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Optics Refraction Dispersion Image formation from Mirrors Lana Sheridan De Anza College June 12, 2017 Last time • speed of light • ray optics • reflection Overview • refraction • dispersion • image formation from mirrors Refraction When light rays pass from one medium into another, they are often observed to bend. 1 Image from Wikipedia, by Zátonyi Sándor. zon due to the curvature of the Earth. Explain this phenomenon. Refraction © Cengage Learning/Charles D. Winters 13. Figure CQ35.13 shows pencil partially immersed a When light rays pass from onea medium into another, they areinoften cup of water. Why does the pencil appear to be bent? observed to bend. 7 ss us re to Figure CQ35.13 Courtesy of Henry Leap and Jim Lehman Refraction a " ! $ % # Refraction action s on strings cident wave s 16.15 and th another nto the secransparent nt medium t enters the mitted wave nal nature es its direcd. The inci- All rays and the normal lie in the same plane, and the refracted ray is bent toward the normal because v2 ! v1. Incident ray Normal Reflected ray A u1 Air Glass u2 u"1 B v1 v2 Refracted ray Refraction Why does this happen? Refraction Why does this happen? Recall that Arago proposed to Fizeau and Foucault that they might measure the speed of light in water as well as air. That is because the answer to the puzzle was already suspected. Refraction Why does this happen? Recall that Arago proposed to Fizeau and Foucault that they might measure the speed of light in water as well as air. That is because the answer to the puzzle was already suspected. Foucault did the experiment in water (1850), and Fizeau (1851) went further investigating light moving water. Both found visible light had a slower speed in water than in air. This agreed with the wave model of light. Refraction Why does this happen? Recall that Arago proposed to Fizeau and Foucault that they might measure the speed of light in water as well as air. That is because the answer to the puzzle was already suspected. Foucault did the experiment in water (1850), and Fizeau (1851) went further investigating light moving water. Both found visible light had a slower speed in water than in air. This agreed with the wave model of light. However, Fizeau could not explain the magnitude of the speed change that occurred for light in moving water... Refraction If a wave enters a medium where it moves more slowly, what happens? 1 the frequency cannot change – the source still “updates” the medium about a new wave front every T seconds. Refraction If a wave enters a medium where it moves more slowly, what happens? 1 the frequency cannot change – the source still “updates” the medium about a new wave front every T seconds. 2 the wavelength changes (v = f λ) When the wavefronts slow, they bend. Refraction 1 c 2002 by The McGraw-Hill Concise Encyclopedia of Physics. McGraw-Hill Companies, Inc. Refraction electrons, and the vertical arrows represent their oscillations. Concrete v1 v2 Grass v2 ! v1 This end slows first; as a result, the barrel turns. 1 Figure Overhead view Serway & Jewett, 9th ed,35.13 page 1066. may encounter a light is absorbed sented by the do an antenna and again absorbed. in terms of quan light passing fro from one atom t place cause the 2 3 108 m/s. On and the light tra A mechanica of the rolling ba on the concrete the barrel to piv Index of Refr In general, the Refraction The same thing happens when waves move into shallower water on a beach. 1 http://astarmathsandphysics.com/o-level-physics-notes/297-refraction-ofwaves.html Refractive Index Light at a particular frequency moves at different speeds in different media. Light interacts with the charges that constitute the medium, and the net effect is a wave that moves more slowly. Refractive index of a medium, n n= c v where v = ω k is the phase velocity of light with angular frequency ω in that medium. Materials with a higher n are said to be more optically dense. Refractive Index The index of refraction also relates to the ratio of the wavelength of light at a particular frequency in the medium, λn , to that same light’s wavelength in a vacuum, λ. n= fλ c = v f λn so, n= λ λn Refractive Index The index of refraction also relates to the ratio of the wavelength of light at a particular frequency in the medium, λn , to that same light’s wavelength in a vacuum, λ. n= fλ c = v f λn so, n= λ λn If the light passes from one medium with refractive index n1 to another