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Optical reconstruction from sampled holograms made with sound waves
Volume24A, number 1O
OPTICAL
PHYSICS LETTERS
REC~)NSTRUCTION MADE WITH
8 May 1967
FROM SAMPLED SOUND WAVES
HOLOGRAMS
A. F. METHERELL, H . M . A . EL-SUM*, J . J . DREHER and L. LARMORE Advanced Research Laboratories. Douglas Aircraft Company. Inc. Huntington Beach. California Received 7 April 1967
Experiments are described in which a reconstructed image was formed from a sampled hologram. The hologram is called a sonoptigram because it was made using sound instead of light.
A r e c o n s t r u c t e d i m a g e has been obtained from a s a m p l e d h o l o g r a m . The h o l o g r a m was made u s i n g sound waves at a f r e q u e n c y of 21 kc, and light f r o m a HeNe l a s e r was u s e d in the r e c o n s t r u c t i o n . To avoid confusing this type of holog r a m with the conventional type where light is used in both stages of the p r o c e s s , it is called a s o n o p t i g r a m - - f r o m sonic optical g r a m . The p r o c e s s is c a l l e d sonoptography. The p r e s e n t e x p e r i m e n t is unique to holography in that v e r y c o a r s e s a m p l i n g of the diff r a c t i o n fringe p a t t e r n s w e r e made in f o r m i n g the s a m p l e d s o n o p t i g r a m . It also d i f f e r s f r o m the only other r e p o r t e d u s e of sound [1] in f o r m ing a h o l o g r a m in that (a) the e x p e r i m e n t s were conducted in a i r , (b) r e l a t i v e l y long wavelengths were used (0.644 in. ), (c) the s o n o p t i g r a m was s e n s e d u s i n g an o r d i n a r y m i c r o p h o n e , (d) the s o n o p t i g r a m was d i s p l a y e d on an o s c i l l o s c o p r e CRT and (e) p e r m a n e n t l y r e c o r d e d by photographing the i m a g e thus f o r m e d , and (f) the n a r r o w angle type of h o l o g r a m (sonoptigram) was made where only the low o r d e r f r i n g e s w e r e r e c o r d e d . The t a r g e t used was a six inch s q u a r e b a s e d p y r a m i d placed d i s t a n c e v (29 in. ) in front of the plane in space s c a n n e d by the m i c r o p h o n e . A sound s o u r c e (high f r e q u e n c y tweeter) was placed d i s t a n c e u (137 in. ) f r o m the s c a n n i n g plane as shown in fig. 1. The m i c r o p h o n e s a m p l e d the a v e r a g e a c o u s t i c i n t e n s i t y over the t6 ,t in. d i a m e t e r of its face, a s it s c a n n e d the plane h o r i z o n tally for a d i s t a n c e of 35.4 in. Between each s c a n the m i c r o p h o n e was r a i s e d 0.5 in. The output f r o m the m i c r o p h o n e was amplified and connected to the cathode input (z-axis) of an o s c i l l o s c o p e so that the spot i n t e n s i t y on the CRT * Consultant. 74 Middlefield, Atherton, California.
