How Does a Rainbow Work?

A rainbow appears when sunlight enters millions of tiny raindrops, bounces inside them, and exits at a precise angle. Each colour of light bends by a slightly different amount, spreading white sunlight into its full spectrum.

Refraction — Bending of Light

Light travels at different speeds in different materials. In a vacuum it moves at c = 3 × 10⁸ m/s. In water it slows down to about 2.25 × 10⁸ m/s. When light crosses from one medium to another at an angle, it changes direction. This bending is called refraction.

The amount of bending is described by Snell's Law:

n₁ sin θ₁ = n₂ sin θ₂

Here n is the refractive index (n = 1 in air, n ≈ 1.33 in water), and θ is the angle from the perpendicular (normal) to the surface.

Inside a Raindrop

When a ray of sunlight hits a spherical raindrop, three things happen:

Refraction at entry. The ray bends as it passes from air into water.
Internal reflection. The ray hits the back of the drop at an angle that causes it to reflect back inward.
Refraction at exit. The ray bends again as it passes back out through the front of the drop.

The net result: the ray exits heading back towards the Sun but deflected by a specific angle — roughly 138° for red light, which means it exits about 42° away from the incoming direction.

Dispersion — Splitting Colours

The refractive index of water is not exactly the same for all colours of light. It is slightly higher for violet (n ≈ 1.343) than for red (n ≈ 1.331). This means each colour bends by a slightly different amount when it enters and exits the drop. This spreading of colours is called dispersion.

The result is that each colour exits the raindrop at a slightly different angle:

Red: ~ 42.3° from the anti-solar point (the point directly opposite the Sun)
Orange: ~ 41.7°
Yellow: ~ 41.4°
Green: ~ 41.0°
Blue: ~ 40.3°
Violet: ~ 40.0°

Because your eye only receives light from the drops that are at the right angle for each colour, you see a distinct band of colour across the sky — a rainbow!

The Magic 42° Angle

Why does red appear at 42° while violet appears at 40°? The angle is determined by the geometry of the drop and the refractive index. René Descartes worked this out geometrically in 1637.

The key insight is that there is a minimum deviation angle — most rays cluster near this angle and very few rays exit at other angles. This concentration of rays at about 138° total deflection (42° from anti-solar) is why you see a bright arc rather than a diffuse glow.

Observable: The sky is brighter inside the rainbow (closer to the anti-solar point) and darker outside. This is because light scattered by all the drops inside the 42° circle adds up — it is called Alexander's dark band outside the primary arc.

Double Rainbows

A secondary rainbow appears at about 51° and has its colours reversed (red on the inside, violet on the outside). It is caused by light that undergoes two internal reflections inside each drop instead of one.

Each internal reflection loses some light intensity, which is why the secondary rainbow is always fainter — and why a third rainbow is almost impossible to see with the naked eye.

Moon rainbows (moonbows): On nights with a bright full moon and light rain, you can see a faint rainbow illuminated by moonlight. It looks colourless to our eyes because the light is too dim to trigger the colour-sensing cones in the retina.

Try It Yourself

On a sunny day, stand with your back to the Sun and spray a fine mist of water from a garden hose. You will see a rainbow in the mist.
Shine a torch through a round glass of water in a dark room — the refracted light will cast a rainbow on the wall.
Use a CD or DVD as a diffraction grating to spread white light into its colours (a different mechanism, but same beautiful result).

Explore further: Try the Rainbow simulation to trace individual light rays through raindrops in real time.

🌈 Open Rainbow Ray Tracer →