## 2 Theory
(**f**) The complex polarization state of the backscattered signal showing the ‘p’ (blue), ‘s’ (orange), and mixed (green) contributions. Quantifying this distribution is the goal of this paper here. \[\alpha_{z}(\varepsilon,h)=\alpha_{\perp}(\varepsilon,h)=\frac{(1+\beta)\alpha_{tip} }{1-\frac{\beta\alpha_{tip}}{16\pi h^{3}}}\enspace, \tag{3}\]  
\[\alpha_{x}(\varepsilon,h)=\alpha_{y}(\varepsilon,h)=\alpha_{\parallel}( \varepsilon,h)=\frac{(1-\beta)\alpha_{tip}}{1-\frac{\beta\alpha_{tip}}{32\pi h^ {3}}}\enspace, \tag{4}\]  
with \(\beta=\frac{\varepsilon-1}{\varepsilon+1}\) the sample response function, and \(\alpha_{\perp}\) and \(\alpha_{\parallel}\) referring to the out-of-plane and in-plane total polarizability components with respect to the sample surface, respectively. The calculation becomes more complex but still achievable when considering polarization angles other than \(\theta=0^{\circ},90^{\circ}\) since the electric field then is interacting with both \(\alpha_{\perp}\) and \(\alpha_{\parallel}\) at the same time. Such a scenario is illustrated in Figure 1c, center. Figure 1d displays the mixing of \(\alpha_{\perp}\) and \(\alpha_{\parallel}\), as calculated with the s-SNOM tip being in firm contact to a fictive sample surface, characterized by \(Im(\varepsilon)=0.2=\text{const}\). for all wavelengths, and the tip-dipole-to-sample distance being \(h=0.785a\). The latter accounts for a shift of the effective tip-dipole position towards the sample upon near-field interaction, as well as for an effective tip-radius of \(a=600\) nm according to literature [28; 35]. Overall, the out-of-plane response (\(\alpha_{\perp}\)) is clearly stronger than the in-plane response (\(\alpha_{\parallel}\)). Nevertheless, for certain incident polarization angles (e.g., \(\theta=75^{\circ}\)), the out-of-plane and in-plane components may reach values that are on the same order of magnitude. Most importantly, both components show resonance peaks at slightly different \(Re(\varepsilon)\); here, \(Re(\varepsilon_{res,\perp})=-3.35\) and \(Re(\varepsilon_{res,\parallel})=-1.72\), with their exact positions depending on the tip-sample distance \(h\). Since \(\varepsilon\) is frequency-dependent, spectrally addressing and separating these peaks into in-plane and out-of-plane sample responses becomes possible. In order to determine the detectability of the local in-plane signals in the far field, the radiation of the point dipole \(\vec{E}_{near}\) is calculated [34] in the backscattering geometry used here (see Figure 1e). Moreover, additional far-field contributions \(\vec{E}_{far}\), e.g., due to direct backscattering, need to be considered, which are superposed to the near-field contribution, resulting in the total scattered field \(\vec{E}_{xz}\) in Jones vector notation, as:  
\[\vec{E}_{sca}=\vec{E}_{far}+\vec{E}_{near}= \begin{pmatrix}p_{far}\\ s_{far}\end{pmatrix} + \begin{pmatrix}p_{near}\\ s_{near}\end{pmatrix} \enspace, \tag{5}\]  
with \(p_{far}\), \(s_{far}\), and \(p_{near}\), \(s_{near}\) describing the p- and s-polarized far- and near-field components, respectively. Note that in this model, we discuss self-homodyne detection by connecting \(\vec{E}_{far}\) to the reflectivity of the sample. Nevertheless, the method is directly applicable to other detection schemes as well by adapting \(\vec{E}_{far}\) accordingly. Investigating the polarization signature of the scattered signal at the detector ultimately requires considering all of the optical elements in the detection path as well. For this purpose, the [PERSON] matrix formalism [33] was applied, representing every element (i.e., mirrors, beam splitters, etc.) in the detection path by a matrix. With that, the polarization signature \(\vec{E}_{det}\) after the detection apparatus for the self-homodyne case is described as:  
\[\vec{E}_{det}=\hat{M}_{mylar}\cdot\hat{M}_{mirror}\cdot\vec{E}_{sca}=\vec{E}_{ far}^{\prime}+\vec{E}_{near}^{\prime}= \begin{pmatrix}p_{far}^{\prime}\\ s_{far}\end{pmatrix} + \begin{pmatrix}p_{near}^{\prime}\\ s_{near}\end{pmatrix}