MICROPHONE SCANNING THE ACOUSTIC PLANE SAMPLES THE ACOUSTICAL T H E A R E A O F IT~
TEST
OBJECT
HIGH-FREQUENCY SOUND SOURCE
Fig. 1. Experimental layout used in making sonoptigr~-ms. c o r r e s p o n d e d to the a c o u s t i c a l i n t e n s i t y s e n s e d by the m i c r o p h o n e . However, it should be e m p h a s i z e d that the c o r r e s p o n d e n c e was not l i n e a r , but m o r e n e a r l y exponential, which i n t r o d u c e s additional d i s t o r t i o n . The h o r i z o n t a l position of the spot was s y n c h r o n i z e d with the microphone by adjusting the oscilloscope time base, which was t r i g g e r e d by a m i c r o s w i t c h on the m i c r o p h o n e c a r r i a g e , such that one inch t r a v e l of the m i c r o phone c o r r e s p o n d e d to 2ram on the CRT. As the m i c r o p h o n e was r a i s e d between s c a n s , the spot on the CRT was r a i s e d p r o p o r t i o n a l l y by m e a n s of the v e r t i c a l position control of the o s c i l l o s c o p e . A photograph of the CRT, which r e s e m b l e s a t e l e v i s i o n p i c t u r e , was made for 40 s c a n s and is shown in fig. 2(a). This i s the s o n o p t i g r a m . Notice that the so n o p t i g r a m of the p y r a m i d , apex up, yields t r i a n g u l a r diffraction f r i n g e s , apex down. 547
Volume24A. number 10
PHYSICS LETTERS
8 May 1967
p h o t o g r a p h i c a l l y to about 0.25 in. width on a P o l a r o i d t r a n s p a r e n c y of fig. 2(a). This was then i l l u m i n a t e d with a HeNe l a s e r . The i m a g e of the p y r a m i d , apex up, was f o r m e d at the t h e o r e t i c a l f o c a l point as c a l c u l a t e d using the conventional f o r m u l a e u s e d in holography. The r e c o n s t r u c t e d i m a g e was r e c o r d e d on a photographic f i l m p l a c e d at the focal point of the r e a l i m a g e . This is shown in fig. 2(b) which is a dark p y r a m i d s i l houetted against a light background. I m a g e s r e c o n s t r u c t e d f r o m s o n o p t i g r a m s of o t h e r o b j e c t s have been m a d e [2]. A m o r e e x t e n s i v e p a p e r on these e x p e r i m e n t s and a d i s c u s s i o n of h o l o g r a p h i c p r i n c i p l e s which may be extended to sonoptography, as well as a d i s c u s s i o n of p o s sible future a p p l i c a t i o n s , will be published s h o r t l y [3]. T h i s work was conducted by the Douglas Adv a n c e d R e s e a r c h L a b o r a t o r i e s under Company s p o n s o r e d r e s e a r c h and d e v e l o p m e n t funds.
Fig.2. Sonoptigram and reconstructed image of pyramid. To r e c o n s t r u c t the i m a g e of the p y r a m i d f r o m the s o n o p t i g r a m , the s o n o p t i g r a m was r e d u c e d
FREQUENCY-DEPENDENT
CONDUCTIVITY
References 1. R.K. Mueller and N.K.Sheridon, Appl. Phys. Letters 9 (1966) 328. 2. A.F.Metherell, H.M.A.E1-Sum, J . J . D r e h e r and L. Larmore. DARL Research Communication No. 25, Douglas Paper No. 4491, (1967). 3° A,F.Metherell, H.M.A.EI-Sum, J . J . D r e h e r and L. Larmore, Introduction to sonoptigraphy, submitted to J. Aeoust. Soc. Am.
OF
METALS
WITH
IMPURITIES
N. M. P L A K I D A Laboratory for Theoretical Physics. Joint Institute for Nuclear Research. Moscow. USSR Received 7 March 1967 A general expression for the frequency-dependent electrical conductivity of metals with impurities is derived in terms of the Green function for displacements of atoms in an imperfect lattice. R e c e n t l y Kagan and Z h e r n o v [1] have d e v e l o p ed a t h e o r y of the s t a t i c e l e c t r i c a l conductivity of m e t a l s with i m p u r i t i e s which t a k e s into account the d e f o r m a t i o n of the phonon s p e c t r u m of an i m p e r f e c t l a t t i c e . In this note we r e p o r t an e x t e n sion of t h e i r r e s u l t s to the c a s e of a p e r i o d i c e l e c t r i c a l field. Our c a l c u l a t i o n s a r e b a s e d on the 548
t w o - p a r t i c l e s G r e e n function method which we have d e v e l o p e d e a r l i e r for s i m i l a r p r o b l e m s [2]. L a t t i c e v i b r a t i o n s in an i m p e r f e c t l a t t i c e a r e e a s i l y d e s c r i b e d by the Hamiltonian in the c o o r dinate r e p r e s e n t a t i o n [1]. In this c a s e the H a m i l tonian of the e l e c t r o n - l a t t i c e i n t e r a c t i o n contains the function vn(q) e x p [ - i q - (/~n + Un)] w h e r e
OPTICAL
PHYSICS LETTERS
REC~)NSTRUCTION MADE WITH
8 May 1967
FROM SAMPLED SOUND WAVES
HOLOGRAMS
A. F. METHERELL, H . M . A . EL-SUM*, J . J . DREHER and L. LARMORE Advanced Research Laboratories. Douglas Aircraft Company. Inc. Huntington Beach. California Received 7 April 1967
Experiments are described in which a reconstructed image was formed from a sampled hologram. The hologram is called a sonoptigram because it was made using sound instead of light.
