Zernike polynomials

---

200px|thumb">
Plots of the values in the unit disk.

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.

Definitions

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

:$Z^\{m\}\_n(\; ho,varphi)\; =\; R^m\_n(\; ho),cos(m,varphi)\; !$

and the odd ones as

:$Z^\{-m\}\_n(\; ho,varphi)\; =\; R^m\_n(\; ho),sin(m,varphi),\; !$

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 "R"^m_n are defined as

:$R^m\_n(\; ho)\; =\; !\; sum\_\{k=0\}^\{(n-m)/2\}\; !!!\; frac\{(-1)^k,(n-k)!\}\{k!,((n+m)/2-k)!,((n-m)/2-k)!\}\; ;\; ho^\{n-2,k\}$

or

:$R^m\_n(\; ho)\; =\; frac\{Gamma(n+1)\{\}\_2F\_\{1\}(-frac\{1\}\{2\}(|m|+n),frac\{1\}\{2\}(|m|-n);-n;\; ho^\{-2\})\}\{Gamma(frac\{1\}\{2\}(2+n+m))Gamma(frac\{1\}\{2\}(2+n-m))\}\; ho^n$

for "n" − "m" even, and are identically 0 for "n" − "m" odd.

For "m" = 0, the even definition is used which reduces to "R"_n⁰("ρ").

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'. [ [http://mathworld.wolfram.com/ZernikePolynomial.html "Zernike Polynomial"] ] 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.

References

* Born and Wolf, "Principles of Optics", Oxford: Pergamon, 1970
* Eric W. Weisstein et al., [http://mathworld.wolfram.com/ZernikePolynomial.html "Zernike Polynomial"] , at MathWorld.
* C. E. Campbell, [http://www.opticsinfobase.org/abstract.cfm?URI=josaa-20-2-209 "Matrix method to find a new set of Zernike coefficients form an original set when the aperture radius is changed"] , J. Opt. Soc. Am. A 20 (2003) 209.
* C. Cerjan, [http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-6-1609 "The Zernike-Bessel representation and its application to Hankel transforms"] , J. Opt. Soc. Am. A 24 (2007) 1609.
* S. A. Comastri, L. I. Perez, G. D. Perez, G. Martin and K. Bastida Cerjan, Zernike expansion coefficients: rescaling and decentering for different pupils and evaluation of corneal aberrations, J. Opt. A: Pure Appl. Opt. 9 (2007) 209.
* G. Conforti, [http://www.opticsinfobase.org/abstract.cfm?URI=ol-8-7-407 "Zernike aberration coefficients from Seidel and higher-order power-series coefficients"] , Opt. Lett. 8 (1983) 407.
* G-m. Dai and V. N. Mahajan, [http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-1-139 "Zernike annular polynomials and atmospheric turbulence"] , J. Opt. Soc. Am. A 24 (2007) 139.
* G-m. Dai, [http://www.opticsinfobase.org/abstract.cfm?URI=josaa-23-3-539 "Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula"] , J. Opt. Soc. Am. A 23 (2006) 539.
* J. Herrmann, [http://www.opticsinfobase.org/abstract.cfm?URI=josa-71-8-989 "Cross coupling and aliasing in modal wave-front estimation"] , J. Opt. Soc. Am. 71 (1981) 989.
* P. H. Hu, J. Stone and T. Stanley, [http://www.opticsinfobase.org/abstract.cfm?URI=josaa-6-10-1595 "Application of Zernike polynomials to atmospheric propagation problems"] , J. Opt. Soc. Am. A 6 (1989) 1595.
* E. C. Kintner, On the mathematical properties of the Zernike Polynomials, Opt. Acta 23 (1976) 679.
* G. N. Lawrence and W. W. Chow, [http://www.opticsinfobase.org/abstract.cfm?URI=ol-9-7-267 "Wave-front tomography by Zernike Polynomial decomposition"] , Opt. Lett. 9 (1984) 287.
* L. Lundstrom and P. Unsbo, [http://www.opticsinfobase.org/abstract.cfm?URI=josaa-6-10-1595 "Transformation of Zernike coefficients: scaled, translated and rotate wavefronts with circular and elliptical pupils"] , J. Opt. Soc. Am. A 24 (2007) 569.
* V. N. Mahajan, [http://www.opticsinfobase.org/abstract.cfm?URI=josa-71-1-75 "Zernike annular polynomials for imaging systems with annular pupils"] , J. Opt. Soc. Am. 71 (1981) 75.
* R. J. Mathar, [http://arxiv.org/abs/0705.1329 "Third Order Newton's Method for Zernike Polynomial Zeros"] , arXiv:math.NA/0705.1329.
* R. J. Noll, [http://www.opticsinfobase.org/abstract.cfm?URI=josa-66-3-207 "Zernike polynomials and atmospheric turbulence"] , J. Opt. Soc. Am. 66 (1976) 207.
* A. Prata Jr and W. V. T. Rusch, [http://www.opticsinfobase.org/abstract.cfm?URI=ao-28-4-749 "Algorithm for computation of Zernike polynomials expansion coefficients"] , Appl. Opt. 28 (1989) 749.
* J. Schwiegerling, [http://www.opticsinfobase.org/abstract.cfm?URI=josaa-19-10-1937 "Scaling Zernike expansion coefficients to different pupil sizes"] , J. Opt. Soc. Am. A 19 (2002) 1937.
* C. J. R. Sheppard, S. Campbell and M. D. Hirschhorn, [http://www.opticsinfobase.org/abstract.cfm?URI=ao-43-20-3963 "Zernike expansion of separable functions in Cartesian coordinates"] , Appl. Opt. 43 (2004) 3963.
* H. Shu, L. Luo, G. Han and J.-L. Coatrieux, [http://www.opticsinfobase.org/abstract.cfm?URI=josaa-23-8-1960 "General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes "] , J. Opt. Soc. Am. A 23 (2006) 1960.
* W. Swantner and W. W. Chow, [http://www.opticsinfobase.org/abstract.cfm?URI=ao-33-10-1832 "Gram-Schmidt orthogonalization of Zernike polynomials for general aperture shapes"] , Appl. Opt. 33 (1994) 1832.
* W. J. Tango, The circle polynomials of Zernike and their application in optics, Appl. Phys. A 13 (1977) 327.
* R. K. Tyson, [http://www.opticsinfobase.org/abstract.cfm?URI=ol-7-6-262 "Conversion of Zernike aberration coefficients to Seidel and higher-order power series aberration coefficiets"] , Opt. Lett. 7 (1982) 262.
* J. Y. Wang and D. E. Silva, [http://www.opticsinfobase.org/abstract.cfm?URI=ao-19-9-1510 "Wave-front interpretation with Zernike Polynomials"] , Appl. Opt. 19 (1980) 1510.
* R. Barakat, [http://www.opticsinfobase.org/abstract.cfm?URI=josa-70-6-739 "Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: Generalizations of Zernike polynomials"] , J. Opt. Soc. Am. 70 (1980) 739.
* A. B. Bathia and E. Wolf, The Zernike circle polynomials occurring in diffraction theory, Proc. Phys. Soc. B 65 (1952) 909.
* T. A. ten Brummelaar, Modeling atmospheric wave aberrations and astronomical instrumentation using the polynomials of Zernike, Opt. Commun. 132 (1996) 329.
* M. Novotni, R. Klein, [http://www.cg.cs.uni-bonn.de/docs/publications/2003/novotni-2003-3d.pdf "3D Zernike Descriptors for Content Based Shape Retrieval"] , in proceedings of The 8th ACM Symposium on Solid Modeling and Applications June 2003.
* M. Novotni, R. Klein, [http://www.cg.cs.uni-bonn.de/docs/publications/2004/novotni-2004-shape.pdf "Shape retrieval using 3D Zernike descriptors"] , in Computer Aided Design, Vol. 36, No. 11, pages 1047-1062, 2004.

ee also

* Pseudo-Zernike polynomials
* Jacobi polynomials

External links

* [http://www.nijboerzernike.nl The Extended Nijboer-Zernike website.]

* [http://www.strw.leidenuniv.nl/~mathar/progs/Zernike.txt Cross-expansions in terms of powers and Jacobi Polynomials.]