Excimer photorefractive keratectomy can correct refractive error by using the ultraviolet radiation of the excimer laser to precisely remove tissue from the central portion of the cornea. It is primarily being used for the correction of myopia, while astigmatic and hyperopic procedures are being developed.
To flatten the corneal curvature, myopic correction requires the removal of more tissue centrally than peripherally. For a given refractive correction, the central depth of tissue removed increases as the diameter of the ablation is increased.1 Thus, there are physiological and anatomical constraints to be considered when determining the optimal size of the ablation diameter.
In addition to these medical factors, the instruments that are commercially available have limited ablation zones.
These procedural constraints limit the size of the ablation zone. When the ablation zone covers the entire entrance pupil, the retinal image is formed by a uniform optical system. However, if the ablation zone does not fully cover the entrance pupil, the image formed on the retina will be produced by a duplex optical system having in-focus and out-of-focus components. For an on-axis target and centered optical system, light passing through the entrance pupil's central ablated area is in-focus, and light passing through the annular untreated area, is out-of-focus.
The perceptual ramifications of the optical image formed by the resulting duplex optical system requires analysis to identify its impact on clinical tests, and provide for test interpretation consistent with accurate diagnoses of the visual complaints. .Also, conventional clinical tests can be evaluated for their sensitivity to the cause of subjective complaints that might anse from duplex imaging phenomena and the requirement for new tests identified. A glossary of some optical and visual science terms used in this article is provided in Table 1.
Four optical variables are evaluated: pupil size, ablation size, refractive error, and the Stiles Crawford effect. From these variables we derive the duplex point-spread function and, by convolution, use it to derive the line-spread function. By convolving the line-spread function with a luminosity step, we obtain the edge-spread function. The optical transfer function is obtained by convolving the linespread function with a sinusoidal light distribution.
In our computer model, we derive the duplex point-spread function by separating the out-of-focus component from the in-focus component, deriving the point-spread functions of each component separately, and then combining them according to the weighted areas of the ablated and unablated zones.
Our analysis is restricted to a centered on-axis optical system. We assume that both the ablated and unablated areas are spherical, ie have uniform power. The far point of the ablated area is assumed to be at infinity and the far point of the unablated area is assumed to be at the pretreatment locus.
Ablated Area Point-Spread Function
Point-spread functions for the human eye for in-focus images have been directly measured or derived from measurements of the eye's modulation transfer function by several investigators.2-5 To a first order approximation, most of these results are similar. They show that for pupils smaller than about 3.00 mm, the point-spread function is essentially diffraction limited. For larger pupils, optical aberrations produce point-spread functions that have a narrow cone surrounded by a broad skirt.
A series of point-spread functions for pupil sizes varying between 1.50 mm and 6.60 mm has been published by Gubisch.3 For pupils between 3.80 mm and 6.60 mm there is little change in the pointspread function. Since we are concerned with the effects of ablation zones that range between 3.00 mm and 6.00 mm, we can approximate the in-focus point-spread function with a single function. Since the annular blur has spatial dimensions that are large compared to the in-focus PSF, we can approximate Gubisch's 4.90-millimeter point-spread function, for computational convenience, by a stacked set of six concentric cylinders that vary in diameter and intensity as given in Table 2.
Unablated Area Point-Spread Function
Geometric optics give the size of the blur patch produced on the retina in the presence of a refractive error as being proportional to the refractive error and the pupillary diameter. For refractive errors greater than 0.75 diopters, measurements of the blurred line-spread function agree with the values calculated from geometrical optics.0 Accordingly, diffraction and third order aberrations may be neglected for images blurred more than 0.75 D, and the size of the blur patch may be calculated from just the refractive error.
In-Focus Component Specifications
To derive the dimensions of the out-of-focus point-spread component, we adapt Gullstrand's simplified schematic eye.7 We assume that Gullstrand's simplified schematic eye corresponds to emmetropic eye dimensions and that myopic eyes are longer in proportion to the amount of myopia.
