Preparation of this chapter was supported by National Institutes of Health (National Eye Institute) grants R01-EY05109 to LNT and R01-EY-08520 to RAA and a unrestricted grant to the Department of Ophthalmology from Research to Prevent Blindness, NY, NY.
Contemporary visual optics research is changing our mindset, our way of thinking about the optical system of the eye, and in the process is re-defining the field of visual optics. In the past, optical imperfections of the eye were conceived as simple refractive errors - defocus, astigmatism, and perhaps a bit of prism. Although clinical students learned about other kinds of optical imperfections, such as spherical aberration, coma, oblique astigmatism and the other Seidel aberrations, those concepts were confined to courses in optical theory, not to clinical practice. And for good reason - these higher-order aberrations of the eye couldn't be measured routinely in the clinic, and even if they could we didn't have the means to correct them optically at a reasonable cost to patients. Furthermore, since the effects of such aberrations on visual function were largely unknown, there was little reason to suppose that correcting them would do any good for the patient's vision. However, the introduction of laser refractive surgery, with its potential for removing as well as introducing unwanted optical aberrations into the eye, demands changes in established ways of thinking and answers to these unresolved issues.
Figure 5-1. Effect of optical defocus (0.25 Diopters, 6 mm pupil diameter) on the aberration map (top) and simulated retinal image of an eye chart (bottom). Method of calculation is to first determine the point-spread function (PSF), which is then convolved with the eye chart to yield the retinal image.
Today, optical imperfections of the eye are being re-examined within a comprehensive theoretical framework which expresses the combined effect of all the eye's aberrations in a two-dimensional aberration map of the pupil plane. An aberration map is similar in concept to corneal topographic maps used to describe the corneal surface. The major difference is that a corneal map describes the curvature of a physical surface, whereas an aberration map describes the difference between a wavefront of light and a reference wavefront. By concentrating our attention on light instead of the refracting surface, we gain an ability to compute image quality on the retina for simple points of light, for clinical test targets, or for any complex object in the real world. For example, Figure 5-1 shows a wavefront aberration map for a defocused eye from which the retinal image of an acuity chart may be computed Such computations are poised to become routine clinical tools of the future for predicting the visual benefit to the patient of aberration correction, and for explaining to patients the risks and visual consequences of unintended increases in optical aberrations following refractive surgery or other forms of treatment. .
Customized corneal ablation is a surgical procedure designed to improve the optical quality of the eye, thereby improving vision. To assess the outcome of this procedure requires measures of the direct effect on retinal image quality, and secondary effects on visual performance and the quality of visual experience. A variety of methods for specifying optical quality are well established in the field of optics and may be readily applied to the optical image on the retina. Similarly, a variety of visual performance measures are sensitive to the optical quality of the retinal image and therefore may be used to assess the effect of refractive surgery on vision. However, optical limits normally imposed by the eye's optical aberrations may recede in the near future if refractive surgery, contact lenses, or intra-ocular lenses (IOL) improve retinal image quality beyond limits imposed by the neural component of the visual system. If this occurs then common measures of visual performance, such as letter acuity, which are traditionally regarded as good measures of optical quality of the retinal image, may no longer be optically-limited. When this happens, visual performance will be limited instead by the neural architecture and physiology of the retina and visual brain, thereby generating a demand for new measures of vision which are sensitive to even the smallest departures from perfect retinal image quality.
The quality of an optical system may be specified in three different, but related, ways. The first method is to describe the detailed shape of the image for a simple geometrical object such as a point of light, or a line. The distribution of light in the image plane is called a point-spread function (PSF) for a point object or a line-spread function (LSF) for a line object. Simple measurements derived from these functions, such as the width (blur circle diameter) or height (Strehl ratio) of the intensity distribution, are taken as figures of merit which capture the blurring effects of optical imperfections.
