fax: (812) 855-7045
email: thibos
indiana.edu
Address for correspondence:
Larry N. Thibos
School of Optometry
Indiana University
Bloomington, IN 47405
voice: (812) 855-9842 or 855-4475
fax: (812) 855-7045
email: thibosindiana.edu
1. Ph.D.
2. Ph.D., F.A.A.O.
The oblique (off-axis) astigmatism of a the Indiana Eye, an aspheric reduced-eye model of ocular chromatic aberration and spherical aberration, is computed across the visual field by using Coddington's equations for non-spherical surfaces of revolution. Our results show that the amount of astigmatism varies significantly with the shape of the refracting surface and with the axial location of the pupil. For a pupil located 1.91 mm from the apex of the refracting surface (as originally specified for the model), the calculated Sturm's interval was larger than that reported in the literature. However, by moving the model's pupil 0.6 mm axially away from the apex towards the nodal point, a close match was achieved between Sturm's interval of the model eye and published data from human eyes for eccentricities up to 60 deg. These results demonstrate that the aspheric reduced-eye model is capable of simultaneously accounting for the chromatic, spherical, and oblique astigmatic aberrations typically found in human eyes.
Key words: oblique astigmatism, schematic eyes, visual optics
In the companion paper1 we describe a family of aspheric reduced eyes parameterized by a single parameter p which determines the shape of the elliptical refracting surface. One particular member of this family, called the Chromatic Eye (p = 0.437), has been developed previously as a model for the longitudinal and transverse effects of ocular chromatic aberration, unencumbered by spherical aberration.2, 3 To introduce a small degree of positive spherical aberration into the model representative of human eyes, a larger value of p is required. Although spherical aberration is reputed to be highly variable in the population, the measurements of spherical aberration from several studies reported in the literature are well fit by the elliptical model eye with p = 0.6. Additional evidence from studies of the eye's transverse chromatic aberration indicates that, on average, the pupil is well centered on the eye's visual axis.4 To produce a model which is free of coma and other odd-symmetric aberrations at the fovea requires the further assumption that the visual axis and optical axis coincide.1 From these observations, a co-axial aspheric reduced-eye model called the Indiana Eye was developed in the companion paper.1 Our purpose here is to assess the off-axis astigmatism of the Indiana Eye to determine if it is possible for a reduced-eye model to simultaneously account for the chromatic, spherical, and oblique-astigmatic aberrations of the eye.
Several studies of the refractive error of human eyes have found that large amounts of oblique astigmatism are normally present in the peripheral visual field,5-7 although the amount and type of astigmatism varies considerably between individuals.7, 8 Previous attempts to model oblique astigmatism with schematic eyes designed to account for spherical aberration have tended to overestimate the amount of astigmatism found in the population.9-12 Alternatively, models which accurately account for oblique astigmatism yield overcorrected (negative) spherical aberration, suggesting that correction of either aberration occurs at the expense of the other.13 A potential solution to this impasse suggested by Dunne, Barnes and Mission is to manipulate the axial location of the pupil in the schematic eye.14 Although pupil location does not affect the degree of spherical aberration, their simulation of the reduced levels of oblique astigmatism found in human aphakic eyes showed that the degree of astigmatism of the Gullstrand-Emsley schematic eye is reduced significantly when the pupil is shifted towards the nodal point.
Since the Indiana Eye has proved to be a useful model of chromatic and spherical aberrations, we wondered whether adjusting the pupil location in this simple, aspheric model might also provide an accurate description of oblique astigmatism in the peripheral field. To test this idea we have computed the oblique astigmatism of the Indiana eye, retaining the shape factor p and the axial location of the pupil as free variables, and compared the results with published measurements of oblique astigmatism of human eyes.
Figure 1. The Indiana Eye model in the Y-Z plane. O is the apex of the refracting surface. F is the axial focal point, (zf, 0). OF is the optical axis of rotational symmetry. Q is the center of the physical (or exit) pupil, (zq, 0), assuming Q is always on the optical axis. P is the center of the entrance pupil, (zp,0). E is a point on the refracting surface, (ze, ye). EP is the incident chief ray. EQ is the refracted chief ray. EN is the surface normal at point E, where N is the intersection of surface normal with the optical axis. N is also the center of curvature for a sagittal section of the surface. The distance between E and N, rS, is the sagittal radius of curvature of the surface at point E. CT is the center of curvature for a tangential section. ECT is the tangential radius of curvature at point E. Angleis the angle of incidence. Angle
is the angle of refraction, which changes with wavelength. Angle
is the angle of eccentricity. Angle
is the exit angle which also changes with wavelength.
