The Optics of Unusual Rainbows

AWhen sunlight encounters falling rain, the classic display of a primary and secondary arc is often treated as a solved puzzle of elementary physics. In the seventeenth century, natural philosophers established that the primary rainbow arises when parallel light rays enter a spherical water drop, undergo refraction at the boundary, experience a single internal reflection at the rear surface, and refract again upon emerging. The secondary bow results from a second internal reflection, creating an inverted sequence of hues at a wider angular radius. However, treating raindrops as uniform, rigid spheres and light purely as geometric rays presents an oversimplified portrait. Observers throughout history noticed that natural displays diverge substantially from this blueprint, displaying subtle fringes, dark zones, and irregular colour distributions that demanded deeper optical investigation.

BOne of the most persistent challenges to early geometric models was the presence of supernumerary bows. These appear as delicate, pastel-toned bands—often pale green, pink, or purple—clinging to the inner boundary of the primary bow, or much more rarely, along the outer edge of the secondary bow. Classical ray-tracing principles predicted that light intensity should fall smoothly towards the interior of the bow without any repeating bands. It was not until the early nineteenth century that natural philosophers recognised light as a wave phenomenon, which provided the key to this mystery. Light rays following slightly different trajectories within a droplet can emerge along parallel paths. Because these waves are out of phase with one another, they undergo constructive and destructive interference, generating alternating rings of heightened brightness and darkness.

CThe manifestation of supernumerary fringes, as well as the overall vividness of the entire display, depends heavily on the diameter of the suspended water droplets. In showers consisting of uniformly sized drops measuring roughly one millimetre across, multiple distinct supernumerary bands can be discerned with remarkable clarity. Conversely, when raindrops vary widely in size across a storm cloud, the differing interference patterns overlap and blur one another, washing out the delicate fringes altogether. At the extreme lower end of droplet dimensions, such as in mist or maritime fog where droplets measure mere micrometres, diffraction dominates entirely. The resulting feature, commonly termed a fog bow or white rainbow, lacks chromatic separation almost completely, appearing instead as a broad, ghostly veil with faint reddish outer fringes and bluish interior tints.

DAnother complication arises because falling raindrops are rarely the perfect spheres assumed in textbook diagrams. As a drop descends through the atmosphere, aerodynamic resistance flattens its bottom, causing it to deform into an oblate spheroid with a rounded top and a broad base. This distortion increases with droplet mass, meaning that larger drops deviate considerably from sphericity. Because the curvature of an oblate drop differs between its horizontal and vertical cross-sections, rays travelling in different planes exit at subtly varied angles. This distortion can generate split or "twinned" rainbows, where two primary arcs branch outward from a single base near the ground, where the largest deformed drops tend to concentrate.

EBeyond the familiar dual arcs lie higher-order rainbows, produced by three, four, or more internal reflections within raindrops. For centuries, tertiary and quaternary rainbows were regarded as theoretical curiosities that could never be witnessed in the natural sky. Each successive internal reflection causes a significant portion of light to be lost via transmission through the droplet wall, rendering the resulting arcs exceedingly faint. Furthermore, tertiary and quaternary bows form in the direction of the sun itself, meaning their feeble radiance is typically obliterated by intense solar glare. Only in the early twenty-first century did dedicated atmospheric photographers, using advanced digital processing and high-contrast filtering, succeed in capturing verified photographs of these elusive higher-order phenomena in natural conditions.

FBetween the primary and secondary arcs lies an area of noticeable gloom known as Alexander's dark band, named after the ancient commentator who first documented it. This gap is not merely a visual illusion caused by contrast against the adjacent bright arcs, but a genuine zone of optical deficit. Because geometric constraints dictate that light deflected by a single internal reflection cannot emerge at an angle greater than approximately forty-two degrees, and rays experiencing two reflections cannot exit at an angle less than roughly fifty degrees, the sky between these two limits receives virtually no singly or doubly scattered light from the raindrops, leaving it substantially darker than the surrounding backdrop.

GIn recent decades, optical physicists have refined these insights by applying sophisticated electromagnetic models, such as Mie scattering theory, to simulate rainbow dynamics with computational precision. Rather than treating rain as an isolated collection of identical drops, modern algorithms can replicate precipitation profiles containing millions of drops of varied shapes, orientations, and temperatures. This computational modelling has practical applications beyond aesthetic curiosity; by analysing the exact spacing of supernumerary fringes and hue gradients in digital photographs of rainbows, atmospheric scientists can remotely infer droplet size distribution and turbulence levels within distant cloud systems.