## 3 Results  
Figure 2 shows the dependency of the normalized transmittance change spectrum \(\Delta T/T\) on the position of the transmitted light. The reproducibility of the experimental results was confirmed in multiple experiments, since a \(\Delta T/T\) spectrum with sufficient S/N (S/N > 100 for the largest signal around 300 nm in Figure 2, \(X\), No. 1) could be measured in only 90 s (Time Constant 30 s), thanks to simultaneous measurement over the entire wavelength range using the multi-lock-in amplifier. The magnitude and phase of the signal changed depending on the position of the transmitted light. The signal was larger in magnitude at both outer sides (No. 1, 5) than at the inner sides (No. 2, 4), and almost no signal was observed at the center (No. 3). The signal phase was reversed on the left and right sides across the center. The signal at No. 5 is consistent with the transmittance change spectra observed on the electrode to which a positive voltage was applied in the experiment with two electrodes (voltage-applied electrode and grounded electrode) [20; 21; 26]. This indicates that the refractive index change is negative, and the response of the refractive index change has a phase delay with respect to the input AC voltage (as evidenced by the fact that for shorter wavelengths, the \(X\) signal is positive and the \(Y\) signal is negative) [21; 24]. In fact, the transmittance change spectrum can be fitted by the first derivative of the transmission spectrum with respect to wavelength, as shown in Figure 3. This represents the shift of the interference fringes of the TCO thin film due to the refractive index change in the interfacial layers [20]. In Appendix A, we report that a similar interfacial Pockels effect was also observed in electroreflectance measurements of a single copper electrode, suggesting that this phenomenon occurs universally on a variety of electrode surfaces. Figure 2: Position dependance of the synchronized phase (\(X\)) and quadrature phase (\(Y\)) components of normalized transmittance change spectra of the TCO in a 0.1 M NaCl aqueous solution, with an applied voltage of 6.28 V (amplitude) at \(f\) = 20.0 Hz. The detected current was 0.54 A (amplitude). Let us explain the physical meaning of the \(X\) and \(Y\) components of the lock-in signal. The multichannel lock-in amplifier functions as a dual-phase lock-in amplifier to yield \(X\) (cos or in-phase) and \(Y\) (sin or quadrature phase) components as output signals. When an AC electric field of frequency \(f\) (\(\omega=2\pi f\)): \(F(t)=F\text{cos}\,\omega t=F\text{Re}\big{(}e^{i\omega t}\big{)}\) is applied to a sample with an impulse response function \(\chi(t)=\theta(t)e^{-t/\tau}\), where \(\theta(t)\) is the step function, the first-order electro-optic (Pockels) signal of the sample is calculated as:  
\[S(t)=\text{Re}\!\int_{-\infty}^{\infty}\!dt^{\prime}\chi\big{(}t^{\prime}\big{)} F\big{(}t-t^{\prime}\big{)}=\text{Re}\frac{\frac{1}{\tau}-i\omega}{\omega^{2}+ \frac{1}{\tau^{2}}}e^{i\omega t}=\text{Re}\frac{X+iY}{\sqrt{\omega^{2}+\frac{1 }{\tau^{2}}}}e^{i\omega t}=\frac{\cos(\omega t-\theta)}{\sqrt{\omega^{2}+\frac {1}{\tau^{2}}}}=\frac{1}{\omega^{2}+\frac{1}{\tau^{2}}}\bigg{(}\frac{1}{\tau} \cos\omega t+\omega\text{sin}\,\omega t\bigg{)} \tag{1}\]  
with \(\cos\theta=\frac{1}{\sqrt{\omega^{2}+\frac{1}{\tau}}}(=X)\), and \(\sin(-\theta)=-\frac{\omega}{\sqrt{\omega^{2}+\frac{1}{\tau}}}(=Y)\). The \(\cos\theta\) and \(\sin\theta\) signals are measured as \(X\) and \(Y\) components in the lock-in outputs. Since sin \(\omega t\) is delayed from \(\cos\omega t\) by a quadrature phase (\(\cos(\omega t-\tau/2)=\sin\omega t\)), or as evident from Figure 4, \(Y<0\) when \(X>0\) (\(Y>0\) when \(X<0\)). Thus, from the largest \(\Delta T/T\) signal at 305 nm for position No. 1 in Figure 2, the phase delay is evaluated as \(\arctan(Y/X)=\arctan(-1.10\times 10^{-3}/4.54\times 10^{-3})=-0.239\) rad \(=-13.7^{\circ}\). The response time \(\tau\) is evaluated from \(-Y/X=\omega\tau=2\pi f\tau=40\pi\tau=0.242\) to be \(\tau=1.9\) ms.