15-3. Using data presented in the text, calculate the focusing power of the cornea and of the crystalline lens. 15.6 LENS SYSTEM OF THE EYE The focusing of the light into a real inverted image at the retina is produced by refraction at the cornea and at the crystalline lens (see Fig. 15.4). The focusing or refractive power of the cornea and the lens can be calculated using Eq. C.9, (Appendix C). The data required for the calculation are shown in Table 15.1. The largest part of the focusing, about two thirds, occurs at the cornea. The power of the crystalline lens is small because its index of refraction is only slightly greater than that of the surrounding fluid. In Exercise 15-3, it is shown that the refractive power of the cornea is 42 diopters, and the refractive power of Cornea Crystalline lens FIGURE 15.4 Focusing by the cornea and the crystalline lens (not to scale). 210 Table 15.1 Parameters for the Eye Radius (mm) Chapter 15 Optics Front Back Index of refraction Cornea 7.8 7.3 1.38 Lens, min. power 10.00 -6.0 Lens, max. power 6.0 -5.5 1.40 Aqueous and vitreous humor 1.33 the crystalline lens is variable between about 19 and 24 diopters. (For a defini- tion of the unit diopter, see Appendix C.) The refractive power of the cornea is greatly reduced when it is in contact with water (see Exercise 15-4). Because the crystalline lens in the human eye cannot compensate for the diminished power of the cornea, the human eye under water is not able to form a clear image at the retina and vision is blurred. In fish eyes, which have evolved for seeing under water, the lens is intended to do most of the focusing. The lens is nearly spherical and has a much greater focusing power than the lens in the eyes of terrestrial animals (see Exercise 15-5). 15.7 REDUCED EYE To trace accurately the path of a light ray through the eye, we must calculate the refraction at four surfaces (two at the cornea and two at the lens). It is possible to simplify this laborious procedure with a model called the reduced eye, shown in Fig. 15.5. Here all the refraction is assumed to occur at the front surface of the cornea, which is constructed to have a diameter of 5 mm. The eye is assumed to be homogeneous, with an index of refraction of 1.333 (the same as water). The retina is located 2 cm behind the cornea. The nodal point n is the center of corneal curvature located 5 mm behind the cornea. 1.5 cm 1.5 cm 0.5 cm FIGURE 15.5 The reduced eye. 15.7 Reduced Eye 211 This model represents most closely the relaxed eye which focuses parallel light at the retina, as can be confirmed using Eq. C.9. For the reduced eye, the second term on the right-hand side of the equation vanishes because the light is focused within the reduced eye so that n = n2. Equation C.9, therefore, simplifies to "L + = P 9 R (15.2) where n₁ =1, n = 1.333, and R = 0.5 cm. Because the incoming light is par- allel, its source is considered to be at infinity (ie., P=00). Therefore, the dis- tance q at which parallel light is focused is given by 1.333 9 1.333-1 5 9 = 1.333 x 5 0.333 = 20mm The anterior focal point F for the reduced eye is located 15 mm in front of the cornea. This is the point at which parallel light originating inside the eye is focused when it emerges from the eye (see Exercise 15-6). Although the reduced eye does not contain explicitly the mechanism of accommodation, we can use the model to determine the size of the image formed on the retina. The construction of such an image is shown in Fig. 15.6. Rays from the limiting points of the object A and B are projected through the nodal point to the retina. The limiting points of the image at the retina are a and b. This construction assumes that all the rays from points A and B that enter the eye are focused on the retina at points a and b, respectively. Rays from all other points on the object are focused correspondingly between these limits. The triangles AnB and anb are similar; therefore, the relation of object to image size is given by Distance of object from nodal point Distance of image from nodal point Object size Image size (15.3) ог AB An ab an B FIGURE 15.6 Determination of the image size on the retina. 212 Chapter 15 Optics Consider as an example the image of a person 180 cm tall standing 2 m from the eye. The height of the full image at the retina is 1.5 Height of image = 180 x = 1.32 cm 205 The size of the face in the image is about 1.8 mm, and the nose is about 0.4 mm.