The Eye: I. Optics of Vision

Physical Principles of Optics

Understanding the optical system of the eye requires familiarity with the basic principles of optics, including such factors as the physics of light refraction, focusing, and depth of focus. A brief review of these physical principles is presented in this chapter, followed by discussion of the optics of the eye.

Refraction of Light

Refractive Index of a Transparent Substance

Light rays travel through air at a velocity of about 300,000 km/sec, but they travel much slower through transparent solids and liquids. The refractive index of a transparent substance is the ratio of the velocity of light in air to the velocity in the substance. The refractive index of air is 1.00. Thus, if light travels through a particular type of glass at a velocity of 200,000 km/sec, the refractive index of this glass is 300,000 divided by 200,000, or 1.50.

Refraction of Light Rays at an Interface Between Two Media With Different Refractive Indices

When light rays traveling forward in a beam (as shown in Figure 50-1A ) strike an interface that is perpendicular to the beam, the rays enter the second medium without deviating from their course. The only effect that occurs is decreased velocity of transmission and shorter wavelength, as shown in the figure by the shorter distances between wave fronts.

Figure 50-1, Light rays entering a glass surface perpendicular to the light rays (A) and a glass surface angulated to the light rays ( B ). This figure demonstrates that the distance between waves after they enter the glass is shortened to about two-thirds that in air. It also shows that light rays striking an angulated glass surface are bent.

If the light rays pass through an angulated interface, as shown in Figure 50-1B , the rays bend if the refractive indices of the two media are different from each other. In this figure, the light rays are leaving air, which has a refractive index of 1.00, and are entering a block of glass having a refractive index of 1.50. When the beam first strikes the angulated interface, the lower edge of the beam enters the glass ahead of the upper edge. The wave front in the upper portion of the beam continues to travel at a velocity of 300,000 km/sec, whereas that which entered the glass travels at a velocity of 200,000 km/sec. This difference in velocity causes the upper portion of the wave front to move ahead of the lower portion so that the wave front is no longer vertical but is angulated to the right. Because the direction in which light travels is always perpendicular to the plane of the wave front, the direction of travel of the light beam bends downward.

This bending of light rays at an angulated interface is known as refraction. Note particularly that the degree of refraction increases as a function of the following: (1) the ratio of the two refractive indices of the two transparent media; and (2) the degree of angulation between the interface and the entering wave front.

Application of Refractive Principles to Lenses

Convex Lens Focuses Light Rays

Figure 50-2 shows parallel light rays entering a convex lens. The light rays passing through the center of the lens strike the lens exactly perpendicular to the lens surface and, therefore, pass through the lens without being refracted. Toward either edge of the lens, however, the light rays strike a progressively more angulated interface. The outer rays bend more and more toward the center, which is called convergence of the rays. Half the bending occurs when the rays enter the lens, and half occurs as the rays exit from the opposite side. If the lens has exactly the proper curvature, parallel light rays passing through each part of the lens will be bent exactly enough so that all the rays will pass through a single point, called the focal point.

Figure 50-2, Bending of light rays at each surface of a convex spherical lens showing that parallel light rays are focused to a focal point.

Concave Lens Diverges Light Rays

Figure 50-3 shows the effect of a concave lens on parallel light rays. The rays that enter the center of the lens strike an interface that is perpendicular to the beam and, therefore, do not refract. The rays at the edge of the lens enter the lens ahead of the rays in the center. This effect is opposite to the effect in the convex lens, and it causes the peripheral light rays to diverge from the light rays that pass through the center of the lens. Thus, the concave lens diverges light rays, but the convex lens converges light rays.

Figure 50-3, Bending of light rays at each surface of a concave spherical lens showing that parallel light rays are diverged.

Cylindrical Lens Bends Light Rays in Only One Plane—Comparison With Spherical Lenses

Figure 50-4 shows both a convex spherical lens and a convex cylindrical lens. Note that the cylindrical lens bends light rays from the two sides of the lens but not from the top or the bottom—that is, bending occurs in one plane but not the other. Thus, parallel light rays are bent to a focal line. Conversely, light rays that pass through the spherical lens are refracted at all edges of the lens (in both planes) toward the central ray, and all the rays come to a focal point.

Figure 50-4, A, Point focus of parallel light rays by a spherical convex lens. B, Line focus of parallel light rays by a cylindrical convex lens.

The cylindrical lens is well demonstrated by using a test tube full of water. If the test tube is placed in a beam of sunlight and a piece of paper is brought progressively closer to the opposite side of the tube, a certain distance will be found at which the light rays come to a focal line. The spherical lens is demonstrated by an ordinary magnifying glass. If such a lens is placed in a beam of sunlight, and a piece of paper is brought progressively closer to the lens, the light rays will impinge on a common focal point at an appropriate distance.

Concave cylindrical lenses diverge light rays in only one plane in the same manner that convex cylindrical lenses converge light rays in one plane. Figure 50-5A shows how light is focused from a point source to a line focus by a cylindrical lens.

Figure 50-5, A, Focusing of light from a point source to a line focus by a cylindrical lens. B, Two cylindrical convex lenses at right angles to each other, demonstrating that one lens converges light rays in one plane, and the other lens converges light rays in the plane at a right angle. The two lenses combined give the same point focus as that obtained with a single spherical convex lens.

Combination of Two Cylindrical Lenses at Right Angles Equals a Spherical Lens

Figure 50-5B shows two convex cylindrical lenses at right angles to each other. The vertical cylindrical lens converges the light rays that pass through the two sides of the lens, and the horizontal lens converges the top and bottom rays. Thus, all the light rays come to a single point focus. In other words, two cylindrical lenses crossed at right angles to each other perform the same function as one spherical lens of the same refractive power.

