Objective The traditional 2D detector-based phase measurement methods are always limited by the specific spectral response range, for which single-pixel wavefront imaging provides a new method. A digital micromirror device (DMD)-based single-pixel common-path interference is established, in which Hadamard basis is employed to modulate the target wavefront and the checkerboard partition on the DMD is done to divide the light field into the signal and reference fractions. Meanwhile, phase image formation is implemented as usual with the mathematical principles of single-pixel imaging and phase-shifting algorithms. The results show that with the four-step phase-shifting, the mean relative error of the calculated focal length is as low as 0.0298% when the phase image resolution is 128 pixelx128 pixel for a lens with a nominal focal length of 1000 mm. This method is characterized by a simple device, low cost, and simple calculation principle. Benefiting from the single-pixel detection advantages, this method is expected to be adopted for wavefront detection of lenses or transparent objects in weak light environments, extreme ultraviolet, and far infrared bands. Further, it expands the application scope of single-pixel wavefront imaging. Methods According to the single-pixel wavefront imaging theory, a DMD-based single-pixel multi-step phase-shifting common-path interferometer is established, in which the Hadamard basis is utilized to modulate the target wavefront, and the checkerboard partition on the DMD is done to divide the light field into the signal and reference fractions and form the interference. Then, the lens wavefront can be reconstructed using the passively detected coefficients correlated with the Hadamard modulation patterns. Finally, the phase and amplitude of the physical lens can be obtained from the reconstructed complex wavefront. For the wavefront reconstruction, a four/three-step phase-shifting and down-sampling strategy is performed. Results and Discussions Two lenses with focal lengths of 1000 mm and 500 mm are selected as the targets under test. According to the phase detection results, one can gain the truncation phase distribution and the 3D display of the unwrapped phase. The cross-sections of the measured phases agree well with the theoretical values [Figs. 3(c) and 3(f)]. The measured focal lengths of the lenses are 1000.2 mm and 499.5 mm. The relative errors of the focal lengths between the theoretical values and the measurement ones corresponding to the two lenses are 0.02% and -0.10%, which proves the reconstructed results agree well with the theoretical values and further demonstrates the availability of the phase-shifting common-path interferometry for lens phase detection. Next, the influence of different pinhole sizes on the experimental measurement accuracy is demonstrated, as shown in Table 1. Considering the realistic factors, the 20 mu m pinhole is finally selected for subsequent experiments. Then, as a proof-of-concept under low-resolution circumstances, the phase image of the 1000 mm lens with 64 pixelx 64 pixel is retrieved by employing the four-step phase-shifting method (Fig. 4). The 1000 mm lens is measured five times continuously at two different resolutions. At the resolution of 128 pixelx128 pixel, the measured focal length results are given in Table 2. According to Table 2, the average focal length is 1000.080 mm, and the mean relative error between the measured value and the nominal value of the focal length is 0.0298%. At the resolution of 64 pixelx64 pixel, the measurement results are shown in Table 3, and the average value of the measured focal length and the mean relative error are 999.422 mm and 0.1603% respectively. Finally, the experiment of improving the measurement speed is carried out. Under the three-step phase-shifting method, the cross-sections of the measured phases are still consistent with the theoretical values ( Fig. 5). The measurement results of the above two lenses are 1000.6 mm and 501.6 mm. Compared to the theoretical values, the relative errors are 0.06% and 0.32%. The lens phase is reconstructed by combining the three-step phase-shifting with the down-sampling strategy. For the 1000 mm optical lens (Fig. 6), the measured focal lengths are 1001.4 mm and 1001.9 mm corresponding to sampling rates of 0.8 and 0.4, leading to relative errors of 0.14% and 0.19%, respectively. Regarding the 500 mm optical lens ( Fig. 7), the calculated focal lengths are 503.1 mm and 503.4 mm when the sampling rates are set as 0.8 and 0.4, bringing about relative errors of 0.62% and 0.68%, respectively. Conclusions As far as we know, DMD-based common-path interference single-pixel imaging is first successfully employed to detect cemented doublet with different focal lengths. Experimental results show that whether it is 1000 mm or 500 mm optical lens, the measured focal lengths are much closer to the theoretical ones by adopting the four-step phase-shifting algorithm. The influence of image resolution on the measurement results is investigated, which helps conclude that the mean relative error is as low as 0.0298% when the 128 pixelx128 pixel phase image measured by a four-step phase-shifting algorithm is chosen to calculate the focal length. Additionally, by exploiting the down-sampling strategy, the imaging time is shortened further when the three-step phase-shifting algorithm is adopted for phase retrieval. Thus, we currently provide a simple and cost-effective way for lens detection and further advance the single-pixel imaging technology toward practical applications.
