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.

Introduction

There are a number of ophthalmic instruments that are designed to assess various aspects of binocular vision and functions, such as

1. (a) simultaneous perception, fusion and stereopsis

2. (b) accommodation/convergence relationships

3. (c) presence and magnitude of phorias.

These will now be discussed in greater detail.

Simultaneous perception, fusion and stereopsis

The simplest level of binocular vision is simultaneous perception, the ability to see two images, one formed on each retina simultaneously. For normal binocular vision, it must be possible to see two images, one formed at the fovea of each retina, and to superimpose them. This is called simultaneous foveal perception.

Fusion is a more refined level of binocular vision, in which two similar images, one formed on each retina, are seen at the same time and are blended as one. That there is a blending, and not just a superposition, can be tested by placing a small prism in front of one eye. This will displace the retinal image from the fovea. If fusion is present, a corrective eye movement will be made to place the image back at the fovea.

When a solid object is viewed, the two retinal images are slightly different because the two eyes view the object from different perspectives. Stereopsis is the ability to see slightly different images and to blend them as one with a perception or appreciation of depth. This is the most refined level of binocular vision.

A more detailed account of these binocular vision levels can be found in texts such as Duke-Elder (1973).

The purpose of this book is to present a thorough description of the construction and image formation of visual optical instruments, ranging from simple magnifiers, through microscopes and telescopes, to the more sophisticated instruments based upon interference and diffraction. There are many other types of optical instruments, such as spectrophotometers and laser systems, but these are not visual optical instruments; that is, they are not used with the eye as an essential component in the imaging process. The only instrument that we include in this book that may not be regarded as a visual optical instrument is the camera. However, we have included the camera, because while one can take a photograph without any “eye” input, the eye is often used to aim the camera and the final image is usually viewed by the eye, either directly or with a projection system.

There are many other textbooks on optics but most of these only briefly discuss visual optical instruments and even more briefly discuss any visual ergonomic aspects of these instruments. We believe that the major strength of this book is its emphasis on the detail of the construction and image formation and most importantly the visual ergonomic aspects. Visual ergonomics is the study or application of the properties of the eye to human performance. In this context, visual ergonomics involves the following factors that may affect vision through an optical instrument: the aberrations of the eye, depth-of-field of the eye, the role of the pupil of the eye, the amplitude of accommodation, refractive errors, visual acuity and the coordination of the two eyes in binocular vision.

Introduction

Telescopes are conventionally designed for viewing distant objects, but are often used for viewing closer objects. Equation (17.6a) shows that for telescopes, as an object comes closer, the image also comes closer, thus requiring the user to accommodate but to a greater level than the vergence of the object. In practice, this accommodation demand can be eased, because most telescopes can be refocussed by adjusting their length. These aspects have been discussed in Chapter 17. The closest working distance will then depend upon the length of adjustment and the maximum possible accommodation that the user can exert.

On the other hand, some specialized “telescopes” are designed for working distances that are relatively short, for example 20 cm to several metres. These have sometimes been called “near point telescopes”, “intermediate distance telescopes” or “tele-microscopes”. Perhaps a better term would be “macroscope,” and this term would cover what are basically two lens magnifiers with working distances intermediate between microscopes and telescopes. However instruments of this type are often not readily available and when working distances are short, it is often more convenient to adapt a conventional (afocal) telescope (see Section 18.5).

The use of terms containing the words “telescope” and “microscope” to describe these instruments leads to confusion in the interpretation of magnification. Telescopes and microscopes use different definitions of magnifications and these in turn will be different from that defined for macroscopes. These points are discussed in greater detail in Section 18.3 and a numerical example will be given showing how the differences in definition of magnification can lead to very different outcomes.

Introduction

In many visual optical systems, prisms play an important role in the formation of the final image. They are used to control or change the direction of the image forming beam and are often used to change the orientation of the image. In these operations, the prism may only refract the beam, may refract and reflect it or may only reflect it but one or more times. Therefore in discussing the structure and properties of prisms, it is convenient to classify prisms according to whether they (a) are purely refracting, (b) combine refraction with internal reflections or (c) are purely reflecting.

