Wavefront basics

Wavefront basics

David R. Williams

Center for Visual Science, University of Rochester, Rochester, NY, USA

This entire meeting will be about wavefront analysis and what this means for refractive surgery. I would like to introduce the concept of the wave aberration to you and give you some information about why we think it is important and why it holds the key to the future of better laser refractive surgery. Let us start with the basics. Everyone knows about rays entering the eye. In a perfect eye, shown in figure 1, if you had parallel rays entering the eye from a star, the optics would bring these rays to a nice crisp point back at the retina. That would be an ideal situation, but aberrated eyes have misshapen corneas and misshapen lenses and so the rays do not converge to a single point (Fig. 1). What does this have to do with wavefront? Wavefronts are really nothing more than surfaces that are perpendicular at every point to the rays I have just mentioned, as shown in figure 2.

Fig. 1. Ray optics.

Fig. 2. Wave optics.

Address for correspondence: David R. Wiliams, PhD, Director, Center for Visual Science, William G. Allyn Professor, Box 270270, Rochester, NY 14627-0270, USA

Wavefront and Emerging Refractive Technologies, pp. 3–16

Proceedings of the 51st Annual Symposium of the New Orleans Academy of Ophthalmology, New Orleans, LA, USA, February 22-24, 2002

edited by Jill B. Koury

© 2003 Kugler Publications, The Hague, The Netherlands

Inside the eye, if you have an ideal eye, the wave front is transformed from a planar wavefront to a spherical one, and this too will collapse to a sharp point on the retina. But, in an aberrated eye, things are not as simple. You start out with a beautiful planar wavefront from the star, but this wavefront becomes warped into a distorted shape. Again, the wavefront is always perpendicular to the rays that are converging toward the retina. However, it departs from the ideal wavefront shown in red. By enlarging it slightly, the wave aberration is a map that shows you the departures between the actual wavefront in blue and the ideal wavefront in red. Figure 3 shows the wave aberration of three real eyes shown on the right and a perfect aberration on the left. The hills, shown in light, correspond to regions where the wavefront is relatively advanced compared to what would be perfect, and the valleys, shown in dark (Fig. 3), correspond to a location in the pupil plane where the wavefront is slightly retarded relative to what it should be in a perfect eye.

In a perfect eye, you get a gray, uniform plot, because the wave aberration is zero everywhere. The measure we commonly use to characterize the roughness of the surface, i.e. how much it departs from a perfect wavefront, is the root mean square error (RMS). The wavefront error is a measure of how bumpy this whole surface is, and to calculate this, you take the square root of the sum of the squares of all the deviations at every point within the pupil. This is expressed in microns. Sometimes you will see it in wavelength, but the most convenient unit is microns. In Figure 3 at the top, you can see that, like fingerprints, each eye has its own unique wave aberration. You will rarely see two very similar eyes when you plot all the aberrations that can contribute to the total wave aberration.

Why measure the wave aberration? If we can measure visual acuity, what is the point of measuring wave aberrations? Visual acuity tells you something about the overall visual performance of the visual system, but it does not give you any information about how you should try to correct the eye in refractive surgery. It does not tell you how much tissue quantitatively you ought to ablate at each point in the cornea to make the wave aberration flat again.

It is actually rather simple to convert a wavefront measurement to tell you what you ought to do first order to correct the corneal surface, and the equation in Figure 4 captures this. Figure 4 shows a wave aberration with the wavefront being retarded at one location and slightly advanced at another. If it is retarded, the path length to the retina should be made slightly shorter, so that more tissue should be removed to fix this problem. You need to remove relatively less tissue. You can never add tissue. So, in fact, you have to incorporate in this equation a constant C, which means you always have to ablate down some minimum amount in order to correct the entire wave aberration. The amount you ablate is related to the index of refraction in the cornea and the error, and so if you had a wave aberration that is a micron, you would need to ablate about 2.8 times as much tissue because of the denominator in the equation, which is the refractive of the index of the cornea minus the refractive index of the error. So you ablate more tissue than the wave aberration shape itself to obtain the appropriate shape in the cornea.

Despite the simplicity of being able to go to ablation parameters directly from the wave aberration, I wish to stress how complicated this is in practice. There are biomechanical factors that we need to understand,1 and a host of other factors, to get this exactly right. But as a first approximation, this is the right approach, and

Fig. 3. The point spread function can be computed from the wave aberration.

Fig. 4. The wave aberration specifies how much tissue to ablate at each point on the cornea.

there is great simplicity in using the wave aberration that you do not have when using other measurements of the eye.

The wave aberration completely describes the eye’s optical performance, with the exception of light scatter, and chromatic aberration, which can be independently assessed. This single function provides a succinct, compact, and complete description of how the eye is going to perform. The wave aberration is defined in the pupil plane, but you can shift rather effortlessly from the pupil plane to the retinal image plane, so that not only do you know what is wrong with the eye’s optics, but you can also compute what the impact of those errors will be in the retinal image. It is the retinal image that we are especially interested in because the ovality of that image controls visual performance. This is illustrated in Figure 3 at the bottom, the point spread function can be computed from the wave aberration

by taking the squared modulus of the fourier transform of the wave aberration defined in the pupil. Point spread functions are light distributions that show you what the retinal image would be like if you were looking at a single star. What is its image going to be on the retina? For a perfect eye, you get the beautiful pattern seen on the lower left in Figure 3. The rings are caused by diffraction in the eye. In an eye that does not have any aberrations, the only thing left to blur the image of the star is diffraction. But in real eyes, the point spread function is complicated, due to the presence of the aberrations which we can measure with a wavefront sensor. So, this is a very important property, being able to shift from the pupil plane to the retinal image plane.

We can take this one step further. People obviously look at many things besides stars, and we would like to be able to compute what the retinal image would be like for any object someone might want to look at. We can do that by taking the point spread function and, using a mathematical operation called a convolution, we can compute what the retinal image would be like for any object. Figure 5 shows the wave aberration on the left, which results in point spread function in the middle column. From the point spread function, you can also compute what any image will be like, as shown on the right.

You can see a clear difference in the best corrected wave aberration of the same eye prior to laser refractive surgery, and after laser refractive surgery. We have done the best we could to correct the eye for defocus and astigmatism, but note that the wave aberration is worse postoperatively than it was preoperatively. One of the real advantages of having wavefront sensing technology is that we can measure aberrations that are actually induced by refractive surgery, and this helps us to decide what we should do to remove them. We will see, for example, that spherical aberration is an important component of this problem. The very first and most important application of wavefront sensing in laser refractive surgery is to try to use it as a tool to remove the aberrations produced by the procedure itself.

Why measure wave aberrations if we can measure corneal topography? What is the importance of wave aberrations if we already have topography instruments in the clinic? Why do we need yet another device to make more measurements that are even more difficult to understand? There is an important reason for measuring wave aberrations instead of corneal topography, and this is that corneal topography cannot capture the aberrations that occur inside the eye. All the corneal topographer can do with the appropriate computations is to calculate what the wave aberration of the cornea is, but the wave aberration of the total eye, upon which vision rests, is determined not only by the first surface of the cornea, but also the second surface of the cornea, the first surface of the lens, the second surface of the lens, not to mention the gradient index refractive optics that occur inside the lens.

The anterior optics have a complex structure and the optical performance of the eye is not simply determined by the corneal surface. This point is brought home by a recent paper by Artal et al.2 (see Fig. 6) in which they measured the wave aberration of the first surface of the cornea (shown on the left), the internal optics inside the eye (i.e., everything except the cornea), and then the whole eye. You can see the corresponding point spread functions for those measurements below. The best performance in these three cases is the whole eye. The whole eye is better than the sum of its parts. It turns out that, at least in young eyes, although perhaps not as much in older eyes, the internal optics have evolved to compensate for aberrations in the cornea. So, if you were to try to base a customized refractive surgical

Fig. 5. The retinal image can be computed from the point spread function.

Fig. 6. The whole eye is better than the sum of its parts. (Reproduced from Artal, Guirao, Berrio and Williams, by courtesy of J Vision 1, 2001.)

procedure on the cornea alone, you would actually make a mistake, because you would not be taking account of these internal optics, that can compensate for aberrations in the cornea. The wave aberration collapses all the errors associated with all the surfaces in the anterior optics into a single measurement, which is the appropriate basis for wavefront-guided ablation.

Another concept that often arises in discussions of wavefront is Zernike modes. Therefore, I would like to give a short explanation of Zernike modes, what they are and why they are useful, and why it is important in refractive surgery to understand these sources of image blur in the eye. Figure 8 shows the complete wave aberration that can be measured with any of a number of commercial wave-front

Fig. 7.

sensors. We can take the wave aberration and decompose it into its fundamental components by means of a technique known as Zernike decomposition. Below you can see individual Zernike modes, which, if you were to add them all together, and there are actually more than are shown in Figure 7, you could recreate this complete wave aberration. This process of Zernike decomposition is rather like taking a voice or an auditory signal and doing a Fourier analysis to decompose the signal into its frequency components. The mathematics behind this are actually quite similar, except that instead of using sine waves, which you use in analyzing sounds, you use Zernike modes. These are basis functions that are like sine waves of different frequency. Figure 8 is a chart illustrating all the different Zernike modes that might occur in eyes. There are actually more than these, but these are the most important. They are plotted in a pyramid. This is in keeping with a standard that has been established by the Optical Society of America, and I think it is a very good way of looking at the Zernike modes.

I would like to make the case that, even to the pure clinician, it is valuable to understand what these modes are because, when you interpret what is happening in a particular laser refractive surgery patient, you will want to understand what these modes mean and what they do. On the left of Figure 8, you will see something called radial order, going from top to bottom, second, third, fourth, and fifth. There is also sixth, seventh, and eighth, and for as high as you want to go, but these are not generally important in normal eyes. There is also a 0th order and a first order, but I have not included those either, because they do not affect image quality. They would top off this pyramid. So it looks like a pyramid that has had its top lopped off, which is because I have removed the three modes that sit right at the very top. This term ‘order’ simply refers to the mathematics used to describe these functions; second order are quadratics, i.e., they have an exponent of 2, and third order have an exponent of 3, fourth order, an exponent of 4, and so

Fig. 8. Zernike modes.

on. It simply indicates how fast the function changes as you go from the center to the edge.

Second order aberrations will be the most familiar to you, because they correspond to defocus and astigmatism, the aberrations that are corrected by conventional spectacles and contact lenses and refractive surgery. They do not quite look like sphere, axis, and cylinder – you need to do simple mathematics to get from the three Zernike modes to the three values of sphere, axis and cylinder. Though Zernike defocus can be transformed into sphere, axis, and cylinder, it is important not to confuse the second order aberrations with a subjective refraction. The values of sphere, cylinder, and axis that give the best subjective image quality also depend on the higher order aberration of the eye. Higher order aberrations are aberrations besides defocus and astigmatism.

Some of these higher order aberrations have familiar names: coma and spherical aberration are common ones; but there are others too: quadrafoil, secondary astigmatism, secondary trefoil, secondary coma, etc. Each of these aberrations has its own point spread function (Fig. 9). Coma, for example, produces a comet-like shape, and spherical aberration is rather similar to defocus in producing this isotropic blur (Fig. 9).

There is another way of looking at the relative importance of these aberrations in the eye. We would like to get a realistic view of how important these aberrations actually are in different eyes. Figure 10 shows a study carried out by Jason Porter at the University of Rochester,3 in which he took 109 people and measured their wave aberrations, did a Zernike decomposition, and plotted how much RMS can be obtained for each of these various Zernike orders. I have labelled some of these aberrations so you can see the familiar ones. What is clear straight away is that defocus and astigmatism account for most of the aberrations. These are the most important aberrations in the eye, accounting for more than 90% of the total wave

Fig. 11. Higher order aberrations limit vision especially when the pupil is large.

aberration. The higher order aberrations actually only correspond to a relatively small percentage (less than 10%) of wave aberrations in this particular population. However, there were many patients in our study with a high refractive error, and so the focus is slightly larger than it otherwise would be. But still, the bottom line is that higher order aberrations expressed in RMS do not amount to very much.

The other thing to bear in mind when thinking about supervision and higher order correction, is that only at large pupil sizes do these higher order aberrations come into play. Figure 11 shows a point spread function for a 2 mm and a 7.3 mm pupil. This is almost entirely blurred by diffraction. There are almost no effects of aberrations on this point spread function and it is severely blurred by higher order aberrations, but it is only at large pupil sizes that these aberrations become important.

To put in perspective how important higher order aberrations are, we can compare the amount of blur you get with a typical amount of higher order aberrations with what you get from just a certain amount of defocus. It turns out that, in the average eye, higher order aberrations are only equivalent to about 0.3 diopters of defocus, a small refractive error (Fig. 12). These are modulation transfer functions, which is another way of showing the quality of the retinal image. The long dashed line in Figure 12 is for a case in which we have just corrected defocus and astigmatism and the dotted case is for a situation in which we have an ideal eye, but have blurred it by 3/10 of a diopter of defocus). The fact that they agree indicates that higher order aberrations are equivalent to about 3/10 of a diopter, which is not clinically particularly significant in a normal eye.4 It may be possible to refine our refractive surgical technique someday to remove those higher order aberrations in normal eyes and to achieve a slightly better result, but there is not really much room to improve things. Nevertheless, I want to argue that there is still a good reason to pursue wavefront technology. We ought to pursue it vigorously because there are many eyes that are not normal eyes, such as keratoconic and corneal transplant eyes, and there are large gains to be achieved in correcting higher order aberrations in these eyes.

In order to emphasize this point, I want to show you a metric that we have been using in Rochester, to define how much benefit you could get from correcting higher order aberrations (Fig. 13).5 We compute this by estimating the modulation

Fig. 12. Higher order aberrations are equivalent to 0.3 diopters defocus. Average eye, 5.7 mm pupil.

Fig. 13. Visual benefit: a metric for assessing the value of correcting higher order aberrations.

transfer function (MTF) of the eye when all the monochromatic aberrations are corrected, compared to when only defocus and astigmatism are corrected. We take the ratio of those two MTFs, which gives us an idea of how much improvement we could get if we corrected higher order aberrations. The ratio is shown on the right. We call this visual benefit. The visual benefit depends on spatial frequency. There are coarse features at the low end of the spatial frequency scale, and fine features at the high end. The curve shows that a visual benefit of a factor of 3 might be obtained in this particular case. But an important point can be gleaned from the population statistics of the visual benefit that you might get from a population of 113 subjects (Fig. 14). You can see that, for rather fine structure in a visual scene that corresponds to a spatial frequency of 16 cycles per degree, the average person only gets a benefit of about a factor of 2, not a huge improvement in the ability to see low-contrast objects. Nevertheless, there are people who would see enormous benefit. A factor of 4 gets into the regime where you have a benefit large enough that higher order correction will provide substantial benefit. And if

Fig. 14. Distribution of visual benefit for 113 subjects. (Reproduced from Guirao et al., by courtesy of J Opt Soc Am A, 2002.)

you look at keratoconic eyes, there is a huge benefit, a factor of 13.5 in one case and about 25 in another. So, there are enormous gains to be had. Perhaps you would not consider refractive surgery in the keratoconic eye, but for many of the people with normal corneas there would be a huge benefit from customized refractive surgery to correct their wave aberration.

Another point to be made in trying to judge how important higher order aberrations are, is to remember that RMS wavefront error is a measure of the physical waviness of a wavefront. It is not a direct measure of how much that wavefront will blur the eye. We need to perform additional experiments to discover just how important all these aberrations are, because some of them may be more important than others. At Rochester, we have been doing some experiments on this using our adaptive optic system. We have an optical system in Rochester that allows a subject to look through the system at a letter E, and we can blur that letter E with any Zernike aberrations that we want, using a deformable mirror in the system (Fig. 15). On the right, you can see what a test wave aberration is like. The subject views the E and looks at it blurred through a wave aberration created in our adaptive optic system using this flexible mirror. You can see where we have introduced a single Zernike mode (in this case vertical coma) in the wave aberration of the eye, and we have corrected all the other aberrations. Thus we can use this mirror to get rid of a person’s wave aberration and replace it with one that corresponds to a single pure mode. The subject adjusts the amplitude of the single mode to match the subjective blur produced by the standard. Figure 16 shows the matching results. The data dip down once in the second order, again in the third order, dip down again in the middle of the fourth order aberrations, and again in the middle of the fifth order aberrations. So it appears that there are particularly potent aberrations in the middle of each Zernike order. The way this matching works is that, if you require only a small amount of an aberration to match the standard, it must be a very strong or potent aberration. Strength increases downward. So some are strong aberrations and other weak ones. Note that defocus and astigmatism, which are the most important aberrations in the eye from the point of view of RMS, are

Fig. 15. Mode blur matching.

Fig. 16. Aberrations differ in their ability to blur.

actually pretty weak aberrations, from the point of view of blurring the retinal image. They are weak because, in this case, they are high on the chart. We blurred Es so as to simulate what happens when a subject looks through the adaptive optic system (Fig. 17). Es are blurred by the same RMS of each Zernike mode for all the modes from the second to the fifth order, and you can see that the Es that get very blurred are in the center of the pyramid. Those on the edge of the pyramid are not particularly blurred and are not worth worrying as much about in refractive sur-

Blur from Zernike modes.

gery. Those like coma, secondary astigmatism, spherical aberration, and the secondary comas and secondary trefoils, are the really significant aberrations. Therefore, not all aberrations are created equal. They do vary, and the higher order modes can be particularly potent, more potent in many cases than defocus and astigmatism. Ray Applegate and his colleagues have found very similar results on acuity in simulations of the blur from single Zernike modes.6

So I conclude that wave aberrations tell us what is wrong with the optics in a quantitative and succinct way. They tell us what effect these errors will have on the retinal image, which leads directly to predictions about what the visual performance will be. They also tell us rather directly, ignoring the complexities of biomechanics, etc., for the moment, how much tissue to ablate at each point on the cornea. And, finally, wave aberrations are especially valuable in improving the optics of eyes with large amounts of higher order aberrations. In all the excitement about super vision, what we really ought to be doing, and what many people are doing, is to try to reduce the increase in aberrations that occurs because of the laser refractive surgery procedure itself, we should also target those eyes that have large amounts of higher order aberrations. Just as some people have astigmatism and others do not, some people have lots of higher order aberrations and others do not. And the ones you want to target are obviously those who have large enough amounts of higher order aberrations to seriously degrade vision.

References

1.Roberts C: The cornea is not a piece of plastic. J Refr Surg 16(4):407-13, 2000

2.Artal P, Guirao A, Berrio E, Williams DR: Compensation of corneal aberrations by the internal optics in the human eye. J Vis (R1):1-8, 2001

3.Porter J, Guirao A, Cox I, Williams DR: Monochromatic aberrations of the human eye in a large population. J Opt Soc Am A 18(8):1793-1803, 2001

4.Guirao A, Porter J, Williams DR, Cox I: Calculated impact of higher-order monochromatic aberrations on retinal image quality in a population of human eyes: erratum. J Opt Soc Am A 19(3):620-628, 2002

5.Williams DR, Yoon GY, Guirao A, Hofer H, Porter J: How far can we extend the limits of human vision? In: MacRae Scott M, Krueger Ronald R, Applegate Raymond A (eds): Customized Corneal Ablation: The Quest for SuperVision, pp 11-32. Thorofare, NJ: Slack, Incorp, 2001

6.Applegate RA, Sarver EJ, Khemsara V: Are all aberrations equal? J Refrac Surg 18:S556-S562, 2002

Understanding wavefront clinically

What it can and cannot do

Raymond A. Applegate

University of Houston, Houston, TX, USA

When discussing refractive surgery and wavefront aberration corrections, image quality is the key. Variables include diffraction, scatter, optical aberrations, and chromatic aberration. It is also important to remember that the quality of the retinal image is the input to the visual system. However, by itself, the quality of the retinal image does not define the visual perception. It is similar to using a camera with excellent optics, and then relying on inferior enlarging optics for developing. Your pictures are not going to come out very well. In the same way, you are not going to end up with good images if the film is old.

In the eye, the optics supply the fundamental information transmitted to the photoreceptors. If the photoreceptors are dead, you are not going to see. And if the neural processing is poor, it may misinterpret the information given to the photoreceptors. You will also have poor perception if all your visual perceptions from the past – that bank of information you use to interpret images – is poor. Overall, you cannot jump immediately from the optics of the eyes to visual perception. Still, there is a correlation, and it is not bad, but it is not perfect and never will be.

You can glean more out of an image with high quality optics. You can take light from optical infinity – from a point source, a distant star – and convert it into waves by drawing small perpendiculars. However, you have to do this at the same phase or at the same time after the light leaves the star. If the optics are perfect, these rays will converge to a point, and so will waves converge to a point. In an eye, parallel rays are preferred, either looking at rays of light or waves of light coming to a perfect focus. Most wavefront devices, particularly the ones using Hartmann-Shack technology, place a point source on the eye.

Because of the reversibility principle, the question then becomes, “Are these waves in fact parallel to one another?” This is a perfectly legitimate way to look at the problem, either an object or image space. You cannot get inside the eye very easily. The wavefront error is the error between the actual wavefront and the ideal wavefront, as a function of location within the pupil. Because wavefront error degrades the optical image, if possible there should be no wavefront error. An

Address for correspondence: Raymond A. Applegate, OD, PhD, University of Houston, 4800 Calhoun Road, Houston, TX 77204, USA

Wavefront and Emerging Refractive Technologies, pp. 17–20

Proceedings of the 51st Annual Symposium of the New Orleans Academy of Ophthalmology, New Orleans, LA, USA, February 22-24, 2002

edited by Jill B. Koury

© 2003 Kugler Publications, The Hague, The Netherlands

example of this is a one-year post-LASIK patient who can read all the way down to 20/20 with best-spectacle correction. He can probably read all the letters correctly, but the aberrations have caused a loss of contrast and fuzziness around the edges.

Wavefront error defines the ideal compensating optic. You simply measure the wavefront error, invert it, and correct for an index change. You then superimpose it in the same plane, and if you have light coming out of the eye, then the plane wavefront should be parallel. Wavefront error can be used to calculate the point spread function and the MTF, as well as numerous metrics. It contains the information necessary to calculate hundreds of metrics of optical quality. One of the more common metrics is the dioptric map, which can be plotted in three dimensions and can be put on color scales. After gaining familiarity with this map, the clinician can determine that the eye has excessive spherical aberration and coma, as in the case of the above-mentioned post-LASIK patient.

The dioptric map can also be used to calculate the point spread function. The image of a point should be as close as possible to a point image in the image plane. This can also be accomplished in a frequency domain, looking at the MTF and asking whether enough information is being transferred from all the different frequencies available to the eye. The perfect modulation transfer function (e.g., for a 6-mm pupil) is a cone. For the MTF realm, the largest cone possible is desirable. But we have also advocated no new aberrations as opposed to no aberrations. Therefore, the point spread function can be used to create simulations of the retinal image.

Pupil size plays an important role. Wavefront error increases as the pupil opens up. However, higher aberrations are not always worse. It depends on the target and how this is processed through the optics of the eye. For instance, a LASIK patient with a natural pupil that dilates to only about 4.5 mm in the dark, and under normal room light to slightly less than 3 mm, is not very happy obviously. But interestingly, even with a drug-dilated pupil at 6.2 mm, the patient’s acuity was about 20/20 (albeit not very clear), indicating that the retinal factors, such as the Stiles-Crawford effect, are playing a helpful role in minimizing the impact of the aberrant rays. This demonstrates one of the inadequacies of wavefront measurements. Wavefront measurements do not include the effects of diffraction by themselves. But you can add it.

To understand diffraction, you need to understand the behavior of the wavefront as it passes through an aperture or by an edge. Wavefronts connect points of similar phase, and rays of light are perpendicular to the wavefront. If light travelled like a bullet along the path of a ray, then the eye could not see a point source, unless a ray from the point source passed through the aperture and into the eye. There is a limit as to where an eye can be positioned if light travels like a bullet. A position is reached in which the eye can no longer detect the point source because of the aperture. But the light can dimly be seen. Light is apparently bent by the aperture.

We postulated that every point on the wavefront was the source of a secondary wavefront. In other words, by placing another infinite point source on all these wavefronts, a secondary wavefront would emanate. For an unbounded wave, the effects of the wavelets are cancelled, except in the original direction where the effect is identical to the original wave motion. Without an aperture, the unbounded wave simply acts like a wave going out. However, for a bounded wavefront, the

effects are not cancelled, because some of the small ‘point sources’ can no longer interact. Light from the wavelets near the edge of the pupil reach the eye, even though a straight line from the eye to the point source does not pass through the aperture.

Moreover, because the wavefront has been bounded, the wavelets interact. This is known as Fresnel diffraction. A special and particularly interesting case of Fresnel diffraction, called Fraunhofer diffraction, occurs at the focal plane of aberrationfree or nearly aberration-free imaging systems. The Fraunhofer diffraction pattern of an axial point source defines the appearance of the point source and image plane. More importantly, Fraunhofer diffraction in an aberration-free imaging system defines the resolution limit. The Fraunhofer diffraction pattern is the best. In an aberration-free system with a circular aperture, the Fraunhofer diffraction pattern is circular, with a central bright spot referred to as an airy disc. For a 6-mm pupil, taking a point source and imaging it achieves a bright spot in the center, but it is not just a spike. It is spread out slightly and has a whole host of light and dark bars reaching out in the periphery.

