Introduction

As magnifying devices, simple magnifying lenses are restricted to short working distances. The magnification and working distance are closely related and therefore single lens magnifiers are only useful for magnifying objects close to the lens and the higher the magnification, the closer this distance has to be. However, if one goes to two lens systems, magnification and working distance can be made independent and in principle, one can design a two lens system to give any magnification for any working distance. The microscope may be regarded as an exception to this rule in the sense that the higher magnifications lead to shorter working distances. Thus microscopes act more like simple magnifiers and do not make full use of the potential independence of working distance and magnification that are possible with a system consisting of two lenses.

In this chapter, another special case will be considered, one in which the working distance is infinite or very long. The completely general case will be left until the next chapter. The class of instruments used for providing magnification for very distant objects are called telescopes. These are usually regarded as afocal; that is their equivalent (refractive) power is zero. Telescopes are not only used as magnifying devices. Their unique properties make them ideal as aligning or focussing devices in a range of optical instruments. Binoculars are made from two identical telescopes with parallel optical axes. Telescopes may be classified as either refracting or reflecting. The distinction is the type of leading component called the objective. In a refracting telescope, the objective is a lens or lens system. In the reflecting telescope, the objective is a mirror or mirror system.

Introduction

Paraxial optics only gives a guide to the image formation by real optical systems. Real imagery is different from the ideal or Gaussian model because of the effects of aberrations and diffraction. Both of these cause the light distribution in the image space, and most importantly in the Gaussian image plane, to be different from that in the object space or plane. Whereas diffraction can only be explained in terms of physical optics, aberrations can be discussed in terms of either geometrical or physical optics. As a general rule, geometrical optics only adequately describes the image plane light level distribution on a coarse scale and this is only accurate in highly aberrated systems. On the other hand, physical optics is more accurate than geometrical optics and so better describes the light level distribution on a fine scale, which is particularly important when the aberrations are small or zero. However, physical optical calculations are usually more complex and difficult and therefore we prefer to use the simpler geometric optical approach as often as possible.

Aberrations may be defined as the factors which cause the departure of real rays from the paths predicted by Gaussian optics. They may be investigated by following the paths of real rays through an optical system, using some suitable ray tracing procedure (e.g. the one described in Section 2.3 of Chapter 2) and comparing their paths with the paths of equivalent paraxial rays.

Aberration of a beam

Beams, not single rays, form images and therefore the quality of an image depends upon the combined aberrations of all the rays in the beam.

Introduction

In this chapter, we will introduce the concept of an image forming system in its most general sense. By tracing rays from an object through the system, using Snell's law at each surface, we will show how to find the image of that object. When we decide to ray trace, there are two types of rays that we can choose, (a) finite or real rays and (b) paraxial rays. A finite or real ray is a general exact ray, and a paraxial ray is a special type of finite ray that is traced very close to the optical axis. One distinct advantage of paraxial rays is that their ray trace equations are much simpler than finite ray trace equations and hence are easier to apply. In this chapter, we will look at each of these two types and use the paraxial rays to develop a concept of the “ideal” image.

In the next chapter, Chapter 3, we will use the behaviour of paraxial rays to explore some of the properties of both simple and more complex optical systems. We will show that given the details of these properties, we can often find the ideal image positions and sizes without recourse to any type of ray tracing.

Image formation

We define an imaging optical system as a system consisting of any number of refracting or reflecting surfaces. Usually the surfaces will be spherical and we will assume that the centres of curvature of each of the spherical surfaces lie on a single line called the optical axis. Such a system is depicted schematically in Figure 2.1, but without any individual surfaces shown.

Introduction

Aberrations were introduced in Chapter 5 but only discussed qualitatively. Now they will be discussed quantitatively and in greater detail. Equations will be presented for calculating aberration levels as a function of system construction parameters, aperture stop size, conjugate plane positions and position of the object point in the field-of-view for any rotationally symmetric system. However, since the derivations of these equations are complex, space consuming and adequately covered in other texts, most of the equations will be presented here without any derivation. The equations will be mostly drawn from two texts: Hopkins (1950) and Welford (1986). Derivations of equations will only be included if the derivations are not adequately or suitably covered elsewhere.

The calculation of exact aberrations requires time consuming and tedious tracing of real rays. On the other hand, an estimate of the aberration levels can be found relatively simply from the results of two suitable paraxial ray traces. For many purposes, these estimates of aberration levels are adequate. The two paraxial rays are the marginal and pupil rays. The ray angles {u} and ray heights {h} along with other system constructional parameters are fed into equations for the calculations of these aberrations. One such set of equations is the Seidel aberration equations and the resulting estimates of aberrations are called Seidel aberrations. These equations will be introduced and discussed in the next section. While these equations are approximate, they have three very useful attributes: (a) they allow the identification and quantification of different aberration types such as spherical aberration and coma, (b) they give the aberration contribution by each surface and (c) they become more accurate the smaller the aperture and field size.

