UBC Theses and Dissertations

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UBC Theses and Dissertations

Giant pulse ruby laser and a holographic system for laser parameter measurement Jones, Douglas Wayne 1975

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A GIANT PULSE RUBY LASER AND A HOLOGRAPHIC SYSTEM FOR LASER PARAMETER MEASUREMENT by DOUGLAS WAYNE JONES B . S c , S t a n f o r d U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e D e p a r t m e n t o f PHYSICS We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December, 1975 In p re sent ing t h i s t he s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I f u r t h e r agree tha t permiss ion fo r ex ten s i ve copying o f t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r ep re sen ta t i ve s . It i s understood that copying or p u b l i c a t i o n o f t h i s t he s i s f o r f i n a n c i a l gain s h a l l not be a l lowed without my w r i t t e n permi s s ion . Department of PHYSICS The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date NOVEMBER 2, 1975 ABSTRACT Three areas concerning a g i a n t pulse ruby l a s e r are d e a l t w i t h i n t h i s work. F i r s t , the s e m i - c l a s s i c a l theory of a g i a n t pulse ruby l a s e r u t i l i z i n g a saturable absorber Q-switch i s developed. Second, a novel holographic system f o r measuring v a r i o u s l a s e r parameters of the g i a n t pulse ruby l a s e r i s d i s -cussed i n d e t a i l from a t h e o r e t i c a l p o i n t of view. T h i r d , a d e s c r i p t i o n i s given of the experimental apparatus constructed, i n c l u d i n g l ) a g i a n t pulse ruby l a s e r , 2) two l a s e r r a d i a t i o n d e t e c t o r s , 3) a. l a s e r beam expansion and alignment system, and the holographic system mentioned above. - i i i -TABLE OF CONTENTS PAGE ABSTRACT i i L I S T OF TABLES v L I S T OF FIGURES v i ACKNOWLEDGEMENTS v i i INTRODUCTION 1 SECTION ONE: SEMI-CLASSICAL THEORY OF A GIANT PULSE RUBY LASER 2 SEMI-CLASSICAL LASER THEORY 6 MAXWELL'S EQUATIONS 8 POLARIZATION OF THE MEDIUM: RUBY ROD 12 • POLARIZATION OF THE MEDIUM: SATURABLE ABSORBER . . . . 19 THE NET POLARIZATION OF THE MEDIUM 24 THE INTENSITY DETERMINING EQUATION 25 THE FREQUENCY DETERMINING EQUATION 30 REFERENCES 37 SECTION TWO: THEORY OF A HOLOGRAPHIC SYSTEM FOR LASER PARAMETER MEASUREMENT 38 THE EXPERIMENTAL SYSTEM. 4-0 THE EXPOSURE PATTERN IN THE PHOTOGRAPHIC EMULSION. . . 42 THE AMPLITUDE TRANSMITTANCE PATTERNS OF THE HOLOGRAMS 54 COUPLED WAVE THEORY OF THE RECONSTRUCTED WAVEFRONTS. . 56 MEASURING THE DEGREE OF TEMPORAL COHERENCE 64 PHASE MEASUREMENTS 68 - i v -PAGE OTHER MEASUREMENTS 74-EXPECTED FORM OF IY (6 ) I 75 SUMMARY 79 REFERENCES 81 SECTION THREE: DESIGN AND CONSTRUCTION OF EXPERIMENTAL APPARATUS 82 A GIANT PULSE RUBY LASER 82 LASER RADIATION DETECTION ELECTRONICS 92 KAPITZA-DIRAC EXPERIMENT OPTICAL SYSTEM 97 HOLOGRAPHIC SYSTEM FOR LASER PARAMETER MEASUREMENT . .101 REFERENCES 107 CUMULATIVE BIBLIOGRAPHY 108 - V -LIST OF TABLES PAGE TABLE I : EXPRESSIONS APPEARING IN EQUATION (59) 28 TABLE I I : EXPRESSIONS APPEARING IN EQUATION (67) 35 TABLE I I I : EXPRESSIONS APPEARING IN EQUATION (172) 67 L I S T OF FIGURES FIGURE PAGE 1 M o d e l f o r r u b y l a s e r c a l c u l a t i o n s 4 2 P r a c t i c a l r u b y l a s e r s y s t e m 5 3 S e m i - c l a s s i c a l l a s e r t h e o r y 7 4 Ruby e n e r g y l e v e l d i a g r a m 13 5 S i m p l i f i e d r u b y e n e r g y l e v e l d i a g r a m 15 6 S i m p l i f i e d c r y p t o c y a n i n e e n e r g y l e v e l d i a g r a m . . . . 20 7 H o l o g r a p h i c s y s t e m f o r l a s e r p a r a m e t e r measurement. . 41 8 I l l u s t r a t i o n o f r e l a t i v e t r a n s v e r s e c o o r d i n a t e X. . .44 9 G i a n t p u l s e r u b y l a s e r c a v i t y 84 10 O v e r h e a d v i e w o f l a s e r head: 88 11 S c h e m a t i c o f l a s e r e l e c t r o n i c s 91 12 P h o t o d i o d e d e t e c t o r 93 13 S c h e m a t i c o f l a s e r powermeter . . . . . 95 14 O v e r h e a d v i e w o f o p t i c a l s y s t e m f o r K-D a p p a r a t u s . . 99 15 O v e r h e a d v i e w o f h o l o g r a p h i c s y s t e m 102 - v i i -ACKNOWLEDGEMENTS I w o u l d l i k e t o t h a n k t h e f o l l o w i n g p e o p l e : D r . I r v i n g O z i e r , whose p a t i e n c e , u n d e r s t a n d i n g , and enc o u r a g e m e n t have b e e n i m m e a s u r a b l y v a l u a b l e ; Mr. Doug S i e b e r g , f o r h i s a s s i s t a n c e i n c o n s t r u c t i n g t h e r u b y l a s e r s y s t e m , and e s p e c i a l l y f o r t h e l a s e r e l e c t r o n i c s ; Ms. K a t h y Huppe, my w i f e , f o r p a t i e n t l y p u t t i n g up w i t h me d u r i n g t h e w r i t i n g o f t h i s t h e s i s . -1-INTRODUCTION I t i s t h e p u r p o s e o f t h i s t h e s i s t o p r e s e n t t h e o r e t i c a l and e x p e r i m e n t a l work c o n c e r n i n g a g i a n t p u l s e r u b y l a s e r w h i c h was a s s e m b l e d b y t h e a u t h o r w h i l e w o r k i n g i n t h e l a b o r a -t o r y o f D r . I r v i n g ' O z i e r a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . The t h e s i s i s d i v i d e d i n t o t h r e e s e c t i o n s . The f i r s t two s e c -t i o n s a r e t h e o r e t i c a l i n n a t u r e , w h i l e t h e f i n a l s e c t i o n i s a summary o f e x p e r i m e n t a l work. S e c t i o n One d e a l s w i t h t h e s e m i - c l a s s i c a l t h e o r y o f a g i a n t p u l s e r u b y l a s e r . The a n a l y s i s i s b a s e d on a p h y s i c a l l y r e a s o n -a b l e m o d el, and l e a d s t o i n t e n s i t y and f r e q u e n c y d e t e r m i n i n g e q u a t i o n s . S e c t i o n Two i s a c o m p l e t e t h e o r e t i c a l d i s c u s s i o n o f a n o v e l h o l o g r a p h i c s y s t e m f o r m e a s u r i n g v a r i o u s l a s e r p a r a m e t e r s o f t h e o u t p u t f r o m a g i a n t p u l s e l a s e r . S e c t i o n T h r e e i s a d i s c u s s i o n o f t h e e x p e r i m e n t a l a p p a r a t u s w h i c h has b e e n c o n s t r u c t e d i n c o n j u n c t i o n w i t h t h e s e a r e a s o f r e s e a r c h . T h i s s e c t i o n d e a l s w i t h t h e c o n s t r u c t i o n and m o d i f i -c a t i o n o f a p r e v i o u s l y d e s i g n e d g i a n t p u l s e r u b y l a s e r o f t h e t y p e d i s c u s s e d i n S e c t i o n One and t h e d e v e l o p m e n t o f 1) a h i g h s p e e d d e t e c t o r and a t h e r m o p i l e l a s e r powermeter f o r use w i t h t h e g i a n t p u l s e r u b y l a s e r , 2) a n o p t i c a l s y s t e m f o r e x p a n s i o n and p r e c i s e a l i g n m e n t o f t h e r u b y l a s e r beam, and 3) a. h o l o -g r a p h i c l a s e r p a r a m e t e r measurement s y s t e m o f t h e t y p e d i s -c u s s e d i n S e c t i o n Two. - 2 -SECTION ONE: SEMI-CLASSICAL THEORY OF A GIANT PULSE RUBY LASER I n t h i s s e c t i o n t h e s e m i - c l a s s i c a l t h e o r y o f a g i a n t p u l s e r u b y l a s e r u t i l i z i n g a p a s s i v e ( s a t u r a b l e a b s o r b e r ) Q-s w i t c h i s d e v e l o p e d , and t h e e q u a t i o n s o f m o t i o n f o r b o t h t h e i n t e n s i t y and t h e f r e q u e n c y o f t h e l a s e r r a d i a t i o n a r e d e r i v e d . I~l 2 l The b a s i c method o f a n a l y s i s f o l l o w s t h a t o f Lamb L ' J q u i t e c l o s e l y . T h i s s e c t i o n d i f f e r s f r o m s i m i l a r work on r u b y l a s e r t h e o r y b y Szabo and S t e i n ^ - ^ i n t h a t t h e r a t e e q u a t i o n s f o r t h e s a t u r a b l e a b s o r b e r a r e i n c l u d e d , t h e e f f e c t s o f s p a t i a l h o l e b u r n i n g a r e n o t i g n o r e d , and t h e " a t o m " - f i e l d i n t e r a c t i o n i s d e v e l o p e d f r o m a quantum m e c h a n i c a l p o i n t o f v i e w w h i c h i s o n l y p a r t i a l l y p h e n o m e n o l o g i c a l . The t r e a t m e n t o f t h e s a t u r -P+l a b l e a b s o r b e r i s b a s e d on t h e work o f G i u l i a n o and H e s s L J , who have shown t h a t o p t i c a l s a t u r a t i o n i n o r g a n i c m o l e c u l e s c a n n o t be m o d e l l e d b y a s i m p l e t w o - l e v e l s y s t e m , and t h a t s a t u r a t i o n i n t h e s e t y p e s o f d y e s i s p r i m a r i l y due t o g r o u n d s t a t e popu-l a t i o n d e p l e t i o n r a t h e r t h a n p o p u l a t i o n e q u a l i z a t i o n . The s p e -c i f i c s a t u r a b l e a b s o r b e r c o n s i d e r e d i s c r y p t o c y a n i n e ( l , l ' d i -e t h y l - 4 , 4 * c a r b o c y a n i n e i o d i d e ) d i s s o l v e d i n a s u i t a b l e s o l -v e n t , s u c h as i s o p r o p y l a l c o h o l o r m e t h a n o l . The r e s u l t s a r e a p p l i c a b l e t o o t h e r s i m i l a r s a t u r a b l e a b s o r b e r s p r o v i d e d t h a t t h e p a r a m e t e r s s p e c i f i c t o t h e dye a r e s u i t a b l y a d j u s t e d . F i n a l l y , t h e t h e o r e t i c a l e x p l a n a t i o n f o r t h e b l u e - s h i f t i n e m i s s i o n f r e q u e n c y , w h i c h has b e e n o b s e r v e d i n r u b y and neo-d y m ium-glass l a s e r s , p u t f o r t h , b y F l a m h o l z and Wolga^-^ i s -3-examined. The laser i s modelled by a plane p a r a l l e l Fabry-Perot type cavity with p e r f e c t l y r e f l e c t i n g ends as shown i n Figure 1. The non-saturable cavity losses, such as the r e f l e c t i v i t i e s of the actual laser r e f l e c t o r s and the l i n e a r absorption losses, are considered to be uniformly d i s t r i b u t e d throughout the length of the cavity, as are the active centers of the laser and saturable absorber mediums. The length of the cavity, L, i s the zero f i e l d o p t i c a l path length of the actual cavity. The transverse dimensions of the cavity are considered to be e s s e n t i a l l y i n f i n i t e i n comparison to an o p t i c a l wavelength. The evolution of the f i e l d from noise fluctuations and mode competition s h a l l be ignored. Ultimately, single (longitudinal) mode operation i n a single plane polarized state s h a l l be con-sidered, as t h i s i s the t y p i c a l operating condition for t h i s type of las e r . The amplitude of the e l e c t r i c f i e l d i s taken to constant over the length of the cavity. In terms of the p r a c t i c a l laser system depicted i n Figure 2, the above considerations amount to the following approxi-mations: l ) the e l e c t r i c f i e l d amplitude i s assumed to be uni-form throughout the length of the ruby rod and the thickness of the dye c e l l ; and 2) t h i s amplitude i s taken to have the same value i n both the ruby rod and the dye c e l l at a l l times. For the case of single mode operation, these approximations should be quite good since s p a t i a l beating i n the cavity does not occur. The selective r e f l e c t i v i t y of resonant r e f l e c t o r s i s taken into account v i a the non-saturable cavity losses. FIGURE 1 Model for ruby laser calculations FIGURE 2 Practical ruby laser system (not to scale) 99.9% reflector (dielectric or resonant) -6-S i n c e s i n g l e mode o p e r a t i o n i s u s u a l l y d e s i r a b l e f r o m b o t h a n e x p e r i m e n t a l and t h e o r e t i c a l p o i n t o f v i e w , i t seems a p p r o -p r i a t e a t t h i s t i m e t o comment upon some o f t h e mode s e l e c t i v e d e v i c e s shown i n t h e l a s e r i n F i g u r e 2 . As i s w e l l known, t h e r e s o n a n t r e f l e c t o r ( s ) and t h e p a s s i v e Q - s w i t c h s e r v e t o i n -c r e a s e t h e mode s e l e c t i o n w i t h i n t h e c a v i t y and, t h e r e b y , im-p r o v e t h e p r o b a b i l i t y o f s i n g l e mode o p e r a t i o n . The t h i c k n e s s o f t h e dye c e l l s h o u l d be s m a l l , on t h e o r d e r o f 2 - 3 mm, t o p r e v e n t s t i m u l a t e d B r i l l o u i n s c a t t e r i n g b y t h e dye s o l v e n t and t h e r e s u l a t a n t d i s t o r t e d , m u l t i - f r e q u e n c y o u t p u t t h i s p r o -duces^-L Due t o t h e s u r f a c e s i n t h e c a v i t y a t B r e w s t e r ' s a n g l e and t h e b i r e f r i n g e n c e o f t h e r u b y c r y s t a l , o n l y t h e p o l a r i z a -t i o n mode p a r a l l e l t o t h e p l a n e o f t h e p a p e r i s c a p a b l e o f h a v i n g a h i g h Q - f a c t o r . F i n a l l y , t h e a p e r t u r e i n c r e a s e s t h e t r a n s v e r s e mode s e l e c t i v i t y o f t h e c a v i t y b y i n c r e a s i n g t h e d i f f e r e n c e b e t w e e n t h e d i f f r a c t i o n l o s s e s o f t h e T E M 0 0 t r a n s -v e r s e mode and t h e d i f f r a c t i o n l o s s e s o f t h e o t h e r , h i g h e r o r -d e r t r a n s v e r s e modes. As shown b y F o x and t h e T E M Q 0 mode i s a l w a y s f a v o r e d , and d e c r e a s i n g t h e a p e r t u r e d i a m e t e r i n c r e a -s e s t h e d i f f e r e n c e b e t w e e n t h e l o s s e s o f t h i s mode and t h e l o s -s e s o f any o t h e r t r a n s v e r s e mode, e v e n t h o u g h t h e r a t i o o f t h e s e l o s s e s i s e s s e n t i a l l y c o n s t a n t . The a p e r t u r e a l s o l i m i t s t h e l a s e r beam t o a r e a s o n a b l y u n i f o r m p o r t i o n o f t h e r u b y r o d . SEMI-CLASSICAL LASER THEORY The e s s e n t i a l f e a t u r e s o f t h e s e m i - c l a s s i c a l l a s e r t h e o r y employed h e r e a r e p i c t o r i a l l y summarized i n F i g u r e 3- A n e l e c -FIGURE 3 Semi-classical laser theory Quantum Mechanics Statistical Summation Maxwel l ' s Equations ^E(r/t) saturable absorber i I Self-Consistency - 8 -t r i c f i e l d , E ( r , t ) , c o u p l e s t o t h e i n d i v i d u a l a c t i v e c e n t e r s i n b o t h t h e r u b y and t h e s a t u r a b l e a b s o r b e r a c c o r d i n g t o t h e l a w s o f quantum m e c h a n i c s . T h i s c o u p l i n g p r o d u c e s m i c r o s c o p i c d i p o l e moments, p^, i n t h e a c t i v e m e d i a . When t h e s e m i c r o -s c o p i c d i p o l e s a r e s t a t i s t i c a l l y summed, t h e y l e a d t o a n e t m a c r o s c o p i c p o l a r i z a t i o n o f t h e a c t i v e m e d i a , P ( r , t ) , w h i c h , i n t u r n , d r i v e s a n e l e c t r i c f i e l d , E * ( r , t ) , d e t e r m i n e d b y M a x w e l l ' s e q u a t i o n s . S e l f - c o n s i s t e n c y t h e n demands t h a t t h e assumed e l e c t r i c f i e l d , E ( r , t ) , must be i d e n t i c a l t o t h e e l e c -t r i c f i e l d g i v e n b y M a x w e l l ' s e q u a t i o n s , E ' ( r , t ) . I n t h e s e m i -c l a s s i c a l t h e o r y t h e "atoms" a r e t r e a t e d quantum m e c h a n i c a l l y , w h i l e t h e f i e l d i s t r e a t e d a s a c l a s s i c a l e l e c t r o m a g n e t i c f i e l d . MAXWELL'S EQUATIONS M a x w e l l ' s e q u a t i o n s f o r t h e l a s e r s y s t e m may be w r i t t e n , i n c g s G a u s s i a n u n i t s , a s V« D = 0 (1) V » B = 0 (2) VxE = -(1/c) | | (3) VxH = (1/C)(J+TTJ + § = ) . (4-') The c o r r e s p o n d i n g c o n s t i t u t i v e r e l a t i o n s a r e D = E + 4-TTP (5) J = o-E (6) H = B . (7) The f i c t i o n a l c o n d u c t i v i t y , a , i s i n c l u d e d t o a l l o w t h e non-s a t u r a b l e c a v i t y l o s s e s t o be t a k e n i n t o a c c o u n t . By t a k i n g t h e c u r l o f (3) and i n s e r t i n g (4) t o g e t h e r w i t h (5)» ( 6 ) , and ( 7 ) , t h e e q u a t i o n r e l a t i n g t h e p o l a r i z a t i o n o f t h e medium t o t h e e l e c t r i c f i e l d i s o b t a i n e d . S u b j e c t t o t h e c o n d i t i o n t h a t t h e p o l a r i z a t i o n v a r i e s s l o w l y i n t h e t r a n s v e r s e d i r e c t i o n s r e l a t i v e t o t h e l o n g i t u d i n a l v a r i a t i o n o f t h e e l e c t r i c f i e l d , t h i s e q u a t i o n becomes, a 2 E 4na aE 1 a 2 E 4-n a 2 p 'dz 2 c 2 s t c 2 a t 2 ~ " c 2 a t 2 (8) F o r a f i e l d w h i c h i s p l a n e p o l a r i z e d i n t h e x d i r e c t i o n , t h e v e c t o r e q u a t i o n , (8), becomes t h e s c a l a r e q u a t i o n , r a 2 , 4-TTa a , 1 a 2 , x 4 n a 2 , , , , [ " a l 2 + ~& at + c 2 a t 2 ^ E ( z ' t } = " ~c7 a t 2 P ( z ' t } ' ( 9 ) where E = E ( z , t ) x . As shown b y K o g e l n i k and L i ^ ^ , t h e p a s s i v e p l a n e p a r a l l e l F a b r y - P e r o t c a v i t y has e q u a l l y s p a c e d r e s o n a n t q u a s i - m o d e s whose ' f r e q u e n c i e s a r e g i v e n b y .<v-T • <«» where n i s t h e number o f a n t i - n o d e s i n t h e c a v i t y , c i s t h e s p e e d o f l i g h t i n vacuum, and L i s t h e z e r o f i e l d o p t i c a l p a t h l e n g t h o f t h e c a v i t y . A f i e l d composed o f r a d i a t i o n i n s u c h q u a s i - m o d e s may be ( s p a t i a l l y ) F o u r i e r decomposed i n t e r m s o f t h e s e n o r m a l modes a s E ( z , t ) = S A it)-U ( z ) , ( 1 1 ) s 11 xi xi where U n ( z ) = s i n ( K n z ) , and K r = Q n / c . ( 1 2 ) F u r t h e r , f o r n e a r l y m o n o c h r o m a t i c modes, t h e n - t h F o u r i e r am-p l i t u d e , A n ( t ) , may he w r i t t e n A n ( t ) = E n ( t ) c o s [ v n t + 0 n ( t ) ] , ( 1 3 ) where E n ( t ) i s t h e s l o w l y v a r y i n g a m p l i t u d e o f t h e n - t h mode, v n + 0 n ( t ) « v n i s t h e o s c i l l a t i o n f r e q u e n c y o f t h i s mode, and 0 n ( t ) i s t h e s l o w l y v a r y i n g phase o f t h e n - t h mode. S i m i l a r l y , t h e p o l a r i z a t i o n o f t h e medium may be w r i t t e n P ( z , t ) = S P n ( t ) U n ( z ) , (14) where P n ( t ) = ( 2 / L ) dz P ( z , t ) U n ( z ) . P n ( t ) i s a l s o v e r y n e a r l y m o n o c h r o m a t i c and, t h e r e f o r e , t a k e s t h e forms P n ( t ) = C n ( t ) c o s [ v n t + 0 n ( t ) ] + S n ( t ) s i n [ v n t + 0 n ( t ) ] . ( 1 5 ) S u b s t i t u t i n g ( l l ) w i t h ( 1 3 ) and (14) w i t h ( 1 5 ) i n t o ( 9 ) , p r o -j e c t i n g o u t t h e n - t h F o u r i e r component, i g n o r i n g n e g l i g i b l y s m a l l t e r m s , and e q u a t i n g t h e s i n e and c o s i n e t e r m s s e p a r a t e l y t o z e r o , t h e b a s i c a m p l i t u d e and f r e q u e n c y d e t e r m i n i n g equa-t i o n s a r e o b t a i n e d * , *The d e t a i l s of t h i s analysis may be found i n either Ref. [ l ] or Ref. [ 2 ] , " -11-[ v n + 0 n ( t ) - n n ] E n ( t ) = - 2nv n C n ( t ) , (16) E n ( t ) + i ( v n / Q n ) E n ( t ) = -2TTV N S n ( t ) . (17) In equation (17)» the f i c t i o n a l conduct iv i ty of the n-th mode has been adjusted to give the damping i n terms of the Q-factor for th i s mode, v n c f n n n (18) n ~ 4nQ n 4-TTL where the second equa l i ty fol lows from the d e f i n i t i o n of the Q-factor. In equation (18), f n i s the s ing le pass attenuation c o e f f i c i e n t * of the non-saturable cav i ty losses for the n-th mode. By def in ing the time-averaged peak anti-node i n ten s i t y ( in ergs/cm 3 /sec), I n ( t ) = (C/8TT) E 3 ( t ) , (19) equations ( l 6 ) and (17) may be wr i t ten i n the a l te rna t i ve forms, [ v n + 0(t) - n n ] I n ( t ) = - | c v n E n ( t ) C n ( t ) . , (16a) I n ( t ) + ( c f N / L ) I n ( t ) = - i c v n E n ( t ) S n ( t ) . (17a) These two equations cont i tute the i n tens i t y and frequency *The s ingle pass attenuat ion c o e f f i c i e n t i s defined to be f n = - ln( I n i t i a l i n t e n s i t y / F i n a l i n tens i t y ), where i n i t i a l and f i n a l r e fe r to just p r i o r and just subse-quent to the s ingle pass through the cav i ty re spec t i ve l y . For small values of f n , i t may be approximated by the f r a c t i o n a l energy loss per pass through the cav i ty . d e t e r m i n i n g e q u a t i o n s f o r t h e l a s e r i n t e r m s o f t h e s p a t i a l F o u r i e r components o f t h e p o l a r i z a t i o n o f t h e a c t i v e medium. POLARIZATION OF THE MEDIUM: RUBY ROD S i n c e t h e a c t i v e c e n t e r s i n t h e r u b y r o d and t h e s a t u r a b l e a b s o r b e r a r e assumed t o i n t e r a c t o n l y t h r o u g h t h e r a d i a t i o n f i e l d , i t i s p o s s i b l e t o w r i t e t h e n e t p o l a r i z a t i o n o f t h e a c t i v e medium as P ( z , t ) = P A ( z , t ) + P R ( z , t ) , (20) where ? A ( z , t ) and P R ( z , t ) a r e t h e c o n t r i b u t i o n s t o t h e p o l a -r i z a t i o n s due t o t h e s a t u r a b l e a b s o r b e r and r u b y r e s p e c t i v e l y . F o r c o m p u t a t i o n a l s i m p l i c i t y , i t i s a d v a n t a g e o u s t o d e a l w i t h e a c h o f t h e s e p a r t i a l p o l a r i z a t i o n s s e p a r a t e l y , t h e n sum t h e r e s u l t s . The p o l a r i z a t i o n o f t h e r u b y medium i s t h e s i m p l e r o f t h e two c a l c u l a t i o n s and i s done f i r s t . The e n e r g y l e v e l d i a g r a m , g i v e n b y M a i m a n ^ J , f o r t h e r u b y l e v e l s r e l e v a n t t o l a s e r a c t i o n a t room t e m p e r a t u r e i s r e p r o -d u c e d i n F i g u r e 4. Pumping o c c u r s f r o m t h e 4 A 2 g r o u n d s t a t e t o e i t h e r t h e 4 F 2 o r 4 F 2 a b s o r p t i o n bands o r t o a h i g h - l y i n g c h a r g e t r a n s f e r b and ( n o t shown i n d i a g r a m ) . T h e s e bands have e x t r e m e l y r a p i d r e l a x a t i o n s t o t h e 2 E d o u b l e t l e v e l s , E and 2A. T h i s r e l a x a t i o n i s so r a p i d , i n f a c t , t h a t , f o r a l l p r a c t i c a l p u r p o s e s , t h e pumping may be c o n s i d e r e d t o be d i r e c t l y f r o m t h e g r o u n d s t a t e t h e t h e 2 E l e v e l s . The f l u o r e s c e n c e l i n e o f i n t e r -e s t h e r e i s t h e R 3 . l i n e c o r r e s p o n d i n g t o t r a n s i t i o n s b e t w e e n t h e E l e v e l o f t h e d o u b l e t and t h e g r o u n d s t a t e . FIGURE 4 Ruby enerqy level diagram (not to scale) -14-The d e g e n e r a c y o f t h e g r o u n d s t a t e i s f o u r due t o l i n e o v e r l a p , w h i l e t h e d e g e n e r a c y o f t h e E l e v e l i s two. As d i s -c u s s e d b y Szabo and S t e i n ^ - ^ , however, t h e E and 2A l e v e l s may be t a k e n t o be i n e q u i l i b r i u m t h r o u g h o u t t h e d u r a t i o n o f t h e l a s e r p u l s e due t o a r e l a t i v e l y r a p i d r e l a x a t i o n («3 n s e c ) be-tween t h e s e l e v e l s . The n e t e