Pascal's ring, cardinal points, and refractive compensation

Radial and Tangential Retinal Magnifications as Functions of Visual Field Angle Across Spherical, Oblate, and Prolate Retinal Profiles

Purpose

To provide a tool for calculating radial and tangential retinal magnifications as functions of field angle and retinal shape and to articulate patterns of magnification across the retina for monocular and binocular combinations of prolate-, oblate-, and spherical-shaped retinas.

Methods

Formulae were derived to calculate radial and tangential retinal magnifications (mm/deg) from field angle (degrees), retinal asphericity (unitless conic constant), retinal vertex radius of curvature (mm), and nodal point position (mm). Monocular retinal magnifications were determined for eyes with prolate, spherical, and oblate retinas as functions of field angle. Bilateral differences in magnifications were examined for combinations of those eyes.

Results

Retinal shape substantially affects magnification profiles even for eyes with the same axial length. Greatest magnification changes across a retina and between eyes, as well as greatest increase in radial-tangential differences (distortion), occur with prolate retinas. Binocular magnification differences were smallest for oblate retinas. Nodal points anterior to the vertex center of curvature and oblate asphericity both cause field-dependent reductions in magnification relative to the fovea (barrel distortion), whereas nodal points posterior to vertex center of curvature and prolate asphericity cause the opposite (pincushion distortion). Retinal magnification differences due to eye shape are much greater than aniseikonia thresholds and chromatic differences in magnification. A spreadsheet tool implements the magnification calculations.

Conclusions

Local retinal magnifications as functions of field angle have substantial effects on objective applications (imaging retinal anatomy) and subjective experiences (aniseikonia) and quantify an ocular property that differs across eye shapes and refractive errors.

Translational Relevance

Methods are provided to customize the calculation of radial and tangential magnifications across the retina for individual eyes, which will bolster the multifactorial study of the effects of foveal and peripheral optics across eye shapes and refractive errors.

Linear optics of the eye and optical systems: a review of methods and applications

The purpose of this paper is to review the basic principles of linear optics. A paraxial optical system is represented by a symplectic matrix called the transference, with entries that represent the fundamental properties of a paraxial optical system. Such an optical system may have elements that are astigmatic and decentred or tilted. Nearly all the familiar optical properties of an optical system can be derived from the transference. The transference is readily obtainable, as shown, for Gaussian and astigmatic optical systems, including systems with elements that are decentred or tilted. Four special systems are described and used to obtain the commonly used optical properties including power, refractive compensation, vertex powers, neutralising powers, the generalised Prentice equation and change in vergence across an optical system. The use of linear optics in quantitative analysis and the consequences of symplecticity are discussed.

A systematic review produced 84 relevant papers for inclusion in this review on optical properties of linear systems. Topics reviewed include various magnifications (transverse, angular, spectacle, instrument, aniseikonia, retinal blur), cardinal points and axes of the eye, chromatic aberrations, positioning and design of intraocular lenses, flipped, reversed and catadioptric systems and gradient indices. The optical properties are discussed briefly, with emphasis placed on results and their implications. Many of these optical properties have applications for vision science and eye surgery and some examples of using linear optics for quantitative analyses are mentioned.

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.