## 2 Materials and Methods
### Practical Calculation of the Polarizabilities of a Single MoS\({}_{2}\) Disk with a Defect  
Since the properties of the metasurface are determined by the properties of individual particles, as well as the collective resonances of the entire structure, one should start with the calculation of the polarizabilities of its building blocks in order to calculate the eigenmodes of a metasurface; see Figure 2. In particular, in order to determine the conditions for collective resonance in a metasurface, it is necessary to determine the polarizability of a building block under its lateral irradiation. Since, in the considered case, we are talking about the metasurface supporting QTMs excited due to bianisotropy of both electric and magnetic types, we consider the following set of equations:  
\[p_{z}^{(\pm k,\mp H,E)} = \alpha_{zz}^{ee}E_{z}\pm a_{zzx}ikE_{z}\mp\alpha_{zy}^{em}H_{y}, \tag{10}\] \[m_{z}^{(\pm k,E,\pm H)} = \pm a_{zz}^{mm}H_{z}\pm c_{zyx}ikE_{y}+\alpha_{zy}^{me}E_{y},\] (11) \[m_{z}^{(E,\pm k,\mp H)} = \mp\alpha_{zz}^{mm}H_{z}\pm c_{zxy}ikE_{x}+\alpha_{zx}^{me}E_{x}. \tag{12}\]Equation (10) corresponds to the irradiation of NP in two opposite directions: forward and backward to the \(x\) axis, while maintaining the direction of the vector \(\mathbf{E}\) along the \(z\) axis. This is sufficient to extract from Equation (10) the expression for electric polarizability of an individual disk in the following form:  
\[\alpha_{zz}^{ee(x)}\equiv\alpha_{zz}^{ee}=\frac{p_{z}^{(k-H,E)}+p_{z}^{(-k,H,E)} }{2E\epsilon_{0}}. \tag{13}\]  
To determine the frequency dependences of the polarizability (13), we used the results of numerical calculations for \(p_{z}^{(k-H,E)}\) and \(p_{z}^{(-k,H,E)}\) when the disk is irradiated from two opposite directions. At the same time, the parameters used in the work resulted from the optimization, for which one of the disk's resonances, satisfying the condition \(\text{Re}\left(1/\alpha_{zz}^{ee(x)}\right)=0\), corresponds to the telecom wavelength of 1550 nm. For deriving the expression (13), we also used the fact that terms with \(a_{zzx}\) and \(\alpha_{zy}^{em}\) in Equation (10) have opposite signs and can be canceled simultaneously. However, this is not sufficient to obtain the direct magnetic polarizability. Therefore, its values, as well as the values for the coefficient of nonlocal response \(c_{zxy}\), are extracted by solving together Equations (11) and (12), and their expressions take the form [8]:  
\[\alpha_{zz}^{mm} =\frac{1}{4H_{z}}\Big{[}\Big{(}m_{z}^{(k,E,H)}-m_{z}^{(-k,E,-H)} \Big{)}+\Big{(}m_{z}^{(E,-k,H)}-m_{z}^{(E,k,-H)}\Big{)}\Big{]}, \tag{14}\] \[c_{zxy} =\frac{1}{2 ik(E_{x}+E_{y})}\Big{[}\Big{(}m_{z}^{(k,E,H)}-m_{z}^{( -k,E,-H)}\Big{)}-\Big{(}m_{z}^{(E,-k,H)}-m_{z}^{(E,k,-H)}\Big{)}\Big{]}. \tag{15}\]  
Note that the occurrence of a nonlocal response of the disk in Figure 2b is associated with a cumulative effect, namely the presence of a shifted hole, which makes NP asymmetrical and is also the consequence of the anisotropy of the material. Formally, we can also include the nonlocal response in effective polarizabilities and use it in the further search for the conditions of QTM excitation:  
\[\alpha_{zz}^{mm(x)}=\frac{m_{z}^{(k,E,H)}-m_{z}^{(-k,E,-H)}}{2H_{z}}. \tag{16}\]  
Figure 2b shows a comparison of the frequency dependences for the inverse polarizabilities \(\alpha_{zz}^{mm}\) and \(\alpha_{zz}^{mm(x)}\). The difference between them is small, especially in the resonance region, where the inverse polarizability becomes zero. In contrast to the case of electric polarizability, there are only two such singular points, but their wavelengths are close to those obtained in Figure 2a.  
Figure 2: Real parts of inverse (**a**) electric and (**b**) magnetic polarizabilities (blue and red curves), as well as nonlocal response (green curve) for MoS\({}_{2}\) disk calculated using Formulas (13), (16) and (14), and (15), respectively. The simulation parameters: \(R_{2}=H=418\) nm, \(R_{1}=133\) nm, \(\Delta_{x}=159\) nm.