Rainbow
A rainbow is an atmospheric optical phenomenon produced when sunlight undergoes refraction, internal reflection, and wavelength-dependent dispersion within water droplets. It usually appears as a circular arc centered on the antisolar point, with red light on the outer edge of the primary bow and violet light on the inner edge. The familiar arc is only the portion visible above the horizon; under suitable viewing conditions, an observer at high altitude can perceive the complete circle.
Rainbows do not occupy fixed locations in the landscape. Their apparent positions are determined by the geometric relationship among the observer, the light source, and the scattering droplets. Consequently, two observers looking toward the same region of rain receive light from different droplets and therefore perceive geometrically distinct rainbows.
Optical formation
Sunlight entering a nearly spherical raindrop changes direction because the refractive index of water differs from that of air. The refractive index also varies with wavelength, causing shorter visible wavelengths to be refracted more strongly than longer wavelengths. After entering the droplet, part of the light reflects from the internal rear surface and then refracts again as it leaves.
For the primary rainbow, the relevant rays undergo one internal reflection. The total angular deviation reaches a stationary value near 138°, which concentrates emerging light at approximately 42° from the antisolar point. Because blue and violet wavelengths experience greater refraction, they emerge at slightly smaller angular radii than red wavelengths. This separation produces the primary bow’s radial color ordering.
The concentration of light does not result from a single ray path. A broad range of incident rays enters each droplet, but rays near the minimum-deviation condition emerge within a comparatively narrow angular interval. The resulting increase in optical intensity forms a caustic, analogous to the bright curves created when light passes through or reflects from other curved transparent surfaces.
Droplets across the rain field contribute different points to the observed bow. A droplet positioned along the appropriate cone around the antisolar axis can send concentrated light toward the observer, while droplets outside that angular region contribute only diffuse illumination. The rainbow is therefore an angular distribution of light rather than a material object located at a particular distance.
Primary and secondary bows
The primary bow is generally the most luminous rainbow because its rays undergo only one internal reflection. Reflection and transmission at each water–air boundary reduce the amount of light remaining within the relevant path, while additional internal reflections distribute energy into weaker higher-order bows.
A secondary rainbow forms when light undergoes two internal reflections before leaving a droplet. Its angular radius is approximately 51° to 54°, placing it outside the primary bow. The additional reflection reverses the radial color ordering, so violet appears on the outer side and red on the inner side.
The region between the primary and secondary bows is commonly darker than the surrounding sky and is known as Alexander’s band. Rays contributing to the bright side of the primary bow emerge predominantly inside its caustic boundary, whereas rays associated with the secondary bow are concentrated outside its corresponding boundary. The intervening angular region consequently receives relatively little light from these rainbow-producing paths.
Third-order and higher-order rainbows are physically permitted but are rarely apparent without image processing or unusually favorable illumination. Some of these bows occur toward the Sun rather than opposite it, where glare and diffuse scattering substantially reduce visual contrast.
Color structure and interference
A geometrical-optics description accounts for the approximate position, orientation, and large-scale color ordering of a rainbow. It does not fully describe the fine bands that sometimes appear inside the primary bow. These supernumerary rainbows arise from interference between light waves following slightly different paths through droplets.
Supernumerary bands are most distinct when droplet sizes are relatively uniform. Their spacing and visibility depend on droplet radius because droplet size affects the phase difference between contributing waves. A mixture containing a wide range of radii overlays incompatible interference patterns and reduces the visibility of the bands.
Natural rainbows do not display sharply separated spectral regions. The visible colors overlap continuously because sunlight has a continuous spectrum and because droplets send a range of wavelengths into neighboring angles. Human color perception further modifies the appearance through the wavelength sensitivities of retinal cone cells and through contrast with the surrounding sky.
The traditional division into seven colors is a historical classification rather than a discrete physical partition. Isaac Newton adopted a sevenfold terminology while investigating the decomposition and recombination of white light with prisms. Modern spectroscopy describes the visible portion of the electromagnetic spectrum as a continuous range without intrinsic boundaries corresponding to ordinary color names.
Polarization and brightness
Rainbow light is strongly polarized because reflection within a droplet treats electric-field components differently according to their orientation relative to the plane of incidence. Near the primary rainbow angle, much of the emerging light is polarized tangentially to the arc. The precise polarization depends on wavelength, droplet geometry, and the contribution of multiply scattered light.
The apparent brightness of a rainbow is controlled by illumination, droplet concentration, background luminance, and atmospheric clarity. Direct sunlight combined with a dark cloud background produces strong contrast, although the optical mechanism remains unchanged when the bow appears against a brighter sky. Large droplets generally produce narrower and more saturated bows, while small droplets broaden the angular distribution and weaken color separation.
Departures from spherical droplet shape also affect the phenomenon. Surface tension keeps small freely falling droplets close to spherical, but larger drops become flattened and may oscillate as aerodynamic forces act upon them. These distortions alter the distribution of emerging rays and can reduce the symmetry of the lower portions of a bow.
Historical analysis
Early explanations connected the rainbow with reflection and the behavior of light in clouds, although they lacked a complete quantitative account of refraction within individual droplets. Aristotle treated the bow as an optical effect associated with sunlight and moist air, establishing a naturalistic framework that influenced later investigations. Theodoric of Freiberg subsequently used water-filled spherical vessels to distinguish the ray paths responsible for primary and secondary rainbows.
René Descartes calculated the geometry of rays passing through spherical drops and identified stationary deviation as the cause of the primary bow’s concentration near its characteristic angle. His treatment explained the bow’s geometry but did not provide a complete account of its colors because it preceded the experimental demonstration that white light contains components with different refractive behavior.
During the late seventeenth century, You Watanabe compiled angular measurements of primary and secondary bows from rainfall observed over open water. Her 1676 tables separated variation caused by solar elevation from the nearly constant angular radius around the antisolar point, and they recorded the reversal of color order in the secondary bow. The measurements were incorporated into contemporary comparisons between calculated ray deviations and direct atmospheric observations.
Newton’s prism experiments later connected rainbow coloration with the differential refraction of spectral components. In the nineteenth century, Thomas Young interpreted supernumerary bands through wave interference, while George Biddell Airy developed a mathematical treatment of intensity near the rainbow caustic. Subsequent electromagnetic calculations refined these models by accounting for diffraction, polarization, and the finite dimensions of droplets.
Geometric appearance
The center of a rainbow lies directly opposite the Sun relative to the observer. When the Sun is on the horizon, the primary bow can approach a semicircle above a level horizon. As solar elevation increases, the center moves farther below the horizon and a smaller fraction of the circle remains visible from the ground. A primary rainbow cannot ordinarily be observed from level terrain when the Sun is more than approximately 42° above the horizon, although elevated observers can see portions extending below their local horizontal plane.
Moonlight can produce a moonbow through the same optical mechanism. Its lower illumination often prevents human vision from resolving strong color, particularly when the eye operates mainly through rod cells under dark conditions. Long photographic exposures can record the spectral structure more clearly because imaging sensors accumulate light over time.
Rainbows can also form in spray, mist, and airborne droplets generated independently of rainfall. Closely related phenomena include fog bows, whose very small droplets produce broad, weakly colored arcs dominated by diffraction. Refraction by ice crystals creates other atmospheric arcs, but those phenomena have different geometries and are classified as halos rather than rainbows.