Chapter 35. The Nature of Light and the Laws of Geometric Optics
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1 Chapter 35 The Nature of Light and the Laws of Geometric Optics
2 Introduction to Light Light is basic to almost all life on Earth. Light is a form of electromagnetic radiation. Light represents energy transfer from the source to the observer. Many phenomena depend on the properties of light. Seeing a TV or computer monitor Blue sky, colors at sunset and sunrise Images in mirrors Eyeglasses and contacts Rainbows Many others Section Introduction 2
3 Light and Optics There are two historical models for the nature of light. The speed of light has been measured in many ways. Reflection and refraction are the fundamental phenomena in ray (geometric) optics. Internal reflection is the basis for fiber optics. Introduction 3
4 The Nature of Light Before the beginning of the nineteenth century, light was considered to be a stream of particles. The particles were either emitted by the object being viewed or emanated from the eyes of the viewer. Newton was the chief architect of the particle theory of light. He believed the particles left the object and stimulated the sense of sight upon entering the eyes. Section
5 Nature of Light Alternative View Christian Huygens argued that light might be some sort of a wave motion. Thomas Young (in 1801) provided the first clear demonstration of the wave nature of light. He showed that light rays interfere with each other. Such behavior could not be explained by particles. Section
6 Christian Huygens Best known for contributions to fields of optics and dynamics He thought light was a type of vibratory motion. It spread out and produced the sensation of light when it hit the eye. He deduced the laws of reflection and refraction. He explained double refraction. Section
7 Confirmation of Wave Nature During the nineteenth century, other developments led to the general acceptance of the wave theory of light. Thomas Young provided evidence that light rays interfere with one another according to the principle of superposition. This behavior could not be explained by a particle theory. Maxwell asserted that light was a form of high-frequency electromagnetic wave. Hertz confirmed Maxwell s predictions. Section
8 Particle Nature Some experiments could not be explained by the wave model of light. The photoelectric effect was a major phenomenon not explained by waves. When light strikes a metal surface, electrons are sometimes ejected from the surface. The kinetic energy of the ejected electron is independent of the frequency of the light. Section
9 Particle Nature Einstein (in 1905) proposed an explanation of the photoelectric effect that used the idea of quantization. The quantization model assumes that the energy of a light wave is present in particles called photons. E = hf (35.1) h is Planck s constant and = J. s Section
10 Dual Nature of Light In view of these developments, light must be regarded as having a dual nature. Light exhibits the characteristics of a wave in some situations and the characteristics of a particle in other situations. This chapter investigates the wave nature of light. Section
11 Measurements of the Speed of Light Since light travels at a very high speed, early attempts to measure its speed were unsuccessful. Remember c = m/s Galileo tried by using two observers separated by about 10 km. The reaction time of the observers was more than the transit time of the light. Section
12 Measurement of the Speed of Light Roemer s Method In 1675 Ole Roemer used astronomical observations to estimate the speed of light. He used the period of revolution of Io, a moon of Jupiter, as Jupiter revolved around the sun. The angle through which Jupiter moves during a 90ºmovement of the Earth was calculated. Section
13 Roemer s Method, cont. The periods of revolution were longer when the Earth was receding from Jupiter. Shorter when the Earth was approaching Using Roemer s data, Huygens estimated the lower limit of the speed of light to be m/s. This was important because it demonstrated that light has a finite speed as well as giving an estimate of that speed. Section
14 Measurements of the Speed of Light Fizeau s Method This was the first successful method for measuring the speed of light by means of a purely terrestrial technique. It was developed in 1849 by Armand Fizeau. He used a rotating toothed wheel. The distance between the wheel (considered to be the source) and a mirror was known. Section
15 Fizeau s Method, cont. d is the distance between the wheel and the mirror. Dt is the time for one round trip. Then c = 2d / Dt Fizeau found a value of c = m/s. Section
16 Example 35.1 Assume Fizeau s wheel has 360 teeth and rotates at 27.5 rev/s when a pulse of light passing through opening A in Figure 35.2 is blocked by tooth B on its return. If the distance to the mirror is 7500 m, what is the speed of light? Solution: 1 rev D Dt 27.5 rev/s 2d 2(7500 m) c Dt s s m/s 16
