# All-Optical Modulation and Ultrafast Switching in MWIR with Sub-Wavelength Structured Silicon
## 3 Optical Model  
Before we proceed to the discussion of the results and analysis, we present the optical model, which aims to approximate the optical response and the experimentally-measured reflectance of the membranes. The main points that should be included in the model are: (1) approximation of the membrane dielectric function by the Maxwell-Garnett mixing rule; (2) incorporation of the contribution of the optically-excited free charge carriers into the dielectric function using the high-frequency conductivity Drude theory; (3) accounting for the variation of charge carrier density along the sample depth by Wentzel-Kramers-Brillouin (WKB) approximation. Figure 1: (**a**) Change of the reflectance, \(\Delta R/R_{0}\), at the temporal overlap between the pump and signal as a function of signal wavelength \(\lambda\) recorded for the pump fluences of 3.8 mJ/cm\({}^{2}\) (red), 5.0 mJ/cm\({}^{2}\) (green), 6.3 mJ/cm\({}^{2}\) (blue) and 8.8 mJ/cm\({}^{2}\) (magenta). The dots denote the experimental results, and the solid lines show the model fitting. The arrows indicate the change of reflectance with increasing pump fluence at a given wavelength, on the shorter and longer spectral ends, blue and red, respectively. (**b**) The carrier concentration, \(N\), (blue) at the sample surface and its scattering rate, \(\gamma\), (red) as a function of the pump fluence. The curves were reconstructed from the experimental data shown in (**a**). The insert shows the top view of the structured silicon used in this work. The reflectance of a single-layer thin slab based on the WKB approximation is given by:  
\[R=\left|\frac{r(0)e^{-i\psi}-r(d)e^{i\psi}}{e^{-i\varphi}-r(0)r(d)e^{i\varphi}} \right|^{2}\, \tag{1}\]  
with the reflection coefficients of the front \(r(0)=\dfrac{k_{z}-q(0)}{k_{z}+q(0)}\) and rear \(r(d)=\dfrac{k_{z}-q(d)}{k_{z}+q(d)}\) interfaces, respectively. The \(z\)-coordinate was chosen to be along the membrane depth. \(k_{z}\) and \(q\) are \(z\)-components of the wavevector of light outside and inside the material, respectively. \(\psi=\int_{0}^{d}dzq(z)\) is the complex phase accumulated by the signal propagating through the sample with the thickness \(d=15\)\(\upmu\)m. \(q(z)=\sqrt{\dfrac{\omega^{2}}{c^{2}}e_{eff}(z)-k_{x}^{2}}\) is the position-dependent wavevector of the signal along the sample depth, while \(k_{x}\) is the \(x\)-component oriented parallel to the interfaces and has the same value of \(\dfrac{\omega}{c}sin\theta_{i}\) outside of the material and at any location inside it; here, \(\theta_{i}\) is the angle of incidence. It can be seen that the model relies on the knowledge of the wavevector \(q(z)\), which is needed for the calculation of the phase, \(\psi\), and the reflection coefficients \(r(0)\) and \(r(d)\). To find \(q(z)\), one needs a good approximation of the effective dielectric function, \(e_{eff}(z)\). The membrane structure has to be taken into account and the contribution of the free carriers excited by the pump should be considered. The later is, in general, non-uniform along the slab depth, but it can be approximated by \(N(z)=\frac{F}{\hbar\omega_{pump}}\alpha_{pump}exp(-\alpha_{pump}2)\), where \(F\), \(\alpha_{pump}\) and \(\omega_{pump}\) are the fluence, absorption coefficient and photon frequency of the pump. The dielectric function, \(\epsilon_{eff}(z)\), which includes both contributions, the structures of the membrane and non-uniform longitudinal carrier distribution, can be calculated as follows:  
\[\epsilon_{eff}(z)=\epsilon_{i}+2 pe_{i}\frac{1-\epsilon_{i}}{1+\epsilon_{i}-p( 1-\epsilon_{i})}-\frac{e^{2}}{\epsilon_{0}m_{opt}}\frac{N(z)}{\omega^{2}+i \omega\gamma}\, \tag{2}\]  
where \(\epsilon_{i}=11.7\) is the intrinsic dielectric function of the bulk silicon, \(p=0.55\) is the volume fraction of the air pores, \(m_{opt}=0.17\) and \(\gamma\) are the optical mass and the scattering rate of the free carriers, and \(\omega\) is the frequency of the probe signal.