Optics of the Human Eye

Butterworth-Heinemann An imprint of Elsevier Science Limited Robert Stevenson House 1-3 Baxter's Place Leith Walk Edinburgh EH1 3AF First published 2000 Reprinted 2002 © Reed Educational and Professional Publishing Ltd 2000 © Elsevier Science Limited 2002. All rights reserved. All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England WlT 4LP. Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publishers.

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress ISBN 0 7506 3775 7 _

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Acknowledgements

Several colleagues have commented on drafts or provided references and advice. In particular, we thank Brian Brown, Niall Strang and Tom Raasch, who read substantial sections of drafts. Others include Ray Applegate, Pablo Artal, Harold Bedell, Arthur Bradley, Neil Charman, Nicolas Chateau, Michael Doughty, Dave Elliot, Richard Guy, Douglas Homer, Tony [oblin, Phil Kruger, Barbara Pierscionek, Katrina Schmid, Lawrence Stark, Peter Swann, Christopher Tyler, Barry Winn, Joanne Wood and Russell Woods.

We are grateful to Pablo Artal, Michael Cox, Larry Thibos and Barry Winn for providing data for some figures; these are also acknowledged at the appropriate figure captions. We thank the American Academy of Optometry, the Optical Society of America, Elsevier Press and the Association for Research in Vision and Ophthalmology for permission to use previously published data; again, these are acknowledged at the appropriate figure captions. Finally, we thank our wives, Janette and Yolette, for their support.

Sign convention and symbols

When we examine image formation and raytracing, we need a sign convention. Although the choice of a sign convention is arbitrary, it must be consistent. In this book we use the standard cartesian and trigonometric sign conventions. Distances to the left of a surface or other reference position or below the optical axis are negative, and those to the right or above are positive. Angles due to anticlockwise rotations of the ray from the optical axis are positive, and those due to clockwise rotations are negative.

Distance notation and sign Points are denoted by Roman letters in bold. Distances are denoted by either a single lower case letter such as 1, or by two upper case letters such as PF. In this example, P and Fare both points and PF denotes the distance from P to F. If F is to the right of P, this distance is positive, and if F is to the left of P, this distance is negative.

Greek alphabet Greek letters are used extensively throughout the book.

a.A alpha M delta T]H eta KK kappa vN nu 1[0 pi t'T tau XX chi

~B beta £E epsilon as theta A.A lambda ~::: xi pP rho uY upsilon 'I''P psi

yr gamma

~Z zeta tI iota JlM mu 00 omicron (J~ sigma phi ron omega

Units and their abbreviations metre centimetre millimetre micrometre nanometre dioptre prism dioptre Joule Watt candela lumens steradian second Hertz Kelvin radian degree minutes of arc

m cm mm Jlm nm D A

J

W cd 1m st

s

Hz K rad 0, deg min. arc

Introduction

The purpose of this book is to describe the optical structure and optical properties of the human eye. It will be useful to those who have an interest in vision, such as optometrists, ophthalmologists, vision scientists, optical physicists, and students of visual optics. An understanding of the optics of the human eye is particularly important to designers of ophthalmic diagnostic equipment and visual optical systems such as telescopes. Most animals have some sort of eye structure or sophisticated light sense. Like humans, some rely heavily on vision, including predatory birds and insects such as honeybees and dragonflies. However, many animals rely much more on other senses, particularly hearing and smell, than on vision. The visual sense is very complex and is able to process huge amounts of information very rapidly. How this is done is not fully understood; it requires greater knowledge of how the neural components of vision (retina, visual cortex, and other brain centres) process the retinal image. However, the first stage in this complex process is the formation of the retinal image. In this text, we investigate how the image is formed and discuss factors that affect its quality. The majority of animal eyes can be divided into two groups: compound eyes (as possessed by most insects), and vertebrate eyes (such as the human eye). Compared with vertebrate eyes, there is considerable variation in the compound eyes. Compound eyes contain a large number of optical elements (ommatidia), each with its own aperture to the

