## Orthonormal polynomials for hexagonal pupils

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

- http://scikits.com/KFacts.html.
- R. Upton and B. Ellerbroek, Opt. Lett. 29, 2840 (2004). [CrossRef]
- 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]
- G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).
- R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976). [CrossRef]
- V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd printing (SPIE, 2004).
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).
- V. N. Mahajan, in Proc. SPIE 5173, 1 (2003). [CrossRef]

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