With the advancement of modern optical technology, more and more optical elements and systems now feature aperture shapes beyond the traditional circular form. For example, rectangular optical elements are used in high-power laser measurements, and elliptical pupils are found in combined magnification extreme ultraviolet lithography objectives used in integrated circuit manufacturing. Lateral shearing interferometers offer advantages such as strong anti-interference capabilities and are not limited by the reference plane aperture. However, the Zernike mode method, commonly used for wavefront reconstruction from shearing interferograms, has typically been limited to wavefronts with circular or annular apertures. In this paper, we address the reconstruction of Zernike modes for wavefronts with arbitrary apertures.

We build upon Zernike circular polynomials as a basis, constructing corresponding Zernike polynomials through matrix transformation, ensuring their orthogonality over regions with arbitrary apertures. By combining the transformation matrix with noise-weighted least squares fitting of multi-directional differential wavefronts, we propose a difference Zernike polynomials fitting method for arbitrary aperture (DZPF-AA). The proposed method is validated through both simulations and experiments.

Simulations are conducted to reconstruct aberrated wavefronts under various aperture shapes. In ideal conditions, the relative reconstruction errors are below 0.21% (Fig. 6). With noise levels up to 50%, the reconstruction errors remain below 1.8% (Fig. 7). The method demonstrates excellent reconstruction performance with different shear amounts (Fig. 8), and the accuracy of single aberration reconstruction is high (Figs. 9?10). The experimental null test results from a quadriwave lateral shearing interferometer show reconstruction errors below 0.0033λ RMS (Fig. 14), confirming the high reconstruction accuracy of the proposed method. A comparison of test results for aberrations between the quadriwave lateral shearing interferometer and a commercial Shack-Hartmann wavefront sensor indicates that wavefronts reconstructed using the DZPF-AA method are approximately consistent with results from the Shack-Hartmann wavefront sensor (Fig. 17), validating that the proposed method effectively reconstructs wavefronts for arbitrary apertures.

A comparative study is conducted to evaluate the performance of this arbitrary aperture wavefront reconstruction method, using four-directional differential wavefronts versus traditional two-directional differential wavefronts, across various aperture shapes, noise levels, and shearing amounts. Simulation results indicate that reconstruction using four-directional differential wavefronts offers higher noise resistance and achieves better accuracy, particularly with small shearing ratios. Experimental results from a quadriwave lateral shearing interferometer’s null test show that the method using four-directional differential wavefronts achieves measurement accuracy better than 0.004λ RMS (λ=635 nm) for wavefronts with annular, square, and rectangular apertures. Comparative aberration testing between the quadriwave lateral shearing interferometer and a commercial Shack-Hartmann sensor reveal consistent measurements of wavefronts with specific aberrations under annular, square, and rectangular apertures, verifying the correctness of the proposed method.