Refraction, Snell’s Law, and Refractive Index

Refraction is the bending of light as it passes from one transparent medium into another because its speed changes; Snell’s law states n₁ sin θ₁ = n₂ sin θ₂, where the refractive index n = c ÷ v is the ratio of the speed of light in a vacuum to the speed in the medium. In brief: light bends toward the normal when it enters a medium with a higher refractive index (slower light) and away from the normal when it enters a medium with a lower one. Glass has n of about 1.5, water 1.33 and air 1.0003. The frequency of the light does not change; its wavelength becomes λ ÷ n.

A straw in a glass of water seems broken, and a pool looks shallower than it is. These are effects of refraction. This guide defines the refractive index, states Snell’s law, explains apparent depth and the path of light through a glass block, compares typical indices in a table, works through calculations and corrects common mistakes.

What is the refractive index?

The absolute refractive index of a medium is n = c ÷ v, where c = 3.00 × 10⁸ m/s is the speed of light in a vacuum and v the speed in the medium. Since v < c, n ≥ 1. A higher n means that light travels more slowly, and that the medium is “optically denser”. The index depends slightly on the wavelength (the cause of dispersion) and on temperature. The relative index from medium 1 to medium 2 is n₂₁ = n₂ ÷ n₁ = v₁ ÷ v₂.

What is Snell’s law?

When a ray passes from medium 1 into medium 2, with angles θ₁ and θ₂ from the normal:

n₁ sin θ₁ = n₂ sin θ₂

• If n₂ > n₁ (air to glass), then θ₂ < θ₁: the ray bends toward the normal.
• If n₂ < n₁ (glass to air), then θ₂ > θ₁: the ray bends away from the normal.
• For θ₁ = 0 (along the normal), there is no bending.
• The ray is reversible: the path in the opposite direction is the same.
• The incident ray, the refracted ray and the normal are in one plane.

The speed and the wavelength change in the new medium (λ₂ = λ₁ × n₁ ÷ n₂), but the frequency does not. Some of the light is also reflected at the boundary.

What are the effects of refraction?

• Apparent depth: viewed from above, an object in water seems closer to the surface. For near-normal viewing, apparent depth = real depth ÷ n. A pool 2.0 m deep seems 2.0 ÷ 1.33 = 1.5 m deep.
• Lateral displacement: a ray passing through a parallel-sided glass block emerges parallel to the incident ray, but shifted sideways.
• Mirages and twinkling stars: caused by gradual changes in the refractive index of air with temperature.
• Lenses and prisms depend on refraction at curved or inclined surfaces.
• Sparkle of diamonds (n = 2.42), whose high index gives a small critical angle and much internal reflection.

Refractive indices of materials

Worked example: refraction calculations

1. Air to water. Light hits the surface of water at θ₁ = 45° from the normal. 1.00 × sin 45° = 1.33 × sin θ₂, so sin θ₂ = 0.7071 ÷ 1.33 = 0.532, and θ₂ = 32.1°. The ray bends toward the normal.
2. Glass to air. Light in glass (n = 1.50) is incident on the glass-air boundary at 20°. 1.50 × sin 20° = 1.00 × sin θ₂, so sin θ₂ = 1.50 × 0.3420 = 0.513, and θ₂ = 30.9°. It bends away from the normal.

3. Speed. The speed of light in glass of n = 1.50: v = c ÷ n = 3.00 × 10⁸ ÷ 1.50 = 2.00 × 10⁸ m/s.
4. Wavelength. Red light of wavelength 650 nm in air enters glass (n = 1.50): λ = 650 ÷ 1.50 = 433 nm. The frequency is the same: f = c ÷ λ_air = 3.00 × 10⁸ ÷ 650 × 10⁻⁹ = 4.6 × 10¹⁴ Hz.
5. Finding n. A ray in air at 60° enters a transparent liquid and is refracted at 40°. n = sin 60° ÷ sin 40° = 0.8660 ÷ 0.6428 = 1.35.

Who discovered Snell’s law?

Ptolemy measured angles of refraction in the second century CE. Ibn Sahl described the law in a manuscript on burning mirrors and lenses in 984. Thomas Harriot found it in 1602, and Willebrord Snellius rediscovered it in 1621. René Descartes published it in 1637, and Pierre de Fermat derived it from his principle of least time in 1662. Christiaan Huygens explained it with the wave theory, which predicts that light is slower in denser media. This was confirmed by the measurements of Léon Foucault in 1850.

Which mistakes are common?

Mistake 1: Measuring angles from the surface

Use angles from the normal in Snell’s law.

Mistake 2: Saying that the frequency of light changes in glass

It does not. The speed and wavelength change.

Mistake 3: Mixing up the direction of bending

Higher index: toward the normal. Lower index: away from the normal.

Mistake 4: Using the real depth as the apparent depth

Apparent depth = real depth ÷ n for viewing near the normal.