Optics: Fermat’s Principle of Least Time – Li Kewei

Introduction

Optics is a complicated subject which involves many levels of approximations. As of now, the full picture of how light behaves is encapsulated in the modern theory of quantum electrodynamics (QED), which is far too advanced for our level at the moment. To study the behaviour of light without invoking QED, we must make some approximations first. When the wavelength of light is comparable to that of other objects concerned, light can be sufficiently approximated to be waves (physical optics). As the wavelength gets shorter, the ray model (geometric optics) becomes more appropriate. When the wavelength gets even shorter than that, the particle (photon) model becomes more accurate, and the light begins to interact with the materials involved, such as through the photo-electric effect. We will only concern ourselves with geometric optics for the time being.

The Principle of Least Time

In geometric optics, Fermat’s principle of least time states that the path taken between two points by a ray of light is the path that can be traversed in the least time. I will first reassure you that the principle is correct by showing that it is consistent with the laws of reflection and refraction.

Consider the following diagram. The line MM’ represents a mirror. Suppose light starts at point A and wishes to go to point B, how should it do it? Well it could just go from A to B in a straight line, so let’s add one more condition: the light must bounce off the mirror. The possible paths that light could take include AD+DB, AE+EB, AC+CB, etc. Fermat’s Principle states that the light will take the path of least time. In this case, that just means light will take the shortest path. How does it do that? Let us reflect the point B to B’, and ask ourselves what is the shortest path from A to B’. You can see from the geometry of the situation that this doesn’t change the problem, since DB=DB’, EB=EB’, CB=CB’, etc. Now it becomes obvious that the shortest path between A and B’ is the straight line joining the two points. From this observation, it is apparent that the angle of incidence, must be equal to the angle of reflection, which is just the normal law of reflection!

Now let us consider refraction. When the time is minimal, the function is stationary, which means that a small change in time is zero around that point. Consider the time T as a function of x. When T is at its minimum:


This result can be generalized to T as a function of many variables. For example T is a now a function of x, y and z (which can represent anything, not necessarily spatial coordinates):

When the function T is at its minimal,


Now consider the following diagram. Remember, when the time is minimal, a small change in the path taken does not change the time. This means other paths around the paths with minimal time also take approximately the same time. The path with minimal time is the path ACB in the following diagram. Path AXB is another path that is very near the path with minimal time.

From Fermat’s principle, the time taken for the two paths must be equal. The time taken on the paths AE and AX is the same. The time taken on the paths FB and CB is the same too. Thus the condition for minimal time should be that the time taken for light to travel through EC and XF must be the same. Now remember that light travels slower in a medium by a factor of 1/n. Thus:

From geometry,


Making the necessary substitutions:

Which is just Snell’s Law of refraction! The reader can humour himself by deriving all the equations for lenses and mirrors through the principle of least time. Here’s a hint: approximate all trigonometric functions to first order, and all the equations of geometric optics can be derived from Fermat’s principle.

There are a few things to be noted about Fermat’s principle. Firstly, is condition for stationary time, not only minimal time: it can be maximal time too. Thus the principle should be more precisely stated as the principle of stationary time, but due to convention we call it the principle of least time.

Range of Validity

Fermat’s principle is not a fundamental principle of nature, and is only most readily applicable in geometric optics. When the other dimensions in a problem become comparable to the wavelength, phenomenon such as diffraction defies the principle of least time. Let’s say that light goes from point A to points B and B’. (see diagram)

Obviously going to B’ takes a longer time than B, but if the path difference is small enough, the light won’t mind taking a bit more time. How small is small enough then? An approximate answer is that the time difference between the two paths is smaller than one period of oscillation of light:

This is in fact, the most general statement of Rayleigh’s criterion.

Philosophical Implications

When Fermat’s principle was first proposed, one of the objections made against it was by Claude Clerselier, an expert in optics and leading spokesman for the Cartesians at that time:

“… Fermat’s principle cannot be the cause, for otherwise we would be attributing knowledge to nature: and here, by nature, we understand only that order and lawfulness in the world, such as it is, which acts without foreknowledge, without choice, but by a necessary determination.”

What he was trying to say is that the concept that light knows that it must take the least time is erroneous. Nature does not know of such things, it merely follows laws such as the law of reflection and refraction blindly. It cannot possibly think about which path takes a longer time. This argument is philosophical in nature, and currently there is no resolution about who’s right or wrong. Although Fermat’s principle is only an approximation, there are other similar principles that are fundamental to physics. One prominent example is the principle of least action in mechanics. This principle reduces to Newton’s laws on macroscopic scales, but it works in quantum mechanics too (where Newton’s laws have failed). It seems that this principle of least action seems to be more fundamental the Newton’s laws which merely dictate how a particle should behave. Clerselier’s argument clearly shows that there are two formulations of physics, one which is more differential (such as Newton’s laws or the laws of reflection and refraction) and one which is more integral (such as the principles of least time and action). All we can say for now is that they both work, but which is more fundamental is a question that has yet to be answered.

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