f f e c t o f t h i s i s a n e f f e c t i v e de-g e n e r a c y o f a b o u t f o u r f o r t h e E l e v e l . T h u s , s i n c e t h e dege-n e r a c i e s o f t h e two l e v e l s d i r e c t l y i n v o l v e d i n t h e l a s e r t r a n -s i t i o n a r e a b o u t e q u a l , d e g e n e r a c y e f f e c t s may be i g n o r e d . The ( s p o n t a n e o u s ) d e c a y c o n s t a n t f o r t h e p o p u l a t i o n o f t h e u p p e r (E) l e v e l , y a , i s a b o u t 3'3'10 3/sec, w h i l e t h e phase c o h e r e n c e d e c a y c o n s t a n t , y» g i v e n b y t h e h o m o g e n o u s l y b r o a d -ened f l u o r e s c e n c e l i n e w i d t h , i s a b o u t 1.5* 1 0 i : L / s e c . * I n v i e w o f t h e above d i s c u s s i o n , t h e c a l c u l a t i o n a l m o d e l f o r t h e r u b y s y s t e m i s a n ensemble o f two l e v e l atoms w i t h t h e s i m p l i f i e d e n e r g y l e v e l d i a g r a m shown i n F i g u r e 5- I n t h i s d i a g r a m , X a and X-^ c h a r a c t e r i z e t h e ( i n c o h e r e n t ) pumping r a t e s t o t h e u p p e r , a, and l o w e r , b, l e v e l s r e s p e c t i v e l y . The d e c a y c o n s t a n t , Yb» w h i c h n o r m a l l y i n c l u d e s s p o n t a n e o u s d e c a y f r o m t h e l o w e r l e v e l , c o r r e s p o n d s h e r e t o r e m o v a l o f atoms f r o m t h e g r o u n d s t a t e b y pumping t o h i g h e r l e v e l s . I~l 2l I t i s w e l l known 1- ' J t h a t t h e e q u a t i o n s o f m o t i o n f o r t h e d e n s i t y m a t r i x r e p r e s e n t i n g a p a r t i c u l a r two l e v e l atom e x c i -t e d t o t h e s t a t e a=a,b, a t t i m e t 0 , and l o c a t e d a t p o s i t i o n z * I n t e r m s o f T x and T 2 , t h e s e v a l u e s c o r r e s p o n d t o T 1 = l / Y a « 3 msec and T 2 = l / y « 7 p s e c . FIGURE 5 Simplified ruby energy level diagram i n the cavity are given "by 3aa ( a » t o »z»"t)=-Y aPaa( a ' to , z , t ) - ( i / t t ) [ V ^ ^ t a , t 0 , z , t ) - c . c ] (21) Pbb(a»"to » z » t )=-Yh^hh( a» to » z»" f c) +(V f t) [ v a h r 3 b a ( a » " t o » z»"t )~c • c • ] (22) t3 ab(a,t 0 ,z,t)=-(iw+Y)P a-b(a,t 0 ,z,t) + ( i / f t O V a b [ j a a ( a , t 0 ,z,t)-P- b- b(a,t 0 ,z,t )] (23) j 3 b a ( a , t 0 , z , t ) P£ b(a,t o lz,t) . • (24) In these equations, " c c . " denotes the complex conjugate of the immediately preceding term. The resonant frequency for the a-b t r a n s i t i o n i s taken to be w, and the matrix element of the inte r a c t i o n p o t e n t i a l , i n the dipole approximation, i s vab= - < a | e r | b >'E . For a f i e l d plane polarized along the x di r e c t i o n , i n a single longitudinal mode (the n-th mode), t h i s i n t e r a c t i o n matrix ele-r 2 i ment may be written, i n the rot a t i n g wave approximation 1- J, as ^ b ^ b V 3 -*P E n ( t } U n ( z ) e x p [ - i ( v n t + 0 n ( t ) ) ] (25) where p=p*= e < a | x j b > i s the x component of the e l e c t r i c dipole matrix element be-tween the a and b l e v e l s . A population density matrix, B(z,t), may be defined by summing over a l l the atoms located i n a small volume about the point z, P(z,t) = % iZ d t o Mz,t 0) J 3 ( a , t 0 , z , t ) (27) The e q u a t i o n s o f m o t i o n f o r t h e p o p u l a t i o n d e n s i t y m a t r i x , a s i m p l i e d "by (21) t h r o u g h ( 2 4 ) , a r e S a a(z,t) ^ a ( z » t ) - Y a 3 a a ( z , t ) - ( i / f t ) [ V a b p b a ( Z | t ) - c . c . ] (28) 3 b b ( z , t ) = ^ b ( z . t ) - Y b 3 b b ( z , t ) H - ( i / i i ) [ V a b g b a ( z , t ) - c . c . ] (29) p a b ( Z f t ) = - ( i w + Y ) P a b ( z , t ) + ( i / « ) V a b [ g a a ( z , t ) - r 3 b b ( z , t ) ] (30) I n t e r m s o f t h e p o p u l a t i o n d e n s i t y m a t r i x , t h e m a c r o s c o p i c p o l a r i z a t i o n o f t h e r u b y medium i s S i n c e t h e pumping t o t h e a l e v e l a p p e a r s t o come d i r e c t l y f r o m t h e b l e v e l , a nd s i n c e t h e b l e v e l i s t h e g r o u n d s t a t e f o r t h e s y s t e m , t h e o b v i o u s r e l a t i o n , X a ( z , t ) = Y b P b b ( z , t ) , i s o b t a i n e d . A l s o , a s s u m i n g t h a t t h e o n l y pumping t o t h e b l e v e l i s due t o s p o n t a n e o u s e m i s s i o n b y atoms i n t h e a l e v e l , i t i s o b v i o u s t h a t Mz,t) = Y a e a a ( z > t ) . Due t o t h e s h o r t d u r a t i o n o f t h e l a s e r p u l s e and t h e l o n g l i f e -t i m e o f t h e a l e v e l , t h e e f f e c t s o f pumping and s p o n t a n e o u s e m i s s i o n may be i g n o r e d o v e r t h e d u r a t i o n o f t h e p u l s e , and e q u a t i o n s (28) and (29) become (31) P R ( z , t ) = p [ p a b ( z , t ) + c . c ] (32) S a a(z,t) = - ( i / t t ) [ V a b P b a ( z , t ) - c . c ] (28a) P b b ( z , t ) = ( i / f t ) [ V a b B b a ( z , t ) - c c ] (29a) D e f i n i n g t h e p o p u l a t i o n d i f f e r e n c e , D ( z , t ) = B a a ( z , t ) - B b b ( z , t ) , (33) t h e e q u a t i o n s o f m o t i o n r e d u c e t o D ( z , t ) = - ( 2 i / f t ) [ V a b p b a ( z , t ) - c c ] (3*0 f 3 a b ( z , t ) = - ( i w + y ) B a b ( z , t ) + ( i / h ) V a b D ( z , t ) .. (35) T h e s e e q u a t i o n s o f m o t i o n may he i n t e g r a t e d b y f i r s t w r i -t i n g t h e f o r m a l i n t e g r a l o f ( 3 5 ) . B a h ( z ^ ) = ( i A ) ^ d t * V a b ( z , t ' ) D ( z , t ' ) e x p [ - ( i w + Y . ) ( t - t ' ) ] . I n s e r t i n g (25) i n t o t h i s e x p r e s s i o n , and a s s u m i n g t h a t E n ( t ) , 0 n ( t ) , and D ( z , t ) do n o t change a p p r e c i a b l y i n t i m e s on t h e o r d e r o f ( l / y ) , t h e i n t e g r a t i o n i s e a s i l y p e r f o r m e d t o y i e l d : f e x p [ - i ( v n t + 0 n ( t ) ) ] ) P a b ( z , t ) = - ( i p / 2 h ) E n ( t ) U n ( z ) D ( z , t ) [ y + i ( w v n ) J " ( 3 6 ) C o m b i n i n g t h i s e x p r e s s i o n w i t h (34) and ( 2 5 ) , t h e e q u a t i o n w h i c h d e t e r m i n e s t h e p o p u l a t i o n d i f f e r e n c e d e n s i t y i s f o u n d t o be b ( z , t ) = - ( l A ) [ p E n ( t ) U n ( z ) / * l ] s D ( z , t ) L ( w - v n ) , (37) where L ( w - v n ) i s t h e d i m e n s i o n l e s s L o r e n t z i a n f u n c t i o n , L ( w - v n ) = i f _ Y 3 + ( w - v n ) 3 . I n t e r m s o f t h e i n t e n s i t y o f e q u a t i o n ( 1 9 ) f and t h e z e r o f i e l d l i m i t o f t h e a b s o r p t i o n c r o s s - s e c t i o n p e r C r + 3 i o n f o r r a d i a -t i o n a t f r e q u e n c y v n , -19-a R ( v n ) = ( 4 n / f t c Y ) v n p L ( w - v n ) e q u a t i o n (37) "becomes D ( z , t ) = -(2 / f t v n ) c r R ( v n ) I n ( t ) D ( z , t ) U 3 ( z ) (37a) The f o r m a l s o l u t i o n o f (37a) i s D ( z , t ) = D 0 ( z ) e x p [ - ( 2 / f t v n ) a R ( v n ) U 3 1 ( z ) J ^ d f I n ( t ' ) ] , (38) where D 0 ( z ) i s t h e p o p u l a t i o n d i f f e r e n c e d e n s i t y a t t h e t i m e , t=0, when a p p r e c i a b l e l a s e r a c t i o n b e g i n s t o t a k e p l a c e . C o m b i n i n g ( 3 8 ) , ( 3 6 ) , and ( 3 2 ) , t h e m a c r o s c o p i c p o l a r i -z a t i o n o f t h e r u b y medium i s f o u n d t o be where t h e r u b y g a i n s a t u r a t i o n f u n c t i o n , B ( t ) , i s d e f i n e d t o be The c r y p t o c y a n i n e s a t u r a b l e a b s o r b e r may be m o d e l l e d b y t h e s i m p l i f i e d e n e r g y l e v e l d i a g r a m shown i n F i g u r e 6 w h i c h f o l l o w s f r o m t h e work o f G i u l i a n o and Hess^-L I n t h i s m o d e l, m o l e c u l e s i n t h e g r o u n d s t a t e , b, a r e e x c i t e d t o t h e l e v e l a x i n t h e a b s o r p t i o n band, A. D e c a y f r o m t h e a x l e v e l t o t h e b o t -tom l e v e l o f t h e band, a, i s e x t r e m e l y r a p i d and c h a r a c t e r i z e d b y Y a x • M o l e c u l e s i n t h e e x c i t e d s t a t e , a, may e i t h e r d e c a y t o t h e g r o u n d s t a t e ( y a ) » o r be e x c i t e d b y a s e c o n d a b s o r p t i o n t o t h e l e v e l c x i n t h e a b s o r p t i o n band, C. I t i s assumed t h a t de-(39) B ( t ) = a R ( v n ) [ l / t t v n ] J ^ d f I n ( f ) POLARIZATION OF THE MEDIUM: SATURABLE ABSORBER FIGURE 6 Simplified cryptocyanine energy level diagram - 2 1 -c a y t o t h e l o w e s t l e v e l , c, o f t h e band, C, i s a l s o e x t r e m e l y r a p i d . S i n c e r e l a t i v e l y l i t t l e i s known a b o u t t h e e x c i t e d s t a t e a b s o r p t i o n , t h e f u r t h e r a s s u m p t i o n i s made t h a t t h i s s e c o n d a r y a b s o r p t i o n does n o t s a t u r a t e . I t s h o u l d be s t r e s s e d t h a t , i n t h i s m o d e l, s a t u r a t i o n i s due t o d e p l e t i o n o f t h e g r o u n d s t a t e p o p u l a t i o n , and i t i s n o t c a u s e d b y p o p u l a t i o n e q u a l i z a t i o n as i s n o r m a l l y t h e c a s e f o r s i m p l e r two l e v e l s y s t e m s . The e q u a t i o n s o f m o t i o n f o r t h e p o p u l a t i o n d e n s i t y m a t r i x r e p r e s e n t i n g t h e s a t u r a b l e a b s o r b e r i n t h e above m o d e l a r e de-r i v e d b y e x t e n d i n g t h e work o f R e f s . [ 2 ] and [4], and a r e flbb(z,t)= Y a f l a a ( z , t ) + ( i A ) [ V a i b B b a . [ z , t ) - c . c . ] B a x a f z , t ) = - Y a i B a i a f z . t ) - ( i / f t ) [ V a i b B b a i ( z , t ) - c . c ] (4l) B a i b ( z , t ) = - ( i w , + Y a 2 f i a 1 b ( z » - t ) + ( i A ) V a i b C f 3 a 1 a 1 ( z ' " t ) - i 3 b b ( z ' " t ) ] (42) B b a ( z , t ) = fi*ib(z,t) (43) ^ a a ( z ' t ^ = V a i ^ a 1 a 1 ( z ' t ) - Y a i 3 a a ( z ' t ) + ( 1 A ) [ V a C : L 1 3 C i a ( z , t ) - c . c . ] (44) ^ c 1 c i z ' t ) = - Y C l f i c 1 c 1 ( z ' t ) - ( 1 / f t ) t V a c 1 f i C i a ( z » " t ) - c - c - ] ^5) flCia(zit)=-(iw«+Yc2BCia(zft)+(i/ft)VCia[BCiCi(zft)-Baa(zft)] (46) B a C i(z,t)= B * i a ( z , t ) (47) B c c ( z , t ) = Y C l G C l C ( z , t ) - Y c B C c ( z ' t ) • ( ^ 8 ) I n t h e s e e q u a t i o n s , w' and w" a r e t h e r e s o n a n t f r e q u e n c i e s f o r g r o u n d s t a t e and e x c i t e d s t a t e a b s o r p t i o n s , r e s p e c t i v e l y . Due -22-t o the e x t r e m e l y r a p i d decays from the a x and c x l e v e l s , e q u a -t i o n s (4-2) and (46) may be i n t e g r a t e d i n t h e same manner as e q u a t i o n (36) t o y i e l d : f e x p [ - i ( v n t + 0 n ( t ) ) ] ^ ) flaib(z,t)=(ip'/2ft)En(t)Un(Z)flbb(Z>t)[ + .(w„ _ j (42a) f e x p [ - i ( v n t + 0 n ( t ) ) ] 1 B0 ( z , t ) = ( i p " / 2 t t ) E ( t ) U ( z ) i 3 a a ( z , t ) - — — C i a n n aa ^ + i ( w » _ V n ) J (46a) where use has been made o f the f a c t t h a t flbb(z,t)»flaia^zft)«0 and B a a ( z , t ) » B C i C ^ z f t)«0 , due t o the e x t r e m e l y r a p i d decays from the a x and cx l e v e l s ; and where p»=p»*= e < &x | x I b > and p"=p"*= e < cx | x | a > , are the e l e c t r i c d i p o l e m a t r i x elements f o r the ground s t a t e and e x c i t e d s t a t e a b s o r p t i o n s r e s p e c t i v e l y . Combining the above r e s u l t s w i t h (40) t h r o u g h ( 4 8 ) , y i e l d s the e q u a t i o n s o f m o t i o n f o r B b b ( z , t ) and fiaa(z»"t)! fibb(z,t)=Yaflaa(z,t) - a«(v n)[l/iiv n]l n(t)U 2 1(z)f3 b b(z,t) (40a) G a a ( z , t ) = - f i b b ( z , t ) , (44a) where c r ' ( v n ) i s the a b s o r p t i o n c r o s s - s e c t i o n per m o l e c u l e f o r the ground s t a t e ^ a b s o r p t i o n , e v a l u a t e d a t the l a s e r f r e q u e n c y . The s a t u r a b l e a b s o r b e r responds q u i c k l y enough t o "keep up" w i t h the changes i n i n t e n s i t y , so e q u a t i o n s (40a) and (44a) may be s o l v e d i n the s t e a d y s t a t e t o g i v e : - 2 3 -A (49) [1 + B- I n ( t ) U 3 ( z ) ] A B« I n ( t ) U=".(z) (50) [1 + B' I n ( t ) U n ( z ) ] where A = B a a ( z » ' t ) + B b h ( z ' t ) i s t h e d e n s i t y o f s a t u r a b l e a b s o r b e r m o l e c u l e s i n t h e c a v i t y ; and where i s t h e s a t u r a t i o n p a r a m e t e r f o r t h e s a t u r a b l e a b s o r b e r . The m a c r o s c o p i c p o l a r i z a t i o n o f t h e s a t u r a b l e a b s o r b e r medium i s g i v e n b y P A ( z , t ) = p ' [ f l a i b ( z , t ) + c . c ] + p " [ l 3 C i a ( z , t ) + c . c ] . (51) S u b s t i t u t i n g (42a) w i t h (49) and (46a) w i t h (50) i n t o (51) , t h e e x p l i c i t f o r m o f t h e p o l a r i z a t i o n i s f o u n d : I n e q u a t i o n (52) , t h e e f f e c t s o f f r e q u e n c y p u l l i n g b y t h e ex-c i t e d s t a t e a b s o r p t i o n have b e e n n e g l e c t e d , and a " ( v n ) i s t h e c r o s s - s e c t i o n p e r m o l e c u l e f o r e x c i t e d s t a t e a b s o r p t i o n e v a l u a -t e d a t t h e l a s e r f r e q u e n c y . B f = ° ' ( v n ) / [ h v n Y a ] (52) - 2 4 -THE NET POLARIZATION OF THE MEDIUM Combining equations (39) and (52) the net p o l a r i z a t i o n of the active medium, P(z,t) = P R(z,t) + P A ( z , t ) i s obtained. Projecting t h i s expression onto U n(z) and iden-t i f y i n g the sine and cosine c o e f f i c i e n t s with those of equa-t i o n (15) y i e l d s C n ( t ) = (2/L)/^dz U j ( z ) [ - 2 a R ( v n ) I n ( t ) D 0 ( z ) e x p [ - 2 B ( t ) U 3 1 ( z ) ] -( w - v n ) / [ y v n E n ( t ) ] + 2 a ' ( v n ) A I n ( t ) [ l + B ' I n ( t ) U 2 1 ( z ) ] - l ( w ' - v n ) / [ Y a i v n E n ( t ) ] ] , (53) and S n ( t ) = (2/L)vf^dz U j ( z ) [ - 2 a R ( v n ) l n ( t ) D 0 ( z ) e x p [ - 2 B ( t ) U 2 ( Z ) ] . l / [ v n E n ( t ) ] + 2 a ' ( v n ) A I n ( t ) [ l + B ' I n ( t ) U ^ ( z ) ] - 1 . [1 + B ' I n ( t ) U 2 1 ( z ) a " ( v n ) / a ' ( v n ) ] / [ v n E n ( t ) ] j . (5*+) Provided that D G(z) does not vary appreciably over the length of the cavity, i t may be replaced by i t s average value, D Q, and taken outside the integrals over z. When t h i s i s done,(53) and (54) may be integrated exactly* to y i e l d : C n ( t ) = - 2 ( l n ( t ) / [ v n E n ( t ) ] ) [ D 0 a R ( v n ) [ ( w - v n ) / Y ] e x p [ - B ( t ) ] -[ l 0 ( B ( t ) ) - l ! ( B ( t ) ) ] - A a ' ( v n ) [ ( w ' - v n ) / Y a i ] -[ 2 / B ' I n ( t ) ] [ l - (1 + B ' I n ( t ) ) - * ] j , (55) *See, for example, Gradshteyn and Ryzhik L J equations 3-915 and 3 . 6 4 7 . -25-and S n ( t ) = - 2 ( l n ( t ) / [ v n E n ( t ) ] ) [ D 0 a R ( v n ) e x p [ - B ( t ) ] [ l 0 ( B ( t ) ) -l ! ( B ( t ) ) ] - A a ' ( v n ) [ [ a " ( v n ) / a ' ( v n ) ] + [ i -a " ( v n ) / a ' ( v n ) ] [ 2 / B ' I n ( t ) ] [ l - ( l + B » I n ( t ) ) " * ] ] ] . (56) I n t h e s e e q u a t i o n s , I 0 ( B ( t ) ) and I 1 ( B ( t ) ) , w h i c h a r e n o t t o he c o n f u s e d w i t h t h e i n t e n s i t y , I n ( t ) , a r e h y p e r b o l i c B e s s e l f u n c -t i o n s o f t h e f i r s t k i n d o f z e r o t h a nd f i r s t o r d e r r e s p e c t i v e l y . THE INTENSITY DETERMINING EQUATION The i n t e n s i t y d e t e r m i n i n g e q u a t i o n f o l l o w s f r o m (17a) w i t h (56): I n ( t ) = ( c / L ) l n ( t ) [ L D 0 a R ( w ) e x p [ - B ( t ) ] [ l 0 ( B ( t ) ) - I x ( B ( t ) ) ' ] - f n - a ( w ) ( [ a " ( w ) / a ' ( w ) ] + [ l - a"(w)/o ' (w)> where i t has b e e n assumed t h a t t h e n - t h mode i s c l o s e t o t h e r u b y l i n e c e n t e r so t h a t v n may be r e p l a c e d b y w i n a l l s l o w l y v a r y i n g f u n c t i o n s o f t h e f r e q u e n c y , t h a t i s , a l l f u n c t i o n s o f th e f r e q u e n c y e x c e p t t h e mode s e l e c t i v e t e r m , f n . I n e q u a t i o n (57)i a(w) i s t h e s i n g l e p a s s a t t e n u a t i o n c o e f f i c i e n t o f t h e s a t u r a b l e a b s o r b e r i n t h e z e r o f i e l d l i m i t , e v a l u a t e d a t t h e mean r u b y f r e q u e n c y , and i s g i v e n b y a(w) = A a'(w) L E v a l u a t i n g (57) i n t h e l i m i t t h a t t goes t o z e r o a nd t h e g a i n j u s t e x c e e d s t h e l o s s e s , t h e p o p u l a t i o n d i f f e r e n c e d en-(57) -26-sity at threshold is found to be Dj = [a(w) + fn]/[L o R(w)] By defining the relative excitation of the ruby for the n-th mode at the onset of laser action, the intensity determining equation may be cast in its final form: Equation (59) represents a complete solution to the time deve-lopment of the intensity of laser radiation in the cavity. None of the approximations made in deriving (59) are very restric-tive. Therefore, this equation should describe passively Q-switched ruby laser systems quite well, provided, of course, the reflectivity of the partial reflector is high enough that the standing wave decomposition of equation (ll) remains valid. Although (59) is not an explicit solution to this problem, the integro-differential form in which it is given is well suited for machine calculation. This is especially true in view of the fact that all the parameters appearing in (59) are either known in the literature, or easily measured for the particular laser system under consideration. To facilitate the use of (59) > the necessary parameters appearing in this equation with their significance and values, where available, are summarized in N = DQ/ Dj = DG L aR(w)/[a(w) + f n] (58) (59) -27-T a b l e I. Q u a l i t a t i v e l y , e q u a t i o n (59) d i s p l a y s t h e e s s e n t i a l f e a -t u r e s o f g i a n t p u l s e l a s e r o p e r a t i o n . F o r r e l a t i v e e x c i t a t i o n s , Nn , l e s s t h a n one ("below t h r e s h o l d ) , t h e r e i s a l w a y s n e t l o s s , and t h e i n t e n s i t y i s a l w a y s n e a r z e r o . I f N n e q u a l s one ( t h r e s -h o l d ) , t h e n t h e i n t e n s i t y i s i n i t i a l l y c o n s t a n t n e a r z e r o . As t h e i n t e g r a t e d i n t e n s i t y i n B ( t ) a c c u m u l a t e s , however, t h e pop-u l a t i o n i n v e r s i o n i s d e p l e t e d , r e d u c i n g t h e g a i n and y i e l d i n g an a l w a y s d e c r e a s i n g i n t e n s i t y . When N n e x c e e d s one (above t h r e s h o l d ) , t h e r e i s n e t g a i n and t h e i n c r e a s i n g i n t e n s i t y de-c r e a s e s t h e t e r m i n c u r l y b r a c k e t s i n ( 5 9 ) , i . e . i t b l e a c h e s t h e d y e . Due t o t h e n e a r l y c o n s t a n t g a i n and t h e d e c r e a s i n g a b s o r p t i o n o f t h e dye, t h e i n t e n s i t y b u i l d s up r a p i d l y . As t h i s o c c u r s t h e i n t e g r a t e d i n t e n s i t y o f B ( t ) i n c r e a s e s r a p i d l y a l s o , c a u s i n g t h e p r o d u c t o f t h e e x p o n e n t i a l and t h e d i f f e r e n c e o f B e s s e l f u n c t i o n s t o d e c r e a s e . T h i s c o r r e s p o n d s t o t h e d e p l e t i o n o f t h e p o p u l a t i o n i n v e r s i o n . E v e n t u a l l y , B ( t ) i n c r e a s e s t o t h e p o i n t t h a t t h e l o s s e s e x c e e d t h e g a i n , t h i s t u r n i n g p o i n t c o r -r e s p o n d s t o t h e peak i n t e n s i t y o f t h e p u l s e . Once t h e l o s s e s e x c e e d t h e g a i n , t h e i n t e n s i t y b e g i n s t o d e c r e a s e , and, as a r e s u l t , t h e a b s o r p t i o n b y t h e dye b e g i n s t o i n c r e a s e a g a i n . The combined a c t i o n o f i n c r e a s i n g dye a b s o r p t i o n and d e c r e a s i n g p o p u l a t i o n i n v e r s i o n , c a u s e s t h e i n t e n s i t y t o d e c r e a s e r a p i d l y t o z e r o . U l t i m a t e l y , t h e i n t e n s i t y i s n e a r z e r o , a s i t was i n i -t i a l l y , t h e dye has r e c o v e r e d , and t h e p o p u l a t i o n i n v e r s i o n i s c o m p l e t e l y d e p l e t e d . Thus, i n t h i s p r o c e s s , a s i n g l e i n t e n s e p u l s e , a g i a n t p u l s e , o f l a s e r r a d i a t i o n has b e e n e m i t t e d . TABLE I i EXPRESSIONS APPEARING IN EQUATION ( 5 9 ) EXPRESSION SIGNIFICANCE I n ( t ) I n t e n s i t y of l a s e r r a d i a t i o n i n c a v i t y i n ergs/cm 3/sec. L O p t i c a l path length of c a v i t y i n zero f i e l d l i m i t . N n R e l a t i v e e x c i t a t i o n of ruby = ( I n i t i a l i n v e r s i o n ) / (Threshold i n v e r s i o n ) . *n S i n g l e pass a t t e n u a t i o n c o e f f i c i e n t f o r non-saturable c a v i t y l o s s e s ( r e f l e c t i v i t i e s , etc.) = - l n [ ( l n t e n s i t y p r i o r to s i n g l e p a s s ) / ( I n t e n s i t y a f t e r s i n g l e p a s s ) ] . a(w) S i n g l e pass att e n u a t i o n c o e f f i c i e n t f o r saturable absorber i n zero f i e l d l i m i t ( c f . previous e n t r y ) . I 0(x),, I i ( x ) Zeroth and f i r s t order hyperbolic B e ssel f u n c t i o n s ( r e s p e c t i v e l y ) of the f i r s t k ind of argument x. B(t) = [a R(w)/hw] J ^ d f I n(t») Ruby gai n s a t u r a t i o n parameter. a R(w) « 2 . 5 ' 1 0 - 2 0 cm3 Peak absorption c r o s s - s e c t i o n per C r + 3 i o n i n ruby medium (value from [ 9 ] ) -TABLE Is (CONTINUED) w ^ 2 . 7 1 * 1 0 1 V s e c Mean a n g u l a r f r e q u e n c y f o r r u b y l a s e r e m i s s i o n . £> R ( w ) / t t w ] w 8 . 