In previous chapters interactions with structureless point charge projectiles have been considered. There are many interactions, however, which involve at least two atomic centers with one or more electrons on each center. In such cases the projectile is not well localized and there is a need to integrate over the non localized electron cloud density of the projectile. Evaluation of cross sections and transition rates for such processes requires a method for dealing with at least four interacting bodies. If multiple electron transitions occur on any of the atomic centers, then some form of even higher order many body theory is required. In general such a many body description is difficult.

In this chapter the probability amplitude for a transition of a target electron caused by a charged projectile carrying an electron is formulated. This probability amplitude may be used for transitions of multiple target electrons if the correlation interaction between the target electrons is neglected. Unless the projectile is simply considered as an effective point projectile with a charge Zeff, the interaction between the target and the projectile electrons may not be ignored. Since this interaction is between electrons on two different atomic centers, the effects of this interaction have been referred to as two center correlation effects (Cf. section 6.2.4).

Introduction

In this chapter interactions of photons with atoms are considered. Here the emphasis is on systems interacting with weak electromagnetic fields so that a single atomic electron interacts with a single photon. Initially interactions with a single electron are considered. In this case the photon tends to probe in a comparatively delicate way the details of the atomic wavefunction (e.g. effects of static correlation in multi-electron atoms). Later two electron transitions are considered. Because these two electron transitions are often negligible in the absence of electron correlation the two electron transitions are usually a direct probe of the dynamics of electron correlation.

In previous chapters the impact parameter (or particle) picture has been used wherever possible in order to recover the product form for the transition probability in the limit of zero correlation. However, here the likelihood of interacting with more than a single photon is quite small since the electromagnetic field of a photon, even for strong laser fields, is almost always small compared with the electric field provided by the target nucleus. Consequently, this independent electron limit is not often useful. Also, photon wavepackets are usually much larger in size than an atom. Consequently the wave picture is used where the electric and magnetic fields of the photon are considered to be plane waves. Transformation to the particle picture may be done using the usual Fourier transform from the scattering amplitude to the probability amplitude (Cf. section 3.3.3).

Introduction

Ophthalmoscopes are optical instruments designed for the visual inspection of the internal structure of the eye, but most commonly the retina. However, because the amount of light reflected from the subject's eye is very low, the observer will only see an image if the subject's retina is well illuminated and thus an auxiliary illuminating system is an essential component of an ophthalmoscope. Thus ophthalmoscopes consist of two main components: a viewing system and an illuminating system.

There are a number of different designs for the viewing system, and these can be divided into two groups, as follows:

1. (a) Direct ophthalmoscopes. These are the simpler of the two types. They are discussed in detail in the next section, Section 29.1.

2. (b) Indirect ophthalmoscopes. The viewing system is more complicated than for direct ophthalmoscopes, having extra lenses between the subject's and observer's eyes. The extra complexity allows independent control over field-of-view and magnification. Indirect ophthalmoscopes are discussed in detail in Sections 29.2 and 29.3.

In this chapter, we will concentrate on the viewing system and only make a brief mention of the illumination system of the direct ophthalmoscope.

The magnification in direct ophthalmoscopy is often quoted as 15. In contrast, the magnification of indirect ophthalmoscopy is quoted as being much lower, usually in the region of about (–)3. However, once we look at the construction of indirect ophthalmoscopes, we will see that there is some potential ambiguity in the way that the magnification of an indirect ophthalmoscope is defined, but for the moment we will leave that problem aside and begin by looking at the properties of the direct ophthalmoscope.

Introduction

A wide variety of visual optical instruments are designed for binocular viewing, but probably the most well known are binoculars. Binocular viewing seems to offer a number of advantages over monocular viewing. Monocular viewing usually requires one eye to be closed, which may often lead to fatigue or discomfort, especially over extended viewing periods, although it may be possible for some people to keep both eyes open when viewing monocularly. This is possible if the image of the other eye is suppressed. In comparison, binocular viewing uses both eyes and hence is probably far less fatiguing. Binocular instruments also have the potential for stereoscopic viewing. Thus superficially, binocular viewing seems to be superior to monocular viewing. However a badly designed, badly manufactured or damaged binocular system can lead to significant problems in binocular viewing. Therefore we should be aware of the need for design and constructional tolerances for binocular instruments.

Stereoscopic and non-stereoscopic constructions

Not all binocular systems provide a stereoscopic image. The production of stereoscopic images requires that the two optical axes at the eyepieces be separated in object space. Let us look at several constructions and see how this requirement may be achieved.

Non-stereoscopic systems

Figure 37.1 shows a possible schematic construction of a binocular instrument, in which the two optical axes are identical in object space and hence the eyes will not see a stereoscopic image. In practical systems, the beam-splitting and deviations are often done with prisms rather than with mirrors as depicted in the diagram.

Introduction

A number of visual optical instruments are designed to measure the angles between two distant objects or the distance of an object. The determination of the angle between two distant objects is frequently done in surveying. Using simple rules of trigonometry, the location of any point can be determined if the direction of two other points and the distances between any two pairs of points are known. The determination of the elevation of celestial objects combined with astronomical tables and an accurate clock can be used to determine a position on the earth's surface, for surveying and navigation.

In many applications, these instruments have been superseded by other quicker or more accurate means. For example distance measurement can now be done with laser rangefinders and satellites are used for routine navigation. However, visual optical instruments are still used in some applications and are still worthy of some attention.

Angle measuring instruments

The theodolite

The theodolite is in essence a telescope of medium magnification with an eyepiece containing an alignment graticule. It is mounted such that it can rotate through horizontal and vertical circles, so that its horizontal and vertical direction of pointing can be measured. The theodolite is fitted with a levelling bubble so that the horizontal scale or table can be accurately made horizontal. If the vertical scale is zero for this horizontal scale then the elevation of an object can be absolutely read from the vertical scale.

Introduction

Optics is the study of light whereas visual optics is the study of the optical properties of the eye and sight. Ancient civilizations such as those of Greece were familiar with some of the properties of light, for example the laws of reflection. However, the Greeks misunderstood the nature of sight and the optical principles of the eye. They believed that light was emitted by the eye and only produced a visual response when the emitted rays struck an object. Many centuries passed before it was realized that light passes from the object to the eye and not from the eye to the object.

We will see later in this book, when we come to look at the optics of the eye, that the ability to sense the visual word around us is limited by the optical properties of the eye and its defects. For example before the advent of optical instruments, the smallest creature that could be seen with the unaided eye was about 0.05 mm in length and the mountains of the moon were unknown. Of particular frustration must have been the deterioration of eyesight with age. For example, as we age, the closest point of clear sight recedes, making it more and more difficult to do some things that we enjoy or need to do, such as reading and fine craft work. The discovery or invention of optical instruments enabled these restrictions to be overcome and allowed mankind to discover a world that was much more complex than ever envisaged, from the discovery of micro flora and fauna to countless galaxies far out in space.

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.