Fermat's Principle

The refractive behaviour of light had been a source of study and consternation for centuries since no simple relationship between the angles of incidence and refraction could be determined. It was in 1621, that the Dutch investigator Willebrord Snell determined that it is the sines of the angles of incidence and refraction that maintain a constant ratio for a given pair of media, an experiment that is worth carrying out yourself. Although Snell is correct, this observation of effects does not address itself to cause. Descartes, insisting that light had to be understood as ballistic particles (in opposition to Leonardo da Vinci, and to keep his own purely mechanical outlook) was forced to conclude, erroneously, that light actually sped up upon entering the water. He also claimed Snell’s discovery as his own. Fermat found this speeding up to be absurd and sought to determine the cause for light’s behaviour.

To note the sine relationship is good, but to actually assert that this trend is a scientific principle would not be an honest blunder, it would be an admission by anyone who would make that statement, that that person believes principles are unknowable.

Fermat’s ought not to describe the motion of the fish, but the shape of the aquarium in which they swam: He returned to the Greek discovery that light reflected off a mirror takes the path of minimal distance, an experiment worth performing on your own. Fermat took up this approach and hypothesized and demonstrated in 1662 that light follows a path of quickest time, rather than shortest distance: As far as the light is concerned, it is always propagating straight ahead by this principle. This hypothesis results in the sine ratio discovered by Snell, but Fermat delivered the child whose form Snell accurately reported.

Applications of Fermat’s Principle

We can make several observations as a result of Fermat’s Principle which will prove useful as we explore the realm of geometric optics:

  1. In a homogeneous medium, light rays are rectilinear. That is, in any medium where the index of refraction is constant, light travels in a straight line.
  1. The angle of reflection of a surface is equal to the angle of incidence. This is the Law of Reflection.

We can also make some interesting and useful observations about conic surfaces. Conic surfaces are particularly useful in mirror optics – for example, the design of telescopes. We consider two conjugate points – two points that are perfect images of each other. A salient property of these conjugate points is that the optical path length of all rays connecting them is equal.

Consider a conic surface such as an ellipse. An ellipse is defined as the locus of all points such that the sum of the distances from each point to two fixed points (the foci) is constant, as in Figure 1. The two foci of a mirrored ellipse must then be optically conjugate points. A point source located at one focus must be imaged perfectly at the other focus.

Fermat’s Principle

An ellipse. The ellipse is defined as the locus of all points such that the sum of the distances from each point to two fixed points is constant: d1 + d2 = c.

In the case of a parabola, one focus has become infinite. This can be interpreted by saying that an aggregate of rays, parallel to one another and to the axis of a paraboloid after being re erected by the paraboloid, will pass through the focus of the paraboloid. The Newtonian telescope leverages this fact in its design to collect and focus light from distant objects. In general, a conic surface can be thought of as having two foci and these foci will be optically conjugate points.


Practise This Question

If there are two objects in thermal equilibrium in one frame, they will be in thermal equilibrium in all frames