A r e c o n s t r u c t e d i m a g e has been obtained from a s a m p l e d h o l o g r a m . The h o l o g r a m was made u s i n g sound waves at a f r e q u e n c y of 21 kc, and light f r o m a HeNe l a s e r was u s e d in the r e c o n s t r u c t i o n . To avoid confusing this type of holog r a m with the conventional type where light is used in both stages of the p r o c e s s , it is called a s o n o p t i g r a m - - f r o m sonic optical g r a m . The p r o c e s s is c a l l e d sonoptography. The p r e s e n t e x p e r i m e n t is unique to holography in that v e r y c o a r s e s a m p l i n g of the diff r a c t i o n fringe p a t t e r n s w e r e made in f o r m i n g the s a m p l e d s o n o p t i g r a m . It also d i f f e r s f r o m the only other r e p o r t e d u s e of sound [1] in f o r m ing a h o l o g r a m in that (a) the e x p e r i m e n t s were conducted in a i r , (b) r e l a t i v e l y long wavelengths were used (0.644 in. ), (c) the s o n o p t i g r a m was s e n s e d u s i n g an o r d i n a r y m i c r o p h o n e , (d) the s o n o p t i g r a m was d i s p l a y e d on an o s c i l l o s c o p r e CRT and (e) p e r m a n e n t l y r e c o r d e d by photographing the i m a g e thus f o r m e d , and (f) the n a r r o w angle type of h o l o g r a m (sonoptigram) was made where only the low o r d e r f r i n g e s w e r e r e c o r d e d . The t a r g e t used was a six inch s q u a r e b a s e d p y r a m i d placed d i s t a n c e v (29 in. ) in front of the plane in space s c a n n e d by the m i c r o p h o n e . A sound s o u r c e (high f r e q u e n c y tweeter) was placed d i s t a n c e u (137 in. ) f r o m the s c a n n i n g plane as shown in fig. 1. The m i c r o p h o n e s a m p l e d the a v e r a g e a c o u s t i c i n t e n s i t y over the t6 ,t in. d i a m e t e r of its face, a s it s c a n n e d the plane h o r i z o n tally for a d i s t a n c e of 35.4 in. Between each s c a n the m i c r o p h o n e was r a i s e d 0.5 in. The output f r o m the m i c r o p h o n e was amplified and connected to the cathode input (z-axis) of an o s c i l l o s c o p e so that the spot i n t e n s i t y on the CRT * Consultant. 74 Middlefield, Atherton, California.
