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Chapter 23: Geometric Optics Review Basic Geometry! 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 Light Rays: Ignore Diffraction and Interference of waves! Diffraction depends on SLIT WIDTH: the smaller the width, relative to wavelength, the more bending and diffraction. We will assume that λ<<d , where d is the diameter of the opening. This approximation is good for the study of mirrors, lenses, prisms, etc. Reflection & Refraction i r n1 sin 1 n2 sin 2 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 35.5 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 θ1 with the normal • The reflected ray makes an angle of θ1’ with the normal 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 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 Law of Reflection i r Why are most materials Opaque? (Opaque – Can’t see through) They absorb light without re-emitting it. Vibrations given by the light to their atoms and molecules are turned into random kinetic energy – they become slightly warmer. Opacity: Mirrors Free electrons in opaque reflective surfaces can vibrate, absorb & re-emit at any frequency. Mirror Mirror Transparency Selective Absorption Glass resonates strongly with UV and infrared, absorbing those frequencies while transmitting visible frequencies. Refraction: Bending Light into Focus Refraction: Bending of Light Transmitted through Materials Light Bends because it Slows Down. Atoms are Optical Tuning Forks Light slows down as it travels through glass because it takes time to be absorbed and re-emitted. Light Slows Down in Materials Light Bends Toward the Normal when going from a medium of lower refractive index to one that has a higher refractive index and visa versa. lower n higher n Index of Refraction c n v n 1 Vacuum: 1 Water: 1.33 Glass: 1.46 Diamond: 2.4 The Index of Refraction • Refraction: Light Bends in Transmission • 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 • For a vacuum, n = 1 – We assume n = 1 for air speed of light in a vacuum c λ n also speed of light in a medium v λn • For other media, n > 1 λ λ in vacuum • n is a dimensionless number greater than unity, not n λn λ in a medium necessarily an integer Some Indices of Refraction Frequency Doesn’t Change! • 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 created or destroyed at the boundary, so ƒ must stay the same Snell’s Law of Refraction Angles are always measured from the normal. n1 sin 1 n2 sin 2 Snell’s Law – Example Light is refracted into a crown glass slab. n1 = 1.00 and n2 = 1.52 If θ1 = 30.0o, θ2 = ? θ2 = sin-1(n1 / n2) sin θ1 = 19.2o The ray bends toward the normal, as expected because n2 > n1 Snell’s Law of Refraction In general: n1 sin 1 n2 sin 2 n11 n22 n1 2 1 n2 If n2 n1 , then 1 2 measured from the normal! Prelab Emerging Beam is Parallel to Incident Beam but offset distance d, called the Lateral Shift and is the subject of this week’s lab! Fig. 35-15, p. 989 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 35.5 Beam & Refraction 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 > n2 Total Internal Reflection 2 90 n1 sin 1 n2 sin 2 n2 sin C n1 The Critical Angle Critical Angle • There is a particular angle of incidence that will result in an angle of refraction of 90° – This angle of incidence is called the critical angle, θC n2 sin θC (for n1 n2 ) n1 • An application of internal reflection • Plastic or glass rods are used to “pipe” light from one place to another • Applications include: – medical use of fiber optic cables for diagnosis and correction of medical problems – Telecommunications • A flexible light pipe is called an optical fiber • A bundle of parallel fibers (shown) can be used to construct an optical transmission line Fiber Optics Critical Angle Sample Problem A ray of light, emitted by a laser located beneath the surface of an unknown liquid with air above it, undergoes total internal refection as shown. What is the index of refraction for the liquid? What is its likely identification? If you pass white light through a prism, it separates into its component colors. long wavelengths short wavelengths R.O.Y. G. B.I.V The index of refraction depends on WAVELENGTH. long wavelengths short wavelengths R.O.Y. G. B.I.V The speed and wavelength change but the FREQUENCY does NOT. Fr Frequency depends on the oscillating source! long wavelengths short wavelengths R.O.Y. G. B.I.V Why does Violet Light bend more than Red Light? Violet light slows down more because the atoms in the material are tuned to higher frequencies. As the violet light travels through glass it takes more time to be absorbed