## 2 Principle  
Leveraging the advantages of the high modulation frequency, large optical aperture, and high modulation purity of photoelastic modulation, and combining them with digital phase-locked data processing technology, a measurement method for waveplate parameters is developed. It is based on differential frequency modulation using dual PEMs. Simultaneous measurements of the waveplate phase retardance and fast-axis azimuth are carried out, and the distribution measurements of the parameters on the aperture are performed by scanning the sample in an electric translation stage. The basic principles of the measurement scheme are illustrated in Figure 1.  
As shown in Figure 1, a polarization analysis system is built using PEM1 and PEM2 as the core. The detection laser passes through the polarizer and the two PEMs in turn. It then passes through the analyzer that will be identified by the detector. The transmission axis directions of the polarizer and analyzer are set to \(0^{\circ}\) and \(45^{\circ}\), respectively, and the modulation fast-axis directions of the two PEMs are set to \(45^{\circ}\) and \(0^{\circ}\), respectively. Throughout the polarization analysis device, the direction of the transmission axis of the polarizer and the direction of the fast axis of the PEMs are sequentially different by \(45^{\circ}\). To realize the simultaneous measurement of the phase retardance and fast-axis azimuth, the two PEMs are designed to have different resonant operating frequencies. They operate in cascade, which forms a differential frequency modulation polarization analysis and measurement system. The sample is placed between the PEMs, and a motorized scanning table is used. The optical signal detection is described by the Stokes vector, and the polarization transmission characteristics of the polarizer, PEM, and sample are described by the Muller matrix [13]. The entire measurement scheme is described by the Stokes vector and Muller matrix as  
\[S_{out}=M_{\mathrm{A}}M_{\mathrm{PEM2}}M_{S}M_{\mathrm{PEM1}}S_{in} \tag{1}\]  
wherein \(S_{out}\) represents the Stokes vector of outgoing light, and \(S_{in}\) denotes the Stokes vector of incident light after passing through the \(0^{\circ}\) polarizer. It is usually expressed as \(S_{in}=I_{0}[1\quad 1\quad 0\quad 0]^{\mathrm{T}}\), where \(I_{0}\) is the total intensity of the incident light transmitted through the polarizer. \(M_{\mathrm{PEM1}}\) and \(M_{\mathrm{PEM2}}\) represent the Muller matrices of PEM1 and PEM2, respectively. The modulation fast-axis azimuth of PEM1 is set at \(45^{\circ}\), and the modulation fast-axis azimuth of PEM2 is set at \(0^{\circ}\). Accordingly, the Muller matrices of the two PEMs can be described as follows [11]. \[M_{\mathrm{PEM1}}= \begin{bmatrix}1&0&0&0\\ 0&\cos\delta_{1}&0&\sin\delta_{1}\\ 0&0&0&0\\ 0&-\sin\delta_{1}&0&\cos\delta_{1}\end{bmatrix} \text{and }M_{\mathrm{PEM2}}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\cos\delta_{2}&-\sin\delta_{2}\\ 0&0&\sin\delta_{2}&\cos\delta_{2}\end{bmatrix} , \tag{2}\]  
Figure 1: Schematic diagram of waveplate parameter measurement. FPGA: field programmable gate array; LC: inductance–capacitance resonant driving circuit; ADC: analog-to-digital converter. where \(\delta_{1}\) and \(\delta_{2}\) represent the modulation phases of PEM1 and PEM2. The modulation phase can be further expressed as \(\delta_{1}=\delta_{10}\sin\omega_{1}t\) and \(\delta_{2}=\delta_{20}\sin\omega_{2}t\), where \(\delta_{10}\) and \(\delta_{20}\) represent the phase modulation amplitudes of the two PEMs, and \(\omega_{1}\) and \(\omega_{2}\) represent the resonant operating frequencies of the two PEMs, respectively. To realize the simultaneous measurement of the waveplate phase retardance and fast-axis azimuth angle, the key to our proposed scheme is that the resonant operating frequencies of the two PEMs are not equal (\(\omega_{1}\
eq\omega_{2}\)), which engenders the polarization analysis measurement system with dual PEM cascaded differential frequency modulation.