1.

Introduction

In recent years, there have been many studies of higher order aberrations associated with foveal vision,¹ but only a few studies of these aberrations are in the periphery. Navarro, Moreno, and Dorronsoro² used a laser ray-tracing technique to measure aberrations at four positions in the nasal visual field, with aberration coefficients presented for only one of four subjects. The rate of change with the angle of the root mean square (rms) higher order aberration of the group was linear. Guirao and Artal³ measured double-pass retinal images with unequal entrance and exit pupil to estimate off-axis astigmatism and defocus (both second-order aberrations) and horizontal coma at three positions in the nasal visual field out to $45\mathrm{deg}$ . The estimates for coma indicated a linear rate of change with visual field angle. Atchison and Scott⁴ used the Hartmann-Shack technique in $5\text{‐}\mathrm{deg}$ steps out to $\pm 40\mathrm{deg}$ along the horizontal visual field, but analyzed the higher order aberrations only in terms of rms aberrations in each order. Each subject showed a linear rate of change of third-order rms aberration, but the rates were different between subjects. Ma, Atchison, and Charman⁵ did a similar study with two patients who had undergone myopic LASIK surgery, but again showed only the rms aberrations in different orders. Atchison ⁶ measured aberrations up to the fourth order across the central $10\mathrm{deg}$ of the horizontal visual field, finding that horizontal coma changed linearly across this region but that there were negligible changes in other higher order aberrations.

Because individual aberrations are of interest in themselves, we have re-evaluated Atchison and Scott’s and Ma, Atchison, and Charman’s data^{4, 5} to investigate how the coefficients of these aberrations change across the central $80\text{‐}\mathrm{deg}$ horizontal visual field. This will enable appreciation of how refractive surgery affects off-axis as well as on-axis aberrations.

2.

Methods

Abbreviated descriptions of the methods^{4, 5} are given here. Untreated right eyes were used for five normal subjects with spherical equivalent refractions between $+1.50$ and $-2.00$ diopters of sphere (DS) and with on-axis astigmatism $⩽0.75\mathrm{D}$ . The apparatus was a laboratory Hartmann-Shack wavefront system.^{6, 7} Pupils were dilated with 1% cyclopentolate. Subjects’ heads were mounted in a movable bite bar and alignment was maintained with the help of a videocamera. Subjects rotated their eyes to view a set of fixation targets viewed through a pellicle beamsplitter placed closed to the eyes. One to four (usually three) images were taken for each fixation target, corresponding to visual field positions out to $40\text{‐}\mathrm{deg}$ temporal∕nasal in $5\text{‐}\mathrm{deg}$ steps, except that no measurements were made at $\pm 35\mathrm{deg}$ for three subjects. For two subjects, the images corresponding to $40\text{‐}\mathrm{deg}$ nasal were too poor to be processed.

Right eyes were used of two subjects a few years after conventional LASIK (Nidek EC-5000 excimer laser with a treated zone of $\approx 5.5\mathrm{mm}$ and a transition zone out to $\approx 7.5\mathrm{mm}$ ). Measurements were made with a complete ophthalmic analysis system (COAS)∕WASCA instrument (Wavefront Sciences, Asclepion). Pupils were dilated with 1% cyclopentolate. Subjects were mounted in a chinrest and headrest assembly. A similar system for varying fixation angle was used as for the untreated eyes.

For all subjects, wave aberration polynomials were analyzed for $6\text{‐}\mathrm{mm}$ -diam pupils up to the sixth Zernike aberration order using the OSA ANSI system,^{7, 8} taking into account the changing shapes of pupils and different sampling across them in horizontal and vertical directions as eccentricity increased.⁴ For one subject, analysis was performed also for a $3\text{‐}\mathrm{mm}$ -diam pupil.

3.

Results

Figures 1a, 1b, 1c show third- to fifth-order Zernike aberration coefficients of untreated subject DAA with $6\text{‐}\mathrm{mm}$ pupils. Note the different vertical scales used for the different orders. For third-order aberrations, the most noted features are the high magnitudes of horizontal coma $c_3^1$ and its linear rate of change with visual field position [Fig. 1a]. For fourth-order coefficients, both spherical aberration $c_4^0$ and secondary astigmatism $c_4^2$ show quadratic dependences on visual field position [Fig. 1b]. There are no particular patterns for the fifth-order coefficients [Fig. 1c].

