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 L<sup>2</sup> 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.
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.
White light-emitting diodes are gradually dominating the illumination markets that new design challenges arise for this emerging source. Based on the white LEDs, an efficient optimization method is presented for integrated reflective optics. During the design process, initial structure of reflective optics is numerically calculated. For further optimization, initial parameters are adjusted by section-modeling method to determine optimal starting point. To complete the design, subsequent spline-modeling method is applied. Design example show that the designed reflective optics for LED illumination could offer both high performance and low space occupancy rate. Comparing to the numerical method, the method offers a 15% uniformity improvement and 6-times rise of processing efficiency. It is believed that the effective optimization method will has practical applications in other integrated optics.