The description Fermat was able to formulate for light is a curious sort of bridge between our simplest minimization law for rubber bands and the one we will eventually learn to use for the paths of objects. This is because, while light travels through space-time, like (other) objects, people usually use light by sending long ``streams'' of it, rather than isolated single ``pieces''. In that sense, a ``stream'' of light occupies all of its path at once, much like whole collection of little bits of rubber that make up the band, considered together, occupy its whole path at once. Therefore, it will be good to keep in mind that it is because we are considering ``many pieces'' of light collectively that we are able to talk about the whole path as if it were a path in space, even though the rule we will find is explicitly a rule that depends on time.
First, we have to understand how the notion of a path even arises when we ask how light behaves. Suppose light shines from some source, like a lamp, to someone's eye. The light typically shines in all directions, and fills up the space around the lamp. Yet we could put a finger at just the right place, between the lamp and the eye, and block ``the light'' that the person sees. Going further, we could even put a box around the lamp with a small hole in it, to prevent not just part, but almost all of the light from getting out. In that case, unless the hole was in just the right place, the eye would not see the light at all. Going even further, we could erect a whole row of baffles (walls with holes in them) between the lamp and the eye. Unless all of the holes are in just the right places fig. 6.4,
the eye cannot see the light. Finally, we note the obvious point that it is these particular holes that matter. We could poke other holes in the baffles wherever else we like, and they wouldn't make any difference.
It becomes clear, then, that in spite of the fact that light from lamps tends to fill up space, there is some notion of a ``particular path'' traveled by whatever light goes from the lamp to the eye, because at any point along the way, we can interrupt it, just as if we were putting a wall in the path of a flying object. The stream of light that travels along one of these paths is usually referred to as a ray. Later we will see to what extent the light that makes up a ray is really like a ``collection of objects''. For now, the fact that rays exist, and that they identify particular paths, will allow us to speak of the two interchangeably.
The question then becomes: what rule identifies the actual rays on which light travels? Whatever rule we use, it must be sufficiently restrictive that it never predicts a ray that we don't see, and sufficiently general to include all those that we do. We will simply consider some real systems, and see what rules are possible.
First, and simplest, is the case of light just shining through empty space. The rays it travels seem to be simple straight lines. What this means is that, if we ask what pattern of holes in baffles is necessary for the eye to see the light, we find that they must all be in a straight line, and any one that isn't blocks the light. So our first guess might be simply that light travels only straight lines.
Now, though, we ask about the same light reflecting from a mirror. As shown in fig. 6.4,
we could block off the stright-line rays from the lamp to the eye, and the light will still be seen in the reflection. The first problem with the mirror is that none of the paths from lamp to eye, by way of the mirror, is a straight line. They are all bent. The second problem is that, even if we consider paths made from pieces of straight lines, there are many such paths leading from lamp to eye (fig. 6.4). Yet, there is one special place on the mirror where, if we put a dot of black paint to keep it from reflecting, we could cut off the reflection seen by the eye (as if we had put a wall there). This path happens to be the one for which the straight line approaching the mirror and the straight line leaving the mirror both make the same angle with the mirror itself. This happens, also, to be precisely the path that has the shortest possible length from the light to the eye, by way of the mirror.
This last fact is not immediately obvious, unless one thinks about rubber. We could imagine doing exactly the same experiment, but instead of using light, we could stretch a rubber band from the lamp to the eye, and pin it against the mirror with something (for example, an oily finger) that is free to move back and forth, and also to let the rubber slide under it without leaving the mirror. This is shown in fig. 6.4.
We know the rubber stretches out to an even tension all along its length, because if it were under higher tension at one place than at another, it could simply move to even out that discrepency (remember, the only place the rubber is ``held'' is at the starting and ending points).