7 » 1 0~^cm 3/erg C o e f f i c i e n t i n r u b y g a i n s a t u r a t i o n p a r a m e t e r . CT' (w) « 5 * 1 0 " l 6 c m 3 A b s o r p t i o n c r o s s - s e c t i o n a t l a s e r f r e q u e n c y p e r m o l e c u l e o f s a t u r a b l e a b s o r b e r f o r g r o u n d s t a t e a b s o r p t i o n ( v a l u e f r o m [ 4 ] i s f o r c r y p t o c y a n i n e i n i s o p r o p y l a l c o h o l ) . a" (w) * l » 1 0 " l 6 c m 3 A b s o r p t i o n c r o s s - s e c t i o n a t l a s e r f r e q u e n c y p e r m o l e c u l e o f s a t u r a b l e a b s o r b e r f o r e x c i t e d s t a t e a b s o r p t i o n ( v a l u e a s p e r p r e v i o u s e n t r y ) . Y a « 2 » l O ^ / s e c D e c a y c o n s t a n t f o r b o t t o m l e v e l o f " f i r s t " a b s o r p t i o n band o f s a t u r a b l e a b s o r b e r ( v a l u e a s p e r above e n t r i e s ) . B» = a»(w)/[Y ahw] « 9 * 1 0 - 1 ^ c m 3 s e c / e r g A b s o r p t i o n s a t u r a t i o n p a r a m e t e r f o r s a t u r a b l e a b s o r b e r a t mean r u b y f r e q u e n c y ( v a l u e a s p e r a b o v e ' e n t r i e s ) . THE FREQUENCY DETERMINING EQUATION I t i s w e l l known t h a t a l l p u l s e d r u b y l a s e r s , b o t h non-Q-s w i t c h e d and Q - s w i t c h e d , d i s p l a y a smooth i n c r e a s e i n e m i s s i o n f r e q u e n c y o v e r t h e d u r a t i o n o f s i n g l e p u l s e s . * T h i s b l u e - s h i f t has b e e n o b s e r v e d d i r e c t l y b y t i m e - r e s o l v e d F a b r y - P e r o t s p e c -t r o s c o p y ^ ^ " ' 1 2 - I , and b y t i m e - r e s o l v e d i n t e r f e r o m e t r y ^ ' ^ » l ^ J , I t has a l s o b e e n m e a s u r e d more i n d i r e c t l y i n h o l o g r a p h i c e x p e r -i m e n t s c o n c e r n i n g c o h e r e n c e l e n g t h ^ ^ - L T h i s b l u e - s h i f t i s n o t l i m i t e d t o r u b y l a s e r s , b u t has a l s o b e e n o b s e r v e d i n p i c o -s e c o n d p u l s e s f r o m m o d e - l o c k e d n e o d y m i u m - g l a s s l a s e r s b y t i m e -r e s o l v e d s p e c t r o s c o p y u t i l i z i n g two p h o t o n a b s o r p t i o n f l u o r e -s c e n c e ^ ' 7 ' ^ - ! . T h e r e has n e v e r b e e n a r e p o r t o f e i t h e r a non-s h i f t e d p u l s e o r a p u l s e d i s p l a y i n g a r e d - s h i f t . The b l u e - s h i f t has b e e n shown t o a f f e c t h o l o g r a p h i c b r i g h t -n e s s and d e p t h o f f i e l d ^ ^ ' , and i n t e r f e r o m e t r i c s t u d i e s o f p u l s e s and p u l s e w i d t h s ^ ^ ] _ I t a l s o e x p l a i n s t h e anomalous l i n e - b r o a d e n i n g o b s e r v e d when F a b r y - P e r o t i n t e r f e r o m e t e r s a r e u s e d i n t h e c o n v e n t i o n a l , n o n - t i m e r e s o l v e d , manner t o measure t h e e m i s s i o n l i n e w i d t h . ^ --^ I n g e n e r a l , t h e e f f e c t s o f t h e b l u e -s h i f t a r e most a p p a r e n t i n p r o c e s s e s w h i c h depend on e i t h e r t h e a u t o - c o r r e l a t i o n f u n c t i o n o f s i n g l e p u l s e s , o r on t h e l i n e w i d t h o f s i n g l e p u l s e s . I n t h e l a t t e r c a s e , c a r e s h o u l d be t a k e n t o d e c i d e w h e t h e r t h e p a r a m e t e r o f i n t e r e s t i s t h e " i n s t a n t a n e o u s " l i n e w i d t h ( r o u g h l y g i v e n b y t h e i n v e r s e o f t h e p u l s e w i d t h ) , o r t h e f r e q u e n c y s w e e p - w i d t h ( t h e " l i n e w i d t h " commonly me a s u r e d b y * U n f o r t u n a t e l y , i t has b e e n t h i s a u t h o r ' s e x p e r i e n c e t h a t t h i s f a c t i s n o t w i d e l y known. -31-F a " b r y - P e r o t i n t e r f e r o m e t e r s , e t c . ) . The f r e q u e n c y s w e e p - w i d t h and t h e i n s t a n t a n e o u s l i n e w i d t h depend, o f c o u r s e , on t h e p a r -t i c u l a r l a s e r s y s t e m b e i n g c o n s i d e r e d , b u t f o r a t y p i c a l g i a n t p u l s e r u b y l a s e r ^ l ' , t h e f r e q u e n c y s w e e p - w i d t h i s on t h e o r d e r o f 100 MHz, w h i l e t h e i n s t a n t a n e o u s l i n e w i d t h : i s o n l y a few m e g a h e r t z . N o r m a l l y , t h e f r e q u e n c y d e t e r m i n i n g e q u a t i o n f o r l a s e r e m i s s i o n i s a r r i v e d a t b y s u b s t i t u t i n g (55) i n t o ( l 6 a ) . T h i s method o f c a l c u l a t i o n , however, does n o t c o n s i s t e n t l y p r e d i c t a b l u e s h i f t f o r e v e r y p u l s e , and p r e d i c t s much s m a l l e r s h i f t s t h a n t h o s e t h a t have b e e n e x p e r i m e n t a l l y o b s e r v e d . As a r e s u l t , t h i s method o f c a l c u l a t i o n must be abandoned. P o h l ^ ^ has shown t h a t t h e r e i s s t r o n g e v i d e n c e t h a t t h e f r e q u e n c y s h i f t i s c l o s e l y r e l a t e d t o t h e p o p u l a t i o n i n v e r s i o n i n t h e r u b y r o d and a t t r i b u t e s t h e f r e q u e n c y sweep t o a change i n t h e p o l a r i z a b i l i t y o f t h e r u b y medium. As d i s c u s s e d b y B e r k l e y and WolgeJ-^-^l, however, a more v i a b l e e x p l a n a t i o n o f t h e b l u e - s h i f t i s a r r i v e d a t b y a s s u m i n g t h a t t h e p h y s i c a l l e n g t h , n o t j u s t t h e o p t i c a l p a t h l e n g t h , o f t h e r u b y r o d i s l i n e a r l y p r o p o r t i o n a l t o t h e i n v e r s i o n i n t h e r u b y medium. I n t h i s m o d e l, a n e x c i t e d C r + 3 I o n c o u p l e s d i f f e r e n t l y t o t h e l a t t i c e s t r a i n s i n t h e h o s t c r y s t a l , and o c c u p i e s a l a r g e r v olume i n c o n f i g u r a t i o n s p a c e t h a n a G r + 3 i o n i n t h e g r o u n d s t a t e . T hus, i n t h i s m o d e l, a r u b y r o d w i t h a s u b s t a n t i a l pop-u l a t i o n i n v e r s i o n i s ( p h y s i c a l l y ) l o n g e r t h a n a r e l a x e d r u b y r o d . The l e n g t h o f t h e r u b y r o d t h e n d e c r e a s e s as l a s e r a c t i o n p r o c e e d s , p r o d u c i n g a c o r r e s p o n d i n g d e c r e a s e i n t h e o p t i c a l -32-p a t h l e n g t h o f t h e c a v i t y a s a whole, w h i c h , i n t u r n , r e s u l t s i n a b l u e - s h i f t i n e m i s s i o n f r e q u e n c y d u r i n g e a c h p u l s e . The d e r i v a t i o n o f t h e b l u e - s h i f t w h i c h i s g i v e n b y F l a m h o l z t h e s i m p l i f i e d l a s e r e q u a t i o n s o f Szabo and S t e i n 1 - ^ - 1 . The f r e -q u e n c y d e t e r m i n i n g e q u a t i o n d e r i v e d h e r e f o l l o w s f r o m t h e a s s u m p t i o n s made b y F l a m h o l z and Wolga, b u t i t i s b a s e d on t h e more e x a c t l a s e r e q u a t i o n s d e r i v e d e a r l i e r i n t h i s s e c t i o n . S i n c e t h e b l u e - s h i f t i s i n d e p e n d e n t o f t h e l o n g i t u d i n a l mode i n d e x , n, and i s much g r e a t e r t h a n t h e mode p u l l i n g e f f e c t s p r e d i c t e d b y ( l 6 a ) w i t h (55)» t h e e m i s s i o n f r e q u e n c y i s t a k e n t o be e s s e n t i a l l y t h e same f o r a l l modes, and i t i s t a k e n t o be t i m e d e p e n d e n t w i t h mean v a l u e g i v e n b y t h e l i n e c e n t e r f r e -q u e n c y o f t h e r u b y f l u o r e s c e n c e l i n e . A l s o , s i n c e t h e n e t ch a n g e s i n t h e f r e q u e n c y , i n t h e l e n g t h o f t h e r u b y r o d , and i n t h e o p t i c a l p a t h l e n g t h o f t h e c a v i t y a r e q u i t e s m a l l i n com-p a r i s o n t o t h e i r r e s p e c t i v e mean v a l u e s , t h e t i m e d e p e n d e n t v a l u e s o f t h e s e v a r i a b l e s i s r e p l a c e d b y t h e i r mean v a l u e s w h e r e v e r p o s s i b l e . S t a r t i n g w i t h t h e a s s u m p t i o n t h a t t h e l e n g t h o f t h e r u b y r o d i s l i n e a r l y p r o p o r t i o n a l t o t h e n o r m a l i z e d p o p u l a t i o n i n -v e r s i o n i n t h e r u b y medium, t h e t i m e d e p e n d e n t l e n g t h o f t h e r u b y r o d i s and Wolga X ( t ) = j e o [ l + P ( l + D ( t ) / N 0 ) ] y i e l d i n g a t i m e r a t e o f change, i ( t ) = X 0 P D ( t ) / N 0 (60) - 3 3 -where ZQ i s t h e mean l e n g t h o f t h e r u b y r o d ; P i s t h e assumed ( d i m e n s i o n l e s s ) c o u p l i n g c o n s t a n t ; N c i s t h e d e n s i t y o f C r + 3 i o n s i n t h e r u b y m a t e r i a l ; and D ( t ) i s t h e a v e r a g e p o p u l a t i o n d i f f e r e n c e d e n s i t y , D ( t ) = (1/L) J ^ d z D ( z , t ) . (61) S u b s t i t u t i n g D ( z , t ) f r o m (38) i n t o ( 6 1) and i n t e g r a t i n g y i e l d s : D ( t ) = D G I 0 ( B ( t ) ) e x p [ - B ( t ) ] w h i c h has t h e t i m e r a t e o f change D ( t ) = - ( 1/ttwL) I n ( t ) N n [ a ( w ) + f n ] exp[-B(t)]« [ l 0 ( B ( t ) ) - I x ( B ( t ) ) ] . (62) E q u a t i o n (58) has b e e n u s e d t o w r i t e (62) i n t e r m s o f t h e r e l a -t i v e e x c i t a t i o n , N n , D e f i n i n g t h e i n t e n s i t y g a i n c o e f f i c i e n t , G ( t ) , b y i d e n t i f y i n g t h e p o s i t i v e c o e f f i c i e n t o f I n ( t ) i n e q u a t i o n ( 5 9 ) i G ( t ) = ( c / L ) N n [ a ( w ) + f n ] [ l 0 ( B ( t ) ) - I 1 ( B ( t ) ) ] e x p [ - B ( t ) ] , ( 6 3 ) e q u a t i o n (62) may be w r i t t e n i n more compact f o r m : D ( t ) = - G ( t ) I n ( t ) / [ t t c w ] . (64) The r a t e o f change o f t h e o p t i c a l p a t h l e n g t h o f t h e c a v i t y c o r r e s p o n d i n g t o t h e r a t e o f change o f t h e l e n g t h o f t h e r u b y r o d (60) i s , f o r t h e t y p e o f c a v i t y shown i n F i g u r e 2 , L ( t ) = ( n R - n A ) X ( t ) , (65) where n R i s t h e i n d e x o f r e f r a c t i o n o f t h e r u b y r o d and n ^ i s t h e i n d e x o f r e f r a c t i o n o f t h e a m b i e n t medium i n t h e c a v i t y , h e n c e f o r t h t a k e n t o be u n i t y . -34-E q u a t i o n ( 6 5 ) , i n t u r n , i m p l i e s a r a t e o f change i n r e s o -n a n t f r e q u e n c y , and t h u s i n e m i s s i o n f r e q u e n c y , o f 3 n(t) = -(w/L) L ( t ) , (66) where 0 n ( t ) i s t h e s l o w l y v a r y i n g p h a s e f r o m e q u a t i o n ( 1 3 ) . C o m b i n i n g ( 6 4 ) , ( 6 5 ) , and (66) w i t h (60) y i e l d s t h e d e s i r e d e q u a t i o n o f m o t i o n f o r t h e s l o w l y v a r y i n g p h a s e o f t h e l a s e r r a d i a t i o n : 3 n(t) = P j > c ( n R - l ) G ( t ) I n ( t ) / [ N 0 L f t c ] . (67) I n t e r m s o f a t i m e d e p e n d e n t e m i s s i o n f r e q u e n c y , v n ( t ) = v n + 0 n ( t ) « w + 0 n ( t ) , i t i s o b v i o u s t h a t v n ( t ) = 0 n ( t ) . Thus, e q u a t i o n (67) t o g e t h e r w i t h (59) p r o v i d e s a c o m p l e t e s o l u -t i o n t o t h e t i m e d e pendence o f t h e e m i s s i o n f r e q u e n c y u n d e r t h e l i n e a r c o u p l i n g h y p o t h e s i s . As w i t h ( 5 9 ) , t h i s e q u a t i o n i s w e l l s u i t e d f o r m a c h i n e c a l c u l a t i o n . T a b l e I I summarizes t h e e x p r e s -s i o n s a p p e a r i n g i n (67) w h i c h have n o t b e e n d e a l t w i t h a l r e a d y i n T a b l e I . Q u a l i t a t i v e l y , i t s h o u l d be n o t e t h a t v n ( t ) , as g i v e n b y ( 6 7 ) , i s p o s i t i v e and, t h u s , p r e d i c t s a smooth i n c r e a s e i n e m i s s i o n f r e q u e n c y o v e r t h e d u r a t i o n o f " t h e p u l s e . A l s o , due t o t h e dependence on t h e i n t e n s i t y , t h e r a t e o f change o f t h e f r e q u e n c y i s l e s s i n t h e "wings" o f t h e p u l s e , w h i c h i s c o n s i s -t e n t w i t h t h e e x p e r i m e n t a l f i n d i n g s o f Chau and L e p p e l m e i e r [ 1 5 ] and o f P o h l ^ 1 ^ . TABLE H i EXPRESSIONS APPEARING IN EQUATION (67) EXPRESSION SIGNIFICANCE P Assumed c o u p l i n g c o n s t a n t r e l a t i n g t h e p h y s i c a l l e n g t h o f t h e r u b y r o d l i n e a r l y t o t h e p o p u l a t i o n i n v e r s i o n i n t h e r u b y medium (no v a l u e p r e s e n t l y a v a i l a b l e ) . K Mean p h y s i c a l l e n g t h o f r u b y r o d . n R « 1 . 7 6 I n d e x o f r e f r a c t i o n o f r u b y . N 0 « 1 . 6 » 1 0 1 9 / c m 3 D e n s i t y o f C r + 3 i o n s i n r u b y ( v a l u e g i v e n i s f o r 0 . 0 5 $ C r , b y w e i g h t ) . N Qttc « 5 . 0 * 1 0 2 e r g / c m 3 C o e f f i c i e n t i n d e n o m i n a t o r o f ( 6 7 ) . G ( t ) = ( c / L ) N n [ a ( w ) + f n ] r I n t e n s i t y g a i n c o e f f i c i e n t [ c f . (59) and T a b l e I ] . [ l 0 ( B ( t ) ) - I 1 ( B ( t ) ) ] e x p [ - B ( t ) ] -36-The r e s u l t o b t a i n e d b y F l a m h o l z and Wolga f o r t h e r a t e o f change o f t h e e m i s s i o n f r e q u e n c y ( e q u a t i o n (29) o f R e f . [5]) may be w r i t t e n i n t h e n o t a t i o n employed h e r e a s v n ( t ) = P' 2 I n ( t ) / [ N 0 L f t c ] (29) o f [5] where P' i s t h e c o u p l i n g " c o n s t a n t " assumed by F l a m h o l z and Wolga. C o m p a r i s o n w i t h e q u a t i o n (67) y i e l d s t h e r e l a t i o n s h i p : = P C i X 0 ( n R - 1 ) G ( t ) ] , (68) and, o t h e r w i s e , t h e two r e s u l t s a r e i d e n t i c a l . The d i s c r e p a n c y b e t w e e n t h e two r e s u l t s i s p a r t i a l l y due t o t h e d i f f e r e n c e s b e tween t h e i n t e n s i t y e q u a t i o n o f Szabo and S t e i n ^ - ^ and t h a t d e r i v e d h e r e , and i t i s p a r t i a l l y due t o t h e " f i r s t o r d e r " a n a l y s i s o f t h i s e f f e c t c a r r i e d o u t b y F l a m h o l z and Wolga. The d ependence o f P* on t h e g a i n c o e f f i c i e n t as shown i n e q u a t i o n (68) m i g h t w e l l e x p l a i n t h e r a t h e r l a r g e r a n g e o f v a r i a t i o n i n t h e e x p e r i m e n t a l v a l u e s o f P' o b t a i n e d b y F l a m h o l z and Wolga: P* « 4-.6»10"-5 cm t o 23-1CT-5 cm ... F i n a l l y , a l t h o u g h (67) a p p e a r s t o be i n q u a l i t a t i v e a g r e e -ment w i t h t h e a v a i l a b l e e x p e r i m e n t a l f i n d i n g s , i t has n o t y e t b e e n q u a n t i t a t i v e l y t e s t e d b y e x p e r i m e n t . F o r t h i s r e a s o n , and f o r t h e sake o f s i m p l i c i t y , t h e f r e q u e n c y sweep i s a p p r o x i m a t e d by a l i n e a r f u n c t i o n o f t i m e i n s u c c e e d i n g s e c t i o n s o f t h i s t h e s i s . The f a c t t h a t t h i s i s a good a p p r o x i m a t i o n o v e r t h e r a n g e o f t i m e s f o r w h i c h t h e i n t e n s i t y i s g r e a t e s t i s n o t r e a d i l y a p p a r e n t f r o m (67). "but a g r e e s q u i t e w e l l w i t h t h e ex-p e r i m e n t a l measurements o f t h e f r e q u e n c y sweep c i t e d a b o v e . - 3 7 -REFERENCES [ I ] W.E.Lamb.Jr., P h y s . Rev. A 134, A1429 (1964). [ 2 ] M . S a r g e n t , I I I , M . O . S c u l l y , and W.E.Lamb,Jr., L a s e r P h y s i c s ( A d d i s o n - W e s l e y , R e a d i n g , M a s s a c h u s e t t s , 1 9 7 4 ) , p. 9 6 . [ 3 ] A.Szabo and R . A . S t e i n , J . A p p l . P h y s . 3 6 , 1562 ( 1 9 6 5 ) . [ 4 ] C . R . G i u l i a n o and L.D.Hess, I E E E J . Quantum E l e c t r o n . Q E - 3 , 358 ( 1 9 6 7 ) . [ 5 ] A . F l a m h o l z and G.J.Wolga, J . A p p l . P h y s . 3 9 , 2723 ( 1 9 6 8 ) . [ 6 ] J . E . B j o r k h o l m and R . H . S t o l e n , J . A p p l . P h y s . 3 9 , 4043 ( 1 9 6 8 ) . [ 7 ] A.G.Fox and T . L i , B e l l S y s t . T e c h . J . 4 0 , 453 ( l 9 6 l ) . [ 8 ] H . K o g e l n i k and T . L i , A p p l . O pt. 5 , 1550 ( 1 9 6 6 ) . [ 9 ] T.H.Maiman, e t . a l . , P h y s . Rev. 1 2 3 , 1151 ( l 9 6 l ) . [ 1 0 ] I . S . G r a d s h t e y n and I.M.Ryzhik, T a b l e s o f I n t e g r a l s , S e r i e s . and P r o d u c t s ( A c a d e m i c P r e s s , New Y o r k , 1 9 6 5 ) , pp. 379 & 4 8 2 . [ I I ] D . J . B r a d l e y , G.Magyar, and M . C . R i c h a r d s o n , N a t u r e (Lond.) 212 (1 O c t . ) , 63 ( 1 9 6 6 ) . [ 1 2 ] V . V . K o r o b k i n , e t . a l . , J E T P L e t t . 3 , 194 ( 1 9 6 6 ) . [ Z h . E k s p . T e o r . F i z . P i s ' m a Red. 3 , 301 ( 1 9 6 6 ) . ] [ 1 3 ] D . A . B e r k l e y and G.J.Wolga, J . A p p l . P h y s . 3 8 , 3231 ( 1 9 6 ? ) . [ l 4 ] D . P o h l , P h y s . L e t t . A 26A, 357 ( 1 9 6 8 ) . [ 1 5 ] H.H.Chau and G.W.Leppelmeier, J . Opt. S o c . Am. 6 l , 998 ( 1 9 7 1 ) . [ 1 6 ] L . D . S i e b e r t , A p p l . O pt. 1 0 , 632 ( 1 9 7 1 ) . [ 1 7 ] E . B . T r e a c y , A p p l . P h y s . L e t t . 14, 112 ( 1 9 6 9 ) . [18] , A p p l . P h y s . L e t t . 17 , 14 ( 1 9 7 0 ) . [ 1 9 ] M . D . C r i s p , O p t . Commun. 3 , 111 ( 1 9 7 1 ) . - 3 8 -SECTION TWO: THEORY OF A HOLOGRAPHIC SYSTEM FOR LASER PARAMETER MEASUREMENT T h i s s e c t i o n d e a l s w i t h t h e t h e o r e t i c a l a n a l y s i s o f a s p e -c i f i c h o l o g r a p h i c s y s t e m t o measure t h e d e g r e e o f t e m p o r a l c o -h e r e n c e and o t h e r l a s e r p a r a m e t e r s o f s i n g l e p u l s e s f r o m a g i a n t p u l s e r u b y l a s e r . A number o f h o l o g r a p h i c s y s t e m s f o r m e a s u r i n g t h e d e g r e e o f t e m p o r a l c o h e r e n c e have b e e n d e v i s e d , and may be d i v i d e d i n t o two g e n e r a l c a t e g o r i e s . F i r s t t h e r e a r e s y s t e m s s u c h as t h o s e o f S i e b e r t ^ - 1 and o f Chau and L e p p e l m e i e r ^ - 2 - l w h i c h u t i l i z e many r e p e t i t i v e p u l s e s t o c o m p i l e a c u r v e o f t h e d e g r e e o f t e m p o r a l c o h e r e n c e w h i c h i s a v e r a g e d o v e r many p u l s e s . The s e c o n d t y p e o f s y s t e m m e a s u r e s t h e f u l l d e g r e e o f t e m p o r a l c o h e r e n c e o f a s i n g l e p u l s e . Systems o f t h i s s e c o n d t y p e i n -c l u d e t h o s e o f S t a s e l k o , D e n i s y u k and S m i r n o v ^ - ^ and o f B l o d -g e t t and P a t t e n ^ - L The s y s t e m d e v e l o p e d i n t h i s s e c t i o n i s o f t h i s s e c o n d t y p e , a l s o , and i s a q u a n t i t a t i v e v e r s i o n o f a q u a l i t a t i v e e x p e r i m e n t p r o p o s e d b y Lehmann^ - J . The S t a s e l k o , D e n i s y u k and S m i r n o v a p p a r a t u s g e n e r a t e s a smooth c o n t i n u o u s c u r v e o f t h e d e g r e e o f t e m p o r a l c o h e r e n c e , w hereas t h e e x p e r i m e n t c o n s i d e r e d i n t h i s s e c t i o n i s c a p a b l e o f m e a s u r i n g a number o f d i s c r e t e p o i n t s on t h i s c u r v e o n l y . The s y s t e m i n t h i s s e c t i o n , however, i s c a p a b l e o f a g r e a t e r r a n g e o f d e l a y t i m e s , o f more a c c u r a t e q u a n t i t a t i v e d a t a , and o f r e c o r d i n g i n f o r m a t i o n on t h e t r a n s v e r s e , a s w e l l a s t h e l o n g i t u d i n a l , s t r u c t u r e o f t h e l a s e r o u t p u t . A c o n t i n u o u s c u r v e -39-o f t h e d e g r e e o f t e m p o r a l c o h e r e n c e i s g e n e r a t e d , a l s o , b y t h e s y s t e m d e v i s e d b y B l o d g e t t and P a t t e n , b u t t h e maximum d e l a y t i m e s o b t a i n a b l e w i t h t h e i r a p p a r a t u s i s a few t e n t h s o f a n a n o s e c o n d . Due t o t h i s n a r r o w r a n g e o f d e l a y t i m e s , t h e B l o d g e t t and P a t t e n a p p a r a t u s i s more a p p l i c a b l e t o v e r y s h o r t c o h e r e n c e l e n g t h l a s e r s , s u c h a s n o n - Q - s w i t c h e d l a s e r s , t h a n t o g i a n t p u l s e l a s e r s . The b a s i c c o n c e p t o f t h e e x p e r i m e n t d i s c u s s e d i n t h i s s e c -t i o n , and, i n e s s e n c e , o f t h e e x p e r i m e n t s d i s c u s s e d a b o v e , i s t o f o r m a number o f s i m p l e h o l o g r a m s w h i c h compare t h e l a s e r o u t p u t t o i t s e l f a t a n e a r l i e r t i m e . The c o m p a r i s o n i s m a n i -f e s t e d i n t h e i n t e r f e r e n c e f r i n g e s t h a t make up t h e h o l o g r a m s . As shown i n t h i s s e c t i o n , t h e d i f f r a c t i o n e f f i c i e n c i e s o f s u c h h o l o g r a m s a r e a d i r e c t measure o f t h e d e g r e e o f t e m p o r a l c o h e r -e n c e . Once t h e d e g r e e o f t e m p o r a l c o h e r e n c e i s known, t h e W i e n e r - K h i n t c h i n e t h e o r e m may be i n v o k e d , f o r c e r t a i n t y p e s o f l a s e r o u t p u t s , t o f i n d t h e power s p e c t r u m o f t h e l a s e r o u t p u t t h r o u g h F o u r i e r a n a l y s i s . The a n a l y s i s o f t h e e x p o s u r e p a t t e r n g i v e n h e r e i s s i m i l a r t o t h a t o f t h e works c i t e d a b o v e . The d e v e l o p m e n t o f t h e f o r m a -t i o n and r e c o n s t r u c t i o n o f t h e h o l o g r a m s , however, d i f f e r s f r o m p r e v i o u s w o r k s . The p a p e r s m e n t i o n e d above have a l l c o n -s i d e r e d o n l y t h e s i m p l e s t m o d e l s f o r h o l o g r a m s : a n i n f i n i t e l y t h i n p h o t o g r a p h i c e m u l s i o n d i s p l a y i n g l i n e a r r e s p o n s e and no volume o r phase e f f e c t s . F o r a c c u r a t e q u a n t i t a t i v e measurements, t h i s s i m p l e model i s a l m o s t c e r t a i n l y i n s u f f i c i e n t . The m o d e l o f t h e h o l o g r a m employed h e r e i s b a s e d on t h e c o u p l e d wave -4-0-theory developed by Kogelnik^-1, and studies of r e a l photo-\~7 8~l graphic materials performed by BuschmannL'' J. This model assumes that the emulsion has a small, but f i n i t e , thickness, d, so that volume grating effects are considered to f i r s t order i n d. Also, i t i s assumed that t h i s emulsion offers l i n e a r re-sponse over some range of exposures. F i n a l l y , the holograms are taken to consist of both an absorption grating and a phase grating, simultaneously. THE EXPERIMENTAL SYSTEM The s p e c i f i c experimental system considered i n t h i s section i s depicted i n Figure 7. The laser output enters the apparatus at the l e f t hand edge of the figure and i s s p l i t into the reference beam and main signal beam by the f i r s t beamsplitter. The reference beam i s directed by mirrors through a two stage beam expander consisting of G a l l i l e a n telescopes. I t str i k e s the front surface of the photographic emulsion, i n the z=0 plane, at an angle of incidence, - 0 . The photographic emulsion i s taken to be p a r a l l e l to the z=0 plane, and has index of re-f r a c t i o n , n (prior to being processed), and approximately uni-form thickness, d. The primed coordinate system has i t s z' axis p a r a l l e l to the wavevector of the reference beam i n the emul-sion, and i s most useful i n formulating the description of the reference beam i n the emulsion. I t i s not made apparent i n Figure 7, but the origins of the three coordinate systems, primed, unprimed and double-primed, a l l coincide as do the x', x and x" axes. FIGURE 7 Holographic system for laser parameter measurement division of main BS=beam splitter signal beam into T=beam expanding telescope upper & lower signal beams - 4 - 2 -The m a i n s i g n a l "beam, m e a n w h i l e , i s d i r e c t e d b y a m i r r o r t o a b e a m s p l i t t e r - m i r r o r c o m b i n a t i o n t h a t s p l i t s t h e m a i n s i g n a l beam i n t o two p a r a l l e l beams ( s e e " S i d e V i e w D e t a i l " i n f i g u r e ) one b e l o w t h e o t h e r i n t h e d i a g r a m . By p r o d u c i n g a n u p p e r and l o w e r s i g n a l beam i n t h i s manner t h e number o f s i g n a l beams s u b s e q u e n t l y s t r i k i n g t h e e m u l s i o n i s d o u b l e d , and more e f f i c i -e n t u s e o f t h e s y s t e m i s p o s s i b l e . The u p p e r and l o w e r s i g n a l beams a r e d i v i d e d i n t o a number o f d e l a y e d s i g n a l