the two layers depend on each other through the interface conditions that enforce conservation of heat flux.

Transducer film {#subsec:film}
---------------

The metal thin film serves as an optical transducer that absorbs the incident optical energy while also enabling the observation of temperature decay through the thermoreflectance coefficient. In our work, we neglect electrons and consider that heat is only carried by phonons in Al. Justification for this approximation is given in Appendix \[sec:electrons\].

Since the system is modulated at a given frequency $\eta$, we can assume that both $g_{1\omega}$ and $\Delta T_1$ are of the form $e^{i\eta t}$ to define $g_{1\omega} = \widetilde{g}_{1\omega}(x_1,\mu)e^{i\eta t}$ and $\Delta T_1 = \Delta \widetilde{T}_1(x_1)e^{i\eta t}$. The volumetric heat generation in thin film is given by $Q_{\omega} = Q^0_{\omega}e^{i\eta t}e^{-x_1/\delta}$, where the amplitude of heating source $Q_0 = \int_0^{\omega_m} Q^0_{\omega} d\omega$. We also assume that phonons are specularly reflected at $x_1 = 0$, *i.e.* $\widetilde{g}_{1\omega}(x_1=0,\mu) =\widetilde{g}_{1\omega}(x_1=0,-\mu)$. Substituting the definition of $\widetilde{g}_{1\omega}$ and $\widetilde{T}_1(x_1)$ and the specular boundary condition at $x_1 = 0$ into Eq. (\[eq:BTE\]) leads to a first-order ODE for $\widetilde{g}_{1\omega}(x_1,\mu)$. Its solution is given by $$\begin{aligned}
\label{eq:ODEsolution_gPlus}\nonumber
\widetilde{g}^+_{1\omega}(x_1,\mu) &=& B_{\omega}e^{-\frac{\gamma_{1\omega}}{\mu}(d+x_1)}+\int_0^{d}\frac{C_{1\omega}\Delta \widetilde{T}(x'_1)+Q^0_{\omega}e^{-x/\delta}\tau_{1\omega}}{4\pi\Lambda_{1\omega}\mu}e^{\frac{\gamma_{1\omega}}{\mu}(x'_1+x_1)}dx'_1 \\ 
&+& \int_0^{x_1}\frac{C_{1\omega}\Delta \widetilde{T}(x'_1)+Q^0_{\omega}e^{-x/\delta}\tau_{1\omega}}{4\pi\Lambda_{1\omega}\mu}e^{\frac{\gamma_{1\omega}}{\mu}(x'_1-x_1)}dx'_1\ (\mu \in (0, 1]) \\ \label{eq:ODEsolution_gMinus}\nonumber
\widetilde{g}^-_{1\omega}(x_1,\mu) 