As defined in the companion paper,1 the Indiana Eye model of spherical and chromatic aberration has a single refracting interface which is a surface of revolution of an ellipse. Figure 1 depicts a cross-sectional diagram of the model and the refraction of the chief ray from an off-axis point source in the plane of the diagram. The refracting surface of the model is a member of the family of ellipses which can be represented by either implicit or explicit equations. In the implicit form,
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where a is the length of semi-major axis in the x-direction, and b is the length of semi-minor axis in the y-direction. In the explicit form,
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where r is the apical radius of curvature and p is a parameter which determines the shape of the ellipse. The relations between a, b, r, and p is given in the following equations
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The refractive index of the model eye for wavelength 
is a modified form of Cornu's formula for water3
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where
is in micrometers.
The physical dimensions of the Indiana Eye model are listed in Table 1 of the companion paper.1 Unlike the case of spherical aberration, the pupil plays a key role in the analysis of oblique astigmatism because the pupil controls the obliquity of the chief rays from off-axis objects. The chief ray is determined by the entrance pupil of the model, which is the virtual image of the physical pupil formed by the refracting surface. Since the refractive index of the model eye varies with wavelength, so does the axial location of the entrance pupil. However, this change is very small (±0.01 mm over the range 380 - 780 nm15 ), so it is ignored by the model. The entrance pupil position also changes slightly with focal power for off-axis objects, but this change is also ignored by the model.
For a spherical refracting surface, the oblique astigmatism can be computed by using the well-known Coddington's equations16
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where i is the angle of incidence; i´ is the angle of refraction; r is the apical radius of curvature of the refracting surface; t and t´ are object and image distances in the tangential plane, respectively; s and s´ are object and image distances in the sagittal plane, respectively. The term on the right hand side of eqn. 5 is often called the oblique power of the surface.
Since the Indiana Eye has an elliptical refracting surface of revolution, the question arises whether it is valid to use Coddington's equations to compute oblique astigmatism for this particular model. Fortunately, the modifications required for aspherical surfaces of revolution are trivial: we simply have to use the appropriate principal radius of curvature instead of the spherical radius of curvature.17 With this change, eqn. 5 becomes
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where rT and rS are the tangential and sagittal radii of curvature, respectively, at the incident point on the refracting surface.
For a given angle of eccentricity, the steps for computing oblique astigmatism of the model eye using eqn. 6 are (i) find the intersection of the incident chief ray with the refracting surface; (ii) determine the tangential and sagittal radii of curvature at the intersection point; (iii) determine cosine values of the angles of incidence and refraction; (iv) apply the modified Coddington's equations to determine tangential and sagittal focal lengths for distant objects; (v) determine oblique astigmatism in diopters in both image and object spaces. Dunne has outlined a similar approach for calculating oblique astigmatism in schematic eyes with aspheric surfaces.18
In Fig. 1, let
be the angle of eccentricity formed by the incident chief ray with
the optical axis for an off-axis object point. Since the incident
chief ray must pass through the known center of the entrance pupil,
we can determine the line equation for the incident chief ray. The
equation for the refracting surface is also known (eqn. 1
or 2). By solving these two equations
simultaneously, we find the intersection point E = (ze,
ye).
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|
where Z is the axial position of the entrance pupil relative to the apex.
As shown in Fig. 1, the sagittal radius of curvature rS is the distance between E and N, where N is the intersection of the surface normal (at point E) with the optical axis. The sagittal radius of curvature is given by19
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The tangential radius of curvature is calculated by the formula
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which simplifies to
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In Fig. 1, the angle of incidence is that between vectors EP and EN. According to a property of the inner product of two vectors, we have
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From Snell's law and the relations of trigonometric functions, the cosine value of the angle of refraction is given by
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Given the above results, we apply Coddington's modified equations
(6) to calculate the tangential and sagittal
focal lengths. Note that for a distant object, t = s =
,
and the corresponding image distances t´, s´
represent the tangential and sagittal focal lengths,
respectively,
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|
where we have substituted n = 1 and n´ = 
. Although eqn. 13 is valid for any
wavelength of light, the numerical results reported in this paper are
for the model's emmetropic wavelength (=589
nm, the sodium D-line), for which the refractive index according to
eqn. 4 is nD = 1.333.
Tangential and sagittal focal powers are
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Sturm's interval in image space is
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and Sturm's interval in object space is smaller by the factor 1/nD.
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Equation 16 is in a form suitable for direct comparison with experimental measurements of refractive error which, by convention, are specified by the spectacle lens power required to correct the refractive error. By our sign convention, Sturm's interval is positive when the eye is more myopic for the tangential meridian than for the sagittal.
