Phaseless inverse problems that use wave interference

We consider the inverse problems for differential equations with complex-valued solutions in which the modulus of a solution to the direct problem on some special sets is a given information in order to determine coefficients of this equation; the phase of this solution is assumed unknown. Earlier, in similar problems the modulus of the part of a solution that corresponds to the field scattered on inhomogeneities in a wide range of frequencies was assumed given. The study of high-frequency asymptotics of this field allows us to extract from this information some geometric characteristics of an unknown coefficient (integrals over straight lines in the problems of recovering the potential and Riemannian distances between the boundary points in the problem of the refraction index recovering). But this is physically much more difficult to measure the modulus of a scattered field than that of the full field. In this connection the question arises how to state inverse problems with the full-field measurements as a useful information. The present article is devoted to the study of this question. We propose to take two plane waves moving in opposite directions as an initiating field and to measure the modulus of a full-field solution relating to interference of the incident waves. We consider also the problems of recovering the potential for the Schrödinger equation and the permittivity coefficient of the Maxwell system of equations corresponding to time-periodic electromagnetic oscillations. For these problems we establish uniqueness theorems for solutions. The problems are reduced to solving some well-known problems.