with index n2 : n1 λ2 = n2 λ1 Snell’s Law: Angle relationship in Refraction How do the refractive indices relate to the angles of incidence and refraction? Let h be the hypotenuse of the two triangles. λ1 λ2 h= = sin θ1 sin θ2 1 Figure by Richard Fitzpatrick, http://farside.ph.utexas.edu/teaching/ Snell’s Law λ1 λ2 = sin θ1 sin θ2 Rearranging, λ2 n1 sin θ2 = = sin θ1 λ1 n2 And rearranging again gives Snell’s Law: n1 sin θ1 = n2 sin θ2 1 Willebrord Snell discovered this law experimentally. Snell’s Law and Refraction If n1 < n2 the bends towards the normal, if n1 > n2 the ray the Principles ofray Ray Optics bends away from the normal. Normal m g the bent Normal u1 u1 u2 ! u1 v1 v2 Air Glass u2 v2 ! v1 Glass Air u2 " u1 v1 v2 v2 " v1 u2 Whe mov into spee ente and ben the ln 5 n 5 1.52 5 388 nm Example 35.4: Light Passing Through a Slab er speed of the light found in part (B). In part (C), we see the air.A light beam passes from medium 1 (refractive index n1 ) to AM medium 2, with the latter medium being a thick slab of material whose index of refraction is n2 . Show that the beam emerging into medium 1 from the other side is parallel to the incident beam. latter medium being 35.15). Show that the to the incident beam. 35.15 (Example 35.4) The ine drawn parallel to the ray out the bottom of the slab ts the path the light would e the slab not there. u1 n1 n2 g t u2 a u3 d m in which apply underonrefraction Also,wefind howthe d wave depends t (and model. θ1 , θ2 ). Measuring Refractive Index way to measure refractive indices of solids is and theAnother Principlesstandard of Ray Optics using a prism: Figure 35.16 The apex angle ! is the angle between the sides of the prism through which the light enters and leaves. ! d A prism efracts a single-wavelength ight ray through an angle of deviation d. is called the angle of deviation. e 35.4,δthe light passes through a slab of material with parallel sides. 1 s when light strikes a prism with sides as shown in Figure See example 35.5, on page nonparallel 1070. Fermat’s Principle Also called the Principle of Least Time. Fermat’s Principle The path a light ray follows between at starting point S and an end point T is the path between S and T that is travelled in the least time. This principle correctly predicts the reflection and refraction equations. This also relates to the calculus of variations, another approach to solving mechanics problems. 1 Do questions 84 and 85 in the textbook. Dispersion We have already said that the speed of light for a given frequency of light is different in different media. However, the speed of light is also different for different frequencies of light in the same medium. This means the refractive index is a function of frequency, n(ω). refraction. sin u 1 Dispersion A plot of refractive index vs. wavelength for some Dividing kinds of the first equat transparent solids: From Equation 35.4, ho n 1.54 Crown glass 1.52 1.50 and Acrylic which is Snell’s law of r 1.48 Fused quartz 1.46 35.7 Dispersio An important propert the index varies with t Figure 35.21 Variation of index Figure 35.21 shows. Th 400 500 600 l (nm) 700 Dispersion All materials exhibit dispersion, to varying degrees, except the vacuum. This has important effects: rainbow creation! Dispersion can also have detrimental effects. In camera lenses, dispersion causes a blurring of the image. 1 Image from http://imaging.nikon.com Prism Dispersion of light both reflects and at the surface between glass as shown in Figure If the refractive index lass is ng , find the angle ence u1 in the air that esult in the reflected ray refracted ray being perlar to each other. θ1 Andy Ryan/Stone/Get am hits a target at the top of the building. al is to find the height of the target above sea ) Draw a diagram of the situation, identifying triangles that are important in finding the . (b) Find the angle of incidence of the beam the water–air interface. (c) Find the angle of on. (d) What angle does the refracted beam ith the horizontal? (e) Find the height of the bove sea level. Figure P35.38 39. The index of refraction for violet light in silica flint M glass is 1.66, and that for red light is 1.62. What is the angular spread of visible light passing through a prism of apex angle 60.0° if the angle of incidence is 50.0°? See Figure P35.39. Deviation of red light ng Figure P35.35 Huygens’s Principle Dispersion ex of refraction for red light in water is 1.331 t for blue light is 1.340. If a ray of white light he water at an angle of incidence of 83.0°, what underwater angles of refraction for the (a) red blue components of the light? beam containing red and violet wavelengths is on a slab of quartz at an angle of incidence The index of refraction of quartz is 1.455 at 1 its index of refraction is 1.468 (red light), and Left, Wikipedia by Spigget; Visible light Angular spread Figure P35.39 R O Y G B V Screen Problems 39 and 40. 