## 3 Results and Discussion
### Evaluation of the Refractive Index Change in the Interfacial Water  
The \(\Delta n\) corresponding to the energy shift was evaluated as follows. We adopted the value \(1.13\times 10^{-4}\) eV for the energy shift at 222 Hz and 2.0 V.  
Firstly, we used the simple analytical formula for the SPP dispersion of Ag, which uses the known bulk plasma frequency \(\hbar\omega_{p}=3.82\) eV (\(\omega_{p}=\sqrt{\frac{n_{e}}{\epsilon_{p}}\frac{e^{2}}{m}}\), screened by the bound electrons) of Ag and the background dielectric constant \(\epsilon_{p}\) of Ag determined from fitting to the SPP resonance energy observed in the present experiment.  
Figure 4: (**a**) Reflection spectrum and (**b**) its first derivative with respect to the photon energy, measured with the cooled spectrometric CCD.  
Figure 5: _Cont._  
The SPP dispersion neglecting dissipation with the wavenumber \(k_{\parallel}=k_{x}\) parallel to the interface is analytically expressed as follows:  
\[k_{x}=\frac{\omega}{c}\sqrt{\frac{\varepsilon_{1}\varepsilon_{2}}{\varepsilon_{1 }+\varepsilon_{2}}} \tag{3}\]  
Figure 6: Dependence of \(\Delta E\) on applied voltage at 222 Hz.  
\[k_{x}=\frac{\omega}{c}n_{g}\sin\theta \tag{4}\]  
with \(\varepsilon_{1}\) and \(\varepsilon_{2}\) being the dielectric constants of dielectric and metal, respectively, the incident angle \(\theta\) in the quartz prism, and the refractive index \(n_{g}\) and \(n_{w}\) of quartz and water. Solving Equations (3) and (4) with \(\varepsilon_{1}=n_{w}^{2}\) and \(\varepsilon_{2}=\varepsilon_{b}\bigg{(}1-\frac{{\omega_{p}}^{2}}{\omega^{2}} \bigg{)}\) for \(\omega\) as a function of \(n_{w}\) and \(\theta\) leads to  
\[\omega=\omega_{p}\sqrt{\frac{\varepsilon_{b}\Big{[}\big{(}n_{g}\sin\theta \big{)}^{2}-{n_{w}}^{2}\Big{]}}{\big{(}\varepsilon_{b}+{n_{w}}^{2}\big{)}\big{(} n_{g}\sin\theta\big{)}^{2}-\varepsilon_{b}{n_{w}}^{2}}} \tag{5}\]  
Table 1 shows the results of fitting of the experimentally observed SPP resonance wavelength at the respective incidence angles \(\theta\) in Figure 2b to Equation (5) to determine \(\varepsilon_{b}\). The value for \(\hbar\omega_{p}\) was fixed at 3.82 eV. The determined value around 5 for \(\varepsilon_{b}\) agrees well with the recent study on the dielectric function of Ag [22].  
From Equation (6), the relation between energy shift in the SPP resonance and the \(\Delta n\) of water is expressed by  
\[\frac{\Delta\omega}{\Delta n_{w}}=-\frac{n_{w}\varepsilon_{b}\frac{1}{2}\big{(} n_{g}\sin\theta\big{)}^{4}}{\Big{[}\big{(}\varepsilon_{b}+{n_{w}}^{2}\big{)} \big{(}n_{g}\sin\theta\big{)}^{2}-\varepsilon_{b}{n_{w}}^{2}\Big{]}^{\frac{3} {2}}\Big{[}\big{(}n_{g}\sin\theta\big{)}^{2}-{n_{w}}^{2}\Big{]}^{\frac{3}{2}} }\omega_{p} \tag{6}\]  
For the energy shift \(1.13\times 10^{-4}\) eV at \(\theta=70.3^{\circ}\), 222 Hz, and 2 V, we deduced \(\Delta n_{w}=-1.08\times 10^{-4}\) for \(\varepsilon_{b}=5.94\).  
Secondly, the SPP dispersion previously determined experimentally by [PERSON] and [PERSON] [23] was employed, and the \(\Delta n\) was determined by numerical calculation as follows.  
In Figure 7, the experimental data are compared with the dispersion relations in Equation (3), obtained from the resonance dip energy as a function of the incidence angle in the reflection spectra calculated by the transfer matrix method [24] using the dielectric constant of Ag in [23] for \(\varepsilon_{2}\) and 1 (air) or \(1.33^{2}\) (water) for \(\varepsilon_{1}\). It was confirmed that the experimental results agree fairly well with the known SPP dispersion relation.  
In order to reliably determine the Pockels coefficient, then, we evaluated \(\Delta nd_{w}\) with \(d_{w}\) being the thickness of the EDL of water where the Pockels effect occurs, as discussed in the following subsections.