How Rainbows Form

FIG. 1 — The visible spectrum spans roughly 380–700 nm. Violet light (shortest wavelength) refracts most; red light (longest) refracts least — producing the rainbow's color order.

01 The Optical Illusion

A rainbow is not an object. It has no fixed position, no physical surface, and no existence independent of the observer. It is an optical phenomenon caused by the interaction of sunlight with water droplets in the air — refraction, internal reflection, and dispersion working together to separate white light into its constituent colors. The result is a multicolored circular arc that appears in the section of sky directly opposite the sun, and its precise position, apparent size, and even whether it is visible at all depend entirely on the geometry between the light source, the droplets, and the observer's eye.

This means that no two people ever see the same rainbow. Each observer sees light reflected from a unique set of droplets at a unique angle, so the rainbow you photograph is your own — a personal projection of atmospheric optics.

02 Refraction and Dispersion

When sunlight enters a spherical water droplet, it passes from air (refractive index ~1.00) into water (refractive index ~1.33). This transition bends the light — refraction — according to Snell's Law. Crucially, the degree of bending depends on wavelength: shorter wavelengths (violet, ~400 nm) refract more than longer wavelengths (red, ~700 nm). This separation of white light into its spectral colors is called dispersion, and it is the same phenomenon that splits light through a glass prism.

Inside the droplet, the refracted light travels to the far side. Because water has a higher refractive index than air, most of this light reflects off the inner surface — a process called total internal reflection — rather than passing through. The light then travels back to the near side of the droplet and exits, refracting a second time as it re-enters the air. By the time the light emerges, the colors have been separated into a fan of wavelengths, each exiting at a slightly different angle.

03 The 42-Degree Geometry

The critical number in rainbow geometry is the angle of deviation — the angle between the incoming sunlight and the outgoing light from the droplet. For a primary rainbow, the maximum deviation occurs at approximately 42 degrees from the antisolar point (the point directly opposite the sun, below the horizon). Red light emerges at about 42.4°, violet at about 40.7°, and the other colors fall between these extremes.

Because droplets across the entire sky contribute, the observer sees a circular arc: every droplet at the correct angle appears to glow with color. The arc is centered on the antisolar point, which is why rainbows appear higher in the sky when the sun is low, and why a full semicircular rainbow requires the sun to be at the horizon. If you could stand high enough — on a mountaintop or in an airplane — you could see the complete circle, since the ground would no longer block the lower half. In fact, from the air, full-circle rainbows are routinely observed.

FIG. 2 — The secondary rainbow involves two internal reflections, producing a fainter, reversed-color arc at 51°. The dark band between the two arcs is called Alexander's band.

04 Double, Supernumerary, and Circular Rainbows

Sometimes a second, fainter rainbow appears outside the primary arc, with its colors in reversed order — red on the inside, violet on the outside. This secondary rainbow is produced by light that undergoes two internal reflections inside the droplet rather than one. Because more light is lost at each reflection, the secondary bow is only about one-tenth as bright as the primary. The angle is also larger: approximately 51° from the antisolar point. Between the two arcs lies Alexander's band, a noticeably darker region of sky named after the Greek philosopher Alexander of Aphrodisias, who first described it around 200 AD.

Supernumerary rainbows are faint, narrow bands of alternating colors that appear just inside the primary arc. They are a wave-optics phenomenon — interference between light rays following slightly different paths through the same droplet — and cannot be explained by geometric optics alone. Their discovery helped establish the wave nature of light in the early 19th century, particularly through the work of Thomas Young and George Airy.

05 The Conditions for Sight

Seeing a rainbow requires a precise alignment of three elements: sunlight (with the sun behind the observer and relatively low in the sky), water droplets in the air ahead (from rain, mist, spray, or even dew), and the observer's position between them. Rainbows caused by sunlight always appear in the portion of sky directly opposite the sun. The lower the sun, the higher the arc — a midday sun produces only a low sliver of rainbow near the horizon, while a setting sun can produce a semicircle towering overhead.

Because the sun must be below 42° above the horizon for any part of the rainbow to appear above the ground, rainbows are most commonly seen in the early morning or late afternoon, or at higher latitudes where the sun stays lower in the sky. They are also common near waterfalls, fountains, and sea spray, where suspended droplets are plentiful and sunlight can reach them at the right angle.

06 Beyond the Visible

Rainbows exist beyond the colors the human eye can detect. Infrared and ultraviolet light also refract through water droplets, producing invisible extensions of the arc. Infrared rainbows have been photographed using specialized sensors, appearing as a faint glow just outside the red end of the visible arc. Ultraviolet rainbows, detectable by certain animals — including birds and insects that see into the UV range — are part of the natural world that we are simply not equipped to perceive.

The study of rainbow optics has deep historical roots. René Descartes published the first geometric explanation of the primary rainbow in 1637, calculating the 42° angle using the laws of refraction. Isaac Newton later demonstrated that white light is composed of the spectrum of colors visible in the rainbow. The complete wave-optics explanation, accounting for supernumerary bows, awaited Airy in 1838. Even today, fine details — such as the polarization of rainbow light, first measured by James Clerk Maxwell — continue to be studied as beautiful demonstrations of the interplay between wave and ray optics in nature.