The diameter of the airy disc varies with pupil diameter. The radius of the airy disc increases as pupil size decreases. In other words, the diameter of the best possible image of a point varies inversely with pupil diameter. Translated into vision terms, the bigger the pupil the better – if there are no aberrations. Precisely the opposite is true for aberrations.

Wavefront measurements do not include the effects of scatter. However, there are instruments that measure aberrations (such as the Hartmann-Shack) in order to evaluate scattering properties. It is possible to generate scattering maps of the optics of the eye, and in the future some of the scattering can be included in modelling.

Meanwhile, wavefront allows us to measure the monochromatic aberrations of the eye. But wavefront measurements do not measure the transmission of light energy. They do not address apodizing functions due to retinal receptor directionality. Light entering different parts of the pupil is not equally sensitive to eliciting a visual response. This is known as the Stiles-Crawford effect and is due to the wave guide properties of the photoreceptors themselves. That is, photoreceptors prefer light coming right down their axial length and not from some angle. In a normal individual, a typical Stiles-Crawford effect will reduce the effects of optical aberrations which are nearer the edge of the pupil.

Moreover, wavefront measurements do not tell us about the neural transfer function or the source of the error. By measuring the total optics, the error could be originating from the cornea (anterior or posterior), the lens, anterior surface or middle of the lens. All we can ascertain is that there is an aberration. Similarly, wavefront measurements do not help with the tilt of the wavefront error. The reason for this is simple: the eye is directed toward a fixation target. Putting a prism in front of the eye, which is what tilt is in optical terms, will make no difference. The eye will simply turn, and the measurement will be the same.

In addition, wavefront measurement techniques cannot accurately measure eyes corrected with aids that have step functions included in their design. For example, some contact lenses have different annular regions of corrections, and there is a step down to the next function. There is only sampling at these fixed locations, so there is no way of measuring these extremely high frequency aberrations with current technologies.

Although the wavefront measurement can be used to define the ideal correction, it does not indicate how to achieve the correction on the cornea. There are biomechanical responses, alignment errors, measurement errors, laser errors, physician and staff errors, plus all kinds of other compounded errors. Regarding biomechanical responses, using an iterative loop – knowing what the procedure did and disturbing the system with the intent of eliminating residual errors – will allow you eventually to minimize residual error. The same is true for coma. Refining the algorithms is also important. There are predictable and unpredictable biomechanical responses to the intervention. Correction of the predictable biomechanical responses can be folded into the ablation design. But correction for the unpredictable biomechanical responses cannot be reliably folded into the ablation design.

A two-step procedure will likely reduce residual refractive error and/or higher order aberration, thus increasing visual performance. Step one would be wavefrontguided surgery, optimized to correct for the predictable biomechanical response that is performed to achieve the desired refractive outcome and to reduce existing higher order aberrations. The most common outcome and the most likely outcome is a happy patient who has a small residual refractive error and increased higher order aberrations. If the patient is not satisfied, then, once the eye is stable, the surgeon and the patient may opt for a second wave-guided optimization procedure to reduce both the residual refractive error and the higher order aberrations. Because the flap has already been made for LASIK, and the largest portion of the ablation performed, unpredictable biomechanical responses should be minimal, and the final outcome very close to the intended outcome.

Careful data collection and analysis of the first procedure will also shed light on how this particular cornea uniquely responds to the ablation, and will allow for customized refinement of the predictable biomechanical responses.

In summary, using state-of-the-art systems is the key. An excellent wavefront sensor, a well-controlled small beam laser, high-speed closed-loop tracking, sophisticated registry, and an intelligent two-step procedure will lead to repeatable excellent outcomes. A key question that remains unanswered, however, is how good is good enough. For the typical patient, our response is wavefront-guided corrections that eliminate the sphere and cylindrical refractive error. The goal is actually to hit the target without inducing new aberrations for all physiological pupil sizes. But the real benefit is for large pupil sizes, so close attention should be paid to large pupil sizes as you evaluate these systems. There is also a commercially available program (CT View) for making the animations, simulations, and graphics of the wavefront error.