## 2 Theory
\enspace, \tag{6}\]  
with \(\hat{M}_{mylar}\)[36] and \(\hat{M}_{mirror}\) being the Jones matrices for the mylar beam splitter and mirror, respectively (see the experimental setup in Figure 2). \(\vec{E}_{far}^{\prime}\) and \(\vec{E}_{near}^{\prime}\) refer to the far- and near-field contributions at the detector. The signal intensity is finally given by:  
\[I=\mid\vec{E}_{det}\mid^{2}=\vec{E}_{far}^{\prime}+2\vec{E}_{far}^{\prime} \vec{E}_{near}^{\prime}+\vec{E}_{near}^{\prime}\enspace. \tag{7}\]  
In order to extract the near-field contribution, typically higher-harmonic demodulation is applied to the detected signal [32]. Therefore, the tip-sample distance \(h\) is modulated at the fundamental cantilever eigenfrequency \(\Omega\), resulting in a modulation of the near field signal at higher harmonics \(n\Omega\) due to the strongly non-linear near-field vs. distance dependence [32]. In general \(|\vec{E}^{{}^{\prime}}_{far}|\gg|\vec{E}^{{}^{\prime}}_{near}|\), but with increasing \(n\), all far-field terms become negligible [32]. Ultimately, it is the mixed term in Equation (7) that is commonly probed by this technique:  
\[I_{n\Omega}\approx 2\vec{E}^{{}^{\prime}}_{far}\vec{E}^{{}^{\prime}}_{near,n \Omega}=2(p^{{}^{\prime}}_{far}p^{{}^{\prime}}_{near,n\Omega}+s^{{}^{\prime}} _{far}s^{{}^{\prime}}_{near,n\Omega})\enspace. \tag{8}\]  
Here, we clearly see that, firstly, p- and s-polarized near-field contributions generally mix with their corresponding far-field component, but secondly, that p-polarization can be clearly separated from s-polarization when using polarizers and analyzers in the overall s-SNOM setup. The experimental confirmation of this finding is discussed below in Section 4.1. The mixing of the far and near field for any polarization is expected to be more complicated. In order to further disentangle the different vectorial near-field components, an analyzer is placed in front of the detector. The analyzer Jones matrix reads as:  
\[\hat{M}_{analyzer}= \begin{pmatrix}cos(\phi)^{2}&cos(\phi)sin(\phi)\\ sin(\phi)cos(\phi)&sin(\phi)^{2}\end{pmatrix} \enspace, \tag{9}\]  
with \(\phi\) denoting the analyzer angle with respect to the p-polarization axis. The optical transmission in the detection path then becomes:  
\[ \begin{split} I_{analyzer}&=|\ \hat{M}_{analyzer}\cdot\vec{E}_{det}\ |^{2}\\ &=cos(\phi)^{2}(p^{{}^{\prime}}_{far}\ {}^{2}+2p^{{}^{\prime}}_{ far}p^{{}^{\prime}}_{near}+p^{{}^{\prime}}_{near}\ {}^{2})\\ &+sin(\phi)^{2}(s^{{}^{\prime}}_{far}\ {}^{2}+2s^{{}^{\prime}}_{ far}s^{{}^{\prime}}_{near}+s^{{}^{\prime}}_{near}\ {}^{2})\\ &+\underbrace{2 cos(\phi)sin(\phi)}_{=sin(2\phi)}(p^{{}^{\prime}}_{ far}s^{{}^{\prime}}_{far}+\underbrace{p^{{}^{\prime}}_{far}s^{{}^{\prime}}_{ near}}+\underbrace{p^{{}^{\prime}}_{near}s^{{}^{\prime}}_{far}}+p^{{}^{ \prime}}_{near}s^{{}^{\prime}}_{near})\enspace.\end{split}  \tag{10}\]  
Here, the total signal intensity has three terms, with the first two terms describing strict p- and s-polarized states. They represent the linear polarized contributions for the trivial cases of \(s_{far}=s_{near}=0\) and \(p_{far}=p_{near}=0\), respectively. However, the third term accounts for the mixing of s- and p-polarized components, which intriguingly includes cross-linked terms of far and near field (see underlined terms in Equation (10)). Moreover, this mixed term is no longer linearly polarized but shows a clover-like intensity distribution with orthogonal maxima at \(\pm 45^{\circ}\) due to its \(2\phi\) periodicity (see again Equation (10)). Hence, for any incident polarization that is not strictly p- or s-polarized, a mixed polarization state can not be avoided. Extraction of these terms, again, is achieved by higher-harmonic demodulation at \(n\Omega\), then reducing Equation (10) to:  
\[