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.

Introduction

Every optical system contains a surface or surfaces which limit the width of the beam passing through the system from each object point. These surfaces may be a lens surface, a face of a prism, a mirror or simply a plate containing an opening of suitable size. Since the amount of light in the beam depends upon the beam width, they control the image brightness. They also affect image quality and to some extent the size of the field-of-view.

The surface that controls the width of the beam from the axial object point is called the aperture stop and because a beam cannot be infinitely wide, every system must have an aperture stop. Typical examples are the iris of the eye and the diaphragm of a camera lens. For off-axis object points, the beam width may be controlled by other surfaces.

Figure 9.1 shows a system with two components, a simple surface that acts as the aperture stop and a simple lens. For the on-axis object point and for object points some distance off-axis, the aperture stop limits the width of the beam. As one moves farther off-axis, the lens mount begins to limit the beam. The obstruction of the rays by a surface other than the aperture stop is called vignetting. As one moves even farther off-axis, the lens mount finally blocks the entire beam passing through the aperture stop and the vignetting has become complete. In this example, the width of the field-of-view is limited by vignetting. In complex optical systems, more than one surface may cause vignetting.

Introduction

So far we have looked at optical systems that we could call “direct vision” systems; that is the optical system is pointed directly at the object and, apart from the effect of image erecting prisms, the optical axis joining the object and image is a straight line. In contrast, in a number of situations, there is a need to bend an image forming beam around a “corner” or several “corners”. Usually this also requires the beam to be restricted to lie within a tube of a certain size and be transmitted over a distance that is long relative to the diameter of the beam. If the beam width is to be so constrained, it is often necessary to form intermediate images along the beam path, which in turn requires the use of what are known as relay lenses or relay systems. The main use of relay systems is to view normally inaccessible or hazardous places or simply to transmit luminous flux (light) to places remote from the source.

The earliest type of relay system was the periscope, which has been developed to a high degree for use in submarines. Much smaller systems have been developed to examine the inside of bodily organs or machines. The most recently developed type of relay system is based upon optical fibres. Optical fibres are usually circular in cross-section, much longer than their diameter and usually made from glass or plastic. They operate on the principle that light entering one end is constrained in the fibre by successive total internal reflections at the internal walls.

Introduction

The eye can be expected to suffer from all the aberrations that we find in other optical systems, but with one essential difference. Man-made systems are usually designed with some symmetry. For example, most visual optical systems have rotational symmetry. In contrast, the eye is not rotationally symmetric, which is mostly due to uneven growth patterns in the different components. The lack of symmetry means that there is a degree of irregularity in the conventional aberrations. These can be readily observed on axis by slightly defocussing the eye while viewing bright point light sources. The irregular star shaped image is due to irregular aberrations, because regular aberrations would produce a uniform circular or elliptical (if astigmatism is present) defocus blur disc.

The significance of optical aberrations, whether regular or irregular, is not at all clear. The eye generally uses only foveal viewing for the discrimination of fine detail and thus it could be argued that the eye only requires good optical image quality over the region of the fovea, which has an angular subtense of about 2°. Since the fovea is about 5° off-axis, one would expect off-axis aberrations, such as coma and transverse chromatic aberration, to be present at the fovea. The other off-axis aberrations (astigmatism, field curvature and distortion) would also be present but perhaps not in significant amounts. This is because, as shown in Table 33.1, these aberrations vary as the square of the field angle or cubic for distortion and therefore build up slowly with field angle for small field angles.

Introduction

Along with interference, diffraction is a manifestation of the wave nature of light and cannot be explained using geometrical optics or ray theory alone. In this chapter, we will explore the nature of diffraction and some applications to visual optical instrumentation, such as the Fresnel zone plate, speckle patterns and Fraunhofer diffraction. In order to explain the theory of these processes fully, we will need to introduce certain aspects of diffraction theory and important equations. The development of some of these equations will be beyond the scope of this book and therefore will be taken from other texts. Unless otherwise stated, the principal source of these equations and the historical development of diffraction will be Born and Wolf (1989).

The nature and cause of diffraction

In a number of situations, it can be observed that when a wave motion passes through an aperture, the waves bend around the edge of the aperture. For example, sea waves on entering an enclosed harbour will bend around the sea wall. Other examples are sound travelling around corners and a number of observable phenomena in the propagation of light. This effect is called diffraction and returning to the above example of sea waves entering a quiet harbour, as shown in Figure 26.1, let us look at what would happen if diffraction did not take place. Let us assume that the waves entering the harbour are parallel. They would remain parallel with a sharp edge at the boundary with the quiet water, producing a vertical wall of water with a sinusoidal profile at the edge as shown in the diagram.

