d q / d z a n d d p / d z t h e r e s p e c t i v e e x p r e s s i o n s f r o m H a m i l t o n ' se q u a t i o n s , t h e c h a i n -l ik e t e r m s s i m p l i f y i n t o t h e P o i s s o n b r a c k e t o f p a n d H :

O p d q O p d p O p O H O p O HOq O; Oq Op Op Oq = { p ' H } , ( 2 . 2. 2 )b y w h i c h t h e e q u a t i o n f o r p ( q , p ; z ) fo l l o ws i n t h e co n c i s e fo rm

d 0d p - p(q , p; {p , H } . (2 .2 .3 )I n g e n e r a l th e z - e v o l u t i o n of a n y d y n a m i c M v ~ r ia b l c p e r t i n e n t t o a s y s t e m ,

w h o s e d y n a m i c s is g o v e r n e d b y H a m i l t o n - l ik e e qu a, i o n s o f e v o l u t i o n , o c c u r sa c c o r d i n g t o E q . ( 2 . 2 . 3 ) .

O n a c c o u n t o f t h e p h ys ic M m e a n i n g o f p ( q , p ; z ) , w e c a n f u r t h e r s p e c i f y i t sv a r i a t i o n w i t h z . I n f ac t , w e m a y o b s e r v e t h a t t h e t o t a l n u m b e r o f r e p r e s e n -t a t i v e p o i n t s i n p h a s e s p a c e r e m a i n s c o n s t a n t , e v e n th o u g h t h e s h a p e o f t h er e gi o n c o n t a i n in g t h e m m a y p r e s u m a b l y c h a n g e a s a c o n s e q u e n c e o f t h e i n -d i v i d u a l m o t i o n s o f t h e e n c l o s e d p o i n t s . I n p a r t i c u l a r , l e t u s c o n s i d e r a f i x edv o l u m e )2 i n p h a s e s p a c e a n d d e n o t e b y S t h e c l os e d s u r f a c e d e l i m i t i n g i t.T h e n u m b e r o f r a y - r e p r e s e n t a t i v e p o i n t s i n s id e ]2 a t a c e r t a i n z is o b t a i n e di n t e g r a t i n g t h e d e n s i t y p ( q , p ; z ) o v e r t h e v o l u m e i t s e l f , n a m e l y

.N " ( z ) - I v p ( q ' p ; z ) d V . (2 .2 .4 )E v i d e n t l y , N "v ( z ) d ep en d s o n 12 an d z . B e i n g 12 f i x ed , N" ( z ) c h an g es o n l y w i t hz , d u e t o t h e m o t i o n o f t h e r e p r e s e n t a t i v e p o i n t s r e f l e c ti n g t h e p r o p a g a t i o no f t h e c o r r e s p o n d i n g r a y s in t h e b e a m . A s n o r a y s a r e c r e a t e d o r d e s t r o y e d ,

a T h e d e r i v a t iv e ~ y i e ld s t h e z - d e r i v a t iv e o f p ( q , p ; z ) i n a f ra m e m o v i n g w i t h t h e p o i n t( q, p ) i n p h a s e s p a c e . I n c o n t r a s t , o y i e l d s t h e z - d e r i v a t i v e o f p ( q , p ; z ) i n a f i x e d f r a m e .


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66 L i n e ar Ra y and W av e Op t i c s i n Pha se Spa c e

t h e n e t r a t e a t w h i c h A /" ( z ) v a r i e s ~ m u s t e q u a l t h e n e t r a t e o f p o i n t s(~Zc r o s si n g t h e b o u n d a r y s u r f a ce S . S o w e w r i t e

O z v ( z ) - - . p u . n d $ . (2 .2 . 5 )t im sm' fa( :c i l l te gr a l I '(~I )r('~s(mting th e ra t ( ; a t wh i( :h I )() ints ( : ross t im s ll rfa ,( :eS () l l twa,r(1. In fa( ' . t , in ( : ( ) l f f() l 'mi ty with the s tan(tar( | ( : ( ) l ivent i ()ns , n is the() , , twa,r(1-(h 'awn uni t n()rmal t , ( ) t l , ( ; s , , r fa , ( :e ( ;h;m(;nt d S , a.,,(1 u = ( d q / d z , d p / d z ) Ti s th e vclo (: i ty vc(: to r in t )ha .s (; st )a .( '(~, s t ) (x: i fying t ,h( ; ra t ( ; 1)y wh i( :h th e ra y a ,ttl~e t)lm.se st)a.('c t)()i~t (q, p) ~() v( ,s t() tt~(; t)()i~t ( q + d q , p + d p ) . Tl~c mimess igI l a , ( : ( 'o lmts for t l lc fa, ('t t lm t , i f t ) ( ) ints lea , re th(~ v()hl~(~ )2, A fv (l(x'rca,scs.

A ( ' ( :or( t i~g to Gm ~ss ' ( liv( '. I'g( ;~ l( :c t l m () rc ~ [5] , t lm s~lrfa .( :c i~ tcgra .1 ~ lay bct~lr~lc(t il~t,() t,h(~ v()hn ~l(~ ixit,(~gra l a,s

. I s t' u . n d S - j v V . (p U ) d r , (2.2.(~)w hc r(, V - ( )(l(~n()t(~s t ln~ (liv(,rg(, , ,(:( , ( , l)(,rat , i()~, an(11,(,n(:c V . ( ) = O / O q + O / O p .I l l t (~res t ingly w(; f i l l ( l

i ) d q 0 d pX 7. (p U ) -- , -7- (P -; -) + ,-:7-( P - ; - ) - { P , I I } .o q a z o p a z (2 .2 . 7 )

B y ~sc of ( 'x l )rcss i() ,~ (2.2. .1) , givi ,~g N"v as t.l~e v()l,n,~,: i,~t('gx'al ,,f p, a,~,l (2 .2. 6)in (2.2 .5) , w(~ cn(t 11t) w i t t l

{ ~ p + v . ( p u )} ~tv - (). (2 .2 . s )w h i ( : h , b e i n g t h e v o l m n c 12 a r l ) i t r a .r i l y ( 'h () s(m , y i e h t s t l l c ( ' o n t i m f i t y e q u a t i o nf()r the t)ha.s(;-st)a(:( ' (h;i ls i ty f l ( q , p ; z ) a.s

0o p + v . ( p . ) - o . ( 2 .2 . 9 )M o re o v e r , u s i n g t h e a t )( )v e r e l a t i o n i n t o E q . ( 2 . 2 3 ) f o r t i l e d e r i v a t i v e dp o n9 d z ~a .( :c o un t o f ( 2 . 2 .7 ) , w e f i n d t h a t t h e c o n t i n u i t y e q u a t i o n is e q u i v a l e n t t o

dd z p - o . ( 2 .2 . 1o )E q u a t i o n ( 2 .2 . 10 ) is t ll e d e s i r ed e q u a t i o n f o r t h e z - e v o l u t i o n o f t h e p h a s e -

s p a c e d e n s i t y p ( q , p ; z ) . W e r e w r i t e i t i n t h e c o n v e n i e n t f o r mO z p - - - { p , H } - - - k H P , (2 .2 .11)


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1 D F i r st - O r d e r O p t ic a l S y s t e m s : T h e R a y - T r a n s f e r M a t ri x 67

w i t h t h e i n i t i a l d a t a b e i n g c o n v e y e d t h r o u g h p ( q , p ; z i ) - p i ( q , p ) . W e r e f e r t o( 2 .2 .1 1 ) a s L i o u v i l l e ' s e q u a t i o n f o r t h e r a y - e n s e m b l e , a s i t d i r e c t l y f o l l o w s b yL i o u v i l l e t h e o r e m ( 2 .2 . 10 ) . I t is e v i d e n t i ts s i m i l a r i ty t o H a m i l t o n ' s e q u a t i o n s( 2 .1 .1 ) f o r t h e l i g h t r ay .E v i d e n t l y , a s t h e p h a s e - s p a c e v e c t o r u ( z ) = ( q ( z ) , p ( z ) ) r m a y b e r e g a r d e da s t h e d e s c r i p t o r o f t h e s i n g le r ay , th e p h a s e s p a c e d e n s i t y p ( q , p , z ) m a y b er a g a r d e d a s t h e d e s c r i p t o r o f t h e r a y - b u n d l e . I t is o f v a l u e t h a t t h e z - e v o l u t i o n

t- .o f b o t h u a n d p is e s s e n t i a ll y g o v e r n e d b y t h e L i e o p e r a t o r L H, e v e n t h o u g hw i t h o p p o s i t e s i g n s d u e t o t h e i n v a r i a n c e o f p ( q , p , z ) a l o n g t h e r a y p a t h , o nw h i c h w e w il l c o m m e n t l a te r . A s a c o n c lu s io n , w e m a y s a y t h a t H a m i l t o n i a no p t i c s d e a l s w i t h e v o l u t i o n e q u a t i o n s o f t h e t y p e

0 ~-0---~v - ? /v , (2 .2 .12 )

w h e r e ~ m a y d e n o t e t h e p h a s e - s p a c e v e c t o r ( q ( z ) , p ( z ) ) r o r th e.. p h a s e - s p a c ed e n s i t y p ( q , p ; z ) a n d a c c o r d i n g l y t i le o p e r a t o r ?-t c o m e s t o b e ? -t - + L H .

T h e f o r m a l s im i l a r i ty o f E q . ( 2 . 2 .1 1 ) t o ( 2 .1 . 1 ) m a k e s e v i d e n t t h a t t h ec o n s i d e r a t i o n s w e d e v e l o p e d i n w 1 .4 .1 a b o u t t h e i n t e g r a t i o n o f ( 2 .1 . 1) a p p l yt o t h e i n t e g r a t i o n o f ( 2 .2 . 1 1) a s w e ll. T h e c r u c i a l p o i n t o f t h o s e c o n s i d e r a t i o n sw a s c o n c e r n e d w i t h t h e o p e r a t o r n a t u r e o f k H a n d i t s p o ss i bl e d e p e n d e n c eo n t h e a,x ia l p a r a m e t e r z . W e f o u n d t h a t t h e s i m p l e s t c a s e p e r t a i n s t o a z -i n d e p e n d e n t H a m i l t o n i a n , i n w h i c h c a s e t h e s o l u t io n t o ( 2 .2 . 11 ) t a k e s t h e fo r m

