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.
Thin lenses of asymmetric power