The second method is a description of the loss of contrast suffered when an image is cast of a sinusoidal grating object. The sinusoidal grating is a very special object in optics because it has the unique property of producing images of the same form. In other words, a sinusoidal grating object forms sinusoidal images with the same spatial frequency (expressed in cycles/degree) and the same orientation1. Thus gratings make if easy to specify the optical effect of the imaging system in terms of just two parameters: spatial contrast and spatial phase. The ratio of image contrast to object contrast captures the blurring effects of optical imperfections and the variation of this ratio with spatial frequency and orientation2 of the grating object is called the modulation transfer function (MTF). The difference between the spatial phase of the image and the phase of the object captures the prismatic displacements induced by optical imperfections. The variation of this phase difference with spatial frequency and orientation of the grating object is called the phase transfer function (PTF). Taken together, the MTF and PTF define the optical transfer function (OTF) of an imaging system. One of the most important results of optical theory in the 20th century was the linking of the PSF, LSF, MTF, PTF and OTF by means of a mathematical operation known as the Fourier transform (Gaskill, 1978; Goodman, 1968). Furthermore, given these characterizations of an imaging system, one may use optical theory to compute the expected retinal image for any visual object, thus overcoming the great handicap imposed on clinicians and visual scientists by the natural inaccessibility of the retinal image.
This chapter examines a third method for specifying optical quality in terms of underlying optical aberrations, rather than the secondary effect of those aberrations on image quality. Such a description may be couched in terms of the deviation of light rays from perfect reference rays (ray aberrations) or in terms of the deviation of optical wavefronts from the ideal reference wavefront (wavefront aberrations). This aberration method is a more fundamental approach to the description of optical imperfections of eyes, from which all of the secondary measures of optical quality described above (PSF, LSF, MTF, PTF and OTF) may be derived. It is also the approach which is most useful for customized corneal ablation since the aberration function of an eye is a prescription for optical perfection.
From a clinical perspective, perhaps the most useful interpretation of optical aberration maps is in terms of errors of optical path length (OPL). The OPL concept specifies the number of times a light wave must oscillate in traveling from one point to another. Since the propagation velocity of light is slower in the watery refractive media of the eye than in air, more oscillations will occur in the eye compared to the same physical distance in air. Thus by defining OPL as the product of physical path length with refractive index, OPL becomes a measure of the number of oscillations executed by a propagating ray of light. This is an important concept because light rays emitted by a point source will propagate in many directions, but if all the rays have the same OPL then every ray represents the same number of oscillations. Consequently, light at the end of each ray will have the same temporal phase and this locus of points with a common phase represents a wavefront of light. Thus a propagating wavefront of light is defined by the locus of points in space lying at the same OPL from a common point source of light. To define the aberrations of an optical system, we compare the OPL for a ray passing through any point (x,y) in the plane of the exit pupil with the chief ray passing through the pupil center (0,0). The result is called the optical path difference (OPD). Thus the aberration structure of the eye's optical system is summarized by a two-dimensional map showing how OPD varies across the eye's pupil.
For a perfect imaging system the OPL is the same for all light rays traveling from the object point to the image point and therefore OPD = 0 for all (x,y) locations in the pupil. In the case of an eye, this means that rays of light from a single point object which pass through different points in the pupil will arrive at the retinal image point having oscillated the same number of times. Such rays will have the same temporal phase and therefore will add constructively to produce a perfect image. If, on the other hand, light passing through different points in the pupil arrive with different phases because they traveled along paths of different OPLs, then the system is aberrated and the quality of the image will suffer. Thus, by conceiving of optical aberrations as differences in optical path length it is easy to see how aberrations might arise due to (1) thickness anomalies of the tear film, cornea, lens, anterior chamber, posterior chamber, etc. or (2) because of refractive index anomalies of the ocular media which might accompany inflammation, disease, aging, etc., or (3) because of decentering or tilting of the various optical components of the eye with respect to each other.
A concrete example of the OPL concept is illustrated in Figure 5-2. For a myopic eye with no other aberrations, the optical path is shorter for rays passing near the pupil margin compared to rays passing through the pupil center. Consequently, the best retinal image of a point object will be formed if we compensate for this variation in optical path length by placing the point source at the eye's far-point. Now the wavefront of light enters the eye as a concave wavefront such that the central rays arrive at the eye before the marginal rays, giving them a head-start so that when they follow a longer optical path through the eye they arrive in-phase with the marginal rays. In short, to obtain the optimum retinal image requires the optical distance from each object point to its image be the same for every path through the pupil. The wavefront aberration map indicates the extent to which this ideal condition is violated.
