Vector space of generalized radii of ellipses for quantitative analysis of blur patches and other referred apertures in the astigmatic eye.

Ellipses are features of several structures in astigmatic eyes; they include retinal blur patches. How can one calculate change or averages or perform other quantitative analyses on such elliptical structures? The matrix A in the equation rTAr=1, commonly used to represent an ellipse, is positive definite; such matrices do not define vector spaces. They are unsuitable, therefore, for quantitative analysis of ellipses. This paper defines a generalized radius R of an ellipse that is positive or negative definite for locally diverging or converging rays, respectively, indefinite between line foci, and singular at foci. Generalized radii of ellipses constitute a vector space and are suitable for quantitative analysis of elliptical ocular structures.

Cardinal and anti-cardinal points, equalities and chromatic dependence.

PURPOSE: Cardinal points are used for ray tracing through Gaussian systems. Anti-principal and anti-nodal points (which we shall refer to as the anti-cardinal points), along with the six familiar cardinal points, belong to a much larger set of special points. The purpose of this paper is to obtain a set of relationships and resulting equalities among the cardinal and anti-cardinal points and to illustrate them using Pascal's ring. METHODS: The methodology used relies on Gaussian optics and the transference T. We make use of two equations, obtained via the transference, which give the locations of the six cardinal and four anti-cardinal points with respect to the system. We obtain equalities among the cardinal and anti-cardinal points. We utilise Pascal's ring to illustrate which points depend on frequency and their displacement with change in frequency. RESULTS: Pascal described a memory schema in the shape of a hexagon for remembering equalities among the points and illustrating shifts in these points when an aspect of the system changes. We modify and extend Pascal's ring to include the anti-cardinal points. We make use of Pascal's ring extended to illustrate which points are dependent on the frequency of light and the direction of shift of the equalities with change in frequency. For the reduced eye the principal and nodal points are independent of frequency, but the focal points and the anti-cardinal points depend on frequency. For Le Grand's four-surface model eye all six cardinal and four anti-cardinal points depend on frequency. This has implications for definitions, particularly of chromatic aberrations of the eye, that make use of cardinal points and that themselves depend on frequency. CONCLUSIONS: Pascal's ring and Pascal's ring extended are novel memory schema for remembering the equalities among the cardinal and anti-cardinal points. The rings are useful for illustrating changes among the equalities and direction of shift of points when an aspect of a system changes. Care should be taken when defining concepts that rely on cardinal points that depend on frequency.

Quantitative analysis of eyes and other optical systems in linear optics.

PURPOSE: To show that 14-dimensional spaces of augmented point P and angle Q characteristics, matrices obtained from the ray transference, are suitable for quantitative analysis although only the latter define an inner-product space and only on it can one define distances and angles. The paper examines the nature of the spaces and their relationships to other spaces including symmetric dioptric power space. METHODS: The paper makes use of linear optics, a three-dimensional generalization of Gaussian optics. Symmetric 2 × 2 dioptric power matrices F define a three-dimensional inner-product space which provides a sound basis for quantitative analysis (calculation of changes, arithmetic means, etc.) of refractive errors and thin systems. For general systems the optical character is defined by the dimensionally-heterogeneous 4 × 4 symplectic matrix S, the transference, or if explicit allowance is made for heterocentricity, the 5 × 5 augmented symplectic matrix T. Ordinary quantitative analysis cannot be performed on them because matrices of neither of these types constitute vector spaces. Suitable transformations have been proposed but because the transforms are dimensionally heterogeneous the spaces are not naturally inner-product spaces. RESULTS: The paper obtains 14-dimensional spaces of augmented point P and angle Q characteristics. The 14-dimensional space defined by the augmented angle characteristics Q is dimensionally homogenous and an inner-product space. A 10-dimensional subspace of the space of augmented point characteristics P is also an inner-product space. CONCLUSIONS: The spaces are suitable for quantitative analysis of the optical character of eyes and many other systems. Distances and angles can be defined in the inner-product spaces. The optical systems may have multiple separated astigmatic and decentred refracting elements.

Pseudophakic monovision: optimal distribution of refractions.

