Zernike PolynomialsΒΆ

The Zernike polynomials Z_j are a complete sequence of polynomials that are orthogonal on the unit disk. Using polar coordinates (\rho, \phi), so that (x, y) = \rho(\cos \varphi, \sin \varphi), the Zernike polynomials are defined as

\begin{eqnarray*}
Z_j(\rho, \varphi) = Z_n^m(\rho, \varphi) =  R_n^m(\rho) Y_m^j(\varphi),
\end{eqnarray*}

with

\begin{eqnarray*}
R_n^m(\rho) & = & \sum_{k=0}^{(n-m)/2} (-1)^k
\left ( \begin{array}{c} n-k \\ k \end{array}  \right )
\left ( \begin{array}{c} n-2k \\ (n-m)/2-k \end{array}  \right )
\rho^{n-2k} \\
Y_m^j(\varphi) & = & \left \{
\begin{array}{ll}
\cos(m\varphi), & \mbox{if}\; m\geq0, \\
\sin(m\varphi), & \mbox{if}\; m<0
\end{array}
\right . ,
\end{eqnarray*}

and where the integer index pair (n, m) is given by

\begin{eqnarray*}
m &  = & \left \{
\begin{array}{ll}
\frac{d^2-j}{2}, & \mbox{if}\; d^2-j\,\mbox{is even}, \\
\frac{-d^2+j-1}{2}, & \mbox{if}\; \mbox{otherwise}
\end{array}
\right . , \\
n & = & 2(d-1) -|m|,
\end{eqnarray*}

where d = \lfloor \sqrt{j-1} \rfloor+1, and \lfloor \cdot \rfloor represents the largest integer that is less or equal to the delimited integer.

Warning

Different orderings (n, m) \rightarrow j of the Zernike polynomials are in use. Here, we followed [1] (page 213). Besides this, different scalings of the Zernike polynomials are used.

In the above, the Fringe convention as been used for scaling (c.f. http://en.wikipedia.org/wiki/Zernike_polynomials or http://mathworld.wolfram.com/ZernikePolynomial.html). For the sake of clarity, the following table lists the leading 36 (Fringe)-Zernike polynomials:

j n m Z_j description
1 0 0 1 piston
2 1 1 \rho \cos(\varphi) x-tilt
3 1 -1 \rho \sin(\varphi) y-tilt
4 2 0 -1+2\rho^2 defocus
5 2 2 \rho^2 \cos(2\varphi) astigmatism
6 2 -2 \rho^2 \sin(2\varphi) astigmatism
7 3 1 (-2\rho+3\rho^3) \cos(\varphi) coma
8 3 -1 (-2\rho+3\rho^3) \sin(\varphi) coma
9 4 0 1-6\rho^2+6\rho^4 spherical aberration
10 3 3 \rho^3 \cos(3\varphi) trifoil
11 3 -3 \rho^3 \sin(3\varphi) trifoil
12 4 2 (-3\rho^2+4\rho^4) \cos(2\varphi) astigmatism
13 4 -2 (-3\rho^2+4\rho^4) \sin(2\varphi) astigmatism
14 5 1 (3\rho-12\rho^3+10\rho^5) \cos(\varphi) coma
15 5 -1 (3\rho-12\rho^3+10\rho^5) \sin(\varphi) coma
16 6 0 -1+12\rho^2-30\rho^4+20\rho^6 spherical aberration
17 4 4 \rho^4 \cos(4\varphi) four wave
18 4 -4 \rho^4 \sin(4\varphi) four wave
19 5 3 (-4\rho^3+5\rho^5) \cos(3\varphi) trifoil
20 5 -3 (-4\rho^3+5\rho^5) \sin(3\varphi) trifoil
21 6 2 (6\rho^2-20\rho^4+15\rho^6) \cos(2\varphi ) astigmatism
22 6 -2 (6\rho^2-20\rho^4+15\rho^6) \sin(2\varphi) astigmatism
23 7 1 (-4\rho+30\rho^3-60\rho^5+35\rho^7) \cos(\varphi) coma
24 7 -1 (-4\rho+30\rho^3-60\rho^5+35\rho^7) \sin(\varphi) coma
25 8 0 1-20\rho^2+90\rho^4-140\rho^6+70\rho^8 spherical aberration
26 5 5 \rho^5 \cos(5\varphi) five wave
27 5 -5 \rho^5 \sin(5\varphi) five wave
28 6 4 (-5\rho^4+6\rho^6) \cos(4\varphi) four wave
29 6 -4 (-5\rho^4+6\rho^6) \sin(4\varphi) four wave
30 7 3 (10\rho^3-30\rho^5+21\rho^7) \cos(3\varphi) trifoil
31 7 -3 (10\rho^3-30\rho^5+21\rho^7) \sin(3\varphi) trifoil
32 8 2 (-10\rho^2+60\rho^4-105\rho^6+56\rho^8) \cos(2\varphi) astigmatism
33 8 -2 (-10\rho^2+60\rho^4-105\rho^6+56\rho^8) \sin(2\varphi) astigmatism
34 9 1 (5\rho-60\rho^3+210\rho^5-280\rho^7+126\rho^9) \cos(\varphi) coma
35 9 -1 (5\rho-60\rho^3+210\rho^5-280\rho^7+126\rho^9) \sin(\varphi) coma
36 10 0 -1+30\rho^2-210\rho^4+560\rho^6-630\rho^8+252\rho^{10} spherical aberration

Bibliography

[1]Gross H. (editor), Handbook of Optical Systems, Volume III, Wiley-VCH 2005