17 The Ray Approximation in Ray Optics Ray optics (sometimes called geometric optics) involves the study of the propagation of light. It uses the assumption that light travels in a straight-line path in a uniform medium and changes its direction when it meets the surface of a different medium or if the optical properties of the medium are nonuniform. The ray approximation is used to represent beams of light. Section
18 Ray Approximation The rays are straight lines perpendicular to the wave fronts. With the ray approximation, we assume that a wave moving through a medium travels in a straight line in the direction of its rays. Section
19 Ray Approximation, cont. If a wave meets a barrier, with l << d, the wave emerging from the opening continues to move in a straight line. d is the diameter of the opening. There may be some small edge effects. This approximation is good for the study of mirrors, lenses, prisms, etc. Section
20 Ray Approximation, final Other effects occur for openings of other sizes. Section
21 If you can't see the image above, please install Shockwave Flash Player. If this active figure can t auto-play, please click right button, then click play. 35.4: The Ray Approximation In the ray approximation, we assume that a wave moving through a medium travels in a straight line in the direction of its rays. If the wave meets a barrier with an opening whose diameter is much larger than the wavelength, the wave continues to move in a straight line. When the diameter of the opening is approximately equal to the wavelength, the waves spread out from the opening in all directions. 21
22 Reflection of Light A ray of light, the incident ray, travels in a medium. When it encounters a boundary with a second medium, part of the incident ray is reflected back into the first medium. This means it is directed backward into the first medium. For light waves traveling in three-dimensional space, the reflected light can be in directions different from the direction of the incident rays. Section
23 Specular Reflection Specular reflection is reflection from a smooth surface. The reflected rays are parallel to each other. All reflection in this text is assumed to be specular. Section
24 Diffuse Reflection Diffuse reflection is reflection from a rough surface. The reflected rays travel in a variety of directions. A surface behaves as a smooth surface as long as the surface variations are much smaller than the wavelength of the light. Section
25 Law of Reflection The normal is a line perpendicular to the surface. It is at the point where the incident ray strikes the surface. The incident ray makes an angle of 1 with the normal. The reflected ray makes an angle of 1 with the normal. Section
26 If you can't see the image above, please install Shockwave Flash Player. If this active figure can t auto-play, please click right button, then click play. 35.6: Reflection According to the law of reflection, the angle of reflection equals the angle of incidence. Explore this law using the animation where you have a beam from a laser pen incident at an angle of your choice on a flat, smooth surface. 26
27 Law of Reflection, cont. The angle of reflection is equal to the angle of incidence. 1 = 1 (35.2) This relationship is called the Law of Reflection. The incident ray, the reflected ray and the normal are all in the same plane. Because this situation happens often, an analysis model, wave under reflection, is identified. Section
28 Law of Reflection, cont. Notation note: The subscript 1 refers to parameters for the light in the first medium. If light travels in another medium, the subscript 2 will be associated with the new medium. Since reflection of waves is a common phenomena, we identify an analysis model for this situation, the wave under reflection analysis model. Section
29 Multiple Reflections The incident ray strikes the first mirror. The reflected ray is directed toward the second mirror. There is a second reflection from the second mirror. Apply the Law of Reflection and some geometry to determine information about the rays. Section
30 Quick Quiz 35.1 In the movies, you sometimes see an actor looking in a mirror and you can see his face in the mirror. During the filming of such a scene, what does the actor see in the mirror? (a) his face (b) your face (c) the director s face (d) the movie camera (e) impossible to determine. Answer: (d) 30
31 Example 35.2 Two mirrors make an angle of l20 o with each other as illustrated in Figure 35.7a. A ray is incident on mirror M l at an angle of 65 o to the normal. Find the direction of the ray after it is reflected from mirror M 2. M Solution: M ' 55 M
32 Retroreflection Assume the angle between two mirrors is 90 o. The reflected beam returns to the source parallel to its original path. This phenomenon is called retroreflection. Applications include: Measuring the distance to the Moon Automobile taillights Traffic signs Section
33 Retroreflection 33
34 Retroreflection 34
35 Retroreflection 35
36 Refraction of Light When a ray of light traveling through a transparent medium encounters a boundary leading into another transparent medium, part of the energy is reflected and part enters the second medium. The ray that enters the second medium changes its direction of propagation at the boundary. This bending of the ray is called refraction. Section
