# Dispersion of the Retardation of a Photoelastic Modulator
## 4 Discussion  
A calibration method for the retardation amplitude of the PEM in a wide wavelength range, which considers the dispersion, can be easily found. The calibration method can be determined as follows: (i) First, the calibration of the dispersion term, \(N_{(\lambda)}=\frac{n_{0}{}^{3}}{\Lambda}\) can be defined by using the Sellmeier equation describing the refractive index of the photoleastical crystal under the incident light wavelength, as illustrated in Figure 5a; (ii) then, the driving term, \(T_{(V_{0})}=KV_{0}\) can be calibrated by obtaining the retardation amplitudes of the PEM at different driving voltages applied on a single wavelength incident light.  
To further verify the validity and maneuverability of the calibration method, the monochromator was replaced by a diode laser at center wavelength of 532 nm as the incident light source. We varied the driving voltage of the PEM from 25.6 V to 95 V, and obtained the PEM retardation amplitudes, as shown in Figure 6a. The PEM retardation shows a good linear relationship with the driving voltage. The equation of the fitting line is \(\delta_{0}=0.0353V_{0}+0.0012\); therefore, the driving efficiency coefficient can be accurately calibrated using Equation (5), where the slope of the fitting line \(N_{(\lambda=0.532\mu\text{m})}K=0.0353\) corresponds to the driving efficiency coefficient \(K=0.0353/N_{(\lambda=0.532\mu\text{m})}=0.0353/(1.461^{3}/0.532)=6.02\times 10^{-3}\). Then, the amplitudes of the PEM retardation can be calibrated for different incident light wavelengths at different driving voltages, by combining the dispersion term and the driving efficiency coefficient. The driving voltage was set to 85.5 V, and the calibration curve of the amplitudes of the  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Driving Voltage (V)**} & \multicolumn{2}{c}{**Driving Term (\(\mu\)m\(\cdot\)rad)**} \\ \cline{2-3}  & **Mean Value** & **Standard Deviation** \\ \hline
64 & 0.38654 & 3.6 \(\times\) 10\({}^{-4}\) \\
70 & 0.42166 & 3.6 \(\times\) 10\({}^{-4}\) \\
82 & 0.49376 & 4.0 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mean value and standard deviation of the driving term.  
PEM retardation was determined by substituting the dispersion term, the driving efficiency coefficient \(K=6.02\times 10^{-3}\), and the driving voltage \(V_{0}=85.5\mathrm{V}\). The calibration curve is shown in Figure 6b. The light source was changed back to the monochromator, the PEM retardation amplitudes were also measured for the incident light wavelength in the range of 0.4-0.8 \(\upmu\)m, and the actual measurement values are also recorded in Figure 6b.  
As can be seen in Figure 6b, the actual measured values of \(\delta_{0}\) are in agreement with the calibration values. The calibration results show that the deviation between the calibration values and the measurement values, \(\Delta\delta_{0}=3.534-3.500=0.034\) rad, is maximal when the wavelength of the incident light is 0.46 \(\upmu\)m. The maximal deviation that corresponds to the relative deviation is \(\frac{\Delta\delta_{0}}{\delta_{0}}=\frac{0.034}{3500}=1\%\), which indicates the calibration method for the PEM retardation is accurate.