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

Magnitude

Table 17.1 shows results of experimental studies of foveal transverse chromatic aberration. Although the wavelength range of measurement must be taken into account, it is reasonable to say that the mean results are about half the 1.05 min. arc predicted for schematic eyes (486-656 nm) with centred

pupils and the fovea 5° to the optical axis. The probable reasons for this discrepancy are that this angle may be larger than that occurring in many people, and that the pupil is usually decentred nasally and its centre is therefore closer to the visual axis. The main concern

with transverse chromatic aberration is for severely decentred natural or artificial pupils. The effect of transverse chromatic aberration

on visual performance is discussed in the next section.

Ogboso and Bedell (1987) made experimental measurements of transverse chromatic

aberration in the peripheral visual field. The results were very different among their four subjects. Out to 40° object eccentricity, all values were less than 7 min. arc and considerably less than the 11 min. arc predicted from

their model eye calculations at 40° eccentricity (wavelength range 435-572 nrn).

Effects of chromatic aberrations on vision

Accommodation

Fincham (1951) introduced step changes in accommodative stimulus to his subjects. Most of them made appropriate accommodation

responses in white light, but 60 per cent of them were unable to do so in monochromatic light. There is now considerable evidence that the longitudinal chromatic aberration helps the accommodation system respond correctly when there is defocus blur. Accommodation responses to steady and moving targets are more accurate in white or broad wavelength light than in monochromatic light, and doubling the eye's normal chromatic aberration has little effect on accommodative accuracy, whereas correcting or reversing the chromatic aberration leads to poor accommodative response (Kruger and Pola, 1986 and 1987; Kruger et al., 1993, 1995 and

1997;Stone ctal., 1993;Aggarwala etal., 1995a and 1995b).

Spatial vision

The effects of chromatic aberration are attenuated by the spectral sensitivity of the eye. For centred pupils, switching from monochromatic light to white light gives a maximum loss to the contrast sensitivity function of only 0.2 log units (Campbell and

Gubisch, 1967), and has a negligible effect on visual acuity (Hartridge, 1947; Campbell and

Gubisch, 1967). The use of achromatizing lenses makes negligible improvement to

visual acuity and contrast sensitivity in white light (Campbell and Gubisch, 1967). Possible

reasons include that the potential improvement is small anyway, some of these lenses

have transverse chromatic aberration when centred (see the following section), and that

additional transverse chromatic aberration is

introduced for any of these lenses if they are not carefully centred (Zhang et al., 1991).

Decentring natural or artificial pupils can have devastating effects on foveal spatial vision in white light because large amounts of transverse chromatic aberration are induced. Experimental studies showing up to twothirds reduction in resolution for es 3 mm displacement of artificial pupils (Green, 1967)

or Maxwellian-view instruments (Bradley et al., 1990; Thibos et al., 1991b) are supported theoretically (Thibos et al.,1990).

Some Maxwellian-view projection systems are used clinically to evaluate potential visual acuity in patients with cataract by positioning them so that the light passes through a relatively clear part of the cataractous lens. The experimental and theoretical results just

mentioned indicate that those instruments using white light are likely to underestimate

potential visual acuity if the light beam enters

the pupil well away from its centre. Theoretically, the transverse chromatic

aberration should increase on going further into the peripheral field (Thibos, 1987), with increasingly deleterious effects on image quality. However, the importance of the aberration is likely to be small because of

decreasing colour sensitivity and spatial resolution due to neural limitations.

Effects of chromatic aberration on retinal

image quality are discussed further in Chapter 18, Retinal image quality.

Chromostereopsis

In Chapter 6 (Binocular vision) we described stereopsis, in which the two eyes provide the potential for seeing depth in a scene. Chromostereopsis is a related binocular phenomenon in which objects at the same distance, but of different colours, are seen in depth. Most commonly, reddish objects appear to be closer than bluish objects.

Chromostereopsis is a consequence of transverse chromatic aberration combined with binocular vision.

