Chapter 33 The Human Eye as an Optical System MILTON KATZ and PHILIP B. KRUGER Table Of Contents 
The eye is a compound optical system comprising a cornea and a lens, as
shown in Figure 1. It is an adaptive optical system because the crystalline lens changes
shape to focus light from objects at a large range of distances on the
retina. Unlike the components of most optical systems, typified in Figure 2, the cornea and lens are not centered on a common axis, nor are they spherically
surfaced. Because a model eye is being treated here, however, it
will be assumed that the surfaces are spherical and that their centers
of curvature lie on the optical axis, a straight line from the
vertex of the cornea to the posterior pole. Furthermore, the incident
rays will be considered paraxial, that is, they lie close to the optical
axis and strike the surfaces with very small angles of incidence (Fig. 3). A bundle of paraxial rays converges to a single point focus. As the
diameter of the bundle of rays grows larger, the incidence angles of the
marginal rays become larger so that they no longer may be considered
paraxial. Spherical aberration forces them to cross the axis at different
points, thus blurring the image, as illustrated in Figure 4. A 2mm maximum pupil diameter for the eye satisfies paraxial assumptions.
Briefly, the corneal portion, including the tear layer, separates air from aqueous humor, and the lens portion separates aqueous from vitreous humor. Rays entering the eye are refracted first and by the greatest amount at the first surface of the cornea because of the large difference in index of refraction at the airtocornea interface. The second surface of the cornea has negative power; nevertheless, the cornea contributes over 70% of the approximately 64 diopters (D) of refractive power of the unaccommodated eye. The crystalline lens supplies the remaining refractive power. During accommodation, additional power is supplied by the lens, which assumes a rounder form. 
HISTORY 
The nature of the ocular image has been studied since the times of the
ancient Greeks. Galen^{1} hypothesized that a psychic spirit moved through a hollow optic nerve
to the retina and crystalline lens into the anterior chamber and was projected
out of the eyes as an emanation of rays that made objects in
space visible. The crystalline lens was the main receptor that somehow
created the visual sensation that traveled back as a visual spirit through
the optic nerve to the brain. Galen's theories were preeminent
in Europe until the Renaissance. After the decline of Greek civilization and during the Dark Ages in Europe, the most important physicist was Alhazen (965–1039), a Persian. He was interested especially in optics and made notable contributions to the study of reflection and refraction of light. He constructed a pinhole camera and parabolic mirrors and studied the rainbow and the properties of lenses. In his Book of Optics, he clearly expressed the view that light emanated from luminous sources such as the sun and was reflected from the object to the eye. Although he believed that an image was formed in the eye, he was unsure of its precise nature because of inadequate appreciation for the refractive properties of the ocular media. Only 600 years later, with the contributions of Kepler, did the understanding of optics progress significantly beyond Alhazen's ideas. Much work was done in optics during the Renaissance. Spectacles were applied to correct vision, the telescope and microscope were discovered, and the belief steadily grew that the eye formed an image in the same way as a pinhole camera. However, the greatest difficulty was experienced in reconciling the fact that the image was inverted with our perception of the world. Even da Vinci, whose notebooks were mirror written, could not accept the idea of an inverted retinal image, and attempts to construct erect images hindered a true understanding of the optics of the eye. The German astronomer Johannes Kepler (15711630) explained the role of the crystalline lens in the imageforming process. Points in space were imaged on the retina to form an inverted, real image caused by refraction by the cornea and lens. Proof of the correctness of Kepler's hypothesis was given by Scheiner (15731650), who removed part of the sclera and choroid from enucleated sheep eyes to reveal the back of the retina. While pointing the eye toward a bright object, he observed a small inverted image on the retina. Once it was realized that the formation of the ocular image depended on the curvatures and spacings of the ocular refracting surfaces and the refractive indices of the ocular media, accurate measurements of these elements became essential if the dioptrics of the eye were to be understood. Although anatomic measurements and index of refraction determinations of cadaver eyes were easy to obtain, severe problems in making in vivo determinations remained. One such measurement was ingeniously made by Scheiner, who recognized that the cornea of the eye was a convex mirror whose reflex image could provide a measure of the curvature of the cornea:
Nevertheless, the lack of sufficiently accurate data, even as late as the 17th century, frustrated Huygens (16291695) in the construction of an optical model of the eye. Although Listing (18081882) conceived a schematic eye in 1851, a major advance in measurements of the dioptrics of the eye was made by Helmholtz, who invented the ophthalmoscope (also independently invented by Babbage) and refined the ophthalmometer. Helmholtz's main difficulties were in obtaining accurate data on the crystalline lens. With the use of an ophthalmophacometer, developed by Tscherning, separate Purkinje images of all refracting surfaces could be formed and aligned at different angles of obliquity. The depth of the anterior chamber and the curvatures of the anterior and posterior crystalline lens surfaces and the lens thickness then could be calculated trigonometrically.^{3} Gullstrand^{4} refined the Helmholtz schematic eye. He invented the photokeratoscope, which he used to photograph the corneally reflected image of a target consisting of concentric circles. Measurements of the spacing of circles in the image reveal whether the cornea is spherical, aspheric, or astigmatic. If the images are elliptic, the cornea is astigmatic, that is, toroidal. Also, the peripheral portions of the cornea could be investigated and its entire contour mapped. Depth or thickness measurements were eased by Gullstrand's invention of the slit lamp. With these advances in instrumentation, Gullstrand developed an authoritative optical model of the eye composed of six spherical refracting surfaces, two for the cornea and four for the crystalline lens. As shown in Figure 1, the lens is seen as a central double convex core surrounded by a cortex that has a lower index of refraction. The 20th century has seen the development of new techniques for in vivo ocular measurements. Rushton^{5} used a roentgenologic technique by which he directed a thin sheet of xrays in a coronal section perpendicular to the visual axis of the dark adapted eye. The xrays produce a phosphene or sensation of light that appears circular because the beam is “slicing” across the eye. As the beam is translated toward the posterior pole of the eye, the section diameter decreases and the circular phosphene seen by the subject constricts. When the posterior pole is reached, the phosphene vanishes. In this way, Rushton could measure the axial length of the eye.^{6} Ultrasonography or echography also can be used to measure the depth of the ocular components. The time taken for highfrequency sound waves to travel and to reflect from the various surfaces of the cornea, lens, and fundus is measured. Distances are obtained from the following formula: distance = velocity × time. Velocities in the media are assumed from measurements on cadavers. No way of measuring the axial length of living eyes existed until the roentgenologic method was applied. Stenstrom^{7} used this method to undertake a statistical study of the length of the eye and its correlation with other optical elements. Although thousands of measurements of optical constants had been made by many researchers, they concentrated on a limited number of parameters in populations of varied sizes and types. Stenstrom noted the importance of finding the correlations among all the various dioptric elements of the eye when all these elements are studied in the same subjects. The corneal refracting power and the length of the living eyeball are the two most important elements in determining the refractive state. Stenstrom was interested in finding the variation of the axial length in various refractive states and, by measurement and calculation, in providing a clear insight into the variation of the other optical elements. Regarding individual optical elements, he concluded that lens refracting power, total refracting power, and corneal radius may be considered normal distributions but that axial length significantly deviated from a normal distribution. Stenstrom also found a slight correlation between the optical elements that resulted in a smaller spread of total refractive power than would be the case if the optical elements varied independently of each other. Finally, he concluded that the depth of the anterior chamber, the radius of the cornea, the refracting power of the lens, and the total refracting power of the eye are uncorrelated with the refractive error. The axial length of the eye showed a pronounced correlation with refractive error, however, which Stenstrom takes as confirmation that the axial length is the most important of those quantities determining the refractive error. 
TERMINOLOGY AND SIGN CONVENTION  
Before the optical system of the eye is described in detail, a review of
a few basic optical terms will be useful. This review presupposes a
knowledge of geometric optics. Light is assumed to travel from left to right. Positive distances are measured from left to right; negative distances are measured from right to left. Object distances are measured from the optical element to the object point. Image distances are measured from the optical element to the image point. In Figure 8, the object distance from the lens to the object point is negative, that is, it is measured from right to left, and the image distance is positive.

FOCAL POINTS AND FOCAL LENGTHS  
When light from an infinitely distant source found to the left of an optical
element strikes the element, the collimated paraxial rays will be
converged to F', the second focal point. This will be a real image point
for positive elements (Fig. 5) and a virtual image point for negative elements (Fig. 6). Distance F'A is the second focal length. Light originating from the
first focal point F will be collimated by the optical element, forming
an image at infinity. FA is the first focal length. The idea of refractive
power is derived from focal length and leads to the idea of vergence.

VERGENCE  
Light diverging from the object point in Figure 7 has negative vergence. The spherical wavefronts grow larger as their radial
distances from the source increase. Because curvature is the reciprocal
of the radius of curvature, the farther the wavefront is from
the object, the smaller its curvature will be. Wavefront vergence in diopters
equals the reciprocal of the radial distance in meters: Vergence = 1/Distance in meters Light that is converging toward an image has positive vergence. The wavefronts become increasingly curved as they approach the image point, and the vergence increases correspondingly. For example, at a distance of 4 meters, the vergence is ¼ = + 0.25 D; at 2 meters, the vergence is ½ = + 0.5 D. Diopters of convergence (+) and divergence () for various distances from object and image are tabulated as shown in Table 1.
TABLE 1. Vergences in Diopters at Various Distances
It is assumed that light travels from left to right. Consequently, a divergent wavefront is centered about a point to the left of the wavefront. Because the wavefront is negative, the distance from the wavefront to the point (which may be a real object or a virtual image point) is negative. Convergent wavefronts are centered about real images or virtual objects to the right of the wavefront. These distances are positive. 
REDUCED VERGENCE  
Objects and images in media of any refractive index have reduced distances. For
objects, the reduced distance is l/n, the reduced image distance is l'/n'. Similarly, the reduced first focal length is f/n and the reduced
second focal length is f'/n'. Just as vergence is the reciprocal of the
distance, reduced vergence is the reciprocal of the reduced distance. L = n/l is the reduced vergence of the object; the reduced vergence of the image
is Optical elements have the power to impress a vergence on incident light. The refracting power F (not to be confused with the first focal point F) is equal to the reciprocal of the reduced focal length, that is, F = n/f = n'/f'. The reduced vergence of the object and image and the refracting power are related in a very simple equation The reduced vergence of the light after refraction (heading toward the image point) is the sum of the reduced vergence of the light from the object when it is incident on the lens or refracting surface plus the power of the lens or surface. For example, an object in air placed 2 meters to the left of a lens will send light to the lens that, when it reaches the lens, has a divergence in diopters If the lens is a positive lens with a focal length of ½ meter the lens power The image vergence becomes L' = 0.5 + 2 = + 1.5 D. This shows that the light leaves the lens with a convergence L' of 1.5 D and will be imaged in air at The positive sign means the image is real and to the right of the lens (Fig. 8). Both the object and the image lie in air. They will lie in different media if the lens is replaced by a single refracting surface. For example, it may be supposed that a convex refracting surface of radius ¼ meter separating water and glass is facing an object 3 meters away (Fig. 9). The object is found in water for which n = 1.33. Consequently, the reduced object vergence
The refracting power of a single refracting surface is the difference between the second and first indices of refraction divided by the radius of curvature in meters: The reduced vergence of the refracted light is L' = L + F = 0.44 + 0.68 = + 0.24 D. By rearranging the equation L' = n'/l' to give l' = n'/L' = 1.5/0.24 = 6.25 meters, we obtain the position of the image. It is real and located 6.25 meters in glass to the right of the refracting surface. 
MAGNIFICATION  
In Figure 10, an object whose height is y is a distance l from a thin positive lens. The lens forms an inverted image at l' with a height of y'. Lateral magnification (m) is the ratio of the image
height divided by the object height, m = y'/y. It also can be shown
that magnification is equal to the reduced vergence from the object
divided by the reduced vergence to the image: m = L/L'.
In the previous lens example, we found L = 0.5 D, and L' = + 1.5 D; consequently, the magnification is m = L/L' = 0.5/150 = 1/3. The image is one third as large as the object, and the minus sign means that the image is inverted. Similarly, in the previous example of a refracting surface separating water and glass, L = 0.44 D and L' = 0.24 D. The magnification is m = 0.44/0.24 = 1.83. 
PRINCIPAL POINTS 
The paraxial characteristics of a complex optical system, such as the series
of lens elements shown in Figure 11a, can be determined readily by reducing the system to six cardinal points. Two
of these points, the first and second focal points, have been
considered already. Another pair, known as the first and second principal
points, will be defined. Suppose an incident paraxial ray from infinity strikes the first lens element at a height h above the axis and is refracted through the lens system. When it strikes positive elements, it is bent toward the axis. Negative elements bend it away from the axis. Ultimately, it emerges from the last element at a height h' above the axis and with a slope that converges it to the second focal point F'. Because this lens system consists of several spacedout elements, it cannot be treated like a thin lens. Consequently, the distance from the last element to F' is not equal to the focal length of the system; it is called the back focal length (BFL) and is of physical rather than optical significance. The focal length of optical significance is the equivalent focal length (EFL). This term implies that the lens system has a focal length that is equivalent to that of a simple “thin” lens. To find the position of this hypothetical “thin”lens with respect to focal point F', the series of lenses is replaced by a black box (see Fig. 11b). What is in the black box is unimportant once the height of incidence h, the height of emergence h', and the slope ' toward the focal point of a paraxial incident ray are known. The second principal plane is located by extending the incident ray forward and the emerging ray backward until they intersect. This plane defines the position of a thin lens that theoretically could replace the lens system. The point H' where this plane crosses the axis is called the second principal point. The distance from F' to H' is the equivalent focal length of the complex lens. If this process is repeated for a paraxial ray from infinity entering the lens system (or black box) from right to left, H, the first principal point, is obtained. The distance from F to H is the same as F' to H', that is, the equivalent focal length is constant no matter how the lens is turned. Usually, H and H' do not coincide. Any ray incident at some height at the first principal plane is transferred to the second principal plane at the same height as if the two planes were coincident. Consequently, the principal planes are considered conjugate planes with a magnification of one, or unit magnification planes (see Fig. 11c). No real thin lens can replace a complex system. Principal planes merely simplify calculating such paraxial quantities as object and image position and magnifications. 
NODAL POINTS 
The remaining two cardinal points are a pair of axial points called the
nodal points. They are extremely useful in calculating image sizes. An
incident ray directed toward the first nodal point will appear to emerge
from the second nodal point with unchanged direction. Therefore, the
nodal points are called points of unit angular magnification. When both the object and image lie in a medium of the same index of refraction, the first and second nodal points coincide with the first and second principal points (Fig. 12c), a lens system surrounded by air. If the lens is a simple thin lens in a uniform medium, the principal and nodal points all coincide at the vertex of the lens (see Fig. 12a). If the image is not in the same medium as the object, as in the single refracting surface of Figure 12b, however, the nodal points do not coincide with the principal points. The two principal points coincide at the vertex of the surface, and the two nodal points coincide at the center of curvature of the surface. For all these cases, the slope of the ray directed toward the first nodal point is the same as the slope of the ray that appears to emerge from the second nodal point. The eye is a complex series of refracting surfaces that forms an image in vitreous of an object in air. Therefore, it has a pair of principal planes and a separate pair of nodal points that can be used to represent it. Their positions are shown in Gullstrand's schematic eye (Fig. 13). 