40. The index of refraction for violet light in silica flint glass S is n V , and that for red light is n R . What is the angular spread of visible light passing through a prism of apex angle F if the angle of incidence is u? See Figure P35.39. Section 35.8 Total Internal Reflection right, Serway & Jewett, page 1083. ow a rainbow is formed, consider FigDispersion in Rainbows wave under reflection and wave under white light) passing overhead strikes a cted and reflected as follows. It is first ith the violet light deviating the most violet light refracts e of the drop, theThe light is reflected and through larger angles undergoes refraction as it moves from than the red light. h that the angle between the incident iolet ray is 40° and the angle between Sunlight e returning red ray is 42°. This small ys causes us to see a colored bow. bow as shown in Figure 35.24. If a rainmost intense red light returning from eviated the least; the most intense vioecause it is deviated 40! 42! the most. Hence, R drop. Similarly, a drop lower in theVsky he observer and appears violet to the is drop passes below the observer’s eye V m other colors of the spectrum reaches R hese two extreme positions. nbow. The secondary rainbow is fainter Figure 35.23 Path of sunlight 35.7 Dispersion 1073 The highest-intensity light traveling from higher raindrops towardPitfall the eyes of the observer Prevention 35.5 is red, whereas the most intense A Rainbow of Many Light Rays light from lower drops is violet. Pictorial representations such as Figure 35.23 are subject to misinterpretation. The figure shows one White ray of light entering the raindrop 42! and undergoing reflection and 40! refraction, exiting the raindrop White in a range of 40° to 42° from the 42! illustration entering ray. This 40! R be interpreted incorrectly as might V meaning that all light entering the raindrop exits V R in this small range of angles. In reality, light exits the raindrop over a much larger range of angles, from 0° to 42°. A careful analysis of the reflection and refraction from the spherical rainFigure 35.24 The formation of a Dispersion Question Figure 35.20 Huygens’s construction for proving Snell’s law of lenses in a camera use refraction. is the w v1 Dt, wh From Quick Quiz 35.41 In photography, refraction to form an image on a light-sensitive surface. Ideally, you want all the colors in the light from the object being photographed Dividing to be refracted by the same amount. Of the materials shown, which would you choose for a single-element camera lens? From Eq n 1.54 (A) crown glass 1.52 (B) acrylic 1.50 Crown glass Acrylic (C) fused quartz (D) impossible to determine Fused quartz 400 Serway & Jewett, page1074. which is 1.48 1.46 1 and 500 Figure 35.21 600 l (nm) 700 Variation of index 35.7 An imp the inde Figure 3 Dispersion Question Figure 35.20 Huygens’s construction for proving Snell’s law of lenses in a camera use refraction. is the w v1 Dt, wh From Quick Quiz 35.41 In photography, refraction to form an image on a light-sensitive surface. Ideally, you want all the colors in the light from the object being photographed Dividing to be refracted by the same amount. Of the materials shown, which would you choose for a single-element camera lens? From Eq n 1.54 (B) acrylic (C) fused quartz Crown glass 1.52 (A) crown glass 1.50 ← (D) impossible to determine Acrylic Fused quartz 400 Serway & Jewett, page1074. which is 1.48 1.46 1 and 500 Figure 35.21 600 l (nm) 700 Variation of index 35.7 An imp the inde Figure 3 Question A water resistant flashlight is switched on under water in a pool. The ray from the flashlight strikes the top surface of the water, and makes an angle of 30◦ with the water’s surface. What is the angle of refraction? The refractive index of water is n = 1.333. Question A water resistant flashlight is switched on under water in a pool. The ray from the flashlight strikes the top surface of the water, and makes an angle of 30◦ with the water’s surface. What is the angle of refraction? The refractive index of water is n = 1.333. Snell’s