## 3 Results  
### Perfect Absorption under Oblique Incidence  
As shown in Reference [15], by tuning the geometrical parameters, the structure described in this work can be impedance matched with the exterior medium, and perfect acoustic absorption can be obtained. For locally reacting materials, this occurs when Equation (4) vanishes, leading to the impedance matching conditions:  
\[\text{Re}(Z)\cos(\theta)=Z_{0}\quad\text{and}\quad\text{Im}(Z)=0. \tag{9}\]where \(Z\) is the acoustic impedance of the rigidly-backed material (i.e., \(Z=iZ_{\text{eff}}\cot k_{\text{eff}}L\)). Thus, for a locally reacting metamaterial whose characteristic effective impedance does not depend on the incident angle, the impedance matching can only be achieved at one single angle for a given frequency. In particular, using Equations (4) and (9), the angle-dependent absorption of a material presenting perfect absorption at a given angle \(\theta_{PA}\) is given by:  
\[\alpha(\theta)=1-\left|\frac{\cos\theta_{PA}/\cos\theta-1}{\cos\theta_{PA}/ \cos\theta+1}\right|^{2}, \tag{10}\]  
where \(\theta_{PA}\) is the angle at which perfect absorption is produced. For \(\theta=90^{\circ}\), this equation is undetermined, but in the limit when \(\theta\rightarrow\pi/2\), the absorption vanishes if not considering other physical phenomena in the model (see Discussion section). To illustrate this, we begin by showing the case of a single HR (i.e, \(N=1\)). The structure was designed to show a peak of perfect absorption at 250 Hz at \(\theta=20^{\circ}\). In particular, the cost using in the optimization was \(\varepsilon=1-\alpha(\theta_{n},f_{n})\) with \(f_{n}=250\) Hz and \(\theta_{n}=20^{\circ}\), and the geometrical parameters of the structure were varied using only the constraint of \(d=5\) cm and \(L=c_{0}/24f_{m}\); i.e., a structure 14 times smaller than the wavelength at 250 Hz. The optimization algorithm was a sequential quadratic programming [27]. The retrieved parameters are given in Table 1 in the Appendix A. Figure 2a-c show the absorption as a function of the angle and frequency. First, for an incidence angle of \(\theta=20^{\circ}\), the structure presents a peak of perfect absorption at \(f_{PA}=250\) Hz, as shown by the red line in Figure 2a. When the incidence angle is increased, the impedance matching condition is no longer fulfilled, and the absorption decreases following Equation (10), as shown by Figure 2b. However, the fact that perfect absorption might not be achieved for other angles at this frequency does not imply that perfect absorption can be obtained at other frequencies. Thus, the mapping of the absorption as a function of the incidence angle and frequency is shown in Figure 2c. It can be observed that by increasing the incidence angle, the absorption peak shifts to higher frequencies: first, at \(\theta=80^{\circ}\) the peak appears at 300 Hz (green curves in Figure 2b,c), and then at \(\theta=87.7^{\circ}\) (blue curves in Figure 2b,c), a peak of perfect absorption is obtained at 350 Hz. Note also that the peak becomes more broadband when the structure is not critically coupled. To interpret the system, it is useful to represent eigenvalues of the scattering matrix (i.e., the reflection coefficient) in the complex frequency plane. Figure 3 shows the complex frequency representation of the reflection coefficient for increasing incidence angles. First, we can see that for \(\theta=20^{\circ}\) (as shown in Figure 3a), a zero of the reflection coefficient is exactly located on the real axis at \(f=250\) Hz; thus, \(\alpha=1\) for this particular frequency and incidence angle. Then, the incidence angle is progressively increased up to \(\theta=80^{\circ}\). Figure 3b shows the reflection coefficient at this angle in the complex frequency plane, where the trajectories of the zero and the pole for the previous incidence angles have been traced (note the colour coding in the corresponding colour bars).