PURPOSE: To determine the optimal distribution of refractions in monofocal, pseudophakic monovision. METHODS: A previously reported mathematical method for describing defocus for a single eye (Acta Ophthalmol, 89, 2011, 111) is expanded to the binocular situation. The binocular distribution of refractions yielding the least defocus over the most extended fixation intervals is identified by mathematical optimization. The results are tested in a group of 22 pseudophakic patients. RESULTS: For the fixation interval 0.25-6.0 m, the optimal refractions are pure spheres of -0.27D for the distance eye and -1.15D for near eye. The mathematically derived defocus structure is confirmed in the clinical series. CONCLUSIONS: The reported mathematical method enables identification of the optimal distribution of refractions over any fixation interval. Monovision with refractions of approximately -0.25 and -1.25D may lead to spectacle independence for distance and intermediate vision. Binocular problems--such as monovision suppression, reduced stereoacuity and binocular inhibition--are likely to be minimal with the suggested anisometropia of 1.0D. This moderate monovision is fully reversible with spectacle correction, as the induced aniseikonia is minimal and it therefore represents a safe alternative to multifocal intraocular lenses (IOLs).

Conditions in linear optics for sharp and undistorted retinal images, including Le Grand's conditions for distant objects.

In 1945 Yves Le Grand published conditions, now largely forgotten, on the 4×4 matrix of an astigmatic eye for the eye to be emmetropic and an additional condition for retinal images to be undistorted. The conditions also applied to the combination of eye and the lens used to compensate for the refractive error. The conditions were presented with almost no justification. The purpose of this paper is to use linear optics to derive such conditions. It turns out that Le Grand's conditions are correct for sharp images but his condition such that the images are undistorted prove to be neither necessary nor sufficient in general although they are necessary but not sufficient in most situations of interest in optometry and vision science. A numerical example treats a model eye which satisfies Le Grand's condition of no distortion and yet forms elliptical and noncircular images of distant circles on the retina. The conditions for distant object are generalized to include the case of objects at finite distances, a case not examined by Le Grand.

Line of sight of a heterocentric astigmatic eye.

BACKGROUND: The line of sight and the corneal sighting centre are important references for clinical work in optometry and ophthalmology. Their locations are not fixed but may vary with displacement of the pupil and other changes in the eye. PURPOSE: To derive equations for the dependence of the locations on properties of an eye which may be heterocentric and astigmatic. METHODS: The optical model used is linear optics. It allows for the refracting surfaces of the eye to be astigmatic and tilted or decentred. Because the approach is general it applies not only to the natural eye but also to a pseudophakic eye and to the compound system of eye and any optical instrument in front of it. The analysis begins with the line of sight defined in terms of the foveal chief ray. RESULTS: Equations are derived for the position and inclination of the line of sight at incidence onto the eye. They allow one to locate the line of sight and corneal sighting centre given the structure (curvatures, tilts, spacings of refracting surfaces) of the eye. The results can be generalized in several ways including application in the case of extra-foveal fixation and when there is a lens or other optical instrument in front of the eye. The calculation is illustrated in the Appendix for a model eye with four separated, astigmatic and tilted refracting surfaces. CONCLUSIONS: The equations allow routine calculation of the line of sight for an eye of known structure and of the eye combined with an optical device such as a spectacle lens. They also allow exploration of the dependence of the line of sight on the location of the centre of the pupil and on other properties in the eye. There is a dependence of the line of sight on the frequency (or vacuum wavelength) of light but this may not be of clinical significance.

Chromatic aberration in heterocentric astigmatic systems including the eye.

PURPOSE: There is inconsistency in the literature in the definitions of longitudinal and transverse chromatic aberration, and there appear to be no definitions that make allowance for astigmatism and heterocentricity. The purpose is to propose definitions of longitudinal and transverse chromatic aberration that hold for systems which, like the typical eye, may be heterocentric and astigmatic and to develop the associated optics. METHODS: Common definitions of longitudinal and transverse chromatic aberration based on Gaussian optics are generalized naturally in terms of linear optics to accommodate heterocentricity and astigmatism. CONCLUSIONS: The definitions offered here apply to systems in general, including the visual optical system of the eye, and hold for homocentric stigmatic systems in particular. Care is advocated in the use of the terms longitudinal and transverse chromatic aberration.

