Reflections of linearly polarized electromagnetic waves from a multilayer periodic mirror

Objectives. The purpose of the article is to carry out a theoretical and experimental study of the angular reflection spectrum of linearly polarized electromagnetic waves from a multilayer periodic mirror on a transparent substrate to exact analytical expressions for reflection and transmission coefficients generalizing the cases of incidence of plane transverse electric (TE) and transverse magnetic (TM) modes on limited periodically structured media with a stepped refractive index profile.

Methods. The theoretical analysis of the reflection problem is based on the search for exact analytical solutions in the form of Floquet–Bloch waves presented in the form of inhomogeneous waves in the domain of periodically structured media. On the basis of the possible existence of a single Floquet–Bloch wave in a limited onedimensional photonic crystal, it is proposed to search for exact solutions of the wave equation in the form of a linear combination of inhomogeneous waves propagating in different directions. By using the canonical forms of the considered periodic structures, it is possible to carry out the simple transition from the case of TE polarization to TM type in dispersion relations and expressions for the angular reflection spectrum.

Results. Cases of reflection of linearly polarized radiation are considered for the following cases: a flat boundary of two dielectrics, a thin plane-parallel plate, and a multilayer dielectric mirror. Exact analytical expressions for the reflection and transmission coefficients generalizing the cases of incidence of TE and TM polarizations waves on a limited one-dimensional photonic crystal are obtained. The transmission coefficients of a plane TE wave from a multilayer dielectric mirror sputtered on thin glass were experimentally measured.

Conclusions. A quantitative and qualitative agreement of experimental measurements of the transmission coefficient of a plane wave incident from a half-space on a confined photonic crystal with theoretical calculations is obtained. The obtained expressions for the transmission coefficient of a confined one-dimensional photonic crystal, which are shown to be determined by the interference of Floquet–Bloch waves presented in the form of inhomogeneous waves, can be reduced to a form analogous to the expression for the value of the transmission coefficient of a traditional Fabry–Pérot interferometer. In the case of TM polarization, when the Brewster condition is fulfilled at the interlayer boundaries, the Floquet–Bloch wave has the form of homogeneous plane waves in the layers of a photonic crystal.

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