Snell's Law of Refraction

By Snell law n₁·sin(α₁) = n₂·sin(α₂). Rearranging for the angle of refraction gives α₂ = arcsin((n₁/n₂)·sin(α₁)). Insert the refractive indices of the two media and the angle of incidence. Example: light passes from air with n₁ = 1.00 into glass with n₂ = 1.50 at α₁ = 30°. Then sin(α₂) = (1.00/1.50)·sin(30°) = 0.333, so α₂ ≈ 19.5°. On entering a denser medium the ray is refracted towards the normal, so the angle becomes smaller. All angles are measured from the normal, not from the surface.

Snell law is defined so that the angles of incidence and refraction are measured between the ray and the normal, that is the perpendicular to the interface. This convention follows from the derivation via Fermat principle and makes the formula unambiguous. A common mistake is to measure the angle to the surface; then you confuse α with 90° − α, and sine and cosine swap, which leads to wrong results. At perpendicular incidence the angle to the normal is zero and the ray is not refracted at all. In every problem check whether the given angle is really measured to the normal, and convert if necessary.

Total internal reflection occurs when light passes from an optically denser into a thinner medium, that is from a large to a small refractive index, and the angle of incidence exceeds a certain critical angle. At the critical angle α_crit the angle of refraction becomes exactly 90°, and the refracted ray runs along the interface. For larger angles there is no refraction any more and the light is fully reflected. You compute the critical angle from sin(α_crit) = n₂/n₁. For the transition glass to air with n₁ = 1.5 and n₂ = 1.0 you get α_crit ≈ 41.8°. This principle is the basis of optical fibres, in which light is guided with low loss through repeated total internal reflection.

The refractive index n states by what factor light propagates more slowly in a medium than in vacuum: n = c/v, where c is the vacuum speed of light and v the speed in the medium. Vacuum has n = 1, air practically also, water about 1.33 and glass about 1.5. A higher refractive index means slower light propagation and stronger refraction at the interface. Because the speed of light depends slightly on the wavelength, n is also slightly wavelength-dependent; this is called dispersion and is the reason why a prism splits white light into its colours. The refractive index is thus the central material quantity of ray optics.

The apparent bend arises from refraction of light at the water surface. Light rays coming from the submerged part of the straw are refracted away from the normal as they pass from the denser water into the thinner air. As a result they reach your eye at a changed angle, and your brain extends them back in a straight line, placing the lower part of the straw at a wrong, shifted position. So the straw appears bent or offset at the water surface. The same principle makes objects in water appear shallower than they really are and is a direct, everyday consequence of Snell law of refraction.