Wavefront technology

As made clear in previous chapters, excimer laser refractive surgery has now reached a stage in its development at which, in carefully selected patients, it offers a realistic, routine alternative to earlier methods used to correct refractive error (spectacles and contact lenses), at a comparable level of cost. Nevertheless, the search continues for ways to improve visual outcomes and further reduce the possibility of post-operative complications.

In this chapter we discuss a relatively new development, wavefront technology, as applied to excimer laser surgery. It is hoped that, as well as aiding the traditional spherocylindrical correction of ametropia, this will lead to reduced post-operative optical aberrations and hence better optical outcomes. The question of how the opportunities offered by the availability of an effective refractive surgical option to correct ametropic patients can best be embraced by the existing ophthalmic professions is also discussed.

Introduction

In the naturally emmetropic eye, the quality of vision achieved depends both on the quality of the optical image on the retina and on the neural properties of the retina and subsequent visual pathways.

The quality of the in-focus foveal optical image depends upon the effects of diffraction, optical aberration and light scatter, at least the first two of which are a function of pupil diameter. With very small pupils (<2mm), the effects of diffraction dominate, but with increases in pupil diameter diffractive blur reduces and the degradative effects of aberration become progressively more important.

Although effects vary between individuals, the optimal optical performance is usually achieved with a pupil diameter of about 3mm, which corresponds to that of the natural pupil under bright photopic conditions.1 A similar general pattern of behaviour is found in ametropes corrected by spectacles or contact lenses, although it is modified slightly by effects such as spectacle magnification and the aberrations of the correcting lenses.

The early years of refractive surgery using radial keratotomy (RK) were dominated by the goal of achieving a tolerably accurate refractive correction. However, it was soon realized that, even when this was achieved, the quality of vision was usually worse than that found in naturally emmetropic eyes or in ametropic eyes corrected with spectacles or contact lenses. Measurements showed that the loss in vision resulted from the much poorer quality of the ‘best-corrected’ retinal image, which was degraded both by light scatter at the corneal RK incisions and by the much-increased levels of aberration associated with the limited optical zone achieved and the discontinuities produced by the pattern of incisions.2

With the advent of methods based on the use of excimer laser ablation it was hoped that, since changes in surface curvature were produced smoothly over a broad area of corneal surface rather than at a limited number of incisions, as in RK, a much improved optical effect would be produced. Again, early efforts with both photorefractive keratectomy (PRK) and laser in situ keratomileusis (LASIK) concentrated on producing a satisfactory spherocylindrical refractive correction, but the final quality of vision was still often disappointing, with a loss in photopic best-cor-

rected visual acuity. Performance under scotopic conditions, when the pupil was dilated, was frequently particularly poor, with complaints of haloes and scatter from lights during night driving.3 While the increased scatter could be attributed to wound-heal- ing problems, many of the optical problems were again found to be derived from the increased levels of aberration in the eye after refractive surgery eye.4

These increases in aberration were associated with a variety of factors. In early PRK procedures using broad-beam lasers, the ablated zone was often of smaller diameter than the natural dilated pupil, which gave a massively undercorrected spherical aberration, while any temporal or spatial beam inhomogeneity led to unwanted changes in the ablation pattern. Decentration of the ablation was a further problem in some cases. Other problems identified included central islands, probably caused by ablation plumes that affected local ablation rates, and irregular ablations caused by changes in corneal hydration. Wider ablation zones, blending zones and other measures have brought some improvement, but they have not eliminated these problems. Thus, one of the goals of more recent developments in refractive surgery is to find ways to reduce post-operative optical aberration. It should, of course, always be remembered that refractive surgery has a negligible effect on the longitudinal chromatic aberration of the eye, which remains essentially unchanged at about 2D across the visible spectrum.

