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problems of synthesis in which ‘the source’ should be found from arbitrarily specified ‘data’. The problem of analysis is a straightforward problem in which an image synthesized by a pregiven ‘source’ is investigated.

Formally speaking, the calculation of the DOE phase is reduced to solving an integral equation. Incorrectness of the problem of synthesis lies in the fact that, first, there may not be a solution; second, if the solution exists it may not be unique; and third, the solution may be unstable. Instability of the solution to the inverse problem means that small deviations of the DOE phase from the calculated one may result in considerable deteriorations of the image formed.

As to non-uniqueness of the solution to the problem of DOE synthesis, as distinct from the field of image reconstruction and phase retrieval [5–7] in which non-uniqueness causes additional problems, in our case nonuniqueness proves to be useful and offers an opportunity to choose the best of several solutions.

In problems of image reconstruction, noise from the measurement data from which the object is reconstructed is responsible for incorrectness. In problems of DOE synthesis data noise is absent because image characteristics are specified uniquely. Note, however, that the desired image can be specified in such a manner that no solution to the inverse problem exists, which means that in the context of the formulated problem it appears to be impossible to calculate a phase DOE that would be able to generate the desired image accurately. That is why the procedure of regularization of the problem of DOE synthesis reduces to the replacement of the desired image by a similar one such that the solution exists and is stable.

The inverse problem of DOE synthesis is, generally speaking, nonlinear because it requires solving a nonlinear integral equation. The reason for nonlinearity is that one has to operate separately with the amplitude or with the phase of a complex function rather than with its real or imaginary part. In an early stage of development of digital holography, this nonlinear task was replaced by a linear one in which the amplitude-phase function of a hologram is derived as a result of conversion of linear integral Fresnel or Fourier transforms (see review in [8]). Next, the necessity of reducing the amplitude-phase function to the phase-only or the amplitude-only function required the elaboration of methods for coding: a method for coding binary amplitude Lohmann holograms [9,10], a Lee method [11], a method of parity sequences [12], etc. The Kirk-Jones method [13] of phase carrier represents a generalization of coding methods and allows a purely phase DOE to be calculated from the amplitude-phase function with a 30–40% efficiency.

The procedure of coding an amplitude-phase function via a phase-only function is carried out successively in the course of iteratively solving a nonlinear integral equation. For the first time, an iterative approach to solving the problem of DOE synthesis was suggested by Hirsh, Jordan and Lesem [2]. Gerchberg and Saxton [14] developed independently an analogous algorithm for reconstructing images. This provides the basic solution for the