Analytical framework for complete pupil function retrieval from single diffraction patterns

We report a framework for complete pupil function retrieval of optical systems (phase and amplitude) based on the analysis of a single intensity pattern arising from point-source imaging. The approach implements unconstrained fits over sets of linearly independent, real-valued, phase and amplitude diffraction modal functions (DMF), defined in the image space of the system. DMFs are derived from explicit expressions involving the optical parameters of the system, and from the base(s) one chooses to represent the amplitude and/or phase pupil maps. Naturally, such bases can be the Zernike basis, although this is not a limitation. Beyond the capability to efficiently retrieve complete pupil functions from single inputs, the approach features other assets by construction, among which: i) it can readily recurse to advanced/rigorous diffraction integrals; ii) it can cope with incomplete datasets, and iii) it allows the evaluation of metrics at any step of the retrieval process, which can be used to assess its physical consistency. This paper gives both theoretical foundations and practical implementation guidelines for the method. Its accuracy and robustness to noise are thoroughly investigated in a series of numerical examples, as preparatory work for experimental applications.