## 4 Microwave Characterization of HR Silicon Material under NIR Illumination  
Using the same experimental set-up, S-parameters measurements of a square HR-silicon sample of with a 2 cm\({}^{2}\) cross section and 500 um thickness are performed at room temperature under optical power up to 10 W and wavelength of 805 nm. Under light illumination, as described in Figure 5, microwave incident plane wave is polarised either perpendicularly or parallel to its plane of incidence (TE or TM mode respectively) onto successively air/plasma, plasma/HR-Si and HR Si/air layers interfaces, synthesized as shunt tranverse impedances representation for each individual propagating medium. \[Z_{2}(z_{1})=Z_{0} \tag{6}\]  
Boundary conditions on tangential fields can be applied at \(z=z_{0}\). Conservation of both tangential/normal electric and normal/tangential magnetic fields through each interface may be achieved by enforcing continuity of transverse impedance \(Z_{1}^{(TM)}(z_{0})=Z_{2}^{TM}(z_{0})\) and \(Z_{1}^{(TE)}(z_{0})=Z_{2}^{TE}(z_{0})\). Then,  
\[Z_{1}^{(\alpha)}(z_{0})=\chi^{(\alpha)}\frac{Z_{0}^{(\alpha)}+\chi^{(\alpha)} \tanh(jk_{z1}\delta)}{\chi^{(\alpha)}+Z_{0}^{(\alpha)}\tanh(jk_{z1}\delta)}, \quad\{\alpha=TM,TE\} \tag{7}\]  
From (7), theoretical respective reflection coefficients \(\Gamma_{0}^{(TM)}(\varepsilon_{r}^{*},\mu_{r}^{*})\) and \(\Gamma_{0}^{(TE)}(\varepsilon_{r}^{*},\mu_{r}^{*})\) can be established and expressed as follows,  
\[\Gamma_{0}^{(\alpha)}(\varepsilon_{r}^{*},\mu_{r}^{*})=\frac{Z_{1}^{(\alpha)} (z_{0})-Z_{0}^{(\alpha)}}{Z_{1}^{(\alpha)}(z_{0})+Z_{0}^{(\alpha)}},\quad\{ \alpha=TM,TE\} \tag{8}\]  
Figure 4: Experimental reflection coefficient values in magnitude and phase ofa 2 cm\({}^{2}\) metallic plate as a function of microwave frequency before and after SSS calibration (\(\theta_{0}=28^{\circ}\) and D = 50 cm)  
Figure 5: Two independent reflection measurements. The free space impedance and plasma characteristic impedance, (\(Z_{0}^{\mathrm{g}}\), \(\chi^{\mathrm{a}}\)), for the TE and TM modes are given for different angles of incidence \(\theta_{0}\) as  
\[ \begin{split} Z_{0}^{(TM)}=\eta_{0}\cos\theta_{0}\\ Z_{0}^{(TE)}=\frac{\eta_{0}}{\cos\theta_{0}}\\ \chi^{(TM)}=\frac{k_{z1}}{\omega e_{r1}^{*}\epsilon_{0}}\\ \chi^{(TE)}=\frac{\omega h_{r1}^{*}\mu_{0}}{k_{z1}}\\ \end{split}  \tag{9}\]  
with \(k_{z1}=\sqrt{k^{2}-k_{0}^{2}\sin^{2}\theta_{o}}\) is the propagation constant along the normal direction to the dielectric. Where \(k=\omega\sqrt{\epsilon_{1}\mu_{1}}\) and \(\eta_{0}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}\) is the free space impedance. From experimental results, photoconductive material complex permittivity is extracted by using a numerical optimization procedure based on genetic algorithm [18]. From (2), photoconductance value \(\Delta\sigma\), attributed to the electron/hole pairs plasma layer presence induced by light absorption inside the semiconducting material, is deduced as a function of microwave frequency. In complement, from shunt impedance de-embedding calculations, a plasma profile \(\Delta\sigma\) is obtained as a function of depth inside the material, as in Figure 6 at a fixed microwave frequency of 50 GHz, and Figure 7 shows the plasma conductivity according the optical power of the NIR source and a fixed plasma depth \(t_{p}\) = 50 \(\upmu\)m. From fitting calculations, material absorption coefficient \(\alpha=234\times 10^{3}\) m\({}^{-1}\), and carrier lifetime of \(\tau=158\times 10^{-9}\) s at 805 nm are simultaneously extracted from measurement results. Figure 6: Extraction of the HR Si substrate photoconductance \(\Delta\sigma\) (left) Real part and (right) imaginary part according to the plasma depth \(t_{p}\) of the HR Silicon illuminated by optical power Popt = 6 W at wavelength \(\lambda=805\) nm and F = 50 GHz.