Nanoscopy of surface polarization with oblique dipole orientations

TL;DR

This work addresses the limitation of conventional IP/OOP dipole descriptions for surface-confined dipoles by introducing a geometry-agnostic polarization-sheet formalism that unifies Felderhof's approach with Feibelman’s SRF. It derives boundary conditions, transfer matrices, and Fresnel coefficients for oblique dipoles and applies the framework to a uniaxial 2D excitonic sheet, revealing polaritonic resonances from both IP and OOP dipole components. The study demonstrates that near-field techniques like s-SNOM can significantly enhance and resolve signatures of dipole orientation, including Fano-like resonances, and provides concrete predictions for reflectance, surface-polariton dispersion, and near-field spectra. Together, these results offer a versatile, interconnected description of anisotropic dipolar responses in 2D materials, thin films, and interfaces, with practical implications for characterizing and engineering vdW heterostructures and plasmonic devices.

Abstract

We present a general electromagnetic description for dipoles confined to surfaces with oblique dipole moment orientations, extending the conventional in-plane (IP) and out-of-plane (OOP) treatments. This description is useful for describing localized polarization in, \textit{e.g.}, van der Waals heterostructures, thin films of molecular aggregates, and metal-dielectric interfaces. The theory is suitable for any material with vanishingly thin thickness relative to the light wavelength, independent of the geometry of the material and the media interfacing it. We apply the formalism to a uniaxial excitonic sheet, covering a large number of two-dimensional (2D) materials and organic thin films. Our theory reveals pairs of polaritonic resonances originating from the IP and OOP components of the excitonic dipole moment. The formalism suggests experimentally accessible signatures of dipole moment orientation, enhanced by near-field probes. This work proposes a unified language for the description of 2D materials, thin films and interfaces with anisotropic dipolar responses.

Figures (4)

• Figure 1: Schematic representation of a surface $S$, representing a 2D sheet of arbitrary geometry, and the optical configurations considered in this work. (a) Oblique and lateral views of the surface, showing the surface polarization $\mathbf P^s$, the outward normal $\hat{\mathbf n}$, and the sheet thickness (much smaller than the light wavelength $\lambda$, thereby allowing the surface treatment). Inset: zoomed-in view of an arbitrary point in the sheet, showing the normal coordinate $r_\perp$ taken along the unit vector $\hat{\mathbf n}$ and the tangential and normal components of a field $\mathbf f$. (b) Same as the inset in panel (a), indicating the susceptibility tensors $\overleftrightarrow{\chi}^\pm$ as defined in the text. (c) Geometry considered for optical response calculations. Plane wave fields impinge on the sheet with a wavevector $\mathbf k$ and incidence angle $\theta_\mathrm{inc}$. (d) Same as panel (c), with the addition of a metallic tip to scatter part of the incident light back to a detector in the far-field region. The tip interacts with the sheet generating a point-dipole $p$, oriented towards the $z$ axis for an elongated tip geometry.
• Figure 2: Reflectance spectrum of excitons confined to a plane, for (a,b) in-plane, (c,d) out-of-plane, and (e) oblique dipole moment orientation. The different curves illustrate the influence of the incidence angle $\theta_\mathrm{inc}$. The insets in panels (b,d,e) show the reflectance of the substrate, which is a sole function of $\theta_\mathrm{inc}$. The oscillator strength is $f_0=2$ nm in panels (a,b,c,d) and $f_0=10$ nm in panel (e) to enhance visualization of the peaks compared to the dip at $\omega=\omega_0$. (f) Visibility, defined as $(|r|^2_\mathrm{peak}-|r|^2_\mathrm{dip})/(|r|^2_\mathrm{peak}+|r|^2_\mathrm{dip})$, as a function of $f_0$ for different incidence angles.
• Figure 3: (a,b) Loss function $\mathop{\mathrm{Im}}\nolimits\{r\}$ close to the excitonic resonance ($\omega=\omega_0$) of a 2D layer of dipoles (a) suspended in air and (b) on top of a thick substrate layer. Red dashed represents the light lines $q=\omega/c$ and $q=\omega\sqrt{\epsilon_\mathrm{subs}}/c$. (c,d,e) Dispersion relation surface modes on a 2D excitonic sheet suspended in air (c) above and (d,e) below the exciton line, for real $\omega$ and complex $q$. Panel (e) shows the modes on a narrower horizontal scale, highlighting the weakly confined modes.
• Figure 4: (a) Modulus and (b) phase of 2nd harmonic demodulated signals of the scattered field, normalized to substrate, for different dipole angles $\theta_{\mathrm{dip}}$. (c,d) Visibility of the left (c) and right (d) peaks of panel (a), defined as $(s_{2,\mathrm{peak}}-s_{2,\mathrm{dip}})/(s_{2,\mathrm{peak}}+s_{2,\mathrm{dip}})$.