Holograms generated by means of a computer can be used to produce wavefronts with any prescribed amplitude and phase distribution; they are therefore extremely useful in applications such as laser-beam scanning and optical spatial-filtering (see sections 13.3 and 14.2) as well as for testing optical surfaces.

The production of holograms using a digital computer has been discussed in detail by Lee [1978], Yaroslavskii and Merzlyakov [1980], and Dallas [1980], and involves two principal steps.

The first step is to calculate the complex amplitude of the object wave at the hologram plane; for convenience this is usually taken to be the Fourier transform of the complex amplitude in the object plane. It can be shown, by means of the sampling theorem, that if the object wave is sampled at a sufficiently large number of points (see Appendix 2), this can be done with no loss of information. Thus, if an image consisting of N × N resolvable elements is to be reconstructed, the object wave is sampled at N × N equally spaced points, and the N × N complex coefficients of its discrete Fourier transform are evaluated. This can be done quite easily with a computer program using the fast Fourier transform algorithm [Cochran et al., 1967] for arrays containing as many as 1024 × 1024 points.

The second step involves using the computed values of the discrete Fourier transform to produce a transparency (the hologram), which reconstructs the object wave when it is suitably illuminated.