# Cascaded Nanorod Arrays for Ultrabroadband, Omnidirectional and Polarization-Insensitive Absorption
## 2 Structure and Design  
Figure 1 shows the schematic map of the CNA-based metasurface under the TM (electric field along \(x\)-axis) wave illumination. The unit cell of the CNAs consists of two types of nanorods. One is the top metal (Ti) nanorod with height \(h\) and smaller diameter \(D_{1}\), the other is the bottom metal/dielectric (Ti/SiO\({}_{2}\)) multilayer nanorods with larger diameter \(D_{2}\), and these two types of nanorods are separated by a dielectric (SiO\({}_{2}\)) buffer layer with thickness \(t_{b}\). The number of the alternating Ti/SiO\({}_{2}\) pairs (\(N\)) of the multilayer nanorods is 5; the thickness of each Ti and SiO\({}_{2}\) film of the multilayer nanorods are \(t_{m}\) and \(t_{d}\), respectively. \(P\) is the period in the \(x\) and \(y\) directions. The background is air. The thickness of Ti substrate is chosen as 200 nm to block the light transmission. The refractive index of SiO\({}_{2}\) is 1.47, and the dielectric constant of Ti is obtained from Palik [52].  
For the CNA-based metasurface at normal incidence, whether a diffraction order propagates or not in the background can be written as [53]  
\[m^{2}+n^{2}<P^{2}/\lambda^{2} \tag{1}\]where \(m\) and \(n\) are the diffracted orders along the \(x\) and \(y\) directions, respectively. \(\lambda\) is the wavelength in free space. To ensure ultrabroadband absorption, a subwavelength unit cell with \(P<\lambda\) should be chosen to avoid the energy transfer via high-order diffraction.  
In simulations, we adopted the three-dimensional finite-difference time-domain (FDTD) method to analyze the absorption performance of the CNA-based metasurface. The absorption of total structure can be simplified as A = 1 - R due to the optically thick Ti substrate, where R is the reflection of the structure. Moreover, the absorption of different parts of the structure can be extracted by integrating power dissipation of every single layer [12; 54; 55]:  
\[\alpha(\lambda) = \frac{1}{2}\varepsilon_{0}\omega\text{Im}\varepsilon(\omega) \int_{V}|E|^{2}dV \tag{2}\]  
where \(\varepsilon_{0}\) is the permittivity of vacuum, \(\omega\) is the angular frequency, \(\varepsilon\) denotes the relative dielectric permittivity of each layer, \(\text{Im}\varepsilon(\omega)\) is the imaginary part of \(\varepsilon\), \(E\) denotes the amplitude of electric field, and the integral is over the desired part of a unit.  
In addition, the average absorption within the operating wavelength region is used to evaluate the overall absorption performance of the CNA-based absorber, which can be calculated as  
\[\text{A}_{av} = \frac{1}{\lambda_{\text{L}}-\lambda_{S}}\int_{\lambda_{S}}^{ \lambda_{\text{L}}}\!\!A(\lambda)d\lambda \tag{3}\]  
where \(\lambda_{L}\) and \(\lambda_{S}\) are the larger and smaller limits of a wavelength region with absorptivity higher than 90%. Furthermore, to compare the absorption bandwidth of the CNA-based absorber with that of other types of broadband absorbers, the RAB is introduced, which is generally defined as RAB = 2(\(\lambda_{L}-\lambda_{S}\))/(\(\lambda_{L}-\lambda_{S}\)) [56].