Description of free-form optical curved surface using two-variable orthogonal polynomials

The orthogonal polynomials of two variables are generated on the unit circle and unit square, and a detailed analysis of the free-form fitting precision is carried out using the orthogonal polynomials with three different sampling grids, which are uniformly pseudo-random grid, array grid and circular grid. To ensure the universality of the fitting analysis, many experiments are conducted on rotationally symmetric aspheric surfaces, free-form surfaces and Peaks free-form surfaces. According to the experiments, among the three sampling grids, the array sampling grid is suitable for most fitting situations. XY-polynomial and orthogonal XY-polynomial give better fitting precision than other surface types in most cases on the wave-front fitting, the orthogonal Zernike polynomial has advantage in circle or square domain and orthogonal Chebyshev is the best polynomial when fitting is required on a square domain using the array sampling grid.