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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Steven A. Burns
  • Vol. 24, Iss. 9 — Sep. 1, 2007
  • pp: 2994–3016

Orthonormal polynomials in wavefront analysis: analytical solution

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


JOSA A, Vol. 24, Issue 9, pp. 2994-3016 (2007)
http://dx.doi.org/10.1364/JOSAA.24.002994


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Abstract

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. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.

© 2007 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
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(220.0220) Optical design and fabrication : Optical design and fabrication
(220.1010) Optical design and fabrication : Aberrations (global)

ToC Category:
Atmospheric and oceanic optics

History
Original Manuscript: March 8, 2007
Manuscript Accepted: April 18, 2007
Published: August 30, 2007

Virtual Issues
Vol. 2, Iss. 10 Virtual Journal for Biomedical Optics

Citation
Virendra N. Mahajan and Guang-ming Dai, "Orthonormal polynomials in wavefront analysis: analytical solution," J. Opt. Soc. Am. A 24, 2994-3016 (2007)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-24-9-2994

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