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

is

D, =05(1-1.0947 X 1(J4 £P + 1.8698X 10-9 (11)

and this equation is shown on Figure 3.7. Assuming that 0 s is the same as 0, this

equation can be converted into the apparent pupil area

Significance of pupil size

Pupil size has a number of effects on vision.

Thepupil 27

image quality. There is an optimum pupil diameter range of 2-3 rom that gives the best balance between these two effects for the corrected eye. The effect of pupil diameter on retinal image quality is discussed in greater detail in Chapter 18.

Campbell and Gregory (1960) and Woodhouse (1975) found that the artificial

pupil size that gives the optimum (corrected) visual acuity is close to the natural pupil size at various background lighting levels. The visual acuity of a defocused eye is strongly dependent upon pupil diameter. This is discussed further in Chapter 9.

Depth-of-field

As with conventional optical systems, the diameter of the pupil affects the depth-of-

field. The larger the pupil, the narrower is the depth-of-field. This is discussed in detail in

Chapter 19.

Retinal light level

Obviously, pupil diameter affects retinal light

level. A detailed analysis of the dependency is left until Chapter 13.

Retinal image quality and visual performance

For large pupil diameters, aberrations cause deterioration in retinal image quality. For small pupil diameters, diffraction limits

Purpose of the pupillary light response

We may expect that pupil size varies in order to maintain a constant light level on the retina,

but this is not so. The variation of pupil size with light level is not sufficient to ensure a constant retinal illuminance, because if the pupil size changes from 2 to 8 mm in diameter, the amount of light entering the

pupil changes by a factor of only 16. By contrast, we operate comfortably over a 1()5

times luminance range, from full moonlight (::: 0.01 cd/m2) to bright daylight (::: 1000 cd /rn-).

The work of Campbell and Gregory (1960) and Woodhouse (1975) suggests that pupil size variations optimize visual acuity for various light levels. This is only applicable for corrected eyes; uncorrected eyes require much smaller pupil sizes. Woodhouse and Campbell (1975) suggested that the purpose of pupil size changes with changes in light

28 Basic optical structureof the Ill/mall eye

level is to reduce retinal illumination and thus help adaptation if there is a return to

darkness.

Measurement of pupil size (pupillometry)

There are a number of experimental and clinical methods for measuring pupil

diameter. These vary both in the level of intrusion to the subject and in accuracy. A

simple clinical device has the construction shown in Figure 3.8. The inside surface of the

end nearest the patient's eye contains a millimetre scale. Parallax is a source of error.

More accurate and less intrusive methods

involve photography, either using standard photography or a video camera. A distinct advantage of a photographic method is that it can be done with infrared radiation in complete darkness, and therefore does not affect pupil size. The use of a video camera offers the further advantage that rapid

changes in pupil diameter can be recorded and monitored.

Loewenfeld (1993) gave a comprehensive review of pupillometry.

Artificial pupils

Artificial pupils are often used to control the effective pupil size of the eye in visual experiments. The pupil of the eye must be dilated first. Artificial pupils can simply be apertures placed immediately in front of the

eye, or they can be projected onto the plane of the actual pupil by a relay system. The artificial pupils may be annular, slit or circular

pupils, and may be decentred relative to the

actual pupil by a controlled amount. Because of aberrations, retinal image quality varies

according 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. When corrected visual acuity does not reach normal

levels, placing a small pupil (e.g. 1 mm diameter) in front of the eye may improve visual acuity markedly if the refraction is not

accurate, but a lack of improvement indicates a pathological basis for the poor vision. Medium-sized artificial pupils (e.g, 3-4 mm)

may be used when the pupil has been

previously dilated by drugs; the pupils reduce the influence of aberrations, which may make refraction difficult or inaccurate.

Summary of main symbols

Ap((J) projected or apparent pupil area in the direction (J

D(entrance) pupil diameter

Ds pupil diameter in the sagittal section D, pupil diameter in the tangential section L scene luminance (cd/rn-)

(J oblique angle

References

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

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

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

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

De Groot, S. G. and Gebhard, J. W. (1952). Pupil size as determined 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 varying perimetric angles. Vision Res.,1,418-28.

Jennings, J. A. M. and Chatman, W. N. (1978). Optical

quality 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 State University Press.

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

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

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

Sloan, L. (1950). The threshold gradients of the rods and cones: in the dark-adapted and in the partially lightadapted 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 the retina. By. J. Ophthal., 32, 340-46.

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

Walsh, G. (1988). The effect of mydriasis on pupillary centration of the human eye. Ophthal. Physiol. Opt., 8,

178-82.

Walsh, G. and Charman, W. N. (1988). The effect of pupil

centration and diameter on ocular performance. Vision Res., 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 illumination and pupil size. OptOIll. Vis. Sci.,69, 129-36.

