# Relationship between Refractive Error and Monochromatic Aberrations of the Eye

Purpose. To examine the relationship between ametropia and optical aberrations in a population of 200 normal human eyes with refractive errors spanning the range from +5.00 to −10.00 D.

Methods. Using a reduced-eye model of ametropia, we tested the hypothesis that the optical system of the eye is uncorrelated with the degree of ametropia. These predictions were evaluated experimentally with a Shack-Hartmann aberrometer that measured the monochromatic aberrations across the central 6 mm of the dilated pupil in well-corrected, cyclopleged eyes.

Results. Optical theory predicted, and control experiments on a model eye verified, that Shack-Hartmann measurements of spherical aberration will vary with axial elongation of the eye even if the dioptric components of the eye are fixed. Contrary to these predictions, spherical aberration was not significantly different from emmetropic eyes. Root mean square of third-order aberrations, fourth-order aberrations, and total higher aberrations (third to 10th) in myopic and hyperopic eyes were also uncorrelated with refractive error. Astigmatic eyes tended to have larger total higher-order aberrations than nonastigmatic eyes.

Conclusions. We conclude that a reduced-eye model of myopia assuming fixed optical parameters and variable axial length is not tenable.

School of Optometry, Indiana University, Bloomington, Indiana

Received May 6, 2002;

revision received August 14, 2002.

**Xu Cheng**

*School of Optometry*

*Indiana University*

*800 E. Atwater Ave*.

*Bloomington, IN 47405*

*e-mail:*xcheng@indiana.edu

Although it is widely accepted that spherical refractive error is due primarily to inappropriate axial length caused by a vitreous chamber that is either too long or too short, ^{1–5} there is some controversial evidence that the optical components may also vary with refractive error. For example, of six studies ^{1, 3, 6–9} examining the relationship between refractive error and corneal radius, only three ^{1, 3, 7} found a significant correlation. The possible role of higher-order aberrations in this controversy is equally unclear. Applegate, ^{10} using a subjective single-pass aberroscope, found dramatically increased coma and spherical aberration in some myopic eyes and reported a mean increase in aberrations in myopes. To the contrary, Collins et al., ^{11} using an objective double-pass aberroscope, reported lower average spherical aberrations in high myopes than emmetropes. However, in the study by Collins et al., at least one third of the myopic eyes had aberrations that were so large that no grid image was observable. Thus, although each study came to opposite conclusions, both the Applegate and Collins et al. studies found high variability in monochromatic aberrations in myopic eyes, with some myopes having significantly elevated monochromatic aberrations.

The recent introduction of Shack-Hartmann wavefront sensing and other ray-tracing technologies has enabled accurate, objective and subjective measurement of higher-order monochromatic aberrations even in highly aberrated eyes. ^{12–16} Several studies have used these newer technologies to examine the relationship between monochromatic aberrations and myopia. ^{17–19} All of these studies reported that myopia is accompanied by slight increases in higher-order aberrations that were either statistically insignificant or barely significant. For example, Paquin et al., ^{19} using a Shack-Hartmann aberrometer, found that optical quality was worse in myopic eyes and coma was more frequent in high myopia. He et al., ^{18} who used a subjective ray-tracing technique, measured aberrations in 146 young adults and found that myopes on average have slightly larger combined fourth-order and higher aberrations than emmetropes (p < 0.01). However, they failed to find a significant correlation between total aberrations and spherical equivalent refractive error. Using an objective ray-tracing method, Marcos et al. ^{17} reported significantly increased third-order aberrations in young myopes (r = 0.82, p < 0.0001). Fourth- and higher-order aberrations were also found to increase with myopia (0.00 to −7.00 D), although not significantly (r = 0.33, p = 0.15; r = 0.32, p = 0.16, respectively).