Although the methods described above permit the optical analysis
of any member of the Indiana Eye family of reduced eyes, we
concentrated our attention upon four specific cases: (1) the
Chromatic Eye, which is free from spherical aberration (p=0.4372) and
has a pupil located at the same axial distance anterior to the nodal
points as in Gullstrand's schematic eye, (2) a model identical to the
Chromatic Eye, except the surface is spherical (p=1), which
introduces a large amount of spherical aberration, (3) Emsley's
reduced eye, 20 which has an implied
pupil at the apex of the spherical refracting surface, (4) an Indiana
Eye model with a small amount of spherical aberration (p=0.6) typical
of eyes reported in the literature. 1 It
is shown that none of these models adequately describes the
oblique astigmatism measured in a large sample of nearly emmetropic
individuals. 7 However, an adequate fit
is achieved if the axial location of the model's pupil model is
shifted slightly posterior towards the nodal point. A sensitivity
analysis is then conducted to determine the effect of changes in
shape parameter p on the axial location of the pupil required
to fit the empirical data. All modeling results described below were
obtained for the emmetropic wavelength, D
= 589 nm.
Figure 2. Tangential and sagittal focal lengths (a) and focal powers (b) as functions of eccentricity angles for the Chromatic-Eye. In (a), t' (solid curve) and s' (dashed curve) are tangential and sagittal focal lengths, respectively. In (b), PT (solid curve) and PS (dashed curve) are tangential and sagittal focal powers in image space, respectively. The axial pupil position used in model computation is 1.91 mm from the apex of the refracting surface.
Figure 2a shows the tangential (t´) and sagittal (s´) focal lengths of the Chromatic Eye as functions of object eccentricity as computed from eqn. (13). When eccentricity = 0 deg, the focal length is 22.22 mm. With the increase of eccentricity, both tangential and sagittal focal lengths are reduced. However, the tangential focal length decreases faster than the sagittal focal length. The separation between t´ and s´ is Sturm's interval in millimeters. As eccentricity increases, the gap between the tangential and sagittal focal lengths increases, i.e., oblique astigmatism becomes larger. Figure 2b shows the corresponding tangential and sagittal focal powers as determined from eqn. (14). When eccentricity = 0, the focal power of the model eye is 60 diopters. With the increase of eccentricity angle, focal powers in both tangential and sagittal planes increase. The difference between tangential and sagittal focal powers is the Sturm's interval in diopters in image space.
Figure 3 compares the empirical results of a population study from Rempt, et al. 7 (triangles) with calculated Sturm's interval in object space for three of the model eyes mentioned above. The Sturm's intervals were computed in terms of refractive correction by using eqn. 16. Clearly, all three eye models overestimate the amount of oblique astigmatism. However, among the models analyzed, the chromatic-eye with elliptical refracting surface gives closest predictions. When the shape of the refracting surface changes from elliptical to spherical, with all other parameters unchanged, the predicted oblique astigmatism increases. Even greater increases in astigmatism result from shifting the pupil forward to the apex in Emsley's reduced eye.
Figure 3. Model predictions of oblique astigmatism. The computed Sturm's interval for the Chromatic Eye (p=0.437) is shown in filled circles. The axial pupil position used in the model computation is Z=1.91 mm from the apex of the refracting surface. Other parameters used are specified in Table 1 of the companion paper (Thibos et al, 1997). Filled diamonds show Sturm's interval for the Indiana Eye with same parameters as the Chromatic Eye but spherical refracting surface (p=1). Inverted triangles are Sturm's intervals for Emsley's reduced-eye (p=1, Z=0). The human data from the population study by Rempt et. al. (1971) are shown by open triangles. Each data point is the mean Sturm's interval of 726 near-emmetropic eyes.
One way to improve the fit of the model to the experimental data is to reduce the degree of oblique astigmatism by changing the axial location of the pupil. As a limiting case, if the entrance pupil is located at the center of curvature of the spherical refracting surface (i.e. the nodal point), then the model will have zero oblique astigmatism. The computational results in Figs. 2 and 3 were obtained with the pupil located 1.91 mm from the apex of the refracting surface, the position originally specified by Thibos for the Chromatic Eye. 15 Figure 4 shows the effect on oblique astigmatism of shifting the axial position of the pupil away from the apex and towards the nodal point. When pupil position is shifted from 1.91 mm to 2.20 mm (Figure 4a), the model predictions are closer to the human data but still overestimate the degree of astigmatism. Shifting the pupil further to 2.60 mm (Figure 4b), the chromatic-eye model underestimates the Sturm's intervals from the population study whereas the model with spherical refracting surface still overestimates the results. From this we conclude that both factors, the shape of the surface as well as the axial location of the pupil, determine the amount of astigmatism of the model.