Figure 5-2. Example of a diverging wavefront from source P being focused to retinal point P' by a myopic eye. Reversing the direction of light propagation, light reflected from retinal point P' emerges from the eye as a wavefront converging on point P. When referenced to the x-y plane of the pupil, the wavefront shape W(x,y) is also an aberration map of the eye.
By reversing the direction of light propagation in Figure 5-2 we achieve a more practical definition of an aberration map for the eye. . It follows from the preceding discussion that if the retinal point P¢ is a source of light reflected out of the eye, then the shape of the emerging wavefront is determined by the variation of OPL across the eye's pupil. If the eye is optically perfect and emmetropic, this reflected wavefront would be a plane wave propagating in the positive z-direction. Thus, for distance vision, any departure of the emerging wavefront from the x-y plane is an optical aberration. On the other hand, for near vision the reflected wavefront emerging from the eye must be compared with a spherical wavefront centered on the fixation point. In practice, the distance W(x,y) between the reflected wavefront emerging from the eye and the corresponding reference sphere is taken as a measure of the wavefront aberration function of the eye for the given viewing distance. By convention, positive aberrations occur when the marginal ray travels a shorter OPL than does the central (chief) ray, as in the case of a myopic eye shown in Figure 5-2. Therefore, by this sign convention, W(x,y) = - OPD(x,y).
In summary, the shape of a wavefront of light reflected out of the eye from a point source on the retina is determined by the OPD for rays passing through each point in the eye's pupil. Therefore, a map of OPD across the pupil plane is equivalent to a mathematical description, W(x,y), of the shape of the aberrated wavefront that emerges from the eye. Either may be used as an aberration map of the eye. Such maps are fundamental characterizations of the optical quality of the eye that may be used to compute other common metrics of image quality (e.g. the PSF or OTF) from which we may compute the expected retinal image of any visual target.
The current explosion of interest in optical aberrations of eyes has been spawned by new technology resting on an ancient principle. Nearly 400 years ago the celebrated Jesuit philosopher and astronomer, Christopher Scheiner, professor at the University of Ingolstadt. and a contemporary of Kepler and Galileo, published his 1619 treatise Optical Foundations of the Eye some 75 years prior to the invention of the wave theory of light by Huygens. This pioneering book (Scheiner, 1619) described a simple device illustrated in Figure 5-3 which is widely known in ophthalmology as Scheiner's Disk. Scheiner reasoned that if an optically imperfect eye views through an opaque disk containing two pinholes, a single distant point of light such as a star will form two retinal images. If the eye's imperfection is a simple case of defocus, then the double retinal images can be brought into register by viewing through a spectacle lens of the appropriate power. This design idea for an optometer for measuring refractive errors of eyes was first proposed by Porterfield in 1747 and afterwards improved by Thomas Young in 1845.
Figure 5-3. Scheiner's disk isolates rays, allowing their aberrated direction of propagation to be traced. An ametropic eye will form two retinal images for each object point when viewing through a Scheiner disk with two apertures.
A simple lens won't always bring the two retinal images into
coincidence, however, so a more general method is needed for
quantifying the refractive imperfection of the eye at each pupil
location. Smirnov (Smirnov, 1961) was first to
extend Scheiner's method by using a fixed light source for the
central reference pinhole and a moveable light source for the outer
pinhole as illustrated in Figure 5-4. By
adjusting the moveable source horizontally and vertically, the
isolated ray of light is redirected until it intersects the fixed ray
at the retina and the patient now reports seeing a single point of
light. Having made this adjustment, the displacement distances
x
and y
are measures of the ray aberration of the eye at the given pupil
point.
Figure 5-4. Smirnov's aberrometer used the principle of Scheiner's disk to measure the eye's optical imperfections separately at every location in the eye's entrance pupil.
Independently of these developments in ophthalmology, Scheiner's simple idea was re-invented by Hartmann for measuring the ray aberrations of mirrors and lenses (Hartmann, 1900). Hartmann's method was to perforate an opaque screen with numerous holes, as shown in Figure 5-5. Each hole acts as an aperture to isolate a narrow bundle of light rays so they could be traced to determine any errors in their direction of propagation. Since rays are perpendicular to the propagating wavefront, any error in ray direction is also an error in wavefront slope. Thus Hartmann's method is commonly referred to a wavefront sensor.
Figure 5-5. The Hartmann screen used to measure aberrations objectively is a Scheiner disk with numerous apertures.
Seventy years later, Shack and Platt invented a better Hartmann screen using an array of tiny lenses which focus the light into an array of small spots, one spot for each lenslet (Shack & Platt, 1971). Their technique came to be known as Shack's modified Hartmann screen, or Shack-Hartmann for short. To see how the array of spot images can be used to determine the shape of the wavefront, we need to look at the wavefront in cross-section as shown in Figure 5-6. For a perfect eye, the reflected plane wave will be focused into a perfect lattice of point images, each image falling on the optical axis of the corresponding lenslet. By contrast, the aberrated eye reflects a distorted wavefront as illustrated in Figure 5-7. The local slope of the wavefront is now different for each lenslet and therefore the wavefront will be focused into a disordered collection of spot images. By measuring the displacement of each spot from its corresponding lenslet axis, we can deduce the slope of the aberrated wavefront as it entered the corresponding lenslet. Mathematical integration of slope yields the shape of the aberrated wavefront which can then be displayed as an aberration map.
Figure 5.6. The Shack-Hartmann wavefront sensor forms a regular lattice of image points for a perfect plane wave of light.
Figure 5-7. The Shack-Hartmann wavefront sensor forms an irregular lattice of image points for an aberrated wavefront of light.
The first use of a Shack-Hartmann wavefront sensor to measure aberrations of human eyes was in 1994 by Liang and colleagues (Liang, Grimm, Goelz, & Bille, 1994), thus completing this historically meandering path to discovery of a fast, objective, reliable method for assessing the aberration structure of human eyes. Liang's concept of a Scheiner-Hartmann-Shack aberrometer is shown schematically in Figure 5-8. Because the shape of an aberrated wavefront changes as the light propagates, it is important to analyze the reflected wavefront as soon as it passes through the eye's pupil. To do this a pair of relay lenses focuses the lenslet array onto the entrance pupil of the eye. Optically, then, the lenslet array appears to reside in the plane of the eye's entrance pupil where it can subdivide the reflected wavefront immediately as it emerges from the eye. The array of spot images formed by the lenslet array is captured by a video sensor and then analyzed by computer to estimate the eye's aberration map.
Figure 5-8. The modern aberrometer built on the Scheiner-Hartmann-Shack principle uses relay lenses to image the lenslet array into the eye's pupil plane. A video sensor (CCD) captures an image of the array of spots for computer analysis.
Four examples of aberration maps for normal healthy eyes are shown in Figure 5-9. By using a grey-scale to encode wavefront height we capitalize on the human visual system's natural ability to infer depth and structure from shading. The maximum difference between the highest and lowest points on each of these maps is about 1 µm, which is a bit more than 1 wavelength of the light used to measure the eyes' aberrations (0.633 µm). Perhaps the most distinctive feature of these maps is the irregular shape of their smoothly varying shapes. Another important feature of aberration maps from normal eyes is the tendency to be relatively flat in the center of the pupil with aberrations growing stronger near the pupil margin. This is consistent with the literature showing that image quality is relatively good for medium sized pupils, but deteriorates as pupil diameter increases (Campbell & Gubisch, 1966; Liang & Williams, 1997).
Figure 5-9. Examples of higher-order aberration maps for four normal, healthy individuals reconstructed from measurements taken with an SHS aberrometer similar to that shown in Figure 5-8. Light areas in the map indicate the reflected wavefront is phase-advanced, dark areas indicate phase-retardance. The maximum difference between high and low points on each map is about 1 micrometer. (Zernike orders 0-2 omitted for clarity).
