Abstract

The use of a phase-modulated reference wave for the electronic heterodyne recording and processing of a hologram is described. Heterodyne recording is used to eliminate the self-interference terms of a hologram and to create a Leith-Upatnieks hologram with coaxial object and reference waves. Phase modulation is also shown to be the foundation of a multiple-view hologram system. When combined with hologram scale transformations, heterodyne recording is the key to general optical processing. Spatial filtering is treated as an example.

© 1978 Optical Society of America

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References

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  1. A. Decker, Ph.D. Thesis, Electronic Heterodyne Recording and Processing of Optical Holograms Using Phase Modulated Reference Waves (Case Western Reserve U., Cleveland, 1977).
  2. L. H. Enloe, J. A. Murphy, C. B. Rubinstein, Bell Syst. Tech. J. 45, 335 (1966).
  3. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [CrossRef]
  4. L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell Syst. Tech. J. 47, 1875 (1968).
  5. A. B. Larsen, Bell Syst. Tech. J. 48, 2507 (1969).
  6. A. Macovski, Ph.D. Thesis, Efficient Holography Using Temporal Modulation (Stanford U., Stanford, Calif., 1968), pp. 71–102.
  7. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, New York, 1971), p. 40.
  8. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 361.
  9. A. Macovski, Appl. Opt. 9, 1906 (1970).
    [PubMed]
  10. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]
  11. E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [CrossRef]

1970 (1)

1969 (1)

A. B. Larsen, Bell Syst. Tech. J. 48, 2507 (1969).

1968 (1)

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell Syst. Tech. J. 47, 1875 (1968).

1967 (1)

1966 (1)

L. H. Enloe, J. A. Murphy, C. B. Rubinstein, Bell Syst. Tech. J. 45, 335 (1966).

1965 (1)

1962 (1)

Champagne, E. B.

Decker, A.

A. Decker, Ph.D. Thesis, Electronic Heterodyne Recording and Processing of Optical Holograms Using Phase Modulated Reference Waves (Case Western Reserve U., Cleveland, 1977).

Enloe, L. H.

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell Syst. Tech. J. 47, 1875 (1968).

L. H. Enloe, J. A. Murphy, C. B. Rubinstein, Bell Syst. Tech. J. 45, 335 (1966).

Jakes, W. C.

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell Syst. Tech. J. 47, 1875 (1968).

Larsen, A. B.

A. B. Larsen, Bell Syst. Tech. J. 48, 2507 (1969).

Leith, E. N.

Macovski, A.

A. Macovski, Appl. Opt. 9, 1906 (1970).
[PubMed]

A. Macovski, Ph.D. Thesis, Efficient Holography Using Temporal Modulation (Stanford U., Stanford, Calif., 1968), pp. 71–102.

Meier, R. W.

Murphy, J. A.

L. H. Enloe, J. A. Murphy, C. B. Rubinstein, Bell Syst. Tech. J. 45, 335 (1966).

Rubinstein, C. B.

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell Syst. Tech. J. 47, 1875 (1968).

L. H. Enloe, J. A. Murphy, C. B. Rubinstein, Bell Syst. Tech. J. 45, 335 (1966).

Upatnieks, J.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, New York, 1971), p. 40.

Appl. Opt. (1)

Bell Syst. Tech. J. (3)

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell Syst. Tech. J. 47, 1875 (1968).

A. B. Larsen, Bell Syst. Tech. J. 48, 2507 (1969).

L. H. Enloe, J. A. Murphy, C. B. Rubinstein, Bell Syst. Tech. J. 45, 335 (1966).

J. Opt. Soc. Am. (3)

Other (4)

A. Macovski, Ph.D. Thesis, Efficient Holography Using Temporal Modulation (Stanford U., Stanford, Calif., 1968), pp. 71–102.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, New York, 1971), p. 40.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 361.

A. Decker, Ph.D. Thesis, Electronic Heterodyne Recording and Processing of Optical Holograms Using Phase Modulated Reference Waves (Case Western Reserve U., Cleveland, 1977).

