The two-step method for determining a piecewise-continuous refractive index of a 2D scatterer by near field measurements

Abstract

The problem of determining an unknown piecewise-continuous refractive index of an inhomogeneous scatterer is considered. The boundary value problem is reduced to the Lippmann–Schwinger integral equation. The solution of the inverse problem is obtained in two steps. At the first step, the integral equation of the first kind is solved in the inhomogeneity region using measured values of the total field outside the region. It is proved that the solution to the integral equation of the first kind is unique in the class of piecewise constant functions. At the second step, the unknown refractive index is explicitly calculated via the found solution of the integral equation and the given total field. The proposed method was implemented and verified by solving a test inverse problem with a given refractive index. Efficiency of the two-step method was approved by comparison between the exact solution and the approximate ones.

Keywords:

• Refractive index reconstruction
• integral equations
• solution uniqueness
• inverse problems
• near field data

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

M. Yu. Medvedik http://orcid.org/0000-0003-4066-1818

Yu. G. Smirnov http://orcid.org/0000-0001-9040-628X

A. A. Tsupak http://orcid.org/0000-0002-4462-4697

Additional information

Funding

This work was supported by the Ministry of Education and Science of the Russian Federation [agreement No. 1.894.2017/4.6] and Russian Foundation for Basic Research [grant No. 18-01-00219 A].