and re-emitted. Variation of Index of Refraction with Wavelength speed of light in a vacuum c λ n speed of light in a medium v λn • This dependence of n on λ is called dispersion • 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 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 35.7 Dispersion via Diffraction constructive : d sin m, m 0,1, 2,3 If you pass white light through a prism, it separates into its component colors. long wavelengths short wavelengths R.O.Y. G. B.I.V The index of refraction depends on WAVELENGTH. long wavelengths short wavelengths R.O.Y. G. B.I.V The speed and wavelength change but the FREQUENCY does NOT. Fr Frequency depends on the oscillating source! long wavelengths short wavelengths R.O.Y. G. B.I.V Why does Violet Light bend more than Red Light? Violet light slows down more because the atoms in the material are tuned to higher frequencies. As the violet light travels through glass it takes more time to be absorbed and re-emitted. Variation of Index of Refraction with Wavelength speed of light in a vacuum c λ n speed of light in a medium v λn • This dependence of n on λ is called dispersion • 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 Angle of Deviation • 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 Dispersion Sample Problem The index of refraction for violet light in silica flint glass is 1.66, and that for red light is 1.62. What is the angular dispersion of visible light passing through a prism of apex angle 60.0° if the angle of incidence is 50.0°? red (660 nm) violet (410 nm) Use Snell’s Law twice and some geometry! Angles are always measured from the normal. n1 sin 1 n2 sin 2 Thin Film Interference Interference in Thin Films When reflecting off a medium of greater refractive index, a light wave undergoes a phase shift of ½ a wavelength. Wave 1 undergoes a phase shift of 180 degrees. From Low to High, a phase change of pi! From High to Low, a phase change? NO! Interference in Thin Films • The wavelength of ray 1 in the film is /n • For constructive interference 2t = (m + ½) /n (m = 0, 1, 2 …) This takes into account both the difference in optical path length for the two rays and the 180° phase change • For destructive interference 2t = m/n (m = 0, 1, 2 …) Problem: Thin Films A thin film of gasoline floats on a puddle of water. Sunlight falls almost perpendicularly on the film and reflects into your eyes a yellow hue. Interference in the the thin gasoline film has eliminated blue (469nm in vacuum) from the reflected light. The refractive indices of the blue light in gasoline and water are 1.40 and 1.33 respectively. Determine the minimum nonzero thickness of the film. What color do you see? Thin Film Interference The light reflected from a soap bubble (n = 1.40) appears red ( = 640 nm). What is the minimum thickness (in nm)? a. 124 b.104 c. 114 d.134 e. 234 How are Rainbows Formed? Dispersion: Raindrops Act like Prisms • 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 The Rainbow • 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 40° – The angle between the white light and the most intense red ray is 42° 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 The droplets form a circular arc, with each droplet within the arc dispersing light and reflecting it back towards the observer with the greatest concentration of outgoing rays found at these 40-42 degree angles of deviation. Every droplet within the arc is refracting and dispersing the entire visible light spectrum (ROYGBIV). Rainbow facts • an observer is in a position to see only a single color from any one droplet of water. • your rainbow is slightly different from the rainbow seen by others • your rainbow moves with you • disk within the bow is brighter because of overlapping of multiple refractions (which don’t occur outside the disk) A line drawn from your eye to the top of the rainbow forms a 42degree angle with the imaginary line from the sun through your eye. (If there is a secondary rainbow, it forms an angle of 51degrees). Because these angles determine the position of the rainbow in the sky, it will sink as the sun rises and rise as the sun sinks. At some points, the entire rainbow, not just the bottom half, will be below the horizon where you can't see it. That's why you'll never see a summer rainbow at midday. Double Rainbow • The secondary rainbow is fainter than the primary • 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 • • Halos are caused by the light of the sun or moon passing through a very thin layer of cirruform (ice-crystal) clouds in the upper atmosphere. The ice crystals refract the light of the moon, similar to the way