Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a Monge-Ampere type equation

摘要

Beam shaping, in other words, the control of both intensity distribution and phase profile, has a wide range of applications. In this paper, double freeform surfaces are utilized to shape collimated beams, realizing an arbitrary output wavefront with desired illumination pattern. Freeform surfaces are designed by solving a second order partial differential equation (PDE) of the Monge-AmpSre (MA) type, without the limitation of symmetry or paraxial approximation. The mathematical derivation of the PDE is based on the Snell's law, the energy conservation law along infinitesimal tubes of rays and the constancy of the OPL. The PDE is discretized with a finite difference scheme into a system of nonlinear equations, which can be numerically solved by Newton's method. Since Newton's method requires a good initialization for the iteration, a simultaneously point-by-point method, based on ray mapping, is employed to find the initial iterate. Different design examples are given to demonstrate the effectiveness and wide application of our method, transforming a collimated Gaussian beam into a spherical wavefront with uniform illumination patterns. Variable-sized uniform illumination pattern can be obtained by moving the observation plane due to a potential benefit of the spherical output wavefront.