# Lattice Resonances in Transdimensional WS\({}_{2}\) Nanoantenna Arrays
## 1 Introduction  
Nanoparticles and nanoantennae resulting from subwavelength patterning of single- and multi-layer structures have been shown to be an effective way to control light at the nanoscale. High-refractive-index nanoparticles support excitation of multipole Mie resonances of different orders whose strength is mainly defined by the nanoparticle size, index of the material, and ohmic losses. While nanoparticles with moderate refractive index \(n\approx\) 1.45-2 (e.g., silicon oxide and nitride) have resonances with relatively high radiative losses and low quality factors, materials with \(n>\) 3.5 (e.g., silicon) enable high mode localization and relatively low radiative losses. Multipole resonances in high-index nanoparticles are comparable to resonances in plasmonic [1] and hyperbolic nanostructures [2; 3], as the main role is played by the index difference at the nanoparticle surface.  
So far, most of the studies have dealt with silicon [4; 5; 6; 7] or III-V nanoantennae [9] because of the readily available fabrication processes and well-established technology. However, recently emerging two-dimensional and layered materials hold great promise to be applied in next-generation optoelectronic devices and more efficient light harvesting [10; 11; 12]. Transition metal dichalcogenides (TMDCs), such as tungsten or molybdenum disulfides and diselenides (WS\({}_{2}\), MoS\({}_{2}\), WSe\({}_{2}\), MoSe\({}_{2}\)) and others, are van der Waals layered materials with high anisotropy, exciton resonances in the visible spectral range, strong nonlinear response, and tunability. Layered material anisotropy results indifferent in-plane and out-of-plane permittivity components, and the excitons in visible range result in a large imaginary part of the permittivity along with the increased real part. A large imaginary part not only causes non-radiative losses but also facilities excitation of Zenneck modes and light confinement within the guiding nanolayer [13] or nanoparticles [14]. In the near-infrared range, the losses in TMDCs are near zero, and the nanoparticle can support Mie resonances with high quality factor [15]. Nanoparticle array can be fabricated by patterning TMDC layer (or flakes) using electron-beam exposure of a negative resist and dry etching, see e.g., [15]. An overview of fabrication techniques of transition metal dichalcogenides heterostructure systems can be found in [16].  
Nanoparticle resonances can be enhanced, and their spectral position can be controlled not only by the nanoparticle size, shape, and material but also their arrangement in the clusters and/or lattices [17; 18; 19; 20; 21; 22; 23; 24]. Nanoparticle clusters facilitate excitation of narrow Fano resonances [1] that are very sensitive to the parameter changes, can be easily tuned in the spectral domain or lost because of the imperfections. In contrast, the periodical arrangement of nanoparticles in the lattice enables excitation of lattice resonances that have a relatively high tolerance to local imperfections. The spectral position of different multipoles can be controlled by mutually perpendicular periods of the nanoparticle array [25; 26].  
In this work, we report on the theoretical prediction of Mie resonances in a periodic array of WS\({}_{2}\) nanodisks and their control by the lattice dimensions. We show excitation of multipole resonances in the proximity to the Rayleigh anomaly \(\lambda_{\rm RA}\) (wavelength equal to the product of surrounding refractive index \(n\) and array period \(p_{\rm x,y}\), \(\lambda_{\rm RA}=np_{\rm x,y}\)) and a different optical response depending on the period changes in such a transdimensional lattice. The lattice resonances can be observed as a peak in reflection and absorption at the wavelength \(\lambda\approx\lambda_{\rm RA}=np_{\rm x,y}\). Because of the high real part of WS\({}_{2}\) in-plane permittivity tensor component in the near-infrared range, the lattice resonance of the electric dipole is excited even for large and elongated periodic cells (\(240\times 700\) nm\({}^{2}\)).