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We have emphasised the synergy between Zernike polynomials and the classical treatment of aberrations in an axially symmetric optical system, i.e. the Gauss-Seidel aberrations. However, in practice, in real optical systems, these axial symmetries are often compromised, either by accident or by design. Some systems are deliberately designed whereby not all optical surfaces are aligned to a common axis. These will inevitably introduce non-standard wavefront aberrations into the system. Most significantly, even with a symmetrical design, component manufacturing errors and system alignment may introduce more complex wavefront errors into the system. Naturally, alignment errors create an off-axis optical system ‘by accident’. Manufacturing or polishing errors might produce an optical surface whose shape departs from that of an ideal sphere or conic in a somewhat complex fashion. For example, the effects of these errors may be to introduce a trefoil term (n = 3, m = 3) into the wavefront error; this is not a standard Gauss-Seidel term.

As argued, Zernike polynomials are widely used in the analysis of wavefront error both in the design and testing of optical systems. From a strictly analytical and theoretical point of view the description of wavefront error in terms of its rms value is the most meaningful. However, for largely historical reasons, wavefront error is often presented as a ‘peak to valley’ error. That is to say, the value presented is the difference between the maximum and minimum OPD across the pupil. Historically, the wavefront error for a system might have been derived from a visual inspection of a fringe pattern in an interferogram. The maximum deviation of fringes is relatively straightforward to estimate visually from a fringe pattern which might have been produced photographically. However, the rms wavefront error is more directly related to system performance. Calculation of the rms wavefront error across a pupil is a mathematical process that requires computational data acquisition and analysis and has only been universally available in more recent times. Therefore, the use of the peak to valley description still persists.

One particular disadvantage of the peak to valley description is that it is unusually responsive to large, but highly localised excursions in the wavefront error. More generally, as a rule of thumb, the peak to valley is considered to be 3.5 times the rms value. Of course, this does depend upon the form of the wavefront error. Table 5.3 sets out this relationship for the first 11 Zernike terms (apart from piston). For comparison, a standard statistical measure is also presented – namely for a normally distributed wavefront error profile, the limits containing 95% of the wavefront error distribution (±1.96 standard deviations).

The values presented in Table 5.3 are simply the ratio of the peak to valley (p-to-v) error for that particular distribution. To overcome the principal objection to the p-to-v measure, namely its heightened sensitivity to local variation a new peak to valley measure has been proposed by the Zygo Corporation. This measure is known as P to Vr or peak to valley robust. In this measure, the wavefront error is fitted to a set of 36 Zernike polynomials. Although this process is carried out by computational analysis, the procedure is very simple. Essentially the calculation process exploits the orthonormal properties of the polynomial set and calculates the contribution of each Zernike term using the relation set out in Eq. (5.12). Following this process, the maximum and minimum of the fitted surface is calculated and the revised peak to valley figure calculated. Of course, the reduced set of 36 polynomials cannot possibly replicate localised asperities with a high spatial frequency content. Therefore, the fitted surface is effectively a smoothed version of the original and the peak to valley value derived is more representative of the underlying physics.

Table 5.3 Peak to valley: Root mean square (rms) ratios for different wavefront error forms.

Noll#	n	m	Description	P_to_V multiplier
2 and 3	1	±1	Tilt	2.83
4	2	0	Defocus	3.46
5 and 6	2	±2	Astigmatism	4.90
7 and 8	3	±1	Coma	5.66
9 and 10	3	±3	Trefoil	5.66
11	4	0	Spherical aberration	3.35
—	—	—	95% Gaussian	3.92

Table 5.4 Comparison of Zernike numbering systems.

n	m	ANSI	Noll	Fringe	n	m	ANSI	Noll	Fringe	n	m	ANSI	Noll	Fringe
0	0	0	1	0	6	−4	22	25	28	8	8	44	44	64
1	−1	1	3	2	6	−2	23	23	21	9	−9	45	55	82
1	1	2	2	1	6	0	24	22	15	9	−7	46	53	67
2	−2	3	5	5	6	2	25	24	20	9	−5	47	51	54
2	0	4	4	3	6	4	26	26	27	9	−3	48	49	43
2	2	5	6	4	6	6	27	28	36	9	−1	49	47	34
3	−3	6	9	10	7	−7	28	35	50	9	1	50	46	33
3	−1	7	7	7	7	−5	29	33	39	9	3	51	48	42
3	1	8	8	6	7	−3	30	31	30	9	5	52	50	53
3	3	9	10	9	7	−1	31	29	23	9	7	53	52	66
4	−4	10	15	17	7	1	32	30	22	9	9	54	54	81
4	−2	11	13	12	7	3	33	32	29	10	−10	55	66	101
4	0	12	11	8	7	5	34	34	38	10	−8	56	64	84
4	2	13	12	11	7	7	35	36	49	10	−6	57	62	69
4	4	14	14	16	8	−8	36	45	65	10	−4	58	60	56
5	−5	15	21	26	8	−6	37	43	52	10	−2	59	58	45
5	−3	16	19	19	8	−4	38	41	41	10	0	60	56	35
5	−1	17	17	14	8	−2	39	39	32	10	2	61	57	44
5	1	18	16	13	8	0	40	37	24	10	4	62	59	55
5	3	19	18	18	8	2	41	38	31	10	6	63	61	68
5	5	20	20	25	8	4	42	40	40	10	8	64	63	83
6	−6	21	27	37	8	6	43	42	51	10	10	65	65	100

It must be stated, at this point, that the 36 polynomials used, in this instance, are not those that would be ordered as in Table 5.1. That is to say, they are not the first 36 ANSI standard polynomials. As mentioned earlier, there are, unfortunately, a number of competing conventions for the numbering of Zernike polynomials. The convention used in determining the P to Vr figure is the so called Zernike Fringe polynomial convention. The logic of ordering the polynomials in a different way is that this better reflects, in the case of the fringe polynomial set, the spatial frequency content of the polynomial and its practical significance in real optical systems.