Chief nodal axes of a heterocentric astigmatic eye and the Thibos-Bradley achromatic axis.

Two kinds of axes are described as achromatic in the vision science literature: those of Le Grand and Ivanoff, originally proposed in the 1940s, and those of Thibos and Bradley proposed in the 1990s. Thibos-Bradley axes are based on chief nodal rays, that is, nodal rays that intersect the pupil at its center. By contrast Le Grand-Ivanoff achromatic axes are pupil independent. The purpose of this paper is to develop the linear optics of Thibos-Bradley achromatic axes and to examine the sense in which such axes can be said to be achromatic. Linear optics is used to define the chief nodal ray of an arbitrary optical system whose refracting elements may be heterocentric and astigmatic and with nonaligned principal meridians. The incident segment of the ray then defines what is called here the incident chief nodal axis and the emergent segment the emergent chief nodal axis. When applied to an eye they become the external and retinal chief nodal axes of the eye. The axes are infinite straight lines. Equations are derived, in terms of the properties of the eye, for the inclination and transverse positions of both axes at incidence onto the eye. An equation is also derived for the position of the retinal chief nodal axis at the retina. The locations of the axes are calculated for a particular model eye in Appendix A. The equations are specialized for the system consisting of an eye and a pinhole in front of it. For a reduced eye the external and retinal chief nodal axes coincide and are independent of the frequency of light; and, hence, the Thibos-Bradley axes are strictly achromatic for that eye. However for more complicated eyes this is not usually the case; the external and retinal axes are usually distinct, dependent on frequency, and, hence, not strictly achromatic. It seems appropriate, therefore, to reserve the term achromatic axis for axes of the Le Grand-Ivanoff type and generalizations thereof, and to call Thibos-Bradley axes chief nodal axes.

Achromatic axes and their linear optics.

If a polychromatic ray segment enters an optical system, is dispersed into many slightly different paths through the system, and finally emerges at a single point, then the incident segment defines what Le Grand and Ivanoff called an achromatic axis of the system. Although their ideas of some 65 years ago have inspired important work on the optics of the eye there has been no analysis of such axes for their own sake. The purpose of this paper is to supply such an analysis. Strictly speaking optical systems, with some exceptions, do not have achromatic axes of the Le Grand-Ivanoff type. However, achromatic axes based on a weaker definition do exist and may for practical purposes, perhaps, be equivalent to strict Le Grand-Ivanoff axes. They are based on a dichromatic incident ray segment instead. The linear optics of such achromatic axes is developed for systems, like the visual optical system of the eye, that may be heterocentric and astigmatic. Equations are obtained that determine existence and uniqueness of the axes and their locations. They apply to optical systems like the eye and the eye in combination with an optical instrument in front of it. Numerical examples involving a four-refracting surface eye are treated in Appendix A. It has a unique achromatic axis for each retinal point including the center of the fovea in particular. The expectation is that the same is true of most eyes. It is natural to regard the Le Grand-Ivanoff achromatic axis as one of a class of six types of achromatic axes. A table lists formulae for locating them.

Aperture referral in heterocentric astigmatic systems.

BACKGROUND: Retinal blur patch, effective corneal patch, projective field, field of view and other concepts are usually regarded as disjoint concepts to be treated separately. However they have in common the fact that an aperture, often the pupil of the eye, has its effect at some other longitudinal position. Here the effect is termed aperture referral. PURPOSE: To develop a complete and general theory of aperture referral under which many ostensibly-distinct aperture-dependent concepts become unified and of which these concepts become particular applications. The theory allows for apertures to be elliptical and decentred and refracting surfaces in an eye or any other optical system to be astigmatic, heterocentric and tilted. METHODS: The optical model used is linear optics, a three-dimensional generalization of Gaussian optics. Positional and inclinational invariants are defined along a ray through an arbitrary optical system. A pencil of rays through a system is defined by an object or image point and an aperture defines a subset of the pencil called a restricted pencil. RESULTS: Invariants are derived for four cases: an object and an image point at finite and at infinite distances. Formulae are obtained for the generalized magnification and transverse translation and for the geometry and location of an aperture referred to any other transverse plane. CONCLUSIONS: A restricted pencil is defined by an aperture and an object or image point. The intersection of the restricted pencil with a transverse plane is the aperture referred to that transverse plane. Many concepts, including effective corneal patch, retinal blur patch, projective field and visual field, can now be treated routinely as special cases of the general theory: having identified the aperture, the referred aperture and the referring point one applies the general formulae directly. The formulae are exact in linear optics, explicit and give insight into relationships.

