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

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

with radial function

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

for $n$ and $m$ integers with $n\geq m\geq 0$ and $n-m$ Even. Otherwise,

$$R_n^m(\rho)=0.$$ (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

$$\int_0^1 R_n^m(\rho)R_{n'}^m(\rho)\rho\,d\rho={1\over 2(n+1)}\delta_{nn'},$$ (4)

where $\delta_{ij}$ is the Kronecker Delta, and are related to the Bessel Function of the First Kind by

$$\int_0^1 R_n^m(\rho)J_m(v\rho)\rho\,d\rho=(-1)^{(n-m)/2}{J_{n+1}(v)\over v}$$ (5)

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

$${[1+z-\sqrt{1-2z(1-2\rho^2)+z^2}\,]^m\over (2z\rho)^m\sqrt{1... ...-2\rho^2)+z^2}} =\sum_{s=0}^\infty z^s R_{m+2s}^{\pm m}(\rho),$$ (6)

and are normalized so that

$$R_n^{\pm m}(1)=1$$ (7)

(Born and Wolf 1989, p. 465). The first few Nonzero radial polynomials are

$$R_0^0(\rho) = 1$$

$$R_1^1(\rho) = \rho$$

$$R_2^0(\rho) = 2\rho^2-1$$

$$R_2^2(\rho) = \rho^2$$

$$R_3^1(\rho) = 3\rho^3-2\rho$$

$$R_3^3(\rho) = \rho^3$$

$$R_4^0(\rho) = 6\rho^4-6\rho^2+1$$

$$R_4^2(\rho) = 4\rho^4-3\rho^2$$

$$R_4^4(\rho) = \rho^4$$

(Born and Wolf 1989, p. 465).

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

$$P_{n'}^{(\alpha,\beta)}(x)=(-1)^{n'}{R_n^m(\rho)\over\rho^\alpha}$$ (8)

and

$$x = 1-2\rho^2$$ (9)

$$\beta = 0$$ (10)

$$\alpha = m$$ (11)

$$n' = {\textstyle{1\over 2}}(n-m).$$ (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

$$F(\rho,\phi)=\sum_{m=0}^\infty \sum_{n=m}^\infty [A_n^m\,{}^oU_n^m(\rho,\phi)+B_n^m\,{}^e U_n^m(\rho,\phi)]$$ (16)

are given by

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

where

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

Let a ``primary'' aberration be given by

$$\Phi=a_{lmn}'{Y_1^{2l+m}}^*(\theta,\phi)\rho^n\cos^m\theta$$ (19)

with $2l+m+n=4$ and where $Y^*$ is the Complex Conjugate of $Y$, and define

$$A_{lmn}'=a_{lmn}'{Y_1^{2l+m}}^*(\theta,\phi),$$ (20)

giving

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

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.

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.