Results and Discussions After the system is built, and the initial calibration is completed, Soleil-Babinet compensator is regarded as a standard sample to be determined. Adjust the displacement of the optical wedge of the compensator from 0 with an interval of about 0.02 mm, the signal amplitude is obtained by digital phase-locked technique, and the retardation and fast axis azimuth angle of the compensator are further solved. The calibration values of retardation are in good agreement with the measured values (Fig. 7), and there is a maximum relative error of 2.3% between the measured value and the calibrated value, which indicates that the measurement accuracy of retardation in this scheme. When the compensator displacement is at 0.500 mm, and the retardation is 151.118 nm, the maximum standard deviation is 0.032 nm, which indicates that the measurement repeatability and sensitivity of this scheme (Table 1). Stress birefringence experiment is carried out by using the BK7 glass specimen. The experimental results show that the stress retardation and birefringence are increased with the increase of the applied pressure (Fig. 8). When the applied pressure is 50.0 N, the standard deviation of stress birefringence is 0.17 nm/cm, which indicates that the measurement repeatability and sensitivity of the stress birefringence (Table 2). We also find that the stress retardation and birefringence of BK7 glass are not zero, when no pressure is applied at the beginning. The small retardation and birefringence values are 0.558 nm and 1.86 nm/cm respectively, which is induced during the production process. This is a defect that must be tested and evaluated in the optical element production. Moreover, the time interval of data measurement in the above experiments is set at 200 ms. The time can be set according to the actual situation via the FPGA digital phase-locked data processing cycle. In the application with no strict requirements on repeatability, the fastest data measurement rate can reach one data point each 10 ms.

The stress parameters of optical materials and optical components are important parameters to evaluate the mechanical strength, thermal stability, imaging quality and beam transmission quality of optical systems. During the growth of optical materials such as glass and optical crystal, structural stress will occur due to defects or physical and chemical changes. In the process of annealing and cooling, the uneven plastic deformation and uneven volume change caused by temperature change will produce residual stress. Cutting, grinding and polishing during the processing of components, as well as external forces during loading and clamping, will generate mechanical stress. When the stress is large, it is easy to cause the materials and components to explode. Even a small stress can cause poor refractive index and birefringence consistency, resulting in imaging distortion and astigmatism. Therefore, it is necessary for the development and production of high-performance optical system to measure the stress of optical materials and optical elements, and the stress should be controlled within the allowable range. Stress induced birefringence become the main index of stress defect evaluation in optical materials and components. Now, methods are applied to research the measurement of stress birefringence, such as polarization interference, polarization compensation, laser feedback, polarization modulation and polarization imaging. Nevertheless, measurement speed and accuracy still need to be further improved. For the needs of rapid and high-precision stress testing and evaluation of optical materials and optical components, a stress birefringence measurement scheme based on double cascaded photoelastic modulation with differential frequencies is proposed in this paper.

Considering the application advantages of photoelastic modulation, such as high modulation frequency, large optical aperture, high modulation purity and stable operation, a novel measurement method using photoelastic modulation is proposed. A simple polarimetry is constructed based on two photoelastic modulators with differential modulation frequencies. The stress birefringence retardation and fast axis azimuth angle are loaded into the differential frequency photoelastic modulation signals, and the digital phase-locked technology is used to extract the differential frequency signals and fundamental frequency signals of photoelastic modulation at the same time, so as to further solve the stress birefringence retardation and fast axis azimuth angle. The principle of the new scheme is analyzed, and an experimental system is built. The initial offset value of the system is calibrated experimentally without any sample. After that, the measurement accuracy and repeatability are measured by using a Soleil-Babinet compensator as standard sample. Finally, a BK7 glass specimen is loaded different stresses, and the measurement of stress birefringence is completed.

In present study, a novel stress birefringence measurement method based on differential frequency modulation with double photoelastic modulators is demonstrated. The principle of the new scheme is analyzed, and an experimental system is built. The initial offset value of the system is calibrated experimentally, and the measurement accuracy and repeatability are measured by using a Soleil-Babinet compensator, and the stress birefringence measurement for a BK7 glass specimen is carried out. The experimental results show that the accuracy of retardation measurement is 2.3%, the repeatability of retardation measurement is 0.032 nm, and the repeatability of birefringence measurement is 0.17 nm/cm. In addition, the measurement time of single data does not exceed 200 ms. Our study realizes simultaneous measurement of retardation and fast axis azimuth angle without any mechanical adjustment. This method has the application advantages of high measurement accuracy, high measurement repetition and fast measurement speed.