THICK LENS EQUATION 
The equivalent refracting power of a thick lens or two separated optical
elements is given by Gullstrand's equation: where: F_{1} is the refracting power of the first element, F_{2} is the power of the second element, and c = d/n is the reduced distance separating the two elements. This equation will prove useful when the combined power of the eye and a spectacle lens is considered. The dioptric power of the spectacle lens will be F_{1} and the dioptric power of the eye will be F_{2}. The distance from the eye at which the spectacle lens is fitted is d; n = 1 because the medium between the lens and the eye is air. Applications of this equation are shown in Figures 67 and 69. 
THE GULLSTRAND SCHEMATIC EYE  
Schematic eyes are models of the optical system of the eye. They are extremely
useful but limited representations of the dynamic living eye. The
schematic eyes developed by Listing, Tscherning, and Helmholtz^{4} greatly advanced the understanding of the optics of the eye. However, it
was Gullstrand who developed the most authoritative model of the eye. His
model was based on the work of many researchers and some very original
experiments and instrumentation of his own. The essential parameters
that Gullstrand needed to find how light travels through the eye
were the curvatures of the surfaces of the cornea and lens, their positions, and
the indices of refraction of the ocular media. The techniques
for making these determinations, described by Helmholtz and Gullstrand,^{4} are experimentally intricate and complicated by considerable trigonometric
calculations. Although the anterior surface of the cornea seems spherical, it is not. Centered about the vertex is an optical zone 2 to 3 mm in diameter through which the rays enter the eye. The optical zone suffers from physiologic astigmatism, that is, it is more steeply curved in the vertical meridian than horizontally. Beyond this zone, the anterior surface of the cornea flattens asymmetrically. It may be less flat in some meridians and asymmetric on opposite sides of the vertex. These topographic features of the anterior surface of the cornea were determined by means of Gullstrand's photokeratoscope, a device that took photographs of the corneally reflected image of an illuminated pattern of circles. Measurements of the photographs were used to calculate the corneal contour. The ophthalmometer also was used for this purpose. Both methods use reflection from the front surface of the cornea. This reflection forms the first of the four Purkinje images. With care, the reflex from the posterior surface of the cornea as well as faint reflections from the anterior and posterior surfaces of the lens may be seen. The anterior surface of the lens is convex and forms a virtual reflex image, but the posterior lens surface is concave and forms a real reflex image of a distant object. Each of these surfaces acts as a simple spherical mirror. Therefore, their radii may be calculated easily, provided that the optical characteristics of the optical elements preceding each surface of interest are known. Measurements of 14 ocular parameters comprising curvatures of surfaces, thicknesses of elements, and indices of the media were required for Gullstrand to define the optical system of a standard eye (see Fig. 13). REFRACTIVE INDICES The indices of refraction of the cornea, aqueous, lens, and vitreous had to be determined. Gullstrand found that the indices of refraction of the aqueous and vitreous humors were both equal to 1.336, which is practically identical to water. The index of refraction of the cornea was higher and is given as 1.376. The crystalline lens structure often is compared with the layers of an onion. This laminar structure has an increasing lens density and index of refraction from the outermost layers to the center. Calculations of ray paths through a lens of continuously varying indices or gradient index are very complicated. Therefore, Gullstrand calculated an equivalent lens made up of a central core with a refractive index of 1.406 surrounded by a cortex of index 1.386. These twoindex lenses closely approximated the size, shape, and power of the average real crystalline lens. Gullstrand's data are presented in Table 2.
TABLE 2. The Gullstrand Schematic Eye
RADII OF CURVATURE OF THE REFRACTIVE SURFACES The radii of the various ocular surfaces were measured with an ophthalmometer. This is an instrument with an illuminated object pattern of known size and position that is reflected by the ocular surface. The reflected pattern is viewed by means of the special optical system of the ophthalmometer that may be adjusted to find the size of the image reflected by the surface of interest. Because the ratio of the image size to the object size is equal to the ratio of the image distance to the object distance, the focal length of the surface, treated as a spherical mirror, can be calculated. The focal length of a spherical mirror is equal to one half of its radius of curvature, so the ophthalmometer can be calibrated directly to read out the radius of curvature of the surface. Clinical ophthalmometers usually are calibrated to provide the refracting power of the front surface of the cornea. Given 1.376, the index of refraction of the cornea, this merely requires solving the equation: Based on Gullstrand's data, the power of the anterior surface of the cornea is Similarly, the power of the posterior surface of the cornea is the result of the difference in the indices of the cornea and aqueous. This difference is much less than at the aircornea interface (the anterior surface) so that the power at the posterior surface of the cornea is weak. In fact, it is a negative or diverging power because the light travels from a higher (cornea 1.376) to a lower (aqueous = 1.336) index medium. Therefore, the numerator is negative and the radius in the denominator is positive: POSITIONS OF THE OCULAR SURFACES Gauges, calipers, and other mechanical devices provided the early data on the thicknesses and positions of the optical elements in the eyes of cadavers. With the invention of the slit lamp, finding these positions optically without dissecting and deforming the eye was possible. Initially, the slit lamp is focused on the anterior surface of the cornea to establish the zero position. The slit lamp then is racked forward to bring the second surface into focus, and the distance that the slit lamp is translated to accomplish this is recorded. This position represents the apparent position of the posterior corneal surface because it is being observed through the anterior surface. The anterior surface is a refracting surface that optically affects the space behind it. Therefore, the actual position of the posterior corneal surface required optical calculation. Similarly, for example, the depth of the anterior chamber would be found by focusing on the anterior surface of the lens or on the edge of the iris. This is also an apparent position. The real position depends on the power of the cornea. GULLSTRAND'S RESULTS Several significant factors should be noted about the Gullstrand schematic eye with relaxed or zero accommodation (Table 3). The first and second principal points at about 1.348 and 1.602 mm behind the vertex of the anterior corneal surface are separated by only about 0.25 mm and shift only approximately 0.4 mm at maximum accommodation. Conversely, the focal points shift more than 6 mm toward each other. Thus, the principal points can be considered fixed points that provide a reference for calculations.^{2} In the reduced eye, to be discussed later, the two principal points are combined into one point that becomes the position of the vertex of a hypothetical single refracting surface. This single refracting surface optically replaces the several corneal and lens surfaces of Gullstrand's eye.
TABLE 3. Complete Optical System of Gullstrand Eye
The power of the cornea is 43.05 D. In the unaccommodated state, the crystalline lens has a power of 19.11 D. Interestingly, although the cortex and core indices are 1.386 and 1.406, respectively, an index of 1.42 would be required if a homogeneous lens were to have the same form and power. The power of the unaccommodated eye is 58.64 D. The data in Table 3 show that the standard eye is axially hyperopic by 1 D; therefore, it has a length of 24 mm from the anterior corneal surface to the fovea. An emmetropic eye would have an axial length of 24.4 mm. Axial myopia and hyperopia are produced by longer or shorter model eyes. Similarly, refractive myopia and hyperopia results if the power of the model eye varies from 58.64 D. These categories of axial versus refractive ametropia really are artificial, but they are useful for illustrating simple optical effects. An optically homogeneous lens with spherical refracting surfaces would produce a significant amount of spherical aberration. As noted, spherical aberration is characterized by rays being brought to a progressively shorter focus as they strike a lens at greater heights from the optical axis. Fortunately, two countereffects exist in the eye. The cornea is not spherical but tends to flatten out at its margins, and thus, marginal rays are not refracted as much as they would have been had the spherical contour of the cornea been maintained. Similarly, the lower index in the outer zones of the lens causes less refraction of the marginal rays. These two effects compensate for spherical aberration and may, in fact, overcorrect it. Constriction of the pupil completes the mechanisms that reduce spherical aberration, at least in bright surroundings. This plays a significant role in increasing visual acuity. During accommodation, the curvatures of the lens become steeper, the axial thickness increases, and the pupil constricts. These changes enable the eye to focus sharply near objects on the retina. The uneven capsule (Fig. 14) allows the front surface of the lens to bulge in the center while keeping the periphery less curved, which helps control spherical aberration as power is increased.
Chromatic aberration is corrected in photographic lenses by appropriately selecting glasses with indices and dispersions that are related so that a combination of positive and negative lenses will have a net refractive power but their opposing dispersions will cancel. The eye has no such combination of optical media. Because the indexes of refraction of the ocular media increase as the wavelength is shortened, short wavelength light is refracted more strongly than long wavelength light, resulting in a corresponding increase in the dioptric power of the eye. Furthermore, the rate of increase in dioptric power increases rapidly as the wavelength is shortened, as shown by the steeper slope of the chromatic aberration curve at short wavelengths compared with the slope at long wavelengths (Fig. 15).
For example, if light of wavelength 550 nm is in focus on the retina, the image in ultraviolet light of wavelength 350 nm will be out of focus by 2.75 D. However, an equal shift to the red (750 nm) produces only 0.75 D of outoffocus blur. The eye effectively becomes more myopic when the wavelength of illumination becomes shorter. The image degrading effect of chromatic aberration of the eye is mitigated by three factors, according to Wald.^{8} First, the crystalline lens acts as a filter that transmits the visible spectrum but absorbs the near ultraviolet light of wavelengths shorter than 400 nm. This region of the spectrum, where chromatic aberration is increasing most rapidly, simply does not reach the rods and cones. Second, the sensitivity of the eye shifts toward the red end of the spectrum as the illumination is increased, and the eye switches from low acuity rod vision to high acuity cone vision. The rods have a peak sensitivity at 500 nm that corresponds to bluegreen, and the foveal cones have a peak sensitivity at 562 nm or in yellowgreen (see Fig. 39). The significance of this shift in sensitivity is that the foveal cones respond more strongly to longer wavelengths where chromatic aberration gradually increases, whereas rods respond more strongly to the region of rapidly increasing chromatic aberration. Thus, the smaller chromatic blur experienced by the cones supports their highacuity function.
Finally, the full chromatic blur of the eye is alleviated by the spectral transmission of the pigmentation of the macula lutea. Wald compared the spectral sensitivities of cones in the macula with cones in a colorless peripheral area of the retina. He found that the maculae pigment “takes up the absorption of light in the violet and blue regions of the spectrum just where absorption by the lens falls to very low values. Thus, the yellow patch removes for the central retina the remaining regions of the spectrum for which the color error is high.”^{8} Despite the “spectrumreducing” factors described by Wald, chromatic aberration has a remarkable influence on reflexive accommodation of the eye. Figure 16 shows the blur patterns for a myopic, emmetropic, and hyperopic eye. Both the red and blue blurs superimpose in the emmetropic eye, producing no noticeable color aberration. In absolute hyperopia, the blue rays focus on the retina and are surrounded by a red blur or halo. In myopia, the red rays are in focus on the retina and some blue halo surrounds. For example, a myope will see red neon signs most sharply, and a hyperope (with no accommodation) will see blue neon signs most sharply. The myope and hyperope will see colored haloes about points emitting a mixture of red and blue light.

MODERN SCHEMATIC EYE  
Modern technological developments in the biometry of the human eye have
resulted in an extensive literature describing its anatomy and optical
properties. Among these are the gradient index of the lens, the aspheric
curvatures of the surfaces of the cornea and the lens, and the dispersive
characteristics of the ocular media. Abetted by powerful optical
ray trace programs, it is easy to evaluate the optical characteristics, especially
the aberrations of model eyes with these properties. One
such model^{9} that predicts spherical and chromatic aberration is shown in Figure 17. Its structural parameters are given in Table 4.
All surfaces are centered on the optical axis. The aperture stop coincides with the front surface of the lens and is decentered by 0.5 mm nasally. Lens surface 2 is an imaginary plane dividing the lens into an anterior and posterior gradient index (Grads A and B). Asphericity Q values between 0 and 1 are ellipsoids about the major axis. The asphericity of the posterior surface of the lens (Q = + 0.96), an ellipsoid about the minor axis, is not a biometric datum. 
REDUCED EYE  
Because the Gullstrand and modern schematic eyes contains six refracting
surfaces, calculations are simplified by treating this eye as a black
box and using the cardinal points for determining objectimage relationships. An
even greater simplification, the reduced eye, was computed
by Listing. He reduced the eye model to a single refracting surface, the
vertex of which coincides with the principal plane and the nodal
point of which lies at the center of curvature (Fig. 18). The justification for this model is that the two principal points that
lie midway in the anterior chamber are separated only by a fraction
of a millimeter and hardly shift during accommodation. Similarly, the
two nodal points lie equally close together and remain fixed near the
posterior surface of the lens. Listing combined the two principal points
and the two nodal points each into a single principal and nodal point. The
single principal point is 1.5 mm behind the cornea. It represents
the vertex of a single refracting surface with a radius of curvature
of 5.7 mm. The nodal point of a single refracting surface is at the
center of curvature so that the nodal point is 1.5 + 5.7 = 7.2 mm
behind the cornea of Gullstrand's eye. Because the emmetropic
schematic eye is 24.4 mm from cornea to fovea and the fictitious refracting
surface of the reduced eye is 1.5 mm behind the cornea, Listing's
reduced eye is 24.4  1.5 = 22.9 mm long. It has a uniform index
of refraction of 1.336.
Retinal image sizes may be determined very easily because the nodal point is at the center of curvature of this single refracting surface. A ray from the tip of an object directed toward the nodal point will go straight to the retina without bending, therefore, object and image subtend the same angle. The retinal image size is found by multiplying the distance from the nodal point to the retina (17.2 mm) by the angle, in radians, subtended by the object. For example, in Figure 19, a ray from an angular direction is shown. The ray goes through the nodal point and is not bent; consequently, the angle subtended at the nodal point by the retinal image is also equal to . The distance F'P from the second focal point to the fictional vertex (which also is the principal point) is the second focal length. It is 22.9 mm and occurs in a uniform index of refraction of the media of 1.336. The first focal length FP is equal to 15.7 + 1.5 = 17.2 mm. The refractive power of the reduced eye is equal to the index divided by focal length. Using the first focal length, the power of the eye is found by dividing it into the first index, which is air:
Using the second focal length, the power of the eye is found by dividing it into the second index: The first focal length in air is optically equivalent to the second focal length in aqueous, and the fictional reduced eye is optically equivalent to Gullstrand's schematic eye, if liberties in rounding off numbers are taken. 
