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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 31, Iss. 7 — Jul. 1, 2014
  • pp: 1604–1613

Zernike expansion of derivatives and Laplacians of the Zernike circle polynomials

A. J. E. M. Janssen  »View Author Affiliations

JOSA A, Vol. 31, Issue 7, pp. 1604-1613 (2014)

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The partial derivatives and Laplacians of the Zernike circle polynomials occur in various places in the literature on computational optics. In a number of cases, the expansion of these derivatives and Laplacians in the circle polynomials are required. For the first-order partial derivatives, analytic results are scattered in the literature. Results start as early as 1942 in Nijboer’s thesis and continue until present day, with some emphasis on recursive computation schemes. A brief historic account of these results is given in the present paper. By choosing the unnormalized version of the circle polynomials, with exponential rather than trigonometric azimuthal dependence, and by a proper combination of the two partial derivatives, a concise form of the expressions emerges. This form is appropriate for the formulation and solution of a model wavefront sensing problem of reconstructing a wavefront on the level of its expansion coefficients from (measurements of the expansion coefficients of) the partial derivatives. It turns out that the least-squares estimation problem arising here decouples per azimuthal order m, and per m the generalized inverse solution assumes a concise analytic form so that singular value decompositions are avoided. The preferred version of the circle polynomials, with proper combination of the partial derivatives, also leads to a concise analytic result for the Zernike expansion of the Laplacian of the circle polynomials. From these expansions, the properties of the Laplacian as a mapping from the space of circle polynomials of maximal degree N, as required in the study of the Neumann problem associated with the transport-of-intensity equation, can be read off within a single glance. Furthermore, the inverse of the Laplacian on this space is shown to have a concise analytic form.

© 2014 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(050.1970) Diffraction and gratings : Diffractive optics
(100.3190) Image processing : Inverse problems
(080.1005) Geometric optics : Aberration expansions

ToC Category:
Geometric Optics

Original Manuscript: April 24, 2014
Revised Manuscript: May 21, 2014
Manuscript Accepted: May 21, 2014
Published: June 25, 2014

A. J. E. M. Janssen, "Zernike expansion of derivatives and Laplacians of the Zernike circle polynomials," J. Opt. Soc. Am. A 31, 1604-1613 (2014)

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