The Physics of Rainbows: Why You Can Never Walk to the End of One

You’ve Seen Hundreds of Them and Don’t Know How They Work

I’m willing to bet you’ve seen a rainbow, thought “oh, nice,” and moved on. Maybe you vaguely remember something about prisms from school. But I’d also bet you can’t explain why it’s an arc, why the colours are in that order, why there’s sometimes a second one above it with reversed colours, or why the sky between the two is darker than the sky outside.

None of this is obvious. Isaac Newton figured out the colour part in the 1660s. René Descartes worked out the geometry in 1637. The full wave-optics treatment wasn’t completed until George Airy’s work in 1836. It took two centuries of physics to fully explain something you can see on a rainy afternoon.

What’s Actually Happening

A rainbow is not an object. It has no physical location. You can’t walk toward it — it recedes as you approach, because it exists only as an angular relationship between you, the Sun, and a population of raindrops.

Here’s the mechanism. Sunlight — white light, containing all visible wavelengths — hits a raindrop. Some light enters the drop. At the air-water boundary, it refracts — bends — according to Snell’s law, because light travels slower in water than in air (the refractive index of water is about 1.33).

Inside the drop, the light hits the back surface. Some of it exits the drop (you never see this light as part of the rainbow). Some reflects off the inner surface — total or partial internal reflection — and heads back toward the front of the drop. At the front surface, it refracts again as it exits back into the air.

The key insight is this: the angle between the incoming sunlight and the outgoing light — after one refraction, one reflection, and one more refraction — is not random. It has a preferred value. For most entry points on the drop, the exit angle varies gradually. But near the edge, many different entry points produce exit angles that cluster around the same value — about 42° from the anti-solar direction. Mathematically, this is an extremum: the exit angle reaches a maximum near 42° and many rays pile up at this angle.

This clustering is why you see a bright arc at 42°, not a diffuse glow. It’s a caustic — a concentration of light rays — and it works the same way as the bright line of light at the bottom of a swimming pool.

Why Colours Separate

White light contains all wavelengths from about 380 nm (violet) to 700 nm (red). Water’s refractive index is slightly different for each wavelength — this is dispersion. Red light (longer wavelength) has a refractive index in water of about 1.331. Violet light (shorter wavelength) has a refractive index of about 1.344.

The difference looks tiny — 1.331 vs. 1.344, barely 1%. But after two refractions and one reflection, this small difference translates into an angular separation of about 1.7° between the red and violet exit angles. Red exits at approximately 42.4°. Violet exits at approximately 40.7°.

At 42°, you see the red from distant, higher raindrops. At 40°, you see violet from closer, lower drops. Every drop in the sky is sending you all colours, but you only see one colour from each drop — whichever colour exits at the angle between that drop, the Sun, and your eyes. The rainbow is built from millions of drops, each contributing a single colour to your eye based on its angular position.

This also means that no two people see the same rainbow. Your rainbow is produced by a different set of raindrops than the person standing next to you. Each observer has their own private rainbow, tailored to their specific position. The person next to you is seeing light from slightly different drops. You’re both looking at the sky and seeing beautiful arcs, but they’re not the same arcs.

Why It’s a Circle (Well, an Arc)

The rainbow is centred on the anti-solar point — the point in the sky exactly opposite the Sun from your perspective. If the Sun is behind you at, say, 40° elevation, the anti-solar point is 40° below the horizon, directly ahead of you. The rainbow is a cone of light at 42° from this point, and the intersection of that cone with the sky is a circle.

From ground level, the anti-solar point is below the horizon whenever the Sun is up. So you only see the part of the circle above the horizon — an arc. When the Sun is low (near sunrise or sunset), the anti-solar point is close to the horizon, and the rainbow arc is tall — a majestic semicircle. When the Sun is high, the anti-solar point is far below the horizon, and the rainbow is a shallow arc low in the sky. When the Sun is higher than 42°, no primary rainbow is visible at all — the entire 42° cone is below the horizon.

This is why you mostly see rainbows in the morning or late afternoon, and never at noon in the tropics. It’s pure geometry.

