Computational calibration method for defocused wavefront aberration

Background

In interferometry, if an axial displacement persists, a defocused wavefront is superimposed on the measurement wavefront, which must be removed to ensure accurate measurements. Power-only removal, which is effective for low-NA optics, becomes inadequate for high-NA optics and potentially leads to false spherical aberrations. To address this issue, correction methods based on Taylor expansion have been proposed. However, they lack precision as they do not provide an accurate and stable expansion of the defocused wavefront.

Results

We aim to provide a rigorous orthogonal expansion, specifically, the Zernike expansion, of the defocused wavefront using the theory of orthogonal polynomials. The Gaussian hypergeometric function allows for the explicit formulation of the inner products between the defocused wavefront and Zernike spherical aberration terms. We evaluated the convergence speed of the proposed method and observed that the expansion converged exponentially to the actual wavefront shape. Using an actual lens design, we compared the results of our new method, the conventional Taylor method, and a numerical method. Our theoretical method demonstrated complete compatibility with the numerical method.

Conclusions

The compatibility between our new method and the numerical method confirms that the Zernike expansion of the defocused wavefront is formulated theoretically. Thus, this theory provides a theoretical foundation for discussions regarding ISO 10110–5 and ISO/TR 14999–2 for handling defocused wavefronts.