## 2 Results  
For the sample being translated through the focus of a beam with a Gaussian profile and considering only the third order nonlinearity, the total time-integrated transmission, \(T_{open}\), for the open aperture can be expressed as [16; 17; 18]:  
\[T_{open}(z)=1-\frac{1}{2\sqrt{2}}\frac{\beta I_{0}L_{eff}}{1+(\frac{z}{z_{0}}) ^{2}}\quad, \tag{1}\]  
where \(I_{0}\) is the on-axis peak intensity and \(L_{eff}=\frac{1-e^{-ad}}{\alpha}\) is the effective optical length of the sample with thickness \(d\) and linear absorption coefficient \(\alpha\). \(z\) is the sample position relative to the beam focal point, and \(z_{0}=2\pi\omega_{0}^{2}/\lambda\) is the confocal parameter of the beam with waist \(\omega_{0}\) and wavelength \(\lambda\). For the closed aperture, the transmittance is given by:  
\[T_{close}(z)=1+\frac{4(\frac{z}{z_{0}})\Delta\Phi}{(1+(\frac{z}{z_{0}})^{2}) (9+(\frac{z}{z_{0}})^{2})}-\frac{\beta I_{0}L_{eff}}{2\sqrt{2}}\frac{(3-( \frac{z}{z_{0}})^{2})}{(1+(\frac{z}{z_{0}})^{2})(9+(\frac{z}{z_{0}})^{2})}\quad; \tag{2}\]  
here, \(\Delta\Phi\) is the time-averaged phase change induced by the nonlinear refraction, which can be approximated as:  
\[\Delta\Phi=\frac{2\pi}{\sqrt{2}\lambda}(1-S)^{0.25}n_{2}I_{0}L_{eff}\quad, \tag{3}\]  
where \(S\) is the transmissivity of the aperture. The Gaussian decomposition method requires the proper determination of the effective optical length, \(L_{eff}\), which can be deduced from the linear optical constants of the \(c\)-Si and \(p\)-Si samples. For the \(c\)-Si samples, the linear absorption coefficient can be estimated by \(\alpha=\frac{4\pi k}{\lambda}\), where \(k\) is the imaginary part of linear refractive index. At an 800-nm wavelength, \(k=0.0065\), as it was estimated by the spectroscopic ellipsometry method by [PERSON] et al. [19], corresponding to \(\alpha=1.03\times 10^{3}\) cm\({}^{-1}\). However, the refractive index and absorption coefficient values for \(p\)-Si are less known since these values depend on the porosity, silicon skeleton dimensionality and surface adsorbates [20]. It has been shown previously that the Maxwell-Garnett mixing rules can be applied in order to calculate the linear complex permittivity of \(p\)-Si at 800 nm [21; 22]:  
\[\epsilon_{p-Si}=\epsilon_{c-Si}+2\epsilon_{c-Si}\frac{f\frac{e_{ air}-\epsilon_{c-Si}}{e_{air}+\epsilon_{c-Si}}}{1-f\frac{e_{air}-\epsilon_{c-Si}}{ e_{air}+\epsilon_{c-Si}}} \tag{4}\]  
where \(\epsilon_{c-Si}\) and \(e_{air}\) are the permittivity of \(c\)-Si and air pores (\(e_{air}=1\)), respectively. \(f\) is the volume fraction of air pores, which was estimated to be 68% for the samples used in this work. Thus, we obtain that \(k=0.0025\) and \(\alpha=386\) cm\({}^{-1}\). It is worth mentioning that the Maxwell-Garnett mixing rules are valid for these samples since the wavelength is about two orders of magnitude greater than the dimensions of the \(p\)-Si components. As for the 2.4-\(\upmu\)m wavelength, there were no explicitly known values for the imaginary part of the refractive index of intrinsic \(c\)-Si. This was due to the insignificant absorption for the incident photon energy much smaller than the band-gap. However, the silicon skeleton of \(p\)-Si that was used in this work was boron-doped, which provided about 3\(\times\)10\({}^{18}\) cm\({}^{-3}\) free carriers [23]. Hence, the free carrier absorption needed to be considered, which further led to an increase of the \(k\) value. The influence of free carriers introduced by boron dopants on the \(c\)-Si dielectric function can be estimated using the Drude model:  
\[\epsilon_{doped}=\epsilon_{undoped}-\frac{Ne^{2}}{(\omega^{2}+i\gamma\omega)}; \tag{5}\]  
where \(N\) is the carrier concentration induced by dopants, \(e\) denotes the electron charge, \(m_{h}=0.36\) is the effective hole mass and \(\epsilon_{0}\) is the vacuum permittivity. The carrier-carrier scattering rate \(\gamma\) was chosen to be \(1\times 10^{13}(s^{-1})\), which is a typical value for doped silicon [24].