Understanding ocular wavefront aberration

The increasing clinical availability of commercial wavefront aberrometers means that many optometrists are or will be attempting to assess the significance of measurements of their patients' wavefront aberration. Such measurements are perhaps most useful for those concerned in the co-management of refractive surgery patients1 but can also be helpful in a wide variety of other problems, from the selection of optimal contact lens corrections to the monitoring of the progress of conditions such as keratoconus.2-4
As is often the case when new instruments producing unfamiliar information are introduced, many of their users (perhaps a majority) may not be entirely confident in their ability to fully understand the information presented. Although they may have a fair idea of what is meant by wavefront aberration, they may be less certain about its more quantitative aspects.
For the true enthusiast, there are already many excellent publications which give a full background to the theoretical basis of wavefront aberration, its analysis in terms of Taylor and Zernike polynomials and the range of aberrometers which have been produced to measure it.5-8 The problem for most of us is that, when we see an aberration map or a list of values of Zernike coefficients produced by the software of one of today's aberrometers we find it difficult to relate this data to the more familiar levels of degradation produced by sphero-cylindrical refractive errors. The aim of this article is to discuss the Zernike approach to wavefront aberration and some ways in which wavefront errors can be at least approximately understood in dioptric terms. Some limitations in the use of wavefront data are also pointed out.

Wave aberration
Let's start by reminding ourselves of what is meant by wave aberration. In any instrument which is free of monochromatic aberration, the rays in the image space ought to converge to the unique Gaussian image point, O'. Correspondingly, the image wavefronts ought to be spheres which are centred on this same image point, the wavefronts always being perpendicular to the rays (Figure 1a). In an aberrated system, the rays either converge to the wrong longitudinal or transverse image point (defocus or distortion, Figure 1b) or fail to converge to any unique point (as in spherical aberration, coma etc, Figure 1c). As well as the normal range of monochromatic aberrations, a biological system like the eye can obviously have other aberrations caused by the variety of tilts, decentrations and other asymmetries that are found in its optical components.
The wavefront aberration is the optical path (ie the product of the distance and the refractive index of the medium in which it is measured) between the ideal wavefront in the exit pupil (called the reference sphere) and its aberrated counterpart. Obviously this difference varies across the pupil, and is often adjusted either to be zero at the pupil centre or to have an average value of zero across the pupil. In the case of the eye, it is the varying optical path difference across the pupil (strictly its entrance pupil, since the exit pupil is not accessible) that is usually plotted to generate the contour map of wavefront aberration as produced by most aberrometers, with the contours joining points which have the same values of wavefront error.
This raises the question of what is the form of the 'ideal' wavefront. In practice the reasonable assumption is that, for the eye with relaxed accommodation, parallel rays (corresponding to plane wavefronts) going into the eye should be brought to an exact focus on the retina (or that a point source on the retina should result in parallel light and plane waves leaving the eye), ie that the eye is emmetropic. In practice, of course, the typical eye tends to show at least some ametropia. Obviously, then, the basic wavefront errors as measured with an uncorrected eye will usually include two contributions: those from what we have traditionally thought of as being 'aberrations' such as spherical aberration, coma and so on and those due to any ametropia. In most instances, the bulk of the total ocular wavefront aberration will be due to the ametropia. As we shall see, for this reason, the aberrometer's software usually allows the user to 'subtract out' the wavefront aberration due to the eye's sphero-cylindrical error, so that the residual aberration can be evaluated. The aberrometer uses the subtracted wavefront error to provide an estimate of refractive error.

The wavefront map -
qualitative aspects
Wavefront aberration maps are usually colour-coded (Figure 2). In an 'ideal' case where the ocular aberration was zero, the map would show no contours. At the present time, there is no agreement between the manufacturers of different aberrometers as to the wavefront map colours or contour intervals used, although it is perhaps true that there is a preference for using green to represent zero aberration, 'hotter' yellow, orange and red colours to indicate where the wavefront is advance of the reference sphere and 'cooler' blues and purples where the wavefront is behind the reference sphere. Attention must be paid to the contour scale used in any display, since a mass of contours may still represent a very small amount of aberration if the contour interval is small. Wavefront aberration is usually expressed in microns, although sometimes multiples of a chosen wavelength are used.
At this stage it is worth emphasising the importance of the pupil diameter over which the wavefront errors are measured. Measurements with most aberrometers are usually made under lighting conditions under which the pupil is at least partly dilated, typically to around 6mm diameter. Since there tend to be greater amounts of aberration in the outer zones of a dilated pupil, the aberration map may, to a casual inspection, suggest that the optical performance of the eye is poor, whereas it might be that, under normal photopic pupil conditions when the pupil diameter was smaller, optical performance was satisfactory. As we shall see, most quantitative summaries of the amounts of aberration present apply only to a specific pupil diameter, so that care needs to be exercised when considering individual results in relation to different illumination conditions under which different pupil diameters may apply, and when comparing data from different authors who may have used different pupil diameters.

Quantitative aspects: small wavefront errors
Given the wavefront aberration for a particular pupil diameter, how do we judge whether the amount of aberration present is large enough to significantly degrade the optical quality of the retinal image? Several methods have been devised to help us make this judgement.

Rayleigh criterion
In an ideal world, the eye should show zero wavefront aberration and all the light disturbances from different parts of the pupil should arrive in phase at the retina. A monochromatic point object would then appear as an Airy diffraction pattern. If there is not zero aberration, constructive addition of the disturbances no longer occurs and image quality suffers. However, over 100 years ago, Rayleigh suggested that it was reasonable to allow some tolerance on the condition for equal optical paths and that the point image would only suffer relatively minor changes if the maximum wavefront error at any point in the pupil was less than a quarter-wavelength, or about 0.14m in the green. Although this Rayleigh criterion is useful, it doesn't tell us much about the relative merits of eyes where the aberrations exceed this limit, as is almost always the case with eyes under mesopic and scotopic conditions when the pupils are larger. We can, however, look at the wavefront map after ametropia has been corrected and see whether the pupil area over which the aberration roughly meets the Rayleigh criterion is comparable with normal photopic pupil diameters, in which case aberrations are likely to have only minor effects on the standard of vision achieved under daylight conditions.

Root-mean-square wavefront error and the Marchal criterion
A useful basic quantitative summary of the distortion of the wavefront, given by all aberrometers, is the root-mean-square wavefront error (RMS error). As its name implies, this is the square root of the mean of the squared difference between the local wavefront error at each point in the pupil, W(xy), and the mean error, Wav, across the pupil, ie

RMS = [(W(x,y) - Wav)2dxdy/dx.dy]1/2

where x and y are the pupil coordinates and the integrals are taken over the area of the pupil. Again, the RMS error will depend on the diameter of the pupil and normally tends to increase with pupil diameter.
Marchal9 pointed out that, in these terms, the point diffraction image suffers from little degradation if the RMS wavefront aberration does not exceed about 1/14 of a wavelength (about 0.04m in the green). Like the Rayleigh limit, this is a fairly tight tolerance and is rarely satisfied for pupil diameters much above 2.5mm.

The Strehl ratio
Yet another way of describing the effect of wavefront aberration on the image of a point is in terms of the Strehl ratio, S, which is the ratio of the retinal illuminance at the centre of the aberrated point image to that in an aberration-free eye under the same pupil and wavelength conditions. The software of aberrometers often gives a value of this ratio for the measured eye. Strehl suggested that there would be little loss in image quality if S> 0.8. It can be shown10 that for small amounts of aberration the Strehl ratio is linked to the RMS wavefront aberration by the following expression:

S 1 - [2(RMS)/<03BB>]2

where <03BB> is the wavelength and is measured in the same units as the RMS wavefront error. Using this expression, it can easily be shown that the Strehl and Marchal criteria are equivalent.

Quantitative aspects: larger wavefront errors
Although it might reasonably be expected that the higher the RMS wavefront error the worse the image would be, we would obviously like to be able to be a little more quantitative than this. We also want to be able to separate the contributions to wavefront error made by simple sphero-cylindrical defocus, which can be corrected by appropriate lenses, from those due to 'classical' aberrations. If we can do this, our aberrometer can also in principle act as an autorefractor, since the defocus term (with the opposite sign) gives the sphero-cylindrical correction required.

