🌈 Atmospheric Optics — Rainbows & Halos

The primary rainbow appears at ~42° from the anti-solar point because that is the angle of minimum deviation for a light ray that enters a spherical water droplet, undergoes one internal reflection, and exits. Red light (n ≈ 1.331) has its minimum deviation at 42.3°, while violet (n ≈ 1.344) exits at 40.6°, spreading the colours across about 1.7°.

The secondary rainbow forms from rays that undergo two internal reflections inside the droplet. Each extra reflection adds a path-reversal, so the colour order is reversed — violet appears on the outside at ~53° and red on the inside at ~51°. Two reflections also cost more light, making the secondary bow about three times fainter than the primary.

Between 42° and 51° there is a region called Alexander's dark band where no rainbow rays reach the observer — primary rays all exit below 42° and secondary rays all exit above 51°. The sky inside the primary bow appears brighter because scattered light from the droplets reaches the observer from those directions, while Alexander's band receives no such light.

The 22° halo arises when sunlight refracts through two prism faces of randomly oriented hexagonal ice crystals. The 60° apex angle of the prism gives a minimum deviation of ~21.8° for water ice (n ≈ 1.31). Rays cannot exit at smaller angles because they undergo total internal reflection, so light accumulates at this minimum, producing a bright ring with a dark interior.

The 46° halo forms when light enters through a prism face and exits through the basal (flat top/bottom) face of a hexagonal ice crystal, giving a 90° prism angle instead of 60°. The minimum deviation for a 90° prism is ~45.7°. This halo is rarely seen because it requires a specific crystal geometry and typically appears much fainter than the 22° halo.

Mie scattering is the exact electromagnetic solution for scattering by spherical particles of any size, including raindrops. For large droplets (radius much greater than wavelength), Mie theory converges to geometric optics and predicts the same 42°/51° rainbow angles. Mie scattering also predicts supernumerary bows — narrow interference fringes just inside the primary rainbow caused by wave interference between rays taking different paths through the droplet.

Rainbows appear in the part of the sky directly opposite the sun — the anti-solar point. The minimum-deviation angles (42° for primary, 51° for secondary) are measured from the anti-solar point outward. If the sun is above 42°, the primary bow falls below the horizon and cannot be seen from the ground; circular rainbows are only visible from aircraft or elevated positions.

Larger droplets (radius above 1 mm) produce vivid, narrow, bright rainbows with well-separated colours. Smaller droplets (drizzle, fog, radius below 0.1 mm) produce fogbows — broad, nearly white arcs — because diffraction becomes significant relative to refraction. Very small droplets produce a featureless white arc as colour spreads merge together.

Yes. Ice has wavelength-dependent refractive index, so red light (n ≈ 1.306) bends less than violet (n ≈ 1.318). At the inner edge of the 22° halo, red appears first, giving the characteristic reddish-orange inner rim. The colour spread in ice halos is similar to that in rainbows, though halos often appear whiter overall because dispersion in ice is weaker than in liquid water.

Rainbow brightness depends on droplet size (larger droplets give a brighter primary), droplet number density (more drops equals brighter bow), the sun's elevation angle, and the refractive index of water. Fresnel reflection losses at each droplet surface reduce brightness further, and the secondary bow loses an additional factor of roughly three due to the second internal reflection and partial polarisation effects.