Introduction

There are a number of different methods for analysing the image quality of an optical system. One of the simplest is to measure the resolving power, which is a measure of the smallest detail that can be resolved in the image. But even with this simple concept there are several different ways of defining resolving power in terms of the type of detail and test method. Some of the most common are listed below.

1. (1) The detail may be in terms of point objects. In this case the resolving power is a measure of the separation at which two points can just be resolved. This definition is used extensively in optical astronomy since the most common requirement here is to be able to resolve two very close stars.

2. (2) An alternative approach is to measure the resolving power of a periodic target, such as a sinusoidal or square wave pattern. The resolving power is the highest spatial frequency of the periodic pattern that can just be resolved.

In practice, the resolution limit of an optical system is most easily measured using a resolving power chart, which consists of a set of patterns containing detail as described above, but repeated at a number of different levels of detail size.

The Rayleigh criterion

The Rayleigh criterion assumes diffraction limited optics and specifies the resolution limit in terms of the minimum separation of point sources. In a diffraction limited optical system, the image of a point source object is the diffraction limited point spread function discussed in Chapters 26 and 34.

Introduction

The measurement of the radii of curvature of surfaces is one of the most common optical metrological measurements. Different methods of measuring the radius of curvature of a spherical or toric surface are presented in Chapter 11. In ophthalmic optics, we are concerned with the radii of curvature of the surfaces of spectacle or contact lenses and the radii of curvature of the anterior corneal surface. Because of the potential toric nature of these surfaces any clinical instrument should be able to measure the radii of curvature in a section and hence the radii of cuvature of the principal meridians of a toric surface. There are three main types of ophthalmic instruments for this task – Geneva lens measures, radiuscopes and keratometers. The Geneva lens measure is designed to measure the surface curvature or power of a spectacle lens and is a contact method based upon the spherometer. The radiuscope is a non-contact or optical method and is based upon the Drysdale principle. The keratometer is also a non-contact or optical method and is designed to measure the surface curvature of the anterior surface of the cornea. The Geneva lens measure has been adequately described in Chapter 11 and we will not spend any more time on discussing it. We will describe the radiuscope and keratometer in the following sections.

The radiuscope

Radiuscopes are instruments designed to measure the radius of curvature, mainly of contact lenses. Because these may be soft and easily deformed, a non-contact or optical method must be used. A typical construction of a radiuscope based upon the Drysdale method is shown in Figure 11.11 and the Drysdale principle is described in Section 11.3.2.1.

Introduction

There are a number of visual optical instruments used in ophthalmic optics, designed to measure such quantities as vertex powers and surface radii of curvature of both lenses and the cornea. The principles of these instruments have been discussed already in Chapter 11. In this and the following five chapters, we will examine these a little further and introduce some other useful ophthalmic instruments. In this chapter, we will look at the measurement of vertex power.

In ophthalmic optics, spectacle lenses are rarely specified by their equivalent power. Instead they are specified in terms of their back vertex power, front surface power, refractive index and thickness. Thus the measurement of vertex powers is perhaps the most common power measurement in ophthalmic optics, and fortunately it is far simpler to measure vertex power than to measure equivalent power. The focimeter is a simple instrument for measuring vertex powers and has already been described in Chapter 11.

Focimeters may be visual, projection or fully automatic. In the visual type shown in Figure 11.16, the image of a target is viewed and focussed through an optical system consisting basically of a collimator and a viewing telescope. The scale reading may also be presented in the eyepiece of the telescope. In the projection type, the telescope is replaced by an optical system which projects the image of the target onto a viewing screen. In both of these types of instruments, the operator has to make some judgement on the best focus of the target. In automatic instruments, the instrument automatically locates the optimum focus and the final powers are given directly on some type of display.

