Ординатура / Офтальмология / Английские материалы / Wavefront Customized Visual Correction The Quest for Super Vision II_Krueger, Applegate, MacRae_2003

Figure 35-8. Tetrafoil greyscale spatial intensity profile programmed into the DMD.

In practice, once the Calhoun LAL has been implanted and refractive stabilization has occurred, a wavefront measurement of the eye’s aberrations is made. The phase conjugate to these aberrations can then be put into the DMD device; the spatial intensity profile is created and then projected onto the LAL to correct the aberration.

A higher-order aberration that is universally common in people is the presence of spherical aberration. The wavefront of this aberration can be represented by the following Zernike polynomial:

C04 = 6 4 - 6 2 - 1

The effect of this aberration is to cause rays of light impinging on the periphery of the lens to come to a different focus than being refracted through the central part of the optic. Applying the principles of geometric optics, one way to minimize spherical aberration is to create an aspheric lens, which in its simplest form would possess a radius of curvature difference between the central and peripheral regions. As an example of this, Figure 35-10A shows the interference fringes from a +20.50 D Calhoun LAL preirradiation. Inspection of this figure shows that the power across the central 4 to 4.5 mm of the Calhoun LAL’s wavefront has been minimized. However, inspection of the interference fringes outside this region to the edge of the Calhoun LAL shows

A B

Figure 35-9. (A) Raw interference fringes of the Calhoun LAL postirradiation with the tetrafoil spatial intensity profile. (B) 3-D wavefront rendering of the interference fringes in A.

A B

Figure 35-10. (A) Preirradiation interferogram of the Calhoun LAL with central 4 to 4.5 mm aperture of the lens at best focus.

(B) Interferogram of the Calhoun LAL postirradiation at the same position along the optical axis of the interferometer.

the presence of approximately four fringes (two waves) due to spherical aberration. Figure 35-10B shows the Calhoun LAL at the same position along the optical axis of the interferometer after it was irradiated on its periphery to effectively change the radius of curvature on the outside portion and reduce the amount of spherical aberration. Comparison of Figures 35-10A and 35-10B shows that the wavefront of the central 4 to 4.5 mm of the Calhoun LAL has remained unchanged due to the irradiation process; however, inspection of the periphery of the Calhoun LAL indicates the removal of two fringes (one wave) of spherical aberration. Therefore, the Calhoun LAL technology shows promise of being able to reduce the amount of spherical aberration of an implanted lens and thus the overall spherical aberration of the entire ocular system.

CONCLUSION

Despite advances in cataract and refractive surgery, imprecise IOL power determination and unpredictable wound healing remain important clinical problems to address. For the first time, the Calhoun LAL provides a tool for the cataract and refractive surgeon to noninvasively correct the refractive power of an IOL postimplantation. LAL formulations are useful in both pseudophakic and phakic IOLs. As additional IOLs, such as accommodative versions, become available, postoperative adjustability of IOL power will enhance value of these and other novel IOL technologies.

REFERENCES

1.Leaming DV. Practice styles and preferences of ASCRS members— 1999 survey. J Cataract Refract Surg. 2000;26(6):913-921.

2.Brandser R, Haaskjold E, Drolsum L. Accuracy of IOL calculation in cataract surgery. Acta Ophthalmol Scan. 1997;75(2):162-165.

3.Olsen T, Thim K, Corydon L. Accuracy of the newer generation intraocular-lens power calculation formulas in long and short eyes.

J Cataract Refract Surg. 1991;17(2):187-193.

4. Olsen T. Sources of error in intraocular lens power calculation.

J Cataract Refract Surg. 1992;18(2):125-129.

5.Pierro L, Modorati G, Brancato R. Clinical variability in keratometry, ultrasound biometry measurements, and emmetropic intraocu- lar-lens power calculation. J Cataract Refract Surg. 1991;17(1):91-94.

6.Mendivil A. Intraocular lens implantation through 3.2 versus 4.0 mm incisions. J Cataract Refract Surg. 1996;22(10):1461-1464.

7.Sun XY, Vicary D, Montgomery P, Griffiths M. Toric intraocular lenses for correcting astigmatism in 130 eyes. Ophthalmology. 2000; 107(9):1776-81.

