Simulation of the thermal and acoustic response of an elastically anisotropic solid to a nanosecond laser pulse in transient grating spectroscopy

TL;DR

This work tackles the challenge of characterizing thermal diffusivity and elastic dispersion in elastically anisotropic solids via transient grating spectroscopy. It introduces a two-dimensional thermomechanical finite-element framework that explicitly couples surface heating, thermoelastic response, and optical heterodyne detection, while handling anisotropy with custom elements. The model uses a Gaussian surface heat source with spatial periodicity and computes the heterodyned signal from surface displacement, enabling direct comparison with time- and frequency-domain TGS data. Validation against Ni(110) shows good agreement for both time-domain evolution and frequency-angular dispersion, including ultra-transient acoustic features, highlighting the method’s potential for in silico exploration of anisotropy and experimental parameter effects.

Abstract

Transient grating spectroscopy (TGS) is a material characterization technique based on laser-induced thermoelastic excitation of thermal and acoustic gratings. On opaque samples, these gratings are dynamic surface displacements that reflect the sample's elastic and thermal properties, enabling both types of parameters to be determined from a single experiment. Here, we develop a detailed finite element model (FEM) simulation of the TGS experiment that fully captures the coupling between the thermal and mechanical fields, as well as the optical detection of surface displacement using a heterodyning approach. Using custom-designed two-dimensional elements, the model is particularly suitable for analyzing TGS measurements on anisotropic media, for which analytical theory is insufficient. The simulation captures not only the anisotropic relaxation of the thermoelastic field but also several acoustic features that arise at very short (ultra-transient) timescales and provide additional information about the elastic properties of the examined material. The model offers new opportunities for the in silico testing of various modifications of TGS experiments and their applications to a broad class of materials.

Figures (5)

• Figure 1: Simplified schematics of the optical paths in the TGS method (a), illustrating the excited grating at the surface (b), and the correspondence with the computational domain (c). The computational domain is shown with illustrated calculated (largely magnified) displacement, where the shading suggests the $y$-axis displacement magnitude.
• Figure 2: Simulated displacement fields ($u_x$,$u_y$, and $u_z$) for the direction 30 off the [001] in Ni(110) at various time steps: reaching the maximum surface temperature (2.2ns, note that the thermal pulse, Eq. \ref{['eq:thermalPulse']}, is centered around $t_P = 2ns$), followed by specific times of the SAW oscillation. Below are the corresponding temperature fields showing the gradual homogenization at the surface and its spread towards the depth. Note the different scales in $x$ and $z$ direction, and also between the displacement and temperature fields.
• Figure 3: (a) Comparison of experimental (blue) and simulated (black) time-domain signals obtained from measurement on Ni(110)[001] at the acoustic wavelength $\lambda = 10µm$ calculated for the element size of 25nm (b) Close-up on the first 10 nanoseconds after the pump pulse. (c) Simulated surface displacement profiles at given time steps marked in (b).
• Figure 4: Simulated time-domain signals comparing attenuation (lifetime) of the acoustic oscillations with mesh sizes of 25, 50, and 100nm shown in black, red, and yellow, respectively. While the acoustic lifetime differs significantly, thermal profile agrees with the quasi-static simulation omitting the acoustic dynamics (denoted in green).
• Figure 5: (a) Frequency spectra of measured (blue) signal and signal obtained with the mesh size of 25nm (black) and 50nm (red), obtained for a direction 0, 30, 60, and 90 off the direction [001]. The amplitudes were adjusted for the best fit of the low-amplitude features. (b) Frequency-angular dispersion maps measured by TGS and obtained by FEM, respectively. Note that the simulated map was obtained with the 50nm mesh for the sake of computational demands.