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Applied Optics

Applied Optics


  • Editor: James C. Wyant
  • Vol. 47, Iss. 19 — Jul. 1, 2008
  • pp: 3433–3445

Orthonormal polynomials in wavefront analysis: error analysis

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

Applied Optics, Vol. 47, Issue 19, pp. 3433-3445 (2008)

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Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.

© 2008 Optical Society of America

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(220.0220) Optical design and fabrication : Optical design and fabrication
(220.1010) Optical design and fabrication : Aberrations (global)
(330.4460) Vision, color, and visual optics : Ophthalmic optics and devices

ToC Category:
Wave-front Sensing

Original Manuscript: April 4, 2008
Manuscript Accepted: May 23, 2008
Published: June 23, 2008

Virtual Issues
Vol. 3, Iss. 8 Virtual Journal for Biomedical Optics

Guang-ming Dai and Virendra N. Mahajan, "Orthonormal polynomials in wavefront analysis: error analysis," Appl. Opt. 47, 3433-3445 (2008)

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