8.Steinert RF, Aker BL, Trentacost DJ, Smith PJ, Tarantino N. A prospective comparative study of the AMO ARRAY zonal-progres- sive multifocal silicone intraocular lens and a monofocal intraocular lens. Ophthalmology. 1999;106(7):1243-1255.

9.Javitt JC, Wang F, Trentacost DJ, Rowe M, Tarantino N. Outcomes of cataract extraction with multifocal intraocular lens implantation - functional status and quality of life. Ophthalmology. 1997;104(4):589599.

10.Negishi K, BissenMiyajima H, Kato K, Kurosaka D, Nagamoto T. Evaluation of a zonal-progressive multifocal intraocular lens. Am J Ophthalmol. 1997;124(3):321-330.

11.Zaldivar R, Davidorf JM, Oscherow S. Posterior chamber phakic intraocular lens for myopia of -8 to -19 diopters. J Refract Surg. 1998;14(3):294-305.

12.Baikoff G, Arne JL, Bokobza Y, et al. Angle-fixated anterior chamber phakic intraocular lens for myopia of -7 to -19 diopters. J Refract Surg. 1998;14(3):282-293.

13.Rosen E, Gore C. STAAR Collamer posterior chamber phakic intraocular lens to correct myopia and hyperopia. J Cataract Refract Surg. 1998;24(5):596-606.

14.Seitz B, Langenbucher A. Intraocular lens power calculation in eyes after corneal refractive surgery. J Refract Surg. 2000;16(3);349-361

15.Seitz B, Langenbucher A. Intraocular lens calculations status after corneal refractive surgery. Curr Opinion Ophthalmology. 2000;11(1): 35-46.

16.Gimbel HV, Sun R, Furlong MT, van Westenbrugge JA, Kassab J. Accuracy and predictability of intraocular lens power after photorefractive keratectomy. J Cataract Refract Surg. 2000;26(8):1147-1151.

17.Azar DT, et al. Intraocular Lenses in Cataract and Refractive Surgery.

Philadelphia, Pa: WB Saunders; 2001.

18.Stamper RL, Sugar A, Ripkin DJ. Intraocular lenses: basics and clinical applications. Paper presented at: the American Academy of Ophthalmology; November 15, 1993; Chicago, Ill.

In a certain sense, all refractive corrections of the eye are customized because each eye has a unique refractive error and it is that unique error that must be corrected. The term customized ablation has come to take on a special meaning to differentiate those procedures that correct more complicated refractive errors from those more common procedures that only correct simple spherocylindrical errors. The first corneal refractive surgical procedures attempted to correct only mean spherical error. As a result, the ablation patterns exhibited a great degree of spatial symmetry and so were fairly insensitive to minor positional errors. When correction for astigmatism was added, rotational symmetry was broken and the relationship of the cylinder axis of the ablation to that of the eye became critical. However, from an optical point of view, minor positional errors in the placement of the ablation pattern on the cornea were not too important.

We find the same thing to be true in the more familiar case of spectacles lenses in which we know that the correction for astigmatism in single vision lenses remains good even as the eye turns to look through different portions of the lens, in effect simulating a decentered ablation pattern. The situation is quite different once we attempt to correct localized refractive errors and so attempt a customized ablation. If such an ablation is not precisely aligned both in position and rotational orientation, the refractive result will quite likely be worse than before the procedure, whereas if alignment is correct, vision will be improved. So when a customized ablation is to be used, it is quite important that the measurements of localized error—both magnitude and location—are precisely made. The alignment requirements will be discussed later.

It is the shape of the cornea that is altered during any type of ablative procedure and so it is only reasonable that we would wish to measure that shape prior to any surgery and use this information to attain our goal: improvement of vision through the removal of refractive error. Corneal topography is a means to measure that shape, and we shall be investigating what role corneal topography can have in obtaining this information. Before we can discuss how the information gained through corneal topography can be used to guide customized ablation procedures, it best to first establish what is measured by corneal topography, how it is presented, and what it means to the refractive state of the eye.