RETINAL IMAGE SIZE  
The size of the retinal image is found easily by using the reduced eye
model. Several rays may be used, but the simplest ray to use is the undeviated
one through the nodal point to the retina. For example, in Figure 19, this ray comes from the tip of a distant tree, which, for example, subtends
an angle = 0.1 radians at the nodal point N. The retinal image
subtends the same angle at the nodal point. Thus, Y' = 17.2 × 0.01 = 1.72 mm. Image point Q' can be constructed with rays other than the one through the nodal point. In Figure 20, two rays from an infinitely distant object that subtends an angle at the eye are incident on the eye. The rays are parallel to each other (collimated). The upper ray is headed toward the nodal point and is identical to the ray previously discussed. It strikes the retina at a height Y' = 17.2 × . The lower ray crosses the axis at the first focal point F. When this ray strikes the eye, it appears to originate at the first focal point and travels parallel to the axis after refraction. Because it travels parallel to the axis after entering the eye, its height remains constant from the point of incidence T, at the refracting surface, to the retina. Triangle FPT is the same as triangle NF'Q' because both triangles have identical angles and sides that are 17.2 mm long. Consequently, height PT must equal height F'Q', and thus, the lower ray will strike the retina at point Q' just as did the ray through the nodal point. The triangles can be visualized better by assuming a flat cornea and retina, which are the conditions for paraxial calculations. That is, the rays are assumed to strike these surfaces so close to the axis that the surfaces practically are flat over this short distance from the axis.
IMAGE SIZE IN AMETROPIA Emmetropia prevails when the refractive power and axial length of the eye are matched properly. Clearly, any number of combinations of power and length will produce emmetropia. The Gullstrand schematic eye with a power of 58.64 D and an axial length of 24.4 mm represents a typical emmetropic eye. Consequently, considering axial ametropia—the condition in which the power of the eye is the normal 58.64 D—is convenient for illustrative purposes, but the length of the eye is not 24.4 mm. Figure 21 illustrates the variation of axial length with ametropia. The easiest way to show the magnitude of the retinal image size changes due to axial ametropia is to use the reduced eye model that has a length of 22.9 mm and a power of about 58 D. It is evident in Figure 21 that for any given angular subtense of the object at the nodal point of the eye, the retinal image will be smallest in hyperopia and largest in myopia. The image size will be in direct proportion to the length of the eye.
IMAGE SIZE OF OBJECT AT FARPOINT IN AXIAL MYOPIA To form a sharp image on the retina of an axial myope, a distant object must be brought toward the eye. As the object nears the eye, the rays strike the eye with increasing divergence. This causes the image, which initially falls in the vitreous at F', to shift back toward the retina (Fig. 22). When the sharp image coincides with the retina of the elongated eye, the object is at the farpoint of the eye. The unaccommodated myopic eye can see clearly up to this distance but no farther. A 5D myope has a farpoint that is 1/5 of a meter in front of the anterior principal point of the eye. The object distance l = 200 mm, and the vergence of the light incident on the eye from this farpoint will be L = 5 D. The distance to the farpoint is reckoned from the principal plane, or refracting surface of the reduced eye, not the nodal point. Consequently, the object subtends a different angle at the principal point than it does at the nodal point. This distinguishes a near object (at the farpoint) from an infinitely distant object for which the rays to these points are parallel. The distinction becomes significant only when the object is very near the eye. An object at the farpoint that subtends, for example, 0.1 radians at the refracting surface of the eye located 200 mm away will have a height of 20 mm because 0.1 × 200 = 20. It will subtend a smaller angle at the nodal point of the eye because the nodal point lies an additional 5.7 mm beyond the refracting surface. This angle will be 20/205.7 = 0.097 radians. The size of the retinal image of this 20mm tall object is given by the product of the angle (0.097) multiplied by the distance of the myopic retina from the nodal point. This last distance is obtained readily when the length of the reduced eye is determined with the equation, L' = L + F, where L = 5 D and F = + 58 D, from which L' = + 53 D. Converting + 53 D to the image distance l' provides the axial length of the 5D myopic eye, viz., We require the distance from the nodal point to the retina or 25.2  5.7 = 19.5 mm. The size of the retinal image is this distance multiplied by the angle subtended by the object at the nodal point, Y' = 0.097 × 19.5 = 1.89 mm. Had the additional distance to the nodal point been neglected, the size of the retinal image would have been Y' = 0.1 × 19.5 = 1.95 mm. This would introduce an appreciable error when compared with the 1.72mm image found in emmetropia. Image Size of Object at FarPoint in Axial Hypermetropia The small globe of the axially hyperopic eye causes the retina to intercept the convergent rays before they come to a focus, as shown in Figure 23. In order for rays from an object to come to a focus on the retina, they must be convergent when they strike the refracting surface. That is, they must appear to originate at a virtual object located behind the retina. When the vergence of the incident light plus the power of the eye cause the image to fall on the retina, the corresponding virtual object is at the farpoint of the eye, behind the retina. A 5D hyperope has a farpoint that is 200 mm to the right of the refracting surface of the reduced eye (Fig. 24). The light has a vergence L of + 5 D at the eye. The image vergence is obtained as before, that is, L' = + 5 + 58 = 63 D. The rays come to a focus at a distance
The distance from the nodal point to the retina is 21.2  5.7 = 15.5 mm. Assuming once again that the virtual object is 20 mm tall, it will subtend 0.1 radians at the principal point. The nodal point is 5.7 mm closer to the virtual object, or 200  5.7 = 194.3 mm; therefore, the angle subtended by the object is 20/194.3 = 0.103 radians. This angle multiplied by the distance from the nodal point to the retina provides the retinal image height, Y' = 0.103 × 15.5 = 1.6 mm. Had no correction been made for the position of the nodal point, the image height would have been Y' = 0.1 × 15.5 = 1.55 mm. The change in image size of a farpoint object in 5 D of axial ametropia compared with emmetropic image size is 
PUPILLARY APERTURE, DIFFRACTION, AND RESOLVING POWER  
Because light consists of rays traveling in straight lines, it works well
for finding object and image positions and sizes. However, are there
rays of light? A direct experiment to elucidate this might be to allow
a bundle of rays to pass through a circular aperture. As the aperture
is made smaller, fewer rays should be transmitted. If these rays are
observed on a screen, the spot diameter also should become smaller. Although
at first, the spot on the screen does constrict, at a certain
point as the aperture continues to shrink, however, the spot on the screen
begins to enlarge (Fig. 25). This phenomenon, known as diffraction, sets a limit to the minimum size
of an image. Diffraction is a bending of light caused by the edge
of an aperture or the rim of a lens. Even a perfect lens, free from aberrations, will
not focus light to a point because of diffraction. Instead
of a point image, a lens with a circular aperture or pupil produces
a blur consisting of a series of concentric bright and dark rings (Fig. 26). At the center of this diffraction pattern is a bright spot known as
the Airy disc because Sir George Airy, (18011892), Astronomer Royal, was
the first to calculate the energy distributions in the pattern. About 84% of
the total energy of the diffraction pattern is in the Airy
disc. The remaining energy is distributed in the surrounding rings that
become progressively fainter. For example, regardless of how well a
telescope may be corrected, this will be the image pattern for a star. The
angular subtense of this pattern depends only on the wavelength of
light and the diameter of the aperture of the telescope or other optical
device. In particular, the radius of the Airy disc subtends an angle
with respect to the lens ofwhere: λ equals the wavelength of light and d is the diameter of
the lens aperture in the same units as wavelength. This equation shows
that the diffraction blur increases directly with the wavelength and
inversely with the aperture diameter. A red light source will be imaged
with nearly twice the diameter of a blue light source. Doubling the
aperture diameter will halve the diffraction blur.
The linear radius R of the Airy disc is obtained by multiplying the angle subtended by the radius of the Airy disc by the focal length f of the imaging lens: R = f · . Figure 27 illustrates the intensity distribution of the diffracted image of a star in the focal plane of a telescope or any wellcorrected optical system. Diffraction limits the resolution of optical systems ranging from the eye to the telescope at Mt. Palomar. Two large diffraction images will run together and appear single at larger angles than two fine diffraction patterns. Therefore, the smaller the aperture, the coarser the resolution is because of diffraction. For example, if a telescope is pointed at two equally bright stars, two diffraction patterns will be produced. The smaller the angle between the stars, the closer the patterns will be imaged. As shown in Figure 28a, the two star images are not resolved when the peaks of their patterns are separated by an angle less than the angle of the radius of the Airy disc. The intensities of the overlapping discs add to form a single peak (the dotted line). Rayleigh showed that stars are resolvable when the two stars were separated by an angle equal to the radius of the Airy disc, that is, the peak _{1}, of pattern 1 falls on the first dark ring of pattern 2, and the peak _{2} of pattern 2 falls on the first dark ring of pattern 1. As illustrated in Figure 28b, the addition of the intensities of the two overlapping patterns results in a dip in the combined patterns (dotted line), which is a sufficient decrease in brightness to be perceived, and makes the patterns resolvable. Other investigators have shown that an even smaller dip is sufficient for resolution and so frequently an angular separation of = 1 λ/d is used instead of 1.22 λ/d. The wavelength is specified in nanometers (nm) which is equal to millimicrons, now an obsolete unit. For example, the angular resolving power (RP) of the eye, given an eye pupil diameter of d = 2 mm and λ = 555 nm = 0.000555 mm, is Because 1 minute of arc equals 0.00029 radians, RP = 0.00034/0.00029 = 1.17 minutes of arc. If we use 1 λ/d, RP = 0.000278 radians and the resolution becomes 0.96 minutes of arc. A resolution of 1 minute corresponds to the separation of the limbs of a 20/10 Snellen E. This shows how remarkably close the resolving power in white light of the eye is to the theoretical limit. The equation shows that the resolving power is inversely related to pupil diameters. In other words, because d is in the denominator, if d becomes smaller as the pupil constricts, then the resolvable angle gets larger, or visual acuity becomes poorer. If the resolving power of the eye was solely dependent on diffraction effects, then a large pupil would improve acuity proportionately. A 6mm pupil should result in a resolving power of 0.782 = 0.39 minutes or 24 seconds. This does not happen because the eye is not a perfect optical instrument. Its aberrations become more pronounced as the pupil dilates. Further complicating the situation is the effect of illumination level on acuity and pupil diameters. As a practical result of the interaction of diffraction, aberrations, and retinal illumination, optimum acuity occurs over an intermediate range of pupil diameters, viz., 2 to 3 mm. A person with uncorrected myopia will obtain improved acuity though an artificial pupil as its diameter is reduced. This is because the blur spot on the retina shrinks as the pupil constricts and in effect, depth of focus increases. However, a reduction in pupil size is reached such that the blur caused by diffraction begins to exceed the geometric blur because of the pinhole pupil. Further reduction of the pupil increases the angular subtense of the Airy disc in accordance with equation 2, and the resolving power of the eye will decrease. 
MODULATION TRANSFER FUNCTIONDIFFRACTION LIMITED 
The imageforming quality of optical systems classically has been expressed
by their resolving power or the limiting resolution below which the
image of a target can no longer be resolved. This is the point at which
the contrast between light and dark regions in the image is so low
that the image appears to be a smear. At best, all images are slightly smeared because of diffraction. Points are imaged as diffraction patterns; fine lines are spread into long blurs (Fig. 29a and b). A plot of the luminance (L) across such smeared images is called the spread function. The spread function causes the sharp edge of an object to have a rounded image luminance profile (see Fig. 29c). This rounded luminance profile on both sides of a coarse square wave or bar target will be a blurred region between a flat maximum (L = 1) and minimum (L = 0) luminance distribution. As progressively finer bar targets are imaged by the optical system, their luminance profiles are progressively imaged closer together and run into each other. In other words, increasingly more light invades what should be the dark areas. As a result, the dark areas gain illumination and the light areas lose illumination. The ideal square wave luminance distribution of the image becomes degraded into a sinusoidal distribution (see Fig. 29e). As the bar targets become more closely spaced (higher spatial frequency), the difference between the maximum and minimum luminance lessens, or the contrast, expressed as a modulation, is reduced. The ideal modulation transfer function (MTF) targets are not bars with square wave luminance distributions but gratings that vary sinusoidally in luminance. Because the luminance distributions in their images always are sinusoidal, their mathematical treatment is facilitated. Modulation of the image usually is described by Michelson contrast, in which the contrasts of grating targets and images are specified by luminance at the peaks and troughs of the grating. The MTF is the ratio of image contrast to object contrast, as a function of spatial frequency. If a curve of modulation versus spatial frequency (lines/mm in the image) is plotted, the curve will start with a modulation of 1 or 100% for very coarse (low spatial frequency) highcontrast targets and fall to zero at some high spatial frequency dependent, for example, on the aberrations characteristics of the optical system. When the modulation decreases to a few percent, the image no longer is resolvable by the eye. This is the limiting resolution and corresponds to the classic criterion of resolving power (see Fig 29f). However, the MTF curve also describes the performance of the system for a whole range of target spatial frequencies, that is, for fine and coarse patterns. The value of this frequency information is shown graphically in Figure 29f which shows two lenses with the same resolving power or limiting resolution. However, lens A provides a higher contrast for lower spatial frequencies and, therefore, is superior to lens B. To properly use MTF, the optical system must be linear.^{10} If the luminance of the object is doubled, for example, the luminance of the image must double. The system also must be spatially homogeneous. A film emulsion that did not have a uniformly distributed grain—but instead had regions of fine grain and regions of coarse grain—would lack spatial homogeneity. Consequently, the MTF of the photograph would vary with the grain density on which the image fell. The retinal cones are distinctly inhomogeneous in distribution and sensitivity. Finally, the characteristics of the optical system must be the same, whatever the orientation of the target. This will be true, onaxis, for a system of rotationally symmetric optical elements even with spherical and chromatic aberration. When a target with vertical and horizontal lines is offaxis, coma and astigmatism will result in different MTF curves as a function of the orientation of the lines. The human visual system is not linear except at threshold luminance levels, and the neuroanatomy of the visual system is nonhomogeneous, except perhaps near the fovea. Although measuring the MTF of the imageforming elements of the eye is possible, perception would depend on retinal and neurological conditions. Consequently, the contrast sensitivity function (CSF), which is discussed later, is used to evaluate the visual system for perceptual responses. 