Breaking down the wave
aberration into components:
Zernike polynomials
Although there are several possible ways in which the complex pattern of wavefront errors might be broken down into simpler components,7,11 that which is currently most popular is to use Zernike polynomials to represent the simpler building blocks (Table 1, Figure 3). These polynomials were devised by Frits Zernike, the Nobel-prizewinning inventor of the phase-contrast microscope. The idea is that our complex wavefront map is equivalent to the sum of appropriate quantities of a set of ingredients, these ingredients being represented by distorted wavefronts defined by the polynomials, Znm, ie
W = C00Z00 + C1-1Z1-1 + C11Z11 + C2
-2Z2-2 + C20Z20 + C3-3Z3-3 + C3-1Z3-1 + C31Z31 + C33Z33 + C4-4Z-44 + C4-2Z4-2 + C40Z40 + C42Z42 + C44Z44 + higher-order terms

where the coefficients C00 etc represent the 'amount' of each polynomial in the case of the particular eye under consideration.
The detailed mathematical properties of these polynomials, some of which are listed in Table 1, need not concern us here, but the key point is that each polynomial is expressed in terms of a normalised radial term <03C1> (= r/rmax, where r is the radial coordinate in the pupil and rmax is the pupil radius) and an azimuthal angle, <03B8>, which is measured following the normal optometric convention for cylinder axes, except that it runs from 0 to 360 degrees, rather than 0 to 180. The polynomials can, of course, alternatively be expressed in terms of Cartesian rather than polar coordinates in the pupil,7 but this is less usual. Each polynomial represents a certain shape of deformed wavefront (Figure 3). The order, n, of each polynomial is given by the maximal power to which the radial coordinate <03C1> is raised, and its azimuthal frequency, m, is given by the multiplier of <03B8> in the sine or cosine terms.
Figure 3 shows the wavefront maps corresponding to the first few orders of polynomials. The zero-order or piston term is just a constant displacement of an otherwise flat wavefront and is not significant from the point of view of image quality. Similarly the two first-order terms represent flat wavefronts which are tilted about either a horizontal or vertical axis (ie they represent prism) and again do not affect image quality. Remembering that the sag formula contains an r2 term, it is not surprising to find that the second-order terms are related to either a spherical error of focus or crossed-cylindrical astigmatic errors with axes at 90/180 or 45/135 (see below). Terms of the third and higher orders are known as higher-order aberrations and are related to (but not exactly equivalent to) our familiar coma, spherical aberration and so on (see commonly-used names in Table 1).
The software of today's aberrometers calculates the values of the coefficients Cnm (normally in microns) from the measured wavefront errors: this can be done for any chosen pupil diameter, up to the maximum diameter at which the measurements were made. The nature of the polynomials for the first order and above is such that the mean wavefront aberration across the defined pupil for each polynomial is zero and, in the normalised form given in Table 1, the coefficient Cnm represents the RMS wavefront error for that polynomial, eye and pupil diameter. Thus, the overall wavefront variance (ie the square of the overall root-mean-square error) is given by the sum of the squares of all the coefficients. It is, then, easy to assess the relative contribution that particular types of aberration make to the overall wavefront aberration and, presumably, to possible degradation of vision. We can also easily evaluate, for example, the relative effects of wear of a particular design of contact lens on residual astigmatism and spherical aberration.
In normal eyes, the values of the coefficients tend to diminish as the order increases, so that, eg, a good approximation to the overall wavefront errors is given in terms of the first six orders of Zernike polynomials.7 However, if the eye shows abrupt local variations in wavefront error across the pupil, as may be found in eyes that have undergone radial keratotomy for example, many more orders of Zernike polynomials are required.

Expressing Zernike
coefficients in
dioptric terms
The problem with Zernike coefficients is that, for most of us, it is not very obvious what they mean in terms of the resultant blur of the retinal image. For example, does an RMS wavefront error of 0.25 micron for a 6mm pupil diameter mean that the patient will experience very blurred or satisfactory vision? A partial solution to this problem is provided by the software now included with many aberrometers which allows the retinal point-spread function (ie the image of a point) to be calculated from the wave aberration data for any particular pupil diameter (Figure 4). This can give a useful subjective impression of the extent to which each image point is blurred, although the fact that most software calculates the monochromatic PSF and fails to allow for the additional effects of chromatic aberration in the white light of the real world is a limitation.
An alternative approach, which will now be discussed, is to use the optometrist's familiarity with the blurring effect of different levels of sphero-cylindrical defocus by expressing the more important Zernike coefficients in appropriate dioptric terms.7,13