THE OPTICAL ROLE OF THE CORNEA

The cornea is one of the principal refractive elements of the eye. It has a highly curved surface and it exists at the interface between the outside world—usually air—and the tissue of the eye itself. We may think of action of any optical refractive surface element to be primarily governed by two parameters: the curvature of the surface and the change index of refraction across that surface. The term optical refractive surface element is used to define an optical element where refraction occurs only at the surface, thus differentiating it from an optical element where the index of refraction changes in some continuous fashion in the bulk of the material, thereby causing refraction to occur throughout the element. This is an important consideration for the eye because the optical action of the cornea may be thought to occur solely on its surfaces whereas the refractive action crystalline lens not only comes from its surfaces but also from an index of refraction gradient throughout the bulk of the lens.

For an optical refractive surface element, the refractive power is directly proportional to the product of the change of index of refraction across the surface and the magnitude of the curvature of the surface. With these thoughts in mind, it is easy to see why the cornea is the major refractive element of the eye. It has a very highly curved surface with a radius of curvature in the vicinity of 8 millimeters (mm) and the refractive index at its anterior surface changes from 1.00 in air to 1.336 in the anterior tear film. The crystalline lens has a curvature very similar in magnitude to that of the cornea on its surfaces, but the index change is much less, changing at the surfaces from 1.336 in the aqueous humor to 1.386 at edge of the cortex of the lens. This change of 0.05 is 15% of that at the anterior tear film interface so the refractive action of the lens surface is much less than that of the cornea because it is the product of the refractive index change and the curvature that determines the refractive power of any optical element. It also suggests that any local curvature changes in lens surfaces will have proportionally less overall effect on the refractive state of the eye than changes of the same magnitude in the curvature of the cornea.

It would seem then that the curvature of the cornea and changes to that curvature are the most important parameters to consider when changing the refractive state of the eye via corneal

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refractive surgery. However, for the purposes of deciding how to control the action of a laser ablation system, curvature is not the most convenient parameter to use. Before explaining why this is the case, let us first consider what the corneal topographer measures at a fundamental level.

Measurements That Can Be Made by a Corneal Topographer: Curvature vs Height

If we were to talk of the topography of the surface of the earth or the topography of some three-dimensional (3-D) object, we would most likely be describing topography in terms of the height or elevation of the surface at designed locations above some fixed plane or reference surface. We would be describing the topography in terms of the 3-D coordinates of the surface. For mostly historical reasons, this is not what is generally meant when one speaks of corneal topography. Corneal measurements were first taken with instruments called ophthalmometers or keratometers. These instruments measure the curvature of small areas of the cornea by measuring the magnification of luminous targets, known as mires, by the highly curved corneal surface. The values thus found are curvature values, and it was in terms of curvature that ophthalmic professionals came to think of characterizing the corneal surface. It was therefore reasonable, when corneal topographers first became available, to present their topographical data in terms of curvature at various locations on the cornea. However, this created a problem in presentation because the curvature at a given location (x, y) on a surface cannot be expressed as a single value as surface elevation (z) can be. Curvature needs three values (ie, sphere, cylinder, and axis) to express it at a given location. Normal topographical maps using contour lines or colors to represent the elevation variable in terms of two position variables are inadequate to fully represent curvature. The compromise that was settled on for corneal topographical curvature maps was to pick one component of the curvature and represent it in a color-coded contour map. The component chosen was the one in the direction of the meridians of a polar coordinate system drawn on the corneal surface and centered on the corneal vertex.

Curvature needs three values (ie, sphere, cylinder, and axis) to express it at a given location. Therefore, typical topographical maps using contour lines or colors to represent the elevation variable in terms of two position variables are inadequate to fully represent curvature.

The types of analyses customarily used in commercial corneal topographers are not the best to use when considering customized ablation. As will be seen from the discussion below, precise measurements of elevation constitute the best form of data with which to begin any analysis. Some corneal topography systems have the ability to export data of this type and only these systems, if their precision of measurement is high enough, are truly useful tools in matters dealing with customized ablation. In addition, the best methods of curvature analysis are not implemented in commercial topography systems. Consequently, to best use available corneal topography data, one needs to employ specialized analysis methods. Some of these will be mentioned next.

Precise measurements of elevation constitute the best form of data with which to begin any corneal analysis.

