Abstract

The holograms considered are formed from opaque or transparent diffracting objects which are contained in an aperture illuminated with a coherent, collimated, quasimonochromatic beam of light. It has been shown that the intensity distribution in the near-field of the aperture, but in the far-field of the individual objects which are contained in the aperture, is given by a function whose essential term represents the interference between the Fraunhofer diffraction pattern from the object and the coherent background. The hologram thus formed is referred to as a Fraunhofer or far-field hologram because of the imposed condition. The reconstruction, which is accomplished by placing the recorded hologram in another coherent collimated quasimonochromatic beam and again going to the far-field of the individual objects, yields an intensity which is essentially the original object distribution. In the far-field region of the individual objects for which this result is valid, the reconstruction is seen to be devoid of the evidence of a virtual image. One advantage of this particular method is that the virtual image which appears in the conventional (Fresnel) hologram method creates no problem here since it reduces to a constant for the far-field approximation.

© 1966 Optical Society of America

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References

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  1. D. Gabor, Proc. Royal Soc. (London) A197, 454 (1949).
    [Crossref]
  2. H. M. A. El-Sum, Ph.D. thesis, Stanford University (1952).
  3. A. U. Baez and H. M. A. El-Sum, X-Ray Microscopy and Microradiography, (Academic Press Inc., New York, 1957).
  4. H. M. A. El-Sum, in Optical Processing of Information, Ed. by D. K. Pollock, C. J. Koester, and J. T. Tippett (Spartan Books, Inc., Baltimore, Md., 1963), p. 85.
  5. H. M. A. El-Sum, J. Opt. Soc. Am. 46, 825 (1956).
    [Crossref]
  6. D. Gabor, Proc. Royal Soc. (London) B64, 449 (1951).
  7. A. Lohmann, Opt. Acta 3, 97 (1956).
    [Crossref]
  8. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1963).
    [Crossref]
  9. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).
    [Crossref]
  10. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  11. A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).
  12. G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
    [Crossref]
  13. G. W. Stroke and D. G. Falconer, in Symposium on Optical and Electro-Optical Information Processing Technology (MIT Press, Cambridge, Mass., 1965).
  14. B. J. Thompson, Soc. Photo-Opt. Instr. Eng. 2, 43 (1964).
    [Crossref]
  15. G. B. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
    [Crossref]
  16. B. J. Thompson, Japan J. Appl. Phys. Suppl. I 4, 302 (1965).
  17. B. A. Silverman, B. J. Thompson, and J. Ward, J. Appl. Met. 3, 792 (1964).
    [Crossref]
  18. G. O. Reynolds, J. Opt. Soc. Am. 55, 595A (1965).
  19. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York and London, 1964), 2nd ed., p. 384.
  20. R. W. Meier, J. Opt. Soc. Am. 55, 595A (1965).
    [Crossref]
  21. J. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
    [Crossref]
  22. G. B. Parrent and G. O. Reynolds, paper given at Soc. Photo-Opt. Instr. Eng. meetingSan Francisco, 26 August 1965.
  23. B. J. Thompson, G. B. Parrent, B. Justh, and J. Ward, Presented to the 237th National Meeting of the American Meteorological Society, Washington, D. C., 19–22 April 1965.

1965 (4)

B. J. Thompson, Japan J. Appl. Phys. Suppl. I 4, 302 (1965).

G. O. Reynolds, J. Opt. Soc. Am. 55, 595A (1965).

R. W. Meier, J. Opt. Soc. Am. 55, 595A (1965).
[Crossref]

J. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

1964 (6)

B. A. Silverman, B. J. Thompson, and J. Ward, J. Appl. Met. 3, 792 (1964).
[Crossref]

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

B. J. Thompson, Soc. Photo-Opt. Instr. Eng. 2, 43 (1964).
[Crossref]

G. B. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

1963 (2)

1956 (2)

1951 (1)

D. Gabor, Proc. Royal Soc. (London) B64, 449 (1951).

1949 (1)

D. Gabor, Proc. Royal Soc. (London) A197, 454 (1949).
[Crossref]

Armstrong, J.

J. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

Baez, A. U.

A. U. Baez and H. M. A. El-Sum, X-Ray Microscopy and Microradiography, (Academic Press Inc., New York, 1957).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York and London, 1964), 2nd ed., p. 384.

El-Sum, H. M. A.