external world. Vertebrate eyes have a single aperture to the external world, which is used by all the detectors. A number of other animals have simple eyes, which can be described as less developed versions of the vertebrate eye. All eyes, of whatever type, involve compromises between the need for detection (sensitivity), particularly at low light levels, and spatial resolving capability in terms of the direction or form of an object. Although this book is about the optics of the human eye we do not wish to consider the optics in complete isolation from the neural components, as otherwise we cannot appreciate what influence changes in the retinal image will have on vision performance. As an example, altering the optics has considerable influence on resolution of objects for central vision but not for peripheral vision. This is because the retina's neural structure is fine enough at its centre, but not in the periphery, for large changes in optical quality to be of importance (Chapter 18). Thus, the neural components of the visual system, particularly the retinal detector, rate some mention in the book. The neural structures of the retina themselves produce optical effects. As an example, the photoreceptors exhibit waveguide properties that make light arriving from some directions more efficient at stimulating vision than light arriving from other directions. Another example is that the regular arrangement of the nerve fibre layers produces polarization effects. While image formation in the eye is similar to that in man-made optical systems such as

xii

Introduction

cameras and must obey the conventional optical laws, there are some interesting differences because of the eye's biological basis. Perhaps the greatest difference is that, as a living organ, the eye responds to its environment, often in an attempt to give the best image under different circumstances. Also, it grows, ages and suffers disease. Unlike most man-made optical systems, the eye is not rotationally symmetrical about a single axis, and different axes must be used to define image formation. There are many interesting and important optical effects associated with ocular diseases such as keratoconus (conical cornea) and cataract. Furthermore, the balance between optical and neural contributions to overall vision performance changes with diseases of the retina and beyond. Although there are some passing references to cataract, we have concentrated on the healthy human eye. We give some prominence to age-related changes in the optics of the eye throughout the book, and devote Chapter 20 to this topic. To make the book easy to read it is divided into a number of short chapters, with each chapter dedicated to a single theme. The most commonly useful topics are at the beginning, and topics with narrower appeal (such as ocular aberrations) are placed towards the

end. Section 1 covers the basic optical structure of the human eye, including the refracting components, the pupil, axes and simple models of the eye. Section 2 is about image formation and refraction of the eye. This includes the refractive errors of the eye, their measurement and correction, and paraxial treatments of focused and defocused image sizes and positions. Section 3 deals with the interactions between light and the eye, considering transmission, reflection and scatter in the media of the eye and at the fundus. Section 4 deals with aberrations and retinal image quality. As well as considering these for real eyes, it covers the modelling of eyes and the performance of a range of schematic eyes of different levels of sophistication. Section 5 considers the topics of depth-of-field and age-related changes in the optics of the eye. While depth-of-field effects could possibly have been placed earlier in the book, understanding them well requires some knowledge about aberration and diffraction. The book concludes with 4 appendices, three of which (Appendices 1, 2 and 4) cover some mathematics relating to paraxial optics, aberrations theory and image quality criteria. Appendix 3 lists construction data, optical parameters and the aberrations of a number of schematic eyes.

1

The human eye: an overview

Introduction

The structure of the human eye is shown in Figure 1.1. The outer layer is in two parts: the anterior cornea and the posterior sclera. The cornea is transparent and approximately spherical with a radius of curvature of about 8 mm. The sclera is a dense, white, opaque, fibrous tissue that is mainly protective in function and is approximately spherical with a radius of curvature of about 12 rom. The centres of curvature of the sclera and cornea

This chapter is a short overview of the optical structure and function of the human eye. It mentions briefly some important aspects such as the cornea, the lens and ocular axes, which are covered in more detail in later chapters. Other important topics, such as the passage of light, aberrations and retinal image quality, are also discussed in later chapters.

