# The circle polynomials of Zernike and their application in optics

```@article{Tango1977TheCP,
title={The circle polynomials of Zernike and their application in optics},
author={William J. Tango},
journal={Applied physics},
year={1977},
volume={13},
pages={327-332}
}```
• W. Tango
• Published 1 August 1977
• Physics, Mathematics
• Applied physics
The Zernike polynomials are orthogonal functions defined on the unit circle, which have been used primarily in the diffraction theory of optical aberrations. A summary of their principal properties is given. It is shown that the polynomials, which are closely related to the general spherical harmonics, are especially useful in numerical calculations. In particular, by using the polynomials as a basis to represent the commonly encountered functions of optical theory, it is often possible to…
Third Order Newton's Method for Zernike Polynomial Zeros
The Zernike radial polynomials are a system of orthogonal polynomials over the unit interval with weight x. They are used as basis functions in optics to expand fields over the cross section of
Quaternion Zernike spherical polynomials
• Mathematics
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• 2015
The Zernikes spherical polynomials within quaternionic analysis ((R)QZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts), are introduced.
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The integrals occurring in optical diffraction theory under conditions of partial coherence have the form of an incomplete autocorrelation integral of the pupil function of the optical system. The
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Several quantities related to the Zernike circle polynomials admit an expression, via the basic identity in the diffraction theory of Nijboer and Zernike, as an infinite integral involving the
Zernike circle polynomials and infinite integrals involving the product of Bessel functions
Several quantities related to the Zernike circle polynomials admit an expression as an infinite integral involving the product of two or three Bessel functions, which is evaluated explicitly for the cases of the expansion coefficients of scaled-and-shifted circle polynnomials, the Fourier coefficients, and the transient response of a baffled-piston acoustical radiator.
Study of the properties of Zernike’s orthogonal polynomials
Wide application of Zernike’s orthogonal polynomials at lens design and optical engineering generates a need of more detailed study of their properties, assemblage new results and results received
Zernike expansion of separable functions of cartesian coordinates.
• Engineering
Applied optics
• 2004
A Zernike expansion over a circle is given for an arbitrary function of a single linear spatial coordinate and a product of two such series can be used to expand an arbitrary separable function of two Cartesian coordinates.
Analysis of primary aberration with the two-dimension discrete wavelet transform
• Physics
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• 2000
As is known, Zernike polynomials find broad application for the solution of many problems of computational optics. The well-known Zernike polynomials are particularly attractive for their unique
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The aberrations of imaging systems with uniformly illuminated annular pupils are discussed in terms of a complete set of polynomials that are orthogonal over an annular region. These polynomials,
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A generalization of the Zernike circle polynomials for expansion of functions vanishing outside the unit disk is given. These generalized Zernike functions have the form Z m,� n (�,#) = R m,� (�)

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