## 2 Materials and Methods
### Features of the Polarizability of a Single MoS\({}_{2}\) Disk with a Defect
The contributions of different multipoles (see Equation (1)) to the total scattering cross section shown in Figure 1 can be written as the sum of their spatial components in the form:  
\[\text{ED}=\sum_{\alpha}\text{ED}_{\alpha}=\frac{k^{4}}{6\pi \epsilon_{0}^{2}|\textbf{E}|^{2}}\sum_{\alpha}\lvert p_{\alpha}\rvert^{2}, \tag{2}\] \[\text{MD}=\sum_{\alpha}\text{MD}_{\alpha}=\frac{k^{4}\epsilon_{d} \mu_{0}}{6\pi\epsilon_{0}|\textbf{E}|^{2}}\sum_{\alpha}\lvert m_{\alpha} \rvert^{2},\] (3) \[\text{EQ}=\frac{k^{6}\epsilon_{d}}{720\pi\epsilon_{0}^{2}|\textbf {E}|^{2}}\sum_{\alpha\beta}\lvert Q_{\alpha\beta}\rvert^{2},\] (4) \[\text{MQ}=\frac{k^{6}\epsilon_{d}^{2}\mu_{0}}{80\pi\epsilon_{0}| \textbf{E}|^{2}}\sum_{\alpha\beta}\lvert M_{\alpha\beta}\rvert^{2},\] (5) \[\text{EOC}=\frac{k^{8}\epsilon_{d}^{2}}{1890\pi\epsilon_{0}^{2}| \textbf{E}|^{2}}\sum_{\alpha\beta\gamma}\lvert O_{\alpha\beta\gamma}\rvert^{2},\] (6) \[\text{MOC}=\frac{k^{8}\epsilon_{d}^{3}\mu_{0}}{1890\pi\epsilon_{0} |\textbf{E}|^{2}}\sum_{\alpha\beta\gamma}\lvert MO_{\alpha\beta\gamma}\rvert^{ 2}. \tag{7}\]  
The detailed analysis of the features of light scattered by an optically anisotropic nanoparticle with a defect taking into account the multipole decomposition is presented in ref. [8]. This analysis is based on the appearance of a new degree of freedom associated with the optical anisotropy of the material. Such a new degree of freedom, together with varying the shape of the nanoparticle and the parameters of the defect in it, allows one to manipulate the spectral position of the main multipole resonances within a wide range. In this work, we use the property of optical anisotropy in order to obtain drastically different scattering patterns when nanoparticle with an irregular shape is irradiated by light with a polarization across (Figure 1c) and along (Figure 1d) the material layers. This is a basis for obtaining resonances of different types at the same wavelength for various irradiation conditions of nanoparticle, which can be used for polarizing control in such systems. In general case, the excitation of minor components of the electric \(p_{z}^{(E,\pm H,z,\pm k)}\) and magnetic \(m_{z}^{(\mp H,E,\pm k)}\) dipole moments of NP irradiated by plane waves \(E=(E_{x},0,0)\exp(\pm ikz)\) or \(E=\left(0,E_{y},0\right)\exp(\pm ikz)\), respectively, can be represented on the basis of theory of nonlocal response as follows [26]:  
\[\sigma_{\text{scat}} \cong \frac{k^{4}}{6\pi\epsilon_{0}^{2}|\textbf{E}|^{2}}|\textbf{p}|^{2}+ \frac{k^{4}\epsilon_{d}\mu_{0}}{6\pi\epsilon_{0}|\textbf{E}|^{2}}|\textbf{m}|^{2} \tag{1}\]0 \[\sigma_{\text{scat}} \cong \frac{k^{4}}{6\pi\epsilon_{0}^{2}|\textbf{E}|^{2}}|\textbf{p}|^{2}+ \frac{k^{4}\epsilon_{d}\mu_{0}}{6\pi\epsilon_{0}|\textbf{E}|^{2}}|\textbf{m}|^{2} \tag{1}\]1  
where \(\alpha_{zx}^{ee}\) (\(\alpha_{zx}^{mm}\)) corresponds to the direct (local) excitation of dipole moments by the electric (magnetic) field of the incident wave; \(a_{zxz}\) (\(c_{zyz}\)) is the nonlocal response of NP, and \(\alpha_{zy}^{em}\) (\(\alpha_{zy}^{me}\)) corresponds to bianisotropic response [1]. The latter effect can occur in the systems with broken rotational symmetry of the particle. As a result, the nonzero longitudinal components of the dipole moments can be excited in the considered disk with a hole, which are parallel to the wave vector of the incident wave [8]. At the same time, the type of bianisotropic response depends on the orientation of the defect relative to the components of the electric and magnetic fields of the incident wave. In particular, if the direction between the defect and center of symmetry of the particle coincides with the electric field strength vector of the incident wave, then the longitudinal component of the electric dipole is excited in the particle (see Figure 1a for ED\({}_{z}\)), and if this direction coincides with the magnetic field strength vector, then the component of the magnetic dipole is excited (see Figure 1b for MD\({}_{z}\)).