Zernike Surface

Zernike Surface Information

KostaCLOUD offers various definitions to Zernike Surfaces, as every field, institution, and researcher has their preference to the ordering of Zernike Polynomials, normalization, etc. For this reason, KostaCLOUD allows you to pick your representation.

In KostaCLOUD Zernike Polynomials are defined as follows:

{NnmRnm(r)cos(mθ)m>0NnmRnm(r)m=0NnmRnm(r)sin(mθ)m<0\begin{cases} \:\:\:\:\: N_n^m R_n^{|m|}(r) \cos(m\theta) & m> 0 \\ \:\:\:\: N_n^m R_n^{|m|}(r) & m = 0 \\ -N_n^m R_n^{|m|}(r) \sin(m\theta) & m < 0 \end{cases}

NnmN_n^m are the Normalization Coefficients. These can be set to 1 to keep the polynomials Un-Normalized. Or for those who want the Zernike Polynomials to remain normalized, these coefficients are:

Nnm=2(n+1)1+δm,0N_n^m = \sqrt{\frac{2(n+1)}{1+\delta_{m,0}}}

Rnm(r)R_n^{|m|}(r) are the Radial Zernike Polynomials, which are defined by the following sum:

Rnm(r)=s=0(nm)/2(1)s(ns)!s!(n+m2s)!(nm2s)!rn2sR_n^{|m|}(r) = \sum_{s=0}^{(n-|m|)/2}\frac{(-1)^s (n-s)!}{ s!(\frac{n+|m|}{2}-s)!(\frac{n-|m|}{2}-s)!}r^{n-2s}

An important consideration when dealing with these polynomials is that the radius (r) of the polynomial function must be normalized to the unit circle, as Zernike polynomials are orthogonal over the unit-circle. Therefore Zernike Surfaces are regularized to their respective aperture maximum radius. An example of this regularization is shown below, where each shape is shown to fit within the unit circle.


  1. J. Schwiegerling, Optical specification, fabrication, and testing, SPIE, Bellingham, Washington (1000 20th St. Bellingham WA 98225-6705 USA) (2014).

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