Biomedical subjects
Publications and source records attributed to M P Keating.
A corrected or uncorrected human eye may be astigmatic and noncoaxial. The first-order (i.e., paraxial) character of arbitrary astigmatic and noncoaxial optical systems can be compared quantitatively in terms of the ray transference of either an anterior or a posterior converter system. This article derives equations that give the relationships between the transferences of the anterior and posterior converter 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 SC x theta 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 SC x theta, 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.
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 x 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 x 2 asymmetric matrix is the equivalent power for a thick bitoric lens with obliquely crossed surface. 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.
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.
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.
It is sometimes assumed that the prismatic effect in any astigmatic lens is orthogonal or perpendicular to the iso-thickness curves. This paper offers a formal proof of the validity of the assumption.
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.
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 micron 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 micron in patients corrected with CR-39 optical plastic and for patients corrected with high-index optical glass, it was between 56 and 78 microns.
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.
Explore the source record for details and available documents.
Explore the source record for details and available documents.
The paraxial 4 x 4 astigmatic system matrix is used to derive a 2 x 2 blurred-image magnification matrix, as well as a 2 x 2 spectacle magnification matrix. The 2 x 2 spectacle magnification matrix describes the meridional magnifications for any spherocyclindrical spectacle correction including a bitoric eikonic lens. The 2 x 2 spectacle magnification matrix can be approximated by the product of 2 x 2 power and shape factor matrices that have algebraic forms exactly analogous to the power and shape factor equations for spherical correcting lenses.
The matrix optics formalism for astigmatic systems is applied to aniseikonia, and an aniseikonic matrix is defined. The aniseikonic matrix is particularly easy to compute even in the case of obliquely-crossed magnificant meridians between the right and left eyes. The straightforwardness of the matrix formalism as applied to aniseikonia provides a conceptually clearer means of understanding aniseikonic optics.
A single 4 X 4 system matrix is used to represent the para-axial properties of optical systems consisting of separated obliquely crossed spherocylindrical lenses. The 4 X 4 system matrix is a generalization and combination of the 2 X 2 Gaussian system matrix for spherical optical systems, and the 2 X 2 dioptric power matrix for a single spherocylindrical lens or for obliquely crossed spherocylindrical lenses in contact with each other. The 4 X 4 system matrix approach simplifies both the conceptual and numerical analysis of complicated astigmatic systems.
The 4 x 4 system matrix is applied to corrected astigmatic systems including a schematic eye in which each surface is astigmatic at a different axis. In addition to representing the eye, the 4 x 4 system generates 2 x 2 magnification matrices which describe the meridional magnifications that occur in the presence of astigmatism including the magnifications that occur with bitoric eikonic correcting lenses, or other meridional magnifying systems.
An effective dioptric power matrix is defined. The effective dioptric power matrix can be calculated directly from the lens dioptric power matrix, its trace, and its determinant, without using the sphere, cylinder, and axis parameters of the lens. This greatly simplifies any calculations in which dioptric power matrices are obtained in intermediate steps, including separated obliquely crossed spherocylindrical lens calculations. The generalization of the effective dioptric power matrix to a reduced vergence matrix is also discussed.
Dr. W. F. Long pointed out that calculations of decentration in spherocylindrical lenses, as well as calculations of combinations of obliquely crossed spherocylindrical lenses, are considerably simplified by the use of matrix methods. In the obliquely crossed lens problem, Long used eigenvalue techniques to obtain the sphere, cylinder, and axis of the equivalent lens. This paper presents an alternative to the eigenvalue method. This alternative method uses the invariance of the trace and determinant of the dioptric power matrix. This alternative is conceptually easier to understand than the eigenvalue method and perhaps will encourage more people to use the matrix methods.
The sine-squared law is shown to describe changes in spherical power which either minimize the blur or maximize the visibility of gratings oriented at various angles to the principal meridians of an astigmatic eye. These results help justify the use of the sine-squared law in applications such as meridional refraction. This derivation and its interpretation is quite different from the usual approach which emphasizes the meridional changes in curvature of a toric surface.