## 4 Discussion  
In order for the Pockels effect to become active, the spatial inversion symmetry of the system must be broken before the application of the electric field [23]. The water at the electrode interface satisfies this condition, but when the applied electric field is parallel to the electrode interface, the two states with reversed directions of the electric field are superimposed by mirroring about the plane perpendicular to the electric field. Consequently, even though the refractive index change is induced by the electric field, the sign reversal due to the reversal of the direction of the electric field cannot occur [35]. Therefore, the observed result seems to contradict physical intuition at first glance, but the position dependence of the magnitude and sign of the observed signal suggests that the application of an in-plane parallel electric field to the electrode causes the distribution of the electric field around the electrode immersed in water and the generation of a component of the electric field perpendicular to the electrode interface (Figure 5), as discussed in Ref. [35]. The behavior of the signal shown in Figure 2 may reflect the spatial distribution and orientation of the electric field perpendicular to the interface. In order to verify the above model, the electric field distribution in the cell was calculated numerically. When a voltage is applied to both ends of the TCO electrode, an in-plane current flows,  
Figure 4: Relation between \(X\) and \(Y\) components in the complex plane when the response function of the sample is \(\chi(t)=\theta(t)e^{-t/\tau}\) and the applied electric field is \(F(t)=F\text{Re}\Big{(}\dot{e}^{i\omega t}\Big{)}=F\text{cos}\,\omega t\). Figure 5: (**a**) Position dependence of refractive index change in magnitude and sign, (**b**) possible current flow. and a potential difference is generated depending on the location. Using the interfacial potential difference as a boundary condition, the two-dimensional Laplace equation:  
\[\mathbf{\
abla}\cdot(\epsilon\mathbf{\
abla}\phi)=0 \tag{2}\]  
was solved using the Finite Difference Method in the horizontal plane, which is perpendicular to the electrode interface and parallel to the direction of the in-plane current to obtain the potential \(\phi\) and the electrostatic field distribution \(\mathbf{E}=-\mathbf{\
abla}\phi\) in the aqueous solution. We also considered the difference in the dielectric constants \(\epsilon\) of the transparent electrode TCO and water. The experimental condition of the dissolved electrolyte was not taken into account. Since the resistance of the electrode and that of the aqueous solution are comparable in size, the electric field distribution on the electrode surface is affected by that in the aqueous solution (the current flowing through the solution cannot be ignored in comparison to the current flowing through the electrode). However, for the purpose of grasping the qualitative characteristics, the following simplified procedure was adopted: first, we calculated the two-dimensional potential distribution when a potential difference was applied between two points on the electrode surface. Then, assuming that the electrode was vertically immersed in water while maintaining that potential distribution, we calculated the potential distribution on the surface at the height where transmitted light measurements were made in water, using the (one-dimensional) potential distribution on the electrode surface at that height as the boundary condition. The dielectric constant of TCO was set to 3.1 [26] and that of water to 80. The potential distribution was smoothed until convergence using a self-made program in the C language, and the image was visualized using Gunplot. The geometric configuration and conditions assumed in the calculation are detailed in Figure 6a.