## 3 Results and Discussion  
### Quasi-Trapped Modes of Metasurfaces Composed of MoS\({}_{2}\) Disks  
The components of electric or magnetic dipoles (8) and (9) excited in each MoS\({}_{2}\) disk normally to the plane of its base can be used to excite the quasi-trapped modes in a metasurface composed of such disks [5]; see Figure 3a. In this case, switching between \(p_{z}\) and \(m_{z}\), related to the electric and magnetic bianisotropy, respectively, can be simply performed by changing the polarization of the incident wave from \(E_{x}\) to \(E_{y}\). In the case where optical losses are not taken into account, the formation of the TM in an infinite periodic array can be realized due to the collective effect of the synchronization of \(p_{z}\) or \(m_{z}\) moments of individual disks. Assuming that the excitation of the corresponding dipole moments \(p_{z}\) or \(m_{z}\) is associated only with the direct polarizabilities \(\alpha_{zz}^{ee(x)}\) and \(\alpha_{zz}^{mm}\), the condition for the existence of trapped electric and magnetic modes corresponds to the solution of transcendental equations written in the form [28]:  
\[S_{z}^{\rm(R)} = \text{Re}\Bigg{(}\frac{1}{\alpha_{zz}^{mm(ee(x))}}\Bigg{)}, \tag{17}\] \[S_{z}^{\rm(I)} = \text{Im}\Bigg{(}\frac{1}{\alpha_{zz}^{mm(ee(x))}}\Bigg{)}, \tag{18}\]  
where \(S_{z}=S_{z}^{\rm(R)}+iS_{z}^{\rm(I)}\) is the dipole sum whose components can be defined as follows:  
\[S_{z}^{\rm(R)} = \frac{k_{d}^{2}}{4\pi}\sum\limits_{l,j}^{\infty}\left(\frac{\cos \Big{(}k_{d}d_{lj}\Big{)}}{d_{lj}}-\frac{\sin\Big{(}k_{d}d_{lj}\Big{)}}{k_{d} ^{2}d_{lj}^{3}}-\frac{\cos\Big{(}k_{d}d_{lj}\Big{)}}{k_{d}^{2}d_{lj}^{3}} \right), \tag{19}\] \[S_{z}^{\rm(I)} = \frac{k_{d}^{2}}{4\pi}\sum\limits_{l,j}^{\infty}\left(\frac{\sin \Big{(}k_{d}d_{lj}\Big{)}}{d_{lj}}+\frac{\cos\Big{(}k_{d}d_{lj}\Big{)}}{k_{d} d_{lj}^{2}}-\frac{\sin\Big{(}k_{d}d_{lj}\Big{)}}{k_{d}^{2}d_{lj}^{3}} \right)\equiv-\frac{k_{d}^{3}}{6\pi}. \tag{20}\]  
Figure 3: (**a**) The model of the metasurface based on MoS\({}_{2}\) disks with a hole shifted from the disk’s center irradiated by normally incident wave \(E_{x}(k_{z})\). (**b**) Dependences of the reflection (R), transmission (T), and absorption (A) coefficients and parameter \(U_{n}=0.5 Ul/\max(U)\), as well as (**c**) the dependences of the \(m_{y}\) component of the magnetic dipole and the \(p_{x}(p_{z})\) component of the electric dipole excited in each disk on the wavelength for a metasurface composed of MoS\({}_{2}\) disks. The simulation parameters: \(P=1125\) nm, \(R_{2}=H=418\) nm, \(R_{1}=133\) nm, \(\Delta_{x}=159\) nm. Here, the parameter \(d_{lj}=P\sqrt{l^{2}+j^{2}}\) is the distance from the Cartesian coordinate system origin, coinciding with the position of a lattice node, to all other nodes of the lattice, which are numbered by indices \(l\) and \(j\), and \(P\) is the period of lattice, \(k_{d}=2\pi/\lambda_{QTM}\). Condition (17) corresponds to the ideal situation of TM excitation without leakage; therefore, it allows one to determine the spectral position of quasi-trapped mode only approximately, while the deviation from condition (18) determines the decrease in the quality factor of the corresponding resonance. In the general case, the solution of Equation (17) determines the exact value of the metasurface's period \(P\) corresponding to the QTM resonance at a given wavelength \(\lambda_{QTM}\). On the other hand, the highest concentration of energy in a metasurface occurs when the resonance conditions are excited in each of its building blocks, which corresponds to the satisfaction of the condition \(P/\lambda_{QTM}=\) 0.72 [1]. For the considered case, we obtain \(P=1125\) nm, and this value is well in agreement with the wavelength \(\lambda_{QTM1}=1553\) nm for one of the resonances in Figure 3b. Note that we simulated an infinite 2D periodic lattice using the unit cell method with Floquet periodicity in Comsol Multiphysics 5.4.