Ординатура / Офтальмология / Английские материалы / Optics of the Human Eye_Atchison, Smith_2000

The fovea is the central part of the macula, whose peripheral limits are where the cells of

the outer nuclear layer are reduced to a single row (Hogan et al., 1971). The macular

diameter is 5.5 mm (19°).

The optic disc and blind spot

The vascular supply to the outer layers of the retina is carried in the choroid, which lies between the retina and the sclera. The vascular supply to the inner retina enters the

eye at the optic disc, whose location is shown in Figure 1.1. There are no cones or rods here, and hence this region is blind. The name given to the corresponding region in the visual field

is the 'blind spot'. The optic disc is approximately 5° wide horizontally and 7° vertically,

and its centre is approximately 15° nasally and 1.5° upwards relative to the fovea. Correspondingly, the blind spot is 15° temporally and 1.5° downwards relative to

the point of fixation. Figure 1.5 provides a demonstration of the blind spot.

The cardinal points

Every centred optical system that has some equivalent power (i.e. is not afocal) has six cardinal points that lie on the optical axis.

These are in three pairs. Two are focal points which we denote by the symbols F and F', two are principal points denoted by the symbols P and P', and two are nodal points denoted by the symbols Nand N'. The positions of these

cardinal points in an eye depend upon its structure and the level of accommodation. For an eye focused at infinity, the approximate positions of these cardinal points are shown in

Figure 1.1.More precise positions are given in Chapter 5, where we discuss schematic eyes

+

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

and their properties. These cardinal points are as follows:

1.Focal points (F and F'). Light leaving the front (also first and anterior) focal point F passes into the eye, and would be imaged at infinity after final refraction by the lens if the retina were not in the way. Light parallel to the axis and coming into the eye from an infinite distance is imaged at the back (also second and posterior) focal point F'. Thus, for the eye focused at infinity, the retina coincides with the back focal point.

2.Principal points (P and P'). These are images of (or conjugate to) each other, such that their transverse magnification is +1. That is, if an object were placed at one of these points, an erect image of the same size would be formed at the other point.

3.Nodal points (N and N'). These are also images or conjugates of each other, but have a special property such that a ray from an off-axis point passing towards N appears to pass through N' on the image side of the system, while inclined at the same angle to the axis on each side of the system. Such a ray is called the nodal ray, and when the off-axis point is the point of fixation, the ray is called the visual axis.

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

The equivalent power and focal lengths

One of the important properties of any optical system is its equivalent power. This is a measure of the ability of the system to bend or deviate rays of light. The higher the power, the greater is the ability to deviate rays. We

denote the equivalent power of an optical system by the symbol F. The equivalent

power of the eye is related to the distances between the focal and principal points by the equation

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

varies greatly from eye to eye. Using this

8Basic optical structure of til"1IIII/lIl1l "!I"

power and the commonly accepted refractive index n' of the vitreous chamber (1.336), the focal lengths of the eye are

While the equivalent power of the eye is a very important property of the eye, it is not easy to measure directly. Its value is usually inferred from the other measurable quantities such as surface radii of curvature, surface separations and eye length, and assumed refractive indices of the ocular media.

However, more important than equivalent power is refractive error. The refractive error can be regarded as an error in the equivalent power due to a mismatch between the

equivalent power and the eye length. For example, if the equivalent power is too high

for a certain eye length, the image is formed in front of the retina and this results in a myopic refractive error. If the power is too low, the

image is formed behind the retina and results in a hypermetropic refractive error. Refractive errors are discussed in Chapter 7.

however, we can nominate a mean position for this point. The rotation of the eye in the horizontal plane was studied by Fry and Hill (1962), who found that the mean centre-of- rotation for 31 subjects was about 15 mm

behind the cornea. Often it is assumed to lie along the optical axis.

Field-of-vision

Examination of the pupil from different

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

Axes of the eye

The eye has a number of axes. Figure 1.1 shows two of these: the optical axis and the

visual axis. The optical axis is usually defined as the line joining the centres of curvatures of

the refracting surfaces. However, the eye is

not perfectly rotationally symmetric, and therefore even if the four refracting surfaces

were each perfectly rotationally symmetric, the four centres of curvatures would not be

co-linear. Thus in the case of the eye, we define the optical axis as the line of best fit through these non co-linear points. The visual axis is defined as the line joining the object of interest and the fovea, and which passes through the nodal points. These and the other axes are discussed in greater detail in Chapter

4.