The monochromatic aberrations of any optical system depend upon the shapes of its surfaces, as well as other factors. In principle, then, aberration might be reduced by reshaping one or more surfaces. Early laser

systems, which involved broad-beam lasers and devices such as expanding diaphragms, could only deliver ‘standard’ ablation patterns. The advent of the new generation of computer-controlled, spot-scanning laser systems, able to ablate different regions of the cornea selectively, rather than merely to produce ‘standard’ ablation patterns, has opened up the intriguing possibility of correcting the axial monochromatic aberrations of the eye by suitably reshaping the corneal surface with an ablation ‘customized’ for each individual. If monochromatic aberration correction can be achieved, then visual acuity should match, or perhaps surpass, that achieved in naturally emmetropic eyes.5

This, then, is the promise of the new generation of laser systems that combine a device to measure the aberration and refractive error of the individual eye, and a spot-scanning or other form of laser that can ablate the cornea selectively, not only to correct the refractive error, but also to minimize the eye’s monochromatic aberrations. Since the ocular aberrations, and also the refractive error, are analyzed in terms of the form of the corresponding wavefront, rather than in terms of the classic Seidel aberrations (spherical aberration, coma, oblique astigmatism, distortion and field curvature), this area is often called wavefront technology.

In the next sections we discuss the concept of wavefront aberration, current methods of measuring it in the individual eye and the limits to the overall quality of vision that might be achievable under ideal circumstances.

Wavefront aberration

Consider the case of a point source of light placed at the first focal point of a converging, positive lens. If the propagation of the light is viewed in terms of rays, in the absence of aberration we can envisage the rays as divergent from the point to emerge from the lens as a parallel bundle. Alternatively, if we think in terms of Huygen’s wave theory, series of concentric spherical wavefronts spread out from the point source and, after refraction by the lens, emerge as plane wavefronts. On both sides of the lens surfaces the local wavefronts are always perpendicular to the local rays (Figure 9.1a).

Now consider the case of a similar, but poorly designed and manufactured, lens with aberration. The rays that emerge from the lens are no longer parallel and the associated wavefronts are no longer spherical (Figure 9.1b).

Evidently, a point object at the first focal point is a special case. In the more general ideal case, when the object is not at the first focal point and the image is not expected to lie at infinity, rays diverge from the object point to converge at a unique Gaussian image point. In wave terms, spherical wavefronts diverge from the object point and the lens reshapes these as a series of spherical wavefronts, all centred at the image point (Figure 9.1c). If there is an aberration, the rays no longer intersect at the unique image point and the imaging wavefronts are no longer spherical (Figure 9.1d).

How can we quantify the aberration? Evidently, we can approach this either in

terms of the characteristics of the emergent rays or of the emergent wavefronts. In principle, we can always derive one description from the other, since wavefronts and rays are always locally perpendicular. For example, we might specify the ray aberrations in terms of the slope of each ray as it leaves the exit pupil as a function of the ray position within the pupil. This, in turn, would allow us to derive a spot diagram that shows the intersection of the rays with any chosen image plane. Alternatively, we could compare the imaging wavefronts with their ideal spherical counterparts. This is normally done in the exit pupil of the system. If the ideal image point is at infinity, the ideal wavefront is plane (i.e., it has an infinite radius of curvature).

We call the ‘ideal’ spherical wavefront, centred at the Gaussian image point O , the reference sphere and take the wavefront aberration at each point in the pupil as the optical path distance between the reference sphere and the aberrated wavefront (Figure 9.2a). In many applications, the radius of curvature of the reference sphere is chosen so that the wavefront aberration on axis at the centre of the exit pupil is zero, but this need not be the case. Clearly, what results is a ‘contour map’ that shows the variation in wavefront aberration across the exit pupil (Figure 9.2b). It is usual to specify the amount of wavefront aberration in microns (or sometimes wavelengths of light) and the position in the pupil in terms of either Cartesian (x, y) or polar (r, θ) coordinates (Figure 9.2a). Not surprisingly, the wavefront aberration varies with the position of the object point in the field

(i.e., the field angle), but in the case of the eye we are almost always concerned with the aberration on the visual axis, when the image point lies on the fovea.

Recalling that to form a diffraction-lim- ited image all parts of the wavefront must arrive in phase at the image point, we can see that it is desirable for the wavefront aberration to remain as small as possible, so the ideal wave aberration map is completely free of contours. Rayleigh suggested that the wavefront aberration should nowhere exceed a quarter wavelength, otherwise the light disturbances from different parts of the exit pupil would start to interfere destructively. Maréchal expressed the same idea in terms of the root mean

square (RMS) value of the wavefront aberration across the pupil and suggested that this should not exceed one-fourteenth of a wavelength (the variance of the wavefront aberration is the square of the local wavefront aberration integrated over the pupil area, divided by the pupil area; the RMS aberration is the square root of the variance).