law: n1 sin θ1 = n2 sin θ2 . Question A water resistant flashlight is switched on under water in a pool. The ray from the flashlight strikes the top surface of the water, and makes an angle of 30◦ with the water’s surface. What is the angle of refraction? The refractive index of water is n = 1.333. Snell’s law: n1 sin θ1 = n2 sin θ2 . What’s happening? Total Internal Reflection When a light ray travels from a medium with a higher refractive index to one with a lower refractive index, and it strikes the interface at a sufficiently large incident angle, there is no valid solution for the refracted ray. In fact, in this case we don’t see one! All of the light is reflected at the boundary. This is called total internal reflection. of incidence u1 increases, The angle of incidence producing refraction u2 increases an angle of refraction equal to 90! ! (ray 4). The dashed line is the critical angle uc . For angles The critical angle,greater θc , isthan the umaximum angle of incidence such t no energy actually c , all the energy of the that there could be a refracted ray. The n this direction. incident light is reflected. ray would just skim along Total Internal Reflection the surface between the media. mal Normal n1 " n2 n1 " n2 2 u2 3 4 5 For even larger angles of In total this internal case, the incidence, reflection occurs (ray 5). n2 n1 uc angle of refraction θ2 = 90◦ . (D) four 7, five light rays enter a glass prism from the left. dergo total internal reflection at the slanted surwo (c) three (d) four (e) five (ii) Suppose the otated in the plane of the paper. For all five rays lection from the slanted surface, should the or (b) counterclockwise? 2 Serway & Jewett, page1075. Courtesy of Henry Leap and Jim Lehman . 1, which is a meaningless result because the sine Total han unity. Internal Reflection (TIR) Question ernal reflection is small when n 1 is considerably e critical angle for a diamond in air is 24°. Any ray hes the surface at an angle greater than 24° is comQuick Quiz 35.52 In the picture, five light rays enter a glass stal. This property, combined with proper faceting, fromare thecutleft. (i) light Howismany of these rays undergo total angles ofprism the facets so that “caught” internal reflection atmultiple the slanted surface of the prism? le internal reflections. These reflections h the medium, and substantial dispersion of colors s through the top surface of the crystal, the rays ave been fairly widely separated from one another. ndex(A) of refraction one and can be made to sparkle very jewel is immersed in corn syrup, the difference in (B) two or the corn syrup is small and the critical angle is escape (C)sooner; three as a result, the sparkle completely not lose all its sparkle when placed in corn syrup. Figure 35.27 (Quick Quiz 35.5) Five nonparallel light rays enter a glass prism from the left. (D) four 7, five light rays enter a glass prism from the left. dergo total internal reflection at the slanted surwo (c) three (d) four (e) five (ii) Suppose the otated in the plane of the paper. For all five rays lection from the slanted surface, should the or (b) counterclockwise? 2 Serway & Jewett, page1075. Courtesy of Henry Leap and Jim Lehman . 1, which is a meaningless result because the sine Total han unity. Internal Reflection (TIR) Question ernal reflection is small when n 1 is considerably e critical angle for a diamond in air is 24°. Any ray hes the surface at an angle greater than 24° is comQuick Quiz 35.52 In the picture, five light rays enter a glass stal. This property, combined with proper faceting, fromare thecutleft. (i) light Howismany of these rays undergo total angles ofprism the facets so that “caught” internal reflection atmultiple the slanted surface of the prism? le internal reflections. These reflections h the medium, and substantial dispersion of colors s through the top surface of the crystal, the rays ave been fairly widely separated from one another. ndex(A) of refraction one and can be made to sparkle very jewel is immersed in corn syrup, the difference in (B) two ←is small and the critical angle is or the corn syrup escape (C)sooner; three as a result, the sparkle completely not lose all its sparkle when placed in corn syrup. Figure 35.27 (Quick Quiz 35.5) Five nonparallel light rays enter a glass prism from the left. 7, five light rays enter a glass prism from the left. dergo total internal reflection at the slanted surwo (c) three (d) four (e) five (ii) Suppose the otated in the plane of the paper. For all five rays lection from the slanted surface, should the or (b) counterclockwise? 