Relationship and comparison between the Monge–Ampère equation and the transport of intensity equation for phase retrieval

Phase retrieval from two or more irradiance (or intensity) distributions is a fundamental and important inverse problem in optics. One of the most widely used phase retrieval methods is based on the transport of intensity equation (TIE). Although simple in form, the TIE method suffers the problems of weak paraxial and weak defocus approximations, etc. The phase retrieval problem can be more generally formulated as a Monge–Ampère equation (MAE) in the framework of geometrical optics. Here, we first prove that the TIE can be derived from MAE when applying paraxial and weak defocus approximations. We then verify through numerical experiments that MAE could sufficiently reduce the phase recovery errors compared with TIE.