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.

1 Definitions
2 Applications
3 References
4 See also
5 External links

[edit] Definitions

There are even and odd Zernike polynomials. The even ones are defined as

$Z^{m}_n(\rho,\phi) = R^m_n(\rho)\,\cos(m\,\phi) \!$

and the odd ones as

$Z^{-m}_n(\rho,\phi) = R^m_n(\rho)\,\sin(m\,\phi), \!$

where $m$ and $n$ are nonnegative integers with $n\geq m$ , $\phi\!$ is the azimuthal angle in radians, and $\rho \!$ is the normalized radial distance. The radial polynomials $R^m_n$ are defined as

$R^m_n(\rho) = \! \sum_{k=0}^{(n-m)/2} \!\!\! \frac{(-1)^k\,(n-k)!}{k!\,((n+m)/2-k)!\,((n-m)/2-k)!} \;\rho^{n-2\,k}$

for $n - m$ even, and are identically 0 for $n - m$ odd.

For $m = 0$ , the even definition is used which reduces to $R^m_n(\rho)$ .

[edit] Applications

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.