Differentiable design of freeform lenses for partially coherent beam shaping

Partially coherent beams (PCBs), with unique properties such as reduced speckles and self-reconstruction, hold great promise for optical communication, particle trapping, and laser material processing. However, designing optical elements for shaping PCBs remains a significant challenge. We present a differentiable design method of freeform lenses for shaping PCBs. This method employs multi-level B-splines to represent the freeform surface. The key issue of our differential design method lies in the integration with a fast-forward simulation approach. The forward simulation approach models the PCBs' diffraction pattern by efficiently computing the convolution of the coherent diffraction pattern and a kernel determined by the degree of coherence. Specifically, for the beam shaping of a "<i>π</i>" pattern, the computational speed of our method is approximately 22 times that of the conventional modal representation method. Furthermore, comparative analysis reveals that the freeform surface represented by multi-level B-splines achieves a remarkable enhancement in shaping accuracy compared to that represented by Zernike polynomials.