Abstract

The operator description of Fourier optics is extended and applied to holography. The existing lens models for ideal holographic processes appear as a self-evident intermediate result; generalization to include apertures, recording- material modulation transfer function, and extended source effects is straightforward. The extended source effect is generally shown to be equivalent to a modification of the actual holographic apertures. The final result is a compact expression for the description of the holographically reconstructed field distribution at an arbitrary plane. A useful, comprehensive list of operator relations is given in two appendixes.

© 1981 Optical Society of America

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  1. M. Nazarathy and J. Shamir, "Fourier optics described by operator algebra," J. Opt. Soc. Am. 70, 150–159 (1980).
  2. A. Vander Lugt, "Operational notation for the analysis and synthesis of optical data processing systems," Proc. IEEE 54, 1055–1063 (1966).
  3. J. Shamir, "Cylindrical lens systems described by operator algebra," Appl. Opt. 18, 4195–4202 (1969).
  4. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
  5. R. F. VanLigten, "Influence of photographic film on wavefront reconstruction, I: plane wavefronts," "II: cylindrical wavefronts," J. Opt. Soc. Am. 56, 1–9, 1009–1114 (1966).
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  7. W. T. Cathey, "Effect of reference and illuminating sources size in holography," J. Opt. Soc. Am. 69, 273–277 (1979).
  8. F. P. Carlson and R. E. Francois, Jr., "Generalized linear processors for coherent optical computers," Proc. IEEE 65, 10–17 (1977).
  9. R. E. Francois, Jr., and F. P. Carlson, "Iterative Fourier approach for describing linear multiple plane, coherent optical processors," Appl. Opt. 18, 2775–2782 (1979).
  10. W. T. Cathey, Jr., "Comparison of single-lens and two-lens coherent imaging of complex distributions," J. Opt. Soc. Am. 56, 1015–1017 (1966).
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  12. M. Nazarathy and J. Shamir, "Effect of wavelength variation in Fourier optics and holography described by operator algebra," Isr. J. Technol. (1981).
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. J. Shamir, "Holography," in Optical Transforms, H. Lipson, ed. (Academic, London, 1972).
  15. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).
  16. S. G. Sandoval and J. O. Castaneda, "Quasi-Fourier transform of an object from a Fresnel hologram," Appl. Opt. 18, 950–951 (1979).
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  19. M. Nazarathy and J. Shamir, "Fourier optics without the Kirchhoff integral," presented at the Eleventh Convention of the Institute of Electrical and Electronics Engineers, Israel, Tel-Aviv, 1979.
  20. Equation (15) of Ref. 5, Part I, essentially states in our notation: ui∝ R[-l]θMTFR[l]uo apart from some phase distortion, shift, and scaling. It is evident that for any value of l (except zero), one of the FPO’s is virtual and nonrealizable.
  21. G. Bonnet, "Introduction a l'optique métaxiale," Ann. Telecommun. 33, 143–165 (1978).
  22. K. W. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

1980 (1)

1979 (3)

1978 (1)

G. Bonnet, "Introduction a l'optique métaxiale," Ann. Telecommun. 33, 143–165 (1978).

1973 (1)

1969 (1)

1968 (1)

1966 (3)

Bonnet, G.

G. Bonnet, "Introduction a l'optique métaxiale," Ann. Telecommun. 33, 143–165 (1978).

Carlson, F. P.

R. E. Francois, Jr., and F. P. Carlson, "Iterative Fourier approach for describing linear multiple plane, coherent optical processors," Appl. Opt. 18, 2775–2782 (1979).

F. P. Carlson and R. E. Francois, Jr., "Generalized linear processors for coherent optical computers," Proc. IEEE 65, 10–17 (1977).

Castaneda, J. O.

Cathey, W. T.

W. T. Cathey, "Effect of reference and illuminating sources size in holography," J. Opt. Soc. Am. 69, 273–277 (1979).

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

Cathey, Jr., W. T.

Francois, Jr., R. E.

R. E. Francois, Jr., and F. P. Carlson, "Iterative Fourier approach for describing linear multiple plane, coherent optical processors," Appl. Opt. 18, 2775–2782 (1979).

F. P. Carlson and R. E. Francois, Jr., "Generalized linear processors for coherent optical computers," Proc. IEEE 65, 10–17 (1977).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Lukosz, W.

Mandelkorn, F.

Nazarathy, M.

M. Nazarathy and J. Shamir, "Fourier optics described by operator algebra," J. Opt. Soc. Am. 70, 150–159 (1980).

M. Nazarathy and J. Shamir, "Effect of wavelength variation in Fourier optics and holography described by operator algebra," Isr. J. Technol. (1981).

M. Nazarathy and J. Shamir, "Fourier optics without the Kirchhoff integral," presented at the Eleventh Convention of the Institute of Electrical and Electronics Engineers, Israel, Tel-Aviv, 1979.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Sandoval, S. G.

Shamir, J.

M. Nazarathy and J. Shamir, "Fourier optics described by operator algebra," J. Opt. Soc. Am. 70, 150–159 (1980).

J. Shamir, "Cylindrical lens systems described by operator algebra," Appl. Opt. 18, 4195–4202 (1969).

M. Nazarathy and J. Shamir, "Effect of wavelength variation in Fourier optics and holography described by operator algebra," Isr. J. Technol. (1981).

J. Shamir, "Holography," in Optical Transforms, H. Lipson, ed. (Academic, London, 1972).

M. Nazarathy and J. Shamir, "Fourier optics without the Kirchhoff integral," presented at the Eleventh Convention of the Institute of Electrical and Electronics Engineers, Israel, Tel-Aviv, 1979.

Vander Lugt, A.

A. Vander Lugt, "Operational notation for the analysis and synthesis of optical data processing systems," Proc. IEEE 54, 1055–1063 (1966).

VanLigten, R. F.

Winthrop, J. T.

Wolf, K. W.

K. W. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

Worthington, C. R.

Ann. Telecommun. (1)

G. Bonnet, "Introduction a l'optique métaxiale," Ann. Telecommun. 33, 143–165 (1978).

Appl. Opt. (3)

J. Opt. Soc. Am. (7)

Other (11)

F. P. Carlson and R. E. Francois, Jr., "Generalized linear processors for coherent optical computers," Proc. IEEE 65, 10–17 (1977).

A. Vander Lugt, "Operational notation for the analysis and synthesis of optical data processing systems," Proc. IEEE 54, 1055–1063 (1966).

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

M. Nazarathy and J. Shamir, "Effect of wavelength variation in Fourier optics and holography described by operator algebra," Isr. J. Technol. (1981).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. Shamir, "Holography," in Optical Transforms, H. Lipson, ed. (Academic, London, 1972).

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

M. Nazarathy and J. Shamir, "Fourier optics without the Kirchhoff integral," presented at the Eleventh Convention of the Institute of Electrical and Electronics Engineers, Israel, Tel-Aviv, 1979.

Equation (15) of Ref. 5, Part I, essentially states in our notation: ui∝ R[-l]θMTFR[l]uo apart from some phase distortion, shift, and scaling. It is evident that for any value of l (except zero), one of the FPO’s is virtual and nonrealizable.

K. W. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

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