## 2 Materials and Methods
### Theoretical Model  
The system is analysed theoretically using the transfer matrix method (TMM). Under the assumption of plane waves travelling inside the metamaterial, which is valid for the low frequency regime, the transfer matrix can be obtained, providing directly the reflection of the metamaterial, as well as its effective parameters. Thus, the relation between the pressure, \(p\), and normal particle velocity, \(v\), at the beginning (\(x_{1}=0\)) and at the end (\(x_{1}=L\)) of the slits is given by the total transfer matrix of the system, \(\mathbf{T}\), as  
\[\left[\begin{array}{c}p\\ v\end{array}\right]_{x_{1}=0}=\mathbf{T}\left[\begin{array}{c}p\\ v\end{array}\right]_{x_{1}=L}=\left[\begin{array}{cc}T_{11}&T_{12}\\ T_{21}&T_{22}\end{array}\right]\left[\begin{array}{c}p\\ v\end{array}\right]_{x_{1}=L} \tag{1}\]  
The total transfer matrix of the whole system can be obtained by the product of the transfer matrices of each layer of the material. Thus, the total transfer matrix method of the system is given by  
\[\mathbf{T}=\left[\begin{array}{cc}T_{11}&T_{12}\\ T_{21}&T_{22}\end{array}\right]=\mathbf{M}_{\Delta l_{\text{slit}}}\prod_{n= 1}^{N}\left(\mathbf{M}_{s}\mathbf{M}_{\text{HR}}^{(n)}\mathbf{M}_{s}\right),\]  
where \(\mathbf{M}_{s}\) is the transmission matrix of half lattice step in the slit, \(\mathbf{M}_{\text{HR}}^{(n)}\) is the transmission matrix of the \(n\)-th resonator, and \(\mathbf{M}_{\Delta l_{\text{slit}}}\) the radiation correction of the slits (i.e., the \"end\" correction [24]) that mimics the effect of the higher-order modes. The transmission matrix for each lattice step in the slit is written as  
\[\mathbf{M}_{s}=\left[\begin{array}{cc}\cos\left(k_{s}a/2\right)&iZ_{s}\sin \left(k_{s}a/2\right)\\ i\sin\left(k_{s}a/2\right)/Z_{s}&\cos\left(k_{s}a/2\right)\end{array}\right], \tag{2}\]  
where \(Z_{s}\) is the slit characteristic impedance, written as \(Z_{s}=\sqrt{\kappa_{s}\rho_{s}}/S_{s}\), \(k_{s}\) is the wavenumber in the slit, and \(S_{s}=h\,a\). The complex and frequency-dependent effective bulk modulus and density accounting for the thermoviscous losses, \(\kappa_{s}\) and \(\rho_{s}\), respectively, are given below. The HRs--introduced as point scatterers--and the radiation correction of the slits are given by the following transfer matrices:  
\[\mathbf{M}_{\text{HR}}^{(n)}=\left[\begin{array}{cc}1&0\\ 1/Z_{\text{HR}}^{(n)}&1\end{array}\right],\quad\mathbf{M}_{\Delta l_{\text{slit }}}=\left[\begin{array}{cc}1&Z_{\Delta l_{\text{slit}}}\\ 0&1\end{array}\right], \tag{3}\]  
with \(Z_{\text{HR}}^{(n)}\) the impedance of the \(n\)-th HR and \(Z_{\Delta l_{\text{slit}}}\) the characteristic radiation impedance given by \(Z_{\Delta l_{\text{slit}}}=-i\omega\Delta l_{\text{slit}}\rho_{0}/\phi_{t}S_{0}\), where \(S_{0}=d\,a\), \(\rho_{0}\) is the air density, and \(\Delta l_{\text{slit}}\) the radiation correction for a periodic distribution of slits [24].  
Figure 1: **(a)** Scheme of the deep-subwavelength absorbing panel under oblique incidence, the inset shows the geometrical parameters of the Helmholtz resonators (HRs); **(b)** Scheme of the unit cell that composes the metamaterial.  
Eventually, after some algebra, it can be shown that under oblique incidence of angle \(\theta\), the reflection coefficient of the rigidly-backed material can be directly calculated from the elements of the matrix \(\mathbf{T}\) as  
\[R(\theta)=\frac{T_{11}\cos(\theta)-Z_{0}T_{21}}{T_{11}\cos(\theta)+Z_{0}T_{21}}, \tag{4}\]  
with \(Z_{0}=\rho_{0}c_{0}/S_{0}\) the impedance of air. Finally, the absorption is obtained as \(\alpha=1-\left|R\right|^{2}\). The effective parameters of each slit can be obtained from the transfer matrix elements as follows  
\[k_{\text{eff}}=\frac{1}{L}\cos^{-1}\left(\frac{T_{11}+T_{22}}{2}\right)\,, \quad Z_{\text{eff}}=\sqrt{\frac{T_{12}}{T_{21}}}. \tag{5}\]  
Note that the TMM using a similar metamaterial has been validated using finite element simulations, modal expansion, effective parameters, and experiments in References [15; 16].