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sampling operator or other phase predistortions. Note that the process of development and application of novel iterative algorithms for calculating DOEs has not yet been finished, and the reader is encouraged to take part in it.

I.2.1 Peculiarities of Calculating Various Types of DOE

No algorithms are available to calculate all the possible types of DOE. For calculating each particular type of DOE one can select an optimal method. Phase DOEs can be conditionally divided into the following types:

1. DOEs generating half-tone images—digital holograms [8,11,39], kinoforms [2,16,19,25,26]

2. DOEs focusing laser radiation into a small spatial area—focusators [33–38]

3. DOEs producing binary images—diffraction gratings [40–42], matrix illuminators [43–46]

4. DOEs forming axial images—multifocus lenses [47,48], axicons [49,50]

5. DOEs generating wavefronts—correctors [51], compensators [52,53]

6. DOEs producing light modes—Bessel modes or nondiffracting beams [54,55], Gauss-Hermite and Gauss-Laguerre modes [56].

Let us take a brief look at the features of the above types of DOE.

1. For digital holograms intended for the generation of three-dimensional (3D) or 2D half-tone images one needs to calculate transmission functions as a matrix characterized by the large number of pixels M×M [57]. For example, the fabrication of a visually perceptible hologram 100×100 mm in size with a resolution of 1 µm calls for the calculation of a pixel matrix with M=100000. In that case, the procedure of interpolation or extrapolation of a hologram datum array has proved successful [58]. In a simple situation, the extrapolation reduces to the replication of a calculated basic array of size N×N, . Note that in this case the hologram features the raster structure of phase, with the image consisting of a regular set of light spots (speckles).

Procedures of phase encoding [8], which can also perform an interpolation operation, because each pixel of the hologram is represented (encoded) as a rectangular sector of an amplitude [10] or phase [13] raster, have also found wide use in the calculation of digital holograms. It is noteworthy that iterative methods have also received wide acceptance in the computation of holograms and kinoforms [2,4,58].

2. Focusators are DOEs focusing laser light into a line with a pregiven energy distribution within this area [33–36]. Such optical elements have found application in laser technology for cutting, welding, branding, and hardening