Original Research

Ray pencils of general divergency

W. F. Harris
African Vision and Eye Health | South African Optometrist: Vol 68, No 2 | a160 | DOI: https://doi.org/10.4102/aveh.v68i2.160 | © 2009 W. F. Harris | This work is licensed under CC Attribution 4.0
Submitted: 13 December 2009 | Published: 13 December 2009

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W. F. Harris, Department of Optometry, University of Johannesburg *PhD FRSSAf, South Africa

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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 such

systems.  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.


asymmetric dioptric power; vergence; step-along procedures; focal lines; divergency; divergence


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