## 1 Introduction
Aluminum (AI) and indium tin oxide (ITO) are used as the metal and lossy dielectric, respectively. We note that lossy dielectric materials other than ITO are also applicable, which may result in more desirable absorption. The metal substrate is 100 nm thick, thus we can assume there is no transmission of visible light. Transverse magnetic (TM) plane waves are incident with angle \(\theta\). Here, we only consider TM polarized incident plane waves since this is necessary to generate GSP modes in the grating cavity at normal incidence (\(\theta=0\)). Due to the polarization selectivity of plasmonic resonance [46], one-dimensional plasmonic gratings have unique applications, such as controlled thermal emitters and polarimetric detectors [47; 48; 49; 50]. To implement polarization-insensitivity, the structure should be extended to two-dimensional configurations with four-fold rotational symmetry [18; 37]. This is because both TM and TE (transverse electric) polarized light can be coupled to GSP modes in grating cavities arranged along two orthogonal directions. Polarization insensitivity based on geometric symmetry can be found in other types of two-dimensional plasmonic structures, e.g., in patterned MIM absorbers [20; 21]. Figure 1: (**a**) A subwavelength metallic grating coated with a dielectric layer. The thicknesses of the upper dielectric layer and grating are \(t_{\text{d}}\) and \(t_{\text{g}}\), respectively. (**b**) The optically equivalent model, in which the grating and substrate are replaced by a semi-infinite effective medium. The simulated spectral reflectance and phase shift maps are shown in Figure 2b,c, where the effect of the upper ITO layer is not considered. The optical response of the grating is calculated using the transfer matrix method (TMM) [51]. In order to reduce the number of design variables, we set the pitch and width of the grating to \(p\) = 160 nm and \(w\) = 40 nm throughout this work, respectively. Although we only investigated the effect of grating thickness, it is notable that the resonance condition in reflective gratings is also closely related to its period and width, as can be found in relevant previous studies [14,18,37]. As shown in Figure 2b, reflection dips in the visible spectrum can be tuned by selecting the proper grating thickness. Since we can ignore the transmitted components, these reflection dips directly correspond to absorption peaks. Thus, the grating thickness can be chosen in accordance with an absorption band of interest. It is expected that these absorption properties will be maintained when the grating is coated with an ITO layer with finite thickness. Absorption in the Al grating coated with ITO of finite thickness can be calculated using the reflection results at the interface between the Al grating and ITO. Figure 3a shows the reflection and transmission coefficients at two interfaces of the structure in which the grating and substrate layers are equivalently replaced by the semi-infinite effective medium. Here, we take \(t_{\text{g}}\) = 30 nm, and the corresponding magnitude and phase profiles of the reflection coefficient from ITO to the effective medium (\(r_{\text{eff}}\)) are shown in Figure 3c,d. Figure 3e-h show the magnitude and phase profiles of the Fresnel reflection and transmission coefficients at the interface between air and the ITO layer. Optical constants for ITO and Al used to calculate the coefficients are presented in Figure 3b and were determined experimentally from ellipsometry measurements. The analytical expression for reflectance from the structure shown in Figure 3a can be calculated using [52].