Basic properties

Thin lenses are made up of elements that are so thin, separated by air gaps that are so small, that all the axial distances may, for all practical purposes, be set equal to zero. Doublets and triplets used as telescope objectives are common examples. For lenses of this nature the effect on the aberrations of introducing the thicknesses is small, so that only small changes in the surface curvatures are needed when, in the end, the finite thicknesses must be accounted for. We will only consider thin lenses used in air.

The rays through the axial point of the lens are undeviated as they emerge into the image space. This point is therefore imaged free from spherical aberration, and, because Abbe's sine rule is clearly satisfied, free from coma as well. If the pupil coincides with the lens, the rays through the center of the lens are the chief rays. As they move on undeviated, there can be no distortion. There is, however, a great deal of astigmatism.

To evaluate the astigmatism, we consider, in the spirit of chapter 23, rays close to one of the chief rays. In the calculation of the four-by-four matrix for these rays the thicknesses are set to zero, so the translation matrices reduce to unit matrices and can be left out of the product. First consider a single thin element. The field angle, i.e. the angle between the axis and the chief ray outside of the lens, is ψ.

Introduction

So far we have only dealt with waves in unbounded spaces. The item now on the agenda is extending the wave theory to encompass the passage of waves through lenses.

When a light wave enters a lens, it is redirected and distorted by the lens surfaces, and truncated by the diaphragm as well as by the lens edges. The calculation of the wave arriving in the image space requires, from the point of view of an uncompromising physicist, the solution of a boundary value problem so complicated that its exact solution is hopelessly intractable except for very simple cases, such as diffraction by a solid homogeneous sphere [18] or the reflection of a plane wave by the exterior of a perfectly conducting paraboloid [26]. Courageous approximations are clearly needed to arrive at a theory that can be used in the daily practice of lens design.

One simplifying feature of the problem is the linear relation between the field in the object space and the field in the image space. The field in the image space created by the sum of several input fields is the sum of the image fields created by the individual inputs. There are, of course, exceptions to this rule. Non-linear media may display frequency doubling; absorption of the input wave may cause thermal effects that change the lens characteristics, etc. Such effects can better be dealt with on a case to case basis after the linear theory is worked out in detail.

Introduction

In this chapter we show how to calculate the paraxial properties of a lens when its construction parameters are specified. Rather than using the eikonal techniques of the previous chapter, we base this work on Snell's law and simple geometry. The results will be quite similar to what we found before; in particular we will rederive the linear equations connecting the ray parameters in the object and image space.

Why then did we bother with the heavy machinery of the previous chapters at all? There are two important reasons. First, in the current chapter we assume right from the start that all angles with the axis are small. Extending this work to larger angles leads to a thicket of thorny mathematics that firmly refuses to provide any general insights. Secondly, the eikonal functions introduced in the previous chapters play, as we shall see later, an indispensable role in the diffraction theory of image formation, where they appear again and again as phase functions in wave propagation integrals. Snell's law is not a suitable tool to clarify the relations between geometrical optics and the wave theory.

Notation and sign rules

An unambiguous notation is needed to keep track of the many parameters involved. The rules we shall use in this book are listed below.

(i) In the object space the light travels from left to right.

(ii) A lower case r denotes the radius of curvature of a lens surface. Its reciprocal 1/r is the curvature R.

The theorem of Malus and Dupin

The vast majority of lenses are used for image formation. Camera lenses, eye glasses, binoculars, and all sorts of other lens systems are useful because they form, each in their own way, an image of an object. To begin a discussion of the image forming process we choose a fixed source point P0 in the object space of a lens that we wish to study, and follow the rays emerging from this point all the way through the lens to the image space. One might hope that the rays emerging into the image space pass through a single point again, but this is rarely the case. Usually each of the emerging rays intersects the nominal image plane at a slightly different point. It is the task of the lens designer to bring, for a specified set of conditions, these intersection points as close together as possible.

Although the emerging rays do not usually pass through one point, the ray patterns in the image space created by a single source point in the object space are not completely arbitrary. Fermat's principle imposes an important restriction. To analyze this restriction we choose in the image space any convenient Cartesian coordinate system (x, y, z) and introduce the path function E(x, y) from the source point P0 to points (x, y, 0) in the z = 0 coordinate plane.

Characterizing a lens

A ray can be specified by one of its points and its direction at that point. In a homogeneous medium we can, for instance, select a ray by specifying the coordinates (x, y) of its point of intersection with the plane z = 0, and the first two components (L, M) of the unit vector in the direction of the ray. The third direction cosine N is not needed because it is the square root of (1 – L2 – M2). The number of parameters needed to specify a ray is clearly four: x, y, L, and M.

Now consider a ray as it approaches a lens. It enters the lens, travels through the lens along a possibly tortuous path, and leaves the lens to enter the image space. Each of the four parameters needed to give a complete description of the ray in the image space is determined by the four parameters that specify the ray in the object space. It seems to follow that four functions of four variables are needed to characterize the lens fully: x′, y′, L′, and M′, each as a function of x, y, L, and M. In this chapter we shall see that this conclusion is incorrect; on account of Fermat's principle one single function of four variables is all that is required. As a result, Fermat's principle puts a severe constraint on the imaging processes a lens can carry out.

Many books on modern optics confine the treatment of lens theory to the paraxial approximation. Aberrations are treated casually as an afterthought, and it usually remains unexplained whether they are due to the laws of physics or due to the limited art of the lens designer. The reader is often left with the notion that lenses must be designed to obey the laws of Gaussian optics as closely as possible. This is regrettable, because it has been known since the eighteenth century that paraxial optics used with finite heights and angles leads to projective geometry, a valuable branch of mathematics which is, however, a poor representation of the behavior of actual lenses.

The development of Fourier optics over the last forty years has brought lens theory and physics much closer together, but again many insights are lost because most authors, in spite of the ubiquity of high aperture lenses in the laboratory, are content with the small angle approximation when dealing with the theory of image formation. Either a clear and convincing demonstration should be given that the small angle approximation can be used with impunity for very large angles, or the theory should be developed honestly, without the small angle approximation. This honest theory exists, but is buried in books and papers providing so much detail that beginners are apt to get lost in the mathematical intricacies.