## 3 Results and Discussion
### Calculation of Reflectivity Modulation  
We calculated the change in the dielectric constant and the resulting modulation of reflectivity due to perturbation of the electron distribution using a model previously applied to gold by [PERSON] et al. [20]. In this model, the onset of inter-band absorption is assumed to be the d-p transition near the L-point as supported by band structure calculations [21]. Within a constant matrix element approximation [21], the imaginary part of the dielectric constant close to this inter-band transition at frequency \(\omega\) is given by:  
\[\epsilon_{2}(\omega)\propto\frac{1}{\omega^{2}}\int\left(1-f(\mathbf{k},\ T_{e}) \right)\delta\left(E_{p}(\mathbf{k})-E_{d}(\mathbf{k})-\hbar\omega\right)\,d^{3 }k \tag{1}\]  
Figure 8: Comparison between plasmon modulation (red, (**b**)), current modulation (at 1.56 A, blue, (**c**)), temperature difference spectra (at \(\Delta\)T = +50 K, thin black line), and the second derivative of the reflectance (green, (**a**)) calculated from the reflectance in Figure 6a. where \(f\) is the electron distribution function in the conduction band at electron temperature \(T_{e}\) and \(E_{p}(\mathbf{k})\), \(E_{d}(\mathbf{k})\) are the band dispersions assumed to be parabolic with:  
\[E_{p}(k) = E_{pL}\ -\frac{\hbar^{2}k^{2}\|}{2m_{p\|}}\ +\ \frac{\hbar^{2}k^{2} \bot}{2m_{p\bot}}\] \[E_{d}(k) = E_{dL}\ -\frac{\hbar^{2}k^{2}\|}{2m_{d\|}}\ -\ \frac{\hbar^{2}k^{2} \bot}{2m_{d\bot}}\]  
Here, \(k_{\|}\) and \(k_{\bot}\) denote the components of \(\mathbf{k}\) along and perpendicular to the [111] direction, taking the L-point as the origin. For silver, we use \(m_{p\|}=0.32\), \(m_{p\bot}=0.17\), \(m_{d\|}=2.08\), \(m_{d\bot}=2.58\), \(E_{pL}=\ -0.155\) eV and \(E_{dL}=-3.818\) eV [13]. The constant of proportionality in Equation (1) is determined by fitting the result to the experimentally-measured dielectric constant of silver [22]. The change in the imaginary part \(\Delta\epsilon_{2}\) due to a change \(\Delta f\) of the electron distribution is given by replacing \((1-f)\) by \(-\Delta f\) in Equation (1). We consider two kinds of perturbations, (i) a temperature change of the electrons by \(\Delta T_{e}\), referred to as thermal and (ii) a rigid shift of the Fermi-sphere by \(\Delta\)k with the shift direction isotropically averaged, referred to as drift. The corresponding change in the real part (\(\Delta\epsilon_{1}\)) is obtained by Kramers-Kronig transformation of \(\Delta\epsilon_{2}\). For the unperturbed dielectric constant, we use the experimental values from [22]. The change in reflectivity is then readily obtained by applying the Fresnel formula to the thin-film geometry. The results are shown in Figure 9.  
The calculated reflectivity spectrum for a thermal distribution with \(\Delta T_{e}=0.16\) K (red line in Figure 9) agrees with the gross features of the experimental results for plasmon and current modulations shown in Figures 5 and 6. Although the drift perturbation gives a spectrum that is weaker and more spread out than the thermal case for the same energy content, the spectral shape is not remarkably different from the latter. Further, the predicted width of the spectrum is mainly determined by the dispersions of the bands and is not strongly dependent on the strength of the perturbation. These facts combined with the uncertainties in band structure parameters makes it difficult to clearly isolate any non-thermal components in the observed spectrum based on calculations alone. We conclude that, in order to unambiguously identify non-thermal electron populations, it is necessary to probe the system in a regime where such contributions dominate, viz., in short time scales immediately following the plasmon excitation. Figure 9: The calculated change in reflectivity for a 50 nm-thick Ag film on a quartz substrate for p-polarized light incident from the air side at an angle of incidence of 55 degrees.