Refraction of light

Refraction is the change in wave direction that occurs when wave speed changes across a boundary. For light, refractive index packages that speed change into one number and Snell's law connects it to the ray angles.

Refractive index and wave speed

n = c/v

A larger n means light travels more slowly in that medium. At a stationary boundary, frequency remains fixed, so wavelength changes with the speed.

Angles are measured from the normal. Entering a higher refractive index bends the ray toward the normal; entering a lower index bends it away.

Snell's law

n₁ sin θ₁ = n₂ sin θ₂

Both angles are measured from the normal. If n₂ > n₁, the ray bends toward the normal; if n₂ < n₁, it bends away. Normal incidence (θ₁ = 0) changes speed and wavelength but not direction.

Snell-law explorer

n₁ n₂ θ₁ 40°

Frequency is fixed by the source and stays continuous across a stationary boundary. Because v = fλ, a change in wave speed changes wavelength.

What does not change at the boundary

The source continues oscillating at the same frequency, so the transmitted light has the same frequency as the incident light. Its speed and wavelength change together according to v = fλ.

The reflected wave remains in the first medium, so its speed and wavelength in that medium match those of the incident wave.

Dispersion and real materials

Refractive index usually depends slightly on wavelength. That is why white light can separate into colors in a prism and why lenses can show chromatic aberration. A single quoted n is therefore tied to a wavelength or wavelength range.

Snell's law remains the starting point, but use the appropriate index for the wavelength being considered.

Ray direction, speed and wavelength are separate questions

Snell's law determines direction. The index relation n = c/v determines speed. Frequency continuity then gives wavelength through v = fλ. Treating those as three separate steps prevents the common mistake of claiming that refraction changes the light's frequency.

Another common mistake is measuring θ from the surface instead of the normal. A ray that makes 20° with the surface makes 70° with the normal. Always sketch the normal before inserting angles into Snell's law.

Worked examples

1. Air to glass

Light enters glass (n = 1.50) from air (n = 1.00) at 30°. Find the refracted angle.

Solution

sin θ₂ = (1.00/1.50) sin 30° = 0.3333, so θ₂ ≈ 19.5°.

2. Speed in a medium

Find light speed in material with n = 1.33. Use c = 3.00 × 10⁸ m s⁻¹.

Solution

v = c/n = (3.00 × 10⁸)/1.33 ≈ 2.26 × 10⁸ m s⁻¹.

3. Wavelength change

A 600 nm wave in air enters glass of n = 1.50. Approximate the wavelength in the glass.

Solution

Frequency stays fixed. With air n ≈ 1, λ₂ ≈ λ₁/n = 600/1.50 = 400 nm.