37 Refraction of Light, cont. The incident ray, the reflected ray, the refracted ray, and the normal all lie on the same plane. The angle of refraction depends upon the material and the angle of incidence. v 1 is the speed of the light in the first medium and v 2 is its speed in the second. sin sin v v (35.3) Section
38 Refraction of Light, final The path of the light through the refracting surface is reversible. For example, a ray travels from A to B. If the ray originated at B, it would follow the line AB to reach point A. Section
39 Following the Reflected and Refracted Rays Ray is the incident ray. Ray is the reflected ray. Ray is refracted into the lucite. Ray is internally reflected in the lucite. Ray is refracted as it enters the air from the lucite. Section
40 If you can't see the image above, please install Shockwave Flash Player. If this active figure can t auto-play, please click right button, then click play : Refraction When a ray of light traveling through a transparent medium encounters a boundary leading into another transparent medium, part of the energy is reflected and part enters the second medium. The ray that enters the second medium is bent at the boundary and is said to be refracted. The angle of refraction depends on the properties of the two media and on the angle of incidence. Explore the properties of refraction with this animation where a beam of light from a laser pointer travels through two media, air and glass. 40
41 Quick Quiz 35.2 If beam is the incoming beam in Active Figure 35.l0b, which of the other four red lines are reflected beams and which are refracted beams? Answer: Beams and are reflected; beams and are refracted. 41
42 Refraction Details, 1 Light may refract into a material where its speed is lower. The angle of refraction is less than the angle of incidence. The ray bends toward the normal. Section
43 Refraction Details, 2 Light may refract into a material where its speed is higher. The angle of refraction is greater than the angle of incidence. The ray bends away from the normal. Section
44 If you can't see the image above, please install Shockwave Flash Player. If this active figure can t auto-play, please click right button, then click play : Reflection Through Three Media When a light beam moves from air into glass, the light slows down on entering the glass and its path is bent toward the normal. When the beam moves from glass into air, the light speeds up on entering the air and its path is bent away from the normal. Explore this animation using different properties of the three media. 44
45 Light in a Medium The light enters from the left. The light may encounter an electron. The electron may absorb the light, oscillate, and reradiate the light. The absorption and radiation cause the average speed of the light moving through the material to decrease. Section
46 A Mechanical Analog of Refraction 46
47 The Index of Refraction The speed of light in any material is less than its speed in vacuum. The index of refraction, n, of a medium can be defined as n speed of light in a vacuum speed of light in a medium c v (35.4) Section
48 Index of Refraction, cont. For a vacuum, n = 1 We assume n = 1 for air also For other media, n > 1 n is a dimensionless number greater than unity. n is not necessarily an integer. Section
49 Some Indices of Refraction Section
50 Frequency Between Media As light travels from one medium to another, its frequency does not change. Both the wave speed and the wavelength do change. The wavefronts do not pile up, nor are they created or destroyed at the boundary, so f must stay the same. Section
51 Index of Refraction Extended The frequency stays the same as the wave travels from one medium to the other. v = f l f 1 = f 2 but v 1 v 2 so l 1 l 2 The ratio of the indices of refraction of the two media can be expressed as various ratios. c l v n n l v c n n (35.6) Section
52 Index of Refraction Extended The index of refraction is inversely proportional to the wave speed. As the wave speed decreases, the index of refraction increases. The higher the index of refraction, the more it slows downs the light wave speed. Section
53 More About Index of Refraction The previous relationship can be simplified to compare wavelengths and indices: l 1 n 1 = l 2 n 2 In air, n 1 = 1 and the index of refraction of the material can be defined in terms of the wavelengths. n l lin vacuum l n l in a medium (35.7) Section
54 Snell s Law of Refraction n 1 sin 1 = n 2 sin 2 (35.8) 1 is the angle of incidence 2 is the angle of refraction The experimental discovery of this relationship is usually credited to Willebrord Snell and is therefore known as Snell s law of refraction. Refraction is a commonplace occurrence, so identify an analysis model as a wave under refraction. Section
55 Quick Quiz 35.3 Light passes from a material with index of refraction 1.3 into one with index of refraction l.2. Compared to the incident ray, what happens to the refracted ray? (a) It bends toward the normal. (b) It is undeflected. (c) It bends away from the normal. Answer: (c) 55