beams o f n e a r l y e q u a l i n t e n s i t y b y t h e m u l t i p l e r e f l e c t i o n c e l l . The many beams l e a v i n g t h e m u l t i p l e r e f l e c t i o n c e l l a r e d e l a y e d r e l a t i v e t o t h e r e f e r e n c e beam and r e l a t i v e t o one a n o t h e r b y d i f f e r i n g amounts o f t i m e c h a r a c t e r i s t i c o f t h e g e o m e t r y o f t h e a p p a r a t u s . T h e s e s i g n a l beams s t r i k e t h e f r o n t s u r f a c e o f t h e e m u l s i o n a t a n a n g l e o f i n c i d e n c e , 0 . The s y m m e t r i c a n g l e s o f i n c i d e n c e o f t h e r e f e r e n c e beam and t h e s i g n a l beams a r e c h o s e n t o m i n i m i z e t h e e f f e c t s on t h e f r i n g e s p a c i n g o f emul-s i o n s h r i n k a g e d u r i n g p h o t o g r a p h i c p r o c e s s i n g . The d o u b l e -p r i m e d c o o r d i n a t e s y s t e m i s u s e f u l i n d e s c r i b i n g t h e s i g n a l beams i n t h e e m u l s i o n , and i t has i t s z" a x i s p a r a l l e l t o t h e w a v e v e c t o r s o f t h e s i g n a l beams i n t h e e m u l s i o n . F o r s i m p l i c i -t y , a l l t h e beams i n F i g u r e 7 a r e t a k e n t o be f u l l y p l a n e p o l a r i z e d w i t h t h e e l e c t r i c f i e l d v e c t o r a l o n g t h e common x a x e s o f t h e t h r e e c o o r d i n a t e s y s t e m s . THE EXPOSURE PATTERN IN THE PHOTOGRAPHIC EMULSION The e x p o s u r e p a t t e r n i n t h e p h o t o g r a p h i c e m u l s i o n i s r e l a -t i v e l y e a s y t o d e r i v e , and f o r a l a r g e c l a s s o f l a s e r o u t p u t - 4 3 -beams i s u n a m b i g o u s l y r e l a t e d t o t h e d e g r e e o f t e m p o r a l c o h e r -ence o f t h e o u t p u t beam. I n p a r t i c u l a r assume t h a t t h e l a s e r beam may be r e p r e s e n t e d b y t h e s c a l a r e l e c t r i c f i e l d , E ( r , t ) = i F ( X , Y ) A ( T 0 ) + c c . ( 6 9 ) f o r a beam p r o p a g a t i n g i n t h e p o s i t i v e z d i r e c t i o n . I n e q u a t i o n ( 6 9 ) . F ( X , Y ) i s a complex f u n c t i o n d e s c r i b i n g t h e t r a n s v e r s e p r o f i l e o f t h e l a s e r beam, and i s w r i t t e n a s a f u n c t i o n o f t h e r e l a t i v e t r a n s v e r s e d i s t a n c e s , X and Y. T h e s e r e l a t i v e t r a n s -v e r s e c o o r d i n a t e s a r e u s e d t o a c c o u n t f o r t h e e x t r e m e l y s l o w l y v a r y i n g z dependence o f t h e t r a n s v e r s e p r o f i l e r e s u l t i n g f r o m t h e d i v e r g e n c e o f t h e beam. The t r a n s v e r s e v a r i a b l e X i s g i v e n oy X = X ( x , z ) = [ x - x 0 ] / r ( z ) , ( 7 0 ) and Y i s g i v e n b y Y = Y ( y , z ) = [ y - y 0 ] / r ( z ) , ( ? l ) where t h e p o i n t ( x 0 , y Q ) , i n t h e c o n s t a n t z p l a n e , i s t h e c o -o r d i n a t e l o c a t i o n o f t h e c e n t e r o f t h e beam and r ( z ) i s t h e r a d i u s o f t h e beam (HWHM) i n t h i s p l a n e . To i l l u s t r a t e t h e d e f i -n i t i o n s , (?0) and ( 7 l ) i F i g u r e 8 shows t h e i n t e n s i t y p r o f i l e o f a d i v e r i n g beam as a f u n c t i o n o f x and X f o r two p o s i t o n s , z x and z 2 , a l o n g t h e beam a x i s . T h r o u g h o u t t h i s s e c t i o n , t h e l a s e r beam i s t a k e n t o be w e l l - c o l l i m a t e d ( d i v e r g e n c e a b o u t 1-10 mrad) so t h a t a l l d e r i v a t i v e s o f r ( z ) a r e n e g l i g i b l y s m a l l . The complex f u n c t i o n A ( T 0 ) i n e q u a t i o n ( 6 9 ) r e p r e s e n t s t h e l o n g i t u d i n a l s t r u c t u r e o f t h e l a s e r o u t p u t . The t e m p o r a l - s p a -t i a l c o o r d i n a t e , T 0 , whose o r i g i n t r a v e l s w i t h a f i x e d p o i n t o f FIGURE 8 I l lustration of relative transverse coordinate, X beam axis Intensity profile at z i hrtensity profile at zo as a function of X & x as a, function of X & x - 4 5 -the. l a s e r p u l s e e n v e l o p e , i s g i v e n "by T 0 = T G ( z , t ) = t - n z / c (72) f o r a beam t r a v e l i n g i n t h e p o s i t i v e z d i r e c t i o n i n t h e u n p r o -c e s s e d e m u l s i o n ; c i s t h e s p e e d o f l i g h t i n vacuum and n i s t h e i n d e x o f r e f r a c t i o n o f t h e u n p r o c e s s e d e m u l s i o n . I n w r i t i n g ( 7 2 ) , t h e p u l s e i s t a k e n t o p r o p a g a t e a t c o n s t a n t v e l o c i t y i n t h e e m u l s i o n , and t h e d i s p e r s i o n o f t h e e m u l s i o n i s assumed t o be n e g l i g i b l e . S i n c e A ( T G ) i s n e a r l y m o n o c h r o m a t i c , i t i s p o s -s i b l e , a l s o , t o w r i t e : where a ( T 0 ) i s a complex, s l o w l y v a r y i n g f u n c t i o n d e s c r i b i n g t h e l o n g i t u d i n a l p u l s e e n v e l o p e , and w i s t h e mean f r e q u e n c y o f t h e l a s e r o u t p u t . W i t h o u t l o s s o f g e n e r a l i t y , t h e f u n c t i o n A ( T Q ) s a t i s f i e s t h e n o r m a l i z a t i o n c o n d i t i o n : I t s h o u l d be n o t e d t h a t c e r t a i n a s s u m p t i o n s a b o u t t h e l a s e r o u t p u t a r e i n t r i n s i c i n e q u a t i o n ( 6 9 ) . I n p a r t i c u l a r , i t has b e e n assumed t h a t t h e t r a n s v e r s e p r o f i l e , F ( X , Y ) , i s i n d e p e n -d e n t o f t i m e and o n l y w e a k l y d e p e n d e n t on z. C o n v e r s e l y , t h e l o n g i t u d i n a l s t r u c t u r e , A ( T 0 ) , i s assumed t o be i n d e p e n d e n t o f t h e t r a n s v e r s e c o o r d i n a t e s , x and y . The t y p e s o f p h y s i c a l l a s e r p u l s e s w h i c h may be d e s c r i b e d b y a n e x p r e s s i o n o f t h e f o r m o f ( 6 9 ) may be f o u n d b y w r i t i n g E ( r , t ) i n a n e x p a n s i o n o v e r t h e n o r m a l q u a s i - m o d e s o f t h e l a s e r c a v i t y : A ( T Q ) = a ( T 0 ) e x p [ i w T G ] (73) j £ d t U ( T 0 ) ( 3 = l (74) E ( r , t ) = | E 5 S F l m ( X , Y ) A l m n ( T 0 ) + c c . (75) -46-T h i s e x p a n s i o n i s q u i t e s i m i l a r t o e q u a t i o n ( l l ) , S e c t i o n One, "but, h e r e , E ( r , t ) d e s c r i b e s a r u n n i n g wave w i t h t r a n s v e r s e s t r u c t u r e . The s u b s c r i p t s , n, 1 and m, c h a r a c t e r i z e t h e T E M ] _ m n quasi-mode o f t h e F o x and L i ^ - 9 ^ t y p e ; n i s t h e l o n g i t u d i n a l i n -dex and 1 and m a r e t h e t r a n s v e r s e i n d i c e s . By c o m p a r i n g (75) and ( 6 9 ) , i t i s e v i d e n t t h a t t h e r e a r e two c a s e s o f p h y s i c a l i m p o r t a n c e f o r w h i c h (75) may be r e d u c e d t o t h e f o r m o f ( 6 9 ) . F i r s t , f o r s i n g l e t r a n s v e r s e mode o p e r a t i o n , a l l t h e modes i n (75) w i l l have i d e n t i c a l 1 and m v a l u e s , and (75) becomes E ( r , t ) = jt F l m ( X , Y ) [S A l m n ( T 0 ) ] + c . c . (75a) T h i s e q u a t i o n i s o f t h e f o r m o f ( 6 9 )• F o r a r e a l l a s e r , t h e most p r o b a b l e t r a n s v e r s e s t r u c t u r e i n t h i s c a s e w o u l d be t h e f u n d a m e n t a l t r a n s v e r s e mode, T E M Q o n . S e c o n d l y , i f t h e c a v i t y i s o f t h e p l a n e p a r a l l e l F a b r y -P e r o t t y p e d i s c u s s e d i n S e c t i o n One, t h e n t h e o s c i l l a t i o n f r e -q u e n c y depends o n l y on t h e i n d e x n, and i s i n d e p e n d e n t o f t h e t r a n s v e r s e i n d i c e s ^ - 1 . I f t h e s l o w l y v a r y i n g e n v e l o p e s o f a l l t h e modes w i t h t h e same v a l u e o f n a r e assumed t o be i d e n t i c a l a l s o , t h e n A l m n ( T 0 ) becomes i n d e p e n d e n t o f 1 and m, and t h e s e i n d i c e s may be s u p p r e s s e d i n t h i s t e r m . F o r t h i s c a s e , (75) be-comes E ( r , t ) = * [5 S F l m ( X , Y ) ] [ 2 A n ( T 0 ) ] + c . c . (75b) w h i c h , a g a i n , i s o f t h e f o r m o f ( 6 9 ) . The s i t u a t i o n d e s c r i b e d b y (75b) a p p e a r s , a t l e a s t f r o m q u a l i t a t i v e o b s e r v a t i o n s , t o be a common o p e r a t i n g r e g i m e o f t h e s i n g l e f r e q u e n c y , g i a n t p u l s e r u b y l a s e r e m p l o y i n g a p l a n e p a r a l l e l F a b r y - P e r o t c a v i t y and a - 4 7 -s a t u r a b l e a b s o r b e r Q - s w i t c h . I n a c c o r d w i t h t h e s e c o n s i d e r a -t i o n s , i t i s assumed t h r o u g h o u t t h e r e m a i n d e r o f t h i s s e c t i o n t h a t t h e l a s e r o u t p u t i s a d e q u a t e l y d e s c r i b e d b y e q u a t i o n ( 6 9 ) . Now t h e e x p o s u r e p a t t e r n i n t h e p h o t o g r a p h i c e m u l s i o n may be e v a l u a t e d . The s c a l a r e l e c t r i c f i e l d o f t h e r e f e r e n c e beam, i n t h e e m u l s i o n , i s w r i t t e n : E R ( r , t ) = £ F ( X ' , Y « ) A ( T ' ) + c c . , (76) where, w i t h r e f e r e n c e t o F i g u r e 7» X' = [x» - x ' ] / r R ( z « ) (77) and Y' = [ y - yS]/rR(z«) . (78) T h e s e e q u a t i o n s may be w r i t t e n i n t e r m s o f t h e u n p r i m e d c o o r d i -n a t e s a s X' = x / r R ( 0 ) (77a) and Y' = y c o s e ' / r R ( 0 ) (78a) t o a good a p p r o x i m a t i o n , and where t h e common o r i g i n o f t h e c o o r d i n a t e a x e s has b e e n l o c a t e d a t t h e p o i n t where t h e c e n t r a l a x i s o f t h e r e f e r e n c e beam i n t e r s e c t s t h e f r o n t s u r f a c e o f t h e e m u l s i o n (z=0 p l a n e ) . A l s o , t h e e x t r e m e l y s l o w z* dependence o f t h e r a d i u s , r R ( z ' ) , h as b e e n u s e d i n r e p l a c i n g t h i s r a d i u s b y i t s v a l u e a t t h e o r i g i n , r R ( 0 ) . F i n a l l y , s i n c e t h e t h i c k n e s s o f t h e e m u l s i o n , d, i s q u i t e s m a l l , t y p i c a l l y l e s s t h a n 10 p,m, t h e z dependence o f (78a) i s n e g l i g i b l e and has b e e n d r o p p e d . The a n g l e , - 0 ' , i s t h e p r o p a g a t i o n a n g l e o f t h e r e f e r e n c e beam -48-in the emulsion r e l a t i v e to the posit i v e z axis and i s given by Snell's law: s i n e 1 = s i n e / n . (79) The longitudinal variable, T*, of the reference beam i s T» = t - nz'/c , (80) or, i n terms of the unprimed coordinates, T» = t - (n/c)[z cose' + y sine'] , (81) so that, as i n (73)» A(T') = a(T) exp[iwT'] (82) where the variable, T = t - (n/c) y sine' , (83) may be used i n the slowly varying part of (82) to a good approx-imation. S i m i l a r l y , the scalar e l e c t r i c f i e l d of the j-th signal beam i s represented by E,(r, t ) = i f . F(X Y ) A(T.) + c c . (84) J J J J J where i t i s assumed that the action of the various beamsplit-ters and mirrors i n Figure 7 may be taken into account v i a the r e a l constant, f . . Thus, the constant f^. i s the o v e r a l l ampli-tude attenuation of the j-th signal beam r e l a t i v e to the r e f e r -ence beam. I t i s assumed that the basic form of the transverse and l o n g i t u d i n a l structure of the laser output i s not altered by the mirrors and beamsplitters. In order for t h i s to be true the various o p t i c a l elements i n Figure 7 must be l i n e a r , uni--49-form and d i spers ion less over the re levant frequency range. I t should he noted that, when th i s i s the case, F(X.,Y.) and A(T.) i n (84) are i d e n t i c a l i n funct iona l form to F (X * ,Y I ) and J A ( T ' ) , re spec t i ve l y , i n (76). In equation (84), the r e l a t i v e coordinates of the j - t h s i g -na l beam, X. and Y., are given by J J and X. = [x" - x'!]/r .(z") (85) = [y" - y]l/r.U") , (86) where x1! and y'! are the coordinates of the' center of the j - t h J J s i gna l beam i n the z=0 plane, and r . ( z " ) i s the beam radius at th i s same locus. Equations (85) and (86) may be wr i t ten i n terms of the unprimed coordinates, making the same approxima-t ions discussed i n wr i t i ng (77a) and (78a), as X = [x - x , ] / r . ( z " ) (85a) J J J and = [y - YAI cose» / r , ( z " ) , (86a) J J J where x- and y . are the coordinates i n the z=0 plane of the center of the j - t h beam, 9* i s the propagation angle of the j - t h beam i n the emulsion from (79)' The argument of the radius func-t i on , r . ( z " ) , becomes J z" = - y . sine* . (87) J The va r i ab le , T . , has the form j T. = t - nz " / c - 6'! , (88) J J where 6'! i s the delay time of the j - t h s i gna l beam r e l a t i v e to J - 5 0 -the reference beam as evaluated at the o r i g i n , z'=z"=0. W r i t i n g equation (88) i n terms of the unprimed coordinates y i e l d s T. = t - (n/c)[z cos9' - y s i n e 1 ] - 6'! . (89) As i n (73) and (82), i t f o l l o w s then t h a t A(T.) = a ( T - 6.) exp[iwT.] (90) J J J where T i s given by ( 8 3 ) , and where &, = & • ! - 2(n/c) y. s i n e ' , (91) J J J to a good approximation f o r small values of d. The delay time, 6 ., shown i n (91) i s the delay time of the j - t h s i g n a l beam r e l a t i v e to the reference beam as evaluated at the p o i n t where the center of the j - t h s i g n a l beam i n t e r s e c t s the f r o n t surface of the emulsion. The t o t a l energy d e n s i t y ( i n ergs/cm 3) exposing the photo-graphic emulsion i n the r e g i o n of the j - t h hologram i s H.(r) = ( n c A r r ) £ d t [ E R ( r , t ) + E . ( r , t ) ] 2 (92) where i t i s assumed th a t the exposure time i s e s s e n t i a l l y i n -f i n i t e i n comparison to the pulse d u r a t i o n . I n terms of the e l e c t r i c f i e l d s , (76) and ( 8 4 ) , the exposure, ( 9 2 ) , becomes H.(r) = ( n c / 8 T T ) [ | F ( X ' , Y ' ) f 3 + f 3. | F ( X . , Y . ) | 2 + [ f . F ( X » , Y ' ) F * ( X . , Y . ) £ d t A(T')A*(T.) + c . c . ] ] (93) where the n o r m a l i z a t i o n , ( 7 4 ) , has been used and the d u r a t i o n of the pulse has been assumed to be much greater than an o p t i -c a l p e r i o d so th a t the i n t e g r a l of r a p i d l y o s c i l l a t i n g terms may be set equal to zero. I n order to s i m p l i f y ( 9 3 ) i the t o t a l - 5 1 -e n e r g y d e n s i t i e s o f t h e r e f e r e n c e beam and t h e j - t h s i g n a l beam t a k e n s i n g l y may be d e f i n e d a s I R = I R ( X ' , Y » ) = (nc/8TT) | F ( X ' , Y ' ) | 2 (94) and I . = I . ( X . , Y . ) = (nc/8Tr) f 2 . | F ( X . , Y . ) I S (95) J J J J J J J r e s p e c t i v e l y . A l s o , t h e t r a n s v e r s e s p a t i a l f u n c t i o n s may be w r i t t e n a s F ( X ' , Y » ) = | F ( X ' , Y ' ) | e x p [ i a ( X « , Y - ) ] (96) and F ( X ,Y ) = | F ( X . , Y )| e x p [ i a ( X , Y . ) ] . (97) J J J J J J S i n c e t h e r e f e r e n c e beam i s g r e a t l y expanded r e l a t i v e t o t h e s i g n a l beams, i t i s c o n v e n i e n t t o t a k e t h e t r a n s v e r s e p r o f i l e o f t h e r e f e r e n c e beam, F ( X * , Y , ) > -to be c o n s t a n t i n a m p l i t u d e and p h a s e o v e r t h e e x t e n t o f e a c h s m a l l h o l o g r a m . When t h i s i s done, e q u a t i o n (96) may be w r i t t e n F ( X ' , Y « ) = V ( 8 n / n c ) I R ; 5 e x p [ i a R j ] (96a) where I R ^ i s t h e c o n s t a n t e x p o s u r e due t o t h e r e f e r e n c e beam a l o n e i n t h e v i c i n i t y o f t h e j - t h h o l o g r a m and a R ^ i s t h e c o r -r e s p o n d i n g c o n s t a n t p h a s e . F u r t h e r , f o r n o t a t i o n a l s i m p l i c i t y , i t i s c o n v e n i e n t t o a b b r e v i a t e t h e argument o f F ( X . , Y . ) , shown i n ( 9 7 ) , t o a , . J I t i s c o n v e n t i o n a l i n h o l o g r a p h y t o u t i l i z e t h e h o l o g r a p h i c d e g r e e o f m o d u l a t i o n w h i c h , i n t h e p r e s e n t n o t a t i o n , i s , w r i t t e n m =m.(X.,Y.) = 2 V l ^ l T / H . , (98) where - 5 2 -H. = H.(X .,Y .) = I p . + I . (99) i s t h e a v e r a g e , o r b i a s l e v e l , e x p o s u r e o f t h e j - t h h o l o g r a m . C o m b i n i n g ( 9 6 a ) , ( 9 7 ) , (98) and (99) w i t h (93) y i e l d s H j ( r ) = H . { l H m . e x p [ i ( a R . - a . ) ] j ; ^ d t A ( T ' ) A * ( T . ) + c . c ] . (100) The i n t e g r a l i n (100) may be s i m p l i f i e d b y u s i n g e q u a t i o n s ( 8 2 ) , (90) and (91) g i v i n g £ d t A ( T ' ) A * ( T . ) = e x p [ - 2 i w ( n / c ) ( y - y . ) s i n e • ]• J ^ d t a ( T ) a * ( T - 6 . ) e x p [ i w 6 . ] , ( l O l ) o r j£~dt A ( T ' ) A * ( T . ) = e x p [ - i K . ( y - y ) ] j;~dT A ( T ) A * ( T - 6 . ) , (102) w i t h A ( T ) = a ( T ) e x p [ i w T ] (103) where T i s g i v e n b y ( 8 3 ) . I n ( 1 0 2 ) , t h e s p a t i a l f r e q u e n c y o f t h e h o l o g r a m , K, i s d e f i n e d b y K = 2w(n/c) s i n e ' . (104) I n w r i t i n g ( 1 0 2 ) , t h e v a r i a b l e o f i n t e g r a t i o n has b e e n t r a n s -f o r m e d f r o m t t o T a t e a c h p o i n t i n t h e e m u l s i o n . E x a m i n a t i o n o f (102) r e v e a l s t h a t t h e i n t e g r a l on t h e r i g h t hand s i d e o f t h i s e q u a t i o n i s s i m p l y t h e s t a n d a r d d e f i n i t i o n o f t h e complex d e g r e e o f t e m p o r a l c o h e r e n c e , Y ( S . ) = f d T A ( T ) A * ( T - 6 . ) (105) J -°° J where t h e n o r m a l i z a t i o n , ( 7 4 ) , has b e e n employed. W i t h t h e above r e s u l t , t h e e x p o s u r e p a t t e r n o f t h e j - t h h o l o g r a m becomes H . ( r ) = H J J 1 + i m^exp^- i [ K ( y - y j ) - a R ^ ]j y (6 . )+c. c. (106) - 5 3 -T i i l M a n d e l and W o l f L -1 have g i v e n a v e r y c o m p l e t e d i s c u s s i o n o f t h e p r o p e r t i e s o f t h e complex d e g r e e o f t e m p o r a l c o h e r e n c e , r-i o i and Heard 1- J has d i s c u s s e d t h e s e p r o p e r t i e s i n t h e c o n t e x t o f l a s e r p a r a m e t e r measurement. I n t h e s e works i t i s shown t h a t q u a s i - m o n o c h r o m a t i c r a d i a t i o n o f t h e t y p e p r o d u c e d b y l a s e r s has a complex d e g r e e o f t e m p o r a l c o h e r e n c e o f t h e f o r m Y ( S J = l Y ( 6 j l e x p [ i w 6 . - i u ( 6 . ) ] , (107) J J d d i n t h e n o t a t i o n employed h e r e ; I -y (S -) I i s t e r m e d t h e d e g r e e o f d t e m p o r a l c o h e r e n c e , and u(6 .) i s t h e s l o w l y v a r y i n g phase o f d t h e n e a r l y m o n o c h r o m a t i c complex d e g r e e o f t e m p o r a l c o h e r e n c e . F o r l a s e r r a d i a t i o n , t h e d e g r e e o f t e m p o r a l c o h e r e n c e i s a l s o s l o w l y v a r y i n g compared t o t h e r a p i d l y o s c i l l a t i n g e x p o n e n t i a l . S u b s t i t u t i n g (107) i n t o (105) y i e l d s t h e f i n a l f o r m o f t h e e x p o s u r e p a t t e r n i n t h e e m u l s i o n f o r t h e j - t h h o l o g r a m : H , ( r ) = H. [1 + m, Iy(6 . ) Icos(Ky + p . ) ] , ( 108) d d d d J where t h e n e t p h a s e , p., i s d e f i n e d w i t h t h e a i d o f (107) and j (91) as p . = p.(X ,Y ,6.) = a ( X . , Y )+ii(6 )-aR.-w6'! . (109) d d d d d d d d d d E q u a t i o n ( 108) d e s c r i b e s a n e x p o s u r e p a t t e r n o f n e a r l y e q u a l l y s p a c e d f r i n g e s o f s p a t i a l f r e q u e n c y , K, and s l o w l y v a r y i n g p h a s e , p.. The v i s i b i l i t y o f t h i s f r i n g e p a t t e r n i s , b y t h e J s t a n d a r d d e f i n i t i o n , V . = m. f Y(6 .) I , (110) J J J and t h e e x p o s u r e o f t h e p h o t o g r a p h i c e m u l s i o n i s b i a s e d a t t h e a v e r a g e e x p o s u r e , H.. -54-THE AMPLITUDE TRANSMITTANCE PATTERNS OF THE HOLOGRAMS When a p h o t o g r a p h i c e m u l s i o n has b e e n s u b j e c t e d t o t h e ex-p o s u r e , ( 1 0 8 ) , and p r o c e s s e d i n t h e u s u a l manner, a s m a l l h o l o -gram o f e a c h o f t h e s i g n a l beams r e s u l t s . F o r r e a l p h o t o g r a p h i c e m u l s i o n s , s u c h as t h o s e s t u d i e d b y B u s c h m a n n ^ ' ^ , t h e s e h o l o -grams a r e m i x e d g r a t i n g s h a v i n g b o t h a b s o r p t i o n and i n d e x o f r e f r a c t i o n m o d u l a t i o n . I n a d d i t i o n t o b e i n g m i x e d g r a t i n g s , t h e s p a t i a l f r e q u e n c y o f t h e m o d u l a t i o n i s g r e a t enough t h a t one must i n c l u d e t h e m o d u l a t i o n t r a n s f e r f u n c t i o n (MTF), o r f r e q u e n -c y t r a n s f e r f u n c t i o n , when a n a l y s i n g t h e t r a n s m i t t a n c e p a t t e r n , r a t h e r t h a n t h e r e s o l u t i o n w h i c h i s a p p r o p r i a t e a t t h e much l o w e r s p a t i a l f r e q u e n c i e s o f c o n v e n t i o n a l p h o t o g r a p h y . F o r s h o r t i n t e n s e e x p o s u r e s , s u c h a s a r e c o n s i d e r e d h e r e , r e c i p r o c i t y f a i l u r e must be t a k e n i n t o a c c o u n t , a l s o , and t h e c h a r a c t e r i s t i c c u r v e f o r t h e p h o t o g r a p h i c e m u l s i o n must be a p p r o p r i a t e l y c o r -r e c t e d t o compensate f o r t h i s e f f e c t . T h u s , i n o r d e r t o f u l l y a n a l y z e t h e p r o c e s s e d h o l o g r a m , i t i s n e c e s s a r y t o know l ) t h e c h a r a c t e r i s t i c c u r v e o f t h e e m u l s i o n f o r t h e r a n g e o f e x p o s u r e s , e x p o s u r e t i m e s , and s p e c t r a l r e g i o n s c o n s i d e r e d ( i n most h o l o -g r a p h i c a p p l i c a t i o n s t h e a m p l i t u d e t r a n s m i t t a n c e , l T a | , a s a f u n c t i o n o f t h e l o g a r i t h m o f t h e e x p o s u r e , H, i s t h e most c o n -v e n i e n t f o r m o f c h a r a c t e r i s t i c c u r v e ) ; 2) t h e p o r t i o n o f t h e t o -t a l m o d u l a t i o n t h a t i s due t o e i t h e r t h e a b s o r p t i o n g r a t i n g o r t h e p h a s e g r a t i n g a l o n e ( u s u a l l y t h e r a t i o o f t h e d i f f r a c t i o n e f f i c i e n c y o f t h e a b s o r p t i o n g r a t i n g a l o n e t o t h e t o t a l d i f f r a c -t i o n e f f i c i e n c y i s s u f f i c i e n t ) ; a nd 3) t h e v a l u e o f t h e MTF o v e r t h e a p p r o p r i a t e r a n g e o f s p a t i a l f r e q u e n c i e s . -55-Following Buschmann L' J, the amplitude transmittance r e s u l -t i n g from the exposure, (108), i s found by assuming that the exposure l i e s within the li n e a r portion of the c h a r a c t e r i s t i c curve, l T a l vs. H, of the emulsion, so that the amplitude transmittance pattern of the j-th hologram may be written l T a j l = l ^ a j l + M(K)[H-(r) - i ^ a i - , ( i l l ) where the average amplitude transmittance, l T a j l , corresponds to the average exposure, Hj, and M(K) i s the value of the MTF at the s p a t i a l frequency, K. Equation ( i l l ) w i l l be v a l i d for average exposures, H j , close .to the middle of the l i n e a r por-t i o n of the c h a r a c t e r i s t i c curve, and for degrees of modulation, m-j, less than about 0.4 for most emulsions. Since i t i s conven-t i o n a l to use a c h a r a c t e r i s t i c curve with a logarithmic expo-sure axis, the approximate, but applicable, r e l a t i o n s h i p h K = a/LHj In 10] (112) with a = i i ^ J ( 1 1 3 ) a(log H ) may be used to write ( i l l ) with (108) as l T a ; j I = | T a j l + M(K)[ct/ln l O ^ M & j ) I cos[Ky + p^] . ( l l 4 ) This equation gives the amplitude transmittance pattern of the processed j-th hologram within the l i m i t a t i o n s of properly chosen exposure parameters. The need for M(K) and a are r e a d i l y apparent i n (114), which, s t r i c t l y speaking, gives only the absorption part of the grating. When the reconstructed wave-front i s considered below, both (114) and the r a t i o , R, of the - 5 6 -d i f f r a c t i o n e f f i c i e n c y o f t h i s a b s o r p t i o n g r a t i n g a l o n e t o t h e t o t a l d i f f r a c t i o n e f f i c i e n c y a r e n e c e s s a r y . COUPLED WAVE THEORY OF THE RECONSTRUCTED WAVEFRONTS The c o u p l e d wave t h e o r y f o r t h i c k h o l o g r a m g r a t i n g s , a s de-v e l o p e d b y K o g e l n i k ^ - ) , ' i s u s e d t o e v a l u a t e t h e r e c o n s t r u c t e d w a v e f r o n t s r e s u l t i n g f r o m r e - i l l u m i n a t i o n o f t h e h o l o g r a m s de-s c r i b e d b y ( 1 1 4 ) . T h i s t h e o r y i s a p p r o x i m a t e , b u t i t i s i n c l o s e a g r e e m e n t w i t h e x p e r i m e n t f o r v a l u e s o f K o g e l n i k ' s Q - f a c -t o r Q = c K 3 d / [ n 0 3 w r ] , ( 1 1 5 ) g r e a t e r t h a n o r e q u a l t o a b o u t 1 0 . I n e q u a t i o n (115) t n 0 i s t h e mean i n d e x o f r e f r a c t i o n o f t h e d e v e l o p e d e m u l s i o n and w r i s t h e f r e q u e n c y o f t h e l i g h t u s e d t o r e c o n s t r u c t t h e h o l o g r a m s . T h i s c o n d i t i o n i s n e a r l y a l w a y s s a t i s f i e d f o r r e a l h o l o g r a p h i c a r r a n g e m e n t s , and, i n p a r t i c u l a r , i t i s s a t i s f i e d b y t h e e x p e r i -ment c o n s i d e r e d h e r e . The c o u p l e d wave t h e o r y b e g i n s w i t h M a x w e l l ' s e q u a t i o n s and t h e c o r r e s p o n d i n g c o n s t i t u t i v e r e l a t i o n s f o r a n e l e c t r o m a g n e t i c f i e l d i n t h e e x p o s e d and p r o c e s s e d e m u l s i o n . I n c g s G a u s s i a n u n i t s , t h e s e e q u a t i o n s may be w r i t t e n V • D = 0 ( 1 1 6 ) V • B = 0 ( 1 1 7 ) V x E = - ( 1 / c ) |= ( 1 1 8 ) V x H = ( 1 / C ) [ > T J + | | ] ( 1 1 9 ) - 5 7 -D = € E (120) J = a E (121) H = B . (122) The e m u l s i o n i s t a k e n t o have r e l a t i v e p e r m e a b i l i t y e q u a l t o u n i t y . By t a k i n g t h e c u r l o f ( 1 1 8 ) , and i n s e r t i n g (119) t o -g e t h e r w i t h t h e c o n s t i t u t i v e r e l a t i o n s , t h e e x p r e s s i o n 3E S 3 E V x V x E = - ( 1 / c ) 3 [4-TTO- + € ~ f ] (123) r e s u l t s . As u s u a l , t h e f i e l d i s t a k e n t o be f u l l y p l a n e p o l a r i -z e d a l o n g t h e x d i r e c t i o n so t h a t t h e s c a l a r e q u a t i o n i s w r i t t e n [ f p - + | p r ] E ( r , t ) = ( 1 / c ) 3 [ 4 i r a ^ + € ^ ] E ( r , t ) . (124) A s s u m i n g t h a t a h i g h l y m o n o c h r o m a t i c c o n t i n u o u s - w a v e l a s e r i s u s e d t o r e c o n s t r u c t t h e h o l o g r a m s , E ( r , t ) i n (124) may be w r i t t e n w i t h a t i m e - i n d e p e n d e n t a m p l i t u d e a s E ( r , t ) = i E ( r ) e x p [ i w r t ] + c c . (125) S u b s t i t u t i n g (125) i n t o (124) l e a d s t o t h e b a s i c d i f f e r e n t i a l e q u a t i o n o f t h e c o u p l e d wave t h e o r y : + + k 3 ^ = 0 » ( 1 2 6 ) where t h e complex p r o p a g a t i o n c o n s t a n t , k, i s g i v e n b y k 3 = € ( w r / c ) 3 - i a ( 4 r r w r/c 3) . (127) I n o r d e r t o d e s c r i b e a g r a t i n g o f t h e t y p e d i s p l a y e d i n ( 1 1 4 ) , s p a t i a l m o d u l a t i o n i s i n t r o d u c e d i n t o € and a: € = € 0 + € x c o s [ K y +• P i ] , (128) - 5 8 -and a = a 0 + a1 c o s [ K y + p ^ ] (129) f o r t h e j - t h h o l o g r a m . As d i s c u s s e d "by K o g e l n i k ^ - 1 , i t i s c o n v e n i e n t , and, i n n e a r l y a l l c a s e s o f i n t e r e s t , a p p l i c a b l e t o c h a r a c t e r i z e t h e p r o c e s s e d e m u l s i o n b y i t s i n d e x o f r e f r a c t i o n , n', and i t s (am-p l i t u d e ) a b s o r p t i o n c o n s t a n t , a, r a t h e r t h a n b y € and o. The a p p r o p r i a t e e q u a t i o n s c o r r e s p o n d i n g t o (128) and (129) a r e n« = n 0 + n x c o s [ K y + P j ] (130) and a = a 0 + a x c o s [ K y + p ^ ] , (131) where n o = ^ o » n i = i € i / n o » a 0 = 2 T T a 0 / ( c n 0 ) , a 1 = 2 r r a 1 / ( c n 0 ) . (132) T h i s d e s c r i p t i o n o f t h e d e v e l o p e d e m u l s i o n may be j u s t i f i e d b y o b s e r v i n g t h a t t h e a m p l i t u d e t r a n s m i t t a n c e o f t h e j - t h h o l o -gram a s shown i n ( l l 4 ) i s | T a ^ | = e x p [ - a d ] (133) b y d e f i n i t i o n . I n s e r t i n g (131) i n t o ( 1 3 3 ) , and a s s u m i n g ( a i d ) t o be s m a l l so t h a t a f i r s t o r d e r s e r i e s e x p a n s i o n i s a p p r o -p r i a t e , (133) becomes | T a j | = [1 - a x d c o s ( K y + p j ) ] e x p [ - a 0 d ] . (134) I t i s a l s o t r u e t h a t l f a J - l = e x p [ - a 0 d ] , (135) and i n s e r t i n g t h i s e x p r e s s i o n i n t o (134) r e s u l t s i n - 5 9 -l T a j ' = l ^ a j l " l T a j l axd cos[Ky + pj] , (136) which i s i d e n t i c a l to (114). This leads to the r e l a t i o n s h i p l T a j I aid = - M(K)[oc/ln 1 0 > j j ) I (137) "between the parameters of the processed emulsion and the expo-sure parameters. I t i s i m p l i c i t l y assumed that there i s a simi-l a r correspondence for the index of r e f r a c t i o n modulation. In terms of (130) and ( l 3 l ) » the complex propagation con-stant, k, becomes k 2 = n 0 k r [ n Q k r - 2 i a 0 + 4C cos(Ky + P j ) ] , (138) where k r = wr/c (139) i s the average, r e a l propagation constant i n vacuum, and C = i [ k r n x - i a i ] (140) i s the coupling constant for the transfer of energy between the reconstructing wave, E r ( r , t ) , and the reconstruction of the j-th signal beam, E s j ( r , t ) . The t o t a l e l e c t r i c f i e l d i n the emulsion, E(r,t.) of ( 1 2 5 ) , w i l l be the superposition of E r ( r , t ) and E s j ( r , t ) , E ( r , t ) = E r ( r , t ) + E s j ( r , t ) , ( l 4 l ) which y i e l d s the complex f i e l d amplitude E(r) = E r ( z ) e x p [ - i n 0 k r ( z cos9£ + y s i n 9 r ) ] +• E g j ( z ) -expj^ipj - i n 0 k r [ z cosQ^ - y(K/n 0k r - sin9£)]j , (142) where the slowly varying complex amplitudes, E r ( z ) and E g j ( z ) , - 6 0 -are assumed to vary s i g n i f i c a n t l y only i n the z d i r e c t i o n . The amplitudes are complex so that va r i a t i ons from the assumed phases i n ( l 4 2 ) may he taken into account. The angle, -9^-, i s the propagation angle of the reconstruct ing beam, E r ( r , t ) , and i s given by the appropriate form of S n e l l ' s law, ( 7 9 ) • Insert ing (14-2) into (126) with (138), the coupled d i f f e ren -t i a l equations, $ 3 ^ i Z ) ~ 2 i n 0 k r [ ( c o s 9 £ |^ + a 0 ) E r ( z ) + i C E s j ( z ) ] = 0 (14-3) and a 2E • (z) a 2 - y | J 2 i n 0 k r [ ( c o s 9 £ ~ +a- o +i0)E s j (z)+iCE r (z)] = 0 , ( l W are obtained by equating terms with the same exponentials separately to zero and by considering only the two beams of i n te re s t . In wr i t ing (144), i t has been assumed that the d e r i -vat ives of pj are n e g l i g i b l y small. The detuning parameter, 0, i s given by 0 = K s in9£ - i K 3 / n 0 k r , (145) and i s a measure of the amount by which the Bragg condi t ion, 1 K = n 0 k r s in9£ (146) or, with (104) and (79), w r s i n 9 r = w sin9 , (l46a) i s v i o l a t e d . Assuming the second der ivat ives i n (143) and (144) are n e g l i g i b l y small, the coupled wave equations r e s u l t : - 6 1 -SEy. (z) - . , „ , , , c o s 6 r — + a o E r ( z ) = -iC E s j ( z ) ; (14?) aEo^z) cos9 r — ^ + [ a 0 + 10] E s j ( z ) = -IC E r ( z ) . (148) As shown hy Kogelnik ^ - J, the general solution of these equa-tions, evaluated for the boundary conditions of a transmission type hologram, i s E s j ( z ) = - i E r(0) exp[-a 0Bz - i£P0z> sin[Bz Vc 3 + i 0 3 ] / Vl + i 0 2 / C 3 , (149) where B = sec9 r , (150) and where i t should be kept i n mind that C 3 i s a complex quan-t i t y . To simplify (l49), i t i s assumed that the photographic plate has been repositioned so that 0 r i s very nearly equal to 9 (within about 100 mrad), and that the wavelength of the cw source used to reconstruct the holograms i s close to the wave-length used to expose the hologram (within about 1000 A). Em-ploying these tolerances and t y p i c a l values^-I of a x (on the order of 400/cm), and k r n x (on the order of 300/cm), i t i s e a s i l y shown that both the r e a l and imaginary parts of the ar-gument of the sine function i n (14-9) are small enough over the thickness of the emulsion that using only the f i r s t term i n the series expansion of the sine i s a reasonable approximation, producing an error of about 10$ at the very most. In t h i s approximation, which i s tantamount to the assumption that mj i s small, the amplitude of the j-th signal beam reconstruction i n the emulsion, and at the interface between the emulsion and - 6 2 -t h e b a c k i n g o f t h e p h o t o g r a p h i c p l a t e ( z = d ) , i s E s j ( d ) = - i B d E r ( 0 ) Id | T a j l P e x p [ - i ( i P 0 d + O ) ] , (151) where D. i s t h e phase a n g l e o f C, g i v e n b y tanQ = a i / k r n x , (152) and where (135) has b e e n u s e d t o i n t r o d u c e t h e a v e r a g e a m p l i -t u d e t r a n s m i t t a n c e , l T a j | . C o m p a r i s o n o f (151) w i t h e q u a t i o n s ( 1 2 5 ) , ( l 4 l ) and (14-2) r e s u l t s i n t h e r e a l s c a l a r e l e c t r i c f i e l d o f t h e r e c o n s t r u c t i o n o f t h e j - t h s i g n a l beam: E s j ( r , t ) | z = d = - i | B d E r ( 0 ) |C| l T a j - | P e x p [ i w r t - i f l + ip-j - i n o k r [ d ( c o s 0 p +• i B 0 / n o k r ) - y ( K / n Q k r - s i n 0 ^ ) ] j + c c . (153) T h i s e x p r e s s i o n may be s i m p l i f i e d somewhat b y d e f i n i n g t h e p r o -p a g a t i o n a n g l e o f t h e j - t h r e c o n s t r u c t e d beam, 8 s j . I n t h e e m u l s i o n , t h i s a n g l e i s 0'^ g i v e n b y s i n e ' j = K / n 0 k r - s i n 0 r (154) w h i c h may be w r i t t e n a l s o a s s i n e g j = V s i n 2 9 £ - 2 0/n ok r . (155) The corresponding cosine i s c o s 0 s j = c o s 9 r V l + 2 0 3 0 / n o k r (156) o r , t o a good a p p r o x i m a t i o n , c o s 9 s j « cos9£ + P 0 / n o k r (157) f o r t h e v a l u e s o f 0 b e i n g c o n s i d e r e d h e r e . - 6 3 -I n s e r t i n g (154) and (157) i n t o (153) y i e l d s E s j ( r , t ) | z = d = -i£ B d E r ( 0 ) Id | T a j | B e x p [ i ( p j + iB0d - Cl) + i w r t - i n 0 k r ( d c o s 9 | j - y s i n 0 s j ) ] + c . c . (158) The wave d e s c r i b e d b y (158) s u b s e q u e n t l y p r o p a g a t e s t h r o u g h t h e b a c k i n g o f t h e p h o t o g r a p h i c p l a t e . T h i s b a c k i n g i s t y p i c a l l y made o f e i t h e r g l a s s o r p l a s t i c and i s t a k e n h e r e t o have t h i c k -n e s s , g, and i n d e x o f r e f r a c t i o n , n g . O u t s i d e t h e p h o t o g r a p h i c p l a t e ( z > d + g ) , t h e e l e c t r i c f i e l d o f (158) becomes E g j C r . t ) = - i i B d E r ( 0 ) Id l T a j I3 e x p [ i ( P j + §B0d - Cl - k r f ) + i w r t - i k r ( z c o s 9 s j - y s i n 9 s j ) ] + c . c . (159) where 9 s j i s t h e p r o p a g a t i o n a n g l e i n f r e e s p a c e o f t h e r e c o n -s t r u c t i o n o f t h e j - t h s i g n a l beam, g i v e n b y s i n 9 s j = n Q s i n 9 s j , ( l 6o) and where f i s t h e c o r r e c t i o n d i s t a n c e t o a c c o u n t f o r t h e op-t i c a l p a t h l e n g t h t r a v e r s e d b y t h e beam i n t h e p h o t o g r a p h i c p l a t e , f = d ( n Q e o s 9 ^ j - c o s 9 s j ) + g ( n g c o s 9 s j - c o s 9 s - j ) , ( l 6 l ) w i t h 9gj» t h e p r o p a g a t i o n a n g l e i n t h e b a c k i n g , g i v e n b y s i n 9 g j = s i n 9 s j / n g . (162) E q u a t i o n (159) c o n s t i t u t e s t h e f i n a l e x p r e s s i o n f o r t h e s c a l a r e l e c t r i c f i e l d o f t h e r e c o n s t r u c t i o n o f t h e j - t h s i g n a l beam. I t s h o u l d be n o t e d t h a t b o t h d and f may be c o n s i d e r e d t o be s l o w l y v a r y i n g f u n c t i o n s o f x and y, t h a t i s d = d ( x , y ) and f = f ( x , y ) , (163) -64-i n o r d e r t o t a k e i n t o a c c o u n t t h e , i n g e n e r a l , inhomogeneous t h i c k n e s s o f r e a l p h o t o g r a p h i c m a t e r i a l s . T h i s may be done p r o -v i d e d , o f c o u r s e , t h a t b o t h d and f may be c o n s i d e r e d e s s e n -t i a l l y c o n s t a n t o v e r d i s t a n c e s i n t h e x - y p l a n e t h a t a r e on t h e o r d e r o f t h e e m u l s i o n t h i c k n e s s . S i m i l a r l y , |c|, l T a j I > and P j a r e f u n c t i o n s o f x and y u n d e r t h e same s l o w l y v a r y i n g r e s t r i c -t i o n s . A l s o , i t may be n o t e d t h a t (159) c a r r i e s t h e ph a s e o f t h e i n t e r f e r e n c e p a t t e r n , p j , g i v e n i n e q u a t i o n (109). U n f o r t u n a t e -l y , t h i s p h a s e i s masked b y o t h e r t e r m s w h i c h a r i s e f r o m t h e r e -c o n s t r u c t i o n p r o c e s s . The r e l a t i o n s h i p b e t w e e n (159) and t h e d e g r e e o f t e m p o r a l c o h e r e n c e , I Y ( 6 ^ ) 1 , and t h e p o s s i b i l i t y o f phase measurements may be d i s c u s s e d now. MEASURING THE DEGREE OF TEMPORAL COHERENCE P e r h a p s t h e s i m p l e s t measurement t h a t may be made w i t h t h e a p p a r a t u s d e s c r i b e d h e r e i n i s t h e d i f f r a c t i o n e f f i c i e n c y , € j . As u s u a l , t h i s q u a n t i t y i s d e f i n e d , f o r t h e j - t h h o l o g r a m , as € d = I s j / l r (164) where I g j i s t h e i n t e n s i t y o f t h e r e c o n s t r u c t i o n o f t h e j - t h s i g n a l beam, and where I r i s t h e i n t e n s i t y o f t h e i n c i d e n t beam u s e d t o r e c o n s t r u c t t h e h o l o g r a m . By r e f e r r i n g t o (159) and (142), i t i s s e e n t h a t I s j = (c/8<rr)8 3d 3 |c| 3 | T a j | 2 P E3(o) , (l65) and I r = (C/8TT) E 3 ( 0 ) , (166) y i e l d i n g - 6 5 -€j = 8 3 d 3 | T a j | 2 p Id 2 . (167) The s h a r e o f t h e d i f f r a c t i o n e f f i c i e n c y due t o t h e a b s o r p t i o n g r a t i n g a l o n e may he f o u n d b y s e t t i n g n x e q u a l t o z e r o i n ( l 4 - 0 ) . S u b s t i t u t i o n o f t h i s v a l u e o f C i n t o (167) y i e l d s € j = ! B 3 d 2 a x 3 lT a < j| 2 P . (167a) I t i s c o n v e n i e n t t o f o r m t h e r a t i o o f t h i s d i f f r a c t i o n e f f i c i -e n c y t o t h e d i f f r a c t i o n e f f i c i e n c y o f t h e g r a t i n g as a w h o l e : R = £ a / € j = i a x 3 / | c | 3 . (168) C o m b i n i n g (137) and (168) w i t h ( 1 6 7 ) , t h e d i f f r a c t i o n e f f i c i -e n c y may be w r i t t e n e- = [ B a M ( K ) / ( 2 V R I n 1 0 ) ] 3 [ | f a j l P _ 1 m - ] 3 I Y ( 6 J ) I 2 . (169) E q u a t i o n ( I 6 9 ) shows t h e c o n n e c t i o n b e t w e e n t h e d e g r e e o f t e m p o r a l c o h e r e n c e and t h e d i f f r a c t i o n e f f i c i e n c y o f t h e j - t h h o l o g r a m . I n t h i s e x p r e s s i o n , t h e f i r s t f a c t o r i n s q u a r e b r a c -k e t s may be c o n s i d e r e d t o be c o n s t a n t o v e r a l l t h e h o l o g r a m s and i s e s s e n t i a l l y a p r o p e r t y o f t h e g e o m e t r y o f t h e a p p a r a t u s and t h e p a r t i c u l a r p h o t o g r a p h i c e m u l s i o n b e i n g e m p l o y e d . Thus, t h e c o n s t a n t , P, w h i c h i s i n d e p e n d e n t , f o r a l l p r a c t i c a l p u r -p o s e s , o f t h e e x p o s u r e i s d e f i n e d t o be P = [ B a M ( K ) / ( 2 V R I n 1 0 ) ] . (170) The v a l u e o f P may be f o u n d b y d i r e c t measurement o f t h e t e r m s i n (170) a s d i s c u s s e d b y Buschmann^ - L I n p r a c t i c e , however, i t i s p r o b a b l y s u f f i c i e n t t o c o n s i d e r P a s a n o r m a l i z a t i o n c o n s t a n t and t o d e t e r m i n e i t s v a l u e b y s e t t i n g IY(O)| = 1 (171) - 6 6 -i n e q u a t i o n ( 1 6 9 ) • The s e c o n d f a c t o r i n s q u a r e b r a c k e t s i n (169) i s a c o r r e c -t i o n f a c t o r t h a t a c c o u n t s f o r t h e a t t e n u a t i o n i n t h e e m u l s i o n and f o r t h e v a r i a t i o n i n t h e d e g r e e o f m o d u l a t i o n o v e r t h e c r o s s - s e c t i o n o f t h e h o l o g r a m . The v a l u e s n e c e s s a r y f o r e v a l u a -t i o n o f t h i s t e r m a r e r e l a t i v e l y s t r a i g h t f o r w a r d , s t a n d a r d pho-t o m e t r i c measurements o f t h e t r a n s m i s s i o n o f t h e d e v e l o p e d emul-s i o n and r e q u i r e a knowledge o f t h e c h a r a c t e r i s t i c c u r v e o f t h e e m u l s i o n f o r e v a l u a t i o n . The v a l u e o f l^aj I f o l l o w s i m m e d i a t e l y f r o m t h e s e measurements, and m-j i s f o u n d w i t h t h e a i d o f t h e c h a r a c t e r i s t i c c u r v e and e q u a t i o n (98). The d e g r e e o f t e m p o r a l c o h e r e n c e i s g i v e n i n t e r m s o f t h e s e m e a s u r a b l e p a r a m e t e r s a s f Y ( 6 j ) | = / [ P m- i T a j P " 1 ] . (172) E q u a t i o n (172) i s one o f t h e m a j o r r e s u l t s o f t h i s s e c t i o n . The s i g n i f i c a n c e o f t h e v a r i o u s p a r a m e t e r s a p p e a r i n g i n t h i s equa-t i o n a r e summarized i n T a b l e I I I . By m a k i n g t h e a p p r o p r i a t e measurements f o r e a c h s m a l l h o l o g r a m on a p a r t i c u l a r p h o t o -g r a p h i c p l a t e , d i s c r e t e v a l u e s o f t h e d e g r e e o f t e m p o r a l c o h e r -ence a t v a r i o u s known d e l a y t i m e s r e s u l t s . I n t e r p o l a t i o n t h e n y i e l d s a n e x p e r i m e n t a l c u r v e o f I Y ( 6 ) | f o r a s i n g l e p u l s e f r o m a g i a n t p u l s e l a s e r . I n v i e w o f t h e r e l a t i v e s i m p l i c i t y o f t h e a p p a r a t u s and t h e e x t r e m e l y s h o r t d u r a t i o n o f s u c h p u l s e s , t h i s method o f m e a s u r i n g t h i s i m p o r t a n t beam p a r a m e t e r i s i m p r e s s i v e . U n f o r t u n a t e l y , t h e phase o f t h e complex d e g r e e o f t e m p o r a l c o h e r e n c e , | i ( 6 j ) , i s n o t known. I n f a c t , a s shown below, t h i s TABLE I I I : EXPRESSIONS APPEARING IN EQUATION ( 1 7 2 ) EXPRESSION SIGNIFICANCE l Y ( 6 j ) l D e g r e e o f t e m p o r a l c o h e r e n c e (modulus o f complex d e g r e e o f t e m p o r a l c o h e r e n c e ) e v a l u a t e d a t d e l a y t i m e , 6 j , [ c f e q u a t i o n s ( 1 0 5 ) and ( 1 0 7 ) ] . D e l a y t i m e o f t h e j - t h s i g n a l beam r e l a t i v e t o t h e r e f -e r e n c e beam as e v a l u a t e d a t t h e p o s i t i o n o f t h e j - t h h o l o g r a m [ c f . e q u a t i o n s ( 8 8 ) and ( 9 1 ) ] . E x p e r i m e n t a l l y measured d i f f r a c t i o n e f f i c i e n c y o f t h e j - t h h o l o g r a m [ c f . e q u a t i o n ( 1 6 4 - ) ] . P h o t o m e t r i c a l l y measured a v e r a g e a m p l i t u d e t r a n s m i t t a n c e o f t h e j - t h h o l o g r a m . B S e c a n t o f 9 r> t h e p r o p a g a t i o n a n g l e o f t h e r e c o n s t r u c -t i n g beam i n t h e e m u l s i o n , [ c f . e q u a t i o n ( 1 5 0 ) ] . H o l o g r a p h i c d e g r e e o f m o d u l a t i o n [ c f . e q u a t i o n ( 9 8 ) ] . p C o n s t a n t f a c t o r t h a t i s c h a r a c t e r i s t i c o f t h e g e o m e t r y o f t h e a p p a r a t u s and o f t h e e m u l s i o n on t h e p h o t o g r a p h i c p l a t e [ c f . e q u a t i o n ( 1 7 0 ) ] . -68-p h a s e i s i r r e t r i e v a b l y l o s t i n t h e r e c o n s t r u c t i o n p r o c e s s . T h i s l i m i t s t h e u s e f u l n e s s o f t h e p r e s e n t e x p e r i m e n t w i t h r e g a r d s t o f i n d i n g t h e power s p e c t r u m o f t h e l a s e r p u l s e . I f , however, t h e l i n e s h a p e o f t h e p u l s e may he assumed t o he s y m m e t r i c a b o u t some mean f r e q u e n c y , w, t h e n u(6) v a n i s h e s and I Y ( 6 ) | i s s u f f i -c e n t t o d e t e r m i n e t h e l i n e s h a p e . I n t h i s c a s e , one m e r e l y a p p l i e s t h e W i e n e r - K h i n t c h i n e t h e o r e m and t a k e s t h e F o u r i e r t r a n s f o r m o f | Y ( 6 ) | rto f i n d t h e power l i n e s h a p e . A t t h e end o f t h i s s e c t i o n a n i l l u s t r a t i v e example f o r a s y m m e t r i c , f r e q u e n c y s h i f t e d p u l s e ( c f . S e c t i o n One, p. 30 e t . s e q . ) i s c a l c u l a t e d w h i c h c l a r i f i e s t h i s p r o c e d u r e and t h e r e s u l t s one may e x p e c t . PHASE MEASUREMENTS A t f i r s t i n s p e c t i o n , i t i s t e m p t i n g t o c o n c l u d e f r o m (159) t h a t i t m i g h t be p o s s i b l e t o e x t r a c t some o r a l l o f t h e p h a s e i n f o r m a t i o n c o n t a i n e d i n p j . I n p a r t i c u l a r , i t w o u l d be most i n t e r e s t i n g t o compare t h e r e l a t i v e p h a s e o f t h e v a r i o u s r e c o n -s t r u c t e d beams i n o r d e r t o measure | i ( 6 ) , t h e phase o f t h e com-p l e x d e g r e e o f t e m p o r a l c o h e r e n c e . As m e n t i o n e d e a r l i e r , t h i s c a n be shown t o be i m p r a c t i c a l . I t i s shown b e l o w t h a t , e v e n w i t h t h e most e x a c t i n g p r o c e d u r e s , t h e o n l y p h a s e v a r i a t i o n t h a t i t i s e x p e r i m e n t a l l y p r a c t i c a l t o measure i s t h e s p a t i a l , t r a n s v e r s e v a r i a t i o n o f t h e p h a s e , CCJ = a ( X j , Y - j ) , f r o m ( 9 7 ) . To see t h e s e c l a i m s e x p l i c i t l y , i m a g i n e t h a t t h e h o l o g r a m s a r e r e c o n s t r u c t e d , a s d i s c u s s e d above, so t h a t (159) r e s u l t s . I n a d d i t i o n t o t h i s , assume t h a t t h e cw s o u r c e u s e d i n r e c o n -s t r u c t i o n i s d i r e c t e d t h r o u g h t h e e n t i r e a p p a r a t u s shown i n - 6 9 -Figure 7 and very near ly retraces the exact path through each o p t i c a l element as the giant pulse used to expose the holograms. When th i s i s done, not only does a reference heam of the type used i n der i v ing (159) s t r i ke the developed emulsion, but cw beams that simulate the o r i g i n a l s i gna l beams also impinge upon the holograms. The reconstructed images formed by the cw s igna l beams are angular ly separated from the beams of i n te res t here and are ignored. Instead, these beams are used as a probe to evaluate the phase of the reconstructed s igna l beams through i n -ter ference. The j - t h of these beams s t r i kes the photographic plate at the l oca t i on of the j - t h hologram at an angle of i n c i -dence, 9 r j » a n d » when i t ex i t s from the photographic p la te , i t i s approximately co l inear with the j - t h reconstructed beam, i . e . 