Focal Length of a Lens

The distance beyond a convex lens at which parallel rays converge to a common focal point is called the focal length of the lens. The diagram at the top of Figure 50-6 demonstrates this focusing of parallel light rays.

$Figure 50-6, The two upper lenses of this figure have the same focal length, but the light rays entering the top lens are parallel, whereas those entering the middle lens are diverging. The effect of parallel versus diverging rays on the focal distance is shown. The bottom lens has far more refractive power than either of the other two lenses (i.e., it has a much shorter focal length), demonstrating that the stronger the lens, the nearer to the lens is the focal point.$

In the middle diagram, the light rays that enter the convex lens are not parallel but are diverging because the origin of the light is a point source not far away from the lens. Because these rays are diverging outward from the point source, they do not focus at the same distance away from the lens as do parallel rays. In other words, when rays of light that are already diverging enter a convex lens, the distance of focus on the other side of the lens is farther from the lens than the focal length of the lens for parallel rays.

The bottom diagram of Figure 50-6 shows light rays diverging toward a convex lens that has far greater curvature than that of the other two lenses in the figure. In this diagram, the distance from the lens at which the light rays come to focus is exactly the same as that from the lens in the first diagram, in which the lens is less convex but the rays entering it are parallel. This demonstrates that both parallel rays and diverging rays can be focused at the same distance beyond a lens, provided that the lens changes its convexity.

Formation of an Image by a Convex Lens

Figure 50-7A shows a convex lens with two point sources of light to the left. Because light rays pass through the center of a convex lens without being refracted in either direction, the light rays from each point source of light are shown to come to a point focus on the opposite side of the lens directly in line with the point source and the center of the lens.

Figure 50-7, A, Two point sources of light focused at two separate points on opposite sides of the lens. B, Formation of an image by a convex spherical lens.

Any object in front of the lens is, in reality, a mosaic of point sources of light. Some of these points are very bright and some are very weak, and they vary in color. Each point source of light on the object comes to a separate point focus on the opposite side of the lens in line with the lens center. If a white sheet of paper is placed at the focus distance from the lens, one can see an image of the object, as demonstrated in Figure 50-7B . However, this image is upside down with respect to the original object, and the two lateral sides of the image are reversed. The lens of a camera focuses images on film via this method.

Measurement of the Refractive Power of a Lens—Diopter

The more a lens bends light rays, the greater is its “refractive power.” This refractive power is measured in terms of diopters. The refractive power in diopters of a convex lens is equal to 1 meter divided by its focal length. Thus, a spherical lens that converges parallel light rays to a focal point 1 meter beyond the lens has a refractive power of +1 diopter, as shown in Figure 50-8 . If the lens is capable of bending parallel light rays twice as much as a lens with a power of +1 diopter, it is said to have a strength of +2 diopters, and the light rays come to a focal point 0.5 meter beyond the lens. A lens capable of converging parallel light rays to a focal point only 10 centimeters (0.10 meter) beyond the lens has a refractive power of +10 diopters.

Figure 50-8, Effect of lens strength on the focal distance.

The refractive power of concave lenses cannot be stated in terms of the focal distance beyond the lens because the light rays diverge rather than focus to a point. However, if a concave lens diverges light rays at the same rate that a 1-diopter convex lens converges them, the concave lens is said to have a dioptric strength of −1. Likewise, if the concave lens diverges light rays as much as a +10-diopter lens converges them, this lens is said to have a strength of −10 diopters.

Concave lenses “neutralize” the refractive power of convex lenses. Thus, placing a 1-diopter concave lens immediately in front of a 1-diopter convex lens results in a lens system with zero refractive power.

The strengths of cylindrical lenses are computed in the same manner as the strengths of spherical lenses, except that the axis of the cylindrical lens must be stated in addition to its strength. If a cylindrical lens focuses parallel light rays to a line focus 1 meter beyond the lens, it has a strength of +1 diopter. Conversely, if a cylindrical lens of a concave type diverges light rays as much as a +1-diopter cylindrical lens converges them, it has a strength of −1 diopter. If the focused line is horizontal, its axis is said to be 0 degrees. If it is vertical, its axis is 90 degrees.

Optics of the Eye

The lens system of the eye ( Figure 50-9 ) is composed of four refractive interfaces: (1) the interface between air and the anterior surface of the cornea; (2) the interface between the posterior surface of the cornea and the aqueous humor; (3) the interface between the aqueous humor and the anterior surface of the lens of the eye; and (4) the interface between the posterior surface of the lens and the vitreous humor. The internal index of air is 1, the cornea, 1.38, the aqueous humor, 1.33, the crystalline lens (on average), 1.40, and the vitreous humor, 1.34.

$Figure 50-9, The eye as a camera. The numbers are the refractive indices.$

Consideration of All Refractive Surfaces of the Eye as a Single Lens—The “Reduced” Eye

If all the refractive surfaces of the eye are added together algebraically and then considered to be one single lens, the optics of the normal eye may be simplified and represented schematically as a “reduced eye.” This representation is useful in simple calculations. In the reduced eye, a single refractive surface is considered to exist, with its central point 17 millimeters in front of the retina and a total refractive power of 59 diopters when the lens is accommodated for distant vision.

About two-thirds of the 59 diopters of refractive power of the eye is provided by the anterior surface of the cornea ( not by the eye lens). The principal reason for this phenomenon is that the refractive index of the cornea is markedly different from that of air, whereas the refractive index of the eye lens is not greatly different from the indices of the aqueous humor and vitreous humor.

The total refractive power of the internal lens of the eye, as it normally lies in the eye surrounded by fluid on each side, is only 20 diopters, about one third the total refractive power of the eye. However, the importance of the internal lens is that in response to nervous signals from the brain, its curvature can be increased markedly to provide “accommodation,” which is discussed later in the chapter.

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