The angle the rainbow has to be, and why nobody chose it
Every rainbow ever seen has had the same radius. Not approximately — the same, to a fraction of a degree, in every country, in every century, from every kind of shower. A bow over a garden sprinkler has it too, and so does one in the spray of a waterfall.
That constancy is the thing to explain, and it is more surprising than the colours. The drops vary in size by a factor of a hundred, the showers vary in density and depth, and none of it makes any difference to the angle. Something in the problem is fixing a number, and it is not a property of water so much as a property of a curve.
Three applications of one law
Nothing happens inside a raindrop that is not already in the law of refraction. A ray does exactly three things.
It refracts on entry, bending toward the normal because water is optically denser than air. The normal at the point of entry is the radius through that point, so a ray striking off-centre meets the surface obliquely and a ray striking dead centre passes straight through.
It reflects from the back, partially — most of the light continues out of the drop and is lost to this story, and the fraction that reflects is what makes a rainbow the faint thing it is — a partial reflection of exactly the kind every boundary produces. This is an ordinary partial reflection rather than a total internal one; the angles inside a drop never exceed the critical angle, so a bow is built from perhaps four per cent of the light that entered.
It refracts again on the way out, bending away from the normal by the same amount it bent on the way in.
Add up the turning. Each refraction turns the ray by , where and are the angles of incidence and refraction; the internal reflection turns it by . So the total deviation is
with fixed by through Snell’s law. The whole of the rainbow is contained in that expression and in the single question of what it does as varies.
The minimum, which is what makes a bow
The curve has a minimum. A ray hitting the very centre of the drop goes straight in and straight back, deviating by a half turn. A ray grazing the edge is deviated less than that, then more again. Somewhere in between the deviation is least, and for water that least deviation is 137.9°, at an impact parameter of 0.861 of the drop’s radius.
Deviation of 137.9° means the light comes back toward the sun’s direction, missing it by . That is the rainbow angle, and it has now been computed rather than looked up.
The reason the minimum produces a bright direction is the part worth dwelling on, because it is a general principle wearing a specific costume. Near a minimum, a function is flat — its slope is zero, and it varies only quadratically. So rays entering over a broad span of impact parameters, roughly from 0.8 to 0.95 of the radius, all emerge within a fraction of a degree of the same angle. Everywhere else on the curve, a spread of entry points produces a comparable spread of exit angles, and the light is smeared thin.
The bow is therefore not a place where light is created. It is a place where light from many different entry points piles up, because the map from entry to exit has a stationary point there. This is the same argument that makes the stationary-time path the one light takes in Fermat’s principle, and the same reason that a caustic on the bottom of a swimming pool is bright: a fold in a mapping concentrates whatever the mapping carries.
The colours, which come from the width of the minimum
Everything so far used one refractive index and produced a white bow. The colours require the index to depend on wavelength, which it does, slightly.
Water’s index runs from about 1.3300 at the red end of the visible range to 1.3406 at the violet end: a difference of eight parts in a thousand. Running the same minimisation at each gives a bow at 42.5° for red and 41.0° for violet — a difference of one and a half degrees, which is about three times the width of the full moon.
Two consequences follow, and the second is the one that catches people out.
Red is on the outside of the primary bow. The colour that deviates least appears at the largest angle from the antisolar point, and violet, deviating most, appears innermost. That is the correct order and it is worth checking against a photograph, because the reasoning inverts once between “deviates most” and “appears where”.
And the bow’s width is set entirely by the dispersion of water. A liquid with less dispersion would produce a narrower, whiter bow at nearly the same angle; a drop of something strongly dispersive would produce a fat, garish one. The angle is set by the index; the width is set by how the index changes with colour; and those are two different properties that happen to be carried by the same material.
Where the bow is, and why it is personal
The construction has said nothing about where the drops are, and it turns out not to need to.
Every drop in the sky is doing this. A given drop sends its concentrated light out on a cone of half-angle 42° about the direction back toward the sun. An observer sees that light only if they happen to lie on that cone — so the drops that contribute are exactly those lying at 42° from the line running from the sun, through the observer’s head, and onward. That direction is the antisolar point, and the set of drops at a fixed angle from it forms a circle.
Three things follow immediately, and each disposes of a common misconception.
A rainbow is not an object and has no location. It is a direction, or rather a set of directions. Walking toward it moves the antisolar point with the observer, so the bow moves too, and it can no more be approached than a horizon can.
Two people never see the same rainbow. Each has their own antisolar point and therefore their own set of contributing drops. Two observers standing a metre apart are seeing light from different water, arriving at the same angle.
And the sun must be low. The antisolar point is as far below the horizon as the sun is above it, so a bow of radius 42° is entirely below the horizon whenever the sun is higher than 42°. That is why rainbows are a morning and evening phenomenon at temperate latitudes and why they are essentially absent at midday in the tropics — and why the bow is a full circle when seen from an aircraft, with nothing to cut it off.
The second bow, and the dark band
A ray can reflect twice inside the drop before leaving, and following that path gives everything else the sky shows.
Running the same minimisation with two internal reflections gives a least deviation of 230.9°, which is 50.9° past a half turn, so the secondary bow sits at 50.9° — outside the primary, and separated from it by about nine degrees.