p ( q , p ; z ) - c - ( z - z ~ ) k , p ( q , p ; z i ) , ( 2 .2 .1 3 )w h e r e p ( q , p ; z i ) d e n o t e s t h e p h a s e - s p a c e d e n s i t y of t h e b e a m a t t h e i n p u t re f-e r e n c e s c re e n . E x p r e s s i o n ( 2 . 2 .1 3 ) r e f l e c ts t h e s a m e v i e w a s E q . ( 1 .4 . 4 ); t h u s ,t h e p h a s e - s p a c e d e n s i t y p ( q , p ; z ) a t z o r i g i n a t e s f r o m t h e a s s i g n e d f u n c t i o np ( q , p ; z i ) a t t h e i n p u t z i - p la n e b y w a y o f t h e L i e t r a n s f o r m a t i o n c - ( z - z ~ ) k H .I n S e c t i o n 1 . 4 w e c o n s i d e r e d L i e t r a n s f o r m a t i o n s a c t i n g d i r e c t l y o n t h e r a yv a r i a b l e s ( q, p ) , a n d , a s s p e c if i c e x a m p l e s i n w 1 .5 . 2 w e d e s c r i b e d t h e L i e t r a n s -f o r i n a t i o n s g e n e r a t e d b y q u a d r a t i c f i m c t i o n s o f q a n d p . H e r e , t h e L i e t r a n s -f o r m a t i o n a c t s o n t h e p h a s e - s p a c e d e n s i t y p ( q , p ; z ) , w h i c h i s a f u n c t i o n o f qa n d p . T o se e w h a t h a p p e n s w e n e e d t o u s e s om e p r o p e r t i e s o f t h e L i e t r a n s -f o r m a t i o n s , o n w h i c h w e w i ll b r ie f ly c o m m e n t i n t h e n e x t p a r a g r a p h . F i r s t l yw e w i ll e l a b o r a t e a g e o m e t r i c a l a n d p h y s i c a l i n t e r p r e t a t i o n o f E q . ( 2 .2 . 1 0) .G e o m e t r y a n d P h y s i c s r e la te d w i t h L i o u v i l l e ' s t h e o r e mE q u a t i o n ( 2. 2. 10 ) s h o w s t h a t t h e d e n s i t y o f p h a s e s p a c e p o i n t s d o e s n o t c h a n g eb y p r o p a g a t i o n o f t h e b e a m t h r o u g h t h e o p ti c al m e d i u m . A s d / d z r e p r e s e n t st h e " m o b i l e " d e r i v a t i v e , w e c a n g i v e ( 2 .2 . 1 0 ) a n i c e p i c t o r i a l v i e w in p h a s es p a c e . S u p p o s e t o i n d i v i d u a l i z e a " p o i n t " P = ( q, p ) i n p h a s e - s p a c e , i . e. , a r a y ,


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6 8 L i n ea r R a y a n d W a v e Opt i c s n P h a s e S p a c e

p - p -

3/,,

F I G U R E 2 . 3 . T l~ c I )lu ~s C- S l) aC c r c g i o ~ ( s t u u l c ( l a r c~ u s) o c c ~ p i c ( t 1 )y t l~ c ~ s s c , n l ) l y o f t h eray-1)~u~( t lc r c I ) r c sc ~ t a t iv c l )o i~d, s c lmx~gcs a s t , i e 1 )eml i I ) roccc r a lo~ lg t l ic z ax is , lnd , ther c l c v m~ t v o h ~ c r c ~ n a i~ s ( : o ~ s tm~ t : ]2 i = ] 2 ( z ) = 1 2,,.

b y s o m e m a r k . L e t p (P ; z i ) l )e th e t ) lm,se s t)a ,( :e ( le ns i ty a , t P for a , ( :c r ta in z i .T lle t )o int P ~les( : r i l)es a t ra je ( : t ( ) r y i ll I ) l la .se sI)a, (:e w i t l i i~l ( ' reasi l lg z > z i . F o rea,(:h z > z ,i w(; (:all (',wdlla,t(; 1,11('~(l( '~llsity p (P ; Z ) at P , l l l()villg w itl l i t, a h)ll t4 i tso w n t) at h. W('~ ti l l(t t l m t i t ( l()(;s 11()1, (:l la.~lg(; , i .e. , p (P ; z) = p (I '; z~) V z > zi . T h enl () ti () ll ()f tll('~ ra,y-('~lls('nfl)l(,~ r('l)l'(;s('~ld,ativ('~ 1)()ild,s ill t) lm s( ' sl)a (:(; is 1,11(~11 lik( ;ttm ,t ()f a,I1 ill( '(nllI)r(;ssit)l('~ tllli(l ill l 'ca l Slm (:e. I ll ()til er w ()r(l s, p ( q , p ; z ) t ) c l m v e slik e a " t)r( )t)( ;rty " ()f th(; t)()i ll l, in t)l las(; sI)a,(:(; , ml(t a,(:(:()I '(t i ligly it, is tra, nst )oI 't t ;dt)y th e t)()i li t, a, l(nig i t s ()w~ l tr aj e( :t () ry wll(;~l it , is r(;t)r(;se~lta,t iv(; ()f a l ig ti t ray .

Tl~ c i~v a,ri ml (:(' of 1,11(; 1)llasc-st)a,('.(; (lc ~l sity p (q , p ; z) in lt) lic s 1,11(; i~ma,l'ian(:(; ofth e t)ha,s(;-st)a(:(; v ()l~tnle 12 ()f 1,11(; regi() l l (m(:lt)sil lg t im ray -l)~ ult l le r( , . t)i 'escnta,t iv(;t)()in ts. In fi~.(: t, s~t)t)()st; t() s(; l( '(: t at a (:(;rtah~ z = z i a ('(;rt, ai ~ ~nm fl)(;r ()fre t ) rescnta t ive I )o ix~ts a roml( t a ( :hose~ t ) ( ) in t , , th rougt l wl l i ( : l l t l l c ( l ( ;~ l s i ty t akes( ) I1 the CO llStall t va,l~lc P i . Let Ni l ) ( ; the volcanic cn( : l ( ) s ing the ( :h( ) s ( ;n t )o in t s .T h e tota,1 Inlnf l ) ( ; r ( )f t )oi~l ts in Ni evi( l t ;~l t ly is

N i = p i N , : . ( 2 . 2 . 1 4 )F o l l o w i n g t h e p o i n t s a s t h e y m o v e t h r o u g h t ) ha , s c- s p ac c , w e s e e t h a, t t h e b o u n d -a r ie s o f t h e e n c l o s i n g v o l m n e c h a n g e w i t h z > z i, b u t b y d e fi n i ti o n t h e t o t a ln u m b e r o f p o i n t s c o n t a i n e d i n s i d e it d o e s n o t va ,r y: N ( z ) = N i . A l s o , the den-s i t y o f p o i n t s k e e p s o n t h e v a l u e p ( z ) = P i b y v i r t u e o f ( 2 . 2 . 1 0 ) . T h u s , f o r a nyz > z i w e c a n w r i t e

=

w h i ch b y comp ar i s on w i th (2 .2 .14 ) i mme d i a te l y l e ad s toV ( z ) = V i . ( 2.2 .1 6)

T h e v o l u m e e n c l o s i n g a f ix e d n u m b e r o f r e p r e s e n t a t i v e p o i n t s in p h as e s p acer e m a i n s c o n s t a n t i n z , a l t h o u g h , d u e to t h e m o t i o n o f t h e e n c l o se d p o i n t s ,


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1D F irst-Order O pt ical Systems: The Ray-Transfer M atr ix 69

i t s s h a p e m a y s e n s ib l y c h a n g e . S o , t h e t o t a l p h a s e - s p a c e v o h l m e 12 o f t il eb e a m r e m a i n s c o n s t a n t in z , r e g a r d le s s o f t h e m o t i o n o f e a c h r a y a n d t h ec o r r e s p o n d i n g m o t i o n o f t h e r e p r e s e n t a t i v e p o i n t s t h r o u g h p h a s e s p a ce : 12o -V ( z ) = lPi (F i g . 2 . 3 ) . T h i s r e s u l t i s r a t h e r i n t u i t i v e i f we t h i n k o f a f lo w i n g" f l u id " . I f t il e t o t a l q u a n t i t y o f f lu id a n d t h e d e n s i t y o f t h e f l u id d o n o t c h a n g et h r o u g h t h e m o t i o n , t h e v o l u m e o c c u p i e d b y t h e f l u i d r e m a i n s c o n s t a n t a sw e l l , e v e n t h o u g h i t m a y d e f o r m s i g n i f i c a n t l y d u r i n g t h e f l u i d ' s f l o w .

T h e i n v a r i a n c e o f t h e p h a s e s p a c e v o l u m e a p p l i e s o b v i o u s ly a l s o t o t h ev o l u m e e l e m e n t s d ip 's : d 1 2 o = d ~ ) i a l o n g a n y r a y p a t h . I n w 2 . 4 we wi l l e l ab o -r a t e a g e o m e t r ic a l i n t e r p r e t a t i o n o f t h e v o l u m e e l e m e n t d V a n d i t s i n v a r i a n c ea l o n g t h e r a y p a t h i n c o n n e c t i o n w i t h t h e c o n c e p t u a l d e f i n i t io n o f t h e i n t u i t i v ep e r c e p t i o n o f t h e e l e m e n t a r y b u n d l e o f r ay s .

F i na l ly , w e n o t e t h a t t h e r a y t r a n s f o r m a t i o n f r o m th e p h a s e - s p a c e p o i n tP~ (q ~ , p ~ ) t o t h e p o i n t P o ( q o , p o ) is a c c o m p a n i e d b y a t r a n s f o r m a t i o n o f t h ep h a s e - s p a c e v o l u m e e l e m e n t s f r o m d l ) i t o d l 2 o ( b ) . W e k n o w n t h a t if S d e n o t e st h e J a c o b i a n m a t r i x a s s o ci a te d w i t h t h e t r a n s f o r m a t i o n P i -- + P o ( see w 1 .3 .1 ) ,t h e r e l e v a n t v o l u m e s d l ) i a n d d l ) o a r e l in k e d b y t h e d e t e r m i n a n t o f S a s d l 2 o -d e t S d l P i . T h e i n v a ri a n c e of t h e v o l u m e e l e m e n t s u n d e r r a y p r o p a g a t i o n i m p l ie st h e r e f o r e t h a t d e t S = 1 , t h u s s o l v i n g b e t w e e n t h e t w o p o s s i b i l i ti e s o f fe r e d b yE q . ( 1 .3 .1 3 ) f or t h e d e t e r m i n a n t o f t h e s y m p l e c t i c m a t r i x a s s o c i a t e d w i t h t h er a y t ra n s f e r . V i c e ve r sa , t h e u n i m o d u l a r i t y o f t h e s y m p l e c t i c J a c o b i a n m a t r i xa s s o c i a te d w i t h t h e r a y t r a n s f o r m a t i o n i m p li e s t h e i n v a r i a n c e o f t h e p h a s es p a c e v o l u m e . T h i s o ff e rs a c le a r g e o m e t r i c a l i n t e r p r e t a t i o i l o f t h e s y I n p l e c t i cn a t u r e o f r a y p r o p a g a t i o n , s i gn i fy i n g t h a t e l e m e n t a r y v o l u m e s a l o ng t h e r a yp a t h i n p h a s e - s p a c e r e m a i n c o n s t a n t . E v i d e n t l y , a r e a s p la y t h e ro le as v o l u m e sw i t h i n a 1 D p i c t u r e , w e a re m a i n l y c o n c e r n e d w i t h ( se e w 1 .7 . 4) .2 . 2 . 2 B a s i c s o f L i e t r a n s f o r m a t i o n sA s n o t e d i n w 1 .3 .2 , g i v e n a d y n a m i c a l v a r i a b l e f ( q , p ) w e c a n g e n e r a t e t h e L i eo p e r a t o r k s b y P o i s s o n b r a c k e t o p e r a t io n "

kf - { -, f} , (2 .2 .17)a n d t h e L i e t r a n s f o r m a t i o n T I b y c o m p u t i n g t h e p o w e r s erie s"

t - cx~ ~ j A j _ _ e ~ f ~T s (~) - - ~ -~ . Ls ~ e R , ( 2 . 2 . 1 8 )j = O

I n t h e p r e s e n t d i s c u s s i o n t h e c o n c e p t o f v o l u m e i n p h a s e s p a c e h a s b e e n l e f t o n t h es a m e i n t u i ti v e f o o t i n g a s t h a t i n p h y s i c a l s p a c e . A c t u a l l y , p h a s e s p a c e is a m a t h e m a t i c a ls p a c e i n w h i c h i n d e e d t h e i n f i n i t e s im a l v o l u m e is d e f i n e d in t e r m s o f t h e l i n e a r a n t i s y m m e t r i cn o n d e g e n e r a t e 2 n - f o r m , a n d f i n i te v o l u m e a r e a c c o r d i n g l y o b t a i n e d f r o m i t b y in t e g r a t i o no v e r t h e d o m a i n o f i n t e r e s t . F o r a n a c c u r a t e a n a l y s i s t h e r e a d e r i s d i r e c t e d t o [3 .3].


