The relaxation of an one-dimensional transient thermal grating (TTG) in a medium with phonon-mediated thermal transport is analyzed within the framework of the Boltzmann transport equation (BTE), with the goal of extracting phonon mean free path (MFP) information from TTG measurements of non-diffusive phonon transport. Both gray-medium (constant MFP) and spectrally dependent MFP models are considered. In the gray-medium approximation, an analytical solution is derived. For large TTG periods compared to the MFP, the model yields an exponential decay of grating amplitude with time in agreement with Fourier's heat diffusion equation, and at shorter periods, phonon transport transitions to the ballistic regime, with the decay becoming strongly non-exponential. Spectral solutions are obtained for Si and PbSe at 300 K using phonon dispersion and lifetime data from density functional theory calculations. The spectral decay behaviors are compared to several approximate models: a single MFP solution, a frequency-integrated gray-medium model, and a “two-fluid” BTE solution. We investigate the utility of using the approximate models for the reconstruction of phonon MFP distributions from non-diffusive TTG measurements.

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