Methods of Analysis Used by Corneal Topographers

How is curvature and elevation information to be obtained by the corneal topographer? Several approaches have been used, among them the keratometric analysis method, the arc step method, and calculations of curvature from analysis of elevation data. They will briefly described next.

Keratometric Analysis Method

It is possible to analyze a multiple ring corneal topographer image (Placido ring system) in the same way one analyzes a keratometer ring image. Each of the multiple rings is considered to be a keratometer ring by itself. Its size, usually expressed as the distance of a location on the ring image from the center of the total image, the corneal vertex, and representing the magnification of the ring by the corneal surface, is converted to a keratometric dioptric value using the logic used in the case of an ophthalmometer. Each chosen location on each ring, defined by the location at which a meridian intersects the ring, is then assigned a curvature value. This method only gives one curvature value at each location, and one finds upon investigation that this is not a true curvature of the surface in any mathematical sense. The keratometric analysis method does not generate corneal surface elevation information.

Arc Step Method

It turns out that there is a better method to analyze the image of a Placido ring system created by the corneal surface to assign the surface curvature than the keratometric analysis method. This method employs a form of ray tracing to follow rays backward from the charge-coupled device (CCD) detector, through the optical system of the topographer to the corneal surface. At the corneal surface, the rays are reflected and pass to their source—the luminous ring. The CCD location, found via image analysis is known as is the geometry of the optical system. This fixes the ray in space both in location and in direction of travel. At some point along its travel, initially an assumed location, the ray strikes the surface of the cornea and is reflected. Having assumed a reflection location and knowing the location of the luminous ring in space, the direction of the ray after reflection is fixed. Then, employing the laws of reflection, the direction of the surface normal (a measure of surface slope) at the point of reflection is found by dividing the angle between the incoming ray and the reflected ray. This slope information is then used to calculate values related to the local curvature of the surface at a great number of points on the corneal surface. If the method used to find these slope values is the iterative method known as the arc step method, the elevation of the surface is also found at each measured location along with the surface slope. The arc step method reconstructs a meridian by piecing together short circular arcs that have the characteristic that each joins to its neighbor in such a fashion that the centers of curvatures of both arcs lie on the common line that is coincident with the radii of both arcs at the point at which they join. This insures that the first derivative of the combined arc is everywhere continuous. The procedure consists of proceeding ring by ring, usually starting in the center of the cornea, in such a fashion that the new arc fulfills the joining criterion to its preceding neighbor while fulfilling the reflec-

Figure 36-1. Because the OPL for edge rays is less than for central rays, they exit the lens sooner and travel some distance beyond the lens before the central rays exit. This causes the refracted wavefront to take a convex curvature, producing the focusing action of the lens.

tion criterion for the ray under analysis at the farther end. This is done with an iterative process of modifying the radius of curvature of the arc until the ray successfully strikes the end point of the arc and its directed to its proper ring to within the tolerance limits chosen for the process. In this process, the end point locations of each arc are precisely found and this provides the necessary high precision elevation information needed for use in planning and analyzing custom ablations.

Note that surface slope information is only found along the meridians and so must be termed the meridional slope instead of the surface slope. Nothing is found as to the surface slope in other directions and hence information on surface curvature is incomplete. This is because only along meridians can curvature be found when present-day corneal topographer analysis methods are employed. Such partial displays of curvature have proved extremely useful to clinicians to get an overall subjective idea of the refractive state of the cornea, but they are not particularly useful for optical analysis of the cornea via ray tracing and other analytical methods.

For full ray tracing, one needs to know the surface normal at various locations. The surface normal is a line passing through a point on the surface perpendicular to a plane tangent to the surface at that point. To know the surface normal, one needs to know not only to the surface slope along a meridian but also the slope at right angles to the meridian. So it seems that corneal topography, as currently done, is not adequate for detailed analysis of refractive error. However, if the elevation information— which is obtained in addition to slope information in certain analysis methods—is considered, the situation changes and there is a way to analyze the optical action of the cornea in great detail.

Curvature From Elevation Data

While the arc step method, as currently implemented, does not give complete curvature information on the corneal surface, it does yield very accurate surface elevation data on the surface. This information is complete in that it completely specifies the corneal shape. If the elevation is of sufficient precision, complete

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curvature information of the corneal surface may be found using well-known methods from differential geometry that involve finding the surface gradient at each desired location from the elevation information and then using the gradient information to find the second derivatives of the surface at each location. The surface gradients and the second derivative values are then used to find all three curvature values at each location.

TREATMENT OF THE OPTICAL ACTION OF A REFRACTIVE ELEMENT IN TERMS OF ITS THICKNESS (OPTICAL PATH LENGTH)

The optical action of a refractive element has thus far been considered to be governed by the change in index of refraction at its surfaces and by the surface curvatures. There is a very different way to treat optical action that is just as valid and useful as the more common approach. Instead of considering the refraction of light rays by a surface, one considers the change in optical path length (OPL) for different rays as they pass through an optical element and imposes the restraint that each ray as it progresses from an initial wavefront to a refracted wavefront has the same OPL. The OPL for a segment of a ray is defined as the physical distance along the ray segment multiplied by the index of refraction found in that segment:

The OPL for a segment of a ray is defined as the physical distance along the ray segment multiplied by the index of refraction found in that segment.

To see how this creates optical action, consider a simple positive lens. Such lenses are thicker in the middle than at the edge. This means that the OPL of a ray passing through the middle of the lens has a longer OPL than does a ray passing through the edge. Now let this positive lens have a plano front surface and a convex back surface and suppose that a plane wavefront enters the flat front surface so that all rays enter the lens at once. Because the OPL for edge rays is less than for central rays, they exit the lens sooner and travel some distance beyond the lens before the central rays exit. This causes the refracted wavefront to take a convex curvature, producing the focusing action of the lens. This is illustrated in the Figure 36-1.

This is a very simple example, but the approach is general and it allows the optical behavior to be expressed in terms of the change in thickness of a lens or other optical element as the position on the element changes. In addition, this method allows aberrations to be treated by considering them to be differences in OPL between the actual wavefront and some ideal wavefront at various positions on the wavefront. We can therefore use distance measures, in the form of elevation changes of a surface, to analyze its optical performance instead of using curvature information or surface slope information. In many ways, this is a better approach because it allows very powerful optical analysis theorems to be used to predict the quality of images without having to resort to ray tracing. Ray tracing is difficult to perform precisely in the case of the eye because of uncertainty regarding the optical characteristics of an individual crystalline lens. There is

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no convenient way to precisely measure the surface profile or the index of refraction gradient within this gradient index lens in the living eye. Even if precise ray tracing could be done, interpretation of ray intercepts with the fovea is not as easy as interpretation of point spread functions (PSFs) and spatial frequency spectrums, which can be directly related to contrast sensitivity and visual resolution (computed directly from wavefront error information).

For the purposes of controlling a laser ablation system, the thickness approach to characterizing a lens is the only practical one. Refractive correction is accomplished when using photorefractive keratectomy (PRK) or laser in-situ keratomileusis (LASIK) procedure by removing a tissue lens with the laser system. This tissue lens will be referred to as the treatment lenticule. The power of this treatment lenticule is the power of the refractive error. As the laser selectively removes tissue to remove the treatment lenticule, it needs to know how much tissue to remove at different locations. When the treatment lenticule is characterized by thickness, the information needed by the laser is immediately available. In fact, for simple refractive errors, there is no need for information on the topography of the cornea. A treatment lenticule with the correct thickness gradient to produce the desired correction is removed and the optical result is the desired one regardless of the shape of the corneal surface.

Refractive correction is accomplished when using PRK or LASIK procedure by removing a tissue lens with the laser system. This tissue lens will be referred to as the treatment lenticule.

CUSTOMIZED ABLATION

We have defined customized ablation as corneal refractive surgical procedures that correct more complicated refractive errors than simple spherocylindrical ones. We may wish to do this for a variety of reasons. An earlier procedure may have induced corneal irregularity that we wish to remove. We may wish to remove higher-order aberrations of the eye in addition to spherocylindrical error, thereby increasing the resolution of the eye. No matter what the reason for the customization, it will be necessary to measure the aberrations of the eye carefully so that they are well characterized not only in magnitude but also in location with respect to the pupil of the eye. This information must then be converted to instructions for the laser ablation system that consist of amounts of tissue to be removed at specified locations.

