Principles of reflection and transmission for two-dimensional polaritons

Understanding polariton behavior at the boundary of two-dimensional (2D) materials is a critical yet unresolved issue. Here, we provide an analytical description of the reflection and transmission coefficients of polaritons under oblique incidence, establishing a Fresnel formula tailored to polaritons. We demonstrate that boundary-scattered fields, whether evanescent or radiative, couple with polaritons, inducing phase shifts in reflected and transmitted polaritons and changing the momentum of polaritons in a direction perpendicular to the boundary by an imaginary amount. These polariton dynamics at the boundary lead to various features, including a Brewster-like effect, the Goos-H"anchen effect without total internal reflection, and the cross-polarization coupling of polaritons. Our analytical understanding of polariton dynamics could be applied to the development of future polaritonic devices utilizing 2D materials.