Second-order defocus, Z20 and mean spherical error
It will be remembered that, when accommodation is relaxed, an emmetropic eye should have zero wave aberration across its pupil. In a myopic eye, the imaging wavefronts are spherical but converge too steeply, that is their outer parts are in advance of the reference wavefront, corresponding to positive wavefront error. In the hyperopic eye, the imaging wavefronts are too flat and the wavefronts lag behind in the periphery of the pupil, giving relatively negative wavefront error in this region.
Starting from the basic ideas of wavefront aberration, a defocus of F dioptres obviously corresponds to a spherical wavefront of radius f m, where f = 1/F (Figure 5a). We can easily see that, by Pythagoras' theorem, the corresponding wavefront aberration Wr at a distance r from the middle of the pupil (Figure 5b) is given by:

f2 = (f-Wr)2 + r2

Expanding the bracket and assuming Wr <<R,F gives:

Wr = Fr2/2 (1)

However, in Zernike terms, the wavefront aberration associated with the j=4,
second-order defocus polynomial Z20 is (Table 1):

W<03C1> = C203.(2<03C1>2-1)

Putting this in terms of r, since <03C1> = r/rmax

Wr = C203.(2(r2/rmax2) -1) (2)

The second term in the bracket corresponds to a piston (constant) term which is included to make the average wavefront aberration zero across the pupil, whereas the first term is a true defocus. We therefore can equate the second-order terms in equations 1 and 2 to find:

Wr = Fr2/2 = C203.(2(r2/rmax2)

This yields, for the dioptric defocus corresponding to a Zernike coefficient C20

F = (C204.3)/rmax2

Since this is the defocus, the corresponding spherical correction, Fs, requires a negative sign:

ie Fs = - (C204.3)/rmax2 (3)

For enthusiasts, the correction given by this formula effectively ensures a least-squares fit between the wavefront and the spherical wavefront corresponding to the defocus F.
An alternative approach, which is more useful in the case of other aberrations, is to differentiate the Zernike expression to give the local slope of the wavefront, ie.

dWr/dr = C2043.r/rmax2

Since the local ray is perpendicular to the wavefront, its slope (ignoring signs) is the reciprocal of that of the wavefront and, from Figure 5b,

dWr/dr = C2043.r/rmax2= r/f = rF
ie F = (C20 4.3)/rmax2

as before, leading directly to equation 3.
However, the overall position regarding spherical defocus is a little more complicated than this, since, as can be seen in Table 1, terms in <03C1>2 (corresponding to a defocus) are found in several even-order polynomials of order higher than 2, eg Z40. Remembering that the overall wavefront error is just the sum of the Zernike components, we can see that if we include the contributions of these additional defocus terms to the overall coefficient for r2, when we equate coefficients or differentiate we will have a spherical correction which includes higher even-order coefficients. The result for our overall spherical equivalent correction, M, is:

M = -(43C20 - 125C40 + 247C60- 409C80 + higher order terms)/rmax2 (4)

In distinction from the correction given by equation 3, this correction is sometimes known as the paraxial correction, since it matches the curvatures in the region of the centre of the pupil, rather than performing a least-squares fit across the whole pupil. M will be in dioptres if the coefficients are measured in microns and the pupil radii in mm.
In practice, if we determine the coefficients for a small pupil diameter, say 3mm, only the contribution of second-order coefficient needs be considered, as the other terms are usually small enough to be neglected. If, however, the coefficients are derived for a larger pupil diameter, eg 6mm, then at least the C40 term needs to be included.7,14 Thibos et al15 found that the mean sphere obtained from paraxial curvature matching (equation 4) with dilated pupils agreed well with the results of subjective refraction, although the least-squares correction (equation 3), using only the second-order C20 coefficient did not, presumably because with large pupils it is essential to include the contributions made by terms from the higher-order polynomials. In contrast, Guirao and Williams16 found rather larger discrepancies between subjective refraction and least-squares or paraxial curvature estimates, although their subjective refractions apparently used charts at the close distance of 6ft (vergence -0.55D) rather than 6m.

References
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<25C6> Neil Charman is professor of optometry at the Department of Optometry and Neuroscience, Life Sciences, University of Manchester