## 2 Materials and Methods
### Characterization Approach  
By virtue of its all-optical nature, laser ultrasonics was chosen to probe the structural condition of the GESA-steel sample. This method uses pulsed illumination to impulsively heat the sample surface and launch SAWs. Both the spatiotemporal evolution of the thermal expansion that witnesses the heat diffusion and the SAW velocity can be measured by monitoring the deflection of a reflected probe laser beam. The gradient in composition is expected to induce acoustic dispersion, with the Rayleigh velocity for the coating layer at high frequencies, which is different from the one of the T91 substrate at low frequencies. Due to the penetration depth of Rayleigh waves being of the order of the acoustic wavelength, and thus decreasing with increasing frequency, the value of the transition frequency from the substrate Rayleigh velocity to the coating Rayleigh velocity is directly linked to the thickness of the coating layer (Figure 2). For theoretical models predicting SAW dispersion in coating-substrate systems, we refer to references [33; 34; 35; 36]. The steepness of the transition is determined by the ratio of the densities of the coating and the substrate, and by the smoothness of the chemical composition gradient [37]. Using those models, the relations between the elastic and thermal properties and respectively the SAW and thermal wave dispersion were numerically implemented as Matlab(r) functions, which were then used for least square fitting the material parameters from the experimental data. Similar to acoustic waves, thermal diffusion is also characterized by a time dependent penetration depth, the so-called thermal diffusion length. In the case of a grating geometry of the generating light pattern, the time needed for the photothermally induced thermal expansion grating to fade away due to thermal diffusion is determined by the thermal diffusivity of the material and by the distance between the hot and cold regions of the grating, i.e., the grating spacing. In the case of an isotropic material, the thermal diffusion tangential to the surface happens at the same rate as the one normal to the surface. Therefore, the decay time of the photothermal part of the displacement signal is determined by the  
Figure 2: Illustration of Rayleigh (bottom Figure) and Sezawa (top Figure) wave phase velocity dispersion [27; 38; 39] for a coating-substrate configuration, for different thicknesses of the coating (1, 10, 100, 1000, 10000 micron). The used material parameters are (\(c_{L1}=4157\) m/s, \(c_{T1}=3473\) m/s, \(\rho_{1}=8100\) kg/m\({}^{3}\), \(c_{L2}=8852\) m/s, \(c_{T2}=6368\) m/s and \(\rho_{2}=3650\) kg/m\({}^{3}\)). The Rayleigh wave velocity saturates at the value of pure coating \(c_{R1}\) and substrate \(c_{R2}\) material in the high and low frequency limit, respectively. The Sezawa wave velocity decreases from the longitudinal velocity of the substrate \(c_{L2}\) in the low frequency range till the longitudinal velocity of the coating \(c_{L1}\), and then further decreases and saturates at the shear velocity of the coating \(c_{T1}\). thermal diffusivity of top layer of the sample till a depth of the order of the grating spacing. As a result, the effective signal decay time in case of a coating-substrate sample is determined by the thermal diffusivity of the coating if the grating spacing is shorter than the coating thickness, and evolves to the thermal diffusivity of the substrate as the grating spacing is increased to beyond the coating thickness. For the theoretical modeling and numerical simulations of the photothermally induced surface grating displacement (versus grating wavelength) in a coating-substrate system, we refer to reference [40] for the photothermally induced temperature depth profile (which is causing the subsequent thermal expansion and diffusion in multilayers in general), and to references [18; 41] for a full model for the temperature, strain depth profile, and surface displacement in a coating-substrate configuration.