modulation frequency $\omega_0$, $\eta$ is chosen to be $\omega_0+n\omega_L$, where integer $n$ is typically ranging from -50 to 50. Using Eq. (\[eq:TDTRSignal\]) yields $Z(T_{12}(\omega), t)$, which is directly compared to experimental data $Z_{exp}(t)$.

Sensitivity of TDTR signal to transmission coefficients
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With the transfer function obtained, it is useful to re-examine the conditions that are required to successfully extract transmission coefficients from TDTR data. First, the number of unknowns should be on the order of the number of data points. As an estimate, consider that we take TDTR data at five modulation frequencies. As in Eq. (\[eq:TDTRSignal\]), in the frequency domain each TDTR signal consists of surface temperature responses at different frequencies, the lowest of which is the modulation frequency. Typically, among those surface temperature responses, the responses at the lowest four to five frequencies contain most of the thermal information.[@Cahill2004] With around five TDTR data sets containing amplitude and phases at five different modulation frequencies, we obtain 40-50 unique data points.

Now consider the number of unknowns. If we take the materials to be isotropic, the transmission coefficients depend only on phonon frequency. In our typical discretization, we find that there around one hundred transmission coefficients that must be 