water droplets in the lower atmosphere can refract sunlight to produce a rainbow. Just like a rainbow, strong halos can have bands of color in them, due to slightly different refractive properties of the ice crystals for different colors. Essentially, halos ARE rainbows caused by primary refraction in ice crystals. Some interesting facts about halos: Halos always occur exactly 22 degrees away from the sun or moon. Occasionally, intense halos can be double halos, just as intense rainbows can be doubled. Intense halos can also produce "moondogs" or "sundogs," very bright regions on the halo evenly spaced at 90 degree intervals around the halo. Physics Fun on an Airplane Always sit on the side opposite the sun when traveling north-south!! Why is the Sky Blue? Galileo In the early 17th century, many scientists believed that there was no such thing as the "speed of light"; they thought light could travel any distance in no time at all. Galileo disagreed, and he came up with an experiment to measure light's velocity: he and his assistant each took a shuttered lantern, and they stood on hilltops one mile apart. Galileo flashed his lantern, and the assistant was supposed to open the shutter to his own lantern as soon as he saw Galileo's light. Galileo would then time how long it took before he saw the light from the other hilltop. The problem was that the speed of light is simply too fast to be measured this way; light takes such a short time (about 0.000005 seconds, in fact) to travel one mile that there's no way the interval could have been measured using the tools Galileo had. The Speed of Light? • • • • 186,000 miles per second 300,000 kilometers per second 3 x 108 m/s first successfully determined by Danish astronomer Ole Roemer in 1675: 2.3 x 108 m/s • First Terrestrial Measurement by Fizeau in 1849: 2.9979 x 108 m/s • In 1926, Michelson used a rotating prism to measure the time it took light to make a round trip from Mount Wilson to Mount San Antonio in California, a distance of about 22 miles (36 km). The precise measurements yielded a speed of 186,285 miles per second (299,796 kilometres per second). Huygens’s Principle Construction for a Plane Wave • Huygens assumed that light is a form of wave motion rather than a stream of particles • 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 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 Huygens’s Principle Prove the Laws of Reflection & Refraction Huygens’s Principle and the Law of Reflection • Triangle ABC is congruent to triangle ADC • cos g = BC / AC • cos g’ = AD / AC • Therefore, cos g = cos g’ and g g’ • This gives θ1 = θ1’ • This is the law of reflection Huygens’s Principle and the Law of Refraction • Ray 1 strikes the surface and at a time interval Δt later, ray 2 strikes the surface • During this time interval, the wave at A sends out a wavelet, centered at A, toward D Huygens’s Principle and the Law of Refraction • 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 BC v1t sin θ1 AC AC sin 1 v1 c n1 n2 sin 2 v 2 c n2 n1 AD v 2t and sin θ2 AC AC n1 sin θ1 n2 sin θ2 Why aren’t images of objects produced on the wall without a lens or hole? Why aren’t images of objects produced on the wall without a lens or hole? Law of Reflection 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 The heating effect of a medium such as glass or the Earth’s atmosphere that is transparent to short wavelengths but opaque to longer wavelengths: Short get in, longer are trapped! Lenses and Mirrors Image Formation Real and Virtual Images Real images can be displayed on screens Virtual Images can not be displayed onto screens. Notation for Mirrors and Lenses The object distance is the distance from the object to the mirror or lens: Denoted by p The image distance is the distance from the image to the mirror or lens: Denoted by q The lateral magnification of the mirror or lens is the ratio of the image height to the object height Denoted by M The focal point: f The radius of curvature: R M Image height h' Object height h 1 1 2 1 p q R f FRONT is on the same side as the object and BACK is the other side! The Principal axis goes through the focal point and the center of curvature of the lens or mirror!! Flat Mirrors Make Virtual Images Virtual Image: An image that cannot be projected onto a surface. A virtual image only appears like light rays came from the location of the image, they are not really there. Flat mirrors make Virtual Images. Images Formed by Flat Mirrors Light rays leave the source and are reflected from the mirror Images are always located by extending diverging rays back to a point at which they intersect One ray starts