Rectangular domain curl polynomial set for optical vector data processing and analysis

Rectangular pupils are employed in many optical applications such as lasers and anamorphic optics, as well as for detection and metrology systems such as some Shack-Hartmann wavefront sensors and deflectometry systems. For optical fabrication, testing, and analysis in the rectangular domain, it is important to have a well-defined set of polynomials that are orthonormal over a rectangular pupil. Since we often measure the gradient of a wavefront or surface, it is necessary to have a polynomial set that is orthogonal over a rectangular pupil in the vector domain as well. We derive curl (called C) polynomials based on two-dimensional (2-D) versions of Chebyshev polynomials of the first kind. Previous work derived a set of polynomials (called G polynomials) that are obtained from the gradients of the 2-D Chebyshev polynomials. We show how the two sets together can be used as a complete representation of any vector data in the rectangular domain. The curl polynomials themselves or the complete set of G and C polynomials has many interesting applications. Two of those applications shown are systematic error analysis and correction in deflectometry systems and mapping imaging distortion.