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

146 Al>ermticm,; alit! retinal iII/as!' qlllliity

Effect of accommodation

Most eyes are considered to suffer from positive spherical aberration when unaccommodated, with a trend to negative

spherical aberration being observed upon accommodation (e.g, Koomen et at., 1949;

Ivanoff, 1956; Jenkins, 1963a; Schober et at., 1968;Berny, 1969;Atchison et al., 1995;Collins

et al., 1995; He et al., 1999). Koomen and colleagues (1949) found that the aberration

reduced considerably for two of their three

subjects by about 2.5 D of accommodation stimulus, and was clearly negative for one of

those subjects at 3.7 D of accommodation stimulus. Ivanoff (1956) and Jenkins (1963a) found the mean aberration was approximately zero by about 3 D and 1.5 D accommodation stimulus, respectively, but with the actual value depending upon the subject and

ray position in the pupil. Atchison et al. (1995) and He et at. (1999) found that the mean

aberration became approximately zero by 2 D and 3 D, respectively, of accommodation

response.

Coma

In a rotationally symmetric system, coma is an off-axis aberration. Figure 15.9 shows the effect of coma on the image of an off-axis

point, providing no other aberrations are present. The image has a straightened-out c?mma shape, with the pointed end facing either 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 produced by coma.

this patch, but is highest at the pointed end.

The complex details of the ray pattern are not shown.

In real eyes, 'coma-like' aberrations are present at the fovea because of the lack of symmetry ~f the eye about an appropriate

reference axis. The terms W6 to W9 in equation (15.1)are these coma-like terms. The values of

these co-efficients in Walsh and Charman's (1985) study varied from -0.128 to

+0.1080101-2, which is much greater than the range of -0.030 to +0.075 mm-3 for the

'spherical aberration-like' terms W to W Chanzes i d .

. anges In ac~ommo ation produce10changes14' In the coma-like aberrations as well as in

spherical aberration (Howland and Buettner, 1989; Lu et at., 1993).

Coma estimated from measured spherical aberration and a pupil decentration factor

Coma can arise on-axis in an otherwise rotationally symmetric system if the pupil is

decentred, and the amount of induced coma

can be p~edicted as follows. For a rotationally symmetnc system containing only the W spherical aberration term, from equati~~ (15.4), the wave aberration W(X,Y) can be

written as

Expanding the brackets and re-grouping the terms

The 'other terms' correspond to field curvature, astigmatism, a lateral shift of focus and a

constant term. These have no application at this point. However, the second term corre-

sponds to coma lying along the X-axis direction, with a wave aberration co-efficient

Once again, we have dropped the digit in

front of the 1 W3 1' because we are now considering all field dependent coma terms.

It is clear from this equation that, if the pupil is decentred by only 0.25mm, the

induced coma co-efficient W3 1 has the same magnitude as the spherical' aberration co-

efficient W4 0, providing the wave aberration and pupil 'co-ordinates are both in milli-

metres.

If we take a pupil decentration (Xo) of 0.5

rom and the value of W4,0 from equation (15.17), we have

W3,1 =0.038X 10-3 (± 0.018 x 10-3) mm-2(15.28)

This is twice the value of the spherical aberration co-efficient W4,0.

We should note that the values of these coefficients are also the values of aberration for a point in the pupil I mm from the pupil centre. At other points in the pupil, the amounts of coma and spherical aberration will be different and the spherical aberration

will increase more rapidly than coma with increase in ray height.

Astigmatism and peripheral power errors

Astigmatism can occur in central vision, due

again to the lack of a rotational symmetry about an appropriate reference axis. Usually the cause is lack of rotational symmetry of at least one surface, most frequently the anterior

cornea. This central astigmatism is conventionally corrected by ophthalmic lenses. In

the aberroscope technique, residual astigmatic effects in foveal vision contribute to the terms W3 to Ws in equation (15.1).

Here we consider the additional astigmatism that arises in the periphery of the visual field. Its effect on the image of a point source is shown in Figure 15.IOa. When astigmatism only is present, a beam of light entering the eye from a point source will be focused down to two mutually perpendicular and distinct focal lines at different positions inside the eye. Vertical sections (or fans) of rays in the beam (i,e. parallel to the plane of the page) focus to points along a focal line at T. Sections of rays perpendicular to the plane of the page focus to points along a focal line at s. The focal line at T is perpendicular to the page (i.e. horizontal), and the focal line at S is perpendicular to the T focal line. The letters T and S have been used because these foci are called the tangential and sagittal foci,

(b) ..,---tl~~I------+-

8T

Figure 15.10. Sagittal, tangential and field curvature errors.

a.The sagittal, tangential and field curvature errors and their effect on the point spread function.

b.Determining the magnitude of the sagittal and tangential power errors in object space.