One of the limitations of prior studies ^{17–19} was the lack of consideration given to a potential misinterpretation of aberrometry results when axial myopia is corrected with trial lenses or by adjusting the reference sphere of the aberrometer. Consider two eyes with different axial lengths but the same dioptric apparatus, as illustrated in Fig. 1 A. Suppose that the dioptric apparatus suffers only from positive spherical aberration, which causes marginal rays to be refracted more than is needed to intersect the paraxial rays at the retinal plane. In the emmetropic eye, the degree of spherical aberration is measured in image space by angle θ between the actual ray and a reference ray that intersects the paraxial rays at the retina. In the myopic eye, the total aberration is measured by angle β, which is the sum of a defocus aberration δ and the original spherical aberration θ. In practice, δ can be substantially larger than θ (e.g., several diopters of myopia vs. ∼1.00 D of spherical aberration), which could make it difficult to measure θ because it is swamped by δ. Experimentally, this measurement problem can be overcome by placing the source conjugate to the retina (i.e., at the eye’s far point), thus nullifying δ and exposing θ, as illustrated in Fig. 1 B. The penalty for adopting this strategy is that it increases the angle of incidence φ, which pushes the eye further into the nonlinear domain of Snell’s law. Snell’s law is a compressive nonlinearity that in this example causes the angle of refraction φ’ to be less than that required for perfect imaging with a spherical surface, thus producing positive spherical aberration. Consequently, the increased angle of incidence caused by correcting axial myopia should lead to an increased amount of positive spherical aberration relative to the emmetropic eye, even though the dioptric apparatus of the two eyes is the same. Conversely, the decreased angles of incidence caused by correcting axial hyperopia will decrease the amount of positive spherical aberration. Similarly, if the emmetropic eye in Fig. 1 B happens to have negative spherical aberration, then correcting axial myopia will cause spherical aberration to change in a positive direction, and correcting axial hyperopia will cause spherical aberration to change in a negative direction. Furthermore, these effects of axial length on spherical aberration cannot be avoided with the objective Shack-Hartmann aberrometer because it performs measurements in object space using light reflected from the retina. If the retina moves axially, then the angles of incidence of reflected rays will change and, therefore, the amount of spherical aberration will change.

The goal of our study was to reexamine the relationship between refractive error and aberrations in a large sample of young adults while taking into account the influence of axial elongation on aberration measurements described above. In short, we aimed to test the null hypothesis that the optical apparatus of the eye is uncorrelated with the degree of ametropia. Paradoxically, this null hypothesis of fixed optical properties leads to the prediction that aberrations will vary with the degree of ametropia when that ametropia is optically corrected.

## METHODS

Aberrometry was performed as part of the Indiana Aberration Study, a large-scale investigation of ocular aberrations and their effect on vision. ^{20} Aberrations of the whole eye were measured with a Shack-Hartmann wavefront analyzer, which is a single pass, objective technique that is more robust to optical defects of the eye than is the Howland aberroscope. ^{21} We measured the aberrations of 200 healthy eyes of 100 young adult subjects with a wide range of refractive errors. By limiting the age distribution we sought to minimize the effect of age on our aberration measurements. ^{22}

### Apparatus

The apparatus and principles of the Shack-Hartmann setup are described elsewhere. ^{13, 23} In short, a He-Ne laser (wavelength, 632.8 nm) was used to create a point source on the retina. Light reflected from this source emerges from the eye as a distorted wavefront. That wavefront is subdivided by a lenslet array (lenslet diameter = 0.4 mm) that is conjugate to the pupil plane of the eye to produce an array of spot images focused onto a CCD video sensor. The displacement of each spot relative to the optical axis of the lenslet that produced the spot is a measure of the local slope of the wavefront in x and y directions. We fit the original slope data to the derivatives of Zernike circle polynomials up to the 10th order to reconstruct the shape of the aberrated wavefront.

### Subjects

We measured the aberrations of 200 eyes of 100 subjects. Accommodation was paralyzed and pupils dilated (mean diameter = 7.58 mm) with the administration of cyclopentalate (0.5%, 1 drop). Most subjects were optometry students at Indiana University. No eyes had pathology. The average age of the subjects was 26.1 ± 5.6 years. The spherical component of refractive error determined from subjective refraction ranged from +5.25 to −9.50 D, and astigmatism ranged from 0 to −3.50 D. For present purposes, we categorized the degree of refractive error into six groups according to the mean sphere equivalence of cyclopleged subjective refraction (Table 1).

Subjects were required to have at least one day free of contact lenses before the Shack-Hartmann test to avoid the possible effects of contact lenses on cornea shape. Spherocylindrical refractive errors were corrected with trial lenses during aberrometry to ensure good quality data images dominated by higher-order aberrations. We aligned the spectacle lenses with the center of the pupil using trial frame adjustments. Three Shack-Hartmann images were taken for each eye.

