# Lattice Resonances in Transdimensional WS\({}_{2}\) Nanoantenna Arrays
## 2 Results  
We consider a periodic rectangular array of WS\({}_{2}\) nanodisks surrounded by a uniform medium with moderate refractive index \(n=1.5\) (equivalent to matched-index substrate and superstrate, Figure 1a). Permittivity components of WS\({}_{2}\) are presented in Figure 1b where \(\varepsilon_{\parallel}\) is in-plane and \(\varepsilon_{\rm zz}\) is out-of-plane tensor components and \(\varepsilon_{\rm WS2}=(\varepsilon_{\parallel},\,\varepsilon_{\parallel},\, \varepsilon_{\rm zz})\). One can see that losses of \(\varepsilon_{\parallel}\) are relatively high for \(\lambda<\lambda_{\rm c}=655\) nm, and nanoparticle resonances in this spectral region are not well defined for a dense array. At the same time, for \(\lambda>\lambda_{\rm c}\), the absorption peak is clearly seen for the electric dipole (ED) resonance \(\lambda_{\rm ED}\approx 697\) nm (Figure 1c, red line and Figure 1d), which indicates excitation of Mie resonance. The E-field profile at \(\lambda=697\) nm in Figure 1d is not the one of a pure ED multipole but the distribution is clearly seen with the main contribution of ED resonance. Absorption is calculated as losses in the nanoparticle regardless of the number of diffraction orders taken into account, and the resonance is associated with an increase in reflection at the wavelength about 690 nm (Figure 1c, blue line).  
We analyze the absorption and reflection properties in the transdimensional lattice [27] with different periods in orthogonal direction \(p_{\rm x}\) and \(p_{\rm y}\). In numerical modeling, we consider the normal incidence of light, one WS\({}_{2}\) nanodisk in the unit cell, periodic boundary conditions in \(x\)- and \(y\)-directions, and the artificial perfectly matched layers in the \(z\)-direction. We perform numerical simulations with finite element method implemented in CST Microwave Studio frequency-domain solver.  
For a spherical particle with isotropic constituent material, each of these periods controls different lattice resonances: under illumination with \(x\)-polarized light, \(p_{\mathrm{x}}\) controls the magnetic dipole (MD) and electric quadrupole (EQ) resonances, and \(p_{\mathrm{y}}\) controls ED and magnetic quadrupole (MQ) [17; 24; 25]. These considerations are approximately valid for disks even though the description of the particle includes a tensorial polarizability. Calculations in Figure 2 show that under change of \(p_{\mathrm{y}}\), the ED resonance shifts to longer wavelength. In Figure 2, calculation results are terminated at Rayleigh anomaly wavelength \(\lambda_{\mathrm{RA}}=n\!p_{\mathrm{x,y}}\) to account for only sub-diffraction effects. Upon further increase in the period (\(p_{\mathrm{y}}>400\) nm), the ED transforms to ED-lattice-resonance (ED-LR) and follows the Rayleigh anomaly. Even for fairly large periods \(p_{\mathrm{y}}>700\) nm, the resonance is well defined, and one can see peaks in absorption and reflection profiles. Similar to the effect of \(p_{\mathrm{y}}\) changes, changes in \(p_{\mathrm{x}}\) result in the excitation of lattice resonances in the proximity to the Rayleigh anomaly, but the resonances do not extend to a longer wavelength. This behavior is typical for the nanoparticles with rapidly decreasing polarizability.  
Figure 1: (**a**) Schematic of the transdimensional lattice under consideration. The parameters are the following: nanodisk radius \(R=100\) nm, height \(h=200\) nm, and refractive index of surrounding \(n=n_{\mathrm{s}}\) = 1.5. (**b**) WS\({}_{2}\) permittivity components. (**c**) Absorption and reflection from the nanoparticle array with periods \(p_{\mathrm{x}}=p_{\mathrm{y}}=240\) nm. Magenta lines on (b) and (c) panels indicate the wavelength \(\lambda_{\mathrm{c}}\) where WS\({}_{2}\) absorption drops significantly, and for \(\lambda>\lambda_{\mathrm{c}}\), the nanoparticles support Mie and lattice resonances. (**d**) E-field distribution at \(\lambda=697\) nm indicating ED contribution of the resonance.