## 3 Experiment
From the calibration method we previously reported [23], the modulation phase retardation can be described as \(A_{0}=N_{(\lambda)}KV_{0}\), where \(N_{(\lambda)}\) is the dispersion term described as \(N_{(\lambda)}=\frac{n_{\text{PEM}}}{\lambda}\), \(K\) is a simplification coefficient representing the driving efficiency of the PEM, and \(V_{0}\) is the amplitude of the driving voltage. The refractive index \(n_{\text{PEM}}\) of the photoelastic crystal depends on the selected crystal material that can be expressed using the Sellmeier formula. The photoelastic crystal of the PEM used in this current study was fused silica, whose dispersion characteristics can be described as \(n_{\text{PEM}}^{2}-1=\frac{0.6961663^{2}}{\lambda^{2}-(0.0684043)^{2}}+\frac{ 0.40794263^{2}}{\lambda^{2}-(0.1162414)^{2}}+\frac{0.89747943^{2}}{\lambda^{2}- (9.896161)^{2}}\)[28]. Firstly, the LN plate sample to be measured is not put in, and the output wavelength of the monochrometer was set to 0.633 \(\upmu\)m as a quasi-monochrome light to be incident. By applying different driving voltages to the PEM, the term \(N_{(\lambda)}K\) can be calibrated by measuring the phase retardation amplitude of the PEM. From references [23,26], we have already known that the harmonic components are proportional to the light intensity and the Bessel function. By using the ratio of the fourth harmonic term to the second harmonic term \(\frac{V_{4f}}{V_{2f}}=\frac{I_{4}(A_{0})}{I_{2}(A_{0})}\), the amplitude \(A_{0}\) can be precisely calibrated [24]. The driving voltage \(V_{0}\) was selected from 25 V to 95 V, and the phase retardation amplitude of the PEM \(A_{0}\) was measured in approximately 5 V steps; the relationship between \(A_{0}\) and \(V_{0}\) are shown in Figure 2a. From the relationship between \(A_{0}\) and \(V_{0}\), which is \(A_{0}=0.0294V_{0}+0.0014\), as shown in Figure 2a, we obtained the term \(N_{(\lambda)}K\) as \(N_{(\lambda=0.633\upmu\text{m})}K=0.0294\). The coefficient \(K=0.0294/N_{(\lambda=0.633\upmu\text{m})}=0.0294/(1.457^{3}/0.633)=6.017 \times 10^{-3}\) can be solved by eliminating the dispersion term \(N_{(\lambda)}\). Therefore, the dispersion of phase retardation amplitude of the PEM at different wavelengths can be solved by combining the dispersion and driving terms, \(A_{0}=6.017\times 10^{-3}N_{(\lambda)}V_{0}\). The amplitude of the driving voltage was set to 85 V, and the dispersion of \(A_{0}\) at different wavelengths was obtained by letting the monochrometer scan from 0.42 to 0.8 \(\upmu\)m with a step of 20 nm. The calibration and measurement of the phase retardation amplitude of the PEM are shown in Figure 2b. The calibration results show that the maximal deviation between the calibration values and the measurement values is \(\Delta A_{0}=2.271-2.260=0.011\)rad when the wavelength of the incident light is 0.7 \(\upmu\)m. The maximal deviation that corresponds to the relative deviation is \(\frac{\Delta A_{0}}{\Delta_{0}}=\frac{0.011}{2.260}=0.5\%\), which indicates the calibration values for the phase retardation amplitude of the PEM at various wavelengths are in good agreement with the measurement values. While keeping the PEM driving voltage setting at 85 V, the Y-cut LN plate sample was added to the system. The sampling frequency of the ADC was set to 3.2 MHz, and the sampling signal data set to every \(1\times 10^{4}\) cycle to acquire one phase-locked data for the phase-locked processing in FPGA, which corresponded to a sampling time of 198.9 ms. The values of the DC voltage applied the EO sample were 200 V, 400 V, 600 V, and 800 V, respectively. Under each DC voltage, the wavelength of the incident light was also set from 0.42 to 0.8 \(\upmu\)m. The monochromator scanned at an interval of 20 nm approximately every 8 s. The DC signal data \(V_{\text{dc}}\) and the harmonics \(V_{1f}\) and \(V_{2f}\) were obtained by the digital phase-locked processing.