The lateral-shear-interference technique is utilized widely for measuring wavefront aberration because of its simple structure, common interference, and minimal susceptibility to external environmental interference. However, the shear interferogram reflects only the gradient of the test wavefront’s shear direction, and the wavefront information to be measured cannot be directly obtained from the interferogram. Wavefront-reconstruction methods based on modal and zonal methods typically depend on orthogonal shear interferograms to reconstruct the wavefront being measured. However, discrepancy exists between the shear direction and the orthogonal coordinate axis of the shear interferogram obtained in an actual experiment. If the wavefront to be measured is reconstructed using the method above, then the accuracy of the reconstruction will be affected. To overcome the limitation of lateral-shear interferometric wavefront-reconstruction technology, which is constrained by the shear direction, and to improve the flexibility of experimental operation, a method for lateral-shear interferometric wavefront reconstruction without directional constraints is proposed. This approach is validated via simulation analysis and experiments.

We introduced a shearing-direction parameter and utilized the underlying algorithm for differential Zernike polynomial wavefront reconstruction to facilitate wavefront reconstruction based on any two directions of lateral-shear interferograms. First, the shear-direction angle θ of the two shear interferograms was calculated using the Radon transform. Subsequently, the differential wavefront information of the measured wavefront was obtained from the shear interferogram in the specified shear direction. The relationship between the differential wavefront and the differential Zernike polynomial in a specific direction was established using θ. The wavefront coefficients to be measured were determined using the least-squares method for wavefront reconstruction. The flowchart of the wavefront-reconstruction algorithm was provided, along with an analysis of the accuracy of wavefront reconstruction for several sets of arbitrary two-direction shear interferograms based on simulation and experiment. The wavefront-reconstruction results were compared with those based on two shear interferograms in the orthogonal directions of the right-angle coordinate axis. The validity of the proposed method was analyzed based on the residual peak-to-valley (PV) and root mean square (RMS) values.

The shear interferogram (Fig.5) of the measured wavefront (Fig.4) along any direction is obtained via simulation. Since the distribution of the measured wavefront is unknown during the experiment, the angle of between the two shear directions used for reconstruction is varied from 2° to 90°. Based on the results, the reconstructed wavefront exhibits the largest PV and RMS residual errors when the angle between the two shear directions is 2°, as compared with the results of a reconstructed shear interferogram in the orthogonal direction. When the angle between the two shear directions exceeds 12°, the wavefront-reconstruction accuracy of the method proposed herein aligns with that of the shear interferogram in the orthogonal direction. Additionally, the accuracy of wavefront reconstruction remains unchanged as the angle between the two shear directions increases (Table 1, Fig.7). Meanwhile, the accuracy of wavefront reconstruction decreases as the noise levels increase at various angles (Fig.8). When the relative noise level reaches 100%, the PV and RMS residual error values of the wavefront reconstruction are λ/40 and λ/285, respectively. Under the same noise level, the wavefront-reconstruction results are consistent regardless of the angle used. When the shear interferogram used for reconstruction presents shear-rate deviation, the reconstruction error of the wavefront to be measured decreases as the angle between the two shear interferograms increases (Fig.9). When the shear-rate deviation is less than 4% and the shear-direction angle exceeds 10°, the reconstructed theoretical PV and RMS residual error values exceed λ/50 and λ/250, respectively, without any noise or systematic error. Furthermore, the accuracy of the method is confirmed experimentally. The lateral-shear interferogram aligns closely with the reconstructed surface-shape features in any two directions with respect to the orthogonal direction (Fig.13), and the measurement results are almost identical. When considering the angle between the two shear interferograms, the accuracies of the self-compared reconstructed PV and RMS exceed 0.029λand 0.0051λ, respectively (Table 2).

This paper introduces an unconstrained wavefront-reconstruction method to overcome the limitations of existing lateral-shear-interferogram reconstruction methods. These methods are constrained by the directionality of the shear interferogram, which restricts their accuracy. The proposed method can achieve wavefront reconstruction using any two lateral interferograms and eliminates the necessity to consider the effect of shear-direction error on the reconstruction accuracy. Moreover, the proposed method is used to process any two directions of lateral-shear interferograms obtained experimentally. When compared with wavefront-reconstruction results based on orthogonal shear interferograms, the accuracy of the reconstructed PV and RMS values are 0.029λ and 0.0051λ higher, respectively, thus validating the accuracy and efficiency of the proposed method.