Zernike polynomials

The Zernike polynomials are named after Frits Zernike orthogonal polynomials, and in particular within the geometrical optics an important role. There are even and odd Zernike polynomials. Straight Zernike polynomials are defined as:

And the odd by

And wherein non-negative integers, in which:. is the azimuthal angle and the normalized radial distance.

The Radialpolynome are as

And if it is odd, defined.

Often you will be normalized.

Properties

Zernike polynomials are a product of a radius-dependent part and an angle-dependent part:

A rotation of the coordinate system by the angle does not change the value of the polynomial:

The radius-dependent part is a polynomial of degree on which no power is smaller. is an even (odd) function if even (odd ) is.

The radius-dependent part represents a special case of the Jacobi polynomials

The series begins with the radius-dependent polynomials

It is generally

Applications

In optics, Zernike polynomials are used to represent the wave fronts, in turn, describe the aberrations of optical systems. This occurs, for example, in adaptive optics applications.

For several years, the use of Zernike polynomials in the optometry and ophthalmology is common. Here, deviations of the cornea or the lens of the ideal shape to imaging errors.