A phaseless inverse problem for electrodynamic equations in the dispersible medium

ABSTRACT

For electrodynamic equations related to non-conducting dispersible medium we consider the inverse problem of recovering two variable coefficients from a given phaseless information of solutions to the equations. One of these coefficients is the permittivity while the second one characterizes the time dispersion of the medium. We suppose that unknown coefficients differ from given constants inside of a compact domain Ω. A plane electromagnetic wave going in the direction ν from infinity fall down on this domain and modulus of the electric strength is measured on a part of the boundary of Ω for all $\nu \in {\mathbb{S}}^2$. The inverse problem consists in determining unknown functions from this information. We reduce the inverse problem to two problems: (1) the inverse kinematic problem for recovering the refractive index and (2) the integral geometry problem for recovering the second coefficient related to the dispersion. An uniqueness theorem for the first problem is stated on the base of known results. The second problem differs from have studied by the more general weight function and it is still open. Then we demonstrate that under some natural assumption the weight function uniformly close to 1. Replacing the weight function by 1, we obtain the integral geometry problem for which the uniqueness theorem and stability estimate are established and some numerical algorithms are proposed.

Keywords:

• Electrodynamic equations
• dispersion
• phaseless inverse problem
• inverse kinematic problem
• integral geometry problem

2010 Mathematics Subject Classification:

• 35R30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by Mathematical Center in Academgorodok, Novosibirsk, at Sobolev Institute of Mathematics, Novosibirsk, Russia. The agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613.