# Dispersion of the Retardation of a Photoelastic Modulator
## 3 Experiment  
In order to study the dispersion of the retardation of the PEM and obtain an accurate calibration method for the PEM retardation, a fused silica PEM was employed, which was widely applied at the visible light range using fused silica as the photoelastic crystal. The fused silica PEM was custom made by our research group, while the photoelastic crystal was a regular octagon fused silica disk with an edge length of 22.50 mm and a thickness of 16.00 mm. The mounted piezoelectric actuator was an \(-18.5^{\circ}\) x-cut quartz plate with a size of 6.48 mm \(\times\) 51.08 mm \(\times\) 19.18 mm. A field programmable gate array (FPGA) provided square wave signals, and an LC oscillator circuit transformed the square wave signals to sinusoidal high voltage for the operation of the PEM. The frequency and amplitude of the sinusoidal driving voltage could be regulated by the FPGA and a power supply, respectively. The FPGA, the LC oscillator circuit, and the power supply constitute the PEM controller while the optical head comprises the fused silica disk and the quartz plate. The experimental system for the measurement and calibration of the PEM retardation is illustrated in Figure 2.  
A xenon lamp monochromator (Omni-\(\lambda\) 200, Zolix Instruments Co., Ltd., Beijing, China), in the light range of 400-800 nm, with a spectral band less than 1 nm, was used as the light source. The light source was collimated to pass through a polarizer, the center of the fused silica PEM, and an analyzer to be detected by a Si photodetector (FPDA510-FC, Thorlabs China, Shanghai, China). Both the polarizer and analyzer were Glan-Taylor polarizers with an extinction ratio greater than 100,000:1, and the transmission axes of the polarizer and the analyzer were orthogonal in an arrangement of \(45^{\circ}\) and \(-45^{\circ}\), respectively. The modulation fast axis of the PEM was set to \(0^{\circ}\). According to the polarization analysis [16], the light intensity arriving at the photodetector can be expressed as  
\[I=\frac{I_{0}}{4}(1-\cos\delta), \tag{6}\]  
where \(I_{0}\) is the total intensity of the incident light. The retardation \(\delta=\delta_{0}\sin(2\pi ft-\varphi)\) can be further expanded using Bessel function of the first kind, and Equation (6) can be rewritten as  
\[I=\frac{I_{0}}{4}(1-(J_{0}(\delta_{0})+2{\sum_{2k}}J_{2k}(\delta_{0})\cos(2k \pi ft-\varphi))), \tag{7}\]  
where \(J_{0}(\delta_{0})\) is the zeroth order of the Bessel function, \(k\) is a positive integer, and \(J_{2k}(\delta_{0})\) is the (2\(k\))th order of the Bessel function. An Altera EP3C FPGA is used to provide the PEM driving signal, to maintain a fast and precise 12-bit analog-to-digital converter clock frequency, and then to complete the digital signal processing [17; 18]. Multiple even harmonic components such as the second, the forth harmonics can be achieved as  
\[\left\{ \begin{array}{c}V_{2f}=\frac{I_{0}}{2}J_{2}(\delta_{0})\\ V_{4f}=\frac{I_{0}}{2}J_{4}(\delta_{0})\\ \ldots\end{array} \right.. \tag{8}\]  
The multiple harmonic components are directly proportional to the light intensity and the Bessel function. By using the ratio of multiple harmonic components, the retardation amplitude of the PEM can be calibrated by  
\[\frac{V_{4f}}{V_{2f}}=\frac{J_{4}(\delta_{0})}{J_{2}(\delta_{0})}. \tag{9}\]  
Figure 2: Schematic diagram of the experimental system. The inset shows a PEM device. The influence of the intensity fluctuation of the light source can be effectively eliminated, and this calibration method is insensitive to the misalignment of the polarizer and analyzer transmission axes. The amplitudes of the driving voltage were set to 82 V, 70 V and 64 V, respectively. The driving voltage and the modulation signals are shown in Figure 3a, and the ratio of the second and forth harmonic components are shown in Figure 3b,c. Figure 3: _Cont._  
The modulation signal in Figure 3a was acquired by the FPGA and the harmonic components obtained by use of the digital phase-locked technique.