A Practical Guide to Digital Holography and Generalized Sampling

The theorems of Nyquist, Shannon and Whittaker have long held true for sampling optical signals. They showed that a signal (with finite bandwidth) should be sampled at a rate at least as fast as twice the maximum spatial frequency of the signal. They proceeded to show how the continuous signal could be reconstructed perfectly from its well sampled counterpart by convolving a Sinc function with the sampled signal. Recent years have seen the emergence of a new generalized sampling theorem of which Nyquist Shannon is a special case. This new theorem suggests that it is possible to sample and reconstruct certain signals at rates much slower than those predicted by Nyquist-Shannon. One application in which this new theorem is of considerable interest is Fresnel Holography. A number of papers have recently suggested that the sampling rate for the digital recording of Fresnel holograms can be relaxed considerably. This may allow the positioning of the object closer to the camera allowing for a greater numerical aperture and thus an improved range of 3D perspective. In this paper we: (i) Review generalized sampling for Fresnel propagated signals, (ii) Investigate the effect of the twin image, always present in recording, on the generalized sampling theorem and (iii) Discuss the effect of finite pixel size for the first time