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.