A more precise statement of Fermat’s principle
Actually, we must make the statement of the principle of least time a little more accurately. It was not stated                         correctly above. It is incorrectly called the                         principle of least time and we have gone                         along with the incorrect description for convenience, but we must now                         see what the correct statement is. Suppose we had a mirror as in                         Fig. 26–3. What makes the light think it has to go to                         the mirror?  The path of least time is clearly $AB$. So some                         people might say, “Sometimes it is a maximum time.” It is not                         a maximum time, because certainly a curved path would take a still                         longer time!  The correct statement is the following: a ray going in a                         certain particular path has the property that if we make a small                         change (say a one percent shift) in the ray in any manner whatever,                         say in the location at which it comes to the mirror, or the shape of                         the curve, or anything, there will be no first-order change in                         the time; there will be only a second-order change in the                         time. In other words, the principle is that                         light takes a path such that there are many other paths nearby which                         take almost exactly the same time.                          The following is another difficulty with the principle of least time, and one which people who do not like this                         kind of a theory could never stomach. With Snell’s theory we can                         “understand” light. Light goes along, it sees a surface, it bends                         because it does something at the surface. The idea of causality, that                         it goes from one point to another, and another, and so on, is easy to                         understand. But the principle of least time                         is a completely different philosophical principle about the way nature                         works. Instead of saying it is a causal thing, that when we do one                         thing, something else happens, and so on, it says this: we set up the                         situation, and light decides which is the shortest time, or the                         extreme one, and chooses that path. But what does it do,                         how does it find out? Does it smell the nearby paths,                         and check them against each other? The answer is, yes, it does, in a                         way. That is the feature which is, of course, not known in geometrical                         optics, and which is involved in the idea of wavelength; the                         wavelength tells us approximately how far away the light must                         “smell” the path in order to check it. It is hard to demonstrate                         this fact on a large scale with light, because the wavelengths are so                         terribly short. But with radiowaves, say $3$-cm waves, the distances                         over which the radiowaves are checking are larger. If we have a source                         of radiowaves, a detector, and a slit, as in Fig. 26–13,                         the rays of course go from $S$ to $D$ because it is a straight line,                         and if we close down the slit it is all right—they still go. But now                         if we move the detector aside to $D'$, the waves will not go through                         the wide slit from $S$ to $D'$, because they check several paths                         nearby, and say, “No, my friend, those all correspond to different                         times.”  On the other hand, if we prevent the radiation from                         checking the paths by closing the slit down to a very narrow crack,                         then there is but one path available, and the radiation takes it! With                         a narrow slit, more radiation reaches $D'$ than reaches it with a wide                         slit! One can do the same thing with light, but it is hard to demonstrate on                         a large scale. The effect can be seen under the following simple                         conditions. Find a small, bright light, say an unfrosted bulb in a                         street light far away or the reflection of the sun in a curved                         automobile bumper. Then put two fingers in front of one eye, so as to                         look through the crack, and squeeze the light to zero very gently. You                         will see that the image of the light, which was a little dot before,                         becomes quite elongated, and even stretches into a long line. The                         reason is that the fingers are very close together, and the light                         which is supposed to come in a straight line is spread out at an                         angle, so that when it comes into the eye it comes in from several                         directions. Also you will notice, if you are very careful, side                         maxima, a lot of fringes along the edges too. Furthermore, the whole                         thing is colored. All of this will be explained in due time, but for                         the present it is a demonstration that light does not always go in                         straight lines, and it is one that is very easily performed.