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problem of synthesis of phase DOE, and is referred to below as an algorithm of error reduction or a Gerchberg-Saxton (GS) algorithm.

This algorithm has been elaborated in a series of subsequent works. In papers by Gallagher and Lin [15,16] the proof of the algorithm convergence is given. Chu and Fienup developed a parametric generalization of the GS algorithm and called it an input-output algorithm [17–19]. According to Fienup [20], the algorithm of error reduction is a special case of gradient methods. The presence of a parameter or a step in the iterative algorithm enables one to govern the convergence rate and to combat the stagnation effect typical of the GS algorithm. In papers by Broja, Bryngdahl, Weissbach and Wyrowski, an iterative error-diffusion algorithm for calculating DOEs is developed [21–23] and an iterative procedure of DOE phase quantization is given [24,25]. In papers by Kotlyar, Soifer and coworkers, an adaptive modernization of the GS algorithm is developed [26,27] and applied to computing radial DOEs [28,29] and iteratively to calculating phase formers of Bessel modes [30] and Gauss-Hermite modes [31,32].

An alternative approach to solving the problem of DOE synthesis is a ray-tracing method based on the analytical solution of the eikonal differential equation and on the construction of the ray path from the points on the DOE surface to the points of a desired image. This approach has been developed in papers by Danilov, Golub, Sisakyan, Soifer and coworkers [33–35]. Optical elements calculated via the above method have been fabricated for the IR and visible ranges of the spectrum and have successfully undergone tests [36–38]. Phase functions of DOEs calculated using the ray-tracing method are characterized by a regular zone structure because the eikonal differential equation can be solved analytically in an explicit form or using series.

As compared with coding methods [13] and a ray-tracing approach [34], iterative algorithms for solving integral equations [14] possess a number of advantages:

• a possibility of obtaining an approximate solution with desired accuracy

• a possibility of obtaining several solutions of the same problem by varying an initial estimate in the iterative process

• a combined solution to the inverse and straightforward problems in one algorithm, which means that a current estimate of DOE phase function derived in the inverse branch of the algorithm is checked and improved in the straightforward branch of the iterative algorithm

• universality and adaptability of iterative algorithms—universality is understood as independence from the kind of required image; adaptability is understood as suitability for computer-aided simulation and ease of programming

• a possibility of accounting for additional limitations imposed on the desired function of the DOE phase—introduction of a quantization or