OSA's Digital Library

Optics Letters

Optics Letters


  • Editor: Alan E. Willner
  • Vol. 35, Iss. 13 — Jul. 1, 2010
  • pp: 2173–2175

Use of Zernike polynomials for efficient estimation of orthonormal aberration coefficients over variable noncircular pupils

Hanshin Lee  »View Author Affiliations

Optics Letters, Vol. 35, Issue 13, pp. 2173-2175 (2010)

View Full Text Article

Enhanced HTML    Acrobat PDF (313 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



An efficient way of estimating orthonormal aberration coefficients on variable noncircular pupils is proposed. The method is based on the fact that all necessary pieces of information for constructing orthonormal polynomials (via the Gram–Schmidt process) can be numerically obtained during a routine least-squares fit of Zernike polynomials to wavefront data. This allows the method to use the usual Zernike polynomial fitting with an additional procedure that swiftly estimates the desired orthonormal aberration coefficients without having to use the functional forms of orthonormal polynomials. It is also shown that the method naturally accounts for the pixelation effect of pupil geometries, intrinsic to recording wavefront data on imaging sensors (e.g., CCDs), making the coefficient estimate optimal over a given pixelated pupil geometry. With these features, the method can be ideal for real-time wavefront analysis over dynamically changing pupils, such as in the Hobby–Eberly Telescope (HET), which is otherwise inefficient with analytic methods used in past studies.

© 2010 Optical Society of America

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(080.1010) Geometric optics : Aberrations (global)
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(220.4840) Optical design and fabrication : Testing

Original Manuscript: February 1, 2010
Revised Manuscript: May 24, 2010
Manuscript Accepted: May 26, 2010
Published: June 22, 2010

Hanshin Lee, "Use of Zernike polynomials for efficient estimation of orthonormal aberration coefficients over variable noncircular pupils," Opt. Lett. 35, 2173-2175 (2010)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Born and E. Wolf, Principles of Optics, 7th ed.(Oxford, 1999).
  2. F. Zernike, Mon. Not. R. Astron. Soc. 94, 377 (1934).
  3. N. Roddier, Opt. Eng. 29, 1174 (1990). [CrossRef]
  4. H. Lee, G. B. Dalton, I. A. Tosh, S.-W. Kim, Opt. Express 15, 3127 (2007). [CrossRef] [PubMed]
  5. W. B. King, Appl. Opt. 7, 197 (1968). [CrossRef] [PubMed]
  6. Hubble Space Telescope, http://hubblesite.org/.
  7. Keck Telescopes, http://www.keckobservatory.org/.
  8. Hobby–Eberly Telescope, http://www.as.utexas.edu/.
  9. South African Large Telescope, http://www.salt.ac.za/.
  10. C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987). [CrossRef]
  11. R. Upton and B. Ellerbroek, Opt. Lett. 29, 2840 (2004). [CrossRef]
  12. G. Dai and V. Mahajan, Opt. Lett. 32, 74 (2007). [CrossRef]
  13. V. N. Mahajan and G. Dai, J. Opt. Soc. Am. A 24, 2994 (2007). [CrossRef]
  14. G. B. Arfken, H. J. Weber, and F. Harris, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).
  15. Maxima version 5.18.1 (a computer algebra system), http://maxima.sourceforge.net/.
  16. H. Lee, M. Hart, and G. J. Hill are preparing a manuscript to be called “Optimal estimation of wavefront slope aberrations on variable non-circular pupils.”

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

Supplementary Material

» Media 1: AVI (701 KB)     

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited