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Zernike Polynomial

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The Zernike polynomials are a set of orthogonal polynomials that arise in the expansion of a wavefront function for optical systems with circular pupils. The odd and even Zernike polynomials are given by

^oU_n^m(rho,phi); ^eU_n^m(rho,phi)=R_n^m(rho)sin; cos(mphi)
(1)

where the radial function R_n^m(rho) is defined for n and m integers with n>=m>=0 by

R_n^m(rho)={sum_(l=0)^((n-m)/2)((-1)^l(n-l)!)/(l![1/2(n+m)-l]![1/2(n-m)-l]!)rho^(n-2l) for n-m even; 0 for n-m odd.
(2)

Here, phi is the azimuthal angle with 0<=phi<2pi and rho is the radial distance with 0<=rho<=1 (Prata and Rusch 1989). The even and odd polynomials are sometimes also denoted

Z_n^(-m)(rho,phi)=^oU_n^m(rho,phi)=R_n^m(rho)sin(mphi)
(3)
Z_n^m(rho,phi)=^eU_n^m(rho,phi)=R_n^m(rho)cos(mphi).
(4)

Zernike polynomials are implemented in the Wolfram Language as ZernikeR[n, m, rho].

Other closed forms for R_n^m(rho) include

R_n^m(rho)=(Gamma(n+1)_2F_1(-1/2(m+n),1/2(m-n);-n;rho^(-2)))/(Gamma(1/2(2+n-m))Gamma(1/2(2+n+m)))rho^n
(5)

for n-m odd and m!=n, where Gamma(z) is the gamma function and _2F_1(a,b;c;z) is a hypergeometric function. This can also be written in terms of the Jacobi polynomial P_n^((alpha,beta))(x) as

R_n^m(rho)=(-1)^((n-m)/2)rho^mP_((n-m)/2)^((m,0))(1-2rho^2).
(6)

The first few nonzero radial polynomials are

R_0^0(rho)=1
(7)
R_1^1(rho)=rho
(8)
R_2^0(rho)=2rho^2-1
(9)
R_2^2(rho)=rho^2
(10)
R_3^1(rho)=3rho^3-2rho
(11)
R_3^3(rho)=rho^3
(12)
R_4^0(rho)=6rho^4-6rho^2+1
(13)
R_4^2(rho)=4rho^4-3rho^2
(14)
R_4^4(rho)=rho^4
(15)

(Born and Wolf 1989, p. 465).

The radial functions satisfy the orthogonality relation

int_0^1R_n^m(rho)R_(n^')^m(rho)rhodrho=1/(2(n+1))delta_(nn^')R_n^m(1),
(16)

where delta_(ij) is the Kronecker delta, and are related to the Bessel function of the first kind by

int_0^1R_n^m(rho)J_m(vrho)rhodrho=(-1)^((n-m)/2)(J_(n+1)(v))/v
(17)

(Born and Wolf 1989, p. 466). The radial Zernike polynomials have the generating function

([1+z-sqrt(1+2z(1-2rho^2)+z^2)]^m)/((2zrho)^msqrt(1+2z(1-2rho^2)+z^2))=sum_(s=0)^inftyz^sR_(m+2s)^(+/-m)(rho)
(18)

(correcting the typo of Born and Wolf) and are normalized so that

R_n^m(1)=1
(19)

(Born and Wolf 1989, p. 465).

The Zernike polynomials also satisfy the recurrence relations

rhoR_n^m(rho)=1/(2(n+1))[(n+m+2)R_(n+1)^(m+1)(rho)+(n-m)R_(n-1)^(m+1)(rho)] R_(n+2)^m(rho)=(n+2)/((n+2)^2-m^2){[4(n+1)rho^2-((n+m)^2)/n-((n-m+2)^2)/(n+2)]R_n^m(rho)-(n^2-m^2)/nR_(n-2)^m(rho)} R_n^m(rho)+R_n^(m+2)(rho)=1/(n+1)(d[R_(n+1)^(m+1)(rho)-R_(n-1)^(m+1)(rho)])/(drho)
(20)

(Prata and Rusch 1989). The coefficients A_n^m and B_n^m in the expansion of an arbitrary radial function F(rho,phi) in terms of Zernike polynomials

F(rho,phi)=sum_(m=0)^inftysum_(n=m)^infty[A_n^m^oU_n^m(rho,phi)+B_n^m^eU_n^m(rho,phi)]
(21)

are given by

A_n^m; B_n^m=((n+1))/(epsilon_(mn)^2pi)int_0^1int_0^(2pi)F(rho,phi)^oU_n^m(rho,phi); ^eU_n^m(rho,phi)rhodphidrho,
(22)

where

epsilon_(mn)={epsilon=1/(sqrt(2)) for m=0, n!=0; 1 otherwise
(23)

Let a "primary" aberration be given by

Phi=a_(lmn)^'Y^__1^(2l+m)(theta,phi)rho^ncos^mtheta
(24)

with 2l+m+n=4 and where Y^_ is the complex conjugate of Y, and define

A_(lmn)^'=a_(lmn)^'Y^__1^(2l+m)(theta,phi),
(25)

giving

Phi=1/(epsilon_(nm)^2)A_(lmn)R_n^m(rho)cos(mtheta).
(26)

Then the types of primary aberrations are given in the following table (Born and Wolf 1989, p. 470).

aberrationlnmAA^'
spherical aberration040A_(040)^'rho^4epsilonA_(040)R_4^0(rho)
coma031A_(031)^'rho^3costhetaA_(031)R_3^1(rho)costheta
astigmatism022A_(022)^'rho^2cos^2thetaA_(022)R_2^2(rho)cos(2theta)
field curvature120A_(120)^'rho^2epsilonA_(120)R_2^0(rho)
distortion111A_(111)^'rhocosthetaA_(111)R_1^1(rho)costheta

See also

Jacobi Polynomial

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References

Bezdidko, S. N. "The Use of Zernike Polynomials in Optics." Sov. J. Opt. Techn. 41, 425, 1974.Bhatia, A. B. and Wolf, E. "On the Circle Polynomials of Zernike and Related Orthogonal Sets." Proc. Cambridge Phil. Soc. 50, 40, 1954.Born, M. and Wolf, E. "The Diffraction Theory of Aberrations." Ch. 9 in Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. New York: Pergamon Press, pp. 459-490, 1989.Mahajan, V. N. "Zernike Circle Polynomials and Optical Aberrations of Systems with Circular Pupils." In Engineering and Lab. Notes 17 (Ed. R. R. Shannon), p. S-21, Aug. 1994.Prata, A. and Rusch, W. V. T. "Algorithm for Computation of Zernike Polynomials Expansion Coefficients." Appl. Opt. 28, 749-754, 1989.Wang, J. Y. and Silva, D. E. "Wave-Front Interpretation with Zernike Polynomials." Appl. Opt. 19, 1510-1518, 1980.Wyant, J. C. "Zernike Polynomials." http://wyant.optics.arizona.edu/zernikes/zernikes.htm.Zernike, F. "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode." Physica 1, 689-704, 1934.Zhang, S. and Shannon, R. R. "Catalog of Spot Diagrams." Ch. 4 in Applied Optics and Optical Engineering, Vol. 11. New York: Academic Press, p. 201, 1992.

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Zernike Polynomial

Cite this as:

Weisstein, Eric W. "Zernike Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ZernikePolynomial.html

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