Abstract

A theory of holographic imaging is formulated in terms familiar from conventional optics. The effects of the curvatures and off-axis angles of the reference and read-out waves are described by equivalent thin lenses and prisms. The formation of the true-image wavefield is found to be completely analogous to the conventional imaging of the object wavefield by the equivalent lenses and prisms. To explain the conjugate image, we introduce the concept of time reversal. The conjugate-image wavefield is the time-reversed object wavefield conventionally imaged by equivalent lenses and prisms (and a plane mirror). The finite size and resolution of the photographic plate are taken into account. The size of the plate determines the effective aperture of the equivalent lenses and prisms, it is equivalent to a diaphragm in the hologram plane. The modulation transfer function of the plate has the same effect as a diaphragm inserted in the imaging bundle during the recording (or the reconstruction) with its center at the reference (read-out) point. The two diaphragms limit the field of view and the resolution.

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  1. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).
  2. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); 53, 1377 (1963); 54, 1295 (1964).
  3. J. A. Armstrong, IBM J. Res. Dev. 9, 171 (1965).
  4. E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).
  5. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
  6. R. W. Meier, J. Opt. Soc. Am. 56, 219 (1966).
  7. That the wavelength ratio between the recording and the reconstructing light in holography is equivalent to the wavelength ratio between the object and the image space in conventional imaging, was first shown by Meier (Ref. 6), by considering both holographic and conventional imaging as projective transformations.
  8. The equivalent lenses of this paper should not be confused with the Fresnel-zone lenses often used to describe holographic imaging [cf. Ref. 1, 2: G. L. Rogers, Nature 166, 237 (1950); Proc. Roy. Soc. (Edinburgh) A63, 193, 313 (1952)]. The Fresnel-zone lenses result from the interaction between the reference wave and the wave issued by an object point. Stroke used the term "equivalent lens" to describe the phase distribution of a spherical wave issued by an object point, which interferes with a plane reference wave [G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966), p. 111]. Our equivalent lenses describe the effect of the curvatures of the reference and the read-out wave, respectively: their properties are independent of the object which they image. This agrees with the basic philosophy of this paper, that the equivalent lenses, prisms, and diaphragms introduced, have exactly the same properties as conventional optical elements.
  9. The position and magnification of the image of the object given by the lens formula agree with the results of Refs. 3-6.
  10. H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).
  11. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 56, 523L (1966).
  12. F. B. Rotz and A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).
  13. H. Frieser, Phot. Korr. 91, 69 (1955); 92, 51, 183 (1956).
  14. L. O. Hendeberg, Arkiv Fysik 16, 417, 457 (1960).
  15. Further references are given by H. F. Gilmore, J. Opt. Soc. Am. 57, 75 (1967).
  16. R. F. van Ligten, J. Opt. Soc. Am. 56, 1, 1009 (1966).
  17. Amplitude transmittance vs exposure curves are given by A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
  18. This can be rigorously proved by describing the propagation of the wavefield bv the Green’s-function method used in the Appendix.
  19. From the analogy between holographic and conventional imaging it is obvious that the evanescent (inhomogeneous) waves behind the object, which correspond to very high spatial frequencies, do not contribute to the image [cf. G. C. Sherman, J. Opt. Soc. Am. 57, 1160 (1967].
  20. A. Sommerfeld, Vorlesungen über Theoretische Physik, Vol. IV, Optik (Dieterich’sche Verlagsbuchhandlung, Wiesbaden, Germany, 1950), p. 202.

Armstrong, J. A.

J. A. Armstrong, IBM J. Res. Dev. 9, 171 (1965).

Friesem, A. A.

F. B. Rotz and A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).

Frieser, H.

H. Frieser, Phot. Korr. 91, 69 (1955); 92, 51, 183 (1956).

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).

Gilmore, H. F.

Further references are given by H. F. Gilmore, J. Opt. Soc. Am. 57, 75 (1967).

Haines, K. A.

E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).

Hendeberg, L. O.

L. O. Hendeberg, Arkiv Fysik 16, 417, 457 (1960).

Kogelnik, H.

H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).

Kozma, A.

Amplitude transmittance vs exposure curves are given by A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).

Leith, E. N.

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 56, 523L (1966).

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); 53, 1377 (1963); 54, 1295 (1964).

E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).

Meier, R. W.

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

R. W. Meier, J. Opt. Soc. Am. 56, 219 (1966).

Rotz, F. B.

F. B. Rotz and A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).

Sommerfeld, A.

A. Sommerfeld, Vorlesungen über Theoretische Physik, Vol. IV, Optik (Dieterich’sche Verlagsbuchhandlung, Wiesbaden, Germany, 1950), p. 202.

Upatnieks, J.

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 56, 523L (1966).

E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); 53, 1377 (1963); 54, 1295 (1964).

van Ligten, R. F.

R. F. van Ligten, J. Opt. Soc. Am. 56, 1, 1009 (1966).

Other

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) B64, 449 (1951).

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); 53, 1377 (1963); 54, 1295 (1964).

J. A. Armstrong, IBM J. Res. Dev. 9, 171 (1965).

E. N. Leith, J. Upatnieks, and K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).

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

R. W. Meier, J. Opt. Soc. Am. 56, 219 (1966).

That the wavelength ratio between the recording and the reconstructing light in holography is equivalent to the wavelength ratio between the object and the image space in conventional imaging, was first shown by Meier (Ref. 6), by considering both holographic and conventional imaging as projective transformations.

The equivalent lenses of this paper should not be confused with the Fresnel-zone lenses often used to describe holographic imaging [cf. Ref. 1, 2: G. L. Rogers, Nature 166, 237 (1950); Proc. Roy. Soc. (Edinburgh) A63, 193, 313 (1952)]. The Fresnel-zone lenses result from the interaction between the reference wave and the wave issued by an object point. Stroke used the term "equivalent lens" to describe the phase distribution of a spherical wave issued by an object point, which interferes with a plane reference wave [G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press Inc., New York, 1966), p. 111]. Our equivalent lenses describe the effect of the curvatures of the reference and the read-out wave, respectively: their properties are independent of the object which they image. This agrees with the basic philosophy of this paper, that the equivalent lenses, prisms, and diaphragms introduced, have exactly the same properties as conventional optical elements.

The position and magnification of the image of the object given by the lens formula agree with the results of Refs. 3-6.

H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 56, 523L (1966).

F. B. Rotz and A. A. Friesem, Appl. Phys. Letters 8, 146 (1966).

H. Frieser, Phot. Korr. 91, 69 (1955); 92, 51, 183 (1956).

L. O. Hendeberg, Arkiv Fysik 16, 417, 457 (1960).

Further references are given by H. F. Gilmore, J. Opt. Soc. Am. 57, 75 (1967).

R. F. van Ligten, J. Opt. Soc. Am. 56, 1, 1009 (1966).

Amplitude transmittance vs exposure curves are given by A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).

This can be rigorously proved by describing the propagation of the wavefield bv the Green’s-function method used in the Appendix.

From the analogy between holographic and conventional imaging it is obvious that the evanescent (inhomogeneous) waves behind the object, which correspond to very high spatial frequencies, do not contribute to the image [cf. G. C. Sherman, J. Opt. Soc. Am. 57, 1160 (1967].

A. Sommerfeld, Vorlesungen über Theoretische Physik, Vol. IV, Optik (Dieterich’sche Verlagsbuchhandlung, Wiesbaden, Germany, 1950), p. 202.

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