Visual Optics

Introduction

The human eye is a remarkable optical instrument ( ). Its performance has been honed by millennia of evolution to meet admirably the needs of the neural system that it serves. At its best, few human-engineered photographic lens systems can match its semifield of more than 90 degrees, its range of f -numbers from about f /11 to better than f /3 and its near-diffraction-limited axial performance when stopped down under photopic light levels. Moreover, the focus of the eye of the young adult can be adjusted with reasonable accuracy for distances between about 0.1 m and infinity. Nevertheless, all eyes suffer from a variety of regular and irregular aberrations, while a substantial subset displays clinically significant spherical and astigmatic refractive errors. In addition, the ability to change the power of the crystalline lens to view near objects is an asset that declines with age, to disappear entirely by the mid-50s, when presbyopia is reached.

The invention of spectacles in the 13th century, and their subsequent relatively slow refinement, followed by the more rapid development of contact lenses in the 20th and 21st centuries, has done much to provide solutions to the problems of both refractive error and presbyopia: improvements in the design of both types of lens continue to be made. Refractive surgical techniques, including both laser-based methods, which modify the corneal contour and intraocular lenses, are beginning to compete with spectacle and contact lens corrections, although unanswered questions still remain concerning the long-term efficacy and safety of some of the procedures used. In this chapter, the basic optics of the eye and its components will first be reviewed. This will be followed by a discussion of the modifications that the correction of refractive error—particularly by contact lenses—produces in factors such as spectacle magnification (SM), accommodation and convergence ( ).

The Basic Optics of the Eye and Ametropia

General Optical Characteristics

The familiar, and deceptively simple, optical layout of the eye is shown in Fig. 3.1 .

About three-quarters of the optical power comes from the anterior cornea, with the crystalline lens providing supplementary power that, in the prepresbyope, can be varied to focus sharply objects at different distances. The actual optical design is, however, subtle, in that all the optical surfaces are aspheric, while the lens, and probably also the cornea ( ), displays a complex gradient of refractive index. There is little doubt that such refinements play an important role in controlling aberration.

The distribution across the population of parameters such as surface radii, component spacing and refractive indices has been studied by a variety of authors ( ; ). Refractive indices of the media vary little between eyes, apart from the nonuniform refractive index distribution within the lens, which changes with age as the lens grows throughout life ( ). Each dimensional parameter appears to be approximately normally distributed among different individuals ( ; ). The values of the different parameters in the individual eye are, however, correlated so that the resultant distribution of refractive error is strongly peaked near emmetropia, rather than being normal ( Fig. 3.2 ).

This correlation is thought to be due to a combination of genetic and environmental factors, visual experience helping to ‘emmetropize’ the eyes actively ( ). The apparently greater incidence of myopia in recent times has been attributed to the greater prevalence of near tasks and other changes in environment and lifestyle ( ; ).

Model Eyes and Ametropia

Many authors have produced paraxial models of the emmetropic eye, based on typical measured values of the ocular parameters ( ). These simplify the optical complexities of the real eye while having approximately the same basic imaging characteristics. Some examples are given in Table 3.1 ; fuller details of these and other more elaborate eye models (e.g. ) are given in the cited references.

Table 3.1

Parameters of Some Paraxial Models of the Human Eye (Charman, 1991)

After Charman, W. N. (Ed.). (1991). Optics of the human eye. In: Vision and visual dysfunction. Vol. 1: Visual optics and instrumentation (pp. 1–26). London: Macmillan.

		Schematic Eye	Simplified Schematic Eye	Reduced Eye
Surface radii (mm)	Anterior cornea	7.80	7.80	5.55
	Posterior cornea	6.50	–	–
	Anterior lens	10.20	10.00	–
	Posterior lens	−6.00	−6.00	–
Distances from anterior cornea (mm)	Posterior cornea	0.55	–	–
	Anterior lens	3.60	3.60	–
	Posterior lens	7.60	7.20	–
	Retina	24.20	23.90	22.22
Refractive indices	Cornea	1.3771	–	–
	Aqueous humour	1.3374	1.333	1.333
	Lens	1.4200	1.416	–
	Vitreous humour	1.3360	1.333	–

Dimensions are in millimetres.

Using the parameters of the model eyes it is straightforward to calculate the positions of the cardinal points, which, in thick-lens theory, can be used to summarize paraxial imagery ( Fig. 3.3 ).

