Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials

Abstract

Purpose: To propose and evaluate Complex Zernike polynomials (CZPs) to represent general wavefronts with non uniform intensity (amplitude) in free-from transmission pupils. Methods: They consist of three stages: (1) theoretical formulation; (2) numerical implementation; and (3) two studies of the fidelity of the reconstruction obtained as a function of the number of Zernike modes used (36 or 91). In the first study, we generated complex wavefronts merging wave aberration data from a group of 11 eyes, with a generic Gaussian model of the Stiles-Crawford effective pupil transmission. In the second study we simulated the wavefront passing through different pupil stop shapes (annular, semicircular, elliptical and triangular). Results: The reconstructions of the wave aberration (phase of the generalized pupil function) were always good, the reconstruction RMS error was of the order of 10−4 wave lengths, no matter the number of modes used. However, the reconstruction of the amplitude (effective transmission) was highly dependent of the number of modes used. In particular, a high number of modes is necessary to reconstruct sharp edges, due to their high frequency content. Conclusions: CZPs provide a complete orthogonal basis able to represent generalized pupil functions (or complex wavefronts). This provides a unified general framework in contrast to the previous variety of ad oc solutions. Our results suggest that complex wavefronts require a higher number of CZP, but they seem especially well-suited for inhomogeneous beams, pupil apodization, etc.

Keywords

• Wavefront encoding
• Complex Zernike polynomials
• Free-form pupil
• Apodization
• Stiles-Crawford effect
• Generalized pupil function