Journal of the European Optical Society - Rapid publications, Vol 8 (2013)

Advanced analytic treatment and efficient computation of the diffraction integrals in the extended Nijboer-Zernike theory

S. van Haver, A. J. E. M. Janssen

Abstract


The computational methods for the diffraction integrals that occur in the Extended Nijboer-Zernike (ENZ-) approach to circular, aberrated, defocused optical systems are reviewed and updated. In the ENZ-approach, the Debye approximation of Rayleigh’s integral for the through-focus, complex, point-spread function is evaluated in semi-analytic form. To this end, the generalized pupil function, comprising phase aberrations as well as amplitude non-uniformities, is assumed to be expanded into a series of Zernike circle polynomials, and the contribution of each of these Zernike terms to the diffraction integral is expressed in the form of a rapidly converging series (containing power functions and/or Bessel functions of various kinds). The procedure of expressing the through-focus point-spread function in terms of Zernike expansion coefficients of the pupil function can be reversed and has led to the ENZ-method of retrieval of pupil functions from measured through-focus (intensity) point-spread functions. The review and update concern the computation for systems ranging from as basic as having low NA and small defocus parameter to high-NA systems, with vector fields and polarization, meant for imaging of extended objects into a multi-layered focal region.

In the period 2002-2010, the evolution of the form of the diffraction integral (DI) was dictated by the agenda of the ENZ-team in which a next instance of the DI was handled by amending the computation scheme of the previous one. This has resulted into a variety of ad hoc measures, lack of transparency of the schemes, and sometimes prohibitively slow computer codes. It is the aim of the present paper to reconstruct the whole building of computation methods, using consistently more advanced mathematical tools. These tools are

  • explicit Zernike expansion of the focal factor in the DI,
  • Clebsch-Gordan coefficients for the omnipresent problem of linearizing products ofZernike circle polynomials,
  • recursions for Bessel functions, binomials and for the coefficients of algebraic functionsthat occur as pre-factors of the focal factor in the DI.
This results in a series representation of the DI involving (spherical) Bessel functions and Clebsch-Gordon coefficients, in which the dependence of the DI on parameters of the optical configuration, on focal values, on spatial variables in the image planes, and on degree and azimuthal order of the circle polynomials are separated. This separation of dependencies, together with bounds on Clebsch-Gordon coefficients and spherical Besselfunctions, facilitate the error analysis for the truncation of series, showing that in the new scheme the DI can be computed virtually without loss-of-digits. Furthermore, this separation allows for a modular implementation of the computation scheme that offers speed and flexibility when varying the various parameters and variables. The resulting scheme is pre-eminently appropriate for use in advanced optical simulations, where large defocus values, high NA and Zernike terms of high order and degree occur.

© The Authors. All rights reserved. [DOI: 10.2971/jeos.2013.13044]

Full Text: PDF

Citation Details


Cite this article

References


F. Zernike, ”Diffraction theory of the knife-edge test and its improved version, the phase-contrast method,” Physica 1, 689–704 (1934).

B. R. A. Nijboer, The Diffraction Theory of Aberrations (Ph.D. thesis, University of Groningen, The Netherlands, 1942).

M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, United Kingdom, 1999).

A. J. E. M. Janssen, ”Extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A19, 849–857 (2002).

J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, ”Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A19, 858–870 (2002).

P. Dirksen, J. J. M. Braat, P. De Bisschop, A. J. E. M. Janssen, C. A. H. Juffermans, and A. Williams, ”Characterization of a projection lens using the extended Nijboer-Zernike approach,” Proc. SPIE 4691, 1392–1399 (2002).

P. Dirksen, J. J. M. Braat, A. J. E. M. Janssen, and C. Juffermans, ”Aberration retrieval using the extended Nijboer-Zernike approach,” J. Microlithogr. Microfabr. Microsyst. 2, 61–68 (2003).

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, ”Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer-Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).

A. A. Ramos, and A. L. Ariste, ”Image reconstruction with analytical point spread functions,” Astron. Astrophys. 518, A6 (2010).

X. Liu, L. Wang, J. Wang, and H. Meng, ”A three-dimensional point spread function for phase retrieval and deconvolution,” Opt. Express 14, 15392–15405 (2012).

P. Riaud, D. Mawet, and A. Magette, ”Nijboer-Zernike phase retrieval for high contrast imaging,” Astron. Astrophys. 545, A150 (2012).

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, and A. S. van de Nes, ”Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. 20, 2281–2292 (2003).

V. S. Ignatowsky, ”Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. 1, 1–36 (1919).

B. Richards, and E. Wolf, ”Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, S. van Haver, and A. S. van de Nes, ”Extended Nijboer-Zernike approach to aberration and birefringence retrieval in a high-numericalaperture optical system,” J. Opt. Soc. Am. A22, 2635–2650 (2005).

S. van Haver, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, ”High-NA aberration retrieval with the Extended Nijboer-Zernike vector diffraction theory,” J. Europ. Opt. Soc. Rap. Public. 1, 06004 (2006).

S. van Haver, J. J. M. Braat, A. J. E. M. Janssen, O. T. A. Janssen, and S. F. Pereira, ”Vectorial aerial-image computations of threedimensional objects based on the extended Nijboer-Zernike theory,” J. Opt. Soc. Am. A26, 1221–1234 (2009).

D. Flagello, T. Milster, and A. E. Rosenbluth, ”Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A13, 53–64 (1996).

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, ”Image formation in a multilayer using the extended Nijboer- Zernike theory,” J. Europ. Opt. Soc. Rap. Public. 4, 09048 (2009).

S. van Haver, The Extended Nijboer-Zernike Diffraction Theory and its Applications (Ph.D. thesis, Delft University of Technology, The Netherlands, 2010).

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, ”Assessment of optical systems by means of point-spread functions,” Progress in Optics, E. Wolf, ed., 51, 349–468 (Elsevier, Amsterdam, The Netherlands, 2008).

O. T. A. Janssen, S. van Haver, A. J. E. M. Janssen, J. J. M. Braat, H. P. Urbach, and S. F. Pereira, ”Extended Nijboer-Zernike (ENZ) based mask imaging: efficient coupling of electromagnetic field solvers and the ENZ imaging algorithm,” Proc. SPIE 6924, 692410 (2008).

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, United Kingdom, 2010).

E. C. Kintner, and R. M. Sillitto, ”A new ”analytic” method for computing the optical transfer function,” Opt. Acta 23, 607–619 (1976).

A. J. E. M. Janssen, J. J. M. Braat, and P. Dirksen, ”On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).

J. Boersma, ”On the computation of Lommel’s functions of two variables,” Math. Comput. 16, 232–238 (1962).

R. M. Aarts, and A. J. E. M. Janssen, ”On-axis and far-field sound radiation from resilient flat and dome-shaped radiators,” J. Acoust. Soc. Am. 125, 1444–1455 (2009).

W. J. Tango, ”The circle polynomials of Zernike and their application in optics,” Appl. Phys. 13, 327–332 (1977).

F. G. Tricomi, Vorlesungen über Orthogonalreihen (Springer, Berlin, 1955).

J. J. M. Braat, and A. J. E. M. Janssen, ”Double Zernike expansion of the optical aberration function from its power series expansion,” J. Opt. Soc. Am. A30, 1213-1222 (2013).


Journal of the European Optical Society:Rapid publications - ISSN 1990-2573. JEOS:RP is subject to the EOS General Terms and Conditions.