### Data Analysis

Analysis was based on a 6-mm-diameter entrance pupil. From the fitted Zernike coefficients (averaged from three trials) we calculated total RMS wavefront error (the square root of the sum of squared Zernike coefficients, excluding first- and second-order aberrations). We also computed the RMS error in third- and fourth-order aberrations that include coma and spherical aberrations, respectively. The two eyes of the same subject were treated separately. The distributions of these various measures across refractive error groups were evaluated statistically with linear regression and one-way analysis of variance (p < 0.05 significance level).

## RESULTS

The average standard deviation of the three repeated measurements for N = 200 eyes was 0.041 μm (0.065 wavelengths at 632.8 nm). This between-trial variability was small compared with between-subject variability and did not contribute significantly to the total variability in the data set (one-way analysis of variance, F_{2, 597} = 0.909, p = 0.404).

The relationships between the magnitude of aberrations (root mean square variance, or RMS of the wavefront error) and spherical refractive error are shown in Fig. 2 for our study population. In each graph, the slope of the best-fitting straight line is approximately zero. Linear regression analysis showed that none of the four slopes were significantly different from zero (p > 0.05). A similar result was found when we analyzed the high astigmatism (> −1.00 D) and low astigmatism (≤ −1.00 D) groups separately (Fig. 2). Regression slopes were almost identical when the hyperopic eyes were excluded from the analysis (m = −0.0002, −0.004, −0.003, and −0.004 for total third, total fourth, spherical aberration, and total higher-order aberrations, respectively), and none differed significantly from zero. As may be seen from Fig. 2 and Table 2, the aberration distributions within each group were highly overlapping. One-way analysis of variance showed no statistical difference in total third-order (p = 0.632), total fourth-order (p = 0.06), spherical aberration (p = 0.504), and total higher-order aberrations (p = 0.118) among the five low-astigmatic and nonastigmatic groups. The only fourth-order mode that differed significantly among refractive error groups (p < 0.05) was *Z*^{+4}_{4}. Although statistically significant, the magnitude of this coefficient was typically very small (mean RMS, 0.009 μm). There was a slight tendency for the highly astigmatic eyes (astigmatism > −1.00 D) to have higher levels of aberrations than seen in the low-astigmatic and nonastigmatic eyes. Independent t-test showed that this difference in total higher-order aberrations was significant (p < 0.05).

### Control for Aberrations of Spectacle Lenses

The results presented above were obtained on eyes rendered emmetropic by trial lenses. Because the power of these trial lenses varies with the degree of myopia in the subject’s eye, it is possible that aberrations of the trial lenses affected our results. To evaluate this potential artifact, we used a commercial ray-tracing program (OSLO) to compute the spherical aberration of a −7.50 D trial lens (the average power used to correct the highly myopic group of subjects) for a 6-mm pupil. The result is compared in Table 3 with the amount of spherical aberration measured experimentally in the highly myopic eyes and in emmetropic eyes. The amount of spherical aberration introduced by the lens is <3% of that measured in myopic eyes, and of opposite sign. From this analysis, we conclude that the negative trial lenses produced negative spherical aberration, which tended to cancel a very small proportion of the positive spherical aberration of the eye. However, the magnitude of this compensation is so small that it could not have been responsible for the nearly zero slope of the regression line in Fig. 2 D.

A similar analysis was performed for coma under the assumption that the trial lenses may have been decentered inadvertently on the patient’s eye. To model the worst-possible case, we assumed that the lens was decentered by 3 mm. The results, shown in Table 3, indicate that coma introduced by decentering the trial lens is also very small, approximately 4% of that observed in myopic eyes. Again, we conclude that the trial lenses did not introduce significant amounts of aberration in our experiment.

### Prediction of the Effects of Axial Elongation

We argued on theoretical grounds in the introduction that spherical aberration should vary with degree of axial ametropia when the ametropia is optically corrected. For the case of an emmetropic eye (Fig. 3 A), the aberration of a marginal ray is specified by the angle θ between the ideal ray and the aberrated ray. Alternatively, the aberration may be specified as a wavefront error that is equal to the difference in optical path length from a distant point source to the paraxial image point along two different paths. The first path (abc) is through a marginal pupil position, and the second path (adc) is through the pupil center. For the case of an ametropic eye (Fig. 3 B), the spectacle lens alters the course of incident rays from a distant point source such that the rays appear to emerge from a virtual object located at the eye’s far point, a. By moving the source point closer to the eye, the difference between optical paths (abc) and (adc) is altered and so is the ray aberration β.

