Computation of Hopkins’ 3-circle integrals using Zernike expansions

The integrals occurring in optical diffraction theory under conditions of partial coherence have the form of an incomplete autocorrelation integral of the pupil function of the optical system. The incompleteness is embodied by a spatial coherence function of limited extent. In the case of circular optical systems and coherence functions supported by a disk, this gives rise to Hopkins’ 3-circle integrals. In this paper, a computation scheme for these integrals (initially with coherence functions that are constant on their disks) is proposed where the required integral is expressed semi-analytically in the Zernike expansion coefficients of the pupil function. To this end, the Zernike expansion coefficients of a shifted pupil function restricted to the coherence disk are expressed in terms of the pupil function’s Zernike expansion coefficients. Next, the required integral is expressed as an infinite series involving two sets of Zernike coefficients of restricted pupils using Parseval’s theorem for orthogonal series. Due to a convenient separation of the radial parameters and the spatial variables, the method avoids a cumbersome administration involving separate consideration of various overlap situations. The computation method is extended to the case of coherence functions that are not necessarily constant on their supporting disks by using a result on linearization of the product of two Zernike circle polynomials involving Wigner coefficients.