For comparison, Figure 5-10 shows four examples of clinically abnormal eyes. Qualitatively these maps have the same irregular, smoothly varying shapes as in normal eyes. The main difference is that the magnitude of the aberrations is about 10-fold larger, and therefore image quality is about 10-fold worse than normal. Another important abnormality of the keratoconic patient (B) and to a lesser extent the dry-eye patient (A) is the tendency to have large aberrations in the middle of the pupil. The implication of this result is that image quality will be sub-normal for small pupils as well as for large pupils.
Figure 5-10. Examples of higher-order aberration maps from eyes with four different clinical conditions. (A) Dry eye, (B) Keratoconus, (C) LASIK surgery, (D) Cataract. The maximum difference between high and low points on each map is about 10 micrometers, except (D) which is closer to 1 micrometer. (Zernike orders 0-2 omitted for clarity).
Classical analysis of data from a Shack-Hartmann wavefront sensor takes no account of the quality of individual spots formed by the lenslet array. Only the displacement of spots is needed for computing local slope of the wavefront over each lenslet aperture. However, experience has shown that the quality of dot images can vary dramatically over the pupil of a human eye as illustrated in Figure 5-11. The presence of blurred spots indicates a violation of the underlying assumption that the wavefront is locally flat over the face of the lenslet. Two possible reasons are illustrated in Figure 5-12.
Figure 5-11. Examples of selective loss of image quality in individual spots for eyes with two different clinical conditions.
The first possibility is that the gross aberrations of the eye are so large that the wavefront is significantly curved over the area of the lenslet. The result is a blurry spot which is difficult to localize. If the aberrations are large enough, neighboring spots can even overlap, which complicates the analysis considerably. The second possible limitation involves irregular aberrations on a very fine spatial scale. Perturbations of the wavefront within the lenslet aperture are too fine to be resolved by the wavefront sensor using classical methods. Rather than displacing the spots laterally, these "micro-aberrations" scatter light and blur the spots formed by the aberrometer. Although these blurry spots are problematic, they nevertheless contain useful information about the degree and location of scattering sources inside the eye which may prove useful in clinical applications (Thibos & Hong, 1999).
Figure 5-12. Blurring of individual spots detected by a Shack-Hartmann wavefront sensor may indicate the presence of gross aberrations of large magnitude, or micro-aberrations. In either case the blur is due to violation of the underlying assumption that the wavefront is locally flat over the lenslet aperture.
One systematic method for classifying the shapes of aberrations
maps is to conceive of each map as the weighted sum of fundamental
shapes or basis functions. One popular set of basis functions
are the Zernike polynomials. This set of mathematical functions are
formed as the product of two other functions, one of which depends
only on the radius r of a point in the pupil plane, and the
other depends only on the meridian 
of a point in the pupil plane. The former is a simple polynomial of
the nth degree and the latter function is a harmonic of a sinusoid or
co-sinusoid. A pictorial dictionary of the first 28 Zernike
polynomials is arranged in the form of a pyramid of basis functions
in Figure 5-13. Every aberration map can be
represented uniquely by a weighted sum of these functions. The
process of determining the weighting coefficient required to describe
a given aberration map is a least-squares curve-fitting process
called Zernike decomposition, which results in a vector of Zernike
coefficients. Mathematical details may be found in several standard
reference works (Born & Wolf, 1999; Malacara,
1992).
Figure 5-13. Pictorial dictionary of Zernike modes used to systematically represent the aberration structure of the eye.
Recently the Optical Society of America sponsored a taskforce of visual optics researchers to develop standards for reporting optical aberrations of eyes. Recommendations of this taskforce were presented at the 2000 topical meeting on Vision Science and Its Applications and will be published in full in a future issue of Trends in Optics and Photonics, published by the OSA (Thibos, Applegate, Schwiegerling, & Webb, 2000).
