Rainbow Physics — Dispersion, Internal Reflection and Supernumerary Arcs

A rainbow is not an object with a fixed location — it is an optical phenomenon that depends on the angle between the Sun, a raindrop, and your eye. Understanding why it is round, why red is always on the outside, and what causes the faint supernumerary arcs beneath it requires Snell's law, calculus, and wave optics.

1. Geometry of a Rainbow

You always see a rainbow as a coloured arc centred on the antisolar point — the point exactly opposite the Sun from your eye. For the primary rainbow this arc is at ~42° from the antisolar point; the secondary at ~51°.

A rainbow is a cone of light: all the raindrops at 42° from your antisolar point simultaneously send red light to your eye. The arc shape is the intersection of this cone with the hemisphere of sky in front of you. From an aeroplane you can see the full circle.

2. Descartes' Ray Theory

René Descartes (1637) computed the path of a ray through a spherical raindrop:

Refraction at entry (air → water): Snell's law, n·sin(r) = sin(i)
One internal reflection
Refraction at exit (water → air)

The total deviation angle D(i) of the outgoing ray depends on the angle of incidence i:

Deviation angle for primary rainbow D(i) = π + 2i − 4·arcsin(sin(i)/n) n = refractive index of water (~1.333 for yellow light) i = angle of incidence at droplet surface Minimum deviation (Descartes' ray) at dD/di = 0: i_min = arccos( √((n²−1)/3) ) ≈ 59.5° D_min ≈ 137.5° → rainbow angle = 180° − 137.5° = 42.5°

At the minimum deviation angle, many rays pile up — that's why the rainbow is bright at one specific angle. Rays at other angles are more spread out and dimmer.

3. Dispersion and Colour Separation

Water is dispersive: its refractive index varies with wavelength. Violet light bends more than red:

Colour	Wavelength (nm)	n (water)	Rainbow angle
Red	650	1.331	42.5°
Yellow	580	1.333	42.0°
Green	530	1.335	41.5°
Violet	400	1.342	40.5°

Red light exits at a larger angle from the antisolar point — so in the sky, red appears at the outer edge (top of the arc when the Sun is low) and violet at the inner edge. The total angular spread of colours is about 2°.

4. Secondary Rainbow

In some raindrops light undergoes two internal reflections before exiting. This forms the faint secondary rainbow at ~51°. Two reflections means:

The colour order is reversed: red on the inside, violet on the outside.
It is ~40% dimmer than the primary (each reflection loses some light to transmission).
The deviation angle formula becomes: D₂(i) = 2π − 2i + 6·arcsin(sin(i)/n), minimum ~51°.

5. Alexander's Dark Band

Between the primary (42°) and secondary (51°) rainbows is a noticeably darker region of sky — Alexander's dark band. This is not shadow; it is a consequence of the geometry. Between 42° and 51° from the antisolar point, no rainbow-deviated rays reach the eye: the primary sends light inside 42° and the secondary sends light outside 51°. The sky outside both bows and inside the primary receives reflected and scattered light; the band between receives neither, and appears darker by contrast.

6. Airy Wave Theory and Supernumerary Arcs

Descartes' ray theory predicts one sharp arc at the minimum deviation angle. In reality, just inside the primary bow you often see faint, closely spaced arcs — supernumerary arcs — alternating pink and green. These are a wave interference effect, not explainable by geometric optics.

George Airy (1838) computed that near the rainbow angle, two ray paths reach the same exit direction (one above and one below the minimum deviation ray). These two rays travel slightly different path lengths through the droplet and interfere constructively or destructively:

Airy function — intensity near rainbow angle I(θ) \propto |Ai(x)|² Ai = Airy function (solution to d²y/dz² = z\cdoty) x = k^(2/3) \cdot (θ - θ_bow) / (some scaling) k = 2π/λ (wavenumber) Supernumerary spacing smaller for larger droplets (larger k)

Supernumerary arcs are most visible in rainbows from small, uniform droplets (fog, drizzle). Large, variable raindrops smear out the interference fringes.

7. Polarisation

Rainbow light is strongly linearly polarised — approximately 90° polarised, tangent to the arc. This is because internal reflection inside the droplet is near Brewster's angle for the refracted ray. If you rotate polarised sunglasses 90° while looking at a rainbow, the rainbow nearly disappears.

Practical use: Photographers use circular polarising filters to enhance rainbow contrast against a sky background. The sky itself is partially polarised (Rayleigh scattering), so the combination of two polarising effects can be tuned for dramatic effect.

8. Other Bows

Tertiary rainbow (3 reflections): At ~140° — so it appears on the same side as the Sun, in the bright sky. Extremely faint, almost impossible to see with naked eye. First photographed in 2011.
Fogbow (white rainbow): Formed by tiny fog droplets (~10–100 µm). Supernumerary fringes overlap across all colours → white or pale pink/blue. Much wider (14–35°) than a normal bow.
Moonbow: Rainbow produced by moonlight instead of sunlight. Same geometry; appears white to dark-adapted human vision (colour vision requires more light). Cameras on long exposures show colour.
Circumzenithal arc: Not a rainbow — formed by hexagonal ice plate crystals, not raindrops. A short, vivid arc near the zenith with colours reversed from a rainbow.