fax: (812) 855-7045
email: thibos
indiana.edu
Larry
N. Thibos Ph.D., F.A.A.O, Ming Ye Ph.D., Xiaoxiao Zhang Ph.D.,
and Arthur Bradley
Ph.D.
School of Optometry, Indiana University, Bloomington, IN
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
Submitted 31-Oct-96. Contains 9 figures (line copy) and 1
table
Resubmitted with revisions 18-March-97
Table Of Contents
Abstract | Introduction
| Quantification of Spherical
Aberration
Spherical Aberration of the Elliptical
Surface | Seidel Aberration
Values
Fitting the Model to
Experimental Data | Appendix
A | Appendix B | Acknowledgments
| References
We extend the single-surface schematic-eye model of ocular chromatic aberration to account for spherical aberration of the eye. This extension is accomplished by allowing the model's single refracting surface to be a member of the family of ellipses with variable shape parameter (eccentricity). The resulting model, dubbed the "Indiana Eye", may have either positive or negative spherical aberration of varying degree depending upon the numerical value of the shape parameter. Spherical aberration of the model eye is well described by third-order optical theory for shape parameters in the range 0 ² p ² 0.7 but requires fifth-order theory for an accurate description over the parametric range 0.7 < p ² 1.0. An improved technique was devised for fitting the model to published measurements of ray aberrations while avoiding errors of estimation of the degree of spherical aberration present in eyes which also manifest odd-symmetric aberrations such as coma. A shape parameter value of about p = 0.6 provided the best fit of the model to selected data from the literature.
Key Words: spherical aberration, schematic eyes, visual optics
In an earlier paper 1 we introduced the "Chromatic Eye", a reduced schematic-eye model of ocular chromatic aberration containing a pupil and a single, aspheric refracting surface separating air from a chromatically dispersive ocular medium. That model was designed to accurately describe the eye’s transverse and longitudinal chromatic aberration while at the same time being free from spherical aberration (at least for the emmetropic wavelength). This design was achieved by employing an elliptical refracting surface which has zero spherical aberration, the so-called Cartesian oval.2
In the present paper we extend the Chromatic Eye to model the combined effects of chromatic and spherical aberrations of the eye. In a preliminary report 3 we pointed out that a controlled amount of spherical aberration could be introduced into the model by altering the shape of the refracting surface away from the Cartesian oval. Our initial attempt in this regard employed a polynomial refracting surface, which unfortunately altered the paraxial properties of the model. In the present paper we take a more satisfying approach of adopting the family of conic sections described by Baker (1943), which includes the Cartesian oval as a special case (see Appendix A). We refer to the new model as the Indiana Eye to distinguish it from the Chromatic Eye. We anticipate that the new model will be useful for predicting the combined effects of defocus, spherical aberration, and chromatic aberration on the quality of retinal images and visual performance.
Figure 1. Comparison of Gullstrand’s schematic eye (top) with the general form of the Chromatic Eye (middle) and the simplified Indiana Eye (bottom). All dimensions are in mm. Circles mark the location of the nodal point (N), center of rotation (C), center of pupil (P), anterior focal point (F), and posterior focal point (F’). The formula for index of refraction of the ocular media assumes wavelength (l) has units of micrometers.
Figure 1 compares the Chromatic Eye and the new Indiana Eye with Gullstrand’s classic schematic eye as described by Emsley.4 The most general form of the Chromatic Eye model is illustrated in the middle diagram, where there is no particular relationship assumed between the optical axis (axis of revolution of the aspheric refracting surface) and the visual axis (join of fovea, nodal point, and distant fixation point) or the fixation axis (path of chief ray from distant fixation point to fovea) or the achromatic axis of the eye (path of chief nodal ray, which is the axis of zero transverse chromatic aberration). Although angle a (the angle between visual and optical axes) figures prominently in the visual optics literature, we have found that angle y (between the visual and achromatic axes) and the closely related angle f (between the fixation and visual axes in image space) are more important for assessing the quality of foveal polychromatic images.5 This is because y and f are sensitive to decentration of the pupil. If the pupil is not well centered on the visual axis (i.e. f 0) , or equivalently, if the visual axis does not coincide with the eye's achromatic axis (i.e. y 0), then the fovea will be subjected to the deleterious effects of transverse chromatic aberration, which can have a dramatic impact on retinal image quality6 and visual performance.7
Recent studies have shown that the statistical mean of angle y in the population is not large8, 9 and therefore the pupil is, on average, well centered on the visual axis. Consequently, the fovea of the hypothetical average eye (representing the population mean of human eyes) is well protected from the effects of transverse chromatic aberration. We make use of these empirical results to simplify the general model and so arrive at the Indiana Eye, shown in the bottom diagram of Fig. 1. The key simplifying assumptions of the Indiana Eye are: (1) the pupil is centered on the visual axis (i.e. achromatic and visual axes coincide) and (2) since this model is intended to model spherical aberration (which in theory has even symmetry) without the contaminating effects of odd-symmetric aberrations, the fovea is assumed to lie on the optical axis of rotational symmetry of the model. This second assumption may be relaxed in a future version of the model designed to account also for coma and other aberrations which depend on field angle. Our purpose here is to explore the consequences of allowing the shape parameter p to vary, and to derive an estimate of p from published measurements of spherical aberration in human eyes.