Temporal side

Vitreous humour Aqueous humour

N

N'

t

Centre of rotation

Nasal side

Figure 1.1. The horizontal section of the right eye as seen from above. The pupil is the opening in the iris. The cardinal points (F, F', P, P', Nand N') are for the relaxed eye.

4

Basic optical structureof thehuman eve

are separated by about 5 mm. More accurate measures of shapes are given in subsequent chapters. The middle layer of the eye is the uveal tract. It is composed of the iris anteriorly, the choroid posteriorly, and the intermediate ciliary body. The iris plays an important optical function through the size of its aperture, the ciliary body is important to the process of accommodation, and both the ciliary body and choroid support important vegetative processes. The inner layer of the eye is the retina, which is an extension of the central nervous system and is connected to the brain by the optic nerve. The inside of the eye is divided into three compartments: 1. The anterior chamber, between the cornea and iris, which contains the aqueous fluid. 2. The posterior chamber, between the iris, the ciliary body and the lens, which contains the aqueous fluid. 3. The vitreous chamber, between the lens and the retina, which contains a transparent colourless and gelatinous mass called the vitreous humour or vitreous body. The internal pressure of the eye must be higher than that of the atmosphere in order to maintain the shape of the cornea, and must be maintained at an approximately constant level in order to maintain the transparency of the ocular media. The pressure is controlled by the production of aqueous fluid in the ciliary body and by drainage of this aqueous

Figure 1.2. Image formation of the human eye.

fluid from the eye. This drainage is through the angle of the anterior chamber (between the cornea and the iris) to the canal of Schlemm (not shown in Figure 1.1) and, finally, to the venous drainage of the eye. The eye rotates in its socket by the action of six extra-ocular muscles. More detailed anatomical descriptions of the human eye can be found in books such as those by Hogan et al. (1971) and Snell and Lemp (1997).

Optical structure and image formation The principles of image formation by the eye are the same as for man-made optical systems such as the camera lens. Image-forming light enters the eye through the cornea, and is refracted by the cornea and the lens to be focused at the retina. Of the two refracting elements, the cornea has the greater power. However, whereas the corneal power is constant, the power of the lens can be changed when the eye needs to focus at different distances. This process is called accommodation, and occurs because of an alteration in the lens shape. It is discussed further in Chapter 20. The diameter of the incoming beam of light is controlled by the iris, which forms the aperture stop of the eye. The opening in the iris is called the pupil. As with all optical systems, the aperture stop is a very important component of a system, affecting a wide range of optical processes, and it is discussed in more detail in Chapter 3.

Thehumall eye: all overview

cussed in greater depth in Chapter 14. The thickness of the retina varies from 50 urn (0.05 mm) at the foveal centre to about 600 um (0.6 mm) near the optic disc. There is a layer of light-sensitive cells at the back of the retina, and the light must pass through the other layers to reach these cells. These receptor cells are of two types, known as rods and cones. The names refer to their shapes, but considerable variations in shape occur with location, and it is not always possible to distinguish between the two types on this basis. Figure 1.4 shows their distribution along the horizontal temporal section of the retina. There are about 100 million rods in the retina, and they reach their maximum density at about 20° from the fovea.

Figure 1.2 shows two light beams from object points forming images on the retina. The image is inverted, as it is for a camera. We discuss this image formation in more detail in Chapter 6.

The retina The light-sensitive tissue of the eye is the retina. It is shown in Figure 1.3, and consists of a number of cellular and pigmented layers and a nerve fibre layer. These layers have varying degrees of optical significance, with the amount of incoming light specularly reflected and scattered by each layer being of particular importance. This aspect is dis-

6. Inner nuclearlayer

7. lnner plexiform layer \ 8. Layerof ganglion cells'\ •

'I {r

....5. Outer plexiform layer

'r

Outer nuclearlayer

, 3. Outer limiting membrane

9. Nervefibre layer '\ I'

I'

2. Photoreceptors (rodsand cones)

10. Inner limiting membrane --~

1. Pigmentepithelium

Bruch's membrane

-LIGHT

•

Sclera

Fovea

'----y

5

---,/

Retina

Figure 1.3. The layers at the back of the human eye (based on Polyak, 1941).

