# Pockels Effect at the Interface between Water and Ti Electrode
## 3 Experimental Results
### Results of Impedance Measurement  
The results of the impedance measurement are shown in Figure 7 as a Nyquist plot (complex plane) and a bode diagram (magnitude and phase as a function of frequency). They were analyzed using the ZVieww 3.5g software. The equivalent circuit was estimated so that the fitting curves matched the Nyquist plot and the Bode diagram. The best fit was obtained using the equivalent circuit in Figure 8. In this case, the two electrodes are symmetrical because they are opposing Ti electrode plates with the same surface area. Therefore, the circuit should be valid for each electrode. Figure 7 shows the results of fitting with the equivalent circuit in Figure 8. The values of each circuit element obtained from the fitting are summarized in Table 1.  
Figure 5: DC voltage dependence of electroreflectance spectrum (\(X\) signal at 2 V AC and 20 Hz. As in Figure 4, the \(Y\) signal was neglected because it is much smaller than the \(X\) signal). Figure 6: Electroreflectance spectra under steady-state UV irradiation. Figure 8: The equivalent circuit assumed for fitting in Figure 7. \(W\) represents the Warburg impedance. The impedance and other information indicated by each element are given in Appendix B. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Resistance** & **Constant Phase Element** & **Warburg Impedance** \\ \hline \multicolumn{3}{c}{Bulk} \\ \hline \(R_{B}[\Omega]\) & \\ \hline
31.9 & \\ \hline \multicolumn{3}{c}{Space Charge Layer (SCL)} \\ \hline \(R_{S}[\Omega]\) & \(CPE_{S}-T\left[\
icefrac{{\mathrm{g}}}{{\mathrm{g}}}\Omega^{-1}\right]\) & \(CPE_{S}-P\) \\ \hline \(1.368\times 10^{6}\) & \(2.276\times 10^{-5}\) & 0.9249 \\ \hline \multicolumn{3}{c}{Electric Double Layer (EDL)} \\ \hline \(R_{EDL}[\Omega]\) & \(CPE_{EDL}-T\left[\
icefrac{{\mathrm{g}}}{{\mathrm{g}}}\Omega^{-1}\right]\) & \(CPE_{EDL}-P\) & \(W-R\)\([\Omega]\) & \(W-T[\mathrm{s}]\) \\ \hline
72.27 & \(1.507\times 10^{-4}\) & 0.6602 & 451.8 & 0.8839 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The values of each element in the equivalent circuit in Figure 8 are determined by fitting in Figure 7. The meaning of each fitting parameter is described in Appendix B.  
Figure 7: Results of the impedance measurement (dashed lines) and fitting curves (solid lines). The Ti surface oxide film is known to be an N-type semiconductor TiO\({}_{2}\) and many impedance measurements have been made on it [21; 22; 23; 24]; the magnitudes of the resistance and capacitance components of the TiO\({}_{2}\) space charge layer (SCL) in Table 1 are in good agreement in order of magnitude with those values reported previously. The magnitudes of the resistance and capacitance components of the EDL of water are also in order of magnitude agreement with previously reported values [25]. From the results in Table 1, the voltage distributed to each layer can be calculated. To find the Pockels coefficient in the EDL of water at the Ti electrode interface, the distributed voltage within the EDL is calculated. The impedances of the SCL and the EDL are \(Z_{S}\) and \(Z_{EDL}\), respectively. \[\frac{1}{Z_{S}}=\frac{1}{R_{S}}+\frac{1}{Z_{CPE-S}},\ \ \frac{1}{Z_{EDL}}=\frac{1}{R _{EDL}+Z_{W}}+\frac{1}{Z_{CPE-EDL}} \tag{1}\]  
\[Z_{all}=R_{B}+Z_{S}+Z_{EDL} \tag{2}\]  
\[V_{S}=\frac{Z_{S}}{Z_{all}}V,\ V_{EDL}=\frac{Z_{DEL}}{Z_{all}}V,\ V_{B}=\frac{R _{B}}{Z_{all}}V \tag{3}\]  
The AC voltage applied is 2.0 V between the electrodes so that the voltage \(V\) at each electrode is 1.0 V. Note that \(V_{S}\) and \(V_{EDL}\) obtained in the above process are complex numbers, and the following distributed voltages are absolute values. The distributed voltage to the EDL at 20 Hz and 2.0 V was calculated from the values of each element obtained by impedance measurement to be \(|V_{EDL}|=0.145\) V. On the other hand, the voltages distributed to the SCL and bulk water + bulk Ti are \(|V_{S}|=0.895\) V and \(|V_{B}|=0.0567\) V, respectively.