The optical construction of most prisms is relatively simple but the ray paths inside a prism, which reflect the beam one or more times, can be complex and some of the properties of the prism are not readily obvious from looking at the ray paths. However these properties are sometimes made clearer by examining the unfolded version. Unfolding a prism involves reflecting each reflecting surface in the next reflecting surface along the ray path. Ideally, this process will become clearer when examples are discussed later in this chapter.

Refracting-only prisms

Refracting prisms occur frequently in visual optical and ophthalmic instruments. They can be used to deviate beams or images, as aids to focussing to form double or split images and often to compensate for eye problems such as phorias and squint.

Just as lenses may be examined using finite (or real) ray tracing on one hand or paraxial theory on the other, so can refracting prisms, but before we examine the paraxial properties of prisms, we will examine their effects on finite rays.

Introduction

The visual performance of a user viewing an image through an optical instrument depends upon a complex interaction between the eye and the instrument. Thus one cannot fully assess an optical instrument without a general understanding of visual optics and occasionally a knowledge of the visual capabilities of the individual user, which must include any anomalies of his or her visual system.

Many visual optical instruments are monocular and some people suffer visual discomfort when using monocular visual instruments for any length of time. Most of this probably stems from the tendency to keep one eye closed while viewing through the instrument. A binocular equivalent instrument will eliminate this problem and offers the possibility of a stereoscopic image. However, binocular instruments may lead to other problems. In this chapter, we will only discuss the ergonomics of monocular instruments and leave the ergonomics of binocular instruments and potential problems until the next chapter.

Instrument focussing

Image vergence

If an instrument is to be correctly focussed, the image must be within the accommodation range of the user. Classically, the ideal image vergence of visual optical instruments is taken as zero, that is the image is at infinity. The reasoning justifying this situation is that it was traditionally believed that the eye prefers viewing with relaxed accommodation. Unfortunately this approach neglects instrument accommodation and any refractive error of the user. The nature of instrument accommodation will be discussed later in Section 36.1.2. The different types of refractive errors and their distribution among the adult population has already been discussed in Chapter 13 and some representative values are shown in Figure 13.9.

Introduction

This chapter deals with the properties of mirrors or reflecting surfaces. Two types of mirrors will be looked at: (a) plane mirrors and (b) curved mirrors. So far, we have tended to concentrate on the paraxial properties of optical systems and therefore looked at such phenomena as image formation in terms of paraxial rays, because these rays are aberration free. When we discuss the properties of plane mirrors, we need not restrict ourselves to paraxial rays because plane mirrors are aberration free and therefore we can use either paraxial or finite rays. However, when we discuss the properties of curved mirrors, these are not free of aberration and therefore we must return to paraxial optics.

Optical systems can be constructed solely of mirrors or reflecting elements and such systems are called catoptric systems. Systems that consist of refracting and reflecting elements are called catadioptric systems. In Chapter 4, we showed how to use paraxial ray tracing to study the properties and image formation in such systems. In this chapter, we will look at the properties of single mirrors and simple mirror systems, starting with plane mirrors.

Plane mirrors

In this section, we will investigate the properties of plane mirrors. These properties are useful in their own right, but are also very useful in understanding the properties of systems of plane mirrors or reflecting surfaces, such as occur in some reflecting prisms, which are discussed in Chapter 8.

The optics of plane mirrors can be analysed using Snell's law. For reflection, Snell's law reduces to the statement that the angle of reflection is equal to the angle of incidence.

Introduction

So far, all the visual optical instruments that have been discussed have been essentially describable in terms of geometrical or ray optics. In this and the following chapter, we will describe some important visual optical instruments and devices based upon physical or wave optics. The essential difference between geometrical and physical optics is the difference between the geometrical description of the propagation of light by rays and the physical optical description using waves. The wave nature of light gives rise to two very important phenomena, interference and diffraction. This chapter is concerned with interference and the next will look at diffraction.

The phenomenon of interference allows us to produce visual stimuli in which the light level varies sinusoidally across the pattern. Since early work by Campbell and Green (1965), the visual system is now regularly analysed in terms of its response to sinusoidally varying luminance profiles. This is often done by measuring the threshold contrast of sinusoidal patterns at a range of spatial frequencies. The resulting function is known as the contrast sensitivity function (see Chapter 35). Sinusoidal patterns can be produced in several ways. They may be produced in printed form, for example as photographs, or can be produced on television screens or by optical interference techniques. This chapter is concerned with the last mode of production.