The increase in axial length that corresponds to a given amount of myopia is calculated from Gullstrand's simplified schematic eye by keeping the optics constant and calculating the image distance for various far points. In so doing, an increase in axial length of 1.00 mm results in about 2.50 D of myopia.
In Gullstrand's simplified schematic eye, the entrance pupil is about 3.05 mm behind the cornea and the exit pupil is about 3.61 mm behind the cornea with the real pupil being located at 3.60 mm behind the cornea. The entrance pupil is about 13% larger than the pupil and the exit pupil is 3.6% larger than the pupil. Thus, the exit pupil equals 91.7% of the entrance pupil.7
Similarly, the size of the ablation zone at the exit pupil plane is approximated by multiplying the ablation zone by 91.7%, since the 3.05-millimeter separation between the entrance pupil and ablation is a small distance in object space.
Blur Patch Calculations
Distribution of Light Between Image Components
Light entering the entrance pupil is distributed between the in-focus and out-of-focus components according to the ratio between the ablated area, Aa, and nonablated area, Am.
Efficacy of Pupillary Entry Position
An adjustment factor for the Stiles-Crawford effect was developed, since, for photopic vision, this phenomenon will tend to reduce some of the influence of the out-of-focus component.
The primary attribute of the Stiles-Crawford effect we are concerned with is related to the relative distribution of light between the ablation zone and the surrounding cornea. Thus, adjustment for the Stiles-Crawford effect consists of reducing the relative quantity of light otherwise assigned to the unablated surrounding cornea.
Since we approximated the in-focus component with a stack of uniform disks, the in-focus linespread function consists of a cascade of similar functions.
The line-spread function is convolved with a step function to derive the edge-spread function, and with a repetitive sinusoid to obtain the optical tz*ansfer function at any particular spatial frequency.
Figure 1 demonstrates the nature of the pointspread, line-spread, edge-spread, and optical transfer functions when annular blur is present. Figure 1 shows the results for a 6.00-millimeter pupil, 4.00millimeter ablation zone, and -2 refractive error. The light distribution between the in-focus and out-of-focus components has not been adjusted for the Stiles-Crawford effect.
The center of Figure 1 shows two plots of the point-spread function.
Above, the concentric circle graph shows the relative spatial arrangements of the duplex components of the point-spread function. The region of annular blur is indicated by the outermost two circles and the stack of concentric luminosity cylinders of the infocus component by the smaller circles in its center.
Below, is the cross sectional intensity graph of the point-spread function.
When the amplitude of the annular surround is less than 1% of the center amplitude, its presence is masked by the resolution of the display. Here, the annular surround amplitude is only 0.30% of the center amplitude.
The abscissa scale is indicated by the calibration bar immediately below the abscissa, which subtends 44.18 min of arc in object space. Each tick mark represents 1/20 of the angle subtended by the outer boundary of the annular blur. The ordinate is a linear plot of the relative amplitude of the pointspread function as indicated by the right hand scale of the graph.
The half-height width of the point-spread function is constant, because the center, in-focus component, is unaffected by annular blur.
Figure 1: Point-spread, line-spread, edge-spread, and optical transfer functions (OTF) for a duplex image formed with a 6. 00-millimeter pupil, 4.00-millimeter ablation, - 2.00-diopter refractive error, and no Stiles-Crawford effect adjustment.
A cross sectional plot of the relative intensity of the line-spread function is shown in the upper right of Figure 1. Its abscissa and ordinate are the same as that of the intensity plot of the point-spread function.
Annular blur has only a small effect on the halfheight width of the line-spread function. Because the in-focus component forms a narrow cone and the out-of-focus component is typically shallow in its center. In Figure 1, the narrow cone of the in-focus line-spread function is seen as superimposed on the center plateau of the out-of-focus line-spread function. To the extent that the out-of-focus component raises the in-focus component of the line-spread function, the half-height width of the line-spread function increases relative to that obtained in the absence of blur. This amount is typically negligible.