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Figures (3)

Fig. 1
Fig. 1

Arrangement for recording a low resolution hologram of a transparency.

Fig. 2
Fig. 2

Reconstruction of a hologram recorded by the heterodyne method.

Fig. 3
Fig. 3

Filtered output in the real image with a virtual image at infinity.

Equations (23)

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U o ( x , y ) = O ( x , y ) cos [ ω t ϕ ( x , y ) ] , U r ( x , y ) = s ( x , y ) cos [ ω t r ( x , y ) + δ sin ω H t ] .
cos ( δ sin ω H t ) = J 0 ( δ ) + 2 l = 1 J 2 l ( δ ) cos ( 2 l ω H t ) , sin ( δ sin ω H t ) = 2 l = 0 J 2 l + 1 ( δ ) sin [ ( 2 l + 1 ) ω H t ] ,
I 0 ( x , y , t ) = O 2 ( x , y ) + s 2 ( x , y ) + 2 J 0 ( δ ) s ( x , y ) O ( x , y ) · cos [ ϕ ( x , y ) r ( x , y ) ] , I 1 ( x , y , t ) = 4 J 1 ( δ ) s ( x , y ) O ( x , y ) sin [ ϕ ( x , y ) r ( x , y ) ] sin ω H t , I 2 ( x , y , t ) = 4 J 2 ( δ ) s ( x , y ) O ( x , y ) cos [ ϕ ( x , y ) r ( x , y ) ] cos 2 ω H t , I 3 ( x , y , t ) = 4 J 3 ( δ ) s ( x , y ) O ( x , y ) sin [ ϕ ( x , y ) r ( x , y ) ] sin 3 ω H t , . . . .
f = V x f x + V y f y ,
f V x f x .
[ O , 2 f m x V x ] [ f H f m x V x , f H + f m x V x ] [ 2 f H f m x V x , 2 f H + f m x V x ] .
f H 3 f m x V x .
f H 2 f m x V x .
i ( t ) = 2 J 1 ( δ ) s ( V x t , y ) O ( V x t , y ) · cos [ ϕ ( V x t , y ) r ( V x t , y ) ω H t ] .
T ( x , y ) = I B + γ 2 J 1 ( δ ) s ( x , y ) O ( x , y ) · cos [ ϕ ( x , y ) r ( x , y ) ω H x V x ] ,
sin θ = [ ( f H ) / ( V x ) ] λ ,
I N = O N ( x , y ) cos [ ϕ N ( x , y ) + k x sin θ N N ω H t ] ,
T ( x , y ) = I B + γ N I N .
I 0 = s 2 ( x , y ) + O 2 ( x , y ) + 2 J 0 ( δ ) s ( x , y ) O ( x , y ) · cos [ ϕ ( x , y ) r ( x , y ) + a ( t ) ] , I 1 = 4 J 1 ( δ ) s ( x , y ) O ( x , y ) sin [ ϕ ( x , y ) r ( x , y ) + a ( t ) ] sin ω H t , I 2 = 4 J 2 ( δ ) s ( x , y ) O ( x , y ) cos [ ϕ ( x , y ) r ( x , y ) + a ( t ) ] cos 2 ω H t .
a ( t ) = π sin ( 2 π V x f m x x ) .
d b = 2 S V S R S R + S V ,
a = ( 2 S ) 1 / 2 ( k k B ) 1 / 2 ( S V S R S V S R ) 1 / 2 ( 1 S d r ) 1 / 2 .
M V = k k B S V S a , M R = k k B S R S a .
H ( x , y ) = exp [ j a ( x , y ) ] ,
E ( x i , y i ) = exp [ j k 2 S ( 1 k B k S S R M R 2 ) ( x 1 2 + y i 2 ) ] .
E ( x i , y i ) = exp [ j k 2 S ( 1 + k B k S S V M V 2 ) ( x i 2 + y i 2 ) ] .
I R = b * ( x R f H k B S R a V x M R , y R M R ) h * ( a x R , a y R ) ,
E ( x i , y i ) = exp [ j k 2 S ( x i 2 + y i 2 ) ] ,

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