Biomedical Engineering Reference
In-Depth Information
Table 4.1
Deinition of the First 22 Zernike Polynomials
i
n
m
Aberration Term
1
0
0
1
Piston
2
1
1
2
r
cos(θ)
Tip
3
1
−1
2
r
sin(θ)
Tilt
4
2
0
Defocus
3(2
r
2
−
1)
5
2
2
6
r
2
cos(2 )
θ
Astigmatism
6
2
−2
Astigmatism
6
r
2
sin(2 )
θ
7
3
1
Coma
2 2(3
r
3
−
2 )cos( )
r
θ
8
3
−1
2 2(3
r
3
−
2 )sin( )
r
θ
Coma
9
3
3
2 2
r
3
cos(3 )
θ
10
3
−3
2 2
r
3
sin(3 )
θ
11
4
0
5(6
r
4
−
6
r
2
+
1)
Spherical (1st)
12
4
2
10(4
r
4
−
3 )cos(2 )
r
2
θ
13
4
−2
10(4
r
4
−
3 )sin(2 )
r
2
θ
14
4
4
10
r
4
cos(4 )
θ
15
4
−4
10
r
4
sin(4 )
θ
16
5
1
2 3(10
r
5
−
12
r
3
+
3 )cos( )
r
θ
17
5
−1
2 3(10
r
5
−
12
r
3
+
3 )sin( )
r
θ
18
5
3
2 3(5
r
5
−
4 )cos(3 )
r
3
θ
19
5
−3
2 3(5
r
5
−
4 )sin(3 )
r
3
θ
20
5
5
2 3
cos(5 )
r
5
θ
21
5
−5
2 3
r
5
sin(5 )
θ
22
6
0
7(20
r
6
−
30
r
4
+
12
r
2
−
1)
Spherical (2nd)
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
-
0.5
-
0.5
-
0.5
-
0.5
-
0.5
-
0.5
1
1
1
1
1
1
-
1
-
0.5
0
0.5
1
-
1
-
0.5
0
0.5
1
-
1
-
0.5
0
0.5
1
-
1
-
0.5
0
0.5
1
-
1
-
0.5
0
0.5
1
-
1
-
0.5
0
0.5
1
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.2
Polar plots of a few Zernike modes: (a)
Z
2
(
r
,θ), tip; (b)
Z
4
(
r
,θ), defocus; (c)
Z
5
(
r
,θ), astigmatism;
(d)
Z
7
(
r
,θ), coma; (e)
Z
11
(
r
,θ), irst spherical; and (f)
Z
18
(
r
,θ), trefoil.
of the polynomials are identical except for a rotation about the origin. For instance, modes 5 and 6
(astigmatism) difer just by a rotation of 45°.
his is, apart from the indexing, equivalent to the deinition of Noll (1976) and has the advantage
of a normalization such that the mean is zero over the unit circle and the variance is unity (except for
n
= 0, where it is zero). All data in this chapter are given in the Zernike coeicient units according to the
deinition in Equation 4.5. By calculating the wavefront using Equations 4.3 and 4.5, one directly gets
the aberration of the wavefront over the normalized pupil of the system in radians.
