## 3 Results and Discussion
### Evaluation of the Pockels Coefficient of the Interfacial Water
#### 3.4.2 Penetration Depth of Surface Plasmon Polaritons  
It has been found that the penetration depth of SPP waves is about \(\delta_{d}\approx\lambda/2\) (about 300 nm on the conditions of Ag, air, and visible light wavelengths) on the dielectric side with \(\lambda\) the wavelength in vacuum of light corresponding to the plasmon resonance frequency, and \(\delta_{m}\approx\lambda_{p}/2\pi\) (\(\lambda_{p}\) is the wavelength in vacuum corresponding to the bulk plasma frequency, 25-30 nm in Ag-the order of the skin depth) on the metal side [29].  
If the penetration depth of SPP into the water side \(\delta_{w}<\) thickness of the EDL \(d_{w}\) of water, the Pockels coefficient \(r_{ij}\) depends on the estimation of the thickness of \(d_{w}\). This is  
Figure 7: Experimental data compared with calculated dispersion relations (Equation (3)) of SPP for Ag (\(\epsilon_{2}\) from [23] in contact with air and water (\(\epsilon_{1}=1\) and \(1.33^{2}\)).  
because (assuming a uniform \(\Delta n\) in the EDL) the SPP feels a uniform \(\Delta n\) over the range of penetration length \(\delta_{w}\), regardless of the thickness of \(d_{w}\), but the magnitude of the \(E\) field in the layer depends on \(d_{w}\). In other words, the estimate of the Pockels coefficient is expressed by  
\[r=\frac{\Delta n}{E}=\Delta n\frac{d}{V}, \tag{8}\]  
which means the thicker \(d\) leads to the larger \(r\) for the same \(\Delta n\). On the other hand, when \(\delta_{w}>d_{w}\), as in the present experiment, the phase change (\(\propto\) spectral shift) of the transmitted light is proportional to the propagation distance \(d\) as follows with \(k\) being the wavenumber in vacuum:  
\[\frac{\Delta T}{T}\propto\Delta\phi=k\Delta nd=krV \tag{9}\]  
(from \(\Delta n=rE\) and \(V=Ed\)), thus  
\[r=\frac{\Delta n}{E}=\frac{k\Delta nd}{kEd}=\frac{\Delta\phi}{kV} \tag{10}\]  
where \(r\) estimated from the shift of the SPP resonance \(\left(\propto\frac{\Delta R}{K}\right)\) is independent of the assumed thickness of \(d_{w}\), as is the case in [3]. In fact, this can be confirmed by calculation on the condition that:  
In a four-layered structure of (semi-infinite) quartz, Ag thin film, EDL of water, and bulk water (semi-infinite), the thickness \(d_{w}\) of the EDL of water, the refractive index \(n\) within the layer (\(\Delta n\), change from the refractive index of 1.33 for bulk water around 600 nm) are assumed to be  
* 300 nm, 1.331 (0.001),
* 100 nm, 1.333 (0.003), and
* 30 nm, 1.34 (0.01),  
respectively. Note that \(\Delta nd_{w}=0.3\) nm for all the cases.  
The SPP resonance spectrum calculated by the transfer matrix method under the same Kretschmann configuration as in the experiment with an incident angle of 70.3\({}^{\circ}\) is shown in Figure 8. The dielectric permittivity of Ag is from the JC's paper [23], and the penetration depths of SPPs using JC's permittivity into the dielectric and metal sides are also calculated. The results show that the energy shift is approximately proportional to \(\Delta n\) when a uniform \(\Delta n\) is assumed as in (E) 1.331, (F) 1.333, and (G) 1.34 for bulk water, whereas when the \(\Delta n\) is limited to within the EDL under the above conditions, \(\delta_{w}(=206\) nm) \(\geq d_{w}\) is approximately satisfied and the spectral shift is determined by \(\Delta nd_{w}\) to be almost the same. It is reasonable that the shift is larger when the \(\Delta n\) is concentrated at the interface even for the same Pockels coefficient, reflecting the fact that the amplitude of SPP is larger near the interface in the range of penetration depth.  
The experimental \(\Delta nd_{w}\) is evaluated as \(\Delta nd_{w}=0.00135\pm 0.000133\) nm. This is derived from 0.3 nm \((\Delta nd_{w})\) multiplied by the ratio of the experimentally observed energy shift (70.3\({}^{\circ}\)2 V,222 Hz) against the averaged energy shift of the calculated SPP resonance of (A), (B), and (C), which agree closely with each other ((E) also agrees), from that with no refractive index change (D) (Table 2).