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In Gaussian optics all observers experience the same magnification, the instrument’s angular magnification, when viewing distant objects though a telescope or other afocal instruments. However, analysis in linear optics shows that this is not necessarily so in the presence of astigmatism. Because astigmatism may distort and rotate images it is appropriate to work with generalized angular magnification represented by a 2 × 2 matrix. An expression is derived for the generalized magnification for an arbitrary eye looking through an arbitrary afocal instrument. With afocal instruments containing astigmatic refracting elements not all eyes experience the same generalized magnification; there is interaction between eye and instrument. Eye-instrument interaction may change as the instrument is rotated about its longitudinal axis, there being no interaction in particular orientations. A simple numerical example is given. For sake of completeness, expressions for generalized magnification are also presented in the case of instruments that are not afocal and objects that are not distant. [ABSTRACT FROM AUTHOR]

Background: Chromatic differences in power, refractive compensation, position and magnification have been described in the literature for Gaussian eyes. Aim: This article explores these definitions and defines chromatic properties for eyes that have astigmatic and decentred or tilted elements. Setting: Linear optics. Methods: The optical model is linear optics and makes use of the ray transference T. Results: Results are presented as either 2 × 2 matrices or 2 × 1 vectors. The dependence of the chromatic properties on the position of an object and the limiting aperture is explored and results are presented as independent of, or dependent on, object and aperture position. Apertures undergo both longitudinal and transverse shifts in position. The results are general and apertures may include pupils or pinholes, either surgically inserted inside the eye or held in front of the eye. Numerical examples are provided for Le Grand's four-surface eye and an arbitrary astigmatic heterocentric four-surface eye. Conclusion: Aperture-independent chromatic properties include chromatic difference in power and refractive compensation, both given as 2 × 2 matrices. Aperture-dependent chromatic properties are all dependent on longitudinal shifts in the plane of the limiting aperture. In addition, chromatic difference in position and inclination depend on both object and transverse aperture position. Chromatic difference in image size or angular spread depends on object position and is independent of transverse aperture position. These four aperturedependent chromatic properties are given as 2 × 1 vectors. Chromatic magnifications are independent of object and transverse aperture position and are given as 2 × 2 matrices. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

: PurposeTo 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.: SettingCorneal Service, Royal Liverpool Hospital, Liverpool United Kingdom, and Optometric Science Group, Rand Afrikaans University, Auckland Park, South Africa.: MethodThe 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.: ResultsThe 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.: ConclusionsThis 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. [Copyright &y& Elsevier]

Because dioptric power matrices of thin systems constitute a (three-dimensional) inner-product space, it is possible to define distances and angles in the space and so do quantitative analyses on dioptric power for thin systems. That includes astigmatic corneal powers and refractive errors. The purpose of this study is to generalise to thick systems. The paper begins with the ray transference of a system. Two 10-dimensional inner-product spaces are devised for the holistic quantitative analysis of the linear optical character of optical systems. One is based on the point characteristic and the other on the angle characteristic; the first has distances with the physical dimension L-1 and the second has the physical dimension L. A numerical example calculates the locations, distances from the origin and angles subtended at the origin in the 10-dimensional space for two arbitrary astigmatic eyes. [ABSTRACT FROM AUTHOR]

The transference defines the first-order character of an optical system; almost all the system's optical properties can be calculated from it. It is useful, therefore, to have some idea of how it depends on the frequency of light. We examine the dependence for two Gaussian eyes. It turns out to be nearly linear for all four fundamental properties. The result is an equation for the dependence of the transference on frequency which is almost symplectic. We also transform the transference into Hamiltonian space, obtain equations for the least-squares straight line for the three independent transformed properties and map them back to the group of transferences. The result is an equation for the dependence of the transference on frequency which is exactly symplectic and therefore representative of an optical system. The results may approximate those of real eyes and give estimates of the dependence of almost all optical properties on frequency. [ABSTRACT FROM AUTHOR]

In the author list of this article published earlier, Tanya Evans' ORCID was unintentionally misprinted as https://orcid.org/0000-0002-5503-0245. [Extracted from the article]

Purpose: Providing vision correction in communities where healthcare facilities are unavailable is problematic. Here, we explore the possibility of using pinholes as near visual aids in the developing world. Method: Uncompensated (without the pinhole) proximal visual acuities (VAs) were measured and recorded for one eye of 45 presbyopic subjects over the age of 50 years in photopic (250 lux) and scotopic (20 lux) conditions. The proximal VAs were then repeated while the subjects looked through nine different pinholes. Results: The VAs varied with the diameter of each pinhole. A comparative measure was devised to determine the relative performance of different diameter pinholes as a near visual aid. The results suggest an improvement of more than 40% in VA in about 67% of subjects in good lighting conditions and about 60% in poorer lighting conditions. Conclusion: By creating more awareness that there are other ways of improving vision that cost almost nothing, visual impairment can be reduced until spectacles become available. [ABSTRACT FROM AUTHOR]

United States demographics are changing, but that change is yet to be recognized in the makeup of the nursing workforce. The underrepresentation of minorities in nursing is a longstanding problem, resulting in missed opportunities for culturally sensitive care that can foster optimal patient care outcomes. This report describes qualitative analysis of leadership opportunities emerging from a collaborative leadership development program between an urban baccalaureate nursing program and a large healthcare system in the northeast United States. The investigative team used qualitative methods to analyze the experiences of 19 participating African American (AA) RN graduates. Findings demonstrated that, for program participants, active engagement in personal and professional nursing leadership activities was enhanced. The academic-practice leadership program was an effective tool for enculturating AA nurses into the practice environment and into eventual leadership in nursing and healthcare.

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.

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.

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.

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.