with respect to the normal in each medium, respectively. The question is: How are θ

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1 Prof. Raghuveer Parthasarathy University of Oregon Physis 35 Winter 8 3 R EFRACTION When light travels from one medium to another, it may hange diretion. This phenomenon familiar whenever we see the bent shape of a straw poking out of a glass of water is known as refration. (The light may also hange its intensity at a boundary between media, whih we ll disuss soon.) Refration, like diffration, is a onsequene of the wave nature of light, and our analysis below applies also to waves in water, sound waves, et. 3. Snell s Law The basi setup for issues of refration is shown in Figure 3.: A ray of light rosses the boundary between two media, with indies of refration n and n, making angles θ with respet to the normal in eah medium, respetively. The question is: How are θ related? The answer is ruial to the propagation of light, and the design of lenses and other optial elements. The answer, as we ll derive, is that light obeys Snell s Law: nsinθ nsinθ. There are several ways to derive this result. We ould diretly examine the wave equations for eletromagneti fields and look for solutions onsistent with the presene of a boundary between two media, but this would be both painful and un-illuminating. There are simpler ways of thinking about light propagation; we ll briefly mention one, and then more fully disuss another. Figure 3.. Refration: The path taken by light traveling between two media bends at the interfae; the relation between the angles θ depends on the indies of refration of the two materials, and is given by Snell s Law. 3. Huygen s Priniple Several enturies ago, Huygens ame up with an interesting way of thinking about propagating waves, realizing that: Eah point in a propagating wave an be treated as a soure Notes on Optis R. Parthasarathy 8 Page

2 of spherial waves. (This idea was later fleshed out by Fresnel and others.) There s nothing too shoking here eah point at a wave is, by definition, a point, and points generate spherial waves, as we noted in Setion.. The envelope of the wavelets generated by all the points defines the shape of the wavefront as the wave progresses. There are some oneptual diffiulties inherent in Huygens idea whih we won t disuss. Keep in mind that it s just a piture that we employ to save us from having to solve the full wave equation. Furthermore, it s often painful to use Huygen s priniple to atually alulate anything quantitative. It does, however, provide a means for deriving Snell s Law, as desribed in last quarter s textbook (A.P. Frenh, Vibrations and Waves, p. 7-74). We won t go into the derivation here, but will rather take a different, and more generally useful, approah. 3.3 Fermat s Priniple 3.3. Minimal time paths and Snell s Law Another general priniple desribing wave motion was put forth by Fermat. It is sometimes stated as light travels from one point to another along the path that takes the minimal amount of time. This isn t quite orret we ll fix it in a few paragraphs but it s a good plae to start. We ll also return to justifying Fermat s priniple shortly. First, let s use it to derive Snell s Law. Imagine you re on a beah, and someone in the oean is drowning. You rush out to help, whih requires both running on land and swimming in the water. You an run faster than you an swim. What path should you take? With a bit of thought, you ll realize that a straight line between you and the drowning person isn t the best idea rather, you should redue the length of the swim to minimize the overall time to your target. How muh should you run and how muh should you swim? The same dilemma is enountered by our light beam, traveling from position A in a medium of index of refration n (your position on the beah, in the above analogy) to position B in a medium of n, (the swimmer s position, in the water) in Figure 3.3. The speed of light in medium is v / n and in medium is v / n. Within eah medium the light travels in a straight line itself a onsequene of Fermat s priniple, as you an onvine yourself later. There are many possible paths between A and B, as illustrated in the figure, that we an label based on the position x at whih they ross the interfae. One of these let s all it the one that goes through position x minimizes the total travel time. What is this path? (In other words, what is x?) The travel time in medium, t, is the distane traveled in medium divided by the speed in n medium : t d/ v y + x ; similarly the travel time in medium is Notes on Optis R. Parthasarathy 8 Page 3

3 n t d / v y + ( L x). (See Figure 3.3 for the geometry.) The total travel time is therefore n n t t+ t y + x + y + ( L x). dt To find the minimal time, we determine the x for whih dx ; all this x : dt n x n ( L x) dx dt dx x x n y + x y + ( L x) x ( L x ) n. y + x y + ( L x) (To show that x is a minimum we should also examine the seond derivative of t, but as we ll see later it does not atually matter if x is the site of a minimum or a maximum. Furthermore, we an intuit from the form of tx ( ) that this extremum is, in fat, a minimum.) Note from geometry that x sinθ y + x ( L x ) sinθ y + ( L x ) Therefore the above ondition beomes nsinθ nsinθ. We ve shown that when the above ondition is met, the travel time for light propagation is minimized. This is Snell s Law! (And a valuable tool for saving drowning swimmers.) Figure 3.3. Light traveling from point A in medium to point B in medium an take infinitely many possible paths onsisting of line segments in eah medium, four of whih are illustrated here. Eah path rosses the interfae at some position x, and hene makes some angles θ with respet to the normal to the interfae, as illustrated for one of the paths. Fermat s priniple states that the atual path the light takes is that for whih the total Notes on Optis R. Parthasarathy 8 Page 4

4 travel time is minimal. (Atually, Fermat s priniple states something slightly more subtle, as we ll disuss shortly.) Applying this priniple, we an derive Snell s Law. We an also use Fermat s priniple to derive Snell s Law of Refletion, whih states that the refleted ray makes the same angle with the interfae as the inident ray Explaining Fermat s Priniple Now let s explain Fermat s Priniple. Suppose light travels along many paths, all of whih interfere with one another. Paths for whih the phase differene is near zero will onstrutively interfere. Consider the minimal time path. As we saw in our derivation of Snell s Law above, this is the path for whih nd is minimal, where the sum runs over all the segments being onsidered (two segments in the above example) and d i is the length of segment i. Reall from Setion.., though, that minimizing nd is the equivalent to minimizing the phase traversed by the wave along the path. Therefore, the minimal time path is also the path of minimal phase, and is also the path of minimal nd all these statements are equivalent. This sum nd is more properly B written as an integral, and is alled the optial path length: OPL n( x) dx. Why should the path of minimal OPL be the path light takes? Let s all this path P. By dopl ( ) onstrution, is zero, where s indiates any variable that haraterizes the paths. d" s" P Therefore nearby paths are similar in phase, and so onstrutively interfere. Consider a path for dopl ( ) whih is not zero moving to a slightly different path, the OPL an hange appreiably, d" s" perhaps higher in one diretion, lower in another, et., and so we would not expet onstrutive interferene. You may be thinking: the minimal OPL path isn t the only one for whih we an guarantee onstrutive interferene. What about the maximal OPL path? This too provides onstrutive interferene. And so the proper formulation of Fermat s priniple is that light travels along paths of extremal optial path length. Typially these are minimal OPL paths, but in various geometries they an be maximal OPL paths as well. As we ll see in lass, Fermat s priniple provides a simple way to think about mirages on hot desert roads. 3.4 Total Internal Refletion n n Let s look more arefully at Snell s Law. What if n > n, is large, so that sin θ >? What θ an satisfy Snell s Law: nsinθ nsinθ? None, sine sin is bounded by A This an be seen as a restatement of Huygen s Priniple. Notes on Optis R. Parthasarathy 8 Page 5

5 ±. This means that there an be no wave transmitted to medium ; the light from medium is totally refleted at the interfae, a ondition known as total internal refletion. Fiber optis, whih underpin muh of modern ommuniation, work beause of total internal refletion. Consider a glass fiber ( n.5) surrounded by air ( n. ). We want the light to travel down the fiber and not leak out into the air (Figure 3.4). This is automatially taken are of by total internal refletion note the higher index of refration of the glass and the large inident angles of light traveling along the fiber (as long as the fiber is not severely bent). In a fiber opti able, light an propagate for kilometers with losses of a fration of a perent! Total internal refletion also has interesting appliations in mirosopy that we will disuss later. Figure 3.4. A fiber opti able. Glass (gray) is surrounded by a material of lower index of refration, e.g. air. Notes on Optis R. Parthasarathy 8 Page 6