Ray pencils of general divergency

W F Harris

doi:10.4102/aveh.v68i2.160

Abstract

That a thin refracting element can have a dioptric power which is asymmetric immediately raises questions at the fundamentals of linear optics. In optometry the important concept of vergence, in particular, depends on the concept of a pencil of rays which in turn depends on the existence of a focus. But systems that contain refracting elements of asymmetric power may have no focus at all. Thus the existence of thin systems with asym-metric power forces one to go back to basics and redevelop a linear optics from scratch that is sufficiently general to be able to accommodate suchsystems. This paper offers an axiomatic approach to such a generalized linear optics. The paper makes use of two axioms: (i) a ray in a homogeneous medium is a segment of a straight line, and (ii) at an interface between two homogeneous media a ray refracts according to Snell’s equation. The familiar paraxial assumption of linear optics is also made. From the axioms a pencil of rays at a transverse plane T in a homogeneous medium is defined formally (Definition 1) as an equivalence relation with no necessary association with a focus. At T the reduced inclination of a ray in a pencil is an af-fine function of its transverse position. If the pencilis centred the function is linear. The multiplying factor M, called the divergency of the pencil at T, is a real 2 2× matrix. Equations are derived for the change of divergency across thin systems and homogeneous gaps. Although divergency is un-defined at refracting surfaces and focal planes the pencil of rays is defined at every transverse plane ina system (Definition 2). The eigenstructure gives aprincipal meridional representation of divergency;and divergency can be decomposed into four natural components. Depending on its divergency a pencil in a homogeneous gap may have exactly one point focus, one line focus, two line foci or no foci.Equations are presented for the position of a focusand of its orientation in the case of a line focus. All possible cases are examined. The equations allow matrix step-along procedures for optical systems in general including those with elements that haveasymmetric power. The negative of the divergencyis the (generalized) vergence of the pencil.

Highlights

The concept of vergence lies at the heart of much optometric thought and practice
In other words our approach allows one to do everything the conventional approach does, and much more; it allows one to handle the optics of systems containing lenses of the new type as well. It represents a generalization of conventional linear optics. We show this by deriving basic equations that represent the behaviour of the pencil and its divergency, first through a thin system and across a homogeneous gap
The first thing we shall do is to examine the case of a pencil with M = O D, that is, the divergency is null, at a particular transverse plane T in an optical system that consists of nothing but a homogeneous medium

Summary

Introduction

The concept of vergence lies at the heart of much optometric thought and practice. It is in terms of vergence that much of the optics is understood. A general definition of a pencil of rays at a transverse plane An abbreviated form of Definition 1 might read as follows: a set of rays in a homogeneous medium is a pencil at a transverse plane if the reduced inclination of every ray is an affine function of its transverse position.

Results

Conclusion