MODULATION TRANSFER FUNCTION OF THE EYE AND RETINAL IMAGE BLUR  
The quality of the image on the retina is altered by four types of blurs: scattering
by small particles in the ocular media, diffraction at the
pupil, inaccurate focus due to accommodation and ametropia, and aberrations
of the eye. Blurring of the retinal image reduces spatial contrast
of the image, and the reduction of contrast increases with spatial
frequency. For the human eye, spatial contrast is transferred from
object to image without significant loss below about 0.5 cycles per degree
or 20/1200, but at higher spatial frequencies, contrast of the image
is reduced by blur. Blur of the retinal image from scattered light results largely from small particles in the ocular media, particularly in the lens and vitreous, that scatter light toward the retina. In the young eye, scattered light is a minor source of retinal blur, but in the aging eye, the lens and vitreous can become major sources of scattered light because of cataracts, vitreous degeneration, and other agerelated disorders. Scattered light reduces contrast of the image by adding a veiling luminance or “wash” over the retinal image. Because the veiling light is added both to the troughs and peaks of grating images, contrast of the image is reduced at all spatial frequencies, including low spatial frequencies (< 0.5 cycles per degree [cpd]). The loss of contrast from scattered light can be debilitating for elderly patients. Diffraction is an interference phenomenon that involves “bending” of rays that pass close to edges like the edges of the pupil. When the pupil is smaller than about 2 mm in diameter, the MTF of the eye is limited by blur from diffraction. As pupil size is reduced, contrast is reduced or eliminated at high spatial frequencies. For pupils larger than about 2 mm in diameter, inaccurate focus and aberrations are the major sources of retinal image blur. We previously found the angular resolving power of the eye with a 2mm pupil to be 0.000278 radians or 0.016 degrees. The reciprocal of this angle is the resolution in cycles per degree (cpd). In this instance, resolution = 1/0.016 degrees 63 cpd. Dividing 63 cpd into 600 converts it into 20/10 Snellen acuity. This is the cutoff or end point of the diffraction limited MTF curve for monochromatic light of 555 nm. The diffraction limited cutoff increases with an increase in the pupil diameter of the eye. If the pupil diameter is doubled to 4 mm, the cutoff spatial frequency doubles to about 126 cpd. However, the blur caused by spherical aberration of the eye increases with the cube of the increased diameter, or becomes nine times greater. Consequently, spherical aberration not merely reduces the cutoff spatial frequency but reduces the contrast at all spatial frequencies. Figure 30 shows the greater contrast in the diffraction limited MTF for perfect eyes with a 4mm pupil compared with a 2mm pupil. Spherical aberration, however, severely reduces the contrast in the MTF of the Gullstrand eye with 4mm pupils, making it worse than the contrast obtained with a 2mm pupil. Note also that the MTF with the 2mm pupil is nearly diffraction limited. Inaccurate focus of the eye (defocus) from uncorrected ametropia and from inaccurate accommodation produces large amounts of defocus blur and loss of contrast at moderate and high spatial frequencies. In the aberrationfree eye, the effect of defocus on blur of the retinal image is similar for myopic and hyperopic focus, and for over and underaccommodation. Figure 31 shows the effect of defocus on MTF for various amounts of defocusblur for an eye with a 3mm pupil. Again, at low spatial frequencies, inaccurate focus does not affect contrast of the image, but by 3 cpd (20/200), the loss of contrast is significant, even for 0.25 D of defocus, and by 10 cpd (20/60), defocus in the amount of 0.75 D reduces contrast close to zero. The effects of defocus blur on contrast are more severe for pupils larger than 3 mm, and uncorrected ametropia and inaccurate accommodation are a primary source of retinal image blur.
In addition to spherical aberration, coma, and oblique astigmatism, the monochromatic aberrations of the eye include field curvature and distortion. Paraxial images lie on a plane image surface. Oblique astigmatism results in tangential and sagittal curved image surfaces. They may be visualized as a teacup and saucer, as shown in Figure 32. When astigmatism is corrected, the tangential and sagittal surfaces coincide to form an image on a curved surface that approaches the paraxial plane. The image suffers from field curvature. The monochromatic aberrations can be divided roughly into those that influence foveal or central vision and aberrations that influence the image of the peripheral visual field. Spherical aberration is the principal onaxis monochromatic aberration of the eye. Because the fovea is 5 degrees offaxis, coma contributes to the blur from spherical aberration at the fovea, especially when the pupil is larger than 2 mm. Unlike spherical aberration, coma produces an asymmetrical blur (Fig. 33). These aberrations determine the monochromatic MTF for the infocus eye as the pupil dilates. Because the blur caused by spherical aberration increases as the cube of the pupil diameter, patients with dilated pupils experience blurred vision.
Spherical aberration usually is undercorrected (peripheral rays focus ahead of central rays) when the eye is focused for infinity, but for some patients, the amount of positive spherical aberration is reduced when the eye accommodates for near (1.5 D), and the aberration can become overcorrected (central rays focus ahead of peripheral rays) with considerable accommodation for near (3 D). In addition to these variations among individuals with regard to spherical aberration and accommodation, wavefront aberration usually is asymmetric across the area of the pupil,^{11} producing an asymmetric comalike blur on the retina that varies extensively from one patient to another. These individual differences make it impractical to fabricate ophthalmic lenses or contact lenses to correct the aberrations of the eye. The effects of offaxis aberrations of the eye are present at the fovea because the visual axis does not coincide with the optical axis. In addition, the pupil often is decentered by a small amount with respect to the visual axis. Some degree of coma is common at the fovea, but the amount of blur from coma is considerably less than blur from spherical or chromatic aberration.^{12} The remaining monochromatic aberrations, oblique astigmatism, field curvature, and distortion, have minimal effect on foveal visual acuity and MTF at the fovea. For peripheral vision, the effects of blur from monochromatic aberration reduce image contrast at high spatial frequencies, but the effects of aberrations on peripheral vision are largely offset by reduced spatial sampling by receptors in the peripheral retina. Chromatic dispersion by the ocular media produces two chromatic aberrations. Longitudinal (axial) chromatic aberration measures more than 2 D across the visible spectrum. In addition to longitudinal chromatic aberration, decentered pupils and the angle between visual and optic axes produce lateral (transverse) chromatic aberration that averages about 30 arc seconds at the fovea. The blur from longitudinal chromatic aberration is significant even at relatively low spatial frequencies (e.g., 1–3 cpd), but blur from lateral chromatic aberration is significant only at higher spatial frequencies (>10 cpd). Figure 34 shows the effects of both defocus and chromatic aberration on MTF of the eye. The figures represent under and overaccommodation in the amount of a halfdiopter with the focus referenced to 550 nm light. Again, at low spatial frequencies (< 1 cpd), contrast of the retinal image is the same for all spectral components of the image, and defocus blur has no effect. At moderate and high spatial frequencies, however, contrast is different for each spectral component of the image. At intermediate spatial frequencies (3–5 cpd), the difference in contrast is about 20% between spectral components that focus a halfdiopter apart (e.g., between 590 and 550 nm. Underaccommodation is specified by higher contrast for the shortwavelength (blue) component of the image than for the longwavelength (red) component, and overaccommodation by higher contrast for the longwavelength components than for the shortwavelength components. Comparison of the contrast of spectral components of the retinal image specifies focus behind or in front of the retina.^{13} At high spatial frequencies, small amounts of defocus have a large effect on contrast, and the effects of chromatic aberration are particularly pronounced. In summary, at spatial frequencies above approximately 0.5 cpd (20/1200), the contrast of the retinal image is different for each spectral component of the image. These chromatic effects control reflex accommodation and are potential stimuli for the process of emmetropization. 
DEPTH OF FOCUS AND DEPTH OF FIELD  
A good 35mm camera has adjustable shutter speed and lensopening settings. When
one wants to stop motion, the shortest shutter speed and largest
lens opening are used, that is, a picture taken at 1/500 of a second
and a large aperture f/1.9. If the subject is 8 feet away, one would
find that everything between about 7.5 and 8.5 feet away would be in
focus on the film, or the depth of field is about 1 foot. At the other
extreme, if one were to photograph a stationary object 8 feet away, one
could get the proper exposure with a long shutter speed setting of 1 second
and a small aperture f/22. The depth of field would range from 5 feet
to about 25 feet. Everything throughout a depth of field of 20 feet
would be in focus on the film when the lens aperture is stopped
down from f/1.9 to f/22. For a 50mm focal length lens, this corresponds
to apertures from 25 mm to a bit more than 2 mm in diameter. That
is, the larger the pupil, the shorter the depth of field becomes. The
rangein axial position of the image point corresponding to the depth
of field is called the depth of focus. The ranges just cited are based
on a 3minute blur tolerance used by the photographic industry. If an eye with fixed accommodation is focused on a point source at M (Fig. 35) and the point source is brought closer to the eye, it will appear just noticeably blurred when it reaches position P. Similarly, when moved farther away, it appears just noticeably blurred when it reaches position R. The distance R P is the total depth of field. It results in a depth of focus R' P' between the conjugate image points that fall in front and behind the retina. The diameter of the blur spot on the retina depends on the size of the pupil. Therefore, depth of field and depth of focus depend on pupil diameters. Comparing the diagrams, the larger pupil produces a just noticeable blur when R and P are moved through shorter distances; thus, the corresponding depths of field and focus are reduced. For an average pupil diameter of 2 to 3 mm, the depth of focus generally is taken to be ±0.25 D because such a change in vergence is tolerated before a blur is perceived.^{14} Two examples will be useful in understanding the depths of field and focus. DEPTHOFFIELD CALCULATION Assume that the eye in Figure 36 is focused on an object point M found 4 meters away. The vergence of the light from this point will be 0.25 D when it strikes the eye and will be brought to a sharp focus on the retina. While accommodating for 4 meters, the subject will not notice a blur at distance farther or nearer than 4 meters until the vergence reaching the eye changes by ±0.25 D. Thus, if the 0.25 D vergence from M at 4 meters is increased by + 0.25 D, one range of the depth of field is found, specifically the far range R
Zero vergence corresponds to an infinitely distant far depth of field. The subject will see clearly from 4 meters to infinity. Similarly, to find the near depth of field P, the 0.25 D tolerance is added to the vergence of the light reaching the eye from point M: A vergence of 0.5 D takes the near depth of field from 4 to 2 meters. The total depth of field is from infinity to 2 meters. The results of this numerical example depended on the selection of a 4meter fixation distance and a ±0.25 D tolerance. Had a tighter tolerance or another fixation distance been chosen, the results would have been different. The depth of focus in the eye simply is the range between the images of the far and near depths of field, that is, the positions of the image conjugates of points R and P. It should be assumed that the power of the eye (F), when it focuses on the point 4 meters away, is 58.25 D. First find the position of the image of object M for which the vergence L_{M} at the eye is 0.25 D. Thus, the vergence after refraction by the eye is and the image of M is at a distance from the refracting surface ofSimilarly, the image distance corresponding to the far depth of field, which is at infinity and for which the vergence at the eye L_{R} is zero, is and the image conjugate of R is at a distance from the refracting surface ofFinally, because the near depth of field extends to within 2 meters of the eye, the vergence L_{p} at the eye will be 0.5D and the vergence after refraction by the eye is and the image conjugate of P is at a distance from the refracting surface of the eye ofThe near depth of focus equals l'_{P}  l'_{M} = 0.13 mm. The far depth of focus equals l'_{R}  l'_{M} = 0.07 mm. Thus, the total depth of focus equals 0.07 + 0.13 = 0.2 mm. Figure 35 illustrates the reduction in depth of field and focus as the pupil enlarges. In a similar way, Figure 37 shows an exaggerated blur B that is just noticeable. For an eye focused at the fixation plane M, point P will produce a blur of some diameter B when it is a distance S closer to the eye with a smaller pupil. When the pupil is enlarged as in the lower illustration, the point may be moved toward the eye, through a distance S' before blur B occurs, where S' is less than S.

VISUAL DISCRIMINATION  
Visual discrimination can be divided into three broad categories: light
discrimination, or the detection of light and color; spatial discrimination, or
the ability to distinguish forms and relationships in space; and
temporal discrimination, concerned with timevarying stimuli. Resolution
and acuity represent visual capacities within the category of
spatial discriminations. The total range of visual capacities are as
follows:
Light discrimination may not seem to be germane to the topic of the eye as an optical system. However, it will be shown that visual acuity, resolution, and spatial contrast sensitivity are related directly to discrimination of brightness. The following discussion will deal with light and spatial discriminations within the context of their influence on acuity, resolution, and spatial contrast sensitivity. The density of rods and cones from the nasal to the temporal edge of the retina (Fig. 38) is basic to an understanding of visual discrimination. Cones are concentrated at the fovea and have a packing density that is about 18 times greater than elsewhere in the retina. No rods occur at the fovea. They reach maximum density at about 18 to 20 degrees from the fovea. Neither rods nor cones are found in the blind spot. Figure 39 illustrates the relative sensitivity to different wavelengths of light of the rods and cones. Photopic vision for highacuity discriminations is mediated by the cones. The rods, sensitive to low light levels, are used for scotopic vision. LIGHT DISCRIMINATION Brightness Sensitivity Brightness sensitivity is the ability to detect a very weak light source in a black background. Hecht and associates^{15} did the classic experiment to find the minimum energy necessary for vision. The subject was darkadapted for maximum sensitivity. They used light of wavelength 510 nm corresponding to the peak spectral sensitivity of the rods shown in Figure 39. A source subtending 10 minutes of arc, put 20' offaxis to correspond with the area of greatest rod density was flashed on for 0.001 second. The detection threshold was found to correspond to approximately 50 to 150 photons striking the cornea. Of these, about 50% are absorbed and reflected by the ocular media, leaving 25 to 75 photons to strike the retina. Only about 20% of these photons are absorbed by the rhodopsin, thus only 5 to 15 photons are left to excite vision. Because they fall on a 10minute area of the retina that contains some 500 rods, the probability of more than one photon falling on any rod is very low. Thus, the light can be seen when each photon is absorbed by a different rod. When one photon is absorbed, it is absorbed by one molecule of rhodopsin. This single molecule initiates the chain of reactions, resulting in nervous stimulation and the perception of light. Amazingly, a retinal rod reaches the absolute limit of sensitivity set by quantum and molecular theories. Brightness Discrimination Brightness discrimination is the ability to detect small differences in brightness between two light sources, or luminance difference thresholds (LDTs). Given two contiguous surfaces of the same color but slightly different luminance, L and L + ΔL, the subject is to detect the brighter surface; ΔL is equal to the LDT. The absolute threshold, described previously, is a special case of LDT where L = 0 and shows the ability to detect light in a dark background. Brightness discrimination, however, determines the ability to distinguish the form and pattern of objects. Weber^{20} formulated a generalized expression applicable to brightness discrimination, stating that for a difference to be detectable, it must be a nearly constant fraction of the background luminance, or ΔL/L = constant. This ratio is about 1% for brightness discrimination and was discovered by Bouguer in 1760. Weber extended it to other senses. Actually Weber's law breaks down at low and high levels of luminance. Brightness Contrast Expressed as a percentage, (ΔL/L) × 100 also is called brightness contrast; ΔL is the difference in luminance between object and background, and L is the brightness of the background. Brightness contrast depends on the size of the object, the background brightness, the wavelength, the region of the retina stimulated, and the shape of the object. Figure 40 shows that smaller differences in brightness are required (lower contrast) as illumination L increases. Larger objects require lower contrast to be discriminated. Spatial Contrast Sensitivity Visual acuity, measured with black Snellen letters on a white background, fails to inform on the individual's ability to see targets of low contrast that commonly make up the real world. A more complete assessment of vision is provided by measuring the contrast sensitivity function (CSF). This usually is done with spatial sinewave patterns that may be varied in contrast and spatial frequency (Fig. 41). Here, we define contrast as a ratio of luminances (L). Spatial frequency is the number of cycles per degree (cpd) in the sinewave pattern, where 30 cpd corresponds to 20/20 or 1 minute acuity. Contrast sensitivity is the reciprocal of the threshold contrast or necessary contrast for seeing the sinewave pattern. A contrast sensitivity curve, based on data by Corwin,^{16} is shown in Figure 42. The normal CSF curve shows that sensitivity falls at both high and low spatial frequencies. Peak sensitivity lies at about 4 to 5 cpd. Thus, the contrast thresholds are low at these spatial frequencies. Visual acuity is the end point of the CSF curve. Two individuals may have the same Snellen acuity but very different sensitivities at intermediate spatial frequencies. Contrast sensitivity decreases with age, cataracts, and ocular and neural pathologies. Claims for using CSF diagnostically have been controversial. However, CSF may be useful in monitoring the efficacy of therapies or following vision changes after corneal surgery. Several tests for clinical CSF measurements are available. The Mentor system displays optotypes and sinewave gratings on a monitor. Thresholds are measured by a blocked up/down psychophysical procedure.^{16} The Arden^{17} and Vision Contrast Test System^{18} shown in Figure 43 comprise several printed sinewave grating charts for distance and near testing that provide a rapid assessment of CSF.
