Snell's Law and Refraction Calculator

Snell's law

When light passes from one transparent medium into another, its speed changes and its path bends. The relation between the angle of incidence and the angle of refraction is Snell's law:

n₁ × sin θ₁ = n₂ × sin θ₂

n₁ and n₂ are the refractive indices of the two media, θ₁ is the angle of incidence and θ₂ the angle of refraction. Both angles are measured from the normal, the line drawn perpendicular to the surface.

Refractive index

The refractive index is the ratio of the speed of light in vacuum to its speed in the medium: n = c / v. It is 1 in vacuum, 1.0003 in air, 1.333 in water, about 1.5 in glass and 2.419 in diamond. Light bends toward the normal when it enters a medium with a higher index and away from the normal when it enters one with a lower index.

Example

For light going from air into water at 45 degrees, sin θ₂ = 1 × sin 45° / 1.333 = 0.5305, so θ₂ = 32.0 degrees. The ray has moved 13 degrees closer to the normal. The speed of light in water is 299 792 / 1.333 = 224 900 km/s.

Critical angle and total internal reflection

Going from a higher index to a lower index, the angle of refraction is larger than the angle of incidence. As the angle of incidence grows, the refracted ray swings toward the surface, and at the critical angle given by sin θc = n₂ / n₁ it skims along it. At larger angles no light gets through at all: it is completely reflected. The critical angle is 48.6 degrees from water to air and 24.4 degrees from diamond to air. The sparkle of a diamond and optical fibres both rely on this effect.

Limits

The refractive indices are for yellow light of 589 nm and vary slightly with colour. The index of glass depends on the type, and the value in the list is approximate. The tool does not calculate the intensity of the reflected light.