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Acknowledgements

Several colleagues have commented on draftsor provided references and advice. Inparticular, we thank Brian Brown, NiallStrang and Tom Raasch, who read substantialsections of drafts. Others include RayApplegate, Pablo Artal, Harold Bedell, ArthurBradley, Neil Charman, Nicolas Chateau,Michael Doughty, Dave Elliot, Richard Guy,Douglas Homer, Tony [oblin, Phil Kruger,Barbara Pierscionek, Katrina Schmid,Lawrence Stark, Peter Swann, ChristopherTyler, Barry Winn, Joanne Wood and RussellWoods.

Weare grateful to Pablo Artal, Michael Cox,Larry Thibos and Barry Winn for providingdata for some figures; these are alsoacknowledged at the appropriate figurecaptions. We thank the American Academy ofOptometry, the Optical Society of America,Elsevier Press and the Association forResearch in Vision and Ophthalmology forpermission to use previously published data;again, these are acknowledged at the appro-priate figure captions.Finally, we thank our wives, Janette and

Yolette, for their support.

Sign convention and symbols

Units and their abbreviations

When we examine image formation and ray-tracing, we need a sign convention. Althoughthe choice of a sign convention is arbitrary, itmust be consistent. In this book we use thestandard cartesian and trigonometric signconventions. Distances to the left of a surfaceor other reference position or below theoptical axis are negative, and those to the rightor above are positive. Angles due toanticlockwise rotations of the ray from theoptical axis are positive, and those due toclockwise rotations are negative.

Distance notation and sign

Points are denoted by Roman letters in bold.Distances are denoted by either a single lowercase letter such as 1, or by two upper caseletters such as PF. In this example, P and Fareboth points and PF denotes the distance fromP to F. If F is to the right of P, this distance ispositive, and if F is to the left of P, thisdistance is negative.

Greek alphabet

Greek letters are used extensively throughoutthe book.

a.A alphaM deltaT]H etaKK kappavN nu1[0 pit'T tauXX chi

metrecentimetremillimetremicrometrenanometredioptreprism dioptreJouleWattcandelalumenssteradiansecondHertzKelvinradiandegreeminutes of arc

~B beta£E epsilonas thetaA.A lambda~::: xipP rhouY upsilon'I''P psi

mcmmmJlmnmDAJWcd1mstsHzKrad0, degmin. arc

yr gamma~Z zetatI iotaJlM mu00 omicron(J~ sigma<1><1> phiron omega

Introduction

The purpose of this book is to describe theoptical structure and optical properties of thehuman eye. Itwill be useful to those who havean interest in vision, such as optometrists,ophthalmologists, vision scientists, opticalphysicists, and students of visual optics. Anunderstanding of the optics of the human eyeis particularly important to designers ofophthalmic diagnostic equipment and visualoptical systems such as telescopes.Most animals have some sort of eye

structure or sophisticated light sense. Likehumans, some rely heavily on vision, includ-ing predatory birds and insects such ashoneybees and dragonflies. However, manyanimals rely much more on other senses,particularly hearing and smell, than on vision.The visual sense is very complex and is able toprocess huge amounts of information veryrapidly. How this is done is not fully under-stood; it requires greater knowledge of howthe neural components of vision (retina,visual cortex, and other brain centres) processthe retinal image. However, the first stage inthis complex process is the formation of theretinal image. In this text, we investigate howthe image is formed and discuss factors thataffect its quality.The majority of animal eyes can be divided

into two groups: compound eyes (as pos-sessed by most insects), and vertebrate eyes(such as the human eye). Compared withvertebrate eyes, there is considerable variationin the compound eyes. Compound eyescontain a large number of optical elements(ommatidia), each with its own aperture to the

external world. Vertebrate eyes have a singleaperture to the external world, which is usedby all the detectors. A number of otheranimals have simple eyes, which can bedescribed as less developed versions of thevertebrate eye. All eyes, of whatever type,involve compromises between the need fordetection (sensitivity), particularly at lowlight levels, and spatial resolving capability interms 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 theoptics in complete isolation from the neuralcomponents, as otherwise we cannot appre-ciate what influence changes in the retinalimage will have on vision performance. As anexample, altering the optics has considerableinfluence on resolution of objects for centralvision but not for peripheral vision. This isbecause the retina's neural structure is fineenough at its centre, but not in the periphery,for large changes in optical quality to be ofimportance (Chapter 18). Thus, the neuralcomponents of the visual system, particularlythe retinal detector, rate some mention in thebook. The neural structures of the retinathemselves produce optical effects. As anexample, the photoreceptors exhibit wave-guide properties that make light arriving fromsome directions more efficient at stimulatingvision than light arriving from other direc-tions. Another example is that the regulararrangement of the nerve fibre layers pro-duces 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 conventionaloptical laws, there are some interestingdifferences because of the eye's biologicalbasis. Perhaps the greatest difference is that,as a living organ, the eye responds to itsenvironment, often in an attempt to give thebest image under different circumstances.Also, it grows, ages and suffers disease.Unlike most man-made optical systems, theeye is not rotationally symmetrical about asingle axis, and different axes must be used todefine image formation.There are many interesting and important

optical effects associated with ocular diseasessuch as keratoconus (conical cornea) andcataract. Furthermore, the balance betweenoptical and neural contributions to overallvision performance changes with diseases ofthe retina and beyond. Although there aresome passing references to cataract, we haveconcentrated on the healthy human eye. Wegive some prominence to age-related changesin the optics of the eye throughout the book,and devote Chapter 20 to this topic.Tomake the book easy to read it is divided

into a number of short chapters, with eachchapter dedicated to a single theme. The mostcommonly useful topics are at the beginning,and topics with narrower appeal (such asocular aberrations) are placed towards the

end. Section 1 covers the basic opticalstructure of the human eye, including therefracting components, the pupil, axes andsimple models of the eye. Section 2 is aboutimage formation and refraction of the eye.This includes the refractive errors of the eye,their measurement and correction, andparaxial treatments of focused and defocusedimage sizes and positions. Section 3 dealswith the interactions between light and theeye, considering transmission, reflection andscatter in the media of the eye and at thefundus. Section 4 deals with aberrations andretinal image quality. As well as consideringthese for real eyes, it covers the modelling ofeyes and the performance of a range ofschematic eyes of different levels ofsophistication. Section 5 considers the topicsof depth-of-field and age-related changes inthe optics of the eye. While depth-of-fieldeffects could possibly have been placed earlierin the book, understanding them well requiressome knowledge about aberration anddiffraction. The book concludes with 4appendices, three of which (Appendices 1, 2and 4) cover some mathematics relating toparaxial optics, aberrations theory and imagequality criteria. Appendix 3 lists constructiondata, optical parameters and the aberrationsof a number of schematic eyes.

1

The human eye: an overview

Introduction

This chapter is a short overview of the opticalstructure and function of the human eye. Itmentions briefly some important aspects suchas the cornea, the lens and ocular axes, whichare covered in more detail in later chapters.Other important topics, such as the passage oflight, aberrations and retinal image quality,are also discussed in later chapters.

Temporal side

Aqueous humour

Nasal side

The structure of the human eye is shown inFigure 1.1. The outer layer is in two parts: theanterior cornea and the posterior sclera. Thecornea is transparent and approximatelyspherical with a radius of curvature of about8 mm. The sclera is a dense, white, opaque,fibrous tissue that is mainly protective infunction and is approximately spherical witha radius of curvature of about 12 rom. Thecentres of curvature of the sclera and cornea

Vitreous humour

NN' t

Centre of rotation

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

4 Basic optical structureof thehuman eve

are separated by about 5 mm. More accuratemeasures of shapes are given in subsequentchapters.The middle layer of the eye is the uveal

tract. It is composed of the iris anteriorly, thechoroid posteriorly, and the intermediateciliary body. The iris plays an importantoptical function through the size of itsaperture, the ciliary body is important to theprocess of accommodation, and both theciliary body and choroid support importantvegetative processes.The inner layer of the eye is the retina,

which is an extension of the central nervoussystem and is connected to the brain by theoptic 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, whichcontains the aqueous fluid.

3. The vitreous chamber, between the lensand the retina, which contains atransparent colourless and gelatinous masscalled the vitreous humour or vitreousbody.The internal pressure of the eye must be

higher than that of the atmosphere in order tomaintain the shape of the cornea, and must bemaintained at an approximately constantlevel in order to maintain the transparency ofthe ocular media. The pressure is controlledby the production of aqueous fluid in theciliary body and by drainage of this aqueous

Figure 1.2. Image formation of the human eye.

fluid from the eye. This drainage is throughthe angle of the anterior chamber (betweenthe cornea and the iris) to the canal ofSchlemm (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 asthose by Hogan et al. (1971) and Snell andLemp (1997).

Optical structure and imageformation

The principles of image formation by the eyeare the same as for man-made optical systemssuch as the camera lens. Image-forming lightenters the eye through the cornea, and isrefracted by the cornea and the lens to befocused at the retina. Of the two refractingelements, the cornea has the greater power.However, whereas the corneal power isconstant, the power of the lens can be changedwhen the eye needs to focus at differentdistances. This process is called accommo-dation, and occurs because of an alteration inthe lens shape. It is discussed further inChapter 20. The diameter of the incomingbeam of light is controlled by the iris, whichforms the aperture stop of the eye. Theopening in the iris is called the pupil. As withall optical systems, the aperture stop is a veryimportant component of a system, affecting awide range of optical processes, and it isdiscussed in more detail in Chapter 3.

Figure 1.2 shows two light beams fromobject points forming images on the retina.The image is inverted, as it is for a camera. Wediscuss this image formation in more detail inChapter 6.

The retina

The light-sensitive tissue of the eye is theretina. It is shown in Figure 1.3, and consistsof a number of cellular and pigmented layersand a nerve fibre layer. These layers havevarying degrees of optical significance, withthe amount of incoming light specularlyreflected and scattered by each layer being ofparticular importance. This aspect is dis-

6. Innernuclearlayer

7. lnner plexiform layer \ 'I8. Layerof ganglion cells'\ •

9. Nervefibre layer '\ I'

10. Inner limiting I'membrane --~

-LIGHT •

Fovea

Thehumall eye: alloverview 5

cussed in greater depth in Chapter 14. Thethickness of the retina varies from 50 urn(0.05mm) at the foveal centre to about 600 um (0.6mm) near the optic disc.There is a layer of light-sensitive cells at the

back of the retina, and the light must passthrough the other layers to reach these cells.These receptor cells are of two types, knownas rods and cones. The names refer to theirshapes, but considerable variations in shapeoccur with location, and it is not alwayspossible to distinguish between the two typeson this basis. Figure 1.4 shows theirdistribution along the horizontal temporalsection of the retina. There are about 100million rods in the retina, and they reach theirmaximum density at about 20° from the fovea.

....5. Outer plexiform layer{r Outer nuclear layer'r,3. Outer limitingmembrane

2. Photoreceptors(rodsand cones)1. Pigmentepithelium

Bruch's membrane

Sclera

'---- ---,/yRetina

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

6 Basic optical structure of the human"ye

central foveaouter fovea

0.2-0.30.7-0.80.46

um lIlil/. arc

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

Polyak (1941) 1.0-1.53.5-4.0

O'Brien (1951)* 2.3

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

field. At high light levels the best resolution isattained by the cones in the fovea, whichoccupies only about l/lOOOth of the totalretinal area. Despite the predominance ofcones at the fovea, it contains only a smallproportion (1 per cent) of the total cones(Tyler, 1996), and an even smaller proportion(0.05 %) of cones is found in the high-resolution foveola. Therefore, the vastmajority of cones are distributed throughoutthe peripheral retina. At low light levels thecones at the fovea do not operate; thus thecentre of the fovea is 'night blind', and it isnecessary to look eccentrically to see objectsusing the rods. At very low light levels,maximum visual acuity and detection abilityoccur 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 ofinterest, the centre of its image is formed onthe foveal centre, which is inclined at about 5°from the 'best fit' optical axis. At the fovea, thelayers overlying the receptor cells are thinnerthan 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, andcorresponds to the rod-free region. Thefoveola is the approximately O.5°-wide avas-cular centre of the foveal pit, and is the regionof highest resolution.The diameters and packing of the foveal

cones affect visual acuity, and we examine thisrelationship briefly in Chapter 18. Estimatesof the diameters of foveal cones are given inTable 1.1.The off-axis position of the fovea is most

intriguing since aberration theory predictsthat the best image of an optical system isusually formed on the optical axis. Thereforethe retinal image quality at the fovea shouldbe worse than at the posterior pole. The off-axis position of the fovea has some interestingvisual effects which we discuss in Chapter 17.

0sterberg (1935)

Curcio andHendrickson ( 1991 )

-,-:

'.

'"

Figure 1.4. The density of cones and rods across theretina in the temporal direction. From 0sterberg (1935),Curcio and Hendrickson (1991).The data from Curcioand Hendrickson (1991) have been converted fromdistances along the retina to angles relative to the backnodal point of the eye, assuming a spherical retina ofdiameter 12 mm and a distance between the back nodalpoint and retina of 17.054 rnm.

10 20 30 40 50 60 70

Angle from fovea (deg.)

160

140

"'E120

E':;s 100'0C...'":> 800

2.::-0:;'; 60c...0

40

20

00

There are approximately 5 million cones in theretina.In general, rods are longer and narrower

than cones. Rods are sometimes described ashighly sensitive low-level light detectors incomparison with cones. However, much ofthis is due to the neural wiring that occursrather than differences between the receptors.The retinal neural network of rods is such thatthe output of about 100 rods can combine onthe way to the brain, so that the rod systemhas very high sensitivity to light but poorspatial resolution. In contrast, the output offewer cones is combined, so the cone systemfunctions at higher light levels and is capableof higher spatial resolution. Cones recoverfrom exposure to light more quickly thanrods. The first stage in colour vision is theexistence of three types of cones, each withdifferent 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 assubtended at the back nodal point N' of theeye. The fovea is free of rods in its central 1°

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

The optic disc and blind spot

The vascular supply to the outer layers of theretina is carried in the choroid, which liesbetween the retina and the sclera. Thevascular supply to the inner retina enters theeye at the optic disc, whose location is shownin Figure 1.1.There are no cones or rods here,and hence this region is blind. The name givento the corresponding region in the visual fieldis the 'blind spot'. The optic disc is approxi-mately 5° wide horizontally and 7° vertically,and its centre is approximately 15° nasallyand 1.5° upwards relative to the fovea.Correspondingly, the blind spot is 15°temporally and 1.5° downwards relative tothe point of fixation. Figure 1.5 provides ademonstration of the blind spot.

The cardinal points

Every centred optical system that has someequivalent power (i.e. is not afocal) has sixcardinal points that lie on the optical axis.These are in three pairs. Two are focal pointswhich we denote by the symbols F and F', twoare principal points denoted by the symbols Pand P', and two are nodal points denoted bythe symbols Nand N'. The positions of thesecardinal points in an eye depend upon itsstructure and the level of accommodation. Foran eye focused at infinity, the approximatepositions of these cardinal points are shown inFigure 1.1.More precise positions are given inChapter 5, where we discuss schematic eyes

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

Tile Ill/mall eye: all (ll'CI'l';ew 7

and their properties. These cardinal points areas follows:1. Focal points (F and F'). Light leaving thefront (also first and anterior) focal point Fpasses into the eye, and would be imagedat infinity after final refraction by the lens ifthe retina were not in the way. Lightparallel to the axis and coming into the eyefrom an infinite distance is imaged at theback (also second and posterior) focal pointF'. Thus, for the eye focused at infinity, theretina coincides with the back focal point.

2. Principal points (P and P'). These are imagesof (or conjugate to) each other, such thattheir transverse magnification is +1. Thatis, if an object were placed at one of thesepoints, an erect image of the same sizewould be formed at the other point.

3. Nodal points (N and N'). These are alsoimages or conjugates of each other, buthave a special property such that a rayfrom an off-axis point passing towards Nappears to pass through N' on the imageside of the system, while inclined at thesame angle to the axis on each side of thesystem. Such a ray is called the nodal ray,and when the off-axis point is the point offixation, the ray is called the visual axis.We make use of these cardinal points

frequently in the following chapters.

The equivalent power and focallengths

One of the important properties of any opticalsystem is its equivalent power. This is ameasure of the ability of the system to bend ordeviate rays of light. The higher the power,the greater is the ability to deviate rays. Wedenote the equivalent power of an opticalsystem by the symbol F. The equivalentpower of the eye is related to the distancesbetween the focal and principal points by theequation

1 n'F =- PF =P'F' (1.1)

where n' is the refractive index in the vitreouschamber. The average power of the eye foradults is about 60 m-l or 60D, but the valuevaries greatly from eye to eye. Using this

8 Basic optical structure of til"1IIII/lIl1l "!I"power and the commonly accepted refractiveindex n' of the vitreous chamber (1.336), thefocal lengths of the eye arePF =-16.7 mm and P'F' =+22.3mm (1.2)While the equivalent power of the eye is avery important property of the eye, it is noteasy to measure directly. Its value is usuallyinferred from the other measurable quantitiessuch as surface radii of curvature, surfaceseparations and eye length, and assumedrefractive indices of the ocular media.However, more important than equivalent

power is refractive error. The refractive errorcan be regarded as an error in the equivalentpower due to a mismatch between theequivalent power and the eye length. Forexample, if the equivalent power is too highfor a certain eye length, the image is formed infront of the retina and this results in a myopicrefractive error. If the power is too low, theimage is formed behind the retina and resultsin a hypermetropic refractive error. Refractiveerrors are discussed in Chapter 7.

Axes of the eye

The eye has a number of axes. Figure 1.1shows two of these: the optical axis and thevisual axis. The optical axis is usually definedas the line joining the centres of curvatures ofthe refracting surfaces. However, the eye isnot perfectly rotationally symmetric, andtherefore even if the four refracting surfaceswere each perfectly rotationally symmetric,the four centres of curvatures would not beco-linear. Thus in the case of the eye, wedefine the optical axis as the line of best fitthrough these non co-linear points. The visualaxis is defined as the line joining the object ofinterest and the fovea, and which passesthrough the nodal points. These and the otheraxes are discussed in greater detail in Chapter4.

Centre-of-rotation

The eye rotates in its socket under the actionof the six extra-ocular muscles. Because of theway these muscles are positioned andoperate, there is no unique centre-of-rotation;

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

Field-of-vision

Examination of the pupil from differentangles shows that the pupil can still be seen atangles 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 thissuggests that the radius of the field-of-viewmay be as great as 105°, the real extent of thefield-of-vision depends upon the extent ofretina in the extreme directions. On the nasalside, vision is cut-off at about 60° because ofthe combination of the nose and the limitedextent of the temporal retina.

Binocular vision and binocularoverlap

The use of two eyes provides better per-ception of the external world than one eye

Figure 1.6. The horizontal field of view in monocularand binocular vision.

alone. Binocular vision improves contrastsensitivity and visual acuity slightly overthose obtained with monocular vision(Campbell and Green, 1965;Home, 1978).Twolaterally displaced eyes give the potential fora three-dimensional view of the world, whichincludes the perception of depth known asstereopsis. The degree of stereopsis dependspartly upon the distance between the twoeyes, which is called the interpupillarydistance. Stereopsis can be improvedconsiderably by optical devices such asrangefinders, which increase the effectiveinterpupillary distance.

Interpupillary distance

The interpupillary distance, or PO, is usuallymeasured by the distance between the centresof the two pupils of the eyes. The distance POis measured for the eyes looking straightahead; that is, the visual axes are parallel.When the eyes focus on nearby objects, theeyes rotate inwards and hence there is acorresponding decrease in interpupillarydistance. The near PO can then be measuredor determined from the distance PO usingsimple trigonometry.Harvey (1982) provided distance POs for

various armed services. Means ranged from

Tile /ullflan eye: an overview 9

61 to 65 mm, with standard deviations of3-4 mm. These values were mainly for men.

Binocular overlapAs shown in Figure 1.6, the total field-of-viewin the horizontal plane is about 2100 with a1200 binocular overlap.

Typical dimensions

All dimensions of the eye vary greatlybetween individuals. Some depend uponaccommodation and age. Representative dataare shown in Figure 1.7. Average values havebeen used to construct representative orschematic eyes, which we discuss in Chapter5. More detailed data are presented in laterchapters.

Summary of main symbols

F equivalent power of the eyen' refractive index of vitreous humour

(usually taken as 1.336)R radius of curvature in millimetresF, F' front and back focal pointsN, N' front and back nodal pointsP, P' front and back principal points

"-+-----16-----I 24

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

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

ReferencesCampbell, F. W. and Green, D. G. (1965). Mono-cular versus binocular visual acuity. Nil/lire, 208,191-2.

Curcio, C. A. and Hendrickson, A. E. (1991). Organizationand development of the primate photoreceptor mosaic.Prog. Retinal Res., 10, 89-120.

Fry,G. A. and Hill, W.W. (1962). The center of rotation ofthe eye. Alii. f. OptOIll. Arc/I. Am. Acad. OptOIll., 39,581-95.

Harvey, R. S. (1982). Some statistics of interpupillarydistance. 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 centralretina. }.Opt. Soc. Alii.,41, 882-94.

0sterberg, G. (1935). Topography of the layers of rods andcones in the human retina. Acta Ophthal. (Suppl.), 6,1-103.

Polyak, S. L. (1941). The Retina, p. 201. University ofChicago Press.

Snell, R.S. and Lemp, M. A. (1997). Clinical Anatomyof theE)/(', 2nd edn. Blackwell Scientific Publications.

Tyler,C. W.(1996). Analysis of human receptor density. InBasic alld Clinical Applications of Vision Science, TheProfessor fay M. Enoch Festschrift Volume (V.Lakshminarayanan, ed.), pp. 63-71. Kluwer AcademicPublishers.

2

Refracting components: cornea and lens

Introduction

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

The refracting elements of the eye are thecornea and the lens. In order to provide agood quality retinal image, these elementsmust be transparent and have appropriatecurvatures and refractive indices. Refractiontakes place at four surfaces, the anterior andposterior interfaces of the cornea and the lens,

CorneaThe majority of the refracting power isprovided by the cornea, the clear, curved'window' at the front of the eye. It has abouttwo-thirds of the total power for the relaxedeye, but this fraction decreases as the lensincreases in power during accommodation.

The schematic cross-sectional structure of thecornea is shown in Figure 2.1. It has a tearfilm at its front surface, and several distinctparts. These are, in order from the outer sur-face of the eye, the epithelium, Bowman'smembrane, the stroma, Descemet's mem-brane and the endothelium. Approximatethicknesses of these components are given inTable 2.1.

Anatomical structure

and there is also continuous refraction withinthe lens.In this chapter we describe the optical

structure of the normal cornea and lens, butwe must be aware that in many eyes there aresignificant departures from these norms.Some of these are due to serious ocular defectsor irregularities, which can have a majorimpact on vision. There are also changes withage, and any descriptions given in this sectionrelate only to mean adult values. More detailon age dependencies is given in Chapter 20.Epithelium

Bowman'smembrane~Stroma

~Decemet's membrane---Endotheliurn

- Aqueoushumour

Tear film

Air

pp'

12 Basic optical structureof the human <'!Ie

Table 2.1. Thicknesses (um) of comeallayers (Hogan etal.,1971).

maintain the stroma at about 78 per centhydration and thus retain transparency.

'The stroma thickens by at least an additional 150urn from the centre to theedge of the cornea.

Tear filmEpitheliumBowman's membraneStroma"Descemet's membraneEndothelium

Total

4-7508-14

500}0-125

- 580

Refractive index

Each corneal layer has its own refractiveindex, but since the stroma is by far thethickest layer, its refractive index dominates.The mean value of refractive index is usuallytaken as 1.376.

The tear film is 4-71J.m thick (Tomlinson,1992). It is composed of oily, aqueous andmucous layers, with 98 per cent of thethickness being provided by the aqueouslayer. The tear film is essential for clear visionbecause it moistens the cornea and smoothsout the 'roughness' of the surface epithelialcells. The tear film does not contributesignificant refractive power itself, since it isvery thin and consists effectively of two veryclosely spaced surfaces of almo~t equal.radi,i.However, the importance of this tear film 1Srealized if it dries out. If this occurs, thetransparency of the cornea decreasessignificantly.The epithelium protects the rest of the

cornea by providing a barrier against water,larger molecules and toxic substances. Itconsists of approximately six layers of cell~,

and only the innermost layer of these cells 1Sable to divide. After cells are formed, theymove gradually towards the surface as thesuperficial cells are shed.Bowman's membrane is 8-141J.m thick, and

consists mainly of randomly arrangedcollagen fibrils.The stroma comprises 90 per cent of the

corneal thickness, and consists mainly of 200or more collagen lamellae. The collagen fibrilswithin each lamella run parallel to each other,and the successive lamellae run across thecornea at angles to each other. This arrange-ment maintains an ordered transparentstructure while enhancing mechanicalstrength.Descemet's membrane is the basement

membrane of the endothelial cells.The endothelium consists of a single layer

of cells, which are hexagonal and fit togetherlike a honeycomb. The endothelium regulatesthe fluid balance of the cornea in order to

Radii of curvature, vertex powers andtotal corneal power

Several studies have measured the anteriorradius of curvature, but there have been farfewer investigations of the rear surface.Experimental distributions of the vertex radii(R) of curvature are given in Table 2.2.Similarresults have been obtained by variousinvestigators for the anterior cornea, butclearly there is a reasonable degree ofvariation between patients, and females haveslightly steeper anterior corneas than males.There is a high linear correlation between theanterior and posterior radii of curvature(Lowe and Clark, 1973; Dunne et ai., 1992;Patel et ai., 1993) and a reasonable fit of thisrelationship isRz=0.81R} (2.1)From these radii of curvature, we cancalculate the surface powers (F) using theequationF=(n' - n)/R (2.2)where nand n' are the refractive indices on theincident and refracted side, respectively. Atthe anterior surface/1 = 1 and /1' = 1.376and at the posterior surfacen =1.376and n' =1.336Surface power values, using these data andequations, are given in Table 2.2.The total power F of the cornea can be

calculated from the 'thick lens' equation:F =F} + Fz - F}Fzd/ J1 (2.3)where F} is the anterior surface power, Fz is

Refracting components: cornea and lens 13

Table 2.2. Population distributions of corneal vertex radii of curvature R and corresponding powers calculated fromequations (2.2)and (2.3),assuming corneal refractive index 1.376, aqueous refractive index 1.336and a cornealthickness 0.5 mm. Where results were provided for more than one meridian, mean values have been used.

AnteriorSIN R(mm) F(D)

Donders (1864)females 38/- 7.80 48.2males 79/- 7.86 47.9Stenstrom (1948) -/1000 7.86±0.26 47.8Sorsby et al. (1957) -/194 7.82±0.29 48.1Lowe and Clark (1973) 46/92 7.65±0.27 49.2Kiely et al. (1982) 88/176 7.72±0.27 48.7Edmund and Sj0ntoft (1985) 40/80 7.76±0.25 48.5Guillon et al. (1986) 110/220 7.78±0.25 48.3Koretz et al. (1989)females 68/· 7.69 ±0.23 48.9males 32/- 7.78 ±0.24 48.3Dunne et al. (1992)females 40/40 7.93±0.20 48.0males 40/40 8.08 ±0.16 47.1Patel et al. (1993) 20/20 7.68±0.40 49.0Mean (unweighted) 7.83 48.0

5 = number of subjects; N = number of eyes; - = number not provided.

Posterior TotalR(mm) F(D) F(D)

6.46 ± 0.26 --6.2 43.2

6.53 ± 0.20 --6.1 42.06.65 ± 0.16 --6.0 41.25.81 ±0.41 --6.9 42.26.34 --6.3

the posterior surface power, d is the vertexcorneal thickness and Jl is the refractive indexof the cornea (usually taken as 1.376). Thepower of the cornea can be estimated from thesum of the surface powersF "" F1 + F2 (2.3a)This value is different from the exact value byan amount FIF2d/u. Using the data given byPatel et al. (1993) in Table 2.2, a cornealthickness of 0.5 mm and a refractive index of1.376, the exact equation (2.3) gives a cornealpower of 42.20 and the approximateequation (2.3a) gives a slightly lower value of42.1 D.The above surface power values apply to

the vertices of the cornea, and would apply toother parts of the corneal surfaces only if theywere spherical. However, neither the anteriornor the posterior surfaces are perfectlyspherical due to both toricity and asphericity.Therefore, the radii of curvature do not fullydescribe the shape of the cornea and itsrefracting properties.

Anterior surface shape

ToricityFrequently the anterior corneal surface

exhibits toricity, which produces astigmatism.In young eyes, the radius of curvature isgenerally greater in the horizontal than in thevertical meridian (referred as 'with the rule'),but this trend reverses with an increase inage.

AsphericityIn general, the radius of curvature increaseswith distance from the surface apex, so thatthe surface flattens away from the apex.Surfaces that are non-spherical in this senseare often described as aspheric.The shape of the anterior corneal surface

has been extensively studied, especially overthe central 8 mm of its approximately 12mmdiameter. This central 'optical' zone is themaximum zone of the cornea through whichlight passes to form the foveal image. Toinvestigate the shape over this zone, cornealsurfaces are often represented by conicoids inthree dimensions or conics in two dimensions.A conicoid can be expressed in the formh2 + (1+Q)Z2 - 2ZR =0 (2.4)

wherethe Z axis is the optical axish2 =)(2 + y2R is the vertex radius of curvature and

14 Basic optical structureof tileIIulllan eye

Q is the surface asphericity, whereQ < -1 specifies a hyperboloidQ =-1 specifies a paraboloid-1 < Q < 0 specifies an ellipsoid, with theZ-axis being the major axisQ = 0 specifies a sphereQ > 0 specifies an ellipsoid with themajor axis in the X-Y plane.

The effect of the value and sign of Qon theshape is shown in Figure 2.2. Sometimesasphericity is expressed in terms of a quantityp,which is related to Q by the equationp= (1+Q) (2.5)

The conicoid form described by equation (2.4)is not the only mathematical representationused in the literature to describe conic(oid)s.Many investigators have measured surfaceshapes separately in different sections, andfitted the data to ellipses, which can bedescribed by the equation

(Z _a)2 + £ = 1 (2.6)a2 b2

where a and b are ellipse axes' semi-lengths.The shape of such ellipses is often describedby the eccentricity e,which is related to a and

b by the equatione2 = 1 - b2/ a2 (2.7)

provided that the Z-axis is the major axis.Equation (2.6) can be transformed easily

into the form of equation (2.4). If we do this,we find that the vertex radius of curvature Ris related to a and b by the equationR = + b2/ a (2.8)and that the asphericity Q is given by theequationQ=b2/ a2-1 (2.9)

It follows from equations (2.7) and (2.9), thatthe quantities e and Q are related by theequationQ =- e2 (2.10)

Specifying asphericity using e is notcompletely satisfactory because e2 may benegative, in which case e cannot have a value.Other forms of representing corneal shape

are described in Chapter 16. More complexforms are important to describe the shape ofthe cornea accurately, especially outside theoptical zone.Measured values of the anterior corneal

yHyperboloids:Q < -I

• Paraboloid: Q = -1.0

Ellipsoids: O>Q>-I

\\

\

z

Figure 2.2. The effect of asphericity on the shape of a conicoid. All the curves havethe same apex radius of curvature.

Refracting components: cornea andlens 15

Table 2.3. Summary of asphericity data for the anterior surface of the cornea.

Lotmar (1971)(using Bonnet (1964)'s data)EIHage and Berny (1973)Mandell and St Helen (1971)Kiely et al. (1982)Edmund and Sj"ntoft (1985)Guillon et al. (1986)'Patel et al, (1993)Lam and Douthwaite (1997)

"The mean of the steepest and shallowest meridians.

No.of subjects/eyes

1/18/888/17640/80110/22020/2060/60

Q

-0.286+0.16-0.23-0.26-0.28-0.18-0.01-0.30

s.d.or range

-0.04 to -0.720.180.130.150.250.13

(2.11a)

asphericity (Q) are given in Table 2.3. Thevalues of Q are usually negative, indicatingthat the cornea flattens away from the vertex.Figure 2.3 shows the profiles of cornealsurfaces, all with a radius of curvature of7.8 mm but with different Q values.

Optical significance of corneal asphericityThere has been considerable speculation as towhy the cornea flattens away from the centre.It has been argued that the cornea flattens inorder to reduce spherical aberration, andcertainly the flattening does lead to a lowerspherical aberration, but the amount ofasphericity in the average cornea is notsufficient to eliminate it. The value of Q

5

<)u§ 2'iiiis~ Q=-o.26

--tr- Q=-0.52---III-- Q=-1.0

o..-.........--,---.-,---.--+-o I 2 3

Distance along optical axis (mm)

Figure 2.3. The anterior surface of a cornea with a vertexradius of curvature of 7.8 rom, with various values ofthe asphericity Q.

required to eliminate spherical aberration atthe anterior surface is -0.528, given a cornealrefractive index of 1.376. Perhaps animportant reason for the flattening is the needfor the cornea to make a smooth join with themain globe of the eye.

Off-axis radii of curvatureThe radius of curvature at any point on aspherical surface and in any meridian is thesame. However, for a conicoid surface, theradius of curvature at off-axis points dependsnot only upon the distance from the vertex,but also on the meridian at that point. Thereare two principal meridians: the tangentialmeridian, which lies along the radius linefrom the vertex, and the sagittal meridian,which is perpendicular to the tangentialmeridian. For conicoids, the correspondingequations for the sagittal radius of curvature(Rs) and the tangential radius of curvature (Rt)areRs =[R2 - Qy2F12andRt = [R2 - Qy2]312 /R2 = Rs3/R2 (2.11b)Figure 2.4 shows changes in Rt and Rswithdistance Y from the corneal vertex for anasphericity Q value of -0.18 (Guillon et al.,1986) and a vertex radius of curvature of 7.8mm.An alternative name for the tangential

radius of curvature is the instantaneousradius of curvature, while the sagittal radiusof curvature is also called the axial radius ofcurvature. Unfortunately, this last term maybe readily confused with the vertex radius ofcurvature.

16 Basic optical structureof the hUlllall eye8.7 +--_...J-__--'-__....L-__-L__-t-

apsule

p'

Lens bulk

p

Lens

Figure 2.5 shows a cross-section of the lens.The lens bulk is a mass of cellular tissue ofnon-uniform gradient index, contained withinan elastic capsule. Wedo not have an accuratemeasure of this index distribution. There is alayer of epithelial cells, extending from theanterior pole of the lens to the equator. Thelens grows continually throughout life, withnew epithelial cells forming at the equator.These cells elongate as fibres that wraparound the periphery of the lens, under thecapsule and epithelium, to meet at sutures.The older fibres lose their nuclei and otherintracellular organelles. Because of thecontinual growth of the lens throughout life,lenticular parameters are age-dependent.The lens capsule plays an important role in

the accommodation process. It is attached tothe ciliary body via the zonules, as shown in asimplified fashion in Figure 2.6. Contractionof the ciliary muscle within the ciliary bodyleads to changes in zonular tension, whichalter lens shape. This process causes a changein the equivalent power of the lens and hence

the anterior and posterior surfaces, thecorneal thickness and the refractive indices.Representative positions are shown in Figure2.1. Note that both principal points are in frontof the cornea.

543

..,

2

Sagittal

Tangential

7.7 +----r----r----,----,----t-o

7.9

7.8

8.6

Posterior surface shape

Positions of the principal points

This is difficult to measure because of theinfluence of anterior surface shape on anymeasurement. It is of lesser significance thanthe anterior surface shape because of the smallrefractive index difference across the posteriorcorneal boundary, but it is not of negligiblesignificance.Patel etal. (1993) estimated the shape of the

posterior surface of 20 corneas from measuredanterior surface shapes and peripheral valuesof corneal thickness. For the 20 subjects, theyfound a mean posterior vertex radius of 5.8mm and a mean asphericity of Q = -0.42.However, Patel and co-workers used a meananterior corneal shape that was almostspherical (Q =-0.01) and therefore much lessaspheric than found in other studies. Lam andDouthwaite (1997) used a similar approachwith a group of 60 young Chinese in HongKong, and obtained asphericity of Q =-0.66 ±0.38. In this case, the anterior surfaceasphericity was Q=-0.31 ± 0.13.

Distance from corneal vertex (mm)

Figure 2.4.The radii of curvature in the sagittal andtangential directions of a cornea with a vertex radius ofcurvature of 7.8mm and a corneal asphericity of Q=-{l.18, from Guillon et al. (1986).

8.5

I 8.4

~ 8.3~~ 8.2=oce 8.1

'"=~ 8.0IX

The positions of the principal points of thecornea depend upon the radii of curvatures of

Figure 2.5. Cross-section of the lens showing theapproximate positions of its principal points.

Refracting components: cornea andlens 17

Table 2.4. Population distributions of in vivo lensvertex radii of curvature (R).

10.29± 1.7812.4± 2.6 -8.1 ± 1.6

in the ocular equivalent power, and allows theeye to focus on objects at different distances.Note that contraction of the ciliary muscledecreases tension on the zonules and thisallows the lens to take up a more curved form,appropriate for near vision.

No. of subjects/ Anterioreyes mm

Lowe (1972) 46/92Brown (1974) 100/-

Posteriormm

Surface radii of curvature and shapes

Some values of the radii of curvature of thelens are given in Table 2.4.These values mustbe treated with caution for three reasons. First,the lens radii of curvature change withaccommodation; second, the values are highlyage-dependent; and third, any measurementsof the lens in vivo (within the eye) dependupon knowing the value of all the opticalparameters that precede the particularsurface. This is a particular problem with theposterior surface because of the uncertainty ofthe refractive index distribution in anyparticular lens.The most common method of determining

the radii of curvature of the lenticular surfacesis by measurement of the Purkinje images,

Accommodated(Near vision)

Figure 2.6.The effect of accommodation on the lensshape and lens position, and on the principal and nodalpoints of the eye.

which are the images of an object formedthrough specular reflection at the ocularsurfaces. Procedures for doing this aredescribed by Tunnacliffe (1993), Smith andGamer (1996), Gamer and Smith (1997) andRabbetts (1998).Howcroft and Parker (1977) provided shape

data for in vitro lenses (outside the eye), butthe lenses were in unknown accommodatedstates and probably did not represent theshape of lenses in vivo. Furthermore, theasphericity values were published as absolutevalues, thus losing the sign. Some attempts toextract asphericity Qvalues from the data aregiven in Table 2.5. There are some data for invivo samples (Brown, 1974) that wereanalysed by Liou and Brennan (1997) to giveQ values. From the results of Smith et al.(1991) it is clear that there is a wide variationin values, and this is probably due to acombination of the difficulty in accuratemeasurement of asphericity and inter-individual variations in asphericity values.

Thickness

The lens thickness is often taken to be about3.6mm in the relaxed state, but the lensthickens upon accommodation and withincreasing age (for example, see Koretz et al.,1989and 1997). These changes in thickness arediscussed further in Chapter 20.

Refractive index distribution

The refractive index within the lens is notconstant, being greatest in the centre and leastin the periphery (Nakao et al., 1969;Pierscionek et al., 1988; Pierscionek and Chan,1989; Pierscionek, 1995). In the nuclear(central) region of the lens the indexmagnitude is almost constant, with thegreatest variations occurring in the cortex

18 Basic optical structure of thehuman eye

Table 2.5. Population distributions of lens surface asphericilies (Ql.

Sample size Anterior Posterior

In vitrovaluesKooijman (1983)a -6.06 -1.19Smith et al. (1991)b 59 -1.08 ± 9.41 --0.12 ± 1.74

In vivo valuesLiou and Brennan (1997)< 100 --0.94 +0.96

'The original values given by Kooijman in his paper were calculated lrom data published by Howcroft and Parker (1977)butwere in error, and the above values are corrected values (private communication, 1985)."calculated from data on 59 eyes. supplied by Parker (private communication. 1985).'Calculated from data of Brown (1974).

(periphery). This variation in index producesa progressive and continuous refraction ofrays, and may improve the quality of theimage by reducing spherical aberration.Gullstrand (1909) gave an equation for the

refractive index distribution within the lens.An equivalent form of his equation isn(Y,2) = 1.406 - 0.0062685(2 - 2 0)2+ 0.0003834(2 - 2 0)3- [0.00052375+ 0.00005735(2 - 2 0) + 0.00027875(2 - 2 0)2]y 2 -o.000066717y4 (2.12)This gives a maximum index of 1.406 at thenominal centre of the lens, which occurs at 2= 20, and an index of 1.386 at the edge of thelens. Gullstrand gave a value for 2 0 of 1.7 mm,and the total lens thickness as 3.6 mm.

Equivalent refractive indexIf the real lens with its gradient index isreplaced by one with the same thickness,same radii of curvature and a uniformrefractive index, this index must be madehigher than the maximum index. Theequivalent refractive index is often taken as1.42, compared with the maximum value ofabout 1.406.

Equatorial diameterThe equatorial diameter of the lens is between8.5 and 10mm (Pierscionek and Augusteyn,1991).

Positions ofprincipal pointsThe positions of the principal points of thelens depend upon the radii of curvature of the

anterior and posterior lens surfaces, the lensthickness and its refractive index distribution.All of these depend upon the level of accom-modation. The positions of the principalpoints depend also on the refractive indices ofthe surrounding media. Representativepositions are shown in Figure 2.5.

Accommodation

During accommodation, when the eye needsto change focus from distant to closer objects,the ciliary muscle contracts and causes thesuspensory ligaments, which support thelens, to relax. This allows the lens to becomemore rounded, thickening at the centre andincreasing the surface curvatures. The frontsurface moves slightly forward. Thesechanges (shown in Figure 2.6) result in anincrease in the equivalent power of the eye.When the eye has to focus from close to moredistant objects, the reverse process occurs.Accommodation is discussed in greater detailin Chapter 20.The stimulus-response mechanism in

accommodation is not fully understood. Forexample, we do not understand how the brainknows which way to change the lens power,although there are some indications thatchromatic aberration is involved (see Chapter17).In a relaxed eye focused for infinity, the

equivalent power of the lens is approximately19 D. In an eye accommodating to a point 10em from the anterior cornea, the lens power isapproximately 30 D.While we measure the level of accom-

modation as the vergence of the 'in focus'object, this vergence should not be mistakenas the power of the eye. For the relaxed eye,

Refracting components: cornea andlens 19

Relaxed staterAccommodation range -I

Far Nearpoint point

Accommodated state

Figure 2.7. The near and far points of the eye. This is a myopic eye, because the farpoint is in front of it. For a hyperopic eye the far point would be behind the eye,and for the emmetropic eye the far point would be at infinity.

the accommodation level is zero, but thepower of the eye is about 60 D. Although theaccommodation level and increase in lenspower are not the same, they are closelyrelated.There are physical limits to the range of lens

shapes, and hence restrictions to changes inlens power and the range of clear vision. Thefurthest and closest object points along thisrange are called the far and near points,respectively, and are shown schematically inFigure 2.7. When the ciliary muscle iscompletely relaxed the eye is focused on thefar point, which is conjugate to the retina.When the ciliary muscle is maximallycontracted or the lens maximally relaxed, theeye has its greatest equivalent power and thenear point is conjugate to the retina.The distance between the far and near

points is called the range of accommo-dation - for example, from infinity to 20 cm.The difference between the vergences of thefar and near points is called the amplitude ofaccommodation.Example 2.1: Calculate the amplitude ofaccommodation of an eye with a far pointof 1.25m and a near point of 10 em.Solution: The vergence of the far point is1/1.25 = 0.8 m-1 or 0.8 D. The veryence ofthe near point is 1/0.10 = 10.0m- or 10.0D.The difference is 10.0- 0.8 =9.2 D.The amplitude of accommodation is

affected by age. It probably reaches a peakearly in the second decade of life, and thengradually declines to become zero atapproximately the middle of the sixth decade.This and other effects of age are discussed inChapter 20.

As the two eyes accommodate to focusclearly on a close object, they also must rotateinwards to fixate on the object. This inwardrotation is called convergence. Accom-modation and convergence are controlled tosome extent by the same nervous pathwayfrom the brain, and there is an interactionbetween accommodation and convergencecalled a synkinesis. A stimulus to eitheraccommodation or convergence can causeboth to change. An example is that of placingan occluder in front of one eye and placing anegative-powered lens in front of the othereye. As well as the negative lens stimulatingaccommodation, the occluded eye turnsinwards.As the accommodation level increases, all

the cardinal points of the eye move towardsthe anterior surface of the lens (Chapter 5).

Summary of main symbols

e eccentricity of an aspheric surfacen, n' refractive indices on incident and

refraction sides of a surfaceQ surface asphericity (=-e2)p surface asphericity (= 1 +Q)R radius of curvatureZ optical axisF equivalent powerX, Y distances perpendicular to optical axis

ReferencesBonnet, R. (1964). La Topographie Corneenne. Desroches

(cited by Lotmar, 1971).Brown, N. (1974). The change in lens curvature with age.

Exp. EyeRes., 19, 175-83.

20 Basic optical structure of thehumaneye

Donders, F.C. (1864). On the Anomalies of Accommodationand Refraction of the Eye(translated by W. D. Moore), p.89. The New Sydenham Society.

Dunne, M. C. M., Royston, J.M. and Barnes, D. A. (1992).Normal variations of the posterior corneal surface. ActaOphthal., 70, 255-61.

Edmund, C. and Sjentoft, E. (1985). Thecentral-peripheral radius of the normal cornealcurvature. Acta Ophthal., 63, 670-77.

EI Hage, S. G. and Berny, F. (1973). Contribution of thecrystalline lens to the spherical aberration of the eye. f.Opt. Soc. Am., 63, 205-11.

Garner, L. G. and Smith, G. (1997).Changes in equivalentand gradient refractive index of the crystalline lenswith accommodation. Optom. Vis. Sci., 74,114-19.

Guillen, M., Lydon, D. P. M. and Wilson, C. (1986).Corneal topography: a clinical model. Ophthal. Physiol.Opt., 6, 47-56.

Gullstrand, A. (1909). Appendix II: Procedure of the raysin the eye. Imagery - laws of first order. In Helmholtz'sHandbuch der Physiologischen Optik, Volume 1 (Englishtranslation edited by J. P. Southall, Optical Society ofAmerica, 1924).

Hogan, M. J., Alvarado, J. A. and Weddell, J. E. (1971).Histology of the Human Eye. W. B.Saunders and Co.

Howcroft, M. J. and Parker, J. A. (1977). Asphericcurvatures for the human lens. Vision Res., 17, 1217-23.

Kiely, P.M., Smith, G. and Carney, L. G. (1982).The meanshape of the human cornea. Optica Acta, 29, 1027-40.

Kooijman, A. C. (1983).Light distribution on the retina ofa wide-angle theoretical eye. f. Opt. Soc. Am., 73,1544-50.

Koretz, J. E, Cook, C. A. and Kaufman, P. L. (1997).Accommodation and presbyopia in the human eye.Changes in the anterior segment and crystalline lenswith focus. Invest. Ophihal. Vis. Sci., 38, 569-78.

Koretz, J. E, Kaufman, P. L., Neider, M. W. and Goeckner,P. A. (1989). Accommodation and presbyopia in thehuman eye - aging of the anterior segment. Vision Res.,29, 1685-92.

Lam, A. K. C. and Douthwaite, W. A. (1997).Measurement of posterior corneal asphericity on HongKong Chinese: a pilot study. Ophthal. Physiol. Opt., 17,348-56.

Liou, H.-L. and Brennan, N. A. (1997). Anatomicallyaccurate, finite model eye for optical modelling.]. Opt.Soc. Alii. A, 14, 1684-95.

Lotmar, W. (1971).Theoretical eye model with aspherics.,. Opt. Soc. Am., 61, 1522-9.

Lowe, R. E (1972). Anterior lens curvature. Comparisons

between normal eyes and those with primary angle-closure glaucoma. Br.]. Dphthal.,56,409-13.

Lowe, R. F. and Clark, B. A. J. (1973). Posterior cornealcurvature. Correlations in normal eyes and in eyesinvolved with primary angle-closure glaucoma. Br. f.Ophthal., 57, 464-70.

Mandell, R. B. and St Helen, R. (1971). Mathematicalmodel of the corneal contour. Br. f. Physiol. Opt., 26,183-97.

Nakao, S., Ono, T., Nagata, R. and Iwata, K. (1969). Thedistribution of refractive indices in the humancrystalline lens.Jp». J. Clin. Ophthal., 23, 903-6.

Patel, S., Marshall, J. and Fitzke, F.W. (1993). Shape andradius of posterior corneal surface. Refract. Com. Surg.,9,173-81.

Pierscionek, B. K. (1995). Variations in refractive indexand absorbance of 670-nm light with age and cataractformation in human lens. Exp. EyeRes., 60, 407-14.

Pierscionek, B. K. and Augusteyn, R. C. (1991). Shapesand dimensions of in vitro human lenses. Clin. Exp.Optom.,74, 223-8.

Pierscionek, B. K. and Chan, D. Y. C. (1989). Refractiveindex gradient of human lenses. Optom. Vis. Sci., 66,822-9.

Pierscionek, B. K., Chan, D. Y. c, Ennis, J. P.et aI. (1988).Non-destructive method of constructing three-dimensional gradient index models for the crystallinelens: 1. Theory and Experiment. Am. ,. Optom. Physiol.Opt., 65, 481-91.

Rabbetts, R. B. (1998). Bennett and Rabbetts' Clinical VisualOptics,3rd edn. Butterworth-Heinemann.

Smith, G. and Garner, L. G. (1996). Determination of theradius of curvature of the anterior lens surface from thePurkinje images. Ophthal. Physiol. Opt., 16, 135-43.

Smith, G., Pierscionek, B. K. and Atchison, D. A. (1991).The optical modelling of the human lens. Ophthal.Pllysiol. Opt., 11,359-69.

Sorsby, A., Benjamin, B., Davey, J. B. et al. (1957).Emmetropia and its Aberrations. A Study in theCorrelationof the Optical Components of the Eye. Medical ResearchCouncil special report series no 293. HMSO.

Stenstrom, S. (1948). Investigations of the variation andcorrelation of the optical elements of the human eye.Part Ill, Chapter III (translated by D. Woolf) Am. ,.Optom. Arch.Am. Acad.Optom.,25, 340-50.

Tomlinson, A. (1992). Tear film changes with contact lenswear. In Complications of Contact Lens Wear (A.Tomlinson, ed.), Ch. 8. Mosby Year Book.

Tunnacliffe, A. H. (1993). Introduction to Visual Optics,4thedn. Association of British Dispensing Opticians.

3

The pupil

Introduction - the iris

The iris forms the aperture stop of the eye. Itsaperture or opening is known as the pupil.The pupil size is determined by twoantagonistic muscles, which are underautonomic (reflex) control:1. The sphincter pupillae, which is a smooth

muscle forming a ring around the pupillarymargin of the iris. When it contracts, thepupil constricts. It is innervated by theparasympathetic fibres from the oculo-motor (3id cranial) nerve by the way of theciliary ganglion and the short ciliarynerves.

2. The dilator pupillae, which is moreprimitive and consists of myo-epithelialcells that extend radially from the sphincterinto the ciliary body. It dilates the pupiland is innervated by sympathetic nervefibres, which synapse in the superiorcervical ganglion and enter the eye by wayof the short and long ciliary nerves.Iris colour varies markedly between

different people, and depends upon theamount of pigmentation within the stromaand anterior limiting layer. Lightly pigmentedirides appear blue, and the more pigmentthere is in the iris, the browner the eyeappears.In this chapter, we discuss the properties of

the aperture stop/pupil, factors that affectpupil size and the effect of pupil size on theretinal image.

Entrance and exit pupils

In general optical systems, the opening in theaperture stop is not referred to as the pupil.The word 'pupil' is used for the images of theaperture stop. The image of the stop formedby the optical elements in front of it is theentrance pupil - in other words, the entrancepupil of an optical system is the image of theaperture stop formed in object space. Theimage of the aperture stop formed by theelements behind it is the exit pupil.Alternatively, we can say that the exit pupil isthe image of the aperture stop formed inimage space. When we look into an eye at theaperture stop, we see the image of the stopformed by the cornea (i.e. the entrance pupil).The exit pupil of the eye is the image of theaperture stop formed by the eye's lens.With respect to the eye, and depending on

the context, the term 'pupil' is generally usedto refer to either the aperture stop opening -the 'real' or 'actual' pupil- or to the entrancepupil that we see. Compared with theentrance pupil, the exit pupil of the eye haslittle practical significance. In the rest of thechapter, when we refer to pupil size we arereferring to the entrance pupil.Given the ocular parameters of an eye (for

example, as given for the schematic eyes inAppendix 3), paraxial optics can be used todetermine the size and positions of theentrance and exit pupils. To determine theentrance pupil position and size, we need totrace a ray from the centre of the iris out of the

22 Basic optical structure of the human eye

li'J;; "'-Entrance pupil

Figure 3.1. The formation of the entrance pupil.

eye. For the exit pupil, we trace a ray from thesame point towards the retina. We show howthis is done in Chapter 5. In such calculations,we assume that the actual pupil lies in thefront vertex plane of the lens.Figure 3.1 shows, in schematic form, a ray

traced from the iris at A through the corneaand out of the eye. This ray appears to crossthe axis at the point E, which locates theentrance pupil. This figure is not to scale, butshows correctly that the entrance pupil isforward to and larger than the aperture. Theentrance pupil is actually only slightlyforward to and larger than the aperture. Inone schematic eye, the Gullstrand number 1relaxed schematic eye, the aperture is 3.6 mmfrom the corneal vertex and the entrance pupilis 3.05 mm from the corneal vertex. The

Temporal side

Nasal side

Figure 3.2.The iris and entrance and exit pupils of theGullstrand number 1 relaxed schematic eye (unit is mm).

entrance pupil is 13.3 per cent larger than theaperture. The exit pupil is 0.07mm behind theaperture and 3.1 per cent larger. Figure 3.2shows the positions and sizes of the apertureand entrance and exit pupils of this schematiceye.

Effect of aberrationsThe above calculations are based uponparaxial optics, which ignore aberrationeffects. The predictions are valid only forsmall pupils observed along or close to theoptical axis. Aberrations of the cornea havesome effect for wide pupils and for obliqueviewing. We discuss the effect of aberrationson pupil magnification in Chapter 15.

Accommodation

Upon accommodation, the anterior surface ofthe lens moves forward. Gullstrand's number1 schematic eye has a highly accommodatedversion in which this movement and that ofthe aperture stop is 0.4mm, The entrance andexit pupils move forward by similar amounts.

The paraxial marginal ray andparaxial pupil ray

If we wish to analyse the optical properties ofthe eye, two useful and special rays are theparaxial marginal ray and the paraxial pupilray (also paraxial chief ray). These can bedefined as follows:• The paraxial marginal ray is the paraxial rayfrom an on-axis object point, which passesthrough the edges of the pupils and theaperture stop and to the image point (whichalso must be on axis).

• The paraxial pupil ray is the paraxial rayfrom an object point, at the edge of anominated field-of-view, which passesthrough the centres of the pupils and theaperture stop.The nominal paths of these two rays are

shown in Figures 3.3a and 3.3b, with theactual paths for schematic eyes found byparaxial ray tracing (Appendix 1). These tworays are useful in various ways. For example,

The pupil 23

Entrance pupil Exit pupil~l:~

Paraxial marginal ray,\ __~.~__

o

(a)

(b)

Q

Paraxialmarginal ray

Paraxialpupil ray

Cornea'"7~r~/ Ins_ "-

Entrance pupil Exit pupil

Figure 3.3. The entrance and exit pupils of theeye and the paraxial marginal and pupil rays.

the path of the paraxial marginal ray and theangle 0.' inside the eye are useful indetermining retinal light level (Chapter 13).The paraxial pupil ray is useful in calculatingthe position of off-axis retinal images (seeChapters 6,9 and 10),and both rays are usefulin estimating the aberrations of the eye.

Pupil centration

In any rotationally symmetric optical system,the pupils are centred. However, the pupils ofreal eyes are usually decentred, often beingdisplaced nasally by about 0.5mm relative tothe optical axis (Westheimer, 1970). Theposition of the (entrance) pupil controls thedirection of the path of a beam passing intothe eye, and therefore affects the amount andtype of aberrations and hence retinal imagequality.The pupil centre may move with change in

pupil diameter. Walsh (1988) found that inboth naturally- and drug-induced pupil

dilations, the pupil centre moved by up to 0.4mm in some subjects. Wilson et al. (1992)confirmed Walsh's findings. Most subjectsshowed temporal movement of the pupilcentre with increase in pupil size.

Pupil size

Some of the factors controlling or affectingpupil size are briefly discussed here.Loewenfeld (1993) gave a comprehensivereview of these factors.

Level of illuminationThis is the most important factor affectingpupil size. The diameter of the pupil may varyfrom about 2 mm at high illumination toabout 8 mm in darkness, corresponding toapproximately 16 times variation in area.At normal levels of photopic illumination,the pupil fluctuates in size at a temporal

24 Basic optical strllcture of the human euc

Drugs that cause pupil dilation are calledmydriatics. These can act by stimulating the

Drugs

Age

Pupil size decreases with increase in age, andpupils react less to changes in light level. Thisis considered further in Chapter 20.

The constriction of the pupil due to directlight stimulation is referred to as the directlight reflex. In a healthy visual system there isalso a consensual light reflex, in which thepupils of both eyes respond equally tostimulation of only one eye. Pupil reactionsare more extensive when both eyes of aperson are stimulated than when only one eyeis stimulated.The pupil decreases in diameter when the

eyes converge or accommodate. This isreferred to as the near reflex.

Influence of binocular vision andaccommodation

and Spencer (1944) reviewed the results ofseveral studies, and proposed an equationwhich, using the unit of cd/m2 instead ofmillilamberts for luminance, isD =4.90- 3.00 tanh{0.400[log10(L) + 1.01l

(3.1a)where D is pupil diameter (mm) and L is fieldluminance (cd/m2). Using all available data,De Groot and Gebhard (1952) proposed anequation which, again using the unit of cd/m2

instead of millilamberts for luminance, is10glO(D) =0.8558 - 4.01x 10-4[loglO(L) + 8.6]3

(3.1b)These two equations are shown in Figure 3.4.The curve fits should be treated with cautionbecause of the wide differences betweensubjects, and because pupil sizes tend toreduce and become less responsive to changesin light level with increasing age (see Chapter20).

3

Reeves (1918 and 1920)

Crawfordo•

7.0

3.0

E.§. 6.0...~0) 5.0~:e 4.0'5.'"c..

I.0 -I-r-T'" r-T'" r-T'" .....-+--8 -7 -6 -5 -4 -3 -2 -I 0 2

LoglO(Luminance) (cd/m2)

Figure 3.4. Pupil diameter as a function of light level fora uniformly extended field. Experimental data fromReeves (1918 and 1920) and Crawford (1936), andmathematical 'mean' curves of Moon and Spencer (1944)and De Groot and Gebhard (1952).

DeGroot and Gebhard (1952), equation O.lb)

8.0 t---:=-_T

2.0 Moon and Spencer (1944). equation (3.la) --__

frequency of approximate 1.4 Hz, exaggeratedcases of which are referred to as hippus.The pupil responds to an increase in

illumination by a decrease in size. When thelight intensity is low, there is a latency of 0.5 sbefore constriction begins. As the stimulatinglight intensity increases, this latency reducesto 0.2-0.3 s. The extent of the response alsodepends on the distribution of light in thefield-of-view. There is less response as a lightsource moves from the central visual field intothe peripheral field (for example, Crawford,1936), which might be due partly to less lightentering the eye because the pupil subtends asmaller angle at the source. Pupil response tochanges in light level is mediated by both rodand cone receptors (Alpern and Campbell,1962).The response to an increase in light level is

usually complete within a few seconds,whereas the response to the withdrawal oflight may take up to a minute to be completed(Reeves, 1920; Crawford, 1936).Reeves (1918 and 1920) and Crawford

(1936) investigated the effect of a large source(==55° diameter subtense) on pupil diameter(Figure 3.4). There was considerable variationbetween subjects, as can be seen from thelarge standard deviations in the figure. Moon

sympathetic division of the autonomicnervous system (sympathomimetics) or byblocking its parasympathic division (para-sympatholytics). Drugs that cause pupilconstriction are called miotics, and can act bystimulating the parasympathic division(parasympathomimetics) or by blocking thesympathetic division (sympatholytics). Somedrugs influence pupil size through theireffects on the central nervous system.Many drugs that affect pupil size also affect

accommodation.

Psychological factorsEmotional states such as fear, joy and surprisecause the pupil to dilate. Hess (1965) foundthat pupil size was affected by mentalactivities. For example, pleasant, arousal-causing mental images increased pupil size,while unpleasant mental images decreasedpupil size.

Optical axis ......

The pupil 25

Shape of the obliquely viewed pupil

So far we have assumed that the pupil iscircular, and we continue to make thisassumption even though this is not true forsome people. If we observe the pupil fromincreasingly oblique angles, the pupilbecomes narrower in the direction of view(the tangential section) but remains approxi-mately the same in the perpendicular section(the sagittal section), as shown in Figure 3.5.Thus, the apparent area of the pupil decreasesas the oblique viewing angle is increased. Thedecrease in tangential diameter with viewingangle has important implications for (a) theoblique aberrations, and hence retinal imagequality, and (b) the amount of light enteringthe eye from oblique angles, and hence thebrightness of a peripheral retinal image(Sloan, 1950;Bedell and Katz, 1982).We can estimate the apparent tangential

diameter and area from simple geometry, asfollows. In the simple geometrical model, acircle appears to be elliptical when viewed

IDs=D

1

Directionof view

Figure 3.5. The shape of the pupil from an oblique direction.

26 Basic optical structureof the human eye

o o

•

o 0o

•

o

•

o

•

Simple geometric model

Sagittal

Tangential

o•

• •10 20 30 40 50 60 70 80 90 100 110

Angle (deg)

o+-,....,-.--r--r-r.,......,-..-,.....,....,.......r-r-r-...-t-.-,....;;'--t-o

6

8

E,5 5

Figure 3.7. Ratio of tangential to sagittal diameters forthe obliquely viewed pupil. Data of Spring and Stiles(l948a), Sloan (1950)and Jay (1962). The solid line is theratio expected from a simple geometric model and thedotted line is the fit to the data given by equation (3.5a).

Figure 3.6. The vertical (sagittal) and horizontal(tangential) diameters of the obliquely viewed dilatedpupil from the data of Spring and Stiles (1948b).Thesolid line is the ratio expected from a simple geometricmodel.

predicted by the simple geometric model.This is most likely to be due to pupilaberrations. A fourth-order equation thatgives a good fit to the experimental results

1.0

0.9 !II'A

O.S +'+0.7

'~Jlct ,0- 0.6 ~,

.S • Spring and Stiles (1948b), "'!E 0.5 large pupil+~... 0 Spring and Stiles (1948b). .;..

" 0..1-.; small pupilE !i!\" 0.3 A Slnan ( 1950)'5 \0'Q. 0.2 + Jay (1962), drug dilated pupil +~:l +8.0e, O Jay (1962). natural pupil

0.1Simple geometric model 0

+0.0 Equation (3.5a)

-0.1·~o -20 0 20 40 60 80 100

Angle (deg)

(3.4)Ap((J) =A(O)p cos ((J)

whereA(O)p =1tD2j4 (3.4a)Thus the apparent area decreases with cos((J),and the ratio of the apparent area at an angle(J to that for axial viewing is also cos(6).This model assumes that the aperture stop

(not the entrance pupil) is plane and does notsuffer any aberration when imaged by thecornea. It assumes also that the iris rim hasminimal thickness. Unless we resort tocomplex ray tracing, we cannot readilypredict the effect of aberrations and determinewhether aberrations increase or decrease thevalues predicted by equations (3.2) to (3.4).However, we would expect the finitethickness of the iris rim to decrease thesevalues slightly.Dimensions of obliquely viewed pupils

when the direction of view is horizontal havebeen determined by Spring and Stiles (1948aand b), Sloan (1950), Jay (1962) and Jenningsand Charman (1978), but facial featuresrestricted measurements from the nasaldirection. The mean horizontal and verticaldiameters from Spring and Stiles' (1948b)photographic study of 13 subjects are shownin Figure 3.6. The sagittal (vertical) diametervaries very little with eccentricity. Thetangential (horizontal) diameter decreasesas expected, but less than predicted byequation (3.2), especially for large angles ofeccentricity.Figure 3.7 shows the ratio of horizontal to

vertical diameters obtained from threestudies. Results are similar. ConsideringFigures 3.6 and 3.7 together, the tangentialdiameter of the pupil is greater than that

obliquely. Thus, if we view an entrance pupilof diameter D from an oblique angle (J in anydirection, the sagittal diameter Ds does notvary with the eccentricity but the tangentialdiameter D, is given by the equationDt = D cos((J) (3.2)which is shown in Figure 3.5. The projected orapparent area A,,((J) is the area of an ellipse,and is given by the equationAp((J) = 1tDP/4 (3.3)Alternatively, this area can be expressed in theform

isD, =05(1-1.0947 X 1(J4 £P + 1.8698X 10-9 (11)(0 in degrees) (3.5a)and this equation is shown on Figure 3.7.Assuming that 0 s is the same as 0, thisequation can be converted into the apparentpupil areaA(O) =A(0)pJ1-1.0947 X 10-4 £p + 1.8698

p x 10-9 lr) (3.5b)

Significance of pupil size

Pupil size has a number of effects on vision.

Depth-of-field

As with conventional optical systems, thediameter of the pupil affects the depth-of-field. The larger the pupil, the narrower is thedepth-of-field. This is discussed in detail inChapter 19.

Retinal light level

Obviously, pupil diameter affects retinal lightlevel. A detailed analysis of the dependency isleft until Chapter 13.

Retinal image quality and visualperformanceFor large pupil diameters, aberrations causedeterioration in retinal image quality. Forsmall pupil diameters, diffraction limits

Subject

Thepupil 27

image quality. There is an optimum pupildiameter range of 2-3 rom that gives the bestbalance between these two effects for thecorrected eye. The effect of pupil diameter onretinal image quality is discussed in greaterdetail in Chapter 18.Campbell and Gregory (1960) and

Woodhouse (1975) found that the artificialpupil size that gives the optimum (corrected)visual acuity is close to the natural pupil sizeat various background lighting levels. Thevisual acuity of a defocused eye is stronglydependent upon pupil diameter. This isdiscussed further in Chapter 9.

Purpose of the pupillary lightresponse

We may expect that pupil size varies in orderto maintain a constant light level on the retina,but this is not so. The variation of pupil sizewith light level is not sufficient to ensure aconstant retinal illuminance, because if thepupil size changes from 2 to 8 mm indiameter, the amount of light entering thepupil changes by a factor of only 16. Bycontrast, we operate comfortably over a 1()5times luminance range, from full moonlight(::: 0.01 cd/m2) to bright daylight (::: 1000cd /rn-).The work of Campbell and Gregory (1960)

and Woodhouse (1975) suggests that pupilsize variations optimize visual acuity forvarious light levels. This is only applicable forcorrected eyes; uncorrected eyes require muchsmaller pupil sizes. Woodhouse andCampbell (1975) suggested that the purposeof pupil size changes with changes in light

Observer

O;~-)O

Figure 3.8. A simple pupillometer.

28 Basic optical structureof the Ill/mall eye

level is to reduce retinal illumination and thushelp adaptation if there is a return todarkness.

Measurement of pupil size(pupillometry)

There are a number of experimental andclinical methods for measuring pupildiameter. These vary both in the level ofintrusion to the subject and in accuracy. Asimple clinical device has the constructionshown in Figure 3.8. The inside surface of theend nearest the patient's eye contains amillimetre scale. Parallax is a source of error.More accurate and less intrusive methods

involve photography, either using standardphotography or a video camera. A distinctadvantage of a photographic method is that itcan be done with infrared radiation incomplete darkness, and therefore does notaffect pupil size. The use of a video cameraoffers the further advantage that rapidchanges in pupil diameter can be recordedand monitored.Loewenfeld (1993) gave a comprehensive

review of pupillometry.

Artificial pupils

Artificial pupils are often used to control theeffective pupil size of the eye in visualexperiments. The pupil of the eye must bedilated first. Artificial pupils can simply beapertures placed immediately in front of theeye, or they can be projected onto the plane ofthe actual pupil by a relay system. Theartificial pupils may be annular, slit or circularpupils, and may be decentred relative to theactual pupil by a controlled amount. Becauseof aberrations, retinal image quality variesaccording to the position of artificial pupils,and careful centration is usually important(Walsh and Charman, 1988).Artificial pupils are often used in clinical

practice as an aid in refraction. Whencorrected visual acuity does not reach normallevels, placing a small pupil (e.g. 1 mmdiameter) in front of the eye may improvevisual acuity markedly if the refraction is not

accurate, but a lack of improvement indicatesa pathological basis for the poor vision.Medium-sized artificial pupils (e.g, 3-4 mm)may be used when the pupil has beenpreviously dilated by drugs; the pupils reducethe influence of aberrations, which may makerefraction difficult or inaccurate.

Summary of main symbols

Ap( (J) projected or apparent pupil area in thedirection (J

D (entrance) pupil diameterDs pupil diameter in the sagittal sectionD, pupil diameter in the tangential sectionL scene luminance (cd/rn-)(J oblique angle

ReferencesAlpern, M. and Campbell, F. W. (1962). The spectralsensitivity of the consensual light reflex. J. Physiol., 164,478-507.

Bedell, H. E. and Katz, L. M. (1982). On the necessity ofcorrecting peripheral target luminance for pupillaryarea. Am. J. Optom. Physiol. Opt., 59, 767-9.

Campbell, F.W. and Gregory, A. H. (1960).Effect of pupilsize on visual acuity. Nature,187, 1121-3.

Crawford, B. H. (1936). The dependence of pupil sizeupon external light stimulus under static and variableconditions. Proc. R. Soc. B.,121, 376-95.

De Groot, S. G. and Gebhard, J. W. (1952). Pupil size asdetermined by adapting luminance. J. Opt. Soc. Am., 42,492-5.

Hess, E. H. (1965). Attitude and pupil size. Sci. Am.,212(4),46-54.

Jay, B. S. (1962). The effective pupillary area at varyingperimetric angles. Vision Res.,1,418-28.

Jennings, J. A. M. and Chatman, W. N. (1978). Opticalquality in the peripheral retina. Am. J. Optom. Physiol.Opt, 55, 582-90.

Loewenfeld, I. E. (1993). The Pupil. Anatomy, Physiology,and Clinical Applications, vols. 1 and 2. Iowa StateUniversity Press.

Moon, P. and Spencer, D. E. (1944). On theStiles-Crawford effect. J. Opt. Soc. Am., 34, 319-29.

Reeves, P. (1918). Rate of pupillary dilation andcontraction. Psychol. Reo.,25, 330-40.

Reeves, P. (1920). The response of the average pupil tovarious intensities of light. J. Opt. Soc. Am., 4, 35-43.

Sloan, L. (1950). The threshold gradients of the rods andcones: in the dark-adapted and in the partially light-adapted eye. Am. J. Ophthal.,33, 1077-89.

Spring, K. H. and Stiles, W. S. (1948a). Variation of pupil

size with change in the angle at which light strikes theretina. By. J. Ophthal., 32, 340-46.

Spring, K. H. and Stiles, W. S. (1948b). Apparent shapeand size of the pupil viewed obliquely. By. J. Ophthal.,32,347-54.

Walsh, G. (1988). The effect of mydriasis on pupillarycentration of the human eye. Ophthal. Physiol. Opt., 8,178-82.

Walsh, G. and Charman, W. N. (1988). The effect of pupilcentration and diameter on ocular performance. VisionRes., 28, 659--65.

Thepupil 29

Westheimer, G. (1970). Image quality in the human eye.Optica Acta.,17, 641-58.

Wilson, M. A., Campbell, M.C. W.and Simonet, P. (1992).Change of pupil centration with change of illuminationand pupil size. OptOIll. Vis. Sci.,69, 129-36.

Woodhouse, J. M. (1975). The effect of pupil size ongrating detection at various contrast levels. Vision Res.,15,645-8.

Woodhouse, J.M. and Campbell, F. W. (1975). The role ofthe pupil light reflex in aiding adaptation to the dark.Vision Res., 15, 649-53.

4

Axes of the eye

Introduction

Most man-made optical systems are rotation-ally symmetric about one line, the optical axis.If the reflecting and refracting surfaces arespherical, this is the line joining the centres ofcurvatures of these surfaces. Some systemscontain astigmatic or toroidal componentsand have two planes of symmetry; the line ofintersection of these two planes is the opticalaxis.By contrast, to fully describe the optical

properties of the eye, we need to introduce anumber of axes. This is because of the lack ofsymmetry of the eye and because the fixationpoint and fovea are not along a best-fit axis ofsymmetry. The optical and visual axes werementioned in Chapter 1. In this chapter wedescribe these and other axes, their signifi-cance, their applications, and how they maybe determined.The validity of some of these axes is

dependent upon some idealized properties ofthe eye. For example, the visual axis requiresthe existence of the nodal points, which onlyexist if the eye is rotationally symmetric. Asecond example is the fixation axis, whichrequires the existence of a unique centre-of-rotation of the eye.Some of the methods for determining the

axes depend upon observing the images of asmall light source formed by specularreflections from the refracting surfaces of theeye. These Purkinje images are discussed ingreater depth in Chapter 12.

Direction of axes

These axes are meaningless without a meansof determining their directions and wherethey enter the eye. The directions are definedrelative to each other, and we often refer to theangles between the axes. Table 4.1 lists someaxes, and the symbols used to denote theangles between them. Three of the axes passthrough the centre of the pupil and, since thepupil centre can change with change indiameter (see Chapter 3), their directionsdepend upon pupil size.

Definitions and significance

Martin (1942) gave an early account of theangles and axes which highlighted theconfusing array of terms that have been usedin this area. The definitions of optical axis, lineof sight, visual axis, pupillary axis andfixation axis given below are similar to thoseprovided in dictionaries of visual science(Cline et at., 1989; Millodot, 1993).

Optical axis

This is the line passing through the centres ofcurvatures of the refracting and reflectingsurfaces of a centred system. The optical axisis not of particular importance by itself, but itis a useful reference for some of the other axesof the eye.

Axes of theeye 31

Table 4.1. Some different axes used for the eye and the symbols used to denote the angles between them.

Optical axis Visual axis Lineof sight Achromatic axis

Visual axis a 'IIPupillary axis x A.Fixation axis r

In a conventional centred optical system,the centres of curvatures of each refracting orreflecting surface lie on one line (i.e. they areco-linear). This line is the optical axis. The eyeis not a centred system, and does not containa true optical axis. The concept of optical axiscan be applied to the eye by defining theoptical axis as the line of 'best fit' through thecentres of curvature of the 'best fit' spheres toeach surface.

Line of sight

This is the line joining the fixation point andthe centre of the entrance pupil.The line of sight is the most important axis

from the point of view of visual function,including refraction procedures, as it defines

Corneal sighting centre

!::--/ .4--0phthalmometnc pole

Visual axis

Line of sightPupillary axis

•Fixation target Visual axis

the centre of the beam of light entering theeye. As mentioned in the previous section, it isunfortunately not fixed because the pupilcentre may alter with fluctuations in pupilsize.The fovea is usually on the temporal side of

the optical axis (Chapter 1). Therefore, thepoint in object space conjugate to the fovea isalso off axis, but on the nasal side of theoptical axis. The line of sight is the central rayof the beam from the fixation point T asshown in Figure 4.1. In paraxial optics, theline of sight is called the paraxial pupil ray,which was defined in the previous chapter(see Entrance and exit pupils). The position atwhich it intercepts the cornea is called thecorneal sighting centre (Mandell, 1995) orvisual centre of the cornea (Cline et al.,1989).

Temporal side

Figure 4.1.Most of the axes and angles referred to in this chapter. The object has been shownextremely close to the eye, thus exaggerating angular differences between visual axis, line of sightand fixation axis.

32 Basic optical structureoi ttu: hunuin 1'.111'

Visual axisThis is the line joining the fixation point andthe foveal image by way of the nodal points.The visual axis is a convenient reference

axis for visual functions, particularly as itdoes not depend on pupil size. It is usuallyclose to the line of sight at the cornea andentrance pupil (see following section, Locatingsome axes).The visual axis is the line segments TN and

N'T' shown in Figure 4.1. This is not a singlestraight line, since the nodal points are notcoincident. Allowing for the exaggeration inFigure 4.1, this shows the proximity of thevisual axis to the line of sight.The foveal achromatic axis is closely

related to the visual axis, and can be definedas the path from the fixation point to the foveasuch that the ray does not suffer from anytransverse chromatic aberration. Ignoring thesmall change in the nodal points that occurswith change in wavelength, this axis is thesame as the visual axis and its definition canbe used as a basis for locating the visual axis.Ivanoff (1953) referred to this axis as simplythe achromatic axis. However, Thibos et al.(1990) redefined this term to refer to the pupilnodal ray, which is the ray passing through

Achromatic axis_.J==-tp--ITFixation target

the centre of the pupil and which has notransverse chromatic aberration. It is similarto the optical axis but, unlike the optical axis,it is dependent on pupil position. WhereIvanoff used the term achromatic axis, we usethe term foveal achromatic axis, and we haveadopted Thibos and co-workers' use ofachromatic axis (Figure 4.2).Rabbetts (1998) criticized the use of the term

'visual axis' for the ray passing through thenodal points, on the grounds that such a ray isnot representative of the beam passing intothe eye from a fixation target. He preferred tocall it the 'nodal axis', and reserved the term'visual axis' for the axis we have defined asthe line of sight.Le Grand and EIHage (1980) referred to the

intersection of the visual axis with the corneaas the ophthalmometric pole (Figure 4.1).

Pupillary axisThis is the line passing through the centre ofthe entrance pupil, and which is normal to thecornea.The pupillary axis is used as an objective

measure to judge the amount of eccentricfixation, the condition in which a retinal point

Temporal side

Nasal side

Figure 4.2.Ocular axes:a. The optical axis, line of sight, visual axis and achromatic axis.b. The angle ",between the visual axis and the achromatic axis.c. The distance d between the visual axis and line of sight at the entrance pupil.

other than the centre of the fovea is used forfixation. Eccentric fixation is an adaptation toheterotropia (squint or turned eye). Wediscuss the measurement of eccentric fixationand heterophoria further in the followingsection (Locating some axes).

If the eye was a centred system and thepupil was also centred, the pupillary axiswould lie along the optical axis. However, thepupil is often not centred relative to thecornea and, furthermore, the cornea may notbe a regular shape. Both these factors causethe pupil axis to lie in some other direction,and in general it does not pass through thefixation point T as shown in Figure 4.1.

Fixation axisThis is the line passing through the fixationpoint and the centre-of-rotation of the eye.The fixation axis is the reference for

measuring eye movements.This axis is shown in Figure 4.1. There is no

unique centre-of-rotation, and estimates of itdepend on the direction of rotation of the eye(Alpern, 1969). Accordingly, the idea of afixation axis is just an approximation, andestimates of it depend also upon the directionof rotation.

Axes of theeye 33

Keratometric axisThis is the axis of a keratometer or video-keratographic instrument, and it contains thecentre of curvature of the anterior cornea.This axis is used for alignment in corneal

topography measurements. In the standardoperation of a corneal topographic instru-ment, the axis intercepts the line of sight at thefixation target (Figure 4.3a), although thefixation target may be moved deliberately sothat this is not the case (Figure 4.3b).According to Mandell (1994), in standard usea small but negligible variation occurs in thisaxis between different instruments, because ofdifferences in the distance of the fixationpoint.When a videokeratographic instrument is

used in standard operation, the keratometricaxis is neither the line of sight nor does it passthrough the apex of the cornea (the point withthe smallest radius of curvature (Mandell andSt Helen, 1969». The point at which the axisintercepts the cornea is sometimes called thevertex normal (Maloney, 1990), and is thecentre of videokeratographs showing cornealcontour. If the vertex normal is sufficientlydecentred from the corneal apex, thevideokeratograph gives a false representationof surface curvature across the cornea(Mandell and Horner, 1995).

(a)

Pupillary axis Fovea

Vertex normal

(b) Figure 4.3.The keratometric axis and the line ofsight of a videokeratographic instrument.a. Standard operation. The line of sight and thekeratometric axis intersect at the fixationpoint.

b. The alignment has been altered so that theline of sight and the keratometric axisintersect at the cornea (corneal sightingcentre). Based on Mandell et al. (1995).

34 Basic opticalstructure (~f the 11l1I/1a1l e!lt'

Locating some axes

The line of sight

Many automated instruments have a videodisplay showing the front of the eye. Toachieve alignment while a patient is fixating areference target, either the patient or theinstrument is moved vertically and hori-zontally until the pupil is correctly centred.Thus the line of sight is made to coincide withthe optical axis of the instrument. Sometimesthis is aided by imaging a centred annulus aswell as the eye.

The visual axis

The intercept of the visual axis at the cornea(ophthalmometric pole) can be determined byhaving the subject view a vernier target, halfof which is blue and half of which is red. Twopossible targets are shown in Figure 4.4. Thesubject views the vernier target through asmall artificial pupil, say 1 mm diameter. Ifthe artificial pupil is not centred on the visualaxis, there is a break in the alignment of theblue and red halves of the target. The positionof the subject's eye is adjusted until alignmentis obtained. The reliability of the methodshould improve as the wavelength bands ofthe target are narrowed.Thibos et al. (1990) indicated a simple way

of estimating the separation of the visual axisfrom the line of sight at the entrance pupil (thedistance d in Figure 4.2). The small artificialpupil, referred to in the previous paragraph,can be scanned across the eye both verticallyand horizontally to find the real pupil limits atwhich the vernier target disappears. Themean of these positions corresponds to the

Figure 4.4. Two targets for locating the visual axis.

corneal sighting centre (on the line of sight).This position is then compared with theophthalmometric pole (visual axis intercept atthe cornea). Thibos and co-workers measuredfive subjects with drug-induced pupildilation, considering only the horizontalmeridian. They obtained values of d between-0.1 mm and 0.4 mm, with a mean of +0.14mm (a positive sign indicating that the visualaxis is nasal to the line of sight in objectspace).Simonet and Campbell (1990) used a

videomonitor to determine the difference inhorizontal position of the visual axis (whichthey referred to as the achromatic axis) andthe line of sight. For natural (althoughgenerally large) pupils, the differences foreight eyes of five subjects were between-0.08 mm and +0.51 mm, with a mean of+0.11 mm.

Keratometric axis

Mandell et al. (1995) described methods ofdetermining the location of the keratometricaxis relative to the corneal sighting centre andthe corneal apex. Using a videokeratographicinstrument in its standard operation and 20normal subjects, they found a mean differencebetween the kera tometric axis and the cornealsighting centre of 0.38 ± 0.10mm, with themajority of subjects having the keratometricaxis below and nasal to the corneal sightingcentre. The mean difference between thekeratometric axis and the corneal apex was0.62 ± 0.23 mm, with the majority of subjectshaving the keratometric axis above thecorneal apex.

