Phase reconstruction from difference equations: a branch point tolerant method

TLDR: In this article, a solution for the phase of a complex field can be decomposed into a regular, single-valued function determined by the divergence of the phase gradient, as well as a multi-value function determined based on the circulation of phase gradient; this second function is called the hidden phase.

Abstract: Numerous optical engineering applications lead to two two- dimensional difference equations for the phase of a complex field. We will demonstrate that, in general, the solution for the phase can be decomposed into a regular, single-valued function determined by the divergence of the phase gradient, as well as a multi-valued function determined by the circulation of the phase gradient; this second function has been called the 'hidden phase.' The standard least-squares solution to the two-dimensional difference equations will always miss this hidden phase. We will present a solution method that gives both the regular and hidden parts of the phase. Finally, we will demonstrate the method with several examples from both speckle imaging and shearing interferometry.