It is, however, important to stress that these eye models are only representative. In practice, an eye of shorter or longer axial length may still be emmetropic. This behaviour and various possible origins of refractive error are easy to understand in terms of these basic models. Consider, for simplicity, the generic reduced eye shown in Fig. 3.4 , with a single refractive surface of radius r , refractive index n ′ and axial length k ′. The power of the eye, F _e , is given by

$F_e=(n^{\prime }-1)/r$

For a distant object (zero object vergence), the image vergence n ′/ l ′ equals F _e . For emmetropia, we require that the image of the distant object lies on the retina, i.e. l ′ = k ′, implying that F _e = n ′/ k ′= K ′, where K ′= n ′/ k ′ is the dioptric length of the eye. There are, then, in principle, an infinite number of matching pairs of values of F _e and K ′ that lead to emmetropia so that eyes that are relatively larger or smaller than the ‘standard’ models may still be emmetropic.

In the case of ametropia, F _e and K ′ are no longer equal. If the power of the eye is too high ( F _e > K ′), we get myopia; if too low ( F _e < K ′), we get hypermetropia. The ocular refraction K is given by

$K=K^{\prime }-F_e$

Thus, for example, myopia ( K negative) can occur if K ′ is too low, corresponding to an axial length k ′ that is relatively too great (axial ametropia), or if F _e is relatively too large (refractive ametropia). A high F _e may arise as a result of either too small a corneal radius r or because n ′ is too large (note, however, that changes in n ′ affect both F _e and K ′). Although more sophisticated eye models are characterized by more parameters, the possible origins of ametropia are essentially the same.

Astigmatism can arise either because one or more of the optical surfaces is toroidal or because of tilts of surfaces with respect to the axis, particularly of the lens.

How accurate do our models and associated calculations have to be? Although in the laboratory it may theoretically be possible to measure all the parameters of an individual eye, in general, all that will be known in the consulting room is that the eye is ametropic. Thus in clinical contact lens practice, precise calculation of the optical effects in the uncorrected or corrected eye is rarely possible: it is more important that the general magnitude of the effects be borne in mind and that the approximate changes brought about by correction be fully understood.

Accommodation and the Precision of Ocular Focus

The decline in the subjective amplitude of accommodation (i.e. the reciprocal of the distance, measured in metres, of the nearest point at which vision remains subjectively clear to the distance-corrected patient) with age is illustrated in Fig. 3.5A .

Few everyday tasks require accommodation in excess of about 4D so that it is normally only as individuals approach 40 years of age that marked problems with near vision start to appear. It is, however, important to recognize that, even for objects lying within the available range of accommodation, accommodation is rarely precise. ‘Lags’ of accommodation usually occur in near vision and ‘leads’ for distance vision ( Fig. 3.5B ). Since the accommodation system is driven via the retinal cones, these lags increase if the environmental illumination is reduced to mesopic levels and the accommodation system is effectively inoperative at scotopic light levels, when the system reverts to its slightly myopic (around −1D) tonic level ( ).

Corneal Topography

It has already been stated that the optical surfaces of the eye are not necessarily spherical. The topography of the anterior cornea is of particular interest, since, as the dominant refractive surface, its form has a major influence on overall refractive error and ocular aberration. In contact lens work, it is of enormous importance to the fitting geometry.

We have already seen ( Fig. 3.2A ) that the radius of curvature over the central region, as measured by conventional keratometers, shows considerable individual variation, and it has been recognized for more than a century that many corneas display marked astigmatism. Corneal astigmatism is not, of course, necessarily equal to the total ocular astigmatism, since additional astigmatism (residual astigmatism) may be contributed by the crystalline lens.

Earlier work on corneal topography using modifications of traditional keratometers concentrated on approximating the form of the corneal surface by a conicoid, in which each meridian is a conic section. In this approach, the anterior corneal surface can be described by the following equation ( ):

where the coordinate system has its origin at the corneal apex, z is the axial coordinate, r ₀ is the radius of curvature at the cornea apex and the shape factor p is a constant parameter characterizing the form of the conic section for the individual eye. Values of p < 0 represent hyperboloids, p = 0 paraboloids, 0 < p <1 flattening (prolate) ellipsoids, p = 1 spheres and p > 1 steepening (oblate) ellipsoids. The same equation is sometimes written in terms of the Q -factor or the eccentricity e of the conic section, where

found mean r ₀ and p values of 7.72 ± 0.27 mm and 0.74 ± 0.18, respectively, these values being supported by the results of , that is 7.85 ± 0.25 mm and 0.85 ± 0.15: broadly similar p values are found in different racial groups (0.70 ± 0.12 in Chinese eyes; ; 0.74 ± 0.19 in Afro-Americans; ). Thus the typical general form of the cornea is that of a flattening ellipsoid, with the curvature reducing in the periphery ( Fig. 3.6A ).

