Rainbow Formation in Mist: Ray Optics & Dispersion (2D)

Scene Geometry

Sun elevation above horizon 20°

Observer position (sideways) 0 m

Droplet diameter 0.60 mm

Show secondary bow (2 internal reflections)

Single-Droplet Ray Trace

Ray colour Red (700 nm)

Incidence angle θᵢ on droplet 59.6°

Live Readout

Deviation D(θᵢ)

137.9°

Viewing angle γ

42.1°

Red primary bow

42.3°

Violet primary bow

40.5°

Diffraction blur

0.05°

Bow visibility

above horizon

How the ~42° angle is derived

A ray hits a spherical droplet at incidence angle θᵢ (from the surface normal). Snell's law gives the refraction angle θᵣ = asin(sinθᵢ / n). Tracing the geometry through entry, one internal reflection and exit gives the total deviation:

D(θᵢ) = π + 2θᵢ − 4θᵣ   (one internal reflection, primary bow)
D(θᵢ) = 2π + 2θᵢ − 6θᵣ  (two internal reflections, secondary bow)

D(θᵢ) is scanned numerically over every θᵢ from 0–90° (2000 samples, live, in this page's JS) rather than solved in closed form. Almost all incoming rays exit at slightly different angles, spreading the light — but near the extremum of D(θᵢ) (a minimum for the primary bow, a maximum for the secondary), many neighbouring rays converge to nearly the same exit angle. That angular pile-up of light is the rainbow. Because water's refractive index n depends on wavelength (dispersion — n≈1.330 for red, n≈1.343 for violet), the extremum sits at a slightly different angle for each colour, which is why the bow separates into a spectrum.

Sun elevation sets the antisolar direction; the bow's peak height above the horizon is (bow angle − sun elevation) — push the sun above ~42° and the primary bow sinks entirely below the horizon, exactly as it does at real local noon.
Droplet diameter adds a diffraction blur ≈ 1.22λ/d (Airy-disk half-angle) around each colour's angle; shrink the droplets toward fog size and the colour bands overlap into a white fogbow, matching real atmospheric optics.
Observer position shifts the eye relative to the fixed mist field, sweeping which droplets satisfy the angle condition — the rainbow is not a fixed object in the mist, it is the set of droplets currently at the right angle for *this* eye.

Primary bow (1 internal reflection) Secondary bow (2 reflections, dimmer, reversed) Ray trace: entry → reflect → exit