The phase retrieval problem: a spectral parameter power series approach

Abstract

This paper presents the application of the spectral parameter power series method [Pauli, Math Method Appl Sci 33:459–468 (2010)] for constructing the Green’s function for the elliptic operator $$-\nabla \cdot I\nabla $$ in a rectangular domain $$\varOmega \subset \mathbb R ^{2}$$ , where $$I$$ admits separation of variables. This operator appears in the transport-of-intensity equation (TLE) for undulatory phenomena, which relates the phase of a coherent wave with the axial derivative of its intensity in the Fresnel regime. We present a method for solving the TIE with Dirichlet boundary conditions. In particular, we discuss the case of an inhomogeneous boundary condition, a problem that has not been addressed specifically in other works, under the restricted assumption that the intensity $$I$$ admits separation of variables. Several simulations show the validity of the method.