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Optics Letters

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  • 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)
http://dx.doi.org/10.1364/OL.31.002462


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Abstract

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

History
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

Citation
Virendra N. Mahajan and Guang-ming Dai, "Orthonormal polynomials for hexagonal pupils," Opt. Lett. 31, 2462-2464 (2006)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-16-2462


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References

  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]

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