Zernike polynomial

Zernike polynomials up to the 4th order and an example of the 6th order

The Zernike polynomials are orthogonal polynomials named after Frits Zernike and play an important role in wave optics in particular . There are even and odd Zernike polynomials. The even Zernike polynomials are defined by:

${\ displaystyle Z_ {n} ^ {m} (\ rho, \ phi) = R_ {n} ^ {m} (\ rho) \, \ cos (m \, \ phi)}$

and the odd through

${\ displaystyle Z_ {n} ^ {- m} (\ rho, \ phi) = R_ {n} ^ {m} (\ rho) \, \ sin (m \, \ phi),}$

where and are non-negative integers, for which: . is the azimuthal angle and is the normalized radial distance. ${\ displaystyle m}$${\ displaystyle n}$${\ displaystyle n \ geq m}$${\ displaystyle \ phi}$ ${\ displaystyle \ rho}$

The radial polynomials are defined according to ${\ displaystyle R_ {n} ^ {m}}$

${\ displaystyle R_ {n} ^ {m} (\ rho) = \! \ sum _ {k = 0} ^ {(nm) / 2} \! \! \! {\ frac {(-1) ^ { k} \, (nk)!} {k! \, ((n + m) / 2-k)! \, ((nm) / 2-k)!}} \; \ rho ^ {n-2 \ , k}}$,

when is even and when is odd. ${\ displaystyle nm}$${\ displaystyle R_ {n} ^ {m} (\ rho) = 0}$${\ displaystyle nm}$

Often they are too normalized. ${\ displaystyle R_ {n} ^ {m} (1) = 1}$

properties

Zernike polynomials are a product of a radius-dependent part and an angle-dependent part : ${\ displaystyle R_ {n} ^ {m}}$${\ displaystyle G ^ {m}}$

${\ displaystyle Z_ {n} ^ {\ pm m} (\ rho, \ phi) = R_ {n} ^ {m} (\ rho) \ cdot G ^ {m} (\ phi) \ !.}$

[For purists it should be pointed out that in physics and optics these functions of two arguments are called polynomials, but depending on the application, only the radial component, i.e. the sine-cosine-shaped azimuth functions, are regarded as too trivial to allow a name extension such as effect on Zernike functions .]

Rotating the coordinate system by the angle does not change the value of the polynomial: ${\ displaystyle \ alpha = 2 \ pi / m}$

${\ displaystyle G ^ {m} (\ phi + \ alpha) = G ^ {m} (\ phi) \ !.}$

The radius-dependent part is a polynomial over of degree , which does not contain a power smaller . is an even (odd) function if is even (odd). ${\ displaystyle \ rho}$${\ displaystyle n}$${\ displaystyle m}$${\ displaystyle R_ {n} ^ {m}}$${\ displaystyle m}$

The radius-dependent part represents a special case of the Jacobi polynomials . ${\ displaystyle P_ {n} ^ {(\ alpha, \ beta)} (z)}$

${\ displaystyle R_ {n} ^ {m} (\ rho) = (- 1) ^ {(nm) / 2} \ rho ^ {m} P _ {(nm) / 2} ^ {(m, 0)} (1-2 \ rho ^ {2})}$

The series of radius-dependent polynomials begins with

${\ displaystyle R_ {0} ^ {0} (\ rho) = 1}$
${\ displaystyle R_ {1} ^ {1} (\ rho) = \ rho}$
${\ displaystyle R_ {2} ^ {0} (\ rho) = 2 \ rho ^ {2} -1}$
${\ displaystyle R_ {2} ^ {2} (\ rho) = \ rho ^ {2}}$
${\ displaystyle R_ {3} ^ {1} (\ rho) = 3 \ rho ^ {3} -2 \ rho}$
${\ displaystyle R_ {3} ^ {3} (\ rho) = \ rho ^ {3}}$
${\ displaystyle R_ {4} ^ {0} (\ rho) = 6 \ rho ^ {4} -6 \ rho ^ {2} +1}$
${\ displaystyle R_ {4} ^ {2} (\ rho) = 4 \ rho ^ {4} -3 \ rho ^ {2}}$
${\ displaystyle R_ {4} ^ {4} (\ rho) = \ rho ^ {4}}$
${\ displaystyle R_ {5} ^ {1} (\ rho) = 10 \ rho ^ {5} -12 \ rho ^ {3} +3 \ rho}$
${\ displaystyle R_ {5} ^ {3} (\ rho) = 5 \ rho ^ {5} -4 \ rho ^ {3}}$
${\ displaystyle R_ {5} ^ {5} (\ rho) = \ rho ^ {5}}$
${\ displaystyle R_ {6} ^ {0} (\ rho) = 20 \ rho ^ {6} -30 \ rho ^ {4} +12 \ rho ^ {2} -1}$

General is ${\ displaystyle R_ {n} ^ {n} (\ rho) = \ rho ^ {n}.}$

Applications

In optics , Zernike polynomials are used to represent wavefronts , which in turn describe the imaging errors of optical systems. This is used, for example, in adaptive optics .

For some years now, the use of Zernike polynomials has also been common in optometry and ophthalmology . Here, deviations of the cornea or lens from the ideal shape lead to imaging errors.

literature

Commons : Zernike Polynomial  - Collection of Images
• Born and Wolf: Principles of Optics . Oxford: Pergamon, 1970.