Few natural phenomena are as immediately striking as a rainbow. Arching across the sky after a rain shower, its colours seem almost impossibly vivid and ordered. Yet a rainbow is not a physical object at all — it has no fixed location, cannot be approached, and disappears the moment the geometry changes. It is an optical phenomenon: a product of light, water, and the particular position of an observer. Understanding it requires optics, geometry, and a careful account of how light behaves inside a droplet of water.
Three things are needed for a rainbow to appear: a source of white light (almost always the Sun), a large number of small spherical water droplets suspended in the air, and an observer whose back is to the light source. Rainbows are therefore always seen in the part of the sky opposite the Sun. On a rainy afternoon in England, for instance, the Sun low in the west and rain falling to the east creates ideal conditions.
The droplets responsible are typically falling rain, though fog, mist, spray from waterfalls, and even garden sprinklers can produce the same effect. The droplets must be roughly spherical — surface tension ensures that small falling drops are very nearly so — and they must be large enough to refract and reflect light efficiently, generally at least 0.1 mm in diameter.
When light passes from one transparent medium into another — say, from air into water — it changes speed. In a vacuum, light travels at approximately 299,792 km/s. In water, it slows to around 225,000 km/s. This change in speed causes the light ray to change direction at the boundary, a phenomenon called refraction, described by Snell's Law:
n1 sin θ1 = n2 sin θ2
where n is the refractive index of each medium and θ is the angle the ray makes with the normal to the surface. The refractive index of water is approximately 1.33, meaning light bends toward the normal when entering water and away from it when exiting.
Crucially, the refractive index of water varies slightly with the wavelength (colour) of light — a property called dispersion. Violet light (shorter wavelengths, around 380 nm) bends more than red light (longer wavelengths, around 700 nm). This differential bending is the fundamental reason a rainbow separates white sunlight into a spectrum of colours.
The geometry of what happens inside a single raindrop was worked out mathematically by René Descartes in 1637 and later given a full physical explanation by Isaac Newton. A ray of sunlight striking a spherical droplet undergoes the following sequence:
The net result is that the ray exits the droplet travelling in a direction substantially different from the direction it entered — and different colours exit at slightly different angles. Red light exits at approximately 42° from the original direction of the incoming sunlight; violet light exits at approximately 40°. The other colours of the spectrum lie between these two extremes.
Not all rays striking a droplet contribute equally to the rainbow. Rays that enter near the edge of the droplet and rays that enter near the centre both exit at various angles, but there is a specific angle — the rainbow angle or Descartes angle — at which a large number of rays emerge clustered together. This concentration of rays is what makes the rainbow bright enough to see; it is an example of a caustic, a region where light rays bunch together.
For red light, this critical angle is about 42° measured from the antisolar point — the point in the sky directly opposite the Sun from the observer's perspective (which lies below the horizon during the day, at the centre of the circular arc). Violet light's critical angle is about 40°. The rainbow therefore appears as an arc at these angles around the antisolar point, with red on the outside and violet on the inside.
This geometry also explains a key observational fact: every observer sees their own private rainbow, formed by different droplets at the precise angles relevant to their position. Two people standing side by side see rainbows formed by entirely different sets of droplets.
Newton identified seven colours in the rainbow — red, orange, yellow, green, blue, indigo, and violet — though the spectrum is in fact continuous, with no sharp boundaries between colours. The choice of seven was partly influenced by Newton's desire to draw an analogy with the seven notes of the musical scale. In practice, most observers see five or six distinct colour bands. The precise appearance depends on the size of the droplets: larger drops produce brighter, more vivid rainbows with well-separated colours; very small droplets (as in fog) produce pale, nearly white bows called fogbows.
| Colour | Wavelength (nm) | Exit angle (primary bow) | Position in bow |
|---|---|---|---|
| Red | ~700 | ~42.3° | Outermost |
| Orange | ~620 | ~41.9° | — |
| Yellow | ~580 | ~41.5° | — |
| Green | ~530 | ~41.2° | — |
| Blue | ~470 | ~40.7° | — |
| Violet | ~380 | ~40.4° | Innermost |
A fainter, larger arc sometimes appears above the primary rainbow — the secondary rainbow. It is produced by light that undergoes two internal reflections inside each droplet rather than one. Each reflection loses some light intensity (some light escapes the droplet rather than reflecting), which is why the secondary bow is considerably dimmer — roughly 43% as bright as the primary.
