26-4 Applications of Fermats principle Now let us consider some of the interesting consequences of the principle of least time. First is the principle of reciprocity. If to go from A to b we have found the path of the least time, then to go in the opposite direction(assuming that light goes at the same speed in any direction), the shortest time will be the same path and therefore, if light can be sent one way, it can be sent the other way An example of interest is a glass block with plane parallel faces, set at an angle to a light beam. Light, in going through the block from a point a to a point B (Fig. 26-6)does not go through in a straight line, but instead it decreases the time ig. 26-6. a beam of light is offset as in the block by making the angle in the block less inclined, although it loses a little it passes through a transparent block bit in the air. The beam is simply displaced parallel to itself because the angles in and out are the same a third interesting phenomenon is the fact that when we see the sun setting it is already below the horizon! It does not look as though it is below the horizon TO APPARENT SUN but it is(Fig. 26-7). The earth's atmosphere is thin at the top and dense at the bottom. Light travels more slowly in air than it does in a vacuum, and so the light LIGHT PATH of the sun can get to point S beyond the horizon more quickly if, instead of just going in a straight line, it avoids the dense regions where it goes slowly by getting through them at a steeper tilt. When it appears to go below the horizon, it is ARTH actually already well below the horizon. Another example of this phenomenon is the mirage that one often sees while driving on hot roads. One sees"water"on the road, but when he gets there, it is as dry as the desert! The phenomenon is the Fig. 26-7. Near the ho What we are really seeing is the sky light"reflected"on the road: parent sun is higher t e true sun by light from the sky, heading for the road, can end up in the eye, as shown in Fig. about 1/2 degree 26-8. Why? The air is very hot just above the road but it is cooler up higher Hotter air is more expanded than cooler air and is thinner, and this decreases the speed of light less. That is to say, light goes faster in the hot region than in the cool region. Therefore, instead of the light deciding to come in the straightforward LIGHT FROM SKY vay, it also has a least-time path by which it goes into the region where it goes faster for awhile, in order to save time. So, it can go in a curve As another important example of the principle of least time, suppose that we would like to arrange a situation where we have all the light that comes out of one 26-8. A mirage point, P, collected back together at another point, P'( Fig. 26-9). That mean of course, that the light can go in a straight line from P to P That is all right But how can we arrange that not only does it go straight, but also so that the light starting out from P toward Q also ends up at Pr? We want to bring all the light back to what we call a focus. How? If the light always takes the path of least time, then certainly it should not want to go over all these other paths. The only way that the light can be perfectly satisfied to take several adjacent paths is to make those times exactly equal! Otherwise, it would select the one of least time. There- OPTICAL SYSTEl fore the problem of making a focusing system is merely to arrange a device so that it takes the same time for the light to go on all the different paths! This is easy to do. Suppose that we had a piece of glass in which light goes Fig. 26-9. An optical"black box slower than it does in the air(Fig. 26-10). Now consider a ray which goes in air he path PoP. That is a longer path than from P directly to P and no doubt takes a longer time. But if we were to insert a piece of glass of just the right thick ai Ss(we shall later figure out how thick) it might exactly compensate the excess time that it would take the light to go at an angle! In those circumstances we can arrange that the time the light takes to go straight through is the same as the time it takes to go in the path PoP. Likewise, if we take a ray Prr'P' which is partI inclined, it is not quite as long as PQP, and we do not have to compensate as much as for the straight one, but we do have to compensate somewhat. We end up with a piece of glass that looks like Fig. 26-10. With this shape, all the ligh which comes from P will go to P. This, of course, is well known to us, and we call such a device a converging lens. In the next chapter we shall actually calculate g.26-10.Afo what shape the lens has to have to make a perfect focus. Take another example: suppose we wish to arrange some mirrors so that the light from P always goes to P'(Fig. 26-11). On any path it goes to some mirror