6

Basic optical structure of the human"ye

There are approximately 5 million cones in the retina. In general, rods are longer and narrower than cones. Rods are sometimes described as highly sensitive low-level light detectors in comparison with cones. However, much of this is due to the neural wiring that occurs rather than differences between the receptors. The retinal neural network of rods is such that the output of about 100 rods can combine on the way to the brain, so that the rod system has very high sensitivity to light but poor spatial resolution. In contrast, the output of fewer cones is combined, so the cone system functions at higher light levels and is capable of higher spatial resolution. Cones recover from exposure to light more quickly than rods. The first stage in colour vision is the existence of three types of cones, each with different wavelength sensitive properties: L (long), M (medium) and 5 (short) cones. The cones predominate in the fovea, which is 1.5 mm or approximately 5° wide as subtended at the back nodal point N' of the eye. The fovea is free of rods in its central 1°

160 0sterberg (1935)

140

"'E

E

':;s

...

'0 C

'"

:> 0

2 .::0:;'; ...c 0

120

Curcio and Hendrickson ( 1991 )

'.

100 -,

80

-:

60 40 20

field. At high light levels the best resolution is attained by the cones in the fovea, which occupies only about l/lOOOth of the total retinal area. Despite the predominance of cones at the fovea, it contains only a small proportion (1 per cent) of the total cones (Tyler, 1996), and an even smaller proportion (0.05 %) of cones is found in the highresolution foveola. Therefore, the vast majority of cones are distributed throughout the peripheral retina. At low light levels the cones at the fovea do not operate; thus the centre of the fovea is 'night blind', and it is necessary to look eccentrically to see objects using the rods. At very low light levels, maximum visual acuity and detection ability occur about 10-15° away from the fovea. The location of the fovea is shown in Figure 1.1. When the eye fixates on an object of interest, the centre of its image is formed on the foveal centre, which is inclined at about 5° from the 'best fit' optical axis. At the fovea, the layers overlying the receptor cells are thinner than elsewhere in the retina (Figure 1.3) and, as a result, the fovea has a pit-like structure. The bottom of this pit is about 1° wide, and corresponds to the rod-free region. The foveola is the approximately O.5°-wide avascular centre of the foveal pit, and is the region of highest resolution. The diameters and packing of the foveal cones affect visual acuity, and we examine this relationship briefly in Chapter 18. Estimates of the diameters of foveal cones are given in Table 1.1. The off-axis position of the fovea is most intriguing since aberration theory predicts that the best image of an optical system is usually formed on the optical axis. Therefore the retinal image quality at the fovea should be worse than at the posterior pole. The offaxis position of the fovea has some interesting visual effects which we discuss in Chapter 17.

'"

0

0

10

20

30

40

50

60

70

Angle from fovea (deg.) Figure 1.4. The density of cones and rods across the retina in the temporal direction. From 0sterberg (1935), Curcio and Hendrickson (1991). The data from Curcio and Hendrickson (1991) have been converted from distances along the retina to angles relative to the back nodal point of the eye, assuming a spherical retina of diameter 12 mm and a distance between the back nodal point and retina of 17.054 rnm.

Table 1.1. Diameters of foveal cones. The angular values are calculated assuming the distance between the back nodal point and the fovea is 17 mm,

Polyak (1941) O'Brien (1951)*

um

lIlil/. arc

1.0-1.5 3.5-4.0 2.3

0.2-0.3 0.7-0.8 0.46

..Actually centre-to-centre spacing between cones.

central fovea outer fovea

Tile Ill/mall eye: all

The fovea is the central part of the macula, whose peripheral limits are where the cells of the outer nuclear layer are reduced to a single row (Hogan et al., 1971). The macular diameter is 5.5 mm (19°).

