Refraction and total internal reflection

Light inside a diamond can only get out if it meets the surface within 24.4° of the normal, which is why diamonds sparkle.

Why light bends

A straw in a glass of water looks broken at the surface, and a fish in a tank can seem to be in two places at once. These are effects of refraction, the change of direction of light as it passes from one material into another. OpenStax explains the cause: light changes speed. Light is slower in glass or water than in air because it interacts with the atoms of the material. How much slower is measured by the index of refraction, n = c/v, where c is the speed of light in a vacuum and v its speed in the material. Since light is never faster than c, n is always at least 1.

Why should a change of speed change the direction? OpenStax uses a lawn mower. Push it at an angle from a footpath onto grass: the wheel that reaches the grass first slows down while the other is still on the path, so the mower swings toward the perpendicular. Going from grass back onto the path, the first wheel speeds up and the mower swings away from the perpendicular. Light does the same. Entering a slower material it bends toward the normal; entering a faster one it bends away. The path is exactly reversible, and light that hits the boundary head on, along the normal, goes straight through without bending.

Snell’s law

The exact rule is the law of refraction, or Snell’s law: n₁ sin θ₁ = n₂ sin θ₂, with both angles measured from the normal. OpenStax notes that it is named after the Dutch mathematician Willebrord Snell (1580 to 1626), although Ibn Sahl had found it in 984 and used it in his work On Burning Mirrors and Lenses. In one OpenStax example, light goes from air into a material at 30.0° and continues at 22.0°; then n₂ = sin 30.0° ÷ sin 22.0° = 1.33, the index of water. In another, light enters diamond at 30.0° and bends to only 11.9°: the bigger the change in speed, the bigger the change in direction.

Total internal reflection

When light goes from a slower material into a faster one, from water into air for example, it bends away from the normal, so the refracted angle is larger than the incident angle. Tilt the incoming ray further and the refracted ray swings toward the surface. At one angle, the critical angle θc, it runs along the surface at 90°. Beyond that no light gets out at all: it is all reflected back inside. OpenStax calls this total internal reflection and gives θc = sin⁻¹(n₂/n₁), which only works when n₁ > n₂. For water to air θc is 48.6°, for polystyrene 42.2°, and for diamond only 24.4°. Going the other way, from air into water, total internal reflection cannot happen.

Total internal reflection is a perfect mirror that needs no silver coating, and it has many uses. In an optical fibre, light enters a thin glass core and keeps striking the side at angles beyond the critical angle, so it is reflected again and again and follows the fibre around bends. A coating of lower index, the cladding, keeps the light in even when fibres touch. OpenStax notes that most telephone and internet traffic now travels as laser light in such fibres, and that bundles of fibres in endoscopes let doctors see inside the body. Binoculars use prisms with total internal reflection instead of mirrors. In a diamond the small critical angle traps light, which bounces around inside and escapes only at a few places: the sparkle.