## 2 Methods
### Spatial Sampling Criteria for Dispersion Relation Recovery  
As stated previously, the SLaTcOW method relies on analyzing the pressure field in the complex wavenumber and frequency domains (\(p(s_{r},s_{i},\omega)\)). Hence, a double transformation of the acoustic pressure is needed to apply the method. First, time to frequency (\(\bar{p}(x,t)\to p(x,\omega)\)) through the temporal FT, then space to wavenumber (\(p(x,\omega)\to P(s_{r},s_{i},\omega)\)) through the spatial LT. While the temporal FT is not limited by or independent from the spatial periodicity of the structure, the spatial LT greatly depends on both the size of the structure, i.e., the number of periodic elements, and the spatial sampling of the acoustic field. To illustrate this, we consider a spatial signal as represented in Figure 2a and applied successively a temporal and a spatial FT. By analogy with temporal signals and following basic sampling theory, the resolution in \(s_{i}\)-space is determined by the length of the signal, \(\Delta s_{i}=1/L_{s}\), where \(L_{s}=\Delta xN\) with \(\Delta x\) the spatial sampling and \(N\) the number of samples of the spatial signal. Lets consider now a finite size one-dimensional periodic structure with lattice constant \(a\) formed by alternating two different materials, as depicted in Figure 2b. From the Nyquist-Shannon sampling theorem [PERSON] (1951) and considering periodicity in \(s_{i}\)-space, the spatial sampling rate \(1/\Delta x\) has to be equal or greater than twice the highest wavenumber component, i.e., \(s_{i}/2\geq k_{max}/2\pi\), where \(k_{max}=\pi/a\). As a result, the following condition for the spatial sampling \(\Delta x\leq a\) should be satisfied to ensure that no information is lost when performing the FT. In summary, for a finite dimension periodic structure comprising a specific number of unit cells \(N_{uc}\) and assuming that the wavefield is measured at a single point per periodic element, the number of these periodic elements (unit cells) determines the number of points and hence the resolution in the \(s_{i}\)-space as follows  
\[\Delta s_{i}=\frac{1}{\Delta xN}=\frac{1}{aN_{uc}}. \tag{5}\]  
Figure 2c-f illustrates the effects of the variation of the number of periodic elements, and therefore of the spatial sampling used on the FT of the pressure field, used to evaluate the dispersion relation of the multilayer structure shown in Figure 2b. A double FT is applied to numerically simulated spatio-temporal acoustic pressure values, \(\bar{p}(x,t)\). Results are presented for positive and negative values of the wavenumber and are compared to the theoretical dispersion relation calculated using the Rytov formula [(23)]. Figures 2c,d analyze the influence of the number of unit cells on the double FT using the critical sampling rate \(\Delta x=a\). The total number of wavevector points in the \(s_{i}\)-space is not high enough to extract accurate information to represent the dispersion relation in the case of a structure containing only \(N_{uc}=10\). Please note that the number of points in the \(s_{i}\)-space is distributed along the positive and negative parts of the wavenumber. Figures 2e,f show the consequences of spatially  
Figure 1: Set-up of the numerical experiments used to calculate the complex dispersion relation: (a) 2D SC composed of \(N\times N\) cylindrical scatterers arranged in a square lattice. (b) 2D Metamaterial composed of \(N\times N\) QWRs.  
under-sampling and over-sampling, respectively. The former causes the FT to be folded, while the latter creates replicates of the FT. However, no gain in information is obtained by increasing the spatial sampling above its critical value. Please note that the wavenumber resolution can be artificially increased by zero-padding the spatial signal in the space domain, although no actual information will be extracted from this process.