Abstract

The copying of holograms by constructing Gabor type holograms from the originals is discussed mathematically. The analysis is based on plane wave expansions, and the paraxial approximation is not used. Two sets of double images of the original scene appear in the reconstruction process with the copy hologram when emulsion thickness is not important. The separation of the double images is equal to twice the separation of the two photographic emulsions during the copying process. The importance of the extra images is discussed and ways to eliminate them are described. Double images are not produced by copies of wide angle holograms with interference fringe spacing that is small compared to the emulsion thickness.

© 1967 Optical Society of America

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References

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  1. G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 193 (1952).
  2. F. S. Harris, G. C. Sherman, B. H. Billings, Appl. Opt. 5, 665 (1966).
    [CrossRef] [PubMed]
  3. F. B. Rotz, A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
    [CrossRef]
  4. H. W. Rose, J. Opt. Soc. Am. 56, 542 (1966).
  5. M. J. Landry, Appl. Phys. Letters 9, 303 (1966).
    [CrossRef]
  6. D. B. Brumm, Appl. Opt. 5, 1946 (1966).
    [CrossRef] [PubMed]
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Co., Inc., New York, 1941), p. 363.
  8. G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966).
  9. Ref. 8, p. 107.
  10. G. C. Sherman, J. Opt. Soc. Am. 57, 1160 (1967).
    [CrossRef]
  11. A. A. Friesem, Appl. Phys. Letters 7, 102 (1965).
    [CrossRef]
  12. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]
  13. H. H. Emsley, Aberrations of Thin Lenses (Constable and Co. Ltd., London, 1956), Sec. 2. 1.
  14. D. B. Brumm, Appl. Opt. 6, 588 (1967).
    [CrossRef] [PubMed]

1967 (2)

1966 (5)

F. S. Harris, G. C. Sherman, B. H. Billings, Appl. Opt. 5, 665 (1966).
[CrossRef] [PubMed]

F. B. Rotz, A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
[CrossRef]

H. W. Rose, J. Opt. Soc. Am. 56, 542 (1966).

M. J. Landry, Appl. Phys. Letters 9, 303 (1966).
[CrossRef]

D. B. Brumm, Appl. Opt. 5, 1946 (1966).
[CrossRef] [PubMed]

1965 (2)

A. A. Friesem, Appl. Phys. Letters 7, 102 (1965).
[CrossRef]

R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
[CrossRef]

1952 (1)

G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 193 (1952).

Billings, B. H.

Brumm, D. B.

Emsley, H. H.

H. H. Emsley, Aberrations of Thin Lenses (Constable and Co. Ltd., London, 1956), Sec. 2. 1.

Friesem, A. A.

F. B. Rotz, A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
[CrossRef]

A. A. Friesem, Appl. Phys. Letters 7, 102 (1965).
[CrossRef]

Harris, F. S.

Landry, M. J.

M. J. Landry, Appl. Phys. Letters 9, 303 (1966).
[CrossRef]

Meier, R. W.

Rogers, G. L.

G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 193 (1952).

Rose, H. W.

H. W. Rose, J. Opt. Soc. Am. 56, 542 (1966).

Rotz, F. B.

F. B. Rotz, A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
[CrossRef]

Sherman, G. C.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Co., Inc., New York, 1941), p. 363.

Stroke, G. W.

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966).

Appl. Opt. (3)

Appl. Phys. Letters (3)

A. A. Friesem, Appl. Phys. Letters 7, 102 (1965).
[CrossRef]

F. B. Rotz, A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
[CrossRef]

M. J. Landry, Appl. Phys. Letters 9, 303 (1966).
[CrossRef]

J. Opt. Soc. Am. (3)

Proc. Roy. Soc. Edinburgh (1)

G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 193 (1952).

Other (4)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Co., Inc., New York, 1941), p. 363.

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966).

Ref. 8, p. 107.

H. H. Emsley, Aberrations of Thin Lenses (Constable and Co. Ltd., London, 1956), Sec. 2. 1.

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Figures (5)

Fig. 1
Fig. 1

Copying process.

Fig. 2
Fig. 2

Construction of the original hologram.

Fig. 3
Fig. 3

Waveforms represented by f0(x,y,z), f0′(x,y,z), and f(x,y,z).

Fig. 4
Fig. 4

Reconstruction from the copy hologram for the case when z2 < 2z1z0.