To quantify the expected change in spherical aberration caused by axial myopia, we computed the wavefront aberration of the Indiana reduced-eye model ^{24} for a source located at the eye’s far point using the method of optical path differences. ^{25} This model was configured with three different levels of asphericity (conic constant values p = 0.4375, 0.5, and 0.6) to vary the amount of spherical aberration in the refracting surface. For each of these three surfaces, we varied the axial length of the model to produce various amounts of axial myopia. The computed results, shown in Fig. 4, indicated that the magnitude of the Zernike coefficient for spherical aberration varies nearly linearly with refractive error. The slope of this linear relationship is approximately the same for all three conic constants, but the lines are displaced vertically more for the larger values of p. This means that the predicted rate of change of spherical aberration with myopia is largely independent of the amount of aberration in emmetropic eyes for the reduced-eye model. Note that the magnitude of change is significant compared with the range of Zernike coefficients in human eyes (Fig. 2 D).

To empirically verify the computer modeling results of Fig. 4, we built a plastic model eye with a single ellipsoidal refracting surface (N = 1.49, r = 7.8 mm, p = 0.5) and variable axial length sufficient to induce up to 4.00 D of myopia. We corrected the model’s axial myopia with a spectacle lens and then measured its aberrations with the same Shack-Hartmann aberrometer used to measure human eyes. These results, shown by the symbols in Fig. 4, agree closely with predictions from the reduce-eye model (dashed curve in Fig. 4) when configured with a 7.8-mm apical radius of curvature (p = 0.5) and refractive index 1.49. The difference in slope of the solid and dashed curves is due primarily to the difference in apical radius of curvature of the two models. We conclude from Fig. 4 that a single-surface eye with fixed optical parameters should exhibit a significant increase in spherical aberration as the axial length of the eye increases.

In Fig. 5, we compare model predictions with the human spherical aberration data (Z^{0}_{4}). Predictions are shown for two versions of a 60.00 D reduced-eye model. Version 1 had a physiological value of refractive index (n = 1.333) but a nonphysiological value of apical radius of curvature (r = 5.55 mm). Version 2 had a nonphysiological value of refractive index (n = 1.468) but a physiological value of apical radius of curvature (r = 7.8 mm). Asphericity of the models were chosen so that when emmetropic, they had the same amount of spherical aberration as the average emmetropic human eye (p = 0.5 for version 1, p = 0.65 for version 2). Thus, under our null hypothesis that the dioptric apparatus of eyes is independent of the degree of myopia, the human data should have followed the trends indicated by the model predictions. In fact, the data for most myopic eyes fall below the model predictions, which we interpret as evidence against our null hypothesis.

## DISCUSSION

Like others before us, we are struck by the large amount of individual variability in the magnitude of higher-order aberrations. Inspection of Fig. 2 suggests that little, if any, of this variability between subjects is correlated with refractive error. Regardless of whether we characterize the eye in terms of total higher-order RMS error (Fig. 2 A) or concentrate on specific Zernike orders (Fig. 2 B and C) or a specific Zernike mode (Fig. 2 D), there is little evidence from our study to suggest that aberrations vary systematically with degree of ametropia. These general impressions were confirmed by statistical tests of correlation and analysis of variance. The only distinction that emerged from that analysis was that astigmatic eyes tended to have slightly larger total higher-order RMS error. Given this large degree of individual variability, it is not surprising that prior studies have reached conflicting conclusions, especially where small sample sizes were involved. It is important to realize that by including both the right and left eyes of our subjects, we effectively exaggerated the power of our statistical tests because the aberrations of the right and left eyes exhibit some degree of correlation. ^{20, 26} Consequently, performing the same analysis separately on the right and left eye data also failed to find any significant correlation between refractive error and higher-order aberrations.

