Stephen Rolt - Optical Engineering Science

As outlined previously, there is a strong connection between Zernike polynomials and primary aberrations when expressed in terms of wavefront error. Table 5.2clearly shows the correspondence between the polynomials and the Gauss Seidel aberrations, with the 3rd order Gauss-Seidel aberrations, such as spherical aberration and coma clearly visible.

The power of the Zernike polynomials, as an orthonormal set, lies in their ability to represent any arbitrary wavefront aberration. Using the approach set out in Eq. (5.13), it is possible to compute the magnitude of any Zernike term by the cross integral of the relevant polynomial and the wavefront disturbance. Furthermore, the total root mean square (rms) wavefront error, as per Eq. (5.14), may be calculated from the RSS (root sum square) of the individual Zernike magnitudes. That is to say, the Zernike magnitude of each term represents its contribution to the rms wavefront error, as averaged over the whole pupil.

The use of defocus to compensate spherical aberration was explored in Chapters 3and 4. In this instance, for a given amount of fourth order wavefront error, we sought to minimise the rms wavefront error by applying a small amount of defocus.

Hence, without defocus, adjustment, the raw spherical aberration produced in a system may be expressed as the sum of three Zernike terms, one spherical aberration, one defocus and one piston term. The total aberration for an uncompensated system is simply given by the RSS of the individual terms. However, for a compensated system only the Zernike n = 4, m = 0 term needs be considered. This then gives the following fundamental relationship:

Table 5.2 First 28 Zernike polynomials.

ANSI#	N	m	N n,m	R(ρ)	G(ϕ)	Name
0	0	0	1	1	1	Piston
1	1	−1		ρ	sin φ	Tilt X
2	1	1		ρ	cos φ	Tilt Y
3	2	−2		ρ 2	sin 2 φ	45° Astigmatism
4	2	0		2 ρ 2− 1	1	Defocus
5	2	2		ρ 2	cos 2 φ	90° Astigmatism
6	3	−3		ρ 3	sin 3 φ	Trefoil
7	3	−1		3 ρ 3− 2 ρ	sin φ	Coma Y
8	3	1		3 ρ 3− 2 ρ	cos φ	Coma X
9	3	3		ρ 3	cos 3 φ	Trefoil
10	4	−4		ρ 4	sin 4 φ	Quadrafoil
11	4	−2		4 ρ 4− 3 ρ 2	sin 2 φ	5th Order astigmatism 45°
12	4	0		6 ρ 4− 6 ρ 2+ 1	1	Spherical aberration
13	4	2		4 ρ 4− 3 ρ 2	cos 2 φ	5th Order astigmatism 90°
14	4	4		ρ 4	cos 4 φ	Quadrafoil
15	5	−5		ρ 5	sin 5 φ	Pentafoil
16	5	−3		5 ρ 5− 4 ρ 3	sin 3 φ	High order trefoil
17	5	−1		5 ρ 5− 4 ρ 3	sin φ	5th Order coma Y
18	5	3		10 ρ 5− 12 ρ 3+ 3 ρ	cos φ	5th Order coma X
19	5	−5		ρ 5	cos 3 φ	High order trefoil
20	5	−5		ρ 5	cos 5 φ	Pentafoil
21	6	−6		ρ 6	sin 6 φ	Hexafoil
22	6	−4		6 ρ 6− 5 ρ 4	sin 4 φ	High order quadrafoil
23	6	−2		15 ρ 6− 20 ρ 4+ 6 ρ 2	sin 2 φ	7th Order astigmatism 45°
24	6	0		20 ρ 6− 30 ρ 4+ 12 ρ 2− 1	1	5th Order spherical aberration
25	6	2		15 ρ 6− 20 ρ 4+ 6 ρ 2	cos 2 φ	7th Order astigmatism 90°
26	6	4		6 ρ 6− 5 ρ 4	cos 4 φ	High order quadrafoil
27	6	6		ρ 6	cos 6 φ	Hexafoil

(5.23)