Abstract

Theory and experiment show that for a hologram object of two or more object points, the nonlinearity of the photographic process causes reconstructed images in addition to both the desired reconstructed image and the higher order reconstructed images. It is theoretically and experimentally shown that for a plane object parallel to the hologram plane, some of these undesired images may be focused in the plane of the desired reconstructed image.

© 1970 Optical Society of America

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References

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  1. D. G. Falconer, Phot. Sci. Eng. 10, 136 (1966).
  2. G. R. Knight, J. Opt. Soc. Amer. 57, 11413 (1967).
  3. A. A. Friesem, J. S. Zelenka, Appl. Opt. 6, 1755 (1967).
    [CrossRef] [PubMed]
  4. J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 59, 1650 (1959).

1967 (2)

G. R. Knight, J. Opt. Soc. Amer. 57, 11413 (1967).

A. A. Friesem, J. S. Zelenka, Appl. Opt. 6, 1755 (1967).
[CrossRef] [PubMed]

1966 (1)

D. G. Falconer, Phot. Sci. Eng. 10, 136 (1966).

1959 (1)

J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 59, 1650 (1959).

Falconer, D. G.

D. G. Falconer, Phot. Sci. Eng. 10, 136 (1966).

Friesem, A. A.

Givens, M. P.

J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 59, 1650 (1959).

Knight, G. R.

G. R. Knight, J. Opt. Soc. Amer. 57, 11413 (1967).

Wyant, J. C.

J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 59, 1650 (1959).

Zelenka, J. S.

Appl. Opt. (1)

J. Opt. Soc. Amer. (2)

G. R. Knight, J. Opt. Soc. Amer. 57, 11413 (1967).

J. C. Wyant, M. P. Givens, J. Opt. Soc. Amer. 59, 1650 (1959).

Phot. Sci. Eng. (1)

D. G. Falconer, Phot. Sci. Eng. 10, 136 (1966).

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Figures (3)

Fig. 1
Fig. 1

Reconstructed image of hologram of two, three, and four object points.

Fig. 2
Fig. 2

Reconstruction of hologram of a piece of ground glass approximately 7 mm square. Flare around square is due to undesired reconstructed points.

Fig. 3
Fig. 3

Location of undesired images as predicted by Eq. (9) for object of length 2a and two objects, each of length a, separated by distance d; extent of objects shown by arrows; extent of external flare due to undesired reconstructions shown by shaded regions.

Tables (2)

Tables Icon

Table I (X,Y) Coordinates of Undesired Reconstructed Points for Hologram of Three Object Points with Coordinates (0,−1), (0,0), and (A,1+B)

Tables Icon

Table II (X,Y) Coordinates of Undesired Reconstructed Points Predicted by Eq. (9) for Holograms of Four Object Points with Coordinates (1,1), (−1,1+A), (−1−B,−1−G), and (1,−1)

Equations (23)

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T A = ( E C ) γ / 2 ( 1 P + Q F + W + ) ,
P = 1 2 γ C ( a + b + d ) , Q = 1 2 γ ( 1 2 γ + 1 ) C 2 ( 1 / 2 ! ) [ a 2 + b 2 + d 2 + 2 ( a b + a d + d b ) ] , F = 1 2 γ ( 1 2 γ + 1 ) ( 1 2 γ + 2 ) C 3 ( 1 / 3 ! ) { a 3 + b 3 + d 3 + 3 [ a 2 ( b + d ) + b 2 ( a + d ) + d 2 ( a + b ) ] + 6 a b d } , W = 1 2 γ ( 1 2 γ + 1 ) ( 1 2 γ + 2 ) ( 1 2 γ + 3 ) C 4 ( 1 / 4 ! ) × ( { a 4 + b 4 + d 4 + 4 [ a 3 ( b + d ) + b 3 ( a + d ) + d 3 ( a + b ) ] + 6 ( a 2 b 2 + a 2 d 2 + b 2 d 2 ) + ( 12 a 2 b d + b 2 a d + d 2 a b ) } + ) ,
C = [ E o + t ( I R + n = 1 N A n 2 ) ] 1 , a = t R * n = 1 N A n , b = t R n = 1 N A n * , d = t i = 1 N j = 1 j i N A i A j * ,
R exp i { k [ ( X R X ) 2 + ( Y R Y ) 2 + Z R 2 ] 1 2 + ϕ }
A n exp i { k [ ( X n X ) 2 + ( Y n Y ) 2 + Z n 2 ] 1 2 + ϕ n } ,
R exp i k [ Z R + ( X R X ) 2 + ( Y R Y ) 2 2 Z R ]
A n exp i k [ Z n + ( X n X ) 2 + ( Y n Y ) 2 2 Z n ] .
2 a d = 2 t 2 R * n = 1 N i = 1 N j = 1 j i j n N A n A i A j * = 2 t 2 R * n = 1 N i = 1 N j = 1 j i j n N A n A i A j exp i k { Z n + Z i Z j + [ ( X n X ) 2 + ( Y n Y ) 2 ] 2 Z n + [ ( X i X ) 2 + ( Y i Y ) 2 ] 2 Z i [ ( X j X ) 2 + ( Y j Y ) 2 ] 2 Z j } .
Z n + Z i Z j + [ ( X n X ) 2 + ( Y n Y ) 2 ] 2 Z n + [ ( X i X ) 2 + ( Y i Y ) 2 ] 2 Z i [ ( X j X ) 2 + ( Y j Y ) 2 ] 2 Z j = const + [ ( X X ) 2 + ( Y Y ) 2 ] 2 Z .
( Y n Y ) 2 2 Z n + ( Y i Y ) 2 2 Z i ( Y j Y ) 2 2 Y j = ( Y Y ) 2 2 Z .
Y n Y Z n + Y i Y Z i Y j Y Z j = Y Y Z .
1 Z n + 1 Z i 1 Z j = 1 Z
Y n Z n + Y i Z i Y j Z j = Y Z ,
Y n + Y i Y j = Y ,
a 2 b = t 3 I R R * n = 1 N i = 1 N j = 1 N A n A i A j *
d 2 a = t 3 R * i = 1 N j = 1 j i N u = 1 N υ = 1 υ u N n = 1 n A i A j * A u A υ * A n .
1 Z i 1 Z j + 1 Z u 1 Z υ + 1 Z n = 1 Z
Y i Z i Y j Z j + Y u Z u Y υ Z υ + Y n Z n = Y Z ,
Y i Y j + Y u Y υ + Y n = Y ,
1 Z i 1 Z j + 1 Z u 1 Z υ + 1 Z s 1 Z t + 1 Z n = 1 Z
Y i Z i Y j Z j + Y u Z u Y υ Z υ + Y s Z s Y t Z t + Y n Z n = Y Z ,
Y i Y j + Y u Y υ + Y s Y t + Y n = Y ,
C = [ E o + n = 1 N t n ( I R n + I n ) ] 1 , a = n = 1 N t n R n A n * , b = n = 1 N t n R n * A n , d = 0 ,

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