148 Aberrationsand retinalimagequality

~espectively. As the direction of the incoming incident beam changes, the sagittal and

tangential foci move over surfaces called the sagittal 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 and tangential foci, are called the sagittal and tangential powers. The corresponding power errors are the differences in the powers from that required to image the focal lines on the retina. As in the case of

~pherical ~berration, sagittal and tangential imagery is usually measured in object

space by the correcting lens powers. These lens powers are the same as lens powers found by conventional refraction, but are functions of the off-axis angle. This approach is the same as finding the sagittal

and tangential vergences of the beam exiting the eye from a point on the retina,

as shown in Figure 1S.lOb. The T and S correcting lens powers are not the same as the image power errors, even with a change of sign, with the discrepancies increasing as the angle increases (Atchison,

1998). In the rest of this chapter and in the next, we will use the term 'power error',

but it is to be understood that this refers to

the correcting lens power. Please note tha t this is the opposite to the convention adopted for spherical aberration.

2.Astigmatism. This is a measure of the distance between the sagittal and tangential foci, either as a distance or as a vergence, but usually the latter. When measured as an object space quantity by refraction, it is the difference in refractive errors between the sagittal and tangential sections. Astigmatism occurs in a rotationally symmetrical system because of the oblique incidence of rays at optical surfaces. The effect is the same as conventional astigmatism associated with the fovea which, as already mentioned, is commonly due to one or more surfaces not being rotationally symmetrical. To make the distinction, the astigmatism associated with peripheral vision is often called oblique astigmatism.

The values of the sagittal and tangential

power errors will be influenced by the

presence of any foveal astigmatism and by the direction of its axis.

The mean results for a number of investigations are shown in Figure 1S.11. The range of individual results within each investigation, and of the means between

inv~st~gati?ns,shows that there is significant variation In the form of the sagittal and

tangential power errors. The measurements were usually made with the fixation or accommodation stimulus at a finite distance. Smith et al. (1988) investigated the effect of accommodation and found that, while sagittal power errors change little with accommodation/ tangential power errors increase considerably. Thus, astigmatism increases with accommodation.

Charman and Jennings (1982) suggested that much of the variation in peripheral

po~er.error.curves between subjects is due to variations In the shape of the retina. If

refractive error is due to a variation in axial length but equatorial diameter is the same

refractive errors should converge towards th~ equator (angle of about 60° in air). This was

demonstrated theoretically by Dunne et al. (1987)/ who modelled the retina as ellipses of varying asphericity.

For a general, rotationally symmetrical,

Angle in air (deg)

Figure 15.11.Mean values of the sagittal and tangential power errors from various sources. The solid lines are the corresponding means fitted by equations (l5.29a)

and (l5.29b).

optical system, aberration theory predicts that the sagittal and tangential power errors denoted here as Ls(0) and Lt( 0) can be expressed in the form of an even order polynomial in the object off-axis angle O. We fitted such polynomials to the experimental data shown in Figure 15.11. Taking only the first two terms of the polynomials, the

equations are

where 0 is in degrees and Ls and Lt are in

dioptres. These equations are shown in the figure (solid curve without symbols).

The astigmatism A( 0) is the difference between sagittal and tangential power errors,

that is

Substituting for Ls(O) and Lt(O) from equations (15.29) gives the equation

Lotmar and Lotmar (1974) used the data of Rempt et al. (1971) for 363 subjects to obtain the equation

where 0 is again in degrees and A( 0) is in dioptres. We should note that this form is not consistent with theory for a rotationally

symmetric system, which predicts that the aberration is an even order polynomial in angle O.

Equations (15.31a) and (I5.31b), and Millodot's (1981) results for 62 eyes, separately in the nasal and temporal meridians, are plotted in Figure 15.12. Our equation (15.31a) predicts larger astigmatism than that of Lotmar and Lotmar (1974) and of Millodot (1981).

