An efficient method for computing nodes and weights of Gaussian-Zernike quadrature

The properties of the underlying Zernike-circle polynomials of the Gaussian-Zernike quadrature are investigated, and an extremely efficient method for computing the nodes and weights is devised. In contrast to the "sequential" method that computes the nodes and weights of a given order based on those of one order lower, the present method eliminates this dependency, and allows direct computation of the nodes and weights for any given order. Therefore, an implementation of the present method can be embedded in application programs in order to calculate the Gaussian-Zernike quadrature formula on the fly. Another advantage of the present method is that it is easy to implement. This new method of evaluating nodes and weights facilitates application of the Gaussian-Zernike quadrature to large antennas, for which the advantages resulted from the reduced number of sampling points become remarkable. Examples of physical optics (PO) analysis for very large reflector antennas are demonstrated.