Total internal reflection and the critical angle

When light can no longer escape

When light passes from a more refractive medium (n1 is large) into a less refractive medium (n2 is small, with n2 < n1), the angle of refraction i2 is always greater than the angle of incidence i1. From a certain angle of incidence, known as the critical angle (denoted ilim), the angle of refraction would reach 90 degrees: the refracted ray would be parallel to the surface.

Beyond this critical angle, there is no refracted ray at all: all the light is reflected back into medium 1. This is total internal reflection.

Calculating the critical angle

The critical angle corresponds to the case where i2 = 90 degrees, so sin(i2) = 1. Snell’s law then gives:

n1 * sin(ilim) = n2 * sin(90) = n2

Therefore: sin(ilim) = n2 / n1

This formula only makes sense if n2 <= n1 (otherwise sin(ilim) > 1, which is impossible).

Practical example: optical fibre

In an optical fibre, light propagates through a high-index core surrounded by a lower-index cladding. Thanks to total internal reflection, the light remains trapped inside the core and travels over very long distances without escaping, which enables high-speed transmission of internet data.

Another example: underwater, a diver looking up towards the surface at too shallow an angle can no longer see the sky but only the reflection of the seabed, due to total internal reflection.

Common misconception

Internal total reflection can ONLY occur when light travels from a more refractive medium to a less refractive medium (n1 > n2). In the opposite direction (n1 < n2), there is no critical angle: the light always enters, but is simply deflected.