3 Serway & Jewett, page1075. Courtesy of Henry Leap and Jim Lehman (TIR) Question . 1,Total which is Internal a meaninglessReflection result because the sine han unity. ernal reflection is small when n 1 is considerably e criticalQuick angle for a diamond Any ray Quiz 35.53 inInair theis 24°. picture, five light rays enter a glass hes the surface at an angle greater than 24° is comprism from the left. (ii) Suppose the prism can be rotated in the stal. This property, combined with proper faceting, of the For light all five rays to experience total internal angles ofplane the facets arepaper. cut so that is “caught” reflection fromThese the slanted the prism should be rotated le internal reflections. multiplesurface, reflections h the medium, and substantial dispersion of colors s through the top surface of the crystal, the rays ave been fairly widely separated from one another. ndex of refraction and can be made to sparkle very jewel is immersed in corn syrup, the difference in (A) clockwise or the corn syrup is small and the critical angle is escape as a result, the sparkle completely (B)sooner; counterclockwise not lose all its sparkle when placed in corn syrup. Figure 35.27 (Quick Quiz 35.5) Five nonparallel light rays enter a glass prism from the left. 7, five light rays enter a glass prism from the left. dergo total internal reflection at the slanted surwo (c) three (d) four (e) five (ii) Suppose the otated in the plane of the paper. For all five rays lection from the slanted surface, should the or (b) counterclockwise? 3 Serway & Jewett, page1075. Courtesy of Henry Leap and Jim Lehman (TIR) Question . 1,Total which is Internal a meaninglessReflection result because the sine han unity. ernal reflection is small when n 1 is considerably e criticalQuick angle for a diamond Any ray Quiz 35.53 inInair theis 24°. picture, five light rays enter a glass hes the surface at an angle greater than 24° is comprism from the left. (ii) Suppose the prism can be rotated in the stal. This property, combined with proper faceting, of the For light all five rays to experience total internal angles ofplane the facets arepaper. cut so that is “caught” reflection fromThese the slanted the prism should be rotated le internal reflections. multiplesurface, reflections h the medium, and substantial dispersion of colors s through the top surface of the crystal, the rays ave been fairly widely separated from one another. ndex of refraction and can be made to sparkle very jewel is immersed in corn syrup, the difference in (A) clockwise or the corn syrup is small and the critical angle is escape as a result, the← sparkle completely (B)sooner; counterclockwise not lose all its sparkle when placed in corn syrup. Figure 35.27 (Quick Quiz 35.5) Five nonparallel light rays enter a glass prism from the left. a ma be su Application of TIR: Optical Fibers a cut is les 1076 Chapter 35 The of Light and the Principle GlassNature or reflec plastic core of inc core if thin fibers are used rathe An cal fiber. If a bundle oftwo pare line, images can be transfer usefu use su Prize in Physics was awarded out m transmit light signals over l Jacket coaxi has led to the development Cladding volum A practical optical fiberFig Figure 35.30 The construction Figure 35.29 Light travels in a material thattravels hasmore i can b ina lower Optical fibers are mainly used for telecommunications: a curved transparent rod by mul- of an optical fiber. Lightmuch the core, which is surrounded ing th be surrounded bybyaaone plastic j information be carried by an optical fiber than an electrical tiple internalcan reflections. cladding and a protective jacket. other in a given amount of time. a cutaway view of this constr trans a curved transparent rod by multiple internal reflections. is less than that of the core Optical Glass or fibers are also used in medicine.reflection if it arrives at the plastic core Ray Optics and Image Formation Simple geometric ray optics can be used to understand how images are formed by simple optical devices: mirrors and lenses. 