Aperture referral in dioptric systems with stigmatic elements

W F Harris

doi:10.4102/aveh.v71i1.60

Abstract

A previous paper develops the general theory of aperture referral in linear optics and shows how several ostensibly distinct concepts, including the blur patch on the retina, the effective cornealpatch, the projective field and the field of view, are now unified as particular applications of the general theory. The theory allows for astigmatism and heterocentricity. Symplecticity and the generality of the approach, however, make it difficult to gain insight and mean that the material is not accessible to readers unfamiliar with matrices and linear algebra. The purpose of this paper is to examine whatis, perhaps, the most important special case, that in which astigmatism is ignored. Symplecticity and, hence, the mathematics become greatly simplified. The mathematics reduces largely to elementary vector algebra and, in some places, simple scalar algebra and yet retains the mathematical form of the general approach. As a result the paper allows insight into and provides a stepping stone to the general theory. Under referral an aperture under-goes simple scalar magnification and transverse translation. The paper pays particular attention to referral to transverse planes in the neighbourhood of a focal point where the magnification may be positive, zero or negative. Circular apertures are treated as special cases of elliptical apertures and the meaning of referred apertures of negative radius is explained briefly. (S Afr Optom 2012 71(1) 3-11)

Highlights

The blur patch on the retina, the effective patch on the cornea, or on any refracting surface, the projective field of a retinal point and the field of view of an optical instrument, all important in vision, are usually thought of as distinct concepts that need to be treated separately
Because of symplecticity, the transference of an optical system simplifies from 20 numbers related by six equations to only eight related by only one equation. (Results concerning simplecticity in the context of linear optics are summarized elsewhere2.) What is a problem in linear algebra becomes, more or less, a problem in the much more familiar vector
Eliminating astigmatism simplifies the linear magnification[8] Xy,1→2 of the general case[1] to the conceptually much simpler and more familiar scalar magnification Xy,1→2 . 2 × 2 matrices have been eliminated in Equation 8 and in Table 1

Summary

Introduction

The blur patch on the retina, the effective patch on the cornea, or on any refracting surface, the projective field of a retinal point and the field of view of an optical instrument, all important in vision, are usually thought of as distinct concepts that need to be treated separately. A recent paper[1] shows, that each is a special case of a general phenomenon described there as aperture referral. The paper[1] uses linear optics to develop a general theory of aperture referral for general dioptric systems whose refracting elements may be heterocentric and astigmatic. Allowance for astigmatism in the general theory means that symplecticity takes its full form. We treat the special case in which all refracting elements are stigmatic. Because of symplecticity, the transference of an optical system simplifies from 20 numbers related by six equations to only eight related by only one equation. Because of symplecticity, the transference of an optical system simplifies from 20 numbers related by six equations to only eight related by only one equation. (Results concerning simplecticity in the context of linear optics are summarized elsewhere2.) What is a problem in linear algebra becomes, more or less, a problem in the much more familiar vector

Objectives

Conclusion