Fresnel Multimodal Scattering Coefficients for Hyperbolic Phonon Polaritons in Uniaxial Slabs at Material Interfaces

Fresnel scattering at a sharp interface is a fundamental process in optics, yet the scattering of hyperbolic phonon polaritons presents a unique challenge because of their inherently multimodal nature. Finite slabs of hyperbolic polar crystals support a dense spectrum of polariton modes whose coherent superposition reconstructs ray-like propagation, enabling multimodal energy transport in deeply subwavelength structures. When the substrate changes abruptly, such as at a dielectric-to-metal transition beneath the slab, the regions on either side support different eigenmode sets, which couple at the boundary. The intermodal coupling accumulated at each discontinuity shapes the overall polaritonic response, but predictive models that capture this cumulative process remain limited. Here, we introduce Fresnel multimodal scattering coefficient matrices at substrate boundaries of uniaxial hyperbolic slabs whose optic axis is normal to the slab. Using coupled-mode theory, we nonperturbatively derive mode-resolved reflection and transmission matrices that explicitly retain the radiation continuum and satisfy reciprocity, without any fitting to full-wave simulations. These interface coefficients serve as fundamental building blocks whose systematic composition efficiently yields the response of complex substrate-defined heterostructures. The theory also remains accurate for lossy media, as its weak-loss extension agrees with full-wave simulations to within a few percent for realistic material parameters. The framework accommodates large modal bases at a computational cost that can be substantially lower than that of full-wave simulations, providing a practical tool for the analysis and design of multimodal polaritonic nanodevices.