Fig. 1

Zernike aberration coefficients for untreated subject DAA as a function of horizontal visual field position in the (a) third, (b) fourth, and (c) fifth orders for a $6\text{‐}\mathrm{mm}$ entrance pupil. For clarity, small horizontal offsets have been made to the data. Errors bars show ranges of two to three measurements for coefficients $c_3^1$ , $c_4^0$ , and $c_5^1$ . Note the differences in vertical scale between (a), (b), and (c).

While there are patterns for combinations of many coefficients and subjects, only three coefficients have similar patterns for nearly all untreated subjects. Figures 2a, 2b, 2c show $c_3^1$ , $c_4^0$ , and $c_4^2$ for the other four untreated subjects with $6\text{‐}\mathrm{mm}$ pupils. As for subject DAA, the most pronounced feature is the linear pattern for $c_3^1$ [Fig. 2a], although this is less pronounced for CSA than for the other subjects. This has the greatest magnitudes of any of the higher order aberration coefficients, reaching absolute values of $1.5\mu \mathrm{m}$ . Slopes vary considerably between the subjects from $-0.021\text{to}-0.032\mu \mathrm{m}∕\mathrm{deg}$ with $R^2$ values of 0.98 ( $4∕5$ subjects) and 0.81 (CSA). $c_4^0$ has peaks with positive values near the center of the visual field for four out of five subjects [Figs. 1b and 2b] and the higher the peak, generally the greater the rate at which the coefficient changes away from the peak. $c_4^2$ shows similar patterns to $c_4^0$ for three out of five subjects (DAA, DHS, CSA), except that rates of change are smaller and in one case, the peak has a negative value [Figs. 1c and 2c].

Fig. 2

Zernike aberration coefficients for four untreated subjects (other than DAA) as a function of horizontal visual field position for a $6\text{‐}\mathrm{mm}$ entrance pupil: (a) horizontal coma, (b) spherical aberration, and (c) secondary astigmatism. For clarity, small horizontal offsets have been made to the data. Note the difference in vertical scale between (a) and (b).

Figures 3a and 3b show third- and fourth-order Zernike aberration coefficients of subject DAA with a $3\text{‐}\mathrm{mm}$ pupil. The coefficients are much smaller than those for a $6\text{‐}\mathrm{mm}$ pupil (Fig. 1). The linear relationship between horizontal coma and visual field position is not as strong as for the large pupil [compare Fig. 3a with Fig. 1a], and other patterns occurring for the large pupil are not apparent.

Fig. 3

Zernike aberration coefficients for untreated subject DAA as a function of horizontal visual field position in the (a) third and (b) fourth orders for a $3\text{‐}\mathrm{mm}$ entrance pupil. Errors bars show ranges of two to three measurements for coefficients $c_3^1$ , $c_4^0$ , and $c_5^1$ . Note the difference in vertical scale between (a) and (b).

Figures 4 and 5 show third- and fourth-order aberration coefficients for two myopic LASIK eyes (symbols joined by solid lines). There are also some theoretical values for $c_3^1$ and $c_4^0$ (diamonds joined by dots) that are mentioned in the discussion in Sec. 4. The presurgical refractions of the eyes were $-8.50\mathrm{DS}∕-1.00\mathrm{DC}\times 85$ (Fig. 4) and $-5.25\mathrm{DS}$ (Fig. 5). As for the untreated eyes, there are obvious patterns for $c_3^1$ , $c_4^0$ , and $c_4^2$ . The shapes for $c_3^1$ are very dissimilar from those for the untreated eyes, being linear but with the opposite slope sign between approximately $25\text{‐}\mathrm{deg}$ temporal and $35\text{‐}\mathrm{deg}$ nasal for subject SR-H [Fig. 4a] and between approximately $20\text{‐}\mathrm{deg}$ temporal and $20\text{‐}\mathrm{deg}$ nasal for subject AW [Fig. 5a], beyond which the slopes changes sign and the aberration coefficients reduce in absolute magnitude. The patterns for $c_4^0$ and $c_4^2$ are similar to those for some of the untreated subjects [Figs. 1b and 2c], but with the patterns being much steeper than for the untreated subjects. Note the high levels of $c_4^0$ in the center of the visual field (0.8 and $0.4\mu \mathrm{m}$ ), which are typical for conventionally treated myopic LASIK eyes.