Binocular vision and binocular overlap

The use of two eyes provides better perception of the external world than one eye

Centre-of-rotation

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

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

alone. Binocular vision improves contrast sensitivity and visual acuity slightly over those obtained with monocular vision (Campbell and Green, 1965;Home, 1978).Two

laterally displaced eyes give the potential for a three-dimensional view of the world, which

includes the perception of depth known as stereopsis. The degree of stereopsis depends partly upon the distance between the two eyes, which is called the interpupillary distance. Stereopsis can be improved considerably by optical devices such as rangefinders, which increase the effective

interpupillary distance.

Interpupillary distance

The interpupillary distance, or PO, is usually measured by the distance between the centres of the two pupils of the eyes. The distance PO is measured for the eyes looking straight ahead; that is, the visual axes are parallel. When the eyes focus on nearby objects, the eyes rotate inwards and hence there is a corresponding decrease in interpupillary distance. The near PO can then be measured or determined from the distance PO using

simple trigonometry.

Harvey (1982) provided distance POs for various armed services. Means ranged from

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

Binocular overlap

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

Typical dimensions

All dimensions of the eye vary greatly between individuals. Some depend upon accommodation and age. Representative data are shown in Figure 1.7. Average values have

been used to construct representative or schematic eyes, which we discuss in Chapter 5. More detailed data are presented in later chapters.

Summary of main symbols

Fequivalent power of the eye

n' refractive index of vitreous humour (usually taken as 1.336)

Rradius of curvature in millimetres

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

References

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

191-2.

Curcio, C. A. and Hendrickson, A. E. (1991). Organization

and development of the primate photoreceptor mosaic.

Prog. Retinal Res., 10, 89-120.

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

581-95.

Harvey, R. S. (1982). Some statistics of interpupillary distance. Optician, 184 (4766), 29.

Hogan, M. J., Alvarado, J. A. and Weddell, J. E. (1971).

Histology of theHuman Eye. W. B.Saunders and Co. Home, R. (1978). Binocular summation: A study of

contrast sensitivity, visual acuity and recognition.

Visioll Res., 18, 579-85.

O'Brien, B. (1951). Vision and resolution in the central retina. }. Opt. Soc. Alii.,41, 882-94.

0sterberg, G. (1935). Topography of the layers of rods and cones in the human retina. Acta Ophthal. (Suppl.), 6,

1-103.

Polyak, S. L. (1941). The Retina,

Chicago Press.

Snell, R. S. and Lemp, M. A. (1997).

E)/(', 2nd edn. Blackwell Scientific Publications.

Tyler,C. W.(1996). Analysis of human receptor density. In

Basic alld Clinical Applications of Vision Science, The Professor fay M. Enoch Festschrift Volume (V.

Lakshminarayanan, ed.), pp. 63-71. Kluwer Academic Publishers.

Introduction

The refracting elements of the eye are the

cornea and the lens. In order to provide a good quality retinal image, these elements must be transparent and have appropriate curvatures and refractive indices. Refraction

takes place at four surfaces, the anterior and posterior interfaces of the cornea and the lens,

- Aqueoushumour

p' p

and there is also continuous refraction within the lens.

In this chapter we describe the optical

structure of the normal cornea and lens, but we must be aware that in many eyes there are

significant departures from these norms. Some of these are due to serious ocular defects or irregularities, which can have a major impact on vision. There are also changes with

age, and any descriptions given in this section relate only to mean adult values. More detail

on age dependencies is given in Chapter 20.

Cornea

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

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

The schematic cross-sectional structure of the cornea is shown in Figure 2.1. It has a tear film at its front surface, and several distinct parts. These are, in order from the outer sur-

face of the eye, the epithelium, Bowman's membrane, the stroma, Descemet's membrane and the endothelium. Approximate

thicknesses of these components are given in Table 2.1.

12 Basic optical structureof the human <'!Ie

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

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

The tear film is 4-71J.m thick (Tomlinson, 1992). It is composed of oily, aqueous and

mucous layers, with 98 per cent of the thickness being provided by the aqueous layer. The tear film is essential for clear vision

because it moistens the cornea and smooths out the 'roughness' of the surface epithelial cells. The tear film does not contribute

significant refractive power itself, since it is very thin and consists effectively of two very

closely spaced surfaces of almo~t equal.radi,i. However, the importance of this tear film 1S

realized if it dries out. If this occurs, the transparency of the cornea decreases

significantly.

