# Dual-Band Light Absorption Enhancement in Hyperbolic Rectangular Array
## 2 Design Principles  
Figure 1a shows the schematic diagram of the HRA, which consists of alternating silver (Ag) and silicon dioxide (SiO\({}_{2}\)) films on top of an Ag substrate. The thickness of the Ag film (\(t_{m}\)) is 10 nm, and its dielectric constant (\(\varepsilon_{m}\)) is described by a multioscillator (Lorentz-Drude model) with the parameters given by [PERSON] et al. [31]. The thickness of SiO\({}_{2}\) (\(t_{d}\)) is 35 nm and its dielectric constant (\(\varepsilon_{d}\)) is 2.16. \(f=t_{m}/(t_{d}+t_{m})=0.22\) is the filling ratio of the Ag film. The number of film stack of alternating Ag/SiO\({}_{2}\) pairs (\(N\)) is 8, and the width of film stack is \(W\). \(P\) and \(D\) are the period and depth of the HRA, respectively. Subwavelength structure (\(P<\lambda\)) is required so as to achieve high absorption efficiency. The Ag substrate is chosen to be optically thick enough (\(t_{s}=100\) nm) to prevent light transmission, thus light absorption (A) can be reduced as A = 1 - R, where R is the reflection of the HRA. In simulation, the 2-dimensional finite-difference time-domain (FDTD) approach is adopted to calculate the absorption performance of the HRA. The reflection of the HRA is defined by the ratio of the reflected power to the launched power. Periodic boundary conditions are used in the \(x\) direction, and perfectly matched layer boundary conditions are used in the \(z\) direction. The grid size is chosen as 2 nm in both the \(x\) and \(z\) directions. In practice, the HRA can be fabricated by using the standard nanolithography processes [32; 33; 33], as shown in Figure 1b. First, a 100-nm-thick Ag layer is deposited on top of the K9 glass. Next, a thick positive photoresist is spin-coated onto the Ag layer. Photolithography is used to create the rectangular array by double-beam interference lithography with the designed exposure process. Then, an electron beam evaporation system is used to deposit the alternating 10 nm Ag and 35 nm SiO\({}_{2}\) multilayer. Finally, the HRA can be obtained after the sample is soaked in acetone to lift-off the photoresist layer. Figure 1: (**a**) Schematic diagram of the HRA illuminated by normal incident TM wave (magnetic field vector lies along the \(y\)-axis). The parameters are: \(P=500\) nm, \(W=375\) nm, \(D=360\) nm, \(t_{m}=10\) nm, \(t_{d}=35\) nm,\(f=0.22\), and \(t_{s}=100\) nm. (**b**) Fabrication process flow of the HRA. The structure of alternating Ag/SiO\({}_{2}\) pairs can be represented as the ultra-anisotropic limit of a traditional uniaxial crystal, and the corresponding dielectric permittivity tensor can be expressed as [4]:  
\[\varepsilon=\left( \begin{array}{ccc}\varepsilon_{\perp}&0&0\\ 0&\varepsilon_{\perp}&0\\ 0&0&\varepsilon_{\parallel}\end{array} \right) \tag{1}\]  
Here, \(\perp\) and \(\parallel\) indicate components perpendicular and parallel to the Ag/SiO\({}_{2}\) interfaces, respectively. Since the thickness of a film stack (\(t_{m}+t_{d}\)) is much smaller than the wavelength, the dielectric constant of the HRA can be estimated using the EMT. According to the EMT of HMM [34], the perpendicular and parallel components of dielectric permittivity can be written as:  
\[\varepsilon_{\perp}=f\varepsilon_{m}(\omega)+(1-f)\varepsilon_{d} \tag{2}\]  
\[1/\varepsilon_{\parallel}=\left[f/\varepsilon_{m}(\omega)\right]+\left[(1-f) /\varepsilon_{d}\right] \tag{3}\]  
For the HRA illuminated by a TM-polarized pane wave, its dispersion relation can be determined by:  
\[\frac{k_{x}^{2}+k_{y}^{2}}{\varepsilon_{\parallel}}+\frac{k_{z}^{2}}{ \varepsilon_{\perp}}=\left(\frac{\varepsilon}{c}\right)^{2} \tag{4}\]  
where \(k_{x}\), \(k_{y}\) and \(k_{z}\) are the components of the wavevector along the \(x\), \(y\) and \(z\) directions, respectively. \(\omega\) is the frequency of the light wave, and \(c\) is the speed of light in vacuum. \(\varepsilon_{\parallel}\varepsilon_{\perp}<0\) is required to achieve hyperbolic light dispersion for the HRA structure with alternating Ag/SiO\({}_{2}\) pairs.