Wave aberration co-efficients W2,0 and W2,2

The sagittal and tangential power errors and astigmatism in a rotationally symmetric system can be converted into wave aberration

co-efficients. W2,o and .W2,~. From the theory and equahons provided by Smith and Atchison (1997), we have

Monochromatic aberrations 149

7 -!-_----I.__-'---_--'-__.l..-_.....I-_-,I,-

6Astigmatism

--0-- Mean. equation (l5.3la)

--0-- Mean. equation (l5.3Ib)

Angle in air (deg)

Figure 15.12. Estimates of astigmatism and field curvature from several sources. Solid lines are mean values obtained from equations (15.31a),(15.31b)and (15.35).

where the values of Wzo and W 2 are for 5° from the optical axis.'While tRis angle is arbitrary, we chose it because the fovea is on average 5° from the best fit optical axis.

From equations (15.29) and (15.32), we have

the following estimates of these wave aberration co-efficients, remembering that the

values of Ls and L, are in dioptres and that the wave aberration will normally be expressed in millimetres

Field curvature

This is another off-axis aberration. If field curvature is the only aberration present, the

image of a point source is imaged as a point, but not on the image plane predicted by paraxial optics (the Gaussian image plane). For optical systems that are composed of mostly positive power components and form a real image (such as the eye), the image is formed in front of the Gaussian image plane. For small angles, the image point falls on a

150 Aberratiolls alld retinalimageql/ality

surface that is behind the retina, called the Petzval surface (Figure 15.lOa). It can be

regarded as a simple defocus or refractive error that increases with distance off-axis.

In the presence of only astigmatism, the surface of best 'general' image quality is usually assumed to be the surface containing the circle of least confusion. This is the smallest circle that encloses all the rays in the beam. This surface lies between the sagittal and tangential surfaces, as shown in Figure 15.10a. When measured outside the eye - that is, as an object quantity - its position can be

found by taking the mean of the sagittal and tangential power errors. That is, field

curvature FC(O) is given by the equation

Substituting for Ls{O) and Lt{e) from equations (15.29) gives the equation

FC(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. The

flatness of the curve indicates that the circle of least confusion lies close to the retina.

Distortion

This is the last of the 'primary' mono-

chromatic aberrations. It is also an off-axis aberration. It is similar to field curvature in that a point source is imaged as a point, this time in the Gaussian image plane but not at the expected position. It is formed either

further away (positive distortion) or closer (negative distortion) to the optical axis. If the

Actual image

eye contains negative distortion, the image position of an off-axis point will be as shown

schematically in Figure 15.13.The presence of distortion in a conventional optical system is best shown by imaging a square, centred on the optical axis, as shown in Figure 15.14. In the presence of distortion, the image of the square is smaller or larger than the expected Gaussian image, and the sides are bowed in or out. This conventional distortion is probably

not meaningful in the eye, because of the curvature of the retina and also because the

change in spatial relationship between image

points, brought on by distortion, is unlikely to have an effect on vision.

Perhaps more important than distortion is

the functional relationship between the actual image position at Q' as shown in Figure 15.13

and the position of the object Q in the object space (or the incident off-axis angle e). The

.£!esence of distortion in the eye means that Q' is different from that at the position Q' expected from Gaussian optics. There are few

studies of this relationship in real eyes. Ames and Proctor (1921) mentioned studies of Donders (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 of

people with exophthalmia. These studies gave the external angle and the position of the

retinal image from the corneal margin, and Ames and Proctor converted these retinal positions to internal angles but without giving any details. Figure 15.15 shows the results.

We can compare such image point positions with those expected from Gaussian optics. If

the eye were free of distortion, then the

positive distortion

" negative distortion

Figure 15.13. Pupil ray angles anddistortion.

Gaussian image

Figure 15.14. The effect ofdistortion onthe image ofa square centred onthe optical axis.

Angle in air (deg)

Figure 15.15. The measured values of internal and external angles, from Ames and Proctor (1921). Predicted

paraxial angles are determined from equation (15.36). The Ames and Proctor data was smoothed but well

fitted by the equation 9' =0.8230859- 0.000753260&. The co-efficient of the 9 term was fixed at the value expected

from the Gullstrand eye, which was used to predict the expected distortion-free values.

paraxial pupil ray angles ti' and u,shown in Figure 15.13, would be related by equation (5.7), i.e,

where these paraxial angles are related to real angles 8 by the equations

and m depends upon the actual construction of the particular eye (0.823085 for the Gullstrand exact relaxed eye). These equations have been used to plot (J against 8 as shown in Figure 15.15 and show that the actual image is closer to the axis than that predicted by Gaussian optics, indicating that there is negative distortion in the eye.

