Snell's Law and the Critical Angle

Uses Snell's law n₁ sin θ₁ = n₂ sin θ₂ to find the angle of refraction. When light passes into a less dense medium, the critical angle for total internal reflection is also given.

Explanation

Light bends when it crosses into a different material. Snell's law fixes the angle.

n_1 \sin \theta_1 = n_2 \sin \theta_2

The refractive index $n$ says how much light slows down in a material, taking vacuum as 1.

Common refractive indices

Air — 1.00
Water — 1.33
Glass — about 1.5
Diamond — 2.42

Example

Light leaves water for air at an angle of incidence of 30 degrees ( $n_1 = 1.33$ , $n_2 = 1$ ).

\sin \theta_2 = \dfrac{1.33}{1} \times \sin 30° = 0.665 \quad \Longrightarrow \quad \theta_2 = 41.7°

Moving into the thinner medium, the ray bends away from the normal.

Total internal reflection

Keep increasing the angle and a moment arrives when $\sin \theta_2$ would have to exceed 1. No such angle exists, so the light cannot refract at all and is entirely reflected. This is total internal reflection.

\theta_c = \arcsin \dfrac{n_2}{n_1}

For water to air that is $\arcsin(1/1.33) = 48.8°$ . Light striking the surface beyond 48.8 degrees can never escape.

It only happens going from the denser to the thinner medium. Air into water cannot do it, and this calculator prints "-" for the critical angle in that case.

What total reflection makes possible

Optical fibre — a core at $n = 1.5$ inside cladding at $n = 1.45$ gives a critical angle of 75 degrees. Light bounces off the wall again and again and stays trapped for tens of kilometres
The fire of a diamond — an index of 2.42 gives a critical angle of only 24.4 degrees, so light that enters struggles to leave. It ricochets inside before bursting out in a few concentrated directions
Looking up from underwater — the whole sky is squeezed into a circular window overhead, and beyond it the surface becomes a mirror