are varied until the optimized transmission profiles are identified that best match the experimental data. Thus, the procedure is identical to that used in traditional TDTR measurements excepting the complications of deriving and inverting the microscopic transfer function.

Derivation of transfer function {#sec:Modeling}
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We now describe the derivation of the microscopic transfer function $Z(T_{12}(\omega), t)$ that maps transmission coefficients to the TDTR amplitude and phase data as a function of time. The result of the derivation is a function for which the only inputs are the phonon dispersions and lifetimes for each material composing the interface, and the only unknown parameters are the spectral transmission coefficients $T_{12}(\omega)$. The output of the function is the TDTR amplitude and phase signal versus delay time. For this work, the phonon dispersion and lifetimes for Si were calculated from first-principles with no adjustable parameters by Lucas Lindsay. We have also used first-principles inputs from Mingo *et al*. with no appreciable difference in results or conclusions.

Let us first briefly review the signal formation in TDTR. Since the thermal response given by the BTE is a linear time-invariant system, the output transient signal $Z(t)$ of TDTR can be represented in terms of frequency response 