On multiple frequency power density measurements II. The full Maxwell's equations

Abstract

We shall give conditions on the illuminations $\varphi_{i}$ such that the solutions to Maxwell's equations \[ \left\{ \begin{array}{l} {\rm curl} E^{i}=i\omega\mu H^{i}\qquad\text{in }\Omega,\\ {\rm curl} H^{i}=-i(\omega\varepsilon+i\sigma)E^{i}\qquad\text{in }\Omega,\\ E^{i}\times\nu=\varphi_{i}\times\nu\qquad\text{on }\partial\Omega, \end{array}\right. \] satisfy certain non-zero qualitative properties inside the domain $\Omega$, provided that a finite number of frequencies $\omega$ are chosen in a fixed range. The illuminations are explicitly constructed. This theory finds applications in several hybrid imaging problems, where unknown parameters have to be imaged from internal measurements. Some of these examples are discussed. This paper naturally extends a previous work of the author [Inverse Problems 29 (2013) 115007], where the Helmholtz equation was studied.