## 2 Theory  
Investigating the near-field interaction as a function of polarization requires a theoretical model that supports the calculation of all vector components, i.e., both in- and out-of-plane. The point-dipole model [32] is uniquely able to accomplish this task and, in a first step, is applied here to evaluate the sample's near-field response for the different linear polarization angles of illumination (see Figure 1). Next, the resulting field scattered off the tip-sample junction into the far field is simulated using the radiating dipole model [34], and the light intensity at the detector position is readily computed when accounting for all electric field components in the detection path. Notably, this signal contains both far- and near-field dependence, as will be shown later. Disentangling these components and correlating them to the sample's local permittivity is thus the exact goal of this paper. Figure 1a sketches the tip-sample system and illumination geometry as used for the study here, both theoretically and experimentally. The electric field at the position of the tip \(\vec{E}_{in}\) is described as:  
\[\vec{E}_{in}= \begin{pmatrix}E_{x}\\ E_{y}\\ E_{z}\end{pmatrix} = \begin{pmatrix}1-r_{p}&0&0\\ 0&1+r_{s}&0\\ 0&0&1+r_{p}\end{pmatrix} \cdot \begin{pmatrix}cos(\gamma)cos(\theta)\\ sin(\theta)\\ sin(\gamma)cos(\theta)\end{pmatrix} \cdot E_{0}\enspace, \tag{1}\]  
with \(E_{0}\) the electric field amplitude of the incident light, \(\theta\) the linear polarization angle of the incident beam (\(\theta=0^{\circ}\) corresponds to p-polarization), \(\gamma\) the incident angle relative to the sample normal, and \(r_{s}\) and \(r_{p}\) the polarization-dependent Fresnel reflection coefficients. Note that any excitation polarization (except for strict s-polarization with \(\theta=90^{\circ}\)) results in both an in- and out-of-plane electric field excitation at the tip-sample junction. Figure 1b illustrates such a state for \(\theta=45^{\circ}\) indicating both p-(blue) and s-polarized (orange) components (see also discussion below). To calculate the near-field interaction of \(\vec{E}_{in}\) with an ideal isotropic sample of permittivity \(\varepsilon\), the point-dipole model is applied [32]. We define the tip polarizability assuming a spherical tip of radius \(a\), as \(\alpha_{tip}=4\pi a^{3}\). It follows:  
\[\vec{P}_{tot}\sim\hat{\alpha}_{tot}\vec{E}_{in}= \begin{pmatrix}\alpha_{x}( \varepsilon,h)&0&0\\ 0&\alpha_{y}(\varepsilon,h)&0\\ 0&0&\alpha_{z}(\varepsilon,h)\end{pmatrix} \cdot \begin{pmatrix}E_{x}\\ E_{y}\\ E_{z}\end{pmatrix} \enspace, \tag{2}\]  
with \(\alpha_{i}\) the tensor elements of the tip-sample-coupled polarizability tensor \(\hat{\alpha}_{tot}\) and \(h\) the tip-dipole-to-sample distance. Using Equation (2), we can separate the near-field coupling into perpendicular (left, blue) and parallel (right, orange)-oriented dipole components as depicted in Figure 1c. The corresponding image dipoles hence will be purely perpendicular and parallel-oriented as well. For non-resonant excitation of that spherical tip positioned above an isotropic sample, the different contributions to \(\hat{\alpha}_{tot}\) are given as [32]:  
Figure 1: (**a**) Light polarization geometry of a standard s-SNOM tip under illumination. (**b**) Linear illumination polarization state with polarization angle \(\theta=45^{\circ}\), separated into individual ‘p’ (blue) and ‘s’ (orange) polarized contributions, according to the Jones matrix formalism [33]. (**c**) Interaction between the tip dipole and image dipole according to the point-dipole model for ‘p’, \(45^{\circ}\), and ‘s’ polarized excitation, respectively. (**d**) Polarizability tensor elements as a function of \(\theta\) and Re(\(\varepsilon\)) (with \(Im(\varepsilon)=0.2\), \(\gamma=60^{\circ}\)) highlighting the in-plane and out-of-plane resonances of the near-field coupling at \(Re(\varepsilon_{res,\parallel})=-1.72\) and \(Re(\varepsilon_{res,\perp})=-3.35\), respectively. (**e**) Backscattering geometry setup.