Fig. 5
Fig. 5

An illustration of a process that produces the same virtual image as is obtained from a copy hologram that was constructed with light differing in wavelength from that used to construct the original. The object at z = z0 is viewed through a transparent plate with a relative index of refraction n = k′/k.

Equations (28)

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f 0 ( x , y , z 1 ) 2 1.
f ( x , y , z ) = - A ( p , q ) exp ( 2 π i { p x + q y ) + ( 1 / 2 π ) [ k 2 - ( 2 π p ) 2 - ( 2 π q ) 2 ] ½ z } ) d p d q ,
B ( p , q ) = - b ( x , y ) exp [ - 2 π i ( p x + q y ) ] d x d y ,
F [ B ( p , q ) ] = b ( x , y ) = - B ( p , q ) × exp [ + 2 π i ( p x + q y ) ] d p d q .
f ( x , y , z ) = F [ A ( p , q ) G ( p , q , z ) ] ,
G ( p , q , z ) = exp { i [ k 2 - ( 2 π p ) 2 - ( 2 π q ) 2 ] ½ z } .
A ( p , q ) = F ( x , y , 0 ) .
s ( x , y ) = 2 - γ f ( x , y , z 1 ) 2 - γ exp ( - i k z 1 ) f ( x , y , z 1 ) - γ exp ( i k z 1 ) f * ( x , y , z 1 ) ,
s ( x , y ) 2 - γ s r ( x , y ) ,
s r ( x , y ) = s P ( x , y ) + s C ( x , y ) ,
s P ( x , y ) = exp ( - i k z 1 ) f ( x , y , z 1 ) ,
s C ( x , y ) = exp ( i k z 1 ) f * ( x , y , z 1 ) .
f B ( x , y , z ) = 4 exp [ i k ( z - z 2 ) ] ,             for z z 2 ,
f P ( x , y , z ) = γ γ exp [ - i k ( z 2 - z 1 ) ] F [ S r ( p , q ) 0 ( p , q , z - z 1 ) ] ,             for z z 2 ,
f C ( x , y , z ) = γ γ exp [ i k ( z 2 - z 1 ) ] { F [ S r ( p , q ) H ( p , q , z - z 1 , z 2 - z 1 ) ] } * , for z z 2 ,
f P ( x , y , z ) = γ γ exp [ - i k ( z 2 - z 1 ) ] [ f P P ( x , y , z ) + f P C ( x , y , z ) ] ,
f C ( x , y , z ) = γ γ exp [ i k ( z 2 - z 1 ) ] [ f C P ( x , y , z ) + f C C ( x , y , z ) ] ,
f P P ( x , y , z ) = F [ S P ( p , q ) G ( p , q , z - z 1 ) ] ,
f P C ( x , y , z ) = F [ S C ( p , q ) G ( p , q , z - z 1 ) ] ,
f C P ( x , y , z ) = { F [ S P ( p , q ) H ( p , q , z - z 1 , z 2 - z 1 ) ] } * ,
f C C ( x , y , z ) = { F [ S C ( p , q ) H ( p , q , z - z 1 , z 2 - z 1 ) ] } * .
S P ( p , q ) = exp ( - i k z 1 ) A ( p , q ) G ( p , q , z 1 ) ,
S C ( p , q ) = exp ( i k z 1 ) A * ( - p , - q ) G * ( - p , - q , z 1 ) .
f P P ( x , y , z ) = exp ( - i k z 1 ) F [ A ( p , q ) G ( p , q , z ) ] = exp ( - i k z 1 ) f ( x , y , z ) , for z z 2 ,
f P C ( x , y , z ) = exp ( i k z 1 ) F [ A * ( - p , - q ) G * ( - p , - q , 2 z 1 - z ) ] = exp ( i k z 1 ) f * ( x , y , 2 z 1 - z ) , for z z 2 ,
f C P ( x , y , z ) = exp ( i k z 1 ) F [ A * ( - p , - q ) G * ( - p , - q , 2 z 2 - z ) ] = exp ( i k z 1 ) f * ( x , y , 2 z 2 - z ) , for z z 2 ,
f C C ( x , y , z ) = exp ( - i k z 1 ) F [ A ( p , q ) G ( p , q , z + 2 z 1 - 2 z 2 ) ] = exp ( - i k z 1 ) f ( x , y , z + 2 z 1 - 2 z 2 ) , for z z 2 .
f P P ( x , y , z ) = F [ A ( p , q ) G k ( p , q , z - z 2 ) G k ( p , q , z 2 - z 1 ) G k ( p , q , z 1 ) ] ,             for z z 2 ,

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