Pascal's ring, cardinal points, and refractive compensation.

Pascal's ring is a hexagon each of whose corners represents one of the six cardinal points of an optical system and whose sides represent relationships of relative axial position of the cardinal points. Changes to the ring represent the axial displacements of the cardinal points of the visual optical system of an eye that are caused when a spectacle lens compensates for the ametropia. Pascal's schema was described some 70 years ago with little theoretical justification. The purpose of this paper is to derive expressions for the axial locations of the cardinal points of a compound system consisting of an optical instrument and a visual optical system and for the shift caused by the instrument, and to provide theoretical justification for Pascal's schema. The cardinal points are treated not as separate entities but in a unified manner as special cases of an infinite class of special points. Expressions are derived using Gaussian optics. The results are specialized for the case of the eye's ametropia compensated by optical instruments in general and by spectacle lenses in particular. Pascal's schema is shown to be broadly correct although some modification is necessary for the effects on the incident cardinal points especially for the myopic eye.

Effective corneal patch of an astigmatic heterocentric eye.

BACKGROUND: The pupil admits to the back of the eye only some of the light arriving from a point in space. As a result only a portion of the cornea is involved when an eye views the point; it is the effective corneal patch for that point. The location, size and shape of the patch are of interest for corneal refractive surgery inter alia. Previous studies have used geometrical optics and a simple model eye (a naked eye with a spherical, single-surface, centred cornea and a concentric circular pupil). Even for the simplest situations geometrical optics provides only implicit formulae which give little insight into relationships and require numerical solution. PURPOSE: To show how linear optics leads to explicit formulae that estimate the geometry of the effective corneal patch in a very wide range of situations. The eye is not restricted to a single refracting surface; the surfaces may be astigmatic and decentred or tilted and the pupil may be decentred and elliptical. The eye may contain implants and it may be looking through a spectacle lens or other optical instrument which may also contain astigmatic and decentred surfaces. METHODS: Linear optics is used to provide general formulae for the geometry of the corneal patch. An appendix illustrates application to some particular cases. RESULTS: General formulae are obtained for the location and geometry of the effective corneal patch for object points that may be near or distant. Formulae are presented in particular for the special case of the naked eye and the case in which all surfaces are spherical and centred on a common axis. Numerical examples in the appendix allow comparison of results obtained via geometrical and linear optics. CONCLUSIONS: In using linear optics one sacrifices some accuracy at increasing angles away from the longitudinal axis but there is considerable gain in the complexity and range of problems that can be tackled, and the explicit formula one obtains clearly exhibit relationships among parameters of clinical relevance.

Transferences of heterocentric astigmatic catadioptric systems including Purkinje systems.

PURPOSE: To develop the linear optics of general catadioptric systems with allowance for both astigmatism and heterocentricity. METHODS: Reflecting elements partition a catadioptric system into subsystems of four distinct types: (unreversed) dioptric subsystems, anterior catoptric subsystems, reversed dioptric subsystems, and posterior catoptric systems. Differential geometry of an arbitrary astigmatic and tilted or decentered surface is used to determine the anterior and posterior catoptric transferences of a surface. RESULTS: The transference of a catadioptric system is obtained by multiplication of the transferences of unreversed and reversed dioptric subsystems and anterior and posterior catoptric transferences of reflecting elements. Formulae are obtained for the transferences of the visual system of an eye and of six nonvisual systems including the four Purkinje systems. CONCLUSIONS: The transference can be calculated for a catadioptric system, and from it, one can obtain other optical properties of the system including the dioptric power and the locations of the optical axis and cardinal structures.

Back- and front-vertex powers of astigmatic systems.

