Rainbow Formation

The intuition — a sky full of tiny prisms

Every raindrop is a microscopic ball of water, and a ball of water is a surprisingly good prism. Sunlight hits the front of the drop and bends inward (refraction). It travels through the drop, hits the curved back surface, and bounces once (internal reflection). Then it bends again on the way out. Three optical events — refract, reflect, refract — and the light comes back toward the Sun's direction, tilted by a fixed angle.

Here's the part that surprises people: a single drop doesn't make a rainbow band. It sends light back in a whole range of directions depending on where the incoming ray strikes it. But the math conspires so that far more light comes back at one particular angle than any other. That angle is about 42° away from the line pointing straight away from the Sun. Pile up millions of drops, and every drop sitting at 42° from that line glows — and the locus of all such drops is a circle. You see an arc.

The colors come for free. Water bends violet light slightly more than red light, so each color's "favorite" exit angle is slightly different. The bow isn't one arc — it's a stack of nested arcs, one per wavelength, smeared into the band you recognize.

How it works — refract, reflect, refract

Follow one ray of sunlight into a spherical drop of radius R. It strikes the surface at an impact parameter b (the perpendicular offset from the drop's center, expressed as b/R = sin i, where i is the angle of incidence).

1. Refraction in. At the front surface the ray bends according to Snell's law, sin i = n · sin r, where n ≈ 1.33 is water's refractive index and r is the refraction angle inside the drop.
2. Internal reflection. The ray crosses the drop and hits the back surface. Most of the light leaks out, but a fraction reflects. By the geometry of a chord in a circle, the reflection turns the ray by (180° − 2r).
3. Refraction out. The ray reaches the front surface again and refracts on exit, bending by the same (i − r) as on entry.

Add up every turn and the total deviation of the ray — the angle between the incoming sunlight and the outgoing ray — is:

D(i) = 2(i − r) + (180° − 2r) = 180° + 2i − 4r

You see the bow at the supplement of this: the angular radius from the antisolar point is 180° − D = 4r − 2i. The whole story of "why 42°" is hidden in how D depends on i.

The rainbow angle — why a caustic at 42°

Plot D against the impact parameter b (or equivalently against i). As b runs from the center of the drop to the grazing edge, D first decreases, reaches a minimum, then increases again. Near a minimum, dD/db ≈ 0, which means a wide band of incoming rays all leave at nearly the same angle. That pile-up of rays is a caustic — a bright fold in the light — and it's why the rainbow direction is so much brighter than its surroundings.

Set the derivative to zero to find the special incidence angle:

dD/di = 0   →   cos²i = (n² − 1) / 3

For n = 1.333 (yellow-green light), this gives i ≈ 59.5°, r ≈ 40.4°, and a minimum deviation

D_min = 180° + 2(59.5°) − 4(40.4°) ≈ 138°
rainbow radius = 180° − 138° = 42°

That's it. The 42° is not a coincidence or a measured fudge — it falls straight out of Snell's law plus the geometry of a sphere, first worked out by René Descartes in 1637 (the ray tracing) and refined by Isaac Newton, who added the color story in the 1660s. Light fainter than the caustic still arrives inside the bow (angles smaller than 42°), which is why the sky inside the arc is brighter than the dark band outside it.

Dispersion — sorting the colors

Water's refractive index is not a single number; it drifts with wavelength. This is dispersion, and it's the same effect that makes a glass prism throw a spectrum. Each color therefore has its own rainbow angle:

Color	Wavelength	Index n (water, 20 °C)	Rainbow radius (primary)
Red	700 nm	1.331	42.4°
Orange	620 nm	1.332	42.1°
Yellow	580 nm	1.333	42.0°
Green	530 nm	1.335	41.6°
Blue	470 nm	1.338	41.2°
Violet	400 nm	1.343	40.6°

The full spread is only about 1.8° — roughly three and a half full-moon widths. Red bends least, so its caustic sits at the largest angle and forms the outer edge of the primary bow; violet bends most and forms the inner edge. Because the bands are only ~1.8° wide and overlap heavily, the colors you actually see are pastel and washed-out, not the saturated bands of a textbook prism — each "pure" color is contaminated by the wings of its neighbors.