For both these criteria it can be seen that, if aberration is to play a negligible role, the typical wavefront aberration across the pupil must remain small (about 0.1μm or less). Referring this tolerance to refractive surgery, this is only a small fraction of the typical total depth of stromal material removed (which is usually of the order of

tens of microns), which implies the need for a very accurate control of ablation depth.

How can we relate these general ideas on wavefront aberration to the measurement of ocular aberration and perhaps combine measurements of refractive or defocus error with those of aberrations like spherical aberration and coma? Consider the ‘ideal’, aberration-free emmetropic eye shown in Figure 9.3a. If, somehow, a point source of light can be produced on the retina (usually with the aid of a low-power laser), all the light rays will emerge parallel from the eye, which corresponds to the ideal, plane reference wavefronts. Suppose, however, that the eye suffers from myopia, but not from aberration. The emergent rays

Optical Society of America.9 This refers the wave aberration to the entrance pupil of the eye (since the exit pupil is not readily accessible) and uses the line of sight as the reference axis. The latter corresponds to the chief ray from the fixation point, which passes through the centres of the entrance and exit pupils to reach the fovea. Since the Zernike polynomials are only orthogonal over a unit circle, the normalized radial distance in the

pupil ρ = r/rmax is used as one polar coordinate, where rmax is the maximum pupil

diameter for the measured wavefront aberration. The azimuthal angle θ is defined in the same conventional way as the cylinder axis in optometry, except that it can have values between 0 and 360° (2π radians). The wavefront aberration W(ρ, θ) is broken down as the sum of the Zernike polynomials, as in Equation (9.1),

where Cnm is the coefficient for each of the Zernike polynomials Znm(ρ, θ) and the coefficients vary with the aberration of the particular eye. The subscript n represents the highest order (power) of the radial parameter ρ contained in the particular polynomial, which also contains a cosine or sine term of a multiple, mθ, of the azimuthal angle θ, so that m is often termed the azimuthal frequency. Note that when, for example, a fifth-order Zernike term is mentioned, it is the value of n that is being referred to (n = 5).

Each polynomial Znm(ρ, θ) is the product of three components: a normalization term, a polynomial in ρ of order n and an azimuthal component of the form sinmθ or cosmθ.

Table 9.1 lists polynomials up to the fifthorder. Details of still higher-order polynomials are given in, for example, Thibos et al.9 As noted earlier, each polynomial essentially describes a particular type of deformation of the wavefront, and the magnitudes of

Figure 9.4

Typical colour-coded contour map of the wavefront aberration of an eye that shows a mixture of defocus errors caused by ametropia and higher-order errors caused by aberrations such as spherical aberration and coma

less degradative effect than individual terms. This is, of course, not surprising. It is, for example, well known in optometry that the effect of a given cylindrical refractive error (i.e., a constant magnitude of second-order wavefront aberration) on visual acuity varies with the orientation of the cylinder axis.

Useful information on the distributions of the values of the various Zernike coef-

ficients in normal (unoperated) eyes is given by Porter et al.12 and Thibos et al.13

For most of the coefficients the values are symmetrically balanced about zero, as shown in Figure 9.7. This suggests a central tendency for natural eyes to be free of most higher-order aberrations, although biological variability means that any individual is equally likely to have a positive or negative aberration. The only clear exception is the C40 coefficient, which is systematically biased towards positive values (undercorrected spherical aberration). The spread of values becomes smaller as the mode order increases.

Wavefront aberration and refractive correction

At first sight, it might appear that if we are to use wavefront measurements as the basis on which to choose a spherocylindrical refractive correction, we need to consider only the values of the second-order defocus coefficients. However, if this approach is used, the derived prescription is usually found to vary with pupil size,14 even though in practice subjective refraction seems to

change little under photopic conditions as the pupil size changes.15,16 The reason for

the failure of the second-order coefficients C2–2, C20 and C22 to provide a good predictor of optimal refractive correction under all circumstances is because of the presence of second-order terms in several of the higherorder Zernike polynomials. As discussed previously, these second-order terms help to balance the degradative effects of spherical and other aberrations. Obviously, the quality of the retinal image is improved by their inclusion and hence they should be incorporated in the correction. Atchison et al.17 give appropriate equations that include all the relevant Zernike coefficients in the calculation of the corresponding objective spherocylindrical prescription. They suggest that an objective refraction should be based on either just the second-order Zernike coefficients for a small pupil (for which the effects of higher-order aberrations are usually small) or on both the secondand fourthorder Zernike aberrations deduced with a larger pupil (say 6mm).

scatter of intersections, or ‘spots’, is obtained (Figure 9.9a), and the greater the concentration of points in the ‘spot diagram’, the more closely the eye corresponds to the ideal case.