Rapid results also are provided by the PelliRobson letter chart shown in Figure 44.^{19} Each chart contains eight lines of six letters in two sets of three. All letters are of the same size. Each set of three letters has a constant contrast; however, the sets decrease in contrast by a factor of 1/ √2 or in steps of 0.15 log units, from 100% at the top left to 0.6% at the bottom right. Administering this test is easy, compared with psychophysical procedures, because it more resembles an acuity test. Color Discrimination Color is classified in terms of its hue, saturation, and brightness. Hue relates to the wavelength of the light that results in the perception of red, green, blue, or other colors. Saturation relates to the purity of the color, that is, how much white light is mixed with the color. Brightness relates to the amount of luminous energy of the color. Hue discrimination varies with the wavelength. It is greatest around 490 nm (bluegreen) and 580 nm (yellow), where differences of about 1 nm can be discriminated (Fig. 45). About 128 hues can be distinguished under good conditions. SPATIAL DISCRIMINATION Minimum Visible Acuity Minimum visible acuity often is called detection and really is an example of brightness discrimination. It is indicative of the smallest area of the retina with which we can merely detect light without regard to form. The quantity of light necessary, that is, the product of the area of the fovea illuminated multiplied by the illumination, was shown by Ricco to be a constant for angles less than 10 arc minutes. The minimum visible acuity is not determined by the size of the object because a point source is not imaged as a point on the retina because of aberrations and diffraction. Furthermore, fixation movements shift the image from area to area. A point object, such as a star, can be seen if the intensity is sufficient. Objects appear larger the higher the intensity becomes because more retinal elements receive more than a threshold stimulus. This explains apparent star magnitude even though all stars are points. Stars of the sixth magnitude are the dimmest visible in the night sky unless special precautions are taken, such as if the observer views the night sky from a dark room through a small opening. Then he or she may see seventh or eighth magnitude stars. This is because it is the brightness of the sky that prevents us from seeing these dim stars. Minimum Perceptible Acuity Minimum perceptible acuity refers to the detection of fine objects, such as dots or lines, against a plain background. The objects may be bright on a dark background, dark on a bright background, or of low contrast, that is, of nearly the same luminance as their background.^{20} This type of acuity depends on brightness sensitivity and discrimination. The object need not be identified, merely detected. A long black line subtending 0.5 seconds of arc is visible against a white background. The image of the line is diffracted and blurred over many cones instead of being imaged as a geometric shadow of approximately 1/60 of the cone diameter. What makes the line perceptible is that the energy on the column of cones under the diffracted image is about 1% less than the background energy. This is evidently the detectable difference threshold for brightness for the given conditions. Thinner lines would produce less than 1% reduction in illuminance on the retina and thus would not be detected. Consequently, detecting black lines on bright backgrounds and detecting a fine white line on a black background differ. The latter type of target will be detected regardless of how thin it is, provided sufficient illumination reaches the retina. Minimum Separable Acuity In the section on pupillary apertures, the resolution of two diffracted images was discussed. It was shown that when the central peak of one pattern was superimposed on the first dark ring of the other pattern, two sources were resolved. The angle between the sources for a 3mm pupil is, according to the Rayleigh limit, = 1.22 λ/d = 0.78 minutes of arc = 0.000225 radians. For best resolution, the expression is = λ/d = 0.64 minutes of arc = 0.000186 radians. These angles are subtended at the nodal point of the eye and correspond to a dimension s on the retina of s = 0.000225 (17.2) = 0.0039 = 3.9 μm for the Rayleigh limit. The smallest separation s' = 0.000186 (17.2) = 0.0032 mm = 3.2 μm. Clearly, if both image patterns fell on one foveal cone, only one object could be seen. If two adjacent cones were stimulated, there still would appear to be one object. The necessary and sufficient condition for resolving two objects is for two cones to be stimulated and separated by a third cone that is subject to a perceptibly lower level of stimulation. Figure 46 illustrates how two diffraction images separated by Rayleigh's limit would appear on the foveal cone mosaic, containing some 147,000 cones/mm^{2}. The diameter of a cone is approximately 2 μm, and the cones are spaced 0.3 μm apart. Cone a lies under the peak intensity of one pattern, and cone c lies under the peak of the other pattern. The reduction in intensity between the peaks which falls on cell b is sufficient to signal the presence of two sources. Although the diffraction patterns cover several cones, their intensities decrease precipitously and are very low in the outlying rings. Nevertheless, twopoint resolution clearly cannot be depicted as simply two geometric point images in cells a and c separated by an unstimulated cell b. Central foveal cones have diameters as small as 1.5 μm or 0.0015 mm. The linear separation of two cones separated by a third is approximately 0.003 mm. This corresponds to a visual angle of This smallest anatomically possible resolution agrees with the best practically achieved resolution of stars. It may be concluded that the limits to twopoint resolution or minimum separable acuity are due to the anatomy of the cone mosaic. In addition to twopoint resolution, the category of minimum separable acuity includes such tests as the Landolt C and grids of equally spaced parallel black and white lines (Figs. 47 and 48). The gap in the C subtends one fifth of the diameter of the ring, and the thickness of the ring is one fifth of the diameter. The position of the gap may be rotated and the rings made progressively smaller until the subject correctly locates the gap more than 50% of the time. Visual acuity is defined as the reciprocal of the gap in minutes of arc. A 1minute gap corresponds to a standard acuity of 1. A 2minute resolvable gap corresponds to a visual acuity of 0.5. Figure 49 illustrates relative visual acuity as a function of the field angle from the fovea. Figure 50 provides visual acuity as a function of background luminance for various field angles.
Parallel line gratings of various angular subtenses provide very precise measures of acuity and have been used to study visual acuity in humans, other mammals, birds, reptiles, amphibians, fish, and insects. For humans, the average angle subtended by the narrowest resolvable lines of a grating is about 1 minute of arc or less. The reciprocal of this angle is, again, a measure of visual acuity. Pirenne^{15} states that humans have a maximum visual acuity of 1.7 versus 0.017 for the bee, or roughly 0.6 minutes versus 60 minutes for the respective angles of resolution. As the resolution limit for a grating consisting of lines that subtend 0.4 minutes of arc is approached, the retinal image of the bars loses the sharpness and contrast dictated by simple geometric optics. Because of diffraction and chromatic aberration of the eye, light from the closely packed white image spaces spills over into the black image spaces. Instead of the theoretical 100% white0% black image pattern of Figure 48b, the distribution calculated by Hartridge occurs as shown in Figure 48c. The highest intensity is about 60%, and the lowest is about 40%. As noted in the discussion on twopoint resolution, there are no unstimulated cones between stimulated cones. Instead, the central cone receives a lower stimulus. Like the grain of photographic film, the cone mosaic is responsible for limiting eye resolution to the angular subtense of these narrowest cones. The limit occurs when the separation of the details in the object results in a diffracted energy distribution at the retina such that the difference in intensity of the middle cone is below the threshold for perception of brightness differences. Visual Acuity and Receptive Fields Implicit in the discussion of the foveal cone mosaic is the idea that each foveal cone is connected to the visual cortex via a single optic nerve fiber. This, of course, is not correct. Rods and cones connect with retinal bipolar cells, which connect with retinal ganglion cells. The ganglion cells send fibers to the brain. These connections are highly complicated.^{21} Several rods and cones form synapses with one bipolar cell, and several bipolar cells may form synapses with one rod or cone. The connections between the bipolar cells and the retinal ganglion cells are similarly complicated. The mosaic of cones and rods that send signals to any particular visual cell, either a retinal ganglion cell or a cell in the visual cortex, is called the receptive field of the cell. Light that stimulates any portion of the receptive field (that is any rod or cone in the receptive field) will elicit a response in the visual cell. Various types of signals are generated, depending on the regions of the receptive field stimulated. Retinal ganglion cells have concentric receptive fields. The size of the centers of these receptor fields varies greatly, but at the fovea, the field centers are about the same size as a single cone. Thus, it is in this sense that the foveal cone mosaic may be considered as the limiting factor in the resolution of the eye. Strictly speaking, the limiting factor is the mosaic of receptive fields of the foveal cones. Hyperacuity and Vernier Acuity Vernier acuity is a hyperacuity test of the ability to detect a break in a line. It is as fine as 3 to 5 seconds of arc. That this is only a fraction of the angular diameter of a foveal cone would appear to contradict the statement that the cone subtense sets a lower limit to the angle of resolution. Anderson and Weymouth^{22} explained this exceptional ability to detect a break in a line on the basis of brightness differences and spatial localization. The cones above the break in the line or edge (Fig. 51) receive a greater stimulus than those below.^{23} This difference is signaled by a greater frequency of nerve fiber impulses. The displacement of the upper line from the lower extension is localized to less than a cone diameter by a refinement of Lotze's local signs, which states that each retinal element corresponds to a specific direction in space (given by the line from the element through the nodal point of the eye). An extended edge (Fig. 52) will stimulate the array of cones marked by dots. Each of these cones corresponds to a different direction in space. The average or mean of these directions determines the local sign of the straight line. A break in the line causes the average local sign for the upper extension to differ from the average local sign of the lower extension by less than the angular diameter of a cone. By averaging the local signs of cones, localization is not restricted to units or increments of cone diameters. Morgan^{24} provides a summary of hyperacuity research and theory. Another form of hyperacuity test is used to evaluate vision through dense cataracts before surgery. Enoch and coworkers,^{25}^{,}^{26} recognizing that patients with dense ocular media disorders retain the ability to project or point to an intense light source, use three bright sources to measure a threepoint vernier acuity. Two of the sources are aligned vertically. A source at their midpoint is randomly offset from the vertical alignment. The patient is required to align it with the two fixed sources. Minimum Legible Acuity In 1862, the Dutch ophthalmologist, Snellen, was the first to devise the familiar eye chart based on findings that most emmetropes had a threshold visual angle of 1 minute of arc for black objects on a white background. He used black block letters to form a chart that has become the basis for the common clinical test of visual acuity. The test requires the identification of letters of the alphabet, the details of which subtend certain angles at specified distances. The process of identifying letters is complicated by experience, familiarity, and psychologic factors that permit some blur interpretation that may be characteristic of the form of the letter. Thus, although the Snellen test is a test of minimum separable acuity, it is not as clear cut, for example, as resolving two points. Nevertheless, it is the clinically preferred acuity test. The form of the Snellen letter corresponding to the 1minute visual angle is illustrated in Figure 53. The letter E subtends 5 × 5 arc minutes. Each bar of the letter subtends 1 arc minute in width. When such a letter is read at a distance of 20 feet, visual acuity is termed 20/20. This is the Snellen fraction, and it is defined as
Thus, 20/20 visual acuity means the subject has read a letter at 20 feet that was designed to be read at 20 feet. A rating of 20/40 means that a letter that normally should be read at 40 feet has to be brought to within 20 feet before it is recognized. The normal rating of 20/20 corresponds to a visual angle of 1 minute arc for the smallest gap in the letter; visual acuity is the reciprocal of this angle. The visual angle for 20/40 visual acuity is 2 minutes of arc, and visual acuity is 20 ÷ 40 = 0.5, which is called decimal acuity. The various letters of a line of Snellen letters are not equally legible.^{27} The B is most difficult. It would have to be increased approximately 1.17 times the letter E to be equally legible. The easiest letter to recognize is the L, which to be as difficult to recognize as the E should be reduced to approximately 0.84 times the E. In other terms, if the ability to read a letter B that subtends 5 minutes is considered to be 20/20 visual acuity, then the letter L can be read with an acuity as poor as 20/30. Snellen test charts cover a range of visual acuity from 20/400 to 20/10. This corresponds to decimal visual acuity of 0.05 to 2. The chart normally is designed for use at 20 feet. If a chart for near vision is required, the line corresponding to 20/20 would contain letters that subtend 5 minutes at, for example, 16 inches, and the letters would have 1minute details. The American Medical Association test chart consists of 17 lines ranging from 20/20 to 20/200. The lines of this chart are designated additionally with a value termed visual efficiency. This is an arbitrary rating evidently used as a basis for industrial compensation for vision impairment. Because the 17 lines are designed in 5% steps of visual efficiency from 100% to 20%, the equivalent Snellen rating is not a round number (Fig. 54). Figure 55 illustrates the relationship between Snellen visual acuity and refractive error for myopes and hyperopes. A 1D refractive error reduces visual acuity to 20/50 for both ametropes. BaileyLovie Chart The BaileyLovie^{28} chart, shown in Figure 56, was designed to overcome deficiencies in the Snellen chart. It has the following design features. All letters are of almost equal legibility and constructed with a 5 × 4 format. At 20 feet, their stroke widths or the angles of resolution of the letters go from 10 to 0.5 minutes of arc, corresponding to an acuity range from 20/200 to 20/10. Each line contains five letters, with betweenletter spacing equal to the width of a letter to avoid crowding. The betweenline spacing above a line is equal to the height of the letters in that line. The 14 lines of letters progress in size geometrically in a ratio of 10 √ 10, or 0.1 log units. Each correctly identified letter on a line of five letters is scored at 0.02 log units. The LogMAR Scale Visual acuity is expressed as the logarithm of the minimum angle of resolution or logMAR.^{28} This is the log_{10} of the stroke width at 20 feet. Table 5 shows the equivalence of logMAR, Snellen, and decimal acuity scales. Also shown is the corresponding angular resolution of a blackandwhite line pair. A 20/20 letter subtends 5 minutes of arc. It has a line stroke of 1 minute of arc. A blackandwhite line stroke subtend 2 minutes of arc. The decimal equivalent of 20/20 = 1. Because the log_{10} 1 = 0, the logMAR value = 0. A 20/200 letter has a line stroke of 10 minutes of arc. Its logMAR value is 1 because log_{10} 10 = 1.