The epithelium protects the rest of the cornea by providing a barrier against water, larger molecules and toxic substances. It consists of approximately six layers of cell~, and only the innermost layer of these cells 1S able to divide. After cells are formed, they move gradually towards the surface as the

superficial cells are shed.

Bowman's membrane is 8-141J.m thick, and consists mainly of randomly arranged

collagen fibrils.

The stroma comprises 90 per cent of the

corneal thickness, and consists mainly of 200 or more collagen lamellae. The collagen fibrils within each lamella run parallel to each other, and the successive lamellae run across the cornea at angles to each other. This arrange-

ment maintains an ordered transparent structure while enhancing mechanical

strength.

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 together like a honeycomb. The endothelium regulates the fluid balance of the cornea in order to

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

Refractive index

Each corneal layer has its own refractive index, but since the stroma is by far the

thickest layer, its refractive index dominates. The mean value of refractive index is usually taken as 1.376.

Radii of curvature, vertex powers and total corneal power

Several studies have measured the anterior

radius of curvature, but there have been far fewer investigations of the rear surface. Experimental distributions of the vertex radii

(R) of curvature are given in Table 2.2.Similar

results have been obtained by various investigators for the anterior cornea, but clearly there is a reasonable degree of

variation between patients, and females have

slightly steeper anterior corneas than males. There is a high linear correlation between the anterior and posterior radii of curvature (Lowe and Clark, 1973; Dunne et ai., 1992; Patel et ai., 1993) and a reasonable fit of this relationship is

From these radii of curvature, we can calculate the surface powers (F) using the equation

where nand n' are the refractive indices on the incident and refracted side, respectively. At the anterior surface

/1 = 1 and /1' = 1.376

and at the posterior surface n =1.376and n' =1.336

Surface power values, using these data and equations, are given in Table 2.2.

The total power F of the cornea can be calculated from the 'thick lens' equation:

where F} is the anterior surface power, Fz is

the posterior surface power, d is the vertex corneal thickness and Jl is the refractive index of the cornea (usually taken as 1.376). The

power of the cornea can be estimated from the sum of the surface powers

This value is different from the exact value by

an amount FIF2d/u. Using the data given by Patel et al. (1993) in Table 2.2, a corneal thickness of 0.5 mm and a refractive index of

1.376, the exact equation (2.3) gives a corneal power of 42.20 and the approximate equation (2.3a) gives a slightly lower value of 42.1 D.

The above surface power values apply to

the vertices of the cornea, and would apply to other parts of the corneal surfaces only if they were spherical. However, neither the anterior

nor the posterior surfaces are perfectly

spherical due to both toricity and asphericity. Therefore, the radii of curvature do not fully

describe the shape of the cornea and its refracting properties.

Anterior surface shape

Toricity

Frequently the anterior corneal surface

exhibits toricity, which produces astigmatism. In young eyes, the radius of curvature is generally greater in the horizontal than in the

vertical meridian (referred as 'with the rule'), but this trend reverses with an increase in age.

Asphericity

In general, the radius of curvature increases with distance from the surface apex, so that the surface flattens away from the apex. Surfaces that are non-spherical in this sense are often described as aspheric.

The shape of the anterior corneal surface has been extensively studied, especially over the central 8 mm of its approximately 12 mm diameter. This central 'optical' zone is the maximum zone of the cornea through which

light passes to form the foveal image. To

investigate the shape over this zone, corneal surfaces are often represented by conicoids in

three dimensions or conics in two dimensions. A conicoid can be expressed in the form

where

the Z axis is the optical axis h2 =)(2 + y2

R is the vertex radius of curvature and

14 Basic optical structureof tileIIulllan eye

Q is the surface asphericity, where

Q< -1 specifies a hyperboloid

Q=-1 specifies a paraboloid

-1 < Q < 0 specifies an ellipsoid, with the Z-axis being the major axis

Q= 0 specifies a sphere

Q> 0 specifies an ellipsoid with the major axis in the X-Y plane.

The effect of the value and sign of Qon the shape is shown in Figure 2.2. Sometimes asphericity is expressed in terms of a quantity p, which is related to Q by the equation

The conicoid form described by equation (2.4) is not the only mathematical representation used in the literature to describe conic(oid)s. Many investigators have measured surface shapes separately in different sections, and fitted the data to ellipses, which can be described by the equation

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

b by the equation

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 R is related to a and b by the equation

and that the asphericity Q is given by the equation

It follows from equations (2.7) and (2.9), that

the quantities e and Q are related by the equation

Specifying asphericity using e is not

completely satisfactory because e2 may be negative, in which case e cannot have a value.

