Volume 26A, number 8 P H Y SIC S L E T T E R S 11 March 1968

S P E C T R U M B R O A D E N I N G IN P H A S E - M O D U L A T E D C O H E R E N T O P T I C A L I L L U M I N A T I O N *

W. L. ANDERSON Southwest Research Inst i tute , San Antonio, Texas 78206, USA

Received 23 January 1968

A method is proposed for broadening the spectrum available from a ,phase-modulator", thereby effect- ing an improvement in two-dimensional image quality when using coherent light.

Upatnieks has repor ted a method whereby imaging of two-d imens iona l t r a n s p a r e n t objects with coherent light may be accompl ished with improved quali ty [1]. The method makes use of a randomly phase -modula ted wave-f ront to i l l u m i - nate the object. The degrading effects of d i f f rac- t ion pa t t e rns f rom defects in the optical sys t em and of g ranu la r i ty a r e overcome to a cons ide rab le extent.

As was d i scussed by Upatnieks, a difficulty r e m a i n s in finding the ideal phase -modula to r , which would have a flat spat ia l f requency spec- t r u m in the range of in te res t . The one he found to most c losely approach this ideal had a more or l ess t r i angu la r spec t rum, dec reas ing l inea r ly f rom a max imum at zero to 5% of max imum at 60 l i ne s /mm, then taper ing off more slowly. The purpose of this le t te r is to point out an approach to the p rob lem of cons t ruc t ing a more un i fo rm and broader spec t rum f rom a re la t ive ly non- un i fo rm and nar row spec t rum.

The approach is i l l u s t r a t ed by cons ider ing f i r s t a s imple one -d imens iona l spec t rum F(~) = : (1 - I~1/~o), I~1 --<~o; F(~) =0 , I~1 >~ ~o (fig. la). Clear ly , a convolution of F(~) with a sum of 2N + 1 de l ta - func t ions , s y m m e t r i c about zero and spaced ~o apart , will give a spec t rum flat over tt~e range l~l --< N~o. Thus, if G(~) is the spec t rum obtained b~ convolution of th ree such de l ta - func t ions , ~ = l _ l 5(~ -n~o) , with F(~), the r e su l t i s the spec t rum shown in fig. lb . The flat region of G(~) can be extended by us ing l a rge r values of N.

The de l ta - funct ions ma___y be cons idered as a r i s i ng f rom the sum ~N=_ N exp (in~oXo) in the i n v e r s e t r a n s f o r m domain; optically, this can be

* Work supported under Grant GM-04256-02 from the National Institutes of Health.

/ Q -~o -Q

(o) (b)

Fig. 1 (a) Example of frequency spectrum requiring broadening; (b) Result of convolution with 3 appropriately-spaced

delta-functions.

cons idered as a sum of plane waves incident on the object plane at va r ious angles s y m m e t r i c with the optical axis. It should be apparent that although the spec t rum of the t r a n s m i s s i o n func- t ion exp [i@(x)] of Upatnieks ' phase modulator is complex, it also can be broadened by i l l umi na - t ion with s eve ra l va r ious ly or iented plane waves. In addition it is evident that these mu l t i - d i r e c t i o - nal plane waves can be obtained by inse r t ing an appropr ia te diffract ion gra t ing adjacent to the phase modulator .

It is impor tan t to recognize that spec t rum broadening is advantageous even if the ideal flat spec t rum is not achieved, and even if different degrees of broadening occur in the two spat ial f requency d imens ions . This has been ver i f ied by exper iments in our labora tory in which even a s ingle off-axis beam, obtained by a b e a m - s p l i t - t e r to supplement the on-ax is beam, was shown to provide image improvement under many c i r - cums tances .

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Volume26A, number 8 P H Y S I C S L E T T E R S l l M a r c h 1968

A d i sadvan tage of the method d e s c r i b e d above o c c u r s if the i n t e r f e r e n c e p a t t e r n s in the ob jec t p lane (and p r e s u m a b l y a l s o in the i m a g e plane) r e p r e s e n t e d by

iV iV exp(in~oX) = 1 + 2 ~ cosn~oX

n=-N n=l

a r e not suf f ic ient ly "f ine" , o r a l t e r n a t i v e l y , if the v a r i a t i o n s in t r a n s m i s s i o n of a d i f f rac t ion g ra t ing used to sp l i t the inc ident b e a m into s e v e - r a l b e a m s a r e not suff ic ient ly c lo se ly spaced . Th is sugges t s the u se of a phase gra t ing , which wil l s e r v e to s p r e a d the s p e c t r u m of the phase modu la to r without affect ing ampl i tude in the i m a g e plane . A f u r t h e r m e r i t cons i s t s in the fac t that , a s sugges t ed by Upatn ieks for the phase modu la to r , an i n v e r s e ope ra t ion can be employed in the i m a g e p lane to r e c o v e r o r ig ina l phase if th i s i s n e c e s s a r y .