The reader may have thought that the null hypothesis presented in the Introduction that the optical apparatus of the eye is uncorrelated with the degree of ametropia is supported by the lack of obvious trends in the data of Fig. 2. However, before we could conduct a rigorous test of this hypothesis, we needed to specify the optical characteristics that we are supposing are shared by all eyes. For this purpose, we chose a reduced-eye model because of its simplicity and proven ability to model the spherical aberration of human eyes. Optical analysis of this model indicates that when axial refractive errors are corrected with spectacle lenses, as we did in our experiments, then the measured amount of spherical aberration should have increased systematically with increasing amounts of myopia. Thus, our null hypothesis, as embodied in the reduced-eye model, predicts a regression line with negative slope as shown in Fig. 5. That prediction was not fulfilled experimentally, so we are obliged to reject the hypothesis that an optically fixed, reduced-eye model with variable axial length accounts for our results.

We can imagine at least two reasons why the null hypothesis was rejected. First, perhaps the hypothesis deserves to be rejected because the optical components of myopic eyes are systematically different from those of emmetropic eyes. To be consistent with predictions from the reduced eye, the positive spherical aberration typically found in human eyes would need to be smaller in uncorrected myopes than in emmetropes. This reduction in aberration would then compensate for the increased aberration caused by correcting the eye’s myopia with spectacle lenses, leaving the corrected eye with no net change. The required decrease in spherical aberration of the myopic optical system could be achieved either by a decrease in the positive spherical aberration of the cornea or by an increase in the negative spherical aberration of the crystalline lens. The available evidence discounts the first of these two possible mechanisms. Two previous studies that examined the relationship between corneal shape and myopia ^{7, 9} both showed a small reduction in corneal peripheral flattering with increased myopia. Because peripheral corneal flattening is a major factor reducing ocular spherical aberration, these two studies suggest that myopic corneas should exhibit increased spherical aberration. Thus, a net reduction in spherical aberration in the myopic eye would require an even larger increase in negative spherical aberration of the lens. A similar argument was recently made by He and Gwiazda. ^{27}

The other possibility is that rejecting the hypothesis was a mistake. If all eyes really are the same but the reduced-eye model misrepresents how the aberrations of eyes vary with axial myopia, then the model would make false predictions that lead to false rejection. If so, then a more sophisticated model, perhaps incorporating a crystalline lens with gradient refractive index profile, might make predictions that are consistent with the experimental data. Evaluation of this possibility will require additional modeling efforts in the future.

Ideally, the question of variation of aberrations with degree of myopia would be addressed by a longitudinal study, rather than a cross-sectional study like ours, to factor out individual variation. It may also be advantageous to measure ocular aberrations in image space rather than object space. The Shack-Hartmann aberrometer measures aberrations of the eye in object space by capturing the reflected light from the retina. To avoid the increased angles of incidence and refraction of the myopic eye depicted in Fig. 1 B would require that somehow the source of light be located in the vitreous chamber at a point optically conjugate to infinity. Instead, objective ^{17} or subjective ^{14} ray-tracing techniques that measure aberrations in the eye’s image space may avoid the additional aberrations associated with changes in the eye’s far point. However, this potential advantage has not yet been realized. Prior ray-tracing studies corrected myopia either with a Badal system ^{18} or with trial lenses. ^{17} Thus, new experimental designs will be needed to avoid the complication of increased angles of incidence in axially myopic eyes.

Experiments on animal emmetropization show that the image quality on the retina has a large impact on how the eye grows. ^{28–30} Similar results have been observed in humans, and they raise the issue of whether normal myopia is also somehow induced by poor image quality. ^{31–37} Our evidence indicates that myopic eyes do not have significantly different amounts of monochromatic aberrations compared with emmetropes. Our observations, therefore, provide no evidence to support the idea that higher-order monochromatic aberrations in humans can have a large impact on the emmetropization process.

## ACKNOWLEDGMENTS

We acknowledge the essential contributions made by the team of investigators who helped conduct the Indiana Aberration Study. This study group includes Prof. D. Miller; research optometrists C. Riley and N. Himebaugh; optometry students E. Agapios, T. Cao, S. Kaluf, J. McKenna, M. Price, P. Quach, J. Rajasansi, J. Tsai, and K. Vicari; and technical staff D. Carter, L. Wagoner, and K. Haggerty. Financial support was provided by National Eye Institute, National Institutes of Health grant R01-EY05109 to LNT and National Eye Institute, National Institutes of Health STTR grant 1R41EY12754-01 to Quarrymen Optical.

## REFERENCES

**Keywords:**

wavefront aberrations; myopia; refractive error