Variational principle in optics II: Dissipative wave equations.

The problem of phase retrieval from intensity measurements is examined for the case of dissipative wave equations. Unlike the conservative case, it is not clear if and when the problem is solvable at all. We provide two solutions. First, it is shown that, for a certain class of dissipating potentials, the problem can be fully solved by converting it through a simple transformation to the framework of the weighted least action principle. Second, for all other dissipating potentials, a deep result from the theory of elliptic partial differential equations is used to show that the problem is always solvable up to a scaling of one of the measured intensities. Moreover, the solution in this general case can be obtained by solving a Monge-Ampere type differential equation. Two numerical examples are given to illustrate some of the theoretical considerations.