A quadrature formula for evaluating Zernike polynomial expansion coefficients (antenna analysis)

Zernike polynomials form a complete orthogonal set that provides a convenient way of expanding an arbitrary function, defined over a circular area, into an infinite series. They provide a numerically efficient way to evaluate the diffraction characteristics of circular aperture antennas and can also be used for surface interpolation. In these applications, a basic step is to expand an appropriate function in a series of Zernike polynomials. Each expansion coefficient is normally determined by evaluating a two-dimensional integral derived using the polynomials´ orthogonal properties. In the present work, an algorithm for numerically performing the integration is presented. This algorithm factors out the oscillatory behavior of the polynomials to provide a fast and accurate procedure. The use of the algorithm to evaluate radiation patterns of reflector antennas is discussed.<>