Angles between axes

Here we define some of the angles betweenvarious axes. In some cases, methods fordetermining these and measurements aregiven. The angles alpha, lambda and kappahave been used differently by differentauthors. In Table 4.2 we indicate some of thevariations from those used here. Thesevariations involve either the line of sight orthe visual axis.

Axes of theeye 35

Table 4.2. Different terms used for angles.

Term usedhere

a

Mil/odot (1993) LeGrand and EIHage (1980)

a

K"

Clineet al. (1989)

a

'between line of sight and optical axis.

Visual axis and optical axis: theangle alpha (a)

Most authors use the term 'angle alpha' torefer to the angle between the optical andvisual axes (Le Grand and El Hage, 1980;Cline et al., 1989; Millodot, 1989). We haveretained this use but, as the technique ofmeasuring this angle involves the subjectfixating on a target, it may be considered thatthe line of sight (i.e, centre of entrance pupil)is involved rather than the visual axis (i.e.nodal points). The distinction is of no practicalimportance.

In optical laboratories, a frequent need is tolocate the optical axis of an optical system.One method of locating this axis is to shine adistant point source into the system andobserve the images of this source of lightreflected from each surface inside the system.In a centred optical system, if the source oflight falls on the optical axis, it and thereflected images are co-linear.Since the eye has four reflecting surfaces,

there are four main reflected (Purkinje)images. However, as the eye is not a centredsystem, there is no position or direction of thelight source that enables the Purkinje imagesto be aligned. All that can be done is to

PatientN

Figure 4.5. The ophthalmophakometer.

minimize the spread of these images, and thecorresponding direction of the source identi-fies the direction of the optical axis.Clinically, the angle a is determined with

the ophthalmophakometer (Figure 4.5). Thisinstrument contains a graduated arc, with anobserving telescope mounted centrally in thearc. The patient's eye is at the centre ofcurvature of the arc. Two small light sourcesare placed on the arc near the telescope, withone slightly above and one slightly below it.This gives pairs of Purkinje images. A smallfixation object T is moved along the arc untilthe observer looking through the telescopejudges that the Purkinje images are in the bestpossible alignment. At this point, the opticalaxis of the eye corresponds with the axis of theinstrument. The angle a is given by the scalereading at the position of the fixation target.The instrument can be rotated through 90degrees to obtain a value in the verticaldirection. The distance between the eye andthe arc is relatively large (typically 86 em) sothat discrepancies between the front nodalpoint and the centre of curvature of the arc arenot critical.The angle between the visual axis and the

optical axis is considered to be positive if thevisual axis is on the nasal side of the opticalaxis in object space. The mean value of anglea is often taken to be about +5° horizontally,but is usually in the range +3 to +5°, and israrely negative. The visual axis is alsodownwards relative to the optical axis by 2-3°(Tscherning, 1990).

Pupillary axis and line of sight: anglelambda 0.)This angle is one of the easiest to determine. Itcan be determined using the opthalmophako-meter just described, but it is determinedeasily with simpler equipment (Figure 4.6).

36 Basic optical structure of tileIll/man eye

Optical axis8 __.-1---1 -

TFixation target

Centre of curvaturer- of cornea1'1 Cc

Anterior cornealsurface

Figure 4.6.The pupillary axis and the line of sight.

In practical terms, this is the same as angle A.It is shown in Figures 4.1 and 4.6.

Pupillary axis and the visual axis:angle /(

Figure 4.7.The Hirschberg test for measuring the angleof heterotropia.a. Appearance of corneal reflexes for no heterotropia.b. Appearance of reflexes for left esotropia (left eyeturned in).

(b) Esotropia(a) Normal

the change in reference axis away from theline of sight.The usual application of these tests involves

the clinician shining a penlight at the patient'seye or eyes. The penlight is just in front of theclinician's face, and the patient is instructed tolook at it. The clinician observes the positionof the corneal reflection (or reflex) in the pupil(Figure 4.7). Usually the reflex is about half amillimetre nasal to the centre of the pupil(Figure 4.7a). Each millimetre change in reflexposition away from this corresponds toapproximately 22 prism dioptres (13°) of eyerotation (Grosvenor, 1996). Care must betaken in the monocular test, as some patientsmay have normal fixation but unusual angles.Results from the two eyes should becompared. Similarly, in the Hirschberg test itis important to compare the difference inreflex positions between the two eyes.

The subject fixates on some suitable target T,and an observer watches the anterior cornealreflection (Purkinje image PI)of a small sourceof light S close to the observer's eye. Theposition of the light is changed, and theobserver's eye is maintained next to the lightsource, until the reflected image S' is seen inthe centre of the pupil E (Figure 4.6). Theobservation axis is now the pupillary axis. Theangle between the line of sight and thepupillary axis at the eye is the angle A.-For most patients, the pupillary axis is

temporal to the line of sight in object space.This is taken as a positive angle. Loper(1959) obtained angles of 1.4 ± 1.6°, andFranceschetti and Burian (1971) obtainedangles of 2.6 ± 1.7°. Furthermore, the anglesshould be similar between the two eyes.The angle A is important for diagnosis of

eccentric fixation and heterotropia. In testingfor the presence of eccentric fixation, angle Aisdetermined monocularly (with the other eyeoccluded). A large angle indicates the likelypresence of eccentric fixation. The line of sightas we have defined it is not being used,because the patient has rotated the eye toalign a retinal point eccentric to the fovea withthe fixation point. Angle A is estimatedbinocularly (with both eyes open) to test fordirection and amount of heterotropia in theHirschberg test. In the presence of hetero-tropia, a large angle Ais observed because oneeye rotates so that its fovea is not being usedto align the fixation target. Either the fovea isbeing suppressed or another retinal point isbeing used for fixation (anomalous corre-spondence), or both are occurring. Again, weare not strictly measuring angle A because of

Axes of tileeye 37

Visual axis and achromatic axis:angle psi (1/1)

Fixation axis and optical axis: anglegamma (}1

Thibos et al. (1990) estimated this angle fromthe equationsin(",) =d/EN (4.1)

where d is the distance between the visual axisand line of sight at the entrance pupil and ENis the estimate of the distance between theentrance pupil and the nodal point (Figure4.2). Based on either the Gullstrand No. 1 orNo. 2 eyes, the distance EN is 4.0mm. Anapproximate, but sufficiently accurate,method for determining dwas given earlier inthis chapter. Using five subjects, Thibos andco-workers determined a range of angles from-1.2° to +5.3°, with a mean of +2.1° (positiveangles indicate the visual axis is inclinednasally to the achromatic axis in object space).

ReferencesAlpern, M. (1969). Specification of the direction of regard.InMuscular Mechanisms, Ch. 2.Vol.3 of TheEye, 2nd edn(H. Davson, ed.), pp. 5-12. Academic Press.

Cline, D., Hofstetter, H. W. and Griffin, J. R. (1989).Dictionary of Visual Science, 4th edn. Chilton Trade BookPublishing.

Franceschetti, A. T. and Burian, H. M. (1971). L'anglekappa. Bull. Mem. Soc. Fr. Ophtalmol., 84, 209-14.

Grosvenor, T. P. (1996). Primary Care Optometry, 3rd edn,Ch. 6. Butterworth-Heinemann.

Ivanoff, A. (1953). Les aberrations de l'oeil. Leur role dansI'accommodation. Editions de la Revue d'OptiqueTheorique et lnstrumentale. Masson and Cie.

LeGrand, Y. and EIHage, S. G. (1980). Physiological Optics(translation and update of Le Grand, Y. (1968). Ladioptrique de I'oeil et sa correction. In OptiquePhysiologique, vol. 1), pp. 71-4. Springer-Verlag.

Loper, L. R. (1959). The relationship between anglelambda and the residual astigmatism of the eye. Am. J.Optom. Arch.Am. Acad. Optom., 36, 365-77.

Maloney, R. K. (1990). Corneal topography and opticalzone location in photorefractive keratectomy. Refract.Corneal SlIrg., 6, 363-71.

Mandell, R. B. (1994). Apparent pupil displacement invideokeratography. CLAD t: 20, 123-7.

Mandell, R. B. (1995). Location of the corneal sighting

Summary of main symbols

C centre-of-rotation of the eyeCc centre of curvature of the anterior

corneal surfaceE, E' centres of entrance and exit pupilsN, N' front and back nodal pointsT fixation targetT' conjugate of T on the retina, i.e. the

foveaS, S' source of light and its image, used to

find the pupillary axisV intersection of the optical axis with the

cornead distance between visual axis and line of

sight at the entrance pupila angle between visual axis and optical

axisr angle between fixation axis and optical

axis/(' angle between pupillary axis and

visual axisA angle between pupillary axis and line

of sight'" angle between visual axis and achro-

matic axis

e]"II

.Y

TFixation target

Figure 4.8. Determination of the angle Yo

IV

Figure 4.8 shows the relationship betweenangles rand a. In this figure, y is the distancebetween the optical axis and the fixationtarget T, N is the front nodal point, C is thecentre-of-rotation of the eye, and w is thedistance from the projection of T, onto theoptical axis, to the cornea at V. We have theequationstan(a) = y/(w +VN) (4.2)andtan(n =y/(w + VC) (4.3)which, combined, givetan(n = tan(a)(w + VN)/(w + VC) (4.4)Angle r is within 1 per cent of angle a forobject distances greater than 50 em.

38 Basic optical structure of thehumall eye

centre in videokeratographyI. Refract. (ameal Surg.,11,253-8.

Mandell, R. B. and Horner, D. (1995). Alignment ofvideokeratographs. In (amen/ Topograplry: Tire State oftire Art (J. P. Gills, D. R. Sanders, S. P. Thornton et al.eds.), Ch. 2. Slack Incorporated.

Mandell, R. B., Chiang, C. S. and Klein, S. A. (1995).Location of the major corneal reference points. Optom.Vis. Sci., 72,776-84.

Mandell, R. B. and St Helen, R. (1969). Position andcurvature of the corneal apex. Am. I. Optom. Arch. Am.Acad. Optom., 46, 25-9.

Martin, F.E. (1942). The importance and measurement ofangle alpha. Br. J. Ophtlral., 3, 27-45.

Millodot, M. (1993). Dictionary of Optometry, 3'd edn,Butterworth-Heinemann.

Rabbetts, R. B. (1998). Bennett and Rabbetts' Clinical VisualOptics, 3,d edn., Ch.12. Butterworths.

Simonet, P. and Campbell, M. C. W. (1990). The opticaltransverse chromatic aberration of the fovea of thehuman eye. Vision Res., 30, 187-206.

Thibos, L. N., Bradley, A., Still, D. L. et al. (1990). Theoryand measurement of ocular chromatic aberration.Visioll Res., 30, 33-49.

Tscherning, M. (1990). Physiologic Optics, pI edn.(translated from the original French edition by C.Weiland). The Keystone.

5

Paraxial schematic eyes

Introduction

We can construct model eyes using popu-lation mean values for relevant ocularparameters. This can be done at differentlevels of sophistication. Ifwe assume that therefractive surfaces are spherical and centredon a common optical axis, and that therefractive indices are constant within eachmedium, this gives a simple family of modelsreferred to as paraxial schematic eyes.Paraxial schematic eyes are only accurate

within the paraxial region. They do notaccurately predict aberrations and retinalimage formation for large pupils or for anglesat more than a few degrees from the opticalaxis. The paraxial region is defined in geo-metrical optics as the region in which thereplacement of sines of angles by the anglesleads to no appreciable error. If we limit theerrors to less than 0.01 per cent, this limitsobject field angles to less than 2° and theentrance pupil diameter to less than 0.5mm.Paraxial schematic eyes serve as a frame-

work for examining a range of opticalproperties. The location of the paraxial imageplane or calculation of the paraxial imageheight has many useful applications. Infor-mation can be obtained from schematic eyesconcerning magnification, retinal illumina-tion, surface reflections (e.g. Purkinje images),entrance and exit pupils, and effects ofrefractive errors. A study of cardinal points of

the systems can also have practical appli-cations, such as the observation that thesecond nodal point moves little on accom-modation and therefore that angular reso-lution is expected to change little withaccommodation. Further applications toretinal image formation are discussed inChapters 6 and 9.For accurate determinations of quantities

such as large retinal image sizes and imagequality due to aberrations, we need morerealistic models than the paraxial schematiceyes. These are referred to as finite or wideangle schematic eyes. These include one ormore of the following features: non-sphericalrefractive surfaces, a lack of surface alignmentalong a common axis, and a lens gradientrefractive index.Historically, paraxial schematic eyes have

had uniform refractive indices, and it mightbe considered that schematic eyes withgradient indices must be finite eyes becausethe gradient index influences aberrations.However, replacing a uniform refractive indexby a gradient index affects the paths ofparaxial rays and hence paraxial properties,and thus gradient indices may be included inparaxial model eyes.This chapter considers paraxial schematic

eyes only. A discussion of finite model eyes isgiven in Chapter 16, following a review of themonochromatic aberrations of real eyes inChapter 15.

40 Basic optical structure of tilellllll/all "y"Development of paraxial schematiceyesThe historical development of the under-standing of the optical system of the humaneye has been described in detail by Polyak(1957). The lens was believed to be thereceptive element of the eye for 13 centuriesfollowing the work of Galen in 200 AD.Leonardo DaVinci (c. 1500AD) proposed thatthe lens is only one element of the refractivesystem which forms a real image on the retina.In 1604, Kepler realized that the image isinverted; this was verified by Scheiner 15years later. The first clear, accurate descriptionof the eye's optical system was given byDescartes in 1637 in his La Dioptrique, whichalso included the first publication of what hasbecome known as Snell's law of refraction.The first physical model of the eye was

probably that of Christian Huygens (1629-95).Smith (1738) described Huygens's eye asconsisting of two hemispheres representingthe cornea and retina respectively, with theretinal hemisphere having a radius ofcurvature three times that of the cornealhemisphere. The two hemispheres were filledwith water and a diaphragm was placedbetween them.Young (1801) discussed the optics of the eye

and presented data, some of which are close topresent day values. He gave the anteriorcorneal radius as 7.9mm, and the anterior andposterior lenticular radii of curvature as 7.6mm and 5.6mm respectively. The anteriorchamber depth was given as 3.0mm. Hisrefractive index for the aqueous and vitreousmedia was 1.333 (water), and that for the lenswas 1.44.According to Le Grand and EIHage (1980),

Moser, in 1844, was the first to construct aschematic eye, but this was hypermetropicbecause it had a very low value for therefractive index of the lens. The first 'accurate'schematic eye has been attributed to Listing.In 1851, he described a three refractingsurfaces schematic eye with a single surfacecornea and a homogeneous lens, with anaperture stop 0.5 mm in front of the lens.Helmholtz (1909, p. 152) modified Listing'sschematic eye by changing the positions of thelenticular surfaces. He also gave this model ina form accommodated to a distance of 130.1mm in front of the corneal vertex. Helmholtz

(1909, pp. 95-96) also described a muchsimpler schematic eye designed by Listing.This contains only one refracting surface (thecornea), and is called a reduced eye.Tscherning (1900) published a four refract-

ing surfaces schematic eye containing aposterior corneal surface, which he claimed tobe the first to measure.Gullstrand (1909) used a comprehensive

analysis of ocular data to construct a sixrefracting surface schematic eye that used afour surface lens with the lenticularcomplexity aimed at accounting for refractiveindex variation within the lens. Thisschematic eye is referred to as Gullstrand'snumber 1 (exact) eye. Gullstrand presentedthis eye at two levels of accommodation.Gullstrand also presented a simplified versionreferred to as Gullstrand's number 2 (simpli-fied) eye, also at two levels of accommo-dation. This simplified eye contains threerefracting surfaces, with only one cornealsurface and a zero lens thickness.Emsley (1952) presented a modified version

of Gullstrand's simplified eye. Emsley gavethe lens the thickness that it has inGullstrand's exact eye, and changed theaqueous, vitreous and lens refractive indices.This modified eye is sometimes called theGullstrand-Emsley eye. Emsley alsopresented a reduced schematic eye.As well as the Gullstrand exact eye, the

Gullstrand-Emsley eye and Emsley's reducedeye, another popular schematic eye is LeGrand's 1945 four refracting surfaces eye,which is referred to as Le Grand's fulltheoretical eye (Le Grand and El Hage, 1980).It is a modification of Tscherning's schematiceye. Le Grand also presented a simplifiedthree refracting surfaces model with a singlecorneal surface and a lens of zero thickness.The lack of lens thickness limits the usefulnessof this particular model.More recently, Bennett and Rabbetts (1988,

1989) presented a modification of theGullstrand-Emsley eye, which they justifiedon the grounds that the data used to constructthe earlier eye was from a restricted numberof eyes and that the mean power is closer to 60D than previously thought. They used thedata from the study of Sorsby et al. (1957),which was based upon 341 eyes (mostly pairsof left and right eyes) with mean equivalentpower of 60.12 ± 2.22D.

Other schematic eyes have been proposedfrom time to time. For example, Swaine (1921)gave details of several eyes referred to asMatthiessen B, D and G eyes, and Laurance Iand II eyes.Blaker (1980) described an adaptive

schematic eye. It is a modified Gullstrandnumber 1paraxial schematic eye, in which thelens has been reduced to two surfaces but isgiven a gradient refractive index. The lensgradient index, lens surface curvatures, lensthickness and the anterior chamber depthvary as linear functions of accommodation.Blaker (1991) revised his model to includeaging effects, with the lens curvatures, lensthickness and anterior chamber depth alteringin the unaccommodated state as a function ofage.Some of the above mentioned eyes are

discussed in greater detail later in this chapter,and constructional details of some eyes aregiven in Appendix 3.

Gaussian properties and cardinalpoints

One of the main applications of paraxialschematic eyes is predicting the Gaussianproperties of real eyes. Of these, probably the

Paraxial schematic eyes 41

most important are the equivalent power F,positions of the six cardinal points (F, F', P, P',Nand N') and the positions and magnifi-cations of the pupils. We can use the paraxialoptics theory described in Appendix 1 todetermine these properties. The Gaussianproperties are given for specific schematiceyes in Appendix 3, and Table 5.1 shows alimited amount of data.

Equivalent power and cardinalpoints

The cardinal points are defined in Chapter 1(Cardinal points). Figure 1.1 shows nominalpositions of these in the emmetropic relaxedeye.There are a number of useful equations

connecting the cardinal points, including:F = -n/PF =n'/P'F' (5.1)PN = P'N' = (n' - n)/F (5.2)

FN =P'F' (5.3a)N'F' =FP (5.3b)where nand n' are the refractive indices ofobject space (air) and image space (thevitreous) respectively.

Table 5.1. Summary of Gaussian data. Distances are in millimetres and powers are in dioptres.

Generallength'

Gullstrand number 1 24.385Le Grand (full theoretical) 24.197Le Grand (simplified) 24.192Gullstrand-Emsley 23.896Bennett and Rabbetts (simplified) 24.086Emsley (reduced) 22.222

Relaxed eyesF VE VN E'F'= E'R' N'F'=N'R' m

Gullstrand number 1 58.636 3.047 7.078 20.720 17.054 0.823085Le Grand (full theoretical) 59.940 3.038 7.200 20.515 16.683 0.813243Gullstrand-Emsley 60.483 3.052 7.062 20.209 16.534 0.818128Bennett and Rabbetts (simplified) 60.000 3.048 7.111 20.387 16.667 0.817532Emsley (reduced) 60.000 0.0 50/9 22.222 16.667 0.750000

Accommodated eyesF Accom. VE VN E'R' N'R' F'R' m

Gullstrand number 1 70.576 10.870 2.668 6.533 21.173 17.539 3.371 0.795850LeGrand (full theoretical) 67.677 7.053 2.660 7.156 20.942 17.041 2.265 0.791122Gullstrand-Ernsley 69.721 8.599 2.674 6.562 20.647 16.987 2.644 0.796683Bennett and Rabbetts (10 D) 71.120 10.192 2.680 6.598 21.140 17.135 3.074 0.791439

42 Basic optical structureo( tI,,'11I11I11I1I eye

Approximate mean valuesSince the mean equivalent power of the eye isclose to 60 D and the values of /1 and /1' are 1.0and 1.336 respectively, we can calculateexpected approximate mean values of theabove quantities. These are:F =60DFP =N'F' = 16.67mmP'F' = FN = 22.27mmPN =P'N' = 5.6 mm.

The aperture stop and entrance andexit pupilsAfter the equivalent power and positions ofthe cardinal points, probably the next mostimportant Gaussian properties of an eye arethe aperture stop and pupil formation. Theaperture stop of an eye is its iris. Reduced eyesdo not have an iris, but we can place anaperture stop in the plane of the cornea or atsome other suitable position. The image of theaperture stop formed in object space, that is,the image of the iris as seen through thecornea, is called the entrance pupil. The imageof the aperture formed in image space iscalled the exit pupil. These concepts arediscussed fully in Chapter 3.

Position and magnification ofentrance pupilFor schematic eyes with a single surfacecornea, the calculations are simple. In thiscase, we can use the lens equation given inAppendix 1,

/1'/I'-/1/I=F (5.4)

Figure 5.1 shows the path of a paraxial raythat can be used to locate the image of the iris.I is the anterior chamber depth, I' is theapparent anterior chamber depth, /1 is therefractive index of the aqueous, and /1' is therefractive index of air (= 1.0). Solving for l'gives

l' =11'1/(11 + IF) (5.5)

and the pupil magnification MEA' defined asthe ratio of the entrance pupil diameter to that

r-/~/,=J1 1

® 0¢-,Entrancepupil

Figure 5.1. The formation of the entrance pupil of theeye and its relationship to the iris in a schematic eyewith a single surface cornea.

of the stop, is given byMEA= /11'/(/1'1) (5.6)The standard sign convention was used in thedevelopment of these equations, withdistances to the left of the refracting surfacebeing negative and distances to the rightbeing positive. Distances I and l' are negative,although usually we express the final answersin a positive form.Example 5.1: Calculate the position andmagnification of the entrance pupil of theGullstrand-Emsley simplified relaxedeye.Solution: From the Gullstrand-Emsleyschematic eye data given in Appendix 3,we have/I = 4/3/I' =1I = -3.6 mm andF = 42.735D.Substituting these data into equations(5.5) and (5.6) gives

I' = 1 x (-3.6) = (-)3.052mm[(4/3) + (-3.6) x 42.735/1000]

and

NT =(4/3) x (-3.052) = 1.1304fA 1 x (-3.6)

Thus the entrance pupil is 3.05mm insidethe eye, compared with a distance of 3.6mm for the actual pupil. The entrance

Paraxial schematic eyes 43

Effect of accommodation

(5.8)

(5.7)

(5.11a)

n'u' - nu ::: - 11F

paraxial pupil ray anglesm::: utt;

N::: __n_n'ME'E

where ME'E is the pupil magnification » exitpupil diameter/entrance pupil diameter.

The angles Il and Il' are the angles of theparaxial pupil ray in object and image spacerespectively, as shown in Figure 5.2. They arerelated by the paraxial refraction equation(Appendix 1)

The cardinal point positions of the relaxed(zero accommodation) and accommodatedversions of schematic eyes can be compared inFigures 5.3, 5.4 and 5.5. Upon accommo-dation, the principal points move away fromthe cornea, the nodal points move towards thecornea, and the focal points move towards thecornea.

where F is the equivalent power of the eyeand 11 is the ray height at the principal planes.Equation (5.8) can be transposed to givem::: [n - (11/u)F]/n' (5.9)

where n has a value of 1 for air.From Figure 5.2, within the paraxial

approximation we have

11/Il ::: - PE ::: - T (5.10)

Therefore we have

N::: [n +TF]/n' (5.11)

which shows that the value of m dependsupon the refractive index of the vitreouswhich is fixed, the distance of the entrancepupilT from the front principal point and theequivalent power F. The values of bothTandF depend upon accommodation level. For atypical schematic eye, T "" 1.5mm, F:::: 60Dand n' ::: 1.336, giving m "" 0.82. Precise valuesfor particular schematic eyes and at differentlevels of accommodation are given in Table5.1. Equation (5.11) can be manipulated intothe following form

Paraxial marginal ray and paraxialpupil ray

pupil is also 13 per cent larger than theactual pupil. The pupil position is shownin Table 5.1, along with the values forother schematic eyes.

These are two special paraxial rays introducedand defined in Chapter 3 (Entrance and exitpupils). As can be seen from Figure 3.3, thepaths of these rays depend upon the positionof the object/image conjugates, field size andthe position of the aperture stop and itsdiameter. Here, as a rule, we denote themarginal ray angles and heights by therespective symbols u and h and the paraxialpupil ray angles and heights by the respectivesymbols Il and 11. The details of these rays(angles and heights) are given by Smith andAtchison (1997) for some schematic eyes withan entrance pupil diameter of 8 mm and afield-of-view of angular radius 5°.

A quantity that is useful in the calculation ofretinal image sizes is the ratio m of the

Paraxial pupil ray angle ratio m

Figure 5.2.The paraxial pupil ray and its use incalculation ofm.

44 Basic optical structureof Ill,' IIIIII/all e!le

'Exact' schematic eyes

In the 'exact' schematic eyes, an attempt ismade to model the optical structure of realeyes as closely as possible while usingspherical surfaces. The minimum requirementof an 'exact' eye is that it must have at leastfour refracting surfaces, two for the corneaand two for the lens.

Gullstrand number 1 (exact) eye

This schematic eye takes into account thevariation of refractive index within the lens(Figure 5.3). It is presented in both relaxed andaccommodated versions. It consists of sixrefracting surfaces; two for the cornea andfour for the lens. The lens contains a centralnucleus (core) of high refractive indexsurrounded by a cortex of lower refractiveindex. The lens can be regarded as acombination of three lenses. The anterior andposterior lenses are thinner in the centre thanat the edge, and may be erroneouslyconsidered to have negative power. However,they have positive power because therefractive index of the core lens is higher thanthat of the cortex.Gullstrand placed the retina 0.39mm short

of the back focal point F' because he thoughtthat the positive spherical aberration wouldlead to the best image plane being slightly infront of the paraxial image. However, this isarbitrary, because the level of sphericalaberration depends upon pupil diameter,with primary wave spherical aberrationdepending upon the fourth power of this

PP' NN""j'j"" ---~"""

,""II II

"PP' NN

Figure 5.4. The Le Grand full theoretical schematic eye.

diameter. Furthermore, the role of sphericalaberration may have been greatly exaggeratedsince real eyes have much less sphericalaberration than schematic eyes. We adopt theusual practice of increasing the length of theeye so that the retina coincides with F'.

Le Grand full theoretical eye

The lens of this eye has a constant refractiveindex, and thus has only two refractingsurfaces (Figure 5.4). The eye is presented inboth relaxed and accommodated forms.

Simplified schematic eyes

For paraxial calculations, the Gullstrandnumber 1 eye and the Le Grand full theo-retical eye are more complex than is requiredfor many optical calculations, such asmeasurement of retinal image sizes. Simpler

l',,,,,,.,,,

,,,,,,,.~

f....,PP' NN7 11_-_"""

Relaxed

""PP' NNFigure 5.3. The Gullstrand number 1 schematic eye. Figure 5.5.The Gullstrand-Emsley schematic eye.

Paraxialschematiceyes 45

eyes are now considered to be adequate. Thisis because errors that arise in using thesesimpler models are usually less than theexpected variations between real eyes.In simplified schematic eyes, the cornea is

reduced to a single refracting surface and thelens has two surfaces with a uniformrefractive index.

Gullstrand number 2 (simplified) eyeas modified by Emsley - theGullstrand-Emsley eye

Emsley (1952) modified Gullstrand's number2 eye in order to simplify computation (Figure5.5). The modifications included altering theaqueous and vitreous refractive indices to4/3, altering the lens refractive index to 1.416for both relaxed and accommodated eyes,thickening the lens and changing theaccommodated lens surface radii of curvatureto ±5.00mm.

Reduced eyes contain only one refractingsurface, which is the cornea. In the exact andsimplified eyes already presented, the twoprincipal points and the two nodal points areeach separated by values in the range0.12--0.37 mm. In reduced eyes, the use of asingle refracting surface means that its vertexmust be at the principal points P(P') and itscentre of curvature must be at the nodalpoints N(N'). To keep a power similar to thatof the more sophisticated eyes, reduced eyesmust have shorter axial lengths. As the corneahas absorbed the power of the lens, the radiiof curvature are much smaller than realvalues. Since reduced eyes do not have a lens,they cannot be used to examine the opticalconsequences of accommodation.

Emsley's reduced eye (1952)

This eye has a corneal radius of curvature of50/9 mm, a refractive index of 4/3 and apower of 60D (Figure 5.6).

Le Grand simplified eye Bennett and Rabbetts (1988, 1989)

Figure 5.6. The Emsley reduced schematic eye.

This eye has a corneal radius of curvature of5.6mm, a refractive index of 1.336 and apower of 60D.

Variable accommodating eyes

While most of the above models have fixedaccommodated forms, none has a variablelevel of accommodation. As mentioned pre-viously, Blaker (1980) presented a variableaccommodating paraxial eye which was latermodified to consider aging effects (Blaker,

pp' NN'... I, ,, ,,,,,,,,,

Reduced schematic eyes

Bennett and Rabbetts simplified eye

Bennett and Rabbetts (1988, 1989) modifiedthe relaxed version of the Gullstrand-Emsleyeye (see Appendix 3). Rabbetts (1998) intro-duced forms for accommodation levels of 2.5,5.0, 7.5 and 10D. He introduced an 'elderly'version of the eye, which has a lower lensrefractive index than do the other forms, andhas a refractive error of 1 D hypermetropia(see Chapter 7).

Further simplifications are possible whichmay give models accurate enough for somecalculations, in particular, estimates of retinalimage size.

Most of the parameters of this eye aredifferent from those of Le Grand's fullschematic eye. The lens is given a zerothickness. The eye has both relaxed andaccommodated forms.

46 Basic optical structureof thehumaneye

VN

VE

Tn

~ --_. ---. -- ---. -------.---. -. -- .. _------------

Summary of main symbols

7r---- 1

A accommodation level at corneal vertexin dioptres

d surface separationsF equivalent power of the eyem ratio u'ln of paraxial pupil ray angles -

this value is a constant for anyparticular eye at any particular level ofaccommodation

MEA pupil magnification, the ratio ofentrance pupil diameter to stopdiameter

Mn pupil magnification, the ratio of exitpupil diameter to entrance pupildiameter

tracing, calculate various quantities. Forexample:Fa(A) =58.636 + 11.940AIAo 0N'R'(A) =17.054 - 0.485AIA omm=11FR - 0.485AIA ommwhere FR is the equivalent power of therelaxed eye. The values of mand the distancesof the entrance pupil and the front nodal pointfrom the anterior corneal vertex (VE and VN)are plotted as a function of accommodation inFigure 5.7.

0.546 - (0.546- 0.6725)x1A o

= 2.419 - (2.419- 2.655)x1Ao

cortical posterior = 0.635 - (0.635thickness - 0.6725)x1A olens anterior 1/10 - (1/10curvature - 1/5.333)xlA olens core anterior = 1/7.911- (1/7.911curvature - 1/2.655)x 1A olens core posterior = -1/5.760 - [-1/5.760curvature - 11(-2.655)]x1Aolens posterior -1/6 - [-1/6curvature -1 1(-5.333)]xlA o

where the distance unit is millimetres and AQis the level of the Gullstrand accommodatedeye in dioptres, that is, 10.87013 D.

and the variable parameters of the eye arerelated to x by the equationsanterior chamber depth =3.1- (3.1- 2.7)xlA olenscortical anteriorthicknesscore thickness

1991). Navarro et al. (1985) presented a finiteaccommodating schematic eye which issuitable for easy paraxial calculations becausethe refractive index of the lens remainsuniform, but we leave discussion of this eyeuntil Chapter 16.We present here a variable version of

Gullstrand's number 1 schematic eye. The eyewas specified at two levels of accommodation(zero and 10.87 D), but we can modify this eyeto have a variable accommodation by assum-ing that the following individual parametersof this eye vary with accommodation:anterior chamber depthlens thicknesseslens cortex anterior curvaturelens core anterior curvaturelens core posterior curvaturelens cortex posterior curvature.To simplify the model, we relate the

accommodation level A, measured at thecorneal vertex, to a parameter x, wherex = 1.052A - 0.00531A2 + 0.000048564A3 (5.12)

o+-....-r-,-,....-..,....-....-r-,-...........,....-I""'"'T'"-,-.....--r-I""'"'T'"-r+o 2 3 4 5 6 7 8 9 10 II

Accommodation (0)

Figure 5.7. The effect of accommodation onm,VE andVN of a variable accommodating version of theGullstrand number 1 schematic eye.

Equivalent power and positions ofcardinal pointsWe can now assemble a schematic eye at anylevel of accommodation and, by paraxial ray-

ReferencesBennett, A. G. and Rabbetts, R. B. (1988). Schematic eyes- time for a change? Optician, 196 (5169), 14-15.

Bennett, A. G. and Rabbetts, R. B. (1989). Clinical VisualOptics, 2nd edn. Butterworths.

Blaker, J. W. (1980). Toward an adaptive model of thehuman eye. J. Opt. Soc. Am., 70, 220-23.

Blaker, J. W. (1991). A comprehensive model of the aging,accommodative adult eye. In Technical Digest onOphthalmic and Visual Optics, vol. 2, pp. 28-31. OpticalSociety of America.

Emsley, H. H. (1952). Visual Optics, vol. 1, 5th edn.Butterworths.

n, n'

ril

il'

E, E'F, F'N,N'0,0'

P,p'

R', R

refractive indices (usually of object andimage space, respectively)radius of curvatureparaxial pupil ray angle in object space(air)paraxial pupil ray angle in image space(in the vitreous)positions of entrance and exit pupilsfront and back focal pointsfront and back nodal pointsgeneral object and correspondingimage pointposition of front and back principalpointsaxial retinal point and correspondingconjugate in object space

Paraxial schematic eyes 47

Gullstrand, A. (1909). Appendix II: Procedure of the raysin the eye. Imagery - laws of the first order. InHelmholtz's Handbuch der Physiologischen Opiik, vol. 1,3rd edn. (English translation edited by J. P. Southall,Optical Society of America, 1924).

Helmholtz, H. von (1909). Handbuch der PhysiologischenOptik,vol. 1, 3rd edn. (English translation edited by J. P.Southall, Optical Society of America, 1924).

Le Grand, Y. and El Hage, S. G. (1980).Physiological Optics(translation and update of Le Grand Y. (1968). Ladioptrique de l'oeil et sa correction. In OptiquePhysiologique, vol. 1), Springer-Verlag.

Navarro, R., Santamaria, J. and Besc6s, J. (1985).Accommodation-dependent model of the human eyewith aspherics. J. Opt. Soc. Am. A, 2, 1273--81.

Polyak, S. L. (1957). The Vertebrate Visual System, Ch. 1.University of Chicago Press.

Rabbetts, R. B. (1998). Bennett and Rabbetts' Clinical VisualOptics, 3rd edn., pp. 209-13. Butterworth-Heinemann.

Smith, R. (1738). A Compleat System of Opticks, p. 25.Cornelius Crownfield.

Smith, G. and Atchison, D. A. (1997). The Eyeand VisualOptical Instruments, Appendix 3. Cambridge UniversityPress.

Sorsby, A., Benjamin, B., Davey, J. B. et al. (1957).Emmetropia and its Aberrations. Medical ResearchCouncil Special Report Series number 293. HMSO.

Swaine, W. (1921). Geometrical and ophthalmic optics-VII: paraxial schematic and reduced eyes. TheOpticianand Scientific Instrument Maker, 62, 133-6.

Tscherning, M. (1900). Physiologic Optics (translated fromthe original French by C. Weiland). The Keystone.

Young, T. (1801).On the mechanism of the eye. Phil. Trans.of theR. Soc. of Lond., 92, 23--88(and plates).

6

Image formation: the focused paraxial•Image

Introduction

In this chapter, we consider in-focus imageformation assuming the image forming raysbehave as paraxial rays. The treatment isapplicable to small angles only, as it ignoresaberrations and the curvature of the retina.The ability to predict retinal image size,

given an object of known size, has manyapplications. For example, if the two eyes of aperson with different levels of refractive errorare corrected with ophthalmic lenses, theretinal image sizes of the two eyes may bedifferent and this difference can lead tobinocular vision problems (see Binocularvision, this chapter). A second example is thecalculation of risks from radiation damagewhere, in the case of thermal damage (dueto wavelengths longer than approximately

500nm), the retinal image size affects the levelof risk.

The general case

Figure 6.1 shows an axial point at 0 and anoff-axispoint at Q on the perpendicular planethrough O. Beams of rays from each of thesepoints pass into the eye through the cornea,iris and lens, and are imaged at 0' and Q'respectively on the retina at R'. All the rays ineach beam are concurrent (i.e. focus) at theappropriate points 0' or Q'. According to therules of paraxial optics, the pointQ' lies on theplane passing through 0' and perpendicularto the optical axis.The points 0 and Q are at the edges of an

object, and 0' and Q' are at the edges of the

Figure 6.1.The general case of formation of the retinal image and image formingbeams.

52 Image formation alld refraction

Point sourceof light

.._------__ 0'----

Figure 6.2. Demonstration that the retinal image is inverted (see text forexplanation).

corresponding image. By noting the relativeorientation of the object and image, we candeduce that the image is inverted. Extendingthis to two dimensions, we would note thatthe image is inverted in both horizontal andvertical directions (equivalent to a 1800

rotation). The inversion is opposite to ourperception, because a further inversionprocess occurs in the brain.This retinal inversion can be demonstrated

simply, without the use of optical ray-tracing.Figure 6.2 shows a slightly out-of-focus pointsource, with its image formed behind theretina. This can be achieved by bringing thepoint source within the nearest point of clearvision. This point source should appear as acircular patch of light. If a sharp edge ismoved upwards as shown, it blocks the lowerpart of the beam. We observe the top edgebeing blocked, indicating that the braininverts the retinal image. Since we observe theimage erect, but the brain is inverting theimage, it follows that the retinal image isinverted.We cannot readily consider the image

formation in a particular eye unless we knowits construction details; in particular, surfaceradii of curvature, surface separations and

refractive indices. These data are not easy todetermine. In many situations, all we need arereasonable estimates from a schematic eye. Wecan determine the image formation in theschematic eye either by tracing suitableparaxial rays or by using known positions ofthe cardinal points and pupils. Since these arereadily available for the standard schematiceyes, we take this approach in the followingdiscussion. The reader should be familiar withthe properties of cardinal points and pupils asdiscussed in earlier chapters.Figure 6.3 shows a beam of rays from an off-

axis point Q being imaged to a point Q' on theretina of an eye with some level ofaccommodation. We have drawn two specialparaxial rays, the nodal ray and the paraxialpupil ray (also paraxial chief ray). The nodalray was introduced in Chapter 1. It is the rayfrom an off-axis object which is inclined at thesame angle to the optical axis in both objectand image spaces. In object space it is directedtowards the first nodal point N, and in imagespace it is directed from the second nodalpoint N'. The paraxial pupil ray wasintroduced in Chapter 3. It is the ray which, inobject space, is directed towards the centre ofthe entrance pupil E, and in image space is

Q

Figure 6.3. The formation of an off-axis image point, the paraxial pupil ray, and thenodal ray for an arbitrary level of accommodation.

Image formation: tilefocused paraxial image 53

The minus sign is present because, in thefigure, 11 is negative and 0 and the distanceON are positive. Combining these twoequations gives

Ii = -1110E (6.7)

Therefore we can finally express the imagesize 11'by the equation11'= - 11m E'R'IOE (6.8)Values form and the positions of the cardinal

directed from the centre of the exit pupil E'.Both of these rays can help us to determinethe size of the retinal image.We can use the nodal ray to find the size of

the retinal image, using Figure 6.3.Within thelimits of paraxial optics, the retinal image size11'isgiven by the equation11' = ON'R' (6.1)

where

We can express the image size 11' in terms ofthe angle Ii the pupil ray is inclined to the axisin object space. The angles Ii' and Ii areconnected by equation (5.7), i.e.Ii'Iii = a constant =m (6.5)

with the value of the constant m dependingupon the schematic eye used and level ofaccommodation. Combining these equationsgives11'=Ii m E'R' (6.6)where

points for different schematic eyes are givenin Table 5.1.Example 6.1: Calculate the retinal imagesize of a letter of height 1 mm, seen atthe near point of the accommodatedGullstrand number 1 schematic eye.Solution: From Table 5.1 we have thefollowing data:N'R' = 17.539mmON = 1000/10.870 + VN = 91.996

+ 6.533= 98.529 mmm = 0.795850E'R' = 21.173 mmOE = 91.996 + VE = 91.996 + 2.668

=94.664mm.Choosing a value of 11 = -1 mm, which isnegative because the object is below theoptical axis, and using equation (6.3) wehave11'= 1 x 17.539/98.529 mm = 0.178mmAlternatively, using equation (6.8) wehave11'= 1 x 0.795850 x 21.173/94.664 mm

=0.178 mmWe note that the two values should beand are the same.

Retinal image size and perceivedangular size in object space

In analysing visual images, we can specify theimage size in two ways; one is the image sizeon the retina, and the other is as a perceivedangular size in object space. These twoquantities are related. Let us consider thesituation in Figure 6.3,where an eye is lookingat an object at 0 and this is imaged at 0' onthe retina at R'. The nodal points Nand N'and the nodal ray are shown. The angle ofinclination of this ray with the axis is the samein both object and image spaces. If the objectsubtends an angle 0at the front nodal point N,then the retinal image subtends the sameangle at the back nodal point N'. For typicalworking distances, the distance to the object islarge in comparison with the dimensions ofthe eye. Therefore, in determining the angularsize of the object, we can take the referencepoint as the corneal vertex, the entrance pupil

(6.4)

(6.2)

(6.3)

11' =i7'E'R'

O=-1110N

11'=-11N'R'IONTo find the image size in a particular case, weneed the object size 11 and distances ON andN'R'. The positions of the nodal pointsdepend upon the level of accommodation ofthe eye, and estimates are available only for alimited range of accommodation forschematic eyes. If the positions are known, theuse of the nodal rays to find the size ormagnification of the image is straightforward.An alternative method uses the paraxial

pupil ray. One advantage of this ray is that italways (by definition) lies in the centre of theimage-forming beam. From Figure 6.3, itfollows that

54 Image formatioll alld refraction

Eye focused at infinity

position or the front nodal point. In summary,we can conclude that the angular size of theperceived image is that angle subtended bythe retinal image at the back nodal point. Thisconclusion is useful in some discussions onimage sizes.

We now use equations (6.11) and (6.13) in anumerical example.Example 6.2: Calculate the retinal imagesize of the moon (angular diameter of0.5°) using the Gullstrand number 1relaxed schematic eye.Solution: For the Gullstrand number 1relaxed schematic eye, the relevant dataare given in Table 5.1; in particularF= 58.6360E'F' =20.720mm andm =0.823085.Substituting the relevant values and 0 =os =0.00872665 radians into equation

If the eye is focused at infinity, we cansimplify the above equations. For an object atinfinity (or very distant), its size can only beexpressed as an angular measure, say O. Forthe eye focused at infinity, the retinal point R'coincides with the back focal point F' and thusN'R' = N'F' (6.9)Combining equations (5.1)and (5.3b)with n =1 givesN'F' = l/F (6.10)

where F is the equivalent power of the eye.Combining equations (6.1), (6.9) and (6.10)gives17'= O/F (6.11)

The values of F for different schematic eyesare given in Table 5.1.Alternatively, we can use the pupil ray

equation (6.8). Because the object is at infinity,its angular size is independent of the pointfrom which this angle is measured. Thereforewe can write

Aniseikonia

Binocular visionStereopsisThe use of two eyes provides the potential forseeing depth in a scene. This perception ofdepth is called stereoscopic vision orstereopsis. The retinal images of that world,while flat two-dimensional images, areslightly different. This is explained with thehelp of Figure 6.4. Figure 6.4a shows, fromabove, the two eyes of a person who is lookingat two points A and B in a horizontal plane.From an observer's point of view, A is imagedto the left of Bby both eyes, and no perceptionof depth occurs. Figure 6.4b shows the personlooking at A and B, which are now in line butat different distances. From the observer'spoint of view, A is imaged to the right of B forthe left eye, but A is imaged to the left of B forthe right eye - i.e. the two retinal images of Aare closer together than are the two retinalimages of B. This difference in relativeposition of the retinal images of A and B leadsto the perception of depth.

Aniseikonia is usually defined as a relative

(6.11) givesr(=0.00872665/58.636 m =0.14883mmSubstituting the relevant values intoequation (6.13)givesn' = 0.00872665 x 0.823085 x 20.720032=0.14883mm

Note that these two solution should beand are the same.

If we take an approximate value of the meanpower of the eye of 60 0, then we can writeequation (6.11) in the form17' "" 0.004850 mm (6.14)where 0 is in minutes of arc. This equation is auseful rule of thumb for working out expectedretinal image sizes and indicates that 1 minarc is approximately equivalent to 0.005mmon the retina, and 1 mm on the retina isapproximately equivalent to 200min arc.

(6.13)

(6.12)0= -17/0EThus equation (6.8) reduces to17'= om-E'F'

left eye right eye

Image formation: thefocused paraxial image 55

Figure 6.4. Binocular vision and the relative positions of the retinal images for two pointobjects.a. Objects side by side.b. One object behind the other.

difference in size and/or shape of the tworetinal images (see, for example, Cline et al.,1989). However, this is a simplification, as it isthe brain which ultimately 'sees', rather thanthe eyes. A person can have different retinalimage sizes and not have any problems, so wewould like to replace the word 'retinal' by'cortical' in the above definition. Mostclinically significant cases of aniseikoniaresult from correcting anisometropia, which iswhere the two eyes have different refractiveerrors. Anisometropia is discussed at greaterlength in Chapter 10.The symptoms of aniseikonia are usually

indistinguishable from those caused by otherbinocular vision problems and by uncorrectedrefractive errors. The classical symptom is adistortion of spatial perception when botheyes are being used, but this occurs inrelatively few patients (Bannon and Triller,1944). Binocular vision must be well devel-oped for aniseikonia to be a problem, andusually the clinician should consider otherpossible causes of symptoms before attempt-ing to correct aniseikonia. Clinical treatmentsof aniseikonia, in order of increasing com-plexity, include the following:1. Altering the prescription to reduce theamount of aniseikonia, for example,reducing high cylindrical corrections,omitting low cylindrical corrections, and

reducing the difference in sphericalpowers.

2. Prescribing contact lenses rather thanspectacles, because the former have muchsmaller effects on retinal image size.

3. Prescribing a pair of spectacle lenses calledisogonal lenses, which have similarmagnifications to each other.

4. Prescribing a pair of aniseikonic correctinglenses called iseikonic (size) lenses,according to results of the eikonometer, aninstrument which measures aniseikonia.Further information on aniseikonia is found

in texts such as Borish (1970) and Rabbetts(1998).

Summary of main symbols

m ratio irm of the angles of inclination ofthe paraxial pupil ray with the axis inimage and object space, respectively.This ratio is a constant for anyparticular schematic eye at a particularlevel of accommodation

1'/, 1'/' object and image sizes(J angular size of object. This angle is

assumed to be small so it can beregarded as a paraxial angle

E, E' positions of entrance and exit pupilsN, N' positions of front and back nodal points

56 Image formation and rejraction

0, 0' general object and corresponding imagepoint

R', R axial retinal point and correspondingconjugate in object space

ReferencesBannon, R. E. and Triller, T. (1944). Aniseikonia - a clinical

report covering a to-year period. Alii. ,. Optom. Arch.Alii. Acad. Optom., 21,171-82.

Borish, I. (1970). Clinical Refraction, 3rd edn., pp. 267-94.Professional Press.

Cline, D., Hofstetter, H. W. and Griffin, J. R. (1989).Dictionary of Visual Science, 4th edn., p. 36. Chilton TradeBook Publishing.

Rabbetts, R. B. (1998). Bennett and Rabbetts' Clinical VisualOptics,3rd edn. Butterworth-Heinemann.

7

Refractive anomalies

Introduction

Ideally, when the eye fixates an object ofinterest, the image is sharply focused on thefovea. In paraxial optical terms, the object andfovea are conjugate. However, the object canonly be focused sharply if it is within theaccommodation range of the eye. If theaccommodation range is inappropriate or toosmall, objects of interest cannot be focusedsharply on the retina. In these cases, theretinal image is out-of-focus or blurred, andvisual acuity is reduced. The effect of thesefocus errors on the retinal image is discussedin Chapter 9 and in Chapter 18.An appropriate range of accommodation

includes all reasonable object distances ofinterest. This includes distant objects,effectively at infinity, down to objects as closeas a few centimetres.An eye with a far point of distinct vision at

infinity is called an emmetropic eye. Theemmetropic eye is regarded as the 'normal'eye, provided that it has an appropriate rangeof accommodation. A refractive anomalyoccurs if the far point is not at infinity. An eyewhose far point is not at infinity is called anametropic eye. The departure from emmet-ropia is often considered to be an error ofrefraction, and ametropias are also referred toas refractive errors. Emmetropia andametropia may be regarded as opposites, butan alternative and more appropriate view isthat emmetropia is part of the distribution ofametropias.

Another refractive anomaly occurs whenthe range of accommodation is reduced sothat near objects of interest cannot be seenclearly. This is called presbyopia and isusually age-related.Defocused retinal images may occur

because the far point is closer to the eye thaninfinity or is beyond infinity. By beyondinfinity, we mean that it is located behind aperson's head. Defocused retinal imagesoccur also when the refractive power of theeye varies with meridian. This is commonlydue to one or more refractive surfaces in theeye being toroidal, transversely displaced ortilted. There are now two far points, onecorresponding to each of two principalmeridians. These errors are referred to asastigmatic or cylindrical refractive errors, incontrast to spherical refractive errors, whichare present when the refractive error is thesame in all meridians.Refractive errors may occur as a result of

surgery. The most obvious example of this isaphakia. In the aphakic eye, the lens has beenremoved, usually because of cataracts (semi-transparent and translucent formations in thelens) which absorb, reflect and scatter theimage-forming light.Whatever the cause of the refractive error, it

can be corrected with appropriate ophthalmiclenses, which include spectacle, contact andintra-ocular lenses. When an ametropic eye iscorrected by an ophthalmic lens, the equiv-alent power of the eye/ophthalmic lenssystem is different from the value of the

58 Image formation ami refraction

uncorrected ametropic eye and hence thereare shifts in the cardinal points. This causeschanges in retinal image sizes, and henceproduces magnification effects, which arediscussed in Chapter 10.Most ophthalmic practitioners do not make

a distinction between the terms refractiveerror and refractive correction, a practice thatwe shall follow in this book, although from apurist perspective they should be opposites.Hence, we may refer to a myope having a-2D refractive error whereas, strictly speaking,we should say either that the myope iscorrected by a -2 0 lens or requires a -2 0correction.For a more intensive review of refractive

error classification, see Rosenfield (1998).

Spherical refractive anomalies

Spherical refractive anomalies are categorizedaccording to the position of the far point (thespherical refractive errors) or of the near point(presbyopia).

Spherical refractive errors

Emmetropia (normal sight>We wrote in the previous section that the farpoint of the emmetropic eye is at infinity. Thisis not a practical definition in terms ofdeciding who is an emmetrope. An emme-tropic range may be considered to include

Far pointof eyeR.

(a) Myopia

-~._-----------

(b) Hypermetropia

refractive errors smaller than the smallestmeasurement interval, which is usually 0.25 0- i.e. the far point is greater than 4 m away.For research purposes, a wider range may beused; for example, -0.25 0 to +0.75D.

Myopia (short sight>If the far point is at a finite distance in front ofthe eye, as shown in Figure 7.1a, the eye ismyopic or is said to suffer from myopia. Thismeans also that the back focal point F' of theeye is in front of the retina as shown in thefigure. An object at infinity is focused in theback focal plane at F' and is out of focus on theretina. This situation can be regarded as dueto a mismatch between the length of the eyeand its power - the eye can be regarded asbeing too powerful for its length, or as beingtoo long for its power. The eye is not able toreduce its power in order to focus distantobjects on the retina.This eye can focus clearly on distant objects

by viewing through a negative poweredophthalmic lens of appropriate power, asshown in Figure 7.2a. This lens forms a distantand virtual image at its back focal plane,which coincides with the far point of the eye.Myopia can be classified in many different

ways (Rosenfield, 1998). These include clas-sification by rate of progression, magnitude(e.g. low, moderate, high), age of onset (e.g.juvenile), and the refractive component con-sidered to be responsible.Uncorrected myopes tend to complain of

blurred distance vision, which is more notice-

Far pointof eye

R:- - - _ _ R.r:

Figure 7.1. The (a) myopic and (b) hypermetropic eyes andtheir far points.

Back focal planeof correcting lens

----~------

Far pointof eye

(~:::8f-------Figure 7.2. The spectacle correction of (a) myopia, (b)hypermetropia, and (c) presbyopia.

able at night. Depending on the degree ofmyopia, they may notice that close objectsappear blurred as well.

Hypermetropia (hyperopia)In a hypermetropic eye, the far point liesbehind the eye and the back focal point F' isbehind the retina, as shown in Figure 7.tb. Anobject at infinity is focused in this back focalplane but, once again, the retinal image isdefocused. This eye can bring the image intosharp focus if there is sufficient amplitude ofaccommodation. Once again, this situationcan be regarded as due to a mismatch betweenthe length of the eye and its power - the eyecan be regarded as being too weak for itslength, or as being too short for its power.This eye can focus clearly on distant objects

by viewing through a positive powered lensof appropriate power, as shown in Figure 7.2b.This lens forms a distant and real image on itsback focal plane, which coincides with the farpoint of the eye.Many young hypermetropes have difficulty

relaxing their accommodation completely. Aresidual tonus in the ciliary muscle produces a

Refractive anomalies 59

degree of latent hypermetropia, which cannotbe determined by subjective refractionwithout the use of cycloplegic drugs. Totalhypermetropia can then be regarded asconsisting of manifest and latent com-ponents. The part of the manifest hyper-metropia that can be overcome byaccommodative effort is referred to as facul-tative hypermetropia. Any deficit remainingis referred to as absolute hypermetropia. Withincrease in age and loss in amplitude ofaccommodation, the manifest component oftotal hypermetropia increases at the expenseof the latent component. Similarly, theabsolute component of manifest hyper-metropia increases at the expense of thefacultative component.Uncorrected hypermetropes tend to

complain of sore eyes and headachesassociated with close visual tasks. This isbecause they must make more accom-modative effort than emmetropes andmyopes to view close objects. In addition, thedegree of convergence they use is inappro-priate for the level of accommodationdemanded. They may also complain ofblurred near and distance vision, dependingon the level of hypermetropia and theamplitude of accommodation. The blurring isgreater at near than at distance, because of thegreater accommodative demand at near.As hypermetropes may not be able to see

distance objects clearly, the term long sight todescribe hypermetropia should be dis-couraged.

Presbyopia

Presbyopia is the difficulty people have inperforming close tasks because of the age-related decrease in amplitude of accom-modation. The near point recedes from theeye so that it is close to or beyond the positionat which a near task is performed. The onsetof presbyopia is related to the degree ofrefractive errors, with uncorrected hyper-metropes likely to have problems earlier inlife than uncorrected myopes. Depending onthe degree of myopia and the near task, thelatter may not suffer from presbyopia. Tocompensate for presbyopia, ophthalmic lensesare required that are more positively powered

60 [mage formation and rcfroction

or less negatively powered than the distancecorrection (Figure 7.2c). We discuss pres-byopia further in Chapter 20.Not surprisingly, uncorrected presbyopes

usually complain of difficulty performingclose tasks.

Astigmatic refractive errors

In many eyes, the refractive error isdependent upon meridian. This type ofrefractive error is known as an astigmaticrefractive error (Figure 7.3). This is usuallydue to one or more refracting surfaces, mostcommonly the anterior cornea, having atoroidal shape. However, it may also be due toone or more surfaces being transverselydisplaced or tilted.There are different types of astigmatism

which may be related to the associatedspherical refractive errors as follows:1. Myopic astigmatism - the eye is too

powerful for its length in one principalmeridian for simple myopic astigmatism,and in both principal meridians forcompound myopic astigmatism (as shownin Figure 7.3).

2. Hypermetropic astigmatism - the eye istoo weak for its length in one principalmeridian for simple hypermetropic astig-matism, and in both principal meridiansfor compound hypermetropic astig-matism.

3. Mixed astigmatism - the eye is toopowerful for its length in one principal

Figure 7.3. An astigmatic eye. The far points Ra andRw + a are imaged at the retinal point R'.

meridian (myopic astigmatism), and tooweak for its length in the other principalmeridian (hypermetropic astigmatism).Astigmatism may also be classified by the

axis direction. With-the-rule astigmatism isusually associated with a cornea that issteeper (i.e. has the greater surface curvature)along the vertical than along the horizontalmeridian. It requires a correcting lens whosenegative cylinder axis is within ±30° degreesof the horizontal meridian. Conversely,against-the-rule astigmatism is usuallyassociated with a cornea that is steeper alongthe horizontal than along the verticalmeridian, and requires a correcting lenswhose negative cylinder axis is within ±30°degrees of the horizontal meridian.Astigmatism with axes more than 30° fromthe horizontal and vertical meridians isreferred to as oblique astigmatism.One further classification of astigmatism is

related to its regularity. Astigmatism isusually regular, which means that theprincipal (maximum and minimum power)meridians are perpendicular to each other,and the astigmatism is correctable withconventional sphere-cylindrical lenses (seeThe power of the correcting lens, this chapter).Irregular astigmatism occurs when theprincipal meridians are not perpendicular toeach other or there are other rotationalasymmetries that are not correctable with theconventional lenses. It may occur in cornealconditions such as keratoconus.Astigmatism is corrected with an astigmatic

ophthalmic lens. Usually this has onespherical surface and one toroidal surface, thelatter generally being the back surface.Astigmatic lenses are often referred to assphere-cylindrical lenses for the historicalreason that at one stage most astigmatic lenseshad one spherical surface and one cylindricalsurface.To make analysis of large scale population

data relatively easy, when astigmatism ispresent an equivalent sphere (also called themean sphere) is used, which is the averagerefractive error of the two principalmeridians.People with astigmatism have blurred

vision at all distances, although this may beworse at distance or near, depending on thetype of astigmatism. They may complain of

sore eyes and headaches associated withdemanding visual tasks.

AnisometropiaThis is the condition in which the refractiveerrors of the two eyes of a person are different.Sub-classifications include anisomyopia, inwhich both eyes are myopic, anisohyper-metropia, in which both eyes are hyper-metropic, and antimetropia, in which one eyeis myopic and one eye is hypermetropic.Spectacle lens correction of anisometropia

may result in different retinal image sizes anddifferent prismatic effects when lookingthrough lens peripheries. These effects maycompromise comfortable binocular vision.Some discussion of these side effects isprovided in Chapter 10.

Distribution of refractive errors andocular components

DistributionThe distribution of refractive errors is stronglyage-dependent, particular in childhood(Hirsch and Weymouth, 1991). Neonates havea normal distribution of refractive errors(Hirsch and Weymouth, 1991; Goss, 1998;Zadnik and Mutti, 1998). From birth tomaturity (about 11-13 years) the ocularcomponents are growing, and the relation-ships among them must be co-ordinated sothat emmetropia can be achieved andmaintained (see review by Troilo, 1992). Theterm emmetropization is given to this fine-tuning of refraction, and it is thought that theprocess is visually regulated. A number ofstudies have been carried out on thedistribution of ametropia in the adultpopulation. Results of Stromberg's (1936),Stenstrom's (1948) and Sorsby et al.'s (1960)studies are shown in Figure 7.4. The meanrefractive error is slightly hypermetropic, andthe distributions are steeper than normaldistributions (leptokurtosis). They have morepronounced tails in the myopic direction thanin the hypermetropic direction (negativeskewness). The distribution of refractive

Refractioe anomalies 61

70

60

50

~ • Stromberg;: 40 • Stenstrom'"c 0 SorsbyIII& 30III...~

20

10

~ ~ ~ ~ ~ ~ ~

r-: '4 v;i 'i ro;i ~ qi 0 M ..; t on -c+ + + + +9 9 9 9 9 9 9 9 9 9 9 9 9 9 90"? r;- '9 ":' -r ":' ~ '7 '+ M .... "$ '" '0+ + + +

Refractive error (D)

Figure 7.4. Population distribution of refractive errors.Data of Stromberg (1936),Stenstrom (1948)and Sorsby etal. (1960). Stromberg's data were obtained fromStenstrom (1948).

errors is fairly stable between the ages of 20and 40 years, after which the distributionbecomes less leptokurtic (Grosvenor, 1991).The distributions of the main ocular

parameters, such as axial length and cornealradius of curvature, are almost normal.Stenstrom (1948) found that the corneal powerand radii were normally distributed, as werethe depth of the anterior chamber, lens powerand total power. He found that axial lengthwas not normally distributed, but Sorsby et al.(1957) claimed that all components includingaxial length were normally distributed.Sorsby et al. (1962) carried out an extensive

study of the relationship between the variousocular parameters, and came to the followingconclusions:1. In emmetropic eyes, wide ranges of corneal

power (39-48 D), lens power (16-24 D) andaxial length (22-26 mm) occur.

2. In ametropic eyes with refractive errors inthe range -4 D to +6 D, the same range ofparameters occurred as for emmetropiceyes, but these were combinedinadequately. Such eyes were referred to ascorrelation ametropic eyes.

3. For refractive errors outside the range of -4D to +6 D, the axial length seemed to be thecause of the ametropia. It was too long in

ast~gmat~sm changes with age. Considerable?stIgmatIsm, usually against-the-rule, exists10 t~e first y~ar of life and decreases quicklydunng early infancy, Most clinically measur-able astigmatism is thereafter with-the-ruleup to the age of 40 years, after which theprevalence of against-the-rule astigmatismincreases (Goss, 1998).

The power of the correcting lens

This. section explores the power of lensesre~)Ulred to .correct a refractive error, startingwlt~ the thin lens case. While this approxi-mation leads to some errors, it is satisfactoryfor simply examining trends. A thick lenstreatment and its implications are supplied ina number of ophthalmic texts (e.g. [alie, 1984;Fannin and Grosvenor, 1996).In ophthalmic practice, the power of the

correcting lens is determined by subjectivetechniques (using trial case lenses or refractorhe~ds), or by objective techniques such asretmoscop~ (described briefly in Chapter 8) orautorefraction, In none of these cases is the farpoint found directly, but mathematicalana.Iysis of the required lens power is madeeasle~ by reference to the far point distance.Consider a hypermetropic eye, as shown inFigure 7.5, in which the far point is a distanced behind the corneal vertex. The value of dm.ust b~ ~iv~n a sign. I~ our sign convention,d IS positive If the far point is to the right of thecorneal vertex and negative if it is to the left ofthe co~neal ve~tex. The~efore, in the myopiceye d ~s neg~t.lve, and 10 the hypermetropiceye d IS positive, If a lens is now placed adistance 11 in front of the corneal vertex, thepower Fs(lI) of the correcting lens, also calledthe spectacle refraction, is a function of vertex

62 Image formation and refraction

myopia and too short in hypermetropia.These eyes were referred to as componentametropic eyes.Component ametropia can be divided into

axial and refractive categories by comparing?imensions of an eye with the range of values10 the emmetropic population or with those ofa schematic eye such as Gullstrand's number1 eye. The refractive error is regarded asrefractive in nature if the axial length is withinan 'emmetropic' range, but the power of theeye or one of its components is outside theemmetropic range. The refractive error is~egarde? as axial in nature if the axial lengthIS outside the emmetropic range, but thepower of the eye and its components arewithin emmetropic ranges. Aphakia, in whichthe lens has been removed, is an obvious case?f ref~active ametropia. Most astigmatisms,including those caused by corneal conditionssuch as keratoconus, can be regarded as casesof refractive ametropia.The ocu~ar component most highly

correlated with refractive error is axial length.Longitudinal studies indicate that axialelongation is the mechanism for juvenile andlate onset myopia, and for the progression ofmyopia (Goss and Wickham, 1995). Thissuggests that most of the myopes that Sorsbyand co-workers (1962) classified as correlationametropes were actually axial componentmyopes.

If an eye is deprived of suitable visualstimuli early in its development, it grows to beunusually long, leading to myopia. Thisgrowth is controlled by local factors withinthe retina (Hodos et al., 1991; Smith, 1991;~allm?n, 1991). Th~s indicates that a change10 axial length IS the main cause ofemmetropization, with the eye increasing itslength until emmetropia is achieved.For re~ie~s of hypotheses concerning

emmetropization and of the possible roles ofgenetics and environmental factors in thedevelopment of refractive errors, see Troilo(1992)and Goss and Wickham (1995).

Astigmatism

---Far pointof eye

:1As with spherical refractive errors (orspherical equivalents), the distribution of

Figure 7.5.The power of the correcting lens (example ofhypermetropia).

Refractive anomalies 63

ORe =-0.144 0

distance h, and is given by the equation

Fs(h) = Ills = I/(h + d) (7.1)where the distance h is always regarded aspositive. We can express equation (7.1) in theform

s, = IId (7.3)is the refractive error (or refractive correction)of the eye at the corneal plane. This is alsoreferred to as the ocular refraction.Example 7.1: Calculate the spectacle lenspower, if the far point is 45 cm in front ofthe eye (i.e. a myopic error) and thespectacle lens vertex distance is to be 15mm.Solution: Here we have h = 15 mm andd = -45 em, Substituting these values inequation (7.1) gives

Fs(15mm) = -2.30 0The sensitivity of the power to changes in

distance h can be investigated by differ-entiating equation (7.1)with respect to h. Ifwedo this, we have

dFs(h)= R/ =-F 2(h) (7.4)dh (1 + hRe)2 s

It follows that a small change lJh in h leads toa change SFs in Fs' given by the approximateequationSFs "'" -Fs2(h)lJh (7.5)The above change in power SFs is equivalentto a change in refractive error ORe' Thus wehaveORe"'" -Fs2(h)lJh (7.5a)Example 7.2: Consider an eye requiring a+12 0 spectacle lens placed at a vertexdistance of h = 12 mm. If the lens isplaced at 13mm, estimate the inducedrefractive error.Solution: In this problem, the change lJh inh is lJh = +1 mm, and substituting Fs =+12 0 and h = 12 mm into equation (7.5a)gives the approximate induced refractiveerror

(7.6)

Example 7.3: Consider the +12 0 lens inExample 7.2. If the eye is to be correctedwith a contact lens, calculate the power ofthat lens.Solution: In this problem, hI is 12 mm, h2is 0 mm and Fs = +12 D. Substitutingthese values into equation (7.6) givesF(O mm) =+14.0 0

Thick lenses and the effect ofthickness

In the simplest mathematical model describ-ing a cylindrical or astigmatic error in the eye,the power F of the eye can be thought of asvarying with azimuth angle 8 in the pupilaccording to the equationF(8) =Fsp + Fey sin2(8 - a) (7.7)

According to this equation, the correction iscomposed of a spherical component of powerFsp' and a cylindrical component of power FcyWIth the cylindrical axis along the direction a.The clinical representation of astigmaticcorrections is

Fsp/Fey x aThe axis notation most commonly usedspecifies the axis by the anticlockwise anglethat it makes with the horizontal meridian.This is taken from the viewpoint of anobserver looking at both lenses as worn infront of the two eyes. The axis varies between0° and 180°. A horizontal axis can beconsidered to be either 0° or 180°, but isusually taken as the latter.

Correcting lenses have some thickness, andtherefore the above theory requires some

Astigmatic corrective powers

In clinical practice it is common to deter-mine the refractive error at one distance (sayhI) and prescribe for another (say h2). Atypical example is changing from spectaclelenses to contact lenses. Knowing F(hI ), it canbe shown using equation (7.1) that F(h2) isgiven by the equation

F(hI )

F(h2) = (h2 - hI)F(h I) + 1

(7.2)Fs(h) = ReI(1 + hRe)

where

64 Image formation and refraction

Refractive error and axial length

Effect of parameter changes onrefractive errors

As already mentioned, spherical refractiveerrors are due to a mismatch between therefractive power of the eye and its axiallength. In some cases, it is important to knowhow a change in axial length can affect thelevel of ametropia. This can be investigatedusing a schematic eye.The lens equation for the eye can be written

as

(7.9)

The power Fc of the anterior surface of thecornea is given by the equation

or

01'=-Q.371Re mm (7.11b)These two equations are useful as a guide inrelating changes in axial length to changes inrefractive error. They assume a value of 60 Dfor the equivalent power of the eye. If a moreaccurate value is available in a particular case,it can be substituted into equation (7.10).

or

Change in corneal curvature

s. = -0I'F2 / n' (7.9a)

This change in object vergence can be equatedto the refractive error Re. ThusRe = -0I'F2/ n' (7.10)which shows that the relationship betweenchanges or error in axial length and refractiveerror depends upon the equivalent power ofthe eye.

If we take an equivalent power for the eyeof 600 and a vitreous humour refractiveindex of 1.336, we haves, =-2.69 01' D (7.11a)

n'o/'---oL=O1'2

Therefore01'=-&1'2In'

Figure 7.6. The change of axial length and correspondingchange in refractive error.

If the eye is initially emmetropic and relaxed,then I =00 and so1'= n'IF

Therefore, equation (7.9) can be written as01'=-cs: I F2

(7.8)n'II'-L=F

modification in the case of real lenses. Inophthalmic optics, it is conventional tomeasure the distance of the lens from the eyeby the back vertex distance, which is thedistance from the back vertex of the lens (thevertex closest to the eye) to the corneal vertex.When this is done, the focal lengthIs in Figure7.5 and in the preceding equations becomesthe back vertex focal length tv' and thus thepower Fs in these equations becomes theback vertex power F'v' Therefore, theequations (7.1) to (7.6) still apply, providingthe distance h is the back vertex distance andthe power Fs is replaced by the back vertexpower r;However, when specifying the retinal

image sizes in eyes corrected with ophthalmiclenses, it is the equivalent power rather thanthe back vertex power that sets the image size.When we go from thin to thick lenses wecannot take all the equations and simplyreplace the equivalent power by back vertexpower. For image size calculations thesituation becomes more complex, and we lookat this problem in Chapter 10.

where n' is the refractive index of the vitreous,F is the equivalent power of the eye and L (=1I I) is the vergence of the object, conjugate tothe retina of the eye, relative to the firstprincipal plane.Suppose that the axial length changes by an

amount &~ as shown in Figure 7.6. Thischange 01' leads to a change oL in theobject vergence, given by differentiatingequation (7.8) and using small changes to get

Refractive anomalies 65

+0.173-{l.814-2.398+1.689

+0.483-{l.056+0.039+0.047

-{l.173-{l.I38-{l.I77-{l.256

distance of far point from the cornealverte~. This distance is negative formyopic eyes and positive for hyper-metropic eyesequivalent power of the eyepower of anterior surface of the cornea

which, being positive, indicates that theinduced refractive error is a hyper-metropic error.

Refractive index +1% increasecorneaaqueouslensvitreous

Radius of curvature +1% increasecornea anteriorcornea posteriorlens anteriorlens posterior

Distance +0.1 mm increasecorneal thicknessanterior chamber depthlens thickness (core)vitreous length

Table 7.1. Effects of small parameter changes onchange in refractive error using the Gullstrand number1 relaxed schematic eye with paraxial ray-tracing.Negative values indicate myopic refractive errors andpositive values indicate hypermetropic refractiveerrors.

Other parameter changes

Using paraxial mathematical modelling andparaxial ray-tracing, we can investigate theeffect of changes in any optical parameter onthe refractive state of an eye. Table 7.1 showsthe results of some calculations based uponthe Gullstrand number 1 relaxed eye. Therefractive error induced by a change of +1 percent in anterior corneal radius is +0.483 0,which is only about 1 per cent different fromthe value given in Example 7.4. The majorsource of the difference is neglecting theeffects of the lens and the posterior cornealsurface in equation (7.15).

Summary of main symbols

d

Fc=(n-1)/r (7.12)

where r is the radius of curvature of thesurface and n is the refractive index of thecornea. A small change or in radius ofcurvature r leads to a change OF in cornealpower Fc given by the equation c

OFc == -(n -1)&/r2 (7.13)

where r is the original corneal radius ofcurvature. This equation can also beexpressed in the formOFc==-F/&/(n-1) (7.14)If we measure the refractive error from theprincipal planes, the change in equivalentpower is the refractive error. Weneed to relatethe change in anterior corneal power to theequivalent power.The equivalent power of the eye is due to

the power of the cornea and lens. The abovepower of the cornea is only the anteriorsurface power, and does not include theposterior surface power. To relate this changein anterior corneal power to a change inequivalent power of the eye is not simple, andinitially we will take a short cut and neglectthe interaction of the posterior corneal powerand the lens power. Later we will examine theerror induced by this simplification.With these approximations, the above

change in corneal power is the same as thecorresponding change in equivalent power,which in turn is the negative value of theinduced refractive error oR . Therefore wehave e

ORe == F/ & / (n - 1) (7.15)

Example 7.4: Using the Gullstrandnumber 1 eye, calculate the inducedrefractive error caused by an increase incorneal radius of 1 per cent.Solution: For this eye, from data given inAppendix 3, r =7.7mm, n=1.376and thepower of the corneal surface F iscFc == (n -1)/r =0.376/7.7 =0.04883 mm-1Also& = 0.01 x 7.7= 0.077mmSubstituting the values for F , & and ninto equation (7.16) gives c

ORe == 0.048832 X 0.077/0.376se 0.000488 mm"! == 0.488 0

66 Image formation and refraction

ReferencesFannin, T. E. and Grosvenor, T. E. (1996). Clinical Optics,2nd edn. Butterworth-Heinemann.

Coss, D. A (1998). Development of the arnetropias.Chapter 3. In Borish's Clinical Refraction. (W.]. Benjamin,ed.), pp. 47-76. W. B.Saunders.

Goss, D. A and Wickham, M. G. (1995). Retinal-imagemediated ocular growth as a mechanism for juvenileonset myopia and for emmetropization. A literaturereview. Doc. Ophthal., 90, 341-75.

Grosvenor, T. (1991). Changes in spherical refractionduring the adult years. In Refractive Anomalies. Researchand Clinical Applications (T.Grosvenor and M. C. Flom,eds), pp. 131-45. Butterworth-Heinemann.

Hirsch, M. J. and Weymouth, F.W. (1991). Prevalence of

image distances (fromplanes of refracting

refractive anomalies. In Refractive Anomalies. Researchand Clinical Applications (T. Grosvenor and M. C. Flom,eds), pp. 15-38. Butterworth-Heinemann.

Hodos, W, Holden, A. L., Fitzke, F. W. et al. (1991).Normal and induced myopia in birds: models forhuman vision. In Refractive Anomalies. Research andClinicalApplications (T.Grosvenor and M.C. Flom, eds),pp. 235-44. Butterworth-Heinemann.

[alie, M. (1984). ThePrinciples ofOphthalmic Lenses, 4th edn.The Association of Dispensing Opticians.

Rosenfield, M. (1998). Refractive status of the eye.Chapter 1. In Borish's Clinical Refraction. (WJ. Benjamin,ed.), pp. 2-29. W B.Saunders.

Smith, E. L. (1991). Experimentally induced refractiveanomalies in mammals. In Refractive Anomalies. Researchand Clinical Applications (T.Grosvenor and M. C. Flom,eds), pp. 246-67. Butterworth-Heinemann.

Sorsby, A., Benjamin, B., Davey, J. B. et al. (1957).Emmetropia and its Aberrations. A Study in theCorrelationof the Optical Components of the Eye. Medical ResearchCouncil Special Report Series no 293. HMSO.

Sorsby, A, Leary, G. A. and Richards, M. ]. (1962).Correlation ametropia and component ametropia.Vision Res.,2, 309-13.

Sorsby, A, Sheridan, M., Leary, G. A. and Benjamin, B.(1960). Vision, visual acuity, and ocular refraction ofyoung men. Br. Med. ,., 9, 1394-8.

Stenstrom, S. (1948). Investigation of the variation andcorrelation of the optical elements of human eyes. PartI and Part III (translated by D. Woolf). Am. J. Optom.Arch.Am. Acad. Optom.,25, 218-32 and 340-50.

Stromberg, E. (1936). Uber Refraktion und Achsenlangedes menschlichen Auges. Acta Ophthal., 14, 281-93(cited by S. Stenstrom, 1948).

Troilo, D. (1992). Neonatal eye growth andemmetropisation - a literature review. Eye, 6, 154-60.

Wallman, ]. (1991). Retinal factors in myopia andemmetropization: Clues from research on chicks. InRefractive Anomalies. Research andClinical Applications (T.Grosvenor and M. C. Flom, eds), pp. 268-70.Butterworth-Heinemann.

Zadnick, K. and Mutti, D. O. (1998). Incidence anddistribution of refractive errors. Chapter 2. In Borish'sClinical Refraction. (W.J. Benjamin, ed.), pp. 30-46. W.B.Saunders.

power of a correcting spherical lens.Also called the spectacle refractionpower of a cylindrical component of anastigmatic or sphero-cylinder lenspower of the spherical component of asphero-cylinder lensvertex distance, i.e. the distance fromthe lens to corneal vertex (alwayspositive)object andprincipalcomponent)corresponding vergence (= n / I) of Irefractive indices (usually of object andimage space)radius of curvature of cornea (anteriorcorneal surface)refractive error measured as the cornealplane. Also called the ocular refractionorientation of a cylindrical lens (usuallythe direction of the cylindrical axis)azimuth angle for a sphero-cylinderlens

a

(J

r

Ln, n'

h

1, l'

8

Measuring refractive errors

Introduction

This chapter is concerned with techniques formeasuring refractive error. This is an area thathas received wide coverage in ophthalmictexts, particularly regarding the subjectivetechniques. The emphasis here is on princi-ples rather than on a detailed investigation oftechniques or instruments. For descriptions ofparticular commercial instruments, readerscan refer to texts such as Henson (1996),Rabbetts (1998) and Campbell et al. (1998).The techniques for determining refractive

error can be classified in three groups: thosethat are subjective, those that may besubjective or objective, and those that can beonly objective. Some of the objective methodsmay be automated.In subjective techniques, the patient makes

a judgement of the correct focus. In theobjective methods, the clinician or an instru-ment makes a judgement of the correct focus.The clinician makes the judgement in the caseof manual or visual instruments, but withautomatic optometers the clinician's role islimited to ensuring the correct alignment ofthe instrument. Automated instruments usean infrared source in the wavelength range80D-1000 nm, and the clinician is replaced byan electronic focus detector. These instru-ments have separate fixation targets, whichare designed to encourage relaxation ofaccommodation.Objective optometers rely upon the fact that

some of the light incident upon the patient's

fundus (including the retina, choroid andsclera) is reflected diffusely. Refractive error isdetermined either by measuring the vergenceof light leaving the eye after fundus reflection,or by adjusting the vergence of light enteringthe eye so that a focused image of a target isformed on the fundus. With respect toobjective refraction, we refer to the fundusimage rather than the retinal image.The measurement of refractive error has a

number of inherent problems and potentialsources of error. These are discussed later inthis chapter (Factors affecting refraction).

Subjective-only refraction techniques

Simple perception of blur

The majority of refraction techniques arebased on asking the patient to observe asuitable target, such as part of a letter chart,and make judgements about the best focus.

Conventional subjective refractiontechniquesConventional subjective refraction usuallyinvolves the patient reporting which of twoviews, seen through two slightly differentcombinations of lenses, gives a 'better' viewof the target. The lens combinations may beplaced as trial lenses into a trial frame, or aslenses in a refractor head (phoropter). Thesetechniques include 'fogging' (also 'fan and

68 Ima8c [annation alld refraction

(b) Hypermetropia

Power F

(8.1)

Re =-(1 + IF)F (8.1a)An alternative version of this equation isRe =-xF2 (8.1b)

For the optometer shown in Figure 8.1a, I isthe required end-point distance, F is thepower of the optometer lens, and d is thedistance between the lens and the eye. Thevergence of the image in the lens, measured atthe eye, is also the refractive error R . This isgiven by e

R = 1 + IFe 1(1- dF) - d

Such a design has several weaknesses. Theideal optometer should have the followingproperties:1. The refractive error should be linearly

related to the target displacement I fromthe lens, which allows easy and accuratecalibration. Equation (8.1) shows that R isnot linearly related to I for the arrangementshown in Figure 8.1.

2. The apparent size of the target should beindependent of its position so that there isno size variation to stimulate accommo-dation.

3. The refractive error measuring rangeshould be adequate.

4. The eye clearance (the distance from theback surface of the optometer to the eye)should be as large as possible so that theoptometer's proximity to the eye does notstimulate accommodation.The first two of these requirements are

satisfied by the Badal optometer (Badal, 1876)which is shown in Figure 8.2. It is essentiallythe same as the optometer shown in Figure8.1, but with the restriction that the eye mustbe placed at the back focal point F' of the lens.Wecan then substitute 1/F for d into equation(8.1) to obtain

T

(a) Emmetropiaf---l----d---l

OptometersAn optometer contains a target moving infront of a suitable optical system, which isplaced close to the eye. A simple optometerconsists of a target T and a single positivelens. The vergence of the image in the lensdepends upon the target position. Under theclinician's instruction, the patient moves thetarget towards the optometer lens from aposition at which it is blurred to a position atwhich it first appears sharp. This point istaken as the measure of refraction. If thepatient is emmetropic with relaxed accommo-dation, this point is at the front focal point F ofthe lens (Figure 8.1). For a hypermetrope, thepoint is farther away from the lens than is F.For a myope, it is closer. In the case ofastigmatism, the refraction can be found byusing a rotatable line target or a targetconsisting of a 'fan' of lines. The patient mustidentify the two positions at which one of thelines is in focus.

block'), the cross-cylinder technique, andbinocular balancing. As these techniques arecovered in considerable detail by texts such asthose by Michaels (1985), Grosvenor (1996),Rabbetts (1998) and Benjamin (1998), they arenot discussed further here.

Figure 8.1. Use of simple optometer in emmetropia,myopia and hypermetropia.

Power F~ - - -,,t--

tr~l----d = I/F-j

Figure 8.2. The simple Badal optometer.

