Laser beam shaping requires controlling the intensity and phase profile of the input laser beam simultaneously. In this paper, a method for designing double freeform surfaces is presented to solve the laser beam shaping problem. Based on Snell’s law and conservation law of energy, a mathematical model is established to convert the double surfaces design problem into an elliptic Monge-Ampère equation with a nonlinear boundary problem by imposing a constraint on the optical path length between the input and output wavefronts. Two different configurations of the beam shaping system are discussed and the good results show clearly the Monge–Ampère equation method provides an effective tool in solving the challenging problem of laser beam shaping.
The Monge–Ampère (MA) equation arising in illumination design is highly nonlinear so that the convergence of the MA method is strongly determined by the initial design. We address the initial design of the MA method in this paper with the L2 Monge-Kantorovich (LMK) theory, and introduce an efficient approach for finding the optimal mapping of the LMK problem. Three examples, including the beam shaping of collimated beam and point light source, are given to illustrate the potential benefits of the LMK theory in the initial design. The results show the MA method converges more stably and faster with the application of the LMK theory in the initial design.
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