Back- and front-vertex powers are concepts of some importance in clinical practice. For example, the former is used for characterizing the typical spectacle lens and the latter for characterizing the addition of a bifocal lens. Typically, they are defined either in terms of vergence or the distance to a focal point. This note argues that current definitions are not as clear as they might be, that there is an unnecessary asymmetry between the definitions of front- and back-vertex powers, and that they are designed primarily for systems that are not astigmatic. The purpose of this note is to offer modified definitions that hold for optical systems in general, that is, for systems that may contain astigmatic and decentered refracting elements. The definitions are conceptually clear and provide a simpler derivation of Keating's elegant and general expressions for back- and front-vertex powers.

Optical axes of eyes and other optical systems.

If a ray enters and leaves an optical system along the same straight line that line is an optical axis of the system. The number of optical axes that a system can have is none, one, or infinity. The purpose of the article is to show how to determine whether a system has an optical axis and to find the optical axis if it is unique and all the optical axes if there are an infinity of them. A simple system may have no optical axis or an infinity of them. A more complicated system is more likely to have a unique optical axis. The optical model is linear optics and the optical system may have refracting elements that are relatively decentered, separated, and astigmatic with non-aligned principal meridians. All the possible types of cases are treated in an appendix. In particular an example examines a simple eye that has an infinity of optical axes and a more realistic eye that has a unique optical axis.

Effect of spectacle and contact lenses on the effective corneal refractive zone.

The effective corneal refractive zone is that portion of the cornea traversed by the light that enters the pupil of the eye from object points at a specified angle from the line of sight. It is of relevance in corneal surgery and for understanding the effect of corneal opacities and lesions on vision. Gaussian optics is used in this paper to obtain explicit equations for the geometry of the effective corneal refractive zone for a simplified eye, when spectacle and contact lenses are worn. The theory shows that lenses of positive power increase the diameter of the effective corneal refractive zone and lenses of negative power decrease the diameter. For axial object points the diameter of the effective corneal refractive zone increases by about 0.015 mm per dioptre increase in the power of the spectacle or contact lens. For object points at 30 degrees from the longitudinal axis, the increase is about twice as much in the case of contact lenses and more than four times as much in the case of spectacle lenses.

Effective corneal refractive zone in terms of Gaussian optics.

The portion of the cornea that transmits light for vision is clinically important in several contexts, including corneal ablation in refractive surgery. In contrast to geometric optics, Gaussian optics allows one to obtain simple, explicit formulas for the geometry of the effective corneal refractive zone for distant object points that are on or off the line of sight. In this article, Gaussian optics was used to derive the formula for the diameter of the zone and, when the zone is annular, the inner and outer diameter, as a function of corneal power, anterior chamber depth, pupil diameter, and angular position of the object point.

Power vectors versus power matrices, and the mathematical nature of dioptric power.

Representation of astigmatic dioptric power as a power vector is satisfactory for basic operations such as summing and averaging powers. However, power vectors do not fully characterize the nature of dioptric power and are, therefore, unsatisfactory for representing power in general. The purpose of this note is to make the case that it is the power matrix instead that is proper for the representation of power in general. The mathematical nature of dioptric power is examined.

Analyzing refractive data.

PURPOSE: To provide a general approach to the analysis of refractive data that overcomes the shortcomings of traditional treatments and can be easily adapted to most spreadsheets. SETTING: Corneal Service, Royal Liverpool Hospital, Liverpool United Kingdom, and Optometric Science Group, Rand Afrikaans University, Auckland Park, South Africa. METHOD: The basis of the analyses is the dioptric power matrix. Using a hypothetical sample of data on pre-event and post-event refractions, the calculation of the mean pre-event and post-event refractions and the effect of an event on the refractive outcome, for example the refractive surgical effect, are illustrated. The most important statistics, the mean and the variance-covariance of refractions, and the formal testing of hypotheses on the mean are provided. RESULTS: The method of analysis demonstrated how an event such as cataract surgery, occlusion treatment of amblyopia, or anisometropia can be evaluated in terms of refractive outcome. Hypothesis testing showed how the significance of this effect may be demonstrated. CONCLUSIONS: This standardized method of analyzing and reporting refractive data enables a quantitative analysis and statistical hypothesis testing of the complete refractive data. This approach has generalized applicability in a range of commonly encountered contexts such as the refractive change after cataract and refractive surgery, corneal transplantation, and treatment for amblyopia or the significance of anisometropia. The method is relatively straightforward and can be adapted to most conventional spreadsheets.