9 r j « 8 s j . Since the cw source used i n th i s reconst ruct ion pro-cedure i s assumed to have a very long coherence length, the j - t h reconstructed beam and the j - t h beam from the mult ip le r e f l e c -t i on c e l l add coherently, and the i n ten s i t y of th i s superposi-t i on d isp lays interference e f fec t s which are a measure of the phase d i f ference between the two beams. Thus i t appears that one might be able to i s o l a te and measure the phase p-j, and perhaps even be able to measure the components that make up p-j as shown i n equation ( 1 0 9 ) . In th i s regard, i t would be p a r t i c u l a r l y i n -te re s t ing to be able to measure oc(Xj,Yj) and |i(6). To see i f th i s i s experimental ly p r a c t i c a l , the angular and frequency deviat ions, A6 r, A 9 r j , and Aw r, are introduced by wr i t i ng 9 r = 9 + A9 r (173) - 7 0 -e r j = e + A 0 r j (174) w r = w + Aw r . (175) T h e s e d e v i a t i o n s a r e assumed t o he s m a l l enough t h a t t h e y n e e d he c o n s i d e r e d o n l y t o f i r s t o r d e r t h r o u g h o u t t h e a n a l y s i s . I n t h e r e c o n s t r u c t i o n p r o c e s s c o n s i d e r e d h e r e , t h e j - t h cw beam f r o m t h e m u l t i p l e r e f l e c t i o n c e l l may be r e p r e s e n t e d b y E r j ( r , t ) = i E r j e x p [ - i ( w r & * + k r f ' ) ] e x p [ i w r t - i k r * (z c o s 9 r j - y s i n 0 r j ) ] + c . c . (176) where 6^ = 6'j + A6'j (177) and f = d ( n 0 c o s 0 ^ j - c o s 0 r ( j ) + g ( n g cos0£j - c o s 0 r j ) (178) a r e i n t r o d u c e d t o a c c o u n t f o r t h e ph a s e s h i f t o f t h i s beam due t o t h e a c t i o n o f t h e m u l t i p l e r e f l e c t i o n c e l l and t h e p h o t o -g r a p h i c p l a t e . The p r i m e s i n (178) have t h e same m e a n i n g a s i n ( l 6 l ) , b u t w i t h r e f e r e n c e t o t h e p r o p a g a t i o n a n g l e 9 r j . The de-l a y t i m e d e v i a t i o n , AS'j> i s i n t r o d u c e d t o a l l o w f o r m i s a l i g n m e n t o f t h e cw beam i n t h e m u l t i p l e r e f l e c t i o n c e l l o r s l i g h t c h a n g e s i n t h e p o s i t i o n o f t h e p l a t e s i n t h i s c e l l d u r i n g t h e p r o c e s s -i n g o f t h e h o l o g r a m s . The e l e c t r i c f i e l d on t h e +z s i d e o f t h e p h o t o g r a p h i c p l a t e i s t h e sum o f (176) and ( 1 5 9 ) . F o r m i n g t h i s s u p e r p o s i t i o n , and e v a l u a t i n g t h e r e s u l t i n g i n t e n s i t y y i e l d s J r j = ^ r j [ 1 + V r j s i n C P j + w r 6 j + k r ( f - f ) + i B 0 d - Cl - k r z * ( c o s 0 s j - c o s 0 r j ) + k r y ( s i n 0 s j - s i n 0 r j ) ] j , (179) - 7 1 -where I r j i s the average i n tens i t y of the f r inge pattern and V r ^ i s the standardly defined f r inge v i s i b i l i t y . Combining (179) with ( 1 7 3 ) , ( 1 7 4 ) , 1 7 5 ) , ( 1 7 7 ) , and ( 1 7 8 ) , y i e l d s , to lowest order i n the deviat ions defined above I r j = f r j £1 + V r j - s in[pj + w6'j + ^B0d - Cl + w6'j(Awr/w + A6'j/6'j) - k r ( z sine + y c o s 9 ) ( A e r + A 6 r j +• 2 tane Aw r /w ) ] j . (180) To s imp l i fy th i s expression i t i s convenient to assume that the holograms have been bleached so that only the phase grat ing i s present and, from ( 1 5 2 ) , Cl = a r c t a n ( a i / k r n i ) = 0 . ( l 8 l ) Also, i t i s poss ib le to assume that the angular and frequency deviat ions are small enough that the term [iB0d] i s n e g l i g i b l y small when compared to the other terms i n the phase. Insert ing (109) into (180) for p-j, and l e t t i n g y* = (y cose + z sine) (182) be the s p a t i a l coordinate perpendicular to the x - d i r e c t i o n and in the plane at which the f r inges are observed, y i e ld s I r j = I r j j \ + V r j sin[otj - a R j + p. (6 j) + w6'j(Awr/w +• A6'j/6'j) - k r y * ( A 6 r + A 6 r j + 2 tane Aw r /w ) ] j . ( I 8 3 ) The phase' term i n ( I 8 3 ) which i s proport iona l to [w6'j] has a devastating e f fec t on the p o s s i b i l i t y of phase measurements. T y p i c a l l y , the value of [w6'-] would be on the order of 10^ im-p l y ing that, i n reconst ruct ing the holograms, the o r i g i n a l f r e -- 7 2 -q u e n c y and d e l a y t i m e s w o u l d have t o he r e p r o d u c e d t o a b o u t one p a r t i n 10? i n o r d e r f o r t h i s t e r m t o have a n e g l i g i b l e e f f e c t . I n p r a c t i c e , t h i s i s c l e a r l y a h o p e l e s s e x p e c t a t i o n s i n c e e v e n a d j a c e n t modes i n t y p i c a l l a s e r s y s t e m s [ c f . S e c t i o n One, equa t i o n ( l O ) ] d i f f e r i n f r e q u e n c y b y a b o u t t h i s amount and t h e b a n d w i d t h o f p o s s i b l e l o n g i t u d i n a l modes f o r most r e s o n a t o r s i s a t l e a s t a n o r d e r o f m a g n i t u d e g r e a t e r t h a n t h i s t o l e r a n c e . I n s h o r t , i t i s e s s e n t i a l l y i m p o s s i b l e t o r e p r o d u c e t h e o r i g i n a l f r e q u e n c y t o t h e d e g r e e o f p r e c i s i o n r e q u i r e d f o r ph a s e measure ments. E v e n i f t h e f r e q u e n c y c o u l d be matched p r e c i s e l y , one w o u l d s t i l l be f a c e d w i t h t h e p r o b l e m o f m a i n t a i n i n g t h e p h y s i -c a l s t a b i l i t y o f t h e a p p a r a t u s and m a t c h i n g t h e o r i g i n a l a n g l e s so t h a t t h e d e l a y t i m e i n t h e m u l t i p l e r e f l e c t i o n c e l l o f e a c h beam i s r e p r o d u c e d t o a b o u t one p a r t i n 10' a l s o . T h i s seems t o be a n e q u a l l y h o p e l e s s e n d e a v o r . I t may be c o n c l u d e d , t h e r e f o r e t h a t c o m p a r i s o n s o f t h e phase o f one p a t t e r n , s a y t h e j - t h , t o t h e p h a s e o f a n o t h e r , s a y t h e j ' - t h , w i l l be d o m i n a t e d b y t h e u n c o n t r o l l a b l e f r e q u e n c y d e v i a t i o n s . T h u s , c a n n o t be mea-s u r e d b y t h e a p p a r a t u s c o n s i d e r e d h e r e . A l t h o u g h i t i s u s e l e s s to- compare t h e ph a s e o f one beam t o t h a t o f a n o t h e r beam, i t i s p o s s i b l e t o measure t h e t r a n s v e r s e s p a t i a l v a r i a t i o n o f t h e r e l a t i v e p hase w i t h i n a s i n g l e beam. T h i s v a r i a t i o n i n p h a s e i n t h e t r a n s v e r s e d i r e c t i o n s i s a d i r e c measure o f t h e r e l a t i v e t r a n s v e r s e p hase v a r i a t i o n o f t h e o r i -g i n a l beam, a-j = a ( X - j , Y j ) , i n t r o d u c e d i n e q u a t i o n ( 9 7 ) . T h i s may be d e m o n s t r a t e d e x p l i c i t l y b y c o n s i d e r i n g o n l y a s i n g l e one o f t h e i n t e r f e r e n c e p a t t e r n s , ( 1 8 3 ) , and n e g l e c t i n g o v e r a l l -73-p h a s e t e r m s f o r t h i s one p a t t e r n t h a t a r e i n d e p e n d e n t o f t h e t r a n s v e r s e s p a t i a l c o o r d i n a t e s , so t h a t , f o r t h e j - t h p a t t e r n , I r j = I r j j \ + V r j s i n [ a ( X j , Y j ) - k r y * ( A 8 r + A 9 r j + 2 tane Aw r/w ) ] j . (184) The h o l o g r a m i s s t i l l assumed t o be b l e a c h e d so t h a t n i s ne-g l i g i b l y s m a l l , and, i n o r d e r t o be a b l e t o n e g l e c t t h e t e r m c o n t a i n i n g 0, i t i s assumed t h a t A 9 r < 5 mrad and t h a t Aw r/w < 0.001 Thes e l i m i t s a r e n o t e x c e s s i v e l y s t r i n g e n t , and, once t h e s e t o l e r a n c e s have b e e n e s t a b l i s h e d , f i n e a d j u s t m e n t s may be made i n e i t h e r t h e a n g l e s o r t h e f r e q u e n c y u n t i l t h e f r i n g e c o u n t a l o n g t h e y * a x i s i s a minimum. A t t h i s n u l l p o i n t , t h e f r i n g e p a t t e r n , ( 1 8 4 ) , r e d u c e s t o I r j = I r j [ l + Y r ; j s i n [ a ( X j , Y ; j ) ] j , (184a) a g a i n , c o n s i d e r i n g t h e j - t h beam o n l y . C l e a r l y , t h e f r i n g e p a t t e r n w h i c h r e m a i n s a t t h e n u l l p o i n t i s a d i r e c t d i s p l a y o f th e t r a n s v e r s e v a r i a t i o n i n r e l a t i v e p h a s e o f t h e g i a n t p u l s e u s e d t o expose t h e h o l o g r a m . I t i s p o s s i b l e t h a t t h e j - t h cw beam f r o m t h e m u l t i p l e r e f l e c t i o n c e l l , (l?6), may have some t r a n s v e r s e v a r i a t i o n i n p h a s e . However, s i n c e t h i s v a r i a t i o n w i l l , i n g e n e r a l , be r e l a t i v e l y s i m p l e f o r cw l a s e r s o u r c e s o p e r a t i n g i n t h e T E M 0 0 t r a n s v e r s e mode, and s i n c e t h i s v a r i a -t i o n may be me a s u r e d b y r e l a t i v e l y s t r a i g h t f o r w a r d i n t e r f e r o --74-m e t r i c means, i t i s n o t d i f f i c u l t t o t a k e t h i s e f f e c t i n t o a c c o u n t and a p p r o p r i a t e l y a d j u s t t h e e x p e r i m e n t a l r e s u l t s t o y i e l d t h e d e s i r e d p h a s e f u n c t i o n , a ( X - j . Y j ) , u n a m b i g u o u s l y . OTHER MEASUREMENTS I n a d d i t i o n t o t h e h o l o g r a p h i c measurements s u g g e s t e d above i t i s p o s s i b l e a l s o t o measure t h e t r a n s v e r s e i n t e n s i t y p r o f i l e o f t h e o r i g i n a l l a s e r p u l s e b y c o n v e n t i o n a l p h o t o m e t r i c methods. The t r a n s v e r s e i n t e n s i t y p r o f i l e o f t h e o r i g i n a l l a s e r p u l s e i s r e c o r d e d i n t h e a v e r a g e t r a n s m i s s i o n o f e a c h s m a l l h o l o g r a m , and i f | T a j | , f o r i n s t a n c e , i s m e a s u r e d as a f u n c t i o n o f X j and Y j w i t h a s t a n d a r d d e n s i t o m e t e r o v e r t h e e x t e n t o f t h e j - t h h o l o g r a m , t h e n t h e t r a n s v e r s e i n t e n s i t y p r o f i l e , I-j (X j , Y-j), o f t h e g i a n t p u l s e beam u s e d t o expose t h e h o l o g r a m i s f o u n d b y e m p l o y i n g t h e c h a r a c t e r i s t i c c u r v e f o r t h e e m u l s i o n . F u r t h e r , s i n c e i n c r e a s i n g d e l a y t i m e s c o r r e s p o n d t o i n c r e a -s i n g p r o p a g a t i o n d i s t a n c e s i n t h e m u l t i p l e r e f l e c t i o n c e l l , t h e d i v e r g e n c e o f t h e g i a n t p u l s e l a s e r beam may be f o u n d b y measur-i n g t h e i n c r e a s e i n s p o t s i z e o f t h e h o l o g r a m as t h e p r o p a g a -t i o n d i s t a n c e i n c r e a s e s . T o g e t h e r w i t h t h e c h a r a c t e r i s t i c c u r v e f o r t h e e m u l s i o n , t h i s d a t a y i e l d s t h e beam r a d i u s , r j ( z " ) , [ c f . e q u a t i o n s (85) and (86)] a s a f u n c t i o n o f t h e p r o p a g a t i o n d i s t a n c e . T h e s e measurements o f t h e t r a n s v e r s e i n t e n s i t y p r o f i l e , o f t h e beam d i v e r g e n c e , and o f t h e t r a n s v e r s e v a r i a t i o n i n r e l a t i v e p h a s e o f t h e o u t p u t f r o m a g i a n t p u l s e l a s e r p r o v i d e a u n i q u e - 7 5 -k e y t o e x p e r i m e n t a l measurements o f t h e c o m p l e t e t r a n s v e r s e s r t u c t u r e o f g i a n t p u l s e f r o m p a s s i v e l y Q - s w i t c h e d l a s e r s . EXPECTED FORM OF | Y ( 6 ) | B e f o r e c o n c l u d i n g t h i s s e c t i o n , i t seems w o r t h w h i l e t o work o u t i n d e t a i l t h e e x p e c t e d f o r m o f t h e d e g r e e o f t e m p o r a l c o -h e r e n c e , I Y ( 6 ) | , f o r a p h y s i c a l l y r e a s o n a b l e m o d e l o f t h e o u t p u t o f a g i a n t p u l s e l a s e r . The l i n e s h a p e i n f o r m a t i o n t h a t c a n he d e r i v e d f r o m t h i s f u n c t i o n s h a l l he d i s c u s s e d b e l o w , a l s o . The m a t h e m a t i c a l m o d e l u s e d h e r e t o d e s c r i b e t h e g i a n t p u l s e i s em-p i r i c a l , b u t i t i s i n c l o s e a g r e e m e n t w i t h e x p e r i m e n t a l o b s e r v a -t i o n s o f t h e o u t p u t f r o m s i n g l e f r e q u e n c y , g i a n t p u l s e l a s e r s . T h i s m o d e l assumes t h a t t h e p u l s e may be r e p r e s e n t e d b y a G a u s s i a n t e m p o r a l e n v e l o p e c o n t a i n i n g a h i g h l y m o n o c h r o m a t i c o p t i c a l f r e q u e n c y o s c i l l a t i o n whose peak f r e q u e n c y d i s p l a y s a s h i f t i n f r e q u e n c y t h a t i s a l i n e a r f u n c t i o n o f t i m e o v e r t h e d u r a t i o n o f t h e p u l s e [ c f . S e c t i o n One, p . 36] s A ( T ) = Nj^dw" exp[-b(w'(T)-W") 2/2(AW) 3] e x p [ i w " T ] , ( 1 8 5 ) where A ( T ) i s t h e n o r m a l i z e d t e m p o r a l f u n c t i o n i n t r o d u c e d i n e q u a t i o n (69). I n t h i s e x p r e s s i o n , t h e t i m e d e pendence i s i n t r o -d u c e d i n t o t h e peak f r e q u e n c y v i a t h e f u n c t i o n w»(T) = w + i s T ( 1 8 6 ) where, as b e f o r e , w i s t h e mean f r e q u e n c y o f t h e l a s e r p u l s e , and s i s t h e f r e q u e n c y sweep r a t e . The o t h e r t e r m s i n ( 1 8 5 ) may be d e f i n e d as f o l l o w s : (Aw) i s t h e ( i n s t a n t a n e o u s ) power l i n e -w i d t h (FWHM); b = 4- I n 2 ; and N i s a n o r m a l i z a t i o n c o n s t a n t - 7 6 -whose v a l u e f o l l o w s f r o m e q u a t i o n (74-). The n e c e s s a r y i n t e g r a t i o n i n (185) i s s t r a i g h t f o r w a r d and y i e l d s A ( T ) = N AwV2<rr/b e x p [ - ( A w ) 3 T 3 / ( 2 b ) ] e x p [ i ( w + £ s T ) T ] . ( 1 8 7 ) I t i s e v i d e n t i n ( 187) t h a t t h e d u r a t i o n (FWHM) o f t h e power p u l s e i s g i v e n "by AT = b/(Aw) . ( 1 8 8 ) I n s e r t i n g ( 1 8 8 ) i n t o ( 1 8 7 ) , and e v a l u a t i n g N e x p l i c i t l y w i t h t h e a i d o f (74) l e a d s t o t h e e x p r e s s i o n A ( T ) = [ b / r r ( A T ) 3 ] * e x p [ - b T 3 / 2 ( A T ) 2 ] e x p [ i ( w + • £ s T ) T ] (189) f o r t h e t i m e d e v e l o p m e n t o f t h e f i e l d i n t h i s m o d e l . The complex d e g r e e o f t e m p o r a l c o h e r e n c e i s c a l c u l a t e d b y i n s e r t i n g ( I 8 9 ) i n t o (105) and p e r f o r m i n g t h e i n t e g r a t i o n t o y i e l d Y(6) = e x p [ - b ( l +• s 3 ( A T ) V b 2 ) 6 3 / 4 ( A T ) 3 ] exp[iw6] . (190) T h i s e x p r e s s i o n i s o f t h e f o r m o f (107) a s e x p e c t e d . A l s o , i t i s e v i d e n t f r o m (190) t h a t t h e c o h e r e n c e t i m e (FWHM) f o r t h i s p u l s e i s r e d u c e d b y t h e f a c t o r [1 + s 3 ( A T ) 4 / b 3 ] " * ( 1 9 1 ) as compared t o t h e c o h e r e n c e t i m e o f a n o n - s h i f t e d (s=0) p u l s e . A c c o r d i n g t o t h e W i e n e r - K h i n t c h i n e t h eorem, t h e n o r m a l i z e d power s p e c t r u m i s t h e F o u r i e r t r a n s f o r m o f (190) m u l t i p l i e d b y t h e a p p r o p r i a t e n o r m a l i z a t i o n c o n s t a n t . E x p l i c i t l y , t h e t h e o r e m g i v e s t h e r e s u l t - 7 7 -| P ( v ) | 3 = e x p [ - b ( w - V ) 3 / [ ( A W ) 3 ( 1 + s 3 ( A T ) 4 / b 3 ) ] ] (192) f o r t h e n o r m a l i z e d power s p e c t r u m . I t s h o u l d he n o t e d t h a t t h i s power s p e c t r u m i s s y m m e t r i c a b o u t t h e mean f r e q u e n c y , w, and, f o r t h i s r e a s o n (i(6) v a n i s h e s [compare e q u a t i o n s (107) and ( 1 9 0 ) ] ' As shown above, however, i t i s t h e d e g r e e o f t e m p o r a l c o h e r e n c e , IY(6) [, t h a t i s e x p e r i m e n t a l l y m e a s u r a b l e w i t h t h e a p p a r a t u s c o n s i d e r e d h e r e r a t h e r t h a n t h e complex d e g r e e o f t e m p o r a l c o h e r e n c e , Y ( 6 ) . I n s p e c t i o n o f (190) and (192) shows t h a t , f o r t h e m o d e l b e i n g c o n s i d e r e d h e r e , t h e F o u r i e r t r a n s -f o r m o f Y(6) d i f f e r s f r o m t h e F o u r i e r t r a n s f o r m o f IY (6)| o n l y b y a c o n v o l u t i o n w i t h t h e D i r a c d e l t a f u n c t i o n , 6(w-v). T h i s d i s t i n c t i o n i s t r i v i a l i n t h e p r e s e n t i n s t a n c e , and, i f o n l y t h e l i n e s h a p e and n o t t h e e x a c t v a l u e o f t h e mean f r e q u e n c y , w, i s o f i n t e r e s t , t h e n a knowledge o f t h e d e g r e e o f t e m p o r a l c o h e r -e n c e , IY(6) I, i s s u f f i c i e n t . I n g e n e r a l , IY(6) I i s s u f f i c i e n t t o c o m p l e t e l y d e t e r m i n e t h e l i n e s h a p e , [ P ( V ' ) I 3 (where v'= w-v), whenever t h e l i n e s h a p e i s s y m m e t r i c . F o r s i n g l e f r e q u e n c y , g i a n t p u l s e l a s e r o p e r a t i o n , t h e a s s u m p t i o n o f s y m m e t r i c l i n e -shape a p p e a r s t o be a r e a s o n a b l e a p p r o x i m a t i o n . I f , however, two o r more l o n g i t u d i n a l modes o s c i l l a t e s i m u l t a n e o u s l y i n a p a r t i c u l a r p u l s e , t h e n t h i s a p p r o x i m a t i o n b r e a k s down, i n g e n e r a l , and c a r e must be t a k e n i n i n t e r p r e t i n g t h e F o u r i e r t r a n s f o r m o f | Y ( & ) I . Under t h e a d d i t i o n a l a s s u m p t i o n t h a t t h e peak power and l i n e s h a p e o f e a c h mode c o m p r i s i n g a m u l t i p l e f r e -q u e n c y o u t p u t a r e i d e n t i c a l , and t h a t t h e s e modes a r e s y m m e t r i -c a l l y p l a c e d a b o u t t h e mean f r e q u e n c y , t h e F o u r i e r t r a n s f o r m o f IY(6) I y i e l d s t h e c o r r e c t l i n e s h a p e s and mode s p a c i n g s . I n - 7 8 -general, though, for multiple frequency operation, the Fourier transform of IY(6) I should be considered to be only a q u a l i t a -t i v e i n d i c a t i o n of the mode structure and lineshapes. In t h i s general case, the linewidths that are obtained are, roughly speaking, "averaged" over a l l the l o n g i t u d i n a l modes present i n the pulse. Returning now to the model single frequency pulse, i t i s evident from (190) that |Y(6)I = exp[-b(l + s 2 ( A T ) 4 / b 2 ) 6 2 / M A T ) 2 ] . (193) The corresponding power lineshape i s | P(v»)| 2 = exp[-b V 2 / [ ( A w ) 2 ( l + s 2 ( A T ) 4 / b 2 ) ] ] , (19^) from which i t i s e a s i l y seen that the action of the frequency sweep i s to increase the linewidth (FWHM) by the factor [1 +• s 2 ( A T ) 4 / b 3 ] * . As discussed at the end of Section One, i t i s appropriate to de-fine both an instantaneous linewidth and a frequency sweep-width f o r giant pulses. The lineshape as determined from Y(&) or IY(6)| c l e a r l y y i e l d s the frequency sweep-width: (Aw)[l + s 2 ( A T ) 4 / b 2 ] * • (195) The instantaneous linewidth, (Aw), as defined i n the integrand of (185), i s most e a s i l y obtained by independently measuring the temporal power p r o f i l e of the pulse and applying equation (188) to give (Aw) = b/(AT) . (188a) - 7 9 -To the e x t e n t t h a t the t e m p o r a l p r o f i l e o f the p u l s e may he a p p r o x i m a t e d by a G a u s s i a n f u n c t i o n , e q u a t i o n (188a) y i e l d s the a p p r o p r i a t e v a l u e o f the i n s t a n t a n e o u s l i n e w i d t h . I n c o n c l u s i o n , the e x p e c t e d form of the degree of t e m p o r a l coherence, as might be o b t a i n e d by the experiment c o n s i d e r e d i n t h i s s e c t i o n has been d e r i v e d . I t has been shown t h a t f o r s i n g l e f r e q u e n c y , g i a n t p u l s e s w h i c h have symmetric power spectrums a knowledge of t h i s f u n c t i o n i s s u f f i c i e n t t o determine the t i m e -average l i n e s h a p e o f the p u l s e , and t h u s y i e l d the f r e q u e n c y sweep-width of the o u t p u t . F o r p u l s e s t h a t a r e a p p r o x i m a t e l y G a u s s i a n , the i n s t a n t a n e o u s l i n e w i d t h may be o b t a i n e d from the p u l s e d u r a t i o n . By c o m b i n i n g t h e s e measurements w i t h e q u a t i o n ( 1 9 5 ) , the f r e q u e n c y sweep r a t e , s, may a l s o be d e t e r m i n e d . SUMMARY I n t h i s s e c t i o n , a n o v e l e x p e r i m e n t a l system f o r m e a s u r i n g v a r i o u s beam parameters of s i n g l e p u l s e s from a g i a n t p u l s e l a s e r system has been t h e o r e t i c a l l y d i s c u s s e d . The advantages and l i m i t a t i o n s of t h i s system have been a n a l y s e d i n some de-t a i l . The r e c o n s t r u c t i o n of the holograms produced by t h i s s y s -tem has been d e v e l o p e d i n the manner s e t f o r t h o r i g i n a l l y by K o g e l n i k ^ - J , and a p p l i e d w i t h i n the framework of the r e a l p h o t o g r a p h i c e m u l s i o n s d i s c u s s e d by Buschmann^-L A l t h o u g h t h i s t h e o r y o f holograms i s a p p r o x i m a t e , i t i s h i g h l y a c c u r a t e f o r n e a r l y a l l c a s e s of p r a c t i c a l i m p o r t a n c e . As d i s c u s s e d i n t h i s s e c t i o n , i t i s p o s s i b l e t o measure, -80-s i m u l t a n e o u s l y , t h e f o l l o w i n g beam p a r a m e t e r s o f s i n g l e g i a n t p u l s e s s l ) t h e t r a n s v e r s e i n t e n s i t y p r o f i l e ; 2) t h e t r a n s v e r s e v a r i a t i o n o f r e l a t i v e p h a s e ; 3) t h e beam d i v e r g e n c e ; 4) t h e d e g r e e o f t e m p o r a l c o h e r e n c e ; and 5) t h e t e m p o r a l p r o f i l e o f t h e p u l s e . W i t h t h e ' e x c e p t i o n o f t h e l a s t p a r a m e t e r l i s t e d above ( w h i c h must be m e a s u r e d i n d e p e n d e n t l y b y any o f t h e s t a n d a r d m e t h o d s ) , a l l t h e s e measurements a r e p e r m a n e n t l y r e c o r d e d on a s i n g l e p h o t o g r a p h i c p l a t e f o r e a c h p u l s e . F o r s i n g l e f r e q u e n c y l a s e r o p e r a t i o n , r e s u l t i n g i n a n a p p r o x i m a t e l y G a u s s i a n p u l s e , i t i s p o s s i b l e t o combine t h e r e s u l t s o f t h e l a s t two measurements above t o f i n d l ) t h e i n -s t a n t a n e o u s l i n e w i d t h , 2) t h e f r e q u e n c y s w e e p - w i d t h , and 3) t h e l i n e a r f r e q u e n c y sweep r a t e ( i n t h e l i n e a r f r e q u e n c y sweep a p p r o x i m a t i o n ) . Thus, n e a r l y e v e r y beam p a r a m e t e r o f i n t e r e s t may be m e a s u r e d s i m u l t a n e o u s l y , f o r s i n g l e p u l s e s b y t h i s r e l a -t i v e l y s i m p l e and s t r a i g h t f o r w a r d method. -81-REFERENCES [ I ] L . D . S i e b e r t , A p p l . Opt. 1 0 , 632 ( 1 9 7 1 ) . [ 2 ] H.H.;Chau and G.W.Leppelmeier, J . Opt. Soc. Am. 6 l , 998 (1971) • [ 3 ] D . I . S t a s e l k o , Yu.N.Denisyuk, and A.G.Smlrnov, Opt. S p e c t r o s c . 