Figure 17.6ashows the pupil ray paths from red and blue targets at position O. The pupil of the eye is decentred temporally relative to the visual axis. The red pupil ray is refracted less than the blue pupil ray, and therefore the red pupil ray intersects the retina to the temporal side of the blue pupil ray. This

Chromatic aberrations 187

Temporal

Nasal

Figure 17.6.Chromostereopsis.

a.An explanation of the source of chromostereopsis.

b.The expected relationship estimating chromostereopsis and transverse chromatic aberration induced with small artificial pupils.

retinal disparity is equivalent to the red and blue rays coming from different distances, as

shown by the dashed lines in the figure. The retinal disparity leads to an apparent longitudinal displacement when the target is viewed binocularly. The red target appears to be closer than the blue target.

The magnitude of chromostereopsis is predicted by the geometry in Figure 17.6b (Ye et al., 1991). Here, L1d is the amount of chromostereopsis as a distance measurement, 2B is the distance between the position of

small pupils in front of the two eyes, d is the viewing distance, and t(A)R and t(A)L are the

right eye and left eye transverse chromatic aberrations. The relationship between these quantities is

(17.13)

By making measurements of transverse chromatic aberration induced by displacement of pinholes from the visual axes

188 Aharatiollsa",l rctinat illlage ql/ality

(Transverse chromatic aberration, this chapter) and comparing these with measurements of chromostereopsis at these displacements, Ye et

al. provided experimental support for this equation and for the theory that chromo-

stereopsis with small pupils can be explained by the interocular difference in monocular transverse aberration.

Chromostereopsis often diminishes as pupil size increases, and may even reverse in direction. For natural pupils, this may be

attributed at least partly to change in pupil centre as pupil size changes (Chapter 3). Ye et al. (1992) attributed the decrease in chromo-

stereopsis, with increase in pupil size, to the

Stiles-Crawford effect (see Chapter 13) acting an as anchor to shift the effective centre of pupils closer to the visual axis.

Aberrations of ophthalmic devices

Longitudinal chromatic aberration of correcting ophthalmic lenses is not considered

important; spectacle and contact lenses

because of their low powers relative to that of the eye, and intraocular lenses because they merely replace the eye's lens. Transverse chromatic aberration is not important in

contact or intraocular lenses, because these move with the eye. However, it blurs foveal vision when an eye looks through peripheral

Figure 17.7. Achromatizing lens of Powell (1981).

parts of a spectacle which is made worse when high index, low V-value materials are used.

Aberration compensation and correction

Natural compensation mechanism

Chromatic aberration effects are attenuated by the non-uniform spectral sensitivity of the eye (Chapter 11). Thibos et al. (1991a) estimated

that, when the wavelength with peak sensitivity is in focus, most of the luminance

in a white light target is less than 0.250 out of focus. The yellow macula pigment contributes to the relative luminous efficiency functions, and it's absence in the periphery may remove some of the attenuation.

Achromatizing correcting lenses

The longitudinal chromatic aberration of the eye can be corrected by a nominally zero power lens, with longitudinal chromatic aberration equal and opposite to that of the eye (van Heel, 1946; Thomson and Wright, 1947; Bedford and Wyszecki, 1957; Fry, 1972; Powell, 1981; Lewis et al., 1982). The Powell (1981) design is given in Table 17.2and Figure 17.7. It is a modification of the Bedford and

Wyszecki triplet design, with a doublet added to reduce the residual longitudinal and transverse chromatic aberration present in the

original lens.

These lenses may be assessed by the level of residual power at the reference wavelength,

and by the residual chromatic aberration when used with the eye. The Bedford and Wyszecki, Fry, Powell and Lewis et al. designs

have less than 0.25 D equivalent power at 587.6nm. The residual variation of power of

one lens/eye system is shown in Figure 17.8. The lenses introduce transverse chromatic aberration, with that of the sophisticated

2.5

Wavelength (nm)

Figure 17.8. The power of the Bedford and Wyszecki (1957)and of the Powell (1981)achromatizing lenses, together with residual chromatic differences of focus for

the Powell lens. The latter was obtained by combining the Powell lens with the Chromatic eye of Thibos and

from mean experimental results of Howarth and Bradley (1986). The theoretical and experimental residuals differ

by 0.1-0.3 D at any wavelength, but neither varies by more than 0.1 D between 400 nm and 700 nm.

