## 3 Results
### Acoustic Iridescence by Perfect Absorption  
In the following, we will analyse a metamaterial using \(N=3\) resonators. The system was designed using an optimization method (sequential quadratic programming) [27] to obtain perfect acoustic absorption at a set of frequencies and angles: \(f_{1}=290\) Hz and \(\theta_{1}=0^{\circ}\); \(f_{2}=350\) Hz and \(\theta_{2}=65^{\circ}\); and \(f_{3}=410\) Hz and \(\theta_{3}=85^{\circ}\). In particular, the cost using in the optimization was \(\varepsilon=1-\sum_{n=1}^{N}\alpha(\theta_{n},f_{n})/N\), and the geometrical parameters of the structure were varied using only the constraint of \(d=5\) cm and \(L=c_{0}/18290\) (i.e., a structure 18 times smaller than the  
Figure 3: Complex frequency representation of the reflection coefficient for the metamaterial with \(N=1\) HR, in logarithmic colour scale, for different angles of incidence. (**a**) \(\theta=20^{\circ}\); (**b**) \(\theta=80^{\circ}\); (**c**) \(\theta=87.7^{\circ}\); (**d**) \(\theta\to 90^{\circ}\). The trajectories of the poles and the zeros of the reflection coefficient as the incidence angle is increased are plotted in coloured lines, corresponding to the blueish and reddish colour bars, respectively. wavelength at 290 Hz). The geometrical parameters are listed in Table 1 in the Appendix A. First, the frequency-dependent absorption produced at the different angles is shown in Figure 4a. We can see that a peak of perfect absorption is produced at the selected frequencies: \(f_{1}=290\) Hz (red curve); \(f_{2}=350\) Hz (green curve), and \(f_{3}=410\) Hz (blue curve). For the selected frequencies, where perfect absorption is obtained, the absorption as a function of the incidence angle is shown in Figure 4b. Again, Equation (10) perfectly agrees with the TMM: the locally reacting system can only be critically coupled at a single angle for each selected frequency. The full map of the absorption as a function of the incidence angle and frequency is given in Figure 4c. It can be observed that when the incidence angle increases, the peaks of absorption shift in frequency, generally to higher ones, producing an acoustic iridescence effect. However, the absorption peak observed at 500 Hz for normal incidence shows a shift towards lower frequencies. In addition, it is worth noting here that the effect of the other resonators is still visible: for each angle, the absorption presents a maximum of \(N\) peaks of absorption. As previously, the system can be analysed for complex frequencies, and the reflection coefficient is represented in the complex frequency plane, as shown in Figure 5. First, the reflection at normal incidence is presented in Figure 5a. We can see that the \(N=3\) zero-pole pairs are visible, one zero being exactly located on the real frequency axis at \(f=290+i0\) Hz. As the reflection vanishes, the absorption is perfect, the metamaterial being perfectly impedance matched at this particular frequency. Note that a second zero of the reflection coefficient is located at \(f=300-i10\) Hz, producing a quasi-perfect peak of absorption (\(\alpha=0.9\)) visible in Figure 4a. The third zero is located around \(f=500+i25\) Hz, producing a third peak of absorption (\(\alpha=0.65\)). When the incidence angle is increased, the positions of the zeros and poles move in the complex frequency plane, thus changing the physical reflection coefficient on the real frequency axis. At \(\theta=65^{\circ}\) (as shown in Figure 5b), the trajectory of a zero crosses the real frequency axis at \(f=350\) Hz, producing perfect absorption. Note the higher frequency zero moves away from the real frequency axis, then its impact on the absorption is reduced. When the incidence angle is tilted up to \(\theta=85^{\circ}\) (as shown in Figure 5c), a zero crosses the real frequency axis at \(f=410\) Hz, and again, perfect absorption is produced. However, it is worth noting that the trajectories of the zeros and poles interact by avoided crossing [14; 28], and do not overlap.