Introduction

Image quality criteria are used to assess how faithfully an optical system can image an object. The aberrations discussed in the previous chapter directly do not give this information, except for the distortion aberration expressed as a fraction or percentage difference in magnification. As will be seen in this chapter, aberration values can be used to calculate some but not all of these image quality criteria. For example, aberrations by themselves do not take into account veiling glare although some of the image quality criteria, using aberrations as input, do take diffraction effects into account. In fact, the quality of the image depends upon the following three factors:

1. (a) Veiling glare, which is due to unwanted reflections or scatter from surfaces

2. (b) Monochromatic and chromatic aberrations, discussed in the previous chapter

3. (c) Diffraction, discussed in Chapter 26

Because aberration and diffraction levels depend upon the size of the pupil and the aberrations also depend upon the position of the image point in the field, the form and magnitude of the image quality criteria depend upon pupil size and field position. Thus to assess fully the quality of a system, the image quality criteria should be evaluated for various pupil sizes and at several field positions.

Many image quality criteria are multi-dimensional functions and as a result are not always easy to use as criteria for comparing two similar systems. Therefore most image quality criteria have a reduced and sometimes approximate equivalent one dimensional form. That is, the image quality can be described by a single number on some suitable scale. Examples will be given at appropriate sections in this chapter.

Introduction

Optometers are instruments designed to determine the accommodative or refractive state of the eye. In Chapter 10, we discussed a number of methods that could be used to measure the accuracy of focus of an optical system. However, in all those cases, the image plane was directly accessible. In the eye, this is not the case and therefore we have to access it indirectly. We can do this in two ways. In one method, we can ask the subject to make some judgement of the quality of the retinal image or focus level and we will call these subjective methods. Alternatively, a second person or observer examines the light reflected from the retina and makes a judgement of the focus error. We will call these objective methods. A number of optometers that are available make the focus judgements automatically, by electronic processing. These are based upon one of the objective methods and are beyond the scope of this book, but have been discussed by Bennett and Rabbetts (1989).

The ultimate accuracy of any optometer method is limited by the depth-of- field of the subject's eye. The depth-of-field of the eye is dependent upon the pupil diameter and typical values have been given in Chapter 13. Errors can also be caused by unwanted accommodation responses of the subject's eye. Many people accommodate when viewing through optical instruments (a phenomenon called instrument myopia) and we discuss this phenomenon further in Chapter 36. This has to be overcome when measuring refractive error and can be done by using fixation targets.

Introduction

In this chapter, we will look further at the optical properties of single or simple lenses, some special lenses and some interesting examples of more complex lens systems.

We study the properties of single lenses to learn more about how they image beams. Such knowledge also helps us to understand the properties of more complex optical systems because these more complex systems are composed of single lenses and an understanding of the role of each single lens helps us to understand the operation of the system as a whole.

There are a wide range of simple lenses. We will initially classify them according to whether they are rotationally symmetric or non-rotationally symmetric and start with the symmetric lenses.

In this chapter, we will assume the lenses are in air, unless it is specifically stated otherwise and there will be such cases. There are some interesting situations where the lenses are not in air and we will look at some examples in Section 6.4.

Rotationally symmetric simple lenses

Rotationally symmetric lenses are constructed with surfaces that are rotationally symmetric and the axes are co-linear. These lenses may be made with spherical or aspheric surfaces. Non-rotationally symmetric lenses will be discussed in the next major section, Section 6.2.

Spherical lenses

Most lenses are constructed using spherical surfaces with co-linear centres of curvature. The line joining the centres is the optical axis. Lenses are usually made with spherical surfaces because of the low manufacturing cost relative to that of other surface forms.

Introduction

In the discussion of image formation so far, we have always assumed that the image is correctly focussed, with only one exception – in the reference to the telecentric pupil in the preceding chapter. However, in this chapter we will show that focussing is not usually exact and therefore in any imaging system, there is some residual defocus. One reason for this focus error is that there is always a range of positions of the “image” plane within which the image appears to be correctly focussed. This range is called the depth-of-field or depth-of-focus. A defocus outside this range will, by definition, be noticeable and hence must have some effect on image quality, but within this range, there will be some loss of image quality that may be detectable under different circumstances.

Sometimes we defocus a system intentionally. For example, we may wish to understand the effect of a defocus on some particular target structure or image quality criteria. In these cases, we may need to control the level of defocus accurately and quantify it.

Therefore, whatever the cause of defocus, it is important to understand the different ways in which a defocus can occur and how to quantify the level of defocus. To meet these needs, in this chapter, we will investigate the defocussing of optical systems, the effect of a defocus on the image, sources of defocussing, and depth-of-field and finally discuss a variety of methods that can be used to focus an optical system accurately.

Quantification of defocus

Any study of defocus must look at the ways of quantifying the level of defocus.