Chromatic aberrations 189

Powell lens being much less than those of the other lenses. For a 5° field of view and between wavelengths from 486 to 656nm, design values are 2.88min. arc (Powell), 5.23

min. arc (Bedford and Wyszecki), 7.33min. arc (Fry) and 9.34 min. arc (Lewis et aI.).

Alignment of such lenses is critical, as otherwise an additional transverse chromatic

aberration is induced which is proportional to the decentration. Zhang et al. (1991) determined that 0.4 mm of misalignment of an achromatizing lens relative to the eye would

cancel any benefit the lens would give to spatial vision.

Modelling chromatic aberrations

Chromatic dispersion

To model the chromatic aberrations of the eye,

a knowledge of its dispersive properties is necessary. This can be described by a formula such as

The advantage of this equation is that it is linear in the co-efficients and therefore, given

any data on the relationship between wavelength and index, the a co-efficients can easily be found by solving a set of simultaneous linear equations. Other formulas include the Cornu dispersion equation

where noe, K and ,1,0 are constants for a particular medium, and Sellmeier's equation

The dispersion of any optical material is fully described by such equations, provided they

are accurate. For some purposes, the dispersion can be reduced to a single number such as the Abbe V-value, which is defined as

where nd, nF and "c are the refractive indices at the wavelengths 589.3nm (A. ), 486.1 nm

(~) and 656.3nm (A.c), respectiv~y, V-values are useful for comparing the amount of

190 AbL'rralioll~ Gild retinal illIGgc '1111//il.1/

dispersion in different optical. media a~d for calculating the Seidel chromatic aberrations.

Bennett and Tucker (1975) used the form of equation (17.14a) to give the refractive index of water as

where the wavelength is in nanometres. Substituting appropriate values obtained from equation (17.16) into equation (17.15) gives Vd =55.15. As shown in Figure 17.6, the dispersion of water is insufficient to account for the longitudinal chromatic aberration of

the eye.

Le Grand (1967) presented data designed for his full theoretical eye. He quantified the

dispersion of the ocular media using t~e Cornu dispersion equation (17.14b). HIs

values of n , K and A. for each ocular medium and corresponding °V-values are given in Table 17.3.

Thibos et al. (1992) used their experimental

results and the Cornu dispersion equation (17.14b) to describe the refractive index distribution of their Chromatic version of Emsley's reduced eye as

corresponding V-value is 50.23.The c~romat~c error of refraction of the Chromatic eye IS

shown in Figure 17.6.

The high dispersion of the eye has been attributed to a high dispersion of the lens. For

example, Sivak and Mandelman (1982) measured V-values of the lens as 29 ± 4 for the

periphery and 35 ± 6 for the core. These are much less than the values of Le Grand (1967), given in Table 17.3.

Table 17.3. The values of the parameters in equation (17.15b) for Le Grand's (1967)dispersion of the ocular

media.

Schematic eyes

Gaussian properties

There are small, wavelength-dependent changes in the principal and nodal points of schematic eyes. For the Gullstrand number 1 eye, over the wavelength range of 400-700 nm, the front principal point moves just less than 0.0007 mm. Over the same wavelength

range, the back principal point moves by 0.013 mm, but this is still small.

Seidel chromatic aberrations

We discussed Seidel monochromatic aberrations in Chapter 16. The Seidel longitudinal (CL) and transverse (CT) chromatic aberrations are described by Smith and Atchison (1997). These can be calculated for schematic eyes from constructional and dispersion data, and the values for some schematic eyes are given in Appendix 3. We use the V-value of 50.23 obtained from equations (17.15) and (17.17). Seidel transverse chromatic aberrations, between 486 and 656 nm, are given as percentages of the image sizes at 589 nm. The full schematic, simplified and reduced eyes have similar Seidel chromatic aberrations.

