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

direction and maximum for the electric vector parallel to the nerve fibres. The arrangement

of nerve fibres is similar to the fibril layers in the cornea (Chapter 12, Scatter), but whereas the corneal fibril diameters are smaller than the wavelength of light, the nerve fibre diameters in the retina are about the same size or 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, van Blokland (1985) and Dreher et al. (1992). Measurements of retinal birefringence must take into account or eliminate any birefringence from the cornea.

Birefringence of the nerve fibre layer has an

application in scanning laser polarimetry (Dreher et al., 1992; Weinreb et al., 1995),which

estimates nerve fibre loss due to diseases such as glaucoma. Red or near infrared light from a

laser, initially linearly polarized, passes into the retina. It is scattered from the deeper

retinal layers and the choroid to pass back out of the eye. The plane of polarization is changed by its passage (twice) through the nerve fibre layer. The degree of polarization change is used to estimate nerve fibre layer thickness, and the thickness is determined at

the retinal area of interest.

References

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Direct measurement of human-cone photoreceptor alignment. 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 the central retina. J. Opt. Soc. Am. A, 14, 2033--40.

Delori, F.C. and Pflibsen, K. P. (1989). Spectral reflectance of the human ocular fundus. Appl. Optics, 28,

1061-77.

Dreher, A W., Reifer, K. and Weinreb, R. M. (1992). Spatially resolved birefringence of the retinal nerve

fiber layer assessed with a retinal laser ellipsometer.

Appl. Optics, 31, 3730-35.

Geeraets, W. J. and Berry, E. R. (1968). Ocular spectral characteristics as related to hazards from lasers and other 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 of human 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 and Co.

Hunold, W. and Malessa, P. (1974). Spectrophotometric determination of the melanin pigmentation of the

human 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 intraocular straylight function in 129 healthy volunteers;

dependence on angle, age and pigmentation. Vision Res., 30, 699-707.

Rodieck, R. W. (1998). Chapters 4 and 6. seeing. Sinauer, pp. 68-87, 122-33.

Ruddock, K. H. (1963). Evidence for macular pigmentation from colour matching data. Vision Res., 3,

417-29.

van Assendelft, O. W. (1970). Spectrophotometry of Haemoglobin Derivatives. Royal VanGorcum.

van Blokland, G. J. (1985). Ellipsometry of the human

retina in vivo: preservation of polarization. f. Opt. Soc. Am. A, 2, 72-5.

van Blokland, G. J. and van Norren, D. (1986). Intensity

and polarization of light scattered at small angles from the 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 fundus reflectometry. Vision Res.,36, 2229-47.

van den Berg, T. J. J. P. (1995). Analysis of intraocular straylight, 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 fiber layer 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.

Introduction

Like defocus, aberrations reduce the image

quality of optical systems such as the eye. When an eye is corrected by ophthalmic

lenses, we are mainly correcting defocus but are actually finding a balance between defocus and aberrations. Defocus is the most

important optical defect of eyes, and it should be appreciated that aberrations are usually only of consequence for an eye well-corrected for defocus.

The representation of aberrations of optical systems depends on what is most convenient

in the particular circumstances. These representations include the following (Figure IS.la):

1.Wave aberration. This is departure of the wavefront from the ideal waveform, as measured at the exit pupil.

2.Transverse aberration. This is the departure of a ray from its ideal position at the image surface.

3.Longitudinal aberration. The departure of the intersection, where this occurs, of a ray with a reference axis (i.e. the pupil ray) from its ideal intersection.

Because we cannot easily measure the aberrations on the image side of the eye's optical system, the aberrations of the eye are usually measured in object space (Figure 15.1b). All but one method of measuring aberrations of the eye described in this chapter work in this fashion. Provided that

the aberration and defocus levels are not

unduly high, these 'object' aberrations are similar to 'image' aberrations. When the aberrations are large, such as in the case of peripheral vision, this is no longer the case.

