# Pockels Effect at the Interface between Water and Ti Electrode
## 3 Experimental Results
### Estimation of Pockels Coefficients  
The Pockels coefficient can be calculated from the following equation [4]. \[r_{13}=-\frac{2\Delta n}{n^{3}E}=-\frac{2\Delta nd}{n^{3}Ed}=-\frac{2\Delta nd }{n^{3}V} \tag{4}\]  
Figure 11: Measured reflection spectra (dashed lines) in Figure 2 at an angle of incidence of 5 degrees in the air of Ti electrodes used in electromodulation experiments for more than 30 h (14 nm), for 3 h (11 nm), and not used (6 nm) and calculated spectra (solid lines) assuming a TiO\({}_{2}\) surface oxide film with the respective thickness. Figure 12: Experimental (dashed line) and calculated (solid line) electroreflectance spectra. (**a**) Assuming refractive index change in the EDL of water (\(\Delta n>0\)) and complex refractive index change in the TiO\({}_{2}\) film (red-shift by 1.5 meV). (**b**) Assuming only the refractive index change (\(\Delta n>0\)) in the EDL of water. (**c**) Assuming only complex refractive index change (red-shift by 1.5 meV) in TiO\({}_{2}\) film. The distributed voltage \(V\) in the EDL was determined from the impedance measurements, and the refractive index change \(\Delta nd\) in the EDL was estimated from the experimental fitting. This allowed us to evaluate the Pockels coefficient at the Ti electrode interfacial water without DC voltage as follows. \[r_{13}=-150\;\mathrm{pm/V}\;(\mathrm{EDL})\]  
As is clear from the derived equation, the assumed thickness of the EDL, \(d_{1}\), cancels out in the numerator and denominator and does not affect the evaluation of the Pockels coefficient. Note that \(+1\) V and \(-1\) V DC voltage application increases the Pockels signal by a factor of about 3 and \(-3\), respectively, so the Pockels coefficient may also increase by a factor of about 3 and \(-3\). Strictly, however, the reliable estimation of the Pockels coefficients under DC bias needs the impedance measurement on the same conditions. From \(|V_{S}|=0.895\) V in the SCL, the Pockels coefficient corresponding to the maximum refractive index change (0.00128) of TiO\({}_{2}\) at photon energy 3.53 eV, as shown in Figure 13, is \(r_{13}=-2.3\) pm/V (SCL). It should be noted that the magnitude of the Pockels coefficient of the Ti electrode interfacial water obtained is the maximum possible value due to the uncertainties in the distributed voltage to each of the layers involved, as shown below. The Pockels effect of water at the electrode interface refers to the Pockels effect of water in the EDL, which is currently understood to consist of three layers [32]. The inner Helmholtz layer, where electrolyte ions are specifically adsorbed on the electrode surface; the outer Helmholtz layer, where hydrated ions are in the closest contact with the electrode; and the diffuse layer, where ion concentration follows a Boltzmann distribution due to equilibration by thermal diffusion of ions at the ambient temperature. The Helmholtz layer is also called the Stern layer or compact layer. At room temperature with 0.1 M electrolyte concentration, the thickness of the diffuse layer is calculated to be about 1 nm in equilibrium (under constant DC voltage application), while the Helmholtz layer is on the order of one water molecule layer (\(\sim\)0.3 nm). Although the equivalent circuit in Figure 8 perfectly fits the impedance measurement result in Figure 7, it only separates the bulk layer, the SCL of the oxide film, and the EDL of water (the amount of information in the impedance measurement data is too small to fit a more multi-layered equivalent circuit), and the Helmholtz layer is not separated, so the distributed voltage to it is undetermined. The maximum magnitude for \(|r_{13}|\) is 150 pm/V if the voltage distributed to the Helmholtz layer is included in \(|V_{EDL}|\)\(=0.145\) V. If it is included in \(|V_{S}|=0.895\) V, \(|r_{13}|\) will be smaller.