Primary, secondary, and Alexander's dark band

Some light reflects twice inside the drop before exiting. That double-bounce path has its own caustic — but it lands at a larger angle and with the color order flipped:

Feature	Primary bow	Secondary bow
Internal reflections	1	2
Angular radius	~42° (40.6°–42.4°)	~51° (50.4°–53.4°)
Color order (outer→inner)	Red → violet	Violet → red
Relative brightness	1 (reference)	much fainter — the extra back-surface reflection keeps only ~6% of the light (the rest escapes)
Width	~1.8°	~3.0° (wider, more washed out)
Discovered/explained	Descartes, 1637	Newton; double-reflection geometry

Between the two bows lies Alexander's dark band, named after Alexander of Aphrodisias around 200 AD. Single-reflection light only ever reaches angles up to ~42° (and floods the region inside), while double-reflection light only appears beyond ~51°. The 42°–51° wedge gets almost no scattered light from either path, so it looks distinctly darker than the sky on either side. Spotting that band is the easiest way to confirm you're looking at a real two-bow rainbow.

By the numbers — what reaches your eye

At each surface, some light transmits and some reflects (Fresnel equations). For unpolarized light hitting water near these angles, only about 4% reflects at the front, but at the curved back surface a useful fraction internally reflects. Tracing the energy budget for the single-reflection (primary) path:

Quantity	Value	Why it matters
Sun's angular size	0.53°	Blurs every color band by ~0.5°, softening the bow
Drop size for sharp bow	0.5–2 mm	Bigger drops = brighter, more saturated colors
Drop size for white "fogbow"	< 0.05 mm	Diffraction smears all colors together into a pale arc
Fraction of incident light in primary bow	~few %	Most sunlight transmits straight through; the bow is the leftover caustic
Max Sun elevation for a ground rainbow	42°	Above this the whole bow sits below the horizon
Supernumerary spacing	fractions of a degree	Interference fringes just inside the violet — wave optics, not ray optics

The ray-optics picture nails the position of the bow but predicts an infinitely sharp edge. Real rainbows have faint pinkish-green supernumerary arcs hugging the inside of the primary bow — these are interference fringes between rays that take slightly different paths to the same exit angle, and only the full wave theory (Airy's 1838 diffraction integral) reproduces them. The drop size sets their spacing, which is why supernumeraries show up best in fine, uniform drizzle.

Where rainbow physics shows up

• Atmospheric optics. Fogbows (tiny drops, no color), moonbows (faint rainbows by moonlight), and sea-spray bows all use the same 42° geometry with different drop sizes.
• Glories and coronae. Backscatter rings around the antisolar point (often seen from planes around the shadow of the aircraft) are the wave-optics cousins of the rainbow caustic.
• Remote sensing. The exact rainbow angle depends on n, and n depends on liquid composition — "rainbow refractometry" measures droplet size and refractive index in fuel sprays and clouds without touching them.
• Optical engineering. Caustics — bright folds where dD/di = 0 — govern lens aberrations, the bright cusp at the bottom of a coffee cup, and the design of reflectors.
• Astronomy and other liquids. "Rainbows" on Saturn's moon Titan would form from liquid methane (n ≈ 1.29), placing the primary bow at a different angle than Earth's water bow.

Common misconceptions and edge cases

• "The rainbow is at a fixed place." No — it's a direction, not an object. It's centered on the antisolar point (your head's shadow), so every observer has their own personal rainbow and you can never walk to its end.
• "The drop splits white light like a prism slit." A single drop sends light back over a continuous range of angles; the bow exists only because the rays pile up at the caustic. It's a statistical brightness peak, not a clean spectrum from one drop.
• "Rainbows are half-circles." They're full 42° circles around the antisolar point. The horizon cuts off the lower half on the ground; from a plane you can see the complete ring.
• "Red is on top because it has the most energy." Red is on the outside because it refracts the least (lowest n), so its caustic sits at the largest angle. Photon energy has nothing to do with the ordering.
• "The second rainbow's reversed colors are an illusion." They're real — the extra internal reflection genuinely flips the geometry, putting red on the inside of the secondary bow.
• "Ray optics explains everything." It gets the 42° angle right but fails on supernumerary arcs and on fogbows, where the drops are small enough that diffraction dominates. You need wave optics for the fine structure.