The Double Rainbow and Alexander’s Dark Band

Sometimes you see a second, fainter rainbow above the primary one. The colours are reversed — red on the inside, violet on the outside. This is the secondary rainbow, produced by light that reflects twice inside the raindrop instead of once.

Each additional reflection loses about 4–5% of the light, so the secondary bow is noticeably dimmer. The double reflection also changes the geometry: the secondary rainbow appears at about 51° from the anti-solar point, roughly 9° above the primary.

Between the two bows is a dark region called Alexander’s dark band, named after Alexander of Aphrodisias, who described it around 200 AD. This region is darker than the sky above the secondary or below the primary because no light from single or double reflections reaches the observer from those angles. Below the primary, scattered light from all the non-rainbow reflections brightens the sky slightly. Above the secondary, the same. But in between — nothing. It’s an angular dead zone, and you can often see it clearly on a good double rainbow day.

I always find Alexander’s dark band the most convincing demonstration that rainbow physics is real. Predicting a bright arc is one thing. Predicting a specific dark band between two arcs — that’s a non-trivial claim. And there it is in the sky, exactly where Descartes’ geometry says it should be.

Supernumerary Bows and Wave Optics

Look closely at a bright rainbow — sometimes you can see faint pastel bands of pink and green just inside the primary bow. These are supernumerary bows, and they cannot be explained by the ray-optics treatment I’ve described so far. You need wave optics.

The rays that cluster near the 42° caustic angle arrive at your eye from slightly different paths. They’ve taken slightly different routes through the drop. In ray optics, this doesn’t matter — they all arrive at the same angle. But in wave optics, the different path lengths mean different phases. Some rays interfere constructively (bright bands), others destructively (dark bands). The result is a series of alternating bright and dark fringes just inside the main bow.

The spacing of supernumerary bows depends on the raindrop size. Smaller drops produce wider fringes (more diffraction). Larger drops produce narrower, more closely spaced fringes. If the drops are too varied in size, the fringes from different drop sizes overlap and wash out — which is why supernumerary bows are best seen when the rain is composed of very uniform droplets, often in mist or drizzle.

George Airy’s 1836 wave-optics theory of the rainbow — the Airy function — describes this pattern beautifully. It was one of the first applications of diffraction theory to a natural phenomenon, and it remains the standard treatment.

Fogbows, Moonbows, and Other Variants

The same physics produces several related phenomena. A fogbow — a white rainbow seen in fog — occurs when the water droplets are so small (less than about 50 micrometres) that diffraction smears out the colours, leaving a broad white arc. The geometry is identical to a normal rainbow; only the drop size differs.

A moonbow is a rainbow produced by moonlight instead of sunlight. The physics is exactly the same, but the light is much dimmer. Moonbows are usually white to the naked eye because the light is too faint to trigger colour vision in the human retina (the cone cells that detect colour require more light than the rod cells that detect brightness). Long-exposure photographs reveal the full spectrum.

A glory — a bright, colourful ring seen around the shadow of your head on a cloud below you — involves different physics entirely (backward scattering and surface waves around the drop). It’s often confused with a rainbow but the mechanism is distinct.

Newton Was Wrong About One Thing

Newton described the rainbow as having seven colours: red, orange, yellow, green, blue, indigo, violet. He chose seven, most historians believe, because of mystical associations with the number — seven notes in the musical scale, seven known planets at the time. But the rainbow is a continuous spectrum. There are no discrete colour bands. Your eye and brain impose boundaries on a smooth gradient.

The number of colours in a rainbow is, strictly speaking, infinite — every wavelength from 380 to 700 nm is present, fading smoothly from one to the next. Labelling them as seven, or six, or four (as some cultures do) is a human convention, not a physical fact. Newton’s seven-colour model has stuck in Western culture for 350 years, but it’s a classification, not a measurement.

The physics doesn’t care how many names you have for colours. The light refracts, reflects, disperses, and arrives at your eye at specific angles for specific wavelengths. Everything else — the beauty, the mythology, the pot of gold — is what you bring to the observation. The physics just provides the geometry.