On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus

We present a new computation scheme for the integral expressions describing the contributions of single aberrations to the diffraction integral in the context of an extended Nijboer-Zernike approach. Such a scheme, in the form of a power series involving the defocus parameter with coefficients given explicitly in terms of Bessel functions and binomial coefficients, was presented recently by the authors with satisfactory results for small-to-medium-large defocus values. The new scheme amounts to systemizing the procedure proposed by Nijboer in which the appropriate linearization of products of Zernike polynomials is achieved by using certain results of the modern theory of orthogonal polynomials. It can be used to compute point-spread functions of general optical systems in the presence of arbitrary lens transmission and lens aberration functions and the scheme provides accurate data for any, small or large, defocus value and at any spatial point in one and the same format. The cases with high numerical aperture, requiring a vectorial approach, are equally well handled. The resulting infinite series expressions for these point-spread functions, involving products of Bessel functions, can be shown to be practically immune to loss of digits. In this respect, because of its virtually unlimited defocus range, the scheme is particularly valuable in replacing numerical Fourier transform methods when the defocused pupil functions require intolerably high sampling densities.