WAVEFRONT ERROR

The most convenient form for the aberration information for the above use is in terms of wavefront error. The wavefront error is the OPL difference at each location between the desired wavefront and the actual wavefront. Because OPL is the product of the index of refraction and the physical path length, a measurement method that measures an actual surface and compares it to a desired surface in essence finds the wavefront error. In the case of the cornea, the index of refraction to use is the index of refraction of the stroma (1.376).

We now can see how information from a corneal topographer can be used if the system can measure the 3-D position of the sur-

face. What the corneal topographer does not measure is the shape of the desired corneal surface. This must be supplied in some fashion. There are a number of reasonable approaches.

One approach uses a combination of the desired spherocylindrical correction and considerations of aberration-free surfaces. Usually, there is a spherocylindrical refractive error to be removed and the ablation profile to accomplish this can be found without any reference to the cornea surface as was noted previously. The elevation values associated with the spherocylindrical correction can then be directly subtracted from the measured corneal elevation values to give the surface that will exist after the basic correction. It is known that surfaces that minimize aberration are very smooth surfaces in the sense that their curvature changes slowly as a function of position. The cornea can exhibit fluctuations in curvature on a very local basis. If these local fluctuations can be removed, the higher-order aberrations will also be reduced. This can be done by fitting the surface found by subtracting spherocylindrical ablation pattern from the measured surface to a smooth function, such a general ellipsoid. The elevation difference between this surface and the best fit smooth surface gives the extra amount of ablation needed to remove corneal irregularities and the resulting higher-order aberrations from the eye.

Approaches to correct the optical errors of the eye, which by design minimize corneal aberration and ignore the contributions of the crystalline lens to the total aberration of the eye, are at risk of increasing the residual higher-order aberration of the eye.

There is an even better way to assess the aberrations introduced into the optical system of the eye due to the cornea alone. This is to find a corneal surface that would introduce no aberrations at all, select this surface as a reference, and find the wavefront error induced by the cornea by subtracting the reference surface from the actual surface to create a tissue error lens. The aberrations created by this tissue error lens are the aberrations induced by the corneal surface. Such aberration-free surfaces have been known since 1637 when they were introduced by Descartes and are known as Cartesian ovals. These ovals take the simple form of an ellipsoid of revolution whose eccentricity is given by the ratio of the index of refraction of the incident media to the index of refraction of the media following the interface. Taking the incident index of refraction to be that of air (1.000) and the exiting index of the refraction to be that of the corneal stroma (1.376), the Cartesian oval desired has an eccentricity of 0.73 and is thus more aspheric than the mean human cornea with an eccentricity (0.55). It now remains to find the apical curvature of this ellipsoid to completely specify it. This may be done by measuring the spherical equivalent refractive error of the eye, subtracting this from the mean surface power of the central cornea and converting the remaining surface power of the cornea to a curvature value. Such a surface will insure that a central collimated pencil of light entering the eye focuses on the retina.

Corneal aberrations found in this way are particularly valuable when they are used in conjunction with aberrations found using a wavefront refractor because they give information on not only the aberrations induced by the corneal surface but also on the aberrations induced by the rest of the optical system of the eye.

When the above method is employed, there are some technical details that need to be considered. The coordinate systems of corneal topography typically use the corneal vertex as the origin. For analysis of wavefront error, it is best to use a coordinate system whose origin is the pupil center. This means that the displacement, if any, of the corneal vertex and the pupil center must be measured and the reference Cartesian oval decentered by this amount when it is subtracted from the corneal elevation data to create the tissue error lens. When this is done, tilt (prism) is invariably introduced into the tissue error lens. This tilt should be removed from the subsequent analysis, as it can have no effect on image formation. There is indeed a prismatic effect introduced by the decentration of the corneal vertex from the pupil center but it is optically removed by a simple eye turn so that the object of interest falls on the fovea.

One eye can adjust for tilt by a simple eye turn; however, under binocular conditions, induced prism could produce a binocular imbalance.