56 Example 35.3 A light ray of wavelength 589 nm traveling through air is incident on a smooth, flat slab of crown glass at an angle of 30.0 o to the normal. (A) Find the angle of refraction. Solution: sin 2 n sin n2 1 n 1 2 sin sin 1 n sin sin 30 =
57 Example 35.3 (B) Find the speed of this light once it enters the glass. Solution: (C) What is the wavelength of this light in the glass? Solution: v c n m/s v m/s l 589 nm ln 388 nm n
58 Example 35.4 A light beam passes from medium l to medium 2, with the latter medium being a thick slab of material whose index of refraction is n 2 (Fig.35.l5). Show that the beam emerging into medium l from the other side is parallel to the incident beam. Solution: n (1) sin sin n2 n (2) sin sin n1 n 2 n 1 sin 3 sin 1 sin 1 n1 n2 58
59 Prism A ray of single-wavelength light incident on the prism will emerge at angle from its original direction of travel. is called the angle of deviation. F is the apex angle. Section
60 Example 35.5 Although we do not prove it here, the minimum angle of deviation min for a prism occurs when the angle of incidence 1 is such that the refracted ray inside the prism makes the same angle with the normal to the two prism faces as shown in Figure 35.l7. Obtain an expression for the index of refraction of the prism material in terms of the minimum angle of deviation and the apex angle F. 60
61 Example 35.5 Solution: F min F min sin 1 (1.00) sin 1 nsin 2 n sin n F sin 2 sin( F / 2) min 2 (35.9) 61
62 Huygens s Principle Huygens assumed that light is a form of wave motion rather than a stream of particles. Huygens s Principle is a geometric construction for determining the position of a new wave at some point based on the knowledge of the wave front that preceded it. All points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate outward through a medium with speeds characteristic of waves in that medium. After some time has passed, the new position of the wave front is the surface tangent to the wavelets. Section
63 Huygens s Construction for a Plane Wave At t = 0, the wave front is indicated by the plane AA. The points are representative sources for the wavelets. After the wavelets have moved a distance cdt, a new plane BB can be drawn tangent to the wavefronts. Section
64 Huygens s Construction for a Spherical Wave The inner arc represents part of the spherical wave. The points are representative points where wavelets are propagated. The new wavefront is tangent at each point to the wavelet. Section
65 Huygens s Principle and the Law of Reflection The law of reflection can be derived from Huygens s principle. AB is a plane wave front of incident light. The wave at A sends out a wavelet centered on A toward D. The wave at B sends out a wavelet centered on B toward C. AD = BC = cdt Section
66 Huygens s Principle and the Law of Reflection, cont. Triangle ABC is congruent to triangle ADC. cos = BC / AC cos = AD / AC Therefore, cos = cos and This gives 1 = 1 This is the law of reflection. Section
67 Huygens s Principle and the Law of Refraction Ray 1 strikes the surface and at a time interval Dt later, ray 2 strikes the surface. During this time interval, the wave at A sends out a wavelet, centered at A, toward D. Section
68 Huygens s Principle and the Law of Refraction, cont. The wave at B sends out a wavelet, centered at B, toward C. The two wavelets travel in different media, therefore their radii are different. From triangles ABC and ADC, we find sin and sin 2 1 BC v1dt AC AC AD v2dt AC AC Section
69 Huygens s Principle and the Law of Refraction, final The preceding equation can be simplified to But sin sin sin sin v v c n c n n n and so n sin n sin This is Snell s law of refraction. Section
70 Dispersion For a given material, the index of refraction varies with the wavelength of the light passing through the material. This dependence of n on l is called dispersion. Snell s law indicates light of different wavelengths is bent at different angles when incident on a refracting material. Section
71 Variation of Index of Refraction with Wavelength The index of refraction for a material generally decreases with increasing wavelength. Violet light bends more than red light when passing into a refracting material. Section
72 Refraction in a Prism Since all the colors have different angles of deviation, white light will spread out into a spectrum. Violet deviates the most. Red deviates the least. The remaining colors are in between. Section
73 The Rainbow A ray of light strikes a drop of water in the atmosphere. It undergoes both reflection and refraction. First refraction at the front of the drop Violet light will deviate the most. Red light will deviate the least. Section
74 The Rainbow, cont. At the back surface the light is reflected. It is refracted again as it returns to the front surface and moves into the air. The rays leave the drop at various angles. The angle between the white light and the most intense violet ray is 40º. The angle between the white light and the most intense red ray is 42º. Section