Klein and colleagues have suggested (Klein &
Garcia, 2000) using the slopes (first partial derivatives in
x- and y-directions) and curvature (average of second partial
derivatives in x and y) of the aberration map to supplement the
interpretation of the wavefront aberration function W(x,y). As
illustrated in Figure 5-14, the slope of
the wavefront aberration map may be interpreted as the transverse
ray aberration, which is defined by angle 
between the aberrated ray and the non-aberrated reference ray. The
associated focusing error is called the longitudinal ray
aberration and is equal to 1/z diopters, which may be computed as
the ratio of transverse aberration to ray height in the pupil
plane.
Figure 5-14. Relationship between wavefront, its first and second derivatives, and measures of transverse aberration (t) and longitudinal aberration (1/z).
To illustrate the derivatives of the aberration map we evaluated the wavefront error for the Indiana Eye, a simple reduced-eye model with aspheric refracting surface which has proved useful on previous occasions for studying the aberrations of the human eye (Thibos, Ye, Zhang, & Bradley, 1992). In the specific example illustrated in Figure 5-15, the pupil diameter was 6 mm and the conic constant of the surface was set to 0.6, a value which generates a degree of spherical aberration that is typical of human eyes (Thibos, Ye, Zhang, & Bradley, 1997). The wavefront aberration map W(x,y) for this model at the wavelength 589 nm, shown Figure 5-15A, is nearly flat in the middle of the pupil but becomes increasingly curved near the pupil margin.
In a symmetrical optical system, the transverse aberration
may be measured in any meridian, but for non-symmetrical systems it
is typically computed as the partial derivative of the aberration
function in the vertical as well as the horizontal directions. One
way to simultaneously visualize the variation of the vertical and
horizontal components of transverse aberration is with a vector field
as shown in Figure 5-15B. Each arrow
represents the transverse ray aberration for the pupil location
marked by the tail of the arrow. The lengths of the horizontal and
vertical components of each arrow give the horizontal and vertical
components of angle ,
respectively. The calibration arrow in the lower left corner is 1
milliradian in length. The arrows all point towards the center of the
pupil for this model eye, indicating that the light reflected out of
the eye forms a converging wavefront. The arrows are longer near the
pupil margin, indicating the steeper slope expected of spherical
aberration.
Figure 5-15. Four methods for displaying an aberration map. (A) height of reflected wavefront from the (x,y) plane perpendicular to the path of the chief ray (line-of-sight, for foveal vision). (B) vector field map showing the displacement of each spot image from the optical axis of the corresponding lenslet. (C) axial power map obtained by dividing average wavefront slope at each pupil location by the radial distance of the point from pupil center. (D) local curvature in the wavefront, obtained by applying the Laplacian operator to the wavefront. Maps in (C) and (D) are calibrated in diopters.
In a perfect optical system, every ray from a point object will intersect the retina at the same location but in an aberrated system the intersection will be displaced by an amount and direction indicated by the arrows in Figure 5-15B. Thus we can visualize where each ray strikes the retina by collapsing all of the arrow so that their tails coincide (the ideal image point) with the head of each arrow showing where the ray from the corresponding pupil location intersects the retina. In optical engineering such a visualization would be called a "spot diagram" and would be taken as a discrete approximation to the continuous PSF.
From the geometry of Figure 5-14 we noted that the ratio of wavefront slope to r, the distance from the pupil center to the given point on the wavefront, is the inverse of the distance between pupil center and the point where the aberrated ray crosses the optical axis. This latter quantity is the traditional definition of longitudinal ray aberration (a.k.a. axial power). In general the longitudinal aberration has horizontal and vertical components, just like the transverse aberration, which reflects the fact that the refracted ray may be skew to the optical axis. In practice we simplify the situation by resolving the transverse ray arrow into radial and tangential components and then using the radial component to compute the longitudinal aberration. This allows us to reduce a vector plot, as in Figure 5-15B, into a scalar plot as in Figure 5-15C.
If the eye suffers only from defocus, then the longitudinal aberration is constant and equal to the spherical refractive error of the eye. For other aberrations, such as coma or spherical aberration, the longitudinal aberration varies with pupil location and thus may be depicted with a longitudinal aberration map like that shown in Figure 5-15C. The scale of this map is calibrated in diopters to enhance its clinical interpretation. For this particular example, the increased power near the margin represents 1.7 diopters of spherical aberration, which is the same result obtained by finite ray tracing (Thibos et al., 1997).
