The accuracy of a quadri-wave lateral shearing interferometer is directly affected by the accuracy of the wavefront reconstruction. Traditional wavefront-reconstruction methods include modal and zonal methods. The modal method expands the wavefront into a set of primary functions to be measured, then fits the coefficients corresponding to the primary functions to reconstruct the measured wavefront. The zonal method discretizes the measured wavefront to establish a mapping relationship between the measured and differential wavefronts for reconstruction. Alternatively, the wavefront can be reconstructed by direct integration in the shearing direction. However, the modal method always uses finite terms to fit the measured wavefront, which directly ignores high-frequency information, reducing the estimated accuracy of the quadri-wave lateral shearing interferometer. The zonal method has a high spatial resolution, but the noise error accumulates along the integrated path during the reconstruction process, forming noise lines, thus, affecting the accuracy of the reconstructed wavefront. To solve this problem, a quadri-wave lateral shearing interferometric wavefront reconstruction method is proposed based on path guidance, which has both high accuracy and spatial resolution.

In this study, a theoretical analysis of the drawbacks of noise error accumulation in wavefront reconstruction using the zonal method without integral-path guidance under noisy environments is carried out. An integral-path evaluation-map model is established based on the deviation of differential phase derivatives, and a flowchart of the wavefront reconstruction algorithm is provided based on integral-path guidance. The proposed method consists of two steps. First, the evaluation model of the differential-phase-derivative deviation is used to count the variational characteristics of the differential phase, identify the noise error, and generate an integral path to avoid noise error. Second, the generated path is used to guide the wavefront reconstruction integral of the Southwell model. Using theoretical simulations, the proposed method could effectively prevent the propagation and accumulation of noise errors compared to the zonal method without integral-path guidance under noisy environments for different signal-to-noise ratios (SNRs). In addition, a verification device having a pure-phase liquid-crystal spatial light modulator (SLM) was set up to experimentally verify the effectiveness of the proposed method. The experimental results of the proposed method were also compared with those of the zonal method without integral-path guidance.

In the simulation, interferograms with a sinusoidal phase distribution are generated (Fig. 4). When the SNR increases from 10 dB to 50 dB, the root-mean-square (RMS) between the wavefront reconstructed by the zonal method without integral-path guidance and the theoretical wavefront decreases from 0.0152λ to 0.0094λ. However, the RMS between the wavefront reconstructed by the proposed method and the theoretical wavefront decreases from 0.0139λ to 0.0041λ. Moreover, the proposed method reduces the RMS of the reconstructed and theoretical wavefronts by a maximum of 55.6% compared to the zonal method without integral-path guidance (Fig. 7). Thus, the proposed method is more robust than the zonal method without integral-path guidance under the Gaussian noise environment with different SNRs (Fig. 6). In the experiment, we measure the random phase generated by the spatial light modulator using the proposed method and zonal method without integral-path guidance (Fig. 10). The results show that the PV value (peak-valley value) of the wavefront reconstructed by the proposed method is 0.7283λ, whereas that of the wavefront reconstructed by the zonal method without integral-path guidance is 2.966λ. The deviation between the PV value of the wavefront reconstructed by the proposed method and that of the theoretical wavefront is 1.6943λ, which is less than the deviation between the wavefront PV value reconstructed by the zonal method without integral-path guidance and the theoretical wavefront PV value (Fig. 13). In addition, the RMS between the wavefront reconstructed by the proposed method and the theoretical wavefront is reduced by 39.7% compared with the RMS between the wavefront reconstructed by the zonal method without integral-path guidance and the theoretical wavefront. In addition, the zonal method without integral-path guidance is used to reconstruct the wavefront, which propagates and accumulates noise points along the shearing direction by forming noise lines. However, the proposed method prevents the propagation of noise points and improves wavefront reconstruction accuracy.

This paper proposes a quadri-wave lateral shearing interference wavefront reconstruction method based on integral-path guidance. The effectiveness of the proposed method is verified through theoretical simulations and experiments. The theoretical simulation results show that the proposed method prevents the propagation of noise errors and improves the wavefront reconstruction accuracy compared to the zonal method without integral-path guidance under noisy environments with different SNRs. The RMS between the reconstructed wavefronts of the proposed method and the theoretical wavefront is smaller than that between the zonal method without integral-path guidance and the theoretical wavefront under the same conditions. Moreover, the experimental results show that the proposed method can effectively prevent the propagation and accumulation of differential phase noise points when measuring the random phase generated by the pure-phase liquid crystal spatial light modulator and reconstructing the wavefront of the random phase. However, the wavefront of the random phase reconstructed using the zonal method without integral-path guidance cannot be accurately reconstructed because of the noise line generated by the accumulation of noise. Therefore, the proposed method has higher accuracy and robustness than the zonal method without integral-path guidance in reconstructing the wavefront in a noisy environment.