Higher order aberrations with foveal vision

In many studies of monocular aberration, the order of aberration terms has been very limited - for example, aberroscope studies have not investigated beyond the fourth

MOllochromatic aberrations 151

power. In equation (15.1), the terms in the

wave aberration polynomial are shown only up to the fourth power in pupil co-ordinates. Using the wave-front sensor method, Liang

and Williams (1997) determined aberration terms up to the 10th order. For 3 mm pupils, they found that the aberrations beyond the fourth order were small and had minimal

effect on image quality. However, for large 7.3 mm pupils, the fifth to eighth orders made substantial contributions to deterioration of image quality relative to diffraction limited performance, according to different image

quality criteria such as the Marachel criterion, the Strehl intensity ratio, and the modulation

transfer function (the latter two are discussed

further in Chapter 18).

He et al. (1999) found that the overall aberrations of fifth and higher orders were at a minimum at approximately 2 D of accommodation and increased for smaller and higher accommodation levels.

Symmetry between fellow eyes

Liang and Williams (1997) found that

aberrations are similar for the two eyes of the same observer.

Significance of monochromatic aberrations

Foveal vision

Spatial visual performance

For corrected vision and moderate photopic luminances, visual acuity is maximum for pupil diameters of 2-3 mm (Atchison et al., 1979). For pupil diameters smaller than 2-3 mm, the decrease in visual acuity is predominantly due to diffraction, although

reduced retinal illuminance may have some

effect. For larger pupil diameters there is a less dramatic decrease in acuity, which must be due to aberrations.

A similar influence of pupil size has been found for the contrast sensitivity function (Campbell and Green, 1965) and the retinal line spread function (Campbell and Gubisch, 1966). (These measures of visual performance are described in Chapter 18).

152 Aberrations and retinal imagequality

No one has yet demonstrated that correcting monocular aberrations of the eye will

improve visual acuity, although Liang et al. (1997)were able to improve grating resolution using an adaptive optics method to correct aberrations. Van Heel (1946) attempted to improve visual acuity with a lens designed to correct the eye's spherical aberration. Some studies comparing contact lenses and spectacle lens attributed better performance with one or the other type of correction to different levels of aberration (Millodot, 1969; Oxenberg and Carney, 1989), but there could have been other factors such as small variations in refraction and spectacle magnification. Collins et al. (1992) demonstrated that it is possible to worsen visual acuity by manipulating the front surface asphericity and, consequently, aberrations of contact lenses. For large pupils, Applegate et al. (1998) found a negative correlation between visual performance as measured by the contrast sensitivity function (see Chapter 18) and the magnitude of corneal aberrations following refractive surgery.

The effect of aberrations on image quality will be considered further in Chapter 18.

Effect of spherical aberration on refractive error

In the presence of spherical aberration, the plane of best image quality is not the paraxial image plane. For positive spherical aberration and a positive powered optical system such as the eye, the plane of optimum image quality lies between the paraxial image plane and the optical system. The position of this best image

plane depends upon the particular criterion used to assess the quality of the image, the level of the aberration (which in turn depends upon pupil size), and the particular form of the detail in the image.

When analysing image quality of conventional optical systems, a range of criteria are used, including minimizing the width of the point spread function, maximizing the Strehl intensity ratio, maximizing the modulation transfer function at set frequencies, and minimizing the variance of the wave aberration function. We do not know if the eye uses any of these for optimum focusing. Burton and Haig (1984) found that the Strehl

intensity ratio and modulation transfer function were not good measures or predictors of perceived image quality. In the absence of definitive information pointing to any particular criterion, the variance of the wave aberration function will be used here

because it is the easiest to manipulate algebraically.

The variance of the wave aberration is very easy to quantify in terms of the wave aberration co-efficients. The variance V is given in terms of the wave aberration W(r,l{» by Smith and Atchison (1997, equation (34.3))

V =Iff [W(r,l{» - (15.37)

E'

where E' indicates integration over the pupil of area A.