MICROPHONE SCANNING THE ACOUSTIC PLANE SAMPLES THE ACOUSTICAL T H E A R E A O F IT~
TEST
OBJECT
HIGH-FREQUENCY SOUND SOURCE
Fig. 1. Experimental layout used in making sonoptigr~-ms. c o r r e s p o n d e d to the a c o u s t i c a l i n t e n s i t y s e n s e d by the m i c r o p h o n e . However, it should be e m p h a s i z e d that the c o r r e s p o n d e n c e was not l i n e a r , but m o r e n e a r l y exponential, which i n t r o d u c e s additional d i s t o r t i o n . The h o r i z o n t a l position of the spot was s y n c h r o n i z e d with the microphone by adjusting the oscilloscope time base, which was t r i g g e r e d by a m i c r o s w i t c h on the m i c r o p h o n e c a r r i a g e , such that one inch t r a v e l of the m i c r o phone c o r r e s p o n d e d to 2ram on the CRT. As the m i c r o p h o n e was r a i s e d between s c a n s , the spot on the CRT was r a i s e d p r o p o r t i o n a l l y by m e a n s of the v e r t i c a l position control of the o s c i l l o s c o p e . A photograph of the CRT, which r e s e m b l e s a t e l e v i s i o n p i c t u r e , was made for 40 s c a n s and is shown in fig. 2(a). This i s the s o n o p t i g r a m . Notice that the so n o p t i g r a m of the p y r a m i d , apex up, yields t r i a n g u l a r diffraction f r i n g e s , apex down. 547
Volume24A. number 10
PHYSICS LETTERS
8 May 1967
p h o t o g r a p h i c a l l y to about 0.25 in. width on a P o l a r o i d t r a n s p a r e n c y of fig. 2(a). This was then i l l u m i n a t e d with a HeNe l a s e r . The i m a g e of the p y r a m i d , apex up, was f o r m e d at the t h e o r e t i c a l f o c a l point as c a l c u l a t e d using the conventional f o r m u l a e u s e d in holography. The r e c o n s t r u c t e d i m a g e was r e c o r d e d on a photographic f i l m p l a c e d at the focal point of the r e a l i m a g e . This is shown in fig. 2(b) which is a dark p y r a m i d s i l houetted against a light background. I m a g e s r e c o n s t r u c t e d f r o m s o n o p t i g r a m s of o t h e r o b j e c t s have been m a d e [2]. A m o r e e x t e n s i v e p a p e r on these e x p e r i m e n t s and a d i s c u s s i o n of h o l o g r a p h i c p r i n c i p l e s which may be extended to sonoptography, as well as a d i s c u s s i o n of p o s sible future a p p l i c a t i o n s , will be published s h o r t l y [3]. T h i s work was conducted by the Douglas Adv a n c e d R e s e a r c h L a b o r a t o r i e s under Company s p o n s o r e d r e s e a r c h and d e v e l o p m e n t funds.
Fig.2. Sonoptigram and reconstructed image of pyramid. To r e c o n s t r u c t the i m a g e of the p y r a m i d f r o m the s o n o p t i g r a m , the s o n o p t i g r a m was r e d u c e d
FREQUENCY-DEPENDENT
CONDUCTIVITY
References 1. R.K. Mueller and N.K.Sheridon, Appl. Phys. Letters 9 (1966) 328. 2. A.F.Metherell, H.M.A.E1-Sum, J . J . D r e h e r and L. Larmore. DARL Research Communication No. 25, Douglas Paper No. 4491, (1967). 3° A,F.Metherell, H.M.A.EI-Sum, J . J . D r e h e r and L. Larmore, Introduction to sonoptigraphy, submitted to J. Aeoust. Soc. Am.
OF
METALS
WITH
IMPURITIES
N. M. P L A K I D A Laboratory for Theoretical Physics. Joint Institute for Nuclear Research. Moscow. USSR Received 7 March 1967 A general expression for the frequency-dependent electrical conductivity of metals with impurities is derived in terms of the Green function for displacements of atoms in an imperfect lattice. R e c e n t l y Kagan and Z h e r n o v [1] have d e v e l o p ed a t h e o r y of the s t a t i c e l e c t r i c a l conductivity of m e t a l s with i m p u r i t i e s which t a k e s into account the d e f o r m a t i o n of the phonon s p e c t r u m of an i m p e r f e c t l a t t i c e . In this note we r e p o r t an e x t e n sion of t h e i r r e s u l t s to the c a s e of a p e r i o d i c e l e c t r i c a l field. Our c a l c u l a t i o n s a r e b a s e d on the 548
t w o - p a r t i c l e s G r e e n function method which we have d e v e l o p e d e a r l i e r for s i m i l a r p r o b l e m s [2]. L a t t i c e v i b r a t i o n s in an i m p e r f e c t l a t t i c e a r e e a s i l y d e s c r i b e d by the Hamiltonian in the c o o r dinate r e p r e s e n t a t i o n [1]. In this c a s e the H a m i l tonian of the e l e c t r o n - l a t t i c e i n t e r a c t i o n contains the function vn(q) e x p [ - i q - (/~n + Un)] w h e r e