Applications of Fermat’s 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 in the block by                         making the angle in the block less inclined, although it loses a                         little 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, 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 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 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 following.                         What we are really seeing is the sky light “reflected” on the road:                         light from the sky, heading for the road, can end up in the eye, as                         shown in Fig. 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 way, 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 point,                         $P$, collected back together at another point, $P'$                         (Fig. 26–9). That means, 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 $P'$? 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. Therefore 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 slower than it does in the air (Fig. 26–10). Now                         consider a ray which goes in air in the path $PQP'$. 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 thickness                         (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 $PQP'$. Likewise, if we take a ray $PRR'P'$ which is partly 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 light 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 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 and comes back, and all times must be                         equal. Here the light always travels in air, so the time and the                         distance are proportional. Therefore the statement that all the times                         are the same is the same as the statement that the total distance is                         the same. Thus the sum of the two distances $r_1$ and $r_2$ must be a                         constant. An ellipse is that curve which has the property that                         the sum of the distances from two points is a constant for every point                         on the ellipse; thus we can be sure that the light from one focus will                         come to the other. The same principle works for gathering the light of a star. The great                         $200$-inch Palomar telescope is built on the following                         principle. Imagine a star billions of miles away; we would like to                         cause all the light that comes in to come to a focus. Of course we                         cannot draw the rays that go all the way up to the star, but we still                         want to check whether the times are equal. Of course we know that when                         the various rays have arrived at some plane $KK'$, perpendicular to                         the rays, all the times in this plane are equal                         (Fig. 26–12). The rays must then come down to the mirror and                         proceed toward $P'$ in equal times. That is, we must find a curve which                         has the property that the sum of the distances $XX' + X'P'$ is a                         constant, no matter where $X$ is chosen. An easy way to find it is to                         extend the length of the line $XX'$ down to a plane $LL'$. Now if we                         arrange our curve so that $A'A'' = A'P'$, $B'B'' = B'P'$, $C'C'' =                         C'P'$, and so on, we will have our curve, because then of course, $AA' +                         A'P' = AA' + A'A''$ will be constant. Thus our curve is the locus of all                         points equidistant from a line and a point. Such a curve is called a                         parabola; the mirror is made in the shape of a parabola. The above examples illustrate the principle upon which such optical                         devices can be designed. The exact curves can be calculated using the                         principle that, to focus perfectly, the travel times must be exactly                         equal for all light rays, as well as being less than for any other                         nearby path. We shall discuss these focusing optical devices further in the next                         chapter; let us now discuss the further development of the                         theory. When a new theoretical principle is developed, such as the                         principle of least time, our first                         inclination might be to say, “Well, that is very pretty; it is                         delightful; but the question is, does it help at all in understanding                         the physics?” Someone may say, “Yes, look at how many things we can                         now understand!” Another says, “Very well, but I can understand                         mirrors, too. I need a curve such that every tangent plane makes equal                         angles with the two rays. I can figure out a lens, too, because every                         ray that comes to it is bent through an angle given by Snell’s                         law.” Evidently the statement of least time                         and the statement that angles are equal on reflection, and that the                         sines of the angles are proportional on refraction, are the same. So is                         it merely a philosophical question, or one of beauty?  There can be                         arguments on both sides.                          However, the importance of a powerful principle is that it                             predicts new things. It is easy to show that there are a number of new things predicted by                         Fermat’s principle. First, suppose that                         there are three media, glass, water, and air, and we perform a                         refraction experiment and measure the index $n$ for one medium against                         another. Let us call $n_{12}$ the index of air ($1$) against water                         ($2$); $n_{13}$ the index of air ($1$) against glass ($3$). If we                         measured water against glass, we should find another index, which we                         shall call $n_{23}$. But there is no a priori reason why there                         should be any connection between $n_{12}$, $n_{13}$, and $n_{23}$. On                         the other hand, according to the idea of least time, there is a                         definite relationship. The index $n_{12}$ is the ratio of two things,                         the speed in air to the speed in water; $n_{13}$ is the ratio of the                         speed in air to the speed in glass; $n_{23}$ is the ratio of the speed                         in water to the speed in glass. Therefore we cancel out the air, and                         get                         \begin{equation}                         \label{Eq:I:26:5}                         n_{23}=\frac{v_2}{v_3}=\frac{v_1/v_3}{v_1/v_2}=\frac{n_{13}}{n_{12}}.                         \end{equation}                         In other words, we predict that the index for a new pair of                         materials can be obtained from the indexes of the individual                         materials, both against air or against vacuum. So if we measure the                         speed of light in all materials, and from this get a single number for                         each material, namely its index relative to vacuum, called $n_i$                         ($n_1$ is the speed in vacuum relative to the speed in air, etc.),                         then our formula is easy. The index for any two materials $i$ and $j$                         is                         \begin{equation}                         \label{Eq:I:26:6}                         n_{ij}=\frac{v_i}{v_j}=\frac{n_j}{n_i}.                         \end{equation}                         Using only Snell’s law, there is no                         basis for a prediction of this kind.1 But of course this prediction                         works. The relation (26.5) was known very early, and was a                         very strong argument for the principle of                         least time.                          Another argument for the principle of least                         time, another prediction, is that if we measure the speed of                         light in water, it will be lower than in air. This is a prediction of                         a completely different type. It is a brilliant prediction, because all                         we have so far measured are angles; here we have a theoretical                         prediction which is quite different from the observations from which                         Fermat deduced the idea of least time. It                         turns out, in fact, that the speed in water is slower than the                         speed in air, by just the proportion that is needed to get the right                         index!                         