However, the line-spread function has two secondary maxima at loci that coincide with the inner diameter of the blur annulus. Because of the convolution process, these secondary maxima can reach significant amplitudes. In Figure 1, they equal 8.68% of the center peak, even though the annular blur in the point-spread function was only 0.30%. The secondary maxima are smaller both with higher refractive errors and greater ablation area to pupillary area ratios.
The edge-spread function is shown in the lower right of Figure 1. Its abscissa and ordinate are the same as for the point- and line-spread functions.
The edge-spread function has a rapid central rise in intensity, which corresponds to the in-focus component. This central intensity jump is flanked by curvi-linear intensity ramps, which are formed by the out-of-focus component. For given pupil and ablation diameters, the slopes of the ramps decrease with an increase in refractive error and with StilesCrawford effect compensation.
Optical Transfer Function
The optical transfer function is shown in the lower left of Figure 1 on log-log coordinates. The duplex optics function is solved at four frequencies, 3, 6, 12, and 18 cpd, as indicated by the circles. A reference curve, the optical transfer function of the in-focus component only, is plotted above the duplex optics function.
Little frequency dependent variance is expected in the duplex optical transfer function, since the duplex line-spread function, which is used to derive the optica] transfer function, is broad relative to the period of the spatial frequencies of concern. For instance, the lowest frequency, ie, 3 cpd, has a period of 20 min of arc, compared to 22 min of arc for the diameter of the smallest possible blur circle of concern, ie, one that is obtained with a 3.1millimeter pupil, 3.0-millimeter ablation, and a refractive error of - 2.00 D.
The observed decrease in modulation is found to correspond closely to the distribution of light between the in-focus and out-of-focus components as defined by Eq 3. For example, if 10% of the incident light falls in the out-of-focus component, we find about a 10% decrease in modulation. Deviations from this first order approximation have been examined and found not to be clinically significant.
To evaluate the influence of pupil size and the Stiles-Crawford effect on image contrast, image contrast functions were graphed for ablation diameters between 3.00 mm and 6.00 mm with and without Stiles-Crawford effect adjustment. The results are shown in Figures 2 and 3 respectively. The abscissas are pupil size, and the ordinates are the relative modulation or contrast shift caused by the presence of the out-of-focus component. Where the pupil is smaller than the ablation, the image is fully modulated.
The Stiles-Crawford effect has a significant, but modest impact on the image contrast functions. The influence of the Stiles-Crawford effect increases with ablation zone size. The Stiles-Crawford effect adjusted functions have a lower slope where they leave the 100% level and a higher asymptotic value for large pupils.
The half-height width of measured in-focus pointspread functions compares well with Snellen visual acuity. Thus, a good predictor of this visual acuity test is the half-height width of the point-spread function.13
In normal subjects, Snellen visual acuity is not expected to be significantly affected by the presence of annular blur, the effect of which is only to reduce the contrast of the test letters. This follows from the annular out-of-focus component of the point-spread function not influencing the distribution of light in the in-focus component. Even in the line-spread function the half-height width is little affected by the out-of-focus component.
However, for patients suffering from reduced image contrast due to cataracts or vitreal clouding, the additional reduction in contrast produced by the out-of-focus annulus may have significant compounding effects on Snellen acuity. These factors should be considered even for young patients, since they may acquire these conditions as they age. Conversely, the decrease in pupil size associated with aging makes it less likely that the older patient will experience a duplex image.
Whenever annular blur is present, halos can be expected. They will be most pronounced at night due to the combined effects of a large pupil, rod function, and the high contrast of isolated lights, such as street lights, viewed against a dark background.
Rod photoreceptors function as almost perfect photodetectors, being able to detect a single photon of light. Therefore, even traces of light surrounding the in-focus image will be perceptually significant under scotopic conditions.
When viewing isolated lights, halos are more likely to be apparent with lower rather than higher refractive errors, because the light is more concentrated at lower refractive errors than at higher refractive errors. For example, with an 8.00-millimeter pupil and 5.00-millimeter ablation, the amplitude of the annular blur with 4.00 D of blur equals 0.05% of the in-focus component, but with 2.00 D the blur equals 0.19% of the in-focus component.
