OSA's Digital Library

Optics Letters

Optics Letters


  • Editor: Anthony J. Campillo
  • Vol. 31, Iss. 16 — Aug. 15, 2006
  • pp: 2462–2464

Orthonormal polynomials for hexagonal pupils

Virendra N. Mahajan and Guang-ming Dai  »View Author Affiliations

Optics Letters, Vol. 31, Issue 16, pp. 2462-2464 (2006)

View Full Text Article

Enhanced HTML    Acrobat PDF (128 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The problem of determining the orthonormal polynomials for hexagonal pupils by the Gram–Schmidt orthogonalization of Zernike circle polynomials is revisited, and closed-form expressions for the hexagonal polynomials are given. We show how the orthonormal coefficients are related to the corresponding Zernike coefficients for a hexagonal pupil and emphasize that it is the former that should be used for any quantitative wavefront analysis for such a pupil.

© 2006 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

ToC Category:
Optical Design and Fabrication

Original Manuscript: March 27, 2006
Revised Manuscript: May 18, 2006
Manuscript Accepted: May 21, 2006
Published: July 25, 2006

Virtual Issues
Vol. 1, Iss. 9 Virtual Journal for Biomedical Optics

Virendra N. Mahajan and Guang-ming Dai, "Orthonormal polynomials for hexagonal pupils," Opt. Lett. 31, 2462-2464 (2006)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. http://scikits.com/KFacts.html.
  2. R. Upton and B. Ellerbroek, Opt. Lett. 29, 2840 (2004). [CrossRef]
  3. C. F. Dunkl, SIAM J. Appl. Math. 47, 343 (1987). From the statement on p. 348 it can be inferred that the real orthonormal polynomials are given by (1/2)(pn,k+pn,−k)/(1/2)||pn,k|| and (1/2i)(pn,k−pn,−k)/(1/2)||pn,k||. However, wrong answers are obtained in some cases, e.g., when k=0 or pn,k is real. For example, 2 does not apply in some cases, such as p2,0 and p4,0. Similarly, p3,3 and p3,−3 are real and unequal, and p3,3+p3,−3 does not yield the correct form of the polynomial. Instead, p3,3/||p3,3|| yields our polynomial H10 and p3,−3/||p3,−3|| yields our polynomial H9. There are other mistakes as well. For example, p4,4 and ||p4,4||2 should equal z4 and 319/3150, respectively. [CrossRef]
  4. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).
  5. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976). [CrossRef]
  6. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE, 2004).
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).
  8. V. N. Mahajan, in Proc. SPIE 5173, 1 (2003). [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

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited