## 4 Discussion
First, the distribution of the potential \(U(x,z)\) in the electrode plane (\(xz\) plane \(0\leq x\leq 60,\ 0\leq z\leq 40\)) standing vertically (\(z\)-axis) was solved with the potential \(U(5,35)=0\), \(U(55,35)=100\), and the boundary condition \(\partial U/\partial n=0\ (x=0,\ 60,\ z=0,\ 40\), where \(n\) is perpendicular to the boundary). Next, the direction perpendicular to the electrode plane was defined as the \(y\)-axis (with the electrode at \(y=24\)), and the distribution of the potential \(\phi(x,y)\) in the \(xy\)-plane perpendicular to the electrode (\(0\leq x\leq 60,\ 0\leq y\leq 24,\ z=25\)) at height \(z=25\) was solved with the boundary condition \(\phi(x,y=24)=U(x,z=25),\ \partial\phi/\partial n=0\ (x=0,60,\ y=0)\). It was assumed that the boundary between the electrode and the water is at \(y=23\) and that \(\epsilon=3.1\) for \(y>23\) and \(\epsilon=80\) for \(y<23\). The numerical solution for the electric field (\(E_{y}\)) distribution perpendicular to the electrode plane is drawn in Figure 6b,c. The perpendicular electric field at the interface (\(y=23.5\)) was zero at the center (\(x=30\)) and increased toward the outside (\(x=10,50\)), and the sign was reversed across the center. The dielectric constant difference between TCO and water induces a polarization charge at the TCO-water interface, weakening the electric field in the water compared to the electric field on the TCO surface. The calculated results qualitatively reproduce the electric flux density, which is high in the EDL at the electrode interface. It is important to note, however, that as a mechanism for the formation of the electric field distribution, the actual polarization caused by the movement of electrolyte ions to the interface is hypothetically replaced by the polarization associated with the dielectric constant difference. This calculation result qualitatively explains the position dependence of the Pockels signal observed in the experiment. When a voltage was applied to both ends of a mono-electrode, the current flowing in the electrode caused a voltage drop, which generated an interfacial perpendicular electric field according to the potential at each position. Depending on its magnitude and direction, the refractive index change in the EDL of water and in the space-charge layer of TCO at the water-electrode interface changed continuously from positive to negative, leading to the observed position dependence. Let us also discuss the quantitative validity of the experimental results. From Figure 6, when a voltage of 6.28 V is applied between the clips, the plane-perpendicular electric field \(E_{y}\) at the TCO electrode interface at position No. 1 (\(x=10\)) is evaluated to be 0.288 V/mm. Therefore, it is reasonable that a signal with a magnitude nearly equal to the experimental results obtained with two electrodes (an electric field of about 0.5 V/mm applied between electrodes, on average) [21; 26] is observed in Figure 2. However, the calculation of the electric field distribution in the present paper assumes that both the TCO electrode and the water are pure dielectrics and does not consider their conductivity, nor the EDL (its dielectric constant and conductivity) generated between the electrode and the bulk water. If the TCO is assumed to be a conductor with conductivity \(\sigma\), the in-plane potential distribution on the TCO is yielded by \(\mathbf{\
abla}\cdot(\sigma\mathbf{\
abla}\phi)=0\), so the shape of the in-plane distribution agrees with the result of Equation (2). Therefore, it is considered that the present calculation qualitatively explainsthe location dependence of the Pockels signal in Figure 2, but a more detailed analysis of the electric field distribution is needed to explain the signal magnitude quantitatively.