Integrable solution for light shaping based on a Fourier-pair mapping

In far-field light shaping, one of the design methods is based on a one-to-one map between the irradiance of the source and target. However, an integrability issue may occur in this kind of algorithms, either in the ray mapping method for designing a freeform surface or in those geometric-optics-based methods for achieving a required output phase. We introduce another mapping-type algorithm to tackle the integrability problem, which instead of establishing a mapping between both the source and target irradiance in the space domain, the mapping is assumed on electric fields of a Fourier pair between the space domain and the spatial-frequency domain. By solving the mapping from the Fourier pair, the gradient of the output phase is achieved, that the gradient is equivalent to the obtained mapping function. Moreover, the existence and the characterization of the mapping guarantees the integrability of the gradient so that a smooth output phase can be directly integrated. Based on the obtained smooth output phase, a freeform surface can then be designed for the light-shaping task. Numerical examples are demonstrated for the comparison of the approaches with different mapping assumptions.