There are two fundamental differences between the production of sinusoidal patterns by interference techniques and those in printed or television screen forms.

Introduction

Camera lenses are optical systems designed to give a real image of distant objects. Therefore they must have a positive power with their aberrations corrected to produce reasonably good image quality for an object plane at infinity, that is for the image formed in the back focal plane of the lens. Some camera lenses known as “macro-lenses” are designed to give optimum image quality at intermediate or close object distances rather than for the object at infinity. Examples of macro-type lenses are those used in photocopiers and for photographing documents.

Camera lenses usually have a variable aperture stop, known as the diaphragm, whose diameter is varied within limits, for control of the image illuminance and hence film exposure (Section 21.4) and the depth-of-field (Section 21.6).

While a camera lens may consist of a single positive power lens, such a lens is not suitable when the focal length is very long or very short. Also a single lens would give less than adequate image quality for many applications, particularly those requiring large apertures or wide fields-of-view. Thus most camera lenses are complex optical systems, often consisting of two or more separate lenses. Designs suitable for very long and very short focal lengths are discussed in Section 21.2 and aberration considerations for wider apertures and field angles are discussed in Section 21.10.

Field-of-view, focal length and image sizes

The field-of-view of a camera system is usually limited by a field stop placed in the image or film plane. In 35 mm cameras, the field stop is 35 × 24 mm; thus the image or frame size on the film has these dimensions.

Introduction

A number of photometric and colorimetric instruments are visual optical systems as these devices have a viewing system to observe a scene either for alignment or to make a subjective judgement of brightness (luminance) or colour.

Photometers

Photometers are either subjective or objective instruments and are designed to measure the absolute or comparative brightness of sources or scenes. In the subjective type, the observer has to make a match between the brightness of a standard scene and that of a second scene. In the objective instruments, an observer uses the viewing system only for alignment.

Subjective photometers

The Lummer–Brodhun photometer

The Lummer–Brodhun photometer is designed to compare the luminous intensities of two light sources. Often one is a secondary standard source of known luminous intensity, and in this case, the luminous intensity of the second or unknown source can be found.

The instrument is shown in Figure 24.1. The light from the two sources illuminates both sides of a white diffuse block as shown in the diagram. Both sides of the block are viewed simultaneously through an optical system which contains a prism assembly and an eyepiece. One side of the block (the left side in the diagram) is seen through the central part of the prism and the other side of the white block is seen through the peripheral part. The two beams are brought together but separated by a sharp boundary by the prisms. The eyepiece is focussed on the sharp boundary. The field-of-view seen through the eyepiece is shown in the diagram.

In a recent series of experimental papers, it has appeared that lasers that are well described by the simplified TSD rate equations (7.42)–(7.44) display what is called, in laser physics, antiphased dynamics. Its simplest manifestation is that when N modes oscillate, the total intensity displays many properties that are those of a single-mode laser. In the simplest experiment, the laser is initially in a steady state. A control parameter is suddenly changed and the relaxation toward the new steady state is recorded. This transient evolution is then Fourier analyzed to evaluate its frequency content. The result of this experiment is that each mode is characterized by as many frequencies as there are modes, whereas the total intensity, which is simply the sum of all modal intensities, is characterized by only one frequency. This frequency is the single-mode relaxation oscillation frequency (7.51) and is also the highest of the N frequencies. This property has been observed with a Nd-doped optical fiber laser [1]. Using a LiNdP4O12 (abbreviated as LNP) laser oscillating on two or three modes, it was shown that the noise spectrum of the laser displayed the same antiphase dynamics [2]. With the same laser, antiphase dynamics was also reported in the case where a feedback loop induces a chaotic output [3]. The feedback loop has the effect of injecting part of the output beam in the laser after each modal intensity has been subjected to a modulation. Antiphase dynamics also plays a significant role when a multimode laser undergoes a Feigenbaum cascade toward chaos. It has been reported that though the total intensity displays this usual route to chaos, the modal intensities do not [4].