CONSIDERATION OF ALIGNMENT PRECISION

NEEDED FOR CUSTOMIZED ABLATIONS

There is no way that a hard and fast rule can be given for necessary alignment precision because of the wide variation in the size and severity of localized refractive errors found in practice. It is helpful, though, to think of the situation in this way. Suppose that there is a local corneal irregularity that degrades vision by some amount. We may think of the correction for this irregularity as the addition of second corneal irregularity that is equal but opposite to the original so that the two cancel one another. Now let us suppose that we misalign the correction so that it does not fall on the original. It then adds a refractive error similar in magnitude to the original error without causing the original to go away. In the worst case, we have essentially doubled the problem for the visual system. On the other hand, if our misalignment is only partial, then some cancellation of error will occur and the net result may be an improvement. This is a very simple approach to what in practice will be a fairly complicated optical situation, but it gives us good direction in our thinking on this matter. It must also be noted that errors of alignment can be rotational as well as translational; a concept well known from our experience in correcting astigmatism. Perhaps the most important idea, as we consider the alignment precision needed for customized ablation, is that the size of the local irregularity is not only important from the point of view of its effect on vision but also on the precision with which we must apply a correction ablation to remove its effect. A second important idea is that the magnitude of the irregularity is also important. To see this, consider the familiar case of correction for simple astigmatism and the need to correctly orient cylinder axis. The residual refractive error when axis is misaligned is proportional to the product of the axis error (the rotational positional error) and the original magnitude of the cylinder error. Much more axis error can be tolerated for a small amount of cylinder power error than can be tolerated for a large amount of cylinder power error.

To treat the requirements for alignment in an analytical manner, the following approach is suggested. Starting with 3-D infor-

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mation on the wavefront error function, a transformation of the coordinate system consisting of a translation and rotation of the desired amount is made, and the wavefront error function is expressed in this new system. Next, the difference between the original and the transformed wavefront error function is found, thus creating a new wavefront error function. This wavefront error function may be considered to be the aberrations induced by the misalignment. To make the approach practical, the initial wavefront error needs to be expressed as an analytical function, and this is most easily done by expressing it in terms of Zernike polynomial functions weighted by their respective coefficients. There exist fairly simple mathematical methods for performing the needed coordinate transformation on a set of Zernike polynomial functions that yield the coefficients for the transformed functions. These may then be directly subtracted from the original coefficient set to give the residual error in terms of its Zernike coefficients.

One interesting effect of applying decentered aberration corrections is that aberrations of different types and symmetries than those found in the original aberration set are generated while at the same time the original aberrations are removed. The simplest example is the well-known effect of introducing prism when a spherocylindrical correction is decentered. When one considers some of the common higher-order aberrations, one finds that decentering a coma correction generates astigmatism and defocus, decentering a trefoil correction generates astigmatism, and decentering a spherical aberration correction generates coma. In general, decentered corrections have the interesting effect of correcting the aberrations they are designed to correct while at the same time generating new aberrations of lower order than the original aberrations.

When aberration corrections are applied with a rotation or axis error, another effect occurs. Now one finds that no aberrations of lower order are introduced, but that original aberrations whose meridional index is non-zero are not completely removed so that a residual magnitude remains with the axis of that aberration (when expressed in magnitude/axis form) rotated.

Decentered corrections remove the original error and generate lower-order aberrations of opposite meridional symmetry. Rotated corrections do not induce aberrations of different order or meridional symmetry but partially remove the original aberration and rotate its axis.

REQUIREMENTS FOR CORNEAL TOPOGRAPHERS

WHEN USED TO GUIDE CUSTOMIZED ABLATION

A corneal topographer must be able to measure the elevation of the corneal surface for it to be useful for planning customized ablation. Systems that only measure curvature may be useful for subjective evaluation of the corneal surface, but they are not adequate tools for planning customized ablation. For those corneal topographers that can make elevation information available, only those with sufficient precision of measurement are truly useful. To guide us on setting limits for this necessary precision, we have the characteristics of laser ablation systems and the magnitude of expected aberrations. It is best to express aberrations in terms of OPL and the appropriate unit is the wavelength of light. Therefore, errors are best expressed in microns (µm), as the midrange wavelength value for the visible spectrum is