The optic disc and blind spot The vascular supply to the outer layers of the retina is carried in the choroid, which lies between the retina and the sclera. The vascular supply to the inner retina enters the eye at the optic disc, whose location is shown in Figure 1.1. There are no cones or rods here, and hence this region is blind. The name given to the corresponding region in the visual field is the 'blind spot'. The optic disc is approximately 5° wide horizontally and 7° vertically, and its centre is approximately 15° nasally and 1.5° upwards relative to the fovea. Correspondingly, the blind spot is 15° temporally and 1.5° downwards relative to the point of fixation. Figure 1.5 provides a demonstration of the blind spot.

The cardinal points Every centred optical system that has some equivalent power (i.e. is not afocal) has six cardinal points that lie on the optical axis. These are in three pairs. Two are focal points which we denote by the symbols F and F', two are principal points denoted by the symbols P and P', and two are nodal points denoted by the symbols Nand N'. The positions of these cardinal points in an eye depend upon its structure and the level of accommodation. For an eye focused at infinity, the approximate positions of these cardinal points are shown in Figure 1.1.More precise positions are given in Chapter 5, where we discuss schematic eyes

+ Figure 1.5. Demonstration of the blind spot. Look steadily at the cross with your right eye (left eye closed) from a distance of about 20 cm. Vary this distance until you find a position for which the spot disappears.

(ll'CI'l';ew

7

and their properties. These cardinal points are as follows: 1. Focal points (F and F'). Light leaving the front (also first and anterior) focal point F passes into the eye, and would be imaged at infinity after final refraction by the lens if the retina were not in the way. Light parallel to the axis and coming into the eye from an infinite distance is imaged at the back (also second and posterior) focal point F'. Thus, for the eye focused at infinity, the retina coincides with the back focal point. 2. Principal points (P and P'). These are images of (or conjugate to) each other, such that their transverse magnification is +1. That is, if an object were placed at one of these points, an erect image of the same size would be formed at the other point. 3. Nodal points (N and N'). These are also images or conjugates of each other, but have a special property such that a ray from an off-axis point passing towards N appears to pass through N' on the image side of the system, while inclined at the same angle to the axis on each side of the system. Such a ray is called the nodal ray, and when the off-axis point is the point of fixation, the ray is called the visual axis.

We make use of these cardinal points frequently in the following chapters.

The equivalent power and focal

lengths

One of the important properties of any optical system is its equivalent power. This is a measure of the ability of the system to bend or deviate rays of light. The higher the power, the greater is the ability to deviate rays. We denote the equivalent power of an optical system by the symbol F. The equivalent power of the eye is related to the distances between the focal and principal points by the equation 1 F =- PF

n'

= P'F'

(1.1)

where n' is the refractive index in the vitreous chamber. The average power of the eye for adults is about 60 m- l or 60 D, but the value varies greatly from eye to eye. Using this

8

Basic optical structure of til" 1IIII/lIl1l

"!I"

power and the commonly accepted refractive index n' of the vitreous chamber (1.336), the focal lengths of the eye are PF =-16.7 mm and P'F'

= +22.3 mm

(1.2)

While the equivalent power of the eye is a very important property of the eye, it is not easy to measure directly. Its value is usually inferred from the other measurable quantities such as surface radii of curvature, surface separations and eye length, and assumed refractive indices of the ocular media. However, more important than equivalent power is refractive error. The refractive error can be regarded as an error in the equivalent power due to a mismatch between the equivalent power and the eye length. For example, if the equivalent power is too high for a certain eye length, the image is formed in front of the retina and this results in a myopic refractive error. If the power is too low, the image is formed behind the retina and results in a hypermetropic refractive error. Refractive errors are discussed in Chapter 7.