12/52

70 L i n e a r R a y a n d W a v e O p t ic s n P h a s e S p a c eAw h e r e ~ is a r e al p a r a m e t e r a n d t h e j - t h p o w e r o f Lf is o b v i o u s l y

A jL / - { { { . . { - , f } . . } , f } , f } . ( 2 . 2 . 1 9 )j t i m e s

A s ev id en ce d | ) y t h e cxponc~I~tia ,1 no t a , t i on on t i l e r i gh t -ha ,nd s ide o f (2 .2 .18 ) ,t he po we r se r i e s i s no th in g t ro t t i l e cxt )( )~ lcnl , ia,1 f lm( '. t ion o f t i l e o t ) c r a to r (k f .Evi (h;,~ tly, 'Y /(( )) is t ,h('~ i(]cld, il,y ()t)('~ra,t()l': "Y /(()) - I [6].

Lie t ra,n sf() rm atio ~ls ha,v(; nin n y i~lI , ( ;r( ;st ing a,n(] ~s(;fl~l t )r() l)crl, i (;s, wh i(:h di-re(:t, ly fo llo w K () ~ t,t~(; t)ro t)e I't ies of t ,l~c Li(; ()t)cra, t, or s a,~(l t11(; cxt)()~m 1~tiation.W e s h a ll ll()I, giv e a. (tct, aih ,(t a.t:(:()ll~d, ()f a, l ti m t)r()t)(;rt, i(;s ()f 1,1~(; Li('~ tra ,lis for lna ,-tio ~s , for wlfi(:t~ tim r( 'a (h, r is (lirc(:t(;(l t ,() [(;], t)~I, slm.ll ( ' ( ) ~ ( , ~ t , i l l Im,rti(:ula,r()~l I,h()s(; l, ha l, a rc ~s(;fifl l,() ga.iIl i n s ig h t in t() t,tl(; })(;tm vi()~ r ()f l,h('~ l)lm s(;-st) a,ce( h q l s i t y f l ( q , p ; z ) t l~r()~gl~ E(t. (2 .2 .1 3) . W c ( 'l~t)lm ,sizc tim, I, I,l~(; ~m l, l~('~m,tica.1s e t t i ~ g } 1 ( ) I ' ( ~ is ~()l, t)r () I) er ly (:( )~t )h ;t(; ; (t~mst, i()l~s r(;h;va,~d,, f()r i~sl, m~(:('~, t() t il ea lm,ly t i ( : i ty ( ) r t , m r a , ( I i ~ s ( ) f ( : () l lV ( ; l' ~ ( ; ll ( :O ( ) f l , l l ( ; t ) ( ) w c r s ( ;I ' i o , ( ; X l ) m ~ s i ( ) I ~ S () r o f()t)era.l,()r i (h ;n tit i(' s, w e w ill ]~(' (:()l~(x,rll(;(l wit, l~, w ill n() t 1)(; aI)t)ro a.(:tm(t.77w. t r l' o d ' m ~ t t n Y ; . s e 'l " tT a t im t , t n ' o p e ' l ' t yWe t)r()w; t i le p t ' o d ' t t , : t p ' l' , '. s r t i o ' u p ' r o p e ' l 't y , wllic :ll t, lu' ()lt gl l l,ll(; r(;la,ti()ll

~ : < ' : h . : / ( , ~< ' : h .) (, .< ': :/), (2.2.20)sta,t(;s tha .t tll(; I,i(; tra ,llsf()nlm ,ti()ll a,(:t, llg ()ll t, l(; l)r() (hlt:t ()f fiU~(:ti()llS l)rotluc csthe sa,nl(; r(;slflt a.s the i)r()(111(:t ()f the Lie tra,nsf()rIlm,ti()IlS a ,(:ting s(;i)a ra tely(m tlm sill gl (; fin~('t,i(m s ii~ t,h(~ I)r(~(111(:t. 'Fh (',ii , (:()nsi(h~.r tll( ; I)r(~(lll('t of tw o(lyna.~fi(:al wuiaI~l('~s h(q,p) an(l g (q, p) for wh ich the Po iss(m ])ra,('k(; ts w ith. f ( q , p ) a,re lll(;a,lliI~gflfl:

v ( q , p ) - - h , (q , p ) g ( q , p ) . ( 2 . 2 . 2 1 )W e w is h t ,o d(;t ,( '~i' inin(~ th e ( ;f lb~c:t ( , f th e Lie t, ra n sf or m a, t i on ~ ( 4 ) ( ) I I v ( q , p ) .( )bvi ( ) l l s ly , w('~ arc fa,(:c(t w i t h t im ( 'Na,hm, ti ()n of rc t )e ate d Po is so n | ) ra ,ckets of fa ,n d v . I n f a c t , iI l a c c o r d w i t h t i m p o w e r s ( 'r i cs r c p r c s e n t a , t i o n ( 2 . 2 . 1 8 ) , o n e h a s

cC Lf v(q 'P ) - - E ~ . LsV (q 'P ) - - v + C{v , f } + -~ -{{v , f } , f } + .. ., ( 2 . 2 .22)j = 0

W e r e c a l l f r o m w 1 .3 .2 t h a t t h e L i e o p e r a t o r a c t s o n t h e p r o d u c t o f f i m c t i o n si n t h e s i m i l a r w a y a s d if f e r e n t ia t i o n . H i g h e r p o w e r s o f t h e L i e o p e r a t o r s a t is f ya n i d e n t i t y a n a l o g o u s t o t h e L i e b n i z r u l e f o r d e r i v a t i v e s , a n d h e n c e

J A l ~ j - - l" LJ hg - ~ ( J~) L / h L / g . ( 2 . 2 . 2 3 )l--O


13/52

1 D Fi r s t-Or der Op t i ca l Sy s tem s: Th e Ray -T r ans fer Matr i x 71W e i n v i t e t h e r e a d e r t o p r o v e t h i s r e l a t i o n b y i t e r a t i o n . W i t h t h i s e x p r e s s i o ni n s e r t e d i n ( 2 . 2 .2 2 ) , w e o b t a i n

v - E (j_~), ,, L~ h L~ g - [ E ~ h i[ ~- ~j=0 l=0 " /=0 Lr . ( j - l ) ! L rC ~ O 0 . ~ . j ~ " " ~ "z=0 L/ j ! L c g ] - ( e c h ) ( c c g )j = 0

(2.2.24)

t h u s p r o v i n g t h e d e s i r e d p r o p e r t y ( 2 . 2 . 2 0 ) . I n t h e s e c o n d s t e p o f ( 2 . 2 . 2 4 ) , w eh a v e m a d e u s e o f t h e p e r m u t a t i o n

co j oc ocE E E Ej=0 l=O l=O j= l

(2.2.25)A c c o r d i n g ly , t h e L i e t r a n s f o r m a t i o n o f p o w e r s o f q a n d p i s

A A Ae C L c q m p n - - ( e C L f q ) m ( e C L r n , (2 .2 .26)a s c a n e a s i l y b e p r o v e d b y a p p l y i n g r e p e a t e d l y t h e r u l e ( 2 . 2 . 2 4 ) .

W e c o n s i d e r n o w a d y n a m i c a l v a r i a b l e g ( q , p ) , a n d s u p p o s e t h a t i t c a n b ee x p a n d e d i n a p o w e r s e r i e s w i t h r e s p e c t t o 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 pa r o u n d a, c e r t a i n p o i n t i n p h a s e - s p a c e , w h i c h f or b r e v i t y w e m a y a s s s u m e t ob e t h e o r i g i n : ( q , p ) = (0 , 0 ) . W e w r i t e t h e r e f o r e

( X ) ( X )

~ (q ,p ) - ~ 5 2 g ..... qmp,. (2.2.27)m = 0 n = O

AW e a p p l y T / ( r t o ( 2. 2. 27 ) a n d u s e t h e l in e a r i t y o f t h e L i e t r a n s f o r m a t i o n t op a s s t h e e x p o n e n t i a l o p e r a t o r t o e a c h t e r m o f t h e s e r ie s . E x p li c it ly , w e h a v e

( X ) O O

m = O n = O (2.2.28)T h e n , u s i n g th e p r o d u c t p r e s e r v i n g p r o p e r t y ( 2 .2 .2 0 ) t o p a s s t il e e x p o n e n t i a lo p e r a t o r i n t o e a c h v a r i a b l e , w e f i n a l l y o b t a i n

m - - 0 n = 0(2 .2 .29)

I t is e v i d e n t t h a t w e c a n r e c o m p o s e t h e f u n c t i o n g a c c o r d i n g t o t h e s e r i e sA A

e x p a n s i o n ( 2 . 2 . 2 7 ) a t t h e p h a s e - s p a c e p o i n t ( c r 1 6 2q , C C L r n a m e l yA A Ae C Lf g ( q , p ) __ g ( e r q , e r p ) . (2.2.30)


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72 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