Figure 4. The effect of changing axial pupil position on oblique astigmatism of the Indiana Eye. (a) axial pupil position is 2.20 mm from the apex of the refracting surface. (b) axial pupil position is 2.60 mm from the apex. Solid curves are the predictions for the Indiana Eye model with zero spherical aberration (p=0.437), dashed curves are the predictions by the Indiana Eye with large amounts of spherical aberration (p=1). Open triangles shows the results from Rempt et al. (1971).
When the physical pupil position is at 2.45 mm from the apex of the refracting surface, the chromatic-eye model yields oblique astigmatism close to the empirical data as shown in Fig. 5. Figure 5 also shows the astigmatism of Lotmar's 4-surface model with aspherics 10, 21 . Lotmar's model can provide a good fit to the data within 30 deg eccentricity. However, when eccentricity is larger than 40 deg, the Sturm's interval computed from Lotmar's model increases rapidly.
Figure 5. Oblique astigmatism of various model eyes compared to the data for human eyes. Solid curve represents the best fitting Chromatic Eye model (p=0.4372, Z=2.45 mm); dashed curve represents the same model but with spherical refracting surface (p=1); symbols represent the data of Rempt, et. al (1971); dot-dashed curve represents Lotmar's 4-surface model with aspherics (Lotmar, 1971; Lotmar & Lotmar, 1974).
In the companion paper we found that a shape parameter value of about p = 0.6 provided the best fit of the model to the spherical aberration data from the literature. Assuming this new p value, we determined that the Indiana Eye continues to provide an excellent fit to Rempt's data, as shown in Fig. 6, provided the pupil is placed 2.55 mm from the apex of the refracting surface. This exercise demonstrates that by introducing a physiologically reasonable amount of positive spherical aberration into the model, the amount of oblique astigmatism increases, but this can be compensated by shifting the pupil away from the apex by an extra 0.1 mm.
Figure 6. Oblique astigmatism of the Indiana Eye with spherical aberration (p=0.6) representative of human eyes. The required axial position of the pupil for best fit is Z=2.55 mm from the apex of the refracting surface. Solid curve represents the model eye, symbols represent the data of Rempt, et al. (1971).
To systematically explore the interaction between the shape
parameter p and pupil position Z, we computed the
optimal pupil location for fitting the entire family of Indiana Eye
models (0 < p 
1) to the mean values of Sturm's interval from the literature.
7, 21
The computational work was done in MATLAB (MathWorks, Inc). Shape
parameter p was varied from 0.0 to 1.0 in steps of 0.05 and
the optimal pupil location was determined by an iterative procedure
(step size 0.01 mm) which minimized the sum-of-squares error
function
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Figure 7 shows the relation between the optimal pupil location Z and the shape parameter p to fit the given dataset. As p increases, the refracting surface approaches a spherical shape, thus introducing more spherical aberration. To compensate for the resulting increase in oblique astigmatism the best-fitting pupil location shifts closer to the nodal point of the model eye. Figure 7 also shows that the quality of fit to the empirical dataset improves for larger p values and residual error is minimized when p = 0.9. A slightly poorer fit is obtained for p = 0.6, which is the value which best represents spherical aberration of human eyes, but inspection of Fig. 6 shows that this slightly less than optimal fit is still excellent.
Figure 7. Interaction between shape parameter p and axial position Z of pupil required for least-squares fit of model to empirical data on human eyes. Surfaces with p < 0.437 have negative spherical aberration, surfaces with p > 0.437 have positive spherical aberration. Solid curve shows the combination of p and Z values required to fit the dataset from Rempt, et al. (1971). Broken curve shows residual (RMS) error for the best fit as computed by text equation (17).
In summary, our computational results demonstrate that any member of the Indiana Eye family of model eyes can be brought into agreement with the oblique astigmatism data of Rempt, et al. 7 by adjusting the axial location of the model's pupil. This indicates that the aspheric reduced eye is capable of simultaneously accounting for the chromatic, spherical, and oblique astigmatic aberrations typically found in human eyes. Because of its flexibility and extreme simplicity, it is conceivable that a customized Indiana Eye could be made to order for individual eyes as required, thus enabling the computation of retinal image quality and its impact on visual performance. Although we have not specifically attempted to reproduce the various types of oblique astigmatism described in the literature (e.g. myopic, hyperopic, mixed), 5, 18 we anticipate that these various forms will arise naturally by adjusting the retinal contour of the Indiana Eye away from its current spherical shape.
This study was supported by grant R01 EY05109 from the National Eye Institute to LNT.