General ray

Wave aberration

Image plane

--+-+---=='1'k-~---+o·

Longitudinal aberration

ITransverse aberration

(a) Image space

Transverse

aberration 1

Longitudinal

aberration

Ot---"""""::~~-+--+--

Object plane

a-eneral ray

(b) Object space

Figure 15.1. Wave, transverse and longitudinal aberrations:

a.For a general optical system.

b.For the eye, when these must be determined in object space.

Note that longitudinal aberration occurs for a ray only when it intersects the reference ray (i.e, pupil ray).

138 A/J,.,.mtiolls alld retinal illlllg<'qualits]

The aberrations of a general optical system with a point object can be represented by a

wave aberration polynomial (or function) of the form

where X and Yare co-ordinates in the entrance pupil. The co-efficients can be described as follows:

tilt (prismatic or distortion)

co-efficients

defocus and astigmatism co-efficients

'coma-like' co-efficients

'spherical aberration-like'

co-efficients

The values of these co-efficients depend upon the position of the object in the field. For a rotationally symmetrical system in which

where the terms still present retain the same meaning as previously. In this case,

W7 =W9 and WlO=W14 =0.5 W12 so we can now write

W(X,y) = W2y + W X2 + Wsy2+ W7Y(X2

+ y2) + WlO(X2 + y~2+ higher order terms

(15.3)

An alternative way of expressing the aberrations of a rotationally symmetrical optical system is

where 1] indicates dependence on the position

of the object in the field. The first five terms in this expansion are known as mono-

chromatic primary aberrations or third order

aberrations, and the co-efficients are as follows:

The significance of the term 'primary' is explained in Appendix 2, where general aberration theory is discussed. Of these five monochromatic aberrations, only spherical aberration occurs on-axis for a rotationally symmetric system. The other four occur only off-axis for a rotationally symmetric system, and generally worsen with increase in distance off-axis. There are two other primary

aberrations called chromatic aberrations, which become manifest when more than one

wavelength is imaged by an optical system. We will leave the chromatic aberrations until Chapter 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 by

equations such as equations (15.2), (15.3) and (15.4), but require the use of an equation such as equation (15.1). Contributors to this lack of rotational symmetry are the difference between the best-fit optical axis and the line- of-sight (because of component tilts and displacements), the lack of rotational sym-

metry of refracting surfaces, and possibly a lack of rotational symmetry of the lens refractive index.

Methods of measuring monochromatic aberrations

Aberrations of the eye have been noted and measured since at least the time of Thomas Young (1801). Some methods have measured the tranverse aberration and others longi-

tudinal aberration, usually in the form of the refraction required. Other aberration forms,

for example the wave aberration, can usually be derived from these values.

Position in pupil (mm)

Figure 15.3.Transverse aberration for one subject at two levels of defocus in the study of Woods et al. (1996).

Object distance 4 m. Modified from Figure Sa of Woods et al. (1996), with kind permission of Elsevier Press.

which can be arranged to

Note that TA(X) has the opposite sign to X in the figure. The longitudinal aberration LA(X) is given, as a difference of vergences, by

LA(X) =1/1-1/10 or

Substituting the right-hand side of equation (15.6) for (I -/0 ) into equation (15.7) gives

The transverse aberration polynomial can be altered also into a wave aberration polynomial W(X) by using the relationship (Smith and Atchison, 1997,equation (33.76))

0:===1Transverse aberration1i)

Figure 15.4.The determination of longitudinal

aberration from transverse aberration, as applied to the study of Woods et al. (1996).

where dW(X)/dX is the derivative of W(X) with respect to X. From this equation,

This is a one-dimensional equivalent of equation (15.1).

The vernier alignment method is slow compared with recent objective techniques, and most studies have considered only one or a few meridians. With improvements in design and in computer-aided data collection and processing, it is possible to measure a

large number of points across the whole pupil in a few minutes (He et al., 1988). In this case,

the equations to derive aberration polynomials from the transverse aberration data will be more complex than indicated above.

