What happens to light passing through two transparent media at the interface of the two media?

When light passes from an optically denser medium to an optically rarer medium, it bends away from the normal in accordance with the Snell's Law of refraction $\mu_1\sin i =\mu_2\sin r$ where $\mu_1$ and $\mu_2$ are the refractive indices of the two media, $\mu_1>\mu_2$, and $i$ and $r$ are the angles of incidence and refraction respectively.

When we increase $i$ gradually, $r$ also increases. Since, $r>i$ as $\mu_1>\mu_2$, $r$ becomes $90^\circ$ before $i$. The angle of incidence for which the angle of refraction is $90^\circ$ is called the critical angle $i_c$. From Snell's law we get, $\sin i_c =\mu_2/\mu_1$. This is how critical angle is explained in almost all sources. Beyond this angle it's said the Snell's law of refraction is no longer valid and refraction is not possible, only reflection or more precisely total internal reflection takes place.

I do not understand why we still consider the refractive index of the medium at grazing refracted ray ($r=90^\circ$) as $\mu_2$. How can we ascertain the speed of light (and hence the refractive index) in the interface separating two media? According to the derivations, the speed of light in the interface is same as that of the medium other than the one from which the ray emerged (here it's the rarer medium). But why do we choose it this way? Why can't it be the other way round?

When we talked about thin film interference, we said that when light encounters a smooth interface between two transparent media, some of the light gets through, and some bounces off. There we limited the discussion to the case of normal incidence. (Recall that normal means perpendicular to and normal incidence is the case where the direction in which the light is traveling is perpendicular to the interface.) Now we consider the case in which light shining on the smooth interface between two transparent media, is not normally incident upon the interface. Here’s a “clean” depiction of what I’m talking about:

and here’s one that’s all cluttered up with labels providing terminology that you need to know:

As in the case of normal incidence, some of the light is reflected and some of it is transmitted through the interface. Here we focus our attention on the light that gets through.

Experimentally we find that the light that gets through travels along a different straight line path than the one along which the incoming ray travels. As such, the transmitted ray makes an angle $\theta_2$ with the normal that is different from the angle $\theta_1$ that the incident ray makes with the normal.

The adoption of a new path by the transmitted ray, at the interface between two transparent media is referred to as refraction. The transmitted ray is typically referred to as the refracted ray, and the angle $\theta_2$ that the refracted ray makes with the normal is called the angle of refraction. Experimentally, we find that the angle of refraction $\theta_2$ is related to the angle of incidence $\theta_1$ by Snell’s Law:

\[n_1 \sin\theta_1=n_2 \sin\theta_2 \label{27-1}\]

where:

$n_1$ is the index of refraction of the first medium, the medium in which the light is traveling before it gets to the interface,
$\theta_1$ is the angle that the incident ray (the ray in the first medium) makes with the normal,
$n_2$ is the index of refraction of the second medium, the medium in which the light is traveling after it goes through the interface, and,
$\theta_2$ is the angle that the refracted ray (the ray in the second medium) makes with the normal.

On each side of the equation form of Snell’s law we have an index of refraction. The index of refraction has the same meaning as it did when we talked about it in the context of thin film interference. It applies to a given medium. It is the ratio of the speed of light in that medium to the speed of light in vacuum. At that time, I mentioned that different materials have different indices of refraction, and in fact, provided you with the following table:

Medium	Index of Refraction
Vacuum	1
Air	1.00
Water	1.33
Glass(Depends on the kind of glass. Here is one typical value.)	1.5

What I didn’t mention back then is that there is a slight dependence of the index of refraction on the wavelength of the visible light, such that, the shorter the wavelength of the light, the greater the index of refraction. For instance, a particular kind of glass might have an index of refraction of 1.49 for light of wavelength 695 nm (red light), but an index of refraction that is greater than that for shorter wavelengths, including an index of refraction of 1.51 for light of wavelength 405 nm (blue light). The effect in the case of a ray of white light traveling in air and encountering an interface between air and glass is to cause the different wavelengths of the light making up the white light to refract at different angles.

This phenomena of white light being separated into its constituent wavelengths because of the dependence of the index of refraction on wavelength, is called dispersion.