Its colours are reversed, and the reason is the extra reflection. Each reflection turns the ray further; the geometry of which colour ends up outermost inverts; red is on the inside of the secondary bow and violet on the outside. This is a genuine prediction that costs nothing to check against any photograph showing both bows, and it is one of those cases where a theory that got the reversal wrong would be immediately dead.
The band between the two bows is darker than the sky either side of it, and has been called Alexander’s dark band since the second century. The explanation is one sentence given the curves above: the primary sends light at 42° and less, since 137.9° is the minimum deviation and every other ray deviates more, appearing at smaller angles from the antisolar point. The secondary sends light at 50.9° and more. Between 42° and 51°, no ray from either family arrives at all. The band is dark because it is a gap in the range of a function — the same kind of gap as the angles at which no refracted ray exists, reached by a different route.
What the ray picture cannot do
The account above is complete, it predicts three angles correctly, and it fails on something visible in a good photograph.
Beneath the primary bow, in fine drops, there are sometimes several further arcs — pale pink and green, close together, fading inward. These are the supernumerary bows, and nothing in the deviation curve can produce them. The curve says light appears at 42° and at every smaller angle; it says nothing about a series of alternating bright and dark arcs.
They exist because light is a wave. Two rays entering a drop at different impact parameters can leave in the same direction — the deviation curve takes each value twice, once either side of the minimum — and having travelled different path lengths they arrive with a phase difference. Where that difference is a whole number of wavelengths they reinforce; where it is a half-integer they cancel. The supernumeraries are an interference pattern between two rays that the ray picture regards as unrelated.
That also explains why they need small drops. The path difference depends on the drop’s size, so large drops produce fringes packed too closely to separate, and only drops below about a millimetre give visible ones. The presence of supernumeraries is a direct reading of the drop size distribution in the shower.
Young used exactly this in 1804 as an argument for the wave theory, and Airy computed the full pattern in 1838 — introducing, in the process, the function now named after him, which also describes the spot a perfect lens makes and the pattern behind a single slit. The rainbow and the diffraction limit of a telescope are the same mathematics, arrived at from two directions that have nothing obviously in common.
What the geometry costs
The drop is doing something remarkable and it is worth counting what it wastes, because the bow’s faintness is not incidental.
Of the light entering a drop, the internal reflection at the back returns only a small fraction — the rest is transmitted straight out and goes nowhere useful. At the angles involved, that fraction is a few per cent. Then the second refraction loses a little more. A primary bow is built from something under five per cent of the light that entered the drops, and a secondary from well under one, which is why the second bow is faint and the third is a photographic achievement rather than a sight.
The second cost is that the concentration is angular rather than spatial. Nothing is focused anywhere; the drop simply maps a range of entry points onto a narrow range of exit directions. So the brightness of the bow is bought entirely from the flatness of the curve near its minimum, and that flatness is the same quadratic behaviour that makes any maximum hard to locate experimentally — a wide range of inputs giving nearly the same output. What is a nuisance when measuring an optimum is the entire mechanism here.
The third is that the whole construction is available only because the drop is a sphere. A sphere has no orientation, so every drop in the sky contributes on the same terms regardless of how it is turned, and the pattern survives being made from millions of independent scatterers. The moment the drops become non-spherical and oriented — as ice crystals are — the calculation has to track orientation as well, and the results are halos and arcs with quite different geometry.
Where the model stops
Beyond the supernumeraries, four assumptions are doing work and each fails somewhere.
The drops are spheres. Small drops are, held that way by surface tension. Large ones — above about a millimetre — are flattened by air resistance into shapes closer to a bun, and their bows are correspondingly distorted, brighter at the sides and weaker at the top.
The sun is a point. It is half a degree across, so every ray arrives with a half-degree spread and every feature of the bow is blurred by that much — the same finite-source blurring that washes out interference fringes. The bow’s colours would be considerably more saturated under a point source, and the sun’s angular size is comparable to the width of a single colour band.
One reflection, or two. Higher-order bows exist — the third and fourth are at 40° and 46° from the sun rather than from the antisolar point, which puts them in the glare, and they are faint enough that the third was not convincingly photographed in nature until 2011.
And the light is unpolarised on arrival. It does not leave that way. The internal reflection happens near the Brewster angle for water, so the light of a rainbow is strongly polarised — about 96 per cent — tangentially around the arc. A polarising filter rotated in front of a rainbow can extinguish it almost entirely, which is the easiest experiment on this page and the most convincing demonstration that a bow is made by reflection rather than by anything the drops are doing on their own.
The ladder from here
Later rungs on this anchor: dispersion measured properly, with the Cauchy and Sellmeier fits that let an index be computed at any wavelength. The prism at minimum deviation, which is the same stationary-point argument in a shape that makes it easier to see. The achromatic doublet, which cancels dispersion while keeping refraction. The Airy theory of the rainbow, and the supernumeraries computed rather than described. Polarisation and Brewster’s angle. Halos and sundogs, which are the same kind of calculation for hexagonal ice rather than spherical water — and which produce 22° rather than 42° for a reason with the same shape. The glory and the fogbow, where the drops are small enough that the ray picture fails outright. And caustics in general, of which the rainbow is the most famous instance and a swimming pool floor the most common.