75 If you can't see the image above, please install Shockwave Flash Player. If this active figure can t auto-play, please click right button, then click play : A Raindrop This animation illustrates how a rainbow is formed using the example of a single raindrop. When a ray of sunlight (which is white light) passing overhead strikes a drop of water in the atmosphere, the ray is first refracted at the front surface of the drop, then reflected at the back surface of the drop, and again refracted as it moves back into the air. A small angular difference between the returning rays causes us to see a colored bow. 75
76 Observing the Rainbow If a raindrop high in the sky is observed, the red ray is seen. A drop lower in the sky would direct violet light to the observer. The other colors of the spectra lie in between the red and the violet. Section
77 Double Rainbow The secondary rainbow is fainter than the primary. The colors are reversed. The secondary rainbow arises from light that makes two reflections from the interior surface before exiting the raindrop. Higher-order rainbows are possible, but their intensity is low. Section
78 Quick Quiz 35.4 In photography, lenses in a camera use refraction to form an image on a light-sensitive surface. Ideally, you want all the colors in the light from the object being photographed to be refracted by the same amount. Of the materials shown in Figure 35.21, which would you choose for a single-element camera lens? (a) crown glass (b) acrylic (c) fused quartz (d) impossible to determine Answer: (c) 78
79 Total Internal Reflection A phenomenon called total internal reflection can occur when light is directed from a medium having a given index of refraction toward one having a lower index of refraction. Section
80 Possible Beam Directions Possible directions of the beam are indicated by rays numbered 1 through 5. The refracted rays are bent away from the normal since n 1 > n 2. Section
81 Critical Angle There is a particular angle of incidence that will result in an angle of refraction of 90º. This angle of incidence is called the critical angle, C. n 2 sin C (for n1 n2) n1 (35.10) Section
82 If you can't see the image above, please install Shockwave Flash Player. If this active figure can t auto-play, please click right button, then click play : Total Internal Reflection When light is directed from a medium having a given index of refraction toward one having a lower index of refraction, an effect called total internal reflection can occur. Explore this animation and determine the relationships between the indexes of refraction of the two media and where the critical angle occurs for the light beam projected from the laser pointer. 82
83 Critical Angle, cont. For angles of incidence greater than the critical angle, the beam is entirely reflected at the boundary. This ray obeys the law of reflection at the boundary. Total internal reflection occurs only when light is directed from a medium of a given index of refraction toward a medium of lower index of refraction. Section
84 Quick Quiz 35.5 In Figure 35.27, five light rays enter a glass prism from the left. (i) How many of these rays undergo total internal ref lection at the slanted surface of the prism? (a) one (b) two (c) three (d) four (e) five (ii) Suppose the prism in Figure can be rotated in the plane of the paper. For all five rays to experience total internal ref lection from the slanted surface, should the prism be rotated (a) clockwise or (b) counterclockwise? Answer: (i) (b) (ii) (b) 84
85 Example 35.6 Find the critical angle for an air water boundary. (Assume the index of refraction of water is l.33.) Solution: sin c n c n
86 Fiber Optics An application of internal reflection Plastic or glass rods are used to pipe light from one place to another. Applications include: Medical examination of internal organs Telecommunications Section
87 Construction of an Optical Fiber The transparent core is surrounded by cladding. The cladding has a lower n than the core. This allows the light in the core to experience total internal reflection. The combination is surrounded by the jacket. Section
88 If you can't see the image above, please install Shockwave Flash Player. If this active figure can t auto-play, please click right button, then click play : A Single-mode, Stepped-index Optical Fiber The upper figure shows a cross-sectional view from the side of a single-mode, stepped-index optical fiber. Light travels through the core by means of a series of total internal reflections. Thus, the core must have a higher index of refraction than the cladding. The intensity curve of a pulse of light entering the core is nearly rectangular. However, as the pulse travels down the core, it broadens and spreads out. The goal of fiber design is to minimize this spreading. Here you can modify three parameters and observe the effect on the spread of the output pulse. 88
89 Fiber Optics, cont. A flexible light pipe is called an optical fiber. A bundle of parallel fibers (shown) can be used to construct an optical transmission line. Section
90 Laser 90
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