If it is necessary to test at less than 20 feet, the logMAR score of the lowest legible line is corrected by adding the log_{10} (20/test distance). For example, if the chart distance is 4 feet, log_{10} 20/4 = 0.7, and if the lowest legible line has a logMAR value of 0.5 (20/63), the total logMAR value is 1.2. This means that the line stroke is 10^{1.2} = 15.85 minutes of arc. Because 1 minute of arc corresponds to 20/20, the denominator of the Snellen fraction will be 20 × 15.85 = 317. Rounding will result in a visual acuity of 20/320. DISTANCE DISCRIMINATION Depth perception may be an estimate of the distance an object is from the observer or a discrimination between the relative distances of two or more objects, particularly which is nearer or farther. Many monocular cues to distance judgments have been learned from experience. These cues frequently are pictorial, such as perspective, light and shadow, overlapping contours, and aerial perspective. Motion parallax, which we may observe from a moving vehicle, will cause near objects to appear to move oppositely when fixating at far, and vice versa. The two binocular cues to distance judgment are convergence of the eyes and stereopsis. Convergence requires muscular action, which poorly provides an indication of the distance of an object. Stereoscopic vision provides a very sensitive cue to depth at near ranges. Approximately 64 mm separate the two eyes, consequently, they receive slightly different views of an object. This creates a retinal disparity that the brain interprets as solidity or depth. If the eyes fixate point M in Figure 57, its image will fall on the fovea of each eye. The images of the more distant point P will fall nasally to the fovea of both eyes. The regions of the retinas that the P images stimulate are not corresponding; however, if the disparity is less than 20 minutes of arc, the P images lie within Panum's area and we will see P as a single point. If the disparity exceeds Panum's area, the more distant point P will appear double when we fixate point M. The right eye will see an image of P to the right of point M and the left eye will see an image of P to the left of point M. We call this uncrossed diplopia. If we shift the fixation point to point P, which is farther than point M, the images of M will appear double when they fall outside Panum's area. In this instance, the doubled images of M will appear crossed. The right eye will see point M to the left of P and the left eye will see point M to the right of P. Retinal disparity decreases with increasing object distance, and beyond about 2000 feet stereoscopic judgments are of little value.
Figure 57 illustrates the geometry of stereoscopic depth perception. The eyes are fixating point M at distance d. Point P at some slightly different distance forms an angle γ at the nodal point of the eye with respect to point M. Angle γ is the angle of depth discrimination. Values of γ as low as 2 seconds of arc have been reported. Stereoscopic depth discrimination, like vernier acuity, is less than the subtense of a cone and is explained similarly. 
REFRACTIVE STATE OF THE EYE 
The image of an infinitely distant object will fall in front of the retina
in myopia, on the retina in emmetropia, and behind the retina in hyperopia, when
these eyes are exerting zero accommodation. Because the
image of an infinitely distant object defines the position of the second
focal point of any optical system, we can restate the previous sentence
to say that the refractive state of the eye depends on the second
focal length. The eye is myopic when the second focal length is shorter
than the length of the eye, it is emmetropic when the two lengths
are equal, and it is hyperopic when the second focal length is greater
than the length of the eye. Twenty feet (6 meters) is the clinical equivalent
of infinity. This introduces a vergence of 1/6 D at the eyes
that we ordinarily neglect. Classification of refractive states into categories of hyperopia, emmetropia, and myopia should not obscure the fact that they represent a continuum of eye growth and changes. Except for the fact that a hyperope requires a plus lens and a myope a minus lens correction, the two eyes are physiologically alike and optically differ only as far as the position of the second focal point is on one or the other side of the retina. As the eyeball grows, some flattening of the cornea and lens offsets the refractive error that otherwise would result. This process of emmetropization tends to restrain the development of high refractive errors and concentrates the distribution of refractive states near emmetropia. 
EMMETROPIA  
Although the emmetropic eye, with relaxed accommodation, forms sharp retinal
images of distant objects, emmetropia is not the statistically normal
refractive state. Studies of refractive state show that the peak
of the distribution curve occurs at about 1 D of hyperopia, although
the frequency of myopia is greater in adults than children (Fig. 58). Most infants are hyperopic, probably because the axial length of their
eyeballs is too short. Consequently, hyperopia decreases with growth. Emmetropia
is considered merely a point on the curve of refractive
status that marks the transition from hyperopia to myopia. It occurs when
the length of the eyeball, the curvature of the cornea, and the power
of the unaccommodated lens all are appropriate for focusing collimated
light on the retina, a remarkable condition to exist in so large
a sample of the population. The young emmetrope with normal amplitude
of accommodation will have distinct distant and near vision, assuming
that there are no problems with binocular vision or amblyopia.

AMETROPIA  
Ametropia exists when distant objects are not focused sharply on the retina
by an eye with relaxed accommodation. The eye is too long or short
for its power or too weak or strong for its length. Whether it is power
or length, that is, whether it is refractive or axial ametropia, depends
on establishing norms for power and length as, for example, those
of Gullstrand's schematic eye. Naturally, any given individual
may suffer from both axial and refractive ametropia. In the interest
of simplicity, the two types will be considered separately. FARPOINT Because the emmetrope forms an image of an infinitely distant object on the retina, we can say that the retina is conjugate with infinity. That is, if the light paths are reversed and the retina is considered an object, then the image of the retina, formed by the emmetropic eye, would lie at infinity. The point in space conjugate to the fovea is called the farpoint. It is the farthest point of distinct vision. Figure 59a illustrates that in emmetropia, the second focal point falls on the retina, therefore, the farpoint is at infinity. To find the farpoint of a myopic eye, it must be remembered that this eye is too strong. As a result, collimated rays focus short of the retina, and the second focal point lies within the vitreous (see Fig. 59b). To compensate for this excessive power of the eye, the object must be brought closer. A closer object sends divergent light to the eye that pushes the focus closer to the retina. When divergence just matches the amount of excessive power of the eye, the image falls on the retina, and the object is at the farpoint of the eye because it is conjugate with the retina. For example, if an object must be brought to within 1 meter of the eye for its image to fall on the retina, the farpoint is at 1 meter and the eye suffers from 1 D of myopia. A myopic eye always has a farpoint at some real distance in front of the eye. The location of the farpoint for the hyperope is precisely the opposite, that is, it is a virtual point behind the eye (see Fig 59c). Because the hyperopic eye has inadequate refractive power, collimated light will appear to focus behind the retina. The light is intercepted by the retina, so it does not actually focus behind the retina. Any object in front of the hyperopic eye will, if brought toward the eye, provide divergent light. However, the hyperopic eye is weak so the image will fall even further behind the retina. The unaccommodated hyperope will not see clearly at any distance in front of him. To move the image from behind the retina onto the retina, the light must be convergent when it strikes the eye or appear to focus behind the eye. Obviously, real objects cannot be seen behind the retina, so a plus lens in front of the eye is needed to achieve this convergence. If a plus lens provides, for example, 1 D of convergence at the eye, the lens will form an image of a distant object 1 meter behind the eye. This image appears to be a virtual object to the eye. If the unaccommodated hyperope's eye can focus this image on the retina, the farpoint of the eye is 1 meter behind the eye, that is, the retina is conjugate with the virtual object. The eye suffers from 1 D of hyperopia. LENGTH OF EYE IN AXIAL AMETROPIA We calculated the size of the retinal image in axial ametropia using reduced eye models with 5D of myopia and hyperopia. Similar calculations could be made with the schematic eye. In this case, the distance from the second principal point H' to the second focal point F' shows the variation in length corresponding to axial ametropia because the principal points are nearly fixed in position compared with the focal points. Figure 60 shows the position of the second principal point. It is 1.6 mm behind the first surface of the cornea. Consequently, the length of the eye will be 1.6 + l'. Length is obtained with the fundamental equation: L' = L + F, where in axial ametropia F is constant (58.64 D), L is the vergence at the eye due to an object at the farpoint, and L' will be the vergence after refraction. Figure 61 presents the calculated length of the schematic eye in axial ametropia.
CORRECTION OF MYOPIA The farpoint of a myopic eye lies in front of the eye. The relaxed myopic eye can see distinctly this point but no farther. To enable the myopic eye distinctly to see infinitely distant objects, it is necessary to make these objects appear to be at the farpoint by altering the vergence of the light from infinity so that it enters the eye with the same divergence as rays from the farpoint. This requires a lens at the eye that will diverge collimated light so that it appears to come from the farpoint. Such a lens is a minus lens and if collimated light strikes this lens, it will appear to be focused at the second focal point of the lens. Consequently, a minus lens with a second focal point that is coincident with the farpoint of the myopic eye will correct that eye. It will cause collimated rays from infinity to enter the eye with the same divergence as rays from the farpoint and be focused on the retina. This is illustrated in Figure 62. The uppermost diagram depicts a myopic eye sharply focusing light from its farpoint on the retina. Light from points beyond the farpoint would have less divergence at this eye and be foused in front of the retina. The action of a properly chosen minus lens is shown in Figure 62b. This lens causes light from infinity to diverge so that on emerging, the rays appear to originate at a virtual image point coincident with the farpoint. If the lens is now combined with the eye as shown in the bottom diagram, the necessary divergence to light from infinity has been provided to enable the eye to focus it on the retina. The combination of minus correction lens and eye system causes the retina to be conjugate with infinity, thus corresponding to the condition of emmetropia. Correction lenses are worn close to the eyes for several very important reasons. First, frames supported by the nose and temples are extremely convenient devices for fitting the lenses in front of the eyes. The resultant closeness of the lenses to the eyes provides large fields of view with smalldiameter lenses. Furthermore, the position of the correction lens mounted in a frame is very near the first focal point of the eye (about 15 mm); as a result, the size of the retinal image of the corrected ametrope practically is the same as the emmetropic image size. The correction lens introduces minimal magnification or demagnification of the retinal image when worn near the anterior focal point of the eye, which is approximately where a frame places the lens. This position for the correction lens is only one of many possible positions at which the correction lens may be placed to produce sharp retinal images. Whatever position is chosen, the second focal point of the correction lens must coincide with the farpoint of the eye. The closer to the farpoint the lens is placed, the shorter its focal length must be. This will, in turn, reduce the size of the image on the myopic retina. CORRECTION OF HYPEROPIA The farpoint of the hyperopic eye lies behind the retina. Although a hyperope can converge light by accommodating to see clearly at far, without accommodation, the hyperope cannot see distinctly any point in front of him or her from near to infinity. Light must strike the eye with convergence, that is, appear to come from the virtual object (farpoint) behind the retina, if it is to be focused on the retina. Consequently, to see clearly at far, a plus lens is needed that has a second focal point coincident with the farpoint of the eye; thus, the image of a distant object is moved to the retina. The combination of correction lens and eye causes the retina to be conjugate with infinity as it would be in emmetropia. The closer this lens is to the eye, the shorter must be its focal length if F' is to be coincident with the farpoint. Just as in the myopic correction, the magnification depends on the position and power of the lens. Figure 63 illustrates how a plus lens corrects the hyperopic eye. The top diagram shows how the eye can focus on the retina rays that initially were converging toward its farpoint behind the retina. The middle diagram shows a plus lens positioned to make its second focal point coincide with the farpoint. The lens converges rays from infinity toward the farpoint; consequently, when these rays strike the eye, as in the bottom diagram, the eye can focus them on the retina. 
SPECTACLE MAGNIFICATION 
A spectacle lens alters the size of the retinal image of a distant object with respect to its size as formed by the eye in its uncorrected state. In this latter state, the image of an extended object is a series of blur circles, and the resultant fuzzy image appears enlarged. Constriction of the pupil of the eye reduces the size of the blur circles by reducing the size of the fuzzy image. Eliminating variation in the size of the image with pupil diameters in the uncorrected eye is necessary if this image is to be compared with the image formed in the corrected eye. Suppose, on a geometric basis, that the pupil constricts until it transmits one ray from every point on the object (this obviously is not possible because of diffraction). These rays are the chief rays. They represent the bundles of rays emitted from each point on the object. Consequently, the chief rays from opposite ends of the object properly delineate the image size on the retina of the uncorrected eye. Spectacle magnification is the ratio of the size of the sharp retinal image of a distant object in the corrected eye to the size of the image delineated by the chief rays in the uncorrected eye. Spectacle magnification is concerned with the change in retinal image size induced by spectacles for a given eye rather than whether the image sizes are normal. The image size in the emmetropic schematic eye is considered normal. SM = 1/(1  dF), where F is the thin lens power and d is the distance in meters from the lens to the eye. 
RELATIVE SPECTACLE MAGNIFICATION  
The ratio of the size of the retinal image formed by a corrected eye to
the size of the image formed by the emmetropic schematic eye (the normal
image size) defines the relative spectacle magnification (RSM). RSM
compares the sizes of sharp retinal images of distant objects formed
by the corrected eye with a standard or normal size. Because the height of the image of a distant object formed by an optical system is directly proportional to its EFL, RSM is the ratio of the EFL of the correction lens and ametropic eye (f_{c}), to the EFL of the emmetropic schematic eye (f_{s}). Refracting power is the reciprocal of focal length (in meters); therefore, the RSM can be expressed as power: The equivalent power of a combination of dioptric elements such as the correction lens and the ametropic eye (F_{c}) is given by the Gullstrand equation where: F_{l} = equivalent power of the correction lens, F_{e} = equivalent power of the ametropic eye, and d = distance from the lens to the first principal point of the eye.If the correction lens is placed at the first focal point of the eye, the distance d equals the first focal length of the eye, that is, d = f_{e} or d = 1/F_{e}. Substituting this into the Gullstrand equation results in This reduces to F_{c} = F_{l} + Fe  F_{l}. The F_{l} terms cancel each other, leaving F_{c} = Fe. In other words, when a correction lens is fitted at the first focal point of the eye, the power of the combined lens/eye optical system is equal to the power of the eye alone. If Fe is substituted for F_{c} in the equation for relative spectacle magnification, then RSM = F_{s}/Fe, which shows that RSM equals the ratio of the power of the emmetropic schematic eye to the power of the ametropic eye. If the ametropia is axial in origin, then the power of the ametropic eye equals that of the emmetropic eye and RSM equals unity (Fig. 64). This is the basis for Knapp's law, which states that a correction lens fitted at the first focal point in axial ametropia will produce an image of the same size as would result in the emmetropic eye. Consequently, spectacle lenses so fitted will not introduce aniseikonia in axial ametropia. Suppose two eyes suffer unequal amounts of axial ametropia. Their first focal points will be equal distances from the principal planes. Lenses fitted at these distances will introduce a relative spectacle magnification of unity, which means that the images of the two eyes will be equal to that of the standard eye and, therefore, equal each other. If the ametropia is refractive in origin and of unequal amounts for the two eyes, however, the first focal points will be at unequal distances from the principal points. Correction lenses in a spectacle frame could not both lie in the respective focal planes of the eyes unless the frame was skewed. Therefore, at least one lens would introduce an RSM of value other than unity, and the images would be unequal.
RETINAL IMAGE SIZE AFTER SPECTACLE CORRECTION Axial Myopia The sole condition to be satisfied by the correction lens is that its second focal point coincides with the farpoint of the eye. Theoretically, the lens can be placed anywhere between the cornea and the farpoint provided that the aforementioned condition is satisfied. The closer the lens is to the farpoint, however, the shorter its focal length must be, and the smaller the image of a distant object will be in the focal plane. It is this image that the eye views. Obviously, as the image at the farpoint becomes smaller, it will subtend a smaller angle at the nodal point of the eye and the retinal image will shrink correspondingly. In Figure 65a, the negative correction lens is a short distance from the eye. It forms a virtual image of the collimated offaxis rays, whose height at the second focal plane (also farpoint plane) is Y_{1'}. In Figure 65b, the eye views this virtual image point whose height Y_{1'} subtends some angle at the eye's nodal point and forms a retinal image of corresponding subtense. Figure 65c illustrates the geometry that results when a negative correction lens of shorter focal length is placed farther from the eye to cause the focal point to coincide with the farpoint. The collimated rays from the object approach this lens with the same slope as before, and after refraction, they virtually are focused at the farpoint plane. The height Y_{2}' in this plane is reduced from that of the previous correction, however, and this image subtends a smaller angle at the nodal point of the eye. Consequently, the retinal image is reduced in size (see Fig. 65d). Usually, the farther a negative correction lens is fitted from a myopic eye, the smaller the retinal image will be. This reduction in image size, as the distance between lens and reduced eye is increased, is shown in Figure 66 and Table 6 for a 5D axial myope and a distant object that subtends 0.1 radians. Table 6 shows how the negative power of the lens must be increased as the lens is fitted farther from the eye and closer to the farpoint. The third column shows the equivalent power of the lens and eye calculated from the equation
TABLE 6. Retinal Image Size and Spectacle Magnification in 5D Axial Myopia
*The location of the first focal point of the eye.