1 Wikipedia user Heptagon. Images Formed by Flat Mirrors When we see an object “in the mirror”, we are not actually seeing something that is behind the mirror. We are seeing an image of an object that is placed in front of the mirror. The image seems to be the same distance behind the mirror as the object is in front of it. The image seems to be the same size as the object. This is not true for all optical devices, but we can work out things about how the image will form for many different optical devices. Image Formation Terminology object distance, p The perpendicular (shortest) distance from the object to the device. image distance, q The perpendicular (shortest) distance from where the image appears to be to the device. (lateral) magnification, M The factor by which the image size exceeds the object’s size M= image height object height For mirrors and lenses M=− q p Image Formation Terminology real image An image that can be displayed on a screen formed when the light rays pass through and diverge from the image point. virtual image An image that cannot be displayed on a screen, but can be seen “in the device” because the light rays appear to diverge from the image point. The image in a flat mirror is virtual. Image Formation Terminology upright image An image that appears to be right-side-up with respect to the object. inverted image An image that appears to be upside-down with respect to the object. The image in a flat mirror is upright. focal length, f The distance from the optical device to where a set parallel rays striking the device head on (perpendicularly) will be focused. For a flat mirror, the f is infinite. path perpen| p | ! | q | and h ! h". Ray Diagram for a Flat Mirror ay follows the law of reflected rays back p Q q P P' 9 behind the object would h' R h u object behind Image Object u , so |p | 5 |q |. irror is as far h equals the have the optical device sketched vertically in cross Ray diagrams Figure 36.2 A geometric confollows: section, the object on the left represented by an arrow pointing up. struction that is used to locate the image of an object placed in front The image is also of arepresented flat mirror.by an arrow, but it may be to the left(36.1) or right, pointing up or down depending on how the image is formed. We sketch rays with paths we know to find the image. a spherical d a concave e R, and its ction, and a 6.6a shows a olid, curved port for the flecting suro a point as Henry Leap and Jim Lehman Images formed by Spherical Mirrors: Concave Mirrors Figure 36.7 focus light. Reflection of paral6.6b,Concave wheremirrorslelcanrays from a concave mirror. ght rays that 6 Images formed by Spherical Mirrors: Concave If the rays diverge from Mirrors O at small angles, they all reflect through the same image point I. Center of curvature Mirror Mirror R C C V O I Principal axis a b (a) A concave mirror of radius R. The center of curvature C is located on th oint object placed at O in front of a concave spherical mirror of radius R, wher principal axis farther than R from the mirror surface, forms a real image at I. Assumption: paraxial rays In studying curved mirrors and thin lenses we assume that all rays are paraxial rays. Paraxial rays are rays close to the principle optical axis of our optical device. (Rays that strike close to the middle.) For a spherical lens, rays that strike further from the axis are not focused to the same point (spherical aberration). Mirror Equation and Thin Lens Equation The same equation can help us to find the location and magnification of the image that will be formed by curved mirrors and thin lenses. 1 1 1 = + f p q Spherical Concave Mirrors The focal point is where the detectors are place on satellite dishes and radio telescopes. 1 Parkes telescope in Australia. Spherical Concave Mirrors Focal Length For a spherical concave mirror of radius R f = R 2 Concave Mirrors and the Mirror Equation Simple geometry shows why the mirror equation is true for a concave mirror. ge Formation The real image lies at the location at which the reflected rays cross. h a O C V I u a h' u Principal axis q R p Figure 36.9 The image formed by a spherical concave mirror when the object O lies outside the center of curvature This this geometric construction is used to derive Equation 36.4. We willC.use ray diagram. Summary • refraction • dispersion • total internal reflection • image terminology • images formed by spherical mirrors 4th Collected Homework! due Monday June 19. Homework Serway & Jewett: • Ch 35, onward from page 1077. OQs: 9, 15; CQs: 7, 9, 13; Probs: 7, 13, 27, 31 37, 39, 41, 47, 59, 69, 75, 84, 85, 64 (in this problem the description means that one side of the mirror reflects 10% of the light and transmits the other 90%, the other side reflects 90%) • Carefully read all of Chapter 36.