Measuring refractive errors 69

where x is the target displacement from thefront focal point F.The apparent angular size () of a target of

physical size 1] is given by the equationo=1]F (8.2)

which is independent of target position.There are different opinions of the appro-

priate ocular reference position for a Badaloptometer, with candidates including thefront focal point of the eye, its front nodalpoint, and the entrance pupil. The authorsfavour the entrance pupil, because theconsistency of angular size occurs not only

when the target is in focus but also when it isblurred.A single lens Badal optometer may not

satisfy requirements of range and eyeclearance. While there is no theoretical upperlimit to the hypermetropic range of the singlelens Badal optometer, the maximum level ofmyopia occurs when the target is at theoptometer lens. Putting 1=0 into equation(8.Ia) or x=-I/F into (8.Ib) shows that themaximum myopic error measurable is -F, thenegative of the power of the Badal lens. If weincrease this power to improve the range ofthe instrument, there is a corresponding

(a)Badal lens

"~u~i~a~y_I~~, -F +FIIII

(b)

F 3F F

Figure 8.3.Variations of the Badal optometer:a. Badal optometer with an auxiliary system toincrease the negative power range.

b. Badal optometer with a second lens of equaland opposite power to the Badal lens toincrease the distance between the Badal lensand the eye. The auxiliary system is neededto provide a negative vergence range.

c. Badal lens system with external principalpoints to increase the range of theoptometer.

d. Badal optometer with a distant target and amovable auxiliary lens to overcome theproblem of poor resolution with computergenerated displays. The auxiliary lenspower and sign can be selected according tothe refractive error range required.

~u~i~a~yJe_n~ Badal lensII

(d)

DistanttargetT _--T---~~~----+

70 Jl1IageformaliOll alld rcfracrio»

decrease in eye clearance (11F), which isclearly undesirable.The negative range of the Badal optometer

can be extended by various modifications.One simple example is by the addition of amoveable auxiliary system consisting of thetarget and a positive lens, which may providea virtual target for the Badal lens (Atchison etai., 1995) (Figure 8.3a). Another alternative isto use a multi-lens Badal system. One suchsolution is the inverse telephoto designsuggested by Gallagher and Citek (1995). TheBadal optometer is combined with a lens ofequal and opposite power, with the secondlens placed in front of the Badal lens - thelenses being separated by the focal length ofthe Badal lens (Figure 8.3b). As the equivalentpower of a system of two lenses of power F1and F2, separated by distance t in air, is givenby

~+~-~~ ~~

the equivalent power for this arrangementwith the Badal lens of power F is-F + F - (1/F)(-F)F =F

- that is, the same as the power of the Badallens. The back principal plane of the systemthrough P' is at a distance 11F from the Badallens. To place the Badal lens at the back focalpoint of the eye, this point must be a distance21F from the Badal lens. The advantage ofsuch a system is that it increases the distancebetween the eye and the Badal lens (Gallagherand Citek, 1995).The two lenses do not give anegative vergence range because the frontfocal point F coincides with the negative lens,but this can be overcome in turn by using amoveable auxiliary system as shown in Figure8.3b. One other solution to increasing thenegative power range that avoids the useof the auxiliary system is to use a sym-metric system, in which the principal pointsare outside the system, as shown in Figure8.3c.In the Badal system and its modifications

described above, the target would usually beplaced close to the system. This requires thetarget to be small. The development of com-puter-generated displays, with the limitedresolution that this entails, requires a modifi-cation that allows a much larger target. Thiscan be achieved by using a fixed, distanttarget and a moveable auxiliary lens (Figure

8.3d). The scale in dioptres is linear withmovement of the lens (Atchison et al., 1995).Another variation of the use of the per-

ception of blur to estimate refractive error is insome telescopic arrangements. Von Graefeused a Galilean telescope, varying the separ-ation between the objective and eyepiece toalter the vergence of the emergent light(Rabbetts, 1998). Moving the eyepiece awayfrom the eye produces a more convergentbeam, while moving the eyepiece towards theobjective produces a more divergent beam.The former movement can be used to measurehypermetropia, and the latter movement tomeasure myopia. The telescopic tube can begraduated to show the effective power as theeyepiece is moved. Unfortunately the scale isnot linear, and the angular subtense of theimage at the eye varies with the position of theeyepiece. Dudragne (1951) described atelescope optometer in which a +200 lens,placed at its focal distance from the eye, wascombined with a moveable -20 0 lens. This isthe same as the Badal optometer variationdescribed above and in Figure 8.3d.

Laser speckle

If visible laser radiation (i.e, highly coherentlight) is incident on a diffusely reflectingsurface, an observer sees a speckle patternthat appears to move as the head moves. Thereflected beam forms an infinite number ofFresnel diffraction (speckle) patterns atdifferent distances from the surface, one ofwhich is conjugate to the retina. If the patternconjugate to the retina is not at the reflectingsurface, any head movement produces aparallax effect. The rate of movement of thespeckle pattern depends upon the position ofthe surface relative to the position at whichthe eye is focused. The rate of movementincreases as the discrepancy of focusincreases, and its direction depends upon thedirection of discrepancy - a myope sees thespeckle pattern moving in the oppositedirection to head movement, while ahypermetrope sees the speckle patternmoving in the same direction as the headmovement. When the surface and the plane offocus coincide, the pattern does not appear tomove in any specific direction.This phenomenon has been used in laser

Measuring refraciioe errors 71

Figure 8.4. The laser optometer.

optometers (Figure 8.4).Instead of moving thehead, the 'target' is moved by using theslowly rotating surface of a drum. Somecorrection needs to be made for the fact thatthe 'plane of stationarity' is inside rather thanon the drum surface. Charman (1979) showedthat 5, the distance of the plane of stationarityfrom the drum rotation axis, is given in termsof did (which is the inverse of the laser beamdivergence at the drum), the radius ofcurvature of the drum r and the angle esubtended by the laser and the eye at thedrum, by the equation

5 = r[dldcos(</» + rcos2(</»] (8.4)dld[1 + cos(</»] + rcos(</»

Laser optometers have had wide researchapplication. They allow the refractive error tobe measured in dim conditions, and thusenable the resting point of accommodation tobe determined easily.

Longitudinal chromatic aberration ofthe eyeThe eye has approximately 2 0 of longi-tudinal chromatic aberration between thewavelengths of 400 nm and 700 nm (seeChapter 17). If a small white light is viewedthrough a piece of cobalt glass, only red andblue light are transmitted, with the blue lightfocused in front of the red light inside the eye(Figure 8.5). If the light is a long distanceaway, an emmetrope sees a purple disc,because the red and blue images are approxi-mately equal in size. A myope of moderatedegree (say 2 D) sees a red central spotsurrounded by a blue annulus. Conversely, ahypermetrope sees a blue central spot sur-

rounded by a red annulus. The strongestpositive or weakest negative lens placed closeto the eye that reduces the coloured fringes toa single purple disc gives the refractivecorrection. With appropriate modification intechnique, astigmatic refractive errors can becorrected.The cobalt filter is rarely used, but the eye's

chromatic aberration is often used in theduochrome test, in which the patientcompares the sharpness of letters presentedon red and green backgrounds. This techniqueis used often near the end of a refraction.Refraction is adjusted until the letters on thetwo coloured backgrounds appear equallyclear, or until the letters on one of thebackgrounds are slightly clearer.

Subjective/objective refractiontechniques

Remote refraction and relay systemsPositioning equipment close to the facemeansthat the clinician is not able to see the patient's

EmmetropianHypermetro ia Myopia

Cobalt filter ~. ~\ \\ \

Figure 8.5. Chromatic optometer, showing theappearance of a distance spot viewed by emmetropes,myopes and hypermetropes.

72 Image formation alldrefraction

face. The patient may also feel uncomfortable.The proximity of the equipment may induce'instrument' myopia in some susceptibleindividuals. To overcome these problems,some instruments use an auxiliary lens ormirror system to image a remote correctinglens at the eye. The auxiliary system is thus arelay system.Figure 8.6a shows a correcting lens of

power F, which is imaged by an auxiliary lensof power Fa to the eye. The target at T for thecorrecting lens may be the real object, or itmay be the image of the real target, in whichcase it may be referred to as the 'intermediate'target. A concave mirror may replace theauxiliary lens (Humphrey, 1976;Bennett, 1977;Alvarez, 1978).Another system that might be considered to

be remote refraction uses a -Ix telescopeconsisting of two lenses of equal powerseparated by twice their focal lengths. Thecorrecting lens is placed at the first focal pointof the first telescope element, and the eye isplaced at the second focal plane of the secondtelescope element (Figure 8.6b).A full description of the imaging properties

of these relay systems is beyond the scope ofthis book.

Auxiliary lensF t;

~.(a)

(b)

Figure 8.6. Remote refraction systems:a. Remote refraction with an auxiliary lens.b. Remote refraction with a -Ix telescope system.

Split image and vernier acuity(coincidence method)

The target using this method has at least onestraight edge, which is split into two.Defocusing moves one part of this edgesideways relative to the other part, so that thepatient (or clinician) sees the straight edgeseparated into two. When the system iscorrectly focused, the two parts are aligned(that is, in coincidence). The principle workswell because the patient (for subjectiveinstruments) or the clinician (viewing lightreflected from the fundus for objectiveinstruments) is very sensitive to verniermisalignment.A subjective instrument using the coinci-

dence principle is the polarizing optometer(Simonelli, 1980). A pair of crossed polarizedfilters is placed over the target, and anotherpair is placed over the pupil of the eye. Thesecond pair is cut at right angles to the firstpair - for example, if the first pair join alongthe horizontal meridian, then the second pairjoin along the vertical meridian (Figure 8.7).The beam from half of the target enters onlyhalf the pupil, and the beam from the otherhalf of the target enters the other half of thepupil. The target is moved until its two halvesappeared aligned.

Scheiner principle

When an unaccommodating, emmetropicpatient views a target through a disc with twoholes in it, the target appears doubled if it isbeyond the focal point of the lens (Figure8.8a). If the pinholes are aligned vertically, theupper hole corresponds to the spot appearinghigher up (lower retinal projection). When thetarget is placed closer than the focal point ofthe lens, the target again appears doubled,with the upper hole corresponding to the spotappearing lower down (Figure 8.8b). Whenthe target is placed at the focal point of thelens, the target appears single (Figure 8.8c).A simple optometer incorporating this

doubling or Scheiner principle consists of amoveable target, which is usually a small lightspot, and a Scheiner disc, which is an opaquedisc with two pinhole apertures about 2 mmapart. The patient moves the target towards

Entrance pupil(with polarized mask)

\.._---~ --------~)YEye

Exit pupil

Measuring refractive errors 73

~fOC"~plane(retina)

Figure 8.7. Coincidence method of focusing using polarized filters.

the optometer lens from a position at whichthe target appears double to a position atwhich it appears single. This position is themeasure of the refraction. For astigmatism,the axis of the two image points coincideswith that of the pinholes only when thepinhole axis coincides with one of theprincipal meridians.The Scheiner principle has been incorpor-

Figure 8.8. Imagery of a point source for anunaccommodating emmetrope using a Scheiner discoptometer:a. Target beyond far point of lensb. Target closer than focal point of lensc. Target at focal point of lens.

ated into some automated optometers. In oneconfiguration, two infrared sources replacethe two pinholes and are imaged in the planeof the patient's pupil. The target is a moveablediaphragm. Two blurred images of the dia-phragm are produced at the fundus atdifferent positions, one from each of the twoinfrared sources, except when the diaphragmis conjugate with the fundus. Radiation fromthe fundus passes out of the eye to a detectorunit, which senses this difference in positionand controls the movement of the target sothat only one fundus image is formed.

Objective-only refraction techniques

RetinoscopyRetinoscopy is probably the most commonmethod of measuring refractive error. In thistechnique, the fundus of the eye acts as ascreen over which a spot or streak of light ismoved by tilting an instrument called aretinoscope back and forth. The clinicianwatches the shape and movement of the patchof reflected light within the patient's pupil.The patch is known as the 'reflex'. By placingtrial lenses in front of the patient's eye, thespeed of movement of the reflex is modifieduntil it moves infinitely fast (or as close to thisas can be judged). This condition is known as'reversal'. At this position, the fundus of thepatient's eye is conjugate with the sight holeof the retinoscope, so that there is an almostinstantaneous cut-off of the return beam

74 Imagejormation alit! nfractioll

Parallax movement between objectand image

entering the examiner's eye as the light patchmoves over the patient's fundus. As this is awell-known clinical technique, we referreaders to texts such as Grosvenor (1996) andRabbetts (1998).

Time or mask displacement(b) In focus

Figure 8.tO. The grating focus principle. See text fordetails.

D etinal image

Correcting d-,.;,· (blurred)lens~:

Target ~ -: . . . . UMask[lr-_ ii~::"!" nnn ~'1 n ~ ~~..-' ..-' UUU~--,lJUi \

-.J Aerial image (blurred)

}k·~~--·/~Time or mask displacement

(a) Out of focus

ORetinal imageCorrecting , .:«: J!,..!': (focused)lens~

Target " TtaskA . I' ~::~ ::~~~:~~Uena Image ~;~; "W, "(focused)

Grating focus

The grating focus principle is similar to theperception of blur, but here an automatedoptometer analyses the intensity of a signal todetermine refractive error. The object is asquare wave infrared intensity pattern(grating) (Figure 8.10). This is imaged into theeye. Radiation reflected by the fundus isimaged on a photodetector through a squarewave grating mask. Either the mask or the

object T and its aerial image T2' laterallydisplaced. Moving T along the axis of theinstrument alters S2' and the angle ofincidence of the ray through T, causing Tt ' tomove across the fundus. At the same time,image T2' moves laterally. When T is focusedon the fundus, Tl' is on the axis of the system,and T2' coincides with T. Thus T2' appears tomove in parallax as T is moved back andforth.

8;

T~ clinician T~

To clinician

s

(b)

The instrument used for this technique is amodification of the single lens indirectophthalmoscope (Figure 8.9).The image S2' ofthe intermediate (off-axis)source SI' is formednear the edge of the patient's pupil. A testobject T on the common optical axis of thepatient's eye and the optometer occludes theilluminating beam along the ray pathTS2'Tt'SI'.Viewing at an angle to the axis of

illumination, the clinician sees the original test

Figure 8.9. The parallax optometer (based on Bennettand Rabbetts, 1989,Figure 18.3).a. Target T is not conjugate with its fundus image at T t ' .The clinician sees T2' displaced laterally relative to T.

b. Target T has been moved so that it is now conjugatewith fundus image Tt'- The clinician sees T2' alignedwithT.

(a)

Automated optometersRetinoscopy has been modified for use insome automated optometers, with a detectorsystem with two photodetectors replacing thesight hole and the clinician. Refractive error isdetermined by the phase shift between thesignals reaching the two detectors from thefundus.

target moves transversely, with the formersituation being shown in Figure 8.10. Thespatial frequency of the mask and the aerialimage are matched. The signal from thephotodetector modulates with the mask (ortarget) movement. The maximum modulationoccurs when the aerial image is focused at themask - this corresponds to the object beingfocused at the fundus (Figure 8.lOb). Thesignal modulation is monitored as anoptometer lens moves along the optical axis.An estimate of the refraction is given by theoptometer lens position corresponding to themaximum modulation. To obtain the fullrefractive state of the eye, measures are madein a number of meridians.

Photography

The use of photography for determining therefraction of the eye is termed photo-refraction (Howland and Howland, 1974). Itsmain application is screening of infants andyoung children. A flash photograph is takenof the eyes, with the flash source near theplane of the camera. The size and location ofthe pupil reflex recorded by the cameraindicate the degree and direction of refractiveerror. Variations of the technique includeorthogonal, isotropic and eccentric photo-refraction (Campbell et al., 1998).In its simplest form, known as isotropic

photorefraction (Howland et al., 1979),a smallflash source of light is mounted in front of acamera with a lens system such that one orboth eyes are imaged with the pupil largeenough to allow analysis. The source isimaged on the fundus of each eye. If the eye isfocused at the source, the light leaving the eyereturns to the source and is occluded from thecamera lens. Thus the pupil appears dark.When the eye is not focused at the source, ablur circle (in spherical errors) or an ellipse (inastigmatism) is formed at the fundus. In tum,this produces an illuminated zone around thesource, so that light enters the camera. Thepupil appears illuminated, with the size of thefilm image depending upon the refractiveerror relative to the plane of the source. Thesign of the refractive error cannot bedetermined by this basic technique, soHowland and co-workers took photographswith the camera focused both in front of and

Measuring rejractioe errors 75

behind the patient's pupil plane. Using simplegeometrical optics, the refractive error can bedetermined from the film image sizes, thepupil size and other dimensions of the camerasetup.The refractive error range that can be

measured by the isotropic method is limited.Larger errors can be measured with eccentricphotorefraction, which is similar toretinoscopy. In this method, an eccentric pointsource of light is used. Some of the lightreflected from the fundus is vignetted by thecamera, and this produces an image shapedlike the profile of a biconvex lens. Inhypermetropia relative to the light source, theimage is on the side of the pupil opposite tothe flash; myopia relative to the light sourceproduces an image on the same side of thepupil as the source. From the placement of thecamera, flash and patient, the refractive errorcan be determined. For measurement ofastigmatism errors, different meridians mustbe investigated. Bobier and Braddick (1985),Howland (1985) and Crewther et al. (1987)evaluated this technique. More recently, thetechnique has been used with Videotape ordigital cameras replacing film (Campbell et al.,1998). This has the advantage that the imagescan be viewed immediately, and retaken ifnecessary.

Visual evoked responseThe visual evoked response (VER) is theresponse of the visual cortex to visualstimulation, primarily reflecting activity at thecentral retinal area. An active electrode isplaced on the occipital scalp region, and aninactive electrode is placed in another part ofthe scalp. Pre-amplifiers, recorders, and acomputer averaging technique are used toobtain electrical responses. With appropriatetargets, the amplitude of the VER dependsupon the refractive status of the eye. It is auseful, but not precise, method for patientswho cannot be assessed by more conventionalmethods.

Factors affecting refraction

The reliability of subjective refraction is about±O.3 0 in young patients with good visual

76 III/age formatiol/ and refraction

acuity (Jennings and Charman, 1973;Rosenfield and Chiu, 1995). Rounding pre-scriptions off to the nearest 0.5 D step had noeffect on their acceptability to Appleton'spatients (Appleton, 1971). Refraction and itsreliability are influenced by various target,optical and neural factors.

Targetfactors

Different refraction methods may givesystematically different results because of thedifferent conditions under which the refrac-tions are determined - for example, differentluminances, spatial frequency distributionsand spectral distributions. These interact withocular factors such as aberrations and pupilsize.

Optical factors

Subjective depth-of-field decreases, orsubjective reliability increases, with increasein pupil diameters up to approximately 5 mm,and then remains fairly constant. This isdiscussed further in Chapter 19.Many automated optometers have a mini-

mum pupil size requirement; the pupil needsto be a certain size to pass the radiation fromthe stimulus into and out of the eye, and theelectronic signals provided by the detectorsneed to have sufficient strength to bedistinguished from background noise.Monochromatic aberrations, particularly

spherical aberration, may affect refractionthrough their interaction with pupil size andtarget factors. With large pupils, sphericalaberration may cause the refraction to bedependent upon target spatial distribution,with positive spherical aberration causing therefraction to move in the myopic direction fortargets having considerable low spatialfrequency components. This is discussedfurther in Chapter 15. Large monochromaticaberrations can decrease the reliability ofrefraction techniques through their influenceon depth-of-field,Refraction is influenced by the spectral

distribution of the target, through its influenceon accommodation. For objective refraction,refraction is also influenced by the spectraldistribution of the radiation reflected by the

fundus of the eye. This is a consequence oflongitudinal chromatic aberration, which isdiscussed in Chapter 17.

Eccentric viewing

Retinoscopy is usually performed at a smallangle to the visual axis. Provided this angle iswithin 10°, the peripheral power errorsintroduced should be less than 0.25D (seeChapter 15). When there is eccentric fixationor heterotropia, the clinician may need todirect the patient's gaze to the mostappropriate axis.

Site offundus reflectance in objectiverefraction

This is affected by the spectral distributionand by any polarization of the radiationsource. The longer the wavelength, the deeperinto the retina and choroid are the mainreflecting layers.

AccommodationIt is important for the eye's accommodation tobe relaxed during refraction. This may beattempted by the following methods:1. Placing positive lenses in front of the non-

tested eye so that accommodation blurs theimage.

2. Ensuring that the axes of both right and lefteye channels are parallel in instruments sothat convergence cannot stimulate accom-modation.

3. Using a blue fixation target, whichprovides a reduced stimulus to accommo-dation because the eye has greater powerfor blue light than for other visiblewavelengths.

4. Using cycloplegic drugs.

Maximum potential visual acuity

Attenuation of high spatial frequency contrastby defocus is generally greater than that oflow spatial frequencies. Thus, high visualacuities may give a greater reliability of

subjective refraction than lower visualacuities.

Discrepancies between subjective andobjective refraction

Discrepancies occur between objective andsubjective measures of refraction if the mainreflecting layers for the radiation used inobjective refraction do not coincide with thesubjectively preferred focal plane, which wemay expect to be at or near the photo-receptors. The dominant reflecting layersdepend on the wavelength of radiation used,whether the radiation is polarized, and theage of patients.A major problem with manual optometers

is that there is not sufficient control over thepatient's accommodation. The targets arevisible to the patient, and the stimulus toaccommodation is altered as vergence fromthe target is altered.The disadvantage of the near-infrared

sources used in automated optometers is thatthere is considerable choroidal scattering ofthe radiation because the retinal pigmentepithelium is fairly transparent at these wave-lengths. This gives the reflection site increaseddepth and size. Because of the longitudinalchromatic aberration of the eye, the eye ismore hypermetropic for the near-infraredwavelengths than for visible wavelengths.This is counteracted to some extent becausethe infrared wavelengths penetrate furtherthan visible wavelengths into the retinal andchoroidal layers. To reduce these sources oferror, manufacturers of automated opto-meters calibrate their instruments usingsubjective refractions.

Summary of main symbols

Re refractive error1 distance from lens to objectd distance from lens to eyeF power of lensFa power of auxiliary lensx distance from front focal point of

lens to object() angular subtense of image of target

at eye

Measuring refractive errors 77

t distance between lenses11 object sizer radius of curvature of drum (laser

speckle optometer)did distance between laser and drums distance of plane of stationarity

from drum axisifJ direction of laser beam from line of

sightF, F' front and back focal points of lens

or optical systemP, P' front and back principal points of

lens or optical system5,51' 52' positions of source and its

conjugatesT, Tl ' T2' positions of target and its images

References

Appleton, B. (1971). Ophthalmic prescription in half-diopter intervals. Patient acceptance. Arch. Ophthal., 86,263-7.

Atchison, D. A., Bradley,A, Thibos, L. N. and Smith, G.(1995). Useful variations of the Badal Optometer.Optom. Vis. Sci., 72, 279-84.

Alvarez, L. W. (1978). Development of variable-focuslenses and a new refractor. J. Am. Optom. Assoc., 49,24-9.

Badal,]. (1876). Optornetre metrique international du DrBadal. Pour la mesure simulanee de la refraction et del'acuite visuelle meme chez les illetres. Ann. Ocul., 5,101-17.

Benjamin, W. ]. (1998). Monocular and binocularsubjective refraction. Chapter 19. In Barish's ClinicalRefraction (I. Borish and W. ]. Benjamin, eds), pp.629-723. W. B.Saunders.

Bennett, A G. (1977). Some novel optical features ofthe Humphrey Vision Analyser. Optician, 173(4481),7-16.

Bennett, A G. and Rabbetts, R. B. (1989). Clinical VisualOptics, 2nd edn., p. 423.Butterworth-Heinemann.

Bobier, W. R. and Braddick, O. ]. (1985). Eccentricphotorefraction: optical analysis and empiricalmeasures. Am. J. Optom. Physiol. Opt.,62, 614-20.

Campbell, C. E., Benjamin, W. ]. and Howland, H. C.(1998). Objective refraction: retinoscopy, autorefraction,and photorefraction. Chapter 18. In Barish's ClinicalRefraction (w. ]. Benjamin, ed.), pp. 559-628. W. B.Saunders.

Charrnan, W. N. (1979). Speckle movement in laserrefraction. I. Theory. Am. ]. Optom. Physiol. Opt., 56,219-27.

Crewther, D. P.,McCarthy, A., Roper,]. and Costello, K.(1987). An analysis of eccentric photorefraction. C/in.Exp. Optom., 70, 2-7.

Dudragne, R. A. (1951). Optometre a variation continue

78 Image formation aIId refraction

de puissance. lnternational Optical Congress 1951, pp.286-98. British Optical Association.

Gallagher, J. T. and Citek, K. (1995). A Badal opticalstimulator for the Canon Autoref R-1 optometer.Optom. Vis. Sci.,72, 276-8.

Grosvenor, T. P. (1996).Primary Care Optometry: Anomaliesof Refraction and Binocular Vision, 3rd edn., pp. 285-307.Butterworth-Heinemann.

Henson, D. B. (1996).Optometric Instrumentation, 2nd edn.Butterworth-Heinemann.

Howland, H. (1985). Optics of photoretinoscopy: resultsfrom ray tracing. Am. j. Optom. Physiol. Opt., 62,621-5.

Howland, H. C. and Howland, B. (1974).Photorefraclion:A technique for study of refractive state at a distance.j.Opt. Soc. Am., 64, 240-49.

Howland, H. C, Atkinson, J. and Braddick, O. (1979). A

new method of photographic refraction of the eye. j.Opt. Soc. Am., 69, 1486.

Humphrey, W. E. (1976). A remote subjective refractoremploying continuously variable sphere-cylindercorrections. Opt. Engr., 15, 286-91.

Jennings, J. A. M. and Charman, W. N. (1973). Acomparison of errors in some methods of subjectiverefraction. Ophthal. Opt., 13, 8-11,18.

Michaels, D. D. (1985). Visual Optics and Refraction: AClinical Approach, 3rdedn. Mosby.

Rabbetts, R. B. (1998). Bennett and Rabbetts' Clinical VisualOptics,3rdedn. Butterworth-Heinemann.

Rosenfield, M. and Chiu, N. N. (1995). Repeatability ofsubjective and objective refraction. Optom. Vis. Sci., 8,577-9.

Simonelli, N. M. (1980). Polarized vernier optometer.Belzat>. Res. Meth. lnst., 12, 29~.

9Image formation: the defocused paraxial•Image

Introduction

In Chapter 6 we examined image formationfor the focused eye. However, often the eye isnot focused on the object of interest. This maybe because of an uncorrected refractive error,Also, when there is a poor stimulus - such asa bright empty field or in low luminances -the accommodation system tends to settletowards an intermediate resting state or tonicaccommodation level (Rosenfield et al., 1993).This corresponds to about 1.5D of accommo-dation for habitually corrected young people,The effect of such focus errors on vision is

important. A focus error affects the quality ofthe retinal image, and hence visualperformance (see, for example, Westheimerand McKee, 1980; Simpson et al. 1986;Pardhan and Gilchrist, 1990). Of particular

clinical importance is the effect of defocus onvisual acuity (Prince and Fry,1956; Atchison etal. 1979; Simpson et al. 1986; Smith, 1991and1996). Its effect on image size is ofconsiderable interest (see, for example, Pascal,1952; Marsh and Temme, 1990; Smith et al.,1992), To fully understand the effect ofdefocus in these situations, the optics ofdefocused vision must be understood, andthis is developed in this chapter.Let us consider the situation shown in

Figure 9,1, which shows the eye focused to theplane at R, but observing an object point Q inthe perpendicular plane at O. Following abeam of rays from Q through the pupils, thisbeam focuses at the point Q' on the per-pendicular plane at 0' in front of the retina,but continues to the retina, Thus, the retinalimage Q' of this object is out of focus. If we

/1

1Figure 9.1.The formation of the blurred image of a distant point by an eye focused at somefinite distance.

80 Image jormation and refraction

ignore the effects of aberrations and diffrac-tion/ the light distribution of Q/ on the retinais uniform and the area illuminated is theprojection of the exit pupil at E', through Q/onto the retina at Qb/. Therefore, the beamcross-section on the retina has the same shapeas the pupils. It is assumed that the normalpupil of the eye is circular, and therefore thelight distribution at Qb' is called the defocusblur disc for the object point Q.Similarly, thecircle centred on R' is the defocus blur disc forthe object point O.The centre of the defocus blur disc is at Qb' ,

and this is the intersection of the paraxialpupil ray (the line of sight) with the retina.We can see from Figure 9.1 that the retinalimage of the object OQ is blurred ordefocused, and it is a different size from thefocused image O/Q/ formed in front of theretina. In this chapter, we explore the effect ofsuch a defocus on the apparent size of theblurred retinal image and the size of thedefocus blur disc. We start with the retinalimage size of the object OQ.

Retinal image size

Consider the points 0 and Q shown in Figure9.1 as the ends of a line object. Each point onthis line is imaged as a defocus blur disc(assuming a circular pupil), but only the blurdiscs for the points 0 and Q are shown. Wecould define the size of the blurred image ofOQ as being measured from the bottom of theblur disc centred on Ob' to the top of the blurdisc centred on Qb/. However, with thisdefinition the de focused image size depends

'1

1

upon pupil size, and it is preferable to have adefinition that is independent of pupil size. Adefinition that satisfies this criterion is theimage size measured from the centres of thedefocused blur discs at Ob' and Qb', that is,the distance Tlb' where the pupil rays from 0and Q meet the retina.In this case of a defocused retinal image we

should not use the nodal ray to determine theimage size, as we did in Chapter 6 for focusedimages. The nodal ray is not the central ray ofthe beam, and may not be part of the image-forming beam. It can be seen in Figure 9.1that, if the pupil diameter is small enough, therays through the nodal points are blocked anddo not reach the retina. The likelihood of thishappening increases as Q moves further awayfrom the axis. When we investigate the size ofa defocused image, we should use pupil raysonly. We can find equations for this image sizeand its magnification relative to the focusedimage.

The size of the defocused image

Figure 9.2 shows the eye focused on the pointO2/ but observing the scene at the plane at 0 1,From this figure, the size Tlb' of the defocusedimage isI1b' =uz/Ez'R' (9.1)Now the angles uz/and Uz shown in Figure 9.2are connected by equation (5.7). Thereforeuz/=mzuz (9.2)

with the value of the constantmz dependingupon the actual schematic eye used, in

Figure 9.2. The retinal image and perceived angular sizes of defocused images.

(9.7a)

(9.7b)

particular the equivalent power and positionsof the pupils and principal planes. Combiningthe above two equations gives1]b' = mzUzEz'R' (9.3)whereliz =-1]/01Ez (9.4)is the angular size of the object, measured atthe entrance pupil. Equation (9.3) shows thatthe retinal image size (whether focused or not)is proportional to the angular size of the objectmeasured at the entrance pupil at Ez.More meaningful than the defocused image

size 1]b' is its size relative to the size 1]' of theimage if it were in focus. This ratio is themagnification MM =1]b'/1]' (9.5)The size of focused images was discussed inChapter 6, and equation (6.6) is relevant. Herewe have1]' =m1u1E1'R' (9.6)where the subscript '1' refers to the quantitiesmeasured for the eye focused on the point 01instead of Oz. Equation (9.5) can now bewritten as

mzuzEz'R'M = E'R'm1ul 1

Using equation (9.4), we can express thisequation in terms of the distance dJ and dz ofthe focused planes from the corneal vertex as

mZ(dl + VE1)Ez'R'M =m1(dz + VEz)E1'R'

llIlage forlllatiol1: tile dejocused paraxial image 81

As the eye changes its refractive state, there isa change in entrance and exit pupil positions.This in turn leads to changes in most of theabove quantities.

An eye focused at a finite distance,looking at an object at infinityBefore deriving an equation for the expectedchange in the image size, we will examine theoptics of this process. As the eye accommo-dates, the equivalent power increases and,according to equation (6.11), the in-focusimage size must decrease. However, becausethe object is at infinity, this focused image isformed in front of the retina as shown inFigure 9.3. Therefore, we may expect that theperceived image also decreases in size.However, in Figure 9.3, the height of theretinal image is set by the position where thepupil ray intersects the retina, and thisposition is higher than the image point Q'.To determine whether the observed image

actually decreases or increases withaccommodation, it is possible to derive asuitable equation directly from first principlesusing Figure 9.3. Equation (9.3) still applies,and for the focused image we can useequation (6.11), with the angle (J replaced bythe angle u1 and the power here written as Fe'Thus1(= u1/ Fe (9.8)

Since the object is at infinity, Ul = uz, and if wenow substitute for 1]b' from equation (9.3) and

6 -~Paraxial pupil ray

Figure 9.3. The retinal image and perceived angular sizes of defocused images(as in Figure 9.Z),but with the object at infinity.

82 Image [ormation and refraction

for 1"1' from the equation (9.8), equation (9.5)becomesM =m2E2'R'Fe (9.9)

Example 9.1: Compare the retinal imagesizes of the moon for the two extremestates of accommodation of theGullstrand number 1 eye.Solution: Equation (9.9) is used, with thefollowing data from Table 5.1,For the relaxed eye: Fe = 58.636100 0For theaccommodated eye: m2 = 0.795850

and E2'R' =21.173464 mm

Substituting these values into equation(9.9) givesM = 0.795850 x 0.058636100 x 21.173464

=0.98807The value of 0.98807 correspond to a 1.2% decrease in image size.Thus, if the eye accommodated by 10 0

while looking at the moon, the moon'simage would decrease by only 1.2%,showing that the retinal image size isremarkably stable with focus error.

The use of artificial pupilsThe results of this section show that the effectof a focus error on retinal image size dependsupon the position of the entrance pupil of theeye. In some experimental situations, the eye'snatural pupil is dilated and artificial pupilsare placed close to the spectacle plane about12-15 mm in front of the eye. The artificialpupil becomes the effective aperture stop andthe entrance pupil of the eye, and its positionand size affect the size of defocused images.

Size of the defocus blur disc

The geometrical aberration-freedefocus blur disc

Calculations of the diameter of the defocusblur disc, whether by physical or geometricoptics, have traditionally used schematic eyemodels (e.g. van Meeteren, 1974; Charman

and Jennings, 1976; Obstfeld, 1982). Thesecalculations use a schematic eye with a givenrefractive power (often that of Gullstrand'snumber 1 model eye) and a specific pupildiameter. Appropriate rays are traced to theretina to determine the diameter of thedefocus blur circle on the retina. The expectedor perceived angular diameter of this blur discis then found by calculating the angulardiameter of the retinal image sub tended at theback nodal point. This method has thedisadvantage that schematic eyes are onlyspecified in the relaxed or in a greatlyaccommodated state, usually about 10 0, andthus defocus calculations at other levels ofaccommodation are not readily found.

If the retinal image diameter is not required,the expected perceived angular diameter canbe found by a very simple equation which,although approximate, may be accurateenough for many circumstances. Smith (1982)showed two derivations of this equation, thesecond and simpler of which is presentedhere.In Figure 9.4, an eye with its front principal

plane at P is focused on a plane at R a distancelR from P. If the eye views a point 0 atdistance 1while still focused on R, the point 0appears as a blur circle or disc superimposedon the plane at R. The longitudinal focus error01 is01 =l-IR (9.10)and the physical linear diameter 1/1, of the blurcircle using similar triangles can be shown tobe1/1,= DpOl/I=Dp(l-IR)/1 (9.11)

where Dp is the diameter of the beam at theprincipal planes. The expected perceived orvisual angular diameter (/>of this disc must bemeasured at the front nodal point N, as shownin the figure. Thus,(/>=iPt,/(lN -IR)that is

(/>= Dp(lR -1) (9.12)1(IN -IR)

In terms of vergences, this equation can bewritten

(9.13)

Image [ormation: thedefocused paraxial image 83

Cornea

o

I:Figure 9.4.The formation of the perceived diameter of the defocus blur disc.

Figure 9.5. The difference between beam diameter at thefront principal plane and the entrance pupil.

The sign is not important since the angle tPcorresponds to a diameter.The beam diameter Dp is not in general the

diameter D of the entrance pupil. The relationbetween these two quantities can be found byreferring to Figure 9.5. From this figure, thetwo quantities are related by the equationD =o.u :T)IIthat is,Dp = D/(I- LT) (9.14)where T is the distance between the frontprincipal plane and the entrance pupil. Thisdistance depends upon the level ofaccommodation, and is 1.7mm in the relaxedGullstrand number 1 eye and 0.9mm in theaccommodated version. If Dp in equation(9.13) is replaced by D using equation (9.14),then

For low to medium levels of refractive error,we can make some useful approximations asfollows:1. IfT is neglected the fractional error is Til,

and its value depends upon the level ofaccommodation and the distance I of theout-of-focus object. However, if thisdistance is infinite, the error is zero.

2. The quantity IN can be neglected if thequantity LR/N« 1.While INdepends uponthe equivalent power of the eye and henceaccommodation level, its value is about 5mm. For a value of LR of 100, the errorinduced in neglecting this term is about5%,but for LR =0, the error is zero.If we accept these two approximations,

equation (9.15) reduces totP= (LR - L)D (9.16)

If (LR - L) is replaced by the focus error L1L, theabove equation can be re-expressed astP=LiL D radian (9.17)which expresses the visual angular subtensetP of the defocus blur disc in terms of thedioptric level of defocus L1L and the pupildiameter D. Thus, within the limits of theapproximations made in deriving thisequation, the angular diameter of the defocusblur disc is independent of the constructionalparameters of the schema tic eye and theacccommodation level.

If we express the pupil diameter in milli-metres and defocus in dioptres and tP inminutes of arc, we have

(9.17a)tP=3.483L1L Dmm min. arc

(9.15)

o

,-

84 Image[ormation and refraction

Experimentally determined angulardiameter of the blur discs

Defocus ratio

The level of defocus can be quantified by thediameter t/Jt,' of the defocus blur disc in theimage, the diameter t/Jt, of the defocus blur discin object space, or its corresponding angularsize eP. Alternatively, if we want to study theeffect of a defocus on the visibility of an objectof a certain size eP, a more appropriatequantity is the defocus ratio, which we defineby the equationDefocus ratio = t/Jt, '/1}'=etv'1J (9.18)In angular terms in object or visual space wehaveDefocus ratio = eP/() (9.19)where ()is the angular size of the object.

separation between the sources is varied untilthe two blur discs appear to just touch, andthe angular diameter of one blur disc is thenequivalent to the angular separation of thesources. At low defocus levels the blur disctakes on an irregular star-shaped pattern,which is due to ocular aberrations, making itdifficult to locate the edge of the blur discs. Athigh levels of defocus, where the defocusdominates over the ocular aberrations, thedefocus blur discs are more regular.Chan et al. (1985) used this technique to test

the accuracy of equation (9.17). Six observerswere used, and since the differences betweenobservers were small, the mean data areshown in Figure 9.6. The spectacle plane lenspower was converted to an equivalent powerat the front principal plane of the eye. There isgood agreement between theory andmeasurements at intermediate levels ofrefractive error (3-9 D). However, at lowerrefractive errors the theoretical values are toolow, while at higher refractive errors thetheoretical values are too high. At the lowerlevels of refractive error, the discrepancy isbecause of the effects of ocular aberrationsand diffraction, which give a wider spread oflight than expected by aberration-freegeometrical optics. At the higher levels ofrefractive error, the deviations can beexplained by equation (9.17) being onlyapproximate. The more accurate equation(9.15) agrees wen with the measured values atthese levels.,,,

2.0 mm, experimental

2.0 mm, equation (9.17a)4.2 mm, equation (9.17a)5.8 mm, equation (9.17a)

4.2 mm, equation (9.15) " .," .,

4.2 mm, experimental ,'"

5.8rom"":?'..//

2 3 4 5 6 7 8 9 10 II 12Defocus (D)

oe

•

.',,~'

;", 0 06 .

• ' ..0'",9.:.:.....0 ........0 ......

oo .....

50

150

100

Figure 9.6. The diameter (/>of defocused blur discsmeasured by Chan et al. (1985),corresponding valuespredicted by the approximate equation (9.17a),andvalues for a 4.2mrn diameter pupil using the exactequation (9.15),with L=O.

Example 9.2: Calculate the defocus blurdisc size for an object at infinity viewedby aID uncorrected myopic eye with a 4mm diameter pupil.Solution: We substitute L!L = 1 and Dmm =4 into equation (9.17a), to giveeP=3.483 x 1 x 4 =13.8min. arc

The accuracy of equation (9.17) relative to themore accurate equation (9.15) is shown inFigure 9.6, where we have plotted the valuesfor a 4.2 mm diameter pupil and an eyefocused at different distances but viewing ascene at infinity - i.e. L=O. The values of Tand IN are taken from the Gullstrand relaxednumber 1 eye data in Appendix 3. Theapproximate equation gives values that aretoo high, but by only a few per cent.Smith (1982) suggested that the perceived

angular diameters of the blur discs can bemeasured using two small light sources, suchas illuminated optical fibres, whose sepa-ration is adjustable. When these sources aredefocused, they are seen as blur discs. The

--~"§ 200s»'":a..::s:is

Example 9.3: Calculate the defocus blurratio for the situation given in Example9.2, if the eye is observing a 6/6 letter, i.e.one that subtends 5 min arc.Solution: Substitute cP = 13.8min arc and(J= 5 min arc into equation (9.19) to giveDefocus ratio = 13.8/5 = 2.76

Other effects of defocus

Alignment of two targets at differentdistancesIn some visual experiments, the eye is alignedby asking the subject to subjectively align twotargets at different distances. Figure 9.7 showspoints Q1 and Q2 aligned along the visualaxis. If these points are so aligned, do theyappear superimposed in the visual field? Inother. words, are their retinal imagessuperimposed? By analysing the situationshown in Figure 9.7, we can deduce that thetargets do not appear to be aligned. Thisshows ~e point Q~ imaged, in focus, at Q2' onthe retina. The other point Q1 is imaged infront of the retina at QI' and is thereforedef<?cused at the retina. The retinal image ofQ1 is the defocus blur disc centred on thepoint Qlb', which is laterally displaced withrespect to the in-focus image Q{ Therefore,while the two points are aligned along thevisual axis, their retinal images are trans-versely displaced.

If the points are to appear superimposed,

Imageformation: thedefocused paraxial image 85

the two targets must lie along the paraxialpupil ray (i.e. the line of sight) rather than thevisual axis.

Effect on visual acuity

Defocus reduces the quality of the retinalimage and hence various aspects of visualperformance. Uncorrected visual acuitycorrelates with refractive error. However, thecorrelation is not perfect, even after pupil sizeis taken into account (Atchison et al., 1979).The relationship has been quantified fromtime to time by various investigators fittingregression equations to clinical data. Perhapsthe most common equation used is of the form10g(A)= a + b 10g(L1L) (9.20)w~e.re A is the (uncorrected) visual acuity inrrurumum angle of resolution in min arc, L1L isthe refractive error (say in dioptres), and aandb are constants. This equation is not basedupon any theoretical optics justification. Incontrast, an argument based upon the defocusblu~ circle model described in the precedingsection suggests that the minimum angle ofresolution should be linearly related torefractive error, except for low levels ofrefractive errors. Smith (1991) used thedefocused blur disc concept and the opticaltran~fer .function to show that, for typicalpupil diameters and for refractive errorsgreater than about 1 0, the expectedrelationship is a simple one of the formA =kDL1L (9.21)

o~

Visual axis(nodal ray)

Figure 9.~. The alignment of two targets along the visual axis. Their retinal images arenot superimposed, and hence they do not appear to be aligned.

86 lmagt; [ormationand refraction

where D is the pupil diameter and k is aconstant whose value depends on thestructure of the acuity target and therecognition rate (e.g. 50 per cent, 75 per cent,etc). For letters of the alphabet and a 50 percent recognition rate, the value of k isapproximately 650 for A in minutes of arc, Din metres and.1L in dioptres (Smith, 1996).For low levels of defocus, aberrations and

diffraction dominate the quality of the retinalimage, but these factors are beyond the scopeof the paraxial optics discussion of thischapter. The influence of these factors isconsidered in Chapter 18.

The value of k and the correspondingdefocus ratioFrom the above value of k,we can estimate thedefocus ratio of alphabetical characters at the50 per cent threshold of visibility.The letter size is regarded as five times the

minimum angle or detail in the letter. Usingthis value, transforming equation (9.21) to thethreshold letter size H in radians givesH = 5A = 5 x 650 x D.1L x tr/(180 x 60)

=0.945D.1L radiansand since

I, l'

L

E, E'N,N'0/0/

p/P/

R/R'

object and image distances from frontand back principal planes, respectivelycorresponding vergence of thedistance 1refractive errordistance of nodal point from principalplanedistance of object, conjugate with theretina, from principal planevergence corresponding to 1distance of entrance pupil from frontprincipal planedefocused image size at retinaobject and image sizesangular size of objectdiameter of defocus blur disc on retinaand its conjugate in object spacecorresponding perceived angulardiameter of defocus blur disc,measured at the nodal points (alwayspositive)position of entrance and exit pupilsposition of front and back nodal pointsgeneral object point and correspond-ing image pointpositions of front and back principalpointsaxial retinal point and correspondingconjugate in object space

Summary of main symbols

cp= D.1L,from equation (9.17)/ it follows thatDefocus ratio =CP/H =1.06 (9.22)which indicates that a letter may be recog-nized at a 50 per cent success level when thedefocus blur disc is about the same diameteras the letter height.

ADDp

visual acuity in minutes of arcentrance pupil diameterequivalent beam diameter at theprincipal planesequivalent power of eye focused oninfinitythe power of the same eye at somelevel of accommodationratio u'lU of angles that paraxial pupilray subtends at axis in image andobject space, respectively

ReferencesAtchison, D. A, Smith, G. and Efron, N. (1979).The effectof pupil size on visual acuity in uncorrected andcorrected myopia. Alii. J. Optom. Physiol. Opt., 56,315-23.

Chan, C, Smith, G. and Jacobs, R. J. (1985). Simulatingrefractive errors: source and observer methods. Am. J.OptOIll. Physiol. Opt., 62, 207-16.

Charrnan, W. N. and Jennings, J. A M. (1976). The opticalquality of the monochromatic retinal image as afunction of focus. Br. J. Physiol. Opt., 31,119-34.

Marsh, J. S. and Temme, L. A (1990). Optical factors injudgements of size through an aperture. HumanFactors,32,109-18.

Obstfeld, H. (1982). Optics in Vision, 2nd edn.Butterworths,

Pardhan, S. and Gilchrist, J. (1990). The effect ofmonocular defocus on binocular contrast sensitivity.Ophthal. Physiol. Opt., 10, 33-6.

Pascal, J. I. (1952). Effect of accommodation on the retinalimage. Br. J. Ophthal., 36, 676-8.

Prince, J. H. and Fry,G. A (1956). The effect of errors ofrefraction on visual acuity. Am. J. Optom. Arch. Am.Acad. Optom.,33, 353-73.

Rosenfield, M., Ciuffreda, K. J., Hung, G. K. andGilmartin, B. (1993). Tonic accommodation: a review. I.Basicaspects. Ophthal. Physiol. Opt.,13, 266-84.

Simpson, T. L., Barbeito, R. and Bedell, H. E. (1986).Theeffect of optical blur on visual acuity for targets ofdifferent luminances. Ophtha/. Physiol. Opt., 6, 279--81.

Smith, G. (1982). Angular diameter of defocus blur discs.Am. J. Optom. Physiol. Opt., 59, 885-9.

Smith, G. (1991). Relation between spherical refractiveerror and visual acuity. Optom. Vis. Sci., 68, 591-8.

Smith, G. (1996). Visual acuity and refractive error. Is

Image [ormation: thedefocused paraxial image 87

there a mathematical relationship? Optometry Today, 36(17),22-7.

Smith, G., Meehan, J. W. and Day, R. H. (1992). The effectof accommodation on retinal image size. HumanFactors, 34, 289-301.

van Meeteren, A. (1974). Calculations on the opticalmodulation transfer function of the human eye forwhite light. Optica Acta,21, 395-412.

Westheimer, G. and McKee, S. P. (1980). Stereoscopicacuity with defocused and spatially filtered retinalimages. J. Opt. Soc. Am., 70, 772--8.

10Some optical effects of ophthalmic lenses

Spectacle magnification

racies, the equations are simple and readilyshow trends. Aberrations associated with thelenses are not considered. We refer readerswho want a full thick lens treatment, or toconsider effects of aberrations, to texts such as[alie (1984)and Fannin and Grosvenor (1996).Most of the equations in this chapter apply

to both spectacle and contact lenses.Exceptions are the equations dealing withrotational magnification and field-of-view inthe section Effects on far and near points andaccommodation demand. The equations assumea rotating eye behind a fixed lens, which is notapplicable for contact lenses, which rotatewith the eye. Most of the effects describedhave much smaller magnitudes for contactlenses than for spectacle lenses, because of thesmall distance between contact lenses andrelevant ocular reference positions such as theentrance pupil.

When an eye is corrected by an ophthalmiclens in the form of spectacles or contact lenses,aspects of the retinal image are changed aswell as the image being focused. Most notice-ably, the lens affects the size of the retinalimage. This change in image size can bemeasured in two different ways. One of theseis the spectacle magnification, which is theratio of the image sizes after and beforecorrection. The second, which is of limiteduse, is called relative spectacle magnifi-cation. This is the corrected or focused retinalimage size compared with that of a 'standard'eye. These magnifications are associated withother effects. For example, a positive powerlens produces a magnified image and a blankpart of the visual field called a scotoma.Image magnification of spectacle lenses isassociated with prismatic effects and changesin the required eye rotation to look at an objectwhich is not on the lens optical axis. There arealso effects on binocular vision. In aniso-metropia the two eyes are corrected by Spectacle magnification SM is defined asdifferent lens powers, and differential effects . l i fbetween eyes, for example different amounts SM - retma Image size a ter correction (101)of eye rotations, may produce eyestrain. - retinal image size before correction .In this chapter we investigate some of these An equivalent definition is

SM = angular sub tense of image in correcting lens at eye (10.la)angular subtense of object at eye before correction

Introduction

effects, but consider only thin lenses. Whilethis leads to some approximation and inaccu-

In both cases, the pupil ray is used as thereference ray because it determines the centre

Some optical effects ofophthalmic lenses 89

(10.9)

(10.8)given bySM =ro'/ roSubstituting the right-hand sides of equations(10.2) and (10.7) for roand ro~ respectively, intoequation (10.8) givesSM = I'q/[I(I' -he)] (10.8a)If we replace I and I' by their correspondingvergences L (= 1/1) and L' (= 1/1'), we obtainSM =qL/(I- heL') (1O.8b)Given that L is known, L' is found simply bythe refraction equation L' = L + Fs' For theobject at infinity, it can easily be shown thatSM=1/(I-heFs) (10.8c)Equations (10.8b) and (10.8c) can be extendedto the thick lens case, with F now theequivalent power of the lens and Land L'determined relative to the front and backprincipal planes, respectively, of the lens.Examination of equation (10.8c) shows:

1. For positive lens powers, spectacle mag-nification is greater than 1, i.e, the observedobject is enlarged. For negative powerlenses, spectacle magnification is less than1.

2. The departure of spectacle magnificationfrom a value of 1 is increased with increasein distance between the lens and the eye.In ophthalmic optics, it is more convenient

to measure the back vertex power rather thanthe equivalent power of a lens. For the objectat infinity, it can be shown (see, for example,Jalie, 1984)that

SM= 1 11 - tFlin 1 - hveFv'

where t is lens thickness, n is lens refractiveindex, hve is the distance between the lensback vertex and the eye's entrance pupil, F1isthe front surface power of the lens, and Fv isthe back vertex power of the lens. Thisequation shows how SM is dependent on theparameters of the lens. The expressions (1 -»v»: and (l-hveF/ l- l on the right-handside of the equation are referred to as theshape factor and power factor, respectively.Example 10.1:An object is 30 em in frontof the corneal vertex of the eye. Acorrecting thin lens of power -S 0 is

(a)

Q r--hc-l"- /--+-- /'---1

(b)

of the defocused image (neglecting aber-rations).The following equation is derived for

spectacle magnification using Figure 10.1. Anobject of height T/ is at a distance q from theentrance pupil of the eye at E (Figure 10.la). Itsubtends an angle ro at the entrance pupil ofthe eye, given byro =T/ / q (10.2)In Figure 10.lb, a correcting lens of power Fshas been placed in front of the eye. The objectis a distance I from the lens, and the eye'sentrance pupil is a distance he from the lens.The object subtends an angle t/J at the lens,given byt/J =T//I (10.3)The image of the object in the lens has heightT/'and is a distance I' from the lens. From thefigure, T/'is given byT/'= t/JI' (10.4)Substituting the right-hand side of equation(10.3) for t/J into equation (10.4) givesT/'=T/I' /1 (10.5)The angle ro'subtended by the image at theentrance pupil E is given byro'=T/'/ (1' - he) (10.6)Substituting the right-hand side of equation(10.5) for T/'into equation (10.6) givesto'=1]1'/[1(1' - he)] (10.7)From the definition of equation (10.la), SM is

Figure 10.1. The optics of spectacle magnification andthe angular size of the images:a. The uncorrected eye looking at an object.b. The eye looking at the object through a correctingspectacle lens.

(10.13)

90 Image formation and refraction

placed 12mm in front of the eye. What isthe spectacle magnification?Solution: Refer the positions of the lensand the object to the entrance pupil of theeye. As this is approximately 3 mm insidethe eye (3.05 mm for the Gullstrandnumber 1 relaxed eye),q=-(0.3 + 0.003)=-0.303 mandhe = 0.012+ 0.003= 0.015mFrom Figure 10.1,I = q+ he = -0.303 + 0.D15 = -0.288 mL is given byL = 1/1 = -1/0.288 = -3.472 0From the lens equation (A1.21)L'=-3.472 - 5 =-8.472 0Substituting the values obtained for q,he'L, and L' into equation (1O.8b) givesSM=(-0.303 x -3.472)/[1- (0.015

x -8.472)] =0.933The object is perceived to be minified by0.067or 6.7 per cent by the spectacle lens.Note: q and I (or L) are always negative ifthe object is not at infinity, and he isalways positive.

Example 10.2: If the object in the previousexample is now at infinity, what is SM?Solution: Using h~ =0.015 and Fs = -5 0,in equation (10.8C)

SM=1/[1- (0.015x -5)] =0.930The object is seen to be minified by 0.003or 0.3 per cent relative to the previousexample.

Pupil position and magnification

Figure 10.2 shows the effect of the correctingophthalmic lens on the position of theentrance pupil of the eye. The ray from thecentre of the actual entrance pupil is refractedby the lens, and the image of this entrancepupil (the new effective entrance pupil of thelens/eye system) is displaced from theoriginal pupil and is also different in size.Application of the lens equation allows us to

Figure 10.2.The effect of the correcting spectacle lens onpupil position and size.

determine the new pupil position and themagnification.The lens equation applied to the ray shown

in Figure 10.2 isur-vn :», (10.10)and solving forT' givesT' = T/(1 + TFs> (10.11)The magnification M of the pupil defined as

M =new pupil diameter (10.12)old pupil diameter

is given by the lens equationM=T'/Tand thereforeM =1/(1 + TFs)

However,T=-Itewith the negative sign present because he isalways positive andT is negative in the figure.It then follows thatM =1/(1 - heFs) (10.14)The right-hand side expression is the same asthat of the spectacle magnification given byequation (lO.8c). Thus pupil magnification isthe same as spectacle magnification of adistant object.

Retinal image illuminanceThe above equations show that, for positivepower lenses, the effective entrance pupil isenlarged, and this allows an increase in theamount of luminous flux entering a system.The amount of luminous flux entering is

50111<' optical effects of ophthalmic lenses 91

~er JT

I--fc----1f-hp---j

Figure 10.3. Parameters for calculation of relativespectacle magnification.

Axial ametropiaIf the difference between an uncorrectedametropic eye and the standard eye is theaxial length only, they have the same powers.Substituting Fe for Fa in equation (10.19) gives

RSM = Fe = 1 (10.20)Fa + r, - hlfs 1 - (hp + fe)Fs

wherefe is the anterior focal length (-liFe) of

Relative spectacle magnification

In this section, for simplification, it is assumedthat the object is at infinity.Relative spectacle magnification, denoted

here by the symbol RSM, is defined as

S=retinal image size in the corrected ametropic eye ( 15)

R M . l' ., d d . 10.retina Image SIze In a stan ar emmetropic eye

proportional to the area of the pupil, and isthus proportional to the square of the pupildiameter - or, in this case, the increase isproportional to the square of the pupilmagnification. However, this does not lead toa change in image brightness, because theimage has the same magnification as thepupil, and the area of the retinal imagechanges by the same amount. Thus the effectsof spectacle magnification and pupil magnifi-cation on retinal illuminance cancel each otherout. However, there is some light loss due tosurface reflections of lenses without anti-reflection coatings (approximately 8 per centfor a lens with a refractive index of 1.5).

with both images in focus and when viewingthe same object at the same distance.The in-focus image size r(of a distant object

of angular subtense rois given by the equation(6.11), i.e,

11'= rolF (10.16)where F is the power of the eye. ThereforeRSM =FeiFI (10.17)where Fe is the equivalent power of theemmetropic eye, and Fl is the equivalentpower of the combined system of theophthalmic lens and ametropic eye. For thecorrected ametrope,Ft =Fa + Fs - hlaFs (10.18)where Fa is the equivalent power of theuncorrected eye, h is the distance betweenthe (back principafpoint of the) ophthalmiclens and the front principal point of theametropic eye, and Fs is the equivalent powerof the correcting ophthalmic lens (Figure10.3). Thus

RSM = Fe (10.19)Fa + t, - hlaFs

the eye. Typical values of hp are 15-20 mm,whileff =-16.67 mm for an eye of power +60D. If IIp is equal and opposite to fe' theprevious equation reduces toRSM =1 (10.21)This is Knapp's law, which states that aspectacle lens placed at the anterior focalpoint of an axially ametropic eye has the sameretinal image size as that of a standardemmetropic eye.

Refractive ametropiaIf the length of an uncorrected ametropic eyeis the same as that of the standard eye, theirpowers must be different. The effective powerof the ophthalmic lens at the eye, togetherwith the power of the ametropic eye, is thesame as that of the standard eye, that is

Fe= F/(l- hls) + Fa = (Fa + Fs -hlaFs)1(1 - hls) (10.22)

Substituting the right-hand side of thisequation for Fe into equation (10.19), after

92 Image [ormation and reiroction

some simplification givesRSM =1/(1-hls) (10.23)RSM is now similar to the spectaclemagnification given by equation (10.8c),except that the distance between the lens andthe entrance pupil of the eye has beenreplaced by the distance between the lens andthe front principal plane of the eye. Thedifference in positions of the entrance pupiland the front principal plane is approximately1.5mm (1.70mm for the Gullstrand number 1relaxed eye).

expected to have RSM values close to 1 asin equations (10.20) and (10.21).Atchison (1996) developed further

equations that give relative retinal image sizesof a pair of corrected eyes. These are useful incalculations involving anisometropia and/orrefractive surgery.

Effects on far and near points andaccommodation demand

Figure 10.4.The effect of the correcting lens on the nearpoint.

(10.24)

I-

The farthest and closest positions of clearvision are called the far point and near point,respectively. These positions are different forthe eye-ophthalmic lens system than for theuncorrected eye alone. The accommodationdemand, that is, the amount of accommo-dation required to focus clearly to an object, isalso affected by a correcting ophthalmic lens.For correcting distance vision, the far point

of the eye-ophthalmic lens system is placed atinfinity. Consider what happens to the nearpoint. Figure 10.4 shows a spectacle lensimaging a point 0 to 0'. Take the point 0' tobe the near point of the uncorrected eye. Theobject and image distances can be relatedusing the lens equation. For the situationshown in the Figure 10.4, the lens equationis

Near point Near point(new) (old)o 0'

1 1-,----,,- - = Fdnp + hv (dnp)new + ltv s

where dnp' and (dnp)new are the near pointdistances in uncorrected and corrected visionrespectively, Fs is the lens power and isassumed to be a function of position, and hv isthe distance from the lens to the cornealvertex. Solving for (dnp)new'

The concept of relative spectacle magnifi-cation has limited use for three reasons:1. It is not always known whether the

ametropia is axial or refractive. Thedistinction is useful for anisometropiaknown to be caused by differences in asingle parameter, for example axial length.

2. The concept of a standard emmetropic eyepower is of limited value. The power ofemmetropic eyes can vary over a widerange (Sorsby et al., 1962). This isunimportant in cases of anisometropia,where the ratio of the relative spectaclemagnifications of the two eyes does noteven require a definition of a standard eye.

3. The relative spectacle magnificationdifference in anisometropia is not necess-arily a good indicator of the likelihood ofmagnification problems. A good exampleof this is axial anisometropia - spectaclesmay provide equal retinal image sizesaccording to Knapp's law, but this does notmean that the brain's cortical images arethe same sizes. A stretching of the retinathat may accompany elongation of the eyein myopia may move the retina's receptorsfurther apart. Alternatively, there may be'rewiring' occurring between the retina andthe brain. Some evidence for thesepossibilities was provided by Winn et al.(1988), who measured aniseikonia, whichis a measure of differences in cortical imagesizes (Chapter 6), in myopic axial aniso-metropes. They found smaller aniseikoniawith contact lenses, for which RSM > 1,than for spectacle lenses, which would be

Further comments

Accommodation through a correctinglensThe presence of an ophthalmic lens in front ofthe eye changes the apparent position of theobject and therefore changes the demand on

SOllie optical effects of ophthalmic lenses 93

Hence the spectacle accommodation demandis simply the absolute value of the inverse ofthe distance of the object from the lens.

Example 10.3: An eye looks at an object30 ern away through a distance correctionof -5 0 placed 12mm in front of thecornea. What are the ocular and spectacleaccommodative demands?Solution: We can make the followingsubstitutions into equation (1O.28b):W= -1/0.3 = -3.333 0, hy = 0.012m andFs =-5 0to give the ocular accommodativedemand as

3.333A(W) = [1_ 0.012x (-5)][1 - 0.012x (-5) - 0.0122 X (-5) x (-3.333)]=2.970

(10.26)

accommodation. In equation (10.25), wereplace dnp and (dnp)new by general dis-tances Wand W' respectively. If we thenreplace these distances by their correspondingvergences Wand W',we can obtain

W' _ (1 + hyW)Fs + W- 21-hyFS -hy FsW

Now the effective spectacle refraction at thecorneal vertex Fso is given by

F = Fs (10.27)so 1-hyFSThe ocular accommodation demand A(W),referenced to the corneal vertex, is simplygiven byA(W) = Fso - W' (1O.28a)which can be shown to be

-WA(W) = 2 (1O.28b)

(1- hyFs)(1- hvFs- hv FsW)Sometimes the accommodation demand isreferred to the spectacle lens plane. In thiscase, we can set h; to zero and replace Win theabove equation by L,which is the vergence ofthe object relative to the spectacle plane. Wethen have the spectacle accommodationdemand A(L), which isA(L) = -L (1O.28c)

The object is a distance -(0.3 - 0.012) =-0.288 m away from the lens, whichmakes the object vergence relative to thelensL = -1/0.288 = -3.47 0from which, using equation (10.28c), thespectacle accommodative demand isA(L) =3.47 0If we repeat the above exercise for a +5 0

lens, we obtain an ocular accommodationdemand of 3.76O. This tells us that correctedhypermetropes have greater ocular accom-modation demands than do correctedmyopes. Thus, all other things being equal,corrected hypermetropes experience pres-byopia earlier in life than do correctedmyopes.

Rotational magnification, field-of-view and field-of-vislon

Rotational magnification

The magnification effect of ophthalmic lensesinfluences the eye's rotation to look at off-axisobjects of regard. We quantify this by therotational magnification, which we define as

RM = angle of rotation of eye looking through lens (8')angle of eye rotation without lens (0) (10.29a)

94 Image jormation and refraction

This is similar to determining spectaclemagnification (SM) at the beginning of thischapter (Spectacle magnification), except thatthe ocular reference point is the centre-of-rotation rather than the entrance pupil. In linewith equation (10.8b),RM is given byRM =qeL/(1 - heL') (10.29b)where qe is the distance from the object to thecentre-of-rotation of the eye, and he is thedistance from the spectacle lens to the centre-of-rotation. Because he is approximately twicethe value of h~ in equation (10.8b), values ofRMare much turther from 1 than are values ofSM. For the object at infinity, it can easily beshown that

(10.29c)

Field-oj-view

Apparent and real fields-of-view are shownfor positive and negative power lenses inFigure 10.5. The apparent field-of-view is themaximum ocular rotation O'ma of the eye,about its centre-of-rotation C, for which theeye can look through a lens at an object ofregard. The real field-of-view is given by theangle 0max' From Figure 10.5, the apparentfield-of-view is

(10.30)

Diplopia~---

Figure 10.5. Real and apparent fields-of-view:a. Corrected hypermetropia.b. Corrected myopia.

Combining equations (10.29a) and (10.29b),but using 0max and O'max instead of 0' and 0,gives the equationomax =O'max(l - hy) / (qeL) (10.31a)Substituting the right-hand side of equation(10.30) for O'max into equation (10.31a) givesomax = Ds(l- heL')/ (2heqeL) (10.31b)For a distant object, this equation reduces to0max =Ds(l-heFs)/(2he) (1O.31c)Comparing the last equation with equation(10.30) shows clearly that the real field-of-view is less than the apparent field-of-viewfor positive lens powers (corrected hyper-metropia). This gives a scotoma (blind area) ofmagnitude (8'max - 8max) in the field of vision(Figure lO.5a). This can be a considerableproblem for moderate and high power hyper-metropes.The real field-of-view is greater than the

apparent field-of-view for negative lenspowers (corrected myopia). This gives aregion of diplopia (double vision) of magni-tude (8max - 8'max) in the field-of-vision(Figure lO.5b), but this does not seem toinconvenience corrected myopes.As these are first order equations, which

ignore the influence of the aberration distor-tion on the real field-of-view, we caution theiruse with large angles.

Field-of-uisionThis is similar to field-of-view, except that theeye is stationary and looking through thecentre of the lens. Limits correspond to thefield seen by the periphery of the eye throughthe lens. A similar set of equations toequations (1O.30)-(10.31c) apply, except thatwe revert to using the distances q and he' thatwere used for the spectacle magnificationcalculations in the Spectacle magnificationsection, instead of qe and he'

Summary of main symbols

distance from ophthalmic lens to theentrance pupil of eye (alwayspositive)distance from ophthalmic lens to the

ReferencesAtchison, D. A. (1996). Calculating relative retinal imagesizes of eyes. Ophthal. Physiol. Opt., 16, 532--8.

Fannin, T. E. and Grosvenor, T. (1996). Clinical Optics,2nd

edn. Butterworth-Heinemann.[alie, M. (1984). Principles of Ophthalmic Lenses, 4th edn.Association of Dispensing Opticians.

Sorsby, A., Leary, G. A. and Richards, M. J. (1962).Correlation ametropia and component ametropia.Vision Res.,2, 309-13.

Winn, B., Ackerley, R. G., Brown, C. A. et al. (1988).Reduced aniseikonia in axial anisometropia withcontact lens correction. Ophthal. Physiol. Opt., 8, 341--4.

l'

L, L'q

centre-of-rotation of eye (alwayspositive)distance from ophthalmic lens to thefront principal point of eye (alwayspositive)distance from ophthalmic lens to thecorneal vertex (always positive)diameter of spectacle lensdistance from ophthalmic lens toobject (always negative)distance from ophthalmic lens toimagevergences corresponding to 1and l'distance between entrance pupil ofeye and the object (always negative)distance between centre-of-rotationof eye and the object (alwaysnegative)distance from corneal vertex to nearpoint of uncorrected eyedistance from corneal vertex to nearpoint of corrected eyeequivalent power of ophthalmic lensequivalent power of the ametropiceyeequivalent power of the emmetropiceye

A(W)ta8max

SMMRSMRM

Some optical effects of ophthalmic lenses 95

equivalent power of ophthalmic lensand ametropic eye togetherocular accommodation demandangular diameter of the objectreal field-of-view through aspectacle lensapparent field-of-view through aspectacle lensspectacle magnificationpupil magnificationrelative spectacle magnificationrotational magnification

11

Light and the eye: introduction

Introduction

Figure 11.1.The electromagnetic spectrum.

If a beam of electromagnetic energy has aspectral radiant flux denoted by FR(A), theamount of radiant power or flux FR in thebeam is given by the integral

FR =JFR(A.)dA. watt (11.1)o

Radio

WOO900

800 E700 Red 5600 ~s::500 ~

400 Blue ~

300200100

Infrared

Visible

X-rays

Gamma rays

Radar

T.V.

Microwave

Ultraviolet

32I0

-I-2

.-.§ -3.c -4Oils:: -5lUQ) -6>'"~ -7'-'s00 -8~

-9

-10-II-12-13-14-15-16-17

Radiation and the electromagneticspectrum

Since the eye is the organ of light sense, howthe eye interacts with light is of great import-ance in understanding the visual process andthe limits to vision. As we see in the next fewchapters, not all the light entering the eyeforms the intended image on the retina. Somelight is reflected, scattered and absorbed, witha small part of the absorbed light being re-emitted in the form of fluorescence.Intense levels of light can cause damage to

the eye. Furthermore, other nearby bands ofthe electromagnetic spectrum (ultraviolet andinfrared radiation) interact with the eye andthese can also cause damage.Since the eye is essentially a detector of

light, we need to understand the nature oflight. In this chapter we discuss the nature oflight, how it is quantified, and differentaspects of it that affect vision.

Light is a small part of the electromagneticspectrum. The complete electromagneticspectrum is shown schematically in Figure11.1. The range of wavelengths that cover theultraviolet, visible and infrared ranges iscalled optical radiation. The limits of theoptical radiation bands are 100-380 nm(ultraviolet), 390-780 nm (visible) and 780-106nm (infrared).

100 Lig/ltIIlld t11~ ~yc

(11.2)

1.0Photopic

0.9 Scotopic

0.8»'Jc: 0.7...'cE 0.6...'"::00 0.5c:'E.2 0.4...>.,. 0.3

<:ic:: 0,2

0.1

the equation00

F =Km f FR(A)V(A)dA lumeno

where the constant Km is known as themaximum spectral luminous efficacy ofradiation for photopic vision. It has a valueof 683.002Im/W (CIE,1983). V(A) is known asthe spectral luminous efficiency function forphotopic vision. It was determined from theaverage response of many subjects across asmall number of studies and was defined bythe Commission lnternationale de l'Eclairage(ClE) in 1924. V(A) has a maximum value of 1at 555 nm. Figure 11.2 shows the V(A)function.The V(A) function is too low below about

450 nm because of shortcomings in the studiesfrom which it was developed, but theadvantages to be gained from correcting it areconsidered to be outweighed by the practicalinconvenience. It should be appreciated thatindividuals' photopic relative sensitivitiesdiffer for a number of reasons, including thefollowing:1. Luminous efficiency is the combined effect

of the three types of cones (containingshort, medium and long wavelengthsensitive photopigments), so that while the

Light

where FR(A) has the unit of watt/unit ofwavelength.

The Penguin Dictionary of Physics (1977)defines light asThe agency that causes a visual sensationwhen it falls on the retina of the eye. Lightforms a narrow section of the electro-magnetic spectrum ...While this is correct, agencies other than the

narrow band of the electromagnetic spectrumthat we normally call light can produce avisual sensation. Our sense of light arisesusually from stimulation of the cones androds in the retina and by signals beingtransmitted from them to the visual centres ofthe brain through a number of different typesof nerve cells and pathways. If any of thesecells or centres are stimulated by other means,a visual sensation still occurs. For example,these cells may be activated by chemicals, X-rays and pressure from a knock on the head.Light can be defined simply as that band of

the electromagnetic spectrum that produces avisual response, and this band is from about380-780 nm. The eye is not equally responsiveto all wavelengths in this band. The spectralvisual response curve is approximately 'bell'«shaped, with its shape and position depend-ing upon the light level. Two extreme forms ofthe spectral response curve are identified; onefor moderate to high light levels, and one forlow light levels. For moderate to high lightlevels, the cones dominate vision, we seecolour and the spectral response is referred toas the photopic response. For low light levels,the rods dominate vision, we are unable todistinguish colour, and the spectral responseis referred to as the scotopic response. Therange of light levels intermediate betweenthese two extremes, where both cones androds operate, is called the mesopic range.Throughout this book, the photopic case isassumed unless stated otherwise.

0.0 ~""""~=;:"""""''''''''TT"T"T'T'CT"T'"rri'''''''"TT''l,..r:;.,.....,...-rt­

350 400 450 500 550 600 650 700 75(

Photopic vision Wavelength (nm)

The amount of light F (luminous flux) in anelectromagnetic beam of radiation is given by

Figure 11.2.The relative luminous efficiency functionsV(A) - photopic and V'(A) - scotopic. The peaks of thesefunctions are at 555nm and 507nm, respectively.

spectral sensitivity of the cones may be thesame for each individual, the relativenumber of cones may vary. In addition,colour defective people have a missingcone type (dichromasy) or a cone typewhose photopigment has an alteredspectral sensitivity (anomalous trichro-mats).

2. Variations in spectral transmittance by theocular media and variations in the densityof a yellow pigment (xanthophyll) in themacula of the retina. In particular, as weage our sensitivity to blue light decreasesbecause our lenses absorb more blue lightand there are neural changes associatedwith the short wavelength-sensitive cones(Werner et al., 1990).

Mesopic visionAs the light level decreases from photopictowards scotopic levels, there is a change inrelative spectral sensitivity accompanying thetransition from cone to rod vision. Thistransition region is called the mesopic regionand the shift in relative spectral sensitivity iscalled the Purkinje shift.

Scotopic visionThe CIE defined the spectral luminousefficiency function for scotopic vision V'(,l) in1951.This function has a maximum value of 1at 507 nm. It is shown, along with V(,l), inFigure 11.2. If we convert radiant energy intolight using V'(,l), we use equation (11.2) withV'(,l) replacing V(,l) and K'm' the maximumspectral luminous efficacy of radiation forscotopic vision, replacing Km . The value ofK' is 1700.061m/W (CIE, 1983), which isde~ived from the definition that 1 photopiclumen =1 scotopic lumen for a mono-chromatic source with a frequency of 540 x1012 Hz. The ratio of the two Km values is theratio of the V(,l) and V'(,l) values at a wave-length of 555.016nm in air with a refractiveindex of 1.00028.Since there is only one class of cells (the

rods) operating at scotopic light levels, wewould expect less variability betweenindividuals for scotopic relative sensitivitythan for photopic relative sensitivity.

Light and tile eye: introduction 101

Photopic, mesopic and scotopiclimitsThere are no sharp divisions between theboundaries of these three ranges. The lowerluminance limit of photopic vision isapproximately 3 cd/m2, with mesopic visionextending from this level to 0.03 cd/m2, afterwhich scotopic vision starts (see next sectionfor unit of luminance).

Photometric quantities, units andexample levels

There are four fundamental photometricquantities: luminous flux, luminous intensity,luminance and illuminance.

Luminous flux (F)

Luminous flux is the measure of the totalamount of light in a beam, and has the unit oflumen (1m).

If we think of a light source as emitting somany watts of electromagnetic radiation, itemits a certain amount of lumens of light. Theratio of lumens to watts of a particular lightsource is known as its luminous efficacy. Forexample, there are about 10 lumens for eachwatt of a tungsten filament light source(giving about 600 lumens for a 60W lightbulb), about 40 lumens per watt for afluorescent tube, and about 95 lumens perwatt for sunlight.

Luminous intensity (1)

Luminous intensity is a measure of thebrightness of a point source of light, and hasthe unit of candela (cd). It is a measure ofluminous flux density. The luminous intensityin a given direction is the quotient of theluminous flux (t5F), contained in an infinitesi-mally narrow cone in the given direction, bythe solid angle (~n) of the cone (Figure 11.3).That isI = t5F / ~n lumen/steradian or candela (cd)

(11.3)In the above example of the 60W light bulb,the 600 lumens is not emitted nor distributed

102 Light and the "If"

Figure 11.3. Luminous intensity I = Dr/on.

Luminance (L)

evenly in all directions in space. For manylight sources, it is more important to know theluminous intensity emitted in a givendirection rather than the total luminous fluxemitted. In practice, luminous intensity isused only for sources of small angularsubtense.The luminous intensity of a common wax

candle is about 1 cd, and this is the origin ofthe word 'candela'. Traffic signal lights haveaxial luminous intensities of about 200-600cd, or even higher. The axial luminous inten-sity of a car headlight on full beam can beabout 20,000 cd. Marine lighthouses haveintensities of millions of candelas.

Some sources have the same 'brightness' orluminance in all directions, i.e. luminance isindependent of 8. These sources are refered toas Lambertian.Ma~y sources of light, such as the sky, are

effective sources because they scatter incidentlight. A perfectly diffusing surface scattersincident light equally in all directions, whichmeans that the surface luminance is the samefor all directions, and it acts as a Lambertiansource. Magnesium oxide, soot, bariumsulphate and roughened chalk (calciumcarbonate) come close to being perfectdiffusers. For a Lambertian source or surfaceL(8) =constant =L (11.5)

and the luminous intensity is given by1(8) =LoA cos(8) (11.6a)For normal viewing, 8 =0° and1= L oA (11.6b)For a perfect specular surface, which is theopposite of a perfect diffusing surface, all theincident light is reflected according to Snell'slaw. All surfaces have both specular anddiffusing properties, thus lying somewherebetween the two extremes.The .luminance of the sun depends upon its

elevation above the horizon and thescattering, reflection and absorption by watervapour, dust and other substances in theatmosphere. The scatter and absorption byatmospheric molecules is wavelength-depen-dent, thus producing the blue sky and thereddish sun at sunset. Under clear atmos-pheric conditions, the luminance of the sun athigh elevations is about 1.5 x 109 cd/m2 andthe luminance of the moon is about 2000cd/rn-.

.. (iF

(11.4)

Luminance is the objective measure of the'brightness' of an extended source, and hasthe unit of candela per square metre (cd/m-),For a small element of an extended source,

luminance can be related to the luminousintensity of the element and a direction.Referring to Figure 11.4, if the source is a smallplane element of area oA with luminance L(8)in a direction 8, the luminous intensity 1(8) inthat direction is given by the equation1(8) =L(8) oA cos(8)That is

_ 1(8) 2L(8) - oAcos(8) cdlm

Source

Illuminance (E)

Figure 11.4.Relationship between luminance (L) andluminous intensity (I).

Illuminance is a measure of the luminous fluxdensity incident on a surface, and has the unitof lumens per square metre (lm/m2 or lux).Since illuminance can vary over a surface, it

is best defined in terms of small elements ofarea oA. ThusE=OF / oA lux (11.7)On a clear day with the sun high in the sky,

the illuminance at the earth's surface can be ashigh as 50,000 lux. The illuminance at a deskin a well-lit office is about 200-1000 lux.

Some useful relationships

The four basic photometric units areconnected within the preceding definitions,but there are other relationships that areuseful in a number of different situations. Wediscuss two of these in this section.

Luminous intensity and illuminance:the inverse square lawThe illuminance E on a surface at a distance dfrom a small source of light, described by itsluminous intensity I, is given by the inversesquare law equation

IE = d2 cos(8) (11.8)

where 8 is the angle of inclination of thesurface normal to the direction of the source.This equation assumes that the source is a

point source. Since all sources have some size,this equation is only an approximation. If theangular subtense of the longest dimensionof the source is less than 5° at the distance d,the error in this equation is less than 1 percent.

Luminance and illuminanceVisual performance in any task is related tolight level, and it is the stimulus luminancethat is the important measure of light level.For many stimuli that reflect light, such as avisual acuity chart, the luminance dependsupon the illuminance in the plane of thestimulus. The relationship between the twoquantities depends on the scattering proper-ties of the stimulus material. If the stimulus isperfectly diffusing and reflects a fraction r ofthe incident light, luminance L is related toilluminance E by the simple equationL = rE/1r (11.9)

Light and theeye: introduction 103

Which quantity to use

Since visual performance is dependent uponlight level, we may need to specify light levelsfor particular tasks. We need to know whichof the four photometric quantities of lumi-nous flux, luminous intensity, illuminanceand luminance is most relevant to each task.

Threshold detection

The threshold light level for detection of alight stimulus with a small angular subtensedepends upon the total amount of lightcollected by the retina - that is, the luminousflux. This is because the light from thisstimulus falls on a few photoreceptors, whichpool their input. This is known as spatialsummation. Since this pooled amount of lightis the product of local illuminance and thearea, the amount of light collected is theluminous flux. Under such circumstances,Ricco's law states thatluminance x area = constantbut more strictly speaking this should belocal illuminance x area =luminous flux

= constantFor a large source, whose retinal image ismuch larger than the region of spatialsummation, detection thresholds dependupon the stimulus luminance (or retinalilluminance) and not on stimulus size. Thedetection threshold depends upon the lumi-nance contrast of the stimulus relative to itsbackground.In between the extremes of size, thresholds

vary approximately with the square root ofthe product of luminance and area (Piper'slaw).

Supra-threshold visibility of sourcesWIth a small angular subtense

For light levels well above threshold, thevisibility of a small 'point' source is related toits luminous intensity. For example, theperformances of signal and warning lampssuch as traffic signal lamps are specified bytheir luminous intensities.

104 Light and the eye

Summary of main symbols

supra-threshold levels depend upon pupildiameter. We discuss how pupil size affectsretinal light level in Chapter 13.

wavelengthspectral radiant fluxradiant power or radiant flux(integrated over all wavelengths)spectral luminous efficiencyfunctions for photopic andscotopic visionmaximum spectral luminousefficacies of radiation for photopicand scotopic visionluminous fluxluminous intensityluminanceilluminancesolid angleareadistancedirection relative to the normal ofa surfacereflectance fractionr

FILEQAde

V(A), V'(A)

Supra-threshold visibility of sourceswith a large angular subtense

In general, visual performance improves withincrease in light level. Most realistic scenescontain many objects of various sizes,luminances and luminous intensities. Weneed a consistent, repeatable method ofmeasuring the ambient light level of suchscenes. The obvious approach is to measurethe amount of light entering an observer'seye, i.e. the luminous flux. However, thisrequires knowing the pupil size of theobserver. This problem can be avoided byusing the illuminance at the pupil plane of theobserver.

The visibility of a source with a large angularsubtense is usually correlated with its lumi-nance. For example, the performances ofpedestrian road crossing signals are specifiedby their luminances.

Measurement of ambient light level

Other comments

While thresholds and levels of visibilitydepend upon the light level reaching theretina, we can measure accurately only thephotometric properties of the source itself andcorresponding light levels (e.g. illuminance)at any distance from the source. However, thelight level reaching the retina is dependentupon pupil size, and therefore threshold and

ReferencesCIE (1983). The Basis of Pilysical Photometry. PublicationCIE No. 18.2 (TC-1.2). Commission Intemationale del'Eclairage.

Penguill Dictiollary of Physics (1977). (Y.H. Pitt, ed.), p. 219.Penguin Books Ltd.

Werner, J. S., Peterzell, D. H. and Scheetz, A. J. (1990).Light, vision, and aging. Optom. Vis. Sci.,67, 214-29.

12Passage of light into the eye

Introduction

Not all the light entering the eye forms theretinal image. A significant amount is lostfrom the beam by the following processes:1. Some light is specularly reflected at thefour major refracting surfaces.

2. Some light is elastically scattered (nochange in wavelength) by the ocularmedia. In some other chapters we refer todiffusely reflected light (e.g. Chapter 8), butin this chapter we refer to the light which isdiffusely reflected as back-scattered light,to distinguish it from forward-scatteredlight.

3. Some light is absorbed and is then eithera. re-emitted at other (longer) wavelengths,which is known as inelastic scattering oras fluorescence, or

b. converted to other forms of energyBoth 2 and 3a above are causes of veiling

glare (also straylight). Veiling glare fromelastically scattered light is not usuallyuniform and has an angular distribution,which we discuss further in Chapter 13. Bycontrast, fluorescence produces a uniformlydistributed veiling glare, and we discuss thesource of this fluorescence later in thischapter.Of the above three causes of light loss,

specular reflection by the refracting surfacesmakes up only a small proportion. Most ofthis occurs at the cornea, and this is useful indetermining the radii of curvature of these

refracting surfaces. However, this may beannoying because it produces veiling glare forthe clinician during direct ophthalmoscopy.Light scattered by the ocular media plays asimilar dual role. Light specularly reflected orback-scattered out of the eye helps cliniciansto delineate the different components of theeye during internal eye examination with theslit-lamp, particularly in examination of thelens and the cornea. However, any forward-scattered light produces a veiling glare at theretina and reduces scene contrast. This is aproblem particularly when bright lights arepresent in otherwise dark fields and in lowcontrast fields, and increases with age. Whilethe loss of light due to absorption reduces theamount of light reaching the retina, it protectsthe retina from light of shorter wavelengths,which may be damaging to ocular structures.Light is able to reach the retina after

passage through the iris and sclera. Van denBerg et at. (1991) determined that, for eyeswith light irises, the effective transmission isbetween 0.2 per cent (green) and 1 per cent(red), and that the eye wall around the iristransmits a significant amount of light. Thislight makes a small contribution to veilingglare at large angles.

Specular reflection

Some light is reflected at each interface in theeye. Since the surfaces are smooth, thespecularly reflected light is image forming.

106 Ligllt and theeye

The fractions of reflected and transmittedlight depend on the refractive indices on eachside of the surface, and these fractions aregiven by the Fresnel equations. If the re-fractive indices are /I and n' on the incidentand refracted sides respectively, for normalincidence the reflectance (i.e. fraction of lightreflected) R and the transmittance (i.e. thefraction of light transmitted) T are given bythe equationsR =[(/I' - n)/(n' + 11)F and T =4nn' /(/1' + 11)2

(12.1 )

implying that there is no absorption of light atthe surface. However, there is someabsorption by the bulk tissue. Since the eyecontains four main reflecting surfaces, thereare four main reflected images. These arecalled Purkinje-Sanson or Purkinje images,and are often denoted by the symbols PI' PI1'Pm and PIV'The above equations apply strictly only to a

simple smooth surface between twohomogeneous media with well-definedrefractive indices nand n', In biologicalsystems, this ideal situation does not exist. Forexample, in the case of the anterior cornealsurface, the boundary consists of a number oflayers (see Chapter 2) of different refractiveindices. The outermost layer (the tear film)has an index of approximately 1.336,compared with a value of 1.376 for the bulk ofthe cornea. The lens surfaces are alsocomplicated by the presence of the capsule,and are not as smooth as the cornea, and thismakes their Purkinje images more diffusethan those of the cornea. A rigorous analysisof the reflectance of surfaces would have totake into account the fine structure of thecapsular matrix.The positions, sizes and brightnesses of the

Purkinje images depend upon the position of

We may note that from these equations

R+T=l (12.2)

the light source and the optical structure ofthe eye. Table 12.1 has these details for theGullstrand number 1 relaxed schematic eyeand an axial, distant light source. Figure 12.1shows the positions and sizes of the images ofthis schematic eye.These images can be used to determine the

positions and curvatures of the intra-ocularsurfaces and, in particular, those of the lens.Measuring Purkinje image sizes allows us tomonitor lens changes due to accommodationand aging in the lens.The Purkinje images are useful also in

locating the different axes of the eye (seeChapter 4) and for monitoring eye move-ments. The brightness of Purkinje images hasalso been used to determine the spectraltransmission of the lens (Said and Weale,1959;Johnson et al., 1993).

Images formed by multiple reflections

The Purkinje images discussed above areformed from single reflections. Some lightsuffers more than one reflection, but theamount of light in a multi-reflected beam israpidly attenuated as the number of reflec-tions increases. If there is an odd number ofreflections, the beam exits the eye. On theother hand, if the beam suffers an evennumber of reflections, it finally reaches theretina and forms an image somewhere alongits path. There are six possible combinationsof double reflection images, and the brightestare the three involving the anterior cornealsurface. However, to see any of these doublereflections, the image has to be formed nearthe retina and has to arise from a bright sourcein a dim or dark field.The degree of focus at the retina of these

higher order Purkinje images depends uponthe distance of the source, the structure of aparticular eye, and the level of accommod-ation. If the source is about 25 em in front of

Table 12.1. The Purkinje images of the relaxed Gullstrand number 1 eye for a distant light source.

Purkinie illlage

P, (anterior cornea)P" (posterior cornea)Pili (anterior lens)PIV (posterior lens)

Relative size

1.0000.8821.967-0.760

Distance froIII corneal pole (mm)

3.8503.76510.6203.979

Relative brightness

1.0000.008260.01280.0128

Passa~... of ligllt info fir... ey... 107

Source atTinfinityI.L... _r«-~~:-~~-~i:

L- -J

Figure 12.1.The positions and relative sizes of the Purkinjeimages for the Gullstrand number 1 relaxed eye, for the lightsource at infinity.

the cornea, the reflections from the posteriorlens surface and the anterior corneal surfaceare in focus close to the retina and form anerect image.Specular reflections may occur that are not

Purkinje images. For example, the lenticularcortex shows a small steady increase in back-scatter with age, which may be because of thespecular reflections from the ever increasingzones of discontinuity in the cortex (Weale,1986).

the transmittance of the eye, particularly ofthe lens. These include psychophysical andphysical methods. Physical methods includecomparing the relative intensities of the thirdand fourth Purkinje images, and in vitromeasurements of individual parts of the eyeor the whole of the eye.

Spectral transmittance of the wholeeye

Transmittance

There have been several studies measuring

Figure 12.2 shows spectral transmittances ofthe whole eye (which includes the cornea,aqueous, lens and vitreous) from the work ofLudvigh and McCarthy (1938), Boettner and

2500

16 mm path in water

--0-- Geeraets et al. ( 1968)

--0-- Ludvigh and McCarthy (1938)

Boettner and Wolter ( 1962)- total

Boettner and Wolter ( 1962)- direct

500

110

tOO

90

~ 80~

'" 70<.Jc:~ 60's

50v:c:E£- 40

30

20

10

00 1000 1500 2000

Wavelength (nm)

Figure 12.2.The spectral transmittance of the whole eye from Ludvigh and McCarthy(1938), Boettner and Wolter (1962), Geeraets and Berry (1968), and 16 mm of water. Thedata of Boettner are for a young child/young adult, except for wavelengths less than 380nrn, where they are those for a young child.

108 Light lind the eye

Wolter (1962) and Geeraets and Berry (1968).The mean age of the four eyes of Ludvigh andMcCarthy was 62 years, but they adjusted thetransmittances as if the eyes contained lensesof a mean age of 21.5 years. Boettner andWolter (1962) investigated nine eyes rangingfrom 4 weeks to 75 years of age. Geeraets andBerry (1968) measured seven eyes, whose ageswere not given.Since the ocular components scatter light,

the measured transmittance depends uponthe amount of scattered light that is collectedby the instrument. Boettner and Wolter (1962)measured the transmittance under twoconditions; first, light collected only within a10 cone centred on the transmitted beam, andsecond, light collected within a 1700 cone. Thefirst condition simulated directly transmittedlight plus a small amount of scattered light,and the second condition included most of theforward-scattered light and measured totaltransmittance. Boettner and Wolter's resultsshow a strong dependency on age. Ludvighand McCarthy (1938) and Geeraets and Berry(1968) did not state their collecting angles.The transmittance of the whole eye can be

found by removing the sclera, choroid and

retina in the region of the fovea. However,Boettner and Wolter calculated their wholeeye values from the transmittances ofindividual components. Their individualcomponent transmittances, as well as datafrom other sources, are discussed below.

Spectral transmittance of each ocularcomponent

The corneaFigure 12.3a shows the spectral transmitt-ances of the cornea (Boettner and Wolter,1962;Beems and van Best, 1990;van den Berg andTan, 1994). Boettner and Wolter stated that thetotal transmittance (scattered plus direct light)was representative of six eyes with no ageeffect, and that the direct (scattered lightexcluded) transmittance for the 53-year-oldeye was close to the mean of eight eyes. Theyfound that the direct transmittance is age-dependent, but Beems and van Best (1990)and van den Berg and Tan (1994) did not findan age dependency.

Boettner and Wolter (1962)-lotal

Boettner and Wolter (1962)-direct

.... 6···· Beerns and van Best (1990)

- -1lI- - van den Berg and Tan (1994)-tolal

- - .- - van den Berg and Tan (\ 994)- direct

0.5 mm path in water

10

20

30

90

~ 80~ 70~e~ 60'E~ soeEo- 40

1l0+-~~...L..o~~....L....~~-'-~~--'-~~-+-

100

2S(K)I<KK) isoo 2<KK)

Wavelength (nrn)

SOOo-f-,""'~"'T"""r-r-"-"--r-""""''''''''''--,-'''''''''"'T-1~'r--r~.:''t­o

(a)

Figure 12.3.The spectral transmittances of the ocular components from various sources:a. Thecornea, from Boettner and Wolter (1962) for a 53-year-old subject, Beems and van Best (1990)mean data for

22-43 years and 67-87 years groups, and van den Berg and Tan (1994) from their equation (1).b. The aqueous, from Boettner and Wolter (1962),both direct and total.c. The lens, from Said and Weale (1959)mean of ages 21-45 years, Boettner and Wolter (1962)for a young child, and

Mellerio (1971)estimated mean of 19-32 years and 46-66 years groups.d. The vitreous, from Boettner and Wolter (1962).

110 Passage of lightintotheeye 109

100 ........................

90

~80

70QJt.) Boettner and Wolter (1962)c'" 60= 3.5 mm path in water·s'" 50c~ 40

30 ~ii

20 Iito! ,

10 /00 500 1000 1500 2000 2500

(b) Wavelength (nm)

110

100 .......................................

90

~80 - - 6- - Said and Weale (1959)

-directQJ 70t.) Boellner and Wolter (1962)c~ 60 -jotat·s

50Boettner and Wolter (1962)

'" -directc'"... Mellerio (1971)~ 40 ····0····

30 f\ 3.5 mm path in water

20 II,10

00 500 1000 1500 2000 2500

(c) Wavelength (nm)

110

100

90

80!~

Boenner and Wolter (1962)

~-tolal

QJ 70 Boettner and Wolter (1962)t.)

-directc~ 60·s 16 mm path in wateree 50c'"...~ 40

30

20

10

00 500 1000 1500 2000 2500

(d) Wavelength (nm)

110 Light and the eye

The aqueousFigure 12.3b shows the mean spectraltransmittance of the aqueous of several eyes,from Boettner and Wolter (1962). They foundno differences in transmission due to age.They found no difference between total anddirect transmittance, indicating that there isno significant scattering in the aqueous.

The lensFigure 12.3c shows spectral transmittance ofthe lens (Said and Weale, 1959; Boettner andWolter,1962; Mellerio, 1971). The data for Saidand Weale (1959) and for Mellerio areestimated means of their results, which showa strong age-dependence. Boettner and Wolter(1962) found that both the total and directtransmittances decrease with age, particularlyat the short wavelengths. This decreasedtransmittance at shorter wavelength producesthe yellowing of the lens with age.

The vitreousFigure 12.3d shows the total and directspectral transmittances of the vitreous(Boettner and Wolter, 1962). There is ameasurable amount of scattering in the

vitreous, which does not seem to be depen-dent upon wavelength. No differences intransmittance due to age were found.

Progressive loss of light as it passesthrough the eye

Figure 12.4 shows the decrease in directtransmittance as light passes through the eye(Boettner and Wolter, 1962). The data showsthe spectral transmittance at the posteriorsurface of each ocular component.

Causes of absorption bands

Since the major component of the eye is water,we may expect that the spectral absorption ofthe ocular media is strongly influenced by theabsorption properties of water. Spectralabsorption data for water have been given byHulburt (1945), Curcio and Petty (1951), andSmith and Baker (1981). These data have beenused to calculate the spectral transmittancesof different thicknesses of water samples, andappropriate curves are plotted on Figures 12.2and 12.3a-d. Since the eye is not made totallyof water, we would expect that, for any

110

100

90

80

~70

.,60(j

c~ 50'EVJc 40<U..Eo-<

30

20

10

00 500 1000 1500 2000 2500

Wavelength (nrn)

Cornea

Aqueous

Lens

vhreous

24 mm path in water

Figure 12.4.The cumulative spectral transmittances at the posterior surfaces of theocular components, from Boettner and Wolter (1962). These data are for directtransmittance and for a young child/young adult, except for wavelengths less than380nm, where they are those for a young child.

component, the best match occurs for a lengthof water shorter than the eye componentthickness. We would expect a good match forthe aqueous, which has the highest concen-tration of water, and the worst match for thecornea and lens. For the eye as a whole, itappears that a 16mm thickness of watermatches the whole eye fairly well.Examination of the transmittance curves for

water in Figures 12.2 and 12.3a-d shows thatthe spectral absorption of the eye is domi-nated by water for wavelengths greater thanabout 600nm. Absorption of energy by waterat these wavelengths leads to heating of thewater and surrounding tissue, and this maypose a risk of thermal damage.For shorter wavelengths, the ocular tissue is

far more absorbing than a similar path lengthof water, indicating that the absorptionproperties of proteins and other cellularcomponents are dominating the absorptionprocess. Since these materials dominate theabsorption, there is a risk that they will bedamaged by the radiation.Figures 12.2-12.4 show that there is strong

absorption for wavelengths less than about400 nm. The cornea absorbs all radiationbelow 290nm. Wavelengths less than 300 nmare potentially damaging to the cornea, and itis most sensitive to damage at a wavelength ofabout 270 nm, where the threshold forobservable damage is about 50J/m2 (Pitts,1978). The lens absorbs radiation stronglybetween 300nm and 400 nm, with the shortestwavelength reaching the retina beingapproximately 380nm.

Luminous transmittanceThe above data describe the spectraltransmittance of the ocular media. Of equalimportance is the transmittance for a givenlight source. This can be calculated, given thespectral transmittance and the spectral outputof the particular light source, as

f S(A)T(A)V(A)dALuminous transmittance =~-:---:-....:.....;:.-:....:..-f S(A)V(A)dA(12.3)

where S(A) is the spectral output of the lightsource, T(A) is the spectral transmittance ofthe eye, V(A) is the photopic relative luminous

Passage of light into theeye 111

efficiency value, A is wavelength and theregion of integration is over the visiblespectrum, typically 380-780 nm.

Scatter

Scatter is due to spatial variations in therefractive index within a medium, usually ona microscopic scale. Scatter is due to acombination of diffraction, refraction andreflection. For example, light incident on atransparent object embedded in a medium ofa different refractive index is scattered. Someof this light reflects from the incident surface,some passes through the surface and isrefracted in a forward direction, and some isreflected inside the object a number of timesand is finally refracted, either backwards orforwards. Finally, light outside the object butnear its edge is diffracted in a forwarddirection.The angular distribution of this scatter

depends upon a number of factors; in partic-ular, the size and shape of the scatteringparticles, the refractive index mismatch, thescale of the inhomogeneities relative to thewavelength, and whether the inhomogenei-ties have any regularity. Thus, the angulardistribution of scattered light may becomplex. The complexity increases if the lightis scattered more than once. In biologicalmedia, the angular distribution can be socomplex that it is not usually possible topredict the amount of forward-scatter frommeasures of back-scatter. This is particularlyimportant in visual optics, because it is easy tomeasure back-scattered light objectively butimpossible to measure forward-scatter objec-tively in the living eye. Forward-scatteredlight is usually measured subjectively usingpsychophysical measures (see Chapter 13).Since the cornea and lens consist of cells

and connective tissue, which containinhomogeneities on the scale of the order ofthe wavelength of light, it is surprising thatthey have a high transparency. By contrast,other cells (such as those in the skin) scatterlight strongly. Weneed to understand why thenormal cornea and lens have such a hightransparency. The aqueous and vitreoushumours are much more homogeneous, andare therefore less likely to scatter light.

112 Light and theeye

Even in healthy eyes scatter within the eyeis usually sufficient to cause a reduction invisual performance in the presence of brightlight sources (veiling glare), but this is muchworse if cataracts are present. Cataracts aredue to an aggregation of lens proteins, whichlead to an increase in inhomogeneity andanisotropy of the lens. The forward-scatterfrom cataracts produces a veiling glare, whileback-scatter from them reduces the amount oflight reaching the retina.The effect of forward-scattered light may be

partly mitigated by the directional sensitivityof the receptors - the Stiles-Crawford effect,which is described in Chapter 13. TheStiles-Crawford effect relates mainly to thecones, and is thus mainly a photopic phe-nomenon. It does not greatly reduce theeffects of scattered light under low light levelconditions.

Scattering theory

For particles whose dimensions are muchsmaller than the wavelengths underconsideration, and for which the scatteringparticles are mutually incoherent andindependent, Rayleigh theory may be used todescribe the scattering process. This theoryassumes that the scattering particles arepolarizable. Incident radiation polarizes theelectronic structure of each particle into theform of a dipole, in which the electrons arepushed towards one side of the particle,leaving a positive change on the opposite side.These dipoles oscillate in time with theincident radiation. According to classicalphysics, oscillating dipoles must radiateenergy; thus they absorb energy from theincident field and re-radiate it. The re-radiated energy is maximally radiated in adirection perpendicular to the dipole axis, andzero energy is radiated along the axis of thedipole.Rayleigh scattering predicts the following:

1. The radiation scattered at 90° to the directlytransmitted beam is completely polarized.

2. The amount of scattered light isproportional to the inverse of the fourthpower of the wavelength. Therefore, bluelight of 400 nm is scattered 9.4 times morestrongly than red light of 700 nm wave-length.

3. The amount of forward- and backward-scattering are the same.

Rayleigh theory does not apply when thesizes of scattering particles are comparable orgreater than the wavelength under consider-ation, and in this case one has to use morecomplex scattering models. The significanceof particle size is as follows:

1. If the oscillating dipole is much longer thanthe wavelength, the calculation of theamount of radiation emitted in anydirection must take into account theradiation from each point of the dipole.Since these are now at different distances,and hence have different phases withrespect to an observer or detector, inter-ference from the different points on thedipole must be taken into account.

2. If a scattering particle is much longer thanthe wavelength along the direction of lighttravel, there are a number of oscillatingdipoles in the particle which interfere in aconstructive manner in the forwarddirection, but tend to interfere destruc-tively in the backwards direction. Thesetrends become more pronounced withincrease in particle size. Therefore, as theparticle size increases, forward-scatterincreases at the expense of backward-scatter.

The prediction of the scattering propertiesof particles of arbitrary size and shape is verydifficult, if not impossible, but solutions areavailable for simple shapes. Because theamount of scattered light is proportional tothe amount of light incident on the particle,the greater the cross-sectional area of theparticle, the greater the scatter. For example,Mie theory explains the scattering by spheresof any size. For very large spheres, thescattering is independent of wavelength, andthe amount of scattered light is twice thatincident on the cross-section of the particle.For example, in liquid water aerosols thescattering is proportional to the total cross-sectional area of the droplets. This means that,for a given volume of water in droplet form,the scattering is inversely proportional to thedroplet radius or proportional to N1/3, whereN is the number of droplets.A third scattering theory is the Rayleigh-

Gans theory, or Rayleigh-Deybe theory

according to Kerker (1969). This theoryapplies when the refractive index of thescattering particles is close to that of themedium in which they are imbedded.Applied to spherical scattering particles, thistheory predicts that, for spheres, forward-scatter increases and back-scatter decreaseswith increase in size.The above scattering theories can be readily

applied when the scattering particles areindependent of each other. This assumes thatthey are weakly scattering and/or randomlyordered. These assumptions do not apply tothe cornea, which has an ordered structure.

CorneaThe bulk of the cornea is the corneal stroma(Chapter 2). The stroma contains 200-250 ormore layers (lamellae) of long cylindricalcollagen fibrils, with the thickness of eachlamella being about 2.0/Jm (Hogan et al.,1971). Fibrils within each lamella havediameters of 32-36 nm, and are separated by20-50 nm. The fibrils within a layer areparallel to each other, and uniform in size andspacing (Hogan et al., 1971). The fibrils withina layer are inclined at large angles to fibrils inadjacent lamellae. The refractive index offibres is about 1.47, and the surroundingground substance has a refractive index ofabout 1.354 (Maurice, 1969).Hart and Farrell (1969), using a theoretical

model of scattering by arrays of cylindricalstructures with the degree of regularity foundin real corneas, argued that the transparencyof the cornea is due to the regularity of thefibril separation. Their theoretical predictionsfor the rabbit cornea closely matchedexperimental measures of transparency. Theirtheory was based upon the diffraction/interference from regular arrays. According totheir theory, for an infinite array of equallyspaced point-scattering particles, with aseparation that is negligible compared towavelength, the scattered light destructivelyinterferes in all directions for all wavelengthsexcept in the direction of the incident beam.However, because the fibrils have finitediameters, their array is not perfectly regularand infinite, and because the spacing is notnegligible compared with the wavelength,there is some residual scattering that is

Passage of light into tile eye 113

wavelength-dependent. It follows that, if thisregularity is disrupted further, scatteringincreases.McCally and Farrell (1988) investigated the

wavelength dependency of scatter within thecornea and concluded that the range of theordering of the fibrils (e.g. short distanceversus long distance order) would affectscatter. They argued that if the order is shortrange, the dependency is proportional to theinverse of the cube of the wavelength (i.e,1/,1.3). This was supported experimentally forthe rabbit cornea. This cubic dependency is incontrast with the Rayleigh fourth power (i.e.1/).,4) dependency, which assumes noregularity of the scattering particles and thatthe particles have negligible dimensionscompared with the wavelength.

Lens

Because the lens is much thicker than thecornea and is composed of cells, we expect itto scatter more light than the cornea. Thishappens, with scatter increasing with age, andthe forward-scatter being greater than theback-scatter (Bettelheim and Ali, 1985).Hemenger (1988, 1992) argued that thescattering is caused by the lens fibre lattice.

Fluorescence

Fluorescence is the absorption of radiation bya medium at one wavelength, and theimmediate re-emission of radiation at longerwavelengths. The re-emission is isotropic andtherefore, in the eye, tends to produce auniform veiling glare on the retina.Fluorescence occurs mostly in the lens, and

increases with age and in people with cataract(Lerman and Borkman, 1976; Bleeker et al.,1986; Siik et al., 1993). The lens contains atleast three fluorescent compounds; one calledtryptophan, with a maximum sensitivity at290 nm, and at least two fluorogens, withmaximum sensitivities at 370 nm and 430 nm.Figure 12.5 shows the spectral emissionproperties of these compounds. Tryptophan ispresent at birth, and does not change greatlywith age, but the two fluorogens are notpresent at birth and accumulate throughoutlife.

114 Light and tileeye

80

TRP 33270

60=0";:'" 50's..,..,

40>.~

1leli'. 30

20

10

o+-O-T"T"1'TT"T""T'"r""f"'T-r-r-..,.,,..,.......,.,.-..,....,..,,.....,...,......,....,.......-.+250 300 350 400 450 500 550 soo

Wavelength (om)

Figure 12.5. The fluorescent emission profiles oftryptophan and two fluorogens for a 78-year-old humanlens (from Lerman and Berkman, 1978),

The veiling glare produced by fluorescencecan lead to a small loss in visual acuity of lowcontrast targets, an effect that increases withage (Elliott et al., 1993). The level of fluor-escence depends upon the spectral sensitivityof the fluorescent compound and the spectralprofile of the incident radiatiori-i.e. howmuch ultraviolet radiation and blue light iscontained in the illuminating source. Thecontribution from tryptophan is minimized,because the cornea absorbs much of theactivating wavelengths.

Birefringence

Optical materials such as water and glass areisotropic, which means that their physicalproperties are the same in all directions. Thisis because of the lack of atomic or molecularorder. In contrast, many crystals have a well-defined atomic order. In some crystals, such ascalcite, the physical properties (in particularthe speed of light) depend upon direction oftravel. This is referred to as anisotropy,Materials are called birefringent when thespeed of light (or refractive index) dependsnot only on direction of travel, but also on theorientation of the electric field. This can occurin glass and plastics when they are placedunder stress.

If unpolarized light enters a birefringentmaterial, a phenomenon called doublerefraction occurs. The incident ray splits intotwo parts on entering the material, with theelectric vectors of the rays at right angles toeach other. These rays are now completelylinearly polarized. One ray is refractedaccording to Snell's law and is called theordinary ray. The other ray is called the extra-ordinary ray, and does not obey this law.Some materials have a particular direction oraxis for which double refraction does notoccur, and crystals with this property arecalled uniaxial crystals. They have tworefractive indices, no for the ordinary ray andlIe for the extra-ordinary ray. The level ofbirefringence is quantified by the difference(l1e - 110 ) between these two refractive indices.In biological materials there is usually little

crystal-like atomic order, but there issometimes order at the tissue structural level.This is called fonn birefringence, to allowdifferentiation from the intrinsic birefrin-gence due to molecular arrangement. The eyeexhibits some birefringence, mostly due toform birefringence. This can occur only wherethere is some regularity to the structure, and itoccurs only in the cornea, lens and retina. Inthis section, we look at birefringence in thecornea and lens. Birefringence of the retina iscovered in Chapter 14.

Cornea

As mentioned in the above Scatter section, thebulk of the cornea consists of the approxi-mately 200 lamellae of the stroma, in whichthe fibrils are cylindrical in shape and areusually regularly spaced. The fibrils inadjacent layers lie at large and differentangles. Thus, we would expect each layer offibrils to exhibit form birefringence.According to Bour (1991), Wiener (1912)showed that an assembly of parallel rodsimmersed in a medium of lower index acts asa uniaxial crystal, with a positive bire-fringence with the optical axis in the directionof the rods. However, because of the largenumber of layers and their wide-rangingorientation, a light beam incident normallyon the cornea would not encounter fibrilswith any overall preferred orientation.Therefore, the cornea would not appear to be

birefringent. On the other hand, a beaminclined to the corneal surface would undergoa slight birefringence. Bour and LopezCardozo (1981) estimated corneal birefrin-gence to be +0.0020.In contrast with the uniaxial model of the

cornea, van Blokland and Verhelst (1987)argued that the cornea acts as a biaxial crystal,with the faster principal axis lying normal tothe cornea and the slower axis lying nasallydownwards.

Lens

The structure of the lens is different from thatof the cornea. The lens consists of fibres (about8 Jim in diameter) that are laid down radially,rather than being parallel as in the cornea.Therefore, we would not expect the same levelof birefringence. Weale (1979) gave valuesvarying between -0.5 x 10-6and -3.5 x 10-6.

Summary of main symbols

n, n' refractive indices on the incident andrefracted side of a surface

R Fresnel reflectance at a surfaceT Fresnel transmittance at a surface

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Bettelheim, F. A. and Ali, S. (1985). Light scattering ofnormal human lens III. Relationship between forwardand backward scatter of whole excised lenses. Exp. EyeRes.,41, 1-9.

Bleeker, J. C; van Best, J. A., Vri], L. et at. (1986).Autofluorescence of the lens in diabetic and healthysubjects by fluorophotometry. Invest. Ophthal. Vis. Sci.,27,791-4.

Boettner, E. A. and Wolter, J. R. (1962). Transmission of theocular media. Invest. Ophihal., 1, 776-83.

Bour, L. J. (1991). Polarized light and the eye. In Vision andVisual Dysfunction, vol. 1 of Visual Optics andInstrumentation rw N. Charrnan, ed.), pp. 310-25.MacMillan.

Bour, L. J. and Lopez Cardozo, N. J. (1981). On thebirefringence of the living human eye. Vision Res., 21,1413-21.

Curcio, J. A. and Petty, C. C. (1951). The near infraredabsorption spectrum of liquid water. J. Opt. Soc. Am.,41,302-4.

Passage of light into the eye 115

Elliott, D. B.,Yang, K.C. H., Dumbleton, K. and Cullen, A.P. (1993). Ultraviolet-induced lenticular fluorescence:intraocular straylight affecting visual function. VisionRes., 33, 1827-33.

Geeraets, W. J. and Berry, E. R. (1968). Ocular spectralcharacteristics as related to hazards from lasers andother light sources. Am. J. Ophthal., 66, 15-20.

Hart, R.W.and Farrell, R.A. (1969). Light scattering in thecornea. J. Opt. Soc. Am., 59, 766-74.

Hemenger, R. P. (1988). Small-angle intraocular lightscatter: a hypothesis concerning its source. J. Opt. Soc.Am. A., 5, 577-82.

Hemenger, R. P.(1992). Sources of intraocular light scatterfrominversion of an empirical glare function. AppliedOptics,31, 3687-93.

Hogan M. J., Alvarado J. A. and Weddell J. E. (1971).Histology of the Human Eye. An Atlas and Textbook, p. 89.W. B.Saunders.

Hulburt, E. O. (1945). Optics of distilled and naturalwater. J. Opt. Soc. Am., 35, 698-705.

Johnson, C. A., Howard, D. L., Marshall, D. and Shu, H.(1993). A non-invasive video-based method ofmeasuring lens transmission properties of the humaneye. Optom. Vis.ScL,70, 944-55.

Kerker, M. (1969). TheScattering of Light, p. 414. AcademicPress.

Lerman, S. and Borkrnan, R. F. (1976). Spectro-scopic evaluation and classification of thenormal, aging and cataractous lens. Ophthal. Res., 8,335-53.

Lerman, S. and Borkman, R. F. (1978). Ultravioletradiation in the aging and cataractous lens. A survey.Acta Ophthal., 56, 139-49.

Ludvigh, E. and McCarthy, E. F. (1938). Absorption ofvisible light by the refractive media of the human eye.Arch. Ophthal, 20, 37-51.

McCally, R. L. and Farrell, R. A. (1988). Interaction of lightand the cornea: light scattering versus transparency. InThe Cornea: Transactions of the World Congress on theCornea 111. (H. D. Cavanagh, ed.), pp. 165-71. RavenPress.

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Mellerio, J. (1971). Light absorption and scatter in thehuman lens. Vision Res.,11, 129-41.

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van Blokland, G. J. and Verhelst, S. C. (1987). Corneal

116 Lightand theeye

polarization in the living human eye explained with abiaxial model. /. Opt. Soc. Am. A, 4, 82-90.

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13Light level at the retina

Introduction

In the previous chapter, we considered thetransmittance of the ocular media. Given thespectral transmittance data of the media, wecan determine the amount of radiation andlight reaching the retina from a source ofknown spectral output. The results shown inFigure 12.2 indicate that 50-90 per cent of thelight entering the eye reaches the retina asimage-forming light (although it is importantto be aware that results from different studiesvary considerably, and there is considerableage-dependence).Ideally, the spectral and spatial light

distribution in the retinal image should beproportional to that in the object. In realitythis is not so, because of the effects of aber-rations, absorption, diffraction and scatter.Absorption and diffraction are wavelength-dependent, and aberrations, diffraction andscatter affect the spatial distribution. In thischapter, we assess the spatial light distri-bution in the image.

Retinal illuminance: directlytransmitted light

In this section, we present equations that canbe used to calculate the retinal light level orilluminance, given the brightness of the object(luminance of large area sources or luminousintensity for effectively point sources). Weassume that the eye is correctly focused on

that object, and therefore the equations are notstrictly valid if the object is out of focus. If thelevel of defocus is small compared with thesize of the object, the defocus has little effecton retinal illuminance except near the edge ofthe object.On the other hand, if the defocus islarge compared to the size of the object, theequations give very misleading results. This isparticularly so for point sources. Campbell(1994) presented a scheme for extendedsources that can be used whether the eye isfocused or not. Campbell's scheme is forcalculating retinal irradiance, not retinalilluminance, but the equations apply equallyto illuminance.In this section, we consider only directly

transmitted light, i.e. image forming light. Todetermine the retinal illuminance of imagesformed from this directly transmitted light, itis most convenient to divide this sectioninto two parts; one dealing with on-axisimagery, and the other dealing with off-axisimagery.

On axisLarge area sourcesFor large area sources, small angle scatter,aberrations and diffraction can be neglected,and then the light distribution in the imagehas the same form as the object. When the eyeis observing an object of large angularsub tense which has a luminance L (cd/rn-),the corresponding retinal illuminance E' is

118 Li~11t and tilee!le

orE'T== 106E'/ (rF2) troland (13.4b)If we now use the approximation that thepower F is 60 0, we can write these equationsas

same units for pupil diameter, and if wereplace the pupil diameter 0 in equation(13.2a) (which is in metres) with the diameter0mm in millimetres, we haveE' == rF2E'T/106Iux (13.4a)

(13.8)

(l3.5a)

(l3.5b)

where d is the distance of the source and I isthe luminous intensity of the source.For the diffraction limited (i.e. aberration-

free) image, the peak value of the illuminanceis given by the equation (Smith and Atchison,1997)

r _ [rrF ]2 4E - tl 41l'dA. D

E'T== (278/r)E' troland

The point source - diffraction limitedThe preceding equations do not apply forvery small sources imaged without anyaberration. In this case, the geometricaloptical image is smaller than the diffractionlimited point spread function, which has anangular radius8 = 1.22A;0 (13.6)

In general, while the point spread function isaffected by aberrations and diffraction, onlythe diffraction effect is easily predicted for anyoptical system.The total luminous flux in the image of a

point source, whether aberrated or not, isgiven by the equation (Smith and Atchison,1997)

mID2Flllx =4T (13.7)

E' == 0.0036rE'Tluxor

The trolandThe retinal illuminance is sometimesexpressed in terms of trolands, which are theproduct of the object luminance L measured incd/m2 and the area A of the pupil measuredin square millimetres. Thus the retinalilluminance E'T in trolands for a source ofluminance L and viewed through a pupil ofarea Amm in square millimetres is give by theequationE'T= LAmm trolands (13.3a)and sinceAmm = rrD2mm/4thenE'T= nl. 0 2mm /4 trolands (l3.3b)

given by the equation (Smith and Atchison,1997)E' =mLIl'2sin2(a') lux (13.1)where r is the transmittance of the ocularmedia ("" 0.6 to 0.9) taking into account lightlosses due to reflection, absorption andscatter, n' is the refractive index of the vitreoushumour (usually taken as 1.336), and a' is thehalf angular subtense of the exit pupilmeasured at the retina. We can re-express thisequation in terms of the (entrance) pupildiameter 0 rather than the angle a' asE' =mL02P/41ux (13.2a)where F is the equivalent power of the eyeand has a value of about 60 O.We can also express the retinal illuminance

in terms of area A of the (entrance) pupil,instead of its diameter 0, asE' = rLAF2 lux (13.2b)Since the power of the average eye is close to60 0, and if we take 0 and A in millimetreunits, the above two equations can beapproximated by the equations

E' "" 0.002830rL02mm or "" 0.003600rLAmm lux(l3.2c)

Relationship between troland and luxWe can relate retinal illuminance E' in luxwith retinal illuminance E'T in troland byequation (13.2a) and equation (13.3b). Inrelating these two equations, we must use the

These equations demonstrate two usefulresults:1. The total luminous flux in a point spread

function is proportional to the square of thepupil diameter (i.e, proportional to pupilarea).

Ligl1t leucl at tire retilia 119

Object

Figure 13.1. Geometry for calculating the retinalilluminance for peripheral large area sources. See textfor details.

an annular zone of this source as shown inFigure 13.1. The luminous flux entering thepupil of the eye is given by the equationFlux =21tLA(0)p sin(O) 80 (13.9)where A(O)p is the projected or apparent areaof the circular pupil in the direction of thesource. This equation shows that theluminous flux incident on the pupil dependsonly upon the solid angle of the annulus, andnot on its distance and shape.

If we now assume that this annular lightsource is focused on the retina, it is imaged asan annular zone on the retina as shown in thefigure. The area of this zone is given by theequationArea= s'80'21tY'/cos(i') (13.10)where the internal and external angles areconnected by some function 9'= f(9). Forsmall angles, paraxial theory predicts thattan(8') =m tan(9) (13.11)where m has a value of about 0.82 (Chapter 5).For larger angles, we should use a more

0'

Entrancepupil of eye

C\

Light entering eye

Exit pupilof eyeThe retinal illuminance of off-axis sources is a

little more complicated than for on-axissources. In conventional optical systems suchas a camera, the image plane illuminancedecreases as the fourth power of the cosine ofthe peripheral angle, a result often known asthe (OS4 law. This is due to a combination ofthe following:1. The reduction in apparent size of the

peripheral pupil (proportional to cos(angle».

2. The increase in distance from the exit pupilto the image on the image plane (theinverse square law, proportional to cos?(angle».

3. The inclination of the image plane to thedirection of the incident beam (tilt, which isproportional to cosianglej).In the case of the eye, the cos" law does not

hold. While the effective pupil area decreaseswith distance off-axis, approximately ascos(angle), the curved shape of the retinameans that the factors 2 and 3 above do notapply. The curved retina puts the imagesurface closer to the exit pupil, and tilts thenormal to the surface in the direction of theexit pupil.How the retinal light level depends upon

peripheral angle is important in the measure-ment of visual fields and the calculation of therisk of the retina being damaged byhazardous radiation sources. Because of thisimportance, there have been a number ofstudies in the area (Drasdo and Fowler, 1974;Bedell and Katz, 1982; Kooijman, 1983;Kooijman and Witmer, 1986;Charman, 1989).Using Figure 13.1, we examine this situ-

ation with a simple model of the eye lookingat a plane Lambertian source of infinitesubtense and of luminance L. Let us consider

Off-axis or peripheral sources

2. The peak illuminance is proportional to thefourth power of the pupil diameter or thesquare of the area.

The point source - aberratedThe light distributions in real eyes and theeffect of aberrations and diffraction arediscussed in detail in Chapter 18.

120 Light and the c!li'

(13.12)

Figure 13.2. Variation of retinal illuminance withperipheral angle for a large area source, predicted fromparaxial raytracing with a simple model eye.

90

cos4 prediction

equation (13.12)

10 20 30 40 50 60 70 80

Angle in air (deg)

.......".

...........

............

1.0

0.9

0.8

llJu 0.7c'"c'E 0.6.2::; 0.5c

~ 0.4llJ;> 0.3~"0e::: 0.2

0.\

0.00

In order to calculate values for this equation,we need to relate s', Y' and i' to O. We can dothis as follows. First, from the figure we haveY' =2' tan(O') (13.13a)Wecan find the intersection (Y', 2') of this linewith the retina, which we will assume to becircular in cross-section and described by theequationy'Z + (2' - a)z =R'z (13.13b)

and then we have

accurate analysis (e.g. ray trace through afinite schematic eye) to determine therelationship, but we will leave this calculationfor Chapters 15 and 16.The retinal illuminance E' is the ratio of flux

to illuminated area and thus is given by theequation

, _ LA(9)psin(O)cos(i')E - s'Y'(oO'loO)

and equation (13.11) gives00'1DO=msecz(0)1sec2(0') (13.14)

Figure 13.2 shows estimates of the relativeretinal illuminance as a function of off-axisangle for a spherical retina (R' =12mm) andthe distance a=8 mm for a projected pupilarea A(O)p falling off as cos(O). The relativeretinal illuminance assuming the cos4 law isshown for comparison.Figure 13.2 predicts that, as an object moves

off-axis, the retinal illuminance decreases,although not as rapidly as predicted by thecos! law. The simple model neglects a numberof important factors:1. The retina is not spherical.2. The effective pupil of the eye is slightly

larger at peripheral angles than thatpredicted by the simple cosine reduction(Chapter 3).

3. The exact relationship between the externalangle (J and internal angle O~ Equation(13.11) is only accurate for small angles. Forlarger angles, the aberrations affect therelationship between 0 and 0'.

s' = -V (Y'z + 2'z)

tan(l/>') =Y'1(2' - a)

andi' = tfJ' - 0'

(13.13c)(13.13d)

(13.13e)

4. Variation of ocular transmittance withperipheral angle. We expect the trans-mittance to decrease as peripheral angleincreases because of the increase in pathlength within the lens.

5. Aberrations of the eye increase withincrease in peripheral angle, and this willaffect the illuminance distribution for smallsources.Kooijman and Witmer (1986) found that the

light level in excised eyes reduces moreslowly with increase in peripheral angle thangiven by the above theoretical predictions(Figure 13.2). In Chapter 16 we use morerealistic models of the eye, with more accurateestimates of apparent pupil area A( O)p. and ofthe relationship between external and internalangles 0 and O~

Retinal illuminance: scattered light

As discussed in the previous chapter, asignificant amount of light entering the eye isscattered forward, but out of the image-forming beam. In this section, we examine thelight distribution due to scatter from a pointsource of light. This can be considered interms of the retinal illuminance or an equiv-alent luminance of the image. Because we do

not yet have accurate models of the complexnature of scatter, and because the light levelon the retina is too low to be measureddirectly, the only determinations that can bemade of the scattered light distribution aresubjective measurements.Several investigators have measured the

light level at the retina due to scatter from aglare source in terms of the equivalentveiling luminance. In the presence of a glaresource, the threshold luminance of a smallpatch of light was determined. The glaresource was then turned off and the luminanceLy of a uniform background determined,which gave the same threshold for the smallpatch of light. This equivalent veilingluminance was fitted at the fovea Ly ( 8) byequations of the form

Ly(8) =KE/8" (13.15)

where E is the illuminance in the plane of theeye of the glare source, 8 is the off-axis anglein degrees of the glare source, and K and n areconstants depending upon the particularinvestigation and the range of values of 8.Setting n to 2, this equation is well known asthe Stiles-Holliday relationship. Some specificequations are as follows:

Ly(8) =9.2 E/82 2.5°~8~25°Holladay (1927)

(13.16a)

1° s 8~ 10°Stiles (1929)

(13.16b)

1° < 8< 8°Vos and Bouman (1959)

(13.16c)

Ly(9) = 29 E/(9 + 0.13)2.8 0.15° < 8 < 8°Walraven (1973)

(13.16d)

with Ly(8) in cd/m2,E in lux and 8 in degrees.Vos (1984) showed that, in young subjects,

the equivalent veiling luminance is producedin approximately equal proportions by thecornea, lens and fundus. Scatter by the corneaand lens is discussed in Chapter 12, andscatter by the fundus is discussed in Chapter14.Various attempts at modelling the equiv-

alent veiling luminance have been made. Forexample, Fry (1954)used a reduced schematic

Light level at the retina 121

eye with a refractive index /1' (1.333)and axiallength t (20mm), and derived the theoreticalequationLv(8) = (2/3)tR E[cot(8/n') - cos(8/n')]ln'2

(13.17)where R is a constant. If E is given a value of 1and R is given a value of 3.39, this equationgives the same value of Ly(8) at 5° asequations (13.16a) and (13.16b).

Effect of position in the lens of ascattering centre

It is often observed that the effect of ascattering centre is strongly dependent uponits position. For example, posterior polarcataracts have more severe effects on visionthan cataracts of similar size that are situatedmore anteriorly. This has often been explainedby the proximity of the scattering centre to thenodal points of the eye, but there is no opticalreason why this should affect the scatterproperties.

Measurement of angular distributionof scattered light

The light distribution of the image of a pointsource is known as the point spread function.The shape and width of the point spreadfunction depend upon the levels of diff-raction, aberrations and scatter, and upon theshape of the pupil. The effects of diffractionand aberration are discussed in Chapter 18,where it is shown that diffraction andaberrations produce a point spread functionthat has a half-width of a few minutes of arc.On the other hand, scatter causes light to bedirected over much wider angles. Therefore,the light in the periphery of the point spreadfunction is easily identified as scattered light.Close to the centre of the point spreadfunction, most of the light is from bothdiffraction and aberrations, and it is difficultto extract the scattered component. For thisreason, some investigators of scattered light(e.g. van den Berg, 1995) looked only at thatcomponent of light scattered through an anglegreater than 1°.Various methods have been developed to

measure the angular distribution of scattered

122 Light and til""!I"

where

Black body source

(13.22)

(13.19)

The form of Lr(A.) in the previous equationdepends upon the nature of the source. Onetype of source of particular interest is theblack body source. For a black body source ofknown temperature T Kelvin, the spectralradiance Lr(A.) from Planck's law is

1 _ cj 3Lr(lI.) - 1t,1,5[eCz/ (XT) _ 1] W I(st. m )

N =JE;("A.) W("A.)"A. axtot he

where W(,1,) may be the photopic or scotopicrelative luminous efficiency functions, i.e.V(,1,) or V'(,1,). Now from equation (13.2a), foran extended sourceE;(A.) = rnLr(A.)D2F214 (13.20)

where Lr(A.) is the corresponding spectralradiance of the source in W I(st. m3). Equation(13.19) can now be written in the form

't1tD2F2JN tot =~ Lr("A.)W("A.)A.dA. photons/rn-Zs

(13.21)

photonsZmc/s in the same bandwidth isgiven by the equation

N("A.)~"A. = E;("A.)~"A. = E;("A.)M"A. (13.18)hv he

where h is Planck's constant (6.62620 x 10-34[.s), c is the speed of light (2.99792 x 108 m/s),and v is the frequency in Hertz. Weighting thephoton density by a weighting function W(,1,)over the entire spectrum, we have

Flicker method (e.g. van den Berg andI}speert, 1992)

The glare source, which consists of annuli ofvarious angular sizes, flickers at about 8 Hz.One of the annuli is illuminated. If the testpatch in the middle of the annuli is very dim,the subject sees the test patch flicker becauselight is scattered onto the subject's fovea. Thesubject increases the luminance of the testpatch, which flickers in counterphase to theglare source. When no flicker is apparent, theamount of luminance modulation of the testpatch equals the amount of light scattered.This is determined for all the annuli of theglare source.

Photon density levels

light. All of these methods assume that thescattering source has a very small angularsubtense, and can therefore be regarded as apoint source of light. Two subjective methodsare described below.

Conventional threshold method

The essence of this method was given near thestart of this section. The threshold luminanceof a small patch of light is determined in thepresence of a glare source. With the glaresource turned off, the luminance of a uniformbackground is determined that produces thesame threshold for the small patch of light.This luminance is the equivalent veilingluminance, and it is determined for a range ofoff-axis angles of the glare source.

(13.23)

Real sources

cj =3.7418 X 10-6W.m2 andc2 =1.4388 X 10-2 m.K.

Sometimes there is a need to express the lightlevel at the retina in terms of photon orquantal density (e.g. photona/rn'[/s), ratherthan in terms of lux or trolands. We can findan equation for this quantity by proceeding asfollows. For real sources, the amount of radiationLet us suppose that the spectral irradiance emitted is always less than that of the black

at the retina is E' (,1,).1,1, in the bandwidth .1,1,. body, and the ratio of the emittances is theThe corresponding number N(,1,).1,1, of emissivity e.That is

output of real sourcee=----:-:---:-:-......:........,...-----------output of the black body at the same temperature

(13.28)

This is usually a function of wavelength andtemperature. However, for approximatecalculations a constant value can be assumed,and values for various materials can be foundin the Handbook ofChemistryand Physics (1975).

Maxwellian view

In many visual optical instruments, a smalllight source S provides. a uniform f~el~-of­

view. If the entrance pupil of the eye coincideswith the image S' of the source, the field-of-view has maximum width and maximumuniformity of luminance. This arrangement ~s

called Maxwellian view (Figure 13.3).The exitpupil of the instrument at E' is unifor~ly

illuminated, and subtends an angular radiusa' to the eye. The eye is focused approxi-mately on this exit pupil. If S' is much smallerthan the entrance pupil of the eye, pupilfluctuations and small eye movements do notaffect retinal illuminance.Illumination and viewing with Maxwellian

view is optically very different from con-ventional viewing. For example, the fact thatthe effective pupil may be much smaller thanthe normal pupil places limits on expectedvisual acuity. The illumination may no longerbe completely incoherent, so that con-ventional incoherent image quality criteriasuch as the point spread and modulationtransfer functions may no longer be valid(Westheimer, 1966). Here, only the light levelsare considered.

Equivalent luminance of aLambertian source

Based on equation (13.1), the light in the exitpupil of the instrument produces an illumi-

II.I

Entrancepupil

Figure 13.3. Maxwellian view.

LiglIt levellit tileretina 123

nance E1 at the plane of the entrance pupil ofthe eye, given byE1 = nl: sinz(a') (13.24)where L is the luminance of the instrument'sexit pupil, and this exit pupil subtends anangular radius a' to the entrance pupil of theeye (Figure 13.3). If the image S' of the sourceS has an area A', the luminous flux Flux;falling on the pla~e of the entrance pupil ofthe eye is given by

Flux; = E1A's = 1tL sinZ(a.')A's (13.25)If S' is smaller than the entrance pupil of theeye, all of this flux falls on the retina, and theexit pupil appears to have a luminance L'.

If the observed bright field of the instru-ment exit pupil is replaced by a Lambertiansource of the same luminance L', theilluminance Ez in the plane of the entrancepupil of the eye isEz=nL'sinz(a') (13.26)Since the new source is Lambertian, light fromit fills the pupil of the eye completely. If theentrance pupil of the eye has area Ap' theluminous flux Fluxz falling on the plane of theentrance pupil of the eye is given byFluxz=EzAp=nL'sinZ(a')Ap (13.27)The illuminated area of the retina is the samein the cases given by equations (13.25) and(13.27), and therefore the ratio of retinalilluminances and hence luminances is theratio of the fluxes entering the eye. Since thetwo luminances are equal, the fluxes areequal, andni. 'sinz(aJAp = ni. sinZ( aJA's

ThereforeL'=LA'/As p

II.I

Exit pupil

0.8

c.2tic

0.6..2"2..2~eu 0.-1Iv:~c'5

0.2

00

124 Light and tileeye

As an example, if the area of the Maxwellianimage is 2 mm2, the area of the entrance pupilof the eye is 10mm- and the luminance at theinstrument exit pupil is 100 cd/m2, theequivalent luminance of a Lambertian sourceis 100x 2/10 =20 cd/rn-,

Adapting pupil size

If the pupil is fully illuminated, pupil sizeaffects retinal illuminance, which in turncontrols pupil size. This feedback systemcannot operate in Maxwellian view, where thesource image is usually smaller than thesmallest pupil size. Palmer (1966) found that,provided the source image in Maxwellianview is smaller than the natural pupil, thenatural pupil is larger than is the case for thesame light flux entering the eye in normalviewing.

The Stiles-Crawford effect

Stiles and Crawford (1933) discovered that theluminous efficiency of a beam of lightentering the eye and incident on the foveadepends upon the entry point in the pupil.This phenomenon is known as the Stiles-Crawford effect of the first kind, but moregenerally as the Stiles-Crawford effect. Later,Stiles (1937) reported that varying the entry ofthe beam also altered the perceived saturationand hue of the light. This colour effect, calledthe Stiles-Crawford effect of the second kind,is not discussed further here.The Stiles-Crawford effect is important to

visual photometry and retinal image quality,It can be considered both as a neural and as an

optical phenomenon, as it is retinal in originand is explained as a consequence of thewave-guide properties of the photoreceptors.It is predominantly a cone phenomenon, andhence predominantly a photopic phenom-enon, A good review of the Stiles-Crawford effect is given by Enoch andLakshminarayanan (1991).A number of different mathematical

functions have been used to describe theStiles-Crawford effect, with the most popularone being a Gaussian distribution as first usedby Stiles (1937). This function is usually anexcellent fit to experimental data out to 3 mmfrom the peak of the function, and has theaddition virtue of simplicity. We describe the

...~ .."'~." '"

" -", -............', ,

" ,..... "' ..

', ,', ,-, ,', ,

', '..... -.'. ,

', ,', ,

', '..... ",', ,.. ,

~ =0.057 mm-2 \'" -,,_f} =O.116 mm-2 •.•• ... ...... .. ...-,~ =0.173 mm-2 ••••••

<,<,

2 3 4Pupil radius (mrn)

Figure 13.4.The Stiles-Crawford function for f3 values of0.057,0.116 and 0.173, which are 2.5 per cent, 50 per centand 97.5 per cent population limits, respectively(Applegate and Lakshminarayanan, 1993).

Table 13.1. Some published values of the Stiles-Crawford f3 parameter and the position of the peak of theStiles-Crawford functions.

Investigation

Dunnewold (1964)bApplegate and Lakshminarayanan (1993)d

No. subjects/eyes

29/4749/49

f3 ± 1 sd (»me2)

0.116 ± 0.02ge

peak ± 1 sd (mm)

0.37 ± 0,78 n, 0.29 ± 0.80 Sc0.47 ± 0.68 n, 0.20 ± 0,64 s

n - nasal; 5 - superior"lhe Il,o values in Applegate and Lakshminorayanan (1993) have beenconverted to Jlvalues using equation (13.29b).bRelative to centre of pupil.'Determined by Applegate and Lakshminarayanan (1993) from figure 45 of Dunnewold (1%1).dRelative to first Purkinje image.'""Mean of horizontal and vertical values.

Light level at tileretina 125

Photometric efficiency and reducedaperture modelPhotometric efficiencyAlthough the Stiles-Crawford effect is aretinal phenomenon, the pupil of an eye witha Stiles-Crawford effect can be treated asbeing less efficient at transmitting light than apupil of the same diameter when there is noStiles-Crawford effect. We can quantify thisusing the concept of photometric efficiency.Following Martin (1961), we denote it as 5(15)and define it as

Stiles-Crawford effect function Le(r) asLe(r)= exp(-f3r2) (13.29)where r is the distance in the pupil from thepeak of the function. The function isnormalized to have a value of 1 at the peak.The Stiles-Crawford co-efficient 13 describesthe steepness of the function, and is assumedto reflect the directionality (variation inalignment) of the photoreceptor populationbeing tested. It may not have the same valuefor measurements in different meridians,particularly in eyes affected by retinalpathology. Measured 13 co-efficients for the

5C) = Effective light collected by a pupil of radius (JJ)P Actual light collected by the same pupil (13.30)

We can write this as

(13.31)

(13.32a)

(13.32b)

In terms of the entrance pupil diameter D, thephotometric efficiency 5(D) is

5(D) =4[1- exp(-f3D2/ 4)]f3D2

P-I2nLe(r)rdr

5(15) =....;:0:....-.....,.....".....-_np2

Ifwe use equation (13.29) for Le(r), the integralis easily solved to give

5C) = 1 - exp(-f3152)p f3p2

large-scale study of Applegate andLakshminarayanan (1993) are given in Table13.1, and Figure 13.4 shows the Stiles-Crawford effect function for different 13 co-efficients. Combining the data across manystudies gives a mean value of 0.12 (Applegateand Lakshminarayanan, 1993).In many papers, the Stiles-Crawford effect

function is given by equations similar toLe(r) = 1O-PJOr2 (13.29a)

The co-efficients in equations (13.29) and(13.29a) are related by13=In(10)f3lO =2.30261310 (13.29b)

Peak of the Stiles-Crawford effect

The position of the peak does not usuallycorrespond to the centre of the pupil (whichitself varies with pupil size), and it also variesbetween individuals. Its position is con-sidered to reflect the overall alignment in thepupil of the photoreceptor population beingtested. The ray distance r in the pupil shouldbe measured from the peak. Measured valuesof the peak from two large-scale studies aregiven in Table 13.1. Combining the data acrossmany studies gives mean values of "" 0.4mmnasal and "" 0.2mm superior relative to thepupil reference, whether this be the pupilcentre or the pupillary axis (Applegate andLakshminarayanan, 1993).

Hence, if we express the luminance of animage in terms of the pupil area, we shouldmultiply the latter by the appropriate factor5(15) or 5(0).

Reduced aperture modelSometimes we may want to know the dia-meter D* of the equivalent, but smaller, pupilthat would collect the same luminous flux ifthere was no Stiles-Crawford effect. Therelationship is simply0* = 0...J[5(15)] or 0* = 0...J[5(0)] (13.33)

Figure 13.5 shows 0* as a function of 0 forvarious values of 13.

126 Ligl1t and the eye8+-_~-..J_~_---'-_~_-'--_~-+ in the mid-periphery of the visual field (Enoch

and Lakshminarayanan, 1991).

AccommodationBlank etal. (1975) reported that the peak of theStiles-Crawford effect shifted between 0.4mm and 1.0 mm nasally in three subjects witha 9 0 increase in accommodative stimulus.They attributed this to retinal stretch duringaccommodation.

LuminanceThe Stiles-Crawford effect exists also underscotopic conditions, but is small comparedwith the effect found under photopicconditions (Crawford, 1937; van Loo andEnoch, 1975). The magnitude of the effectmakes a gradual change over the mesopicrange.

" ., '.~.....

,:.:....,.'

.,,:.~.,

\3= 0.057 111111-2

\3=0.116111111-2

\3=().I73 111111-2

0+--..----,-----,.---,.--...,...--,---.---+o

EE 6

246Pupil diameter D (rnm)

Figure 13.5.The photometric equivalent pupil diameterD' as a function of pupil diameter D for f3 values of0.057, 0.116 and 0.173.

Some factors influencing theStiles-Crawford effect

WavelengthThe Stiles-Crawford effect varies with wave-length. For the fovea, it shows a minimum inthe green and larger effects at both the shortand long ends of the visible spectrum (Stiles,1937 and 1939; Enoch and Stiles, 1961;Wijngaard and van Kruysbergen, 1975).Parafoveally (50 from the fovea) measuredfunctions are minimal for blue and greenregions of the spectrum, and increase only forthe long wavelengths (Stiles, 1939). As anexample, Stiles (1937) found for his own eye(foveal vision) that f3 varied between 0.13 and0.17 over the visible spectrum. If allowance ismade for lens absorption, estimates of theStiles-Crawford effect increase, particularly atwavelengths less than 450 nm (Weale, 1961;Mellerio,1971).

Theory

Since the discovery of the Stiles-Crawfordeffect, a number of explanations have beenoffered for it. Most theories are based aroundthe assumption that the photoreceptors act aswave-guides.O'Brien's (1946)geometric optical theory of

the Stiles-Crawford effect considered that thecones act as wave-guides, using total internalreflection to channel the light to the photo-pigment. As the ray position in the pupilincreases away from the centre, the rays enterthe cones at a larger angle to increase theangle of incidence on the internal wall of thecones and hence increase the probability oflight loss through the walls. His theory doesnot explain the influence of wavelengthadequately. A comprehensive theory waspresented by Snyder and Pask (1973). Theymodified the wave-guide model by using aphysical optical approach, which betterpredicts the Stiles-Crawford effect, includingthe effect of wavelength.

EccentricityUnder photopic conditions, the Stiles-Crawford effect increases from the fovealcentre out to approximately 2-30 eccentricity,where the value of f3 may have doubled, afterwhich it declines slowly to reach foveal levels

Measurement

The Stiles-Crawford effect has been measuredby a number of subjective and objectivetechniques (Enoch and Lakshminarayanan,

Light level at the retina 127

1991). The values of f3 and the peak positionmentioned in this section are from subjectivemethods. Subjective techniques includeflicker photometry and photometric match-ing. Objective methods include fundusreflectometry, consensual pupillary response,the electroretinogram, and the visual evokedresponse.

Retinal illuminance: directly transmitted lightE'T retinal illuminance in trolandsI luminous intensity of a point

source in cd

Retinal illuminance: scattered lightE illuminance in plane of eyeLv(8) equivalent veiling luminance in a

direction ()

Summary of main symbols

Role of the Stiles-Crawford effect

As mentioned at the start of this section, theStiles-Crawford effect is important in visualphotometry and retinal image quality.Concerning visual photometry, it reduces theeffective retinal illuminance at photopiclighting levels. As discussed earlier, and asgiven by equations (13.32) and (13.33), this isequivalent to having a smaller effective pupilsize.The Stiles-Crawford effect reduces the

detrimental effects of scattered light on retinalimage quality at photopic lighting levels,although the extent to which this improvesvisual performance is not known (Enoch,1972). In a similar fashion, it reduces theeffects of defocus and aberrations on retinalimage quality and thus reduces theirinfluence on visual performance. The Stiles-Crawford effect can be included in opticalmodelling of the eye as an apodization, whichmeans that it can be treated as an optical filterof variable density placed at the pupil. Itsinfluence in this respect, as discussed furtherin Chapter 18, seems to be small.

0*

f35(p)

5(0)

Stiles-Crauiford effectLe(r) luminous efficiency for a ray

entering pupil at a height rStiles-Crawford co-efficientphotometric efficiency for pupilradius pphotometric efficiency for pupildiameter 0diameter of equivalent pupil

Photon density levelsN(A).1A photon density in photons/rn-ys

Maxwellian viewA area of pupil of eyeA~ area of image of light source,

imaged in the pupil plane of theeye

L' apparent luminance of field

ReferencesApplegate, R. A. and Lakshminarayanan, V. (1993).Parametric representation of Stiles-Crawfordfunctions: normal variation of peak location anddirectionality. J. Opt. Soc. Am. A, 10, 1611-23.

Bedell, H. E. and Katz, L. M. (1982). On the necessity ofcorrecting peripheral target luminance for pupillaryarea. Am. J. Optom. Pilysio/. Opt., 59,767-9.

Blank, K., Provine, R. P. and Enoch, J. M. (1975). Shift inthe peak of the photopic Stiles-Crawford function withmarked accommodation. Vision Res., 15, 499-507.

Campbell, C. (1994). Calculation method for retinalirradiance from extended sources. Ophthal. Physiol.Opt., 14, 326-9.

Charman, W. N. (1989). Light on the peripheral retina.Ophtha/. Pilysio/. Opt., 9, 91-2.

Crawford, B. H. (1937). The luminous efficiency of lightrays entering the eye pupil at different points and itsrelation to brightness threshold measurements. Proc. R.Soc. B., 124, 81-96.

Drasdo, N. and Fowler, C. W. (1974). Non-linearprojection of the retinal image in a wide-angleschematic eye. Br. J. Ophtha/., 58, 709-14.

Dunnewold, C. J. (1964). On the Campbell andStiles-Crawford effects and their Clinical Importance.

area of the pupil in square metresarea of the pupil in squaremillimetrespupil diameter in metrespupil diameter in millimetresretinal illuminance in luxequivalent power of the eye indioptres (i.e. rrr")luminance of an extended sourcein cd/m2

ray height in pupilmean luminous transmittance ofocular media of the eye

rr

L

o°mmE'F

GeneralAAmm

128 Light 11Ild Ihl'eye

Institute for Perception. The Netherlands NationalDefence Research Organisation RVOTNO.

Enoch, J. M. (1972). Retinal receptor orientation and therole of fiber optics in vision. Am. J. Oplom. Arch. Am.Acad. Optom., 49, 455-71.

Enoch, J. M. and Lakshminarayanan, V. (1991). Retinalfibre optics. In Visual Optics alld 1I1slrumePllalioll, vol. 1of Cronly-Dillon, J.R. Visioll alld Visual Dysfl/llclicm (W.N. Charrnan, ed.), Macmillan Press.

Enoch, J. M. and Stiles, W.S. (1961). The colour change ofmonochromatic light with retinal angle of incidence.Oplica Acta,8, 329-58.

Fry, G. A. (1954). A re-evaluation of the scattering theoryof glare. Illuminating Engineering, 49, 98-102.

Handbook of ClTemislry and Pltusic« (1975). Weast, R. C.(ed.). 56th edn. CRC Press.

Holladay, L. L. (1927). Action of a light-source in the fieldof view in lowering visibility. J. 01'1. Soc. Am., 14, 1-15.

Kooijman, A. C. (1983). Light distribution on the retina ofa wide-angle theoretical eye. J. 01'1. Soc. Am., 73,1544-50.

Kooijman, A. C. and Witmer, F. K. (1986). Ganzfeld lightdistribution on the retina of human and rabbit eyes:calculations and ill vilromeasurements. J. 01'1. Soc. Am.A,3, 2116-20.

Martin, L C. (1961). Tcch II ical Optics Vol II, pp. 247-8.Pitman and Sons.

Mellerio, J. (1971). Light absorption and scatter in thehuman lens. Visioll Res.,11, 129-41.

O'Brien, B. (1946). A theory of the Stiles and Crawfordeffect. J. 01'1. Soc. Am., 36, 506-9.

Palmer, D. A. (1966). The size of the human pupil inviewing through optical instruments. Visioll Rcs., 6,471-7.

Smith, G. and Atchison, D. A. (1997). ti« Eye alld VisualOplical IllslrumCllts, pp. 286, 308, 547. CambridgeUniversity Press.

Snyder, A. W. and Pask. C. (1973). The Stiles-Crawford

effect - explanation and consequences. Visioll Rcs., 13,1115-37.

Stiles, W. S. (1929). The effect of glare in the brightnessdifference threshold. Proc. R. Soc. 8.,104,322-51.

Stiles, W. S. (1937). The luminous efficiency ofmonochromatic rays entering the eye pupil at differentpoints and a new colour effect. Proc. R. Soc. B., 123,90-118.

Stiles, W.S. (1939). The directional sensitivity of the retinaand the spectral sensitivities of the rods and cones. Proc.R. Soc. B.,127, 64-105.

Stiles, W. S. and Crawford, B. H. (1933). The luminousefficiency of rays entering the eye pupil at differentpoints. Proc. R. Soc. B.,112, 428-50.

van den Berg, T. J. J. P. (1995). Analysis of intraocularstraylight, especially in relation to age. Oplom. Vis. Sci.,12,52-9.

van den Berg, T. J. T. P. and Ijspeert, J. K. (1992). Clinicalassessment of intraocular stray light. Applied Oplics, 31,3694-6.

Van Loo, J. A. and Enoch, J. M. (1975). The scotopicStiles-Crawford effect. Visioll Res.,15, 1005-9.

Vos. J.J. (1984). Disability glare - a state of the art report.ClE-I, 3(2), 39-53.

Vos, J. J. and Bouman, M. A. (1959). Disability glare;theory and practice. Proc. CIE,Brussels 1959,298-306.

Walraven, J. (1973). Spatial characteristics of chromaticinduction; the segregation of lateral effects fromstraylight artefacts. Visioll Res., 13, 1739-53.

Weale, R.A. (1961). Notes on the photometric significanceof the human crystalline lens. Vision Res., 1, 183-91.

Westheimer, G. (1966). The Maxwellian View. Vision Rrs.,6,669-82.

Wijngaard, W. and van Kruysbcrgen, J. (1975). Thefunction of the non-guided light in some explanationsof the Stiles-Crawford effects. In Plwloreceplor Oplics(A. H. Snyder and R. Menzel. eds), pp. 175-83.Springer-Verlag.

14Light interaction with the fundus

Introduction

As well as the absorption of light by theretinal photoreceptors, which initiates theneural processes of vision, other interactionsof light with the fundus of the eye (a term thatgenerally includes the retina, choroid andsclera) are important. Light from the fundusthat passes out of the eye is essential for thediagnosis of ocular disease in ophthalmos-copy.Light from the fundus passing out of theeye is also important for determination ofrefractive error by retinoscopy and otherobjective techniques (Chapter 8). The spectralabsorption by the fundus is important forunderstanding the processes of retinaldamage by excessive light levels. In thischapter, we examine the specular reflection,scatter and absorption of light at the fundus.These properties are affected by the optical

properties of the fundus layers (see Figure1.3). Of particular importance are fourabsorbing pigments: macular pigment in theretina, visual pigments in the photoreceptors,melanin mainly in the pigment epithelium,and haemoglobin mainly in the choroid.There are individual and racial differences inthe amount of the pigments, particularlymelanin (Hammond et al., 1997). We willconsider specular reflection, scatter andabsorption at the layers along a typical raypath.

Inner limiting membrane tophotoreceptors (six layers>

Light is first incident on the retina at the innerlimiting membrane, and there will be somespecular reflection at this boundary. The sixlayers between this boundary and thephotoreceptors are highly transparent, but theregular array of the fibres in the nerve fibrelayer has some effect on polarized light. Thereis a yellow pigment called xanthophyll in themacula (the macula pigment), and the amountof this pigment varies greatly betweenindividuals (Ruddock, 1963; Bone andSparrock, 1971).

The photoreceptorsSome light is then absorbed by the visualpigments in the photoreceptors. There is littledata available on the proportion of lightincident on the retina which is absorbed bythe visual pigments and, therefore, contri-butes to visual perception, but the proportionwill vary with state of light adaptation, retinallocation and spectral light distribution.Rodieck (1998) made some estimations of theproportion of light which stimulates vision fora large pupil (0:: 7mm diameter) when lookingdirectly at a star. Ninety-two per cent of thelight incident at the inner limiting membranereaches the cones after absorption by themacular pigment. Of this amount, 53%reaches the cone outer segments which

130 Li~ht and me eye

~ Fovea

400o-t-----,._liEtsI!!"i---,----r---,...----+200 600 800 1000 1200 1400

Wavelength (nm)

Figure 14.1.Spectral reflectance of the fundus fromGeeraets and Berry (1968)- reflectance of pigmentepithelium and choroid, Hunold and Malessa (1974)-derived from their extinction values, and Delori andPflibsen (1989)- illl'illo measurements on 10 normalsubjects at different retinal locations.

--<>-- Nasal fundus40 -0-- Parafovea

10

50

-- Geeraets and Berry ( 19681

~ Hunold and Malcssa (1974)

60-t-----1---'-----J----l.-__L-_-+

~c 30"t;'"t;::

~ 20

Delori and Pflihsen (1989)

light emerging from the eye. However, this iscomplicated by a number of factors:1. III iiioo spectral reflectance measurements

involve the passage of light through theocular media twice. We know fromChapter 12 that the ocular media absorblight, and that this absorption is notspectrally neutral. The effect of the spectralabsorption of the ocular media is thusdoubled.

2. III vivo results are contaminated byspecular reflections from the refractingsurfaces.

3. Reflectance measurements must use areference baseline, which is often taken tobe a white tile of high, diffuse reflectance,such as one based upon magnesium oxide.

4. Pigment absorption by the photoreceptorsdepends upon the level of bleaching andhence their immediate previous lightexposure.Measured values of fundus reflectance are

shown in Figure 14.1. Reflectance is low atshort wavelengths, and gradually increaseswith increase in wavelength. The high reflect-ance at the longer wavelengths is attributed tothe blood in the choroid.

The term fundus reflectance refers to the lightspecularly reflected and scattered at thefundus and which eventually passes back outof the eye. Fundus reflectance can bemeasured by illuminating the fundus with alight source and analysing the spectrum of

Fundus reflectance

The sclera

The thin (350-450 urn) choroid is highlyvascularized, but contains some melanin.Thirty to sixty per cent of the choroid is blood(Delori and Pflibsen, 1989). This containshaemoglobin, which is mainly oxygenatedand strongly absorbs short wavelength lightand back-scatters longer wavelengths. Asmall amount of light penetrates the choroidto reach the sclera.

The pigment epithelium

This dense, whitish tissue back-scatters lightstrongly, so that most light reaching the sclerapasses back through the retina.

The choroid

The remaining light passes into the pigmentepithelium, where there is strong absorptionand scatter by melanin granules. Some lightpasses through the pigment epithelium andenters the choroid.

contain the visual pigments. Of this amount,38% is absorbed by visual pigment. Finally,67%of this amount results in a photochemicalreaction. Combining these proportions gives aretinal 'efficiency' of 12%. Combining this12%with the 54% of the light incident on thecornea which reaches the retina, approxi-mately 7%of the light incident at the cornea isresponsible for initiating the neural responsesin the retina and beyond. Rodieck madeestimations at 15° eccentricity to obtain asimilar overall result for the rods of the darkadapted retina.

Polarized light

Some early workers in ocular polarizationconcluded that most polarized light incidentupon the fundus is depolarized upon returnout of the eye. This was because conversion oflinear polarized light into elliptical polarizedlight was mistaken as depolarization (vanBlokland, 1985). More recent studies havefound that most polarization is retained,although the orientation of polarization maychange (birefringence). Van Blokland (1985)found about 90 per cent retention (fovea, 514nm) and Dreher et al. (1992) found 5O-S0 percent retention (close to the optic disc, 633nm).Van Blokland and van Norren (1986) foundthat the retained polarization at the foveadecreases with increase in wavelength ("" 90per cent at 488nm and "" 40 per cent at 647nm).

Guided and unguided light

Some of the light incident on the retinalpigment epithelium is returned within theouter segments of photoreceptors, and can beconsidered to be guided or directed (vanBlokland and van Norren, 1986). The rest ofthe returned light may be considered to beunguided. Bums et al. (1995 and 1997)developed and used an objective instrumentfor measuring cone photoreceptor alignmentby measuring the distribution of lightreturning from the retina corresponding todifferent positions of a small light source atthe pupil. The distribution is affected by theguiding of light along the photoreceptors. Forbleached retinas, this produced functions withpeak pupil positions that match thoseobtained by psychophysical measurements ofthe Stiles-Crawford effect (see Chapter 13,TheStiles-Crawford effect).

Layers responsible for the fundusreflectance

On the basis that different components ofreflectance (polarized versus unpolarized,and guided versus unguided) showed similardependencies on wavelength, van Bloklandand van Norren proposed a simple model in

Lixilt interaction untl: tile [undus 131

which one layer is mainly responsible for thefundus reflectance. Because the wavelengthdependence of the fundus reflectance wassimilar to that of the retinal pigmentepithelium, they suggested that the retinalpigment epithelium is this layer. Thisargument was supported by the fact that theguided component increased with increasedbleaching of retinal pigment, an effect thatwould not occur if the responsible layer wasin front of the receptors. More sophisticatedmodels have been developed since with vande Kraats et al. (1996) proposing that the outeraspects of the cones are responsible for theguided component of reflectance.

Veiling glare

Some of the light scattered by the fundus willcontribute to veiling glare, because light thatis scattered sideways (halation) and back-wards can illuminate photoreceptors, some along way from the original site of incidence.The Stiles-Crawford effect may reduce theluminous efficiency of this scattered light.Veiling glare is reduced for people with darkirides compared with people with light irides,and this has been attributed largely to thelevel of fundus pigmentation (IJspeert et al.,1990; van den Berg, 1995).

Absorption

Light that is not specularly reflected orscattered back out of the eye from the fundusis either absorbed or scattered in otherdirections within the eye. Since scattered lighthas the potential to excite photoreceptors inother parts of the fundus, one would expectthe fundus to preferentially absorb rather thanscatter. The layers of the retina in front of thecones and rods are highly transparent.Absorption is due mainly to visual pigmentsin the photoreceptors, melanin in the pigmentepithelium, and haemoglobin in the choroid.Figure 14.2a shows spectral absorption

of the pigment epithelium and choroid(Geeraets and Berry,1968). Figure 14.2bshowsspectral absorption of oxygenated hemo-globin (van Assendelft, 1970), and Figure

132 Lightand tire eye

90 ((XlRaw values r80 <)(J

Corrected for reflectance

70 80

7060

~ ""60

50e c0 .s 50.~ 40 Q.

I-0 0 40'" v:.c s:-e 30 < 30

20 20

10 10

0 0 \

200 400 600 800 1000 1200 1400 200 aoo 600 800 1000 1200 1400

(a) Wavelength (nrn) (b) Wavelength (nm)

70

60

50

~40c.52e. 300'".c«

20

10

0200 400

(c)600 800 1000 I200 1400Wavelength (nm)

Figure 14.2. Absorption of the fundus:a. Pigment epithelium and choroid (Geeraets and Berry,

1968).b. Oxygenated hemoglobin (van Assendelft, 1970).This

is for a concentration of 0.00233moles/litre and 0.1mm thickness.

c. Macular pigment (Wyszecki and Stiles, 1967).

14.2c shows spectral absorption of themacular pigment (Wyszecki and Stiles, 1967).

Birefringence

Birefringence has been discussed in Chapter12 (Birefringence> with regard to the media ofthe eye. In birefringent materials, refractiveindex depends upon the direction of the lightbeam and upon the direction of the electricvector. The retinal nerve fibre layer exhibitsbirefringence because of the structure of the

nerve fibres and their regular arrangement(Dreher et al., 1992). The nerve fibres form aradial pattern centred on the optic disc but,because over short distances neighbouringfibres can be regarded as parallel, the nervefibres can be regarded as a layer of longparallel cylinders perpendicular to the retinalsurface. Such an arrangement acts as auniaxial crystal, with its optic axis beingparallel to the axis of the cylinders andperpendicular to the incident light. Therefractive index will be at minimum for theelectric vector perpendicular to the nerve fibre

direction and maximum for the electric vectorparallel to the nerve fibres. The arrangementof nerve fibres is similar to the fibril layers inthe cornea (Chapter 12, Scatter), but whereasthe corneal fibril diameters are smaller thanthe wavelength of light, the nerve fibrediameters in the retina are about the same sizeor larger than the incident wavelengths(0.6-2.0 urn, Hogan et al., 1971).There have been several in vivo studies of

retinal birefringence, for example, vanBlokland (1985) and Dreher et al. (1992).Measurements of retinal birefringence musttake into account or eliminate any bire-fringence from the cornea.Birefringence of the nerve fibre layer has an

application in scanning laser polarimetry(Dreher et al., 1992; Weinrebet al., 1995),whichestimates nerve fibre loss due to diseases suchas glaucoma. Red or near infrared light from alaser, initially linearly polarized, passes intothe retina. It is scattered from the deeperretinal layers and the choroid to pass back outof the eye. The plane of polarization ischanged by its passage (twice) through thenerve fibre layer. The degree of polarizationchange is used to estimate nerve fibre layerthickness, and the thickness is determined atthe retinal area of interest.

ReferencesBone, R. A. and Sparrock, J. M. B. (1971). Comparison ofmacular pigment densities in human eyes. Vision Res.,It, 1057-64.

Bums, S. A, Wu, S., Delori, F. and Elsner, A. E. (1995).Direct measurement of human-cone photoreceptoralignment. J. Opt. Soc. Am. A., 12, 2329-38.

Burns, S. A., Wu, S., He, J. C. and Elsner, A E. (1997).Variations in photoreceptor directionality across thecentral retina. J. Opt. Soc. Am. A, 14, 2033--40.

Delori, F.C. and Pflibsen, K. P. (1989). Spectral reflectanceof the human ocular fundus. Appl. Optics, 28,1061-77.

Liglrt interaction with thefundus 133

Dreher, A W., Reifer, K. and Weinreb, R. M. (1992).Spatially resolved birefringence of the retinal nervefiber layer assessed with a retinal laser ellipsometer.Appl. Optics, 31, 3730-35.

Geeraets, W. J. and Berry, E. R. (1968). Ocular spectralcharacteristics as related to hazards from lasers andother light sources. Am. J. Ophthal., 66, 15-20.

Hammond, B. R., Wooten, B. R. and Snodderly, D. M.(1997). Individual variations in the spatial profile ofhuman macular pigment. f. Opt. Soc. Am. A, 14,1187-96.

Hogan, M. J., Alvarado, J. A. and Weddell, J. E. (1971).Histology of the Human Eye, p. 483. W. B. Saunders andCo.

Hunold, W. and Malessa, P. (1974). Spectrophotometricdetermination of the melanin pigmentation of thehuman fundus in vivo.Ophthal. Res.,6, 355-62.

Ijspeert, J.K., de Waard, P.W. T., van den Berg, T. J.J. P.and de [ong, P. T. V. M. (1990). The intraocularstraylight function in 129 healthy volunteers;dependence on angle, age and pigmentation. VisionRes., 30, 699-707.

Rodieck, R.W. (1998). Chapters 4 and 6. Thefirst steps inseeing. Sinauer, pp. 68-87, 122-33.

Ruddock, K. H. (1963). Evidence for macularpigmentation from colour matching data. Vision Res., 3,417-29.

van Assendelft, O. W. (1970). Spectrophotometry ofHaemoglobin Derivatives. Royal VanGorcum.

van Blokland, G. J. (1985). Ellipsometry of the humanretina in vivo: preservation of polarization. f. Opt. Soc.Am. A, 2, 72-5.

van Blokland, G. J. and van Norren, D. (1986). Intensityand polarization of light scattered at small angles fromthe human fovea. Vision Res.,26, 485-94.

van de Kraats, J., Berendschot, T.T.J. M. and van Norren,D. (1996). The pathways of light measured in fundusreflectometry. Vision Res.,36, 2229-47.

van den Berg, T. J. J. P. (1995). Analysis of intraocularstraylight, especially in relation to age. Optom. Vis. Sci.,12,52-9.

Weinreb, R. N., Shakiba, S. and Zangwill, L. (1995).Scanning laser polarimetry to measure the nerve fiberlayer of normal and glaucomatous eyes. Am. J. Ophthal.,lt9,627-36.

Wyszecki, G. and Stiles, W. S (1967). Color Science.Concepts and Methods, Quantitative Data and Formulas,pp. 218-19. John Wiley and Sons.

15Monochromatic aberrations

(b) Object space

(a) Image space

a-eneral ray

Wave aberrationImage plane

Transverseaberration 1

Longitudinalaberration

Ot---"""""::~~-+--+--

Object plane

--+-+---=='1'k-~---+o·

Longitudinalaberration

ITransverseaberration

General ray

Figure 15.1.Wave, transverse and longitudinalaberrations:a. For a general optical system.b. For the eye, when these must be determined in objectspace.

Note that longitudinal aberration occurs for a ray onlywhen it intersects the reference ray (i.e, pupil ray).

the aberration and defocus levels are notunduly high, these 'object' aberrations aresimilar to 'image' aberrations. When theaberrations are large, such as in the case ofperipheral vision, this is no longer the case.

Introduction

Like defocus, aberrations reduce the imagequality of optical systems such as the eye.When an eye is corrected by ophthalmiclenses, we are mainly correcting defocus butare actually finding a balance betweendefocus and aberrations. Defocus is the mostimportant optical defect of eyes, and it shouldbe appreciated that aberrations are usuallyonly of consequence for an eye well-correctedfor defocus.The representation of aberrations of optical

systems depends on what is most convenientin the particular circumstances. Theserepresentations include the following (FigureIS.la):

1. Wave aberration. This is departure of thewavefront from the ideal waveform, asmeasured at the exit pupil.

2. Transverse aberration. This is the departureof a ray from its ideal position at the imagesurface.

3. Longitudinal aberration. The departure ofthe intersection, where this occurs, of a raywith a reference axis (i.e. the pupil ray)from its ideal intersection.