The reciprocal of F is the equivalent focal length and is tabulated in the fourth column. Because the image size is proportional to focal length, the reduction in image size that clearly occurs, as the correction lens is fitted farther from the eye, is the result of this reduction in equivalent focal length. Tabulated in the fifth column is the actual image size. It is the product of the equivalent focal length and the angle subtended by the object that is, in this case, 0.1 radians multiplied by EFL. To find the spectacle magnification, finding the size of the image in the uncorrected eye is necessary because spectacle magnification is given by the ratio of the corrected image size to the uncorrected size. Chief rays that in turn travel through the center of the pupil of the eye delineate the size of the uncorrected image. The real pupil lies a few millimeters behind the cornea. It will simplify matters to consider the pupil as coincident with the refracting surface of the reduced eye. The various image size constructions for determining spectacle magnification and relative spectacle magnification are shown in Figures 67 and 69. An emmetropic reduced eye is shown in Figure 69a. The chief ray strikes the vertex of the refracting surface, which may be considered the center of the pupil, with a slope . The ray is refracted into the eye following Snell's law, which, for small angles, means that angle It then travels 22.9 mm to the retina. If is set equal to 0.1 radians, then ' = 0.1/1.336 = 0.075 radians, and the intersection height at the retina is 0.075 × 22.9 = 1.72 mm. Relative spectacle magnification is calculated with respect to this height. A 5D axially myopic eye is shown in Figure 67a. The same chief ray will determine the size of the blurred retinal image. Spectacle magnification is calculated with respect to this uncorrected image. Because the eye is 25.2 mm long, the chief ray, which has a slope of 0.075 radians in the eye, will intersect the retina at a height of 0.075 × 25.2 = 1.89 mm. When this axially myopic eye is corrected by a negative lens fitted at its first focal point, as in Figure 67b, the chief ray is refracted by the lens so that its slope ω on striking the eye is less than the initial slope of 0.1 radians. After refraction by the eye, the slope of the ray (ω') for this position of the lens is reduced exactly so that the product of this slope multiplied by the length of the elongated eye results in a retinal image height of 1.72 mm. The image is the same size as in emmetropia because, according to Gullstrand's equation, the equivalent power of the combined lens and eye is 58 D, and this is precisely the same as the power of the reduced emmetropic eye. Because they have equal powers, the focal length of the eye with the correction lens at its first focal point must be 22.9 mm. The second principal plane H' has been shifted from the vertex of the reduced eye to a position 22.9 mm from the retina; it lies beyond the optical system consisting of correction lens and refracting surface of the eye. Finally, it may be noted that because the focal lengths of the corrected eye and the emmetropic eye are equal, their images also must be equal. In Figure 67c, the correction lens is placed in contact with the refracting surface of the reduced myopic eye. The chief ray is undeviated by the lens because it crosses its optical center. It is refracted into the eye with an angle of 0.075 radians and intersects the retina at a height of 1.89 mm, as in Figure 67a. However, the combined power of the lens and the eye is 5 + 58 = 53 D. This lower power corresponds to a second focal length for lens and eye of 1336/53 = 25.2 mm that exactly matches the elongation of the eye. The principal plane remains at the refracting surface; consequently, the uncorrected image that formerly fell in the vitreous is shifted back to the retina. Compared with the emmetropic image, the corrected image is enlarged in proportion to the focal lengths of the corrected eye to the emmetropic eye. The correction lens really can be no closer than 1.5 mm to the reduced eye because this is where the cornea lies (see Fig. 67d). A small reverse telephotographic effect caused by the spacing of the lens and principal plane of the eye results in a retinal image slightly less than 1.89 mm long. Consequently, a spectacle magnification of unity cannot be obtained exactly for a real eye. Spectacle and relative spectacle magnification values for a 5D axial myope for correction lens positions of 0 to 150 mm are shown in Table 7. At zero distance (position C), the spectacle magnification is zero, but RSM is + 9.9%; when the lens is at the first focal point (position F), spectacle magnifcation = 9%, but RSM = 0.
TABLE 7. Comparison of Magnification Effects and Retinal Image Sizes in 5D
Myopia
F, lens at first focal plane; C, contact lens.
Axial Hyperopia As expected, plus lenses will produce image size effects opposite that of negative lenses. The plus lens correction also must fulfill the condition of superimposing its second focal point on the farpoint of the eye. For the hyperope, however, the farpoint is behind the eye. The plus correction lens can be fitted no closer to the farpoint than the vertex of the eye. Theoretically, of course, the lens can be fitted any distance in front of the eye. However, the closer to the eye the lens is fitted, the shorter its second focal length must be to maintain coincidence with the farpoint. Concomitantly, the image formed by the lens at the farpoint plane will be smaller. The eye, in effect, views this image, which will subtend correspondingly smaller angles at the nodal point. Figure 68a shows the correction lens relatively close to the vertex of the eye. An image of height Y_{1}' is formed at the farpoint plane that, when viewed by the eye, subtends an angle , as shown in Figure 68b. When a longer focal length lens is fitted farther from the eye as in Figure 68c, the image Y_{2}' at the farpoint plane enlarges and subtends a larger angle ' at the nodal point of the eye. If the plus lens is fitted at the first focal point of the eye, according to Knapp's rule, the retinal image size will be 1.72 mm, as in emmetropia. RETINAL IMAGE IN REFRACTIVE AMETROPIA A 5D refractive myope may be illustrated by a reduced eye with a length of 22.9 mm but with a refractive power of 63 D (see Fig. 68b). A distant object that subtends 0.1 radians will have an uncorrected image height of 0.075 × 22.9 = 1.72 mm because that is where the chief ray intersects the retina. The focal point lies in the vitreous humor at a distance equal to the second focal length, that is, The uncorrected image size in refractive ametropia is equal to the standard image size. This means that spectacle and relative spectacle magnification will be identical because spectacle magnification is referred to as the uncorrected image size and RSM is referred to as the standard image size. A correction lens placed at the first focal point of this eye results in an equivalent power for the lens and eye combination that is equal to the power of the eye alone, that is, 63 D (see Fig. 69c). The second principal plane of the combination is at the cornea, that is, 21.2 mm from the retina. A sharp image is formed on the retina. Because the first focal length is 15.9 mm, the size of the sharp retinal image is (15.9)(0.1) = 1.59 mm. The magnifications are If the lens is placed in contact with the refracting surface, the net power is 5 + 63 = 58 D, as in Figure 69d. The power, focal length, and axial length of the corrected refractively myopic eye is the same as the standard eye. The image will be 1.72 mm long, and the magnifications are Usually, the closer a lens is fitted to the refractive ametrope, the less magnification there will be. The magnification effects for a correction lens at the first focal point of the eye compared with a contact lens are summarized in Table 7. RELATIVE SPECTACLE MAGNIFICATION AND ANISEIKONIA Aniseikonia is produced by RSM in anisometropia. For example, let the myopic conditions of Table 7 apply to the right eye and let the left eye be emmetropic. The left eye retinal image size of a 0.1radian target will be 1.72 mm. Axial myopia corrected at the first focal point results in a 1.72mm image. Refractive myopia that is corrected with a lens in contact with the reduced eye also results in a 1.72mm image. These right eye corrections produce no aniseikonia or RSM. However, the axial myopia correction at the eye produces a 1.89mm image. This results in + 9.9% aniseikonia and RSM. Similarly, the refractive myopia correction at the first focal plane produces a 1.59mm image or 7.6% aniseikonia and RSM. 
ASTIGMATISM  
Paraxial theory indicates that spherical refracting surfaces form point
images because these surfaces have constant curvatures in all meridians. Cylinders, conversely, have a maximum curvature along their circumferential
direction and zero curvature along their length, that is, parallel
to the cylinder axis. The zero curvature is 90 degrees to the maximum
curvature. A cylindrical refracting surface will form a line image
of a point parallel to the cylinder axis. If the cylinder is bent
into a doughnut shape, then the meridian that formerly had zero curvature
takes on a curvature. This curvature is less than the circumferential
curvature and is at 90 degrees to the latter. Thus, a toric surface
results, which forms two line images of a point at right angles to each
other and at different distances along the axis. The distance between
these line foci is called the interval of Sturm in honor of the mathematician
who investigated it in 1838. The interval of Sturm is shown
in Figure 70 as formed by a toroidal lens that is curved more deeply vertically than
horizontally. Figure 71 illustrates the same lens with a circular aperture. The bundle of light, as
it traverses the interval of Sturm, has its crosssection transformed
from a horizontal line to a horizontal ellipse. Then it becomes
circular in section, and this position is known as the circle of least
confusion. As the light progresses, the section becomes elongated into
a vertical ellipse. This narrows to a vertical line at the end of the
interval. The eye becomes astigmatic when any of its refracting surfaces
assume a toroidal shape. The astigmatism is termed regular if the
meridians of maximum and minimum curvature are at right angles to each
other. These meridians are called principal meridians.
WITH AND AGAINSTTHERULE ASTIGMATISM Astigmatism of the eye may be hyperopic or myopic depending on whether the astigmatic foci fall behind or in front of the retina. Corneal astigmatism is most pronounced, although lenticular astigmatism often is manifested when a spherical contact lens eliminates the toricity of the cornea. The condition in which the meridian of greatest power is vertical, or within 30 degrees of the vertical, is most common and is called withtherule astigmatism. It is corrected with a minus cylinder at axis 180 degrees. When the corneal curvature of greatest power is horizontal ±30 degrees, it is called againsttherule astigmatism. In the following classification, accommodation is inactive and the object point is very distant. REGULAR ASTIGMATISM Simple Hyperopic Astigmatism In simple hyperopic astigmatism, the meridian of maximum power is emmetropic; therefore, it forms a line image of a point on the retina. If that meridian is at 90 degrees, the astigmatism is with the rule, and a horizontal line is focused on the retina. The meridian of minimum power is hyperopic. It will form a vertical line image behind the retina. The interval of Sturm extends from the retina to this image behind it (Fig. 72). Suppose an individual has 1 D of againsttherule corneal astigmatism. Using the reduced eye as a model, the power in the 180degree meridian is 58 D. The power in the 90degree meridian is 57 D because the hyperopic astigmatism is refractive in nature. The positions of the line images with respect to the principal point P of the reduced eye are: Vertical line image: f = 1336/58 = 23.0 Correction will be obtained with a + 1.00 DC axis 180 degrees, or with + 1.00 DS  1.00 DC axis 90 degrees. Compound Hyperopic Astigmatism Both the maximum and minimum meridional powers are refractively hyperopic in compound hyperopic astigmatism. In Figure 73, the meridian of maximum power (57 D) is at 90 degrees. This is withtherule astigmatism, and this meridian forms a horizontal line image at a distance of 23.4 mm. The more hyperopic (weaker) meridian at 180 degrees has a power of 56 D in this example. It forms a vertical line image along the 90degree meridian, at a focal distance f = 1336/56 = 23.8 mm. The interval of Sturm is 23.8  23.4 = 0.4 mm, calculated in vitreous, but it would extend from 0.4 to 0.8 mm behind the retina. Although accommodation will allow the movement of the line foci toward the retina, both line foci cannot be placed simultaneously on the retina. Accommodation here is an attempt to find the best compromise focus, which usually is the circle of least confusion. A + 1.00 + 1.00 × 90degree lens or + 2.00  1.00 × 180degree lens will correct this eye. Simple Myopic Astigmatism In this type of astigmatism, the eye is emmetropic in one meridian. A distant point will be imaged as a line on the retina by the power in this meridian. Figure 74 illustrates simple myopic withtherule astigmatism. The power in the 90degree meridian is 59 D or myopic by 1 D. The power in the 180degree meridian is a normal 58 D. The image conjugates with respect to the principal point of the reduced eye are as follows: Horizontal line image: f = 1336/59 = 22.64 mm; Compound Myopic Astigmatism This condition occurs when the meridians of maximum and minimum power are both too strong, that is, when they are refractively myopic. Both line images fall short of the retina, and the interval of Sturm is displaced forward of the retina, within the vitreous. In Figure 75, the meridian of greatest power is horizontal, therefore, the astigmatism is against the rule. The interval of Sturm is 22.27  22.64 = 0.37 mm. The myope is unable to shift the circle of least confusion back to the retina. A minus sphere combined with a minus cylinder will bring both foci to a common point on the retina. Mixed Astigmatism As the name implies, this condition has a myopic and a hyperopic component. The meridian of weakest power forms a line conjugate to a distant point that lies behind the retina, whereas the meridian of greatest power produces a line image within the vitreous humor. Thus, the interval of Sturm straddles the retina. In Figure 76, the vertical meridian is hyperopic. The meridian of greatest power is horizontal; therefore, the astigmatism is against the rule. The interval of Sturm is 0.43 + 0.36 = 0.79 mm. Irregular Astigmatism In the previous examples of types of regular astigmatism, the axes were at 90 and 180 degrees. In reality, the axes may be at any meridian. If the maximum and minimum curvatures are 90 degrees apart, the astigmatism is regular—for example, 45 degrees and 135 degrees, or 65 degrees and 155 degrees. If, however, the two principal meridians of curvature are not 90 degrees apart or the corneal curvature is not axially symmetric, the condition is irregular astigmatism. This may be due to injury, corneal diseases that leave scars, keratoconus, or congenital abnormalities. Because spectacle lenses necessarily are ground to uniform curves, they cannot properly correct axial asymmetry of the cornea. If grinding an asymmetric correcting lens was possible, it would be proper only for a fixedeye position because the mobile eye would view through different portions of the lens that would vary in power. The ideal solution to irregular corneas and irregular corneal astigmatism is to use contact lenses that replace the cornea with a spherical refracting surface and move with the eye so that they always are centered. Retinal Image Size Because astigmatism is a refractive type of error, the refracting powers of the two principal meridians differ. Consequently, the cardinal points for the two principal meridians will not coincide. Separate first focal points, corresponding to these meridians, exist. Fitting a correction at both focal points evidently is impossible. If the lens is fitted at one of the first focal points, it does not introduce any relative spectacle magnification for that meridian. Because the lens is not at the first focal point of the other meridian, however, it will produce magnification here. As a result, the retinal image will suffer meridional magnification that distorts it. 
