# All-Optical Modulation and Ultrafast Switching in MWIR with Sub-Wavelength Structured Silicon
## 4 Wavelength Dependence and Ultrafast Temporal Response of Optically-Pumped Membranes  
Figure 1a shows the reflectance change, \(\Delta R/\mathrm{R_{0}}=(\mathrm{R_{e}}-\mathrm{R_{0}})/\mathrm{R_{0}}\), measured at the temporal overlap between the pump and signal pulses, as a function of the signal wavelength and for the different fluences of the pump, spanning between 3.8 and 8.8 mJ/cm\({}^{2}\). \(\mathrm{R_{e}}\) and \(\mathrm{R_{0}}\) denote the reflectance with and without application of the pump pulse, respectively. Figure 1a presents the experimental results as the dots and the model fitting, described in the previous section, as the solid lines. The corresponding carrier concentration used in the calculation of the optical model, \(N\), and scattering parameter, \(\gamma\), as a function of the pump fluence are shown in Figure 1b. Note that \(N\) and \(\gamma\) are functions of the fluence only and were not influenced by the probe, as its fluence was too weak and the photon energy too small to alter them. In general, it can be seen that the optical model can be adequately fitted to the experimental results, providing a background to discuss the physics behind the findings and the feasibility to use the membranes for a reflective-type light modulator. It can be seen that at certain conditions, the modulation efficiency can reach beyond 80%, yet it is not uniform over the entire range of wavelengths. At the lower fluence of about 4 mJ/cm\({}^{2}\), the greatest modulation was observed at the longer wavelength of the spectrum around 6 \(\upmu\)m, while at the highest fluence, the situation was reversed: it was the shorter wavelength around 4 \(\upmu\)m that was the most affected. Such dependence of the reflectance spectra on the pump fluence can be considered for use in tunable spectral filters. At the lower pumping, the membrane acted as a shortwave reflecting filter,i.e., at 3.8 mJ/cm\({}^{2}\), almost 70% of the 4 \(\upmu\)m probe was reflected back, while for 6 \(\upmu\)m, it was only 15%. At the highest fluence, the filter reflectance as a function of the wavelength was flipped over: it behaved as a a long-wavelength reflective filter with 90% of 6 \(\upmu\)m and 25% of 4 \(\upmu\)m reflected back, respectively. Overall, there are two critical details that need to be explained, which are the sign of the reflectance change as a function of the fluence, F, for a given wavelength \(\lambda\), \(\frac{\Delta R}{R_{0}}\big{(}F\big{)}_{\lambda}\), indicated by the arrows on Figure 1a, and its absolute value characterising the modulation contrast. It can be seen that \(\frac{\Delta R}{R_{0}}\big{(}F\big{)}_{\lambda}\) has the opposite sign: it was negative for the shorter wavelength and reversed to the opposite for the longer wavelength, while its absolute value was greater. To understand this behaviour, one should examine Equation (2) in terms of the evolution of \(\epsilon_{eff}\) as a function of the injected free carrier density, \(N\), and relate it to the reflectance. When the carrier density, \(N\), increased, the dielectric function, \(\epsilon_{eff}\), monotonically dropped, yet the reflectance behaviour was more complex than that. At the lower carrier injection, when the third term in Equation (2) was smaller than the sum of the first two, the increase of \(N\) led to the decrease of the reflectance, while at a certain point where \(e_{eff}\to 1\), the minimum was reached, and the reflectance started to increase, reaching the maximum at \(e_{eff}=0\). The change of the sign at \(\frac{dR}{dc}\big{|}_{\epsilon=1}\) corresponds to the onset of the transition of the optical response from dielectric to metal, also known as the plasma edge.