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Zernike Polynomial

Orthogonal Polynomials which arise in the expansion of a wavefront function for optical systems with circular pupils. The Odd and Even Zernike polynomials are given by

\matrix{{}^o U_n^m(\rho,\phi)\cr {}^e U_n^m(\rho,\phi)\cr}=R_n^m(\rho)\matrix{\sin\cr\cos\cr}(m\phi)
\end{displaymath} (1)

with radial function
$R_n^m(\rho)=\sum_{l=0}^{(n-m)/2} {(-1)^l (n-l)!\over l![{\textstyle{1\over 2}}(n+m)-l]![{\textstyle{1\over 2}}(n-m)-l]!}\rho^{n-2l}$

for $n$ and $m$ integers with $n\geq m\geq 0$ and $n-m$ Even. Otherwise,
\end{displaymath} (3)

Here, $\phi$ is the azimuthal angle with $0\leq\phi<2\pi$ and $\rho$ is the radial distance with $0\leq\rho\leq 1$ (Prata and Rusch 1989). The radial functions satisfy the orthogonality relation
\int_0^1 R_n^m(\rho)R_{n'}^m(\rho)\rho\,d\rho={1\over 2(n+1)}\delta_{nn'},
\end{displaymath} (4)

where $\delta_{ij}$ is the Kronecker Delta, and are related to the Bessel Function of the First Kind by
\int_0^1 R_n^m(\rho)J_m(v\rho)\rho\,d\rho=(-1)^{(n-m)/2}{J_{n+1}(v)\over v}
\end{displaymath} (5)

(Born and Wolf 1989, p. 466). The radial Zernike polynomials have the Generating Function
{[1+z-\sqrt{1-2z(1-2\rho^2)+z^2}\,]^m\over (2z\rho)^m\sqrt{1...
=\sum_{s=0}^\infty z^s R_{m+2s}^{\pm m}(\rho),
\end{displaymath} (6)

and are normalized so that
R_n^{\pm m}(1)=1
\end{displaymath} (7)

(Born and Wolf 1989, p. 465). The first few Nonzero radial polynomials are
$\displaystyle R_0^0(\rho)$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle R_1^1(\rho)$ $\textstyle =$ $\displaystyle \rho$  
$\displaystyle R_2^0(\rho)$ $\textstyle =$ $\displaystyle 2\rho^2-1$  
$\displaystyle R_2^2(\rho)$ $\textstyle =$ $\displaystyle \rho^2$  
$\displaystyle R_3^1(\rho)$ $\textstyle =$ $\displaystyle 3\rho^3-2\rho$  
$\displaystyle R_3^3(\rho)$ $\textstyle =$ $\displaystyle \rho^3$  
$\displaystyle R_4^0(\rho)$ $\textstyle =$ $\displaystyle 6\rho^4-6\rho^2+1$  
$\displaystyle R_4^2(\rho)$ $\textstyle =$ $\displaystyle 4\rho^4-3\rho^2$  
$\displaystyle R_4^4(\rho)$ $\textstyle =$ $\displaystyle \rho^4$  

(Born and Wolf 1989, p. 465).

The Zernike polynomial is a special case of the Jacobi Polynomial with

\end{displaymath} (8)

$\displaystyle x$ $\textstyle =$ $\displaystyle 1-2\rho^2$ (9)
$\displaystyle \beta$ $\textstyle =$ $\displaystyle 0$ (10)
$\displaystyle \alpha$ $\textstyle =$ $\displaystyle m$ (11)
$\displaystyle n'$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(n-m).$ (12)

The Zernike polynomials also satisfy the Recurrence Relations

$\rho R_n^m(\rho)={1\over 2(n+1)} [(n+m+2)R_{n+1}^{m+1}(\rho) +(n-m)R_{n-1}^{m+1}(\rho)]$ (13)
$R^m_{n+2}(\rho)={n+2\over(n+2)^2-m^2}\left\{{\left[{4(n+1)\rho^2-{(n+m)^2\over n}}\right.}\right.$
$ \left.{\left.{-{(n-m+2)^2\over n+2}}\right]R_n^m(\rho)-{n^2-m^2\over n}R_{n-2}^m(\rho)}\right.\quad$ (14)
$R_n^m(\rho)+R_n^{m+2}(\rho)={1\over n+1}{d[R^{m+1}_{n+1}(\rho)-R^{m+1}_{n-1}(\rho)]\over d\rho}$ (15)
(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}^\infty \sum_{n=m}^\infty [A_n^m\,{}^oU_n^m(\rho,\phi)+B_n^m\,{}^e U_n^m(\rho,\phi)]
\end{displaymath} (16)

are given by
\matrix{A_n^m\cr B_n^m\cr}={(n+1)\over{\epsilon_{mn}}^2\pi}
...n^m(\rho,\phi)\cr {}^e U_n^m(\rho,\phi)\cr}\rho\,d\phi\,d\rho,
\end{displaymath} (17)

\epsilon\equiv {1\over\sqrt{2}} & for $m=0$, $n\not=0$\cr
1 & otherwise\cr}
\end{displaymath} (18)

Let a ``primary'' aberration be given by

\end{displaymath} (19)

with $2l+m+n=4$ and where $Y^*$ is the Complex Conjugate of $Y$, and define
\end{displaymath} (20)

\Phi={1\over{\epsilon_{nm}}^2} A_{lmn}R_n^m(\rho)\cos(m\theta).
\end{displaymath} (21)

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

Aberration $l$ $m$ $n$ $A$ $A'$
spherical aberration 0 4 0 $A_{040}'\rho^4$ $\epsilon A_{040}R_4^0(\rho)$
coma 0 3 1 $A_{031}'\rho^3\cos\theta$ $A_{031}R_3^1(\rho)\cos\theta$
astigmatism 0 2 2 $A_{022}'\rho^2\cos^2\theta$ $A_{022}R_2^2(\rho)\cos(2\theta)$
field curvature 1 2 0 $A_{120}'\rho^2$ $\epsilon A_{120} R_2^0(\rho)$
distortion 1 1 1 $A_{111}'\rho\cos\theta$ $A_{111}R_1^1(\rho)\cos\theta$

See also Jacobi Polynomial


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.

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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© 1996-9 Eric W. Weisstein