OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 19, Iss. 5 — May. 1, 2002
  • pp: 858–870

Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions

Joseph Braat, Peter Dirksen, and Augustus J. E. M. Janssen  »View Author Affiliations

JOSA A, Vol. 19, Issue 5, pp. 858-870 (2002)

View Full Text Article

Enhanced HTML    Acrobat PDF (274 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We assess the validity of an extended Nijboer–Zernike approach [J. Opt. Soc. Am. A 19, 849 (2002)], based on recently found Bessel-series representations of diffraction integrals comprising an arbitrary aberration and a defocus part, for the computation of optical point-spread functions of circular, aberrated optical systems. These new series representations yield a flexible means to compute optical point-spread functions, both accurately and efficiently, under defocus and aberration conditions that seem to cover almost all cases of practical interest. Because of the analytical nature of the formulas, there are no discretization effects limiting the accuracy, as opposed to the more commonly used numerical packages based on strictly numerical integration methods. Instead, we have an easily managed criterion, expressed in the number of terms to be included in the Bessel-series representations, guaranteeing the desired accuracy. For this reason, the analytical method can also serve as a calibration tool for the numerically based methods. The analysis is not limited to pointlike objects but can also be used for extended objects under various illumination conditions. The calculation schemes are simple and permit one to trace the relative strength of the various interfering complex-amplitude terms that contribute to the final image intensity function.

© 2002 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.4430) General : Numerical approximation and analysis
(050.1960) Diffraction and gratings : Diffraction theory
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(110.2990) Imaging systems : Image formation theory
(110.3960) Imaging systems : Microlithography

Original Manuscript: June 21, 2001
Revised Manuscript: October 31, 2001
Manuscript Accepted: October 31, 2001
Published: May 1, 2002

Joseph Braat, Peter Dirksen, and Augustus J. E. M. Janssen, "Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions," J. Opt. Soc. Am. A 19, 858-870 (2002)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
  2. B. Richards, 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). [CrossRef]
  3. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).
  4. A. J. E. M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002). [CrossRef]
  5. SOLID-C, a software product (release 5.6.2) (SIGMA-C GmbH, Thomas-Dehlerstrasze 9, D-81737 Munich, Germany).
  6. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  7. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980). [CrossRef] [PubMed]
  8. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994). [CrossRef] [PubMed]
  9. C. J. Progler, A. K. Wong, “Zernike coefficients, are they really enough?” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 40–52 (2000). [CrossRef]
  10. G. Szegö, Orthogonal Polynomials, 4th ed. (American Mathematical Society, Providence, 1975).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited