Zernike polynomials
From Wikipedia, the free encyclopedia
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after Frits Zernike, they play an important role in geometrical optics.
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[edit] Definitions
There are even and odd Zernike polynomials. The even ones are defined as
and the odd ones as
where m and n are nonnegative integers with n≥m, φ is the azimuthal angle in radians, and ρ is the normalized radial distance. The radial polynomials Rmn are defined as
or
for n − m even, and are identically 0 for n − m odd.
For m = 0, the even definition is used which reduces to Rn0(ρ).
[edit] Applications
In precision optical manufacturing, Zernike polynomials are used to characterize higher-order errors observed in interferometric analyses, in order to achieve desired system performance.
In optometry and ophthalmology the Zernike polynomials are used to describe aberrations of the cornea or lens from an ideal spherical shape, which result in refraction errors.
They are commonly used in adaptive optics where they can be used to effectively cancel out atmospheric distortion. Obvious applications for this are IR or visual astronomy, and spy satellites. For example, one of the zernike terms (for m = 0, n = 2) is called 'de-focus'.[1] By coupling the output from this term to a control system, an automatic focus can be implemented.
Another application of the Zernike polynomials is found in the Extended Nijboer-Zernike (ENZ) theory of diffraction and aberrations.
Zernike polynomials are widely used as basis functions of image moments.
[edit] References
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