Neural spatial summation may mitigate some of the optical differences between low and high refractive errors. Area-intensity reciprocity extends to about 30 min of arc for uniform circular fields viewed under photopic conditions, and to about 10° for uniform circular fields viewed under scotopic conditions.10 The significance of these mitigating factors will depend upon the state of light adaptation, the area of the retina upon which the image falls, and the absolute size of the annular surround and its relationship to the central peak.
Optical Ghost Images
The secondary maxima produced in the linespread function, if high enough, could manifest perceptually as a pair of faded lines paralleling a bright primary image. As shown in Figure 1, these secondary maxima are much higher than the related annular surround in the point-spread function. For the conditions of Figure 1, ie, a 6.00-millimeter pupil, 4. 00-millimeter ablation, and 2.00-diopter refractive error, the line-spread secondary maxima equal 8.68% of the intensity of the in-focus image.
Figure 2: Relative image contrast versus pupil diameter for ablation diameters between 3.00 and 6.00 mm in 0.50millimeter steps. The Stiles-Crawford effect has not been taken into consideration.
Examples of conditions in which ghost images may be experienced include the viewing of neon signs, especially at night, and watching lines on highways, especially white lines on black road surfaces.
Neural Ghost Images
The curvi-linear luminosity ramps observed in the duplex edge-spread function may lead to Mach band type illusions, since this perceptual phenomenon has been experimentally associated with blurred images.11 For example, ghost images could occur with the edge-spread function of Figure 1 at locations X1 and X2.
Watching television in a dark room may be particularly annoying because of the illumination step at the edge of the screen combining with large pupils to produce Mach band illusions around the edge of the screen. Because Mach band illusions occur under a wide range of photopic viewing conditions, they are not only likely to be annoying, but perhaps debilitating to individuals in certain occupational groups such as artists, photographers, and radiologists.
If the proportion of light contributing to the out-of-focus component is sufficiently small or disburse, as with high refractive errors, those aspects of the duplex image giving rise to ghost images are diminished, and the primary visual complaint may be restricted to just haze or fog, because of the overall decrease in contrast. For example, if pupil size, ablation size, and Stiles-Crawford effect adjustment are the same as in Figure 1, but the refractive error is - 6.00 D, instead of - 2.00 D, the secondary line-spread maxima are just one third of the height found with -2.00 D. Also, the edge-spread curvilinear intensity ramps are distributed over 66 min of arc, versus 22 min of arc, thereby reducing the robustness of Mach band like phenomena.
Contrast Sensitivity Testing
Contrast sensitivity testing should detect the reduction in modulation observed in the optical transfer function. Therefore, an overall decrease in a patient's contrast sensitivity function may provide a measure of the distribution of light between the ablated and untreated corneal areas for the test conditions.
Figure 3: Relative image contrast versus pupil diameter tor ablation diameters between 3.00 and 6.00 mm in 0.50millimeter steps. The ablation and pupillary areas have been adjusted for the Stiles-Crawford effect.
However, since the changes induced in the optical transfer function are not frequency dependent, the contrast sensitivity function would not provide insight into the spatial distribution of the annular blur. Accordingly, contrast sensitivity testing will not be able to predict the occurrence of ghost images.
Our analysis is restricted to on-axis viewing with a coaxial optical system. If the object were off-axis or the ablation were not centered on the entrance pupil, asymmetries in the point-spread function would be generated that would shift the location of the artifactual image components as the object of regard moved in the visual field.12 These effects would aggravate any symptoms related to the visual artifacts caused by a static duplex image.
We have assumed uniform power within the ablated and unablated areas. Actually, these conditions are not likely to exist. Nonuniform power in the ablation zone will tend to aggravate the influence of the contrast reducing annular blur, since the infocus image will itself will be lower in contrast. Nonuniform power in the surrounding unablated cornea is not likely to affect the conclusions of our analysis, since the primary geometric optical blur will remain paramount.
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In-Focus Component Specifications