Because we cannot easily measure theaberrations on the image side of the eye'soptical system, the aberrations of the eye areusually measured in object space (Figure15.1b). All but one method of measuringaberrations of the eye described in thischapter work in this fashion. Provided that

138 A/J,.,.mtiolls alld retinal illlllg<'qualits]

OW4,O spherical lW3,1 comaaberration co-efficientco-efficient

2W2,O field curvature 2W2,2 astigmatismco-efficient co-efficient

3Wl,1 distortionco-efficient

The significance of the term 'primary' isexplained in Appendix 2, where generalaberration theory is discussed. Of these fivemonochromatic aberrations, only sphericalaberration occurs on-axis for a rotationallysymmetric system. The other four occur onlyoff-axis for a rotationally symmetric system,and generally worsen with increase indistance off-axis. There are two other primaryaberrations called chromatic aberrations,which become manifest when more than onewavelength is imaged by an optical system.We will leave the chromatic aberrations untilChapter 17.Eyes are not rotationally symmetric about

an appropriate reference axis such as the line-of-sight or the visual axis. Consequently,ocular aberrations are not described well byequations such as equations (15.2), (15.3) and(15.4), but require the use of an equation suchas equation (15.1). Contributors to this lack ofrotational symmetry are the differencebetween the best-fit optical axis and the line-of-sight (because of component tilts anddisplacements), the lack of rotational sym-metry of refracting surfaces, and possibly alack of rotational symmetry of the lensrefractive index.

chromatic primary aberrations or third orderaberrations, and the co-efficients are asfollows:

tilt (prismaticor distortion)co-efficientsdefocus andastigmatismco-efficients'coma-like'co-efficients'sphericalaberration-like'co-efficients

The values of these co-efficients dependupon the position of the object in the field. Fora rotationally symmetrical system in whichthe object lies along the Y-axis, this poly-nomial can be reduced to

W(X,Y) =W2Y+ W3X2 + Wfy2 + W7X2y+ W9y3 + WlOX~ + W12X2y + W14y4+ higher order terms (15.2)

where the terms still present retain the samemeaning as previously. In this case,

W7 =W9 and WlO=W14 =0.5 W12

so we can now write

The aberrations of a general optical systemwith a point object can be represented by awave aberration polynomial (or function) ofthe form

W(X,Y) =W1X + W2Y+ W3X2 + W4XY+ W y2 + W X3+ W X2y + w: Xy2 + W y3

S 4 6 3 7 2 ~ 93+ WlOX + WllX Y + W12X Y + W13XY+ W14y4+ higher order terms (15.1)

where X and Yare co-ordinates in theentrance pupil. The co-efficients can bedescribed as follows:

W(X,y) = W2y + W X2 + Wsy2+ W7Y(X2+ y2) + WlO(X2 + y~2 + higher order terms

(15.3)

An alternative way of expressing theaberrations of a rotationally symmetricaloptical system is

W(1];X,Y) = oW4Q(X2+ y2J2 +0W3 11](X2 + y2)y...2 2 '2 ...2' 3+ 2W2,O" (X + Y ) + 2W2,2't- + 3Wl,11J Y

+ higher order terms (15.4)

where 1] indicates dependence on the positionof the object in the field. The first five termsin this expansion are known as mono-

Methods of measuringmonochromatic aberrations

Aberrations of the eye have been noted andmeasured since at least the time of ThomasYoung (1801). Some methods have measuredthe tranverse aberration and others longi-tudinal aberration, usually in the form of therefraction required. Other aberration forms,for example the wave aberration, can usuallybe derived from these values.

Subjective methods

Vernier alignmentTransverse aberrations are measured in thistechnique. The technique is closely related tothe coincidence technique for refractiondescribed in Subjective/objective refractiontechniques in Chapter 8. Light from a referencetarget passes through a reference part of thepupil, and light from a test target passesthrough another pupil location. If there is noaberration of the ray bundle associated withth~ second pupil location, the targets appearaligned when they have the same location inspace. If there is aberration of the ray bundle,the targets do not appear aligned. The linearor transverse movement of one target neces-sary to achieve apparent alignment is then ameasure of tranverse aberration. Typically thetwo targets are thin lines, to take advantage ofthe visual system's ability to make precisevernier alignments.There have been many variations of the

technique - for example, Ames and Proctor(1921), Ivanoff (1953), Smirnov (1962), Jenkins(1963a), Schober et al. (1968), Campbell et al.(1990), Woods et al. (1996), Cui andLakshminarayanan (1998), and He et al.(1998). One variation by Woods et al. (1996) ofthe technique is shown in Figure 15.2. The

MOIwclmllllatic aberratiolls 139

subject aligns a laser spot projected onto adiffuse reflector with the gap between a co-linea.r pair of lines displayed on a computermomto!. The ~ertica.llines are visible throughthe entire pupil, while the spot is visible onlythrough a 0.75mm diameter aperture in apolaroid filter. The aperture is translatedhor~zonta~lyacross the pupil in steps, and thesubject adjusts the location of the vertical linesuntil these are aligned subjectively with thespot. Woods and colleagues fitted thetranverse aberrations to a polynomial up tothe fifth power of the formTA(X) =Ao+ AtX + A2X2 + A3X3 + A4X4

+AsXs (15.5)where TA(X) is the transverse ray aberrationat the computer monitor, X is the horizontalray location in the entrance pupil, and A toAs are co-efficients. Figure 15.3 shows °theresults for one subject.Transverse ray aberration measurements

along a meridian can be converted intolongitudinal aberrations LA (X). If the dis-tance from the eye's entrance pupil to thetargets is 10 and the distance from the entrancepupil to where the test ray intersects thereference axis is 1, from similar triangles inFigure 15.4TA(X)/ X = (1-1

0)/1

Figure 15.2. Vernier alignmentapparatus for measuring transverseaberration. The optical arrangementuses a mirror and a cube beam-splitter, so that the subject sees abr.ight laser spot aligned horizontallyWIth the gap between two verticallines. The subject controls the lines'horizontal movement so that the spotis aligned in the gap. Modified fromFigure 2 of Woods et al. (1996),withkind permission of Elsevier Press.

140 Aberrations and retinal imageqllality30+-_.l....-.---l~--I._......L._....l..-_-'-_.l....-.-+-

or

Figure 15.4.The determination of longitudinalaberration from transverse aberration, as applied to thestudy of Woods et al. (1996).

Scheiner disc and annulus methodsVon Bahr (1945) used the Scheiner principle(Chapter 8, Subjective/objective refractiontechniques) to measure the subjective re-fraction (longitudinal aberration) corre-sponding to different diameters in the pupil,in both vertical and horizontal meridians.For the annulus method, the Scheiner discs

are replaced in front of the eye by aperturesconsisting of annuli of various radii. Theobject can be moved, or correcting ophthalmiclenses are placed in front of the eye. Koomenet al. (1949) kept the area of all annuli thesame. Like most other subjective techniques, itis only applicable for foveal vision. Because itconsiders all meridians at once, it is only ableto measure the rotationally symmetricalaberration known as spherical aberration.

Telescope focusingVan den Brink (1962) measured longitudinalaberrations by isolating small parts of thepupil and adjusting the eyepiece of atelescope for optimum focusing for a target.

Refraction in the peripheryOccasionally, subjective refraction techniques(measuring longitudinal aberration) havebeen modified to measure astigmatism andfield curvature in the periphery (Millodot et

TA(X) = -/odW(X)/dX (15.9)where dW(X)/dX is the derivative of W(X)with respect to X. From this equation,W(X) = -(A X + A1X2/2 + A2X3/3 + A3X4/4+ higher or~er terms)/Io (15.10)This is a one-dimensional equivalent ofequation (15.1).The vernier alignment method is slow

compared with recent objective techniques,and most studies have considered only one ora few meridians. With improvements indesign and in computer-aided data collectionand processing, it is possible to measure alarge number of points across the whole pupilin a few minutes (He et al., 1988). In this case,the equations to derive aberration poly-nomials from the transverse aberration datawill be more complex than indicated above.

in-focus

y =-0.860 - 0.595x~ + 0.501x3

which can be arranged toI -/

0= TA(X)1/X (15.6)

Note that TA(X) has the opposite sign to X inthe figure. The longitudinal aberration LA(X)is given, as a difference of vergences, byLA(X) =1/1-1/10

-20

LA(X) =-(1-/0 ) / (lIo) (15.7)

Substituting the right-hand side of equation(15.6) for (I -/0 ) into equation (15.7)givesLA(X) = - TA(X)/(X/o) (15.8)

The transverse aberration polynomial can bealtered also into a wave aberration poly-nomial W(X) by using the relationship (Smithand Atchison, 1997,equation (33.76))

-30+-.L-.r----r---r-.....--,--.---r-_t_-4 -3 -2 -I 0 I 2 3 4

Position in pupil (mm)

Figure 15.3.Transverse aberration for one subject at twolevels of defocus in the study of Woods et al. (1996).Object distance 4 m. Modified from Figure Sa of Woodset al. (1996), with kind permission of Elsevier Press.

Transverse aberration1i)

0:===1

20 y '" -4.357 - 7.024x - 0.411x~ + 0.5 13x3

E -20-5 10e.20;~ 0..coso'"~ -10'"=~

al., 1975; Wang et al., 1995). These are veryslow.

Objective testsKnife-edge testsThe basic principles of retinoscopy werediscussed in Chapter 8, Objective-onlyrefraction techniques. Retinoscopy was used insome of the earliest attempts to measurelongitudinal monochromatic aberrationsassociated with foveal vision, by comparingthe state of refraction through various regionsof the pupil (jackson, 1888; Pi, 1925; Stine,1930; Jenkins, 1963a). These studies showedthat, for the relaxed eye, peripheral pupilregions require usually more negative cor-rection (i.e. are more myopic) than the centralpart of the pupil. Retinoscopy has also beenused for determining the peripheral refractionof the eye, taking the whole of the pupil intoaccount, and hence determining fieldcurvature and oblique astigmatism (see, forexample, Rempt et al., 1971; Millodot et al.,1975). Here retinoscopy can be considered tobe a longitudinal aberration measure, becauseit determines the lens power required tocorrect the aberrations of the eye.Berny (1969) and Berny and Slansky (1970)

developed a technique based on the Foucaulttest. Light returning from the retina of the eyewas intercepted by a knife edge near the focalplane. The image of the pupil was photo-graphed and its intensity pattern analysed todetermine transverse aberrations, which werethen transformed to give the correspondingwave aberrations.

Aberroscope techniqueA possible set up of the aberroscope is shownin Figure 15.5. The method employs a distantlight source and an aberroscope placed closeto the eye. The aberroscope consists of asquare (or nearly square) grid sandwichedbetween positive and negative plano-cylindrical lenses of equal power (typically 50), with the cylindrical axes perpendicularand at 45° to the vertical. Additional correct-ing lenses are added as required to place theretinal shadow of the grid midway between

Monochromatic aberrations 141

/\-~J@il Crossed-cylinder with grid

/ ~Laser with expanded beam

Figure 15.5. A setup for the aberroscope technique,showing the retinal image of a 7 x 7 grid in the presenceof primary spherical aberration.

the two focal lines produced by the cylindricallenses. Without the lenses, the grid shadowintersection points would be too close toresolve. Aberrations are revealed by distor-tions away from a square shadow pattern. Forgrid intersection separations of 1 mm and foran 8 mm pupil, the grid shadow subtends anangle at the nodal point of approximately 2°.The central part of the grid shadow is used todetermine intersection points that the wholegrid shadow would have if it were aberration-free. The departures of the actual intersectionpoints from the aberration-free points aretransverse ray aberrations. From thetransverse ray aberrations, a wave aberrationpolynomial is derived of the form of equation(15.1). Figure 15.6shows the variation in waveaberration across the pupil of one eye.The original aberroscope technique

involved subjects drawing the grid image.Later, Walsh et al. (1984) and Walsh andCharman (1985) made the technique objectiveby photographing the grid shadow. Furtheradvances in imaging methods were describedby Atchison et al. (1995) and Walsh and Cox(1995). It may be thought that this is now a'double pass' technique, in which the aber-rations of the eye are involved for the passageof light both into and out of the eye. However,it is only for the passage of light into the eyethat each grid intersection point is 'distorted'in the image by a small part of the pupil. Forthe passage out of the eye, light from eachintersection point passes through the wholepupil - the aberrations of the eye for thisdirection serve only to blur the final imagerather than to distort it.