Introduction

It is sometimes necessary to determine the structural properties of a single lens or lens system. This information may be needed to understand the optical properties which can be further investigated by ray tracing. For example, we can use the knowledge of the refractive indices, surface curvatures and surface separations to determine the Gaussian properties such as the powers and positions of the cardinal points by paraxial ray tracing, using techniques described in Chapter 3. The ray trace results can also be used to calculate the primary or Seidel aberrations using the equations given in Chapter 33. The powers and cardinal points can also be measured directly by laboratory techniques without the need to take the system apart.

Thus this chapter is concerned with the analysis of optical systems that have been constructed and for which we do not have the constructional details. That is we do not know the refractive indices of the materials, the surface separations or the surface curvatures. If we wish to examine the optical properties of such a system, we can solve the problem in two ways.

1. (1) We can take the system apart, measure the refractive indices, surface separations and surface curvatures. Using paraxial ray tracing, we can determine the Gaussian properties such as equivalent power and positions of the cardinal points.

2. (2) We can measure the Gaussian properties directly. For example, in this chapter we will describe several methods for measuring the equivalent power of an optical system in the laboratory.

The powers and cardinal point positions are not the only Gaussian parameters that occasionally have to be verified.

Introduction

Collimators are optical systems designed to produce a reasonable quality image of a target (or light source or some other object) at optical infinity. The angular size of the image is usually small; therefore the field-of-view is small and thus the system is relatively simple. Since the target has to be imaged at optical infinity, it must be placed at the front focal point of the collimator lens.

Collimated light is often referred to incorrectly as parallel light. No doubt, the term arises because paraxial or unaberrated real rays from a single point in the object or target are all parallel to each other in image space. This term often leads to the misunderstanding that a collimated beam has parallel sides. If this were true, a collimated beam would have zero divergence. In reality, a collimated beam diverges and there are three causes of this divergence: (1) the finite size of the source or target, (2) aberrations and (3) diffraction. Diffraction usually only dominates the divergence if the beam has a small diameter, say several millimetres or less. The diameter of collimators used in visual optics is usually much wider than this and therefore source size and aberrations are the dominant causes of beam divergence. Let us look at these in turn.

Effect of source size

In Gaussian optics, the beam must diverge and the amount of divergence is proportional to the size of the source or target. This can be easily demonstrated using Figure 23.1, which shows a source or target of radius η at the front focal point F of the collimating lens.

Introduction

This chapter is a brief introduction to the paraxial theory of reflecting optics. The term “mirror” has not been used because although all refracting surfaces also act as reflecting surfaces, they cannot be classified as mirrors. Here mirrors are defined as reflecting surfaces where there is no transmission of rays.

Although reflecting optics do not have a very large role to play in the optics of the eye or visual and ophthalmic instruments, they are very useful and important in some cases. The reflections from the four refracting surfaces of the eye are most useful as they can be used to measure the radii of curvature of these surfaces. Measurement of the radius of curvature of the cornea is a special case and is called keratometry. Reflections from the refracting surfaces of the eye are known as Purkinje images.

Many optical situations involving reflections also involve some refraction. For example in the measurement of the radius of curvature of the front surface of the crystalline lens of the eye, the beam is refracted by the cornea, reflected from the front surface of the lens and then refracted by the cornea once again. Optical systems that are a mix of refracting and reflecting elements are called catadioptric systems (see Section 4.2). Those that are purely reflecting are called catoptric. However, very few optical systems are catoptric.

Reflecting components are often used instead of refracting components because they can be made with a smaller mass and they have no chromatic aberration. They are also useful with high energy beams, where the smallest amount of absorption would damage a lens.

Introduction

The primary role of an ophthalmic lens is to correct a refractive error of the eye, thus allowing the eye to clearly see objects at a chosen distance. The refractive error may be due to myopia, hyperopia, presbyopia or astigmatic errors and these have been explained in Chapter 13. However, optical refractive corrections have some side effects such as altering the effective positions of the near and far points, altering retinal image sizes, making it more difficult to satisfactorily use visual instruments and finally their aberrations may lead to reduced visual performance. These different aspects of ophthalmic lenses will now be discussed in detail.

Spectacle lenses, contact lenses or intra-ocular lenses

Either spectacle or contact lenses may be used to correct refractive errors. Both have their advantages and disadvantages. For example, spectacle lenses have little or no biological interaction with the tissues of the eye and therefore cause less or no biological reaction. However, they have a more restricted visual field and affect the size of the retinal image. This change in retinal image size is called spectacle magnification and the magnitude increases with lens power and distance of the lens from the eye or more strictly, from the entrance pupil. Contact lenses also have spectacle magnification but since these are much closer to the pupil, their spectacle magnification is much less than that of spectacle lenses. Intra-ocular lenses are artificial lenses inserted in the eye to replace the original lens after it has been removed, usually because of a cataract. Because these lenses are placed close to or in the same position as the original lens, that is close to the pupil, their spectacle magnification is almost zero.

Introduction

Projection systems are optical systems designed to project images of solid objects and photographic objects such as transparencies, usually with some magnification, onto an observation screen. Typical uses are as profile projectors in engineering, 35 mm photographic slide projectors, motion picture projectors, microfilm and microfiche readers and photographic enlargers. They consist of a projection lens, an illumination system and a screen on which a real image is observed. A typical projection system is shown in Figure 22.1. Some projection systems, though very few, have no screen because the image is virtual and this is projected directly into the eye using an eyepiece. Figure 22.1 shows the object being trans-illuminated, but some objects are opaque and the reflected light is used to form the projected image. In the common photographic projector, the object is commonly a piece of photographic film either in positive or negative form.

The projection lens

The projection lens is a positive equivalent power lens, usually well corrected for aberrations. It usually does not contain an aperture stop. Instead, the aperture stop is provided by the illuminating system with the image of the light source being imaged into the projection lens and acting as the effective entrance pupil of the projection lens. We will discuss the pupils of a projection system in Section 22.3.

The optics of the projection system are shown in Figure 22.1, where the projection lens is depicted as a thin lens. In reality, the projection lens will be more complex mainly because of the need to give good image quality for a wide aperture and over a wide field.

Introduction

The performance of visual optical instruments cannot be fully assessed without some knowledge of the anatomy and functions of the eye, working either monocularly or binocularly. This chapter describes the optics of the eye, but its interaction with visual instruments is covered later in Chapters 36 and 37.

A cross-section of the human eye is shown in Figure 13.1, giving only the most relevant optical components. A more detailed anatomical description can be found in a number of textbooks, for example Davson (1990). Image forming light enters the eye through and is refracted by the cornea. It is further refracted by the lens, bringing it to a focus on the retina. Of the two refracting elements, the cornea has the greater refractive power. However, whereas the power of the cornea is constant, the power of the lens depends upon the level of accommodation, which is the process by which the refractive power of the eye changes to allow closer or more distant objects to be sharply imaged on the retina. The diameter of the incoming beam of light is controlled by the iris, which is the aperture stop of the eye.

The dimensions of the eye and its optical components vary greatly from person to person and some further depend upon accommodation level, age and certain pathological conditions. In spite of these variations, average values have been used to construct representative or schematic eyes. These are discussed further in Section 13.6.

The refractive components

The relaxed eye has an equivalent power of about 60 m–1. The corneal power is about 40 m–1, which is two-thirds of the total power.

Introduction

The simple magnifier has an upper limit of magnification of about 20. Above this value, the lens becomes too small and the aberrations become too high to form a useful image. When higher magnifications are required, they must be achieved by a two stage process. Two stage magnification is possible by using two lenses as shown in Figure 16.1 and the extra complexity allows more freedom to control the aberrations. The first stage magnification is done by the objective and magnifications of between 10 and 100 are achieved depending upon the equivalent power of the objective. The objective forms a real, inverted and magnified image of the object. This image is further magnified by the eye lens. The eye lens is effectively a simple magnifier and therefore the upper limit of magnification is that of a simple magnifier, that is about 20. Therefore the upper limit of the magnification of the microscope as a whole is about 2000. Thus the extra magnification gained by a two component microscope over the simple magnifier is just that gained by the magnification due to the objective.

Construction and image formation

A microscope basically consists of two positive power lenses: the objective and the eye lens, as shown in Figure 16.1. The objective carries out the first stage of magnification and produces a real image of the object. The second lens (the eye lens) further magnifies the image. The objective is the aperture stop. Usually a field lens and field stop are used at or near the intermediate image plane in order to reduce vignetting and hence provide a wider field-of-view.

Introduction

Ray tracing, with either paraxial or real rays, is insufficient to assess the efficiency of optical systems or indicate the quality of the final image. Apart from the effect of aberrations, it is also necessary to understand basic photometric principles, photometric quantities such as source luminance and surface illuminance, and a number of other factors such as vignetting which affect the image plane illuminance in optical systems.

Before beginning a study of the photometry of optical systems, it is necessary to understand the four basic photometric quantities, namely luminous flux, luminous intensity, luminance and illuminance. For a long time, photometry was regarded as an independent field of study with its own fundamental units and international standards. For example, luminous intensity has been a fundamental basic physical quantity for many years with a physical standard using a sample of thorium oxide held at the melting point of platinum (2042 K) as the light source (Sanders and Jones 1962). This light source had a defined luminance of 60 x 104 cd/m2. More recently, however, there has been a trend to regard photometry as a branch of radiometry and derive all photometric quantities from radiometric quantities.

Radiometry may be defined as the measurement of the energy or power in an electromagnetic beam, measured over the entire spectrum. The division of the entire electromagnetic spectrum is shown in Figure 1.2, Chapter 1. Light is only a very small part of this spectrum.

The nature of light

Light is that part of the electromagnetic spectrum that elicits a visual response in the eye and is in the range of approximately 400–700 nm. The eye is not equally responsive to electromagnetic energy in this range.