Michael P. Keating

Dioptric power in an off-axis meridian: the torsional component

ABSTRACT There is much confusion about whether or not dioptric power exists in an off‐axis meridian of a spherocylindrical lens. One source of this confusion is an overly simplified definition of dioptric power. This paper proposes the conditions that a good definition of dioptric power should meet. Under these conditions, dioptric power does exist in an off‐axis meridian. However, that dioptric power has two components—a curvature component and a torsional component. A second source of confusion has been the neglect of the torsional component.

A MATRIX FORMULATION OF SPECTACLE MAGNIFICATION

Abstract The paraxial 4 × 4 astigmatic system matrix is used to derive a 2 × 2 blurred‐image magnification matrix, as well as a 2 × 2 spectacle magnification matrix. The 2 × 2 spectacle magnification matrix describes the meridional magnifications for any spherocyclidrical spectacle correction including a bitoric eikonic lens. The 2 × 2 spectacle magnification matrix can be approximated by the product of 2 × 2 power and shape factor matrices that have algebraic forms exactly analogous to the power and shape factor equations for spherical correcting lens.

Asymmetric dioptric power matrices and corresponding thick lenses.

Background When thickness or separation is taken into account, the dioptric power of systems with obliquely crossed toric surfaces (OCTS), including some corrected and uncorrected human eyes, can be represented by a four-dimensional vector space, provided that an arbitrary 2 × 2 asymmetric matrix is the equivalent power matrix of some optical system. The provision is crucial for the statistical validity of dioptric power matrix methods that take account of thickness. Previous efforts to verify the provision have not been satisfactory. Purpose. To verify the provision. Methods. Prove that an arbitrary 2 × 2 asymmetric matrix is the equivalent power for a thick bitoric lens with obliquely crossed surfaces. Results.The procedure and new equations needed to find the lens. Conclusions. The provision holds, which is an important foundation block for the validity of the statistical matrix methods for thick astigmatic systems. The resulting equations also have optical uses other than statistics.

Oblique central refraction in spherocylindrical corrections with both faceform and pantoscopic tilt.

Thin lens equations, accurate to third order, are presented for the effective spherocylindrical parameters for oblique central refraction (OCR) through spherocylindrical lenses with both faceform and pantoscopic tilt. The equations can also be used to find the parameters of a lens that compensates for the tilts. Accuracy of the equations was checked by exact ray trace results.

Equivalent dioptric power asymmetry relations for thick astigmatic systems.

For optical systems consisting of separated obliquely crossed toric interfaces, the equivalent dioptric power has principal meridians that are not necessarily orthogonal to each other. In this case it takes four parameters to specify the equivalent power. A set of parameters convenient for ophthalmic optics consists of three traditional spherocylindrical parameters S C x 6 β together with a dioptric asymmetry parameter g. The parameter g has been described as “so far” being entirely mathematical in nature. The purpose of this paper is to develop further optical knowledge about the equivalent power asymmetry g. The method was a theoretical and numerical study involving optics and the dioptric power matrix theory. Among the results of this study are a number of new equations involving g that clarify the relationships between the nonorthogonal principal meridians and the power and axis meridians of S C x β, as well as explicitly illustrating the parameters that can increase or decrease g. It is also pointed out that the asymmetry g is formally identical to the circular astigmatism that has previously been presented in discussions of ray vector deflection fields (and is used in the Humphrey Lens Analyzer measurements). In conclusions, the theoretical relations presented here provide optical insight into the equivalent dioptric power asymmetry and the parameter g. The relations and insight can assist further developments.

Oblique central refraction in spherocylindrical lenses tilted around an off-axis meridian

Thin lens equations accurate to third order are presented for the effective spherocylindrical parameters for oblique central refraction through spherocylindrical lenses that are tilted around an off-axis meridian. This situation occurs in either pantoscopic or faceform tilt for spectacle corrections for oblique astigmats. The thin lens equations appear to be good approximations for oblique central refraction through actual spectacle lenses.

The aniseikonic matrix.

Abstract The matrix optics formulism for astigmatic systems is applied in aniseikonia, and aniseikonic matrix is defined. The aniseikonic matrix is particularly easy to compute even in the straight forwardness of the matrix formalism as applied to aniseikonia provides a conceptually clearer means of understanding aniseikonic optics.

Retinal images and astigmatic distortions in the individual corrected aphakic eye

For paraxial objects viewed at finite distances from the correction plane, retinal image size and retinal distortions due to astigmatism are obtained for spectacle- or contact-lens-corrected aphakic individuals with astigmatism. Our methods are based not on the analysis of schematic or reduced eyes but on clinical measurements (corneal, refractive, pachometer, and correction parameters) that correspond to the individual patient.

Clinical determination of the corrected retinal image size in spectacle-corrected aphakes.

ABSTRACT For individual spectacle‐corrected aphakic patients, a method of estimating the size of the corrected retinal image corresponding to distant objects smaller than Snellen 6/360 (20/1200) is developed. Our results indicate that an estimate of the corrected retinal image size corresponding to a distant Snellen 6/12 (20/40) object in an individual spectacle‐corrected aphake can be obtained to within plus or minus 1.0 &mgr;m of error using a multivariate regression model based on the measurement of six patient factors: axial length, anterior corneal power, spectacle lens back vertex power, lens thickness, lens back curve, and lens vertex distance. Clinical application of these results are discussed along with comparisons to other results based on traditional spectacle magnification formulas. The size of the corrected retinal image corresponding to a Snellen 6/12 (20/40) object was found to vary between 60 and 84 &mgr;m in patients corrected with CR‐39 optical plastic and for patients corrected with high‐index optical glass, it was between 56 and 78 &mgr;m.

Clinical prediction of postsurgical aphakic refractive correction.

ABSTRACT A simple clinical predictor of the amount of spectacle convex lens power that will be needed to correct the refractive error of an aphakic patient after cataract surgery is developed based on clinical measures of anterior corneal surface power (keratometer measurements) and axial length. This predictor derived from matrix optics is compared to previous ones developed by Binkhorst and Sanders et al. using different methodologies.