T h e a c t i o n o f a L i e t r a n s f o r m a t i o n o n a t ) h as e - s t) a c e f l m c t i o n is t h e n t o p e r -f o r m a, L ie t r a n s f o r m a t i o n o n it s a r g u m e n t s . I n o t h e r w o r d s , t h e a b o v e r e s u lte s t a b l is h e s a p a r M l el b e t w e e n t h e t r a n s f o r m a t i o n e x p e r i e n c e d b y t h e p h a s e -s t )a ( 'e v e ( : t o r u = ( q , p ) T a ,n d t h a , t e x t ) e r i e n c e d t )y t h e p h a s e - s p a , c e f l m c t i o n.q(q, p) = . q ( ~ ) , i .~ . ,

phase-space vector phase-space function. . . . . . > ~- (2 .2 .3 1 )u ~ , ,~ ~ u - - v ~ / ( u V ) ~ , .~'~ ~(uV ) - - . q ( v V )T h e ( :() i~l)()si ti () l~ I )r()I)c rty (2.2.3 ()) ()f t t~(; I , i( ; t ra.~sfi)Hlm .t i ()~s is ( 'ss( ;~t ia,1 forth e la.te r (:~ llsi(lcra.ti(n~s rc g a r( li ~ g tt~(; ~;v(~l~lt, ()~ (ff tl~c l)l~asc-st)a(:(; (lc~ sit y.T h e t ~ o i . ~ s o ' n - b ' r a c k e t p ' r ~ :. s cr v a ti o 'n p ' ~ ' o p e r t yA n( ) th c r r e~ m rk a l ) l e I ) r (~ l )c r ty (~f t h e L ie . t rm~ sf ( ) rnmt i (n~ s i s t h e l ~ o i s , s o ' n b ' ~ ' a c k ( ' . tp r e s e r ' t ~ i n g p r o p e r t y , wl~i(:l~ i~ a sc ~s e ret)ro(l~l(:e.s (2 .2.2 ()) witl~ t,l~('. (w(ti~m,l 'yt ) ro(h~( ' t h .q i )e in g I 'et) la~:c(l ])y t ,h(; P() iss()~ia.n I)r()( t~( ' t { h, g} , i .e. ,

T h e L ic l,rallS fi)rllm , l,i()ll a,(:l,illg (m 1,11~', P(~iss(~ll l~ra,('k~'~t (~f fllll(:t,i(nlS g iv es th esa.llw, re sl ll t as til e t~(~iss~ll })ra,(:k(;t ()f tll~' IA e i,ra.llsfiwllm.l,i()lls a(:l , i l lg s~'.l)a rate ly()111;t1(; sillg l(; fllll(:ti()llS i~v(~lv(;(l in. N(;e,(tl(;ss t,() sa y, t, lw. P()iss()x~ t)ra ~:k et t)r(;s(;r-w~,tio~ t)r ()I)(;rty lm s tt~(; sa.~ m l)r()()f as (2 .2 .2 0) , t)(;i~g |)as(;(t ()11 tl~(; I~('~it)~dz rill(;f i )r 1,11(; ld g h er t )()w(;rs (ff Lie (~I)( ;rat()rs a, ( : t ing ( ~ a. P(f is s(m t)ra,( :ket :

J- . j , -/ ~ j - /L , { / , , : ~ } - Z ( ~ ) { L , t , , , : ~ } , ( 2 . 2 . a a )I =0

w h i c h , p a ,r a ,l lc l ii lg ( 2 . 2 . 2 3 ) , f o l l o w s f r o n l t h e d e r i v a, t i o n - l i k e b c h a v i ( ) u r o f I_/w i t h rcs t )c( : t to t im' Poissox l t )ra , ck et nnl l t i t) l i (:a , t ion .

E v i ( t c l l t l y t l lc t )r ( ) t) e r ty ( 2 . 2 . 3 2 ) e s t a ,t ) li s h c s t h a t e v e r y L i c t ra , n s f o r m a t i o n i ss y m p l c ( : t i( : . I n fa ,( :t, t h e L i c t r a n s f o r m a , t i o n T / ( ( ) g c n c r a , t e s a, m a , p p i n g w h i c ht a k e s t h e ( :o n j u ga , te v a r i a b l e s q a n d p t o t h e n e w v a r i a b l e s ~ a n d 7) a,s

A

q ~ 0 - e~L/^ q , (2 .2 .34)p ---+ ~fi - - e C t- p .

T h e P o i s s o l l b r a c k e t o f h a ,n d g a s f u n c t i o n s o f ~ a n d p is t h e nA A

{ h ( q , p ) , 9 ( q , ~ ) } - { e C k f h ( q , p ) , e r { h , 9 } ( ~ , ~ ) , ( 2 . 2 . 3 5 )t h u s e v i d e n c i n g t h e p r e s e r v a t i o n o f t h e P o i s so n b r a c k e t u n d e r t h e m a p p i n g( 2 .2 . 3 4 ); u s e h a s b e e n d o n e o f t h e c o m p o s i t i o n p r o p e r t y ( 2 .2 . 3 0) t o p a s s t h e


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1 D First-Order Op t ical Systems: The Ray- Transfer Ma tr ix 73Ao p e r a t o r e ~ kf f r o m t il e a r g u m e n t s t o t il e r e s p e c t i v e f u n c t i o n s h a n d g , a n d

v i c e v e r s a , a f t e r a p p l y i n g ( 2 . 2 . 3 2 ) , f r o m t h e f l m c t i o n { h , . q } t o t h e r e s p e c t i v ea r g u m e n t s . I n p a r t i c u l a r ,

(2 .2 .36)I n w 1 .3 .2 w e p r o v e d t h a t t h e p r e s e r v a t i o n o f t h e P o i s s o n b r a c k e t o f a n y t w od y n a m i c M v a r i a b l e s is a n e c e s s a r y a n d s u ff ic ie n t c o n d i t i o n f o r a t r a n s f o r m a t i o nb e s y m p l e c t i c .L i e t r a n s f o r m a t i o n s a s a L i e g ro u pW e n o w t a k e u p t h e q u e s t i o n o f t h e g r o u p s t r u c t u r e o f t h e s e t of L i e t r a n s f o r -m a t i o n s { c ; L f ; ~ E N } g e n e r a t e d b y t h e d y n a m i c a l v a r i a b l e f ( q , p ) . A c c o r d -i n g l y w e w i l l l i n g e r o n s o m e p r o p e r t i e s o f L i e a l g e b r a s a n d L i e g r o u p s t o h e l pt h e r e a d e r t o g a i n s o m e f a m i l i a r i t y w i t h t h e s e c o n c e p t s ( s e e a l s o A p p e n d i xA ) . A l t h o u g h t h e d i s c u s s io n b e l o w w il l b e c o n c e r n e d w i t h t h e L i e a l g e b r a o ft h e o p e r a t o r s ( 2 .2 .1 7 ) a n d t h e c o r r e s p o n d i n g L i e g r o u p o f t h e t r a n s f o r m a -t i o n s ( 2 . 2 . 1 8 ) , i t h a s a g e n e r a l v a l i d i t y , b e i n g r e l e v a n t t o t i l e l i n k b e t w e e n t h ec o n c e p t o f L i c a l g e b r a a n d t h a t o f t h e a s s o c i a t e d L i e g r o u p [7 ].

A s r e m a r k e d i n w 1 .3 .2 , t h e s e t o f L i e o p e r a t o r s { k I } , g e n e r a t e d b y t h ed y n a m i c a l v a r i a b l e s { f ( q , p ) } t h r o u g h ( 2 . 2 . 1 7 ) , d o f o r m a L i e a l g e b r a w i t hr e s p e c t t o t h e L i e b r a c k e t [, ] o f o p e r a t o r s

[ A , - - ( 2 . 2 . 3 7 )

w h i c h , e~s p r o v e d b y ( 1 .3 . 4 3 ), d e f i n e s a b i n a r y o p e r a t i o n t a k i n g t w o L i e o p e r -a t o r s t o a L i e o p e r a t o r a s w e l l :

[ , ] " (Lh , L9 ) E { L I ) { L s ) ~ [Lh ,Lg ] - L{ g,h } C { LI ) " (2 . 2 .3 8 )E v i d e n t l y , if { g, h } = 0, t 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 s c o m m u t e w i t h e a c ho t h e r " ]Lh , Lg ] - -0 .i - - , . . 1

L .1T h e n o t i o n o f g r o u p a s t h a t o f a l g e b r a i m p l i es t h e e x i s te n c e o f a b i n a r yo p e r a t i o n s e n d i n g p a i r s o f e l e m e n t s o f t h e s e t i n to e l e m e n t s o f t h e s e t. W h e nd e a l i n g w i t h o p e r a t o r s t h e b i n a r y o p e r a t i o n t o s t r u c t u r e a s e t a s a L i e a l g e b r ao r a L i e g r o u p i s j u s t t h e c o m m u t a t i o n p r o d u c t ( 2 .2 .3 7 ) b e t w e e n e l e m e n t s i nt h e a l g e b r a a n d t h e s e q u e n t i a l a p p l i c a t i o n , i.e . t il e m u l t i p l i c a t i o n , o f o p e r a t o r si n t h e g r o u p .

I t is a n o t a b l e p r o p e r t y o f L i e a l g e b r a s t h a t b y e x p o n e n t i a t i n g t h e e l e m e n t so f t h e a l g e b r a t h r o u g h a re a l p a r a m e t e r C w e m a y f o rm t h e a s s o c i at e d g r o u p .

A " ~ "A n e l e m e n t 7 " o f a L ie g r o u p is s a i d to b e g e n e r a t e d b y t h e e l e m e n t L o fa L ie a l g e b r a i n t h e s e n s e t h a t f o r a n i n f i n i t e s i m a l v a l u e d ~ o f t h e g r o u p


16/52

7 4 L i n e ar R a y a n d W a v e Opt i c s n P h a s e S p a c eA A Ap a r a m e t e r w e h a v e T ( d ~ ) - I + d ~ L . T h u s , t h e L i e a l g e b r a o f a g r o u p is a

d e s c r i p t io n o f t h a t g r o u p n e a r t h e i d e n t it y e l e m e n t . S e q u e n t i a l n m l t ip l i c a t io n so f i n f i n i t e s i m a l - s t e p e l e m e n t s T ( d ~ ) l e a d b y i n d u c t i o n t o f i n i t e - s t e p e l e m e n t sT ( ( ) , f or w h i c h t h e e x p o n e n t i a l r e p r es e .n t a ti o n i n t e r m s o f t h e g e n e r a t o r L isc o n s e q u e n t l y o b t a i n e d . I n f a,( :t, i t is p r o v e d b y i n ( h m t i o n t h a t

A A A t~ ~ AT ( ( + d ( ) - T ( ( ) T ( d ~ ) - T ( ( ) ( I + d(L), (2.2.39)whi(: l~ ~ ea ,ns

T ( ( ) + d T - T ( ( ) + d(LT(( ) ,ttn~s y ie] (li~ g tl~e fiI'st-(w(h :r (liff(:l'(:~tia] e(t~m.t,i()~ fi)r 7 -(( )"

' Z , - / - ( ( ) _ C rd (

(2 .2 .40 )

(2 .2 .41)"l'l~(: (:Xl)()~(:xd,ia,l.r(:l)r(:s(:~d,a,(,i()~ ()f (,l~(' gI'() ii ) (:h :lll(:id, 'T ( ( ) i~ (,(:n~ s of th ealg(: | )ra, ( : I(:~(:~(, k i~(:(l ia,( , ( : ly f()lh)ws:

Aq - ( ( ) _ , . r k , ( 2 . 2 . 4 2 )w itl l 1)eillg T(( )) - I fin" (;w;ry L ill (,ll(' a lg(;lwa..

