# An Integration Strategy for Acoustic Metamaterials to Achieve Absorption by Design
## 4 Integration Strategy for Achieving Absorption by Design  
In order to be concrete, in the following we will consider the use of Fabry-Perot (FP) resonators to realize our integration strategy. A simple-minded approach to putting together multiple resonances for achieving, e.g., broad-band absorption, is by integrating within a single metamaterial unit multiple resonators with equally-spaced resonance frequencies. However, we show below that there is a much better strategy than using equally-spaced resonances. To see how such a strategy can be derived, let us consider an idealized structure with a continuum of resonances so that its impedance can be written as [12]  
\[Z(\omega)=i\lim_{\beta\to 0}\frac{Z_{0}d}{\omega v_{0}}\left[\int_{0}^{ \infty}\frac{\alpha(\Omega)D(\Omega)}{\Omega^{2}-\omega^{2}-i\beta\omega}d \Omega\right]^{-1} \tag{7}\]  
where \(D(\Omega)\) is the density of the resonant modes, and \(\beta\) is the dissipative constant that controls the width of the absorption peak. It should be noted that letting \(\beta\to 0\) does not imply the system to have zero dissipation when integrated over frequency, since a zero \(\beta\) means that the imaginary part of the quantity in the square bracket becomes a delta function with zero width, but which has a non-zero integrated effect. It follows that from Equation (7) we have  
\[Z(\omega)\simeq\frac{Z_{0}d}{\omega v_{0}}\left[-i\mathrm{P}\int_{0}^{\infty} \frac{\alpha(\Omega)D(\Omega)}{\Omega^{2}-\omega^{2}}d\Omega+\int_{0}^{\infty} \pi\alpha(\Omega)D(\Omega)\delta(\omega^{2}-\Omega^{2})d\Omega\right]^{-1} \tag{8}\]  
Here, the symbol \(\mathrm{P}\) denotes taking the principal value. In Equation (8), the integrant in the imaginary part is oscillatory in nature. Hence, it can be small. If for the moment we ignore the imaginary part in the square bracket and focus only on the real part, then  
\[\frac{Z(\omega)}{Z_{0}}=\frac{2d}{\pi v_{0}}[\alpha(\omega)D(\omega)]^{-1} \tag{9}\]  
Here, the mode density can be written as \(D(\Omega)=\left(d\Omega/d\overline{n}\right)^{-1}\), where \(\overline{n}\) is a continuous variable that indexes the resonance modes, defined as \(\overline{n}=\lim\limits_{N\rightarrow\infty}(n-1)/N\), with \(n\) being an integer ranging from \(1\) to \(N\). By recognizing that \([\alpha(\Omega)]^{-1}=\pi v_{0}/4\phi\Omega d\) for the lowest-order FP resonances, from Equation (9) we obtain the following simple differential equation for determining the mode density from the target impedance:  
\[\frac{d\Omega}{d\overline{n}}=2\phi\left[\frac{Z(\Omega)}{Z_{0}}\right]\Omega \tag{10}\]  
Here, \(\phi\) is the area fraction of the FP resonators' openings in the total surface area of the metamaterial unit. It will be seen that the determination of \(\phi\) requires the causal relationship to achieve both minimum sample thickness as well as consistency with the target absorption spectrum. Two comments are in order. First, since we want to target a certain absorption spectrum, then \(Z(\Omega)/Z_{0}\) can be directly obtained from Equation (2) and the fact that \(A=1-|R|^{2}\). Second, the mode density obtained by solving Equation (10) cannot exactly reproduce the target absorption spectrum because in the derivation of Equation (10) we have ignored the imaginary part of the impedance. In order to illustrate this point, let us choose for the target absorption spectrum a flat, perfect absorption spectrum starting at a lower cutoff frequency \(\Omega_{\mathrm{c}}\). In that case \(Z(\Omega)/Z_{0}=1\) above \(\Omega_{\mathrm{c}}\) and we have \(\Omega_{\overline{n}}=\Omega_{\mathrm{c}}\exp[2\phi\overline{n}]\). Then, \(\alpha(\Omega)D(\Omega)=2d/(\pi v_{0})\) is a constant, and Equation (7) can be integrated to obtain:  
\[\frac{Z(\omega)}{Z_{0}}=\frac{\pi}{\pi-2i\mathrm{tanh}^{-1}(\Omega_{\mathrm{c }}/\omega)} \tag{11}\]  
This expression is graphically illustrated in Figure 1 below. From Equation (11) one can easily evaluate \(d_{\min}\) as defined by Equation (1).