Snell’s law explained: how light bends between materials

Snell's law describes how light changes direction when it passes from one transparent material into another: n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the two materials and θ₁ and θ₂ are the angles the ray makes with the normal, an imaginary line perpendicular to the surface. Light entering water from air at 45° continues at about 32°. The Snell's law calculator solves it for common materials; this guide explains why light bends, and what happens when it cannot get out.

Refractive index

Light travels at about 299,792 km/s in a vacuum but more slowly in matter. The refractive index n is the ratio of the two speeds, n = c ÷ v. Typical values for yellow light:

• air 1.0003, so close to 1 that it is often taken as exactly 1;
• water 1.333, where light travels at about 225,000 km/s;
• crown glass about 1.52, or about 197,000 km/s;
• diamond 2.417.

The index varies slightly with wavelength, an effect called dispersion. Violet light is slowed a little more than red light in glass, which is how a prism spreads white light into a spectrum and how raindrops make rainbows. Even air bends light measurably: near the horizon, refraction lifts the Sun's image by about half a degree, as How sunrise and sunset times are calculated — latitude, the seasons, and the equation of time explains.

Why light bends

Picture a wide beam of light hitting a water surface at an angle. One edge of the beam reaches the water first and slows down while the other edge is still moving through air at full speed, so the wavefront swings round, the way a marching column turns when one side steps onto mud. Going into a slower material (higher n), light bends towards the normal; going into a faster one (lower n), it bends away from it. A ray that meets the surface head-on, at 0°, does not bend at all. The law is named after the Dutch astronomer Willebrord Snellius, who worked it out in 1621, although the Persian scientist Ibn Sahl had described it in Baghdad in 984.

Worked examples

• Air to water at 45°: sin θ₂ = 1.0003 × sin 45° ÷ 1.333 ≈ 0.531, so θ₂ ≈ 32.0°. This is why a straw in a glass looks bent and a swimming pool looks shallower than it is.
• Air to crown glass at 30°: sin θ₂ = sin 30° ÷ 1.52 ≈ 0.329, so θ₂ ≈ 19.2°.
• Water to air at 30°: sin θ₂ = 1.333 × 0.5 ≈ 0.667, so θ₂ ≈ 41.8°. The ray bends away from the normal as it leaves the water.

Lenses are carefully curved surfaces that use exactly this bending to bring rays to a focus. The lens equation calculator takes over from there, working with focal lengths rather than individual rays.

The critical angle

The last example points to something interesting. When light goes from a higher index to a lower one, the refracted angle is always larger than the incident angle, so at some incident angle the refracted ray reaches 90° and skims along the surface. That incident angle is the critical angle. Setting θ₂ = 90° in Snell's law, where sin 90° = 1, gives sin θc = n₂ ÷ n₁. For water to air the critical angle is about 48.6°, for crown glass to air about 41.1°, and for diamond to air only about 24.4°. The calculator reports it automatically whenever light is heading into a lower-index material.

Total internal reflection

Beyond the critical angle Snell's law has no solution, because sin θ₂ would have to be greater than 1. No light passes through at all; it is all reflected back inside. This total internal reflection really is total, more efficient than a metal mirror, and technology relies on it. Optical fibres carry telephone and internet traffic by trapping light in a glass core surrounded by cladding of slightly lower index, so light striking the boundary at a shallow angle bounces along the fibre for kilometres. Prisms in binoculars use it to fold the light path. Gemstone cutters exploit it too: diamond's high index and small critical angle mean that light entering the top of a well-cut stone is reflected around inside and sent back out through the front, which is a large part of its brilliance.

You can see total internal reflection for yourself underwater. Look up at the surface at a shallow angle and it acts like a silvery mirror; only within a cone of about 48.6° from straight up can you see out into the air. For converting the wavelengths of different colours into frequencies, the wavelength and frequency calculator is related.