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.
| 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 | – |
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
Fe=(n′−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′−Fe
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 ( ):
x2+y2+pz22r0z
ρ=1+Q=1−e2
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
d=ΔFD/K′
We can express the blur circle diameter in angular terms as
α=ΔFD10−3rads=3.44ΔFDminarc
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):
MAR=0.65ΔFDminarc
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 .
| Condition | Pupil diameter mm Ages 40–49 | Pupil diameter mm Ages 50–59 |
|---|---|---|
| Night driving |
|
|
| Reading (low illumination, 215 lx) |
|
|
| Reading (high illumination, 860 lx) |
|
|
| Outdoors (indirect sunlight, 3400 lx) |
|
|
| Outdoors (direct sunlight, 11,000 lx) |
|
|
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.
Diffraction
If the optical performance of the eye was limited only by diffraction, the in-focus retinal image of a point object would be an Airy diffraction pattern. The angular radius of the first dark ring in this pattern would be
θmin=1.22λ/Dradians=4194λ/Dminarc
Monochromatic Aberrations
Aberration obviously acts to introduce additional blur into both in- and out-of-focus images. Monochromatic aberration can arise from a variety of causes. The eye would be expected to display the classical Seidel aberrations (spherical aberration, coma, oblique astigmatism, field curvature and distortion) inherent in any system of spherical centred surfaces but, due to the various asphericities, tilts, decentrations and irregularities that may occur in its optical surfaces ( Fig. 3.6B ), its aberrational behaviour is much more complex than that which would be expected on the basis of simple schematic eye models of the type illustrated in Fig. 3.3 and Table 3.1 .
Early authors attempted to analyse ocular aberration in terms of the individual Seidel aberrations. However, these attempts were of limited value because of the lack of rotational symmetry in the system. Monochromatic aberration is now most commonly expressed in terms of wavefront aberration ( ). The behaviour of a ‘perfect’ optical system, according to geometrical optics, can either be visualized as involving rays radiating from an object point to be converged to a unique image point, or as spherical wavefronts diverging from the object point to converge at the image point, so that the object point is the centre of curvature of the object wavefronts and the image point that of the image wavefronts ( Fig. 3.9A ).
The rays and wavefronts are everywhere perpendicular to one another. If we have aberration, the image rays fail to intersect at a single image point. Similarly, the wavefronts, which are still everywhere perpendicular to the rays, are no longer spherical ( Fig. 3.9B ). It is usual to express the wavefront aberration at any point in the pupil as the distance between the ideal spherical wavefront, centred on the Gaussian image point, and the actual wavefront, where both are selected to coincide at the centre of the exit pupil ( Fig. 3.9C ).
Recent years have seen an explosion of interest in ocular aberrations, largely fuelled by the realization that the earlier excimer-laser refractive surgical techniques often resulted in poor vision because these procedures introduced unacceptably high levels of aberration. As a result, a variety of commercial aberrometers have become available for measuring the wavefront aberration of the eye ( ). One of the more elegant designs involves the use of a Hartmann–Shack wavefront sensor ( ). A hexagonal array of identical microlenses allows the slope of the wavefront across a lattice of points in the pupil to be determined. The principle can be understood with reference to Fig. 3.10 .
Suppose we have a point source on the retina of a perfect emmetropic eye. The light leaving the eye can be envisaged either as a bundle of parallel rays or as a series of plane wavefronts ( Fig. 3.10A ). We now place our array of microlenses in the path of the emerging light. Evidently, each lens will converge the parallel rays to its second focal point so that in the common focal plane we shall see an absolutely regular array of image points. If now the eye suffers from aberration, the emergent rays are no longer parallel and the associated wavefronts are no longer flat ( Fig. 3.10B ). Thus the rays no longer come to a focus on the axes of the lenses: the lateral displacement from the focal point of each lens is directly proportional to the local inclination of the ray or the slope of the wavefront. It is, then, easy to calculate the form of the emergent wavefronts and the wavefront aberration from the distorted pattern of image points.
Examples of some typical axial results for normal eyes corrected for any spherocylindrical refractive error are shown in Fig. 3.11 . The wavefront error is usually expressed in microns.
Departures from the reference sphere (in this case of infinite radius) of more than a quarter of a wavelength (i.e. around 0.14 µm for the green region of the spectrum) would be expected to degrade image quality. What is striking is the wide variation between the aberrations shown by different eyes. The aberration in the central 2–3 mm of the pupil is usually modest but much larger amounts may be found in the periphery of dilated pupils. On the basis of wavefront aberration results, it is possible to calculate monochromatic point and line spread functions and also the ocular modulation and phase transfer functions for any pupil diameter ( ).
Note that the wavefront maps shown in Fig. 3.11 were obtained on axis with the eyes under cycloplegia. In each case, ocular aberrations get worse nearer to the peripheral pupil, as with most optical systems. In practice, the aberrations on the visual axis of each individual eye vary slightly with time due to factors such as accommodation fluctuations and tear layer changes after a blink ( ). There will also be variation in the measured wavefront errors due to the limited reliability of any aberrometer. It can be seen, for example that in Fig. 3.11A–C the signal-to-noise of the Hartmann–Shack point images is poor in some cases: this may lead to errors in the estimates of the corresponding local slope and form of the wavefront.
