Refraction and Snell's law

In vacuum every electromagnetic wave travels at c = 2.998 × 10⁸ m s⁻¹. Inside matter the oscillating field drives the bound electrons, they re-radiate, and the sum of the original wave and everything they emit advances more slowly. Call that speed v and define the refractive index as n = c/v. It is a pure number, it is at least 1 for ordinary transparent media, and it is the only property of the material that plain refraction cares about. Air is 1.0003, near enough to 1 for any bench work; water is 1.333, so light crosses a swimming pool at 2.25 × 10⁸ m s⁻¹; crown glass is about 1.52; diamond is 2.42, or 1.24 × 10⁸ m s⁻¹. Nothing here strains relativity: c is the ceiling on the speed of a signal, and n > 1 says the wave in the medium runs below that ceiling, not above it.

Take a plane wavefront — a line of crests — running at angle θ₁ to a flat boundary, so one end of it reaches the surface before the other does. Let the trailing end need a time Δt to arrive. In that Δt it covers v₁Δt in medium 1, while the end that crossed first has already run v₂Δt into medium 2. Both of those distances hang off the same stretch L of boundary between the two crossing points, so sin θ₁ = v₁Δt/L and sin θ₂ = v₂Δt/L. Divide the two, and Δt and L both vanish: sin θ₁/sin θ₂ = v₁/v₂ = n₂/n₁, which rearranges to n₁ sin θ₁ = n₂ sin θ₂. The front has pivoted because one end was held back, and the ray turns with it. Air into water at θ₁ = 40°: sin θ₂ = 0.6428/1.333 = 0.4822, so θ₂ = 28.8°. Fermat's principle reaches the identical equation by minimising the travel time instead of tracking wavefronts.

Three conventions travel with the equation. Measure every angle from the normal — the line perpendicular to the surface at the point of incidence — never from the surface itself. That is not cosmetic: a ray described as 60° to a water surface is 30° to the normal, and sin 30° = 0.500 against sin 60° = 0.866, so confusing them scales the answer by 1.7. Second, the incident ray, the normal and the refracted ray all lie in one plane, the plane of incidence; refraction never throws a ray sideways out of it, because the boundary looks the same everywhere along itself. Third, normal incidence is the case that catches people out: θ₁ = 0 gives sin θ₂ = 0 and θ₂ = 0, so a ray entering a window face-on slows down and shortens its wavelength but does not bend at all. A speed change is necessary for bending; it is not sufficient.

Direction first. If medium 2 is the slower one — larger n — then sin θ₂ < sin θ₁ and the ray bends towards the normal; crossing the other way, into the faster medium, it bends away from it. The size of the bend is where intuition fails, because it is the sines that stand in proportion, not the angles. Take air into glass of n = 1.50. At θ₁ = 30° Snell gives θ₂ = 19.5°; double the incidence to θ₁ = 60° and θ₂ climbs only to 35.3°, well short of the 38.9° that doubling would demand, while the deviation θ₁ − θ₂ swells from 10.5° to 24.7°. Each extra degree of incidence buys less turning inside than the degree before it. Small angles are the exception worth keeping: for θ in radians sin θ ≈ θ, so n₁θ₁ ≈ n₂θ₂, and that paraxial form is the seed the refracting-surface and thin-lens equations later in this section grow from. The law also carries a ceiling: sin θ₁ ≤ 1 forces sin θ₂ ≤ n₁/n₂, so a ray arriving from air can never be more than 41.8° from the normal inside n = 1.50 glass, however grazing its approach.

Swap the labels 1 and 2 in n₁ sin θ₁ = n₂ sin θ₂ and the same equation comes back. The path is therefore reversible: send a ray backwards along the refracted direction and it leaves along the original incident line, angle for angle. Two consequences. A ray crossing a slab with parallel faces is bent at entry and bent back by exactly as much at exit, because the second refraction is the first one run in reverse, so it emerges parallel to the direction it arrived on. What is not restored is position. The ray spent its time inside the slab on a tilted path, so it steps sideways by Δ = t sin(θ₁ − θ₂)/cos θ₂. A 5.0 mm window of n = 1.52 struck at 45° refracts to 27.7° inside and shifts the emerging ray by 1.7 mm — enough to matter when that slab sits in front of a detector.

Crests can neither pile up at the boundary nor disappear into it: whatever arrives each second must leave each second, so the frequency f is the same on both sides, fixed by the source and not by the material. Since v = fλ and v falls by the factor n, the wavelength has to fall with it: λ = λ₀/n. Sodium light of λ₀ = 589 nm runs at 442 nm inside water. Colour follows f rather than λ, which is why nothing underwater changes hue. One caveat closes this topic and opens a later one. Treating n as a single number is an approximation: in BK7 crown glass n is 1.5224 at 486 nm and 1.5143 at 656 nm, so blue is bent a little more than red by the same surface. That drift is dispersion, what a prism lives on, and this section's final topic takes it up alongside the point where the ray model itself gives out.