Search

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 r T Ar=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.

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., (Ophthalmic & Physiological Optics © 2012 The College of Optometrists.)

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., (Copyright © 2012 Elsevier Ltd. All rights reserved.)

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.

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., (Ophthalmic & Physiological Optics © 2011 The College of Optometrists.)

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., (© 2010 The College of Optometrists.)

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 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.

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.

. 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 −0.27 D for the distance eye and −1.15 D 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 −1.25 D 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 1.0 D. 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). [ABSTRACT FROM AUTHOR]

Abstract: 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 65years 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 . 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. [Copyright &y& Elsevier]