Tlle~l, ew'~ ry ()l)(;rat(n" ill tile a.lge l)ra g elle ra, tes a s llt)gr()lll) ()f exl)(nlelltia.1-ty t)e ()I)era,t(ws {('( L; ( E R} , w lli(:ll is (:(nlt ilnu )lM y l)armll(;triz(;(1 1)y th e tea,1t )a raI ~e ter ( . ( )n( : t ,yI) i( :a .l ly sp ea ks ( )f a o n c - p a r a m , c t e r s~d)gr()~l), t ' ;ach valueof th e t)a,ra,n~cter ( i~(livi(l~m ,lizes ()11( . ()t)e rat() r i~ th e s~fl)gr()~l). ()t )er a,to rs('orrest)()~ (til~g 1,()(lifli;rel~t val~u;s ()f 1,11e t) a ra ,~ et er t '(n~n~ml,e witl~ (;a(:t~ o th er ;na,n~ely

7 ( ( , ) 7 ( ( : ~ ) - 7 - ( ( 2 ) 7 - ( ( , ) , ( 2 . 2 . 4 3 )w hi ch (l ire(: t ly follow s fr()~l~ (,l~e t)a,s i(: I 'ela,t io~ (2 .2.3 9) (:lea,rly i~l) lyil~ g in finit et e r m s t h a t

- r + G ) - + q ) - " - 2 ( G ) r ( 2 .2 . 4 4 )AT h e s u b g r o u p {c~k; ~ E ]~} is t he r e f o r e a be l i a n . A l s o , ( 2 . 2 .43 ) a l l ow s u s t o

i d e n t i f y t h e i n v e rs e to T ( ( ) a,s t h e o p e r a t o r i n d i v i d u a l i z e d b y t h e o p p o s i t ev a lu e - ( o f t h e p a r a m e t e r , n a m e l y

~' -1 (~ ) __ r __ G-~L. (2.2.45)I n t h e l ig h t o f t h e s e g e n e r a l c o n s i d e r a t i o n s , t h e L i e t r a n s f o r m a t i o n s f or a

g i v en L ie o p e r a t o r k s ar e se e n t o fo r m t h e o n e - p a r a m e t e r s u b g r o u p { ~ ( ~ )Ac r k s ; ~ E R } , f o r w h i c h

( 2 . 2 . 4 6 )


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1D First-Order O pt ical Systems: The R ay- Transfer M atr ix 7 '5

a n d_ r ( 2 . 2 . 4 7 )

E a c h o p e r a t o r in th e s u b g r o u p p r o d u c e s a s y m p l e c t i c m a p p i n g o f t h e c o n j u g a t ev a r i a b l e s q a n d p t o t h e n e w v a r i a b l e s ~ a n d p a s g i v e n i n ( 2 . 2 . 3 4 ) , i . e . ,

{ q , P } {-0 c ~ L I c ~ L I 9q , p - p } (2 . 2 .4 8 )C l e a r l y , s u c c e s s i v e L i e t r a n s f o r m a t i o n s p r o d u c e a s y m p l e c t i c t r a n s f o r m a t i o nAa s w e l l . T h u s , t h e c o m p o s i t e o p e r a t o r e (1 kf e~ 2 Ig g e n e r a t e s t h e t w o - s t e p s y m -p l e c t i c m a p p i n g

A A A A{ q , P } ~ { q -- er Lgq, iO -- er L~P} ~ { q - I ~ l L f q - . ~ _ e ~ 1 L f / , l ~ } ( 2 . 2 . 4 9 )H o w e v e r i t c a n n o t i n g e n e r a l b e a s s o c i a t e d w i t h a s i n g l e L i e t r a n s f o r m a t i o n ,u n l e s s, a c c o r d i n g t o ( 2 . 2. 4 6 ), g = f , o r m o r e i n g e n e r a l , a s w il l b e p r o v e db e l o w , { . q , h } = O .

N o t a b l y , n e a r t h e i d e n t i t y , i. e. , f o r s u f f ic i e n t ly s m a l l v a l u e s o f t h e p a r a m e -t e r s, L i e t r a n s f o r m a t i o n s m u l t i p l y i n to L i e t r a n s f o r m a t i o n s . N a m e l y , w e h a v e

z ~ f ( ~ l ) Z ~ g ( ~ 2 ) - - 7 " ~ h ( ~ ' ) , ( 2 . 2 . 5 0 )fo r ~1 an d ~2 s u f f i c i en t l y s m a l l . T h e ex p l i c i t ex p re s s i o n o f t h e c l o s u re r e l a t i o n( 2 . 2 . 5 0 ) f o r L i e t r a . n s f o r m a t i o n s , i . e . , f o r e x p o n e n t i a l o p e r a t o r s , w i t h i n a s u f -f ic i cn t ly s m a l l n e i g h b o r h o o d o f t h e i d e n t i t y i s k n o w n a s t h e B a k e r - C a m p b e l l -H a u s d o r f f ( B C H ) f o r m u l a . I t d i c t a t e s t h e c o n d i t i o n s a n d t h e r u l e s t o u n i t t w oe x p o n e n t i a l o p e r a t o r s i n to a s in g le e x p o n e n t i a l o p e r a t o r [8 ]. T h e e x p o n e n t s o ft h e f a c t o rs o n o n e s id e c o m b i n e i n t o a s in g le e x p o n e n t o f t h e p r o d u c t o n t h eo t h e r , w h i c h i s u l t i m a t e l y d e t e r m i n e d b y t h e u n d e r l y i n g a l g e b r a i c s t r u c t u r e ,d i r e c t l y i n v o l vi n g t h e L i e p r o d u c t s o f t h e e x p o n e n t s o f t h e f a c t o rs .

H e r e w e w i ll p r o v e ( 2 .2 . 5 0) i n a r a t h e r n aY v e w a y , e x p l o i t i n g t h e p o w e r s e r i e se x p a n s i o n ( 2 .2 .1 8 ) . F i r s t l y w e c o n s i d e r t h e i n t e r e s t i n g c a se t h a t { f , g } = 0an d s o [L / , Lg] - 0 . I n t h a t c a s e , i n f ac t , t h e c l o s u re r e l a t i o n (2 . 2 .50 ) o f L i et r a n s f o r m a t i o n s h o l d s fo r f i n it e v a l u e s of t h e p a r a m e t e r s ~ , a n d ~2, w i t h t h eI_ _lm u l t i p l i c a t i o n l a w t u r n i n g i n to t h e a d d i t i v i t y o f t h e e x p o n e n t s a s

~ f ( ~ l ) z ~ g ( ~ 2 ) - - " ~ l f + ~ 2 g ( 1 ) - ' T g ( ~ 2 ) ~ f ( ~ l ) ~ ( 2 . 2 . 5 1 )w h i c h f u r t h e r b e c o m e s m e r e l y a d d i t i v e w i t h r e s p e c t t o t h e p a r a m e t e r s ~ 1 a n d~2 i f a lso g = f , i .e . ,

~ f ( ~ l ) ' T f ( ~ 2 ) - - ~ f ( ~ l - J l - ~ 2 ) - - z ~ f ( ~ 2 ) ' T f ( ~ l ) , ( 2 . 2 . 5 2 )t - . .t h u s r e c o v e r in g t h e L / - g e n e r a t e d s u b g r o u p r e l a t i o n ( 2 .2 .4 6 ).


18/52

7 '6 Linear Ray and Wave O ptics in Phase 5paceAI t w il l b e e v i d e n t t h a t t il e h y p o t h e s i z e d c o n u n u t a t i v i t y o f t il e o p e r a t o r s L f

a,n d L.q is c ru ci a,1 to e x t e n d t h e a d d i t i v i t y o f t h e e x p o n e n t s f r o m e x p o n e n t i Ms c a l a r s t o e x p o n e n t i a l ( ) p e r a t o r s , t h u s y i e h t i n g t h e s ( 'a ,l a r- l ik e r e l a t i o n ( 2 . 2 . 5 1 ) .

W e c o n s i d e r t h e e x p o n e nt ia , 1 o t ) e r a t o r o n t h e r i g h t o f ( 2 . 2 .5 1 ) , w h i c h e x p l i c -i t l y wr i t e s a , s~ ~ ~ 1 ^ ) j, T C , / ~ < ~ : , ( 1 ) _ , , < , L , + ( : ~ L :, _ Z . ~ . ( ( , k f + ( 2 " L . q . ( 2 . 2 . 5 3 )

j --()

a , n ( t l l ( ;X l ( :e ( h ; l l m l l ( l s ( ; v a h l a t , i l l g t ) ( ) w ( ; r s ( ) f t h e ( ) t ) e r a t ( ) r ( ~ L I + ( ~ L , / . T l l( '~ S ( llm , r ('~i s , f l ) r i l l s l , m l ( : ( ; ,( ~ , L ~ + ~ L , , ) " - ( r L ~ + C ~ L , , ) ( C , L ~ + ~ L , , )

.,. 2 . . . . . ' )( 2 L s + ( , ( 2 L ~ L : , + ( . , ( , L : , L s + q;. L ; ( 2 " 2 ^ ^ . .,"" . - L s + 2 ( s ( 2 L.r L:I + (,~ L ~ .I t sll()lll(1 I)(~ ll()t,(',l ill ti le lnlfi )l(le r (~xl)r(~ss i~nl a.1)r t, ln ' two) 1)rr LI L.qa,u(1 k: /L I, wl~i(:l~ a r~', i~ g(',~nwal ~lifli~r~',~l,. thn~('~', t, l~ y (:m~ 1~(~s~n~n~(l I ,~ 2 k~ L.qon ly ~ ( l ( w l, ln~ a, s s ~ l) t i ( )~ ()f t l~(~ (: () ~n lt a , t i vi l , y ()f k z toni L:t . l~ l ,lm.t ( :a .s(; ,t,l~(: S(l~m r(', ~f t, ln' (~l)(Wat,~w (, LI ~ (~ L:~ (:() ~('s t,~ l~aw ' t,l~(' st ai n, ('xl)r (;ssi ()na,s tl~a,t (~f t,l~(, s(l~m,r~' ~f a I~y~ n~ia,1 ~f s(:a lm's. A (:('~ r(li~ gly , ti n' j-tl ~ I)(~wer((1 LI + (~ L.,~)J (:m ~ 1~', (:al(:~ lla,t(~! l)y 1,1~(~well k ~ ) w ~ fin'~n~la, f i~r t ln~ j-t l~ l)()wer~f a, t)y~r

J. . . . A j _ !9 L / ( . j - ' L : / ( 2 . 2 . 5 5 )C , L / + ( ~ L : ,) ~ - ~ ( 1 ) ( [ /l = ( )

UI)()n s~ fl) st i t~t i~g (2. 2.5 5) i~d,(~ t lm t)()w(;r ser ies (2 .2. 53 ), w(; ~4)l,a i~^ A o ~ j 1 l - - - I A j - I( " ~ 1 Lf-[-Q2 L 'q---- j--()~ l-- 0 ( J - l ) ! l ' Q l " k s ( j - z k . ~

= Z ~ ( I L / ( j - z ) ! L . q . ,/ = 0 j /

t h u s t ) r () v i~ g r e l a t i o n ( 2 . 2 . 5 1 ) .W e ( ;~ t )ha , s i ze aga i n tha , t (2 .2 .51 ) g r ou nd s ( )n the ex t ) r( ; ss i ()~ (2 .2 .55 ) fo r

p o w e r s ()f a, b y n o m i a l o f o t ) e r a, t o r f u n c t i o n s , w h i c h r e p r o d u c e s t h a ,t f or p o w e r so f a, b y n ( )n f i al o f s c a J a r f im ( : t io n s o n l y f o r c o m m u t i n g o p e r a t o r s ( P r o b l e m 3 ).

