## 2 Methods  
### SLatCoW  
The SLatCow method relies on the analysis of the spatial LT of the acoustic field \(p(\mathbf{x},\omega)\) in the frequency domain. The analysis is performed in the linear harmonic regime at the circular frequency \(\omega\) with the implicit time dependence \(e^{-\mathrm{i}\omega t}\). For two-dimensional problems, the pressure field is recorded on a surface map of dimensions \(x\in[0,L_{s}]\) and \(y\in[0,L_{s}]\), as depicted in Figure 1. The excitation is assumed to take the form of a plane incident wave along the positive \(x\)-axis. Therefore, the pressure field takes the form  
\[p(\mathbf{x},\omega)=\sum_{m\in\mathcal{M}}\bar{p}_{m}\mathrm{e}^{\mathrm{i} \left(k_{x}^{m}x+k_{y}^{m}y\right)}\Pi\left(x,L_{s}\right)\Pi\left(y,L_{s} \right), \tag{1}\]  
where \(\bar{p}_{m}\) is the complex amplitude of the \(m\)-th mode, \(\mathbf{K}^{m}=\left(K_{x}^{m},K_{y}^{m}\right)\) is the complex wavenumber of the \(m\)-th mode, \(\mathcal{M}\) is the set of modes, and \(\Pi\left(z,L\right)\) is the gate function equal to 1 when \(z\in[0,L]\) and equal to 0 elsewhere. For the sake of explanation the \(\omega\) dependence is further dropped. The complex wavenumber is also defined as \(\mathbf{K}^{m}=\mathbf{k}_{r}^{m}+\mathrm{i}\mathbf{k}_{i}^{m}\), with \(\mathbf{k}_{r}^{m}\) and \(\mathbf{k}_{i}^{m}\) are two component vectors, one along the \(x\)-axis and the other along the \(y\)-axis. Please note that along the specific direction \(\mathbf{X}\), \(\mathbf{K}^{m}\cdot\mathbf{X}=k_{r}^{m}+k_{i}^{m}\), with \(k_{r}^{m}\geq 0\) and \(k_{i}^{m}\geq 0\) ensuring that Equation (1) only involves forward propagating waves along positive \(\mathbf{X}\). Applying the usual spatial FT only enables the recovery of \(\mathbf{k}_{r}^{m}\). In order to recover both real and imaginary parts of \(\mathbf{K}^{m}\), a spatial LT is applied to \(p(\mathbf{x})\) denoted\(P(\mathbf{s})=\iint_{-\infty}^{\infty}p(\mathbf{x})e^{-\mathbf{s}\cdot\mathbf{x}}d \mathbf{x}\), where \(\mathbf{s}\) is a complex valued vector \(\mathbf{s}=\mathbf{s}_{r}+\mathbf{i}\mathbf{s}_{i}\), with \(\mathbf{s}_{r}\) and \(\mathbf{s}_{i}\) being two component vectors, again one along the \(x\)-axis and the other along the \(y\)-axis. The spatial LT takes the form  
\[ \begin{split} P(\mathbf{s})&=\sum_{m\in\mathcal{M}} \bar{p}_{m}\int_{0}^{L_{x}}\int_{0}^{L_{y}}e^{\mathrm{i}\left(\left(K_{x}^{m} -s_{x}\right)x+\left(K_{y}^{m}-s_{y}\right)y\right)}dxdy\\ &=L_{\mathrm{s}}^{2}\sum_{m\in\mathcal{M}}\bar{p}_{m}e^{( \mathbf{i}K_{x}^{m}-s_{x})I_{\mathrm{s}}/2}\frac{\sinh\left(\left(\mathrm{i}K_ {x}^{m}-s_{x}\right)L_{\mathrm{s}}/2\right)}{\left(\mathrm{i}K_{x}^{m}-s_{x} \right)L_{\mathrm{s}}/2}e^{\left(\mathrm{i}K_{y}^{m}-s_{y}\right)L_{\mathrm{s }}/2}\frac{\sinh\left(\left(\mathrm{i}K_{y}^{m}-s_{y}\right)L_{\mathrm{s}}/2 \right)}{\left(\mathrm{i}K_{y}^{m}-s_{y}\right)L_{\mathrm{s}}/2}\.\end{split}  \tag{2}\]  
In what follows, we are only interested in specific directions, namely \(\Gamma X\) and \(\Gamma M\), obtained by rotating the structure in the area \([0,L_{\mathrm{s}}]\times[0,L_{\mathrm{s}}]\). For a structure of infinite lateral extent, i.e., along the \(y\)-direction, the continuity of the real part of the \(K_{y}^{m}\) is ensured between the outer and the inner domain. In the present case of finite lateral dimensions, every modes are excited because of wave diffraction by the edges. Fixing \(s_{y}=0\) ensures that we are only looking for modes propagating along the \(x\)-axis.