Scheiner disc and annulus methods

Von Bahr (1945) used the Scheiner principle (Chapter 8, Subjective/objective refraction techniques) to measure the subjective re-

fraction (longitudinal aberration) corresponding 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 apertures consisting of annuli of various radii. The object can be moved, or correcting ophthalmic lenses are placed in front of the eye. Koomen et al. (1949) kept the area of all annuli the same. Like most other subjective techniques, it

is only applicable for foveal vision. Because it considers all meridians at once, it is only able to measure the rotationally symmetrical

aberration known as spherical aberration.

Telescope focusing

Van den Brink (1962) measured longitudinal aberrations by isolating small parts of the pupil and adjusting the eyepiece of a telescope for optimum focusing for a target.

Refraction in the periphery

Occasionally, subjective refraction techniques (measuring longitudinal aberration) have been modified to measure astigmatism and field curvature in the periphery (Millodot et

Objective tests

Knife-edge tests

The basic principles of retinoscopy were discussed in Chapter 8, Objective-only refraction techniques. Retinoscopy was used in some of the earliest attempts to measure longitudinal monochromatic aberrations

associated with foveal vision, by comparing the state of refraction through various regions of the pupil (jackson, 1888; Pi, 1925; Stine,

1930; Jenkins, 1963a). These studies showed that, for the relaxed eye, peripheral pupil

regions require usually more negative correction (i.e. are more myopic) than the central part of the pupil. Retinoscopy has also been used for determining the peripheral refraction of the eye, taking the whole of the pupil into account, and hence determining field curvature and oblique astigmatism (see, for example, Rempt et al., 1971; Millodot et al., 1975). Here retinoscopy can be considered to be a longitudinal aberration measure, because it determines the lens power required to

correct the aberrations of the eye.

Berny (1969) and Berny and Slansky (1970) developed a technique based on the Foucault test. Light returning from the retina of the eye was intercepted by a knife edge near the focal plane. The image of the pupil was photographed and its intensity pattern analysed to determine transverse aberrations, which were then transformed to give the corresponding wave aberrations.

Aberroscope technique

A possible set up of the aberroscope is shown in Figure 15.5. The method employs a distant light source and an aberroscope placed close to the eye. The aberroscope consists of a

square (or nearly square) grid sandwiched between positive and negative planocylindrical lenses of equal power (typically 5 0), with the cylindrical axes perpendicular and at 45° to the vertical. Additional correcting lenses are added as required to place the retinal shadow of the grid midway between

/\-~

J@ilCrossed-cylinder with grid

/ ~Laserwith expanded beam

Figure 15.5. A setup for the aberroscope technique, showing the retinal image of a 7 x 7 grid in the presence of primary spherical aberration.

the two focal lines produced by the cylindrical lenses. Without the lenses, the grid shadow

intersection points would be too close to resolve. Aberrations are revealed by distortions away from a square shadow pattern. For grid intersection separations of 1 mm and for an 8 mm pupil, the grid shadow subtends an

angle at the nodal point of approximately 2°. The central part of the grid shadow is used to determine intersection points that the whole

grid shadow would have if it were aberrationfree. The departures of the actual intersection points from the aberration-free points are transverse ray aberrations. From the transverse ray aberrations, a wave aberration polynomial is derived of the form of equation (15.1). Figure 15.6shows the variation in wave aberration across the pupil of one eye.