I n g e n e r a l , t o p r o v e ( 2 . 2 . 5 0 ) w e e x p l o i t t h e p o w e r s e r i e s e x t ) a n s i o n ( 2 . 2 .1 8 ) .W e s u t ) t ) o se n o w t h a t b o t h t h e rea,1 n u m b e r s (~ a n d (~, a r e s u f f i c i e n tl y s m a l lt o j u s t i f y t h e t r u n c a t i o n o f t h e s e ri e s e x p a n s i o n s o f b o t h T f ( ( l ) a,nd T~( (~) a tt h e f i r s t o r d e r o f (1 a n d ( ~ . W e w r i t e t h e n

A A A' T f ( ~ "_ ~ O ) - - [J l - t . . ~, .k~' ( 2 . 2 . 5 7 )% ( q ~ o ) - I + Lg,


19/52

1D First -Orde r Op t ica l Systems: The R ay- Transfer M atr ix 77'A As o t h a t t h e p r o d u c t o f T I f f1 ) a n d T g ( ( 2 ) i s

A A A A AT I ( r - I + r k f + < 2 L g, < 1, ~ 2 ~ 0 ( 2 . 2 . 5 8 )w h e r e , o f c o u r s e , o n l y t h e f ir s t o r d e r t e r m s i n t h e p a r a m e t e r s ~1 a n d .(2 h a v eb e e n r e t a i n e d . R e s o r t i n g a g a i n t o t h e p o w e r s e r i e s e x p a n s i o n , w e c a n r e c o v e rt 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 a s

A A AT f ( ~ 1) ~ 9 < 2 ) c ( l k f ~ 2 k g. . t - . _ _ _ _ _ G k C l f + ~ 2 g _ ~ ( 1 / + ~ 9 ( 1 ) ' (2 . 2 . 59 )fo r r an d r s u f f i c i en t ly s m a l l , an d h en ce n ea r t h e i d e n t i t y ]'.

A s p r e v i o u s l y r e m a r k e d , t h e v a l i d i t y of t h e m u l t i p l i c a t i o n l aw ( 2 .2 .5 1 ) f orL i e t r a n s f o r l n a t i o n s n e a r t h e i d e n t i t y g r o u n d s o n t i l e B C H t h e o r e l n , w h i c hp r o v i d e s t h e r u l e t o c o m b i n e t h e e x p o n e n t s o f t i le f a c t o r s o n t h e l e ft a s

A A A A AC


20/52

7 8 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 5 p a c e

W h e n g e o m e t r i c a l o p t i c s is c o n c e r n e d a n d t h e L ie t r a n s f o r m a t i o n is a s s o c i a t e dw i t h t h e H a m i l t o i f i a n f lo w , t h e e x p o n e n t i a t i o n p a r a m e t e r ~ t a k e s th e ro l e a st h e a x i a l d i s t a n c e Z o - z i b y w h i c h t h e l i g h t r a y p r o p a g a t e s a n d a c c o r d i n g l yt h e L i e tr a, n s f o r m a t i o n i s t h e o p t i c a l " p r o t ) a g a , t o r " o f t h e r a y f r o m z,i to Zo.W e no te th at , a ,s th e ot) ti (:a,1 Haini l tonia .x~ is ~m it less , t i le ( :orre s t )on (t ing Lie~ ) I )c rator t ins th e ( t im ens i (m ( )f l e i~g th -1 .

2 . 2 . 3 R a y - b u ' n , d h : t r m p a g a t i o n : th , c r a y t 'r a' n ,s fc r o p e r a t o r a n d m a tri:r~A l)t) lyi ~g tl~(; r(;s~ lt (2.2.3 ()) t() E(t . (2 .2. 13 ) we ii~ul t lm.l, p( q, p; z) (;v()lv(;s fi 'om1,11(; ilfi tia .l (lis l,ri l)~ l,i() ~ p ( q , p ; z i ) a.(:(:()r(til~g 1,()

t , ( q , p ; z ) - t , ( , , - ( ~ - ~ , ) t . , , q , ` - ( : - : , ) , , , p; z~) . ( 2 . ~ . ~ )W e re(:all l , l~a,t i~ .~ 1.4.1 wl~e~ i~lv estig a,t i~g th e z-ev~fll~l, i~n~ ~f t ln; l)o sit io na,n d (tire('l,i{) l~ (x)~n'{li~m,tes ~)f ti m ra y , w~: i~d,Ix)~tlu'(:~l l, w, ra,y-l.ra.l~sf(;r ~)l)e I'a,torA9 Y l ( z , z i ) a~{t l)r{)wxt t l~a.t i~ t im ca se (~f a z-iluh ;l)~;~{te~t I la~ ilt~n ~ia ,~ it ta k esl,h(; exi)()~n;~d, a.l,e(t fi)rx~

~ l ( z , z i ) - ,'(~ '- :~ ) L , , . ( 2 . 2 . 6 3 )A (:(:()r(l i~g t(~ (2.2. ,17 ), t in: ()t)(;ra, t()r (:~t(:ri~ g tl~(; a r g ~ ( ; ~ t s (~f p i~ (2. 2.6 2)

a.t)I)(;a.rs t() I)(: tl~(; i~v(;rs(; ()f tl~(; ra,y -tra ,~sf i;r ()l)(;rat()r (2 .2 .6 3) , ~m~l(;lyA

, - ( ~ - ~ . , ) t _ , , _ . ~ - ~ ( z , z i ) . ( 2 . 2 . ~ 4 )In fa,(: t , if t l ,(; ray w,.ria, l)h;s u(z) - ( q ( z ) , p ( z ) ) T (;v()lv(: fix),,, t l,(; i,,I ),,t (ta, a,u ( z i ) t ) y 9 Y t ( z , z i ) , (:()l~v(;rs.(;ly, th(; int)~lt (la, a u (z i) (: a, ~ t)(; r(;(:()v(;r(;(t fr()m u ( z )1)y th e i~lv(;rs(; ()t)e ra,to r 9)1 - l (z, zi)"

u ( ~ ) - @ - ~ (~ , ~ ) u ( ~ ) . (2 .2 . (~5 )R c la ,t i~ m (2 . 2 . 6 2 ) s ~ g g e s t s t h e r e fo re t he ; f o l l o w i n g t ) i c t ~ r e . L e t ( q, p ) b e t h ec o o r d i n a t e s o f a r a y a t z , t l m n ( c - ( z - ~ ' ) e - q , c - (~ -~ ~ ) e , p ) m a , y t) e i n t e r p r e t e d

a s t h e s t a r t i n g d a, a f o r t h e ( 't~ ose n r a y ( q, p ) c o n s i s t e n t l y w i t h t h e H a m i l t o n i a nd y n a m i c s , i d e n t i f ie d b y t h e r e l ev a n t L i e o p e r a t o r [ , . E v i d e n t l y , t h i s is a m a n -i f e s t a ti o n o f t h e p r e v i o u s l y e s t a b l i s h e d i n v a r i a n c e o f t h e p t m s e - s p a c e d e n s i t yM o n g t h e p h a s e - s p a c e t r a j e c t o r y .

AR e m a r k a b l y , a c c o r d i n g t o ( 2 .2 . 62 ) , t h e L i e t r a n s f o r m a t i o n c (z -z ~) e - , w h i c his b a s i c t o t h e s i n g l e - r a y t r a c i n g , b e c o m e s a s w e l l b a s i c t o t h e r ~ y - e n s e m b l ep i c t u r e o f l i g ht p r o p a g a t i o n r u l i n g t h e e v o l u t i o n o f t h e p h a s e - s p a c e d m ~sit y.

I n w 1 . 5 .3 w e m e n t i o n e d t h a t i n ge ne ra .1 t h e r a y t r a n s f e r o p e r a t o r 0 Jr (z , z i)c a n b e r e p r e s e n t e d i n t h e f a c t o r e d f o r m a s t h e p r o d u c t o f L i e t r a n s f o r m a t i o n s .T h e n , a p p l y i n g p r o p e r t y ( 2 .2 .3 0 ) to e a c h t e r m i n t h e p r o d u c t o r e x p l o i t i n g th e


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1D F irst-Order Opt ical Systems: The Ray-Transfer Ma tr ix 7 '9

g r o u p c l os u r e p r o p e r t y ( 2. 2.5 0 ) o f L ie t r a n s f o r m a t i o n s , w e m a y r e s t a t e r e l a t io n(2 . 2 . 6 2 ) i n t h e g en e ra l fo rmf l ( q , p ; Z ) - - p(@ -I(z, z i ) q , @ -I(z, z i ) p ; Z i ) - - pi(@ I(z, z i ) q , @ -I(z, Z i ) p ) ,

(2 .2 .66)i n v o l v in g t h e r a y - t r a n s f e r o p e r a t o r , o f w h i c h t h e s i n g l e - e x p o n e n t i a l f o r m ( 2 .2 . 63 )is a sp e ci fi c d e t e r m i n a t i o n , s u i te d t o a z - i n d e p e n d e n t H a m i l t o n i a n .

I n p a r t i cu l a r , s i n ce i n t i l e l i n ea r ap t ) ro x i I n a t i o n a 4 x 4 o r 2 x 2 m a t r i xr e p r e s e n t a t i o n i s a l l o w e d f o r t h e r a y t r a n s f e r o p e r a t o r , p r o p e r t y ( 2 . 2 . 6 2 ) c a . nb e f o r m u l a te d in t e r m s o f t h e r a y - t r a n sf e r m a t r i x M ( z , z i ) as we l l . Th u s , u s i n gt h e h a n d y v e c t o r n o t a t i o n , w e c a n w r i t e

p(uT; Z )- p ( v T ; z i ) - pi(vT), V - M - l ( z , z i ) u , (2 .2 .67)w h i c h i n p a r t i c u l a r s h o w s t h a t

p(0 , 0; z) = p~(0, 0), (2.2 .68 )b e i n g t h e p h a s e p l a n e o r ig i n m a p p e d i n to i t se l f b y M .

