# Uniqueness of refractive index

Let $$A$$ and $$B$$ be two media. Let $$\alpha$$ be the angle of incidence of light on medium $$B$$ and let $$\beta$$ be the angle of refraction. Let $$_{_{A}}n_{{_B}}$$ be the refractive index of $$B$$ with respect to $$A$$.

We know that $$_{_{A}}n_{{_B}} = \dfrac{\sin\alpha}{\sin\beta}$$

$$_{_{A}}n_{{_B}}$$ can also be expressed in terms of the speed of light in $$A$$ and $$B$$.

Let the speed of light in $$A$$ be $$c_{_{A}}$$ and in $$B$$ be $$c_{_{B}}$$

So, $$_{_{A}}n_{{_B}} = \dfrac {c_{_{A}}}{c_{_{B}}}$$

As the refractive index is a constant, this means that both the values of the refractive index must be the same.

This means that for any two media, the ratio of the sines of the angles of incidence and refraction will be equal to the ratio of the speed of light in both the media.

Is there some proof other than experimental proof for this?

How was it discovered?

The key to this formula is Fermat's principle of least time. That is, the path given by

$$n=\frac{\sin\alpha}{\sin\beta}$$

ensures that the light travel time is minimized.

Fermat's principle is itself a special case of the principle of stationary action, which has widespread application throughout physics.

Fermat's principle is of historical significance as the first example of the principle of stationary action. Fermat himself couldn't fully justify why the principle of least time had to apply in the refraction of light, but it seemed reasonable at the time to assume that it did so. It wasn't until much later that Fermat's principle was given a solid theoretical basis as a case of the principle of stationary action.

You can read about more about the historical details in the linked Wikipedia articles. But I'd like to mention a couple of them here because this issue was rather important in the history of physics.

Ptolemy stated that the refraction law is (in modern notation)

$$n=\frac{\alpha}{\beta}$$

He claimed that he had empirically verified this theory, but it seems pretty clear that he fudged the data to fit his theory. To be fair, Ptolemy didn't have access to a modern optics lab, and it's a reasonable approximation for small angles, since $$\sin\theta \approx \theta$$ when $$\theta$$ is small. (That's the well-known small angle approximation which is often useful, eg in approximating the period of a simple pendulum.)

About 800 years later, the Persian mathematician and physicist Ibn Sahl discovered the true law of refraction. Eventually, about 500 years later, this became known in Europe as Snell's law.

• Thanks, so Fermat, basically knew the how, but not the why. This sounds very interesting... Apr 22, 2020 at 13:51
• @Rajdeep_Sindhu It is interesting! You could write whole books about this. :) It's a great example of how scientific concepts can evolve over many centuries. The people on the History of Science and Mathematics can probably give you more details, but the info on Wikipedia is quite good and in-depth. Apr 22, 2020 at 13:58
• So, I read online that Snell's law was discovered in 1621. Since then, the term refractive index has been in use, right? And the formula for refractive index as ratio of velocities was given later. And the refractive index for a medium was same by both the formula. Was this just a coincidence or not? I'm sorry if you already covered this in your answer and I missed something (maybe I did). This was actually what I wanted to know... Thanks Apr 22, 2020 at 14:02
• @Rajdeep_Sindhu The early theorists didn't know about the velocity of light in the medium. They just knew that the refractive index of a medium was constant. As the Wikipedia article on Snell's law says, Descartes assumed the velocity of light was infinite. But Fermat assumed it was finite, "and his derivation depended upon the speed of light being slower in a denser medium". So Fermat was the first person to equate the ratio of the refractive indices of the media to the ratio of the speeds of light in the media. Apr 22, 2020 at 15:47
• That clears all doubts, thanks again Apr 22, 2020 at 18:24