at point P, travels to Q and reflects back on itself Another ray follows the path PR and reflects according to the law of reflection h’ = h for all images Flat mirrors make virtual images Reversals in a Flat Mirror A flat mirror produces an image that has an apparent left-right reversal For example, if you raise your right hand the image you see raises its left hand Properties of the Image Formed by a Flat Mirror – Summary The image is as far behind the mirror as the object is in front |p| = |q| The image is unmagnified The image height is the same as the object height The image is virtual The image is upright h’ = h and M = 1 It has the same orientation as the object There is a front-back reversal in the image Mirror Reflection Convex & Concave “Object” on the left, image on the right. Convex Mirror Convave Mirror Lateral Magnification Image height h' M Object height h Magnification does not always mean bigger, the size can either increase or decrease. M>1: Increase Positive: Upright M<1: Decrease Negative: Inverted Focal Length Shown by Parallel Rays Focal Length& Radius of Curvature When the object is very far away, then p → ∞ and the incoming rays are essentially parallel In this special case, the image point is called the focal point The distance from the mirror to the focal point is called the focal length The focal length is ½ the radius of curvature R = 2f Ray Diagrams:Concave Mirrors Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected through the focal point, F Ray 2 is drawn from the top of the object through the focal point and is reflected parallel to the principal axis Ray 3 is drawn through the center of curvature, C, and is reflected back on itself The intersection of any two of the rays at a point locates the image. The third ray serves as a check of the construction Concave Mirror, p > R The center of curvature is between the object and the concave mirror surface (f >0) The image is real (q>0) The image is inverted (M<0) The image is smaller than the object (absM<1) 1 1 2 1 p q R f Concave Mirror, p < f The object is between the mirror surface and the focal point (p>0) The image is virtual (q<0) The image is upright (M>0) The image is larger than the object (M>1) Ray Diagrams:Convex Mirrors Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected away from the focal point, F Ray 2 is drawn from the top of the object toward the focal point and is reflected parallel to the principal axis Ray 3 is drawn through the center of curvature, C, on the back side of the mirror and is reflected back on itself Convex Mirror The object is in front of a convex mirror (p>0) The focal point distance q is negative (q <0) The image is always virtual and upright (M>0) As the object distance decreases, the virtual image size increases The image is smaller than the object (0<M<1) Sign Conventions: Mirrors 1 1 2 1 p q R f Optics Activity Fun with Mirrors!!! Lenses Image formation is a consequence of light traveling in straight lines The first camera—the pinhole camera— illustrates this fact. Lenses A lens nicely bends the straight-line paths of light. Lenses A converging lens can project an image. Lenses Key features of lenses principal axis focal point line joining the centers of curvature of the two lens surfaces point at which all the light rays come together focal length distance between the center of the lens and either focal point Lens Refraction Converging & Diverging Converging Lens Diverging Lens Lenses Lenses two common types converging (convex) lens thicker at the center than edges converges light diverging (concave) lens thinner at the center than edges diverges light Focal Length:Converging Lens Focal Length:Diverging Lens Converging Thin Lens Shapes These are examples of converging lenses They have positive focal lengths They are thickest in the middle Diverging Thin Lens Shapes These are examples of diverging lenses They have negative focal lengths They are thickest at the edges Signs for Thin Lenses 1 1 2 1 p q R f h' q M h p Compare Signs for Mirrors and Thin Lenses Thin Lenses Ray Diagram for Converging Lens, p > f The image is real (q>0) The image is inverted (M<0) The image is on the back side of the lens (q>0) Ray Diagram for Converging Lens, p < f The image is virtual (q < 0) The image is upright (M>0) The image is larger than the object (M>1) The image is on the front side of the lens (q<0) Ray Diagram for Diverging Lens For a diverging lens, the image is always virtual and upright (M>0) This is regardless of where the object is placed The image is on the front side of the lens (q<0) Optics Activity Fun with Lenses!!!! Combination of Thin Lenses, example