Quantification of Spherical Aberration
Figure 2. Spherical aberration of the reduced eye depicted in image space (upper diagram) and in object space (lower diagram). Transverse spherical aberration (TSA) is defined as the angular error between the refracted ray (marked with arrow head) and the un-aberrated reference ray. Longitudinal spherical aberration (LSA) is defined by the vergence error of the refracted ray. Q is the point of intersection of the ray with the refracting surface and has coordinates (0,y,z) with respect to the reference frame shown. N is the paraxial nodal point, which lies at the distance r from the vertex point O. The z-axis is the axis of symmetry of the refracting surface and the x-axis is orthogonal to the plane of the diagram at point O.
Although numerous previous experiments have been designed to measure spherical aberration of the eye,10-16 the interpretation of those measurements has not always been consistent with optical theory. Therefore, we begin with a brief review of the geometrical optics formulation of the problem of quantifying the spherical aberration of the eye. Consider Fig. 2, which depicts spherical aberration of a symmetrical refracting surface in image space (top) and in object space (bottom). To quantify the degree of spherical aberration in image space, a ray parallel to the optical axis, but displaced by the distance y, is traced from left to right. In the absence of spherical aberration, the ray would intersect the optical axis at point F in the paraxial focal plane. An aberrated ray, on the other hand, will deviate from the reference ray by the angle t, and will cross the optical axis at distance S instead of F from the vertex at O. We take the angular error t as a measure of the transverse spherical aberration (TSA) and the vergence error (n/S - n/F) as a measure of the longitudinal spherical aberration (LSA). (Although most optics textbooks define TSA as the linear distance in the focal plane from the optic axis to the point of intersection of the marginal ray, the choice of an angular measure for TSA is common in visual optics because it fits more naturally with the design and interpretation of experiments which measure the aberration.) A proportional relationship between LSA and TSA is easily derived for small angles a,b for which the approximation x=tan(x) applies (and neglecting the small distance z=OA)
(1)
An equally valid formulation of the problem depicts the spherical aberration in object space, as shown in the lower diagram of Fig. 2. Again neglecting the small distance OA, the vergence error of the exiting ray is 1/T, which we take as a measure of LSA in object space. From the geometry of Fig. 2 we conclude that the transverse aberration t in object space is
(2)
Notice that if TSA is specified in radians, and y is in meters, then LSA will have units of diopters. Equations 1,2 indicate that TSA is larger in object space than in image space by the factor n, the index of refraction of the ocular medium. Exact ray tracing confirms this result for small angles of incidence, and shows further that the ratio grows larger than n as ray height increases.
With the exception of the study by Koomen, Tousey, and Scolnik,11 who used a method of best focus for viewing through annular pupils, most of the classic studies of ocular spherical aberration adapted Scheiner’s method to estimate LSA from object-space measurements of TSA according to eqn. 2. In its simplest form, Scheiner’s technique is to have the experimental subject view one point source through a pinhole pupil centered on a reference axis while simultaneously viewing a second point source through a displaced pinhole. If the two sources physically coincide, the presence of ocular spherical aberration will cause their images to fall upon different retinal locations, resulting in diplopic vision. To compensate for the eye’s spherical aberration, the second object is then displaced from the first until the subject reports that they appear to be aligned (i.e. retinal images are coincident). The angular separation between the two objects required to achieve single vision in such an experiment is represented by t in Fig. 2, which is interpreted as a measure of TSA which may be used to estimate LSA according to eqn. 2.
Although the preceding line of reasoning is valid for eyes suffering solely from spherical aberration, the presence of defocus or other aberrations, such as coma or curvature of field, will influence the experimental measurement of transverse aberration. This situation could yield misleading conclusions if all of the measured transverse aberration is ascribed to spherical aberration. Howland and Howland17 commented that this problem of interpretation of transverse aberration data was ignored in the early visual optics literature, an oversight which may have introduced errors into the estimated magnitude of LSA for human eyes and may also have contributed to the large amount of individual variability in LSA reported in the literature. They introduced to visual optics a sophisticated mathematical analysis which resolves aberrations into various Zernike terms from which the true amount of spherical aberration can be inferred. More recently, Campbell and colleagues have illustrated the appropriate use of simpler polynomial functions to achieve the same end.18 Further on we present a simple, alternative method for avoiding such potential errors by separating the effects of even- and odd-symmetric aberrations on experimental data.