Hybrid design of diffractive optical elements for optical beam shaping

Hybrid methods combining the geometrical-optics and diffraction-theory methods enable designing diffractive optical elements (DOEs) with high performance due to the suppression of stray light and speckles and, at the same time, with a regular and fabrication-friendly microrelief. Here, we propose a geometrical-optics method for calculating the eikonal function of the light field providing the generation of a required irradiance distribution. In the method, the problem of calculating the eikonal function is formulated in a semi-discrete form as a problem of maximizing a concave function. For solving the maximization problem, a gradient method is used, with analytical expressions obtained for the gradient. In contrast to geometrical-optics approaches based on solving the Monge–Ampére equation using finite difference methods, the proposed method enables generating irradiance distributions defined on disconnected regions with non-smooth boundaries. As an example, we calculate an eikonal function, which provides the generation of a "discontinuous" irradiance distribution in the form of a hexagram. It is shown that the utilization of the hybrid approach, in which the obtained geometrical-optics solution is used as a starting point in iterative Fourier transform algorithms, enables designing DOEs with a quasi-regular or piecewise-smooth microrelief structure. The calculation results are confirmed by the results of experimental investigations of a DOE generating a hexagram-shaped irradiance distribution.