## 3 Results and Discussion
### Evaluation of the Pockels Coefficient of the Interfacial Water
#### 3.4.3 Magnitude and Phase of Voltage Distributed in the Electric Double Layer of Water
In fact, in the Pockels effect of interfacial water on transparent electrode ITO and metal electrode Pt, \(Y\) is inverted (delayed) with respect to \(X\) at frequencies of this range, as shown in Figure 11, being consistent with the impedance measurement. Note that the results in Figure 11b are the same in mechanism as the present signal in that the resonance reflectance dip is generated by coherent perfect absorption [20; 21]. \begin{table}
\begin{tabular}{c c c c} \hline \(d_{w}\) & \(\Delta n\) & \(\Delta nd_{w}\) & Energy Shift (eV) \\ \hline
300 & 0.001 & 0.3 & 0.022 \\ \hline
100 & 0.003 & 0.3 & 0.025 \\ \hline
30 & 0.01 & 0.3 & 0.028 \\ \hline \end{tabular}
\end{table} 
Table 2: The energy shift of the SPP resonance from that with n = 1.330 in Figure 8c. Figure 9: (a) Cole–Cole plot (\(-\)Im \(Z\) against Re \(Z\)). The green circle shows the impedance at 224 Hz, closest to 222 Hz. (b) Equivalent circuit assumed for the electrolyte bulk solution and the electric double layer (EDL). Although it is difficult to find a reason, if the phase of \(r_{EDL}\) is advanced by about \(\frac{\pi}{2}\), as shown in Figure 10, we can fairly well reproduce the behavior observed in the experiment, where \(Y\) is dominant at lower frequencies and \(X\) and \(Y\) have the same sign. Alternatively, if the phase is delayed by about \(\frac{\pi}{2}\), the behavior can also be explained as \(-X\) and \(-Y\), which are 180\({}^{\circ}\) reversed. In the former case, the resonance dip is red-shifted, i.e., \(\Delta n\) is positive (with applied voltage positive; \(r_{ij}\) is negative from the formula \(\Delta n=n_{i}-n_{w}=-n_{w}^{3}r_{ij}E_{j}/2\)). On the other hand, in the latter case, the opposite is true. Presently, the mechanism of this additional phase change with respect to the phase of the distributed voltage is not obvious. Here, we just list three possible reasons for the additional phase. 1. It may be a unique effect of the plasmon. However, to the best of our knowledge, there have been no experimental and theoretical reports about the additional phase of the plasmon in response to the dynamical refractive index change. It is unlikely that the plasmon does not instantaneously respond to the refractive index change at the interfacial dielectric around this low frequency range. 2. The \(\Delta n\) of the interfacial water on Ag surface rises with the additional phase (nearly \(\pm\frac{\pi}{2}\)) in response to the electric field falling in the EDL. As shown in Figure 11, however, as long as the surface on ITO and Pt electrodes are concerned, the Pockels effect of interfacial water nearly instantaneously responds to the field around this low frequency range. 3. The equivalent circuit was not correctly determined due to the imperfect impedance measurement. As shown in Figure 10, the present parameters reproduce the overall decaying behavior of the experimental signal magnitude with frequency, but the  
Figure 10: Frequency dependence of (**a**) \(X\) and (**b**) \(Y\) components in the electro-modulation spectra and in the calculation from Equation (12) with phase advanced by \(\frac{\pi}{2}\). For both \(X\) and \(Y\), the magnitude is normalized to 24 Hz. (**c**) Frequency dependence of \(X\) in the electro-modulation spectra and in the calculation from Equation (12) with phase advanced by \(\frac{\pi}{2}\), normalized to the magnitude of \(Y\) at 24 Hz. phase is inconsistent and there is discrepancy in the ratio of \(X/Y\) even after adjusted by \(\pm\frac{\pi}{2}\) phase. We used an Au electrode for the impedance measurement, because with the Ag electrode, the Cole-Cole plot showed anomalous behavior for lower frequencies, and it is difficult to be fit by the usual equivalent circuit. One of the possible reasons is the effect of the electrode reaction with electrolyte ions. The result for Ag is shown in Figure 12, where the same equivalent circuit is used for fitting.