How does principle of least time suggest a relation between three indices of refraction?

Question

In the 26th Feynman Lecture, Fermat's principle of least time is discussed and this point about refractive index is brought up:

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 n23. 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.

And, Feyman goes into discussing how the refractive indexs have to be related to the ratio of speed of light, however I don't get the point of how the existence of such a relation was predicted by principle of least time. I'm looking for an explanation on this.

Answer 1

Feynman is trying to say that the principle of least time implies that the indices $n_{ij}$ are ratios of speeds, probably by a reasoning similar to the following:

If the paths of the light in the two materials are $s_1$ and $s_2$ and the speeds of light in the materials are $v_1$ and $v_2$, then the principle of least time says that the total travel time $\frac{s_1}{v_1} + \frac{s_2}{v_2}$ is minimized for varying $s_i$, and so equivalently $s_1 + \frac{v_1}{v_2}s_2$ is minimized. Experimentally, you can observe that $s_1$, $s_2$ and $n_{12}$ are exactly such that $s_1 + n_{12}s_2$ is minimal for varying $s_i$. So if you believe in Fermat's principle, you conclude that $n_{12} = \frac{v_1}{v_2}$.

Answer 2

This answer expands on the answer by ACuriousMind.

The total travel time:

$$ \frac{s_1}{v_1} + \frac{s_2}{v_2} \qquad (1) $$

Let $S_i$ represent sweeping out a range of possible paths

Let's first consider the two contributions separately.

Both $\tfrac{s_1}{v_1}$ and $\tfrac{s_2}{v_2}$ have a minimum of their own; those two minima do not occur at the same point in the variation space.

So there is a point in variation space where one contribution is already ascending again, whereas the other one is still descending.

The minimum of the sum $\tfrac{s_1}{v_1} + \tfrac{s_2}{v_2}$ is at the cross-over point.

At the overall minimum the derivative of (1) with respect to the variation is zero.

$$ \frac{d(\frac{s_1}{v_1} + \frac{s_2}{v_2})}{dS_i} = 0 \qquad (2) $$

So: if you take derivatives of the two contributions separately then those two derivatives match each other at the overall minimum

$$ \frac{d(\frac{s_1}{v_1})}{dS_i} = - \frac{d(\frac{s_2}{v_2})}{dS_i} \qquad (3) $$

The Wikipedia article about Snell's law gives a derivation of Snell's law from Fermat's least time using the above differentiation relation as starting point.

The speed of light is too fast to be observable (at least it was until recently). The angles described by Snell's law are static: very accessible to observation.

With Snell's law corroborated with experiments the relation with Fermat's least time facilitates interpreting the angles of Snell's law in terms of ratio of velocities.