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

Woodhouse, J.M. and Campbell, F. W. (1975). The role of

the pupil light reflex in aiding adaptation to the dark.

Vision Res., 15, 649-53.

Introduction

Most man-made optical systems are rotationally symmetric about one line, the optical axis. If the reflecting and refracting surfaces are spherical, this is the line joining the centres of curvatures of these surfaces. Some systems contain astigmatic or toroidal components and have two planes of symmetry; the line of intersection of these two planes is the optical axis.

By contrast, to fully describe the optical properties of the eye, we need to introduce a number of axes. This is because of the lack of symmetry of the eye and because the fixation point and fovea are not along a best-fit axis of symmetry. The optical and visual axes were mentioned in Chapter 1. In this chapter we describe these and other axes, their significance, their applications, and how they may be determined.

The validity of some of these axes is dependent upon some idealized properties of the eye. For example, the visual axis requires the existence of the nodal points, which only exist if the eye is rotationally symmetric. A second example is the fixation axis, which requires 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 a small light source formed by specular reflections from the refracting surfaces of the eye. These Purkinje images are discussed in greater depth in Chapter 12.

Direction of axes

These axes are meaningless without a means of determining their directions and where they enter the eye. The directions are defined relative to each other, and we often refer to the angles between the axes. Table 4.1 lists some axes, and the symbols used to denote the angles between them. Three of the axes pass through the centre of the pupil and, since the pupil centre can change with change in diameter (see Chapter 3), their directions depend upon pupil size.

Definitions and significance

Martin (1942) gave an early account of the angles and axes which highlighted the confusing array of terms that have been used in this area. The definitions of optical axis, line of sight, visual axis, pupillary axis and fixation axis given below are similar to those

provided in dictionaries of visual science (Cline et at., 1989; Millodot, 1993).

Optical axis

This is the line passing through the centres of curvatures of the refracting and reflecting surfaces of a centred system. The optical axis is not of particular importance by itself, but it is a useful reference for some of the other axes of the eye.

In a conventional centred optical system, the centres of curvatures of each refracting or reflecting surface lie on one line (i.e. they are co-linear). This line is the optical axis. The eye is not a centred system, and does not contain a true optical axis. The concept of optical axis can be applied to the eye by defining the optical axis as the line of 'best fit' through the

centres of curvature of the 'best fit' spheres to each surface.

Line of sight

This is the line joining the fixation point and the 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

Pupillary axis Line of sight

the centre of the beam of light entering the eye. As mentioned in the previous section, it is unfortunately not fixed because the pupil

centre may alter with fluctuations in pupil size.

The fovea is usually on the temporal side of the optical axis (Chapter 1). Therefore, the point in object space conjugate to the fovea is also off axis, but on the nasal side of the optical axis. The line of sight is the central ray of the beam from the fixation point T as

shown in Figure 4.1. In paraxial optics, the line of sight is called the paraxial pupil ray,

which was defined in the previous chapter (see Entrance and exit pupils). The position at which it intercepts the cornea is called the

corneal sighting centre (Mandell, 1995) or visual centre of the cornea (Cline et al., 1989).

Temporal side

Visual axis

This is the line joining the fixation point and the foveal image by way of the nodal points.

The visual axis is a convenient reference axis for visual functions, particularly as it does not depend on pupil size. It is usually

close to the line of sight at the cornea and entrance pupil (see following section, Locating some axes).

The visual axis is the line segments TN and N'T' shown in Figure 4.1. This is not a single

straight line, since the nodal points are not coincident. Allowing for the exaggeration in Figure 4.1, this shows the proximity of the visual axis to the line of sight.

The foveal achromatic axis is closely related to the visual axis, and can be defined as the path from the fixation point to the fovea such that the ray does not suffer from any transverse chromatic aberration. Ignoring the

small change in the nodal points that occurs with change in wavelength, this axis is the

same as the visual axis and its definition can be used as a basis for locating the visual axis. Ivanoff (1953) referred to this axis as simply the achromatic axis. However, Thibos et al. (1990) redefined this term to refer to the pupil nodal ray, which is the ray passing through

Achromatic_.J==axis -tp--I

the centre of the pupil and which has no transverse chromatic aberration. It is similar

to the optical axis but, unlike the optical axis, it is dependent on pupil position. Where Ivanoff used the term achromatic axis, we use the term foveal achromatic axis, and we have adopted Thibos and co-workers' use of achromatic axis (Figure 4.2).

Rabbetts (1998) criticized the use of the term 'visual axis' for the ray passing through the nodal points, on the grounds that such a ray is

not representative of the beam passing into the eye from a fixation target. He preferred to

call it the 'nodal axis', and reserved the term 'visual axis' for the axis we have defined as the line of sight.

