# All-Optical Modulation and Ultrafast Switching in MWIR with Sub-Wavelength Structured Silicon
## 4 Wavelength Dependence and Ultrafast Temporal Response of Optically-Pumped Membranes
Furthermore, it can be shown that in semiconductors at the conditions around the plasma edge for a given signal wavelength, \(\lambda\), the absolute value of reflectance change as a function of the carrier concentration has the relation of \(\big{|}\frac{\Delta R}{R_{0}}\big{(}N\big{)}\big{|}_{\lambda_{i}>0<e_{eff}<1} >\big{|}\frac{\Delta R}{R_{0}}\big{(}N\big{)}\big{|}_{\lambda_{i}<e_{eff}>1}\). This discussion equally applies to \(\frac{\Delta R}{R_{0}}\big{(}F\big{)}_{\lambda}\), assuming that \(N\) is directionally proportional to \(F\). Thus, the opposite sign and the difference of the absolute values of the arrows in Figure 1a indicated that the system was pumped in the region of dielectric-metal transition, where at the shorter wavelength \(\epsilon_{eff}>1\), while at longer wavelength approaching 6 \(\upmu\)m \(\epsilon_{eff}<1\) and at the crossover around 5 \(\upmu\)m \(e_{eff}=1\). We note that, compared to the bulk semiconductor, less pumping was required to achieve this transition with structured membranes, as their dielectric function was lower due to the second negative term in Equation (2), related to the dilution of \(\epsilon_{i}\) by the holes. Figure 1b shows the fitting parameters used to calculate \(\Delta R/R_{0}\). To improve the accuracy, we used a calculation routine that iteratively determines the real and imaginary parts of the effective dielectric function, \(e_{eff}\), in Equation (2) from the amplitude and slope of the experimentally measured \(\Delta R/R_{0}\) as a function of \(\lambda\)[24]. The fitting results suggested that with the carrier concentration above \(8\times 10^{19}\) cm\({}^{-3}\) and scattering rate of about 1.5 \(\times\)\(10^{14}\) s\({}^{-1}\), dielectric-metal transition occurred for signals with \(\lambda>5\)\(\upmu\)m. Moreover, it can be seen that \(N\) is a non-linear function of \(F\); therefore, we had to correct our initial assumption of the linear absorption of the pump pulse. We note that a non-linear absorption in silicon optical membranes is not well-understood and reported, and we leave this point out of the arguments of this work. In addition, the scattering parameter, \(\gamma\), seemed to saturate at the higher fluences. We did not investigate this observation at a greater depth; however, the most likely reason for this is the saturation of the carrier-carrier collisions at the non-equilibrium conditions [28]. To further investigate the optical response and the temporal evolution of the spectra at different pump fluences, we performed ultrafast pump-probe measurements over a few picoseconds' delay time between the pump and probe pulses. During this time, the injection of the charge carries reached saturation as the excitation process had been completed, but the recombination had not developed yet to affect the carrier density and optical response. We note that the temporal resolution was degraded by the convolution of the ultrafast excitation and somewhat longer probe pulse. The results are shown in Figure 2a-d for the pump fluences of 3.8, 5.0, 6.3 and 8.8 mJ/cm\({}^{2}\), respectively. These measurements can be viewed as the evolution of the curves shown in Figure 1a as a function of time (along the horizontal axis). Figure 2a-c show that after the excitation completed, the spectra did not change significantly after 1 ps as a function of the delay time, \(\Delta t\). That is, once the equilibrium was achieved, the optical properties remained stable. However, in Figure 2d, we observe a peculiar feature around 6 \(\upmu\)m and the delay time of 1 ps, as indicated by the dashed rectangular. Unlike anywhere else, in this region, the signal passed through the minimum before it recovered and stabilised. This curious behaviour was interesting for both understanding the physics behind the process and the possible development of applications. Indeed, such an ultrafast drop and recovery of the reflectance signal at a particular wavelength can be used for the extremely fast reflective-type all-optical switching.