Whereas the spot diagram relies purely on geometrical optics, the point-spread function (PSF) or image of a point includes the effects of diffraction (Figure 9.9b). When the amount of aberration is large, there is little difference between the spot diagram and the PSF, but for eyes with very little aberration the PSF is more realistic. The PSF is calculated by first using the wave aberration to determine the variations in the phase of the light disturbance across the pupil. The local phase is simply the wave aberration in microns, divided by the wavelength of the light in use and multiplied by 360°. The amplitude across the pupil is either treated as being constant or can be weighted to allow for the Stiles–Crawford effect. From the amplitude and phase distribution across the pupil (the pupil function), the PSF can be calculated.7

A third descriptor of image quality that is often of interest is the modulation transfer function (MTF), which shows how the contrast of sinusoidal gratings is degraded in the image as a function of spatial frequency. When the wavefront aberration, and the associated spot diagram and PSF, lack rotational symmetry, the MTF varies with the orientation of the gratings (Figure 9.9c). The ocular MTF can also be calculated from the pupil function for the eye, as the real part of its Fourier transform.

Lastly, software often generates a representation of the retinal image of a chosen object (e.g., a Snellen E) in the presence of a known ocular wavefront aberration. This may be obtained by convolution of the object luminance distribution with the appropriate PSF (i.e., by blurring each point in the object so that it appears as a PSF of appropriately weighted illuminance and then summing the combined PSFs to give the overall image). Alternatively, the same calculation may be carried out by Fourier methods. Care must be exercised when interpreting these calculated images, since neural and other factors may mean that they do not correspond very closely to what the patient actually sees.

Using the wavefront information in refractive surgery

Wavefront-guided surgery involves applying an ablation that attempts to neutralize the measured wavefront aberration of the original eye, that is both the ametropia

and the higher-order aberrations. Success in this balancing procedure is not easy to achieve. Not only must the original wavefront measurements be valid and reliable, but also many laser beam, tissue ablation and healing characteristics will affect the final result. It is, for example, obvious that laser-spot diameters of less than 1mm are necessary if higher-order aberrations are to be corrected. Yet this, in turn, means that more spot pulses are required to cover the full ablation area, so that at any given pulse frequency the ablation takes longer than with a larger spot and the problems of maintaining patient alignment and corneal hydration tend to increase. In LASIK, the variables introduced by the need to replace the flap offer further difficulties. This may favour the development of alternative methodologies, such as LASEK. Calculation and control of the number and position of the laser spots required must be extremely accurate. Fast eye tracking is likely to be necessary to avoid problems caused by eye movement. A further constraint on the degree to which aberration can be corrected is set by the need to preserve an adequate thickness of undisturbed cornea (at least 250μm): this is a particular problem for higher refractive corrections, for which the diameter of the ablation zone may have to be limited, even though this leaves an aberrated eye for larger pupil diameters.

At the present time, most manufacturers of laser systems have begun to offer wavefront measurement integrated with their ablation systems, and experience in their combined use is beginning to build up. Early results show promise, but indicate that, although it can be demonstrated that eyes with less post-oper- ative aberration tend to show higher levels of visual performance, low levels of ocular aberration cannot be achieved routinely at present. The main immediate value of the technique is undoubtedly that, since preand post-operative wavefront maps can be compared, the exact effects of the particular ablation pattern can be assessed in relation to the other parameters of the individual eye. Such comparisons should lead to a fuller understanding of the factors involved and to the development of improved ablation algorithms able to produce more consistent refractive outcomes.

In the longer term, however, it is difficult to see how the various factors involved in a single excimer laser ablation procedure can be controlled routinely to the degree of precision required to correct accurately not only spherical and astigmatic errors (second-order wavefront errors), but also the higher-order

aberrations. It seems reasonable to hope, however, that higher-order aberrations may be kept at or even below the levels found in naturally emmetropic eyes, rather than being substantially higher, as is the case without wavefront technology.

