Truncation of the Series Expressions in the Advanced ENZTheory of Diffraction Integrals
Abstract
The pointspread function (PSF) is used in optics for design and assessment of the imaging capabilities of an optical system. It is therefore of vital importance that this PSF can be calculated fast and accurately. In the past 12 years, the Extended NijboerZernike (ENZ) approach has been developed for the purpose of semianalytic evaluation of the PSF, for circularly symmetric optical systems, in the focal region. In the earliest ENZyears, the Debye approximation of the diffraction integral, by which the PSF is given, was considered for the very basic situation of a lowNA optical system and relatively small defocus values, so that a scalar treatment was allowed with a focal factor comprising a quadratic function in the exponential. At present, the ENZmethod allows calculation of the PSF in low and highNA cases, in scalar form and for vector fields (including polarization), for large wavefront aberrations, including amplitude nonuniformities, using a quasispherical phase focal factor in a virtually unlimited focal range around the focal plane, and no limitations in the offaxis direction. Additionally, the application range of the method has been broadened and generalized to the calculation of aerial images of extended objects by including the finite distance of the object to the entrance pupil. Also imaging into a multilayer is now possible by accounting for both forward and backward propagation in the layers.
In the advanced ENZapproach, the generalized, complexvalued pupil function is developed into a series of Zernike circle polynomials, with exponential azimuthal dependence (having cosine/sine azimuthal dependence as special cases). For each Zernike term, the diffraction integral reduces after azimuthal integration to an integral that can be expressed as an infinite double series involving spherical Bessel functions, accounting for the parameters of the optical system and the defocus value, and Jinc functions comprising the radial offaxis value. The contribution of the present paper is the formulation of truncation rules for these double series expressions, with a general rule valid for all circle polynomials at the same time, and a dedicated rule that takes into account the degree and the azimuthal order of the involved circle polynomials to significantly reduce computational cost in specific cases. The truncation rules are based on effective bounds and asymptotics (of the Debye type) for the mentioned spherical Bessel functions and Jinc functions, and show feasibility of computation of practically all diffraction integrals that one encounters in the ENZpractice. Thus it can be said that the advanced ENZtheory is more or less completed from the computational point of view by the achievements of the present paper.
 Publication:

Journal of the European Optical Society
 Pub Date:
 September 2014
 DOI:
 10.2971/jeos.2014.14042
 arXiv:
 arXiv:1407.6589
 Bibcode:
 2014JEOS....9E4042V
 Keywords:

 Diffraction integral;
 pointspread function (PSF);
 advanced ENZtheory;
 double series;
 Jinc functions;
 Debye asymptotics of Bessel functions;
 Physics  Computational Physics;
 Mathematical Physics;
 Physics  Optics
 EPrint:
 67 pages, 20 figures