## 3 Experiment  
Lithium niobate (LiNbO\({}_{3}\), LN) crystals have excellent performances, which include large electro-optic (EO) coefficients, simple production of large crystal, good processability, excellent work stability, and relatively low cost. LN is one of the most widely used EO materials for storage, processing, and transmission of information [27]. There have been many reports on the measurement of the EO coefficients of LN; however, the results of each method are moderately different. Therefore, an efficient and accurate measurement method for the EO coefficients and a study on its spectral dispersion are crucial for the development of the EO devices for wide spectral applications. To verify the feasibility of this method, a Y-cut type LN plate is selected as the sample, and the dispersion of the EO coefficient \(\gamma_{22}\) is studied. Under the external electric field applied along the Y axis, the two principal axes of the refractive index, the crystal X axis and Y axis, are not rotated, but a change in their values occurs. When the incident light passes through the sample, the phase retardation of the polarization components along the components of the two refractive index principal axes is as follows:  
\[\delta_{(\gamma,\lambda,\
u)}=\frac{2\pi}{\lambda}n_{0}^{3}\gamma_{22}\frac{L _{z}}{L_{y}}V \tag{4}\]  
where \(\lambda\) is the wavelength of the incident light in vacuum, \(n_{0}\) is the refractive index of the ordinary light before applying the external electric field, \(L_{y}\) and \(L_{z}\) are the lengths of the LN plate in the direction of Y axis and \(Z\) axis, respectively, and \(V\) is the DC voltage applied to the LN plate. Combining Equations (3) and (4), the EO coefficient at each wavelength can be further calculated as follows:  
\[\gamma_{22}=\frac{\lambda L_{y}}{2\pi n_{0}^{3}V_{Lz}}\text{arctan}(\frac{V_{ 1f}f_{2}(A_{0})}{V_{2f}f_{1}(A_{0})}) \tag{5}\]  
To conduct an experimental research under this scheme, a dispersion measurement system for the EO coefficients was built according to the schematic shown in Figure 1. The light source is a grating monochromator (Omni-\(\lambda\)200i, Zolix Instruments Co., Ltd., Beijing, China) equipped with a Xenon lamp source. The monochromator can achieve automatic scanning with flexible adjustment of wavelength step, with a full width at half maximum of spectrum of less than 1 nm. The polarizer and the analyzer were both Glan-Taylor calcite polarizers with an extinction ratio exceeding 10\({}^{5}\):1. The PEM was a symmetrical octagonal structure fused quartz PEM constructed by us, and its resonant frequency is 50.269 kHz. The detector is a Si avalanche photodetectors (APD410A, Thorlabs Inc., Newton, NJ, USA). An Altera EP3C FPGA was used to provide the driving signal of the PEM and to control the clock frequency of a fast and high-precision 12-bit ADC. The digital signal processing was also completed in the FPGA. By Compiling the Labview program in a computer and establishing communication with the FPGA, the PEM frequency can be adjusted, the driving sinusoidal voltage amplitude can be adjusted through computer visual input, the modulation phase retardation amplitude of the PEM can be adjusted, and the phase-locked data completed by the FPGA can be retrieved. The EO sample was a Y-cut LN crystal plate, with plate size (\(L_{x}\), \(L_{y}\), \(L_{z}\)) of 11.20 mm \(\times\) 7.66 mm \(\times\) 17.00 mm. The incident light passed through the \(Z\) direction (optical axis) of the EO sample, and the electric field was applied in the Y direction. To obtain the EO coefficients, the amplitude of modulation phase retardation \(A_{0}\) of the PEM must be first calibrated at each wavelength.