A rainbow is not an object. It has no physical location, no fixed position in space, and no substance you could ever touch. It is an optical phenomenon — a pattern of light created by millions of individual water droplets, each refracting, reflecting, and dispersing sunlight at precise angles that happen to direct different wavelengths of colour toward your eyes. Every rainbow you see is uniquely yours: a person standing even a few metres away sees a different rainbow, created by different droplets at slightly different angles. The rainbow that appears to arch across the sky is not "out there" in any meaningful sense — it exists only in the geometry between the sun, the rain, and your eyes. And yet the physics that creates it is among the most elegant in all of optics.
TL;DR: Rainbows form when sunlight enters water droplets, refracts (bends), reflects off the back surface, and refracts again on exit — dispersing white light into its component colours. The primary rainbow appears at 42° from the anti-solar point, with red on the outside and violet on the inside. Secondary rainbows (51°) reverse the colour order due to a second internal reflection. Supernumerary bows, fogbows, and moonbows are variations caused by different droplet sizes and light sources.
42°
Angle between incoming sunlight and the primary rainbow — constant for all observers
51°
Angle of the secondary rainbow — wider, fainter, and colour-reversed
1.33
Refractive index of water — the number that determines the rainbow's geometry
7
Traditional colour bands — actually a continuous spectrum with infinite gradation
Refraction: Why Light Bends in Water
The rainbow begins with refraction — the bending of light as it passes from one medium to another of different optical density. When a ray of sunlight strikes a water droplet, it slows down from approximately 300,000 km/s to about 225,000 km/s. This change in speed causes the ray to bend toward the normal (a line perpendicular to the droplet surface) as it enters. The amount of bending depends on the wavelength of the light: shorter wavelengths (violet, blue) slow down more and bend more sharply than longer wavelengths (orange, red). This wavelength-dependent bending is called dispersion, and it is the mechanism that separates white sunlight into the spectrum of colours we see in a rainbow.
The refractive index of water — 1.33 for visible light — is the fundamental number that determines the geometry of every rainbow. This single value, combined with the laws of reflection and refraction derived by Snell in 1621 and explained by Descartes in 1637, is sufficient to predict the exact angle at which each colour appears. The physics of the rainbow was one of the earliest triumphs of mathematical optics, and it remains one of the most beautiful demonstrations of how simple physical laws produce complex natural beauty.
Each rainbow is unique to the observer — created by millions of individual droplets refracting light at precisely the right angle toward your eyes
Inside the Droplet: Refraction, Reflection, Refraction
The journey of light through a single raindrop follows three steps that together create the rainbow. First, the ray refracts as it enters the front surface of the droplet, bending inward and beginning to separate into component colours. Second, it reflects off the back interior surface of the droplet — not all light reflects; some passes straight through, but a significant fraction bounces back. Third, the reflected ray refracts again as it exits the front surface, bending once more and further separating the colours.
The critical insight, first calculated by René Descartes, is that rays entering the droplet at different heights exit at different angles — but there is a minimum deviation angle at which rays concentrate. For red light, this angle is approximately 42.4° from the anti-solar point (the point directly opposite the sun from the observer's perspective). For violet light, it is approximately 40.7°. This concentration of light at specific angles is what creates the bright, well-defined arc of the rainbow rather than a diffuse wash of colour across the entire sky. The rainbow is bright precisely because many rays converge at nearly the same exit angle — a mathematical phenomenon called a caustic.
The Secondary Rainbow and Alexander's Dark Band
Look carefully above a bright primary rainbow and you may see a second, fainter arc with its colours reversed — red on the inside, violet on the outside. This is the secondary rainbow, formed by light that undergoes two internal reflections inside the droplet before exiting. Each additional reflection weakens the light (some escapes at each bounce), making the secondary rainbow dimmer. The two reflections also reverse the colour order and shift the arc to a wider angle — approximately 51° from the anti-solar point.
Between the primary and secondary rainbows lies Alexander's dark band, named after Alexander of Aphrodisias who first described it in 200 AD. This region appears noticeably darker than the sky above and below the rainbows because no light from single or double reflections is directed into this angular range. Light that enters droplets at angles corresponding to Alexander's band is directed either below the primary bow or above the secondary bow, leaving the band between them depleted of rainbow-scattered light. The effect is subtle but unmistakable once you know to look for it — the sky inside the primary rainbow is brighter than the sky outside it, and the dark band between the two bows is the darkest region of all.
Why Rainbows Are Circular: A rainbow is actually a full circle — the 42° cone of refracted light extends in all directions around the anti-solar point. From the ground, you see only the upper arc because the lower portion is blocked by the horizon. From an aircraft or a tall mountain, however, you can see a complete circular rainbow — a rare and spectacular sight. The higher the sun in the sky, the lower the rainbow appears; when the sun is above 42° elevation, no primary rainbow is visible at ground level because the entire arc falls below the horizon. This is why rainbows are most common in the morning and late afternoon, when the sun is low.
Unusual Rainbows: Fogbows, Moonbows, and Supernumeraries
Fogbows (white rainbows) form in fog, where water droplets are much smaller than raindrops — typically 10 to 50 micrometres compared to 0.5 to 5 millimetres for rain. These tiny droplets produce the same refraction and reflection, but the small size causes significant diffraction that smears the colours together, producing a broad white or very faintly coloured arc. Fogbows are most commonly seen from mountains, coastal cliffs, and aircraft, and they are eerie and beautiful — a ghost rainbow stripped of its colour.