Fig. 4

Zernike aberration coefficients for myopic LASIK subject S-RH as a function of horizontal visual field position in (a) third and (b) fourth orders for a $6\text{‐}\mathrm{mm}$ entrance pupil (symbols joined by solid lines). Zernike aberration coefficients $c_3^1$ and $c_4^0$ for a Navarro model eye, modified to have an anterior cornea with the subject’s conicoid parameters ( $R$ $8.43\mathrm{mm}$ , $Q+1.49$ ), and with a compensating change in axial length, are also shown (diamonds joined by dots).

Fig. 5

Zernike aberration coefficients for myopic LASIK subject AW as a function of horizontal visual field position in (a) third and (b) fourth orders for a $6\text{‐}\mathrm{mm}$ entrance pupil (symbols joined by solid lines). Zernike aberration coefficients $c_3^1$ and $c_4^0$ for a Navarro model eye, modified to have an anterior cornea with the subject’s conicoid parameters ( $R$ $8.38\mathrm{mm}$ , $Q+0.73$ ), and with a compensating change in axial length, are also shown (diamonds joined by dots).

4.

Discussion

For the untreated eyes, the interesting findings are that horizontal coma is linearly affected by horizontal visual field position for all subjects, and that spherical aberration and secondary astigmatism are influenced quadratically by position for some subjects. Previously, Atchison ⁶ found linearity for horizontal coma to occur across the central horizontal $10\mathrm{deg}$ of the visual field, and this study extends this finding to the central $80\mathrm{deg}$ . The Seidel theory⁹ predicts that coma is linearly affected by field position, and this study bears this out for the eye. It is expected that a similar relationship will hold for the vertical visual field and vertical coma. Rate of change of coma with visual field position varies by a factor of 1.5 times across the five untreated subjects. Using eye models, it was shown previously that the rate of change of coma is affected considerably by the asphericity of ocular surfaces but not by magnitudes of surface tilts and decentrations.⁶

Patterns of higher order aberrations and visual field position are not as distinct with small pupils as with larger pupils (compare Figs. 3 and 1). This is due at least in part to the proportionately higher noise with the small pupil than for larger pupils.

For the two myopic LASIK eyes, horizontal coma is linear within a certain range of visual field positions, but its sign is reversed compared with those of the untreated eyes at similar positions. As for some of the untreated eyes, spherical aberration and secondary astigmatism are influenced quadratically by position. An eye modeling exercise⁶ was undertaken for the two myopic LASIK subjects. The ellipsoidal parameters of vertex radius of curvature and asphericity were found by fitting the central $6\mathrm{mm}$ of anterior corneal topographic data for the two subjects. These were substituted into Navarro’s model eye^{10, 11} with a compensating alteration in vitreous chamber length. As the model is rotationally symmetric and the investigation is along the horizontal meridian, only the coefficients for which the superscript is zero or positive have nonzero values. The theoretical results for $c_3^1$ and $c_4^0$ (diamonds connected by dots) are included with the experimental results for the two LASIK eyes in Figs. 4 and 5. While the rates of change for experimental and theoretical results are not in perfect agreement, the overall patterns are similar.

Studies going back more than $50\text{years}$ have shown most people to have positive spherical aberration in the unaccommodated state for foveal vision.¹ This study finds that spherical aberration reduces in magnitude into the horizontal periphery and may become negative. The Seidel theory predicts no change in spherical aberration with visual field position. The Zernike spherical aberration polynomial includes a ${\rho }^2$ term ( $\rho$ being the relative position in the pupil relative to its center) as well as a ${\rho }^4$ term. In the Seidel theory, the ${\rho }^2$ term is defocus, and it may be that there is some interaction between the Zernike spherical aberration and defocus terms that changes with visual field position both for the untreated and treated eyes. The quadratic pattern for the secondary astigmatism coefficient for untreated and treated eyes may be due to an interaction with the astigmatism, which is the dominating off-axis second-order aberration (ignoring on-axis defocus accompanying spherical refractive errors).

There is considerable emphasis in ophthalmology on the importance of higher order aberrations to vision, particularly following refractive surgery. However, the emphasis has been limited to central (foveal) vision. This study shows that the higher order aberrations can be very large in the peripheral visual field. Although these are likely to be small relative to the second-order aberrations of defocus and astigmatism,⁴ their correction may be necessary if near-diffraction-limited fundus imagery is to be obtained well into the periphery.