ANISOMETROPIA  
Anisometropia is the condition in which the refractive error of one eye
differs from the other. It may be characterized by unequal amounts of
myopia or hyperopia, or one eye may be myopic and the other hyperopic, to
which the special term antimetropia is applied. When the inequality
is greater than 2 D, the anisometropia is considered of high degree. The
correction of anisometropia obviously requires lenses of unequal
power. If the lines of sight of both eyes pass through the optical centers
of these lenses, prismatic imbalances will not occur. Lenses cause
prismatic effects, that is, displacements of the field of view, when
the line of sight passes eccentrically through them. If both lenses
have the same power, the prismatic effects will be identical for the two
eyes, whatever the direction of gaze. When anisometropia exists and
the correction lenses have unequal powers, prismatic imbalances may result
that interfere with single binocular vision. This is particularly
true for vertical imbalances induced when the eyes look down to read. The
lines of sight will pass through points on the lens that are several
millimeters below the optical centers, and the vertical displacement
of the reading matter will be greater for one eye than the other. Prismatic effect is calculated according to Prentice's rule, which states that prismatic deviation is equal to the lens power multiplied by the distance offcenter through which the line of sight passes. P = dF, where P equals deviation in prism diopters, d equals the offcenter distance in centimeters, and F equals the power of the lens in diopters. A prism diopter (Δ) is a measure of ray deviation given by a linear displacement in centimeters on a screen placed 1 meter from the prism (Fig. 77). A displacement Y of 1 cm at 1 meter equals 1Δ. An example of an induced vertical imbalance follows for the following prescription: OD = + 1 D; OS = + 3 D. The corresponding prismatic effects for a line of sight 1 cm below the optical centers are P_{OD} = 1 × 1 = 1Δ baseup and P_{OS} = 1 × 3 = 3Δ baseup. The vertical imbalance is 3  1 = 2Δ baseup on left eye.
Light is deviated toward the base of a prism. In the aforementioned example, the prism is baseup because a plus lens is thickest at the optical center and the line of sight passes 1 cm below the center (Fig. 78). Myopic corrections induce basedown prismatic effects when the line of sight is depressed because minus lenses are thickest at the edge. Vertical disparities are poorly tolerated. They may cause reading problems because of difficulties in maintaining single binocular vision. Ultimately, suppression of vision in one eye may result.

APHAKIA  
Inequality of images in refractive anisometropia can be reduced by fitting
the lenses as close to the principal planes as possible, that is, with
contact lenses. Although assessing the degree to which ametropia
may be axial or refractive in origin is difficult, it is a good assumption
to consider differences in the corneal powers of the two eyes as
measured by an ophthalmometer as indicative of refractive ametropia. Unilateral
aphakia results in refractive ametropia and the formation of
unequal images. Extraction of the crystalline lens reduces the power of the eye and changes the position of the far point. The dioptric farpoint L of the aphakic eye can be found with the vergence equation L' = L + F where F = 43 D and the image distance from the cornea to the retina, l' must be 24.4 mm. Therefore, the image vergence is L' = 1336/24.4 = 54.75 D. Substitution in the vergence equation will result in 54.75 = L + 43. Thus, L = 11.75 D. The farpoint distance is l = 1000/11.75 = 85.1 mm behind the cornea. Figure 79 compares the optical systems of the schematic and aphakic eyes. In the emmetropic schematic eye, ray 1 crosses the optical axis with a slope at the anterior focal point of the eye F_{1}. After refraction it is collimated and travels parallel to the optical axis, with a height Y_{e}. Ray 2, which has the same slope and is directed toward the nodal point, will be undeviated by the cornea and also will strike the retina at a height Y_{e}. The image formed on the retina where these two rays intersect has a height Y_{e} corresponding to a field angle equal to 0.1 radian or
In the aphakic eye, the solid lines correspond to ray paths without the correction lens. Ray 1, with slope , crosses the axis at the first focus of the aphakic eye, which lies 23.2 mm in front of the cornea and after refraction emerges parallel to the axis. Ray 2 is directed toward the nodal point with a slope and intersects ray 1 at a distance of 31.2 mm behind the eye. The retinal image is blurred. Ray 1 has a constant height at both the retina and the focal plane, which is 6.8 mm behind the retina. When the correction lens is introduced at the first focus of the eye, ray 1 is not deviated by the lens because it crosses its optical center (assuming thin lenses). Therefore, it travels to the retina along the same path as formerly. Ray 2, however, is bent (dashed lines) by the lens and intersects ray 1 at the retina at height Y_{a}. The height of the sharp retinal image is Thus, the size of the corrected image in aphakia compared with the size of the emmetropic image, or the relative spectacle magnification is The aphakic image is 35% larger when the lens is at the first focal point. If the correction lens is fitted in contact with the cornea of the aphakic eye, its second focal point must coincide with the farpoint 85.1 mm away; therefore, the power of the lens is The power of the lens combined with the aphakic eye is 11.75 + 43 = 54.75 D, which corresponds to a focal length of 18.3 mm. The size of the retinal image is the product of focal length multiplied by the angle subtended by the object or 18.3 × 0.1 = 1.83 mm. The RSM = 1.83/1.72 = 1.064 or + 6.4% when the lens is at the cornea, as is a contact lens. If the aphakia is monocular and the other eye is emmetropic, the RSM of 35% and 6.4% also will be equal to the aniseikonia induced by a spectacle and contact lens correction, respectively. 
ACCOMMODATION  
Normally one can sharply see distant scenes and objects held close to the
eye without awareness of any focusing by the eye. As a result, the
noncritical observer assumes that all distances are simultaneously in
focus for the eye. Scheiner^{23} showed with his twohole disc that, in fact, when a distant scene is sharply
seen, a fine pin held close to the eyes appears double. Similarly, if
the eye is focused on the pin, a distant scene appears double. As
seen in Figure 80, the two pinholes transmit two small bundles of the rays that otherwise
would enter the eye and come to a focus. If this focus falls on the
retina, the two bundles converge to one point and the observer perceives
one point. If the focus falls in front or behind the retina, the two
bundles intersect the retina at two separate points and a double image
is perceived. Scheiner used this to measure the accommodation of the
eye. Simply stated, a fine distant object was brought closer to the
eye. As long as the eye accommodated, it maintained the object in focus, and
it was perceived singly. Accommodation was exerted maximally when
the object had reached the nearpoint. Further approach caused the
object to appear double. The point at which doubling is first perceived
is called the nearpoint of accommodation. The sequential occlusion of the two holes of the Scheiner disc can be used to find the refractive state of the eye. For example, the pencil of rays through the lower hole will strike the hyperopic retina below the axis and the myopic retina above the axis. The hyperope will note a disappearance of the upper image (opposite), and the myope will not see the lower image (same) when the lower hole is occluded. (Hint: project the retinal point back through the nodal point for the apparent direction of the same source.) Thomas Young (17731829) pioneered the investigation of the accommodative mechanism of the eye. Theoretically, the eye may change its focus in several ways. It may change its axial length. This is essentially how a camera is focused for near objects. The lens simply is moved farther from the film plane by means of a focusing ring built into the lens barrel or, in extreme closeup work, the lens is attached to a bellows that permits positioning the lens through a much greater range. A natural question, then, is whether the eye elongates to focus on near objects. A young emmetrope can focus easily on an object 10 cm from the eye. We can find the elongation of the eye required to place the retina in the image plane of this object with the vergence equation. The eye must elongate to 27.7 mm. When the object is at infinity, the position of the image was shown to be 22.9 mm, thus, the growth in axial length of the eyeball is 27.7  22.9 = 4.8 mm. This is a change in axial length of more than 20% of the emmetropic length of the eye, and it must be accomplished almost instantly, imperceptibly, and unflaggingly during the waking hours. Clearly, the human eye does not accommodate in this manner. Accommodation also may be mediated through an increase in the power of its refractive elements. The eye theoretically may increase its power through a shortening of the radius of curvature of the cornea or the lens or by an axial shift of the lens. Young did an experiment in which he immersed his eyes in water. By nearly matching the refractive indices at the water/cornea interface, he practically eliminated the corneal refractive power. He became an extreme refractive hyperope. To see distant objects clearly, Young introduced a positive lens to replace the lost corneal power and could accommodate for near objects. Thus, he showed that accommodation could be exerted despite the neutralization of the cornea. Although the crystalline lens remained the most logical agent of accommodation, the question to be resolved was whether the crystalline lens moved axially or changed shape. Axial movement was eliminated by the constraints on how much the lens could move within the anterior chamber. Calculations show that the depth of the anterior chamber is not sufficient for maintaining the focus of near objects. The steepening curvatures of the crystalline lens account for the ability to accommodate. Helmholtz concluded that the zonule maintains shallow lens curvatures in the unaccommodated state. Relaxation of the fibers, when the ciliary body constricts, allows the elastic capsule of the lens to assume a rounder form. Tscherning noted that the central portion of the lens became more deeply curved whereas the peripheral zone of the lens surface flattened during accommodation. He concluded that this occurred because of an increase in tension by the zonule during accommodation. It generally is accepted that Helmholtz is correct in concluding that relaxation of tension by the zonule allows the lens to assume a more deeply curved form. Fincham concluded that it was the nonuniform thickness of the lens capsule that caused the bulge in lens curvatures in accommodation rather than capsule elasticity.^{23} Gullstrand provides a radius of curvature of 10 mm for the anterior surface of the crystalline lens when relaxed and a radius of curvature of 5.33 mm when accommodated by nearly 10 D. The axial thickness of the lens slightly increases because of the forward bulge of the anterior surface. Mainly because of the bulge, the power of the lens increases from 19 to 33 D, and the power of the eye correspondingly increases from 58.64 to 70.57 D. STIMULUS FOR ACCOMMODATION The nature of the optical stimulus for reflex changes in accommodation has been debated for more than 50 years. The standard view is that accommodation is a closedloop negative feedback system that alters focus to maximize or optimize the luminance contrast of the retinal image. In this view, contrast is reduced both for underaccommodation and overaccommodation, and feedback from changes in defocus blur is an essential part of the accommodative process. However, recent experiments confirm that accommodation responds in the absence of blur feedback^{13} and that the stimulus on the retina has directional quality that distinguishes myopic from hyperopic focus. Along these lines, Fincham^{29} suggested that accommodation responds directly to the vergence of light at the retina, using the effects of chromatic aberration and the StilesCrawford effect. At spatial frequencies above approximately 1 cpd, chromatic aberration ensures that the contrasts of long, middle and shortwavelength components of the retinal image are different. For example, the relative contrasts red>green>blue specify focus in front of the retina. In one model of the process, the refractive state of the eye is determined by comparing relative cone contrasts, measured separately by L, M and Scone classes. In addition to the effects of chromatic aberration, the waveguide nature of directionally sensitive foveal cones could play a role in the accommodative process, but the notion remains largely unexplored. AMPLITUDE AND RANGE OF ACCOMMODATION When the eye is accommodated fully, the point in space conjugate to the retina is the nearpoint of the eye. It is the nearest point of distinct vision. How much accommodation is exerted from the relaxed state to full accommodation is termed the amplitude of accommodation. If the distances of the far and nearpoints from the first principal point of the eye are denoted by r and p and the corresponding reduced vergences are denoted by R and P, the difference R  P = A, in diopters, is the amplitude of accommodation. The corresponding distance of distinct vision from the nearpoint to the farpoint of p  r = a is termed the range of accommodation (Fig. 81).
The range of accommodation for a given amplitude of accommodation depends on the refractive state of the eye (Table 8). An emmetrope, a myope, and a hyperope may each have the same amplitude of accommodation, but their ranges of accommodation will differ greatly. To find the range of accommodation, we need to know the amount of ametropia and the amplitude of accommodation. From these, the far and nearpoints can be calculated. The range of accommodation is the distance between these points. For example, of three individuals, each with an amplitude of accommodation of A = 10 D, one is an emmetrope, the second is a 5D hyperope, and the third is a 5D myope. What are their ranges of accommodation?
TABLE 8. Range of Accommodation for a Given Amplitude
The emmetrope's farpoint r is at infinity, therefore, R = 1/∞ = 0. Substitution into the equation R  P = A, results in 10 = 0  P, or the dioptric value of the nearpoint is P = 10 D. This corresponds to a nearpoint distance p = 1/P, or p = 1/10 meter = 10 cm. The emmetrope's range of accommodation is from 10 cm in front of the first principal plane of the eye to infinity. The hyperope has a farpoint R = + 5 D, that is, the farpoint distance r is 1/5 meter behind the eye. P = R  A = 5  10 = 5 D. The nearpoint distance p is one fifth of a meter in front of the eye. In effect, 5 D of accommodation were exerted to overcome the hyperopia, to see sharply at infinity, and the remaining 5 D determined the nearpoint. The range of accommodation of the hyperope is from 20 cm in front of the eye, through infinity, to 20 cm behind the eyes (see Fig. 81). The dioptric value of the farpoint of the 5D myope is R = 5. Thus, P = 5  10 = 15 D. The range of accommodation of the myope is a = p  r = 100/15 + 100/5 = 6.7 + 20 = + 13.3 cm, or from 6.7 cm to 20 cm in front of the eyes. These examples illustrate the limited range of accommodation available to the myope who otherwise has an amplitude of accommodation equal to that of an emmetrope and hyperope. 
PRESBYOPIA  
The amplitude of accommodation decreases from childhood to 75 years of
age. When the reduction in amplitude causes the nearpoint to move beyond
the comfortable reading distance, the condition is termed presbyopia. Presbyopia
has its onset at about 45 years of age when, according
to Donders, the amplitude of accommodation is 3.5 D. If a 45yearold
individual is emmetropic, his nearpoint will be R  P = 3.5 D. Therefore, p = 100/3.5 = 28.5 cm = 11 inches. At 55 years of age, his amplitude
has dropped to 1.75 D and his nearpoint is p = 100/1.75 = 57 cm = 22.5 inches. Ten
years later, at 65 years of age, he has only 0.5 D of
amplitude and his nearpoint is now p = 100/0.5 = 200 cm = 80 inches. Finally, at 75 years
of age, he has zero amplitude and his nearpoint
is at infinity along with his farpoint. His range of accommodation
also is zero. Because of a ¼ D depth of focus, he still can see
clearly from 4 meters out to infinity. The relationship between amplitude of accommodation and age was investigated by Donders and for a long time was used as a basis for prescribing a near add. Measurements of monocular amplitude of accommodation were made on 4000 eyes by Duane. The results, shown as an average middle curve and upper and lower ranges on amplitude at any given age, are plotted in Figure 82 and listed in Table 8.
Various causes have been proposed to account for the reduction in accommodative amplitude. Accommodation has two parts. One is physical and concerns the change in shape of the lens during accommodation. In presbyopia, the physical part is related to hardening or sclerosis of the crystalline lens that reduces the elasticity of the lens capsule and the plasticity of the lens core. The physiologic part of accommodation is the innervation and contraction of the ciliary muscles. Some hold that sclerosis of the ciliary body reduces its ability to constrict, and the lens does not sufficiently obtain the conditions required for changing its shape. If most of the cause of presbyopia is physical, that is, it is related to the inability of the crystalline lens to alter its shape to bring near objects into focus, then the lens is an indicator of age and may be considered a biological clock. 