We will demonstrate a Runge Kutta based algorithm for solving the Mone-Ampère-Problem. The method is utilizing its parabolic type. We will show the exact numerical solving method down to each step. The derived solution is used in the context of laser beam shaping to reshape any given input distribution to arbitrary output distributions free of any constraints other than diffraction, which includes a true black in said distribution. The presented algorithm will give the required phase which will be used to verify the solution theoretically via diffraction integral and practically via Spatial Light Modulator (SLM).
Alexander Kratsch, Markus Olbrich, Steffen Weißmantel, and Alexander Horn, "Solving the logarithmic Monge-Ampère equation with a RK4-algorithm for beam shaping purposes of femtosecond laser beams with spatial light modulators," Proc. SPIE 10518, Laser Resonators, Microresonators, and Beam Control XX, 105181L (Presented at SPIE LASE: February 01, 2018; Published: 16 February 2018); https://doi.org/10.1117/12.2287454.
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