Two internal reflections reverse the order of colours: in the secondary bow, red appears on the inside and violet on the outside, the opposite of the primary. The secondary bow appears at roughly 51° from the antisolar point. The sky between the two bows — between about 42° and 51° — appears noticeably darker than the sky outside the primary or inside the secondary. This dark band is called Alexander's dark band, named after Alexander of Aphrodisias who described it around 200 AD. It arises because no rainbow rays from either the primary or secondary bow reach that angular region, so the sky there receives less scattered light.
Just inside the primary rainbow's violet band, careful observers sometimes notice a series of faint, closely spaced bands of pink and green called supernumerary arcs. These cannot be explained by geometric optics — the ray-tracing model of Descartes and Newton — and were a long-standing puzzle. Their explanation requires wave optics.
Light is a wave, and rays exiting a droplet at slightly different angles can interfere with one another. Where wave crests align (constructive interference), the light is bright; where a crest meets a trough (destructive interference), the light cancels out. Supernumerary arcs are the result of this interference pattern. They are more prominent when the droplets are uniform in size — variation in droplet size smears the interference pattern and washes the arcs out. Their discovery in the 19th century by Thomas Young provided early evidence that light behaves as a wave.
A rainbow is, geometrically, a full circle centred on the antisolar point. From ground level, the horizon cuts off the lower half, so we see only an arc. The higher the Sun is in the sky, the lower (and smaller) the visible arc; when the Sun is above 42°, the primary rainbow's arc lies entirely below the horizon and cannot be seen at all. From an aircraft or a mountain above a rain shower, the full circular rainbow is sometimes visible. Pilots occasionally observe complete circular rainbows when flying over clouds illuminated from above.
Light from a rainbow is strongly polarised. When sunlight reflects inside a raindrop, it preferentially reflects light whose electric field oscillates in a particular orientation. Rainbow light is polarised tangentially — that is, the electric field oscillates parallel to the arc of the bow. Holding a polarising filter (such as polarised sunglasses) and rotating it reveals this: at one orientation the rainbow dims significantly and at the other it appears brighter. This polarisation is a direct consequence of the physics of reflection at the water–air interface, governed by Fresnel's equations.
Rainbows can also be produced by moonlight. Called moonbows or lunar rainbows, they are formed by exactly the same mechanism but are far dimmer, since the Moon reflects only a small fraction of sunlight. Moonbows appear white or very faintly coloured to the naked eye because the light level is too low to activate the colour-sensitive cone cells in the human retina — only the more sensitive rod cells respond, and these do not distinguish colour. Long-exposure photographs, however, reveal the full spectrum of colours in a moonbow.
The rainbow has driven centuries of scientific progress. Descartes used it to demonstrate the power of geometric reasoning about light. Newton used it to prove that white light is a mixture of colours. Young used supernumerary arcs to establish the wave nature of light. George Biddell Airy developed a more precise mathematical theory in 1838 — Airy's rainbow integral — which accurately describes the intensity distribution across the bow, including the supernumerary fringes, using wave theory. Modern treatments apply Mie scattering theory, which gives a complete electromagnetic description of how light interacts with a sphere of arbitrary size, reproducing all of the observed features of rainbows with high precision.
What began as a wonder in the sky has become, over four centuries, one of optics' most richly understood phenomena — a demonstration that beauty and rigorous physical explanation are not opposites, but often the same thing seen from different angles.
This document provides a general scientific overview of the physics of rainbows for educational purposes.