Le Grand and EI Hage (1980) referred to the intersection of the visual axis with the cornea as the ophthalmometric pole (Figure 4.1).

Pupillary axis

This is the line passing through the centre of the entrance pupil, and which is normal to the cornea.

The pupillary axis is used as an objective measure to judge the amount of eccentric fixation, the condition in which a retinal point

Temporal side

other than the centre of the fovea is used for fixation. Eccentric fixation is an adaptation to heterotropia (squint or turned eye). We discuss the measurement of eccentric fixation and heterophoria further in the following section (Locating some axes).

If the eye was a centred system and the

pupil was also centred, the pupillary axis would lie along the optical axis. However, the

pupil is often not centred relative to the cornea and, furthermore, the cornea may not

be a regular shape. Both these factors cause the pupil axis to lie in some other direction, and in general it does not pass through the fixation point T as shown in Figure 4.1.

Fixation axis

This is the line passing through the fixation point 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 it depend on the direction of rotation of the eye (Alpern, 1969). Accordingly, the idea of a

fixation axis is just an approximation, and estimates of it depend also upon the direction

of rotation.

Axes of theeye 33

Keratometric axis

This is the axis of a keratometer or videokeratographic instrument, and it contains the centre of curvature of the anterior cornea.

This axis is used for alignment in corneal topography measurements. In the standard

operation of a corneal topographic instrument, the axis intercepts the line of sight at the

fixation target (Figure 4.3a), although the fixation target may be moved deliberately so that this is not the case (Figure 4.3b). According to Mandell (1994), in standard use a small but negligible variation occurs in this axis between different instruments, because of

differences in the distance of the fixation point.

When a videokeratographic instrument is used in standard operation, the keratometric axis is neither the line of sight nor does it pass

through the apex of the cornea (the point with the smallest radius of curvature (Mandell and

St Helen, 1969». The point at which the axis intercepts the cornea is sometimes called the vertex normal (Maloney, 1990), and is the centre of videokeratographs showing corneal contour. If the vertex normal is sufficiently

decentred from the corneal apex, the

videokeratograph gives a false representation of surface curvature across the cornea (Mandell and Horner, 1995).

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

Locating some axes

The line of sight

Many automated instruments have a video

display showing the front of the eye. To achieve alignment while a patient is fixating a

reference target, either the patient or the instrument is moved vertically and horizontally until the pupil is correctly centred. Thus the line of sight is made to coincide with the optical axis of the instrument. Sometimes this is aided by imaging a centred annulus as well as the eye.

The visual axis

The intercept of the visual axis at the cornea (ophthalmometric pole) can be determined by having the subject view a vernier target, half of which is blue and half of which is red. Two possible targets are shown in Figure 4.4. The subject views the vernier target through a small artificial pupil, say 1 mm diameter. If

the artificial pupil is not centred on the visual axis, there is a break in the alignment of the

blue and red halves of the target. The position

of the subject's eye is adjusted until alignment is obtained. The reliability of the method

should improve as the wavelength bands of

the target are narrowed.

Thibos et al. (1990) indicated a simple way of estimating the separation of the visual axis from the line of sight at the entrance pupil (the distance d in Figure 4.2). The small artificial pupil, referred to in the previous paragraph, can be scanned across the eye both vertically and horizontally to find the real pupil limits at which the vernier target disappears. The mean 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 the

ophthalmometric pole (visual axis intercept at the cornea). Thibos and co-workers measured five subjects with drug-induced pupil

dilation, considering only the horizontal meridian. They obtained values of d between

-0.1 mm and 0.4 mm, with a mean of +0.14 mm (a positive sign indicating that the visual

axis is nasal to the line of sight in object space).

Simonet and Campbell (1990) used a videomonitor to determine the difference in horizontal position of the visual axis (which they referred to as the achromatic axis) and the line of sight. For natural (although generally large) pupils, the differences for eight 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 of determining the location of the keratometric axis relative to the corneal sighting centre and the corneal apex. Using a videokeratographic

instrument in its standard operation and 20 normal subjects, they found a mean difference

between the keratometric axis and the corneal sighting centre of 0.38 ± 0.10 mm, with the

majority of subjects having the keratometric axis below and nasal to the corneal sighting

centre. The mean difference between the keratometric axis and the corneal apex was

0.62 ± 0.23 mm, with the majority of subjects having the keratometric axis above the corneal apex.

Angles between axes

Here we define some of the angles between various axes. In some cases, methods for determining these and measurements are given. The angles alpha, lambda and kappa have been used differently by different

authors. In Table 4.2 we indicate some of the variations from those used here. These

variations involve either the line of sight or the visual axis.

