Nanoscopy of surface polarization with oblique dipole orientations
The boundary conditions imposed by confined dipoles with arbitrary orientation on surfaces are presented, extending the conventional in-plane (IP) and out-of-plane (OOP) treatments, here applied for planar and cylindrical sheets. Examples include van der Waals heterostructures, thin films of molecular aggregates, and metal-dielectric interfaces. For large dipole strengths, the reflectance peak associated with the dipole oscillation frequency splits into two, revealing the presence of oblique dipoles. The loss function for the dipole sheet reveals pairs of polaritonic resonances originating from the IP and OOP dipole components, accessible through near-field probes. The point-dipole model for s-SNOM shows two distinct peaks, revealing higher sensitivity to dipole obliqueness than reflectance experiments. We apply the model to monolayer WSe$_2$, showing that the oblique dipole formulation with a dipole angle of $5.7^\circ$ yields significant improvements in the qualitative and quantitative agreement with an experiment reported in the literature. This work proposes a unified language for the description of two-dimensional materials, thin films, and interfaces with anisotropic dipolar responses and shows that near-field methods, sensitive to high in-plane momenta, are suitable for measuring such oblique dipoles.