Moonbows (lunar rainbows) form by exactly the same physics as solar rainbows but using moonlight instead of sunlight. Because moonlight is roughly 500,000 times dimmer than sunlight, moonbows are extremely faint and typically appear white to the naked eye — human colour vision requires a minimum light intensity that moonbows rarely provide. Long-exposure photography reveals that moonbows contain the full spectrum of colours, proving their optical equivalence to solar rainbows. The best locations to see moonbows are near large waterfalls on clear, full-moon nights — Victoria Falls and Yosemite Falls are renowned moonbow sites.
Supernumerary bows are faint pastel-coloured bands that sometimes appear just inside the primary rainbow, alternating between pink-green and blue-violet. They are caused by wave interference — a phenomenon that cannot be explained by the geometric ray optics of Descartes and requires the wave theory of light developed by Thomas Young in the early 1800s. When two rays that have taken slightly different paths through a droplet exit at nearly the same angle, they can constructively or destructively interfere, creating the alternating bright and dark fringes of the supernumerary pattern.
Higher-Order Rainbows and Modern Discoveries
Beyond the primary and secondary bows, light can undergo three, four, or even more internal reflections inside a droplet, producing third-order, fourth-order, and higher rainbows. For centuries, these were theoretical predictions that no one had actually observed — each additional reflection weakens the light dramatically, and third- and fourth-order rainbows appear on the sunward side of the sky, washed out by direct sunlight. It was not until 2011 that the first verified photographs of a natural tertiary (third-order) rainbow were published, captured by dedicated observers who used specific camera settings and image processing to enhance the faint signal against the bright sky.
In laboratory settings, rainbows of up to the 200th order have been created using precisely controlled water droplets and laser light. Each order appears at a different angle, and the colour separation increases with order number, producing increasingly wide spectral bands. These laboratory experiments confirm the mathematical predictions of Mie scattering theory — the complete wave-optics treatment of light interacting with spherical particles — and demonstrate that the humble raindrop contains an almost infinite optical complexity. The primary rainbow we see casually during a rain shower is merely the simplest manifestation of a phenomenon that extends to hundreds of orders, each one fainter but each one present in every raindrop that catches the light.
The Rainbow in Science and Culture
The scientific study of rainbows has driven fundamental advances in optics. Theodoric of Freiberg correctly described the basic geometry of rainbow formation in 1304 — through refraction and internal reflection in individual droplets — making it one of the earliest correct physical explanations of a complex natural phenomenon. Descartes refined the geometric analysis in 1637, calculating the precise angles. Newton used the rainbow to demonstrate that white light is composed of a spectrum of colours, publishing his findings in Opticks in 1704. And Young's explanation of supernumerary bows in the early 1800s provided key evidence for the wave nature of light, contributing to one of the most important debates in the history of physics.
Culturally, rainbows appear in the mythology of virtually every civilisation. In Greek mythology, Iris was the goddess of the rainbow and a messenger between gods and humans, travelling along the rainbow bridge between Olympus and Earth. Norse mythology featured Bifröst, a burning rainbow bridge connecting Midgard (Earth) to Asgard (realm of the gods). The biblical Book of Genesis describes the rainbow as God's covenant with Noah. In Irish folklore, a pot of gold sits at the rainbow's end — a fitting metaphor for something you can see but never reach, since the rainbow moves with you, forever maintaining its 42° geometry relative to your position.
The Observer Paradox: No two people ever see the same rainbow. Because the rainbow's position depends on the angle between the sun, the droplets, and the observer's eyes, each observer sees light refracted from a completely different set of water droplets. Your rainbow and someone else's rainbow are created by different physical objects — they simply look similar because the geometry is the same. You cannot walk toward a rainbow; it recedes as you advance, maintaining its angular distance. You cannot photograph the "same" rainbow someone else sees. Every rainbow is a private optical event between you and the atmosphere.
How to See the Best Rainbows
Sun position: The sun must be behind you and below 42° elevation — low sun means high, bright rainbows.
Best times: Morning and late afternoon provide the lowest sun angles and the tallest rainbow arcs.
Look for the dark band: Alexander's dark band between the primary and secondary rainbow is visible once you know where to look.
Check inside vs outside: The sky inside the primary rainbow is noticeably brighter than outside — scattered light concentrates there.
Supernumeraries: Look for faint pastel bands just inside the primary bow, especially when rain droplets are uniform in size.
Full circles: From aircraft windows, you may see a complete 360° rainbow — one of optics' most spectacular sights.
A rainbow is proof that beauty and physics are not separate categories. Every arc of colour you see is the visible consequence of Snell's law, total internal reflection, and the wavelength-dependent refractive index of water — principles that can be described in a few equations but that produce a phenomenon of extraordinary visual power. The rainbow taught Newton that white light is composite. It taught Young that light is a wave. It taught Descartes that natural phenomena could be explained through mathematics rather than mythology. And it continues to teach anyone who looks up during a rain shower that the universe operates on principles that are simultaneously simple enough to calculate on paper and beautiful enough to inspire awe in every culture that has ever seen the sky.