Visual axis and optical axis: the angle alpha (a)

Most authors use the term 'angle alpha' to refer to the angle between the optical and

visual axes (Le Grand and El Hage, 1980; Cline et al., 1989; Millodot, 1989). We have retained this use but, as the technique of measuring this angle involves the subject

fixating on a target, it may be considered that the 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 practical importance.

In optical laboratories, a frequent need is to

locate the optical axis of an optical system. One method of locating this axis is to shine a

distant point source into the system and observe the images of this source of light reflected from each surface inside the system.

In a centred optical system, if the source of light falls on the optical axis, it and the reflected 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 centred system, there is no position or direction of the light source that enables the Purkinje images to be aligned. All that can be done is to

Patient

N

Figure 4.5. The ophthalmophakometer.

minimize the spread of these images, and the

corresponding direction of the source identifies the direction of the optical axis.

Clinically, the angle a is determined with the ophthalmophakometer (Figure 4.5). This

instrument contains a graduated arc, with an

observing telescope mounted centrally in the arc. The patient's eye is at the centre of curvature of the arc. Two small light sources

are placed on the arc near the telescope, with one slightly above and one slightly below it. This gives pairs of Purkinje images. A small fixation object T is moved along the arc until

the observer looking through the telescope judges that the Purkinje images are in the best

possible alignment. At this point, the optical axis of the eye corresponds with the axis of the

instrument. The angle a is given by the scale reading at the position of the fixation target. The instrument can be rotated through 90

degrees to obtain a value in the vertical direction. The distance between the eye and

the arc is relatively large (typically 86 em) so that discrepancies between the front nodal point and the centre of curvature of the arc are not critical.

The angle between the visual axis and the optical axis is considered to be positive if the visual axis is on the nasal side of the optical axis in object space. The mean value of angle a is often taken to be about +5° horizontally,

but is usually in the range +3 to +5°, and is rarely negative. The visual axis is also downwards relative to the optical axis by 2-3° (Tscherning, 1990).

Pupillary axis and line of sight: angle lambda 0.)

This angle is one of the easiest to determine. It can be determined using the opthalmophakometer just described, but it is determined easily with simpler equipment (Figure 4.6).

36 Basic optical structure of tileIll/man eye

Optical axis

8 __.-1---1 -

T

Fixation target

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

Centrer- of curvature of cornea

1'1 Cc

Anterior corneal surface

The subject fixates on some suitable target T, and an observer watches the anterior corneal

reflection (Purkinje image PI)of a small source of light S close to the observer's eye. The

position of the light is changed, and the observer's eye is maintained next to the light source, until the reflected image S' is seen in the centre of the pupil E (Figure 4.6). The observation axis is now the pupillary axis. The

angle between the line of sight and the

pupillary 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°, and Franceschetti and Burian (1971) obtained angles of 2.6 ± 1.7°. Furthermore, the angles should be similar between the two eyes.

The angle A is important for diagnosis of eccentric fixation and heterotropia. In testing

for the presence of eccentric fixation, angle Ais determined monocularly (with the other eye

occluded). A large angle indicates the likely

presence of eccentric fixation. The line of sight as we have defined it is not being used,

because the patient has rotated the eye to align a retinal point eccentric to the fovea with the fixation point. Angle A is estimated

binocularly (with both eyes open) to test for direction and amount of heterotropia in the Hirschberg test. In the presence of hetero-

tropia, a large angle Ais observed because one eye rotates so that its fovea is not being used

to align the fixation target. Either the fovea is being suppressed or another retinal point is

being used for fixation (anomalous correspondence), or both are occurring. Again, we are not strictly measuring angle A because of

the change in reference axis away from the line of sight.

The usual application of these tests involves

the clinician shining a penlight at the patient's eye or eyes. The penlight is just in front of the clinician's face, and the patient is instructed to

look at it. The clinician observes the position of the corneal reflection (or reflex) in the pupil (Figure 4.7). Usually the reflex is about half a millimetre nasal to the centre of the pupil

(Figure 4.7a). Each millimetre change in reflex position away from this corresponds to

approximately 22 prism dioptres (13°) of eye rotation (Grosvenor, 1996). Care must be taken in the monocular test, as some patients may have normal fixation but unusual angles. Results from the two eyes should be compared. Similarly, in the Hirschberg test it is important to compare the difference in reflex positions between the two eyes.

Pupillary axis and the visual axis: angle /(

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

Figure 4.7. The Hirschberg test for measuring the angle of heterotropia.

a.Appearance of corneal reflexes for no heterotropia.

b.Appearance of reflexes for left esotropia (left eye turned in).