mechanism. As a result, we assume that heat is carried across the interface solely by phonons.

The second assumption is the neglect of mode conversion, or the change of phonon polarization as phonons transmit or reflect at the interface. In Appendix \[sec:ModeConversion\], we explicitly calculate the transfer function including this mechanism, showing that it has very little impact on the results. Therefore, fitting the data with or without mode conversion yields identical results.

Overview of calculation
-----------------------

We now describe how these pieces fit together to provide the transfer function $Z(T_{12}(\omega),t)$. At any given frequency $\eta$, the analytical expression of the unknown distributions from both sides, $g_{1\omega}^{\pm}$ and $g_{2\omega}^{\pm}$ can be obtained according to Sec. \[subsec:film\] and Sec. \[subsec:substrate\], which then are evaluated at $x_1 = d$ and at $x_2 = 0$, respectively. Given the values of $T_{12}(\omega)$, the values of $R_{12}(\omega)$, $T_{21}(\omega)$ and $R_{21}(\omega)$ can be inferred, and the unknown coefficients $P_{\omega}$ and $B_{\omega}$’s are obtained by plugging Eqs. (\[eq:ODEsolution\_gPlus\]), (\[eq:ODEsolution\_gMinus\]), and (\[eq:SubstrateDistribution\]) into Eqs. (\[eq:IC\_1\_special\]) and (\[eq:IC\_2\_special\]) and solving the linear system.

Once $B_{\omega}$ is known, the temperature profile in the transducer film is solved using Eq. (\[eq:FilmTemp\]) as well as the surface temperature $H(\eta)$. At a given 