Other forms of representing corneal shape are described in Chapter 16. More complex forms are important to describe the shape of

the cornea accurately, especially outside the optical zone.

Measured values of the anterior corneal

Ellipsoids: O>Q>-I

\

\

\

z

asphericity (Q) are given in Table 2.3. The

values of Q are usually negative, indicating that the cornea flattens away from the vertex.

Figure 2.3 shows the profiles of corneal surfaces, all with a radius of curvature of 7.8 mm but with different Q values.

Optical significance of corneal asphericity

There has been considerable speculation as to why the cornea flattens away from the centre.

It has been argued that the cornea flattens in order to reduce spherical aberration, and

certainly the flattening does lead to a lower spherical aberration, but the amount of asphericity in the average cornea is not sufficient to eliminate it. The value of Q

5

<)

u

§ 2

'iii

is

~Q=-o.26

--tr- Q =-0.52

---III-- Q =-1.0

o..-.........--,---.-,---.--+-

o I 32

Distance along optical axis (mm)

Figure 2.3. The anterior surface of a cornea with a vertex radius of curvature of 7.8 rom, with various values of the asphericity Q.

required to eliminate spherical aberration at the anterior surface is -0.528, given a corneal refractive index of 1.376. Perhaps an important reason for the flattening is the need for the cornea to make a smooth join with the main globe of the eye.

Off-axis radii of curvature

The radius of curvature at any point on a

spherical surface and in any meridian is the same. However, for a conicoid surface, the

radius of curvature at off-axis points depends not only upon the distance from the vertex, but also on the meridian at that point. There are two principal meridians: the tangential meridian, which lies along the radius line from the vertex, and the sagittal meridian,

which is perpendicular to the tangential meridian. For conicoids, the corresponding equations for the sagittal radius of curvature

Figure 2.4 shows changes in Rt and Rswith distance Y from the corneal vertex for an asphericity Q value of -0.18 (Guillon et al.,

1986) and a vertex radius of curvature of 7.8 mm.

An alternative name for the tangential radius of curvature is the instantaneous radius of curvature, while the sagittal radius

of curvature is also called the axial radius of curvature. Unfortunately, this last term may

be readily confused with the vertex radius of curvature.

16Basic optical structureof thehUlllall eye

8.7+--_...J-__--'-__....L-__- L__-t-

Distance from corneal vertex (mm)

Figure 2.4. The radii of curvature in the sagittal and tangential directions of a cornea with a vertex radius of

curvature of 7.8 mm and a corneal asphericity of Q= -{l.18, from Guillon et al. (1986).

Posterior surface shape

This is difficult to measure because of the influence of anterior surface shape on any measurement. It is of lesser significance than

the anterior surface shape because of the small refractive index difference across the posterior corneal boundary, but it is not of negligible

significance.

Patel etal. (1993) estimated the shape of the posterior surface of 20 corneas from measured

anterior surface shapes and peripheral values of corneal thickness. For the 20 subjects, they

found a mean posterior vertex radius of 5.8 mm and a mean asphericity of Q = -0.42. However, Patel and co-workers used a mean

anterior corneal shape that was almost spherical (Q =-0.01) and therefore much less

aspheric than found in other studies. Lam and Douthwaite (1997) used a similar approach with a group of 60 young Chinese in Hong

Kong, and obtained asphericity of Q =-0.66 ± 0.38. In this case, the anterior surface

asphericity was Q=-0.31 ± 0.13.

Positions of the principal points

the anterior and posterior surfaces, the corneal thickness and the refractive indices.

Representative positions are shown in Figure

2.1. Note that both principal points are in front of the cornea.

Lens

Figure 2.5 shows a cross-section of the lens. The lens bulk is a mass of cellular tissue of non-uniform gradient index, contained within an elastic capsule. We do not have an accurate measure of this index distribution. There is a layer of epithelial cells, extending from the

anterior pole of the lens to the equator. The

lens grows continually throughout life, with new epithelial cells forming at the equator.

These cells elongate as fibres that wrap around the periphery of the lens, under the

capsule and epithelium, to meet at sutures. The older fibres lose their nuclei and other intracellular organelles. Because of the continual 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 to the ciliary body via the zonules, as shown in a simplified fashion in Figure 2.6. Contraction

of the ciliary muscle within the ciliary body leads to changes in zonular tension, which

alter lens shape. This process causes a change in the equivalent power of the lens and hence

apsule

Lens bulk

p'

p

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

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