H. M. A. El-Sum, J. Opt. Soc. Am. 46, 825 (1956).
[Crossref]

A. U. Baez and H. M. A. El-Sum, X-Ray Microscopy and Microradiography, (Academic Press Inc., New York, 1957).

H. M. A. El-Sum, in Optical Processing of Information, Ed. by D. K. Pollock, C. J. Koester, and J. T. Tippett (Spartan Books, Inc., Baltimore, Md., 1963), p. 85.

H. M. A. El-Sum, Ph.D. thesis, Stanford University (1952).

Falconer, D. G.

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

G. W. Stroke and D. G. Falconer, in Symposium on Optical and Electro-Optical Information Processing Technology (MIT Press, Cambridge, Mass., 1965).

Gabor, D.

D. Gabor, Proc. Royal Soc. (London) B64, 449 (1951).

D. Gabor, Proc. Royal Soc. (London) A197, 454 (1949).
[Crossref]

Justh, B.

B. J. Thompson, G. B. Parrent, B. Justh, and J. Ward, Presented to the 237th National Meeting of the American Meteorological Society, Washington, D. C., 19–22 April 1965.

Leith, E. N.

Lohmann, A.

A. Lohmann, Opt. Acta 3, 97 (1956).
[Crossref]

Meier, R. W.

R. W. Meier, J. Opt. Soc. Am. 55, 595A (1965).
[Crossref]

Parrent, G. B.

G. B. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

G. B. Parrent and G. O. Reynolds, paper given at Soc. Photo-Opt. Instr. Eng. meetingSan Francisco, 26 August 1965.

B. J. Thompson, G. B. Parrent, B. Justh, and J. Ward, Presented to the 237th National Meeting of the American Meteorological Society, Washington, D. C., 19–22 April 1965.

Reynolds, G. O.

G. O. Reynolds, J. Opt. Soc. Am. 55, 595A (1965).

G. B. Parrent and G. O. Reynolds, paper given at Soc. Photo-Opt. Instr. Eng. meetingSan Francisco, 26 August 1965.

Silverman, B. A.

B. A. Silverman, B. J. Thompson, and J. Ward, J. Appl. Met. 3, 792 (1964).
[Crossref]

Stroke, G. W.

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

G. W. Stroke and D. G. Falconer, in Symposium on Optical and Electro-Optical Information Processing Technology (MIT Press, Cambridge, Mass., 1965).

Thompson, B. J.

B. J. Thompson, Japan J. Appl. Phys. Suppl. I 4, 302 (1965).

B. A. Silverman, B. J. Thompson, and J. Ward, J. Appl. Met. 3, 792 (1964).
[Crossref]

G. B. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

B. J. Thompson, Soc. Photo-Opt. Instr. Eng. 2, 43 (1964).
[Crossref]

B. J. Thompson, G. B. Parrent, B. Justh, and J. Ward, Presented to the 237th National Meeting of the American Meteorological Society, Washington, D. C., 19–22 April 1965.

Upatnieks, J.

Vander Lugt, A.

A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

Ward, J.

B. A. Silverman, B. J. Thompson, and J. Ward, J. Appl. Met. 3, 792 (1964).
[Crossref]

B. J. Thompson, G. B. Parrent, B. Justh, and J. Ward, Presented to the 237th National Meeting of the American Meteorological Society, Washington, D. C., 19–22 April 1965.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York and London, 1964), 2nd ed., p. 384.

IBM J. Res. Develop. (1)

J. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

IEEE Trans. (1)

A. Vander Lugt, IEEE Trans. IT-10, 139 (1964).

J. Appl. Met. (1)

B. A. Silverman, B. J. Thompson, and J. Ward, J. Appl. Met. 3, 792 (1964).
[Crossref]

J. Opt. Soc. Am. (6)

Japan J. Appl. Phys. Suppl. I (1)

B. J. Thompson, Japan J. Appl. Phys. Suppl. I 4, 302 (1965).

Opt. Acta (2)

G. B. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

A. Lohmann, Opt. Acta 3, 97 (1956).
[Crossref]

Phys. Letters (1)

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

Proc. Royal Soc. (London) (2)

D. Gabor, Proc. Royal Soc. (London) B64, 449 (1951).

D. Gabor, Proc. Royal Soc. (London) A197, 454 (1949).
[Crossref]

Soc. Photo-Opt. Instr. Eng. (1)

B. J. Thompson, Soc. Photo-Opt. Instr. Eng. 2, 43 (1964).
[Crossref]

Other (7)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York and London, 1964), 2nd ed., p. 384.