A major routine application of wavefront technology may be to help identify why some post-operative eyes have poor acuity and to help plan enhancement procedures to reduce unusual levels of aberration.

Super vision

Let us suppose, perhaps optimistically in the light of the previous section, that understanding of the factors involved in aberration correction improves to the extent that ablation-corrected eyes can give diffraction-limited performance in monochromatic light for distance vision (or, if required, for any other distance). With such excellent optics, what improvements in visual performance over those achieved by natural emmetropes could be expected? As normal eyes suffer from aberration, which blurs the retinal image, will the ideal aberration correction yield ‘super vision’ with levels of acuity much better than the values of 6/4 usually achieved by normal, young adults.18

The key factor here is that visual performance is not just limited by optics: it also depends upon the retina and subsequent stages of neural processing. It is clear that the sampling limitations imposed by the finite size of the cone photoreceptors that form the foveal retinal mosaic would set a limit on achievable acuity, even if there were no optical degradation whatsoever, as

Wavefront technology ■ 73

illustrated in Figure 9.10. Although this sampling limit varies somewhat with the individual, the spacing of the foveal cones is such that it typically lies at about 60 cycles per degree (c/°) for grating objects, corresponding to about 6/3 Snellen equivalent. Finer gratings may detected, but will appear as some other form of coarser pattern, typically as ‘zebra stripes’. This phenomenon is known as aliasing.

We can see, then, that the limits set by retinal neural factors mean that a reduction in the natural level of aberration in the eye is unlikely to produce high-contrast Snellen acuities much better than 6/3. This is, of course, much better than the levels typically achieved after current refractive surgery (in which only about 80% of patients achieve uncorrected acuities of 6/6), but not much better than that achieved by the best of natural eyes.

What combined optical and neural performance levels might be achieved in practice if higher-order monochromatic aberrations were eliminated? Since in the natural and current post-surgical eye the degradative effects of aberration worsen as the pupil diameter rises, while diffractive blur reduces, the greatest improvements in optical retinal image quality are potentially obtainable when the pupil diameter is large. However, unless the pupil is artificially dilated, large pupils only occur when light levels are low and the spatial resolution achieved by the retina is degraded because of a shift towards rod vision and increased spatial integration. We must remember, too, that even if the monochromatic aberrations are corrected, the retinal image will still be blurred by longitudinal chromatic aberration.

Figure 9.10

Resolution limit set by the sampling limit of the foveal cone mosaic. To resolve the bars of a Snellen ‘E’, there must be unstimulated cones between the stimulated cones

To illustrate these effects, Figure 9.11 shows the MTF for the eye when the entrance pupil diameter is either 3.0 or 6.0mm. Three MTFs are shown for each pupil diameter:

•A typical experimentally measured MTF for a natural eye with normal levels of aberration;

•MTF for an ideal aberration-free eye working in monochromatic light of wavelength 555nm; and

•The same eye working in white light with the degradative effects of chromatic aberration included.

Also shown in Figure 9.11 is the photopic contrast threshold at the retinal level. If the MTF falls below this threshold level, the grating cannot be resolved, even at maximal contrast. The highest spatial frequency of high-contrast grating that can be resolved lies at the intersection of the MTF with the threshold curve. As discussed earlier, it can be seen that to eliminate optical aberration gives much more benefit at the larger pupil diameter, but only if the retinal threshold is unchanged (i.e., the pupil is artificially dilated). With smaller, normal photopic pupils, it can be seen that it is likely that the main benefit would be an overall improvement in image contrast at spatial frequencies below the cut-off, with only minor improvements in the cut-off frequency itself.

Finally, even if the technical problems associated with the corrective ablation and possible regression effects can be overcome, it is likely to be impossible to correct monochromatic aberration fully at all times by surgical means.19 The natural aberrations vary as a function of accommodation (i.e., object distance) and age, while in any case lags and leads of accommodation are known to typify the accommodation system, and introduce variable second-order defocus aberrations. Further problems are caused by the fast (0.1–2.0Hz) fluctuations in accommodation that occur with amplitudes of 0.1–0.2D: these demand a dynamic correction of aberration.20 There is also evidence that aberrations may change as a result of tear film changes, prolonged near work or diurnal variation in corneal thickness and curvature.21,22

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