## 2 Measurement Principle  
EO effect is an intrinsic property of EO materials, where variations in the refractive index of the materials occur under an external electric field. When a light passes through, the light is decomposed into two polarization components along the refractive index principal axes, and thereby inducing phases. It is also the fundamental principle of the application of the EO effect. The difference of the phases between the two polarization components is generally recorded as phase retardation \(\delta_{(\gamma,\lambda,V)}\), where \(\gamma\) is the EO coefficient, \(\lambda\) is the wavelength of the incident light, and \(V\) is the voltage of the external electric field. Therefore, the EO coefficients can be obtained by measuring the phase retardation of the EO sample, when the wavelength of the incident light and the electric field voltage are known. This gives a good inspiration to determine the EO coefficients and study their dispersion. A fast, highly precise, and sensitive spectroscopic polarimetry for dispersion measurement of the EO coefficients through measuring the phase retardation \(\delta_{(\gamma,\lambda,V)}\) can be conducted using the advantages of photoelastic modulation, as shown in the schematic diagram of the experimental setup (Figure 1). The detection light passes through a 45\({}^{\circ}\) polarizer and enters the PEM. The modulation axis of PEM is set to 0\({}^{\circ}\) direction. The polarization axes of the polarizer and analyzer are arranged at 45\({}^{\circ}\) and -45\({}^{\circ}\), respectively, relative to the modulation axis direction of the PEM. As the PEM is working, the polarization of the incident light is temporally modulated. Subsequently, the phase retardation induced by the EO sample under the external electric field is loaded into the photoelastic modulation polarization signals, and the signals finally pass through the analyzer to be detected. The monochromator automatically scans the wavelength of the light source to change the incident light to quasi-monochromatic light. The phase retardations of the EO sample at different wavelengths are measured and the spectral dispersions of the EO coefficients are further studied. As shown in Figure 1, neglecting the loss of the detected light in the propagation process, the emitted light from the analyzer can be expressed using Stokes parameters as follows [25]. \[S_{out}=M_{A}M_{sample}M_{PEM}S_{in} \tag{1}\]  
where the polarization transmission characteristics of the analyzer, PEM, and EO sample can be described using the Muller matrices \(M_{A}\), \(M_{sample}\), and \(M_{PEM}\), respectively. The Stokes vectors of the incident light can be described as \(S_{in}=I_{0}\Big{[} \begin{array}{cccc}1&0&1&0\end{array} \Big{]}^{T}\) after passing through the polarizer at the 45\({}^{\circ}\) direction, where \(I_{0}\) represents the total light intensity. The temporal photoelastic modulation is expressed as \(X=\frac{2\pi}{\lambda}\mathrm{d}\Delta n=\frac{2\pi}{\lambda}\mathrm{d} \Delta n_{0}\sin\omega t=A_{0}\sin(2\pi f_{0}t)\), where \(A_{0}\) is the modulation phase retardation amplitude of the PEM, and \(f_{0}\) is the work frequency. All the Muller matrices and the Stokes vectors of the incident light are substituted in Equation (1), according to reference [20] and considering that the detector can only detect the first component of Stokes vector, the light intensity detected by the detector can be expressed as:  
\[ \begin{split} I_{(t)}&=\frac{I_{0}}{2}(1-\cos\delta_{( \gamma,\lambda,V)}(J_{0}(A_{0})+2\sum J_{2m}(A_{0})\cos(4m\pi f_{0}t))+\\ &\quad\quad+\sin\delta_{(\gamma,\lambda,V)}(2\sum J_{2m-1}(A_{0}) \sin((4m-2)\pi f_{0}t)))\end{split}  \tag{2}\]  
In the measurement system, a FPGA is applied to generate the driving signal of the PEM, then the signal is amplified via the LC oscillating circuit to drive the PEM to work. Meanwhile, the sampling frequency of the analog-to-digital converter (ADC) is controlled via the FPGA.