Now, where would the rubber finally come to rest? Suppose, for instance, that we started it with the finger anywhere except at the point of ``equal angles'' (of the approaching and the leaving lines). Specifically, suppose we started it off to one side, as in the second part of fig. 6.4. Look at how the tension from the two parts of the band is directed. The half that is stretched more vertically from the mirror pulls mostly ``straight out'' from its surface, and not very much to the side. Since the finger won't let it leave that surface, though, this tension accomplishes nothing. On the other side, where the rubber is more horizontal, much more of the tension is pulling to the side, and less of it straight out from the mirror. Since we already know that the absolute tension (within the rubber itself) is the same in both halves (because it is free to slide under the oily finger), which ever side has the greater fraction of that tension pulling horizontally has the greater absolute tension pulling horizontally as well. Therefore the finger is, in total, pulled more sideways by the horizontal section than by the vertical section, and the finger can ease that tension by moving toward the side on which the rubber is horizontal, to make it less so. The fact that the rubber is pulling the finger that way must mean that its length can be reduced by moving the finger that way. The only place the two tensions exactly even out, and the finger is pulled neither one way nor the other, is the place where both angles are the same. At this point, since the absolute tensions are the same, and the angles are the same, the fractions are the same, and so the sideways tensions on the finger are also the same. There is no incentive for the finger to move one way or the other. Thus, this must be the shortest length the rubber can achieve.
Since light takes the same path, it appears to behave much like rubber, and indeed gives us a new guess for the rule for light: that it takes the path of minimum length. Not only does this work for mirrors, it also continues to work fine for empty space, because it just chooses straight lines, for light or rubber.
This does not quite suffice, though. Now we consider light that starts in one kind of material and travels across a boundary into another. For instance, we could be looking at a ``very short person'' standing in a swimming pool, or at a ``bent pencil'' in a glass of water. There is no mirror in this case, but clearly the light paths are not straight lines, which is why the person looks short and the pencil looks bent.
This system is more complicated, but even it had been understood empirically prior to Fermat. The problem is that, depending on where we stand relative to the person in the water, there can be many different angles made by the rays of light. We need a rule that is consistent with all of them, but that correctly predicts the relation in any given case between the light ray approaching the water surface from below, and the one leaving it from above. Such a law had been found by ? ? Snell, and it is this:
Snell's Law: The sine of the angle made by the ray below and the sine of the angle made by the ray above, as measured from the vertical, always have a fixed ratio.
This seems a bit obscure at first, but it is really almost the same thing as we found for the mirror. The sine of the angle is just the ratio of the horizontal distance it covers, between any two points, and the length of the whole path (along the diagonal) between those two points. Thus it is exactly the same ratio as our ratio of tensions in the case of the rubber band. The tension in the rubber itself lies along the diagonal, so the sine of the angle gives the part of it that is pulling along the horizontal. The only difference is that for the mirror, the two ``horizontal tensions'' (arriving and departing) were always the same. Here they are always scaled by some fixed ratio. That ratio happens to be determined by what the substances are that the light is traveling through, such as air and water, air and glass, water and glass, etc.
It will not be surprising, then, that this ``fixed sine ratio'' law of Snell is closely related to the length-minimization rule we guessed for the mirror. In fact, it is exactly what we would have had if the ``tensions in the two halves of the rubber'' (above or under the water surface) had not been equal, but had been some fixed ratio of each other, just as if, somehow, the piece of the path had been ``stretched longer than we thought'' in one of the two places, to increase the tension. For rubber, we associate path length with tension. If two pieces of rubber are stretched the same amount, they have the same tension. And in rubber, it is the tensions that pull on things to eventually place them on the actual path.
Yet, in the end, we do not need to consider how the tensions achieve the actual path, because we know that it is minimization of length that specifies that path, all by itself. So, having used it to understand what light is doing, we can now stop using the language of tensions, and simply say that, for mirrors, the actual path of light was the path of minimum length. Now, considering the water/air system, the length that is minimized seems not to be the length that we measure, but some other quantity that is proportional to it by an overall re-scaling, depending on whether the light is traveling in air, under water, in glass, or wherever. In particular, the ``effective length'' in air seems to be shorter than the length we measure, and in water it seems to be longer. In other words, light bends because, by doing so, it can get more rapidly to the surface, and so put less of its path length underwater. The price is that more path length must be traveled in air. This is not done, though, to the absolutely greatest extent possible, but only in a measured ratio.
Fermat's observation was that the simplest thing to assume would be that light consisted of some objects ``traveling over these paths'', but with a speed that depended on what the light was traveling through. In that case, the time taken to travel a path would always be proportional to the length of the path in each substance , but the proportionality factor would be different in water than in air, or glass, or whatever. As such, if the ratios of speeds were chosen correctly, and the time minimized along the actual path, this would produce exactly the sine-ratio law of Snell.
If the speed of light in water were slower than that in air by a fixed amount, light could always bend in such a way that less water-path could be traveled, in exchange for more air-path. Even though the total length traveled would be greater, the saving of time by traveling less far under water would outweigh the extra time needed to travel slightly further in air, and the total time would be reduced from that of a straight-line path. For the simpler case of the mirror, where the light travels through the same medium on both halves of the path, this reduces to precisely the result of minimum length that we had before, and of course it works perfectly in empty space, because minimum length and minimum time both identify the straight line paths. This, then, is Fermat's conclusion:
Fermat's Least Time Principle for Light Light traveling through some substance has a speed which is determined by the substance. The actual path taken by light between any two points, in any combination of substances, is always the path of least time that can be traveled at the required speeds.
To notice one more aspect of Fermat's rule, we will look at one of its more useful applications: that of making lenses.
We will only worry about lenses whose purpose is to focus light, that is, to gather as many of the outgoing rays from some source as possible, and bring them back together at a single point. In other words, the function of a focusing lens is to create many actual paths from a starting to an ending point. How this can be possible is shown in fig. 6.4.
By experimenting with light traveling in air and glass, we can determine from Fermat's principle that light travels more slowly in most glasses than in air. (Are there even any exceptions?) Therefore, by making a lump of glass thick in the middle and thin at the edges, and placing it as shown between a lamp and an eye, there can be many paths, all of least time. The paths in the center travel short distances in air, but must then travel long distances more slowly in the glass. A little to the side, if the lens is correctly shaped, there are paths of greater overall length, with all of the increase taken up in air, and a decrease in the length traveled through glass, which can again minimize time.
This points out two interesting properties of Fermat's principle, that can be seen at work in real physical systems. Often, one tends to think of the starting conditions of a path as the only way to specify what that path will be. From this point of view, it seems strange that a useful description of paths even could be based on information about their starting and ending positions alone. Yet for light, it is not even clear ahead of time which of the starting rays (with which direction) will arrive at the final destination, until it has been specified what combinations of air, water, glass, etc., lie between. This is why a minimization rule that depends on starting and ending data can still work as an identifier of paths.
The second point arises because we have usually spoken of identifying, from among all imaginable paths, ``the actual path'' that an object takes in a given circumstance. With the lens, though, we see that in fact there are systems with a large ambiguity, in which there can be many actual paths that an object can consistently take, even given the same starting and ending points.
Finally, before we go on to look at general properties of minimization rules, we list one caveat concerning this example. Light is unusual stuff. We will find that out, again and again, as we understand it in more and more ways. It has already been noted that, because we tend to work with it in long streams, it behaves more like rubber bands, literally, than many of our other examples will. It is not quite like rubber, though, because the rule we have uncovered identifies a path through space by means of a criterion of time. However, it will also not be quite like other objects we will discover, either, because the path we have found is still a path through space. In later examples, pictures of paths through space alone will not be enough to even frame the questions correctly, and time will not merely be a criterion, but rather an essential ``part of the map'' needed to identify different imaginable paths. Therefore, the results we will find will again be somewhat different, in the particulars, from Fermat's principle. These latter differences arrise because light has no mass, which is an extremely common characteristic of most of the other systems we will consider. That difference alone is enough to make most of the rules that can be phrased for light, even if they come from the same principles as similar rules for other objects, look quite different in the details.
With these examples in mind, we now return to the relation of minimization rules to equations of motion, in the general case.