Reflection and transmission of light by a half-space with a nonlocal response: A complex variable approach

Abstract

A solution of the electromagnetic reflection and transmission problem by a half-space of a simple material with a nonlocal response to external electric fields is investigated with the aid of several mathematical results of the complex variable theory, the Sokhotski relations, and the Hilbert transform. We found that the problem of reflection and transmission of electromagnetic radiation in the Fourier space is equivalent to an integral equation arising in the Riemann boundary problem. The classical results of Fresnel’s formulas for S polarization are recovered, and the special case of the reflection and transmission of light by a half-space of small spheres in the diluted regime of particle concentration is found to also follow Fresnel’s formula with effective optical properties.