Fresnel-like formulas for the reflection and transmission of surface phonon-polaritons at a dielectric interface

Abstract

The reflection and transmission coefficients of a surface phonon-polariton propagating along the surface of a thin film of ${\mathrm{SiO}}_{2}$ and crossing the interface of two dielectric media are analytically determined. Based on the expansion of the electrical and magnetic fields in terms of normal modes, explicit expressions for the reflectivity and transmissivity of the radiation fields generated at the dielectric interface are also obtained. Symmetrical and simple Fresnel-like formulas are derived for nanofilms. For the dielectric interfaces of $\mathrm{air}/{\mathrm{BaF}}_{2}$ and $\mathrm{air}/{\mathrm{Al}}_{2}{\mathrm{O}}_{3}$, it is shown that: (i) The polariton reflectivity (transmissivity) decreases (increases) as the film thickness increases, while its radiation equivalent follows the opposite behavior. (ii) In the polariton and radiation fields, the transmissivity is significantly more sensitive than the reflectivity to the changes on the permittivity mismatch of the dielectric interface. For a 143-nm-thick film, the polariton transmissivity (reflectivity) changes $13.2%$ ($1.9%$), when this mismatch varies by $50%$. (iii) The reflectivity and transmissivity of the radiation fields are smaller than their polariton counterparts, which together account for around $82%$ of the total energy. The proposed formalism accurately fulfills the principle of conservation of energy for describing the reflection and transmission of both the polariton and radiation fields generated at a dielectric interface.