The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field

Under consideration is the stationary system of equations of electrodynamics relating to a nonmagnetic nonconducting medium. We study the problem of recovering the permittivity coefficient $\varepsilon$ from given vectors of electric or magnetic intensities of the electromagnetic field. It is assumed that the field is generated by a point impulsive dipole located at some point $y$. It is also assumed that the permittivity differs from a given constant $\varepsilon_0$ only inside some compact domain $\Omega\subset\mathbb R^3$ with smooth boundary $S$. To recover $\varepsilon$ inside $\Omega$, we use the information on a solution to the corresponding direct problem for the system of equations of electrodynamics on the whole boundary of $\Omega$ for all frequencies from some fixed frequency $\omega_0$ on and for all $y\in S$. The asymptotics of a solution to the direct problem for large frequencies is studied and it is demonstrated that this information allows us to reduce the initial problem to the well-known inverse kinematic problem of recovering the refraction index inside $\Omega$ with given travel times of electromagnetic waves between two arbitrary points on the boundary of $\Omega$. This allows us to state uniqueness theorem for solutions to the problem in question and opens up a way of its constructive solution.