## 3 Results
### Analysis Methods
For a detailed explanation of this part, please refer to Refs. [12; 13; 16]. Each \(\Delta n\) assumed in the calculation is shown in Figure 3. Here, all \(\Delta n\) values are based on the assumption that the EDL thickness is 2 nm (from the Debye-Huckel length scale of 0.1 M electrolyte ionic strength, although the Debye length of ionic liquids is much shorter). This is different from the actual thickness of the EDL under an AC voltage of 210 Hz. Therefore, the \(\Delta n\) displayed is an indication of the relative magnitude of the signal between the solutions, but the absolute magnitude itself is not quantitatively correct. However, since the intra-layer electric field is also calculated from the intra-layer voltage using the same EDL thickness of 2 nm, the magnitude of the Pockels coefficient does not directly depend on the assumed layer thickness, so that the magnitude of the Pockels coefficient is quantitatively justified. That is, taking the thickness of the EDL as \(d\) and the appropriate proportionality constant as \(C\), \(\Delta T/T=C\Delta nd\) and the Pockels coefficient \(=\Delta n/E\). Then, the Pockels coefficient is independent of \(d\) since \(\Delta n=(\Delta T/T)/(Cd)\) and the electric field \(E=V/d\)[13]. For the calculation of \(T\), the same complex refractive index of ITO as in Ref. [16], 1.52 for the refractive index of the glass, and the values in Table 1 for those of the solvents were  
Figure 4: The models of the samples for calculating \(\Delta T/T\)=\((T^{\prime}-T)/T\). Left: The single layer model (bulk ITO) without applied voltage for calculating \(T\). Right: The three layer model (EDL, SCL, bulk ITO) with applied voltage for calculating \(T^{\prime}\). used. In the SCL, it is assumed that \(\Delta n_{\text{SCL}}+i\Delta n_{\text{SCL}}\) has the same wavelength dependence as in Refs. [13; 16], \(\Delta n_{\text{SCL}}=-0.00042\) at 500 nm for water with the SCL thickness being 30 nm [13; 16], and \(|\Delta n_{\text{SCL}}|\) is reduced in the other solvents so that \(\Delta n_{\text{SCL}}/\Delta n\) is the same. The thickness of ITO (commercially specified as 300 \(\pm\) 20 nm), which determines the interference fringe patterns observed in \(\Delta T/T\), is taken to be 320 nm for FA, NMF, DMF and 285 nm for water and IL for the best fit. 1.2 Estimation of Electric Field in Electric Double Layer by Equivalent Circuit and Evaluation of Pockels Coefficient  
Next, let us consider the electric field in each layer from the results of the impedance measurement. We applied the equivalent circuit in Figure 5 to the result of each solvent. The parallel circuit on the right in the red frame is configured to make the impedance of the SCL, and the middle part in the blue frame to make the impedance of the EDL. Using this equivalent circuit, the results of the impedance measurement were fit with the red lines in Figure 11, and the voltage distributed to the EDL and the electric field strength there were estimated at 210 Hz. The impedance values for elements in the equivalent circuit are determined as given in Table 11. From the \(\Delta n\) and \(E\), the Pockels coefficients were derived from the following equation [13] as listed in Table 2. The accuracy of the coefficient values is estimated to be about \(\pm 10\%\). \[r_{13}\Big{[}\frac{\text{pm}}{\text{V}}\Big{]}=-\frac{2\Delta n}{n^{3}E\Big{[} \frac{\text{V}}{\text{m}}\Big{]}}\times 10^{12} \tag{1}\]