### INTRODUCTION

This book represents an alternative approach to that found in most standard texts to the study of image formation by spherical surfaces. As such it complements the usual treatment and may well be used along with another text. The text "Fundamentals of Optics" by F. A. Jenkins and H. E. White (Toronto, McGraw-Hill, 3rd ed., 1957) has been referred to throughout as it uses a sign convention which is consistent with that used in this book.

The subject of geometrical optics starts with the laws of refraction and reflection for transparent media. It is then a question of using these laws to discover the properties of various optical systems which may contain any number of curved refracting and reflecting surfaces. In principle we could solve any optical problem by the exact application of the basic laws. However, it is possible to derive some very useful general results by using approximate forms of the laws which treat only rays, called paraxial rays, which make small angles with respect to an optic axis. Also, it is usual to confine the treatment to spherical surfaces. This approximate form of the basic laws as applied to paraxial rays in an optical system consisting of spherical surfaces is called the first order theory and constitutes a major part of the subject of geometrical optics.

The first order theory forms the basis for obtaining the equation describing image formation which relates image position to object position and defines magnification for the case of a single spherical refracting surface between two transparent media. For an extensive optical system this equation can be applied repeatedly to each surface of the system in turn. However the equations so derived quickly become involved algebraically for all but the simplest cases, such as, for example, the single thin lens in air.

The matrix method is another approach to the study of optical systems using the first order theory. By properly specifying the rays passing through the optical system it is possible to reduce a problem to a succession of matrix operations. This not only simplifies the algebraic manipulation but more importantly reveals some very general properties of optical systems not easily obtainable in other ways.

From the mathematical point of view, the first order theory can be treated in terms of linear transformations. And, since matrix algebra is the most suitable way of handling linear transformations, it is obvious that we should apply the results of matrix algebra to treat the first order optical problem.

The matrix method then represents a basic approach to the analysis of optical systems which results in discovering the cardinal points of a lens or mirror system from a more general point of view. The familiar lens equations for an optical system also result in both the Newtonian and Gaussian forms. This much is accomplished part way through chapter 5. Included in this is a chapter which discusses the properties of two by two matrices as they describe the linear transformation involved in an optical system. The rest of chapter 5 shows a number of alternative ways of expressing image formation in terms of different cardinal points of a lens system. In chapter 6 some practical optical systems are discussed in terms of the matrix approach.

Chapter 7 is a discussion of spherical mirrors by the matrix method. In the Appendix is given a simple computer programme for calculating the matrix for a lens system and some of the characteristics of the system.