•• ENGINEERING APPLICATION An optical fiber allows raysof light to propagate long distances by using total internal reflec-tion. Optical fibers are used extensively in medicine and in digitalcommunications. As shown in Figure 31-58 the fiber consists of acore material that has an index of refraction n, and radius b sur-rounded by a cladding material that has an index of refractionn3 < n2. The numerical aperture of the fiber is defined as sine,where 0, is the angle of incidence of a ray of light that impingeson the center of the end of the fiber and then reflects off the core-cladding interface just at the critical angle. Using the figure asa guide, show that the numerical aperture is given bysine, = Vn3 - n? assuming the ray is initially in air. Hint: Use of thePythagorean theorem may be required.44

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•• ENGINEERING APPLICATION An optical fiber allows rays of light to propagate long distances by using total internal reflec- tion. Optical fibers are used extensively in medicine and in digital communications. As shown in Figure 31-58 the fiber consists of a core material that has an index of refraction n, and radius b sur- rounded by a cladding material that has an index of refraction n, < n2. The numerical aperture of the fiber is defined as sin6,, where 6, is the angle of incidence of a ray of light that impinges on the center of the end of the fiber and then reflects off the core- cladding interface just at the critical angle. Using the figure as a guide, show that the numerical aperture is given by sin6, = Vn- n assuming the ray is initially in air. Hint: Use of the Pythagorean theorem may be required. 44

•• ENGINEERING APPLICATION An optical fiber allows rays of light to propagate long distances by using total internal reflec- tion. Optical fibers are used extensively in medicine and in digital communications. As shown in Figure 31-58 the fiber consists of a core material that has an index of refraction n, and radius b sur- rounded by a cladding material that has an index of refraction n, < n2. The numerical aperture of the fiber is defined as sin6,, where 6, is the angle of incidence of a ray of light that impinges on the center of the end of the fiber and then reflects off the core- cladding interface just at the critical angle. Using the figure as a guide, show that the numerical aperture is given by sin6, = Vn- n assuming the ray is initially in air. Hint: Use of the Pythagorean theorem may be required. 44