The second derivative of the aberration function measures the rate of change of slope of the wavefront, i.e. local curvature. The average curvature in the horizontal and vertical directions is called the Laplacian of the aberration map. An example of the Laplacian curvature map for the Indiana Eye model is shown in Figure 5-15D. The scale of this map is also calibrated in diopters to enhance its clinical interpretation. Notice that the longitudinal aberration underestimates local curvature, which means that local segments of the wavefront come to focus before the rays intersect the optical axis. Some difference is to be expected because the traditional measure of longitudinal aberration is proportional to wavefront slope, whereas curvature is a measure of the rate of change of slope.
The point-spread function is computed as the squared magnitude of the Fourier transform of a complex-valued pupil function built from the aberration map. Since the PSF represents the intensity distribution of light in the image of a point source, it should be a highly localized, bright spot. Diffraction sets a lower limit to the diameter of the spot and an upper limit to the intensity in the center of the spot. A common metric of image quality called Strehl's Ratio is computed as the ratio of the actual intensity in the center of the spot to the maximum intensity of a diffraction-limited spot. As pupil diameter increases, the intensity of a diffraction-limited spot increases faster than the intensity of an aberrated spot, which tends to reduce Strehl's Ratio. It is not uncommon for the PSF of human eyes to have multiple peaks, which complicates the simple notion of Strehl's Ratio. More importantly, it signals the formation of two or more point images for a single point object. This condition of di- or poly-plopia has great clinical significance because of its implications for visual performance and the quality of visual experience (Woods, Bradley, & Atchison, 1996a; Woods, Bradley, & Atchison, 1996b).
Figure 5-16. Point-spread functions (left panels) and simulated retinal images of an eye chart (right panels) for the eye of Figure 5-10C, analyzed for two pupil diameters (2 mm, 6 mm).
Examples of PSFs computed from the wavefront aberration function displayed in Figure 5-10C are shown for small and large pupils in Figure 5-16. In the dark this patient's natural pupil diameter was 6 mm, which was large enough to expose significant amounts of aberration introduced by refractive surgery. The effect of these aberrations was to blur the central spot while simultaneously spreading some of the light into a long, secondary, wispy tail. Since each point of light in an object will produce this same type of pattern, the retinal image of an eye chart will be blurred and contain a second, lower-contrast ghost image which hampers legibility. This situation may be contrasted with the much improved image quality for daylight viewing through a 2mm pupil, which is small enough to exclude most of the aberrations of this patient's eye. Under these conditions the PSF is more compact, which results in a clearer, more focused retinal image.3
The optical transfer function (OTF), comprised of the modulation transfer function (MTF) and phase transfer function (PTF), is computed as the inverse Fourier transform of the PSF. The MTF component represents the contrast of the retinal image for a sinusoidal grating target of 100% contrast. Diffraction sets an upper limit to the contrast of the retinal image and therefore the ratio of MTFs for a real eye to a diffraction-limited optical system is a measure of the losses of contrast due to aberrations. One implication of the Fourier transform relationship between the OTF and the PSF is that Strehl's Ratio, defined as the ratio of intensities at the center of the test PSF and a diffraction-limited PSF, is equal to the volume under the OTF of an aberrated system divided by the volume under the OTF for the same system without aberrations. In general, the OTF, PSF, and MTF are 2-dimensional functions of spatial frequency and orientation, or equivalently, of spatial frequency in the x- and y-directions. Such functions may be reduced to 1-dimensional graphs by averaging across orientation, a process called radial averaging.
Figure 5-17. Radially-averaged modulation transfer functions for the eye of Figure 5-10C, analyzed for two pupil diameters (2mm, 6 mm).
Radially averaged MTFs for the LASIK patient of Figure 5-16 are shown in Figure 5-17 for small and large pupils. The transfer of contrast from object to image is from 3 to 5 times lower for the 6mm pupil condition compared to the 2mm condition over most of the visible range of spatial frequencies. It is to be expected that these optical losses would be reflected directly in visual performance measurements of contrast sensitivity for grating targets.
