## Introduction

This review introduces the reader to linear optics, which, at its heart, is the study of paraxial optical systems that include elements that may be astigmatic, tilted and decentred, such as the eye. Linear optics allows one to obtain explicit formulae for optical properties and give insight into the relationships and dependencies among the optical properties. An advantage of linear optics is that it gives insight into understanding clinical phenomena in astigmatic heterocentric optical systems, such as the eye, and as a first step towards the design process. Linear optics is based on first-order optics to give information about the position of the image and the geometry of the blur patch,1 whereas higher-order aberrations and exact ray tracing techniques give information about the quality of an image. Together, defocus and astigmatism provide the greatest contribution to the image quality and are the optical components typically measured and corrected by conventional lenses.

Linear optics makes use of two concepts, the ray state or ray state vector and the transference of an optical system, also known as the system matrix,2–4 ray transfer matrix,5 ray transference,6 system matrix7 or *ABCD* matrix.5 8 9 The transference is a symplectic matrix that represents all the fundamental first-order optical properties of an optical system.6 10 For a Gaussian optical system that has elements that are centred and rotationally symmetric, the order of the transference is
11–14; where the elements are astigmatic, the transference is ,6 15 and where, in addition, the elements are decentred or tilted, such as in the eye, a augmented transference can be used.2 16

The ray vector6 17 or ray state18 is a matrix that defines the ray in terms of its transverse position and reduced inclination at a transverse plane. The transference operates on the incident ray state to provide the emergent ray state from a system. The arrangement of the entries of the transference or system matrix and corresponding ray state vector may differ2–4 6 7 16 19–21 from the arrangement in this review.

The Results explain how to define an optical system, obtain the transference and how the transference operates on the incident ray state to obtain the emergent ray state. Four special systems are presented, leading to a discussion on how to obtain the most commonly used derived properties. In linear optics, familiar optical properties such as power, front-vertex and back-vertex powers, refractive compensation, thick lens power, transverse and angular magnification, neutralising powers and prismatic effect are derived from the transference and are generalised to three-dimensional and four-dimensional concepts for thick systems. The transference is a matrix that belongs to the real symplectic Lie group,22–24 which poses restrictions for quantitative analysis of linear optical systems because transferences do not constitute a linear (or vector) space; for example, they cannot be added. Options that are currently available for quantitative analysis of optical systems using transformed transferences are mentioned.

The Discussion reviews a range of topics where linear optics has been applied, including magnification, referred apertures, cardinal points, axes of the eye, chromatic aberrations, flipped, reversed and catadioptric systems, Gradient Indices (GRINs) and intraocular lenses (IOLs). Studies using linear optics to analyse their data are mentioned. Most of the Discussion is summarised and given without proof; however, the proofs are available in the relevant referenced papers.