Gradient-based strategy of designing cascaded phase holograms for arbitrary distribution behind arbitrary aperture

Aperture, flexibly defined as common-seen periphery-blocked structures, or special-shaped patterns, or even with artificial-designed complex amplitude modulating function, are inevitably a core factor of optical system that forces the light path deviate from the unconstrained situation. To bridge the gap between the ideal simplified phase modulation system and the real aperture-constrained system, we propose an inverse design algorithm (GOODBAP) which uses cascaded phase holograms to achieve arbitrary intensity distribution behind arbitrary aperture. The algorithm exploits a parallel optimization procedure based on numerical gradient, and the phase patterns on the two holograms are comprehensively updating using algorithmic differentiation. Multiple examples demonstrate that the GOODBAP algorithm can achieve high-fidelity (above 0.98) and high-diffractive-efficiency (above 0.98) intensity distributions behind seemingly unreasonable apertures. Furthermore, we expand the scope of this algorithm to cascaded modulation with more holograms. Also, an application is demonstrated about deceiving multiple cameras behind separate complex coded apertures. We expect that the GOODBAP algorithm can be applied to more practical scenarios of beam modulation, including laser imaging, illumination, annealing, etc.