For the Gullstrand number 1 eye at 5° angle,

Seidel transverse chromatic aberration is -0.35 per cent, and this can be converted to an angular value of 1.05 min. arc. This is a close

approximation to the exact value, and can be used as an estimate of transverse chromatic

aberration at the fovea, assuming that the line

of sight is 5° to the optical axis and the pupil is centred on the optical axis. This is consider-

ably larger than mean experimental measures

(Table 17.1), and reasons for this were discussed earlier in this chapter (Measurement of iransoerse chromatic aberration).

Chromatic difference of power and chromatic difference of refraction

Figure 17.1b shows a general schematic eye and the retinal conjugates for wavelengths A.

and l. We have

11 '(A.) / I'- L(A.) =F(A.) and 11 '(l)/ I' - L(l) =F(l)

(17.18a and b)

For both situations, the distance I' is the distance from the back principal point to the

retina and is here assumed to be independent of wavelength. The chromatic difference of

refraction RE(.:t) is given by equation (17.1), that is

The chromatic difference of power M'(.:t), the

difference between the equivalent power F(.:t) at wavelength .:t and the equivalent power F(X) at the reference wavelength X, is given by

Subtracting equation (17.18b) from (17.18a),

and after a small amount of manipulation involving equations (17.1) and (17.19), we

have

If we replace the length I' using equation (17.18b), equation (17.20) can be written in the

form

For an emmetropic eye focused on infinity, L(X) =0 and equation (17.21) reduces to

This equation shows that there is a difference between the chromatic difference of refraction RE(.:t) and the chromatic difference of power M'(.:t), apart from a change in sign.

Using appropriate chromatic dispersions of the media, paraxial schematic eyes are excellent at estimating the chromatic difference of refraction of real eyes. As an example, Atchison et al. (1993) used the Gullstrand number 1 eye with its refractive

indices, but scaled the variations in refractive index so that the chromatic dispersion was the same as for equation (17.18). This gave a range of chromatic difference of refraction of 1.940

between 400 and 700nm. This is probably a slight underestimation of chromatic difference

of focus in most eyes.

Atchison et al. (1993) determined the effect

of refractive error on change in range of chromatic difference of refraction for schematic eyes, modified to have axial ametropia and refractive ametropia. The latter was

Chromatic aberrations 191

modelled by altering corneal curvature. For the Gullstrand number 1 eye, changes occurring were 0.0120 (0.56 per cent) per

dioptre of axial ametropia and 0.0470 (2.4per cent) per dioptre of refractive ametropia. Similar results were obtained for reduced

eyes. The accommodated version of the

Gullstrand eye gave a range of chromatic difference of refraction 0.55 0 greater than the

relaxed, emmetropia eye, which corresponds to 0.0500 (2.6per cent) increase per dioptre of

accommodation.

Chromatic difference of refraction of reduced schematic eyes

If the corneal radius of curvature of the

reduced eye is r, the equivalent power F(.:t) as a function of wavelength is simply

where n'(.:t) is given by equations such as equation (17.17). The radius can be given in

terms of the reference power F(X) as

The equivalent power of the eye F(.:t) as a function of wavelength is then

and the chromatic difference of power M'(.:t) is

Substituting the right-hand side of this equation (17.27) for F(X) into equation (17.21)

gives

For an emmetropic eye focused on infinity, L(X) =0 and equation (17.28) reduces to

Using a value of 1.333for n'(X) shows that the chromatic difference of refraction is threequarters of the chromatic difference of power.

192 Abl.'rmliollsalltl r<'lillal illlagl.' qllalily

Chromatic and Indiana reduced eyes of Thibos et al. (1992,1997)

Some of the details of these reduced eyes were provided in Chapter 16. The refractive index distribution is given by equation (17.17), and the chromatic difference of refraction (see Figure 17.6) is given in dioptres by

where the reference wavelength is 589 nm and the wavelength A. is in nanometres. To obtain good predictions of transverse chromatic

aberration of their eyes according to equations (17.2) and (17.5), Thibos et al. placed the stop

1.91 mm inside the eye so that the distance EN of 3.98mm between the entrance pupil and the nodal point was similar to that of more

sophisticated schematic eyes.

