Issue |
J. Eur. Opt. Soc.-Rapid Publ.
Volume 9, 2014
|
|
---|---|---|
Article Number | 14042 | |
Number of page(s) | 13 | |
DOI | https://doi.org/10.2971/jeos.2014.14042 | |
Published online | 27 September 2014 |
Regular paper – Invited publication
Truncation of the series expressions in the advanced ENZ-theory of diffraction integrals
1
S[&]T Experts Pool (STEP), P.O. Box 608, 2600 AP Delft, The Netherlands
2
Optics Research Group, Faculty of Applied Sciences, Technical University Delft, Van der Waalsweg 8, 2628 CH Delft, The Netherlands
3
Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
* svenvanhaver@gmail.com
** a.j.e.m.janssen@tue.nl
Received:
13
August
2014
Revised:
17
September
2014
The point-spread 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 Nijboer-Zernike (ENZ) approach has been developed for the purpose of semi-analytic evaluation of the PSF, for circularly symmetric optical systems, in the focal region. In the earliest ENZ-years, the Debye approximation of the diffraction integral, by which the PSF is given, was considered for the very basic situation of a low-NA 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 ENZ-method allows calculation of the PSF in low- and high-NA cases, in scalar form and for vector fields (including polarization), for large wave-front aberrations, including amplitude non-uniformities, using a quasi-spherical phase focal factor in a virtually unlimited focal range around the focal plane, and no limitations in the off-axis 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 multi-layer is now possible by accounting for both forward and backward propagation in the layers.
In the advanced ENZ-approach, the generalized, complex-valued 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 off-axis 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 ENZ-practice. Thus it can be said that the advanced ENZ-theory is more or less completed from the computational point of view by the achievements of the present paper.
Key words: Diffraction integral / point-spread function (PSF) / advanced ENZ-theory / double series / Jinc functions / Debye asymptotics of Bessel functions
© The Author(s) 2014. All rights reserved.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.