Gaussian quadrature for optical design with noncircular pupils and fields, and broad wavelength range

Forbes introduced the usage of Gaussian quadratures in optical design for circular pupils and fields, and for a specific visible wavelength band. In this paper, Gaussian quadrature methods of selecting rays in ray-tracing are derived for noncircular pupil shapes, such as obscured and vignetted apertures. In addition, these methods are generalized for square fields, and for integrating performance over arbitrary wavelength bands. Integration over wavelength is aided by the use of a novel chromatic coordinate. These quadratures achieve low calculations with fewer rays (by orders of magnitude) than uniform sampling schemes.