(\omega) C^{i}_{2\omega}v^{i}_{2\omega}.$$ Therefore, once $T^{ii}_{12}(\omega)$ is specified, all the other transmission and reflection coefficients are determined. For now, we only consider this special case and neglect the mode conversion in our BTE simulations. Later, we show that the mode specific transmission coefficients cannot be resolved by the TDTR measurements and the measurable quantity is $\sum_{j}T^{ij}_{12}(\omega)$ instead of individual transmission coefficients. For simplicity, we will use $T_{12}(\omega,p) $ rather than the summation.

Justification of approximations {#sec:ApproxJustification}
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In this section, we justify the approximations made in the derivation of the transfer function. First, we have neglected the role of electrons, either in carrying heat in the metal or in carrying heat across the interface by direct coupling to substrate phonons. For the first point, in Appendix \[sec:electrons\], we solve the BTE explicitly including electron-phonon coupling and show that it has negligible effect on the TDTR signal and hence the transmission coefficients. For the second point, our approach cannot rule out this mechanism. However, the available evidence in the literature [@Lyeo2006; @lee_phonon_2013; @giri_mechanisms_2015] shows that the metal electron-substrate phonon coupling effect is too small to be observed for several material systems. Therefore, little evidence exists to support a large contribution from this 