A range of topographic instruments is available, marrying optical with electronic and computer technology, that can routinely give a much fuller picture of the corneal contour (see Chapter 34 ). These corneal topography and scanning-slit instrument show that the conicoidal model is only a first approximation to corneal shape and that individual eyes show a wide range of individual asymmetries. In particular, the rate of corneal flattening is often different in different meridians ( Fig. 3.6B ), while the corneal cap of steepest curvature may be displaced with respect to the visual axis, on average lying about 0.8 mm below ( ). More elaborate models have been devised to describe these asymmetries in corneal shape ( ).

Currently the most popular form of output for the topographic data is probably a colour-coded map of the cornea, showing regions of different axial (sagittal) power (see Chapter 34 ). This may be slightly misleading, since each local area of the cornea is toroidal rather than spherical. For this reason, both sagittal and tangential power maps are often used ( ). It is possible that other forms of representation, such as those which plot local departures in height from a best-fitting sphere, will ultimately prove more useful, particularly in relation to the fitting of rigid contact lenses ( ; ). The contribution of the cornea to the overall ocular wave aberration can be deduced from corneal topography measurement ( ). Scanning slit instruments, such as the Orbscan and Pentacam, allow the form of the posterior surface of the cornea to be deduced, as well as that of the anterior surface (see Chapter 34 ).

Pupil Diameter and Retinal Blur Circles

As will be discussed below, although the retinal image is always blurred by both aberration and diffraction, in ametropia and presbyopia it is often defocus blur that is the major source of degradation. Defocus will occur whenever the object point lies outside the range of object distances embraced by the far and near points of the individual. As noted earlier, even within this range, small errors of focus will normally occur due to the steady-state errors that are characteristic of the accommodation system. Using a reduced eye model and simple geometric optical approximations ( ; ; )—which are normally valid for all errors of focus over about 1D—such blur depends on the dioptric error of focus and the pupil diameter. From Fig. 3.7 , it can be seen that for any object point and assuming that the eye pupil is circular, spherical defocus produces a ‘blur circle’ on the retina.

Using similar triangles, it is easy to show that the diameter ( d , in mm) of this blur circle is

where Δ F is the dioptric error of focus with respect to the object point, D is the pupil diameter in millimetres and K ′ is the dioptric length of the eye. If astigmatism is present, the blur patch is an ellipse, with major and minor axes corresponding to the focus errors in the two principal meridians.

We can express the blur circle diameter in angular terms as

Thus for a 3-mm diameter pupil, the blur circle diameter increases by roughly 10 min arc per dioptre of defocus. measured blur circle diameters experimentally and found that results for pupil diameters between 2 and 6 mm and defocus between 1 and 12D were quite accurately predicted by Eq. (3.1) .

The impact of blur on visual acuity depends somewhat on the acuity target chosen and the criteria and observation conditions used. We would expect the minimum angle of resolution (MAR) to be somewhat smaller than the blur circle diameter. suggests that for errors of focus above about 1D, letter targets, a 50% recognition rate and normal chart luminances of about 150 cd/m ² (giving pupil diameters of about 4 mm):

With errors of focus smaller than about 1D, diffraction, aberration and the neural capabilities of the visual system are more important than defocus blur and the MAR exceeds that predicted by Eq. (3.2) .

The natural pupil diameter is chiefly dependent on the ambient light level. Fig. 3.8 shows typical results for this relationship in young adults.

Pupil diameters at any light level tend to decrease with age (senile miosis: ) and with accommodation, as well as varying with a variety of emotional and other factors ( ). Some typical values for older eyes under different lighting conditions are given in Table 3.2 .

Clearly, reducing the pupil size results in smaller amounts of blur in the retinal image for any given level of defocus, and thus the depth-of-focus is increased. For example, an uncorrected low myope may experience minimal levels of distance blur under good photopic levels of illumination but may notice considerable blur when driving at night, when the pupil is large. Pupil diameter strongly influences the design and performance of bifocal and other types of contact lens for the presbyope ( , see Chapter 22 ).

Effects of Diffraction and Aberration

As noted above, these are chiefly important when the eye is close to its optimal focus. The point image for a spherical error of focus then no longer approximates to a blur circle (or a point in the absence of defocus) but has a more complex form.