G. B. Parrent and G. O. Reynolds, paper given at Soc. Photo-Opt. Instr. Eng. meetingSan Francisco, 26 August 1965.

B. J. Thompson, G. B. Parrent, B. Justh, and J. Ward, Presented to the 237th National Meeting of the American Meteorological Society, Washington, D. C., 19–22 April 1965.

H. M. A. El-Sum, Ph.D. thesis, Stanford University (1952).

A. U. Baez and H. M. A. El-Sum, X-Ray Microscopy and Microradiography, (Academic Press Inc., New York, 1957).

H. M. A. El-Sum, in Optical Processing of Information, Ed. by D. K. Pollock, C. J. Koester, and J. T. Tippett (Spartan Books, Inc., Baltimore, Md., 1963), p. 85.

G. W. Stroke and D. G. Falconer, in Symposium on Optical and Electro-Optical Information Processing Technology (MIT Press, Cambridge, Mass., 1965).

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Figures (6)

Fig. 1
Fig. 1

Fraunhofer holograms: (a) formation of hologram and (b) reconstruction process.

Fig. 2
Fig. 2

Intensity distribution in reconstructed image of a uniform object for various values of γ.

Fig. 3
Fig. 3

Intensity distribution in the reconstructed image of a uniform object with (——) and without (- - - -) g2; z=a2/λ.

Fig. 4
Fig. 4

Illustration of the reconstruction process: (a) original object, (b) Fraunhofer hologram, and (c) reconstructed image.

Fig. 5
Fig. 5

Illustration of the reconstruction process: (a) original object, (b) hologram, and (c) reconstructed image.

Fig. 6
Fig. 6

Illustration of the reconstruction process: (a) original object, (b) hologram, and (c) reconstructed image.

Equations (17)

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z ( ξ 2 + η 2 ) max / λ ; z ( ξ 2 + η 2 ) max / λ ,
I ( x ¯ ) = U ( x ¯ ) 2 = 1 - k / π z D ˜ ( x ¯ / λ z ) sin ( k x ¯ 2 / 2 z ) + ( k 2 / 4 π 2 z 2 ) [ D ˜ ( x ¯ / λ z ) ] 2 ,
t ( x ¯ ) = 1 + ( γ k / π z ) D ˜ ( x ¯ / λ z ) sin ( k x ¯ 2 / 2 z ) .
U ( ξ ¯ ) = 1 4 π t ( x ¯ ) G n ( ξ ¯ , x ¯ ) d x ¯ ,
2 G + k 2 G = δ ( ξ ¯ - x ¯ ) .
U ( ξ ¯ ) = g 1 + g 2 + g 3 ,
g 1 = - i k 2 π z e i k r d x 1 d x 2 ,
g 2 = - γ k 2 4 π 2 z 2 D ˜ ( x 1 λ z , x 2 λ z ) e i k ( x 1 2 + x 2 2 2 z ) e i k r d x 1 d x 2 ,
g 3 = γ k 2 4 π 2 z 2 D ˜ ( x 1 λ z , x 1 λ z ) e - i k ( x 1 2 + x 2 2 2 z ) e i k r d x 1 d x 2
r = [ z 2 + ( x 1 - ξ 1 ) 2 + ( x 2 - ξ 2 ) 2 ] 1 2 .
g 2 = ( - i γ / 4 ) e i k z e i k [ ( ξ 1 2 + ξ 2 2 ) / 2 z ] F ( ξ ¯ ) ,
F ( ξ ¯ ) = f 1 ( ξ ¯ ) f 2 ( ξ ¯ ) e ( - i π / 2 ) α ¯ 2 d α ¯
f 1 ( ξ ¯ ) = ( ξ ¯ - a / 2 ) / ( λ z ) 1 2 ,             f 2 ( ξ ¯ ) = ( ξ ¯ + a / 2 ) / ( λ z ) 1 2 ,
α ¯ = ( 1 / ( λ z ) 1 2 ) ( ξ ¯ + σ ¯ ) .
g 3 = ( γ / 2 ) e i k z e i k [ ( ξ 1 2 + ξ 2 2 ) / 2 z ] D ( ξ 1 , ξ 2 ) .
I ( ξ ¯ ) = U ( ξ ¯ ) U * ( ξ ¯ ) = 1 + γ D ( ξ 1 , ξ 2 ) cos k [ ( ξ 1 2 + ξ 2 2 ) / 2 z ] + ( γ 2 / 4 ) [ D ( ξ 1 , ξ 2 ) ] 2 .
L 2 l ,