The original aberroscope technique involved subjects drawing the grid image. Later, Walsh et al. (1984) and Walsh and Charman (1985) made the technique objective by photographing the grid shadow. Further advances in imaging methods were described by Atchison et al. (1995) and Walsh and Cox (1995). It may be thought that this is now a 'double pass' technique, in which the aberrations of the eye are involved for the passage

of light both into and out of the eye. However, it is only for the passage of light into the eye that each grid intersection point is 'distorted'

in the image by a small part of the pupil. For

the passage out of the eye, light from each intersection point passes through the whole pupil - the aberrations of the eye for this direction serve only to blur the final image rather than to distort it.

butions to the aberration cannot be determined. Williams et al. (1994) and Liang and

Williams (1997) suggested that the low sampling density leads to an apparent over-

estimation of the image quality for this method compared with some other methods (see Chapter 18, Retinalimage quality).

Wave-front sensor (Liang et al., 1994; Liang and Williams, 1997)

A narrow beam (e.g. 1 mm wide) from a point light source is imaged by the eye, and the light passed back from the fundus travels through a wave-front sensor consisting of an array of micro-lenses (typically of the order of 0.5 mm

diameter) and onto the array of a CCO camera. Each micro-lens isolates a small

bundle of light passing through a small region of the pupil. The transverse ray aberration

associated with each microlens can be determined from the departure of the centroid of its corresponding image from the ideal

image position. Using an adaptive mirror, this technique has been used to correct the

aberrations of the eye (Liang et al., 1997).

Objective optometers

Objective optometers have been used to

measure longitudinal peripheral aberrations of astigmatism and field curvature - see, for example, Ferree et al. (1931, 1932 and 1933),

Millodot (1981) and Smith et al.

Choice of reference point and/or axis

With all methods, there is a problem of choice of the reference point and axis in the pupil that will be designated as aberration free. This point and axis are critical to the determined aberrations. In conventional optical systems, this is usually the pupil ray. Koomen et al. (1956) showed the effect of a different choice

of the reference axis on asymmetries of measured aberrations. The reference point

and!or axis do not appear to have been well defined or well monitored in many early studies. The most obvious choices are the

centre of the entrance pupil as the reference

point and the line-of-sight as the reference axis. However, this has the disadvantage that

the pupil centre shifts as pupil size changes under the influence of lighting level and drugs (Walsh, 1988; Wilson et al., 1992). Some investigators have used the visual axis (e.g. Ivanoff, 1953; Millodot and Sivak, 1979; Woods et al., 1996) (see Chapter 4, Axes of the eye). This will not change with pupil size and

has the added advantage of being the reference axis for describing chromatic aberrations (see Chapter 17, Chromatic aberrations).

Types and magnitudes of monochromatic aberrations

Early investigators noted that the monochromatic ocular aberrations were not usually rotationally symmetrical within the pupil. However, many have considered the aberration measured to be conventional spherical aberration, which is a rotationally symmetrical aberration. Because of this, care needs to be taken when interpreting their

Monochromatic aberrations 143

results, particularly where measurements have been made along only a single meridian. Later authors (for example, Smimov, 1962; Howland and Howland, 1976 and 1977)

emphasized the asymmetric nature of the eye's aberrations.

Spherical aberration

The effect of spherical aberration is shown in Figure 15.7a, which shows a set of rays from an axial point at O. Its paraxial focus is at 0' on the retina. When spherical aberration is present, non-paraxial rays do not intersect at

the paraxial focus. The further a ray is from the optical axis, the further its axial crossing

point is from the paraxial focus. The figure

shows the case of positive spherical aberration. For negative aberration, the rays (if projected) cross the axis beyond the retina.

The aberration for each ray may be expressed in terms of the longitudinal aberration distance 8' as a function of the height 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 positive spherical aberration by a negative power lens

-.1F(r).

c.Determination of spherical aberration by finding the retinal conjugate for a ray. The longitudinal spherical aberration is -L(r)

(= -l//(r».

144 Aberrations and retinal image quality expressed as a power error M(r), where

where n'is the refractive index of the vitreous and I'is the distance from the back principal plane of the eye at P' to the paraxial image at 0'.