142 Aberrations and retinal iII/age qlltllity

~ 3

~5E 2 I.fKlE

'.0.75

~ --{150 -10 25 0.50=s.. I::se,c ,;-c 0.g 0'Vi 0.25~ -I

~~~0

OJ 025tJ (/ft3""O€

~11:75QJ -2> /'1.25c~ -3

"0 I i i I I I-3 -2 -I 0 2 3subject's left subject's right

Horizontal position in pupil (mm)

Figure 15.6. Example of the wave aberration across thepupil of a subject's right eye, as determined with theaberroscope technique. Contour intervals are in units of10-3 mm. Data kindly provided by Michael Cox.

The original aberroscope grid was 'pre-distorted', with the intention that its pro-jection onto the entrance pupil would besquare so that orthogonal fitting techniquescould be used to derive the aberrations. Smithet al. (1996) showed that this pre-distortion isunnecessary if a least squares fitting techniqueis used.This method is distinct from most other

methods described here in that it is effectivelya ray trace from a point object into the eye,rather than a ray trace from a retinal point outof the eye (however, note that Navarro andcolleagues (Navarro and Losada, 1997;Navarro et al.,1998)have recently described atechnique in which a laser beam is projectedinto the eye through different points on thepupil and the aerial image of the retinal spotis recorded). There are a number of calibrationissues with the aberroscope technique, whichare discussed by Smith et al. (1998). The levelof defocus must be within approximately ± 0.5O. Highly aberrated eyes and those withhigher degrees of ametropia are very difficultto measure (Collins et al., 1995), and it isnecessary to use a coarser grid for such eyes,thereby yielding less information. Thesampling rate across the pupil is relativelylow, usually of the order of one sample per1 mm. This suggests that high order contri-

butions to the aberration cannot be deter-mined. Williams et al. (1994) and Liang andWilliams (1997) suggested that the lowsampling density leads to an apparent over-estimation of the image quality for thismethod compared with some other methods(see Chapter 18, Retinal image quality).

Wave-front sensor (Liang et al., 1994; Liangand Williams, 1997)A narrow beam (e.g. 1 mm wide) from a pointlight source is imaged by the eye, and the lightpassed back from the fundus travels througha wave-front sensor consisting of an array ofmicro-lenses (typically of the order of 0.5 mmdiameter) and onto the array of a CCOcamera. Each micro-lens isolates a smallbundle of light passing through a small regionof the pupil. The transverse ray aberrationassociated with each microlens can bedetermined from the departure of the centroidof its corresponding image from the idealimage position. Using an adaptive mirror, thistechnique has been used to correct theaberrations of the eye (Liang et al., 1997).

Objective optometersObjective optometers have been used tomeasure longitudinal peripheral aberrationsof astigmatism and field curvature - see, forexample, Ferree et al. (1931, 1932 and 1933),Millodot (1981)and Smith et al. (1988).

Choice of reference point and/or axis

With all methods, there is a problem of choiceof the reference point and axis in the pupilthat will be designated as aberration free. Thispoint and axis are critical to the determinedaberrations. In conventional optical systems,this is usually the pupil ray. Koomen et al.(1956) showed the effect of a different choiceof the reference axis on asymmetries ofmeasured aberrations. The reference pointand!or axis do not appear to have been welldefined or well monitored in many earlystudies. The most obvious choices are thecentre of the entrance pupil as the referencepoint and the line-of-sight as the referenceaxis. However, this has the disadvantage that

the pupil centre shifts as pupil size changesunder the influence of lighting level anddrugs (Walsh, 1988; Wilson et al., 1992). Someinvestigators have used the visual axis (e.g.Ivanoff, 1953; Millodot and Sivak, 1979;Woods et al., 1996) (see Chapter 4, Axes of theeye). This will not change with pupil size andhas the added advantage of being thereference axis for describing chromaticaberrations (see Chapter 17, Chromaticaberrations).

Types and magnitudes ofmonochromatic aberrations

Early investigators noted that themonochromatic ocular aberrations were notusually rotationally symmetrical within thepupil. However, many have considered theaberration measured to be conventionalspherical aberration, which is a rotationallysymmetrical aberration. Because of this, careneeds to be taken when interpreting their

Monochromatic aberrations 143

results, particularly where measurementshave been made along only a single meridian.Later authors (for example, Smimov, 1962;Howland and Howland, 1976 and 1977)emphasized the asymmetric nature of theeye's aberrations.

Spherical aberration

The effect of spherical aberration is shown inFigure 15.7a, which shows a set of rays froman axial point at O. Its paraxial focus is at 0'on the retina. When spherical aberration ispresent, non-paraxial rays do not intersect atthe paraxial focus. The further a ray is fromthe optical axis, the further its axial crossingpoint is from the paraxial focus. The figureshows the case of positive spherical aber-ration. For negative aberration, the rays (ifprojected) cross the axis beyond the retina.The aberration for each ray may be

expressed in terms of the longitudinalaberration distance 8' as a function of theheight r of the ray entering the eye. It may be

o(a)

o(b)

o(c)

Figure 15.7. Positive spherical aberration:a. The effect of positive spherical aberration on

rays passing through the eye.b. 'Compensation' or 'correction' of positivespherical aberration by a negative power lens-.1F(r).

c. Determination of spherical aberration byfinding the retinal conjugate for a ray. Thelongitudinal spherical aberration is -L(r)(= -l//(r».

144 Aberrations and retinal image quality

The spherical aberration term in the waveaberration functionIf we wish to use this experimentallydetermined value of spherical aberration to

the spherical aberration as the sign of thepower error, and not that of the correctinglens. Discrepancies will arise when the cor-recting lens is not placed at the first principalplane of the eye but at a short distance in frontof it, and if the object vergence is notreferenced to the first principal plane. Also,increasing discrepancies result as aberrationlevels become very high, as noted by Thibos etal. (1997).The mean results of some investigations are

plotted in Figure 15.8 in terms of the powererror. All of these investigations found meanspherical aberration was positive for therelaxed eye.Aberration theory predicts that, for a

rotationally symmetric optical system, thepower error can be expressed as an even orderpower function in ray height r, and can thusbe expressed in the formM(r) = br2 + terms of order y4 and higher

(15.13)If the eye contains only primary aberration, allthe terms can be neglected except the first sothat the power error is quadratic in ray heightr in the pupil. Some investigators haveassumed that this rule can be applied to thereal eye, at least out to some specified pupildiameter (e.g. van Meeteren, 1974;Charman etal., 1978).The b values determined from somepublished values of spherical aberrationshown in Figure 15.8 are listed in Table 15.1.Following re-analysis of data, the values areslightly different from those given in Table35.2 of Smith and Atchison (1997). Thesevalues of b show a large variation, which isdue partly to the variation in sphericalaberration between subjects. From Table 15.1,the mean weighted value of b (for non-aberroscopic studies) isb=0.076 X 10-3 ± 0.035x 10-3 mrrr-' (15.14)Thus, we can write equation (15.13) asM(r) = 0.076x 10-3 (± 0.035x 1O-3)r2 (15.15)This equation is plotted, as well as theexperimental data, in Figure 15.8.

4O.O+--F=---,,-......--,-......--,..--r--+

o I 2 3Ray height in pupil (mrn)

2.0Koomen et al. (1949)~

----<>- Ivanoff (1956)

--0- Jenkins (1963a)

1.5 ~ Schober et al. (1968)

e --e-- Charman et al. (1978)

c --+-- Millodot and Sivak ( 1979)0.~

1.0ll,l

.D

'"<;(J

"Cll,l.c~ 0.5

Figure 15.8.Some experimental measures of sphericalaberration of the relaxed eye. The thick curve is theweighted mean fitted by a quadratic function of theform of equation (15.15).

expressed as a power error M(r), whereM(r) =n'l(/'+ 01') - n'II' (15.11)

where n' is the refractive index of the vitreousand I'is the distance from the back principalplane of the eye at P' to the paraxial image at0'.In object space, the longitudinal spherical

aberration may be expressed as the power of acorrecting lens -M(r) placed in front of theeye, but only covering a narrow region of thepupil at a distance r from the centre, as shownin Figure 15.7b.This lens will alter the path ofthe ray so that it crosses the axis at the retina.The power of this lens will depend uponvertex distance. The longitudinal sphericalaberration may be expressed also in terms ofthe object vergence-M(r) = L(r) = 111(r) (15.12)

of the retinal conjugate for a ray, as shown inFigure 15.7c.If the excess power is positive, asit usually is for the unaccommodated eye, thecorrecting lens will have a negative power.The power error M(r) is an image space

quantity that should be similar to the objectspace quantities of the correcting lens andchange in object vergence L(r), except for thechange of sign. We usually define the sign of

Monochromatic aberrations 145

Table 15.1. Values of the co-efficients b of r2 and W4,ocalculated from experimentally determined sphericalaberration from various sources.

Sample size

Studies not usingtheaberroscope techniqueKoomen et al. (1949)Ivanoff (1956)Jenkins (1963a)Schober et al. (1968)Charman et al. (1978)*Millodot and Sivak (1979)

Mean (weighted)

Studies usingtheaberroscope techniqueWalsh and Charman (1985)Walsh and Cox (1995)

'Charman et al.quoted an approximate value of 0.12x Hr3.

0.188 X 10-3 30.0876 X 10-3 100.0589 X 10-3 120.163 x 10-3 10.104 X 10-3 20.0552 X 10-3 20

0.076 X 10-3 ± 0.035 x 10-3

1110

0.019±O.009 X 10-3

0.0084X 10-3 ± 0.0134x 10-30.0091 X 10-3 ± 0.0127 x 10-3

further examine its effect on retinal imagequality, using criteria such as the point spreador optical transfer functions, we need to knowthe value of the corresponding wave sphericalaberration co-efficient. In equation (15.4), thisco-efficient is denoted as oW4J) ' which is onlythe axial term. However, the fovea is off-axis,and therefore measured values of sphericalaberration include field dependent values,which would be summed. Wedenote this sumby a single co-efficient W4 c- This is related tob by the equation '

W4,0 = b/4 (15.16)Using the value of b given by equation (15.14),we haveW4 0 = 0.019X 1O-3(± 0.009x 10-3) mm-3, 05.1n

Studies of the aberrations using the aberro-scope technique (see Methods of measuringmonochromatic aberrations, this chapter) haveused an equation of the form of equation(15.1), with truncation of the series at the W14term. We refer to the co-efficients WlO to W 14as 'spherical aberration-like' terms. If theaberration present were purely sphericalaberration, we would haveWlO = W14 = W 12/2 = W4,o (15.18)with all of the other co-efficientsequal to zero.However, because of the asymmetrical natureof the aberrations, this does not occur in realeyes. Howland and Howland (1997) showed

that, in these cases, the spherical aberrationcan still be derived. Using their equations andthose of Atchison (1995), we can obtainW4,o= (3W10 + W12 + 3W14)/8 (15.19)Applying this equation to the data of Walshand Charman (1985) givesW4,o=0.0084 x 1O-3(± 0.0134 x 10-3) mm-3

(15.20a)and applying the equation to the Walsh andCox (1995) data givesW4 0 = 0.0091 x 1O-3(± 0.0127x 10-3) mm-3

, (IS.20b)The mean value of the spherical aberration forthe above two sets of results isW4 0 = 0.0088X 10-3 (± 0.0131 x 10-3) mm-3

, (15.21)Substituting this mean value into equation(15.16), we have a predicted b value ofb=0.035(±0.052) mm-3 (15.22)and hence, from equation (15.13), the powererror at the edge of an 8 mm diameter pupil isM(r =4 mm) =0.56(±0.84)D (15.23)The value of W4 0 in equation (15.21) is abouthalf the value predicted from combining theprevious studies and given by equation(15.17), but they are probably not statisticallysignificantly different because of the largestandard deviations.

146 Al>ermticm,; alit! retinal iII/as!' qlllliity

Effect of accommodationMost eyes are considered to suffer frompositive spherical aberration when unac-commodated, with a trend to negativespherical aberration being observed uponaccommodation (e.g, Koomen et at., 1949;Ivanoff, 1956; Jenkins, 1963a; Schober et at.,1968;Berny, 1969;Atchison et al., 1995;Collinset al., 1995; He et al., 1999). Koomen andcolleagues (1949) found that the aberrationreduced considerably for two of their threesubjects by about 2.5 D of accommodationstimulus, and was clearly negative for one ofthose subjects at 3.7 D of accommodationstimulus. Ivanoff (1956) and Jenkins (1963a)found the mean aberration was approxi-mately zero by about 3 D and 1.5 D accom-modation stimulus, respectively, but with theactual value depending upon the subject andray position in the pupil. Atchison et al. (1995)and He et at. (1999) found that the meanaberration became approximately zero by 2 Dand 3 D, respectively, of accommodationresponse.

Coma

In a rotationally symmetric system, coma is anoff-axis aberration. Figure 15.9 shows theeffect of coma on the image of an off-axispoint, providing no other aberrations arepresent. The image has a straightened-outc?mma shape, with the pointed end facingeither towards or away from the optical axis.The light distribution is not uniform across

Point spread functiono

(o

Figure 15.9.The retinal image of a point object producedby coma.

this patch, but is highest at the pointed end.The complex details of the ray pattern are notshown.In real eyes, 'coma-like' aberrations are

present at the fovea because of the lack ofsymmetry ~f the eye about an appropriatereference axis. The terms W6 to W9 in equation(15.1)are these coma-like terms. The values ofthese co-efficients in Walsh and Charman's(1985) study varied from -0.128 to+0.1080101-2, which is much greater than therange of -0.030 to +0.075 mm-3 for the'spherical aberration-like' terms W to WChanzes i d . 10 14'. anges In ac~ommo ation produce changesIn the coma-like aberrations as well as inspherical aberration (Howland and Buettner,1989; Lu et at., 1993).

Coma estimated from measured sphericalaberration and a pupil decentration factorComa can arise on-axis in an otherwiserotationally symmetric system if the pupil isdecentred, and the amount of induced comacan be p~edicted as follows. For a rotationallysymmetnc system containing only the Wspherical aberration term, from equati~~(15.4), the wave aberration W(X,Y) can bewritten asW(X,Y)=W4 0(X2 + y2)2 (15.24)

If the pupil is decentred to the point X =Xthe wave aberration becomes 0'

W(X,Y) =W4,o[(X - Xo)2 + y2J2 (15.25)Expanding the brackets and re-grouping thetermsW(X,Y) =W4,O(X2 + y2)2 - 4W4,OXo(X2 + y2)X+ other terms (15.26)The 'other terms' correspond to field curva-ture, astigmatism, a lateral shift of focus and aconstant term. These have no application atthis point. However, the second term corre-sponds to coma lying along the X-axisdirection, with a wave aberration co-efficientW3,1 =-4W4,oXo mm-

2 (15.27)Once again, we have dropped the digit infront of the 1W3 1' because we are nowconsidering all field dependent coma terms.

It is clear from this equation that, if thepupil is decentred by only 0.25mm, the

Monochromaticaberrations 147

induced coma co-efficient W3 1 has the samemagnitude as the spherical' aberration co-efficient W4 0, providing the wave aberrationand pupil 'co-ordinates are both in milli-metres.

If we take a pupil decentration (Xo) of 0.5rom and the value of W4,0 from equation(15.17), we haveW3 1=0.038X 10-3 (± 0.018x 10-3) mm-2

, (15.28)This is twice the value of the sphericalaberration co-efficient W4,0.We should note that the values of these co-

efficients are also the values of aberration fora point in the pupil I mm from the pupilcentre. At other points in the pupil, theamounts of coma and spherical aberrationwill be different and the spherical aberrationwill increase more rapidly than coma withincrease in ray height.

Astigmatism and peripheral powererrors

Astigmatism can occur in central vision, due

again to the lack of a rotational symmetryabout an appropriate reference axis. Usuallythe cause is lack of rotational symmetry of atleast one surface, most frequently the anteriorcornea. This central astigmatism is con-ventionally corrected by ophthalmic lenses. Inthe aberroscope technique, residual astigmaticeffects in foveal vision contribute to the termsW3 to Ws in equation (15.1).Here we consider the additional astig-

matism that arises in the periphery of thevisual field. Its effect on the image of a pointsource is shown in Figure 15.IOa. Whenastigmatism only is present, a beam of lightentering the eye from a point source will befocused down to two mutually perpendicularand distinct focal lines at different positionsinside the eye. Vertical sections (or fans) ofrays in the beam (i,e. parallel to the plane ofthe page) focus to points along a focal line atT. Sections of rays perpendicular to the planeof the page focus to points along a focal line ats. The focal line at T is perpendicular to thepage (i.e. horizontal), and the focal line at S isperpendicular to the T focal line. The letters Tand S have been used because these foci arecalled the tangential and sagittal foci,

Figure 15.10. Sagittal, tangential and field curvature errors.a. The sagittal, tangential and field curvature errors and theireffect on the point spread function.

b. Determining the magnitude of the sagittal and tangentialpower errors in object space.

T

(b) ..,---tl~~I------+-

8

S . I rf Petzval surfaceagrtta su ace\

Tangential surface.~ p Circle of least'T- - - '-1- - .., -, 8 confus~'on\ 8I angentra I - - __

I meridian'---t I ~ TI Sagittal I T~ _ !JI.-erldia'l _ ~

(a) -H-~;L..Jf------+

605010

---0- Ferree et at.(1931): Type A subjects

-0--- Jenkins ( 1963b)_ Rempterat. (1971)

-...- Smith et al, (1988)

-4.0+-----y---.----,r----r--.......liL--~o 20 30 40

Angle in air (deg)

Figure 15.11.Mean values of the sagittal and tangentialpower errors from various sources. The solid lines arethe corresponding means fitted by equations (l5.29a)and (l5.29b).

~E 2.0.,..~ 1.08.] o.o~~~~~/C.,011S -1.0"0c::.:: -2.0g.Oil~ -3.0

0' 3.0

power errors will be influenced by thepresence of any foveal astigmatism and by thedirection of its axis.The mean results for a number of

investigations are shown in Figure 1S.11. Therange of individual results within eachinvestigation, and of the means betweeninv~st~gati?ns,shows that there is significantvariation In the form of the sagittal andtangential power errors. The measurementswere usually made with the fixation oraccommodation stimulus at a finite distance.Smith et al. (1988) investigated the effect ofaccommodation and found that, while sagittalpower errors change little with accommo-dation/ tangential power errors increaseconsiderably. Thus, astigmatism increaseswith accommodation.Charman and Jennings (1982) suggested

that much of the variation in peripheralpo~er.error.curves between subjects is due tovariations In the shape of the retina. Ifrefractive error is due to a variation in axiallength but equatorial diameter is the samerefractive errors should converge towards th~equator (angle of about 60° in air). This wasdemonstrated theoretically by Dunne et al.(1987)/ who modelled the retina as ellipses ofvarying asphericity.For a general, rotationally symmetrical,

148 Aberrationsand retinalimagequality

~espectively. As the direction of the incomingincident beam changes, the sagittal andtangential foci move over surfaces called thesagittal and tangential surfaces.The aberration effect can be measured in

two ways:1. In terms of sagittal and tangential power

errors. The. principal powers of the eye,corresponding to the sagittal andtangential foci, are called the sagittal andtangential powers. The correspondingpower errors are the differences in thepowers from that required to image thefocal lines on the retina. As in the case of~pherical ~berration, sagittal and tangentialimagery is usually measured in objectspace by the correcting lens powers. Theselens powers are the same as lens powersfound by conventional refraction, but arefunctions of the off-axis angle. Thisapproach is the same as finding the sagittaland tangential vergences of the beamexiting the eye from a point on the retina,as shown in Figure 1S.lOb. The T and Scorrecting lens powers are not the same asthe image power errors, even with achange of sign, with the discrepanciesincreasing as the angle increases (Atchison,1998). In the rest of this chapter and in thenext, we will use the term 'power error',but it is to be understood that this refers tothe correcting lens power. Please note tha tthis is the opposite to the conventionadopted for spherical aberration.

2. Astigmatism. This is a measure of thedistance between the sagittal andtangential foci, either as a distance or as avergence, but usually the latter. Whenmeasured as an object space quantity byrefraction, it is the difference in refractiveerrors between the sagittal and tangentialsections. Astigmatism occurs in a rotation-ally symmetrical system because of theoblique incidence of rays at optical sur-faces. The effect is the same as conventionalastigmatism associated with the foveawhich, as already mentioned, is commonlydue to one or more surfaces not beingrotationally symmetrical. To make thedistinction, the astigmatism associatedwith peripheral vision is often calledoblique astigmatism.The values of the sagittal and tangential

Monochromatic aberrations 149

60

(15.33a)

(15.33b)

50

....0····0'··

...(r··e Field curvature

--e-- Mean. equation (15.35)

10

Astigmatism

--0-- Mean. equation (l5.3la)

--0-- Mean. equation (l5.3Ib)

····0···· Millodot (1981), nasal

....{>.... Mil1odot(1981).temp

6

-I +-----.--~-_r--r--_._--+­o 20 30 40

Angle in air (deg)

Figure 15.12. Estimates of astigmatism and fieldcurvature from several sources. Solid lines are meanvalues obtained from equations (15.31a), (15.31b)and(15.35).

Field curvature

7 -!-_----I.__-'---_--'-__.l..-_.....I-_-,I,-

W2,2 = [Ls(5°) - Lt(5°)]/2 (15.32b)where the values of Wzo and W 2 are for 5°from the optical axis.'While tRis angle isarbitrary, we chose it because the fovea is onaverage 5° from the best fit optical axis.From equations (15.29) and (15.32), we have

the following estimates of these waveaberration co-efficients, remembering that thevalues of Ls and L, are in dioptres and that thewave aberration will normally be expressed inmillimetresW2,O=-7.91 X 10-6 mm"!

WZ,2 =3.32 X 10-5 mm"!

optical system, aberration theory predicts thatthe sagittal and tangential power errorsdenoted here as Ls(0) and Lt ( 0) can beexpressed in the form of an even orderpolynomial in the object off-axis angle O. Wefitted such polynomials to the experimentaldata shown in Figure 15.11. Taking only thefirst two terms of the polynomials, theequations areLs(O) =6.31x 10-402 + 7.64x 1O-8Ql (15.29a)

Lt(O) =-2.03 x 10-302 + 2.86x 1O-7Ql (15.29b)where 0 is in degrees and Ls and Lt are indioptres. These equations are shown in thefigure (solid curve without symbols).The astigmatism A( 0) is the difference

between sagittal and tangential power errors,that isA( 0)=Ls(0) - Lt( 0) (15.30)

Substituting for Ls(O) and Lt(O) from equations(15.29) gives the equationA(O) = 2.66X 10-302 - 2.09x 1O-7Ql (15.31a)Lotmar and Lotmar (1974) used the data ofRempt et al. (1971) for 363 subjects to obtainthe equationA(O) = 01.5 X 10-2 (15.31b)where 0 is again in degrees and A( 0) is indioptres. We should note that this form is notconsistent with theory for a rotationallysymmetric system, which predicts that theaberration is an even order polynomial inangle O.Equations (15.31a) and (I5.31b), and

Millodot's (1981) results for 62 eyes, sepa-rately in the nasal and temporal meridians,are plotted in Figure 15.12. Our equation(15.31a) predicts larger astigmatism than thatof Lotmar and Lotmar (1974) and of Millodot(1981).

Wave aberration co-efficients W2,0 and W2,2

The sagittal and tangential power errors andastigmatism in a rotationally symmetricsystem can be converted into wave aberrationco-efficients. W2,o and .W2,~. From the theoryand equahons provided by Smith andAtchison (1997), we haveW2,o=-Ls(5°)/2 (15.32a)

This is another off-axis aberration. If fieldcurvature is the only aberration present, theimage of a point source is imaged as a point,but not on the image plane predicted byparaxial optics (the Gaussian image plane).For optical systems that are composed ofmostly positive power components and forma real image (such as the eye), the image isformed in front of the Gaussian image plane.For small angles, the image point falls on a

150 Aberratiolls alld retinal imageql/ality

surface that is behind the retina, called thePetzval surface (Figure 15.lOa). It can beregarded as a simple defocus or refractiveerror that increases with distance off-axis.In the presence of only astigmatism, the

surface of best 'general' image quality isusually assumed to be the surface containingthe circle of least confusion. This is thesmallest circle that encloses all the rays in thebeam. This surface lies between the sagittaland tangential surfaces, as shown in Figure15.10a.When measured outside the eye - thatis, as an object quantity - its position can befound by taking the mean of the sagittal andtangential power errors. That is, fieldcurvature FC(O) is given by the equationFC(O) == [Ls(O) + Lt{e»)/2 (15.34)Substituting for Ls{O) and Lt{e) from equations(15.29) gives the equationFC(e) == -7.00 x lO-4lP + 1.81 x 10-7ot (15.35)where eis in degrees and FC(e) is in dioptres.This equation is plotted in Figure 15.12. Theflatness of the curve indicates that the circle ofleast confusion lies close to the retina.

DistortionThis is the last of the 'primary' mono-chromatic aberrations. It is also an off-axisaberration. It is similar to field curvature inthat a point source is imaged as a point, thistime in the Gaussian image plane but not atthe expected position. It is formed eitherfurther away (positive distortion) or closer(negative distortion) to the optical axis. If the

eye contains negative distortion, the imageposition of an off-axis point will be as shownschematically in Figure 15.13.The presence ofdistortion in a conventional optical system isbest shown by imaging a square, centred onthe optical axis, as shown in Figure 15.14. Inthe presence of distortion, the image of thesquare is smaller or larger than the expectedGaussian image, and the sides are bowed in orout. This conventional distortion is probablynot meaningful in the eye, because of thecurvature of the retina and also because thechange in spatial relationship between imagepoints, brought on by distortion, is unlikely tohave an effect on vision.Perhaps more important than distortion is

the functional relationship between the actualimage position at Q' as shown in Figure 15.13and the position of the object Q in the objectspace (or the incident off-axis angle e). The.£!esence of distortion in the eye means thatQ' is different from that at the position Q'expected from Gaussian optics. There are fewstudies of this relationship in real eyes. Amesand Proctor (1921) mentioned studies ofDonders (1877), who carried out measure-ments on a person who had exophthalmia(protruding eyeballs), and of Drault (1898),who examined extracted eyes and two eyes ofpeople with exophthalmia. These studies gavethe external angle and the position of theretinal image from the corneal margin, andAmes and Proctor converted these retinalpositions to internal angles but without givingany details. Figure 15.15 shows the results.We can compare such image point positions

with those expected from Gaussian optics. Ifthe eye were free of distortion, then the

Figure 15.13. Pupil ray angles anddistortion.

o

Actual imageParaxial image--..... . ~

O'

positive distortion" negative distortion

Gaussian image

Figure 15.14. The effect ofdistortion on the image ofasquare centred onthe optical axis.

MOllochromatic aberrations 151

paraxial pupil ray angles ti' and u, shown inFigure 15.13, would be related by equation(5.7), i.e,

power. In equation (15.1), the terms in thewave aberration polynomial are shown onlyup to the fourth power in pupil co-ordinates.Using the wave-front sensor method, Liangand Williams (1997) determined aberrationterms up to the 10th order. For 3 mm pupils,they found that the aberrations beyond thefourth order were small and had minimaleffect on image quality. However, for large 7.3mm pupils, the fifth to eighth orders madesubstantial contributions to deterioration ofimage quality relative to diffraction limitedperformance, according to different imagequality criteria such as the Marachel criterion,the Strehl intensity ratio, and the modulationtransfer function (the latter two are discussedfurther in Chapter 18).He et al. (1999) found that the overall

aberrations of fifth and higher orders were ata minimum at approximately 2D of accom-modation and increased for smaller andhigher accommodation levels.

Symmetry between fellow eyesLiang and Williams (1997) found thataberrations are similar for the two eyes of thesame observer.

Paraxial prediction, equation (15.36)

Real eyes: Ames and Proctor (1921)

10 20 30 40 50 60 70 so 90

Angle in air (deg)

Figure 15.15. The measured values of internal andexternal angles, from Ames and Proctor (1921). Predictedparaxial angles are determined from equation (15.36).The Ames and Proctor data was smoothed but wellfitted by the equation 9'=0.8230859- 0.000753260&. Theco-efficient of the 9 term was fixed at the value expectedfrom the Gullstrand eye, which was used to predict theexpected distortion-free values.

90

so

70

'@ 60...~ 50...»....5 40...boc 30-e

20

10

00

Higher order aberrations with fovealvisionIn many studies of monocular aberration, theorder of aberration terms has been verylimited - for example, aberroscope studieshave not investigated beyond the fourth

where these paraxial angles are related to realangles 8 by the equationsti = tan (8) and ti' = tan(8) (15.36a)and m depends upon the actual constructionof the particular eye (0.823085 for theGullstrand exact relaxed eye). These equa-tions have been used to plot (J against 8 asshown in Figure 15.15 and show that theactual image is closer to the axis than thatpredicted by Gaussian optics, indicating thatthere is negative distortion in the eye.

-I --u=mu (15.36) Significance of monochromaticaberrations

Foveal visionSpatial visual performanceFor corrected vision and moderate photopicluminances, visual acuity is maximum forpupil diameters of 2-3 mm (Atchison et al.,1979). For pupil diameters smaller than 2-3mm, the decrease in visual acuity is pre-dominantly due to diffraction, althoughreduced retinal illuminance may have someeffect. For larger pupil diameters there is a lessdramatic decrease in acuity, which must bedue to aberrations.A similar influence of pupil size has been

found for the contrast sensitivity function(Campbell and Green, 1965) and the retinalline spread function (Campbell and Gubisch,1966). (These measures of visual performanceare described in Chapter 18).

152 Aberrations and retinal imagequality

No one has yet demonstrated that correct-ing monocular aberrations of the eye willimprove visual acuity, although Liang et al.(1997)were able to improve grating resolutionusing an adaptive optics method to correctaberrations. Van Heel (1946) attempted toimprove visual acuity with a lens designed tocorrect the eye's spherical aberration. Somestudies comparing contact lenses andspectacle lens attributed better performancewith one or the other type of correction todifferent levels of aberration (Millodot, 1969;Oxenberg and Carney, 1989), but there couldhave been other factors such as smallvariations in refraction and spectacle magnifi-cation. Collins et al. (1992) demonstrated thatit is possible to worsen visual acuity bymanipulating the front surface asphericityand, consequently, aberrations of contactlenses. For large pupils, Applegate et al. (1998)found a negative correlation between visualperformance as measured by the contrastsensitivity function (see Chapter 18) and themagnitude of corneal aberrations followingrefractive surgery.The effect of aberrations on image quality

will be considered further in Chapter 18.

Effect of spherical aberration on refractiveerror

In the presence of spherical aberration, theplane of best image quality is not the paraxialimage plane. For positive spherical aberrationand a positive powered optical system such asthe eye, the plane of optimum image qualitylies between the paraxial image plane and theoptical system. The position of this best imageplane depends upon the particular criterionused to assess the quality of the image, thelevel of the aberration (which in turn dependsupon pupil size), and the particular form ofthe detail in the image.When analysing image quality of con-

ventional optical systems, a range of criteriaare used, including minimizing the width ofthe point spread function, maximizing theStrehl intensity ratio, maximizing the modu-lation transfer function at set frequencies, andminimizing the variance of the waveaberration function. We do not know if the eyeuses any of these for optimum focusing.Burton and Haig (1984) found that the Strehl

intensity ratio and modulation transferfunction were not good measures or pre-dictors of perceived image quality. In theabsence of definitive information pointing toany particular criterion, the variance of thewave aberration function will be used herebecause it is the easiest to manipulatealgebraically.The variance of the wave aberration is very

easy to quantify in terms of the waveaberration co-efficients. The variance V isgiven in terms of the wave aberration W(r,l{»by Smith and Atchison (1997, equation (34.3))

V =If f [W(r,l{» - wj2 r dr dl{>} / A (15.37)E'

where E' indicates integration over the pupilof area A.Let us suppose that the wave aberration

W(r) is purely spherical and consists of onlyone spherical aberration order, the sphericalaberration term W4,o. If we defocus thesystem, the wave aberration polynomial canbe written as

W=W(r)=W2,orz + W4,or4 (15.38)where r is the ray height in the pupil and Wzois the term allowing for defocus. The value ofW4 0 is constant, but the value of Wzo can bemanipulated by refocusing to minimize thevariance of the wave aberration given byequation (15.37).This value is minimized withrespect to Wz,o whendV/dWz,o = 0Differentiating V with respect to Wz0 and thensetting the result to zero gives the optimumdefocus occuring at a defocus level of

Wz,o =- W4,oPZ (15.39)

where p is the pupil radius. The corre-sponding change in refractive error ~x(p) isgiven by

L1Rx(j5) = 2Wz,o (15.40)

ThusD.Rx(j5) = -2W4,opz (15.41)

Thus the amount of refocusing depends uponthe pupil radius p. It should be rememberedthat this equation applies if the waveaberration function consists of only thespherical aberration term W4 o- If more thanone term exists, all terms should be included

Monochromatic aberratiolls 153

till.x(j5) = -0.0176 X 1O-3(± 0.0262X 1O-3)p2 mm"!

60

6mm

2mm

20 30 40 50Spatial frequency (c/deg)

10

/ ./ .

.'..•.........

-0.8 +---,-----r-.....--r---.-----r----r-r-~.,...---r"_;_

o

RetinoscopyThis objective refraction technique was des-cribed in Chapter 8 (Objective-only refractiontechniques), and its use in early studies of theaberrations was mentioned earlier in thischapter (Methods of measuring monochromaticaberrations). If an eye is aberration-free, at thereversal point the complete pupil will appeareither dark or light as the light returning fromthe patient's pupil is scanned across the sight-hole of the retinoscope. In the presence of

high spatial frequencies are not available fordetermining refraction, and the naturalincrease in pupil size (and increase inaberrations). Green and Campbell (1965) andCharman et al. (1978) showed that optimalfocus did not depend upon spatial frequencyfor small pupils, but would shift considerablyfor large pupils - for example, Green andCampbell found that optimum refraction for a7 mm diameter pupil shifted approximately0.8 D between 1.5and 26 cycles/degree in twosubjects. Charman et al, (1978) and Atchison(1984a) showed that this shift could beexplained by the presence of sphericalaberration (Figure 15.16).

Figure 15.16.Theoretical optimum refraction as afunction of spatial frequency for 2 mm and 6 mm pupilsizes. Wavelength 605nm, with primary sphericalaberration of 0.11D and 1.0D at the edge of 2 mm and 6mm pupils. respectively.

-0.2§c.2~..:: -0.4II)

IXE::0.§8' -0.6

(15.42b)

These values indicate that a change in pupildiameter from 2 mm to 6 mm would lead to achange in focus of 0.29D according toequation (15.42a) and of 0.14D according toequation (15.42b).The above results neglect the Stiles-

Crawford effect, which reduces the influenceof spherical aberration towards the edge ofthe pupil and hence reduces the defocusingprediction given by the above equations.However, its influence is likely to be small(Atchison et al., 1998).Under photopic conditions, refraction using

letter targets alters little with changes in pupilsize (Koomen etal., 1949and 1951;Charman etal., 1978; Atchison et al., 1979). For example,Charman et al, (1978) found less than 0.3Dvariation in refraction with pupil size, even inthe presence of greater than 1D sphericalaberration, while Atchison et al. (1979) did notobserve variation in refraction across 1-8 mmpupil diameters for any of their 22 subjects(refraction interval 0.25D). Charman et al.(1978) and Atchison (1984a) showed that theoptimum refraction for high frequencysinusoidal targets (~ 20 cycles/degree) shouldbe little affected by changes in pupil size, evenin the presence of substantial sphericalaberration (Figure 15.16). The small variationin refraction with pupil size is thus explainedby the use of small letters containing arelatively high proportion of high spatialfrequency information.When luminance is lowered, there is often

a substantial negative (myopic) shift inrefraction (Koomen et al., 1951). A largeamount of this is due to the combination ofdecrease in visual acuity, which means that

in the integration of the variance equation(15.37).We can use equation (15.41) to predict the

effect of spherical aberration on refractiveerror. Substituting the spherical aberration co-efficients from the non-aberroscope studiesand from the aberroscope studies, providedby equations (15.17) and (15.21), respectively,into equation (15.41) givestill.x(j5) = -0.038 X 1O-3(± 0.018X 1O-3)p2 mm"

(15.42a)

154 Allamtiol/s and retinal image quality

aberration, the shadow movement is morecomplex. Quite common is the situation inwhich the retinoscope reflex corresponding toperipheral parts of the pupil moves 'against'while the central part of the reflex is 'with',corresponding to positive spherical aber-ration. Roorda and Bobier (1996) related theappearance of the retinoscopic reflex to avariety of aberrations.

Peripheral vision

At angles beyond about 20° the peripheralrefractive errors, that is, sagittal andtangential power errors and the associatedastigmatism, are the dominant aberrations.These affect the quality of the retinal imageduring ophthalmoscopy. Correcting periph-eral refractive errors has little effect on visualacuity beyond a few degrees from the fovea(MiIlodot et al., 1975; Rempt et al., 1976), butimproves peripheral detection (Leibowitz etal., 1972; Wang et al., 1995; Williams et al.,1996). This is discussed further in Chapter 18.

Ocular component contributionsThere is considerable interest in under-standing the corneal and lens contributions tothe aberration of the eye. This interest hasincreased with advances in the treatment ofrefractive errors by surgically manipulatingthe anterior corneal surface. It is important toknow the best shape to make the cornea andthe visual consequences of poor post-surgicalshapes (Applegate and Howland, 1997;Applegate et al., 1998;Martinez et al., 1998).There have been several attempts to

identify the magnitude of the corneal andlenticular contributions to ocular aberrations(Jenkins, 1963a; El Hage and Berny, 1973;Millodot and Sivak, 1979; Sivak and Kreuzer,1983). These studies suffered from problemssuch as inaccurate measurements of thecornea and measuring the lens in vitro, whereits shape will be different from the naturalstate or its optical conjugates have beenaltered. The emphasis has been on sphericalaberration and the cornea has often beenmodelled as a rotationally symmetrical sur-

face, which is a considerable over-simpli-fication.The approach to determining the relative

contribution of the components is to measurethe total aberration of the eye and determinethe aberrations of either the cornea or lens.The easiest component to measure is theanterior cornea. The shape of this can bemeasured with corneal topographicalinstruments. The aberration of the cornea isgiven by its departure from the ideal shape,which would give no aberration about thechosen reference axis. For a distant object, theideal cornea would have a front surfaceconicoid asphericity Q, independent ofradius, related to the cornea's refractive indexII byQ = -1/112 == -1/1.376 ee -0.53 (15.43)Q was explained in Chapter 2. The cornealaberration in wave aberration terms for anyray traced through the cornea is given bycorneal aberration == (11 -1)z (15.44)where z is the departure of the anterior corneafrom its ideal shape for the particular ray. Thelenticular contribution is then obtained bysubtracting the corneal contribution from thetotal aberration. In determinations such asthese, the aberration contribution of theposterior cornea is ignored. This is assumed tobe small because of the small refractive indexdifference between cornea and aqueous.The corneas of most eyes have negative

corneal asphericities, but the corneas usuallycontribute positive aberration because theasphericity is not high (Kiely et al., 1982;Guillon et al., 1986; see Chapter 2). Thedirection and magnitude of the aberration oflenses of eyes is still unclear, with theliterature giving results of both positive andnegative spherical aberration (Jenkins, 1963a;El Hage and Berny, 1973; Millodot and Sivak,1979; Sivak and Kreuzer, 1983; Tomlinson etal., 1993). There is more about this in the nextchapter.Correction of myopia by refractive surgery

increases the aberrations associated with thecornea, with the fourth order (sphericalaberration-like) aberrations increasingconsiderably relative to the third-order (coma-like) aberrations represented in equation(16.1) (Oliver et al.,1997;Applegate et al.,1998;Martinez et al., 1998; Oshika, 1999).

Monochromatic aberrations 155

........•...........-.

_____ Alexandridis and Baumann(1967)

Gullstrandnumber 1eyeGullstrandnumber 1eye (asphericcornea)

looking at close objects. The aberrations of theeye and instrument must be combined. Wheninstruments are aberration-free or have largeexit pupils relative to that of the eye, the eyelimits image quality. If alignment with the eyeis poor, additional aberrations will occur.These matters are covered by Smith andAtchison (1997).The most common correcting ophthalmic

devices are spectacle lenses. Optical design isconcerned with ensuring that good fovealvision is maintained for various rotations ofthe eye for which the line of sight will notcoincide with the lens optical axis. Thespectacle lens is designed as a wide angle,small pupil system whose stop is at the centreof rotation of the eye (approximately 15mmbehind the corneal apex). Accordingly, themonochromatic aberrations considered to beimportant are the sagittal and tangentialpower errors of the lens (and their combi-nations, i.e, astigmatism and field curvature).The aberrations of the eye itself are ignored,together with spherical aberration and comaof the lens, because these are small. Aber-rations associated with peripheral vision are

234567 g

Apparent pupil diameter (mm)

Figure 15.17.The magnification of the pupil frommeasurements of Alexandridis and Baumann (1967),paraxial prediction of the Gullstrand number 1 eye, andtheoretical results using Gullstrand's number 1schematic eye with an anterior corneal surface ofasphericity Q=-0.2, The experimental measurementsare the means from three excised corneas, which werepositioned approximately 3.6mm in front of a grid.

1.14

1.13

c0.~ 1.12u

l.;:'abIl...E 1.11's..::>P-

LIO

1.090

Aberrations of ophthalmic devices

Many optical instruments are used in combi-nation with the eye, including telescopes andmicroscopes and clinical instruments such asoptometers, keratometers and ophthalmo-scopes. There are ergonomic issues relating tothe way in which instruments must be used,e.g. the most appropriate alignment of the twotubes of binocular instruments so that the twoeyes are not placed under undue accom-modative and convergence stress when

Pupil aberration

Ocular aberrations affect not only the qualityof the retinal image, but also the formation ofthe entrance pupil. The entrance pupil is theimage of the aperture of the iris (the actualpupil), and ocular aberrations arising from thecornea will affect the appearance of the pupil.Of the five monocular aberrations, the aber-ration of most significance will be distortion,which affects the apparent size of the pupil(i.e. the position of the pupil margin).According to Gaussian optics, the entrance

pupil is about 13 per cent larger than theactual pupil (seeAppendix 3). Since distortionis a non-linear magnification effect thatincreases with image size by at least thesquare of the image size, we expect pupildistortion to have a greater effect, the largerthe pupil. The pupil distortion arising at thecornea is negative, and therefore as the realpupil becomes wider, distortion leads to asmaller magnification than the 13 per centpredicted from Gaussian optics.Alexandridis and Baumann (1967) exam-

ined the effect of pupil size on the apparentpupil diameter using excised corneas. Theyshowed that, for large pupil sizes, the pupilmagnification was less than the expectedGaussian value of about 1.13. We havedetermined the expected magnification effectfor the Gullstrand number 1 relaxed sche-matic eye with the anterior corneal surfaceaspherized with a Q value of -0.2, which isclose to the means of the studies of Kiely et al.(1982) and Guillon et al. (1986). Figure 15.17compares the experimental and theoreticalresults. They show similar trends, althoughthe magnification changes more quickly forthe former.

References

156 Aberratiolls alld retinal image ql/ality

also ignored because of the eye's poorperipheral resolution.The sagittal and tangential power errors are

minimized by varying the bending of thelenses and aspherizing one or both surfaces.The aberrations and design of spectacle lensesare covered by [alie (1984) and Atchison(l984b,1992).Design issues of contact lenses are very

different from those of spectacle lenses. Acontact lens rotates with the eye so that theoff-axis aberrations of the lens are not ofconsiderable concern. The surfaces are verycurved so that they fit well to the eye, whichmeans that spherical aberration becomes amajor aberration. The aberrations introducedby the contact lens must be combined withthose of the eye. Because the back surface ofthe contact lens must closely match thecornea, the degree of freedom provided byaltering lens bending is no longer available.The major design variable is aspherizing. Thelarge difference between the refractive indicesof the cornea and air (=; 0.376) is replaced by amuch larger difference between the refractiveindices of the contact lens and air (=; 0.45-0.49). This has a considerable effect on thecombined aberrations. When rigid contactlens are placed on eyes, aberrations change todifferent extents in different eyes for tworeasons (Atchison, 1995): (1) different eyeshave different anterior corneal shapes, and (2)approximately 90 per cent of the departure ofthe anterior corneal surface away from asphere is neutralized by the tear film betweenthe lens and eye.An artificial intraocular lens can replace the

natural lens of the eye when the latter isremoved, usually because of cataract. Like thecontact lens, the intraocular lens moves withthe eye, and its aberrations must be combinedwith those of the eye. Degrees of freedom arelens bending and surface aspherizing.Spherical aberration is an important aber-ration, but considerable sagittal and tan-gential power errors may result if the lensesare tilted or decentred. The aberrations anddesign of intraocular lenses are covered byAtchison (1990and 1991).

TA15DrbL1F(r)

W(X,Y)

W(r)

W2,O

W2,2

v8Ls(8)

Lt(8)

A(8)

FC(8)

n, Ii'

Qn

transverse aberrationentrance pupil radiusentrance pupil diameterray height in the pupilco-efficient in equation (15.13)spherical aberration as a power errorfor a ray entering the eye at a heightr. Also used for correcting power tominimize wavefront variance inequation (15.41)wave aberration for a ray passingthrough a point (X,Y) in the pupilwave aberration for an axial point ina rotationally symmetric systemgeneral wave aberration co-efficientsprimary spherical aberration waveaberration co-efficientspherical aberration wave aberrationco-efficient, which includes all fieldvarying termsprimary coma wave aberration co-efficientcoma wave aberration co-efficient,which includes all field varyingtermswave aberration polynomial co-efficient for defocuswave aberration polynomial co-efficient for astigmatismvariance of wave aberration overpupilangular distance off-axis in airsagittal power error for off-axisdirection 8tangential power error for off-axisdirection 8astigmatism corresponding to Ls(8)and Lt(8)field curvature corresponding toLs(8) and Lt(8)paraxial pupil ray angles in objectand image space respectivelyratio of paraxial pupil ray angles,defined by equation (15.36)asphericityrefractive index

Summary of main symbols

LA longitudinal aberration

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16Monochromatic aberrations of schematiceyes

Introduction

The monochromatic aberrations of schematiceyes are considered in this chapter. We beginwith paraxial model eyes, which were intro-duced in Chapter 5, and show that thesemodels predict ocular aberrations poorly. Theconstruction of more accurate schematic eyes,known as finite or wide angle schematic eyes,will then be considered. These models haveuseful applications, such as predicting retinalimage sizes (Drasdo and Fowler, 1974),predicting light levels on the retina(Kooijman, 1983), and predicting the effects ofchanges in any ocular structure on ocularaberrations and hence image quality. The lastof these is particularly relevant now thatcorneal shapes are being surgically modifiedto reduce refractive errors.Aberrations of schematic eyes have been

the subject of many investigations. Simpleequations do not exist for their exact calcu-lation from the system parameters. Useful,although approximate, equations exist in theform of Seidel aberration equations, and thesewill be used in this chapter, as well asdetermining exact aberrations. Seidel equa-tions increase in accuracy with decrease inpupil and field size; that is, as aberrationlevels decrease. Seidel aberration equationsprovide surface contributions to the totalaberrations. The equations indicate how thesesurface contributions depend upon theposition of the aperture stop and surfaceasphericity, factors that will be explored when

finite schematic eyes are discussed. Many ofthe Seidel aberration equations that appear inthis chapter were explained in detail by Smithand Atchison (1997a).The aberrations of the Gullstrand number 1

eye are representative of those of the LeGrandfull theoretical, Gullstrand-Emsley andBennett and Rabbetts eyes, at least in the caseof the relaxed (unaccommodated) versions(Appendix 3). Thus in this chapter, theaberrations of the Gullstrand number 1relaxed eye are compared with those of finiteschematic eyes and real eyes.

Note on notation for the waveaberration polynomial co-efficientsIn Chapter 15, we dropped the fielddependent sub-prefix before the 'w' of thewave aberration co-efficients. The reason forthis was that, since the aberrations of real eyesare measured at the fovea (which is about 5°off axis), the sum of all field dependent termsis actually being measured. In this chapter,where we use Seidel aberration theory topredict the corresponding wave aberration co-efficients, we must retain the field dependentprefixes.

MOlloclzrolllatic aberratiolls of schematic eyes 161

Real eyes····0···· Non-aberroscope studies

----6---- Aberroscope studies

Gullstrand number I eye

--0-- Seidel approximation

........<>........ Exact ray-tracing

2

8

9

e 7

5 6.~

5~<ii 4()'C{ 3ell

...-::::::~::::::t::....:-~:::····

I 2 3 4

Ray height in pupil (mm)

Figure 16.1. Spherical aberration of the Gulistrandnumber 1 schematic eye and of real eyes. Power errorsof the Gullstrand eye were calculated both from thevalues in equation (16.1a) derived from finite ray-tracing, and by using the value of b in equation (16.2a)derived from the Seidel approximation. Power errors ofreal eyes were derived from non-aberroscope studies asgiven by equation (15.14), and by aberroscope studies asgiven by equation (15.22).

equation. The value of oW4 0 can be foundfrom either a finite ray trace or from the Seidelspherical aberration 51 using equation(A2.14a). Values of 51 are given in Appendix 3for a number of schematic eyes. The value of bpredicted by Seidel theory for the Gullstrandnumber 1 eye is 0.000359 mm-3, which is closeto the value derived from the finite ray trace.

It follows from equations (15.13) and (15.16)thatM(r) '" br2 =40W4,or

2 (16.2)This equation and variants have appeared inthe literature several times - for example,Charman et al. (1978). For the Gullstrandnumber 1 schematic eye,M(r) '" 0.000359 y2 (16.2a)Equation (16.2) is approximate because thehigher order terms in equation (16.1) havebeen neglected. Such an approximation can becalled a Seidel approximation. The errorinduced by ignoring the higher order termswill become less important as the ray height rdecreases.

Aberrations of paraxial schematiceyesSpherical aberrationFor an object at infinity, a positive powerspherical surface or a positive power singlelens with spherical surfaces has positivespherical aberration, and the aberration levelincreases with the power. Since paraxialschematic eyes have spherical surfaces andthe powers of all surfaces except the posteriorcornea are positive, it may be expected thatrelaxed paraxial schematic eyes have positivespherical aberration that will increase withincrease in accommodation. While the backsurface of the cornea has negative power andis therefore expected to have negativespherical aberration, it has the lowest of thesurface powers of the eyes and thereforecannot cancel the positive aberration of theother surfaces. These predictions areconfirmed by Seidel aberration results inAppendix 3.The exact spherical aberration can be

calculated as a power error M(r) by finite(exact) ray-tracing. This is done by tracingrays from the axial point on the retina,through and out of the eye into air, anddetermining the vergence of these rays at thefront principal plane. Such a ray is shown inFigure 15.7c. Apart from the sign, thevergence of this ray at the front principalplane is the power error M(r).Aberration theory predicts that the power

error (F(r) can be expressed as an even powerpolynomial in ray height r at the eye. That is,M(r) = br2 + cr4 + dr6 + e,s + frIO (16.1)In Chapter 15, we assumed for the real eyethat c =d=e=f =O. Fitting this polynomial tothe Gullstrand number 1 schematic eye'spower error values givesb=0.355 X 10-3 mrrr',c =0.0128 X 10-3 mm-5,d = -0.00166 X 10-3 mm",e = 0.000242 x 10-3 mrrr",f =-0.00000857X 10-3 mm" (16.1a)The value of b can be used to calculate theprimary wave aberration co-efficient oW4 0 bythe simple equation (15.16). Alternatively, ifthe value of oW4 0 is known, this can be usedto predict the value of b, using the same

162 Aberratio1!s and retinal illlase quality

Figure 16.1 shows the power errors of theGullstrand number 1 eye, with exact powererrors derived from the values in equation(16.la) after finite ray-tracing, and approxi-mate power errors derived from equation(16.2a). The Seidel approximation is in errorby -8 per cent at a ray height of 2 mm.

Surface contributionsAlthough the Seidel approximation is notaccurate for the larger ray heights (Figure16.1), the Seidel equations indicate the relativesurface contributions to the sphericalaberration. In the Gullstrand number 1 eye,the anterior corneal surface provides most ofthe Seidel spherical aberration (== 60 per cent)with the posterior surface of the lens being thenext largest contributor (== 30 per cent)(Appendix 3). The accommodated version ofthe Gullstrand number 1 eye has about 3times the spherical aberration of that of therelaxed eye, with the increase being duemainly to a 4.6 times increase in the aberrationof the lens.

Comparison with real eyesAs well as showing the power errors of theGullstrand number 1 eye, Figure 16.1 showsmean real ele power errors. The b value 0.354x 10-3 mm" ,given by equation (16.1a) for theGullstrand schematic eye, is 5-10 times the bvalues for real eyes: 0.076 x 10-3 mrrr-' fornon-aberroscope studies as given by equation(15.14) and 0.035x 10-3 mm-3 for aberroscopestudies as given by equation (15.22). Clearlyfrom the figure and the values of b, real eyeshave on average much less sphericalaberration than the paraxial schematic eyes.

ComaFor a rotationally symmetric eye, coma occursonly off-axis and increases with distance off-axis. It will be calculated at the fovea,assuming this to be 5° from the optical axis,and with the pupil centred on the optical axis.The exact amount of coma in a schematic eyecan be determined by tracing two finite raysin the tangential section of the pupil,

calculating the wave aberrations of each, andtaking the difference. Ifwe trace a finite ray ata height of 1 mm in the upper and lowertangential sections of the pupil and denotethese aberrations as W(I,OO) and W(I,1800),respectively, providing there are no other oddoff-axis aberrations present, the coma waveaberration co-efficient W3,1 (including all fielddependent terms) will beW3,1 = [W(I, 0°) - W(l, 180°)]/2 (16.3)For the Gullstrand number 1 schematic eye,W(l, 0°) = 0.000300 mm and W(I,1800) =0.000194 mm. Thus,

W3,1 =+0.0000530 mm-2 (l6.4a)Since this value is positive, the coma flare ison the axis side of the image point Q' asshown in Figure 15.9.The Seidel coma 52 is given in Appendix 3,

and the corresponding primary waveaberration co-efficient 1W3 1 is related to it byequation (A2.14b) to give'

1W3,1 =0.0000573 mrrr? (16.4b)The coma co-efficients in equations (16.4a)and (16.4b) are similar, showing that Seideltheory predicts accurately the level of exactcoma for a ray height of 1 mm and at the foveaof a schematic eye. This is probably becausethe higher order terms are not significant at a5° off-axis angle.

Comparison with real eyesThere are measures of coma-like aberrationterms from aberroscope studies. Walsh andCharman's (1985) determined values of W6 toW9 co-efficients, as given in equation (15.1), inthe range -0.000128 mm-2 to +0.000108 mm-2.The above estimates for the Gullstrandnumber 1 schematic eye lie within this range,and therefore it is possible that the coma ofreal eyes is greater than expected fromschematic eyes.Some of the coma found in the aberroscope

studies may be because of inaccurate pupilcentre location (Smith et al., 1998), andbecause the pupil is probably decentred fromthe best fit optical axis. In Chapter 15 wedetermined the amount of coma arising frompupil decentration in the presence of sphericalaberration. For a transverse pupil shift L1X, we

can rewrite equation (15.27), giving thechange in coma L1W3,1 asL1W3,1 =-4 L1X W4,o (16.5)If the pupil shifts longitudinally from thenodal point by the distance EN and the pupilray is inclined at an angle (Jto the optical axis,the centre of the pupil is transversely shiftedby a distance L1X given by the equationL1X =(JEN (16.6)for small angles. Substituting this value of L1Xinto equation (16.5) gives the expected changeL1W3,1 in coma ofL1W3,1=-4(JENW4,o (16.7)Guidarelli (1972) argued that the eye is almost'homocentric', because the centres ofcurvature of the corneal surfaces and theretina lie almost on the nodal points. If theaperture stop was also at the nodal points, theeye would have no off-axis aberrations. If theaperture stop was not at the nodal points, thevalue of L1W3,1 in equation (16.7) would be theabsolute value of coma, not merely a change.van Meeteren (1974) relied on Guidarelli'sargument that the eye is homocentric to

0.20 -1--'----'----'--..1...--'----'----'--+

4

6..'-0.10 -I--r----r--.--,------,r----r-.y.-+o 2 345 678Aperture stop distance from corneal vertex (rnm)

Figure 16.2. Primary wave coma co-efficient of theGullstrand number 1 eye for different positions of theaperture stop relative to the corneal vertex (off-axisangle 5°). The nodal points Nand N' are also indicated.The numbers represent the positions of surfaces: 1 -anterior cornea; 2 - posterior cornea; 3 - anterior lens; 4- anterior lens core; 5 - posterior lens core; 6 - posteriorlens.

MOllocllroma/ic aberra/iolls of schematic eyes 163

predict the level of coma in real eyes frommeasured values of spherical aberration andthe above equation.

If we are to rely on this argument forpredictions of coma, we need to confirmwhether equation (16.7) predicts accuratelythe level of coma in the eye. While the centresof curvatures of the corneal surfaces do lienear the nodal points, those of the lens andretina do not. The primary coma co-efficient1W3 1 of the Gullstrand number 1 eye isplotted against aperture stop position inFigure 16.2. This eye has zero coma if theaperture stop is 1.5mm in front of the frontnodal point, with the corresponding entrancepupil position 3.8mm from the corneal vertexinstead of the usual 3.05 mm. Therefore,placing the aperture stop at the front or backnodal point does not eliminate coma, and theparameters of Gullstrand number 1 schematiceye do not support Guidarelli's hypothesis ofhomocentricity.

Astigmatism and peripheral powererrors

Sagittal and tangential power errorsIf we trace thin beams of light (pencils) raysinto a schematic eye as shown in Figure15.10a, the fans in the sagittal and tangentialsections focus on the sagittal and tangentialimage surfaces, respectively. In general, thesagittal image surface is behind the retina andthe tangential surface is in front of the retina.For real eyes, we measure the vergences inobject space of the sagittal and tangential fociconjugate with the retina (Figure 15.10b), soperipheral power errors in a schematic eye arecalculated by tracing finite or exact rays out ofthe eye. Figure 16.3 shows exact power errorsLs((J) and Lt( (J)of the Gullstrand number 1 eyeas measured from the corneal vertex plane.The results depend upon the shape of theretina, which has been assumed to bespherical with a radius of curvature of-12mm.We can investigate the effect of the retinal

radius of curvature using Seidel aberrationtheory. According to Seidel theory, the sagittaland tangential image surfaces shown inFigure 15.lOa are spherical, and the radii ofcurvature can be calculated from equations

164 Aberrations lind retinal ill/age quality

605040302010

Field curvature

----lIl-- Gullstrand number I: exact results__ Mean of real eyes. equation (15.35)

-10+--,....-.,--..---.----r--.,-..---,---r---,r--..,.--+o

Angle in air (deg)

Figure 16.4. Astigmatism and field curvature of theGullstrand number 1 eye and of real eyes.

Astigmatism

--iJ- Gullstrand number I eye: exact results5 15 _ Mean of real eyes. equal ion (I5.31aJ

AstigmatismAs in Chapter 15, astigmatism A(8) isquantified as the difference between thesagittal Ls(8) and tangential L t(8) powererrors. Recalling equation (15.30),A(8) =Ls(8) - Lt(8) (16.10)

Figure 16.4 shows exact astigmatism for theGullstrand number 1 eye from the data shownin Figure 16.3. Combining equations (16.9aand b) and equation (16.10), the Seidelapproximation for astigmatism is

(fl 1 1A(8) == -- [- - -) (16.11)

2nvit rs r tThis equation shows that, within the Seidelapproximation, astigmatism is independent ofthe value of the radius of curvature rR of theretina. For the Gullstrand relaxed eye,

Comparison with real eyesAs well as showing the power errors of theGullstrand number 1 eye, Figure 16.3 showsmean real eye power errors. The sagittalpower errors of the schematic eye are similarto those of real eyes. However, the tangentialpower errors of the schematic eye are muchlarger than those of real eyes .

60

(16.9a)

5040302010

....0 .... ~

-....0-- Ls

-15 +--,....~-.----r--r--.-..---r---.----,r--..,.--+o

...."'B::::f$"'Q·"id:.:::.O'·..o, ••• Tangential

Gullstrand number I eye .•~... 0"'0•••--0- t, Seidel ····~>.O 0 ...

....[J.... t, Seidel ···-,Q.:·..·.·O"'0.

---<>- Lsexact ..~ .

····0···· L,exact

Real eyes ~.

(fl 1 1Lt(8) == -- [- - -) (16.9b)

2nvit rt rRwhere rR is the radius of curvature of theretina, 8 is the off-axis angle in radians, andnvit is the refractive index of the vitreous.Figure 16.3 compares the power errors

obtained from these equations with the exactpower errors for the Gullstrand number 1 eye.The Seidel equations are reasonably accuratewithin about 15-20° from the optical axis.

(A2.15)and the Seidel data given in Appendix3. For the Gullstrand number 1 eye, thesagittal radius rs and tangential radius rt havethe valuesrs = -13.89 mm and rt = -9.77 mm (16.8)These values were used to plot the sagittaland tangential surfaces shown in Figure15.lOa and so predict the astigmatism. Thepower errors can be expressed in terms of theobject space field angle 8 and the sagittal andtangential surface radii of curvature usingequations (A2.19),which are Seidel equations(A2.16) modified to take into account thecurvature of the retina. These equationsare:

Angle in air (deg)

Figure 16.3.The sagittal and tangential power errors ofthe Gullstrand number 1 eye and of real eyes.

MOllochromatic aberratiolls of schematic eyes 165

Distortion

given bylP 1 1 2

Field curvature (0) '" -- [ - + - --]2nvit rs rt rR

As discussed in Chapter 15 (Types andmagnitudes of monochromatic aberrations),distortion has little meaning for the eyebecause of the curvature of the retina. Of

Real eyes

Ames and Proctor ( 1921)

Gullstrand number I eye--0-- Exact ray-tracing

20

80

10

70

~6Oee~~ 50'">.

'".5 40

'"~ 30«

Comparison with real eyesAs well as values of astigmatism, Figure 16.4shows exact values of field curvature of theGullstrand number 1 schematic eye and themean field curvature of real eyes. Theschematic eye values are similar to those ofreal eyes to about 30°,but at higher angles theschematic eye values are greater.

o(:F----,..-----r---r----.-....,...-..----,..--r-___+_0102030405060708090

Angle in air (deg)

Figure 16.5. Internal and external angles for the pupilray of the Gullstrand number 1 eye and of the 'real eye'results of Ames and Proctor (1921).

(16.15)which shows that field curvature is depen-dent upon the shape of the retina. For theGullstrand number 1 schematic eye, thisequation reduces toField curvature(0) ee -8.73 X 10-4 lPdeg D

(16.15a)

Petzval and field curvatureIn equation (15.34), field curvature wasdefined as the mean of the sagittal andtangential power errorsField curvature(O) = [Ls(O) + Lt(O)]12 (16.13)If astigmatism is zero, the sagittal andtangential surfaces coincide. In Seidel theory,this surface is called the Petzval surface(Figure 15.lOa). The radius of curvature rp ofthe Petzval surface is related to the value ofthe Seidel aberration 54 by equation (A2.15c)and, using the data given in Appendix 3, forthe relaxed Gullstrand number 1 eye:rp =-17.85 mm (16.14)which is considerably larger than the retinalradius of curvature, assumed here to be-12mm.Using the definition of field curvature given

by equation (16.13) and the expressions forLs(O) and Lt(O) given by equations (16.9a andb), the Seidel estimate of field curvature is

Comparison with real eyesAs well as showing the astigmatism of theGullstrand number 1 eye, Figure 16.4 showsthe mean astigmatism of real eyes as takenfrom equation (15.31a). Once again, theschematic eye has worse aberration than realeyes.

equation (16.11) predicts that the astigmatismvaries with field angle asA(Odeg) '" 0.198 lPdegD (16.11a)where 0deg is now the off-axis angle indegrees.Equations (A2.17) can be used to predict the

expected value of the two wave aberration co-efficients Wz,o and W2,7' Using the data inFigure 16.3, for 5° off-axis,W2,0 =-1.61 X 10-5 mm! (16.12a)andW2,2 =4.33 X 10-5 mm! (16.12b)At 5°, the exact and Seidel values are the sameto three significant figures.

(16.16)

166 Abam/illl/> IIl/d retinal iII/age'1l/a/i/y

importance is the relationship between theangles of the pupil ray (J'inside the eye andthat () outside the eye. We can calculate theexact relationship by tracing exact rays. Theresults for the Gullstrand number 1 schematiceye from exact ray-tracing are shown inFigure 16.5, together with the data of Amesand Proctor (1921), which have been replottedfrom Figure 15.15. The two curves are similar,suggesting that paraxial schematic eyes canpredict retinal image positions accurately,even for large off-axis angles.

Summary

The reader is reminded that there areconsiderable inter-individual variations inaberrations of real eyes, and the comparisonsgiven here are based on the authors'estimations of mean levels of aberrations forreal eyes.Paraxial schematic eyes do not predict

accurately the spherical aberration of realeyes, with the predictions being 5-10 timesthat occurring in real eyes. The paraxialschematic eyes do better at predicting the off-axis sagittal and tangential power errors, withthe sagittal power error prediction beingmuch better than tangential power error. Theastigmatism of paraxial schematic eyes isapproximately twice that of real eyes, and thefield curvature of paraxial schematic eyes isaccurate to about 30° off-axis angle. Based onsome indirect experimental data, the paraxialeyes would seem to be accurate in predictingthe position of the off-axis retinal image,which is related to the distortion aberration.Little can be said about coma, because of thelack of definitive experimental data, except tostate that its value will depend upon anytransverse placement of the pupil.We have already mentioned that approxi-

mate Seidel aberration theory is useful inshowing trends in aberrations and indicatingthe surface contributions to aberrations. Alsoas mentioned previously, although in thereciprocal sense, Seidel aberration approxi-mations are increasingly inaccurate as pupilsize or off-axis angle increases. Seidel aber-ration theory underestimates the amount ofspherical aberration in the schematic eye, byabout a factor of two at a ray height of 4 mm.In the case of sagittal and tangential power

errors, the Seidel aberrations are reasonablyaccurate to about 15-20° from the optical axis,after which the Seidel astigmatism becomestoo large, particularly for the tangential errors.This leads to over-estimations of bothastigmatism and field curvature.

Modelling surface shapes

Having shown that paraxial schematic eyespredict some of the aberrations of real eyespoorly, particularly spherical aberration, wewill investigate the ways in which the paraxialmodels can be improved to more accuratelyrepresent real eyes. The properties of asphericsurfaces, the ways of representing thegradient refractive index of the lens, and howthese affect the aberrations, will be con-sidered. In this section, surface asphericity isconsidered.All paraxial schematic eyes have spherical

surfaces, but surfaces of real eyes are non-spherical. More accurate model eyes shouldinclude aspheric refracting surfaces. As theword 'aspheric' simply means 'non-spherical',there is an infinite range of aspheric surfaces.However, we will restrict this range torotationally symmetric aspheric smooth sur-faces. In conventional optics, the asphericalsurface most frequently found to be useful isthe conicoid.

Conicoid surfaces

Equation (2.4) gives the form of a conicoidsurface as1z2 + (1 + Q)Z2 - 2ZR =0wherethe Z axis is the optical axis1z2 = X2 + y2R is the vertex radius of curvature andQ is the surface asphericity whereQ < 0 surface flattens away from its vertexQ =0 specifies a sphereQ > 0 surface steepens away from its

vertex.The effect of the value and sign of Q on

surface shape is shown in Figure 2.2.Asphericity is sometimes expressed in termsof a quantity p called the shape factor, which

(16.24a)

is related to Q by the equationp=1 + Q (16.17a)Another quantity which has often been usedis the eccentricity e.This is related to Q by theequationQ = -e2 (16.17b)but while e2 is always positive, e by itself onlyhas meaning if Q is negative. For ellipses(ellipsoids in one section only), the followingequation is also popular(Z-a)2/a2+ y2/b2= 1 (16.18)where a and b are the ellipse axes semi-lengths. The vertex radius of curvature R isrelated to a and b by the equationR = +b2/a (16.19)and the asphericity Q is related to a and b byQ = b2/a2 -1 (16.20)

Effect of conicoid asphericity on SeidelaberrationsHopkins (1950) and Welford (1986) defined asurface aspheric aberration contributionfactor k, which isk = C2h4 FQ (16.21)where C is the surface curvature, h is theparaxial marginal ray height at the surface,and F is the surface power. In terms of thisfactor k, aspherizing produces the followingchanges in the Seidel aberrations:L1S1 = k, L1S2 = Ek, L1S3 = E2k and L1Ss = E3k

(16.22)where E is the stop shift factor defined as h/h,h is the marginal ray height at the surface, andh is the height of the paraxial pupil ray at thesurface.Equations (16.21) and (16.22) show that

aberrations are likely to change more quicklywith asphericity as surface power increases -e.g. asphericity is likely to have its greatestinfluence at the anterior corneal surface.However, for off-axis aberrations this isdependent on the stop shift factor E such thataspherizing is more effective the further asurface is away from the aperture stop.Aberrations such as astigmatism will not be

Monocttromatic aberratiol1s of schematic eyes 167

affected by aspherizing the lens anteriorsurface where E has a value of zero.

Figured conicoid surfaces

Another type of aspheric surface is given bythe polynomialZ = v1h

2 + v2h4 + v3h

6+ ... (16.23)A conicoid can always be expressed in thisform, but not all polynomials of this form areconicoids. A conicoid can be expressed in thisform by firstly solving equation (16.16) as aquadratic equation in Z to give

Z = R- V[R2 - (1 + Q)h2](1 + Q)

and then using the binomial theorem toexpand the square root term to express thisequation for Z in polynomial form. However,equation (16.24a) is not useful for a flatsurface (i.e. R = 00) or paraboloids (Q = -1),and an alternative form is

h2

Z = R + ~[R2 _ (1 + Q)h2] (16.24b)

which is solvable for all values of theparameters.Some non-conicoid surfaces can be

approximated by a conicoid and a smallsurface adjustment. These surfaces are calledfigured conicoids, and can be described by theequation

Z =Zconicoid +14h4 +16h6 +Ishs + etc. (16.25)where Zconicoid is the value of Z given byequation (16.24a or b) and the co-efficients14,16, etc are called figuring co-efficients. Thevalues of the figuring co-efficients dependupon the aspheric surface being modelled. Ifthe surface curvature is not zero, the term 14can be omitted provided that the value of Qand all other figuring co-efficients arechanged appropriately.Usually, an exact representation of a non-

conicoid aspheric in equation (16.25) requiresan infinite number of figuring. terms. Inpractice, the figuring power series isterminated at a finite number of terms,leaving a residual error. The required numberof figuring co-efficients depends upon thepermissible error.

168 Aberrations and retinal image quality

Modelling the anterior cornealsurface

There is considerable published data on theanterior corneal asphericity. Many investi-gators fitted their corneal shape data to eithera conic in certain sections or to conicoidsusing three-dimensional data. They expressedthe results in terms of the asphericity Q, shapefactor p or the eccentricity e. Some results, allin terms of Q, are given in Table 2.3.The meanvalues obtained by the larger scale studies arevery similar, being in the approximate range-0.2 to -0.3. The effect of the value of Q oncorneal shape is shown in Figure 2.3 forcorneas with a vertex radius of curvature of7.8mm. Equations (16.21) and (16.22) showthat negative values of Q will reduce sphericalaberration.Other types of equations have been used to

describe corneal shape, e.g. those of Bonnetand Cachet (1962) and Bonnet (1964).

Modelling the posterior cornealsurface

In contrast with the anterior corneal surface,the posterior corneal surface has not attracteda great deal of attention. It is neglected inSimplified eye models (Chapter 5). This isusually justified on the grounds that it hasmuch less power than the anterior surface(about one-tenth of the power).

It is difficult to measure accurately theshape of this surface, because it is so close tothe anterior surface and its apparent shape isinfluenced by the shape of the anteriorsurface. As mentioned in Chapter 2, Patel et al.(1993) measured the shape of the anteriorsurface and the corneal thickness, and thendeduced the posterior surface from that data.They found a mean posterior surface aspheri-city Q value of -0.42. However, their resultsmust be viewed with caution because theiranterior mean asphericity was -0.03, which isvery different from most other values shownin Table 2.3.Equations (16.21) and (16.22) show that the

above negative values of Q would increasepositive spherical aberration, but since thepower of this surface is small, the effect wouldbe small.

Modelling the lenticular surfaces

Attempts to extract asphericity Q values fromthe in vitrodata of Howcroft and Parker (1977)are given in Table 2.5, together with Q valuesderived from the data of Brown (1974) by Liouand Brennan (l997). Probably there is a widevariation in values due to a combination ofthe difficulty in accurate measurement ofasphericity and of inter-individual variationsin asphericity values (Smith et al., 1991).

Modelling the lenticular refractiveindex distribution

The refractive index of all ocular media exceptthe lens can be regarded as uniform, and thusonly the lens needs special treatment. It hasbeen known for well over a hundred yearsthat the lens has a varying refractive index,and Gullstrand's (l909) representation of therefractive index is given in equation (2.12).In modelling the internal structure of the

lens, two types of lens models have been used.These are a multiple layered shell structure(e.g. Gullstrand, 1909; Pomerantzeff et al.,1984; Raasch and Lakshminarayanan, 1989;Mutti et al., 1995) and a continuously varyingindex (e.g. Gullstrand, 1909; Blaker, 1980 and1991;Smith et al., 1991). The advantage of theshell model is that it allows conventionalparaxial ray-tracing procedures to be used toexamine powers, but has the disadvantagethat it is more difficult to analyze itsaberrations, because conventional Seideltheory and finite ray-tracing routines cannotbe used with such shell structures. Bycontrast, while ray-tracing through a gradientrefractive index and aberration analysis ismuch more complex, routines for performingthese calculations are well established. Wewill look at both these models, beginning withthe continuous refractive index model.

Continuous refractive index model

A general mathematical form for representingthe refractive index distribution N(Y,Z) in theY-Z section, assuming rotational symmetryabout the optical (Z) axis, isN(Y,Z) = No(Z)+ N1(Z)y2 + N2(Z)y4 ... (16.26)

The power of the lensThe power of the lens can be divided into twocomponents: (a) one arising from the frontand back surfaces, and (b) one arising fromthe gradient index alone.

The values of the Nj,j co-efficients set therefractive index distribution within the lens.Smith et al. (1991) presented a gradient

index model of the lens in which the indexalong any line from the centre of the lens tothe edge and at a relative distance r from thelens centre can be described byN(r) = Co + c1r

2 + c2r4 + c3r6 (16.27)

They divided the refractive index distributioninto two half ellipses (Figure 16.6). The frontellipse has a semi-axis a1 along the optical axisand a semi-axis b along the equatorialmeridian. The back ellipse has a semi-axis a2along the optical axis and semi-axis b alongthe equatorial meridian. For the front half ofthe ellipse, some of the N, j co-efficients aregiven by ,

No,o = Co + c1 + c2+ c3NO,1 =(-2c1 - 4c2 - 6c3) / a~NO,2 =(c1 + 6c2 + 15c3)/a1Nl,O = (c1 + 2c2 + 3c3)/ b2 (16.28a)For the back half of the ellipse, some of the N, jco-efficients are given by ,

No,o= CoN0 1 =0, 2NO,2 = c/aZN1,o=c/lr (16.28b)A full set of co-efficients was given by Smithet al. (1991).The refractive index distribution has been

investigated in many vertebrate lenses andhas been often fitted by a parabolic form ofequation (16.27) in which c2 and c3 are set tozero. However, Pierscionek and Chan's (1989)results with human lenses indicate that thedistribution is more complex and may requirea higher order polynomial description.

y

z

z

v

Figure 16.6. The gradient index of the lens modelled astwo half ellipses joined at the equator. Some iso-incidallines are shown.

Figure 16.7.The power of the gradient index of the lensmodelled as a 'slab' lens.

The gradient index powerWe can study the gradient index contributionto the total refractive power of the lens byregarding the lens bulk as a slab as shown inFigure 16.7. The slab of gradient indexmaterial has thickness t and is immersed in amedium of refractive index J.l. We can find anapproximate equation for the power of thisslab by tracing the ray AA', assuming it is aparaxial ray and that the change of height of

Surface powersThe power of the surfaces can be found fromthe equationF = (n' - n)/R (16.29)

where R is the radius of curvature of a surfaceand nand n' are refractive indices on eitherside of the surface.

(16.26a)

where

No(Z) = Noo+ No lZ +N0 2Z2 ...1 , , 2N1(Z) = N10 + N1 1Z + N1 2Z '", , , 2N2(Z) = N2,0 + N2,1Z + N2,2Zetc....

170 Aberratiolls mId retinal illlo:\<, quolity

this ray in the lens is negligible, and findingwhere this ray crosses the optical axis at F'.This lens will then have a back vertex focallength f'v and a corresponding back vertexpower F'v' which are related by the equationt ; =I1If'v (16.30)The paraxial ray must pass through the pointF' for all ray heights Y. According to Fermat'sprinciple, this will be so providing the opticalpath lengths for the two ray paths VV'F' andAA'F' are equal. Denoting the optical pathlength by square brackets, we must have[VV'] + l1f'v =[AA'] + I1 V'( y2+ f'})Atchison and Smith (1995) have shown thatthis equation reduces to

F'; se -2(N~,ot + N1,lt Z/2 + Nu t3/ 3 + N1,3t4I 4+ N1,i 15 + ...) (16.31)

Within the approximation, this equationshows that this power is dependent only uponthe co-efficients IN] j' j =0, 1... 1which are theco-efficients of yZ' in the refractive indexfunction N(Y,Z) given by equations (16.26).Therefore, the power does not depend onterms of higher order than Yz in the N(Y,Z)function. These higher order terms are, ineffect, aberration terms.

Total lens power and positions of thecardinal pointsEquation (16.31) does not give the equivalentpower or any information that allows us tocalculate the positions of the cardinal points.If the equivalent power and cardinal pointpositions are needed, a paraxial ray tracesuitable for gradient index media can be used,such as those of Moore (1971) or Doric (1984),or a finite ray traced which simulates aparaxial ray by being close to the axis e.g.Sharma et al. (1982). Smith and Atchison(1997b) extended the method used to deriveequation (16.31), and showed that theequivalent power and cardinal point positionscan be predicted from a knowledge of the N, .co-efficients. oj

Aberrations of the lens due to the gradientindex lensSands (1970) presented a set of equations fordetermining the Seidel aberrations of gradient

index media. To make these equationsconsistent with the Seidel aberrations ofHopkins (1950) and Welford (1986), Sands'aberration values must be multiplied by afactor of two and the signs changed. Themodified equations are given in Appendix 2.As argued in the appendix, these equationsshow that NZ,j terms are likely to be the maincontributors to the Seidel or primary sphericalaberration, with a smaller contribution fromthe NOj and N1j terms. The spherical aber-ration has the opposite sign to that of the Nz .terms, so the sign of these co-efficients musfbe positive to reduce the spherical aberrationof the eye.

The shell model

While the lens has a continuous gradientindex structure, the refractive index has oftenbeen represented by a shell structure, with therefractive index in each shell kept constant,but varying from shell to shell. In theconstruction of such a model, it is necessary todecide the number of shells, how therefractive index varies from shell to shell, andthe value of the curvatures of the surface ofeach shell.Once the shell structure is established,

paraxial ray-tracing can determine the lenspower. Gullstrand (1909)presented a model ofthe lens with only two shells, but if it isintended to represent accurately the con-tinuously varying index by such a shellstructure, a larger number of shells should betaken. For example, Mutti et al. (1995) used 10shells and Pomerantzeff et al. (1984) used 200shells.

The power of the lensThe advantage of the shell model over thegradient index model is that conventionalparaxial ray-tracing can be used to find thelens equivalent and vertex powers. Given ashell structure with a large number of shells,Atchison and Smith (1995) derived anequation for the power of the bulk medium,according to the same principles that theyused to derive equation (16.31). They showedthat the back vertex power F'; is given by the

Monoc!lTvmatic aberrations of schematic eyes 171

equationr; == IN'(Z)C(Z) dZ (16.32)where N'(Z) is the derivative of the refractiveindex with respect to the distance Z along theoptical axis and C(Z) is the shell curvature asa function of Z. This equation shows that theshell model will have some power providingN(Z) is not constant and C(Z) is not zero.

Aberrations of the lensAs stated above, aberration analysis of shellstructures is not possible with conventionalSeidel theory or conventional ray-tracing, andthe authors are not aware of any proceduresfor such calculations.

gradient index (Liou and Brennan, 1997).Schematic eyes of Pomerantzeff et al. (1971,1972 and 1984) and Wang et al. (1983) are notdiscussed further because they did notprovide full constructional details.Many finite eye models have been

developed for a restricted set of conditions.For example, the eye of Liou and Brennan wasdesigned to give realistic spherical aberration,and off-axis monochromatic aberrations werenot considered. It is unfair to be critical of amodel if it does not predict quantitiesaccurately that are outside the scope of itspurpose.

Lotmar (1971)

Survey of finite eye models

Modelling the retina

Table 16.1. Schematic eye values of retinal radius ofcurvature.

Comprehensive details of the more anatomi-cally accurate schematic eyes are given inAppendix 3. Most of the finite model eyes arebased upon established paraxial schematiceyes, improved by aspherizing one or more ofthe refracting surfaces, but one includes a lens

Lotmar (1971) modified the Le Grand fulltheoretical eye, a four surface schematic eye.He aspherized the anterior surface of thecornea using the data of Bonnet and Cochet(1962) and Bonnet (1964), who used a specialtype of function that was not readily suitablefor ray-tracing. Lotmar represented the frontsurface of the cornea by equation (16.23) up tothe sixth power. This function can be inter-preted as a figured paraboloid, that is with Q=-1, and figuring co-efficients of /4 and f,.Smith and Atchison (1983) showed that the!.term can be eliminated, providing the value 01Q and all the other figuring co-efficients areadjusted suitably. If this transformation isdone for the Lotmar cornea, we have Q=-0.286 and new figuring co-efficients as givenin Appendix 3. There are two advantages ofusing this ellipsoid model. First, calculationsshow that this ellipsoid gives a better fit thanthe paraboloid to Lotmar's full polynomialform. Second, it allows a comparison with thedata of other investigators, who fitted ellip-soids or ellipses to the cornea and foundsimilar values of Q (Table 2.3). Lotmar alsomodified the lens by replacing the posteriorspherical surface by a paraboloid.This schematic eye is not an accurate

representation of real eyes. While the cornealasphericity is close to that of real eyes, the lensis not accurately modelled. Lotmar ignoredthe aspheric nature of the anterior lenssurface, and as yet there is no firm evidencethat the posterior surface is parabolic.Lotmar determined the spherical aberration

and astigmatism of his model. He claimed

Radius of retina (mm)

-11.06-12-12.3-10.8-14.1

Source

Stine (1934)Drasdo and Fowler (1974)Lotmar (1971)Kooijman (1983)(ellipsoidal with Q= 0.346)

Compared with many other ocular para-meters, the shape of the retina has not beenextensively studied because of its inaccess-ability. Some schematic eye values are listed inTable 16.1. The retina is not necessarilyspherical.Usually, Seidel theory assumes that the

image surface is flat. For analysing the sagittaland tangential power errors, we took intoaccount the curved nature of the retina bymodifying the respective aberrations, as givenby equations (16.9a and b).

172 Aberrations and retinal imag« quality

that the spherical aberration was of the sameorder as found experimentally, but did notshow any comparison. Lotmar also claimedthat the astigmatism was in good agreementwith experimental data from Ferree et al.(1931) and Rempt et al. (1971).

Drasdo and Fowler (1974)

This is a modified form of a schematic eyeattributed by Stine (1934) to Cowan (1927),which is the same as the Gullstrand-Emsleysimplified eye except that it uses 1.336 for therefractive index of the aqueous and vitreousand 1.43 for the refractive index of the lens.Stine added a retinal radius of curvature of-11.06 mm. The corneal surface has a radiusof curvature of 7.8 mm and an asphericity ofQ = -0.25, obtained from published data ofPrechtel and Wesley (1970) and Mandell andSt Helen (1971).The purpose of this model was to determine

retinal projection from the visual field. WhileDrasdo and Fowler (1974)acknowledged thatreal crystalline lenses have aspherical sur-faces, they argued that the level of asphericitywould make little difference to the path ofprincipal rays, and therefore used a simplifiedlens with spherical surfaces. This argument issupported by the data in Figure 16.5.

Kooijman (1983)

Kooijman modified the Le Grand fulltheoretical eye model for the purpose ofpredicting retinal illumination. Both cornealsurfaces have a Q value of -0.25, with thevalue of the anterior corneal surface takenfrom Mandell and St Helen (1971). Theanterior lens surface is hyperbolic with a Qvalue of -3.06, taken from Howcroft andParker (1977). The posterior lens surface isparabolic (Q = -1), again taken from Howcroftand Parker (1977). Two forms of the retinalshape were obtained from Krause (Helmholtz,1909). One is spherical with a radius ofcurvature of -10.8 mm and the other iselliptical with a radius of curvature of -14.1mm and an asphericity of Q= +0.346. Theaberration results presented in the nextsection and in Appendix 3 were obtained withthe spherical retina.

Navarro, Santamaria and Bescos(1985)

Navarro et al. (1985) used a variable accom-modating model, in which the crystalline lensparameters and the distance between thecornea and lens are expressed as functions ofthe level of accommodation. Two paraxialvariable accommodating eyes have beendiscussed already in Chapter 5.The base paraxial model is the Le Grand

schematic eye, except for a slightly differentanterior corneal radius and corneal index. Theanterior corneal surface and the lenticularsurfaces are conicoids. The lens surface curva-tures and asphericities, the lens thickness, theanterior chamber depth and the lens refractiveindex vary as functions of accommodation,most in a logarithmic manner. The combi-nation of parameters lead to sphericalaberration decreasing with accommodation,as in real eyes (see Chapter 15, Types andmagnitudes of monochromatic aberrations), and itis approximately zero at an accommodationlevel of 5 0 out to approximately 2 mm pupilheight.Escudero-Sanz and Navarro (1999) added a

spherical retina of -12 mm radius ofcurvature. The aberration results presented inthe next section were obtained with thisretina.

Liou and Brennan (1997)

This model includes conicoid corneal andlenticular surfaces and a gradient index lens.Liou and Brennan selected, where possible,anatomical values based on 45-year-old eyes.The primary purpose of this schematic eyewas to model the spherical aberration of realeyes, and it has a level of 1.0 0 longitudinalspherical aberration at a ray height of 4 mm. Itwas also intended that the model should havenormal levels of chromatic aberration;however all the chromatic aberration of themodel occurs at the anterior corneal surfaceand its range of chromatic difference ofrefraction is only 1.1 0 (400-700 nm), which isabout half the normal level (see Chapter 17).The lenticular gradient index was based onthe model of Smith et al. (1991)as described byequations (16.27), (16.28a) and (16.28b). Liouand Brennan used a parabolic gradient in

which Co = 1.407, cl = -0.039 and c2 and c3 areset to zero in equation (16.27). To obtain theN. .co-efficients of the front and back halves ofth~ lens, the values of aI' a2and bare 1.59mm,2.43 mm and 4.4404 mm, respectively, inequations (16.28a)and (16.28b).The aperture stop is displaced 0.5 mm from

the optical axis to the nasal side and the anglebetween the line of sight and the optical axisin object space is 5° (also the angle betweenvisual axis and optical axis).The model does not specify a retinal shape.

The aberration results presented in the nextsection were obtained with a spherical retinaof -12 mm radius of curvature.

Reduced eye models of Thibos andcolleagues

Unlike the above eye models, there is nointention of anatomical accuracy in thesemodels. However, they are included becausethey demonstrate features of real eyes such asaxes and aberration levels.Thibos et al. (1992) developed their

'Chromatic' eye based on Emsley's reducedeye, with an aspheric surface of Q= -0.56 inorder to correct spherical aberration. This eyecontains an aperture stop 1.91mm from thecornea so that the entrance pupil is a similardistance from the nodal point as that in moresophisticated models. It has longitudinalchromatic aberration similar to that of realeyes (see Chapter 17). The Chromatic eye isnot a rotationally symmetrical eye, but has anumber of axes with no fixed relationshipbetween them.Thibos et al. (1997) developed a second

schematic eye called the 'Indiana' eye. Theasymmetries of the Chromatic eye wereremoved. The asphericity of the eye isvariable, but they selected Q= -0.4 as givingthe best fit to experimental results oflongitudinal spherical aberration. Thisasphericity gives .. +1.2D aberration at a rayheight of 2.5 mm.By moving the stop further towards the

nodal point, Wang and Thibos (1997) claimedthat it was possible to obtain reasonable levelsof both spherical aberration and peripheralastigmatism, for example with Q=-0.4 andthe stop 2.55mm inside the eye. However, thisallows no control over coma, or over the

Monochromatic aberrations of schell/aticeyes 173

sagittal and tangential powers. Atchison(1998) pointed out that their results forastigmatism are not comparable withexperimental results at larger angles becausethey calculated the astigmatism inside the eyerather than outside. If Q =-0.4, the aperturestop is moved from 2.55mm to 2.9mm insidethe eye, and the retinal radius of curvatureis changed from its value of -11 mm to-13.25 mm, then the sagittal and tangentialpower errors and astigmatism better matchthe mean experimental values used by Wangand Thibos.The aberration results presented in the next

section were obtained with the Indiana eyewith Q = -0.4 and a -11 mm retinal radius ofcurvature.

Performance of finite eye models

Seidel aberrations offinite schematiceyes

The Seidel aberrations were calculated foreach of the above models. These values weretransformed to wave aberration co-efficientsusing equations (A2.14) and (A2.20), and theresults are listed in Table 16.2, along withmean values for real eyes in the case ofspherical aberration, astigmatism and fieldcurvature.

Spherical aberrationThe spherical aberrations of the finite eyes are,apart from the Drasdo and Fowler eye,considerably smaller than those of paraxialeyes. However, only the Liou and Brennaneye has a level similar to that of real eyes. Thehigh level of the Drasdo and Fowler eye mayseem surprising, since the anterior cornea isaspherized at the level expected in real eyes.However, the rear lenticular surface of theDrasdo and Fowler eye has more aberrationthan the rear two surfaces of the Gullstrandeye.

ComaComa is similar for both paraxial and finiteeyes, except that the Thibos et al. eyes haveextremely high levels and the Liou andBrennan eye has extremely low levels.

174 Aberrations and retinal image quality

Table 16.2. The mean wave aberration co-efficients of real eyes and the wave aberration co-efficients ofunaccommodated schematic eyes. The latter were determined using Seidel theory. All off-axis co-efficients are for5° off-axis.

W4•0 (mrrr') W3•1 (mm-2) W2•0 (mm-l ) aW2•0 (mrrr') W2•2 (rnm:')

Real eyes bl.89 x 10-5 -7.91 X 10-i> 3.32 X 10-5c8.8X 10-6

Paraxial schematic eyes for reference5.73 x 10-5Gullstrand No.1 8.96 X 10-5 1.031 X 10-4 -1.57 X 10-5 4.34 X 10-5

Le Grand Fulltheorelical 9.01 x 10-5 5.76 X 10-5 1.038 X 10-4 -1.56 X 10-5 4.14 X 10-5

Finite schematic eyes4.67 x 10-5 6.54 X 10-5 9.56 X 10-5 -2.08 X 10-5 2.50 X 10-5Lotmar (1971)

Drasdo and Fowler (1974) 7.99 x 10-5 7.38 X 10-5 1.06 X 10-4 -1.34 X 10-5 3.84 X 10-5Kooijman (1983) 3.63 x 10-5 5.78 X 10-5 9.65 x 10-5 -3.61 x 10-5 2.69 X 10-5Navarro et al. (1985) 3.19 x 10-5 6.50 x 10-5 9.64 x 10-5 -2.24 X 10-5 2.52 X 10-5Liou and Brennan (1997) 1.70 x 10-5 1.54 X 10-5 8.76 X 10-5 -3.11 X 10-5 3.99 X 10-5Thibos et al. (1997)d 3.96 x 10-5 2.45 X 10-4 1.16 X 10-4 -3.78 X 10-i> 5.92 X 10-5Thibos et al. (1997)e 3.96 x 10-5 2.36 X 10-4 1.03 X 10-4 -1.63 X 10-5 3.43 X 10-5

"For real eyes, the measured values. For schematic eyes, adjusted W2.lI' taking into account the radius of the retina using equation (A2.20).bNon.aberroscope studies. equation (15.14).'Aberroscope data studies. equation (15.22)."Aperture stop 1.91 mm inside eye.'Aperture stop 2.55 mm inside eye.

Astigmatism and field curvatureAstigmatism is a similar level for real,paraxial and finite eyes, with a W2,2 co-efficient range of 2.5x 10-5 mm! to 5.9 x 10-5mm". The change in the aperture stopposition of 0.64mm of the Thibos et al. eyealters the astigmatism by a factor of two. TheWzo(sagittal) co-efficient is 1.6-5 times higherfor'the schematic eyes than for real eyes.

Sagittal and tangential power errorsFigure 16.9 shows sagittal and tangentialpower errors. Sagittal and tangential powererrors are sensitive to retinal shape, and thespread of the plots is partly because of thedifferent retinal radii specified for the eyes.Comparing Figure 16.9 with Figure 16.3shows that the finite models are generally

4

Kooijman (1983)

l.otmar (1971)

Drasdo and Fowler (1974)

......?Navarro "'(11. (1985)

Liou and Brennan (1997) ~

Mean of real eyes. equation (15.151/

:,P:0'

.c/ .,/... ·'16

D .4-' ..'.0'" ...~.:..:'8.....*....0" -IA-;;ti~

.. 0 ,n,,,,,g: ... '.' ,.0... .-8".......

10

9········0····

····0.. ··8 ..../;....

C 7 ',,16--,c0 6 ..........~]'"::; 4.~II.l.c 3c..

CIl

2

123Ray height at corneal vertex (mm)

Figure 16.8.Spherical aberration of finite schematic eyesand of real eyes.

Exact aberrations offinite schematiceyesAberrations of the models, apart from thereduced eye of Thibos et al., are shown inFigures 16.8 to 16.11.

Spherical aberrationFigure 16.8 shows spherical aberration.Comparing this figure with Figure 16.1showsthat, with the exception of the Drasdo andFowler model, the finite schematic eyesperform much better than the Gullstrandschematic eye. The Liou and Brennan sche-matic eye has the lowest level of sphericalaberration, which is close to mean real eyevalues. The Kooijman, Navarro et al. andLotmar models all have similar levels ofaberration.

Monochromatic aberrations of schematic eyes 175

Lotmar(1971 ). t,Lotmar(1971).L1Drasdoand Fowler (1974).L,

Drasdoand Fowler(1974)~Kooijman(1983), t,Kooijman(1983), t,Navarroet ol. (1985). L,Navarroetal. (1985). L,Liouand Brennan(1997),L,

Liou and Brennan(1997), t.MeanLs real eyes.equation(l5.29a)MeanLt real eyes•equation (l5.29b)

----¢--

........<> .---0--

........0 ........

---tr-........6 ........

----l&-........18........

~

.................

6050

-~~~......................q

\ .•.•.••.•..

10-IO+----.--~.,...__,-..,.....---.--~.,...__,-..,..........,..-t_

o 20 30 40

Angle in air (deg)

Figure 16.9. Sagittal and tangential power errors of finite schematic eyes and of real eyes.

much better than the Gullstrand number 1 eyefor estimating mean tangential power errors,but not for estimating mean sagittal powererrors. The Lotmar, Navarro et al. and Liouand Brennan eyes have reasonable estima-tions of both these errors. The Drasdo andFowler eye estimates sagittal power errorswell, but is inaccurate for the tangentialpower errors beyond approximately 40°, TheKooijman eye values for both sagittal and

tangential power errors are much too positive.If the flatter retina proposed by Kooijman isused, the estimations with his eye improveconsiderably to be similar to those of the Liouand Brennan eye.

Astigmatism and field curvatureFigure 16.10 shows astigmatism and fieldcurvature. The astigmatism, at least for small

'0. .....

20+------'---'-----'--------'---'----+ ----¢-- Lotmar(1971 ). astigmatism

........<>........ Lotmar(1971),fieldcurvature

---0-- Drasdoand Fowler(1974).astigmatism........0........ Drasdoand Fowler(1974). fieldcurvature

---tr- Kooijman (1983).astigmatism·......·6........ Kooijman(1983). field curvature----l&- Navarroetal. (1985).astigmatism........18........ NavarroetaI. (1985). fieldcurvature

~ Liouand Brennan(1997).astigmatism................. Liouand Brennan(1997), fieldcurvature

Astigmatism. meanof realeyes•equation (15.31a)

Fieldcurvature.meanof realeyes.equation (15.35)

§ 15e::l~~ 10::lu

"0'iil.:: 5"0

: ol-a_QIml!!Ct~~~..·F:....o~·::::~:::a~.~.....~..g~""·,,,~:::::::a:::::::~::::::..~ ·"""....···"""::l'6::;;;;;.iF:::~~::.:.::::;::::: ...2!l~ -5

605010-IO+-----,r----.--.,-----,r----.--+

o 20 30 40Angle in air (deg)

Figure 16.10.Astigmatism and field curvature of finite schematic eyes and of real eyes.

176 Aberratiolls and retinal image quality

Angle in air (deg)

Figure 16.11. Internal and external angles of the pupilray for the Gullstrand number 1 eye, some finiteschematic eyes and 'real eyes'.

we estimated the relative retinal illuminanceas a function of off-axis angle using paraxialdata, a spherical retina and equation (13.12).The estimations were smaller thanmeasurements with excised eyes (Figure 13.2).More accurate estimates require accurateestimates of the apparent pupil area A(8)r. andthe quantity 88'/88 used in equation (13.12).An accurate estimate of the apparent pupil

area was given by equation (3.5b), which is

A(O)p = A(O)1'(1-1.0947 x 10-4 fP + 1.8698X 10-9 Ol) (8 in degrees)... (16.33)

where A(O)p is the on-axis pupil area.Accurate estimates of 68'/88 can be found byexact ray-tracing through finite schematiceyes (Figure 16.11). For the Liou and Brennan(1997) eye, ray trace results were fitted to asecond order polynomial giving8'=0.822198- 2.8689x 10-4& (16.34)Estimates of retinal illuminance for this modeleye are shown in Figure 16.12, using thederivative of equation (16.34) together withresults using the approximate model inChapter 13 and measurements of Kooijmanand Witmer (1986). The model eye showshigher values of retinal illuminance thangiven by both the approximate model and theexperimental data.

90Schematic eyes

80 --0-- Gullstrand number I eye

70 ~ Lotrnar ( 197I)

---t:r-- Kooijman (1983)60

~ ----lII-- Navarro et al. (1985)

'"~ 50 ---e--'";.,'" 40.s'"0iJ 30"«

20 Real eyes

Ames and Proctor (1921)10

0() 10 20 30 40 50 60 70 80 90

Retinal illuminanceClose to the optical axis and for small pupils,paraxial schematic eyes give an accurateestimate of retinal illuminance. For largepupils and point sources, the effect ofaberrations must be included.For off-axis points, we must consider the

influences of distortion, retinal shape, and thesize of the oblique pupil. In Chapter 13(Retinal illuminance: directly transmitted light),

Retinal image positionFigure 16.11 shows the relationship betweeninternal and external angles of the pupil rayfor the Gullstrand number 1 eye, four finiteschematic eyes and 'real' eyes. Thisrelationship is similar for all these eyes.

angles, is not dependent upon the retinalshape and therefore the astigmatism showsless spread than the sagittal and tangentialpower errors shown in Figure 16.9. The fieldcurvature curves are more spread out, andthis is explained in part by their dependenceupon retinal shape.Comparing Figure 16.10 with Figure 16.4

shows that, beyond about 30° object angle, allthe finite schematic eyes, except the Drasdoand Fowler eye, are much better than theGullstrand number 1 eye at estimating astig-matism of real eyes. The Lotmar eye providesexcellent estimation of the astigmatism (asclaimed by him), and the Kooijman, Navarroet al., and Liou and Brennan eyes givereasonable estimates. If the flatter retinaproposed by Kooijman is used, the estima-tions with his eye are worse beyond 40° objectangle.Improvement in predictions of field

curvature of the majority of finite schematiceyes, compared with the Gullstrand number 1eye, are only apparent beyond about 50°. Fieldcurvature estimation is reasonable for theLotmar, Navarro et al. and Liou and Brennaneyes. The Drasdo and Fowler eye hasreasonable estimates, except beyond about 40°object angle. The Kooijman eye estimates thefield curvature poorly, but if the flatter retinaproposed by Kooijman is used, the estim-ations improve considerably to be similar tothose of the Liou and Brennan eye.

Monochromatic aberrations of schematic eyes 177

Summary ()

h

Ames, A. and Proctor, C. A. (1921). Dioptrics of the eye. ].Opt. Soc. Am., 5, 22-84.

Atchison, D. A. (1998). Oblique astigmatism of theIndiana eye. Optom. Vis. Sci.,75, 247-8.

Atchison, D. A. and Smith, G. (1995). Continuousgradient index and shell models of the human lens.Vision Res., 35, 2529-38.

Blaker, J. W. (1980). Toward an adaptive model of thehuman eye. J. Opt. Soc. Am., 70, 220-23.

Liou and Brennan's (1997) model (with aretina having a -12 mm radius of curvature)appears to give the best overall estimations ofaverage monochromatic aberrations of realeyes.

Summary of main symbols

refractive index of the vitreoushumourray height in pupil (say inmillimetres)direction of a point in the objectfield

Seidel aberrationsS1 spherical aberrationS2 comaS3 astigmatismS4 Petzval curvatureS5 distortion

Ls((), Lt ( () sagittal and tangential powererrors at off-axis angle ()

A(() astigmatism at off-axis angle ()rs' rt, rp' rR radii of sagittal, tangential,

Petzval and retinal surfacesW(r) wave aberration for ray passing

through the pupil at a height roW4,o etc. wave aberration co-efficients (for

the wave aberration in milli-metres)

n refractive indicesR radius of curvatureC surface curvature (= l/R)e eccentricity of an aspheric surfaceQ surface asphericity (=-e2)p surface asphericity (= 1 + Q)M power error measure of spherical

aberrationa, b elliptical parameters

ReferencesEquation (13.12). simple eye model

----e---- Kooijman and Witmer (1986). excised eyes

----6---- Liou and Brennan (1997)

1.1

1.0

0.9

u 0.8(Jcosc 0.7's~ 0.6-;c 0.5.~

IlJ 0.4>'i 0.3s

0.2

0.1

0.00

The reader is again reminded to take intoaccount the purpose of a schematic eye beforecriticizing it for failure for accurate estimationin other areas. The majority of schematic eyesthat were surveyed give better estimationsthan a representative paraxial schematic eyeof mean levels of spherical aberration,tangential power errors, astigmatism andretinal illuminance. The retinal shape of theseeyes could be modified to make someimprovement in estimations of the sagittaland tangential power errors, astigmatism andfield curvature. Despite being intended onlyfor the estimation of spherical aberration,

The quantity o()'/o() is sensitive to smallchanges in the relationship between () and ()'(see Figure 16.11), and will vary between thedifferent eye models. Kooijman (1983) showedthat retinal illuminance depends upon theretinal shape, i.e, its radius of curvature andasphericity. The effect of retinal asphericityhas not been included in the equations thataffect some of the parameters in equation(13.12), but a more sophisticated analysisshould take this into account.

10 20 30 40 50 60 70 80 90Angle in air (deg)