Axes of the eye The eye has a number of axes. Figure 1.1 shows two of these: the optical axis and the visual axis. The optical axis is usually defined as the line joining the centres of curvatures of the refracting surfaces. However, the eye is not perfectly rotationally symmetric, and therefore even if the four refracting surfaces were each perfectly rotationally symmetric, the four centres of curvatures would not be co-linear. Thus in the case of the eye, we define the optical axis as the line of best fit through these non co-linear points. The visual axis is defined as the line joining the object of interest and the fovea, and which passes through the nodal points. These and the other axes are discussed in greater detail in Chapter 4.

however, we can nominate a mean position for this point. The rotation of the eye in the horizontal plane was studied by Fry and Hill (1962), who found that the mean centre-ofrotation for 31 subjects was about 15 mm behind the cornea. Often it is assumed to lie along the optical axis.

Field-of-vision Examination of the pupil from different angles shows that the pupil can still be seen at angles greater than 90° in the temporal field. Light is able to enter the pupil from about 105° to the side, as shown in Figure 1.6. While this suggests that the radius of the field-of-view may be as great as 105°, the real extent of the field-of-vision depends upon the extent of retina in the extreme directions. On the nasal side, vision is cut-off at about 60° because of the combination of the nose and the limited extent of the temporal retina.

Binocular vision and binocular overlap The use of two eyes provides better perception of the external world than one eye

Centre-of-rotation The eye rotates in its socket under the action of the six extra-ocular muscles. Because of the way these muscles are positioned and operate, there is no unique centre-of-rotation;

Figure 1.6. The horizontal field of view in monocular and binocular vision.

Tile

alone. Binocular vision improves contrast sensitivity and visual acuity slightly over those obtained with monocular vision (Campbell and Green, 1965;Home, 1978).Two laterally displaced eyes give the potential for a three-dimensional view of the world, which includes the perception of depth known as stereopsis. The degree of stereopsis depends partly upon the distance between the two eyes, which is called the interpupillary distance. Stereopsis can be improved considerably by optical devices such as rangefinders, which increase the effective interpupillary distance.

Interpupillary distance The interpupillary distance, or PO, is usually measured by the distance between the centres of the two pupils of the eyes. The distance PO is measured for the eyes looking straight ahead; that is, the visual axes are parallel. When the eyes focus on nearby objects, the eyes rotate inwards and hence there is a corresponding decrease in interpupillary distance. The near PO can then be measured or determined from the distance PO using simple trigonometry. Harvey (1982) provided distance POs for various armed services. Means ranged from

/ullflan

eye: an overview

9

61 to 65 mm, with standard deviations of 3-4 mm. These values were mainly for men.

Binocular overlap As shown in Figure 1.6, the total field-of-view in the horizontal plane is about 2100 with a 1200 binocular overlap.

Typical dimensions All dimensions of the eye vary greatly between individuals. Some depend upon accommodation and age. Representative data are shown in Figure 1.7. Average values have been used to construct representative or schematic eyes, which we discuss in Chapter 5. More detailed data are presented in later chapters.

Summary of main symbols F

equivalent power of the eye refractive index of vitreous humour (usually taken as 1.336) R radius of curvature in millimetres F, F' front and back focal points N, N' front and back nodal points P, P' front and back principal points n'

" - + - - - - - 1 6 - - - - - I 24

Figure 1.7. Representative dimensions (millimetres) and refractive indices of the (relaxed) eye. The starred values depend upon accommodation.

10

Basic optical structure of II/(' 11/111/1111 e)/e

References Campbell, F. W. and Green, D. G. (1965). Monocular versus binocular visual acuity. Nil/lire, 208, 191-2.

Curcio, C. A. and Hendrickson, A. E. (1991). Organization and development of the primate photoreceptor mosaic. Prog. Retinal Res., 10, 89-120. Fry, G. A. and Hill, W. W. (1962). The center of rotation of the eye. Alii. f. OptOIll. Arc/I. Am. Acad. OptOIll., 39, 581-95.