E q u a t i o n ( 2.2 .6 6 ) a n d t h e c o r r e s p o n d i n g l i n e a r m o d e l f o r m ( 2.2 .6 7 ) s ig n if yt h e r e l e v a n c e o f t h e r a y - t r a n s f e r o p e r a t o r 9 2 ~ ( z , z i ) a n d t h e r a y - t r a n s f e r m a -t r i x M f or b o t h t h e s i n lg e - r a y a n d r a y - e n s e m b l e p i c t u r e s o f l i g ht p r o p a g a t i o nt ) e r t i n en t l y t o t h e g eo m e t r i ( : a l-o t ) t i c s v i ew . E q u a t i o n s (2 . 2. 6 6 ) an d (2 . 2 .6 7 )c l e a r l y s h o w , i n f a c t , t h a t a n y d i s t r i b u t i o n f u n c t i o n i s p r o p a g a t e d a l o n g t h er a y - t r a j e c t o r i e s i n p h a s e s p a c e . I n p a r t i c u l a r , t h e s i n g l e - r a y p i c t u r e i s e a s i ly re -c o v e r e d f ro m ( 2 .2 .6 6 ) b y n o t i n g t h a t i f t h e p h a s e - s p a c e d i s t r i b u t i o n is i n i t i a ll yr e p r e s e n t e d b y a si n g le ra y , a n d s o m a t h e m a t i c a l l y m o d e l l e d b y a (5 f u n c t i o n ,

p ( q , p ; z i ) = ( 5 ( q - q i ) r S ( p - p i ) , (2 .2 .69)i t r em a i n s a (5 - f i m c t i o n a t an y z > z i , ev o l v i n g i n t o

p ( q , p ; z ) = g i ( q - q ( z ) ) r S ( p - p ( z ) ) . (2 .2 .70)H e r e , q ( z ) a n d p ( z ) a r e t h e r a y v a r i a b l e s a t z p r o p a g a t e d f r o m t h e i n p u t d a t aA(q~ , p~ ) t h r o u g h t h e t r an s fe r o p e r a t o r 9 )l ( z , z~) o r , i n t h e l i n ea r m o d e l , t h r o u g ht h e t r a n s f e r m a t r i x M ( z , z~) (s ee a ls o P r o b l e m 4 ). A c c o r d in g l y , t h r o u g h o u tt h e r e m a i n i n g o f t h e c h a p t e r w e w i ll w o r k w i t h i n t h e s i n g l e - r a y p i c t u r e o f l in -e a r g e o m e t r i c a l o p t i c s . I n d u e a c c o u n t o f t h e e s t a b l i s h e d s i g ni f ic a n c e o f t h er a y - t r a n s f e r m a t r i x , i n t h e n e x t s e c t i o n s w e w i ll p r o c e e d d e r i v i n g t h e r a y m a -t r i c e s f o r a s s i g n e d z - i n d e p e n d e n t m o d e l H a m i l t o n i a n s . T h e i r c o r r e s p o n d e n c et o r e c o g n i z a b l e o p t i c a l s y s t e m s w i l l t h e n b e i n v e s t i g a t e d .

B e f o r e c l o s in g t h i s s e c t i o n w e w i s h t o i l l u s t r a t e t h e r e l a t i o n o f t h e p h a s es p a c e d e n s i t y p ( q , p ; z ) t o t h o s e m e a s u r a b l e q u a n t i t i e s t h a t t h e t r a d i t i o n a lr a d i o m e t r y u s e s t o c h a r a c t e r i z e t h e l i g h t d i s t r i b u t i o n a n d i t s p r o p a g a t i o n i nc o n n e c t i o n w i t h t h e r a d i a t i v e p r o p e r t i e s o f t h e s o u r c e.


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8 0 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 S p a c e

2 . 2 . 4 P h a s e s p a c e d e n s i t y a n d r a d i a n c eA s a n n o u n c e d , i n t h is p a, r a gr a, p h w e e l u c i d a t e t il e c o n n e c t i n g l in k b e t w e e nt h e b a ,s ic q u a n t i t y o f t h e p h a s e s pa,(:(; r e p r e s e n t ~ t i o n o f g c o m e t r ic a , 1 o p t i c s ,i . e . , t i l e t )hase s t )a , ( 'e dens i ty p ( q , p ; z ) , a,n (t t h e b a s i c q u a n t i t y o f r a d i o m e t r y ,i . e ., t i m r ad i an ( : e ( ~) . T h e ( ti s r i s s t r i c t l y a i m c( t a ,t g i v i ng a, f e e l i ng f o r t het)h ys i(:a ,1 c o n ( : r c t e n e s s o f t h e t )h a ,s c s p a ( :c d e n s i t y . A c c o r d i n g l y w e m e r e l y d e r i v eth e f() rnm,1 relat i ( )~l of p t ( ) t im ra, ( t imlcc f ll~l( :t ,i r a( l ( l re ss in g t im tea ( l et to [9]fo r a fl l l le r a(:(:()l l ld, ()f ])() t l l t , t l(; ra(tio~ lletri(" l l l()(l(;1 a,ll(l t l l(; re la ti v el y re( '.cn tlyinv( ; s t iga , t e ( t l i l lk wi t, t l t il t , s t a t i s t i ( : a l wave; I , t l ( ; (~ry.

W(; ( 'l l lI)t lasiz (; t im .t , a,s g(;()Il l(; tri(:al ()t)ti(:s, ra(ti t)l l l t ; tI ' y a n(t t h e r(; la. tc(t th e-o ry ()f ra.(l ia. t iv e (u le rg y l, ra,l lSl)t)rt ar e |)a,se(l ()ll 1,11(' ( :(nlt:el)t ()f t, m lig ht ra y,lm(t(;rst()()(t a,s t l l( ' gt;(n ll( 'tr i(:a l | , l 'aj(;(t i,()ry a l(nlg wlli(: l l t , ll(' ()l)t i( 'al t ; ll t ;I 'gy, ra( ti-a , te(t | )y t i le s(nlr( :e , I) r t ) lm ,gates [ l () ] . T ln ls , ar t ( ) I, m ra( l i ( )m et l ' i ( : nl ( ) ( tel ,t l l e rar (; lm rg y is l()( 'alize(I |)()t ,l~ i~ sI)a( '(; a~ (l i~ (l ir(;( 't i()~. W(; will s('(; t l m tt h e rar is (h;fil~(;(l as a fi m (' ti (m ()f |)()t,l~ l)()sil,i()l~ a,l~(l (lire (:ti()~ . P()sit, i()na,i~(t r162 a,r(; tl~(; ln~ il(li~ g W tl'ial)lt;s ()f tl~(, g( ;()~ ('tr i(:a l-( )I )tir I)lm,s(' sI)a,(:e,w hi( :h th t;n s(;( ;~s i ,()()f[t ;r 1,1~(; lm t~ ra l fi 'an~(; ft)r t im ra.(l i tn~m l,ri(' th(:() ry a.s we ll .

I{ a,r t~as I)(; liev(;(l I,()t)r()t)(; l ' ly (h;st 'r i l)(; l , l~e I)el~av i()r ()f ra,


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1 D First-Order Op t ical Systems: The Ray-Transfer M atr ix 8 1w a v e - o p t i c a l p h a s e - s p a c e d e s c r i p t i o n o f l ig h t f ie ld s r e l a t e t o t h e g e n e r a l i z e dr a d i a n c e f u n c t i o n s p e r t a i n i n g t o r a d i o m e t r y w i t h p a r t i a l l y c o h e r e n t s o u rc e s .M o r e o v e r , b o t h t h e g e n e r a l iz e d r a d i a n c e f l m c t i o n s o f r a d i o m e t r y a n d t h e s p a c e-f r e q u e n c y d i s t r i b u t i o n f l m c t i o n s o f w a v e o p t i c s d o n o t r e t a i n i n g e n e r a l a l l t h ep r o p e r t i e s e x p e c t e d f o r t h e m in t r a d i t i o n a l r a d i o m e t r y a n d i n t h e o r d i n a r yp r o b a b i l i t y t h e o r y , r e s p e c ti v e ly . T h i s f a c t i n d i c a t e s t h a t n e i t h e r t h e f o r m e r sn e i t h e r t h e l a t te r s c a n i n g e n e r a l f o r m u l a t e d in a m a n n e r t h a t is s t r ic t l y s im i l a rt o t h e t r a d i t i o n a l r a d i o m e t r i c a n d g e o m e t r i c - o p t ic a l d e s c r i p t i o n o f l ig h t fie ld s[9.9]. H o w e v e r , i t w a s s h o w n t h a t a ll t h e g e n e r a l i z e d r a d i a n c e f u n c t i o n s r e g a i nt h e p r o p e r t i e s p o s t u l a t e d f or t h e r a d i a n c e in c o n v e n t i o n a l r a d i o m e t r y , i n t h ec a se o f p l a n a r s e c o n d a r y q u a s i - h o m o g e n e o u s s o u r c e s (a) a n d in th e l i m i t o fs h o r t w a v e l e n g t h s . T h i s r e s u l t u n e q u i v o c a l l y e s ta b l i sh e s t h e p l ac e a n d t h e r o leo f c l a s s ic a l r a d i o m e t r y w i t h i n t h e s t a t i s t i c a l w a v e t h e o r y o f o p t i c s [9.1 1, 9.1 2].R a d i a n c e : b a s i c d e f i n i t i o nT h e r a d i a n c e (o r b r i g h t n e s s ) is t h e a r e a a n d s o li d a n g le d e n s i t y o f t h e r a d i a n tp o w e r . I t d e p e n d s in g e n e r a l o n th e p o i n t i n s p a c e a t w h i c h i t is e v a l u a t e da n d o n t h e d i r e c t i o n t h r o u g h t h a t p o i n t . I n o r d e r t o v i s u al iz e t h e d e f i n it io n o fr a d i a n c e w e c o n s id e r a p l a n a r r a d i a t i n g s u r fa c e S in a h o m o g e n e o u s m e d i u mo f r e f r a c t i v e i n d e x n 0. S m a y b e t h e s u rf a, ce o f a r e a l s o u r c e o r o f a n o p t i c a le l e m e n t l ik e a m i r r o r , a le n s o r a l i m i t i n g a p e r t u r e ; a l so , i t : n a y b e a , f i c t i ti o u ss ur fa ,c e s i t u a t e d e v e r y w h e r e i n t h e r a d i a t i o n f ie ld . W e m a y l o c a t e S in t h e p l a n ez - 0 , a n d s u p p o s e i t r a d i a t e s i n t o th e h a l f - s p a c e z > 0 . T h e n , f o r a n y p o i n tP s o n S , s p e c i f i e d b y t h e p o s i t i o n v e c t o r q - (q x, q y) i n t h e s o u r c e p l a n e , w ed e n o t e b y d S t il e e l e m e n t o f a ,re a a r o u n d P s , a n d b y d ~ t il e e l e m e n t o f s o li da n g l e a r o u n d a d i r e c t i o n s p e c i f i e d b y ti le u n i t v e c t o r v o r i g i n a t i n g f r o m P s( F ig . 2.4 .a ,) ). I t is a s s u m e d i n r a d i o m e t r y t h a t t il e p o w e r d P [W ] r a d i a t e d b yt h e s o u r c e e l e m e n t d S w i t h i n t h e s o l id a n g l e e l e m e n t d r2 is o b t a i n a b l e a c c o r d i n gt o t h e e l e m e n t a r y l a w

d P - / 3 ( % v ) c o s O d S d f ~ . ( 2 . 2 . 7 1 )H e r e 0 is t il e a n g l e t h a t t h e v - d i r e c t i o n m a k e s w i t h t h e p o s i t i v e z a x is ; e v-i d e n tl y , 0 < 0 < 7 : /2 . T h u s , t h e p r o d u c t d A - d S c o s 0 y i e l d s t h e p r o j e c t e de l e in e n t o f a r e a p e r p e n d i c u l a r t o v . T h e f u n c t i o n B ( q , v ) is k n o w n a s t il e r a -

d The opt ica l in tens i ty across a planar q u a s i h o r n o g e n e o u s source changes so slowly withposi t ion in the source plane tha t i s remains essent ia l ly constan t thro ugh dis tances com parablewi th the cor re la t ion leng th of the radia t ion f ie ld across the source plane. A lso, i t i s supposedtha t the l inear dimensions of the source are large com pared w i th both the m ean wav elengthof the rad ia ted l ight and the co r re la tion length. Ev ident ly quas ihoinogeneous sources maybe spat ia l ly coherent and incoherent in the " local" sense , but are a lways spat ia l ly ra therinco here nt in the "glo bal" sense [9.8, 9.4].Final ly we recal l that a s e c o n d a r y source radiates after direct or indirect ( i .e. , thr ou gh anopt ical sys tem) i l luminat ion by a p r i m a r y source.


