## 4 Results  
### Impact of Illumination Polarization  
As discussed in Section 2, the in-plane near-field response is strongly connected to resonant excitation at certain values of the sample permittivity. The correlation between in- and out-of-plane near-field signals can therefore be directly connected to the sample's local permittivity. In the following, we thus compare the near-field response of a gold sample at 10.6 um (\(Re(\varepsilon)=-5146.8\)[37]), as a metallic non-resonant reference sample, to 3C-SiC excited resonantly at 10.6 um (\(Re(\varepsilon)=-1.40\)[38]) and 10.7 um (\(Re(\varepsilon)=-1.94\)[38]). Note that here, gold as a spectrally flat, non-resonant sample has been chosen as its metallic character suppresses any in-plane electric field components at its surface, whereas, on the other hand, the undoped, isotropic 3C-SiC shows a phononic-based near-field resonance at CO\({}_{2}\)-laser wavelengths that supports local near-field signals at all polarization angles (see Section 2 and Figure 1). In order to first investigate the role of in-plane excitation experimentally, the illumination polarization is controlled via the 180\({}^{\circ}\) phase retardation plate (see Figure 2), whereas detection is realized without using any analyzer. This corresponds to an intensity distribution as mathematically described by Equation (8). Figure 3 shows this intensity distribution for the second-harmonic demodulation, plotted vs. \(\theta\). Each data point represents the integrated signal of all polarization components of the scattered signal. The data sets in Figure 3 were then fitted using Equation (8) and normalized individually for better comparison. For the non-resonant excitation of gold (Figure 3a), a strong signal excited by p-polarized illumination (\(\theta_{p}=0^{\circ}\), 180\({}^{\circ}\), and 360\({}^{\circ}\)) is observed together with an expected drop in intensity when rotating towards s-polarization (for \(\theta_{s}=90^{\circ}\) and 270\({}^{\circ}\)) (see Figure 3a). At \(\theta_{s}\), the signal reduces to \(I_{s}=0\), matching well to the dipole-model-based simulations. This behavior can be understood as a consequence of the boundary conditions of the dipole field at the metallic sample surface, where only electric field components perpendicular to the sample surface are allowed, hence suppressing any in-plane contributions. This condition changes for the resonantly excited 3C-SiC sample, for which in-plane field components at the boundary are supported. Consequently, when exciting 3C-SiC at 10.7 um (Figure 3b), for s-polarized excitation, a non-zero near-field signal is observed, matching the theoretical prediction. Furthermore, when tuning the wavelength closer to the spectral resonances at 10.6 um, the signal for s-polarized excitation increases (see Figure 3c) to about 25% of the intensity at p-polarized excitation. This clearly underlines that in-plane near-field excitation indeed is possible, leading to a quite strong and measurable s-SNOM  
Figure 3: Second harmonic demodulated signal for rotating illumination polarization \(\theta\) for different materials and illumination wavelengths. Here, \(\theta_{p}=0^{\circ},180^{\circ},360^{\circ}\) and \(\theta_{s}=90^{\circ},270^{\circ}\). Data are fitted using Equation (8) (black line), and p- and s-polarization-dominated contributions are shaded in blue and orange, respectively. (**a**) The gold sample illuminated with \(\lambda=10.6\) μm shows no response for s-polarized excitation. (**b**) 3C-SiC illuminated at \(\lambda=10.7\) μm shows a weak response to s-polarized illumination only, whereas (**c**) an increased s-polarized response is observed when the illumination wavelength hits the sample’s in-plane phonon resonance at \(\lambda=10.6\) μm. All three datasets are fitted for their local sample permittivity (see Table 1). signal. Furthermore, the intensity of the in-plane signal correlates with the resonance as predicted by the point-dipole model.