Harvey, R. S. (1982). Some statistics of interpupillary distance. Optician, 184 (4766), 29. Hogan, M. J., Alvarado, J. A. and Weddell, J. E. (1971). Histology of theHuman Eye. W. B.Saunders and Co. Home, R. (1978). Binocular summation: A study of

contrast sensitivity, visual acuity and recognition.

Visioll Res., 18, 579-85.

O'Brien, B. (1951). Vision and resolution in the central retina. }. Opt. Soc. Alii., 41, 882-94. 0sterberg, G. (1935). Topography of the layers of rods and cones in the human retina. Acta Ophthal. (Suppl.), 6, 1-103.

Polyak, S. L. (1941). The Retina, p. 201. University of Chicago Press. Snell, R. S. and Lemp, M. A. (1997). Clinical Anatomyof the E)/(', 2nd edn. Blackwell Scientific Publications. Tyler,C. W. (1996). Analysis of human receptor density. In

Basic alld Clinical Applications of Vision Science, The Professor fay M. Enoch Festschrift Volume (V. Lakshminarayanan, ed.), pp. 63-71. Kluwer Academic Publishers.

2

Refracting components: cornea and lens

Introduction The refracting elements of the eye are the cornea and the lens. In order to provide a good quality retinal image, these elements must be transparent and have appropriate curvatures and refractive indices. Refraction takes place at four surfaces, the anterior and posterior interfaces of the cornea and the lens,

Tear film

Epithelium Bowman'smembrane ~Stroma ~Decemet's

Air

membrane

--- Endotheli urn - Aqueous humour

p'

p

and there is also continuous refraction within the lens. In this chapter we describe the optical structure of the normal cornea and lens, but we must be aware that in many eyes there are significant departures from these norms. Some of these are due to serious ocular defects or irregularities, which can have a major impact on vision. There are also changes with age, and any descriptions given in this section relate only to mean adult values. More detail on age dependencies is given in Chapter 20.

Cornea The majority of the refracting power is provided by the cornea, the clear, curved 'window' at the front of the eye. It has about two-thirds of the total power for the relaxed eye, but this fraction decreases as the lens increases in power during accommodation.

Anatomical structure

Figure 2.1. The structure of the cornea and the approximate positions of the principal points.

The schematic cross-sectional structure of the cornea is shown in Figure 2.1. It has a tear film at its front surface, and several distinct parts. These are, in order from the outer surface of the eye, the epithelium, Bowman's membrane, the stroma, Descemet's membrane and the endothelium. Approximate thicknesses of these components are given in Table 2.1.

12

Basic optical structureof the human of defocused blur discs measured by Chan et al. (1985),corresponding values predicted by the approximate equation (9.17a),and values for a 4.2 mrn diameter pupil using the exact equation (9.15),with L =O.

12

(9.18) Defocus ratio = t/Jt, '/1}'= etv'1J In angular terms in object or visual space we have Defocus ratio = eP/() (9.19) where ()is the angular size of the object.

Imageformation: thedefocused paraxial image

Example 9.3: Calculate the defocus blur ratio for the situation given in Example 9.2, if the eye is observing a 6/6 letter, i.e. one that subtends 5 min arc.

Solution: Substitute cP = 13.8 min arc and (J = 5 min arc into equation (9.19) to give Defocus ratio = 13.8/5 = 2.76

Other effects of defocus

Alignment of two targets at different distances In some visual experiments, the eye is aligned by asking the subject to subjectively align two targets at different distances. Figure 9.7 shows points Q 1 and Q 2 aligned along the visual axis. If these points are so aligned, do they appear superimposed in the visual field? In other. words, are their retinal images superimposed? By analysing the situation shown in Figure 9.7, we can deduce that the targets do not appear to be aligned. This shows ~e point Q~ imaged, in focus, at Q2' on the retina. The other point Q 1 is imaged in front of the retina at QI' and is therefore def