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As outlined previously, there is a strong connection between Zernike polynomials and primary aberrations when expressed in terms of wavefront error. Table 5.2 clearly 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 3 and 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	Nn,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 2φ	45° Astigmatism
4	2	0		2ρ2 − 1	1	Defocus
5	2	2		ρ 2	cos 2φ	90° Astigmatism
6	3	−3		ρ 3	sin 3φ	Trefoil
7	3	−1		3ρ3 − 2ρ	sin φ	Coma Y
8	3	1		3ρ3 − 2ρ	cos φ	Coma X
9	3	3		ρ 3	cos 3φ	Trefoil
10	4	−4		ρ 4	sin 4φ	Quadrafoil
11	4	−2		4ρ4 − 3ρ2	sin 2φ	5th Order astigmatism 45°
12	4	0		6ρ4 − 6ρ2 + 1	1	Spherical aberration
13	4	2		4ρ4 − 3ρ2	cos 2φ	5th Order astigmatism 90°
14	4	4		ρ 4	cos 4φ	Quadrafoil
15	5	−5		ρ 5	sin 5φ	Pentafoil
16	5	−3		5ρ5 − 4ρ3	sin 3φ	High order trefoil
17	5	−1		5ρ5 − 4ρ3	sin φ	5th Order coma Y
18	5	3		10ρ5 − 12ρ3 + 3ρ	cos φ	5th Order coma X
19	5	−5		ρ 5	cos 3φ	High order trefoil
20	5	−5		ρ 5	cos 5φ	Pentafoil
21	6	−6		ρ 6	sin 6φ	Hexafoil
22	6	−4		6ρ6 − 5ρ4	sin 4φ	High order quadrafoil
23	6	−2		15ρ6 − 20ρ4 + 6ρ2	sin 2φ	7th Order astigmatism 45°
24	6	0		20ρ6 − 30ρ4 + 12ρ2 − 1	1	5th Order spherical aberration
25	6	2		15ρ6 − 20ρ4 + 6ρ2	cos 2φ	7th Order astigmatism 90°
26	6	4		6ρ6 − 5ρ4	cos 4φ	High order quadrafoil
27	6	6		ρ 6	cos 6φ	Hexafoil

(5.23)

The rms wavefront error has thus been reduced by a factor of six by the focus compensation process. Furthermore, this analysis feeds in to the discussion in Chapter 3 on the use of balancing aberrations to minimise wavefront error. For example, if we have succeeded in eliminating third order spherical aberration and are presented with residual fifth order spherical aberration, we can minimise the rms wavefront error by balancing this aberration with a small amount of third order aberration in addition to defocus. Analysis using Zernike polynomials is extremely useful in resolving this problem:

As previously outlined, the uncompensated rms wavefront error may be calculated from the RSS sum of all the four Zernike terms. Naturally, for the compensated system, we need only consider the first term.

(5.24)

For the fifth order spherical aberration, the rms wavefront error has been reduced by a factor of 20 through the process of aberration balancing. In terms of the practical application of this process, one might wish to optimise an optical design by minimising the rms wavefront error. Although, in practice, the process of optimisation will be carried out using software tools, nonetheless, it is useful to recognise some key features of an optimised design. By virtue of the previous example, optimisation of spherical aberration should lead to an OPD profile that is close to the 5th order Zernike term. This is shown in Figure 5.5 which illustrates the profile of an optimised OPD based entirely on the relevant fifth order Zernike term. The graph plots the nominal OPD again the normalised pupil function with the form given by the Zernike polynomial, n = 6, m = 0.

In the optimisation of an optical design it is important to understand the form of the OPD fan displayed in Figure 5.5 in order recognise the desired endpoint of the optimisation process. It displays three minima and two maxima (or vice versa), whereas the unoptimised OPD fan has one fewer maximum and minimum. Thus, although the design optimisation process itself might be computer based, nevertheless, understanding and recognising the how the process works and its end goal will be of great practical use. That is to say, as the computer-based optimisation proceeds, on might expect the OPD fan to acquire a greater number of maxima and minima.

Figure 5.5 Fifth order Zernike polynomial and aberration balancing.

One can apply the same analysis to all the Gauss-Seidel aberrations and calculate its associated rms wavefront error.

(5.25a)

(5.25b)

(5.25c)

(5.25d)

θ represents the field angle

Equations (5.25a)–(5.25d) are of great significance in the analysis of image quality, as the rms wavefront error is a key parameter in the description of the optical quality of a system. This will be discussed in more detail in the next chapter.

Worked Example 5.2 A plano-convex lens, with a focal length of 100 mm is used to focus a collimated beam; the refractive index of the lens material is 1.52. It is assumed that the curved surface faces the infinite conjugate. The pupil diameter is 12.5 mm and the aperture is situated at the lens. What is the rms spherical aberration produced by this lens – (i) at the paraxial focus; (ii) at the compensated focus? What is the rms coma for a similar collimated beam with a field angle of one degree?

Firstly, we calculate the spherical aberration of the single lens. With the object at infinity and the image at the first focal point, the conjugate parameter, t, is equal to −1. The shape parameter, s, for the plano convex lens is equal to 1 since the curved surface is facing the object. From Eq. (4.30a) the spherical aberration of the lens is given by:

r_max = 6.25 mm (12.5/2); f = 100 mm; n = 1.52; s = 1; t = −1

By substituting these values into the above equation, the spherical aberration may be directly calculated:

where A = 4.13 × 10⁻⁴ mm ρ = r/r_max

From Eq. (5.23), the uncompensated rms wavefront error is A/√5 and the compensated error is A/√180. Therefore the rms values are given by:

Φ _rms (paraxial) = 185 nm; Φ _rms (compensated) = 30.8 nm

Secondly, we calculate the coma. From (4.30b), the coma of the lens is given by:

Again, substituting the relevant values for f, n, r_max, s, and t, we get:

where A = 3.24 × 10⁻³ mm ρ = r/r_max r_y = r sin ϕ

From (5.25b)

We are told that θ = 1° or 0.0174 rad. Therefore, Φ_rms = 6.66 × 10⁻⁶or 6.66 nm