Image formation with arbitrary holographic type surfaces

Volume 29A, number 4

IMAGE

PHYSICS LETTERS

FORMATION

5 May 1969

WITH ARBITRARY TYPE SURFACES

HOLOGRAPHIC

R. P. PORTER Northeastern University, Boston, Mass., USA and the MITRE Corporation, Bedord, Mass., USA* Received 14March 1969

Image formation achieved by reversing the direction of propagation of a diverging wave is studied. Surface sources capable of generating the field are found for arbitrary surfaces. The sources are compared with those used Co generate the holographic field. Holographic i m a g i n g s y s t e m s a r e e x a m p l e s of two step s y s t e m s which achieve the f o r m a t i o n of a r e a l image by e s t a b l i s h i n g s u r f a c e s o u r c e s that a r e complex c o n j u g a t e s of the m e a s u r e d field i n f o r m a t i o n . The technique i s u n d e r s t o o d in m o s t g e n e r a l f o r m by studying the complex conjugate of the d i v e r g i n g wave r a d i a t i n g f r o m a point s o u r c e . The a n a l y s i s and r e s u l t s p r e s e n t e d differ s i g n i f i c a n t l y f r o m o t h e r work b e c a u s e a r b i t r a r y recording surfaces are considered. T h i s work h a s c e r t a i n s i m i l a r i t i e s with the w a v e f r o n t continuation a n a l y s i s of L u n e b e r g [1]. It also b e a r s on the infinite p l a n a r h o l o g r a m t r e a t e d by M l t t r a and R a n s o m [2]. C o n s i d e r an object s o u r c e d i s t r i b u t i o n whose r a d i a t i o n obeys the m o n o c h r o m a t i c t i m e i n d e p e n dent s c a l a r wave equation. The r a d i a t i o n of a v o l u m e point s o u r c e , located at r ' , i s given by

G(r,

r') - exp['Jk

I r'r'l]

(1)

where ] r - r ' I i s the d i s t a n c e between r and r ' . C o n s i d e r the i m a g e c o n s t r u c t i o n p r o b l e m where s o u r c e s a r e placed on a closed s u r f a c e S s u r r o u n d i n g a s o u r c e l e s s r e g i o n which o r i g i n a l l y contained all of the object. Let the object be the above point s o u r c e whose r a d i a t e d field in G( r , r ' ) . Suppose the s o u r c e s on S launch a p u r e i n w a r d wave G* ( r , r ' ) which i s the c o m p l e x c o n jugate of G. T h i s field, a s seen f r o m eq. (1), r e q u i r e s a sink at r ' which l s c o n t r a r y to the r e q u i r e m e n t that the r e g i o n i s s o u r c e l e s s . A d i v e r ging wave r a d i a t i n g out f r o m r ' and p a s s i n g u n i m p e d e d through S, when added to the c o n v e r g i n g wave, is capable of s a t i s f y i n g the h o m o g e n e o u s wave equation i n s i d e S. In p a r t i c u l a r , c o n s i d e r the i m a g e k e r n e l

K(r,

r') =

I

r') - G ( r , r')

G*(r,

rin S (2)

- G ( r, r')

r out S .

T h e k e r n e l in the r e a l i m a g e space i s , f r o m eqs. (I) and (2), •

K ( r , r ' ) =-~

sin(k r-r'l)

{r-r'{

(3)

which in the l i m i t of zero wavelength a p p r o a c h e s a one d i m e n s i o n a l i m p u l s e . The s o u r c e v a l u e s which e s t a b l i s h t h i s field a r e of s i g n i f i c a n c e to holographic imaging. The i m a g e field in t e r m s of i t s d i s c o n t i n u i t i e s a c r o s s the s u r f a c e can be shown to be

K( r, r') = Jf { [ v " K r - V K v ] c( r, r " ) + s -

(4)

[Kr -Kv]V "C,(r, r")}. ndS"

w h e r e :~ i s the n o r m a l pointing out of S and the r e a l image space. Note t h a t / ~ v i s the image field outside of S. The d i s c o n t i n u i t i e s a r e found, f r o m eq. (2), to be n . V " G * ( r " , r ' ) for the n o r m a l g r a d i e n t difference and G*( r " , r ' ) for the field difference. T h i s r e s u l t can be g e n e r a l i z e d for an a r b i t r a r y object. The definition of the i m a g e field for an open S is given by eq. (4) subject to the above field d i s c o n t i n u i t i e s . Significantly, the s i m p l e r e s u l t of eq. (2) does not follow. It can be d e m o n s t r a t e d that i m a g i n g s t i l l r e s u l t s . It i s of s i g n i f i c a n c e in applying t h e s e r e s u l t s to holography that the image d e r i v e d h e r e r e q u i r e s the m e a s u r e m e n t of both the f i e l d and i t s n o r m a l gradient. M e a s u r i n g both p a r a m e t e r s i s * The author is a MITRE Fellow presently attending Northeastern University.