Let us suppose that the wave aberration W(r) is purely spherical and consists of only one spherical aberration order, the spherical

aberration term W4,o. If we defocus the system, the wave aberration polynomial can be written as

where r is the ray height in the pupil and Wzo is the term allowing for defocus. The value of W4 0 is constant, but the value of Wzo can be

manipulated by refocusing to minimize the variance of the wave aberration given by

equation (15.37).This value is minimized with respect to Wz,o when

dV/dWz,o = 0

Differentiating V with respect to Wz0 and then setting the result to zero gives the optimum defocus occuring at a defocus level of

where p is the pupil radius. The corresponding change in refractive error ~x(p) is given by

Thus the amount of refocusing depends upon the pupil radius p. It should be remembered that this equation applies if the wave aberration function consists of only the

spherical aberration term W4 o- If more than one term exists, all terms should be included

in the integration of the variance equation

(15.37).

We can use equation (15.41) to predict the effect of spherical aberration on refractive error. Substituting the spherical aberration coefficients from the non-aberroscope studies and from the aberroscope studies, provided by equations (15.17) and (15.21), respectively,

into equation (15.41) gives

till.x(j5) = -0.038 X 1O-3(± 0.018 X 1O-3)p2 mm" (15.42a)

till.x(j5) = -0.0176 X 1O-3(± 0.0262

X 1O-3)p2 mm"!

(15.42b)

These values indicate that a change in pupil diameter from 2 mm to 6 mm would lead to a

change in focus of 0.29 D according to equation (15.42a) and of 0.14 D according to

equation (15.42b).

The above results neglect the StilesCrawford effect, which reduces the influence of spherical aberration towards the edge of the pupil and hence reduces the defocusing prediction 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 pupil size (Koomen etal., 1949and 1951;Charman et al., 1978; Atchison et al., 1979). For example, Charman et al, (1978) found less than 0.3 D

variation in refraction with pupil size, even in the presence of greater than 1 D spherical aberration, while Atchison et al. (1979) did not observe variation in refraction across 1-8 mm pupil diameters for any of their 22 subjects (refraction interval 0.25 D). Charman et al. (1978) and Atchison (1984a) showed that the optimum refraction for high frequency sinusoidal targets (~ 20 cycles/degree) should be little affected by changes in pupil size, even in the presence of substantial spherical aberration (Figure 15.16). The small variation

in refraction with pupil size is thus explained by the use of small letters containing a

relatively high proportion of high spatial frequency information.

When luminance is lowered, there is often a substantial negative (myopic) shift in refraction (Koomen et al., 1951). A large

amount of this is due to the combination of decrease in visual acuity, which means that

Spatial frequency (c/deg)

Figure 15.16. Theoretical optimum refraction as a

function of spatial frequency for 2 mm and 6 mm pupil sizes. Wavelength 605 nm, with primary spherical

aberration of 0.11D and 1.0 D at the edge of 2 mm and 6 mm pupils. respectively.

high spatial frequencies are not available for determining refraction, and the natural increase in pupil size (and increase in

aberrations). Green and Campbell (1965) and Charman et al. (1978) showed that optimal

focus did not depend upon spatial frequency for small pupils, but would shift considerably for large pupils - for example, Green and Campbell found that optimum refraction for a 7 mm diameter pupil shifted approximately 0.8 D between 1.5and 26 cycles/degree in two subjects. Charman et al, (1978) and Atchison (1984a) showed that this shift could be explained by the presence of spherical aberration (Figure 15.16).

Retinoscopy

This objective refraction technique was described in Chapter 8 (Objective-only refraction techniques), and its use in early studies of the aberrations was mentioned earlier in this chapter (Methods of measuring monochromatic aberrations). If an eye is aberration-free, at the

reversal point the complete pupil will appear either dark or light as the light returning from the patient's pupil is scanned across the sighthole of the retinoscope. In the presence of

154 Allamtiol/s and retinal image quality

aberration, the shadow movement is more complex. Quite common is the situation in which the retinoscope reflex corresponding to peripheral parts of the pupil moves 'against' while the central part of the reflex is 'with', corresponding to positive spherical aberration. Roorda and Bobier (1996) related the appearance of the retinoscopic reflex to a variety of aberrations.

