## Zernike expansion of derivatives and Laplacians of the Zernike circle polynomials

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

http://dx.doi.org/10.1364/JOSAA.31.001604

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### Abstract

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

© 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

**History**

Original Manuscript: April 24, 2014

Revised Manuscript: May 21, 2014

Manuscript Accepted: May 21, 2014

Published: June 25, 2014

**Citation**

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)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-7-1604

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