The s p e c t r u m - b r o a d e n i n g ac t ion of a phase g r a t i ng through convolut ion with de l t a - func t ions in the F o u r i e r - t r a n s f o r m p lane can be eas i ly v e r i f i e d by cons ide r ing the s i m p l e example of a g r a t i ng with t r a n s m i s s i o n function

exp (iA cos b~ox) , (A, b r e a l , > 0). The t r a n s - t oo - f o r m i s of he f o r m ~ m = - ~ o imJm(A)5(mb~ o ~),

where in the de l t a - func t i ons now have v a r i o u s a mp l i t ude s dependent on the B e s s e l funct ions Jm(A) and v a r i o u s phases dependent on im. A v a r i e t y of " s p r e a d s p e c t r a " i s ava i l ab l e through cho ices of A and b; for example , if b = 1 and A ~ 1.5, the z e r o and f i r s t o r d e r t e r m s wil l be a p p r o x i m a t e l y equal and for an F(~) a s in fig. l a wil l y i e ld a s p e c t r u m with magni tude subs t a n t i - a l ly f la t f r o m - ~o to ~o"

The extens ion of the concept to two d imens ions can obvious ly be a c c o m p l i s h e d in v a r i o u s ways . I t may a l so be noted that phase g ra t i ngs of the type r e q u i r e d can p robab ly be c o n s t r u c t e d us ing pho tograph ic emu l s ions exposed to a p p r o p r i a t e p a t t e r n s and then b l eached [2].

Refe~'enccs 1. J . U p a t n i e k s , Appl . Opt ics 6 (1967) 1905. 2. N. George and J. W. Matthews, Appl. Phys. Letters

9 (1966) 212.

ON T H E C A L C U L A T I O N O F N O R M A L L Y O R D E R E D C O R R E L A T I O N F U N C T I O N S F O R T H E E L E C T R O M A G N E T I C F I E L D BY M E A N S O F T H E Q ( ~ ) - F U N C T I O N

R. GRAHAM and F. HAAKE Institut f l i t Theorettsche Phys/k der Universititt Stuttgart, Germany

Received 5 February 1968

We give a new formula permitting to calculate multitime correlation functions of the electromagnetic field. This formula even applies when the density operator does not allow the diagonal representation with respect to coherent states.

In the l a s t y e a r s the s t a t i s t i c a l p r o p e r t i e s of a monoc h roma t i c l ight mode have been s tud ied ex ten- s ive ly . The mode i s d e s c r i b e d by the Bose annih i la t ion and c r e a t i o n o p e r a t o r s b, b +. The s t a t i s t i c a l p r o p e r t i e s a r e d e t e r m i n e d by the c o r r e l a t i o n funct ions

g = ((b+(t l ) )Vl . • • (b+(tn))Vn(b(tn))~n... (b(tl))/~ 1) , (1)

with t i >~ ti_ 1. To so lve the equation of motion f_o~ tl~e s t a t i s t i c a l o p e r a t o r p of the mode w i d e s p r e a d use has been made of the P - r e p r e s e n t a t i o n [1] p(t) - j d2fi P(~,t)]~X31 with r e s p e c t to the coheren t s t a t e s bl~) = ~1~). As has been shown by s e v e r a l au tho r s [2,3] the c o r r e l a t i o n funct ions (1) may then be c a l - cu la ted with the he lp of the P-funct ion by s i m p l e c - n u m b e r ope ra t i ons . However , i t i s wel l known that the P - func t ion need not n e c e s s a r i l y ex i s t [4]. Indeed, t h e r e have r e c e n t l y a p p e a r e d s e v e r a l i m p o r t a n t phys i ca l p r o b l e m s for which i t evident ly does not ex i s t [5,6,7]. F o r m a l equat ions of motion for P have

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