Figure 16.12. Retinal illuminance obtained from paraxialcalculations with a simple eye model (shown also inFigure 13.2), exact ray-tracing calculations with the Liouand Brennan eye, and experimental results of Kooijmanand Witmer (1986).

178 Aberrations and retinal imagC' quality

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Stine, G. H. (1934).Tables for accurate retinal localization.AIIl. t. Ophtlwl., 17, 314-24.

Thibos, L. N., Ye, M., Zhang, X. and Bradley, A. (1992).The chromatic eye: a new reduced-eye model of ocularchromatic aberration in humans. Applied Optics, 31,3594-600.

Thibos, L. N., Ye, M., Zhang, X. and Bradley, A. (1997).Spherical aberration of the reduced schematic eye withelliptical refracting surface. Optom. Vis. Sci.,74, 548-56.

van Meeteren, A. (1974). Calculations on the opticalmodulation transfer function of the human eye forwhite light. Optica Acta,21, 395-412.

Walsh, G. and Charman, W. N. (1985). Measurement ofthe axial wavefront aberration of the human eye.Ophthal. Physiol. Opt., 5, 23-31.

Wang, G., Pomerantzeff, O. and Pankratov, M. M. (1983).

Monochromatic aberrations of schematic eyes 179

Astigmatism of oblique incidence in the human eye.Vision Res.,23, 1079-85.

Wang, Y-Z. and Thibos, L. N. (1997). Oblique (off-axis)astigmatism of the reduced schematic eye withelliptical refracting surfaces. Optom. Vis.Sci.,74, 557-62.

Welford, W.T. (1986). Aberrations of Optical Systems. AdamHilger.

17Chromatic aberrations

Introduction

Like other optical systems, the eye suffersfrom chromatic aberration as well as frommonochromatic aberrations. There are twotypes of chromatic aberration, longitudinaland transverse, both of which are mani-festations of the dispersion (variation ofrefractive index with wavelength) of therefracting media of an optical system.The first two sections of this chapter

describe longitudinal and transverse chro-matic aberration, and the following twosections discuss their measurement. Theeffects of these aberrations on visual perfor-mance are then discussed, followed by asection on compensation for these effects.Finally, the inclusion of chromatic aberrationsin eye modelling is discussed, and this shouldbe considered in conjunction with Chapter 16on modelling the monochromatic aberrations.

Longitudinal chromatic aberration

Longitudinal chromatic aberration can beexplained as follows. Figure 17.1a shows abeam of light from an axial point 0 enteringthe eye. Because the refractive indices insidethe eye vary with wavelength, the pathfollowed by a ray inside the eye dependsupon wavelength. As a rule, refractive indicesdecrease with increase in wavelength, so theeye has lower power as wavelength increases.Regarding the eye as focused on the point 0

o(a)

(b)

I·

Figure 17.1. Longitudinal chromatic aberration.a. General effect of longitudinal chromatic aberration.b. Measuring longitudinal chromatic aberration as achromatic difference of refraction.

for a yellow wavelength, rays of longerwavelength (e.g. red) are focused behind theretina and shorter wavelength rays (e.g. blue)are focused in front of the retina.The longitudinal chromatic aberration of

the eye can be quantified as the variation inpower with wavelength. Thibos et al. (l991a)referred to this as chromatic difference ofpower. The aberration can also be quantifiedas the vergences of the source for which thesource is focused at the retina for a range ofwavelengths (Figure 17.1b). Thibos et al.referred to this as chromatic difference ofrefraction. As for other aberrations of the eye,such as spherical aberration and astigmatism,the second method is how longitudinalchromatic aberration is measured experi-

Chromaticaberrations 181

Figure 17.2. Transverse chromatic aberration (greatlyexaggerated).a. Centred pupil and an off-axis object point.b. Decentred pupil and an on-axis object.c. Measuring transverse chromatic aberration in objectspace.

Nodalaxis

(b)

(a)

that the pupil is decentred. The object point ison the optical axis. The small pupil in thisfigure has been decentred so that its positioncoincides with the top of the pupil in Figure17.1a.As happens for longitudinal chromatic

aberration, and generally for the mono-chromatic aberrations, transverse chromaticaberration must be measured outside the eye.This is shown in Figure 17.2c. Two rays, one ofwavelength Aand the other of the referencewavelength I, originate from differentpositions in object space but pass through thesame point in the pupil and intersect at theretina. It can be seen that the transversechromatic aberration associated with a heighth of the rays relative to the nodal ray is givenbyteA) = b-awhere a and b are the angles subtended by thetwo rays with the nodal ray in object space.

Transverse chromatic aberration is demon-strated in Figure 17.2a for the case of an eyethat is a centred optical system (includingpupils), and for an off-axis object point at Q.Because of longitudinal chromatic aberration,the different wavelength images of the pointare defocused by different amounts relative tothe retina. Also, because the power of the eyeis less for long wavelengths than for shortwavelengths, longer wavelength rays aredeviated less than shorter wavelength rays,and meet the retina further from the opticalaxis.Transverse chromatic aberration is also

demonstrated in Figure 17.2b for the case ofan eye that is a centred optical system, except

Transverse chromatic aberration

mentally (see Measurement of longitudinalchromatic aberration, this chapter).A formal definition of chromatic difference

of refraction is:For any level of ametropia and accommo-dation, chromatic difference of refraction isthe difference between the vergences of theretinal conjugates for a wavelength Aand areference wavelength 1Figure 17.1bshows a general schematic eye

and the retinal conjugates for wavelengths Aand 1 These conjugates are at distances I(A)and 1(1) from the eye. Replacing thesedistances by their corresponding vergencesL(A) and L(I), the chromatic difference ofrefraction RE(A) isRE(A) =L(A)- L(A) (17.1)In measurements of chromatic difference ofrefraction, rather than comparing the resultsat a wavelength with those of the referencewavelength, it is common to comparemeasurements between a short and a longwavelength. When doing so, we shall refer tothe range of chromatic difference of refraction.It is important to be careful when comparingthe ranges obtained in different studies, asthese may have used different wavelengthranges.Longitudinal chromatic aberration has been

explained by considering an axial object point,but it should be realized that it is still presentas the object moves off-axis.

182 Aberratiolls and retinal ill/age qllality

Chromatic magnification

line appears deviated as it crosses theboundary between red and blue (Figure17.3b). A similar effect occurs for the verticalblack line when the artificial pupil isdecentred horizontally.

where () is the angular size of the objectsubtended at the eye's nodal point. Forexample, if an object has 100 angular size andthe angular transverse chromatic aberrationfor the edge of the object is 0.10 (0.0017 rad),the chromatic difference of magnification is 1per cent.The chromatic difference of magnification

can be related directly to the chromaticdifference of refraction. In Figure 17.2c

As well as an angle, transverse chromaticaberration can be measured as a wavelength-dependent variation in image size of extendedobjects. Thibos et al. (1991a) referred to this asthe chromatic difference of magnification(COM). In practice, this must be measured inobject space.The chromatic difference of magnification is

the transverse chromatic aberration in angularterms t().), divided by the angular size of theobject, that is

(17.3)COM =t().)/ ()

t().) '" h[L()') - L(A:)]Using equation (17.1), it can easily be seen thatt().) '" hRE().) (17.2)

which establishes a linear relationshipbetween the transverse chromatic aberrationas given by t().) and longitudinal chromaticaberration as given by RE(). ) .The transverse chromatic aberration that is

of most interest is that associated with fovealvision. In this case, the nodal ray becomes thevisual axis, and the pupil location of interest isthat representative of the light beam - i.e. thecentre of the pupil. A method for measuringtransverse chromatic aberration associatedwith foveal vision is described in Measurementof transverse chromatic aberration, this chapter.Thibos et al. (1991a) referred to the angular

measure of transverse chromatic aberrationt().) as a chromatic difference of position.Transverse chromatic aberration in the eye

can be demonstrated by viewing a black-white edge through a small artificial pupilthat is decentred. Another way to observe it isto look at a black cross on a pattern consistingof a central red area surrounded by a blueregion, as in Figure 17.3a. If the artificial pupilis decentred vertically, the horizontal black

Asa'" hL(X) and b '" hL().),

we have

()",Il/EN (17.4)

Figure 17.3. Demonstration of transverse chromaticaberration.a. A black cross is placed on a central red area and ablue surround.

b. Appearance of the target when a small pupil isdecentred downwards in front of an observer's eye.

where EN is the distance between theentrance pupil at E and the front nodal pointat N, and h is the displacement of the entrancepupil from the visual axis. We can substitutethe right-hand side of equation (17.4),together with 11 / RE(). ) for t().) from equation(17.2), into equation (17.3) to give

COM=RE().)EN (17.5)

Based on a range of chromatic difference ofrefraction of 2.1 0 across the visible spectrum(see next section) and a distance EN of 0.004m(Gullstrand number one eye), the range ofCOM across the visible spectrum is less than0.01 (1 per cent). This may rise considerably ifartificial pupils are used in visual experi-ments. Zhang et al. (1993)described a methodfor measuring COM.

Chromatic abamtions 183

Some techniques

Vernier methodTwo narrow test targets of differentwavelengths are imaged on the fovea, butlight from them is restricted to pass onlythrough a.small aperture in front of the eye(Figure 17.4b). The small aperture can bedisplaced across the pupil perpendicularly tothe length of the target. There is one positionin the pupil for which the targets are bothaligned and appear aligned - this locates the'foveal achromatic axis', which is usuallytaken to be the visual axis (seeChapter 4).Oneof the targets can be displaced perpen-dicularly to its length. For chosen aperturepositions relative to the visual axis, this isdone so that the targets appear again to bealigned. If the aperture displacement is h, thetarget displacement is e, the target distancefrom the eye is p and the target distance fromwhere the test wavelength ray intersects theaxis is x, from similar trianglese/h=x/(x-p) (17.6)

Wecan replace pby its vergence P whereP=l/p (17.7)

The chromatic difference of refraction RE()..) isgiven byRE(A) =l/(p - x) -l/p =x/[(p -x)p] (17.8)

Using the previous two equations (17.7) and(17.8)-RE(A)/P = x/(x - p) (17.9)

The left-hand side of this equation (17.9) canbe substituted for the right-hand side of

Laser speckleWhen viewing a laser reflected diffusely froma rotating drum, a speckle pattern is seen thatgenerally moves in the same or oppositedirection to the drum rotation (see Chapter 8).However, when an eye is focused at the drum,the pattern appears merely to 'boil'. Lasers ofdifferent wavelengths are used, and focus isachieved for each wavelength by moving thedrum or using auxiliary trial lenses (Gilmartinand Hogan, 1985).

altering target position (the small chromaticaberration of the lenses should be taken intoaccount).

TriallensesBeam

sp~.itter..'

,-===--J Badal lens

Photomultiplier tube/CCD camera

'::=1====-1 r IActual positions of targets A )..when they appear aligned = =

e

(b)

(c)

Colouredfilters

Lightsource

(a)

Targetimage

Colour fiIter

t()..)

Figure 17.4.Some techniques for measuring chromaticdifference of refraction:a. Best focus.b. Vernier alignment.c. Double-pass technique.

Best focus methodA target with fine detail, back-illuminated bylight of various wavelengths, is movedforwards and backwards in front of anobserver until it is judged to be in focus(Figure 17.4a). This can be done using anachromatic Badal lens (e.g. Howarth andBradley, 1986), which means that the imagealways subtends the same angle at the eyeand that the chromatic difference of refractionis linearly related to the position of the target(see Chapter 8). A clinically related variant ofthis method is to use trial lenses of differentpowers in the spectacle plane rather than

Colour filter

Five methods of measuring chromaticdifference of refraction are described here.

Measurement of longitudinalchromatic aberration

184 Aberralitm, and retinal ill/age ql/alily

equation (17.6) to obtain-elh =RE()..)/P (17.10)

A plot of e as a function of h is only linear inits central region because of the influence ofmonochromatic aberrations. RE(A.) can beobtained from the slope of this linear section(Thibos et al., 1990).This seems a complicatedway to measure the longitudinal chromaticaberration of the eye, but has the advantagethat it can be used to determine the transversechromatic aberration at the same time.

Double pass techniqueThe image of a narrow illuminated slit isformed on the fundus, which reflects aportion of the light (Figure 17.4c). An aerialimage forms outside the eye. Correcting triallenses can be used to minimize the width ofthis image for various wavelengths. Charmanand Jennings (1976) used this method andfound good agreement with their subjectivemeasurements.

Chromo-retinoscopyBobier and Sivak (1978, 1980) used retino-scopy and narrow wave-band filters placed in

1.0

§ 0.5

c.2 0.0Uco..;:~ -0.5'+-0...uc ·1.0... ..'...~ ........'+-:;.S! -1.5 //';E0 -2.0.....cU

front of the tested eye to measure thelongitudinal aberration in a number ofsubjects.

Magnitude

Figure 17.5 shows experimental subjectiveresults of chromatic difference of refractionfrom several studies. These results are for alow level of accommodation stimulus, orunder cycloplegia. The data have beenadjusted for a common reference wavelengthof 589 nm. The figure shows results also for areduced eye filled with water and a reducedChromatic eye, which Thibos et al. (1992)derived from their experimental data (seeModelling chromatic aberrations, this chapter).There is approximately a 2.1D range of

chromatic difference of refraction between 400and 700 nm. Although several techniqueswere used, it is most noticeable that there islittle variation between the majority ofsubjective studies. Only the Gilmartin andHogan (1985) study, which used the methodof laser speckle and obtained results of 1.87 ±0.260 between 488 and 633 nm, has resultsvery different from other studies. The inter-subject difference in the studies was alsosmall. This small variation is in contrast to the

• Wald and Griffin (1947)

o Bedford and Wyszecki (1957)

0 Ivanoff (1953)

il. Millodot and Sivak (1973)

IB Charman and Jennings (1976)

• Powell (1981)

• Lewis er"I. ( 1982)

'" Ware (1982)

til Mordi and Adrian ( 1985)

~ Howanh and Bradley (1986)

lIE Cooper and Pease ( 1988)

e- Thibos etal, (1992)

Watereye

Chromatic eye-2.5 -I-l'--~-,....--r---,----r--.-~--+

350 400 450 500 550 600 650 70t) 750

Wavelength (nrn)

Figure 17.5. Results of experimental studies of chromatic difference of refraction as afunction of wavelength. Also shown are the results for an Emsley reduced eye filled withwater (Water eye) and the Chromatic eye. Data for Figure 6 of Thibos et al. (1992)kindlyprovided by Larry Thibos, and with permission from The Optical Society of America.

large variation in monochromatic aberrations(see Chapter 15).The small variation in chromatic aberration

is because the main constituent of the ocularmedia is water, whose dispersion cannot varybetween subjects. Despite this, the water eyehas insufficient dispersion to fit the experi-mental results well (Figure 17.5). The Chro-matic eye of Thibos et al. (1992) provides anexcellent fit.

Wavelength in focusAn issue related to longitudinal chromaticaberration is the wavelength at which a whitetarget is in focus at various levels of accom-modation. Accommodation response isusually in excess ('lead') for low stimuluslevels, while the response is insufficient ('lag')for higher levels of accommodation - forexample, Charman and Tucker (1978).Corresponding to this, a long wavelength isusually in focus for low accommodationstimuli and short wavelengths are in focus forhigher accommodation stimuli.

Effect of accommodation and refractiveerrorFor optical systems of the same chromaticdispersion, longitudinal chromatic aberrationis related linearly to power. Modellingpredicts =2.5 per cent increase in longitudinalchromatic aberration of eyes for each 1 D ofaccommodation or for each 1 D of refractiveerror when this is caused by an increase inocular power (see Modelling chromaticaberrations, this chapter).Jenkins (1963) claimed an early study by

Nutting (1914) showed an increase inaberration with accommodation, but theauthors believe that Nutting provided insuffi-cient detail to support this claim. Somestudies investigating this relationship wereflawed in that wavelength-dependent accom-modation may have affected measurements(Ienkins, 1963;Millodot and Sivak, 1973;Sivakand Millodot, 1974). Channan and Tucker(1978) found an increase in chromaticaberration of =0.2 D for a 4 D increase inaccommodation for one subject (= 3 per centper dioptre accommodation) between 442 and

Chromatic aberrations 185

633 nm. Wildsoet et al. (1993) measuredchromatic difference of refraction in right eyesof 34 young subjects consisting of 12 myopes(-3.41 ± 2.62 D), 12 emmetropes (-0.04 ± 0.23D) and 10 hypermetropes (+2.28 ± 1.43D), butfound no significant differences betweengroups.

Measurement of transversechromatic aberrationCompared with longitudinal chromaticaberration, there have been relatively fewstudies of transverse chromatic aberrationassociated with foveal vision (Kishto, 1965;Ogboso and Bedell, 1987; Simonet andCampbell, 1990;Thibos et al., 1990;Rynders etal., 1995;Marcos et al., 1999).

Technique

The vernier method used by Thibos et al.(1990) can be used as described above (Figure17.4b). The small aperture displacement h istaken as the distance between the visual axisand the line of sight. The line of sight may belocated by some suitable method, such asdetermining the edges of the pupil byscanning across the pupil with the apertureuntil the target disappears, and obtaining themid-point of these limits. The angular trans-verse chromatic aberration t(A) associatedwith the line of sight ist(A) = -eP (17.11)

Combining equation (17.11) with equation(17.10) givest(A) = hRE(A) radians (17.12)which is the same as equation (17.2).More sophisticated variations of the vernier

method have been used by Simonet andCampbell (1990). Other studies have appliedvernier alignment to the whole pupil ratherthan isolating the line of sight (Hartridge,1947;Ogboso and Bedell, 1987;Rynders et al.,1995). Such measurements may be influencedby the Stiles-Crawford effect if its peak isdecentred from the line-of-sight (Rynders etal., 1995;Marcos et al., 1999).

186 AI'c/Taliolls alld retinal illlllSC qllillily

Table 17.1. Studies of transverse chromatic aberration associated with the fovea. When signs are given, a positivesign indicates that the line of sight is nasal to the optical axis in object space.

Alltlwrs

Hartridge (1947)Kishto (1965)Ogboso and Bedell (1987)Thibos et 111. (1990)Simonet and Campbell (1990)Rynders ct al. (1995)Gullstrand No.1 eye (relaxed)

Ml'Illllllld /'IlI/SC (mill. arc)

+0.6"+0.3+0.9, +0.6 to +1.2+0.61, -0,36 to +1.67+0,43, -0.20 to +1.280.8, ur to 2.7+1.05

No. sll/ljccts

1135585

Wa(Je!ellsllI rallse (11m)

48fHl56not given435-572433-62248fHl56497-605486-656

"Hartridge gave his fl'SUItS in 'cone units' {I cone unit = O.hLJ min. arc), He obtained .1 n·rtk.ll component also of (I.X min .•trc."Calculated from Seidel aberration C,. in Appendix 3, llsinH <1n~k' (l of S'.

MagnitudeTable 17.1 shows results of experimentalstudies of foveal transverse chromatic aber-ration. Although the wavelength range ofmeasurement must be taken into account, it isreasonable to say that the mean results areabout half the 1.05min. arc predicted forschematic eyes (486-656 nm) with centredpupils and the fovea 5° to the optical axis. Theprobable reasons for this discrepancy are thatthis angle may be larger than that occurring inmany people, and that the pupil is usuallydecentred nasally and its centre is thereforecloser to the visual axis. The main concernwith transverse chromatic aberration is forseverely decentred natural or artificial pupils.The effect of transverse chromatic aberrationon visual performance is discussed in the nextsection.Ogboso and Bedell (1987) made experi-

mental measurements of transverse chromaticaberration in the peripheral visual field. Theresults were very different among their foursubjects. Out to 40° object eccentricity, allvalues were less than 7 min. arc and consider-ably less than the 11 min. arc predicted fromtheir model eye calculations at 40° eccentricity(wavelength range 435-572 nrn).

Effects of chromatic aberrations onvision

AccommodationFincham (1951) introduced step changes inaccommodative stimulus to his subjects. Mostof them made appropriate accommodation

responses in white light, but 60 per cent ofthem were unable to do so in monochromaticlight. There is now considerable evidence thatthe longitudinal chromatic aberration helpsthe accommodation system respond correctlywhen there is defocus blur. Accommodationresponses to steady and moving targets aremore accurate in white or broad wavelengthlight than in monochromatic light, anddoubling the eye's normal chromaticaberration has little effect on accommodativeaccuracy, whereas correcting or reversing thechromatic aberration leads to pooraccommodative response (Kruger and Pola,1986 and 1987; Kruger et al., 1993, 1995 and1997;Stone ctal., 1993;Aggarwala etal., 1995aand 1995b).

Spatial visionThe effects of chromatic aberration areattenuated by the spectral sensitivity of theeye. For centred pupils, switching frommonochromatic light to white light gives amaximum loss to the contrast sensitivityfunction of only 0.2 log units (Campbell andGubisch, 1967), and has a negligible effect onvisual acuity (Hartridge, 1947; Campbell andGubisch, 1967). The use of achromatizinglenses makes negligible improvement tovisual acuity and contrast sensitivity in whitelight (Campbell and Gubisch, 1967). Possiblereasons include that the potential improve-ment is small anyway, some of these lenseshave transverse chromatic aberration whencentred (see the following section), and thatadditional transverse chromatic aberration isintroduced for any of these lenses if they arenot carefully centred (Zhang et al., 1991).

Chromatic aberrations 187

Temporal

(17.13)

Nasal

Redandblue

T28

1

blue 0 red

(b)

(a)

retinal disparity is equivalent to the red andblue rays coming from different distances, asshown by the dashed lines in the figure. Theretinal disparity leads to an apparentlongitudinal displacement when the target isviewed binocularly. The red target appears tobe closer than the blue target.The magnitude of chromostereopsis is

predicted by the geometry in Figure 17.6b (Yeet al., 1991). Here, L1d is the amount ofchromostereopsis as a distance measurement,2B is the distance between the position ofsmall pupils in front of the two eyes, d is theviewing distance, and t(A)R and t(A)L are theright eye and left eye transverse chromaticaberrations. The relationship between thesequantities is

By making measurements of transversechromatic aberration induced by displace-ment of pinholes from the visual axes

Figure 17.6.Chromostereopsis.a. An explanation of the source of chromostereopsis.b. The expected relationship estimating chromo-stereopsis and transverse chromatic aberrationinduced with small artificial pupils.

Decentring natural or artificial pupils canhave devastating effects on foveal spatialvision in white light because large amounts oftransverse chromatic aberration are induced.Experimental studies showing up to two-thirds reduction in resolution for es 3 mmdisplacement of artificial pupils (Green, 1967)or Maxwellian-view instruments (Bradley etal., 1990; Thibos et al., 1991b) are supportedtheoretically (Thibos et al., 1990).Some Maxwellian-view projection systems

are used clinically to evaluate potential visualacuity in patients with cataract by positioningthem so that the light passes through arelatively clear part of the cataractous lens.The experimental and theoretical results justmentioned indicate that those instrumentsusing white light are likely to underestimatepotential visual acuity if the light beam entersthe pupil well away from its centre.Theoretically, the transverse chromatic

aberration should increase on going furtherinto the peripheral field (Thibos, 1987), withincreasingly deleterious effects on imagequality. However, the importance of theaberration is likely to be small because ofdecreasing colour sensitivity and spatialresolution due to neural limitations.Effects of chromatic aberration on retinal

image quality are discussed further inChapter 18,Retinal image quality.

ChromostereopsisIn Chapter 6 (Binocular vision) we describedstereopsis, in which the two eyes provide thepotential for seeing depth in a scene.Chromostereopsis is a related binocularphenomenon in which objects at the samedistance, but of different colours, are seen indepth. Most commonly, reddish objectsappear to be closer than bluish objects.Chromostereopsis is a consequence oftransverse chromatic aberration combinedwith binocular vision.Figure 17.6ashows the pupil ray paths from

red and blue targets at position O. The pupilof the eye is decentred temporally relative tothe visual axis. The red pupil ray is refractedless than the blue pupil ray, and therefore thered pupil ray intersects the retina to thetemporal side of the blue pupil ray. This

188 Aharatiollsa",l rctinat illlage ql/ality

(Transverse chromatic aberration, this chapter)and comparing these with measurements ofchromostereopsis at these displacements, Ye etal. provided experimental support for thisequation and for the theory that chromo-stereopsis with small pupils can be explainedby the interocular difference in monoculartransverse aberration.Chromostereopsis often diminishes as pupil

size increases, and may even reverse indirection. For natural pupils, this may beattributed at least partly to change in pupilcentre as pupil size changes (Chapter 3). Ye etal. (1992) attributed the decrease in chromo-stereopsis, with increase in pupil size, to theStiles-Crawford effect (see Chapter 13) actingan as anchor to shift the effective centre ofpupils closer to the visual axis.

Aberrations of ophthalmic devicesLongitudinal chromatic aberration of correct-ing ophthalmic lenses is not consideredimportant; spectacle and contact lensesbecause of their low powers relative to that ofthe eye, and intraocular lenses because theymerely replace the eye's lens. Transversechromatic aberration is not important incontact or intraocular lenses, because thesemove with the eye. However, it blurs fovealvision when an eye looks through peripheral

Figure 17.7. Achromatizing lens of Powell (1981).

parts of a spectacle which is made worsewhen high index, low V-value materials areused.

Aberration compensation andcorrection

Natural compensation mechanismChromatic aberration effects are attenuated bythe non-uniform spectral sensitivity of the eye(Chapter 11). Thibos et al. (1991a) estimatedthat, when the wavelength with peaksensitivity is in focus, most of the luminancein a white light target is less than 0.250 out offocus. The yellow macula pigment contributesto the relative luminous efficiency functions,and it's absence in the periphery may removesome of the attenuation.

Table 17.2. Achromatizing lens of Powell (1981).The aperture diameter of bothcomponents is 15 mm. Units: millimetres.

glass II V d surface

air0.0

LaF21 1.788310 47.39 1.015.5 2

SF56 1.784700 26.08 3.00.0 3

air 1.0 14.50.0 4

SF56 1.784700 26.08 1.015.5 5

LaF21 1.788310 47.39 5.0-15.5 6

SF56 1.784700 26.08 1.00.0 7

air 1.0 17.00.0 8 pupil

air 1.0

Achromatizing correcting lenses

The longitudinal chromatic aberration of theeye can be corrected by a nominally zeropower lens, with longitudinal chromaticaberration equal and opposite to that of theeye (van Heel, 1946; Thomson and Wright,1947; Bedford and Wyszecki, 1957; Fry, 1972;Powell, 1981; Lewis et al., 1982). The Powell(1981) design is given in Table 17.2and Figure17.7. It is a modification of the Bedford andWyszecki triplet design, with a doublet addedto reduce the residual longitudinal and trans-verse chromatic aberration present in theoriginal lens.These lenses may be assessed by the level of

residual power at the reference wavelength,and by the residual chromatic aberrationwhen used with the eye. The Bedford andWyszecki, Fry,Powell and Lewis et al.designshave less than 0.25D equivalent power at587.6nm. The residual variation of power ofone lens/eye system is shown in Figure 17.8.The lenses introduce transverse chromaticaberration, with that of the sophisticated

2.5

0 Bedford and Wyszecki lens

8 2.0 0 Powell lensc.g • Powell lens + Chromatic eyeu 1.5~ ---0--- Residual with Powell lens~ (Howarth and Bradley. 1986)... 1.00

'"uC~ 0.5~...:.0.::! 0.0-;Ee..c -0.5U

-1.0+--,---y-----,r----r---.--,---,---+350 400 450 500 550 600 650 700 750

Wavelength (nm)

Figure 17.8.The power of the Bedford and Wyszecki(1957)and of the Powell (1981)achromatizing lenses,together with residual chromatic differences of focus forthe Powell lens. The latter was obtained by combiningthe Powell lens with the Chromatic eye of Thibos andfrom mean experimental results of Howarth and Bradley(1986). The theoretical and experimental residuals differby 0.1-0.3 D at any wavelength, but neither varies bymore than 0.1 D between 400 nm and 700 nm.

Chromatic aberrations 189

Powell lens being much less than those of theother lenses. For a 5° field of view andbetween wavelengths from 486 to 656nm,design values are 2.88min. arc (Powell), 5.23min. arc (Bedford and Wyszecki), 7.33min. arc(Fry) and 9.34min. arc (Lewis et aI.).Alignment of such lenses is critical, as

otherwise an additional transverse chromaticaberration is induced which is proportional tothe decentration. Zhang et al. (1991) deter-mined that 0.4 mm of misalignment of anachromatizing lens relative to the eye wouldcancel any benefit the lens would give tospatial vision.

Modelling chromatic aberrations

Chromatic dispersion

Tomodel the chromatic aberrations of the eye,a knowledge of its dispersive properties isnecessary. This can be described by a formulasuch asn2(A.) =a} + a2A.

2 + a3 / ,1,2 + a4 / ,1,4 (17.14a)The advantage of this equation is that it islinear in the co-efficients and therefore, givenany data on the relationship betweenwavelength and index, the a co-efficients caneasily be found by solving a set of simul-taneous linear equations. Other formulasinclude the Cornu dispersion equationn(A.) = noe + K/(A.-A.o) (17.14b)where noe, K and ,1,0 are constants for a partic-ular medium, and Sellmeier's equation

2 1 _ B}A.2 B2A.2

n (I\.) - 1 + ,1,2 _ C + ,1,2 _ C +... (17.14c)} 2

The dispersion of any optical material is fullydescribed by such equations, provided theyare accurate. For some purposes, the disper-sion can be reduced to a single number suchas the Abbe V-value, which is defined as

nd -1Vd = (17.15)

nF-nCwhere nd, nF and "c are the refractive indicesat the wavelengths 589.3nm (A. ), 486.1 nm(~) and 656.3nm (A.c), respectiv~y, V-valuesare useful for comparing the amount of

190 AbL'rralioll~ Gild retinal illIGgc '1111//il.1/

dispersion in different optical. media a~d forcalculating the Seidel chromatic aberrations.Bennett and Tucker (1975) used the form of

equation (17.14a) to give the refractive indexof water asn2(A.) =1.7642- 1.38 x 10-8,12 + 6.12

x 10+3/ ,12 + 1.41 X 10+8/ ,14 (17.16)

where the wavelength is in nanometres.Substituting appropriate values obtainedfrom equation (17.16) into equation (17.15)gives Vd =55.15. As shown in Figure 17.6, thedispersion of water is insufficient to accountfor the longitudinal chromatic aberration ofthe eye.Le Grand (1967) presented data designed

for his full theoretical eye. He quantified thedispersion of the ocular media using t~e

Cornu dispersion equation (17.14b). HIsvalues of n , K and A. for each ocular mediumand corresponding °V-values are given inTable 17.3.Thibos et al. (1992) used their experimental

results and the Cornu dispersion equation(17.14b) to describe the refractive indexdistribution of their Chromatic version ofEmsley's reduced eye as

4.685n(A.) =1.320535 + (,1- 214.102) (17.17)

where A. has the unit of nanometres. Thecorresponding V-value is 50.23.The c~romat~c

error of refraction of the Chromatic eye ISshown in Figure 17.6.The high dispersion of the eye has been

attributed to a high dispersion of the lens. Forexample, Sivak and Mandelman (1982)measured V-values of the lens as 29 ± 4 for theperiphery and 35 ± 6 for the core. These aremuch less than the values of Le Grand (1967),given in Table 17.3.

Table 17.3. The values of the parameters in equation(17.15b) for Le Grand's (1967)dispersion of the ocularmedia.

II K All V~

Cornea 1.3610 7.4147 130.0 56.01Aqueous 1.3221 7.0096 130.0 53.00Lens 1.3999 9.2492 130.0 50.01Vitreous 1.3208 6.9806 130.0 53.00

Schematic eyes

Gaussian propertiesThere are small, wavelength-dependentchanges in the principal and nodal points ofschematic eyes. For the Gullstrand number 1eye, over the wavelength range of 400-700nm, the front principal point moves just lessthan 0.0007mm. Over the same wavelengthrange, the back principal point moves by 0.013mm, but this is still small.

Seidel chromatic aberrationsWe discussed Seidel monochromatic aber-rations in Chapter 16. The Seidel longitudinal(CL) and transverse (CT) chromatic aberrationsare described by Smith and Atchison (1997).These can be calculated for schematic eyesfrom constructional and dispersion data, andthe values for some schematic eyes are givenin Appendix 3. We use the V-value of 50.23obtained from equations (17.15) and (17.17).Seidel transverse chromatic aberrations,between 486 and 656 nm, are given aspercentages of the image sizes at 589 nm. Thefull schematic, simplified and reduced eyeshave similar Seidel chromatic aberrations.For the Gullstrand number 1 eye at 5° angle,

Seidel transverse chromatic aberration is -0.35per cent, and this can be converted to anangular value of 1.05min. arc. This is a closeapproximation to the exact value, and can beused as an estimate of transverse chromaticaberration at the fovea, assuming that the lineof sight is 5° to the optical axis and the pupilis centred on the optical axis. This is consider-ably larger than mean experimental measures(Table 17.1), and reasons for this werediscussed earlier in this chapter (Measurementof iransoerse chromatic aberration).

Chromatic difference of power andchromatic difference of refractionFigure 17.1b shows a general schematic eyeand the retinal conjugates for wavelengths A.and l. We have11 '(A.) / I' - L(A.) =F(A.) and 11 '(l) / I' - L(l) =F(l)

(17.18a and b)For both situations, the distance I' is thedistance from the back principal point to the

Chromatic aberrations 191

(17.20)

Chromatic difference of refraction ofreduced schematic eyesIf the corneal radius of curvature of thereduced eye is r, the equivalent power F(.:t) asa function of wavelength is simply

modelled by altering corneal curvature. Forthe Gullstrand number 1 eye, changesoccurring were 0.0120 (0.56 per cent) perdioptre of axial ametropia and 0.0470 (2.4percent) per dioptre of refractive ametropia.Similar results were obtained for reducedeyes. The accommodated version of theGullstrand eye gave a range of chromaticdifference of refraction 0.550 greater than therelaxed, emmetropia eye, which correspondsto 0.0500 (2.6per cent) increase per dioptre ofaccommodation.

where n'(.:t) is given by equations such asequation (17.17). The radius can be given interms of the reference power F(X) asr = [n'(X) -I]1F(X) (17.24)

The equivalent power of the eye F(.:t) as afunction of wavelength is then

F(.:t) = [n'(.:t) -1]F(X) (17.25)[n'(X) - 1]

and the chromatic difference of powerM'(.:t) is

F(' ) = F(X)[n'(.:t) - 1l'(X)]d II. n'(X) _ 1 (17.26)

Equation (17.26) can be rearranged to give

F(X) = M'(.:t)[n'(X) - 1] (17.27)n'(.:t) - n'(X)

Substituting the right-hand side of thisequation (17.27) for F(X) into equation (17.21)gives

R (.:t) = [n'(.:t) - n'(X)]L(X) - M'(.:t) (17.28)E n'(X)

For an emmetropic eye focused on infinity,L(X) =0 and equation (17.28) reduces toRE(.:t) = -M'(.:t)/n'(X) (17.29)

Using a value of 1.333for n'(X) shows that thechromatic difference of refraction is three-quarters of the chromatic difference of power.

(17.23)F(.:t) = [n'(.:t) -1]/rIf we replace the length I' using equation(17.18b), equation (17.20) can be written in theform

R (.:t) = [n'(.:t) - n'(X)][F(X)+ L(X)] _ M'(.:t)E n'(X)

(17.21)For an emmetropic eye focused on infinity,L(X) =0 and equation (17.21) reduces to

RE(.:t) = [n'(.:t) ~~;\X)]F(X) _ M'(.:t) (17.22)

This equation shows that there is a differencebetween the chromatic difference of refractionRE(.:t) and the chromatic difference of powerM'(.:t), apart from a change in sign.Using appropriate chromatic dispersions of

the media, paraxial schematic eyes areexcellent at estimating the chromaticdifference of refraction of real eyes. As anexample, Atchison et al. (1993) used theGullstrand number 1 eye with its refractiveindices, but scaled the variations in refractiveindex so that the chromatic dispersion was thesame as for equation (17.18). This gave a rangeof chromatic difference of refraction of 1.940between 400 and 700nm. This is probably aslight underestimation of chromatic differenceof focus in most eyes.Atchison et al. (1993) determined the effect

of refractive error on change in range ofchromatic difference of refraction for sche-matic eyes, modified to have axial ametropiaand refractive ametropia. The latter was

retina and is here assumed to be independentof wavelength. The chromatic difference ofrefraction RE(.:t) is given by equation (17.1),that isRE(.:t) =L(.:t) - L(X) (17.1)

The chromatic difference of power M'(.:t), thedifference between the equivalent power F(.:t)at wavelength .:t and the equivalent powerF(X) at the reference wavelength X, is given byM'(.:t) = F(.:t) - F(X) (17.19)

Subtracting equation (17.18b) from (17.18a),and after a small amount of manipulationinvolving equations (17.1) and (17.19), wehave

RE(.:t) = n'(.:t) ~n'(X) _ M'(.:t)

192 Abl.'rmliollsalltl r<'lillal illlagl.' qllalily

Summary of main symbols

Chromatic and Indiana reduced eyes ofThibos et al. (1992,1997)Some of the details of these reduced eyes wereprovided in Chapter 16. The refractive indexdistribution is given by equation (17.17), andthe chromatic difference of refraction (seeFigure 17.6) is given in dioptres by

633.46RE(A.) =1.68524 - (A. _ 214.102) (17.30)

where the reference wavelength is 589nm andthe wavelength A. is in nanometres. To obtaingood predictions of transverse chromaticaberration of their eyes according to equations(17.2) and (17.5), Thibos et al. placed the stop1.91 mm inside the eye so that the distance ENof 3.98mm between the entrance pupil andthe nodal point was similar to that of moresophisticated schematic eyes.To allow for variations in transverse

chromatic aberration, the pupils of thechromatic eye do not need to be on the opticalaxis. Thibos et al. (1992) included a number ofadditional axes: line of sight, visual axis andachromatic axis (see Chapter 4). The corneahas an asphericity Q of -0.56 to correctspherical aberration at the referencewavelength of 589 nm.The Indiana eye varies from the chromatic

eye in that the optical axis, visual axis and lineof sight are now coincident and the cornea hasa variable asphericity (Thibos et al., 1997).

t(A.)

CDMEN

F(A.)

L1F(A.)

(angular) transverse chromaticaberrationchromatic difference of magnificationdistance between entrance pupil at Eand front nodal point at Nwavelength, usually expressed innanometresreference wavelength at which thereference power F(~) of the eye isdefinedequivalent power of a schematic eyeas a function of wavelengthchromatic difference of power; thedifference in equivalent powerbetween that at the wavelength A. andthat at the reference wavelength Achromatic difference of refraction

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18

Retinal image quality

Introduction

The quality of the visual system dependsupon a combination of optical and neuralfactors, with subjective measures dependingalso upon psychological factors.The optical factors that determine the

retinal image quality and affect visual systemquality are refractive errors, ocular aber-rations/ diffraction and scatter. The levels ofthe first three depend upon wavelength andpupil size, and can be easily quantified.Scatter is complex, and depends upon thelevel and nature of the turbidity of the ocularmedia, in particular the size and spatialdistribution of the scattering centres.The neural factors affecting visual system

quality include sizes and spacing of retinalcells, the degree of spatial summation at thevarious levels of processing from the retina tothe visual cortex, and higher level processing.The relative influences of optical and neural

factors upon visual system quality vary withretinal position and the criterion used forassessing quality. In the foveal region, the in-focus retinal image quality appears wellmatched to the neural network's resolution atoptimum pupil sizes of 2-3 mm (Campbelland Green, 1965; Campbell and Gubisch,1966)/ but resolution in the peripheral visualfield is limited much more by neural factorsthan by optical factors. For example, Green(1970) found that 'bypassing' the optics didnot improve the resolution of sinusoidalgratings beyond about 5° degrees from the

fovea. Bycomparison, the quality of the opticshas a large influence on detection in theperiphery (Wang et al., 1997).Direct measurement of retinal image

quality is not possible because of the inacces-sibility of the retina. The retinal image qualitycan be estimated from aberrations. The aerialimage of the retinal image can be analysed;techniques using this approach are referred toas double-pass techniques, and also asophthalmoscopic techniques. Another possi-bility is psychophysical measurement ofvisual performance in which two similarmethods are used, one of which bypasses theoptics of the eye; comparison of the resultsfrom the two methods yields the retinal imagequality.The retinal light distribution is not the same

as the perceived light distribution, partlybecause of the Stiles-Crawford effect, whichdescribes the luminous efficiency Le(r)of raysentering the eye through different heights r inthe pupil (Chapter 13). The Stiles-Crawfordeffect is a retinal phenomenon, but for somepurposes it can be considered as a pupilapodization; that is, as a filter of transmittanceLe(r)placed over the pupil (Westheimer, 1959).As such, it is often included in calculations ofretinal image quality.In this chapter we examine retinal image

quality criteria that are common in theanalysis of general optical systems. These arethe point and line spread functions and theoptical transfer function. Equations forcalculating the point spread function and the

optical transfer function from the aberrationsof an optical system are given in Appendix 4.Relationships between image quality criteriaand measurements are shown in Figure 18.1.The chapter is completed with an evaluationof the retinal image quality of the eye.

The point and line spread functions

The point spread function is the illuminanceor luminance distribution in the image of apoint source of light, while the line spreadfunction is the distribution in the image of aline (of zero width) source of light. Theabbreviations PSF and LSF are often used forthe point spread function and line spreadfunction, respectively.The form of the PSF depends upon

diffraction, defocus, aberrations and scatteredlight. In the absence of defocus, aberrationsand scatter, the PSF is called the diffraction-limited PSF. Defocus, aberrations andscattered light broaden the PSF. The form ofthe PSF also depends upon the shape anddiameter of the aperture stop. In the following

Retinal image quality 195

discussion, it is assumed that the pupil iscircular.

The diffraction-limited PSF(monochromatic light)

For a circular aperture or pupil, thediffraction-limited PSFis a radially symmetricfunction. For a monochromatic source, therelative light level L(Qat a distance' from thecentre of the PSF is given by the equationU,J =[2Jl(QF/~ (18.1)where MQ is a Bessel function. Tabulatedvalues ofMQcan be found in books such asAbramowitz and Stegun (1965), and it is wellapproximated by polynomials such as thatgiven in Table 18.1.In object space with the object at infinity,

, = 2n()p/A = n()D/A (18.2)

where

. 'A 'A() =angular distance (radian) =-2- =-Dnp x.

(18.2a)

integration

Stiles-Crawford function*I

squareroot

+Amplitude function A(X, y) Ware abcmltion function IV(X, Y)

multiplication

+ F .Phase transfer function (PrF) Pupil function P(X, Y) - tr~7,~!~~~1 -. Ampli~udePSF

.l --:-- complexT autocorrelation squarep/:ase.,.----

----. Intensit· PSF

Aerial (external) PSF*asymmetric (pupil) double pass

IFouriertransform ofaerialPSF

Fouriertral/sform ofrf.frafliol/-fimitetl PSF

Aerial (external) PSF*symmetric double pass

1..---------- Fourier transform. squareroot ---------'

Figure 18.1.Relationships between image quality criteria. Main image quality criteria are in boxes, operations are initalics, and quantities obtained by measurement are marked by *.

196 Aberrations and retinal imogequo/ify

Table 18.1. Polynomial approximation to the Bessel function h(~, from Abramowitz and Stegun (1965).

If (, < 3) or (,=3), theny ='/3x = 0.5 - 0.56249985y2 + 0.21093573y~ - 0.03954289t/' + 0.00443319y8 - 0.00031761y lO + 0.00001l09yI2/1 =zx

Otherwisey =3/'g =+0.79788456 + 0.00000156y + 0.01659667y2 + 0.00017105y3 - 0.00249511y~ + 0.001l3653ys - 0.00020033)1w =z - 2.35619449 + 0.12499612y + 0.0000565y2 - 0.00637879y3 + 0.00074348t + 0.00079824ys- 0.00029166)1/1 =g cos(w)/V,

The aberrated PSF

The outer limit of the Airy disc occurs whenthe above function (that is, the light level) firstgoes to zero, which occurs at

From equation (18.2),the corresponding valueof 8 at which this zero occurs is given by theequation

(18.5)

(18.4)

8 =1.22 A.ID rad

For pupil diameters greater than about 2 mm,or in polychromatic light, aberrations cannotbe neglected, and the point spread function isnot accurately described by the above diffrac-tion-limited equation. The effect of anyaberration is to spread light out more thanpredicted by diffraction. Therefore, the aber-rated PSF is broader and the peak is lowered,relative to the diffraction-limited case.

This is the angular radius of the Airy disc.Considering only the effect of diffraction,equation (18.5) shows that the PSF decreasesin width as the aperture stop diameterincreases in size.The diffraction-limited PSF can be observed

by looking at a bright monochromatic pointsource against a dark background through asmall artificial pupil. To be able to resolve theAiry disc and the surrounding rings, the discmust be several times larger than the smallestresolvable detail, which is often taken as 1min. arc. Assuming that the disc is 4 min. arcin diameter, equation (18.5) predicts that thepupil diameter should be about 1.2 mmdiameter at a wavelength of 550 nm.

,= 3.8317

o.0 ...j::::=:;:::::.....-'-T"""'r=;::.-,f""-,---,---.~.::::;:::::;.-..T'"""f'O:::;;;:"'+

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0

Distance in image plane ~

0.7

0.8

co'8 0.6c.E-0 0.5tU~~ 0.4C'0 0.3Q.

0.2

Figure 18.2.The monochromatic diffraction limited PSFsfor two light sources. The sources are separated so thatthe first zero of one source falls on the peak value of theother source (Rayleigh criterion). The half-width is alsoshown.

0.1

0.9

and j5 is the radius of the entrance pupil ofeye, D is the diameter of the entrance pupil(=2j5),and A. is the wavelength in a vacuum. Ifwe wish to convert the angular distance 8 inobject space to a distance on the retina, we canuse the conversion equationretinal distance =8F (18.3)where F is the equivalent power of the eye.This diffraction-limited PSF is plotted in

Figure 18.2. The figure shows that the lightdistribution consists of a central peak andcentral disc of light, known as the Airy disc.This is surrounded by a number of rings oflight of ever-decreasing light level.

The PSF and its use in quantifyingimage quality

The PSF of the eye is affected by defocus,aberrations and scatter, and is usually formedby polychromatic light. Thus it is notgenerally well described by the diffraction-limited PSF. Furthermore, because of asym-metries in the function, it is difficult often tocompare the PSF under different conditions.Comparisons are much easier if the PSFcan bereduced to a single number that specifies theimage quality on some meaningful scale.Three of these are the Rayleigh criterion,half-width and the Strehl intensity ratio.

The Rayleigh criterion (diffraction-limitedand monochromatic sources)The Rayleigh resolution criterion applies onlyto monochromic point sources and states that,for a diffraction-limited system, two pointsources can just be resolved if the peak of oneof the PSFs lies on the first minimum of theother. For two sources of the same wave-length, the situation is shown in Figure 18.2.Since the radius of the first dark ring of thediffraction-limited PSF is given by equation(18.5), it follows that this is the minimumangular resolution according to the Rayleighcriterion.

The half-widthOnce the influence of aberrations, scatter andpolychromatic light are included, the Airydisc ceases to exist because the PSF has, ingeneral, no well-defined zeros. The width ofthe PSF is then often taken as the half-width,which is the width at half the peak height. Ifthe diffraction-limited PSF is analyzed, wefind that the half-width .1' is.1'= 3.2327 (18.6)which is less than the diameter of the Airydisc (equation (18.4)). With this value, thediffraction-limited angular half-width .10 isgiven by the equation

l.029A. 3537A..18= -D- rad or = -D- min. arc (18.7)

The half-width of the diffraction-limited PSF

Retinal image quality 197

is shown in Figure 18.2. If aberrations areintroduced, the half-width increases.When astigmatism and coma are present,

the half-width should be considered in twomutually perpendicular directions, where thewidth is minimum and maximum. These canbe reduced to a single number by taking thearithmetic or geometric mean of the twowidths.

The Strehl intensity ratioThe Strehl intensity ratio is a measure of theeffect of aberrations on reducing themaximum or peak value of the PSF. It isdefined as follows:E = maximum light level valueof aberrated PSF

maximum lightlevelvalueofunaberratedPSF(18.8)

The Strehl intensity ratio has an advantageover the half-width by always being a singlenumber, even if the PSF is not rotationallysymmetric.Since the effect of aberrations is to spread

out the PSF and decrease the maximum peakheight, the Strehl intensity ratio is always lessthan or equal to one. The greater the aber-rations, the lower the value of the Strehlintensity ratio and the poorer the imagequality. A criterion for a good, near todiffraction-limited system is that the Strehlintensity ratio has a value of ~ 0.8.

The PSF and LSF ofeyesThe light distribution of a point source at theretina cannot be measured directly, but thelight passing back out of the eye (the aerial orexternal image) is measurable. Because thelight has passed twice through the eye'soptical system, this method is known as adouble-pass method. The light is doublyaberrated, and the aerial PSFis wider than theretinal PSF. It has long been thought that theretinal PSF is y,iven by the inverse Fouriertransform (F'l) of the square root of theFourier transform (FT) of the aerial PSF, whichis expressed mathematically asPSF= F'll..J[FT(aerial PSF)] (18.9a)

198 Aberrations and retinal illlage quality

This is not. correct,. because the aerial imagelos~s the information of asymmetric aber-rations, such as coma, which results in loss ofphase information (Artal et al., 1995a). Thisdoes not affect the determination of themodulation transfer function, but it doesaffect the phase transfer function so thatearlier determinations of this are incorrect(e.g. Artal et al., 1988). To overcome thisproblem, Artal and co-workers used differentdiam~ters of the. aperture stops for lightentering and leaving the eye, one of whichwas small enough to be considered diffractionlimited (e.g. 1 mm diameter). They called thisa 'one-and-a-half pass' method (Artal et a1.,1995b; Navarro and Losada, 1995) and an'asymme.tric (pupil) double-pass method'.Using this method, the retinal PSF with thelarger a~erture stop is obtained by dividingthe F.ouner transform of the aerial PSF by theFourier transform of the diffraction-limitedPSF, and then obtaining the inverse Fouriertransform of this result. This is expressedmathematically as

Early measurements of the eye's LSF weremade by Flamant (1955), Krauskopf (1962,196:4)' and Westheimer and Campbell (1962).Estimates of the modulation transfer function(see next section) were made by taking thesquare root of a one-dimensional Fouriertransform of the aerial LSF. LSF measure-ments preceded measurements of the PSF byabout 30 years (Santamaria et a1., 1987)because of the problem of low light levels.With the development of lasers and moresensitive detectors, it is now possible tomeasure PSFs in monochromatic light, anddetectors are sensitive enough for accuraterecording within a few minutes of arc of thecentre of the PSF. This is the region influencedby diffract.ion, a~errations and small anglescatter. Diffraction and aberrations havealmost no influence on image quality at angleslarger than a few minutes of arc, and lightbeyond this distance is due to scatter alone.Technical issues involved in the measurementof the PSF are described by Artal et al. (1993),Navarro et a1. (1993) and Williams et al. (19941996). 'Sophisticated techniques are being devel-

PSF=Fl1[FT(aerial PSF)IFT(diffraction-limited PSF)] (18.9b)

F.igure.18.3.A on~-dimensional pattern with asinusoidally varymg light level.

oped to derive the aberrations of the eye fromthe PSF (Iglesias et al., 1998a and b).

The optical transfer function

Figure 18.3 shows a one-dimensional gratingpattern with a light level that variessi~usoi~ally. The variation of this light levelWith distance x can be described by theequationLight level (x) =A o + A sin(21lXlp + 0) (18.10)

where A is the amplitude of the variation andits valu~ may be negative, 8 is a phase factor,an~ p IS ~he period of this pattern whosereciprocal is the spatial frequency a, that isa> lip (18.lOa)In object .space, the period is in units of angle(e.g. radians, degrees, minutes of arc), andthus the units of spatial frequency arecycles/units of angle (e.g. c/rad, c/deg, andc/min. arc).

If this pattern. i~ imaged by an opticalsystem, and providing the aberrations do notchange too rapidly over the field of thes!nuso.idal pattern, the image is alsosinusoidal but now has amplitude A', phasefactor 8 and period p', and can be representedby the equationLight level (x') =A o' + A' sin(21lX'I p' + 8?

(18.11)The modulation transfer function is definedas the amplitude A' of the image divided bythe amplitude A of the object, and it is af~nction of spatial frequency. In one dimen-sion, we denote the modulation transferfunction by the symbol G(a) and soG(a) =A'IA (18.12)

Determination of the OTFThe OTF can be determined in three ways(Figure 18.1).

amax=2p/).or=D/). c/rad (18.15)and this equation can be used to predict thetheoretical OTF resolution limit of the eye forany pupil size and wavelength.

1.0waves

2.0 waves

0.5 waves

Diffractionlimited

1.0

0.8

...~ 0.6<J>c<U1::c.~ 0.4~"'5-00

::E 0.2

0.0

-0.2 -t-,.........--r-.........,r-r--.--..--.--,--r-,.............,....,-..................,....+0.0 0.5 1.0 1.5 2.0

Reduced spatial frequency s

Figure 18.4. Examples of the defocused modulationtra~sfer function for three values of oW2•0 in wavelengthUnits.

Retinal image quality 199

frequency's, which is defined in terms of theactual spatial frequency a, by the equations =aA/p (18.14)

Equation (18.13) can still be used, but withG(a) replaced by G(s) to calculate the OTF interms of 5, and the equation of rbecomesr= 2 cos-1(s/2) (18.14a)This is an example in which there is no phaseshift with spatial frequency, and the MTF andthe OTF are the same.The function G(s) giving the diffraction-

limited modulation transfer function isplotted in Figure 18.4, and it follows from thisfigure that the upper limit of 5 is 2.0, whichcorresponds to the resolution limit of theoptical system. The corresponding actualspatial frequency or resolution limit 0". isthus max

Because this will depend upon lightattenuation in the system, the modulationtransfer function is normalized so thatG(O) =1.Different aberrations have different effects

on the image. Spherical aberration anddefocus cause a decrease in the amplitude, butcoma causes both a decrease in the amplitudeand a transverse shift from the Gaussian (oraberration-free) image position. Astigmatismis a defocus that varies with azimuth(meridian) in the pupil, and field curvature isa defocus that is independent of this azimuth.Distortion causes a transverse shift only.Those aberrations that produce a transverseshift produce an effective phase shift in theimage - i.e, 0and o'have different values. Thephase shift across a range of spatialfrequencies is the phase transfer function.The optical transfer function (OTF) is a

complex quantity that includes both themodulation transfer function (MTF) and thephase transfer function (PTF). In somesituations, for example on-axis in a rotation-ally symmetric optical system, there is nochange in phase with spatial frequency. Inthese situations the OTF is identical to theMTF, and we can use the terms inter-changeably.The MTF is closely associated with the

contrast threshold function, which is thevisual threshold contrast of sinusoidalpatterns as a function of spatial frequency.This function depends upon neural factors aswell as optical effects. The reciprocal of thecontrast threshold function is called thecontrast sensitivity function (CSF).

The diffraction-limited OTF with noStiles-Crawford effect

Using the theory presented in Appendix 4,one can easily show that the monochromaticdiffraction-limited optical transfer function,without any Stiles-Crawford effect, is givenby the equationG(a) = [r- sin(J)]I Jr (18.13)wherer= 2 cos-1[aM(2p)] (18.13a)It is common practice to express the diffrac-tion-limited OTF in terms of a 'reduced spatial

200 Aberrations and retinal image quality

Determining the OTF from the measuredwave aberrationsThe OTF can be calculated theoretically fromthe aberrations, and the mathematical basis ofthis is described briefly in Appendix 4.

Determining the OTF from the aerial PSF(See, for example, Artal et al., 1995a.)The OTF is the Fourier transform of the PSF.

However, as described in the previous section,with a single aperture stop for both ingoingand outgoing light, phase information is lost.The full retinal OTF cannot be determined,but the MTF is the square root of the Fouriertransform of the aerial PSF. Using the asym-metric double-pass method in which eitherthe ingoing or outgoing light beam isdiffraction limited, the OTF is the ratio ofthe Fourier transforms of the aerial anddiffraction-limited PSFs.

Determining the OTF from psychophysicalcomparison(See, for example, Arnulf and Dupuy, 1960;Campbell and Green, 1965; Bour, 1980;Williams et al., 1994.)The CSF is measured for sinusoids viewed

naturally (e.g. created on a video screen), andby using an interferometer with Maxwellianview which bypasses the optics and projectssinusoidal fringes directly on the retina. Thefirst function includes both the optics andneural factors, and the second functioninvolves only the neural factors.For the second function, two mutually

coherent point sources are produced near thenodal points, and the two resulting beamsoverlap on the retina to produce a series ofparallel fringes with angular separationY= AIa, where Ais the source wavelength anda is the source separation in air (Le Grand,1935 (translated Charman and Simonet, 1997);Smith and Atchison, 1997). Assuming the twosources have equal intensity, the fringes havea contrast of 1. The fringe contrast is reducedby adding light from an incoherent source ofthe same wavelength.Denoting the CSF determined with both

optics and neural factors as CSF0 + n and theCSF determined with only the neural factorsas CSFn' the modulation transfer function is

simply the ratio of these two functions; that is

MTF = CSF0 + n/CSF n (18.16)CSF0 + n must be measured for each pupil sizeof interest. Campbell and Green's (1965) CSFresults for one subject are shown in Figure18.5a, with the derived MTFs for a range ofpupil sizes appearing in Figure 18.5b.

OTF in the presence of defocus

To examine the optical transfer function withsome defocus, we must add a W2 0 term to thewave aberration function (Appendix 2), andthis is related to the defocus expressed as apower M by equation (A2.5). This is anotherexample where the MTF and OTF areidentical.Results are shown in Figure 18.4 for

different levels of defocus. For higher levelsof defocus, the MTF eventually becomesnegative, and then has an oscillatory naturewith a decreasing amplitude. When the MTFis negative, the image pattern has reversedcontrast compared with that of the object. Thismeans that the brighter parts of the objectbecome the darker parts of the image, and thedarker parts of the object become the brighterparts of the image. The spatial frequency atwhich the modulation transfer function firstgoes to zero is the resolution limit. Anyresolution of higher frequency patterns iscalled spurious resolution.Spurious resolution can be seen with the

Siemen's star pattern in Figure 18.6 when thisis defocused severely by using lenses or byviewing from a close distance within theaccommodation limit. Some regions in thepattern show contrast reversal. If eyes wererotationally symmetric, the regions of contrastreversal would form concentric annuli.However, because real eyes are asymmetric,particular the aberrations, these annuli haveirregularities. The presence of theasymmetries means that the phase transferfunction is not zero.

The geometrical optical approximation fordefocusThe phenomenon of spurious resolution canbe investigated by a much simpler but

Retinal image qllality 201

2.0mm

--+-- 2.8mm

10 20 30 40 50 60

Spatial frequency (c/deg)

2.5 1.0

Neural system alone 0.9.............. Neural system plus optics

2.0 . 0.8~ .~'> . 0.7.;: . ....;;;

~c 1.5 . '" 0.6'" c'" . s:;;~ e 0.5C .S0 1.0 «i 0.4U -;'0 -ebll . 0 0.30 ~

...l0.5 0.2

0.1

0.0 0.00 10 20 30 40 50 60 0

(a) Spatial frequency (c/deg) (b)

Figure 18.5.a. Results of Campbell and Green (1965) showing the contrast threshold functions for the whole eye; that is, the

combined neural and optical systems, and the neural system alone. For the former the subject viewed sinusoidalgratings on a television monitor with a green phosphor through a 2 mm artificial pupil, while for the latter thefringes were generated by interferometry using monochromatic light (wavelength 633 nm).

b. Modulation transfer functions from a range of pupil sizes derived from the ratio of the two types of curves shownin Figure 18.5a.

approximate process if it is assumed that thedefocus is large. In this case, the PSF is auniformly illuminated disc. In Chapter 9, weshowed that the angular diameter (/> of thedefocus blur disc was related to the level of

refractive error t:..L and pupil diameter D, byequation (9.17), that is

(/>=.1L D (18.17)

In the simple aberration and diffraction-freedefocused system, the PSF is the defocus blurdisc of diameter (/>. The OTF is the Fouriertransform of this PSF. For a circular uniformfunction with a diameter (/>, the OTF G(O') isgiven by the equation

G(0') = J1(tr(/>O') (18.18)tr(/>O'

where II is the same Bessel function as inequation (18.1). For distance viewing, i.e. (/>isin radians, the spatial frequency 0' is incycles/radian. The first zero of this equationoccurs at

tr(/>O' = 3.83

and thus

(18.19)

Figure 18.6. The Siemen's star, which can be used forobserving spurious resolution.

3.83 1.22G=--=--

tr(/> (/>

Replacing the blur disc diameter (/> by theright-hand side expression from equation

202 Aberratiollsand retinal image ql/alily

To appreciate the limits that the optics of the

Retinal image quality

This value of (J is the resolution limit.As an example, let us calculate the

resolution limit of an eye with a pupildiameter of 4 mm and 1 0 of defocus. Inequation (18.20), we put L1L = 1 0 and D =0.004m, and thus the resulting resolutionlimit is(J= 305 c/rad = 5.32 c/degFigure 18.7 shows the expected region ofspurious resolution in an aberration-free eyeaccording to physical and geometric opticspredictions (Smith, 1982). Geometric opticsapproximations become more accurate forhigher levels of defocus and lower spatialfrequencies.

(18.17), we have1.22

(J = t:.LD c/rad (18.20)

eye place on VISIOn, we also need someunderstanding of the limits provided by theretina. To be able to resolve the detailprovided by a pattern of light imaged uponthe retina, the adjacent 'receptor units' mustbe sufficiently close together to correctlyinterpret the pattern. In Figure 18.8, obliquesquare wave light patterns imaged on asquare array of receptor units are shown. InFigure 18.8a, the pattern repeats every fourunits in the vertical meridian. Another way ofputting this is that the receptor units are aquarter of a cycle apart. The visual systeminterprets the spatial frequency and orienta-tion of the pattern correctly (the termveridically is sometimes used). In Figure18.8b, the pattern repeats every two units inthe vertical direction. For every two receptorunits in the vertical direction struck byadjacent light bars, there is a receptor unitbetween them which is struck by a dark bar(and vice versa). The visual system is just ableto resolve the pattern; that is, vision is againveridical. In Figure 18.8c, the sinusoidalpattern is yet finer, repeating every receptor

(d)

(b)

(c)

(a)

~-~

• .+.. •

~10.0

Regionofspuriousresolution

2mm

physicaloptics

geometricoptics

1.0

Defocus (0)

I -1----.--r-.,......,-.-,..,...,"'T""""---.--"'-~.,.......;:....._::l'_T_.,.f_

0.1

Regionfreeofspurious resoluuon

Figure 18.7. Physical and geometrical optical predictionsof the boundaries of spurious resolution for anaberration-free system and for different pupil diametersas marked (mm). Spurious resolution occurs above theboundaries. The physical optical predictions are givenby the curves. The geometrical optics approximationsgiven by the straight lines are increasingly accurate aspupil size increases, defocus increases, and spatialfrequency decreases. Wavelength is 550 nm (based onSmith,1982).

Figure 18.8. Oblique square wave patterns are imagedon a square array of receptor units. The receptor unitsare excited by the bright bars and inhibited by dark bars.a. The pattern repeats itself every four units in the

vertical direction.b. The pattern repeats itself every two units in thevertica I direction,

c. The pattern repeats itself every unit in the verticaldirection.

d. An aliased perception of the pattern in (c).

unit in the vertical meridian, and now thesampling rate of the receptor units is inad-equate to correctly interpret the pattern. Thevisual system 'undersamples' the pattern,which may be perceived to have a much lowerspatial frequency and a different orientationsuch as that in Figure 18.8d. This incorrectinterpretation of the pattern is referred to asaliasing. Another way of considering this isthat the visual system cannot distinguishbetween the light patterns shown in Figures18.8cand d.Figure 18.8b shows the finest light pattern

that can be correctly resolved by the retina. Itsspatial frequency is called the Nyquist limit(NL), and is given byNL=1/(2cs) (18.21a)where Cs is the centre-to-centre spacing of thereceptor units.The 'receptor unit' that correlates best with

resolution is the spacing between ganglioncells (Thibos et al., 1987). At the fovea there isa one to one correspondence between conesand ganglion cells, so the cone spacing can beused to determine resolution limits. Becauseof the tight hexagonal packing of cones, theprevious equation needs to be modified toNL=1/('-I3cs) (18.21b)Williams (1985) calculated the Nyquist limit tobe 56c/deg in object space (or subtense atback nodal point), by assuming that theclosest spacing of human foveal cones is 3 urnand that 0.29mm on the retina corresponds to1°. The resolution limit drops quickly awayfrom the fovea because of the rapid decreasein ganglion cell density (Curcio and Allen,1990).There are two major types of spatial tasks

that are performed by the visual system.Resolution has already been referred to; theother type is detection. Perimetry is anexample of a detection task in which a patientmust detect that a spot of light is present;details such as the shape of the spot areusually unimportant. Although in Figure18.8c the pattern is not seen veridically, it isstill seen, provided that the contrast issufficiently high. It is likely that the highestspatial frequency that can be detected by theretina is limited by the size of photoreceptors(Still, 1989). Visual acuity involves resolution,as a patient must be able to resolve detail of a

Retinal image quality 203

target. It can be argued that visual acuity ismore than a resolution task, as identifyingthe arrangement of resolved elements isimportant for identification of a letter.The relative importance of optical and

neural limitations to visual performance canbe determined by using the CSF both as aresolution task and as a detection task. In thedetection task, the subject is asked which oftwo presentations contains a grating. In theresolution task, the subject is asked todetermine the orientation of a sinusoidalgrating; for example, whether it is horizontalor vertical. For normal presentation of thegratings on a monitor, when the two CSFs areessentially the same, the optics act as a filter toprovide the main limitation to both detectionand resolution (see central curves in Figure18.9). However, if the detection CSF issuperior to the resolution CSF, thenresolution, although not necessarily detection,is limited by the sampling rate of the receptorelements (see peripheral curves in Figure18.9).This issue was discussed in detail by Thibos

and Bradley (1993), who recommended using

3.0

Detection

2.5

~,::.2.0.:;

'p';jiClU

'" 1.5~

'"..E0~ 1.0b/)0

..J

0.5

10 100Spatial frequency (c/deg)

Figure 18.9. Normal contrast sensitivity functions forresolution and detection tasks for the centre andperiphery (300 nasal in object space) of the visual field.The region between the two peripheral vision curvesindicates the spatial frequencies for which aliasingoccurs. Representation is based on results of Thibos etal.(1996) in which measurements in the periphery weredone after careful refraction.

204 Aberratiolls and retina! imageqllality

60

Williams et ul, (1994).3 subjects

Walsh and Charman (1985l.10 subjects

0.0 +---r-----r-..--,..-........--,r-...---r----.--r--r-_+_o

0.2

0.8

10 20 30 40 50

Spatial frequency (c/deg)

Figure 18.10.Comparison of MTFs obtained for 3 mmdiameter pupils from the psychophysical comparisonmethod (Williams et al.,1994)and the aberroscopetechnique (Walsh and Chatman, 1985).Results showmeans and standard deviations. Vertical gratingorientation. For the Walsh and Charman study, theMTFs were determined from the 10 subjects' aberrationco-efficients.

...~~ 0.6gc.2] 0.4:::l

'8:E

imaging manipulations to remove influencesof forward- and back-scattered light.Williams et al. (1994) found that the double-

pass method produced slightly lower MTFsthan the psychophysical method in red light(633 nm) for three subjects with 3 mmdiameter pupils. They attributed the differ-ence to light scattered back from the choroid,because repeating the PSF measurements ingreen light (543 nm), which reduces thechoroidal contribution to the fundus reflect-ance, gave results which were consistent withthe psychometric results. Their MTFs weremuch lower than studies using the aberro-scope technique to determine MTFs (Figure18.10), and were lower than a subsequentinvestigation by Liang and Williams (1997)using the wave-front sensor technique.There is considerable variation in retinal

image quality between different eyes, asshown in Figure 18.10 for 3 mm diameterpupils (Walsh and Charman, 1985).Although much of our emphasis is on the

modulation transfer function, the phasetransfer function is an aspect of image qualitythat should not be neglected. Phase transfer

Despite being carried out in the mid-1960s,the classic work of Campbell and colleagues isstill considered to provide good data on thequality of the optics of the eye. Figure 18.5bshows the psychophysically determinedmodulation transfer functions of Campbelland Green (1965) for one subject. Opticalperformance is near diffraction-limited for 2mm diameter pupils. As pupil size increases,aberrations cause performance to decreaserelative to diffraction-limited performance,with optimum quality occurring for 2.0-2.8mm pupil diameters. The maximum spatialfrequency at which resolution of the pattern ispossible is approximately 60 cldeg. Using animproved technique with reduced spatialnoise masking effects, which can affect theCSF measured by bypassing the optics,Williams (1985)showed that the neural abilityis better than that given by Campbell andGreen beyond 40 c/deg. This indicates thatthe retinal image quality beyond 40 c/deg isworse than calculated by Campbell andGreen. Although both the optics and neuralfactors contribute to the normal CSF, theoptics are the major limitation to centralvision because both detection and resolutionCSFs give similar results - that is, there is noaliasing (Thibos et al., 1996).The aberration methods of determining

image quality might be expected to over-estimate image quality because mediascattering effects are ignored. The psycho-physical comparison method has severalconsiderations. Again, the scattering effectsmay be ignored, assuming that they affect thetwo CSFs equally. There is also the problem ofmaking the two contrast sensitivity measure-ments equivalent for subjects to perform. Theophthalmoscopic techniques involve assump-tions, including that the retina acts as a diffusereflector and that scatter is similar for bothdirections of light. They may be affected by aretinal effect similar to the Stiles-Crawfordfunction (Artal, 1989). They also involve

Central vision

the distinction between resolution anddetection to design clinical tests to determineat which stage (detection, resolution oridentification) an abnormal visual system isbreaking down.

Retinal imagequality 205

Figure 18.11.Measured monochromatic CSFs for onesubject for in-focus condition (0D) and defocuscondition (-3 D). Also shown is the predictedmonochromatic CSF for the defocus condition. This wasobtained by multiplying the CSF for the in-focuscondition by the ratio of MTFs for the defocused and in-focus conditions. The MTFs were themselves obtainedfrom aberration measurements. The arrows indicatespatial frequencies corresponding to four 'notches' in theCSF; these correspond well with those predicted fromtheory. Unpublished data of the study of Atchison et al.(1998b).

Polychromatic lightOn the basis of theoretical considerations, vanMeeteren (1974) considered that chromaticaberration is the major optical limitation to in-focus retinal image quality. Thibos et al. (1991)showed theoretically that, with 2.5 mmcentred pupils, the longitudinal chromaticaberration would have similar effects in whitelight to 0.2D defocus in monochromatic light(Figure 18.12)..They obtained a maximumcontrast sensitivity loss of 0.2 log unit and avisual acuity loss of - 10 per cent « 0.05logunit) relative to those in monochromatic light,similar to the experimental findings ofCampbell and Gubisch (1967).As mentioned in Chapter 17 (Effects of

chromatic aberrations on vision), decentration ofnatural or artificial pupils in white light canhave devastating effects on retinal imagequality. Large levels of transverse chromaticaberration produce wavelength-dependentspatial phase shifts of the image of a

15

in-focus

··..····0········ -3D

predicted. -3D

5 10

Object spatial frequency (c/deg)

! !

6mmpupil

~ 1.5~

:~'Vic~

~Eou'"6bll.3 0.5

DefocusThe most important optical defect affectingretinal image quality is defocus. Fluctuationsin the modulation transfer function (e.g.Figure 18.4) produced by defocus can beexpected to be accompanied by fluctuations inthe CSF. Few studies dealing with defocushave shown dips (or notches) in the CSF,probably because of limitations in techniquesuch as sampling rate and quality ofequipment. Apkarian et al. (1987) and Bourand Apkarian (1996) showed notches in theCSF in the presence of astigmatism. Woods etal. (1996) and Atchison et al. (1998b) were ableto show up to four notches in the CSF (Figure18.11) which generally coincided well withpredictions from MTFs. Even in less well-controlled experiments, which are muchcloser to the usual clinical situation, it is stillpossible to show these fluctuations in the CSF(Woods et al., in press). Even small levels ofdefocus (0.5D) can produce quite markedlosses, demonstrating that it is importantcarefully to correct even small refractiveerrors to prevent incorrect attribution of lossesto retinal/neural pathological causes.

functions of eyes are very different from zero,corresponding to the presence of coma-likeaberrations (Walsh and Charman, 1985;Charman and Walsh, 1985). This is veryimportant in recognition of complex objects,which do not look like the originals because ofdifferent phases at different spatial fre-quencies.Liang et al. (1997) used adaptive optics in

conjunction with the wave-front sensortechnique to correct the aberrations of foursubjects. They were able substantially tocorrect the aberrations of the eyes, obtainingnear diffraction-limited performance with 6mm diameter pupils. Contrast sensitivity andresolution limits were improved markedlywith the adaptive optics, and retinal conescould be seen easily with a fundus camerathrough the adaptive optics. An interestingphenomenon they noticed illustrates theirsuccess in correcting the eye's optics. Asteadily fixated red laser light appeared tofluctuate in colour between red and green -this was due to fixational eye movementsshifting the retinal location between long andmedium wavelength cones.

206 Aberrations and retinal intag): qllality

6010 20 30 40 50Spatial frequency (c/deg)

-2.5 -+---..----,r---r---r-L....,--r..l~---,--r--r__.-+o

Figure 18.13.White-light MTFs for the Chromatic eye,which is free of monochromatic aberrations, for variousdisplacements of a 2.5mm diameter pupil perpendicularto the grating orientation (white light provided by a P4phosphor, reference wavelength 589 nm). Also shown isthe neural contrast threshold of one subject fromCampbell and Green (1965). Intersection of the MTFcurves with the neural contrast threshold predicts thecut-off spatial frequency. Based on Figure 9 of Thibos etal. (1991), with data kindly provided by Larry Thibosand with permission from The American Academy ofOptometry.

-0.5

...~

'"c~ -Ic.2co:;-g -1.5E'0OJ)

j -2

Pupil decentrationDecentration of the eye's pupil inducesadditional optical aberrations, such as trans-verse chromatic aberration and coma, whichdecrease spatial visual performance (Green,1967; van Meeteren and Dunnewold, 1983;Artal et al., 1996). The Stiles-Crawford effectmay be of assistance to spatial vision byreducing the influence of the aberrations ofthe parts of the pupil furthest from the peak ofthe Stiles-Crawford effect; Bradley and

Krakau, 1974; van Meeteren, 1974; Carroll,1980). Even in the presence of defocus, theinfluence of the Stiles-Crawford functionshould be small (e.g. van Meeteren, 1974;Atchison et al., 1998a) (Figure 18.14). TheStiles-Crawford effect is expected to influenceoptimal refraction in the presence ofaberrations, but the magnitude is likely to besmall- for example, approximately 0.20 at 10c/deg in Figure 18.14.

60-2.5 -+---r----,--r--r---,--r---.----,--r--r--.-+

o

.........

···...··)~~7~~~.:.7:.7.7.7.7.7:.... ....../0.25 D ........~"........... .

.....;;,;~~ -l .White light ..0- .

:" -e

..................

.............

•••....~oveal neural threshold.'.......................

sinusoidal target, leading to loss of imagecontrast. This has its greatest effects for targetorientation at right angles to the decentrationdirection, with up to three times loss ofresolution for 3 mm displacements of smallartificial pupils (Figure 18.13)(Green, 1967;Thibos etal., 1991).

The Stiles-Crawford effectThe Stiles-Crawford effect has often beenimplicated by vision researchers to explain thedifference between expected and actualfindings - for example, the failure of depth-of-field to decrease with increase in pupil size asquickly as expected (Campbell, 1957; Tuckerand Charman, 1975;Charman and Whitefoot,1977; Legge et al., 1987). However, the resultsof theoretical investigations using the apodiz-ation model suggest that the Stiles-Crawfordeffect is not important for spatial resolutionfor in-focus imagery, even in the presence ofconsiderable aberrations (Metcalf, 1965;

10 20 30 40 50Spatial frequency (c/deg)

Figure 18.12.Comparison of in-focus white-light MTFwith defocused monochromatic MTFs for the Chromaticeye, which is free of monochromatic aberrations (pupildiameter 2.5mm, white light provided by a P4phosphor, reference wavelength 589 nm). Also shown isthe neural contrast threshold of one subject fromCampbell and Green (1965). Intersection of the MTFcurves with the neural contrast threshold predicts thecut-off spatial frequency. Based on Figure 4 of Thibos etal. (1991), with data kindly provided by Larry Thibosand with permission from The American Academy ofOptometry.

-0.5

-;:-~

'"C<U -It:c.2';l:;

-1.5-gE'0OJ)0 -2...J

Thibos (1995) refer to this as an anchoringrole. A number of studies have shown thatsubjective transverse chromatic aberrationdeclines as pupil size increases; for example,Ye et al. (1992). This effect reduces as lumi-nance is reduced, thus indicating that it is aretinal effect and due to the Stiles-Crawfordeffect.

Figure 18.14.Modulation transfer as a function ofdefocus at various object spatial frequencies(cycles/ degree) when there is +1.0 0 of primaryspherical aberration at the edge of a 6 mm diameterpupil in 605nm wavelength light. Results are shownwith and without Stiles-Crawford apodization of f3 =0.17,which is near the 97.5 per cent upper limit(Applegate and Lakshminarayanan, 1993).TheStiles-Crawford effect has only a small influence onimage quality, which is greater when the defocus andspherical aberration are in the same direction than whenthey are opposed. Data from Figure 6 of Atchison et al.(1998a),with permission from The Optical Society ofAmerica.

fovea, circle of least confusion

fovea, astigmatism corrected

- - -e- - - 20°, circle of least confusion_ 20°, astigmatism corrected

- - -c- - - 40°, circle of least confusion

--0-- 40°, astigmatism corrected

0.8

0.2

..<2:!'"c 0.6fJc.2(;j:; 0.4'8::E

1.0.~-'--~--'----'-~-'--~--'---+

Retinal image quality 207

improves considerably (Jennings andCharman, 1978 and 1981; Still, 1989; Williamset al., 1996). Navarro et al. (1998) found thatthe root-mean-squared wave aberrationincreases by a factor of only two from thecentral visual field out to 40° in the peripherywhen astigmatism and field curvature/defocus are corrected (6.7 mm diameterpupil). The importance of the peripheraloptics has often been discounted becauseimproving them has given little improvementin resolution (see, for example, Green, 1970).However, marked improvement in detectionoccurs (Williams etal., 1996;Wang et al., 1997).

Curve fitting of modulation transferfunction resultsJennings and Charman (1997) listed a numberof mathematical fits that have been proposedfor experimentally determined modulationtransfer functions, both for the centre andperiphery of the visual field. As an example,Jennings and Charman (1974) examined

2

fJ= 0 mm- 2

fJ=0.17 mm-2

-I 0 IDefocus (D)

0.8

.. 0.6<2:!'"c~ 0.4c0.~

:; 0.2'8::E

0

-0.2

-3 -2

600.0+--'--r--r-~-r--,---'--r--.-~-""'-+

o 10 20 30 40 50

Spatial frequency (cycles/degree)

Figure 18.15. Fits of computed MTFs, averaged over twosubjects, for different horizontal object angles, from theresults of Williams et al. (1996). MTFs were determinedfrom PSFs.The circle of least confusion plots areorientation-averaged MTFs obtained without correctingoblique astigmatism at a refractive state correspondingto the circle of least confusion. The astigmatism-corrected plots are effectively those obtained whendefocus and oblique astigmatism are corrected. The off-axis correction gives considerable improvement in imagequality. Pupil diameter 3 mm, wavelength 543 nm.

Peripheral visionThe optics associated with the peripheralretina are poor, mainly because of focusingerrors in the form of oblique astigmatism andfield curvature (see Chapter 15). Retinalimage quality declines steadily with objectangle (Jennings and Charman 1978, 1981;Navarro et al., 1993 and 1998). However, asshown in Figure 18.15,when the periphery iscarefully refracted, the image quality

208 Aberrations and retinal illlage quality

References

Summary of main symbols

Abramowitz, M. and Stegun, I. A. (1965). Handbook ofMatltematical Functions with Formulas, Graphs alldMathematical Tables, p. 370. USGovernmeni Print Office.

Apkarian, P., Tijssen, R., Spekreijse, H. and Regan, D.

where a is again the spatial frequency incycles/degrees and A, Band C are constantsfor a particular object angle for a range ofobject angles. The results of these are shownin Figure 18.15.

(1987). Origin of notches in CSF: Optical or neural?lmvst, 01'1111/111. Visual. Sci., 28, 607-12.

Applegate, R. A. and Lakshminarayanan. V. (1993).Parametric representation of Stiles-Crawfordfunctions: normal variation of peak location anddirectionality. f. Op«. Soc. Alii. A, 10,1611-23.

Arnulf, A. and Dupuy, O. (1960). La transmission descontrastes par Ie systeme optique de I'CEiI et II'S seuilsdes contrastes retiniens. C. R. Acad. Sci. Paris, 250,2757-9.

Artal, P. (1989). Incorporation of directional effects of theretina into computations of optical transfer functions ofhuman eyes. J. Opt. Soc. Alii. A, 6, 1941-4.

Artal, P., Santamaria, J. and Bescos, J. (1988). Phase-transfer function of the human eye and its influence onpoint-spread function and wave aberration. J. Opt. Soc.Alii. A., 5, 1791-5.

Artal, P., Ferro, M., Miranda, I. and Navarro, R. (1993).Effects of aging in retinal image quality. J. Opt. Soc. Am.A., 10, 1656-62.

Artal, P., Marcos,S., Navarro, R. and Williams, D. R.(1995a). Odd aberrations and double-pass measure-ments of retinal image quality. J. Opt. Soc. Am. A., 12,195-201.

Artal, P., Iglesias, I. and Lopez-Gil, N. (1995b). Double-pass measurements of the retinal-image quality withunequal entrance and exit pupil sizes and thereversibility of the eye's optical system. f.Opt. Soc. Am.A., 12, 2358-66.

Artal, P.,Marcos.B, Iglesias, I. and Green, D. G. (1996).Optical modulation transfer and contrast sensitivitywith decentered small pupils in the human eye. VisiollRes., 36, 3575-86.

Atchison, D. A., Smith, G. and [oblin, A. (1998a).Influence of Stiles-Crawford apodization onspatial visual performance. f. Opt. Soc. Alii. A, 15,2545-51.

Atchison, D. A., Woods, R. L. and Bradley, A. (1998b).Predicting the effects of optical defocus on humancontrast sensitivity. f. Opt. Soc. Am. A, 15, 2536-44.

Bour, L. J. (1980). MTF of the defocused optical quality ofthe human eye for incoherent monochromatic light. f.Opt. Soc. Am., 70, 321-8.

Bour, L. J. and Apkarian, P. (1996). Selective broad-bandspatial frequency loss in contrast sensitivity functions.lnucsi, Ophthal. Vis. Sci.,37, 2475-84.

Bradley, A. and Thibos, L. N. (1995). Modelling off-axisvision - I: The optical effects of decentring visualtargets or the eye's entrance pupil. In Vision Models forTarget Detection and Recogllition, pp. 313-37. WorldScientific Press.

Campbell, F.W. (1957). The depth of field of the humaneye. Optica Acta, 4, 157--M.

Campbell, F. W. and Green, D. G. (1965). Optical andretinal factors affecting visual resolution. f. Pltysiol.iLond.), 181, 576-93.

Campbell, F.W.and Cubisch, R.W. (1966). Optical qualityof the human eye. f. Physiol. tLond.), 186, 558-78.

Campbell, F.W. and Cubisch, R. W. (1967). The effect of

point spread functionline spread functioncontrast sensitivity functionmodulation transfer functionoptical transfer functionphase transfer functionoptical transfer functionspatial frequency (c/radian)corresponding modified spatialfrequency, called 'reduced spatialfrequency', related to a by equation(18.14)angular diameter of defocus blur discradius of entrance pupildiameter of entrance pupil

epPD

PSFLSFCSFMTFOTFPTFG(a)as

fitting central modulation transfer functionsof the eye by the functionMTF(a) =exp[-(a/ac)IlJ (18.22a)where a is the spatial frequency, ac is a spatialfrequency scaling constant at each pupil size,and n is a shape factor constant. Their fits areaccurate only for lower spatial frequencies. Athigher frequencies, errors arise because actualmodulation transfer functions have a finiteresolution limit and the above function doesnot. Later, Jennings and Charman (1997)found that this approximation was useful forthe peripheral field, with n remainingrelatively constant at about 0.9 out to 40°eccentricity and ac declining steeply over thisrange. Navarro et al. (1993) and, later,Williams et al. (1996) used the fitMTF(a) =(1 - C)exp(-Aa) + Cexp(-Ba)

(18.22b)

chromatic aberration on visual acuity. J. Phvsiol. (Lond.),192,345-58.

Carroll, J. P. (1980). Apodization model of theStiles-Crawford effect. J. Opt. Soc. Am., 70, 1155-6.

Charman, W. N. and Simonet, P. (1997). Yves Le Grandand the assessment of retinal acuity using interferencefringes. Ophthal. Physiol. Opt., 17, 164-8.

Charman, W.N. and Walsh, G. (1985).The optical transferfunction of the eye and perception of spatial phase.Visioll Res.,25, 619-23.

Charman, W.N. and Whitefoot, H. (1977). Pupil diameterand the depth-of-focus of the human eye for Snellenletters. Optica Acta,24, 1211-16.

Curcio, C. A. and Allen, K. A. (1990). Topography ofganglion cells in human retina. J. Compo Neurology, 300,5-25.

Flamant, F.(1955).Etude de la repartition de lumiere dansI'image retienne d'une fente. Rev. Opt. Theor. lnsirum.,34,433-59.

Green, D. G. (1967). Visual resolution when light entersthe eye through different parts of the pupil. J. Physiol.(Lond.), 192, 345-58.

Green, D. G. (1970). Regional variations in the visualacuity for interference fringes on the retina. J.Physio/.(Lond.), 207, 351-{).

Iglesias, 1., Lopez-Gil, N. and Artal, P. (1998a).Reconstruction of the point-spread function of thehuman eye from two double-pass retinal images byphase retrieval algorithms. J. Opt. Soc. Am. A., 15,326-39.

Iglesias, 1.,Berrio, E. and Artal, P. (1998b). Estimates of theocular wave aberration from pairs of double passretinal images. J. Opt. Soc. Am. A, 15, 2466-76.

Jennings, J. A. M. and Charman, W. N. (1974). Analyticapproximation of the off-axis modulation transferfunction of the eye. Br. J. Physiol. Opt., 29, 64-72.

Jennings, J. A. M. and Charman, W. N. (1978). Opticalimage quality in the peripheral retina. Am. J. Optom.Physiol. Opt., 55, 582-90.

Jennings, J. A. M. and Charman, W. N. (1981). Off-axisimage quality in the human eye. Visioll Res., 21,445-55.

Jennings, J. A. M. and Charrnan, W. N. (1997). Analyticapproximation of the off-axis modulation transferfunction of the eye. Visioll s«, 37, 697-704.

Krakau, C. E. T. (1974). On the Stiles-Crawfordphenomenon and resolution power. Acta Ophthalmol.,52,581-3.

Krauskopf, J. (1962). Light distribution in human retinalimages. J. Opt. Soc. Am., 52, 1046-50.

Krauskopf, J. (1964). Further measurements of humanretinal images. J. Opt. Soc. Am., 54, 715-16.

Legge, G. E., Mullen, K. T., Woo, G. C. and Campbell, F.W. (1987). Tolerance to visual defocus. J. Opt. Soc. Am.A, 4, 851-{)3.

Le Grand, Y. (1935). Sur la mesure de l'acuite visuelle aumoyen de fringes d'interference. C. R. Acad. Sci. Paris,200, 490-91 (translated by W. N. Charman and P.Simonet, 1997).

Retinal image quality 209

Liang, J. and Williams, D. R (1997). Aberrations andretinal image quality of the normal human eye. J. Opt.Soc. Am. A., 14, 2873-83.

Liang, J., Williams, D. R and Miller, D. T. (1997).Supernormal vision and high-resolution retinalimaging through adaptive optics. J. Opt. Soc. Am. A., 14,2884-92.

Metcalf, H. (1965). Stiles-Crawford apodization. J. Opt.Soc. Am., 55, 72-4.

Navarro, R and Losada, M. A. (1995). Phase transfer andpoint-spread function of the human eye determined bya new asymmetric double-pass method. J. Opt. Soc. Am.A, 12, 2385-92.

Navarro, R, Artal, P. and Williams, D. R (1993).Modulation transfer of the human eye as a function ofretinal eccentricity. J. Opt. Soc. Am. A., 10, 201-12.

Navarro, R, Moreno, E. and Dorronsoro, C. (1998).Monochromatic aberrations and point-spread functionsof the human eye across the visual field. J. Opt. Soc. Am.A., 15, 2522-9.

Santamaria, J., Artal, P. and Besc6s, J. (1987).Determination of the point-spread function of thehuman eyes using a hybrid optical-digital method. J.Opt. Soc. Am. A., 4, 1109-14.

Smith, G. (1982).Ocular defocus, spurious resolution andcontrast reversal. Ophthal. Physiol. Opt., 2, 5-23.

Smith, G. and Atchison, D. A. (1997). The Eyeand VisualOptical Instruments, pp. 517-24. Cambridge UniversityPress.

Still, D. L. (1989). Optical limits to contrast sensitivity inhuman peripheral vision. Unpublished PhD thesis,University of Indiana.

Thibos, L. N. and Bradley, A. (1993). New methods ofdiscriminating neural and optical losses of vision.Optom. Vis. Sci.,70, 279--87.

Thibos, L. N., Bradley, A. and Zhang, X. (1991). Effect ofocular chromatic aberration on monocular visualperformance. Optom. Vis. Sci.,68, 599-{)o7.

Thibos, L. N., Cheney, F. E. and Walsh, D. (1987). Retinallimits to the detection and resolution of gratings. J. Opt.Soc. Am. A., A4, 1524-9.

Thibos, L. N., Still, D. L. and Bradley, A. (1996).Characterization of spatial aliasing and contrastsensitivity in peripheral vision. Vision Res., 36,249-58.

Tucker, J. and Charman, W. N. (1975).The depth-of-focusof the eye for Snellen letters. Am. J. Optom. Physiol. Opt.,52,3-21.

van Meeteren, A. (1974). Calculations on the opticalmodulation transfer function of the human eye forwhite light. Optica Acta,21, 395-412.

van Meeteren, A. and Dunnewold, C. J. W. (1983). Imagequality of the human eye for eccentric entrance pupils.Vision Res.,23, 573-9.

Walsh, G. and Charman, W. N. (1985). Measurement ofthe axial wavefront aberration of the human eye.Ophthal. Physiol. Opt., 5, 23-31.

Wang, Y.-Z.,Thibos, L. N. and Bradley, A. (1997). Effectsof refractive error on detection acuity and resolution

210 Aberrations and retinal image quality

acuity in peripheral vision. Invest. Ophthaimol. Vis. Sci.,38, 2134-43.

Westheimer, G. (1959). Retinal light distribution forcircular apertures in Maxwellian view. J. Opt. Soc. Am.,49,41-4.

Westheimer, G. and Campbell, F. W. (1962). Lightdistribution in the image formed by the living humaneye. J. Opt. Soc. Am., 52, 1040-45.

Williams, D. R. (1985). Visibility of interference fringesnear the resolution limit. J. Opt. Soc. Am. A., 2, 1087-93.

Williams, D. R., Artal, P.,Navarro, R.et al. (1996). Off-axisoptical quality and retinal sampling in the human eye.Vision Res.,36,1103-14.

Williams, D. R., Brainard, D. H., McMahon, M. J. and

Navarro, R. (1994). Double pass and interferometricmeasures of the optical quality of the eye. J. Opt. Soc.AII/. A., 11, 3123-35.

Woods, R. L., Bradley, A and Atchison, D. A. (1996).Consequences of the monocular diplopia for thecontrast sensitivity function. Vision Res.,36, 3587-96.

Woods, R. L.,Strang, N. C. and Atchison, D. A (in press).Measuring contrast sensitivity with inappropriateoptical correction. Opllth. PllysioJ. Opt.

Ye, M., Bradley, A, Thibos, L. N. and Zhang, X. (1992).The effect of pupil size on chromostereopsis andchromatic diplopia: interaction between theStiles-Crawford effect and chromatic aberrations.Vision Res., 32,2121-8.

19Depth-of-field

Introduction

In any optical system, the ultimate precisionin focusing is set by the ability to detect errorsin focus. The range of distances over whichthe system's detector cannot detect anychange in focus is called the depth-of-field,and this range may be specified by amovement of the object plane or by the corre-sponding movement of the image plane.Because these two distances are usuallydifferent, some textbooks differentiatebetween depth-of-field, a movement of theobject plane, and depth-of-focus, a movementof the image plane. In vision science, depth-of-field is usually expressed as a change invergence, which has the same value in bothobject and image space. Distinctions may stillhave to be made between object and imagespace quantities in some circumstances, suchas when using visual optical instruments.However, in this chapter there is no need todistinguish between object and imagesituations.The definition adopted here for depth-of-

field is the vergence range of focusing errort1L, which does not result in objectionabledeterioration in retinal image quality. This issometimes referred to as the total depth-of-field. This can be determined according to anumber of criteria (Experimental results, thischapter). Some studies mentioned in thischapter used half the total depth-of-field andexpressed values as ±.:1L (e.g. Campbell, 1957).When referring to such studies in this chapter,

their numbers have been doubled. This issatisfactory where depth-of-field is measuredin just one direction from a focus position, orwhere simple theory is used in which thedepth-of-field is symmetrical about theposition of focus.The depth-of-field sets the precision to

which refractive state, including the ampli-tude of accommodation, can be measured bysubjective methods. It also determines thedistance range for which a target can be seenclearly when using visual optical instruments,such as simple magnifiers, microscopes andtelescopes. For example, the depth-of-field ofa simple magnifier or microscope of magnifi-cation M can be given as a total distance &min object space. This distance is related to thedepth-of-field of the eye t1L by the approxi-mate equation (Smith and Atchison, 1997)t1lm =t1L/(l6M2) (19.1a)As another example, the depth-of-field of atwo-lens afocal telescope of magnification Mcan be given as a total vergence in object spacet1Lt' and this is related to the depth-of-field ofthe eye & by

t1L I = t1L/M2 (19.1b)Increasing the depth-of-field is advantageousin some circumstances. For example, the age-related reduction in amplitude of accommo-dation can be ameliorated by increasingdepth-of-field. One way to do this is to intro-duce additional aberrations. This approachhas been used with contact lenses for

214 Miscd/all('(1I1S

Criterion 1: the range offocusingerrors for which no perceptible blur ofa target is noticeable

This criterion is relevant to subjectiverefraction and determining the amplitude ofaccommodation. The vergence of the target isvaried, and the extremes at which the targetfirst appears to be blurred are measured. The

may depend upon different optical, neuraland psychological factors, it is not alwaysmeaningful to try to compare depth-of-fieldresults based on different criteria. The com-mon feature of experimental results accordingto most criteria is that depth-of-field decreasesas pupil size increases, at least out to 5-6 mmpupil diameters.

87

--+- Campbell (1957)

_ Ogle and Schwanz (1959)

....Q.... Charman andWhite!!}(,l (1977)

....0.... Atchison<,Iat. (1997)

~ * 5 6Pupil diameter (mm)

:1.5

s.o

2.5

c~

~,O

~

J:r.s

15."c I.ll

o.s

u.oII

Figure 19.1. Depth-of-field as a function of pupil size.Details of studies are as follows:Campbell (1957): threshold blur, retinal illuminanceconstant (corresponding to 318 cd/rn? at 1 mm pupildiameter), one subject.

Ogle and Schwartz (1959): 50 per cent probability limitsof correctly resolving checkerboard with equivalentletter size 6/7.5, subject jTS.

Charman and Whitcfoot (1977): limits of depth-of-fieldgive 95 per cent correct identification of the directionof movement of laser speckle, mean of six subjects,error bars indicate ±1 standard deviation of subjects.

Atchison 1.'1 al. (1997): threshold blur, 6/7.5 letter E, meanof five subjects, error bars indicate ±1 standarddeviation of subjects.

There are several criteria for measuringdepth-of-field according to a focusing errorrange that do not cause an 'objectionable dete-rioration in retinal image quality'. Six of theseare considered here. Because these criteria

Experimental results

presbyopes but, unfortunately, this reducespeak visual performance (Bradley et al., 1993;Plakitsi and Charman, 1995).Depth-of-field depends upon several

factors, including the following:1. Optical properties of the eye

pupil diameter (interacts with the otheroptical properties)accommodation levelmonochromatic and chromaticaberrationsdiffraction.

2. Retinal and visual processing propertiesphotoreceptor size and ganglion celldensityvisual acuity and contrast thresholdsdisease in ocular pathway.

3. Target propertiesluminancespatial detailcontrastspectral profile, e.g. colour.

Depth-of-field in the eye can be explainedat a simple level using the defocus blur discmodel of defocused systems (Chapter 9) andthe size of the detector elements in the imageplane. In an aberration and diffraction-freesystem, the image of a defocused point is adefocus blur disc. If this disc is smaller than adetector element, the system will not be ableto detect defocus. Defocus will be detectableonly once the defocus blur disc overlaps atleast two detectors. Because this modelneglects aberrations, diffraction and how thevisual system processes retinal images (e.g.interactions between adjacent receptors), it isa crude model and cannot be expected topredict accurately the depth-of-field of theeye.In the following sections we look at

experimentally determined values, andconsider models, such as the above defocusblur disc model, that can be used to predictdepth-of-field.

Del'fll·(lf-field 215

1.5-0.5 0 O.SDefocus (D)

·1·1.5

·0.2

o.x

... 0.6~

'"c:es0.4l:

c:.2

""3 0.2'tl0~

O+--+---+--.J--I------\-~~::-=.,..+

This criterion has two aspects, which can beexplained with reference to Figure 19.3. Thisfigure shows theoretical 'through-focus'modulation transfer as a function of twolevels of spherical aberration for spatialfrequencies of 5 and 10 c/deg. Consideringjust the 10 c/deg plots and adopting anabsolute level of depth-of-field as that forwhich modulation transfer values are greaterthan 0.5, the no-aberration condition has adepth-of-field of 0.4 D. However, the aber-ration condition has a depth-of-focus of 0 Dbecause its modulation transfer is always lessthan 0.5. If the modulation transfer require-ment is lowered to 0.3, the aberration condi-tion has greater depth-of-field (0.7D) thandoes the no-aberration condition (0.5D).

If we adopt either a relative loss in modu-lation transfer or a particular proportionalloss in modulation transfer, relative to the

Criterion 2: the range of focusingerrors for which the visual acuity orcontrast sensitivity does not decreasebelow a particular level or by morethan a certain amount

Figure 19.3.Theoretical modulation transfer as afunction of defocus at 5 and 10 c/deg when there is ODor +1.0 D of primary spherical aberration at the edge of a6 mm diameter pupil in 605 nm wavelength light.Results are shown without Stiles-Crawford apodization.Data of Figures 5 and 6 of Atchison et al. (1998), withpermission from The Optical Society of America.

·0.2 0 0.2 0.4 0.6 O.llTarget detail size (log min arc)

0.0 +-....,....--,--r----r--.-,-....-,....-..---r-...,........,...--r_t_·0.4

................. Jacobs rt al. (19119)4.2 mmpupildiameter

1.0 ---..- Atchison et at, (199714 mmpupildiameter

~ O.He'tl

~ 0.6..:.oJ:.Q..~ 0.4

0.2

Figure 19.2. Depth-of-field as a function of target size.Jacobs et al. (1989)measured the depth-of-field in onlyone direction from the optimum focus, and their resultshave been doubled so that they are comparable withthose of Atchison et al. (1997).

details of presentation can vary. The target canbe moved backwards and forwards to locatethe range within which it appears to be infocus (Campbell, 1957; Atchison et al., 1997).Two targets may be presented, either insuccession or simultaneously side-by-side,one in focus and the other with various levelsof defocus, and the subject is asked to decidewhich is not in focus (Jacobs et al., 1989).Studies using this criterion show that the

depth-of-field decreases with increase in pupildiameter (Campbell, 1957; Atchison et al.,1997) (Figure 19.1), increasing targetluminance and correction of longitudinalchromatic aberration of the eye (Campbell,1957). As an example of the dependence onpupil size, in Campbell's study the depth-of-field decreases from 1.7 D to 0.3 D betweenthe pupil diameters of 1 mm and 7 mm(Figure 19.1). Depth-of-field is smallest fortarget sizes near the visual acuity limit, andincreases slowly with increase in target size(Jacobs et al., 1989; Atchison et al., 1997)(Figure 19.2). Jacobs and colleagues (1989)measured the threshold of just detectablechange in defocus of an already defocusedtarget and found that the threshold wasslightly less than the depth-of-field.

216 Miscellalleous

peak value, the no-aberration conditionalways has smaller depth-of-field than theaberration condition (considering only thedefocus region within which the modulationtransfers first fall to zero). This is because therate of loss with modulation transfer awayfrom the peak value is always slower for theformer than for the latter condition. Thisaspect of criterion 2 has been used in somestudies comparing the depths-of-field fordifferent bifocal contact lenses, e.g. Plakitsiand Charman (1995).A couple of studies are now mentioned

which use the first aspect of criterion 2. Ogleand Schwartz (1959) determined the defocusproviding 50 per cent and 99 per cent proba-bility levels for correct recognition of checker-board patterns of various sizes. As expected,depth-of-field was larger for the 50 per centlevel. The depth-of-field decreased withincrease in pupil size (Figure 19.1) and withdecrease in target size. Tucker and Charman(1975) found similar results with lettertargets.Tucker and Charman (1986) estimated

depth-or-field using the visibility of sinu-soidal modulated (80 per cent) luminancetargets as the criterion. As spatial frequencyincreased, the depth-of-field decreased. Forexample, at a 10 cd/m2 luminance level, thedepth-of-field was zero for 30 c/deg, 1 0 at 20c/deg, and 5 0 for 10 c/deg. Increasing targetluminance from 0.001 to 10 cd/m2 increaseddepth-of-Held. Tucker and Charman foundlittle difference in depth-of-field between 3mm and 7 mm pupil diameters.

Criterion 3: the range offocusingerrors for which changes in contrastare not detected for a target inlongitudinal sinusoidal motion

The subject views a target through a Badaloptical system. The target is usually periodic -i.e. the luminance profile is generally that of asine wave or square wave. As measured as animage vergence at the eye, the target is movedforwards and backwards in sinusoidalmotion. The peak-to-peak amplitude of themovement is varied until the subject candetect apparent variation in the target'scontrast (for non-periodic targets some other

criterion can be used, such as the appearanceof blur or changes in target shape).Similar to the results of Jacobs et al. (1989)

with criterion 1/ the depth-of-field is aminimum when the centre of the range isslightly off-set to one side of the optimal focus(Campbell and Westheimer, 1958; Walsh andCharman, 1988). Typically, the depth-of-fieldat optimal focus is 0.6 0/ while the minimumdepth-of-field is approximately 0.20 (Walshand Charman, 1988). The effect of increasingpupil size is to decrease the differencebetween the optimum focus and the centre ofthe range at which the minimum depth-of-field occurs. Walsh and Charman investigateda range of variables, including target colour,luminance and temporal frequency.

Criterion 4: the range offocusingerrors for which a laser specklepattern appears to be stationaryThe laser speckle method for measuringrefractive errors was described in Chapter 8.Briefly, a subject views a speckle patternproduced by a laser beam reflected diffuselyfrom a moving surface such as a rotatingdrum. The speckle pattern appears to move atincreasing velocity as the refractive errorrelative to the surface increases. This tech-nique can be adapted to depth-of-fieldmeasurement by determining the focus rangeover which the subject cannot reliablydistinguish the pattern's direction of move-ment.Results with this technique show that pupil

size has a strong effect on depth-of-field(Ronchi and Fontana, 1975; Charman andWhitefoot, 1977). Charman and Whitefoot(1977) found that depth-of-field decreasedwith increase in pupil size up to about 5 mmdiameter; their values are slightly smallerthan Campbell (1957) using criterion 1 fornearly all pupil sizes (Figure 19.1).

