Why a Rainbow Has a Dark Band Beside It
The Rainbow Lights the Other Side
Imagine a double rainbow with its colors temporarily removed. What remains is not two empty outlines. The drops contribute light inside the smaller bow and outside the larger one, with a relatively dark interval between them. Clouds and background scenery can complicate that pattern, but the rainbow itself is changing more than two narrow stripes.
The colors get the applause. The dark space explains something they can distract us from: a rainbow is not just sunlight sorted into wavelengths. It is also sunlight redistributed among directions.
That interval is called Alexander's dark band. Its explanation begins with the light on either side of it—not with something absorbing light in the middle.
A bright edge, not a painted stripe
Sunlight entering a water drop bends, some reflects inside, and the returning portion bends again as it leaves. The primary rainbow comes from light that has undergone one internal reflection. Different wavelengths bend by slightly different amounts, which helps separate the colors. But that does not yet explain why the light gathers into an arc.
The missing ingredient is where the sunlight enters the drop.
Even for one wavelength, rays entering at different distances from a spherical drop's center emerge in different directions. As the entry position moves outward, their total turning first decreases, reaches a minimum, and increases again. Near that minimum, a relatively wide range of entry positions feeds a narrow range of exit directions. Light crowds into an angular neighborhood.
That concentration makes the bright edge. Les Cowley's primary-ray explanation follows the turning point and the light that emerges on its other side. The important surprise is that the colored edge is not the entire illuminated region. Other one-reflection rays also reach the sky inside the primary bow. There, contributions from different wavelengths overlap, giving a more nearly colorless brightening rather than another clean spectrum.
It is better to picture the primary as the conspicuous edge of an illuminated angular region than as a colored ribbon pasted onto otherwise unaffected sky.
Forty-two degrees from what?
There are two angles here, and confusing them makes the next step unnecessarily mysterious.
One is the accumulated turning along the light's path, counting each bend and reflection—not necessarily the smaller angle between its initial and final directions. The other is where we look relative to the antisolar point—the direction directly opposite the Sun, around which the rainbow is centered.
For the primary bow, a ray turned through about 138 degrees appears about 42 degrees from that center: 180 minus 138. Greater turning sends the corresponding sight line closer to the center. Thus the ray family's minimum turning creates its outermost radius.
I checked the geometry for ideal spherical drops with a single illustrative refractive index of 4/3. Snell's law gives a primary limiting radius of about 42.03 degrees. That is a calculation under stated assumptions, not a measurement of a rainbow or an exact radius for every color. Real water's refractive index varies with wavelength; that variation puts the primary red edge farther out than the blue edge.
The angle belongs to a viewing direction. It does not locate a solid colored object at one distance in the rain.
The second bow brightens the outside
The secondary rainbow comes from light that reflects twice inside a drop. The Met Office's rainbow overview identifies that extra reflection and the reversed color order. Cowley's secondary-ray account supplies the crucial geometry: the rays now turn through more than 180 degrees.
Their limiting turn is about 231 degrees. The radius is now the amount beyond 180 degrees—about 51 degrees—not 180 minus the turn. The same sort of minimum-turning condition therefore marks an inner edge rather than an outer one. The two-reflection family supplies light outside that edge.
The color reversal follows the same geometry. Red light is refracted less and has a smaller minimum turning angle. Subtract that smaller angle from 180 degrees for the primary, and red lies outside. Subtract 180 degrees from it for the secondary, and red lies inside. The water has not changed its preference; the relation between turning and viewing direction has changed.
Using the same illustrative index, my secondary calculation gives about 50.98 degrees. Between that and the primary's 42.03 degrees lies an interval of roughly nine degrees. These are monochromatic model boundaries, not a calibrated width for the colored display. Their job is to make the opposing bright sides concrete.
Darker does not mean unlit
Now the gap has an explanation.
For a given wavelength in this spherical-drop ray account, one-reflection light reaches directions inside the primary rim. Two-reflection light reaches directions outside the secondary's inner rim. Between those edges, neither of those families supplies a geometrical path to the observer. In the ordinary visible display, the broad gap lies beyond the primary's outer red edge and inside the secondary's inner red edge. The neighboring sky gets contributions that the interval lacks. That is the structure described in Cowley's dark-band explanation.
Nothing has to swallow the sunlight in the middle. It is relatively dark because of where these contributions go.
But “these contributions” matters. The interval is not a photon-free hole. Background atmospheric scattering remains; higher-order reflected light is not excluded by a calculation that follows only one and two reflections. Even the Met Office's shorthand that no sunlight is scattered toward the observer there needs this qualification. The ray account excludes specified paths, not every way a real sky can send us light.
Nor is the broad dark band simply the two rainbows canceling each other. That would be a different explanation, requiring wave phase rather than just ray directions.
Where the drawing runs out
Rays do not explain every feature of a rainbow. Fine extra bands called supernumeraries, often inside the primary, require the wave nature of light. As Cowley's supernumerary explanation shows, contributions following different paths can reinforce or cancel according to their relative phase. Adding their brightness without accounting for phase misses those fringes and the bow's broadening.
Drop size and shape also limit the simple spherical-ray picture. My calculation does not predict the exact contrast of a particular sky; it contains no measured droplet distribution, background brightness or full wave-scattering solution. This essay uses published teaching accounts and an ideal calculation, not a new sighting, photograph analysis or experiment.
Still, the model changes what is worth looking for. The next time the colors command the view, the larger question is what happens beside them: which side receives the extra light, and which side does not?
Beyond that is another puzzle. If the drops become very small, the crisp colored edge can give way to a broad, pale fogbow. At what point does the neat drawing of rays stop being the most useful picture? The dark band has already supplied a reason to ask: the picture explains a contribution, not the whole sky.