## 4 Discussion  
The Pockels coefficient of water is 165 pm/V, which is smaller than the previously estimated values of about 200 pm/V [13] and 230 pm/V [16]. This can be attributed to the improvement in the impedance measurement and the equivalent circuit. In this paper, the electrodes were irradiated with white light during the impedance measurement as in the electric-field modulation experiments (no irradiation in Refs. [13; 16]). As the equivalent \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Solvent** & **FA** & **NMF** & **DMF** & **[BMIM]** & **[BF\({}_{\text{s}}\)]** & **Water** \\ \hline Relative permittivity & 109.5 & 186.0 & 37.4 & 13.9 & 80.2 \\ Density [g/cm\({}^{3}\)] & 1.133 & 1.011 & 0.944 & 1.21 & 0.998 \\ Viscosity [mPa-s] & 3.23 & 1.72 & 0.71 & 132 & 1.00 \\ Refractive index & 1.447 & 1.43 & 1.43 & 1.41 & 1.333 \\ Refractive index change & \(-1.14\times 10^{-2}\) & \(-1.14\times 10^{-2}\) & \(-1.52\times 10^{-2}\) & \(-8.00\times 10^{-3}\) & \(-2.00\times 10^{-2}\) \\ Intralayer electric field [V/m] & \(9.11\times 10^{7}\) & \(9.74\times 10^{7}\) & \(8.66\times 10^{7}\) & \(1.05\times 10^{8}\) & \(1.02\times 10^{8}\) \\ Pockels coefficient [pm/V] & 83 & 80 & 120 & 55 & 165 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Pockels coefficient determined for each solution in the EDL with the refractive index change \(\Delta n\) and the electric field \(E\) in the EDL when the thickness of the EDL is assumed to be 2 nm. Figure 5: Equivalent circuit diagram with resistance (R) and constant phase element (CPE) (see Appendix A). circuit, a constant phase element (CPE) was used instead of capacitance (capacitance used in Ref. [12]), and the measurement frequency range was widened compared to Refs. [13; 16]. Furthermore, the addition of CPE in series to represent the EDL as in Figure 5 has improved the fit in the low-frequency range (the line running diagonally up in the Cole-Cole plot). As a result, while the fit was not perfect in Refs. [13; 16], an almost perfect fit was obtained in the present study, resulting in a more accurate estimate of the electric field in the layer. The estimated Pockels coefficients of FA, NMF, and DMF are 83, 80, and 120 pm/V, respectively, all of which are lower than the 165 pm/V of water. The Pockels coefficient of IL is 55 pm/V, which is also not higher than that of water. From the Pockels coefficient of the three amides, one can say that the static dielectric constant of the solvent is not an important factor in determining the magnitude of the Pockels coefficient. It is also interesting to note that the \(\Delta T/T\) spectrum of IL is nearly the same in shape as that of water despite the fact that they are completely different types of liquids, considering the following differences. In case we regard the ionic liquid as an 100% electrolyte solution, the molar concentration of the electrolyte in [BMIM][BF\({}_{4}\)] is calculated to be 5.3 M (density 1.2011 [g/mL] \(\times\) 1000 [mL/L]/molecular weight 226.03 = 5.3139), but as shown in Appendix A, the impedance measurements show that the resistance of the bulk liquid part is not much different from other liquids of 0.1 M solution. On the other hand, the thickness of the EDL is on the order of 0.1 nm if the Debye length is applied as it is [8], which is 1/10 of the size of the constituent ions of IL, so it is obvious that the Debye length cannot be applied as it is, and the structure of the EDL is still one of the controversial topics [8; 9; 10]. It has been previously reported [16] that MeOH and EtOH, which are both polar protic solvents with relatively strong hydrogen-bonding, have relatively large Pockels coefficients, although they are smaller than that of water. In that paper, it was concluded that the Pockels coefficient of water, MeOH, EtOH, and DMSO at the interface of transparent electrode scales with (hydrogen-bond strength)/(viscosity). FA and NMF are polar protic solvents, and it is known that strong hydrogen bonding is the reason for their large static dielectric constants [23].