Criterion 5: the range offocusingerrors for which the accommodationresponse does not change

Accommodation response can occur for targetmovements as small as 0.10 (Ludlum et al.,

Depth-of-fiet« 217

that the system is aberration- and diffraction-free and, therefore, a point is imaged as apoint. When this point is defocused, its imageis a uniformly illuminated blur disc as alreadyreferred to in the previous section.In Chapter 9, equation (9.17), the angular

diameter tPof the defocus blur disc was givenas

where 0 is pupil diameter and .1L is refractiveerror. Using (total) depth-of-field to replacethe refractive error, this equation becomestP= 0.1L/2 (19.2a)Wewill now assume that the depth-of-field isset by the range over which this defocus blurdisc is smaller than a certain threshold dia-meter. If this diameter is tPth' the depth-of-field is.1L =2tPth / 0 (19.3)This equation predicts that the depth-of-fieldis inversely proportional to the pupil dia-meter.The smallest meaningful estimate of tPth is

obtained using the diameter of a foveal cone,which is approximately 0.003mm (Polyak,1941). At the back nodal point, a distance of0.003mm subtends an angular diameter of-= 0.003/17 -= 0.000176 rad (-= 0.61 min. arc)where 17mm is the approximate distancefrom the back nodal point to the retina.Substituting this value for tP\ll in equation(19.3) gives the threshold of defocus as.1L =0.000352/0 (19.3a)If we use a pupil diameter of 3 mID, i.e.D =0.003m, this equation gives a depth-of-field of 0.1160, which is much less than theexperimental values given in the precedingsection. Therefore, there are serious flaws inthis model.

1968), even if this produces no perceptibleblur (Kotulak and Schor, 1986).

Criterion 6: the range offocusingerrors which degrades retinal imagequality below a particular level or bymore than a certain amount

The retinal image quality referred to here maybe measured by the point spread function,line spread function or modulation transferfunction, or by their derivations, such as half-width of the point spread function (seeChapter 18).The modulation transfer functionis closely related to the contrast sensitivityfunction mentioned with criterion 2, and weused the modulation transfer function toexplain two aspects of criterion 2.Using the double-pass point spread

function technique, Artal et al. (1995) deter-mined through-focus modulation transfer at2.6cldeg in patients with monofocal andmultifocal intraocular lenses. Marcos et al.(1999) used the criterion of a quantity, similarto the peak of the double-pass intensity pointspread function, decreasing to 80 per cent ofits value at optimum focus. They found thatrather than decreasing, the average depth-of-field for three subjects increased slightly(although non-significantly) for increase inpupil diameter from 4 mm to 6 mm.

Modelling depth-of-field

Various models will be discussed here thatcan be used in understanding the factors thataffect depth-of-field, by examining the opticsof defocused images using two image qualitycriteria at various levels of complexity.

tP= O.1L (19.2)

Criterion 1: The range offocusingerrors for which no perceptible blur ofa target is noticeable

The threshold of blur can be modelled byexamining the image of a defocused pointsource of light and making assumptions aboutthe threshold of defocus. We will begin byconsidering the simplest of models, assuming

Effect of diffraction and aberrationsBecause of diffraction and aberrations, thefocused image of a point is not a point but apatch of light - the point spread function asdiscussed in Chapter 18.The size of this patchdepends upon the pupil diameter, since bothdiffraction and aberrations depend upon thesize of the pupil.

218 Miscelloll"ollS

In the presence of diffraction and aber-rations, the point spread function is muchmore complex than a uniform disc (seeChapter 18), and this complexity presentsproblems in measuring its diameter. In suchcases, the diameter or width is often repre-sented by its half-width, which is the width atwhich the light level drops to half the centralor maximum value. In-focus point spreadfunctions have half-widths of about 2-3minutes of arc.We will now adopt the defocus, which

would produce a blur disc equal to the halfwidth of the in-focus point spread function,for our blur detection model. Any defocuswith a blur disc less than this size can beexpected to have little chance of beingdetected. Taking the threshold of blur disc size4>th as 2 min. arc (i.e. 0.000582rad), we put4>th= 0.000582 in equation (19.3) to obtain.& =0.00116/D (19.3b)For a pupil diameter of 3 mm this equationgives a depth-of-field value of 0.380, which ismuch closer to experimental values (Figure19.1).

Influence of diffraction alone at small pupildiametersAt small pupil diameters, e.g. 2 mrn, thethreshold value 4>th is governed by diffractionand not by aberrations. Diffraction theorypredicts that the width of the point spreadfunction is proportional to wavelength andinversely proportional to pupil diameter(equation (18.5». Thus, on assuming that 4>thfollows the same trend, equation (19.3)becomes

(19.4)This equation predicts that, for small pupilsizes, the depth-of-field is inversely pro-portional to the square of pupil diameter.

Influence of aberrations alone at large pupildiametersAs pupil diameter increases, the effects ofdiffraction become less and the influence ofaberrations increases. For larger pupildiameters, the point spread function can beregarded as being affected by aberrationsalone.

As aberrations usually increase with pupildiameter, the size of the focused point spreadfunction is expected to increase with pupildiameter, and the threshold diameter 4>th ofthe defocus blur disc is expected to increase.In the presence of primary sphericalaberration, the transverse aberration at theedge of the pupil is proportional to the cube ofthe pupil diameter. Therefore, we couldexpect that 4>th would show some higherorder dependence on pupil diameter. Forlarge pupil sizes, from equation (19.3), it isexpected that depth-or-field increases withincrease in pupil diameter. However, from theresults of Figure 19.1, this does not happenout to at least 5-6 mm in diameter. This maybe because the aberrations of the eye are toolow to have a large effect, or because theaberrations are too irregular. The Stiles-Crawford effect (Chapter 13) may playa smallrole here by reducing the effect of aberrationsand defocus at larger pupil sizes.

More complex objectsWe have so far examined the effect of defocusonly on the image of a point object. This is notrealistic, since few scenes are composed ofpoint sources. On the other hand, edges arevery common, and we will consider a sharpluminous edge of high contrast. Using thedefocus blur disc model, a luminance profileslope forms at the edge, and the width of thisedge is equal to the width of the corre-sponding defocus blur disc. The threshold fordefocus will depend in some complex manneron the width of the image edge and on theimage's contrast.Now suppose that the edge is the edge of a

bar of a finite width (e.g. in a letter). Thedefocus now has two phases. In the first phase(low levels of defocus), only the edge of theimage is affected. The second phase beginsonce the defocus reaches the level when thetwo sloping edges (on either side of the bar) ofthe image meet in the centre. Once this occurs,the contrast of the bar is reduced. Thenarrower the bar, the sooner this second phaseoccurs. Depending upon how the visualsystem detects defocus, it may be moresensitive to this reduction in image contrastthan on the profile of the image's edge.These considerations suggest that depth-of-

field depends on target size. Atchison et al.

Depth-of-field 219

CRITERIONIn focus Blur limit

Change inimage form

Change inimage contrast

I:~~~ I:t I I1W 111111I

_ A A' A A'Distance vertically along IcttcrE_In focus Blur limit

r~~~~:; -, n n rr vUU 11I111

A A' A A'Distance vertically along letter E_

Change inedgegradient

IAIn focus Blur limit

.----1[l Ci-A A

Distance vertically along letter E _

Figure 19.4.A model of a person's determination of what is perceptible blur of a letter. For a small letter (top), the lowspatial frequency information in the letter is used; for an intermediate-sized letter (middle), the contrast between lightand dark bars in the image is used; for a large letter (bottom), the luminance slope at the image's edges is used.

(1997) developed a model to describe how thevisual system might change its criterion ofwhat is perceptible blur, according to the sizeof letter targets (Figure 19.4). According to thismodel, for very small letter sizes (e.g. < 0.0 logmin. arc of target detail), the perception ofblur is based on low spatial frequencyinformation in the letters. This low spatialfrequency information affects mainly theoverall contrast of the retinal image of theletter against its background. As letter sizeincreases, the fundamental spatial frequencyof the pattern moves into the spatial fre-quency range at which modulation transfer isvery sensitive to defocus, and thus thecontrast between dark and light bars of animage is important. For even larger letter sizes(e.g. > 1.0 log min. arc of target detail), thespatial frequency of the fundamental is so lowthat its modulation transfer is relativelyunaffected by defocus. However, higher orderharmonics in the letter will be at spatialfrequencies whose modulation transfer isaffected by defocus. These are important to

the detection of edges, so the perception ofblur may be now based on edge sharpnessrather than bar contrast.

Criterion 6: the range offocusingerrors which degrades retinal imagequality below a particular level or bymore than a certain amountThe point spread function, line spreadfunction and the modulation transfer functionwere discussed in Chapter 18. Using themodulation transfer function image qualitycriterion, we can investigate the effect of smalllevels of defocus on the contrast of sinusoidalpatterns as a function of spatial frequency.This is a much more advanced approach thanthe simple defocus blur disc model, since itcan readily take into account aberrations(including chromatic), diffraction and theStiles-Crawford effect. Furthermore, with aknowledge of the retinal contrast sensitivity

220 Miscetlalleolls

function we can readily calculate the thres-hold of visibility of sinusoidal patterns(criterion 2) as a function of defocus andspatial frequency, and the results can becompared with experimental measures (see,for example, Tucker and Charman, 1986).As an example, we consider the depth-of-

field for sinusoidal patterns in Figure 19.3,where theoretical through-focus modulationtransfer is shown for spatial frequencies of 5and 10 c/deg (this example was introduced inthe previous section). Here, there is either noaberration or 1 0 primary spherical aberrationat the edge of a 6mm diameter pupil in 605nm wavelength light. If we consider thedepth-of-focus as the defocus range for whichthe modulation transfer is greater than acertain proportion of the peak modulationtransfer for a set of conditions, the figureshows the general finding that aberrationsincrease depth-of-field. Using half the peakvalue as this proportion, depths-of-field at 10c/deg are 0.4 0 for the no-aberration condi-tion and 0.8 0 for the aberration condition,respectively. A similar pattern occurs for 5c/deg to that for 10 c/deg. The Stiles-Crawford effect increases these ranges slightlyfor most spatial frequencies (Figure 19.4). Wenote, however, that using experimentally-determined wave aberrations and a criterionbased on the point spread function, Marcos etal. (1999) calculated that the Stiles-Crawfordfunction decreased depth-of-field at 6 mm by3 per cent and 7 per cent in two subjects.

Summary of main symbols

D pupil diameterL1L (Total) depth-of-fieldfP angular diameter of defocus blur discfPth detection threshold value of fP

ReferencesArtal, P., Marcos. S; Navarro, R. 1'1 al. (1995). Through-focus image quality of eyes implanted with monofocaland multifocal intraocular lenses. Opl. Eng., 34, 772-8.

Atchison, D. A., Charman, W. N. and Woods, R. L. (1997).Subjective depth-of-focus of the eye. 01'10111. Vis. Sci., 74,511-20.

Atchison, D. A., Smith, G. and [oblin, A (1998). Influenceof Stiles-Crawford apodization on spatial visualperformance. J. Opl. Soc. Alii. A, 15,2545-51.

Bradley, A, Rahman, H. A, Soni, P. S. and Zhang, X.(1993). Effects of target distance and pupil size on lettercontrast sensitivity with simultaneous vision bifocalcontact lenses. Oplolll. Vis. Sci., 70, 476-81.

Campbell, F. W. (1957). The depth of field of the humaneye. Optica Aria, 4, 157-64.

Campbell, F. W. and Westheimer, G. (1958). Sensitivity ofthe eye to differences in focus. J. Physiol., 143, 18.

Charman, W. N. and Whitefoot, H. (1977). Pupil diameterand the depth-of-field of the human eye as measuredby laser speckle. Optica Acla, 24, 1211-16.

Jacobs, R. J., Smith, G. and Chan, C. D. C. (1989). Effect ofdefocus on blur thresholds and on thresholds ofperceived change in blur: comparison of source andobserver methods. 0l'lom. Vis. Sci., 66, 545-53.

Kotulak, J.C. and Schor, C. M. (1986). The accommodativeresponse to subthreshold blur and to perceptual fadingduring the Troxler phenomenon. Perception, IS, 7-15.

Ludlum, W. M., Wittenberg,S., Giglio, E. J. andRosenberg, R. (1968). Accommodative responses tosmall changes in dioptric stimulus. Am. J. OpIOIll. Arch.Alii. Acad. 0l'lom., 45, 483-506.

Marcos S., Moreno, E. and Navarro, R. (1999). The depth-of-field of the human eye with polychromatic lightfrom objective and subjective measurements. VisiollRes.,39, 2039-49.

Ogle, K. N. and Schwartz, T. J. (1959). Depth of focus ofthe human eye. J. °1'1.Soc.Alii., 49, 273-80.

Plakitsi, A and Charman, W. N. (1995). Comparison ofthe depths of focus with the naked eye and with threetypes of presbyopic contact lens correction. J. Br. COlli.Lells Assor., 18, 119-25.

Polyak, S. L. (1941). The Retina. University of ChicagoPress.

Ronchi, L. and Fontana, A. (1975). Laser speckles and thedepth-of-Held of the human eye. Optica Acta, 22, 243-6.

Smith, G. and Atchison, D. A (1997). The Eye and VisllalOptical Illstrlllllellls, pp. 703-4. Cambridge UniversityPress.

Tucker, J. and Charman, W. N. (1975). The depth-of-focusof the human eye for Snellen letters. Alii. f. OptOIll.Pllysiol. 01'1., 52, 3-21.

Tucker, J. and Chatman, W. N. (1986). Depth of focus andaccommodation for sinusoidal gratings as a function ofluminance. Am. f. Oplolll. Physiol. Opt., 63, 58-70.

Walsh, G. and Charman, W. N. (1988). Visual sensitivity totemporal change in focus and its relevance to theaccommodation response. vision Res., 28, 1207-21.

20

The aging eye

Introduction

Age-related optical changes in the eye werementioned briefly in earlier chapters, but herewe will give a fuller account, with emphasison changes occurring in the adult eye. Manyof these changes produce progressive reduc-tion in visual performance. A number ofchanges in the optical properties may beregarded as pathological, e.g. cataract, butthese will not be considered here. Neuralproperties also change with age, but these arebeyond the scope of this book. For thoseparticularly interested in the subject of theaging eye, we recommend two books byWeale (1982 and 1992).

Cornea

With increasing age, there is decreasedspacing between collagen fibrils of the stroma,some fibre degeneration, and increases in thecross-sectional area of collagen fibres (Kanaiand Kaufman, 1973; Malik et al., 1992).Descemet's membrane increases in thicknesswith age (Cogan and Kuwabara, 1971).Possibly the most important age-relatedchange is endothelial degeneration. The sizeof endothelial cells becomes more variable(polymegathism), because some cells eitherincrease in size or fuse (Daus and Volcker,1987). Eventually endothelial function may beimpaired, and then aqueous humour may

seep into the cornea, disrupting the structuralorder and increasing light scatter (Pierscionek,1996).

Corneal thickness

There is no clear trend of thickness changeswith increasing age. Findings include nochange (Kruse Hansen, 1971; Olsen, 1982; Siuand Herse, 1993), increase (Koretz et al., 1989),and slow decrease (Alsbirk, 1978; Olsen andEhlers, 1984).

Corneal shape

In young eyes the curvature of the anteriorsurface is usually greater in the verticalmeridian than that in the horizontal meridian,but this tends to reverse with increase in age(Phillips, 1952; Lyle, 1971; Anstice, 1971;Baldwin and Mills, 1981; Kame et al., 1993;Goh and Lam, 1994; Lam et al., 1994). Kiely etal. (1984) and Hayashi et al. (1995) found thatthe cornea becomes more curved with age,but more so in the horizontal meridian than inthe vertical meridian.The aberrations due to the cornea seem to

increase with age, and this is due to the coma-like aberration contribution rather thanspherical aberration-like contribution (Oshikaet al., 1999).

222 Miscdlalll'''IIS

TransmittanceMost studies of corneal transmittance havefound that there is no significant variationwith age (Boettner and Wolter, 1962; Beernsand van Best, 1990; van den Berg and Tan,1994). However, Boettner and Wolter found adecrease in the direct (non-scattered) lightwith increased age.

Lens

The most dramatic age-related opticalchanges in the eye occur in the lens. Its shape,size and mass alter markedly, its ability tovary its shape (i.e, accommodation) dimin-ishes, and its light transmission reducesconsiderably, particularly at short wave-lengths.

Lenticular transmittance, scatter andfluorescence

The transmittances of both ultraviolet andvisible wavelengths decrease with increase inage, with the lens becoming more yellow,particularly in the nucleus (Said and Weale,1959;Boettner and Wolter, 1962;Mellerio, 1971and 1987; Pokorny et al., 1987; Sample et al.,1988). There is an increase in both forwardand backward scattered light with age, partic-ularly after the age of 40 years (Bena-Sira etal.,1980; Ilspeert et aI., 1990; Fujisawa and Sasaki,1995). There is increased fluorescence withincrease in age (Hockwin et al., 1984).

Thickness

The lens increases in volume and massthroughout life, with most of this being due toincrease in axial thickness of the cortex(Brown, 1973a; Niesel, 1982;Cook et al., 1994).Koretz et al. (989) found an increase in axialthickness of 13llm per year. The anteriorchamber depth decreases throughout lifeapproximately at the same rate as the lensaxial thickness increases (Koretz et al., 1989).Brown 0973a) and Weekers et al. (973), butnot Koretz ct al. (1989), found a small decreasein vitreous chamber depth with increase inage.

ShapeBrown (1974) found a linear reduction in thecentral anterior lens radius of curvature ofemmetropic, unaccomrnodated eyes withincrease in age. This changed from 16.0 mm at8 years to 8.3 mm at 82 years of age. Changesin radius of the posterior surface with increasein age were less marked, decreasing fromapproximately 8.6 rnm at 8 years to 7.5 mm at82 years. Cook et al. (1991) obtained similarresults.As age increases, the maximum possible

changes in lens shape decline (Koretz et al.,1997), reflecting the decrease in amplitude ofaccommodation with age.Magnetic resonance imaging shows no

change in unaccommodated lens diameter,but increase in accommodated lens diameter,with age (Strenk et aI., 1999).

Refractive index distribution

Models for the refractive index distribution ofthe lens were discussed in Chapters 2 and 16.Based on the increasing thickness and surfacecurvatures of the lens with age, it would beexpected that eyes should become moremyopic. However, the general shift is in theother direction, with Saunders (1986a) findinga mean hypermetropic shift of 20 betweenthe ages of 30 and 60 years (Figure 20.1). Thisis the 'lens paradox', and it is not known whythis occurs. Grosvenor (1987) proposed thatthere is a decline in axial length during adultlife, but another possibility is a change in therefractive index distribution of the lens. Thiscould be by a reduced magnitude of refractiveindex variation across the lens as proposed byKoretz and Handelman (1988), or by a changein the refractive index pattern with increasingage (Pierscionek, 1990). The latter was mod-elled by Smith et al. (992), with the gradientindex profile becoming flatter near the middleof the lens and then increasing in steepnesstowards the edge (see Schematic eyes, thischapter), and there is some clinical support forthis (Hemenger et al., 1995).The zones in the lens seen with the slit-

lamp, which become more obvious andgreater in number with increasing age, are ofconsiderable interest. Although severalreasons for their appearance have been

TIlt.' agillg t.'yt.' 223

Fledelius, 1984). Most measurable astigma-tism is with-the-rule up to 40 years of age,after which the prevalence of against-the-ruleastigmatism increases (Coss, 1998).Grosvenor (1987) re-analyzed Sorsby and

co-workers' (1957, 1962) data, and found thatthe mean axial length of emmetropic eyesdecreased after the age of approximately 20years, with a difference of 0.6 mm between a20-29 years age group and a 50+ age group.He cited other studies with similar results.However, Koretz et al. (1989) did not findsignificant change in the axial length ofemmetropes with age. Also, Ooi andGrosvenor (1995) did not find a significantdifference between the axial lengths of youngand old adult groups, in which the twogroups were matched on the basis of sex andrefractive error.

...........

Longitudinal (Saunders. 19M6)

Cross-sectional (Saunders. 19M I )

10 20 30 40 50 60 70 MO

Age (years)

o

2

...g 1......>u..<l: 0~cosQ)

~ -I

c

Figure 20.1. Change in mean refractive error with age.Cross-sectional data of Saunders (1981)and longitudinaldata of Saunders (l986a). 'Cross-sectional' means thatdata are collected from each subject once, while'longitudinal' means that each subject is followed over aperiod of time.

suggested, including refractive indexdiscontinuities, the nature and origin of theseis not yet known (Pierscionek and Weale,1995).

Refractive errors and axial length

Refractive errors are relatively stable betweenthe ages of 20 and 40 years (Grosvenor, 1991),after which there is a shift in the hyper-metropic direction (Saunders, 1986a) (Figure20.1). After the approximate age of 70 years,there is some shift of the mean refractive errorin the myopic direction associated with thedevelopment of nuclear cataract (Brown andHill, 1987). The distribution of refractiveerrors of a young adult population is steeperthan a normal distribution (leptokurtosis),The distribution of refractive errors becomesmore normal with increasing age, withincreases in the proportions of both myopesand hypermetropes (Grosvenor, 1991).Age-related changes in the corneal shape,

discussed earlier in this chapter, are reflectedin the astigmatism of the eye (e.g. Hirsch,1959;Anstice, 1971;Saunders, 1981 and 1986b;

Accommodation and presbyopia

Accommodation is the ability of the eye tochange its power to bring objects of interest atdifferent distances into focus. As such, itmakes an essential contribution to visualperformance. Our understanding of its mech-anism is based on Helmholtz's (1909) theory.In the unaccommodated form, with the focusof the eye at its far point, the zonulesconnecting the lens and ciliary body pull onthe lens and flatten it. When changing focusfrom far to near vision, the ciliary musclecontracts, thus reducing the tension on thezonules. Because of the elastic properties ofthe capsule of the lens, the lens takes on amore rounded shape. Thus, both the lens andthe eye increase in refractive power.When the eye accommodates, the anterior

surface moves forward and takes on a morehyperboloid form (Brown, 1973b). Axialthickness changes are confined to the nucleus(Brown, 1973b; Koretz et al., 1997). Theequatorial diameter of the overall lens and thenucleus are reduced (Brown, 1973b). Most ofthe curvature change occurs at the frontsurface, with Brown (1973b) obtainingchanges in anterior and posterior radii ofcurvature of 12.8mm to 7.7 mm and 7.1 mmto 6.5mm, respectively, in the eye of a 19-year-old subject upon accommodation of 6 D. Asage increases, the maximum possible changein lens movement declines (Koretz et al., 1997).

224 Miscelltll/('oIlS

Uncertainty still exists about the exactinteraction between the zonules and theciliary body (Atchison, 1995).

PresbyopiaThe range or amplitude of accommodationreaches a peak early in life, then graduallydeclines. The decline in accommodationbecomes a problem for most people in theirforties, when they can no longer see clearly toperform near tasks. This condition is calledpresbyopia. The age at which this occurs forany particular person depends on the refrac-tive error, nature of close work and, probably,on genetic and environmental factors.Accommodation is completely lost in thefifties, well before most other physiologicalfunctions are appreciably affected. However,as discussed below, most clinically relatedstudies do not show this complete loss.There have been many studies of the

reduction in the amplitude of accommodationwith age. Figure 20.2 shows the data ofUngerer (1986). The authors chose theseresults because they are for a large number ofpeople (1285) who were not part of a clinicalpopulation. They were predominantly civilservants, mainly Anglo-Celtic, in the coastalAustralian city of Melbourne (latitude 38°5).

Ungerer fitted an exponential function for theexpected mean between the ages of 20 and 60years ofamplitude (dioptres) = exp[1.93 + 0.0401 age

- 0.00119 (age)2](20.1)

This equation predicts mean amplitudes ofaccommodation of 9.5 0 at 20 years, 3.70 at45 years, 2.6 0 at 50 years and 1.00 at 60years. Using a 3.750 amplitude of accom-modation limit as the criterion for presbyopia,she found that only 2 per cent of those over 50years were not presbyopic.Studies such as Ungerer's overestimate the

amplitude of accommodation due to depth-of-field effects. Figure 20.3 shows Hamasakiand co-workers' (1956) results for subjectsbetween the ages of 42 and 60 years, using astigmatoscopic method and the conventionalclinical push-up method. Linear fits areshown also for Sun and co-workers' (1988)results for subjects between the ages of 13and46 years using the stigmatoscopic method andthe push-up method. The stigmatoscopicmethod, designed to eliminate depth-of-fieldeffects, reduced the estimate of amplitude byapproximately 2 0 across the range of ages.Studies such as those carried out by Hamasakiet at. and Sun et al. also indicate that accom-

605030 40Age (years)

20

~ Hamasakiet al, (1956). push up

__ Hamasakiet al, (1956). stigrnatoscopy

--0-- Sun et al, (1988,. push up

_____ Sun et al. (1988). stigmaroscopy

Figure 20.3. Amplitude of accommodation with age.Results of Hamasaki et al. (1956) and Sun and co-workers' (1988)data fit.

6050

95 fh prediction limit

--0-- 5 fh'predictionlimit

Mcan

30 40Age (years)

20o-+----r--.-----.--r----.--r-----,..---.---~_+

10

Figure 20.2. Amplitude of accommodation with age.Results of Ungerer (1986).

modation does not occur beyond the early50s.Cross-sectional studies mask the age-

related decline in amplitude of individualsubjects. They show a non-linear trend in themean amplitude with age, but with the rateof decline decreasing as presbyopia isapproached (Figure 20.2). Two small-scalelongitudinal studies (Hofstetter, 1965;Ramsdale and Charman, 1989) found thatindividual subjects have a linear decrease ofaccommodation with age. The non-lineartrend in cross-sectional studies is probablydue to artefacts introduced by the averagingprocess (Charman, 1989). At any age beyondthat at which some individuals no longeraccommodate, the distribution of amplitudesis truncated, because these individuals cannotcontribute negative values to it.The rate of amplitude decline varies

considerably, and may be affected by severalfactors. The rate of progression of presbyopiais faster the closer people live to the equator,and faster for people living at low altitudesthan at higher altitudes. These findingsindicate that ambient temperature may affectthe progression of presbyopia (Miranda, 1979;Weale, 1981).

Presbyopia theories

Current theories of the development ofpresbyopia can be categorized as follows(Atchison, 1995):1. Lenticular theoriesa. Mechanical changes in lens and capsulei. Hess-Gullstrand theoryii. Fincham theory

b. Geometric theory.2. Extra-lenticular theoriesa. Changes in ciliary muscle - Duane theoryb. Changes in elastic components of zonuleand/or ciliary body.

The development of presbyopia is generallyregarded as originating in the 'plant' of theaccommodative system, either within the lensand its capsule or within their support struc-tures. Because the optical parameters of theeye are involved in several of these theories,they are discussed briefly below. For a moreextensive coverage and full reference details,see Atchison (1995).

Tile agingeye 225

Lenticular theoriesMechanical changes in lens and capsuleThe lens is purported to become more rigidwith age and to be increasingly resistant to theelastic forces of the capsule upon it. RonaldFisher's in vitro measurements of the mech-anics of the various parts of the accom-modative 'plant' provide evidence for thisgroup of theories: the ciliary muscle does notlose its power with age; the lens behaves as asimple elastic body, which requires moreenergy to deform it with increasing age; andthe capsule's elasticity declines so that it isless able to provide this deforming energy.

Hess-Gullstrand theoryThe amount of muscle contraction requiredfor a given change in accommodation ispurported to remain constant throughoutlife (Hess, 1904 (cited by Alpern, 1962);Gullstrand, 1909). This theory predicts that anincreasing proportion of ciliary musclecontraction will be 'latent', i.e, it does notaffect accommodative status. This theory isdistinct from all other theories of presbyopia,which predict that the maximum possibleamount of ciliary muscle contraction isrequired to produce maximum accommo-dation at any age.The Hess-Gullstrand theory would be

supported by evidence that the ciliary muscleincreases its contraction at accommodationstimulus levels beyond those which producethe maximal accommodation response, butthis evidence is limited. Some studies havefound electrical changes in the region of theequator of the eyeball for stimuli beyondthose producing the maximal response, butwhether this indicates ciliary muscle activityhas been disputed.The Hess-Gullstrand theory predicts that

zonule slackness should be more apparent inolder subjects trying to accommodate beyondthe amplitude limit, but the reverse has beennoted in monkeys without irises. Similarly,the predicted increased zonule slacknessupon accommodation in older subjects shouldallow the lens to be more influenced bygravity for older rather than younger subjects,and again the reverse seems to occur.

Fincham theoryAccording to the mechanical change theories,

226 Miscl'llmlcol/s

the lens becomes more resistant to change inshape with age. Fincham (1937) believed thatgreater pressure from the capsule, necessarywith increase in age to achieve a given level ofaccommodation, can only be achieved byfurther releasing the zonular tension on thecapsule. This means that the changes respon-sible for the decline in accommodation residein the lens and capsule, but ciliary musclecontraction required for a given change inaccommodation increases throughout life.This theory is supported by studies of the

effect of age and drugs on the accommo-dation-convergence synkinesis. For example,the response AC/A ratio (amount of con-vergence induced by 1 D change in accom-modation response) increases with age, aspredicted if increasing innervation to theciliary muscle is required to produce a unitchange in accommodation response as ageincrease. By contrast, the Hess-Gullstrandtheory predicts that this ratio should beunaffected by age. The evidence supportingthe Fincham theory, at the expense of theHess-Gullstrand theory, also supports thefollowing theories.

Geometric theoryThis attributes the decline in amplitude withage mainly to increased size and curvature ofthe lens (see Lens, this chapter). Koretz andHandelmann (1986 and 1988) suggested thatthe increasing curvature and the likely changein orientation of the zonules due to shiftingzonule insertions means that zonules applytension less radially to the capsule's surface.This means that, upon ciliary musclecontraction, the zonule relaxation may havesmaller effects on the lens shape.Schachar (1992) developed a variation of

the geometric theory. He rejected theHelmholtzian explanation for accommo-dation, believing instead that the ciliary bodymoves away from the lens upon increasedaccommodation, so that zonular tensionincreases and the lens becomes spindle-shaped. Increasing lens diameter with agewill restrict the ability of the zonules toprovide this tension. Schacher has published anumber of experimental and theoreticalpapers claiming to support his theory. Theexperiments of Fisher (1977) and Glaser andCampbell (1998), which showed that stretch-

ing human lenses decreases the power ratherthan increasing it, would seem to provideoverwhelming evidence against Schachar'stheory.

Extra-lenticular theoriesThese theories attribute accommodationamplitude decline either to weakening of theciliary muscle or to loss of elasticity of zonulesor ciliary body components. Workers usingthe rhesus monkey as an animal model forhuman accommodation found that electricallystimulating the mid-brain region thatinfluences accommodation produced axialthickening of the lens, narrowing of the ciliaryring, and zonule slackening at high ampli-tudes. As the monkeys' age increased, thesefindings were less noticeable. Lenticulartheories do not predict the decline in thenarrowing of the ciliary ring.

Changes in ciliary muscle - Duane theoryDuane (1922 and 1925) believed that theciliary muscle weakens with increased age.However, Fisher (1977) found that thestrength of muscle contraction should declineonly slightly before 45-50 years of age, andanatomical studies of the ciliary muscle andsurrounding tissue suggest that its possiblemovement should not decrease markedlywith increase in age.

Changes in elastic components of zonulesandlor ciliary bodyBito and Miranda (1989) claimed thatpresbyopia is a loss of the ability to relaxaccommodation, rather than loss of the abilityto increase accommodation, which is oppositeto the usual way in which presbyopia isconsidered. This occurs through deteriorationof elastic components in the ciliary body andchoroid. As age increases, the lens takes up amore curved shape under its elastic forces,even when ciliary muscle contraction does notoccur, because the elastic antagonists of theciliary muscle in the ciliary body and choroidare not doing their work. The theory ignoreschanges in lens and capsule elasticity.Some anatomical studies have found age-

related changes in the attachments to theciliary muscle. As discussed earlier in thischapter, the lens certainly does become morecurved with increasing age.

SummaryPresbyopia possibly has more than onecause - for example, changes in capsule andlenticular elasticity combined with changes inlens geometry. There is considerable evidenceagainst the Hess-Gullstrand theory, and it isunlikely that changes in ciliary musclecontractility contribute significantly to pres-byopia. A lenticular origin of presbyopiaaccording to the Fincham theory seems mostlikely, with recent experiments indicating thatit is possible to restore a measure of accom-modation by an intraocular lens, which can bemoved or adjusted in shape within the lenscapsule under the influence of the ciliary body(Koprowski, 1995; Cumming and Kamman,1996). Glasser and Campbell (1998) haveconfirmed Fisher's (1977) finding that, withincreasing age, the ability to change in vitrolens power by stretching will decrease at arate which is similar to the decrease inamplitude of accommodation. Although theforces acting on the ill vitro lenses are notnecessarily the same as those occurring in theeye, this evidence tends to discount the needfor an extra lenticular role in presbyopia.

Pupil diameter

Pupil size decreases with increased age(Birren et al., 1950;Kumnick, 1954;Kadlecovaet al., 1958; Leinhos, 1959; Said and Sawires,1972; Winn et al., 1994) (Figure 20.4). This isreferred to as senile miosis. In addition, thespeed and extent of pupillary reactionsdecrease with increase in age (e.g. Kumnick,1954). The maximum pupil size in dark-adapted eyes is reached in the teenage years,after which it declines (Kadlecova et al., 1958;Said and Sawires, 1972). For example,Kadlecova et al. (1958) found maximumdiameters ranging from about 7.5 mm at 10years of age to about 5 mm at 80 years of age.As age increases, the variation in pupil dia-meter with change in luminance decreases(Figure 20.4).

Theas;IIS eye 227

y = S.W6 - n.O~.h r~ = 0.557 • 9 cd!m1

o 4400cd!m1

y = ~.070 - o.n 15x

O-f-"T'"'""",--..--,-r-r-"'c--,---.---.,----..-,--,--,---r-+10 20 30 ~O 50 6() 70 SO 90

Age (years)

Figure 20.4. Effectof age on pupil size. Ninety-onesubjects viewed a 10° field monocularly in Maxwellianview with relaxed accommodation. Results, includingregression equations, are shown for luminance levels of9 cd/m2 and 4400cd/m2. Data from Figures 2a and 2eof Winn et al. (994), kindly provided by Barry Winn andwith permission from the Association for Research inVision and Ophthalmology.

Aberrations and retinal imagequality

Jenkins (1963) used retinoscopy to investigatethe direction of spherical aberration of 133subjects between the ages of 2 and 60 years.He found that negative spherical aberrationpredominated before the age of 6 years, butafter this age, a dramatic shift occurred topositive spherical aberration. He claimed that,after about 35 years of age, there was aconsiderable increase in positive sphericalaberration. In support of increasing positivespherical aberration with increase in age,Chateau et al. (1998) found that optimumcontrast sensitivity (at 12 c/deg) was pro-duced with contact lenses which increased innegative asphericity as age increased.Artal et al. (1993) and Guirao et al. (1999)

determined modulation transfer functions(MTFs) from point spread functions forsubjects in different age groups and foundmuch poorer retinal image quality for theoldest group at any particular pupil size(Figure 20.5). Having corrected their resultsfor scatter, Artal et al. attributed the

228 Miser/lmlcolls

0.8

...~~ 0.6

~e.s:] 0.4::I-gz

0.2

10 20 30 40 50 60

Spatial frequency (c/dcg)

Figure 20.5. Modulation transfer functions (averaged inall meridians) for a young subject group (20-30 years ofage) and for an old subject group (60-70 years of age)using 543 nm laser light and a 4 mm pupil. The dashedlines indicate 1 standard deviation from the mean. Dataof Guirao et al. (1999) kindly provided by Pablo Artaland with permission from the Association for Researchin Vision and Ophthalmology.

age-related decline to increase in aberrations.Using the aberroscope technique, Calver et al.(1999) found MTFs to be lower in an old (68 ±5 years) than in a young age group (24 ± 3years) at any particular pupil size, but senilemiosis caused the older eyes to have the loweraberration levels at natural pupil sizes.Some studies have found a decrease in

longitudinal chromatic aberration with age(Millodot and Sivak, 1973; Millodot, 1976;Mordi and Adrian, 1985), but a majority ofstudies have found no change (Lau et al., 1955;Ware, 1982;Pease and Cooper, 1986;Howarthet al.,1988;Morrell et al., 1991).

Photometry

Retinal illumination decreases with age due totwo factors. One is the reduction of pupildiameter with age, particularly at low lightlevels. The other factor is the decrease inocular transmittance with age (Chapter 12).The data of Winn et al. (1994), shown in

Figure 20.4, can be used to predict the effect of

changing pupil size. At 9 cd/m2 backgroundluminance, the pupil diameter decreasedapproximately linearly with age from 20 yearsto 60 years by about 25 per cent. This corre-sponds to a light loss at the retina of morethan 43 per cent at low light levels .The decrease in ocular transmittance with

age is due mainly to the lens, particularly forthe shorter wavelengths (Figure 20.6). Saidand Weale (1959) found that the lenstransmittance decreases by about 25 per centbetween the ages of 20 and 60 years (558nm)(Figure 20.6).Combining the pupil size and ocular

transmittance changes with age given aboveindicates a reduction in light level reachingthe retina, between the ages of 20 and 60years, of approximately 60 per cent at lowerlight levels. This is similar to an earlierdetermination by Weale (1961).

Effect of light loss on visualperformanceSpatial visual performance decreases withincrease in age (Blackwell and Blackwell,

100 - 596nm90 ---t!r- 558 nm

80 -0--- 532 nm

--<>--- 440 nm

70 398 nm

~<.l 60:.JCg

50·E'"ee 40f-

.~o

20~10

0 10 20 30 40 50 60 70Age (years)

Figure 20.6. Variation in lens transmittance with age atdifferent wavelengths. Data of Said and Weale (1959),after conversion from densities to transmittances. Thetechnique was based on the relative brightnesses of thethird and fourth Purkinje images.

1971; Richards, 1977; Owsley et al., 1983;Elliot, 1987; Elliot et al., 1993 and 1995;Haegerstrom-Portnoy et al., 1999). This hasboth optical and neural causes (Elliot, 1987;Sloane et al., 1988).The loss in performance ismuch more marked at low than at higherluminances (Weston, 1948; Guth, 1957;Blackwell and Blackwell, 1971;Richards, 1977;Sloane et al., 1987), so that transmittancedecreases at lower light levels can be compen-sated to a large extent by increasing the lightlevel. An example of this is. given by Guth(1957), concerning visibility of words. Thegreater loss in transmittance at the shorterwavelengths affects colour perception, reduc-ing the ability to discriminate shades of greensand blues (Knoblauch et al., 1987).The decrease in transmittance has two

contributions; absorption and backwardscatter. Unlike these, increasing the light levelcannot compensate for the forward scatter.Forward scatter produces a veiling glare overthe retina, which reduces the contrast of theretinal image. Low-contrast objects maybecome invisible. This becomes worse at lowambient light levels when bright lights are inthe field of view, e.g. on dark streets withstreetlamps.With increase in age, there is an increase in

the amount of forward light scatter from thelens, so that K increases in the equivalentveiling luminance equation (13.15)Lv(8) = KEIll! (20.2)although the angular dependency remainssimilar (Fisher and Christie, 1965; Ilspeert etal., 1990). For example, Christie and Fisherfound that K increased linearly with age, butthat 11 was essentially independent of age. Kincreased by a factor of 1.9 to 3.3 in theirexperiments between the ages of 20 and 70years.

Theasi"x eye 229

Stiles-Crawford effect

Two longitudinal studies have shown that the~ co-efficient (equation (13.29» is relativelyunaffected by age for healthy eyes (Rynderset al., 1995;DeLint et al., 1997).

Schematic eyes

The structures of paraxial and finite schematiceyes were discussed in Chapters 5 and 16,respectively, and the dimensions of many ofthese eyes are listed in Appendix 3. We foundin the preceding section that many dimen-sions of the eye depend upon age, butdesigners of most schematic eyes gave noindication that a particular age was modelled.An exception is the finite eye of Liou andBrennan (1997), which contains ocular para-meters for an eye near the age of 45 years.Also, Blaker (1991), Smith et al. (1992) andSmith and Pierscionek (1998) designed modeleyes adapted for age. Rabbetts (1998) pre-sented an 'elderly' version of the Bennett andRabbetts' simplified eye. Any selection ofparameters is complicated by other factors,such as sex and race. As examples, femaleeyes are shorter and hence have higherpowers than male eyes (Koretz et al., 1989),and there are many racial variations includingpupil size (Said and Sawires, 1972)and refrac-tive error distributions. A very sophisticatedschematic eye would be adaptable for theseeffects.We include a brief description of an age-

dependent, relaxed, emmetropic, paraxialschematic eye based on Smith et al. (1992).Age-dependent parameters are shown inTable 20.1 for 20-, 40- and 60-year-old eyes.The eye has an equivalent power of 60 D atthe age of 20 years.

Table 20.1.Age-dependent parameters in age-dependent schematic eye.

20Ase (years)

40 60

Anterior chamber depth (mm)Lens anterior radius of curvature (mm)Lens posterior radius of curvature (mm)Lens thickness (mm)Refractive index of lens

3.3814.735-8.4193.721.4506

3.1212.655-8.1193.981.4398

2.8610.575-7.8194.241.4280

230 Miscetllllleolls

RadiiCorneaSome decrease in radius occurs with age, asdiscussed earlier in this chapter, but this issmall, and the Gullstrand number one sche-matic eye values of 7.7 mm and 6.8mm areused for the anterior and posterior surfaces,respectively.

LensThe lens radii are taken from Brown's (1974)study:R} (mm) =16.815- 0.104x age (years) (20.3a)

Rz (mm) =-8.719 + 0.015x age (years) (20.3b)

Gullstrand number one schematic eye. Theseare 1.376 for the cornea and 1.336 for theaqueous and vitreous.

LensWe use an age-dependent uniform index.From the data given for radii and distances,the eye will remain emmetropic ifrefractite index of lens =1.4608- 0.000488x age - 0.00000097 x (age)z

(20.6)This is a high refractive index compared withother schematic eyes, indicating that some ofthe other parameters in the model may not bevery accurate.

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Leinhos, R. (1959>' Die Altersabhangigheit desAugenpupillendurchmessers. Oplik, 16, 669-71.

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Mellerio, J. (1987). Yellowing of the human lens: nuclearand cortical contributions. Vision Res., 27, 1581-7.

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Olsen, T. and Ehlers, N. (1984). The thickness of thehuman cornea as determined by a specular method.Acta Opllilral., 62, 859-71.

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AtParaxial optics

Introduction

The study of the image formation by opticalsystems can be reduced to the imagery ofselected points in the object space or field. Thestudy of the image formation of an objectpoint can be reduced to tracing a number ofrays from this point, through the system, andexamining their paths in image space. Such asituation is shown in Figure A1.I, whichshows an object point Q and a set of threeimage-forming rays. Ideally, these rays shouldbe concurrent at some point - say Q/ - in theimage space, as shown in the figure. However,they are not usually concurrent, and this isdue to what are known as aberrations. Thegreater the spread of the rays in the imageplane, the greater the aberrations. Usually theaberrations increase as the light beam widensand the object point Q moves further awayfrom the optical axis. If rays are traced very

Q'

Object space

Q?-----'\\Optical system

Figure At.t, Ideal imagery.

close to the axis, aberrations are reduced theray-trace equations can be simplified bymaking some simple approximations calledparaxial approximations.In this book, exact or actual rays are

referred to as finite rays, and rays tracedusing the paraxial approximations as paraxialrays.

Finite ray tracing

The tracing of a finite ray through an opticalsystem involves a number of steps:1. Choosing an origin or starting point and a

direction for the ray, such as the point 0and angle u, shown in Figure A1.2(sometimes the starting point may be off-axis).

2. Locating the point of intersection B of theray with the surface, using trigonometry,algebra and a knowledge of the surfaceposition and shape.

3. Determining the angle of incidence i at thissurface.

4. Refraction at B by the application of Snell'slaw, which connects the angles of incidencei and refraction i' with the refractiveindices, nand /1// by

n' sinu') = /I sin (i) (AU)to find the angle of refraction i' and angle u'the ray makes with the optical axis (Figure

238 Al'l'eIld ices

Tangent plane-:

bRefracting surface

o

r- r(+)---l

\ ,,Next surfaceIB'

II

I

Figure Al.2. Refraction by a surface, showing important variables and the sign convention.

Paraxial ray-trace equations

Definition of a paraxial ray

There are two paraxial ray-trace equationsthat are used for steps 2-4 in the previoussection.

A paraxial ray is a finite ray traced close to theoptical axis, in which the angles involved aresufficiently small that replacing the sines andtangents of the angles by the angles them-selves (in radians) in ray-trace equationsproduces a negligible error.

(A1.3)lI'i' =IIitions, Snell's law reduces to

The useful outcome of this paraxial approxi-mation is that, if all the rays within the beamshown in Figure A1.l are traced as paraxialrays, they are all concurrent at some point inimage space. In this sense, paraxial rays areaberration-free rays.

A1.2). These angles specify the direction ofthe refracted ray. This is called the 'refrac-tion' step. If there is more than one surfacein the system, the above steps 2-4 arerepeated until the last surface is reached.Each return to step 2 requires the point ofintersection with the next surface to befound. Figure A1.2 shows a ray-trace to thepoint B' on the next surface. The process oftracing to the next surface is called the'transfer' step.

5. Locating the point of intersection with theoptical axis or the expected image surface.The sign convention used for tracing rays is

that shown in Figure A1.2. Distances to theleft of a surface or below the optical axis arenegative and those to the right or above arepositive. Angles due to an anticlockwiserotation of the ray from the optical axis arepositive, and those due to a clockwise rotationare negative. The origin for axes at eachsurface is the vertex V. The signs enclosed inbrackets indicate the signs of the quantitiesshown in the figure.

is the refractive power of the surface, C is thesurface curvature (= l/r), and the othervariables are shown in Figure A1.2.

Paraxial refraction equationThis equation is used to determine the newdirection of the ray after refraction:

The paraxial approximations andparaxial rays

If rays are traced very close to the optical axis,then all the angles shown in Figure A1.2 aresmall. For small angles x expressed in radians,tan (x) '" sin(x) '" x (A1.2)This approximation improves as the size ofthe angle decreases. When we apply thisparaxial approximation to ray-tracing equa-

lI'U' - /111 =-hF

whereF = C(II' - II)

(AlA)

(Al.S)

Step 2: refraction at the jth surface

Starting at the first surface (i.e, j = 1), we useequation (Al.4), i.e.n'·u'·-n.u.=-hF (A1.8)

J J ) J J Jwhere

_---d----~

Figure Al.3. Ray-tracing: the transfer step.

which satisfy the equationly=-h]/u]

Paraxial optics 239

(A1.7)

Step 3: transfer to the next (j + 1)thsurface

Paraxial transfer equationThis equation is used to locate the intersectionpoint or height at the next surface:h' = h + u'd (A1.6)This equation can be derived with thequantities shown in Figure A1.3.

F·= c.(H' . - H .)J J J )is the power of the jth surface.

(Al.9)

A paraxial ray-tracing scheme

Since most optical systems consist of morethan one surface, equations (Al.4) to (A1.6)are used repeatedly. The following schemeshows how this is done, with a small changein notation. The variables given in the follow-ing equations are defined in Figure Al.4.

Step 1: choosing a rayWe assume that the position of the axial point0, which is the origin of the ray or where anoff-axis ray crosses the axis, is known. Let thedistance of this point from the first surfacevertex be Iv.We now choose a direction of theray by selecting a pair of values u] and h]

Here we use equation (A1.6), i.e.

hj + 1 =hj + u'Jdj (Al.l0)Steps 2 and 3 are repeated at each surface,using the following equivalences:n'j =Hj + 1 and U'j =uj +] (Al.ll)until the last (kth) surface is reached. At thisstage, the axial distance I'v from the lastsurface vertex of the point where this raycrosses the axis is given by the equationl'y=-hk/u'k (A1.l2)

Image size and magnificationUsually, we need to know the transversemagnification M of the image, which is

Figure Al.4. The tracing of a paraxial ray through a general system consisting of ksurfaces.

240 Appel/dices

defined as

M = image size (T1') (A1.B)object size (11)

where the object value (11) and image size (11)are shown in Figure AI.4. The value of thismagnification can be found from the aboveray-trace and is given by the equation

f'

Figure A1.S.Particular ray-traces for determining thepositions of the back cardinal points.

(A1.14)

Special case of the object at infinity

If the object is at infinity, equation (A1.7)cannot be used to generate a ray because thevalue of Iv is infinite. In this case, Il] =0, andwe can choose any suitable value of h).

Choice of ray

Cardinal points and equivalentpower

The location of the back cardinal points P' andF' can be found from the results of a paraxialray-trace from object space to image space,initially parallel to the axis (i.e, lI] = 0), asshown in Figure AI.5. Given the initial rayheight 11), final ray height Itk and final angleu 'k' the positions of these points can be foundfrom the equationsP'F' = -It)/u'k (A1.16)

If the positions of the cardinal points and theequivalent power of an optical system areknown, the position of an image can be foundwithout the above detailed ray-tracing.Instead, it can be found by the direct appli-

(A1.20)

(A1.17)power F is given by the

PN =P'N'

The lens equation

V'F' =-/zk/lI'k

The equivalentequationF= l1'k/P'F' (A1.18)The position of the back nodal point N' isgiven by the equationP'N'=(II'k-II))/F (A1.19)The positions of the front cardinal points P, Fand N can be found by tracing a similarparaxial ray from the image space (i.e. ll'k = 0)back through the system into object space. Theequivalent power can also be calculated fromthis ray-trace and, if the equivalent ofequation (A1.18) is used, it gives the samenumerical value. It can also be found that

The optical invariantIf two distinct paraxial rays, denoted by A andB, are traced through any optical system, thequantityI1A(uAlt B -IlBltA)

at any surface and on either side of the surfacehas the same value throughout the system,and thus its value is invariant. Using thesymbol H to denote its value,

H = I'lA(IlAhB - "B/I A) (A1.IS)This quantity is useful in aberration theory,which we discuss in Appendix 2.

Since paraxial rays are aberration-free rays, allparaxial rays in a beam arising from an objectpoint are concurrent at the image point. Tolocate the image point, we need in principle totrace only two rays in the beam to find thepoint of concurrency. However, for an axialobject point, one of these rays may be taken asthe optical axis. Therefore, for axial objects weneed to trace only one ray, and where this rayintersects the optical axis is the image pointand the location of the paraxial image plane.

Paraxial optics 24t

11

Figure At.6. The general system and the lens equation.

Sometimes it is useful to assume that rays thatare beyond the paraxial region, according tothe approximation in equation (A1.2), never-theless behave as if they are paraxial rays (i.e,are aberration-free). The application of par-

where n' and n are the object and image spacerefractive indices, L is the object vergence(= n/I) and L' is the image vergence (= n' /1'),the distance I is measured from the frontprincipal plane at P and the distance l' ismeasured from the back principal plane at P'.F is the equivalent power of the system,defined by equation (Al.18), and is the powerof an equivalent thin lens placed at theprincipal planes (Figure A1.6).In this situation, equation (A1.14) for the

transverse magnification is still applicableand, using the symbols shown in Figure A1.6,this equation can be written asM =nu/(n'u') (A1.22)It can also be expressed in terms of thedistances I and l' of the object and imagesfrom the respective principal planes by theequation

cation of the lens equationn' / l' - n / I = L' - L = F

M =nl' /(n'/)

Gaussian optics

(A1.21)

(A1.23)

axial approximations beyond the paraxialregion is called Gaussian optics.

Summary of main symbols

j angle of incidenceu paraxial ray angle11 paraxial ray heightn refractive indexF (equivalent) power of a surface or lensC surface curvature (reciprocal of radius of

curvature)d separation between adjacent refracting

surfacesFor thin components, this is the distancefrom the vertex V of the component to theaxial object point O. For thick components,the distance is measured from the frontprincipal plane P to the axial object pointo

L corresponding vergence (= n/ I)M transverse magnificationH optical invariant11 object sizeF focal pointN nodal pointP principal pointQ off-axis object pointo on-axis object pointV vertex pointA prime (') superscript following the abovesymbols (except F,C, d,M, and H) means thata quantity occurs after refraction or in imagespace.

A2

Introduction to aberration theory

Wave (path length)aberration

Quantification of aberrations

In the absence of aberrations, diffraction andscatter, all rays from any point in object space,that are refracted by an optical system, arefocused to one point in the image plane - thatis, they are concurrent. The position of thispoint can be predicted by using the paraxialequations described in Appendix 1. In thepresence of aberrations, the rays are notconcurrent at the expected image point, butintersect the paraxial image plane in a spread-out pattern.There are three common ways of quantify-

ing the aberrations of an optical system;namely wave, transverse and longitudinal.Aberrations of a system are often determinedby calculating the aberrations of repre-sentative rays. For each ray, there is always awave and a transverse aberration, and some-times a longitudinal aberration. Figure A2.Ishows a schematic system and typical raystraversing the system. The wave, transverseand longitudinal aberrations of this ray aredescribed as follows.

Rays from an axial pointFigure A2.Ia shows a particular real rayOBB'H in a beam from the point O. This rayshould cross the optical axis and the paraxialimage plane at the point 0' but, because ofaberrations, it intersects the axis at the point Gand the paraxial image plane at the point H.

The path OBO' indicates the route of thecorresponding unaberrated ray. The aber-ration may be specified in terms of any of thefollowing quantities:Longitudinalaberration O'G =81'

Transverse aberrationO'H

[OEE'O'] - lOBO'](A2.Ia)

where the square brackets refer to optical pathlengths, which are products of physical pathlengths and refractive indices. The waveaberration is the difference in optical pathlength between that of the pupil ray (thecentral ray of the beam) and that of any otherray of the beam. In this case, it is assumed thatthe pupil ray of the beam travels along theoptical axis.

Rays from an off-axis point

Figure A2.Ib shows a general ray from an off-axis point Q. If such a ray does not intersectthe optical axis or the pupil ray of the beam,longitudinal aberration is not applicable to theray. The transverse aberration is Q'H. Thewave aberration is similar to that given byequation (A2.Ia), but now the pupil rayfollows the path QEE'Q' instead of the opticalaxis. In summary,

Introduction to aberration theory 243

Figure A2.1. Three forms for quantifying the aberrations ofa ray: wave, transverse and longitudinal.

Wave (path length)aberration

Transverse aberrationQ'H

[QEE'Q'] - [QBQ'](A2.1b)

There are other methods of quantifying thelevel of aberrations of rays and beams thatdepend upon the particular aberration. Forexample, the spherical aberration of the eye isoften expressed as an equivalent power error.

The wave aberration function

The aberration of a particular ray dependsupon the co-ordinates of the point B, where

the ray passes through the entrance pupil. Ifwe set up a (X, Y) co-ordinate system in theentrance pupil, as shown in Figure A2.1b, wecan express the wave aberration as a functionof X and Y. The wave aberration for the raythrough B is also a function of the position 11of the object point Q in the field. The waveaberration W is thus a function of X, Y, and 11and can be expressed as a power series inthese variables. For a rotationally symmetricsystem, it has the form

W(1];X,Y)=0W~O(X2 + yz)z+ 1W3,1 11(XZ

+ Y.')y + zWzof (X2 + y2)+ zWz,zfy2';' 3Wl,1~Y+ higher order aberrations (A2.2)

The first five terms in this expansion are the

244 Al'l'c/ldices

primary aberrations or third order aber-rations, and are known asW (X2 + y2)2 spherical aberrationo 4,0 2 2lW3111(X + Y)Y coma

, 2 2 v')2W2 011 (X + I -) field curvature

, 2 22W2211 Y astigmatism, 3

3Wl,l11 Y distortionIt is often more convenient to express thisfunction in terms of polar co-ordinates (r,q,)whereX =r sin(q,) and Y =r cos(q,) (A2.3)and q, is the angle between the line EBand theY axis, as shown in Figure A2.1b. The abovepolynomial can then be expressed in the form

W(11;r,q,) = oW400 + 1W3 l11rJ cos(q,)+ 2W2,o112r2 +'2W2,2112r2 cos2(q,)+ 3W 1113r cos(q,)+ higher order aberrations (A2.4)

The level of the wave aberration need only beas large as a wavelength to have a significanteffect on image quality.

Units of aberrationsAll the aberrations defined above aredistances or differences in distances, andtherefore have units of length. Because evenvery small values can be significant, it is

4FYl

Q

common practice to express wave aberrationsin units of the wavelength. For example, adistance of 0.001 mm is equivalent to twowavelengths if the wavelength is 500 nm.

Defocus and wave aberration

The wave aberration polynomial is veryuseful in studying the effects of a defocus aswell as aberrations on an image. If a defocus ispresent, we can add the following term:

oW2,Or2

to the wave aberration polynomial given byequation (A2.4), and the value of °W2 0 isrelated to the defocus M by the equation'

M =20W2,o (A2.5)where M is the change in power required toproduce that level of defocus.

Calculation of the wave aberrationfunctionThe definition of wave aberration given byequations (A2.1a and b) is ideal for thetheoretical analysis of, and deriving explicitequations for, the wave aberration, but onlyfor relatively simple systems. For complexsystems, it is easier in practice to determine

r:

bit pupil

Figure A2.2. The distortion of a wavefront passing through an optical system andthe wave aberration [B'B")

(A2.7)

the wave aberration by numerically ray-trac-ing specific rays. Such ray-traces do not givethe path QBQ' but the path QBB'H, so wecannot determine the wave aberration usingequation (A2.1a or b). Therefore, in practice,we need a different and more practicaldefinition of wave aberration.

In the ideal optical system, a point sourcecan be thought of as emitting light withspherical wavefronts as shown in Figure A2.2.These wavefronts enter the system throughthe entrance pupil. If the imagery is perfect,these wavefronts exit the system also with aspherical shape. However, if aberrations arepresent, the wavefronts are distorted. Thewave aberration is a measure of the distortion.Figure A2.2 shows a distorted or aberrated

wavefront leaving the exit pupil, and alsoshows an undistorted wavefront (the refer-ence sphere). The wave aberration of any rayis the optical distance along the ray betweenthe distorted wavefront and the referencesphere. Arbitrarily, the two wavefronts aredefined to coincide at the centre of the exitpupil.W= [QEE'] - [QBB'] (A2.6)As[QEE'] = [QBB'B"],equation (A2.6) reduces toW=[B'B"]Since for any specific optical system theposition and shapes of all the surfaces areknown, the ray intersection points with eachsurface can be found by conventional geo-

Table A2.1. The seven Seidel aberrations.

Introduction toaberratioll theory 245

metrical and trigonometrical rules and theabove path differences can be calculated forany specific ray.The two definitions of wave aberration

used (equations (A2.1 and A2.7)) are slightlydifferent, but for low levels of aberrations willlead to very similar results. They will givedifferent results for large levels of aberrations,but in this case, some other equations used inthe chapters covering aberrations will also bein error; for example, the relationship betweenwave, transverse and longitudinal aberrationsused in Chapter 15.

Seidel aberrations

There are seven Seidel aberrations, and theseare listed in Table A2.1 along with some oftheir properties. Only five of these are mono-chromatic aberrations, and these are related tothe primary wave aberrations given inequation (A2.2). The Seidel aberrations can becalculated from the paths of the paraxialmarginal and paraxial pupil rays.Seidel aberrations are usually calculated as

wave aberrations, although they areoccasionally calculated in the transverse orlongitudinal aberration forms. Equations fortheir calculations can be found in texts byWelford (1986) and Smith and Atchison(1997). They are calculated from the systemconstructional parameters (refractive indices,surface curvatures and shapes and surfaceseparations), and depend upon the beamwidth and the field-of-view, which in turn are

AberrationDependence on'

Symbol Imagery point to point aperture (p) field (q)

MonochromaticSpherical 51 no 4 0Coma 52 no 3 1Astigmatism 53 no 2 2Field curvature 54 yes 2 2Distortion 55 yes 1 3% distortion 0 2

ChromaticLongitudinal CL no 2 0Transverse Cr no 1 1

-These values are for wave aberrations, except for distortion expressed as a percentage. For transverse aberrationsthe p power dependence reduces by 1, and for longitudinal aberrations the p power dependence reduces by 2.

246 Appendices

Q

Figure A2.3. Field points and field angles.

The combination of beam width and field sizeis embodied in the optical invariant H, whichwe introduced in Appendix 1. Using theparaxial marginal and pupil rays, we have

H = n(uh - u1i) (A2.9)

where u and h are the marginal ray angle andheight at any plane in the system, uand 1i arethe corresponding pupil ray values, and 11 isthe refractive index on the side of the planewhere the above paraxial angles are mea-sured. At the entrance pupil plane,H =nulP (A2.9a)

defined by the paraxial marginal and paraxialpupil rays. Values are given in Appendix 3 forselected schematic eyes, and are calculated fora beam width limited by the paraxial marginalray passing through the edge of an 8 mmdiameter entrance pupil (that is, through theedge of pupil of radius p=4 mm), and for afield width specified by the paraxial pupil rayinclined at angle of 5° to the optical axis.Figure A2.3 shows a beam arising from thepoint Q, which is at the edge of the nominalfield of view and subtends an angle °1 at theentrance pupil at E. The pupil ray sub tends aparaxial angle ul to the axis, and this paraxialangle is the tangent of the real angle accordingto

spherical aberration(A2.13a)

Ul =tan(Ol)Thus, in this case,°1 =5° and soul = tan(5°) = 0.087488663

(A2.8)

(A2.8a)

Therefore, for the data in Appendix 3,H = 1 x 0.0874887 x 4 = 0.349955 (A2.1O)For a beam with a different width and from adifferent field point, the aberrations aredifferent and can be found fromNew aberrationvalue = old aberration value

x (fractional pupil radius)Px (fractional field position r)q (A2.11)

where the values of p and q are given in TableA2.1.The fractional value r can be used to specify

the position of some other field point, say thepoint Q r shown in Figure A2.3, and this pointis off-axis by an angle Of' measured at theentrance pupil. The fractional field value r isdefined asr= tan(Or)/tan(Ol) (A2.12)where Or is an off-axis angle point inside oroutside the 5° field.

Seidel aberrations and the primarywave aberration co-efficientsThe five monochromatic Seidel aberrationsare directly related to the wave aberration co-efficients of equations (A2.2) and (A2.4) asfollows:

51 =80W4,OP4

Illtrodllctioll to aberratioll tlleory 247

Sagittal, tangential and Petzvalsuifaces

(A2.17a)(A2.17b)

(A2.18a)(A2.18b)

Ls((J) = -2W2,oLt((J) =-2(W2,o+ W2,2)

or

In visual optics, the peripheral power errorsLs(8) and Lt(8) are more meaningful than theSeidel or wave aberration values. However, ifwe wish to calculate the point spread oroptical transfer functions, we need to knowthe corresponding wave aberration co-efficients W2,oand W2,2' We haveW2,o= -Ls((J)/2W2,2 = [Ls(9) - Lt(9)l!2

The use of Seidel aberrations assumes usuallythat the image surface is flat. Some of theabove equations can be modified by takinginto account the radius of curvature of theretina. The equivalent of equations (A2.16aand b), for the sagittal and tangential surfacevergences, now become

Ls((J) ""~ [~_2-] (A2.19a)2nvit rs rR

Modifications for a curved retina

(jlLt(8) ""-- (A2.16b)

2nvi{ t

where 8 is the off-axis angle in radians. Thesevergences are the same as peripheral powererrors as used in Chapters 15 and 16.

Wave aberration co-efficients W2 0and W2,2 '

be represented by the corresponding ver-gences L and Lt. This is particularly useful ifthe imag~ plane is at infinity, because then theimage surfaces cease to have any meaning.Using equations in Smith and Atchison (1997),for ray-tracing out of the eye and for the object(instead of the image) at infinity, we can showthat the vergences are related to the off-axisangle 8by the equations

(jlLs(8) ""-- (A2.16a)

2nvi{ s

(A2.14a)(A2.14b)

astigmatism(A2.14d)

distortion (A2.14e)

comaspherical

coma (A2.13b)astigmatism

(A2.13c)

54=(42W2,O-22W2,2)p 2 Petzval curvature(A2.13d)

55= 23Wl,lp distortion (A2.13e)The Seidel aberration values can be foundfrom the wave aberration polynomial co-efficients, if the latter are known, as well asfrom the Seidel formulae given in texts suchas Welford (1986) and Smith and Atchison(1997). If the Seidel aberrations are known, thewave aberration co-efficients can becalculated after re-arranging the aboveequations asoW4,o = 5/(8p4)

1W3,1 =52/(2p3)

2W2,O = (53+ 54)/(4p2) field curvature(A2.14c)

52= 21W3,1p 3

53= 22W2,2p2

In the Seidel approximation, the sagittal andtangential surfaces are spherical with radii ofcurvatures given by the following equations(Smith and Atchison, 1997)

H2rs = - n'(5

3+ 54) sagittal surface (A2.15a)

H2rt =- '( tangential surface (A2.l5b)n 353 + 54)In the absence of astigmatism, i.e. 53=0, wecan see from the above equations that the tworadii of curvatures are the same. The corre-sponding surface is known as the Petzvalsurface, and it follows that its radius ofcurvature, denoted as rp, is

H2rp =- --'--5Petzval surface (A2.15c)

n 4

The sagittal and tangential image surfaces can

(A2.20)

(A2.19b)

248 Appendices

(j[11]Lt(8) == - - - -2nvit rt rRwhere rR is the radius of curvature of theretina.In the Seidel approximation, the wave

aberration W2 2 is independent of the retinalradius, but the co-efficient W2,ochanges by anamount L1W2,owhere(jaW2,o= - -4nvitrR

Seidel aberrations of a gradient indexmedium

Sands (1970) presented a set of equations fordetermining the Seidel aberrations of gradientindex media. To make these equations con-sistent with the Seidel aberrations of Hopkins(1950) and Welford (1986), Sands' aberrationvalues must be multiplied by a factor of twoand the signs changed. The aberrations arebroken up into two types: the refractivecontribution arising at the surfaces, and thetransfer contribution arising from the passageof the rays through the lens from the anteriorto the posterior surface.

Refractive contributionK= +2CtI2Nt(0) + O.5CtdNo(O)/dZ}

- 2C212Nt(d) + 0.5C2dNo(d)/dZI (A2.21)Spherical aberration (5t) KJl4Coma (52) KhJTiAstigmatism (53) KJI'lJz2Field curvature (54) 0 _Distortion (55) KhliJwhere Ct and C2are the front and back surfacecurvatures of the lens, d is its thickness,dNo(Z)/dZ is the differential of No(Z) withrespect to Z, h is the paraxial marginal rayheight at the surface, and h is the paraxialpupil ray height at the surface.

Transfer contributionSpherical aberration (5t)- INo(d)h(d)u

3(d) - No(0)h(0)u3(0)}

- 12S[4N2(Z)h4(Z) + 2Nt(Z)h

2(Z)1I2(Z)o

- 0.5No(Z)u4(Z)jdZI

Coma (52)

- INo(d)h(d)1I2(d)u(d) - No(0)h(0)u2(0)U(0)1

d- 12J[4N2(Z)h

3(Z)h(Z)

o+ Nt(Z)h(Z)u(Z)[h(Z)u(Z) +h(Z)u(Z)j

- 0.5No(Z)1I3(Z)U(Z)]dZI

Astigmatism (53)

- INo(d)h(d)u(d)u2(d) - No(0)h(0)u(0)u2(0)l

- 121[4N2(Z)h2(Z)1i2(Z)o

+ 2Nt(Z)h(Z)h(Z)u(Z)u(Z)- 0.5NO(Z)1I2(Z)u2(Z)]dZI

Distortion (55)- INo(d)h(d)u3(d) - No(0)h(0)u3(0)l

-121[4N2(Z)h(Z)h3(Z)o+ N; (Z)h(Z)u(Z)[h(Z)u(Z) + h(Z)II(Z)]

- 0.5No(Z)II(Z)u3(Z)]dZI

Relevance of gradient index expressions tothe eyeIf a ray passing through the lens of the eye hasa shallow trajectory, the angle u(Z) is small. Inthe expressions above, the effects of the No(Z)and Nt(Z) co-efficients are attenuated by thevalue of II, and so their contributions are alsolikely to be small. The N2(Z) co-efficients arelikely to be the main gradient index contri-butors to Seidel aberration of the lens. Theequations show that the spherical aberrationhas the opposite sign to that of the N2(Z) co-efficients. Thus, if the gradient index mediumis to reduce the spherical aberration of the eye,the sign of these co-efficients must be positive.

Summary of main symbols

A. vacuum wavelengthn' refractive index of the vitreous

humour15 radius of entrance pupil

r ray height in pupil (say in milli-metres)

r relative field position defined byequation (A2.12)

H optical invariant defined byequation (A2.9)

(J direction of a point in the objectfield

1] distance of object from axisSeidel aberrations

Sl spherical aberrationS2 comaS3 astigmatismS4 Petzval curvatureS5 distortionCL longitudinal chromatic

aberration

Introduction to aberration theory 249

CT transverse chromatic aberrationW(r) wave aberration for ray passing

through the pupil at a height roW4,o etc. wave aberration co-efficients

References

Hopkins, H. H. (1950). The Wave Theory of Aberrations.Clarendon Press.

Sands, P. ]. (1970). Third order aberrations ofinhomogeneous lenses. J. Opt. Soc. Am., 60, 1436--43.

Smith, G. and Atchison, D. A. (1997). In TheEyeand VisualOptical Instruments, Ch. 33. Cambridge UniversityPress.

Welford,W.T. (1986). Aberrations ofOptical Systems. AdamHilger.

A3Schematic eyes

Introduction

This appendix lists constructional data,Gaussian constants such as the pupil positionsand sizes, the positions of the cardinal points,and the Seidel aberrations of several paraxialand finite schematic eyes. The constructionaldata were taken from references, but Gaussianand aberration values were determined by theauthors.Seidel aberrations were evaluated for the

schematic eyes using an 8 mm entrance pupildiameter, a semi-field angle of 5° and areference wavelength of 589 nm. The opticalinvariant H has the value 0.349955. The V-value of all ocular media is taken as 50.23,based on equations (17.15) and (17.17).

UnitsDistances are in millimetres and are generallymeasured from the front surface vertex of thecornea.Powers are expressed in units of dioptres (0

or m").

Accommodation levels are measured at theanterior corneal surface vertex.The Seidel aberrations and the primary

wave aberration co-efficients are given inunits of wavelengths (A. =589 nm), exceptfor S5and Cp which are given as percentages.

Paraxial schematic eyesList of eyesGullstrand exact (Gullstrand, 1909)relaxedaccommodated 10.8780

Le Grand theoretical (Le Grand and ElHage,1980)relaxedaccommodated 7.0530

Gullstrand-Emsley simplified (Emsley,1952)relaxedaccommodated 8.599 0

Bennett-Rabbetts simplified (Rabbetts,1998)relaxed

Emsley reduced (Emsley, 1952).

Schematic eyes 251

Relaxed 'exact' schematic eyes

Gullstrand exact or number 1

Medium n R d Equivalent powersSurface Component Whole eye

Air 1.0007.700 48.831

Cornea 1.376 0.500 43.0536.800 -5.882

Aqueous 1.336 3.10010.000 5.000 58.636

Lens:Cortex 1.386 0.546

7.911 2.528Core 1.406 2.419 19.111

-5.760 3.472Cortex 1.386 0.635

-6.000 8.333Vitreous 1.336 17.18540

Le Grand full theoretical eye

Medium n R d Equivalent powersSurface Component Whole eye

Air 1.00007.800 48.346

Cornea 1.3771 0.550 42.3566.500 -6.108

Aqueous 1.3374 3.050 59.94010.200 8.098

Lens 1.4200 4.000 21.779-6.000 14.000

Vitreous 1.3360 16.59655

Relaxed simplified schematic eyes

Gullstrand-Emsley

Medium n R d Equivalent powersSurface Component Whole eye

Air 1.0007.800 42.735 42.735

Cornea 4/3 3.6 60.48310.000 8.267

Lens 1.416 3.6 21.755-6.00 13.778

Vitreous 4/3 16.69620

252 Appendices

Bennett-Rabbetts

Medium 11 R d Equivalent powersSurface Component Whole eye

Air 1.0007.800 43.077 43.077

Cornea 1.336 3.611.000 7.818 60.000

Lens 1.422 3.7 20.828-6.47515 13.280

Vitreous 1.336 16.78627

Reduced eyes

Emsley

Medium 11 R d

Air 1.0005.555

Vitreous 4/3 22.22222

SurfaceEquivalent powers

Component Whole eye

60.00

Accommodated 'exact' schematic eyes

Gullstrand (accommodation distance = -92.00 mm ('" 10.878 0»Medium 11 R d Equivalent powers

Surface Component Whole eye

Air 1.0007.700 48.831

Cornea 1.376 0.500 43.0536.800 -5.882

Aqueous 1.336 2.7005.333 9.376 70.576

Lens:Cortex 1.386 0.6725

2.655 7.533Core 1.406 2.6550 33.057

-2.655 7.533Cortex 1.386 0.6725

5.333 9.376Vitreous 1.336 17.18540

Schematic eyes 253

Le Grand full theoretical eye (accommodation distance =-141.793 mm (== 7.0530»Medium II R d Equivalent powers

Surface Component Whole eye

Air 1.00007.800 48.346

Cornea 1.3771 0.550 42.3566.500 -6.108

Aqueous 1.3374 2.650 67.6776.000 14.933

Lens 1.4270 4.500 30.700-5.500 16.545

Vitreous 1.3360 16.49655

Accommodated simplified schematic eyesGullstrand-Emsley (accommodation distance =-116.298 mm (== 8.5990»Medium n R d Equivalent powers

Surface Component Whole eye

Air 1.0007.8 42.735 42.735

Cornea 4/3 3.2 69.7215.0 16.533

Lens 1.416 4.0 32.295-5.0 16.533

Vitreous 4/3 16.69621

Gaussian properties

Relaxed eyes Accommodated eyes'Exact' Simplified Reduced 'Exact' Simplified

GulLI Le Grand Gull-Ems B-R Emsley Gull. LeGrand Gull-Ems

Power (D) 58.636 59.940 60.483 60 60 70.576 67.677 69.721Eye length 24.385 24.197 23.896 24.086 22.222VV' 7.2 7.6 7.2 7.3 7.2 7.7 7.2OV 92.000 141.792 116.298Accom, level (D) 0 0 0 0 0 10.870 7.053 8.599

Cardinal point positionsVF -15.706 -15.089 -14.983 -15.156 -16.667 -12.397 -12.957 -12.561VF' 24.385 24.197 23.896 24.086 22.222 21.016 21.932 21.252VP 1.348 1.595 1.550 1.511 0.000 1.772 1.819 1.782VP' 1.601 1.908 1.851 1.819 0.000 2.086 2.192 2.128VN 7.078 7.200 7.062 7.111 5.556 6.533 6.784 6.562VN' 7.331 7.513 7.363 7.419 5.556 6.847 7.156 6.909PN = P'N' 5.730 5.606 5.511 5.600 5.556 4.761 4.965 4.781FP=N'F' 17.054 16.683 16.534 16.667 16.667 14.169 14.776 14.343P'F'=FN 22.785 22.289 22.045 22.267 22.222 18.930 19.741 19.124N'R' 17.539 17.041 16.987F'R' 3.370 2.264 2.644

Continued overleaf

254 AI'I'<'IldiC<'s

Gaussian properties continued

Relaxed<'l/<'S'Exact' Simplified

Gull. 1 LeGrand Gull-Ems B-RReducedEmsley

Acc(1/1l1/lOdall'd cue»'Exact' Si/ilpl!fied

Gull. LeGrand Gull-Ems

Pupils: rNA for 8 /III/I diameter PI/Pi/)VE 3.047 3.038 3.052 3.048 0.000 2.668 2.660 2.674VE' 3.665 3.682 3.687 3.699 0.000 3.212 3.255 3.249MEA 1.133 1.131 1.130 1.131 1.000 1.117 1.115 1.114ME'A 1.031 1.041 1.036 1.036 1.000 1.051 1.055 1.049NA 0 0 0 0 0 0.0423 0.0277 0.0375NA' 0.2345 0.240 0.242 0.240 0.240 0.2374 0.2414 0.2710m 0.823085 0.813243 0.818128 0.817532 0.750000 0.795850 0.791122 0.796683E'R' 20.720 20.515 20.209 20.387 22.222 21.173 20.942 20.647

Seidel aberrations

S/8 S2/ 2 S~/4 S~/4 %SS CL/2 %CT

Relaxed eyesGullstrand exact eye (by surface contribution)1 23.6327 9.6202 0.4895 1.8447 -0.0800 9.5960 -0.32872 -2.0030 -0.7207 -0.0324 -0.1663 0.0060 -0.8665 0.02623 0.1577 0.2090 0.0346 0.1404 -0.0195 0.4821 -0.05384 0.2057 0.2100 0.0268 0.0675 -0.0081 0.2489 -0.02145 5.1871 -1.0374 0.0259 0.0926 0.0020 0.5175 0.00876 11.7803 -2.0552 0.0449 0.2340 0.0041 1.2479 0.0183Total 38.9605 6.2258 0.5893 2.2127 -0.0955 11.2259 -0.3506

Le Grand 39.1445 6.2642 0.5619 2.2578 -0.1012 11.3592 -0.3694Gullstrand-Emsley 41.2324 5.8719 0.5901 2.2731 -0.0990 11.5061 -0.3603Bennett-Rabbetts 38.5293 5.8075 0.5819 2.2534 -0.0997 11.4031 -0.3622Emsley 59.4091 28.8757 1.7543 2.3392 -0.1674 12.1682 -0.4977

Accommodated eyesGullstrand exact eye (by surface contribution)1 29.6779 12.4057 0.6482 1.8447 -0.0877 9.8212 -0.34552 -2.5355 -0.9513 -0.0446 -0.1663 0.0067 -0.8856 0.02803 5.6696 2.8705 0.1817 0.2632 -0.0190 1.2617 -0.05384 22.8734 6.7077 0.2458 0.2009 -0.0110 1.0154 -0.02515 34.7899 1.8831 0.0127 0.2009 -0.0010 1.0154 -0.00466 15.8359 -1.4826 0.0173 0.2632 0.0021 1.3966 0.0110Total 106.3111 21.4332 1.0613 2.6066 -0.1098 13.6247 -0.3900

LeGrand 53.5239 12.0929 0.8439 2.510-1 -0.1126 12.9305 -0.4044Gullstrand-Emsley 65.9357 14.0976 0.9216 2.5765 -0.1092 13.4028 -0.3921

Finite schematic eyes

List of eyes

Lotmar (1971)Kooijman (1983)Navarro et al. (1985)Liou and Brennan (1997)

Further information about these model eyes is given in Chapters 16 and 17.

Schematic eyes 255

Lotmar (1971) - same as Le Grand eye except for surface asphericities

Medium /I R Q d Equivalent powersSurface Component Whole eye

Air 1.00007.8 -D.286* 48.346

Cornea 1.3771 0.55 42.3566.5 0 -6.108

Aqueous 1.3374 3.05 59.94010.2 0 8.098

Lens 1.4200 4.00 21.779-6.0 -1.0 14.000

Vitreous 1.3360 16.59655-12.3

"Lotmar represented the front surface of the cornea by equation (17.23), and the above value of Q is the value for a figured conicoid filled tu equation (17.25).The conicoid aspheridty and figuring co-efficients are:Q = ..j).2857143,f. = 0.0./. = -2.547626E.()6'/. = -8.104263E-tJ9./", = -6.660308E-ll./12 = -5.864599E-13The termination of the figuring co-efficients at the "'2 term gives an error of less than 2E.()6at a ray height of 1/= 4 mm.

Kooijman (1983) - same as Le Grand eye except for surface asphericities

Medium /I R Q d Equivalent powersSurface Component Whole eye

Air 1.00007.8 -D.25 48.346

Cornea 1.3771 0.50 42.3566.5 -D.25 --{;.108

Aqueous 1.3374 3.05 59.94010.2 -3.06 8.098

Lens 1.4200 4.00 21.779--{;.O -1.0 14.000

Vitreous 1.3360 16.59655

Two models of retinal radius were offered; one with a radius of curvature of -10.8 mm and a Q value of 0, and the other with a radius of curvature of -14.1 mmand a Q value of 0.346.

Navarro et al. (1985)

Medium /I R Q d Equivalent powersSurface Component Whole eye

Air 1.00007.72 -D.26 48.705

Cornea 1.376 0.55 42.8826.5 0 -5.983

Aqueous 1.3374 (3.05)d2 60.416(1O.2)R3 (-3.1316)Q3 8.098

Lens (1.42)/13 (4.oo)d3 21.779(--{;.0)R4 (-1.0)Q4 14.000

Vitreous 1.3360 16.40398-12.0

The bracketed values are for the relaxed condition.

256 Appendices

The model is set any level of accommodation by the following equations:R3 =10.2 - 1.75 In(A + 1)R4 =-6.0 + 0.2294 In(A + 1)dz 3.05 - 0.05 In(A + 1)d3 = 4.0 + 0.1 In(A + 1)n3 = 1.42 + 9x10-5 (lOA + AZ)Q3 =-3.1316 - 0.34 In(A + 1)Q4 =-1.0 - 0.125 In(A + 1)

where A is the accommodation level (dioptres).For example, at 10 D of accommodation,

Medium II R Q d Equivalent powersSurface Component Whole eye

Air 1.00007.72 -0.26 48.705

Cornea 1.376 0.50 42.8826.5 0 -5.938

Aqueous 1.3374 2.930110 71.1456.00368 -3.94688 16.756

Lens 1.438 4.23979 34.548-5.44992 -1.29974 18.716

Vitreous 1.3360 16.28415-12.0

Liou and Brennan (1997)

Medium n R Q d Equivalent powersSurface Component Whole eye

Air 1.0007.77 -0.18 48.391

Cornea 1.376 0.55 42.2626.40 -0.60 -6.250

Aqueous 1.336 3.1612.40 -0.94 2.581 60.314

Lens Grad A 1.59 6.28322.134

Lens Grad P 2.43 9.586-8.10 +0.96 3.950

Vitreous 1.33616.23883

Gradient index details of lens:

Grad A

NO•O= 1.368No.1 = 0.049057NO•2=-0.015427N1•o= -0.001978

No retinal radius of curvature was provided.The stop is displaced 05 mm from the optical axis to the nasal side,

Grad P

1.4070.000000-0.006605-0.001978

Schematic eyes 257

Gaussian properties

Lotmar', Navarro Liou andKooijman' Relaxed Accomm (10 D) Brennan

Eye length 24.196552 24.003979 24.003979 23.950014Power 59.940 60.416 71.145 60.343

Cardinal pointpositions-12.051 -15.040VF -15.089 -14.969

VF' 24.197 24.004 21.172 23.950VP 1.595 1.583 2.005 1.532VP' 1.908 1.890 2.393 1.810VN 7.200 7.145 6.727 7.100VN' 7.513 7.452 7.116 7.378PN = P'N' 5.606 5.561 4.723 5.568FP=N'F' 16.683 16.552 14.056 16.572P'F'=FN 22.289 22.114 18.779 22.140F'R' 0.000 0.000 2.832 0.000

Pupils:3.098VE 3.038 3.042 2.928

VE' 3.682 3.682 3.551 3.720MEA 1.133 1.133 1.128 1.133ME'A 1.041 1.041 1.058 1.035ill 0.813243 0.814493 0.797668 0.819238

'Same as LeGrand full theoretical eye.

Seidel aberrations

51/8 52/2 5/4 54/4 %55

Relaxed eyesLotmar (1971) 20.3144 7.1107 0.3392 2.2578 -0.1001Kooijman (1983) 15.7668 6.2829 0.3648 2.2578 -0.0995Navarro et al. (1985) 13.8628 7.0582 0.3421 2.2775 -0.0991Liou and Brennan' (1997) 7.3745 1.6754 0.5424 1.8371 -0.2064

Accommodated eyesNavarro et al. (1985) (10 D) -26.1827 9.0316 0.4876 2.6315 -0.1019

"The stop is displaced 0.5 mm from the optical axis to the nasal side, but this has been ignored for the analysis.

Summary of symbols

nrdE, E'F, F'P, piN,N'V, V'

refractive indexradius of curvaturesurface separationsEntrance and exit pupilsFront and back focal pointsFront and back principal pointsFront and back nodal pointsFront and back vertex points

NA,NA'

Position of retina (at axial pole)magnification of entrance pupiland exit pupilratio of the paraxial pupil rayangles in image and objectspacenumerical apertures in objectand image spaces

258 Appelldict',;

ReferencesEmsley, H. H. (1952). Visual Optics, vol. 1, 5th edn, pp.

4Q.41, 346. Butterworths.Gullstrand, A. (1909). Appendix 11: Procedure of rays inthe eye. Imagery -laws of the first order. In Helmholtz'sHandbucll der Pllysiologiscllell Opti«, vol. 1, 3rd edn(English translation edited by J. P. Southall, OpticalSociety of America, 1924).

Helmholtz, H. von (1909). Halldbllcll dcr PllysiologischellOptik, vol. 1, 3rd edn (English translation edited by J. P.Southall, Optical Society of America, 1924).

Kooijrnan, A.C. (1983). Light distribution on the retina ofa wide-angle theoretical eye. J. 01'1. Soc. Am., 73,1544-50.

LeGrand, Y. and ElHage. S. G. (1980). Physiological Optics.Translation and update of Le Grand Y. (1968). Ladioptriquc de l'oeil e! sa correction, vol. I of Optiqucpll)fSiologiql/e, pp. 65-7. Springer-Verlag.

Liou, H.-L. and Brennan, N. A. (1997). Anatomicallyaccurate, finite model eye for optical modeling. f. Opt.Soc. Am. A., 14, 1684-95.

Lotrnar, W. (1971). Theoretical eye model with asphericsurfaces. J. 01'1. Soc. Am., 61, 1522-9.

Navarro, R., Santamaria, J. and Bescos, J. (1985).Accommodation-dependent model of the human eyewith aspherics. f. Opt. Soc. Am. A., 2, 1273-81.

Rabbetts, R. B. (1998). Benllettalld Rabbett,' Clinical VisualOptics, 3rd edn, pp. 209-13. Butterworth-Heinemann.

A4Calculation of PSF and OTF fromaberrations of an optical system

The point spread function (PSF)

Weshow how to calculate the PSF, taking intoaccount diffraction, defocus, aberrations,polychromatic light, the photopic luminousefficiency function V(It) and the Stiles-Crawford effect. Scatter is more difficult toinclude in these calculations, so we willneglect it.The PSF can be calculated if the wave

aberration in the pupil of the eye is known.From optical image formation theory (seeSmith and Atchison, 1997), the PSF is relatedto the wave aberration via a Fourier trans-form. The background for the followingequations is taken from the above reference.Rather than express the PSF as a light distri-

bution at the retina, we will express it as theequivalent distribution projected back intoobject space. The major difference will be that,in the first case spatial co-ordinates will be inlinear quantities such as millimetres, and inthe second case they will be in angular units.Calculating the point spread function back inobject space, taken as air, also avoids the needto use the image space refractive index in thediffraction integral.Before we present equations for calculating

the PSF from the wave aberration in the pupil,we must first distinguish the amplitude PSFfrom the intensity PSF. The amplitude PSF isthe complex amplitude of the light distri-bution, whereas the intensity PSF is the actuallight distribution that we measure with a lightmeter. When we refer to the PSF, we mean the

intensity PSF unless otherwise indicated.If we denote the amplitude PSF as ga(u,v),

where u and v are the directions in objectspace, and the PSF as g(u,v), these twoquantities are related by the equationg(u,v) =ga(u,v)g*a(u,v) (A4.1)where * refers to the complex conjugate. Theamplitude PSF is related to the waveaberration W(X,Y)by the equationga(u,v) =CJJEP(X,y)e-i2lt(UX+vY)dXdY (A4.2)

whereC is a constant = (II It) (A4.3)E implies integration over the pupil of radius15, (u, v) are related to the actual angles (inradians) Ox and 0y in the X and Y directions,respectively, by the equationsOx = Itu and 0y = ltv (A4.4)(X, Y) are the cartesian co-ordinates in thepupil, and P(X, Y) is the complex amplitude inthe pupil known as the pupil function, whichis mathematically defined asP(X, Y) = A(X,Y)e[-ikW(X,Y)) (A4.5)

The constant k =2rrl It, A(X, Y) is the ampli-tude transmittance at the point (X, Y) in thepupil, and is included to allow for theStiles-Crawford effect. W(X, Y) is the waveaberration as an optical path difference, and isexpressed in normal units of distance and notwavelength.Equation (A4.2) is a Fourier transform of

260 Appel/dices

the pupil function P(X/ Y)/ and is zero outsidethe pupil. However, it would not be a Fouriertransform if the object space variables werethe real angles Ox and 0y/ instead of II and v. Inother words, the use of II and v and not Ox and0y allows the amplitude point spread functionto be expressed as a Fourier transform.Wewill regard the Stiles-Crawford effect as

rotationally symmetric in the pupil, and canwrite it asA(X, Y) =eHfJ/Z)(X

2+ ¥2)j (A4.6)

where f3 is the Stiles-Crawford attenuationfactor to base e and the //2/ factor is includedbecause we must use an 'amplitude'Stiles-Crawford effect and not the normal'intensity' effect (Krakau, 1974). Typicalvalues for f3 are given in Chapter 13.

The wave aberration functionW(X, Y)

The wave aberration function W(X/ Y) is des-cribed in detail in Appendix 2. It is one way ofquantifying the level of aberrations. Thefunction is a polynomial in X and Y/ and thedifferent terms represent the differentaberrations - e.g. spherical aberration andcoma. It can also incorporate defocus andchromatic aberrations.

DefocusIf the eye is defocused by an amount L1F/ e.g.0.5 0/ a defocus term WZO(X2 + y2) can beadded to the wave aberration polynomial,where L1F and the co-efficient Wz0 are relatedby equation (A2.5), i.e. '

Wz,o = L1F/2 (A4.7)For example, a defocus of +0.5 0 givesWz,o = +0.00025mm".

Chromatic aberrationAs discussed in Chapter 17/ longitudinal andtransverse chromatic aberration arise becauseof the dispersion of the ocular media. Thedispersion leads to a chromatic change inpower L1F(,t) given by the equationL1F(,t) = F(,t) - F(X) (A4.8)

where F(A) is the power at the wavelength Aand X is the reference wavelength for zerochromatic aberration. Equations for F(,t) aregiven in Chapter 17.For on-axis and off-axis point spread

functions, longitudinal chromatic aberrationis taken into account with a wave aberrationterm Wz,o(,t)(Xz + yZ). The co-efficient isrelated to the chromatic change in powerL1F(A) by the equationWzo(,t) =L1F(,t)/2 (A4.9)For off-axis point spread functions, transversechromatic aberration is taken into accountwith a wave aberration term W11(,t)Y orW1,1(A)X. The co-efficient is related to thechromatic change in power L1F(,t) by theequationW1,1(,t) =OENL1F(,t)/n(,t) (A4.1O)where 0 is the angular distance off-axis, EN isthe distance from the entrance pupil to thefront nodal point, and n(,t) is the refractiveindex of the vitreous medium. Transversechromatic aberration leads to a PSF havinga transverse shift that is wavelength depen-dent.

Polychromatic sourcesFor polychromatic light sources, the PSF iscalculated at a number of wavelengths. Thechromatic aberrations are included in thewave aberration function. Each point spreadfunction is weighted by both the relativesensitivity of the eye V(,t) (Chapter 11) and therelative radiance of the source S(,t). Thepolychromatic point spread function isformed by adding the weighted individualpoint spread functions, but only after thesehave been expressed in terms of the real?ngles Ox and 0y/ instead of 1I and v. That1S,

(A4.11)

Computation checksFor any computation of the point spreadfunction via equation (A4.2), it is goodpractice to have independent checks of thefinal result. The following three conditions

Calculation of PSFand OTFfrom a/Jerratiolls ofalloptical system 261

can be used:1. If the system is free of aberration:

g(O, 0) = Iga(O, 0) 12 = IcIJEP(X, y)dXdY I 2={(n/ A){1-exp[-(/312)p211/(/312)}2

(A4.12)If no Stiles-Crawford apodization ispresent, i.e, /3 =0,g(O, 0) = [(nl A)p2]2 (A4.12a)

2. Volume under PSF=effective flux passingthrough pupil= n(1- exp(-/3p2)] I /3 (A4.13)If no Stiles-Crawford apodization ispresent, i.e, /3 = 0,Volume =np2 (A4.13a)

3. For an off-axis calculation on a rotationallysymmetric system and a single wave-length, the peak of the point spreadfunction should occur at the angle inequation (A4.l0), i.e,

0peak = "0 (A4.14)

The optical transfer function (OTF)

A sinusoidal pattern with a real spatialfrequency (Jand an orientation", is shown inFigure A4.1. The spatial frequency com-ponents in the X and Ydirections are(Jx =(Jcos(l/f) and (Jy =(Jsin(",) (A4.15)The two dimensional OTF,which we write asG«(Jx,(Jy)' is the Fourier transform of the PSF

y

y

xo

Object (or image) plane

Figure A4.1. Sinusoidal pattern of grating inclined at anangle IJI.

g(0x,Oy)' that is+00+00

G(crx' cry) = f fg(Ox' Oy)e-i2n(8x<1x +8y<1y)dOxdOy-<»-00 (A4.16)

However, calculation of the OTF by this directrelationship is not necessarily the bestapproach, because the PSFhas no bounds andthe integral should therefore be carried outover an infinite range. In practice we must usefinite bounds but, depending upon how muchof the PSF is outside these bounds, there willbe errors in the final results. A better alter-native is to use a Fourier transform identity(Bracewell, 1986), that states that the Fouriertransform of the product of the transforms of

y

x

(a) (b)

Figure A4.2. The sheared pupils used to calculate the OTF from equations (A4.17) and (A4.18).

262 Appendices

References

Summary of main symbols

Point spread function (PSF)angles in X and Ydirectionscartesian co-ordinates in the actualpupilStiles-Crawford parameter to baseeoptical transfer function (OTF)spatial frequency (c/rad)components of 0" in the X and Ydirectionscorresponding modified spatialfrequencies, related to (O"x,O") byequation (A4.17a) .

Bracewell, R. (1986). Tile Fourier Transform and itsApplications, revised edn, p. 112.McGraw-Hill.

Krakau, C. E. T. (1974). On the Stiles-Crawfordphenomenon and resolution power. Acta Oplllilal., 52,581-3.

Macdonald, J. (1971). The calculation of the opticaltransfer function. Optica Acta,18, 269-90.

Smith, G. and Atchison, D. A. (1997). Image qualitycriteria. In The Eye and Visllal Optical Instrllmellts.Cambridge University Press pp. 647-72.

f3wherelOx =a). and lOy =O"y'A (A4.17a)replace o"x and O"y in G and the integrand.Since the pupil function P(X, Y) is zero

outside the pupil, the limits of integrationreduce to the region 'c' common to the twosheared pupils shown shaded in Figure A4.2a,and we do not have to write the integral limitsfrom -00 to +00, A useful alternative andequivalent form of this integral is

G(lOx'lOy)=IfcP(X + lO)2, Y+ lOyl2)P*(X- lO)2, Y- lOyl2)dXdY (A4.18)

with the region of integration now being thatshown in Figure A4.2b.Macdonald (1971) has described such a

method for the routine calculation of the OTFbased upon the above auto-correlationintegral, but with the amplitude functionA(X, Y) = 1.

It is common practice to normalize the OTFso that G(O, 0) =1.

two functions is the convolution of twofunctions. In this case, equation (A4.16)reduces to a convolution of the pupil functionP(X, Y) with its complex conjugate, as follows:

G(lOx'lOy)=Ifcp(X, Y)P*(X - lOx' Y- lOy)dXdY(A4.17)

Index

Abbe V-value, 189Aberrationsand age, 227and depth-of-field, 217, 220higher order, 151longitudinal, 137, 242primary (or third order), 244-8Seidel, 245-8transverse, 137, 242wave, 137, 242

Aberrations, chromaticlongitudinal, 180effect of accommodation, 185, 191effect on accommodation, 186effect of age, 228effect of refractive error, 185, 190-2effect on spatial vision, 186,205magnitude, 186measurement techniques, 183and refraction, 71and wavelength in focus, 185

Seidel, 190,245transverse, 181magnitude, 186measurement techniques, 185

Aberrations, monochromaticastigmatism, 147-9, 164coma, 146, 162ocular component contributions, 154distortion, ISO, 165field curvature, 149, 165higher order, 151measurement techniques, 138-42sagittal and tangential errors, 148, 163schematic eyes, 161-6, 173-6,254,257

Seidel, 161-5, 167, 170, 171, 173,245-8spherical aberration, 143-6, 161

Aberrations, ophthalmic devices, ISS, 188Aberrations, pupil, 155Aberroscope technique, 141Absorptionfundus, 129, 131ocular media, 107-11

Accommodation, 18, 223amplitude, 19and age, 224

effect of longitudinal chromatic aberration,186

effect on longitudinal chromatic aberration,185, 191

mechanism, 223Accommodation demandocular, 93spectacle, 93

Achromatizing correcting lenses, 189Ageand aberrations, 227and amplitude of accommodation, 224and cornea, 221and lens, 222, 230and photometry, 228and pupil size, 227and refractive errors, 61, 223and schematic eyes, 229and Stiles-Crawford effect, 229and visual performance, 228

Airy disc, 196Aliasing, 203Ametropia see Refractive errorsAmplitude of accommodation, 19,224

264 Index

Anglealpha (a), 35gamma ()1, 37kappa (K"), 36lambda (it), 35psi (If/), 37

Aniseikonia, 54, 92Anisometropia, 55, 61Annulus method, 140Anterior chamber, 4Aperture stop, 21Aphakia, 57Aqueous, 4transmittance, 110

Artificial pupils and defocus, 82Aspheric surfaceconicoid, 13, 166figured conicoid, 167

Asphericityof cornea, 13-6,168of lens, 17, 168

Astigmatism (refractive error), 60Axisachromatic, 32, 37fixation, 33, 37foveal achromatic, 32keratometric, 33, 34line of sight, 31, 34, 35optical, 8, 30, 35pupillary, 32-33, 35visual, 8, 32, 34, 35, 37

Badal optometer, 68-70Binocular overlap, 9Binocular vision, 8-9Birefringence, 114cornea, 114form, 114intrinsic, 114lens, 115nerve fibre layer, 132

Black body source, 122Blind spot, 7Blur disc see Defocus blur discBowman's membrane, 12

Capsule of lens, 16Cardinal points, 7, 240of finite schematic eyes, 257focal, 7, 240nodal, 7, 52-4, 80, 82, 240of paraxial schematic eyes, 41, 253

principal, 7, 240Centre-of-rotation, 8, 33Choroid, 4, 7absorption, reflectance, and transmission

130,131Chromatic aberration see Aberration,

chromaticChromatic difference of magnification, 182Chromatic difference of position, 182Chromatic difference of power, 181, 190Chromatic difference of refraction, 181, 190Chromatic dispersion, 189Chromo-retinoscopy,184Chromostereopsis, 187Ciliary body, 4, 223Circle of least confusion, 150Component ametropia, 62Cones,S, 127Conicoid, 14-6, 166ellipsoid, 14hyperboloid, 14paraboloid, 14

Contact lensesaberrations, 156

Contrast sensitivity function (CSF), 199,200,203,204

effect of defocus, 205Cornea, 3, 11-6and age, 221asphericity, 13-6, 168birefringence, 114power, 12radii of curvature, 12, 15refractive index, 12sagittal (axial) radius of curvature, 15scatter, 113surface powers, 12tangential (instantaneous) radius of

curvature, 15transmittance, 108toricity, 13

Corneal astigmatism, 60, 223Corneal sighting centre, 31Correlation ametropia, 61coso! law, 119Crystal,biaxial, 114uniaxial, 115

Cylindrical lens and axis, 60, 63

Defocusand alignment of objects at different

distances, 85

effect on visual acuity, 85and wave aberration, 244, 260

Defocus blur disc, 82-5, 214, 217Depth-of-field, 213-20and aberrations, 217and pupil size, 214and target size, 215, 218effect of luminance, 215geometrical approximation, 217

Depth-of-focus see Depth-of-fieldDescemet's membrane, 12and age, 221

Detection task, 203Diffraction, 194, 195, 198and depth-of-field, 217laser speckle, 70, 183

Diplopia with ophthalmic lenses, 94Double pass techniques, 194, 200Duochrome test, 71

Eccentric fixation, 32, 36Eccentricity, 14Eikonometer, 55Emmetropia, 57, 58Emmetropization, 61Entrance and exit pupils, 21and accommodation, 22of paraxial schematic eyes, 42

Equivalent focal length, 7Equivalent power, 7, 240Equivalent sphere (mean sphere), 60Equivalent veiling luminance, 121, 122and age, 229

Far point, 19, 57, 58, 59, 62, 92Field-of-view of ophthalmic lenses, 94Field-of-vision, 8of ophthalmic lenses, 94

Figured conicoids, 167Finite (real) ray, 237Fluorescencelens, 113

Fluorogens,113Focal points, 7, 240Fovea, 6image quality, 204-7

Foveola, 6Fundus, 67, 129absorption, 129, 131reflectance, 130scatter, 131

Index 265

Gaussian optics, 241Gradient refractive index of the ocular lens

17,39,41,168-71and age, 222, 229

Grating focus, 74

Halation, 131Half-width of point spread function, 197Heterotropia (tropia), 36Hippus, 24Hirschberg test, 36Hypermetropia (hyperopia), 59

Illuminance, 102retinal, 117-22

Image quality criterialine spread function, 195, 198modulation transfer function, 198,200,204,

205,207,208optical transfer function, 198-202,261phase transfer function, 199, 204point spread function, 195-98, 201, 204,

259-61wave aberration function, 138, 243-5

Inter-pupillary distance (PO), 9Intraocular lensesaberrations, 156, 188

Inverse square law, 103Iris, 4, 21Iseikonic lenses, 55Isogonal lenses, 55

Knapp's law, 91Knife edge tests, 141

Lambertian sources, 102Laser speckle, 70and measuring chromatic aberration, 183and measuring depth-of-field, 216

Lens equation, 240Lens of the eye, 16-8and age, 222asphericity, 17, 168birefringence, 115diameter, 18fluorescence, 113-4power, 18, 169-71principal points, 18radii of curvature, 17

266 lndcx

Lens of the eve, (cont.)refractive i~dex distribution, 17, 168-71,

222scatter, 113thickness, 17transmittance, 110

Lensesthick, 62, 63

Lenses, ophthalmiccontact, 156, 188intraocular, 156, 188iseikonic, 55isogonal, 55spectacle, 60, 63, 155, 188

Line spread function (LSF), 195, 198Luminance, 102Luminous efficacy, 100, 101Luminous flux, 101Luminous intensity, 101Luminous transmittance, 111

Macula, 7Magnificationpupil,90relative spectacle, 91rotational,93spectacle, 88transverse (lateral), 239, 241

Maximum spectral luminous efficacies ofradiation

for photopic vision, 100for scotopic vision, 101

Maxwellian view, 123Mean sphere (equivalent sphere), 60Mesopic vision, 100, 101Mie theory, 112Modulation transfer function (MTF), 198,200,

204,205,207,208Myopia, 58

Near point, 19,58,59,92Nerve fibre layer (retinal), 5, 132Nodal points, 7, 52-4, 80, 82, 240of schematic eyes, 41, 45, 253, 257

Nyquist limit, 203

Ophthalmic lenses see Lenses, ophthalmicOphthalmometric pole, 32, 34

Ophthalmophakometer, 35Optic disc, 7Optical invariant, 240, 246Optical path length, 242Optical radiation, 99Optical transfer function (OTF), 194, 198-200,

261calculation from diffraction and

aberrations, 261diffraction limited, 199effect of defocus, 200-2

OptornetersBadal, 68-70incorporating Scheiner principle, 72laser speckle, 70polarizing, 72telescopic, 70

Parallax movement between object andimage, 74

Paraxial approximations, 238Paraxial marginal ray, 22, 43, 246Paraxial optics, 237-41Paraxial pupil ray, 22, 243, 246Paraxial pupil ray angle ratio, 43, 53Paraxial refraction equation, 238Paraxial transfer equation, 239PO (inter-pupillary distance), 9Phase transfer function (PTF), 199, 204Photography (photorefraction), 75Photometric efficiency and Stiles-Crawford

effect, 125Photometryand age, 228equivalent veiling luminance, 121, 122illuminance, 101inverse square law, 103luminance, 102luminous efficacy, 100, 101luminous flux, 101luminous intensity, 101photon density levels at retina, 122reflectance, 105-7relation between photometric quantities,

103-4seaHer, 111transmittance, 107-11troland, 118units, 101-3

Photon density levels at retina, 122Photopic vision, 100Photorefraction, 75Piper's law, 103

Point spread function (PSF), 121, 195-8,201,204,259-61

diffraction limited, 195half-width, 197polychromatic sources, 260and Rayleigh criterion, 197and Strehl intensity ratio, 197

Polarizationretina, 131, 132

Polymegathism, 221Posterior chamber, 4Powerequivalent, 240of a surface, 238thick lens, 62, 88thin lens, 62vertex, 89

Presbyopia,57,59,224-7theories, 225-7

Principal points, 7, 240of lens, 18

Pupiland accommodation, 22artificial, 28, 82auto-correlation of function of, 260centration, 23entrance, 21exit, 21function, 259magnification, 90senile miosis, 227shape, 25-7size, 23-5and age, 24, 227, 228and binocular vision and

accommodation, 24and depth-of-field, 214-8and drugs, 24and illumination level, 23measurement, 28and retinal image quality, 204, 208and retinal light level, 118, 123significance, 27

Pupillometry, 28Purkinje images, 106Purkinje shift, 101

Radiant power (radiant flux), 99Rayextra-ordinary, 114finite (real), 237ordinary, 114paraxial, 238

Index 267

paraxial marginal, 22, 43, 246paraxial pupil, 22, 43, 246pupil nodal, 32

Rayleigh criterion, 197Rayleigh-Gans (Rayleigh-Debye) theory, 112Rayleigh scattering, 112Ray tracingfinite, 237paraxial, 238

Reduced aperture model and Stiles-Crawford effect, 125

Reference sphere, 245Reflectance, fundus, 130Reflection, specular, 105-7Refractiondiscrepancies between subjective and

objective refraction, 77factors influencing, 75-7in presence of spherical aberration, 152ocular, 63spectacle, 62

Refraction techniquesobjective-only, 73-5grating focus, 74parallax movement between object and

image, 74photography, 74retinoscopy, 73visual evoked response (VER),75

subjective-objective, 71-3remote refraction and relay systems, 71Scheiner principle, 72split image and vernier acuity

(coincidence method), 72subjective only, 67-71laser speckle, 70longitudinal chromatic aberration, 71simple perception of blur, 67-70

Refractive errorsand age,61anisometropia, 61astigmatism, 60component ametropia, 62axial,62refractive, 62

correlation ametropia, 61distribution, 61effect of ocular parameter change, 64emmetropia, 57, 58hypermetropia (hyperopia), 59and longitudinal chromatic aberration, 185,

190-2myopia, 58population distribution, 61

268 Index

Refractive indexof the cornea, 12of the lens, 17continous refractive index model, 168-70shell model, 170

Relative spectacle magnification, 91and axial ametropia, 91and Knapp's law, 91and refractive ametropia, 91

Remote refraction and relay systems, 71Resolution limit, 199, 203Resolution task, 203Retina, 5-7birefringence, 132fovea, 6-7macula, 7polarization, 131, 132shape, modelling, 171

Retinal illuminance, 117-22Retinal image quality, 194-208fovea, 204-7effect of defocus, 205effect of polychromatic light, 205effect of pupil decentration, 206

peripheral vision, 207Retinal image sizedefocussed image, 79-86defocus blur disc, 82-5defocus ratio, 84-6

focused image, 52-4eye focused at infinity, 54

Retinoscopy, 73, 153Ricco's law, 103Rods, 5-6Rotational magnification, 93-94

Sagittal power error, 148, 163,247Sagittal section, 147Scatter, 111-3cornea, 113fundus, 131lens, 113Mie theory, 112Rayleigh, 112

Scheiner principle, 72and measuring monochromatic aberration,

140Schematic eyes, finite (wide angle)aberrations, 173-6, 257Chromatic eye, 173, 192Drasdo and Fowler, 172, 173-6Indiana eye, 173, 174, 192Kooijman, 172, 174-6,255,257

Liou and Brennan, 172-173, 174-7, 256, 257Lotmar, 171-172, 174-6, 255, 257Navarro et al., 172, 174-6,255-7retinal illuminance, 176

Schematic eyes, paraxial, 40-6aberrations, 161-6, 190, 254and age, 229Bennett and Rabbetts reduced, 45Bennett and Rabbetts simplified, 45, 250,

252-4development, 40Emsley reduced, 45, 252-4entrance and exit pupils, 42-43Gaussian properties and cardinal points,

41-42Gullstrand number 1 (exact), 44,161-6,

190,251-4Gullstrand-Emsley (simplified), 45, 251,

253-4Le Grand full theoretical, 44, 251, 253-4Le Grand simplified, 45variable accommodating, 45

Sclera, 3seatter, 130

Scotoma with ophthalmic lenses, 94Scotopic vision, 100, 101Seidel aberrations, 245-8influence of asphericity, 167influence of gradient index, 170, 248influence of retinal curvature, 163-5, 171,

247of finite schematic eyes, 173,257of paraxial schematic eyes, 161-165,

190-191,254Seidel approximation, 161, 162, 164Senile miosis, 227Simple perception of blur, 67-70Snell's law, 237Spectacle lensesaberrations, 155-6, 188

Spectacle magnification, 88-90Spectral luminous efficiency functionfor photopic vision, 100for scotopic vision, 101

Spherical aberration, 143-6, 161, 244effect on depth-of-field, 220effect on refractive error, 152effect on retinoscopy, 153

Split image and vernier acuity (coincidencemethod), 72

Spurious resolution, 200Stereopsis, 9, 54Stiles-Crawford effect, 124-7, 260, 261and accommodation, 126

and age, 229and eccentricity, 126and influence on image quality, 206and luminance, 126measurement, 126and photometric efficiency, 125and pupil function, 260and reduced aperture model, 125role, 127theory, 126and wavelength, 126

Strehl intensity ratio, 197Symbols, Greek, ix

Tangential section, 147Tangential power error, 148, 163, 247Thick lenses, 62, 63Transmittance of eye, 107-11aqueous, 110cornea, 108direct, 108lens, 110luminous, 111total,108vitreous, 110whole eye, 107

Index 269

Transverse aberrations, 137, 242Troland, 118Tropia (heterotropia), 36Tryptophan, 113

V-value (or Abbe V-Value), 189Vernier alignmentand chromatic aberration, 183, 185and monochromatic aberration, 139and refraction, 72

Vertex distance, 62Vertex normal of cornea, 33Visual centre of the cornea, 31V~sual Evoked Response (VER),75VIsual performance, 151,203and age, 228

Vitreous chamber, 4Vitreoustransmittance, 110

Wave aberration, 137fun~tion (polynomial), 138, 243-5vanance, 152

Wave-front sensor, 142