## 3 Theoretical Framework
### Simulation  
To simulate the frequency spectra of the laser-generated SAWs, the BVP is formulated with respect to the complex displacement amplitude \(\mathbf{u}=(u_{1},u_{2},u_{3})\) of the time-harmonic oscillation \(\mathbf{u}(\mathbf{x})e^{-i\Omega t}\); \(\Omega=2\pi f\) is angular frequency, \(f\) is frequency, \(\mathbf{x}=(x,y,z)\) is a point in the Cartesian coordinate system; plane \((x,y)\) and axis \(z\) are aligned with the sample's surface and outward normal (Figure 5b). The components of the displacement vector \(\mathbf{u}\) obey the elastodynamics equations  
\[C_{ijkl}u_{l,jk}+\rho\Omega^{2}u_{i}=0,\quad i=1,2,3. \tag{1}\]  
The elastic stiffness tensor \(C_{ijkl}\) and the density \(\rho\) are piecewise constant functions of the transverse coordinate \(z\), thereby keeping constant values within the sublayers of thickness \(h_{m}\), \(m=1,...,M\). The bottom substrate is a half-space (\(h_{M}=\infty\)). The outer surface \(z=0\) is stress free, except within the loading region \(D\):  
\[\tau|_{z=0}=\mathbf{q},\quad(\mathbf{q}(x,y)\equiv 0\text{ for }(x,y)\
otin D), \tag{2}\]  
and the sublayers are perfectly bonded with each other. Here \(\tau=\{\tau_{xz},\tau_{yz},\tau_{z}\}\) is a tension vector at a horizontal surface area \(z=\text{const}\), \(\mathbf{q}\) is a given load simulating the action of the laser beam generating SAWs. The solution to this BVP is obtained in terms of the [PERSON]'s matrix \(k(\mathbf{x})\) and the vector of load \(\mathbf{q}\) applied to a surface area \(D\), or equivalently, via their Fourier symbols \(K(\alpha,\alpha_{2},z)=\mathcal{F}_{xy}[k]\) and \(\mathbf{Q}(\alpha_{1},\alpha_{2})=\mathcal{F}_{xy}[\mathbf{q}]\)[16; 17; 20]:  
\[ \begin{array}{l}\mathbf{u}(\mathbf{x})=\int\int\limits_{D}k(x-\xi,y-\eta,z) \mathbf{q}(\xi,\eta)d\xi d\eta\\ \phantom{\int\int\limits_{D}k(x-\xi,y-\eta,z)\mathbf{q}(\xi,\eta)d\xi d\eta} =\frac{1}{(2\pi)^{2}}\int\limits_{\Gamma_{1}}\int\limits_{\Gamma_{2}}K(\alpha _{1},\alpha_{2},z)Q(\alpha_{1},\alpha_{2})e^{-i(\alpha_{1}x+\alpha_{2}y)}d \alpha_{1}d\alpha_{2}.\end{array}  \tag{3}\]  
Here \(\mathcal{F}_{xy}\) is the Fourier transform with respect to \(x\) and \(y\) variables; the Fourier parameters \(\alpha_{1}\) and \(\alpha_{2}\) play the role of wavenumbers for the waves propagating along the \((x,y)\) surface. The integration paths \(\Gamma_{1}\) and \(\Gamma_{2}\) go along the real axes, deviating from them into the complex planes \(\alpha_{1}\) and \(\alpha_{2}\) for rounding the real poles \(\zeta_{n}\) of the matrix \(K\) elements. The frequency domain [PERSON]'s matrix \(k=(\mathbf{k}_{1}\mathbf{:}\mathbf{k}_{2}\mathbf{:}\mathbf{k}_{3})\) is formed from the solution vectors \(\mathbf{k}_{j}\) corresponding to the point loads applied along the basic coordinate vectors \(\mathbf{i}_{j}\). In the case under consideration, \(\mathbf{k}_{j}=\mathbf{u}\), where \(\mathbf{u}\) is the solution to BVP (1) and (2) with \(\mathbf{q}=\mathbf{i}_{j}\delta(x,y)\) in the boundary conditions at the surface \(z=0\); \(\delta(x,y)\) is [PERSON]'s delta-function. The exhaustive description of the algorithms of matrix \(K\) calculation can be found in Refs. [16; 20]. Ibid, as well as in Reference [17], the derivation of asymptotic representations for SAWs generated by the surface load \(\mathbf{q}\) is described in details. The SAWs are extracted from the path integrals of Equation (3) as the residues from the real and nearly real poles \(\zeta_{n}\). Since the action of the laser beam on the sample's surface can be simulated by a vertical load \(\mathbf{q}=(0,0,q)\), only the third column of matrix \(K\) is involved in the representation of the laser-generated wave field \(\mathbf{u}\). Moreover, since only the third (vertical) component \(u_{3}=u_{z}\) of the excited displacement \(\mathbf{u}\) is laser acquired as the sample's response, only one element \(K_{33}\) of the matrix \(K\) is needed to simulate the experimental measurements. Under these conditions, the SAW asymptotics of Reference [16] is reduced to the form  
\[