2 6 , 225 ( 1 9 6 9 ) . [Opt. S p e k t r o s k . 2 6 , 413 ( 1 9 6 9 ) . ] [ 4 ] J . A . B l o d g e t t and R . A . P a t t e n , A p p l . Opt. 1 2 , 2147 ( 1 9 7 3 ) . [ 5 ] M.Lehmann, Hol o g r a p h y . Technique and P r a c t i c e ( F o c a l P r e s s , London and New York, 1 9 7 0 ) , p, 6 3 . [ 6 ] H . K o g e l n i k , B e l l S y s t . Tech. J . 48, 2909 ( 1 9 6 9 ) . [ 7 ] H.T.Buschmann, P h o t o g r . S c i . Eng. IjS, 425 ( 1 9 7 2 ) . [ 8 ] , i n O p t i c a l and A c o u s t i c a l H o l o g r a p h y ( E . C a m a t i n i , ed., Plenum P r e s s , New York and London, 1 9 7 2 ) , pp, 151 - 7 2 . [ 9 ] A.G.Fox and T . L i , B e l l S y s t . Tech. J . 4 0 , 453 ( l 9 6 l ) . [ 1 0 ] H . K o g e l n i k and T.Li,- A p p l . Opt. 5 , 1550 ( 1 9 6 6 ) . [ I I ] L.Mandel and E.Wolf, Rev. Mod. Phys. 3 7 , 231 ( 1 9 6 5 ) . [ 1 2 ] H.G.Heard, L a s e r Parameter Measurements Handbook (RADC-TR-6 6 - 7 0 4 , AD 650 8 7 2 , F i n a l R e p o r t , Rome A i r Development C e n t e r , R e s e a r c h and Technology D i v i s i o n , A i r F o r c e Sys-tems Command, G r i f f i s s A i r F o r c e Base, New Y o r k , 1 9 6 7 ) , V o l . I l l , Chap. 7. -82-SECTION THREE s DESIGN AND CONSTRUCTION OF EXPERIMENTAL APPARATUS T h i s s e c t i o n d e a l s w i t h t h e e x p e r i m e n t a l a p p a r a t u s c o n s t r u c -t e d i n c o n n e c t i o n w i t h t h e e x p e r i m e n t a l p r o b l e m s d i s c u s s e d h e r e . T h i s e x p e r i m e n t a l work may be d i v i d e d i n t o r o u g h l y f o u r m a i n a r e a s s l ) c o n s t r u c t i o n and m o d i f i c a t i o n o f a g i a n t p u l s e r u b y l a s e r ; 2a) a s s e m b l y o f a h i g h s p e e d l a s e r p u l s e d e t e c t o r , and 2b) d e s i g n and c o n s t r u c t i o n o f a t h e r m o p i l e l a s e r powermeter; 3) d e s i g n and c o n s t r u c t i o n o f a n o p t i c a l s y s t e m f o r t h e K a p i t z a -D i r a c e f f e c t ^ ' 2 ' - ^ e x p e r i m e n t ; and 4 ) d e s i g n and c o n s t r u c t i o n o f a h o l o g r a p h i c l a s e r p a r a m e t e r measurement d e v i c e o f t h e t y p e d e s c r i b e d i n S e c t i o n Two. The e l e c t r o n i c s d i s c u s s e d i n c o n j u n c -t i o n w i t h a r e a s l ) and 2a) have b e e n d e v e l o p e d by Mr. D. S i e b e r g o f t h e P l a s m a P h y s i c s Group a t t h e U n i v e r s i t y o f B r i t i s h Colum-b i a . B e c a u s e o f d e l a y s r e s u l t i n g f r o m a c c i d e n t a l f a i l u r e o f p a r t o f t h e s y s t e m , i t was n o t p o s s i b l e t o c o m p l e t e t h e a c t u a l ex-p e r i m e n t s d i s c u s s e d h e r e w i t h i n t h e t i m e a v a i l a b l e . However, t h e a p p a r a t u s has b e e n c o m p l e t e d and i t s h o u l d be p o s s i b l e t o c a r r y o u t t h e measurements i n t h e f u t u r e . A GIANT PULSE RUBY LASER The g i a n t p u l s e r u b y l a s e r i s o f t h e t y p e d i s c u s s e d i n S e c -t i o n One o f t h i s t h e s i s , and by B j o r k h o l m and S t o l e n ^ ^ . The d e t a i l s o f t h e d e s i g n have b e e n d e v e l o p e d b y t h e P l a s m a P h y s i c s -83-Group a t the U n i v e r s i t y o f B r i t i s h Columbia, and a r e d i s c u s s e d by C h u r c h l a n d ^ - 1 . The l a s e r a l s o employs some of the m o d i f i c a -t i o n s and improvements d e s c r i b e d by A l b a c h ^ - L A l t h o u g h t h i s l a s e r i s n o t e x c e p t i o n a l l y r e l i a b l e or e f f i c i e n t , i t p o s s e s s e s the v i r t u e s o f easy m o d i f i c a t i o n t o s u i t p a r t i c u l a r e x p e r i m e n t a l r e q u i r e m e n t s and o f low c o s t (as compared t o c o m m e r c i a l u n i t s ) . The d e s c r i p t i o n g i v e n here i s b r i e f and i s n o t i n t e n d e d t o be complete. R a t h e r i t s t r e s s e s the m o d i f i c a t i o n s t o the b a s i c de-s i g n w h i c h are employed i n t h i s p a r t i c u l a r l a s e r . The l a s e r was chosen t o meet the r e q u i r e m e n t s of the n e u t r a l a t o m i c beam K a p i t z a - D i r a c e x p e r i m e n t . G e n e r a l l y s p e a k i n g , t h e s e r e q u i r e m e n t s a r e l ) h i g h peak power d e n s i t y (about 1 - 100 Mw/cm 2), 2) s p e c t r a l p u r i t y (about ±0 . 0 5 & @ 69^3 A or l e s s ) , 3) s i n g l e f r e q u e n c y o p e r a t i o n , 4) good c o l l i m a t i o n (< 1 mrad), and 5) r e p r o d u c i b i l i t y . The l a s e r c a v i t y i s o f the p l a n e p a r a l l e l F a b r y - P e r o t type and i s shown r o u g h l y t o s c a l e , w i t h the a p p r o p r i a t e d i m e n s i o n s n o t e d , i n F i g u r e 8. The r u b y r o d i s mounted i n the l a s e r head d i s c u s s e d below. The r e s t o f the o p t i c a l components are mounted i n o p t i c a l c a r r i e r s w h i c h r i d e on a double r a i l o p t i c a l bench w h i c h a f f o r d s easy adjustment of the d i m e n s i o n s of the c a v i t y . The s a p p h i r e e t a l o n and the back m i r r o r a r e mounted i n double g i m b a l l e d h o l d e r s w h i c h a l l o w a n g u l a r a d j u s t m e n t about two o r t h o g o n a l axes t h a t are p e r p e n d i c u l a r t o the beam a x i s . The s t a n d a r d g i m b a l l e d h o l d e r s were m o d i f i e d t o a c c e p t m i c r o meter heads ( M i t u t o y o #148-112) r a t h e r t h a n screws f o r f i n e FIGURE 9 Giant pulse ruby laser cavity (scale: 2:5) 34.5° 3cm 13.5cm - 8 5 -a d j u s t m e n t . The s a p p h i r e e t a l o n i s a l a s e r q u a l i t y L i n d e Cz s a p p h i r e ( U n i o n C a r b i d e C o . ) . I t i s 1" i n d i a m e t e r and 1/8" t h i c k w i t h a f l a t n e s s o f 1/10 wave and a p a r a l l e l i s m o f 5 s e c o n d s o v e r 90$ o f t h e d i a m e t e r . The e t a l o n s e r v e s t o enhance t h e f r e q u e n c y s e l e c t i v i t y o f t h e c a v i t y b y a c t i n g a s a s e c o n d , c o u p l e d F a b r y -P e r o t c a v i t y . The peak r e f l e c t i v i t y o f t h e e t a l o n i s a b o u t 26f», and i t has a f r e e s p e c t r a l r a n g e o f a b o u t 0.4-3 • N e i t h e r s u r -f a c e o f t h i s e t a l o n i s c o a t e d . The dye c e l l i s p l a c e d i n t h e c a v i t y a t B r e w s t e r ' s a n g l e t o enhance t h e p o l a r i z a t i o n s e l e c t i v i t y o f t h e c a v i t y and t o m i n i -m i z e t h e r e f l e c t i o n l o s s e s w i t h i n t h e c a v i t y . I t c o n s i s t s o f two one s u r f a c e f u s e d q u a r t z o p t i c a l f l a t s (Edmund S c i e n t i f i c Co. #1914) w h i c h a r e 2" i n d i a m e t e r and i " t h i c k . E a c h i s f l a t t o l / l O wave on i t s f i r s t s u r f a c e . The f l a t s a r e mounted w i t h t h e i r f l a t s i d e s o u t on a t e f l o n s p a c e r a b o u t 0 . 2 5 cm i n t h i c k n e s s so t h a t a t h i n s e a l e d chamber i s f o r m e d b e t w e e n t h e f l a t s . A s m a l l h o l e w i t h a s t a i n l e s s s t e e l p l i g a l l o w s a c c e s s t h r o u g h t h e t e f -l o n s p a c e r so t h a t a h y p o d e r m i c n e e d l e may be u s e d t o f i l l and empty t h e c e l l . The e n t i r e c e l l a s s e m b l y i s mounted on a p i v o t a b o u t a h o r i z o n t a l a x i s ( p e r p e n d i c u l a r t o t h e beam a x i s ) a l l o w -i n g i t t o be r a i s e d i n t o a v e r t i c a l p o s i t i o n f o r f i l l i n g and e m p t y i n g . The r e l a t i v e l y t h i n c e l l t h i c k n e s s i s u s e d t o a v o i d s t i m u l a t e d B r i l l o u i n s c a t t e r i n g i n t h e dye s o l v e n t ^ -L C r y p t o -c y a n i n e ( l , l ' d i e t h y l - 4 , 4 ' c a r b o c y a n i n e i o d i d e , E a s t m a n Kodak) d i s s o l v e d i n m e t h a n o l i s u s e d a s t h e s a t u r a b l e a b s o r b e r . The c o n c e n t r a t i o n o f t h e dye i n t h e c e l l , a t n o r m a l Q - s w i t c h e d - 8 6 -o p e r a t i n g c o n d i t i o n s , i s v e r y s l i g h t , and i s most e a s i l y a d -j u s t e d t h r o u g h a t r i a l a nd e r r o r p r o c e s s . One s e e k s t o m a x i -m i z e t h e o u t p u t power, w h i l e a t t h e same t i m e p r e s e r v i n g s i n g l e f r e q u e n c y o p e r a t i o n . S i n g l e f r e q u e n c y o p e r a t i o n i s m o n i t o r e d b y t h e method o f B j o r k h o l m and S t o l e n ^ - I : t h e t e m p o r a l e n v e l o p e o f t h e p u l s e i s a smooth, b e l l s h a p e d c u r v e f o r s i n g l e f r e q u e n c y o p e r a t i o n , and i t i s wavy o r s p i k e d ( i n d i c a t i n g b e a t f r e q u e n c y v a r i a t i o n s ) f o r m u l t i p l e f r e q u e n c y o p e r a t i o n . The dye c e l l i s l o c a t e d a s c l o s e a s p o s s i b l e t o t h e f r o n t r e f l e c t o r o f t h e c a v i -t y t o a v o i d mode l o c k i n g . The r u b y r o d i s a n SIQ g r a d e L i n d e Gz r u b y ( U n i o n C a r b i d e Co.) w i t h B r e w s t e r - B r e w s t e r end f a c e s f o r p o l a r i z a t i o n s e l e c -t i v i t y and m i n i m a l r e f l e c t i o n l o s s e s . The c r y s t a l o p t i c a x i s i s p e r p e n d i c u l a r t o t h e p l a n e o f F i g u r e 8 . The r o d i s 6" l o n g and ^" i n d i a m e t e r , and i t has a s u r f a c e f l a t n e s s o f 3/4 f r i n g e p e r i n c h . The r u b y i s doped w i t h 0 . 0 5 $ C r + 3 b v w e i g h t . The a p e r t u r e c o n s i s t s o f l / l 6 " s t a i n l e s s s t e e l s h e e t m e t a l w i t h a s m a l l h o l e a c c u r a t e l y d r i l l e d t h r o u g h i t . A number o f i n t e r c h a n g e a b l e a p e r t u r e s ( a b o u t 1 - 4 mm 0) may be s u b s t i t u t e d e a s i l y i n t o t h e a p e r t u r e h o l d e r . T h i s h o l d e r has t r a n s l a t i o n a l a d j u s t m e n t s a l o n g two o r t h o g o n a l a x e s i n t h e p l a n e p e r p e n d i c u l a r t o t h e beam a x i s . I n o p e r a t i o n t h e l a r g e s t a p e r t u r e w h i c h i s c o m p a t i b l e w i t h a homogeneous beam c r o s s - s e c t i o n i s employed t o maxmize t h e o u t p u t power. As m e n t i o n e d a t t h e b e g i n n i n g o f S e c -t i o n One ( p . 6 ) , t h e a p e r t u r e i n c r e a s e s t h e t r a n s v e r s e mode s e l e c t i v i t y o f t h e c a v i t y , and i t l i m i t s t h e l a s e r beam t o a r e a s o n a b l y homogeneous p o r t i o n o f t h e r u b y r o d . T h i s l a t t e r -87-a c t i o n i s i m p o r t a n t s i n c e i t i s p r i m a r i l y t h e h o m o g e n e i t y o f t h e r u b y r o d w h i c h u l t i m a t e l y l i m i t s t h e beam d i v e r g e n c e and t h e t r a n s v e r s e q u a l i t y o f t h e l a s e r o u t p u t . The t r a n s v e r s e u n i -f o r m i t y o f t h e l a s e r beam i s m a x i m i z e d b y a d j u s t i n g t h e p o s i t i o n o f t h e a p e r t u r e u n t i l a "good" p a t h t h r o u g h t h e r o d i s f o u n d . The b a c k m i r r o r o f t h e c a v i t y i s a m u l t i l a y e r d i e l e c t r i c c o a t e d BSC g l a s s s u b s t r a t e ( L a s e r O p t i c s I n c . #108 - 6 9 5 0 0 ) . I t i s a f r o n t s u r f a c e r e f l e c t o r w i t h a f l a t n e s s o f 1/10 wave, a r e f l e c t i v i t y o f 99 ± i f> @ 69^3 A, and a damage t h r e s h o l d o f g r e a t e r t h a n 1000Mw/cm 2. The p a r a l l e l i s m o f t h e c a v i t y end r e f l e c t o r s i s a d j u s t e d b y use o f t h e m i c r o m e t e r heads on t h e g i m b a l l e d mounts, and i t i s m o n i t o r e d w i t h a l a s e r a l i g n m e n t a u t o - c o l l i m a t o r ( A m e r i c a n Op-t i c a l I n c . M o d e l AC - 3 ) w h i c h has a r e s o l u t i o n o f 0.2 mrad. T h i s r e s o l u t i o n i s much l e s s t h a n t h e i n t r i n s i c beam d i v e r g e n c e o f t h e t y p i c a l l a s e r o u t p u t ( a b o u t 1 - 0 . 5 mrad) so t h a t p r e c i s e c a v i t y a d j u s t m e n t i s e a s i l y o b t a i n e d . The l a s e r h e a d i s d e p i c t e d i n F i g u r e 9 w i t h i t s c o v e r and t h e t o p h a l f o f t h e pump c a v i t y m i r r o r removed. The pump c a v i t y m i r r o r c o n s i s t s o f two h a l v e s , e a c h i s m i l l e d f r o m a s o l i d b l o c k o f aluminum t o f o r m a d o u b l e e l l i p t i c a l m i r r o r . T h i s m i r r o r f o -c u s e s t h e o u t p u t o f t h e two l i n e a r f l a s h l a m p s o n t o t h e r u b y r o d w h i c h i s l o c a t e d a l o n g t h e c e n t r a l , common f o c a l l i n e s o f t h e e l l i p s e s . The f l a s h l a m p s , o f c o u r s e , a r e a l i g n e d a l o n g t h e two e x t e r i o r f o c a l l i n e s . I n o r d e r t o m a i n t a i n e f f i c i e n t o p e r a t i o n , t h e pump c a v i t y m i r r o r must be p e r i o d i c a l l y c l e a n e d and p o l i s h e d w i t h a good q u a l i t y m e t a l p o l i s h ( e . g . " S i l v o " ) . FIGURE 10 Overhead view of laser head (scale: 1:2) cooling water flow white o-rinq seals  ruby rod cooling jacket  ruby rod pump cavity mirror (lower half) flashlamp mountXA o-rinq seals - \ flashlamp cooling jacket\ flashlamp flashlamp mount (qnd. electrode) f P (pos. electrode) -89-The r u b y r o d i s h e l d i n p l a c e b y w h i t e o - r i n g s i n t h e end-c a p s o f t h e r u b y r o d c o o l i n g j a c k e t . T h e s e e n d c a p s a l s o h o l d t h e f u s e d s i l i c a r u b y r o d c o o l i n g j a c k e t i n p l a c e w i t h w h i t e o - r i n g s e a l s . W h i t e o - r i n g s a r e u s e d i n t h i s a s s e m b l y t o r e t a r d t h e d e t e r i o r a t i o n o f t h e s e a l s due t o t h e i n t e n s e i r r a d i a t i o n f r o m t h e f l a s h l a m p s . The e n d c a p s o f t h e r u b y r o d c o o l i n g j a c k e t a r e s o l i d l y mounted t o t h e l a s e r head b a s e b y means o f b r a c k e t s a t e i t h e r end o f t h e a s s e m b l y . T h e s e b r a c k e t s a l l o w r o t a t i o n a l a d -j u s t m e n t o f t h e r u b y r o d a b o u t i t s a x i s and have s e t s c r e w s t o m a i n t a i n t h i s a d j u s t m e n t . The f l a s h l a m p s a r e l i n e a r , h i g h e n e r g y , x e n o n f l a s h l a m p s w i t h f u s e d s i l i c a j a c k e t s (EG&G I n c . #FXS-47C-6.5). T h e y a r e h e l d i n p l a c e b y t h e i r e l e c t r o d e s w h i c h a r e f i r m l y c o n n e c t e d t o t h e b r a s s e n d c a p s o f t h e f l a s h l a m p c o o l i n g j a c k e t s . T h e s e end-c a p s a r e h o l l o w and s e r v e as t h e e n t r a n c e and e x i t d u c t s f o r t h e c o o l i n g w a t e r . They a l s o s e r v e a s t h e s e a l s f o r t h e f l a s h l a m p c o o l i n g j a c k e t s and as t h e e l e c t r i c a l c o n n e c t o r s f o r t h e f l a s h -lamp power s u p p l y . T h e s e e n d c a p s mount s o l i d l y i n t o b r a c k e t s a t e a c h end o f t h e l a s e r h e a d . The b r a c k e t s a r e d i r e c t l y c o n n e c t e d t o t h e e n e r g y s t o r a g e c a p a c i t o r s and must be g r o u n d e d whenever t h e head i s open f o r s e r v i c i n g . T r a n s l a t i o n a l a d j u s t m e n t o f t h e e n d c a p s i n t h e b r a c k e t s a l l o w s f o r a c c u r a t e a l i g n m e n t o f t h e f l a s h l a m p s a l o n g t h e f o c a l a x e s o f t h e e l l i p t i c a l m i r r o r . S i n c e t h e c o o l i n g j a c k e t s a r e s e a l e d b y a n i n t e r n a l e x p a n s i o n o - r i n g , c a r e must be t a k e n t o i n s u r e a w a t e r t i g h t s e a l w i t h o u t damaging t h e c o o l i n g j a c k e t s . The f l a s h l a m p c o o l i n g j a c k e t s a r e f u s e d s i l i c a t u b i n g f o r l o w pump e n e r g y l o s s , low s u s c e p t i b i l i t y t o - 9 0 -t h e r m a l s h o c k , and h i g h s t r e n g t h . As i n d i c a t e d i n F i g u r e 9» t h e d i r e c t i o n o f c o o l i n g w a t e r f l o w i s a l w a y s f r o m t h e l o w e r end o f t h e l a s e r head t o w a r d t h e u p p e r end. T h i s e l i m i n a t e s a i r h u b b i e s , and r e s u l t i n g h o t - s p o t s f r o m t h e c o o l i n g s y s t e m . The w a t e r f l o w i s c o n n e c t e d i n s e r i e s t h r o u g h a l l t h r e e c o o l i n g j a c k e t s , and i s a c o n t i n u o u s f l o w o f t a p w a t e r . P r o b l e m s w i t h c o n d e n s a t i o n on t h e o u t s i d e o f t h e j a c k e t s may be e l i m i n a t e d e i t h e r b y f l o w i n g d r y a i r i n t o t h e l a s e r head, o r b y u s i n g a m i x i n g t a p t o r a i s e t h e t e m p e r a t u r e o f t h e c o o l i n g w a t e r s l i g h t l y . Not shown i n F i g u r e 9 a r e t h e t r i g g e r w i r e s w h i c h a r e wrap-p e d a r o u n d t h e o u t s i d e s o f t h e f l a s h l a m p c o o l i n g j a c k e t s . The t r i g g e r w i r e s may a l s o be c o n n e c t e d t o t h e pump c a v i t y m i r r o r , b u t t h i s may r e s u l t i n a r c i n g t o t h e r u b y r o d b r a c k e t s and l e s s r e l i a b l e t r i g g e r i n g . The l a s e r e l e c t r o n i c s a r e s c h e m a t i c a l l y d e p i c t e d i n F i g u r e 1 0 . The c h a r g i n g u n i t and t r i g g e r u n i t a r e s t a n d a r d i z e d assem-b l i e s d e s i g n e d b y t h e P l a s m a P h y s i c s Group a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , and t h e d e t a i l s o f t h e c i r c u i t d e s i g n s s h a l l n o t be d e a l t w i t h i n t h i s work. I t s h o u l d be n o t e d , however, t h a t some s l i g h t m o d i f i c a t i o n s t o t h e o r i g i n a l u n i t s have b e e n made b y D r . T. F. K n o t t t o a l l o w f o r a u t o m a t i c a l l y c o n t r o l l e d r e p e t i t i v e f i r i n g o f t h e l a s e r . The c h a r g i n g u n i t has a v a r i a b l e v o l t a g e c o n t r o l and v o l t m e t e r s f o r e a c h o f t h e c a p a c i t o r b a n k s . I t a l s o f e a t u r e s a n a d j u s t a b l e o v e r v o l t p r o t e c t i o n t o p r e v e n t e x c e e d i n g t h e r a t i n g s o f t h e s t o r a g e c a p a c i t o r s , and a dumping F I G U R E 11 Schemat ic of l a s e r e l ec t ron i c s charging unit 0-2.5kVDC trigger unit  20-30 kV  pulsed energy storage capacitors flashlamps ti 0 0 i - 9 2 -u n i t ( n o t shown) t o d i s c h a r g e t h e c a p a c i t o r s w i t h o u t f i r i n g t h e f l a s h l a m p s . The e n e r g y s t o r a g e c a p a c i t o r s c o n s i s t o f f o u r 560 | i f c a p a c i t o r s (Sangamo #782013-5207), two c o n n e c t e d i n p a r a l l e l t o e a c h f l a s h l a m p . These c a p a c i t o r s a r e r a t e d a t 2.5 kVdc and e a c h i s c a p a b l e , t h e r e f o r e , o f s t o r i n g a b o u t 2,6 k J o f e n e r g y . T h u s , a t o t a l maximum emergy o f 5-6 k J may be s u p p l i e d t o e a c h f l a s h l a m p . The t r i g g e r u n i t i s a s o l i d s t a t e c i r c u i t e m p l o y i n g a n SCR f a s t s w i t c h i n g c i r c u i t . I t s u p p l i e s a 20 - 30 kV p u l s e o f a few m i c r o s e c o n d s d u r a t i o n t o t h e e x t e r n a l l y c o u p l e d t r i g g e r w i r e s . LASER RADIATION DETECTION ELECTRONICS I n o r d e r t o m o n i t o r t h e o u t p u t o f t h e g i a n t p u l s e r u b y l a s e r , two s e p a r a t e d e t e c t i o n u n i t s were c o n s t r u c t e d . F i r s t , a f a s t p h o t o d i o d e u n i t was a s s e m b l e d and mounted b e h i n d t h e b a c k m i r r o r o f t h e l a s e r c a v i t y . U t i l i z i n g t h e l e a k a g e o f r a d i a t i o n t h r o u g h t h e b a c k m i r r o r i t g i v e s a t i m e r e s o l v e d o u t p u t w h i c h i s p r o p o r t i o n a l t o t h e p u l s e i n t e n s i t y . S e c o n d , a l a s e r power-m e t e r o f t h e t h e r m o p i l e v a r i e t y , a s d e s c r i b e d b y R(5ss^-^, was d e s i g n e d and c o n s t r u c t e d t o a l l o w f o r a b s o l u t e measurement o f t h e t o t a l p u l s e e n e r g y . F i g u r e 11 shows t h e c i r c u i t d i a g r a m o f t h e p h o t o d i o d e d e t e c -t o r a nd a c r o s s - s e c t i o n a l v i e w o f t h e d e t e c t o r h e a d . The p h o t o -d i o d e i s a P I N t y p e ( H e w l e t t - P a c k a r d #5082-4220) f e a t u r i n g h i g h s e n s i t i v i t y a t t h e r u b y l a s e r w a v e l e n g t h and a b o u t 0.1 n s e c r i s e t i m e . T h e r e a r e no e l e c t r i c a l c o n n e c t i o n s b e t w e e n t h e d e t e c t o r FIGURE 12 Photodiode detector shielding .90V PIN photodiode (HP 5082-4220) \"""" !RG'58A/U O.ljuf 100V 50ohm termination(?W) (i)-^to oscilloscope (Tektronix475)  neutral density filter interference filter shielding photodiode diffuser rom laser (back mirror) detector head -94-c i r c u i t and t h e s h e i l d i n g , t h e o n l y g r o u n d i n t h e d e t e c t o r c i r -c u i t b e i n g a t t h e o s c i l l o s c o p e c o n n e c t i o n . T h i s i s t o p r e v e n t r i n g i n g o r r e f l e c t i o n s i n t h e c i r c u i t . F o r t h i s same r e a s o n , and f o r b e t t e r v o l t a g e r e g u l a t i o n , a b a t t e r y r a t h e r t h a n a power s u p p l y i s u s e d . The l a s e r l i g h t e n t e r s t h e d e t e c t o r head t h r o u g h a d i f f u s e r o f e i t h e r t e f l o n o r t r a n s l u c e n t t a p e , t h e n p a s s e s t h r o u g h a n i n t e r f e r e n c e f i l t e r ( K a r l F e u e r FL - 1 2 ) w h i c h has peak t r a n s m i s -s i o n a t 694-3 A" and a b a n d w i d t h o f a b o u t 100 1. The i n t e r f e r e n c e f i l t e r s e r v e s t o e l i m i n a t e s t r a y l i g h t f r o m t h e f l a s h l a m p s . F o l l o w i n g t h i s f i l t e r i s a h o l d e r f o r n e u t r a l d e n s i t y f i l t e r s w h i c h a d j u s t t h e l i g h t l e v e l so t h a t t h e d i o d e o p e r a t e s i n a l i n e a r r e g i m e . F i n a l l y , t h e l a s e r l i g h t e n t e r s t h e window o f t h e p h o t o d i o d e . To m o n i t o r t h e t o t a l p u l s e e n e r g y i n a b s o l u t e u n i t s , a c a l i -b r a t e d t h e r m o p i l e l a s e r powermeter was d e s i g n e d and c o n s t r u c t e d . The g e n e r a l d e s i g n o f t h i s u n i t ( c f . R e f . 