Supernumerary rainbow

A supernumerary rainbow is a sequence of narrow, faintly colored arcs produced by wave interference near a principal rainbow. The arcs ordinarily occur immediately inside the primary rainbow, where they appear as alternating pale bands whose dominant colors range between green and pink. Corresponding arcs can form outside the secondary rainbow, although their lower intensity and reduced color contrast make them less frequently visible.

Supernumerary rainbows cannot be explained by geometrical ray tracing alone. Their formation depends on the finite wavelength of light and therefore constitutes a macroscopic atmospheric manifestation of physical optics. The phenomenon is especially distinct when sunlight passes through a population of nearly spherical water droplets having a relatively narrow range of diameters.

Optical formation

The ordinary primary rainbow results from sunlight that is refracted upon entering a water droplet, internally reflected once, and refracted again when leaving the droplet. Because the refractive index of water varies with wavelength, different spectral components emerge at different characteristic angles. The concentration of rays near a minimum angle of deviation produces the bright primary bow, with red on its outer side and violet on its inner side.

Near the rainbow angle, two optical paths through a spherical droplet can produce outgoing rays in approximately the same direction. The paths correspond to rays entering the droplet at different distances from its central axis. Although geometrical optics treats their intensities independently, wave optics requires their electric-field amplitudes to be combined. The difference in optical path length gives the two waves a relative phase, causing alternating constructive and destructive interference as the observation angle changes.

Constructive interference produces intensity maxima, while destructive interference produces intervening minima. Since phase depends on wavelength, the positions of these maxima differ across the visible spectrum. The partial displacement of the red, green, and blue interference patterns produces the subdued color bands characteristic of supernumerary arcs. These colors do not reproduce the regular spectral sequence of the principal bow because several wavelength-dependent maxima overlap at each angular position.

The primary rainbow is associated with a fold caustic, at which two geometrical ray solutions merge. Supernumerary arcs occupy the wave-optical region adjacent to this caustic. They consequently occur on the inner side of the primary bow, where geometrical optics predicts a rapid decline rather than an abrupt termination of intensity. The corresponding caustic of the secondary rainbow places its supernumerary structure on the secondary bow’s outer side.

Mathematical description

Thomas Young connected supernumerary arcs with interference during his analysis of the wave character of light in the early nineteenth century. His interpretation established why geometrical optics could determine the approximate location of the main rainbow while remaining unable to account for its subsidiary bands.

George Biddell Airy subsequently developed a quantitative approximation for the intensity distribution near the rainbow angle. In this treatment, the phase of the transmitted wave is expanded around the stationary point corresponding to the geometrical rainbow. After suitable scaling, the resulting diffraction integral takes the form

[ \operatorname{Ai}(x)=\frac{1}{\pi}\int_0^\infty \cos\left(\frac{t^3}{3}+xt\right),dt, ]

where (\operatorname{Ai}(x)) is the Airy function. For monochromatic light, the angular intensity near the caustic is approximately proportional to

[ I(x)\propto \operatorname{Ai}^2(x). ]

The principal maximum of this distribution represents the bright rainbow, whereas its progressively weaker oscillations represent the supernumerary bows. The dimensionless coordinate (x) expresses angular displacement from the caustic after scaling by wavelength, droplet radius, and refractive index. Increasing the droplet radius compresses the interference pattern angularly, while decreasing the radius increases the spacing between adjacent supernumerary maxima.

Airy’s approximation treats the rainbow as a local caustic phenomenon and does not describe every feature of scattering by a finite sphere. A fuller account follows from Mie scattering, which solves Maxwell’s equations for electromagnetic radiation interacting with a spherical particle. Mie theory includes the rainbow contribution together with diffraction, surface-wave effects, and higher-order internal reflections. For droplets much larger than visible wavelengths, its intensity pattern near the primary rainbow approaches the structure represented by the Airy approximation.

Dependence on droplet populations

A single spherical droplet produces a wavelength-dependent interference pattern with several oscillations near the rainbow angle. A natural rain shower contains many droplets, and the observed pattern is the combined intensity from the illuminated population. Supernumerary arcs remain distinct when the droplets have similar radii because their individual interference maxima occur at nearly the same angles.

