Rainbows appear so reliably after storms that we forget how strange they are. The primary bow has a nearly fixed angular size, colors in a reliable order, and a shape that vanishes when you try to approach it. Rainbow formation physics explains all of it, and the explanation turns out to be more elegant than the phenomenon itself.
How Rainbow Formation Physics Works
Every rainbow starts with a single raindrop. When sunlight enters a water droplet, it slows down and bends, a process called refraction. The light then bounces off the back of the droplet and bends again as it exits.[s] Each droplet acts as a tiny prism that both disperses the light and reflects it back to your eye.[s]
But not all light exits in the same direction. Rays hitting different parts of the droplet emerge at different angles. The crucial discovery, made by Descartes in 1637, is that rays tend to concentrate near one particular angle. Near this maximum deflection, the angle changes slowly, so there is an “accumulation” of outgoing rays.[s] This concentration creates the bright arc we see.
Why 42 Degrees Matters
The rainbow formation physics comes down to geometry. When light enters a droplet at approximately 59.4 degrees relative to the surface normal, it exits at a minimum deviation angle that produces a rainbow angle of 42 degrees for red light.[s] This angle appears at a sharp boundary between a dark and a bright region where light rays concentrate, corresponding to what physicists call a caustic.[s]
Red light and blue light bend by slightly different amounts because water’s refractive index varies with wavelength. Red light, with a refractive index of 1.332, produces a rainbow angle of 42.2 degrees. Violet light, with a refractive index of 1.343, produces an angle of 40.6 degrees.[s] This 1.6 degree spread separates the colors into the familiar spectrum.
Red light appears at the top and on the outer perimeter of a rainbow, while blue light appears on the bottom and the inner perimeter.[s] The colors always appear in this order because shorter wavelength blue light refracts more than longer wavelength red light.[s]
The Secondary Rainbow and the Dark Band
Sometimes a fainter, larger arc appears above the primary rainbow with its colors reversed. This secondary rainbow forms when light bounces twice inside the droplet instead of once.[s] The extra reflection inverts the color order and sends the light out at a wider angle of about 51 degrees.
Between the two rainbows, the sky appears noticeably darker. No light scatters between 42 and 51 degrees after one or two reflections, creating this phenomenon known as Alexander’s band.[s] With each reflection, light intensity decreases, so the second rainbow is less vivid than the first. Third and higher order rainbows theoretically exist, but they are not visible under normal conditions because too much light is lost.[s]
Your Rainbow Is Yours Alone
The angle of scatter from raindrops is different for everyone, which means that every rainbow is unique to the observer.[s] The primary rainbow lies on the surface of a cone with its vertex at the observer, its axis pointing along the antisolar direction, and an aperture that varies by color: about 42 degrees for red and about 41 degrees for violet.[s] Move your head, and the cone moves with you, selecting a different set of droplets.
Understanding rainbow formation physics is part of a broader pattern in science: the physics of everyday phenomena often reveals surprising mathematical elegance. Related optical effects, such as the shimmering caustics on the bottom of pools, are governed by the same kind of ray-focusing geometry.[s]
Variations on the Arc
Droplet size affects what you see and complicates rainbow formation physics. Large droplets, about 1 millimeter or larger in diameter, produce brighter and more vivid rainbows with sharply defined color bands and more noticeable separation between red and violet.[s] In fog or very fine mist, the raindrops are extremely small, and diffraction dominates over refraction, resulting in a pale, nearly white rainbow known as a fogbow.[s]
Rainbows can last longer than most people assume. On 30 November 2017, Chinese Culture University documented a rainbow at Yangmingshan, Taipei, lasting 8 hours and 58 minutes; Guinness World Records lists it as the longest lasting rainbow observation.[s]
Rainbow Formation Physics: Geometric Optics
In the geometric optics description, the primary rainbow forms when sunlight undergoes two refractions and one reflection inside a spherical water droplet.[s] Each droplet acts as a tiny prism that both disperses the light and reflects it back to the observer’s eye.[s]
Let the dimensionless impact parameter be b = d/R, where d is the distance of the incident ray from the droplet center and R is the droplet radius. Using Snell’s law at each interface, the total deviation angle D(b) varies non-monotonically with b. The rainbow appears at an angle corresponding to the extremum of this function, where dD/db = 0. This angle corresponds to a caustic, a sharp boundary where light rays concentrate.[s]
Near the maximum deflection, the angle changes slowly, creating an accumulation of outgoing rays.[s] The scattering intensity I diverges as |theta – theta_0|^(-1/2), where theta_0 is the rainbow angle. This inverse-square-root singularity is characteristic of ray-optics caustics.[s]
The Descartes Ray and Minimum Deviation
The rainbow formation physics centers on a specific geometric configuration. The minimum deviation occurs when the angle of incidence alpha is approximately 59.4 degrees, leading to a rainbow angle of 42 degrees for red light.[s] This critical ray is called the Descartes ray.