Peripheral vision

At angles beyond about 20° the peripheral refractive errors, that is, sagittal and

tangential power errors and the associated astigmatism, are the dominant aberrations. These affect the quality of the retinal image during ophthalmoscopy. Correcting peripheral refractive errors has little effect on visual acuity beyond a few degrees from the fovea (MiIlodot et al., 1975; Rempt et al., 1976), but improves peripheral detection (Leibowitz et al., 1972; Wang et al., 1995; Williams et al., 1996). This is discussed further in Chapter 18.

Ocular component contributions

There is considerable interest in understanding the corneal and lens contributions to the aberration of the eye. This interest has increased with advances in the treatment of refractive errors by surgically manipulating the anterior corneal surface. It is important to know the best shape to make the cornea and the visual consequences of poor post-surgical

shapes (Applegate and Howland, 1997; Applegate et al., 1998; Martinez et al., 1998).

There have been several attempts to identify the magnitude of the corneal and lenticular contributions to ocular aberrations

(Jenkins, 1963a; El Hage and Berny, 1973; Millodot and Sivak, 1979; Sivak and Kreuzer,

1983). These studies suffered from problems

such as inaccurate measurements of the cornea and measuring the lens in vitro, where

its shape will be different from the natural state or its optical conjugates have been altered. The emphasis has been on spherical aberration and the cornea has often been modelled 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 measure the total aberration of the eye and determine the aberrations of either the cornea or lens. The easiest component to measure is the anterior cornea. The shape of this can be measured with corneal topographical instruments. The aberration of the cornea is given by its departure from the ideal shape, which would give no aberration about the chosen reference axis. For a distant object, the ideal cornea would have a front surface conicoid asphericity Q, independent of radius, related to the cornea's refractive index

II by

Q was explained in Chapter 2. The corneal aberration in wave aberration terms for any ray traced through the cornea is given by

where z is the departure of the anterior cornea from its ideal shape for the particular ray. The lenticular contribution is then obtained by subtracting the corneal contribution from the total aberration. In determinations such as

these, the aberration contribution of the posterior cornea is ignored. This is assumed to

be small because of the small refractive index difference between cornea and aqueous.

The corneas of most eyes have negative corneal asphericities, but the corneas usually

contribute positive aberration because the asphericity is not high (Kiely et al., 1982; Guillon et al., 1986; see Chapter 2). The

direction and magnitude of the aberration of lenses of eyes is still unclear, with the literature giving results of both positive and

negative spherical aberration (Jenkins, 1963a; El Hage and Berny, 1973; Millodot and Sivak, 1979; Sivak and Kreuzer, 1983; Tomlinson et

al., 1993). There is more about this in the next chapter.

Correction of myopia by refractive surgery increases the aberrations associated with the cornea, with the fourth order (spherical aberration-like) aberrations increasing considerably 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).

Pupil aberration

Ocular aberrations affect not only the quality of the retinal image, but also the formation of

the entrance pupil. The entrance pupil is the image of the aperture of the iris (the actual

pupil), and ocular aberrations arising from the cornea will affect the appearance of the pupil. Of the five monocular aberrations, the aberration 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 the actual pupil (see Appendix 3). Since distortion

is a non-linear magnification effect that increases with image size by at least the

square of the image size, we expect pupil distortion to have a greater effect, the larger the pupil. The pupil distortion arising at the cornea is negative, and therefore as the real pupil becomes wider, distortion leads to a smaller magnification than the 13 per cent

predicted from Gaussian optics. Alexandridis and Baumann (1967) exam-

ined the effect of pupil size on the apparent pupil diameter using excised corneas. They showed that, for large pupil sizes, the pupil magnification was less than the expected Gaussian value of about 1.13. We have

determined the expected magnification effect for the Gullstrand number 1 relaxed schematic eye with the anterior corneal surface

aspherized with a Q value of -0.2, which is close to the means of the studies of Kiely et al. (1982) and Guillon et al. (1986). Figure 15.17

compares the experimental and theoretical results. They show similar trends, although the magnification changes more quickly for the former.

Aberrations of ophthalmic devices

Many optical instruments are used in combination with the eye, including telescopes and microscopes and clinical instruments such as

optometers, keratometers and ophthalmo-

scopes. There are ergonomic issues relating to the way in which instruments must be used, e.g. the most appropriate alignment of the two tubes of binocular instruments so that the two eyes are not placed under undue accommodative and convergence stress when