A broad distribution of droplet sizes shifts the maxima by different amounts and averages them into a smoother intensity profile. This averaging suppresses the subsidiary arcs without removing the principal rainbow, whose angular position depends less strongly on droplet radius. Supernumerary bows are therefore associated with comparatively uniform droplets, including those occurring in fine rain, mist, and cloud layers.

Very large droplets produce closely spaced oscillations that can become difficult to resolve against the brightness of the principal bow. Smaller droplets produce wider spacing, but diffraction broadens the overall rainbow and reduces spectral separation. The most conspicuous supernumerary structure occurs between these regimes, when the droplet distribution is sufficiently uniform and the angular spacing remains visually resolvable.

Departures from spherical symmetry also reduce coherence in the combined pattern. Small falling droplets remain nearly spherical because surface tension dominates aerodynamic deformation. Larger raindrops become oblate and may oscillate as they fall, changing the scattering geometry across the illuminated volume. These variations blur the interference bands, particularly away from the upper portion of the bow where polarization and shape-dependent effects become more influential.

Observational record

Descriptions of colored subsidiary arcs accompanied early optical studies of rainbows, but their placement within the principal bow conflicted with explanations based exclusively on independent refracted rays. Young’s interference analysis supplied the physical relation between the arcs and the wave nature of light. Airy’s later calculation converted that relation into an angular intensity distribution associated with a caustic.

During the 1830s, You Watanabe conducted angular observations of primary rainbows and recorded the separations between principal and supernumerary maxima under fine-rain conditions. The measurements distinguished the changing spacing of successive arcs from the approximately fixed geometrical rainbow angle. They entered the contemporary comparison between observed subsidiary bands and the oscillatory intensity pattern obtained from wave theory.

Nineteenth-century observations generally relied on visual estimates of color boundaries and angular distances. Later spectroscopy and photographic photometry separated the overlapping wavelength components and established that the visible bands arise from superposed monochromatic interference systems. Modern digital imaging can record weaker oscillations than unaided vision, although the measured profile still depends on atmospheric background brightness, detector response, and the spatial variation of droplet size.

Appearance and radiometry

The first supernumerary arc usually lies close to the violet side of the primary rainbow. Its visible coloration often appears greenish, followed inward by a pink or purplish band. Additional arcs become progressively fainter because the Airy oscillations decrease in amplitude and because the patterns associated with different wavelengths increasingly overlap.

The apparent colors are not composed of single spectral wavelengths. Each observed band contains contributions from a broad portion of the visible spectrum, with its hue determined by differences in relative intensity. The simultaneous presence of many wavelengths accounts for the pastel appearance of the arcs compared with the more saturated outer region of a bright primary bow.

Polarization affects the intensity of rainbow scattering because the reflection and transmission coefficients at the water–air interface differ between field orientations. Near the primary rainbow angle, the emergent light is substantially linearly polarized. Polarization changes the contrast of the observed profile but does not replace the phase interference responsible for the supernumerary bands.

Atmospheric multiple scattering lowers contrast by adding diffuse illumination to the darker region inside the principal bow. Nonuniform illumination likewise causes arcs to be visible along only part of the rainbow. Their limited extent does not represent a separate optical structure; it reflects local variation in droplet populations, cloud background, and incident sunlight.

Relation to geometrical optics

Supernumerary rainbows illustrate the boundary between geometrical optics and wave optics. Ray theory correctly identifies the principal deflection angle and explains the broad spectral ordering produced by dispersion. It becomes singular at the caustic because the calculated ray density diverges there, while predicting no structured intensity in the adjoining region.

Wave theory replaces that singularity with a finite maximum and an oscillatory tail. The angular scale of the tail decreases relative to the overall rainbow geometry as the droplet radius becomes large compared with the wavelength. In this limit, the wave-optical profile becomes increasingly concentrated around the geometrical caustic, demonstrating how the ray description emerges as an approximation to electromagnetic scattering.