Optical Mechanics of Rainbows

A secondary rainbow occasionally appears outside the primary bow, presenting a broader, fainter band with its colour sequence reversed. This optical phenomenon arises when incident sunlight enters the lower half of a spherical droplet and undergoes two internal reflections before emerging back towards the observer. Each internal reflection entails an inevitable loss of energy, as a portion of the light refracts out of the drop rather than reflecting internally. As a consequence of this double reflection, the secondary bow is markedly dimmer than the primary bow, possessing only about a tenth of its luminance.

The additional reflection also alters the exit geometry of the light rays. The minimum deviation angle for a secondary rainbow is roughly 51 degrees for red light and 54 degrees for violet light. This inversion of angles causes violet to form the outer edge of the arc, with red appearing along the inner boundary.

Between the primary and secondary arcs lies a conspicuously dim region of the sky known as Alexander's dark band. This phenomenon is directly caused by the directional nature of droplet scattering. In the primary bow, light rays are concentrated at angles below 42 degrees, illuminating the sky within the inner circle. Conversely, secondary bow rays are directed outward beyond 51 degrees. Because geometric optics prevents light from being deflected into the intermediate angular zone between 42 and 50 degrees, this intervening band appears exceptionally dark compared to the surrounding sky.

Standard ray optics, which models light strictly as straight geometric trajectories, successfully explains the primary and secondary rainbow arcs but fails to account for supernumerary rainbows. These features appear as a series of faint, narrow, pastel-coloured bands situated directly along the inner edge of the primary bow. To explain their formation, classical physics had to incorporate the wave nature of light, demonstrating that emergent rays follow slightly different path lengths through the raindrop, resulting in constructive and destructive interference.

When sunlight traverses a water droplet, rays that enter at marginally different points can emerge along identical trajectories. Because these paired rays traverse slightly different path lengths within the droplet, their phase relationship shifts. If the wave peaks align, constructive interference occurs, producing an intense fringe of light. Conversely, if a crest meets a trough, destructive interference cancels the light. Because the condition for interference varies according to wavelength, multiple overlapping fringes of alternating colours are produced.

The appearance of supernumerary bows is exquisitely sensitive to droplet size. They are most pronounced when raindrops are uniform in diameter and relatively small, typically measuring less than one millimetre across. If the precipitation consists of droplets of widely varying sizes, the interference patterns generated by individual drops overlap randomly and cancel each other out, washing away the supernumerary arcs and leaving only the smooth spectral gradient of the primary bow.

Under typical daylight conditions, a rainbow displays a full spectrum spanning from violet to red. However, when a rain shower occurs during sunrise or sunset, observers may witness a monochrome or entirely red rainbow. This dramatic alteration in appearance is driven by Rayleigh scattering within the Earth's atmosphere rather than any change in the optical properties of the raindrops themselves.

When the sun is positioned extremely low on the horizon, its rays must travel through an atmospheric path length that is significantly thicker than at midday. As light traverses this extended column of air, molecular scattering preferentially disperses shorter wavelengths, including blue, green, and yellow light, out of the direct beam. By the time the sunlight reaches the distant raindrops, the incoming beam consists almost entirely of longer, red and deep-orange wavelengths.

Because the illuminating light is deprived of its shorter-wavelength components, the raindrops have only red light to refract and internally reflect. Consequently, the resulting rainbow lacks the concentric rings of blue and violet, taking on a uniform crimson or bronze hue. Furthermore, because scattering reduces the total intensity of the incoming sunlight, red rainbows often appear accompanied by a bright, diffused glow across the horizon, as the scattered red light illuminates the surrounding rain sheet without the sharp chromatic transitions seen under high-sun conditions.

A fogbow, sometimes referred to as a white rainbow, is an optical arc produced when sunlight illuminates fog or cloud rather than conventional raindrops. While the underlying geometric principles of refraction and internal reflection remain identical to those of a standard rainbow, the dramatic difference in appearance stems directly from the minute scale of the water droplets involved.

Typical raindrops measure between one and several millimetres across, allowing geometric optics and wavelength dispersion to dominate. In contrast, fog droplets generally range from ten to thirty micrometres in diameter. At this microscopic scale, diffraction becomes the predominant optical mechanism, fundamentally altering how light emerges from the droplets. Diffraction causes light waves to spread out widely upon exiting the drop, causing the individual colours to overlap and blend back together into white light.

As a result, a fogbow appears as a broad, ghostly white band rather than a distinct series of coloured stripes. The bow is noticeably wider than a standard rainbow because the diffraction effect broadens the angular distribution of the emergent light. Although predominantly white, a well-defined fogbow often displays subtle colouration, featuring a faint reddish tint on its outer perimeter and a soft bluish fringe on its inner boundary. In exceptionally fine mists, even these pale margins disappear, leaving an entirely colourless white arc against the grey mist.