For the primary bow, the total deviation D(alpha) = pi + 2*alpha – 4*beta, where beta is the refraction angle inside the droplet. Setting dD/d(alpha) = 0 and applying Snell’s law (n1*sin(alpha) = n2*sin(beta) with n2 approximately 1.33) yields the extremum condition.
Dispersion separates the colors. The refractive index n varies with wavelength: n = 1.332 for red light (rainbow angle 42.2 degrees) and n = 1.343 for violet light (rainbow angle 40.6 degrees).[s] Red light appears at the top and outer perimeter of the rainbow, blue light at the bottom and inner perimeter, because shorter wavelengths refract more than longer ones.[s]
Secondary Bow and Alexander’s Band
The secondary rainbow forms via two internal reflections, with deviation D(alpha) = 2*alpha – 6*beta + 2*pi. The minimum deviation angle is approximately 129 degrees, producing an observed rainbow angle of 51 degrees.[s] Rays of light are refracted as they enter the drop and then reflected inside the drop at the air-water interface: once for the primary rainbow, twice for the secondary bow.[s]
The extra reflection inverts the color sequence and reduces intensity. As no light scatters between 42 and 51 degrees for one or two reflections, the sky appears dark in this angular range, a phenomenon called Alexander’s band.[s]
Higher-order rainbows from three, four, or more reflections exist theoretically but produce extremely weak singularities. The third order bow appears at approximately 40 degrees from the sun, while the fourth order bow lies a few degrees further out at around 45-46 degrees from the sun, both essentially invisible against its glare. With each reflection, intensity decreases, rendering these higher-order bows unobservable under normal conditions.[s]
Polarization at the Brewster Angle
Close to the caustic singularity, the scattering in the primary bow is 96% s-polarised, and in the secondary 90%. This polarisation results from the fact that the angle at which the light is reflected inside the drops is close to the Brewster angle, where the reflection coefficient for p-polarised light approaches zero.[s]
Observer Geometry
Each observer sees a unique rainbow because the angle of scatter from raindrops differs for each viewing position.[s] The rainbow lies on the surface of a cone with its vertex at the observer, its axis extending along the antisolar direction, and an aperture that varies by color: about 42 degrees for red and about 41 degrees for violet.[s] The physics of everyday phenomena often follows similarly compact geometric or angular rules, even when the mechanism is different.
Limits of Geometric Optics
Airy’s theory extends ray optics by accounting for wave interference. This approach has its limitations, especially when the size of raindrops becomes comparable to the wavelength of the incident light.[s] For droplets much larger than visible wavelengths, geometric optics suffices. For smaller droplets, diffraction effects dominate.
Large droplets (diameter approximately 1 mm or larger) produce brighter and more vivid rainbows with sharply defined color bands.[s] In fog or very fine mist, diffraction dominates over refraction, producing fogbows: pale, nearly white arcs where color separation fails.[s]
Supernumerary arcs, the faint additional bands sometimes visible inside the primary bow, arise from wave interference between rays emerging at the same angle from different impact parameters. These features, first explained by Thomas Young and later refined by George Airy, mark the boundary between ray optics and wave optics. The full theory of rainbow formation physics requires moving beyond geometric optics into wave mechanics.
