OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 27, Iss. 2 — Feb. 1, 2010
  • pp: 218–237

Derivation of the refraction equations for higher-order aberrations of local wavefronts at oblique incidence

G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler  »View Author Affiliations

JOSA A, Vol. 27, Issue 2, pp. 218-237 (2010)

View Full Text Article

Enhanced HTML    Acrobat PDF (803 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



From the literature the calculation of power and astigmatism of a local wavefront after refraction at a given surface is known from the vergence and Coddington equations. For higher-order aberrations (HOAs) equivalent analytical equations do not exist. Since HOAs play an increasingly important role in many fields of optics, e.g., ophthalmic optics, it is the purpose of this study to extend the “generalized Coddington equation” to the case of HOA (e.g., coma and spherical aberration). This is done by local power series expansions. In summary, with the results presented here, it is now possible to calculate analytically the local HOA of an outgoing wavefront directly from the aberrations of the incoming wavefront and the refractive surface.

© 2010 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(080.2720) Geometric optics : Mathematical methods (general)
(330.4460) Vision, color, and visual optics : Ophthalmic optics and devices
(080.1005) Geometric optics : Aberration expansions
(080.1753) Geometric optics : Computation methods
(080.7343) Geometric optics : Wave dressing of rays

Original Manuscript: June 30, 2009
Revised Manuscript: October 30, 2009
Manuscript Accepted: November 18, 2009
Published: January 20, 2010

Virtual Issues
Vol. 5, Iss. 4 Virtual Journal for Biomedical Optics

G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, "Derivation of the refraction equations for higher-order aberrations of local wavefronts at oblique incidence," J. Opt. Soc. Am. A 27, 218-237 (2010)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Born and E. Wolf, “Foundations of geometrical optics, geometrical theory of optical imaging and geometrical theory of optical aberrations,” in Principles of Optics (Pergamon, 1980), pp. 109-232.
  2. V. N. Mahajan, “Gaussian optics, optical aberrations and calculation of primary aberrations,” in V.MahajanOptical Imaging and Aberrations Part I: Ray Geometrical Optics (SPIE, 1998), pp. 91-361.
  3. R. Shannon, “Geometrical optics,” in The Art and Science of Optical Design (Cambridge Univ. Press, 1997), pp. 25-105.
  4. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389-1401 (2005). [CrossRef]
  5. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26, 1090-1100 (2009). [CrossRef]
  6. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26, 1503-1517 (2009). [CrossRef]
  7. R. Krueger, R. Applegate, and S. MacRae, Wavefront Customized Visual Correction (Slack, 2004).
  8. J. Porter, H. Quener, J. Lin, K. Thorn, and A. Awwal, Adaptive Optics for Vision Science (Wiley, 2006). [CrossRef]
  9. J. Porter, A. Guirao, I. Cox, and D. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793-1803 (2001). [CrossRef]
  10. R. Applegate, “Glenn Fry Award Lecture 2002: Wavefront sensing, ideal corrections, and visual Performance,” Optom. Vision Sci. 81, 137-177 (2004). [CrossRef]
  11. R. Blendowske, “Wieso funktionieren Gleitsichtgläser? Über Aberrationen in der Progressionszone,” Deutsche Optikerzeitung 2, 60-64 (2007).
  12. R. Blendowske, “Brillengläser und die Korrektion der Abbildungsfehler höherer Ordnung,” Deutsche Optikerzeitung 6, 18-25 (2007).
  13. W. Wesemann, “Korrektion der Aberrationen höherer Ordnung des Auges mit Brillengläsern--Möglichkeiten und Probleme,” Deutsche Optikerzeitung 9, 44-49 (2007).
  14. Rodenstock, “Spectacle lens with small higher order aberrations,” U.S. patent 7,063,421 B2, June 20, 2006.
  15. Rodenstock, “Method for computing a progressive spectacle lens and methods for manufacturing a spectacle lens of this kind,” U.S. patent 6,832,834 B2, December 21, 2004.
  16. Rodenstock, “Method for calculating an individual progressive lens,” U.S. patent application 2007/0132945 A1, June 14, 2007.
  17. J. Landgrave and J. Moya-Cessa, “Generalized Coddington equations in ophthalmic lens design,” J. Opt. Soc. Am. A 13, 1637-1644 (1996). [CrossRef]
  18. D. Burkhard and D. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897-909 (1981). [CrossRef] [PubMed]
  19. O. Stavroudis, “Surfaces,” in O.Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), pp. 136-160.
  20. W. Becken, A. Seidemann, H. Altheimer, G. Esser, and D. Uttenweiler, “Spectacle lenses in sports: optimization of the imaging properties based on physiological aspects,” Z. Med. Phys. 17, 56-66 (2007). [PubMed]
  21. E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vision Sci. 84, 72-78 (2007). [CrossRef]
  22. W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the paraxial case,” Optom. Vision Sci. 85, 581-592 (2008). [CrossRef]
  23. W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the general case of strongly oblique incidence,” Optom. Vision Sci. 85, 593-604 (2008). [CrossRef]
  24. C. Campbell, “Generalized Coddington equations found via an operator method,” J. Opt. Soc. Am. A 23, 1691-1698 (2006). [CrossRef]
  25. M. A. Golub, “Analogy between generalized Coddington equations and thin optical element approximation,” J. Opt. Soc. Am. A 26, 1235-1239 (2009). [CrossRef]
  26. R. Dorsch, W. Haimerl, and G. Esser, “Accurate computation of mean power and astigmatism by means of Zernike polynomials,” J. Opt. Soc. Am. A 15, 1686-1688 (1998). [CrossRef]
  27. E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vision Sci. 82, 923-932 (2005). [CrossRef]
  28. W. Harris, “Dioptric power: its nature and its representation in three- and four-dimensional space,” Optom. Vision Sci. 74, 349-366 (1997). [CrossRef]
  29. W. Harris, “Power vectors versus power matrices, and the mathematical nature of dioptric power,” Optom. Vision Sci. 84, 1060-1063 (2007). [CrossRef]
  30. L. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997). [CrossRef]
  31. G. Esser, Derivation of the Imaging Equations for the Calculation of the Higher Order Aberrations of a Local Wavefront after Refraction (Hieronymus, 2008). [PubMed]
  32. R. K. Tyson, “Conversion of Zernike aberration coefficients to Seidel and higher-order power-series aberration coefficients,” Opt. Lett. 7, 262-264 (1982). [CrossRef] [PubMed]
  33. K. Dillon, “Bilinear wavefront transformation,” J. Opt. Soc. Am. A 26, 1839-1846 (2009). [CrossRef]

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.


Fig. 1 Fig. 2 Fig. 3
Fig. 4 Fig. 5

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited