Refraction of a plane wave

A plane wavefront AB strikes the boundary PP′ between medium 1 (speed v₁) and medium 2 (speed v₂) at an angle of incidence i. Let τ be the time for the end B of the wavefront to reach the boundary at C, so BC = v₁τ.

In that same time the wavelet from A travels a distance AE = v₂τ into medium 2. The tangent plane CE drawn from C onto this wavelet is the refracted wavefront.

Triangles ABC and AEC share the side AC. So sin i = BC/AC = v₁τ/AC and sin r = AE/AC = v₂τ/AC, which gives sin i/sin r = v₁/v₂.

If the ray bends towards the normal (r < i), then v₂ < v₁: light is slower in the second medium. This is the wave-model prediction that Foucault's water measurement confirmed.

With refractive indices n₁ = c/v₁ and n₂ = c/v₂, where c is the speed of light in vacuum, the same result reads n₁ sin i = n₂ sin r. This is Snell's law.

If BC is one wavelength λ₁ in medium 1, then AE is one wavelength λ₂ in medium 2, because the crest from B reaches C just as the crest from A reaches E. So λ₁/λ₂ = v₁/v₂: entering a denser medium, speed and wavelength drop in the same ratio.

The frequency ν = v/λ does not change. Atoms of the medium are set into forced oscillation by the incoming light and re-emit at the frequency that drives them, so reflected and refracted light keep the incident frequency (Example 10.1).

Slowing down costs the wave no energy: the energy a wave carries depends on its amplitude, not on its speed (Example 10.1).

Exercise 10.1: 589 nm light passes from air into water, n = 1.33. The reflected light keeps 589 nm, ν = 3 × 10⁸/(589 × 10⁻⁹) = 5.09 × 10¹⁴ Hz and 3 × 10⁸ m/s. The refracted light has the same ν, speed 3 × 10⁸/1.33 = 2.26 × 10⁸ m/s and wavelength 589/1.33 = 443 nm.

Exercise 10.3: in glass of refractive index 1.5, v = 3.0 × 10⁸/1.5 = 2.0 × 10⁸ m/s. The refractive index depends on colour; it is larger for violet than for red, so violet light travels more slowly in a glass prism.