In object space, the longitudinal spherical aberration may be expressed as the power of a correcting lens -M(r) placed in front of the

eye, but only covering a narrow region of the pupil at a distance r from the centre, as shown

in Figure 15.7b.This lens will alter the path of the ray so that it crosses the axis at the retina.

The power of this lens will depend upon

vertex distance. The longitudinal spherical aberration may be expressed also in terms of

the object vergence

of the retinal conjugate for a ray, as shown in Figure 15.7c.If the excess power is positive, as

it usually is for the unaccommodated eye, the correcting lens will have a negative power.

The power error M(r) is an image space quantity that should be similar to the object space quantities of the correcting lens and change in object vergence L(r), except for the change of sign. We usually define the sign of

Ray height in pupil (mrn)

Figure 15.8.Some experimental measures of spherical aberration of the relaxed eye. The thick curve is the weighted mean fitted by a quadratic function of the

form of equation (15.15).

the spherical aberration as the sign of the power error, and not that of the correcting lens. Discrepancies will arise when the cor-

recting lens is not placed at the first principal plane of the eye but at a short distance in front of it, and if the object vergence is not referenced to the first principal plane. Also, increasing discrepancies result as aberration levels become very high, as noted by Thibos et

al. (1997).

The mean results of some investigations are

plotted in Figure 15.8 in terms of the power error. All of these investigations found mean

spherical aberration was positive for the relaxed eye.

Aberration theory predicts that, for a rotationally symmetric optical system, the power error can be expressed as an even order power function in ray height r, and can thus be expressed in the form

M(r) = br2 + terms of order y4 and higher (15.13)

If the eye contains only primary aberration, all the terms can be neglected except the first so that the power error is quadratic in ray height r in the pupil. Some investigators have assumed that this rule can be applied to the real eye, at least out to some specified pupil diameter (e.g. van Meeteren, 1974;Charman et al., 1978).The b values determined from some published values of spherical aberration shown in Figure 15.8 are listed in Table 15.1. Following re-analysis of data, the values are slightly different from those given in Table 35.2 of Smith and Atchison (1997). These values of b show a large variation, which is due partly to the variation in spherical aberration between subjects. From Table 15.1, the mean weighted value of b (for non-

aberroscopic studies) is

This equation is plotted, as well as the experimental data, in Figure 15.8.

The spherical aberration term in the wave aberration function

If we wish to use this experimentally determined value of spherical aberration to

further examine its effect on retinal image

quality, using criteria such as the point spread or optical transfer functions, we need to know the value of the corresponding wave spherical aberration co-efficient. In equation (15.4), this

co-efficient is denoted as oW4J)' which is only the axial term. However, the fovea is off-axis, and therefore measured values of spherical aberration include field dependent values,

which would be summed. Wedenote this sum

Using the value of b given by equation (15.14), we have

W4,0 = 0.019X 1O-3(± 0.009x 10-3) mm-305.1n

Studies of the aberrations using the aberroscope technique (see Methods of measuring monochromatic aberrations, this chapter) have used an equation of the form of equation (15.1), with truncation of the series at the W14 term. We refer to the co-efficients WlO to W14

as 'spherical aberration-like' terms. If the aberration present were purely spherical aberration, we would have

with all of the other co-efficientsequal to zero. However, because of the asymmetrical nature of the aberrations, this does not occur in real eyes. Howland and Howland (1997) showed

that, in these cases, the spherical aberration

can still be derived. Using their equations and those of Atchison (1995), we can obtain

Applying this equation to the data of Walsh and Charman (1985) gives

W4,o=0.0084 x 1O-3(± 0.0134 x 10-3) mm-3 (15.20a)

The mean value of the spherical aberration for the above two sets of results is

Substituting this mean value into equation (15.16), we have a predicted b value of

and hence, from equation (15.13), the power error at the edge of an 8 mm diameter pupil is

The value of W4 0 in equation (15.21) is about half the value predicted from combining the previous studies and given by equation (15.17), but they are probably not statistically significantly different because of the large standard deviations.