Although the basic wavefront map gives a useful impression of the form and extent of the wavefront errors, it is helpful to be able to quantify this in some way. Various methods are available but those which are the most popular at the present time are the total root-mean-square (RMS) wavefront error and the values of the Zernike coefficients for the wavefront error.
The basic method for obtaining RMS wavefront error for any diameter of the pupil is easily understood. We divide the pupil into equal small areas and sum the squared values of the wavefront error for each small area. This sum is then divided by the number of areas and the square root of this result gives the RMS wavefront error. It can be shown that, if the RMS aberration is less than a 14th of a wavelength, i.e. about 0.04 µm, there is a negligible loss in retinal image quality in comparison with an eye whose performance is limited only by diffraction. Obviously, for any eye, the RMS error will vary with pupil diameter: in general, since the wavefront aberration tends to increase in the outer zones of the dilated pupil, the RMS aberration increases with pupil diameter.
investigated axial RMS wavefront errors as a function of pupil diameter and age in a large sample of normal eyes, which were corrected for spherical and cylindrical refractive error. Table 3.3 gives the means and standard deviations of their data for subjects aged 30–39 years.
| Pupil diameter (mm) | Typical luminance level (cd/m 2 ) | RMS wavefront error (µm) | Equivalent defocus (D) |
|---|---|---|---|
| 3 | 400 | 0.052 ± 0.022 | 0.16 |
| 4 | 70 | 0.102 ± 0.041 | 0.18 |
| 5 | 7 | 0.174 ± 0.062 | 0.19 |
| 6 | 0.1 | 0.289 ± 0.091 | 0.22 |
| 7 | 0.0005 | 0.513 ± 0.138 | 0.29 |
It is interesting to note that the typical axial RMS wavefront error for a 3-mm pupil (see Table 3.3 ) is close to the limit at which the image differs negligibly from that from an aberration-free system (about 0.04 µm). The luminance at which this pupil diameter is found, a few hundred cd/m 2 , corresponds to that found on cloudy days in the United Kingdom. Thus in most eyes, wavefront aberration can only play a minor role in vision under daylight conditions.
To give some clinical insight into the image degradation caused by these levels of RMS wavefront aberration, we can roughly evaluate the blurring effect of the RMS aberration by equating it with those of an ‘equivalent defocus’, that is the spherical error in focus, which produces the same magnitude of RMS aberration for the same pupil size. The equivalent defocus is given by
Equivalentdefocus(D)=4.31/2[RMSerror]/R2
The second common way of specifying aberrations is in terms of Zernike coefficients ( ). The idea here is that, since very different forms of the wavefront can have the same total RMS error yet still produce somewhat different effects on vision, it is better to break the complex wavefront patterns of the type shown in Fig. 3.11 into a set of simpler ‘building blocks’. Each ‘block’, mathematically described by a Zernike polynomial, corresponds to a specific type of wavefront deformation: some of these are closely related to the traditional Seidel aberrations. The set of polynomials, named after their originator Fritz Zernike (1888–1966), has the advantage that the individual polynomials are mathematically independent of one another. The overall complex wavefront can then be specified in terms of the size of the contributions made by each of these constituent wavefront deformations: the size of the contribution that each makes is given by the value of the coefficient of the corresponding polynomial. In the recommended formulation in current use, each coefficient gives the RMS wavefront error (in microns) contributed by the particular Zernike mode ( ; ): the overall RMS wavefront error is given by the square root of the sum of the squares of the individual coefficients. The relative sizes of the different Zernike coefficients thus give detailed information on the relative importance of the different aberrational defects of any particular eye.
The Zernike polynomials can be expressed in terms of polar coordinates ( ρ , θ ) in the pupil, where ρ = R / R max is the relative radial coordinate, R max being the maximum pupil radius. θ is the azimuthal angle, defined in the same way as in the optometric notation, except that it can rise to 360 degrees. Each polynomial, or wavefront building block, is defined by the highest power ( n ) to which ρ is raised (the radial order) and the multiple ( m ) for the angle θ (the angular frequency): m = −2, for example means that θ appears as sin2 θ , while m = +3 means that it appears as cos3 θ . The polynomials and coefficients are, then, conveniently described as <SPAN role=presentation tabIndex=0 id=MathJax-Element-10-Frame class=MathJax style="POSITION: relative" data-mathml='Znm’>𝑍𝑚𝑛Znm
Z n m
and <SPAN role=presentation tabIndex=0 id=MathJax-Element-11-Frame class=MathJax style="POSITION: relative" data-mathml='Cnm’>𝐶𝑚𝑛Cnm
C n m
, respectively. Fig. 3.12 shows the first few levels of the ‘Zernike tree’ formed by the different polynomials, the levels corresponding to successively greater powers of n .
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