To allow for variations in transverse

chromatic aberration, the pupils of the chromatic eye do not need to be on the optical axis. Thibos et al. (1992) included a number of

additional axes: line of sight, visual axis and achromatic axis (see Chapter 4). The cornea has an asphericity Q of -0.56 to correct spherical aberration at the reference wavelength of 589 nm.

The Indiana eye varies from the chromatic eye in that the optical axis, visual axis and line of sight are now coincident and the cornea has a variable asphericity (Thibos et al., 1997).

Summary of main symbols

between that at the wavelength A. and that at the reference wavelength A chromatic difference of refraction

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Introduction

The quality of the visual system depends upon a combination of optical and neural factors, with subjective measures depending also upon psychological factors.

The optical factors that determine the

retinal image quality and affect visual system quality are refractive errors, ocular aber-

rations/ diffraction and scatter. The levels of

the first three depend upon wavelength and pupil size, and can be easily quantified. Scatter is complex, and depends upon the

level and nature of the turbidity of the ocular

media, in particular the size and spatial distribution of the scattering centres.

The neural factors affecting visual system quality include sizes and spacing of retinal

cells, the degree of spatial summation at the various levels of processing from the retina to

the visual cortex, and higher level processing. The relative influences of optical and neural factors upon visual system quality vary with retinal position and the criterion used for assessing quality. In the foveal region, the infocus retinal image quality appears well matched to the neural network's resolution at optimum pupil sizes of 2-3 mm (Campbell and Green, 1965; Campbell and Gubisch, 1966)/ but resolution in the peripheral visual field is limited much more by neural factors

than by optical factors. For example, Green (1970) found that 'bypassing' the optics did

not improve the resolution of sinusoidal gratings beyond about 5° degrees from the

fovea. Bycomparison, the quality of the optics has a large influence on detection in the periphery (Wang et al., 1997).

Direct measurement of retinal image quality is not possible because of the inaccessibility of the retina. The retinal image quality can be estimated from aberrations. The aerial image of the retinal image can be analysed;

techniques using this approach are referred to as double-pass techniques, and also as

ophthalmoscopic techniques. Another possibility is psychophysical measurement of visual performance in which two similar

methods are used, one of which bypasses the

optics of the eye; comparison of the results from the two methods yields the retinal image quality.

The retinal light distribution is not the same as the perceived light distribution, partly because of the Stiles-Crawford effect, which describes the luminous efficiency Le(r) of rays entering the eye through different heights r in the pupil (Chapter 13). The Stiles-Crawford effect is a retinal phenomenon, but for some purposes it can be considered as a pupil apodization; that is, as a filter of transmittance Le(r)placed over the pupil (Westheimer, 1959). As such, it is often included in calculations of retinal image quality.

In this chapter we examine retinal image quality criteria that are common in the

analysis of general optical systems. These are the point and line spread functions and the

optical transfer function. Equations for calculating the point spread function and the

optical transfer function from the aberrations

of an optical system are given in Appendix 4. Relationships between image quality criteria

and measurements are shown in Figure 18.1. The chapter is completed with an evaluation of the retinal image quality of the eye.

The point and line spread functions

The point spread function is the illuminance or luminance distribution in the image of a

point source of light, while the line spread function is the distribution in the image of a line (of zero width) source of light. The abbreviations PSF and LSF are often used for the point spread function and line spread function, respectively.

The form of the PSF depends upon diffraction, defocus, aberrations and scattered light. In the absence of defocus, aberrations and scatter, the PSF is called the diffraction-

limited PSF. Defocus, aberrations and scattered light broaden the PSF. The form of

the PSF also depends upon the shape and diameter of the aperture stop. In the following