the substrate ballistically. As illustrated in Fig. \[fig:Schematics\] (a), when MFPs are much shorter than the characteristic length scale of the thermal gradient, information about the phonon distribution at the interface is lost due to scattering. In this case, the TDTR signal is largely insensitive to the transmission coefficient profile so long as the overall interface conductance remains fixed.

On the other hand, if some phonons have sufficiently long MFPs, the non-equilibrium phonon distribution penetrates into the substrate. In this case, as we will show in subsequent sections, the TDTR signal depends not only on the magnitude of the interface conductance but on the spectral profile of the transmission coefficients. It is this sensitivity that we will exploit to retrieve the transmission coefficients from the TDTR data sets.

This discussion implies that not every substrate will be suitable for our approach as sufficiently long phonons MFPs compared to the induced thermal gradient are required. Fortunately, many experimental reports have demonstrated clear evidence of this quasiballistic heat transport regime in different material systems [@Koh2007; @Siemens2010; @Minnich2011a; @Regner2012; @Johnson2013; @Vermeersch2015a]. In particular, MFPs in Si are generally accepted to exceed one micron at room temperature. [@Cuffe2014] Considering that thermal penetration depths in 