193

Volume 29A, number 4

PHYSICS LETTERS

not r e d u n d a n t b e c a u s e they a r e used to specify d i s c o n t i n u i t i e s a c r o s s S. It i s i m p l i e d by t h e s e r e s u l t s that the photographic h o l o g r a m i s l e s s than ideal. The fact that a h o l o g r a m i s e s s e n t i a l l y m o d e l e d a s specifying G* on S by M i t t r a a n d R a n s o m [2] and o t h e r i n v e s t i g a t o r s , a s opp o s e d to the f i e l d d i s c o n t i n u i t i e s n - V ' G * and G*, s u g g e s t s that holographic r e a l i m a g e s can be i m p r o v e d . Work i s in p r o g r e s s on d e r i v i n g the

5 May 1969

f o r m of the image k e r n e l f o r open s u r f a c e s and c o m p a r i n g it to the holographic image. References 1. R.K. Luneberg, Mathematical theory of optics, (University of California Press, Berkeley, 1964) p. 305-311. 2. R. Mittra and P. L. Ransom, Modern optics, ed. J. Fox, (Polytechnic Press, Brooklyn, 1967, J. Wiley and Sons Inc., New York, distr.), p. 619.

* * * * *

THE

RESISTIVE

TRANSITION

IN W E A K L Y

COUPLED

SUPERCONDUCTORS*

R. S. THOMPSON, M. STRONGIN, O. F. KAMMERER and J. E. CROW Brookhaven National Laboratory, Upton, New York, USA Received 10 March 1969

Films of weakly connected particles have narrower transitions above the half resistance point than uniform films of the same resistance. In contrast, the region of finite resistivity below the halfway point remains very broad.

The A s l a m a z o v - L a r ! ~ i n (AL) t h e o r y [1] i s g e n e r a l l y taken as a b a s i s for u n d e r s t a n d i n g r e s i s t i v e t r a n s i t i o n . Recent e x p e r i m e n t a l r e s u l t s [2-3] g e n e r a l l y a p p r o x i m a t e AL with v a r y ing d e g r e e s of s u c c e s s . We have obtained r e s u l t s which differ m a r k e d l y f r o m AL in Pb f i l m s made by depositing P b onto g l a s s o r p r e v i o u s l y d e p o s i t e d L i F , and held at 4.2°K, that b e c o m e e l e c t r i c a l l y continuous only at about 75 A. In c o n s t r a s t Pb f i l m s deposited onto other s u b s t r a t e s can be made continuous at about 10 A. In f i l m s that b e c o m e e l e c t r i c a l l y continuous at about 75 ,~ we b e l i e v e the f i l m is c o m p o s e d of weakly coupled s u p e r c o n d u c t i n g g r a i n s , p r o b a b l y n e a r 75 A in d i a m e t e r . T h i s i n t e r p r e t a t i o n i s c o n s i s t e n t with the e x t r e m e l y high f i l m r e s i s t a n c e and with s t u d i e s of the i n i t i a l growth stages of f i l m s [5]. In fig. 1 we can see the s t r u c t u r e i s u n s t a b l e and that a n n e a l i n g a l r e a d y o c c u r r s n e a r 7°K. The t r a n s i t i o n has two i n t e r e s t i n g f e a t u r e s . F i r s t , the width f o r t e m p e r a t u r e s above the ½ r e s i s t a n c e point is much s m a l l e r than the AL p r e diction. Secondly, s i g n i f i c a n t r e s i s t a n c e extends in some c a s e s even four d e g r e e s below the t e m p e r a t u r e at half r e s i s t a n c e , which we loosely * This work was performed under the auspices of the U. S. Atomic Energy Commission. 194

a s s o c i a t e with Tc. We suggest both f e a t u r e s can be m a i n l y explained by the weak coupling between p a r t i c l e s , and that d i s t r i b u t i o n s of p a r t i c l e s i z e s with somewhat different Tc'S which would be expected to influence the t r a n s i t i o n width do not play the m a j o r role. S i m i l a r work on 1 m m tin s p h e r e s has been done by Clark and T i l l e y [6]. F i r s t we show how above Tc the AL r e s u l t may be modified to apply to our f i l m s . F r o m AL the ratio of the additional conductivity ~ ' due to f l u c t u a t i o n s above Tc to the n o r m a l state conductivity ~ can be given as ~ ' / ~ = (2ro/~2(0)) ; k 2 dk2/(k 2 + ~ -2(T))3 , o where ~ (T) is the t e m p e r a t u r e dependent coherence length for the a s s u m e d u n i f o r m m a t e r i a l , and To is p r o p o r t i o n a l to the film r e s i s t a n c e p e r s q u a r e a r e a : r o = R o × 1 . 5 2 × 1 0 - S H -1. If we have weakly coupled p a r t i c l e s of d i a m e t e r d, we m u s t c o n s i d e r s e p a r a t e l y f l u c t u a t i o n s over d i s t a n c e s g r e a t e r and l e s s than d. Short wave length fluctuations a r e g o v e r n e d by the c o h e r e n c e length of the m e t a l in the p a r t i c l e , which is given by ~ 2v(T) -- 0.7 ~ o l / ~ where "r = IT - TcI/Tc and l i s the m e a n f r e e path inside the p a r t i c l e , which is p r o b a b l y ~ d . The effective c o h e r e n c e length for long wave f l u c t u a t i o n s ~eff is defined in the same