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8 2 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

J q.l'

qxs d S

q )

q x

( a ) ( b )

dr)

FIG UII , I" 2 .4 . I~a .s ic e le i~e ld, S i l l~ s t ra t i~ g t im (h; li~f i( ,ioI~ , ) f r a r (a ) T i m l ) la lmr so~ rceS i~ t l~e pira te z = () ra ( l i at i ~g i l~to t im h alf- l) lmm z > () m~(l ( b ) t im g( :o~mtr ica l q~mi~t i t icsi~fl~crei~( , iI~ Eq. (2.2 .7 1) , i .e . , t lw ar ea el(~lt'~H, d,S ' aro~n~(l t t~e s(n~rce l) () i~l, l ' s( q ) aim th ee le m en t d ~ of soli ( l m~gh', a r ()~n~( l t lw ray- ( l i r t :c t io~ v.

( l i m i t : ( : ( ) r l w i g l , l l l l C S s ( ) f l , l l ( : s ( ) l l r c ( : . I t , r t : l ) r t : s t : l l l , s 1 , 1 1 ( : r a l , ( : a l , w l l i ( : l l e l l t : r g y i sr a , t t i a t c d i n ( , ( ) 1 ,1 1 ( : l l l l i ( , s ( ) l i ( l a.l l gh ; a r ()111l(l l , l l ( : s t ) ( : ( : i f i ( : ( 1 ( t i r ( : t : l , i ( n l v t ) y (,hi: s()lxr(:(:( : l( : l l le ld, ( ) f 1111i1 l) r()j( :( :( ,( :(l a r ( : a ( t) ( ;r l ) ( ql ( l i( ' l l la r 1,() v ) a l ' ( ) l l l l ( l I l l ( : Sl)( ' ( ' i I i ( :( l I)O ild,- '> - I ( ,, )P s ( q ) ; l l( '~ ll (:e i (, i s l l l( ;a,S lU ' (; ( l i l~ I V . ~ - . . s t

A ( : ( : ( ) r ( l i~ g 1 ,( ) i t s l ) lWs i ( : a ,l s i g~ l i f i ( 'm 1( : ( : , ( ,h ( : r a ( l i a ,n ( : ( : i s ~m ( : ( : s sa r i l y ~ ( ) m m g a , -(,iv(:,

B ( q , v ) > () ( 2 . 2 . 7 2 )f i n " a l l l ) ( ) s s i ] ) I ( : w d ~ ( : s ( ) f i t s a r g ~ m ~ ( : ~ ( , s , m 1 ( l s ( ) f i ) r a , l l l ) ( ) s i ( , i ( ) l l v ( : ( : t ( ) r s q = ( q : ~ , % )i ~ i t , h ( : s ( ) ~ r c ( : l ) l a ,~ 1 ( : m ~ ( l ( l i r ( : c ( , i o ~ i s v l ) ( ) i ~ t i ~ g i ~ ( ; ( ) t , h ( : I m . l f - s l ) a , ( : ( : z > ( ) . A l s o ,i t , v a n i s h ( : s a . ( , l ) ( ) i n t s q i ~ ( , h ( : s ( ) ~ n ' c ( : l ) l a , n ( : l y i n g ( ) ~ I t , s i ( l ( : ( , I ~ ( : s ( ) ~ r ( : ( : a , r ( : a , S , i . e . ,

/ 3 ( q , v ) - - () i f q ~ S , ( 2 . 2 . 7 3 )f ( ) r a , l l t l l ( : t ) ( ) ss i t ) l ( : ( l i r ( : ( ' t il l r ( : ( : sur fa , ( : ( : S .E v i ( t ( : n ( , l y , ( ,t 1( : t o t a . 1 t ) () w ( : r r a , ( t i a t e ( l | ) y t h e s o u r c e i s

p - co.. 0 s v).S , 74)e O i l a c c o u n t o f t h e s p e c t r a l c o n t e n t o f t h e e m i t t e d r a d i a t i o n , i t is c u s t o i n a r y t o c o n s i d e rt h e c o n t r i b u t i o n s t o t h e r a d i a t e d p o w e r fo r e a c h f r e q u e n c y c o m p o n e n t s e p a r a t e l y . T h e s y m b o ld P ~ [ W - r i m - 1 ] is t y p i c a l l y u s e d w i t h t h e m e a n i n g a s t h e l i m i t i n g v a h le o f t h e r a t io A P / A C ef o r f r eq l x e n c y i n t e r v a l A c e b e i n g m a d e i n f i n i t e s i m a l l y s m a l l ; A P i s c l e a r l y i n t e n d e d a s t i lea m o u n t o f r a d i a n t p o w e r c o n t a i n e d w i t h i n A w a r o u n d t h e s p e ci fi c f r e q u e n c y co. N e e d l e ss

t o s ay , d P ~ o b e y s t h e l a w ( 2 .2 . 7 1 ) i n t e r m s o f t h e s p e c t r a l r a d i a n c e B ~ ( q , p ) [ W . m - 2s r - 1 . r i m - l ] . W e w i ll i g n o r e h e r e t h e s e f r e q u e n c y - r e l a t e d a s p e c t s o f t h e r a d i o m e t r i c m o d e l ;a c c o r d i n g l y w e w i ll n o t c o n s i d e r t h e p o s s i b l e f r e q u e n c y - d e p e n d e n c e o f t h e r e f r ac t i v e i n d e x .


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1 D First-Order Op tical Systems: The Ray-Transfer M atrix 8 3

t h e i n t e g r a t i o n s e x t e n d i n g o v e r t h e s o u r c e a r e a S a n d o v e r t h e 2 7 r -s olid a n g l ef o r m e d b y a ll t h e d i r e c t i o n s v p o i n t i n g i n to t h e h a l f - s p a c e z > 0.T h e r a d i a n c e a s a p h a s e - s p a c e f u n c t i o nB e i n g a f u n c t i o n o f b o t h p o s i t io n in s p a c e a n d d i r e c t i o n f r o m t h a t p o s i t io n ,t h e b r i g h t n e s s c o m e s n a t u r a l l y t o b e a k in d o f p h a s e s p a c e f u n c t io n . E v i d e n t l yt h e p a i r ( q , v ) i d en t if ie s a r a y e m e r g i n g f r o m t h e s o u r c e p o i n t P s ( q ) i n t o t h ev d i r e c t io n ( F ig . 2 .4 . b )) , t h e r a y m o m e n t u m p b e i n g o b t a i n e d j u s t s c a l in gt h e u n i t v e c t o r v b y t h e r e f r a c t iv e i n d e x v a l u e n o o f t h e m e d i u m . T h u s t h er a d i a n c e c a n a s w e ll b e u n d e r s t o o d a s a f lm c t i o n o f t h e p r o p e r r a y - v a r i a b l e s( q , p ) : B ( q , v ) ~ , B ( q , p ) .

R e s o r t i n g t o th e p h a s e - s p a c e p i c t u r e o f l ig h t d i s t r i b u t i o n s i n t e r m s o f t h ed e n s i t y o f r a y s i n p h a s e s p a c e , i t is e v i d e n t t h a t t h e r a d i a t i n g s u r f a c e S c o r re -s p o n d s t o a p h a s e - s p a c e d e n s i t y f u n c t i o n , w e m a y d e n o t e a s Ps ( q , P ) - L i k ew i s et h e s a m p l e r a y t h r o u g h t h e s o u r c e p o i n t P s ( q ) e x t e n d i n g in t h e v d i r e c ti o nc o r r e s p o n d s t o t h e p o in t ( q , n 0 v ) i n p h a s e s p a c e. T h e n , t h e r a y s e m i t t e d b yt h e s u r f a c e e l e m e n t d S a,t P s w i t h i n t h e s o l id a n g l e d f~ a r o u n d v i n d i v i d u a l -iz e t h e e l e m e n t d l ; o f v o l u m e i n p h a s e s p a c e , w h i c h e n c l o s e s t h e s a m p l e r a y.A c c o r d i n g t o t h e g e o m e t r i c -o p t i c a l p i c t u r e o f u n c o r r e l a t e d r a y s, g i v in g i n de -p e n d e n t e n e r g y c o n t r i b u t io n s , w e c a n s a y t h a t ea,ch ra y r a d i a t i n g f r o m t h es ur fa , c e c~ I 'r ie s a c e rt a, in a m o u n t o f p o w e r i n d e p e n d e n t l y o f t h e o t h e r e m i t t e dr a y s . E v i d e n t l y , d P i s t h e p o w e r c a r r i e d b y th e p h a s e - s p a c e v o l u m e d I ;, a n dso i t i s a l so ob t a in~ tb l e as

d P = (5 P ps ( q , p ) d l 2 , ( 2 . 2 . 7 5 )H e r e (SP r o u g h l y d e n o t e s t h e a v e r a g e p o w e r c a r r i e d b y e a c h r a ,y r a d i a ,t e d b yt h e su r fa c e S . T h e p r o d u c t ~ P P s ( q , P ) a c q u i r e s th e m e a n i n g a s t h e o p ti c a lp o w e r d e n s i t y i n p h a s e s p a c e o f t h e l ig h t f ie ld a c r o s s t h e r a ,d i a, ti ng s u r f a c e . I tis w o r t h n o t i n g t h a t t h e l i ne a r a d d i t i o n o f p o w e r r e p r e s e n t e d b y t h e i n t e g ra l( 2 .2 . 7 4 ) f o r t h e t o t a l p o w e r is c o n s i s t e n t w i t h t h e v i e w o f i n c o h e r e n t e n e r g yc o n t r i b u t i o n s e m e r g i n g f r o m a l l e l e m e n t s o f t h e s o u r c e i n t o a .ll d i r e c t io n s , t h u si m p l y i n g t h a t t h e r e a r e n o i n t e r f e r e n c e e f f e c t s p r e s e n t .

W e c o n s i d e r i n s o m e d e t a i l t h e p h a s e - s p a c e v o l u m e d12 = d q x d q y d p x d p y ,w h i c h w e m a y r e w r i t e a s

d )2 = d S d p x d p y , (

2b 1d first-order optical systems the ray-transfer matrix

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