## 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 addition, information on the voltage falling on the EDL of water at the Ag electrode interface is needed to evaluate the Pockels coefficient. Therefore, complex impedance measurements were performed using an impedance/gain-phase analyzer (Model 1260, Solatron, Victorville, CA, USA) and a potentiostat/galvanostat (Model 1287, Solatron). The results for the same solution and electrode configuration (but Au was used instead of Ag because it was more robust) as in the \(E\) field modulation experiments are shown as a Cole-Cole plot in Figure 9a, where the circled data point in the graph corresponds to a frequency of 222 Hz. The fitting curve for the graph was calculated using the equivalent circuit shown in Figure 9b. Here, the contribution from the electrode surface is taken as the constant phase element (CPE) [30]. At a frequency of 222 Hz, the voltage distribution ratio to the EDL is 0.22, so the corresponding voltage drop is 0.44 V for the applied voltage of 2 V. For simplicity, the distribution ratio to the Pt- and Ag-electrode interfaces is assumed to be equal, which is 0.22 V. From the above, the Pockels coefficient \(r_{33}\) of the interfacial water on Ag can be evaluated as follows:  
\[n_{33}(\approx-r_{33})=\frac{\Delta n}{E}=\frac{\Delta adv}{V}=\frac{0.00135 \pm 0.000133}{0.22}=0.0061\pm 0.0006\;\mathrm{nm/V}=6.1\pm 0.6\;\mathrm{pm/V}\]  
\[\Delta n=n_{3}-n_{w}=-n_{w}^{3}r_{33}E_{3}/2\]  
\[r_{33}=-\frac{2\Delta n}{n_{w}^{3}E_{3}}=-\frac{2}{1.33^{3}}\frac{\Delta n}{E} =-5.2\;\mathrm{pm/V}\]  
where the subscript 33 indicates that the photoelectric field is a p-polarized evanescent wave parallel to the applied \(E\) field normal to the interface. From the following discussion about the sign of the coefficient and reliability of the impedance measurement, we conclude:  
\[|r_{33}|=5\pm 1\;\mathrm{pm/V}.\]  
Figure 8: (**a**) Penetration depth of SPP as a function of photon energy. (**b**) Enlarged graph for the penetration depth into the metal side. (**c**) SPP resonance spectra calculated by the transfer matrix method under the same Kretschmann configuration as in the experiment with an incident angle of 70.3\({}^{\circ}\). Let us consider the phase of the lock-in signal. Figure 10 shows the frequency dependence of the \(X\) and \(Y\) signals of the experiment. From the fitted impedance \(Z\),  
\[Z=R_{s}+\frac{1}{\frac{1}{R_{p}}+(j\omega)^{p}T}, \tag{11}\]  
the distribution ratio \(r_{EDL}\) of the voltage to the EDL of water is expressed by  
\[r_{EDL}=\frac{\left(1+\frac{R_{s}}{R_{p}}+R_{s}T\omega^{p}\cos\frac{\pi}{2}p \right)-j\left(R_{s}T\omega^{p}\sin\frac{\pi}{2}p\right)}{\left(1+\frac{R_{s}} {R_{p}}+R_{s}T\omega^{p}\cos\frac{\pi}{2}p\right)^{2}+\left(R_{s}T\omega^{p} \sin\frac{\pi}{2}p\right)^{2}}=\frac{X-iY}{\sqrt{X^{2}+Y^{2}}}. \tag{12}\]  
The real and imaginary parts are considered to be reflected in the \(X\) and \(Y\) of the lock-in signal. The negative sign of the imaginary part indicates that the voltage distribution to the interfacial layer is delayed due to the time required to charge the EDL capacitance, i.e., due to the formation of the EDL, and this is expressed as a phase delay. From Equation (12), 0.14:\(-\)0.17 is the ratio of the real and imaginary parts of \(Z\) at 222 Hz, which is inconsistent with about the 2:5 signal amplitude ratio for \(X:Y\) of the lock-in amplifier in the raw experimental data in Figure 3a,b. What is more puzzling in the electromodulation experiment is that \(Y\) becomes more dominant at lower frequencies, and that \(X\) and \(Y\) are not sign-reversed, but are of the same sign, being inconsistent with the impedance measurement where the \(r_{EDL}\) in Equation (12) indicates that \(X\) is dominant at low frequencies and \(|-Y|\) increases with frequency, and the \(Y\) signal should be inverted with respect to \(X\).