7) i s shown i n F i g u r e 12. The d e v i c e i s e s s e n t i a l l y a W h e a t s t o n e b r i d g e t h a t compares t h e r e s i s t a n c e o f a t h e r m i s t o r i n c o n t a c t w i t h a n i r -r a d i a t e d l i g h t a b s o r b i n g cone t o t h e r e s i s t a n c e o f a s e c o n d t h e r m i s t o r t h a t i s i n c o n t a c t w i t h a n i d e n t i c a l b u t n o n - i r r a -d i a t e d c o n e . The c o n e s have a 60° i n t e r i o r a n g l e and a r e h a r d b l a c k a n o d i z e d on t h e i r i n t e r i o r s u r f a c e s . B o t h c o n e s w e i g h a b o u t 1 gm. The p r o b e cone i s h e a t e d b y a b s o r p t i o n o f l a s e r l i g h t ( o r b y ohmic h e a t i n g due t o a c u r r e n t p a s s e d t h r o u g h t h e h e a t i n g w i r e f r o m t h e c a l i b r a t i o n u n i t ) , t h i s h e a t i n g , i n t u r n , p r o d u c e s a change i n t h e r e s i s t a n c e o f t h e t h e r m i s t o r mounted F1GURE13 Schematic of laser powermeter black anodized interior - 9 6 -i n t h e b a s e o f t h e c o n e . T h i s change i n r e s i s t a n c e and t h e c u r -r e n t i t c a u s e s t o f l o w t h r o u g h t h e c e n t e r o f t h e b r i d g e a r e r o u g h l y p r o p o r t i o n a l t o t h e t o t a l amount o f e n e r g y a b s o r b e d b y t h e c o n e . The t h e r m i s t o r s a r e b o t h r a t e d a t 2 kQ @ 25°C ( F e n w a l l #GB32P2) and a r e e p o x i e d i n t o s m a l l t h r e a d e d s t u d s w h i c h s c r e w i n t o t h e b a s e s o f t h e c o n e s a s shown i n t h e f i g u r e . The am-m e t e r a c r o s s t h e c e n t e r o f t h e b r i d g e has a r a n g e o f 0-10 p,Adc and a n a c c u r a c y o f a b o u t 1 % ( H i o k i # R - 6 5 ) . The r e s i s t o r , R M , d e t e r m i n e s t h e r a n g e o f t h e power measurement, and, on t h e a c -t u a l u n i t , t h r e e r a n g e s a r e b u i l t i n , 0 - 10 J , 0 - 5 J » and 0 -2 . 5 J ( r e s p e c t i v e v a l u e s o f R M a r e 4-5 kQ, 20 kQ, and 5-5 kn). The r e s i s t o r R C a d j u s t s t h e c a l i b r a t i o n o f t h e m e t e r and t h r e e o f t h e s e a r e b u i l t i n t o a l l o w i n d e p e n d e n t c a l i b r a t i o n o f e a c h r a n g e ( r e s p e c t i v e v a l u e s o f R C , c o r r e s p o n d i n g t o t h e above r a n g e s , a r e 0 - 2 0 kH, 0 - 1 0 kO, and 0 - 5 kO, a l l t h r e e a r e 20 t u r n , m e t a l f i l m p o t . s ) . A l l r e s i s t o r s and p o t e n t i o m e t e r s i n t h e c i r c u i t a r e p r e c i s i o n m e t a l f i l m u n i t s f o r h i g h e r a c c u r a c y . M e r c u r y b a t t e r i e s a r e u s e d i n t h i s u n i t f o r t h e same r e a s o n . C a l i b r a t i o n o f t h e powermeter i s a c h i e v e d b y d i s s i p a t i n g a known amount o f e n e r g y i n t o t h e h e a t i n g w i r e w h i c h i s e p o x i e d t o t h e o u t s i d e o f t h e p r o b e c o n e . T h i s h e a t i n g c o i l c o n s i s t s o f a b o u t 2' o f 31 ft/foot c o n s t a n t a n w i r e . A t i g h t , e v e n s p i r a l o f h e a t i n g w i r e on t h e o u t s i d e o f t h e cone i s somewhat d i f f i c u l t t o o b t a i n . However, b y u s i n g a l e n g t h o f t e f l o n i n s u l a t e d w i r e a s a s p a c e r d u r i n g t h e e p o x y i n g p r o c e s s , a t i g h t , e v e n l y s p a c e d c o i l o f t h e c o n s t a n t a n w i r e i s e a s i l y f o r m e d . A f t e r t h e e poxy has -97-s e t , t h e t e f l o n w i r e i s e a s i l y r emoved. F o u r c a l i b r a t i o n e n e r -g i e s a r e b u i l t i n t o t h e powermeter: 10 J ; 5 J» 2 . 5 J ; and 1 .25 J . T h ese e n e r g i e s a r e o b t a i n e d b y c h a r g i n g a c c u r a t e l y m e a s u r e d c a p a c i t o r b a n k s t o a w e l l - k n o w n v o l t a g e , and t h e n "dumping" t h i s s t o r e d e n e r g y t h r o u g h t h e h e a t i n g c o i l . F o u r d i f f e r e n t c a p a c i t o r b anks a r e u s e d t o o b t a i n t h e f o u r c a l i b r a t i o n e n e r g i e s . The c a -p a c i t a n c e o f e a c h bank i s most e a s i l y m e a s u r e d b y m o n i t o r i n g t h e dumping c y c l e on a n o s c i l l o s c o p e and s o l v i n g f o r t h e c a p a c i t a n c e f r o m t h e RC t i m e c o n s t a n t . The c a l i b r a t i o n v o l t a g e , V c , i s de-r i v e d f r o m a b a t t e r y t h r o u g h a v a r i a b l e v o l t a g e d i v i d e r . T h i s v o l t a g e may be i n d e p e n d e n t l y a d j u s t e d and m o n i t o r e d so t h a t t h e powermeter c a l i b r a t i o n may be made t o c o n f o r m w i t h a n i n d e p e n -d e n t s t a n d a r d . When c a l i b r a t e d i n t h i s manner, t h e powermeter has a n a c c u -r a c y o f b e t t e r t h a n 5 i° and a p r e c i s i o n o f b e t t e r t h a n 1 % on a l l r a n g e s . T h e s e f i g u r e s i n c l u d e t h e d e v i a t i o n s f r o m l i n e a r i t y o f t h e b r i d g e c i r c u i t and t h e t h e r m i s t o r s . A l s o , t h e powermeter has c o n n e c t o r s so t h a t an e x t e r n a l ammeter o r a n e x t e r n a l c a l i -b r a t i o n - e n e r g y may be u s e d . KAPITZA-DIRAC EXPERIMENT OPTICAL SYSTEM The K a p i t z a - D i r a c e x p e r i m e n t r e q u i r e s a r e l a t i v e l y l a r g e d i a m e t e r ( a b o u t 1 - 4 cm), h i g h i n t e n s i t y ( a b o u t 0 .1 - 10 Mw/cm2 a f t e r e x p a n s i o n ) l a s e r beam w h i c h i s w e l l - c o l l i m a t e d (< 1 mrad) and a c c u r a t e l y a l i g n e d (< 1 mrad) w i t h r e s p e c t t o a p l a n e m i r r o r t o f o r m s t a n d i n g l i g h t waves w i t h maxima l y i n g i n p a r a l l e l p l a n e s . To a c h i e v e t h i s end, w i t h t h e l a s e r d i s c u s s e d a b o v e , a - 9 8 -s i m p l e o p t i c a l s y s t e m has b e e n d e s i g n e d and c o n s t r u c t e d . An o v e r h e a d v i e w o f t h i s o p t i c a l s y s t e m i s d e p i c t e d i n F i g u r e 13-The m i r r o r , l o c a t e d i n s i d e t h e vacuum chamber o f t h e a t o m i c beam m a c h i n e , i s a h i g h q u a l i t y , f i r s t s u r f a c e m u l t i l a y e r d i -e l e c t r i c m i r r o r . I t was s p e c i a l l y f a b r i c a t e d b y L a s e r O p t i c s I n c . o f l a s e r g r a d e BSC g l a s s , and i t i s 3 " i n d i a m e t e r b y 5 / 8 " t h i c k w i t h a . l / 1 0 wave f i r s t s u r f a c e . I t i s c o a t e d t o have a r e -f l e c t i v i t y o f 9 9 . 9 % @ 694-3 A and 0° a n g l e o f i n c i d e n c e . The m i r r o r mount f e a t u r e s m e c h a n i c a l vacuum f e e d - t h r o u g h s ( d e s i g n e d and c o n s t r u c t e d b y D r . T. F. K n o t t ) w h i c h a f f o r d t r a n s l a t i o n a l a d j u s t m e n t ( b o t h f i n e and c o a r s e ) a l o n g t h e l a s e r beam a x i s , and a n g u l a r a d j u s t m e n t a b o u t two o r t h o g o n a l a x e s p e r p e n d i c u l a r t o t h e l a s e r beam a x i s . The window i s f a b r i c a t e d f r o m t h e same s u b s t r a t e m a t e r i a l a s t h e m i r r o r and has t h e same d i m e n s i o n s . I t has a f l a t n e s s o f l / l O wave on b o t h s u r f a c e s and a p a r a l l e l i s m o f 2 s e c o n d s . I t i s f i r m l y mounted on t h e o u t s i d e o f a f l a n g e on t h e vacuum chamber w i t h a n o - r i n g s e a l . L e n s 2 i s a p l a n o - c o n v e x l e n s w i t h f o c a l l e n g t h f 2 - I t r i d e s i n a t e l e s c o p i n g mount f i x e d t o t h e same f l a n g e ; a s t h e window. I t has t r a n s l a t i o n a l a d j u s t m e n t a l o n g t h e l a s e r beam a x i s o n l y . L e n s 1 i s a p l a n o - c o n c a v e l e n s w i t h f o c a l l e n g t h f ± . T h i s l e n s r i d e s on a n o p t i c a l b e n c h c a r r i e r t h a t i s f i r m l y a t t a c h e d t o t h e o p t i c a l b e n c h c a r r y i n g t h e r u b y l a s e r . L e n s 1 has c o a r s e t r a n s -l a t i o n a l a d j u s t m e n t a l o n g t h e l a s e r beam a x i s , f i n e t r a n s l a -t i o n a l a d j u s t m e n t a l o n g two o r t h o g o n a l a x e s t h a t a r e p e r p e n d i -FIGURE 14 Overhead view of optical system for K-D apparatus (not to scale) atomic beam  vacuum system \ mirror - 1 0 0 -cular to the laser beam axis, and fin e angular adjustment about two orthogonal axes perpendicular to the laser beam axis. The plano-concave - plano-convex geometry i s used to minimize spherical aberation by making the angular deviations of each ray of the laser beam approximately equal at each interface. When the lens system i s properly adjusted, the beam expansion r a t i o (linear) i s given by the r a t i o of the f o c a l lengths f 2 / f j _ . The beam collima t i o n i s improved by t h i s same r a t i o , also. The most important feature of t h i s lens system i s the loca-t i o n of the common f o c a l point of the lenses over the pivot axis of the laser o p t i c a l bench. In t h i s configuration, with lens 1 fixed with respect to the laser and with lens 2 f i x e d with re-spect to the plane mirror, the alignment of the laser o p t i c a l bench i s far less c r i t i c a l than i f the two lenses were fixed with respect to each other. In fa c t , analysis of the various de-viat i o n s produced by misalignments shows that i f the common fo-ca l point i s within about 1 cm of the pivot axis, i f other t r a n s l a t i o n a l adjustments are within about 0 .1 mm of the correct positions, and i f the angular adjustments of the lenses are within 10 mrad of the correct orientations, then the collimation deviation, and the angular deviation of the laser beam, i n the vacuum chamber, are well within the natural l i m i t s imposed by the i n t r i n s i c divergence i o f the laser beam. The alignment of the plane mirror, of course, remains c r i t i c a l . Two plano-convex lenses with f o c a l lengths of 4 0 0 mm and 300 mm respectively, and three plano-concave lenses with f o c a l lengths of - 1 0 0 mm, - 5 0 mm, and - 2 5 mm res p e c t i v e l y are available. This y i e l d s a selec-- 1 0 1 -t i o n o f s i x d i f f e r e n t e x p a n s i o n r a t i o s r a n g i n g f r o m 3*1 t o 1 6 : 1 . HOLOGRAPHIC SYSTEM FOR LASER PARAMETER MEASUREMENT A l l t h e n e c e s s a r y a p p a r a t u s t o p e r f o r m e x p e r i m e n t s o f t h e t y p e d i s c u s s e d i n S e c t i o n Two has b e e n d e s i g n e d and a s s e m b l e d . F i g u r e 7 shows t h e g e n e r a l l a y o u t o f t h i s a p p a r a t u s , and F i g u r e lk i s a s c a l e d r a w i n g o f t h e a c t u a l e x p e r i m e n t a l a p p a r a t u s w i t h t h e d i m e n s i o n s and components n o t e d . As i n F i g u r e 7t t h e l a s e r beam e n t e r s t h e a p p a r a t u s f r o m t h e l e f t hand edge o f t h e f i g u r e and f i r s t s t r i k e s a b e a m s p l i t t e r , BS^ . T h i s b e a m s p l i t t e r , and a l l o t h e r m i r r o r s and b e a m s p l i t t e r s i n t h e s y s t e m were f a b r i -c a t e d b y L a s e r O p t i c s I n c . (LOI) and a r e m u l t i l a y e r d i e l e c t r i c c o a t e d l a s e r g r a d e s u b s t r a t e s . BS± i s a n i n t e r f e r o m e t e r f l a t (LOI #102-25010) t h a t has a n a n t i - r e f l e c t i o n c o a t i n g (AR) on one s u r f a c e and a 2k % r e f l e c t i v e c o a t i n g on t h e o t h e r s u r f a c e ( a t 694-3 A, 4-5° a n g l e o f i n c i d e n c e , and S p o l a r i z a t i o n ) . M i r r o r Mi i s a f r o n t s u r f a c e r e f l e c t o r (LOI #110-25220) f e a t u r i n g a 9 9 - 9 7° r e f l e c t i v e c o a t i n g a t k5° a n g l e o f i n c i d e n c e and S p o l a r i z a t i o n . M i r r o r s M2 a r e i d e n t i c a l f r o n t s u r f a c e r e -f l e c t o r s (LOI #110-25220) c o a t e d f o r 9 9 . 9 % r e f l e c t i v i t y a t k0° a n g l e o f i n c i d e n c e and S p o l a r i z a t i o n . M i r r o r M3 and b e a m s p l i t t e r BS2 d i v i d e t h e s i g n a l beam i n t o t h e u p p e r and l o w e r s i g n a l beams ( c f . F i g u r e 7» " s i d e v i e w de-t a i l " ) . T h e y a r e b o t h c o a t e d f o r P p o l a r i z a t i o n and 1 5 ° a n g l e o f i n c i d e n c e . M3 i s a f r o n t s u r f a c e m i r r o r (LOI #110-25220) t h a t i s 9 9 . 9 % r e f l e c t i v e . BS2 i s a n i n t e r f e r o m e t e r f l a t (LOI FIGURE 15 Overhead view of holographic system (scale: 1:5, all dimensions in centimeters) input beam i ^ ^ L i BSi i 17.5 •8.5 M 1 15c 15° ,375° M -i o p r - 1 0 3 -#102-25010) w i t h a n AR c o a t i n g on one s u r f a c e and a 52 $ r e f l e c -t i v e c o a t i n g on t h e o t h e r s u r f a c e . The m u l t i p l e r e f l e c t i o n c e l l , MRC, i s composed o f two s p e -c i a l l y f a b r i c a t e d f u s e d s i l i c a p l a t e s t h a t measure 5 " b y 2" b y 5 / 8 " . T h e s e p l a t e have a f l a t n e s s o f 1/10 wave and a r e wedged 30* ± 5 ' ' E a c h p l a t e has a n AR c o a t i n g on one s u r f a c e , and a 9 9 - 3 $ r e f l e c t i v e ( a t 1 5 ° a n g l e o f i n c . ) c o a t i n g on t h e o t h e r s u r f a c e . The q u a l i t y o f t h e s e p l a t e s f a r e x c e e d s t h e r e q u i r e -ments o f t h i s e x p e r i m e n t , and t h e c o s t o f t h e a p p a r a t u s may be s u b s t a n t i a l l y r e d u c e d b y u s i n g l o w e r q u a l i t y p l a t e s . The p l a t e s d e s c r i b e d h e r e were c h o s e n so t h a t t h e y m i g h t be e mployed i n a s t a n d a r d , l a s e r F a b r y - P e r o t i n t e r f e r o m e t e r ^ ^ o r , and e v e n more a d v a n t a g e o u s l y , i n a l a r g e s e p a r a t i o n t i l t e d p l a t e i n t e r f e r o m e -t e r o f t h e t y p e d e s c r i b e d b y Moos, e t . a l . ^ - L T h i s i n c r e a s e i n v e r s a t i l i t y o f f s e t t h e added expense o f i n t e r f e r o m e t r i c q u a l i t y p l a t e s . As shown i n F i g u r e 14, t h e p l a t e s a r e p a r a l l e l and a r e t i l t e d 1 5 ° o u t o f p e r p e n d i c u l a r w i t h r e s p e c t t o t h e beam a x i s . The c h a r a c t e r i s t i c d e l a y t i m e o f e a c h s u c c e s s i v e beam l e a v i n g t h e MRC ( r e l a t i v e t o t h e beam a d j a c e n t t o i t ) i s g i v e n b y 6 = [ 2 d/c] cos9 ^ (196) where d i s t h e p l a t e s e p a r a t i o n and 0 i s t h e a n g l e o f i n c i -d ence o f t h e l a s e r beam. The p o s i t i o n and o r i e n t a t i o n o f e a c h o f t h e p l a t e s i s c o n t i n u o u s l y v a r i a b l e , b u t f o r t h e e x e m p l a r y c o n f i g u r a t i o n shown i n F i g u r e 14, d e q u a l s 2 cm and 0 i s 1 5 ° • T h e s e v a l u e s l e a d t o a c h a r a c t e r i s t i c d e l a y t i m e o f a b o u t -104-130 p s e c . C h a r a c t e r i s t i c d e l a y t i m e s r a n g i n g f r o m a b o u t 20 p s e c t o a b o u t 2 n s e c a r e p o s s i b l e w i t h t h e p r e s e n t a p p a r a t u s . I n t h e c o n f i g u r a t i o n shown i n t h e d i a g r a m , 22 d e l a y e d s i g n a l beams a r e p r o d u c e d , 11' i n t h e u p p e r p l a n e and 11 i n t h e l o w e r p l a n e . E a c h s e t o f 11 beams c o v e r s a t o t a l r a n g e o f d e l a y t i m e s o f a b o u t 1.3 n s e c ( t h i s c a n be v a r i e d f r o m a b o u t 200 p s e c t o a b o u t 20 n s e c b y v a r y i n g t h e p l a t e s e p a r a t i o n ) . A l s o , t h e amount o f o v e r -l a p b e t w e e n t h e u p p e r beam d e l a y t i m e s and t h e l o w e r beam d e l a y t i m e s may be i n d e p e n d e n t l y v a r i e d b y v a r y i n g t h e s e p a r a t i o n be-tween and BS2. I n t h i s manner, one may e i t h e r d o u b l e t h e number o f d a t a p o i n t s w i t h i n a c e r t a i n r a n g e o f t i m e s , o r one may d o u b l e t h e t o t a l r a n g e o f t h e d a t a p o i n t s . P r o p e r s t a g g e r i n g o f t h e u p p e r and l o w e r d e l a y t i m e s w o u l d a l l o w t h e a t t a i n m e n t o f a b o u t a 10 p s e c t e m p o r a l r e s o l u t i o n . The r e l a t i v e l y h i g h r e f l e c t i v i t y o f t h e MRC p l a t e s was cho-s e n so t h a t t h e a t t e n u a t i o n p e r r e f l e c t i o n w o u l d be m i n i m a l , w h i l e s t i l l a l l o w i n g enough r a d i a t i o n t o p a s s t h r o u g h t h e p l a t e s t o f o r m t h e h o l o g r a m s . T h i s makes t h e i n t e n s i t i e s o f a l l s i g n a l beams s t r i k i n g t h e f i l m a b o u t e q u a l . The f o r m u l a f o r t h e i n t e n -s i t y o f t h e j - t h s i g n a l beam, r e l a t i v e t o t h e f i r s t s i g n a l beam ( t h e s t r a i g h t - t h r o u g h beam) i s I./I± = R2(M) ( 1 9 7) where R i s t h e r e f l e c t i v i t y o f t h e p l a t e s (99-3 $ i n t h i s c a s e ) and where I j and I x a r e measured t o t h e r i g h t o f t h e MRC i n F i g u r e 14. F o r t h e a p p a r a t u s shown, t h e r a t i o f o r t h e e l e v e n t h beam i s a b o u t 0.87. so t h e 11 beams v a r y o n l y s l i g h t l y i n r e l a -- 1 0 5 -t l v e i n t e n s i t y . B o t h beam e x p a n d e r s i n F i g u r e 14 a r e s i m p l e G a l l i l e a n t e l e -s c o p e s e m p l o y i n g p l a n o - c o n c a v e - p l a n o - c o n v e x g e o m e t r i e s t o r e -duce s p h e r i c a l a b e r a t i o n . A two s t a g e s y s t e m was c h o s e n t o a l l o w t h e m a i n e x p a n d e r , L 3 and L j ^ , t o r e m a i n f i x e d , w h i l e l e n s e s L^ and L 2 may be v a r i e d t o a d j u s t t h e beam e x p a n s i o n . F o r t h e l e n s e s shown, t h e l i n e a r e x p a n s i o n r a t i o i s 8 0 : 1 . An-o t h e r l e n s I 4 ( f o c a l l e n g t h - 2 5 mm) and t h r e e o t h e r l e n s e s L 2 ( f o c a l l e n g t h s 150 mm, 100 mm, and 70 mm) a r e a v a i l a b l e t o y i e l d a r a n g e o f 8 e x p a n s i o n r a t i o s f r o m 2 8 : 1 t o 1 6 0 : 1 . A l l l e n s e s a r e f r o m O p t i c a l I n d u s t r i e s I n c . , and a r e f a b r i c a t e d f r o m BSC g l a s s . L e n s e s ~L± and L 2 a r e 2 2 . 4 mm i n d i a m e t e r , and, a s shown i n t h e f i g u r e , have f o c a l l e n g t h s o f - 5 0 mm and 20 mm r e s p e c t i v e l y . L e n s L3 i s 25 mm i n d i a m e t e r w i t h a f o c a l l e n g t h o f - 2 5 mm, and l e n s L 4 i s 145 mm i n d i a m e t e r w i t h a f o c a l l e n g t h o f 500 mm. The p h o t o g r a p h i c p l a t e P may be a n y h i g h q u a l i t y , f a i r l y h i g h s p e e d h o l o g r a p h i c e m u l s i o n . D e p e n d i n g on t h e l a s e r o u t p u t e n e r g y , p o s s i b l e c h o i c e s a r e Kodak F E - 3 8 0 7 - I , A g f a I O E 7 5 , o r A g f a 8 E 7 5 , a r r a n g e d i n o r d e r o f d e c r e a s i n g s e n s i t i v i t y and i n -c r e a s i n g r e s o l u t i o n . The f i l m h o l d e r a c c e p t s 4" b y 5" g l a s s p l a t e s a t t h e p r e s e n t t i m e . The e m u l s i o n s a r e p r o c e s s e d ( i n -c l u d i n g b l e a c h i n g ) b y t h e m a n u f a c t u r e r ' s recommended methods. I n a d d i t i o n t o t h e a p p a r a t u s shown i n F i g u r e 14, t h e r e a r e a l s o two m o v e a b l e b e l l o w s w i t h c o n v e n t i o n a l 4" b y 5 " f i l m h o l d e r s l o c a t e d t o t h e r i g h t o f t h e h o l o g r a p h i c p l a t e . T h e s e s two f i l m h o l d e r s may be l o a d e d w i t h c o n v e n t i o n a l s h e e t f i l m , - 1 0 6 -and a r e u s e d t o measure t h e i n t e n s i t i e s o f t h e r e c o n s t r u c t e d beams, e t c . , b y s t a n d a r d p h o t o m e t r i c means. A l l t h e a p p a r a t u s shown i n F i g u r e lk i s mounted on a l a r g e r e i n f o r c e d aluminum p l a t e (k0" b y 2k" b y 5 / 8 " ) w h i c h r i d e s on a v i b r a t i o n damping s u p p o r t . F o r t h e e x t r e m e l y s h o r t e x p o s u r e t i m e s u s e d h e r e , t h i s b a s e has s u f f i c i e n t s t a b i l i t y f o r h o l o -g r a p h y . F i n a l l y , s i n c e t h e h o l o g r a p h i c d e g r e e o f m o d u l a t i o n and t h e b i a s l e v e l e x p o s u r e p e r u n i t e n e r g y d e n s i t y i n p u t a r e b o t h p r o -p e r t i e s o f t h e a p p a r a t u s o n l y , t h e y may be computed. F o r t h e s y s t e m shown, t h e h o l o g r a p h i c d e g r e e o f m o d u l a t i o n [ c f . S e c -t i o n Two, e q u a t i o n ( 9 8 ) ] i s f o u n d t o be a b o u t O.kO ± 0 . 0 3 f o r a l l 22 beams. I n t h i s same c o n f i g u r a t i o n , t h e t o t a l e n e r g y den-s i t y a t t h e f i l m p l a n e p e r u n i t l a s e r o u t p u t e n e r g y d e n s i t y i s ( 1 . 2 5 ± 0 . 0 1 ) » 1 0 - \ w h i c h d e t e r m i n e s t h e b i a s l e v e l e x p o s u r e u n i q u e l y f o r any g i v e n i n p u t e n e r g y d e n s i t y . 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