The fringes below the rainbow

The rainbow’s angle is forced by the geometry of a sphere and the refractive index of water, and the calculation that gives it also gives an infinite brightness at that angle. This essay is about what is actually there.

Where the ray calculation breaks #

A ray entering a spherical drop, reflecting once inside and leaving is deviated by an angle that depends on where it struck. That deviation has a minimum — the Descartes angle, 42.08° from the antisolar point for water — and near a minimum a large range of impact parameters gives nearly the same deviation.

The intensity is the incoming flux divided by the outgoing angular spread, and at a minimum the spread goes to zero. So the intensity goes to infinity — which is not a prediction but a notice that the approximation has failed. What has failed is the assumption that light travels along rays at all, and the repair is to go back to the wave.

Near the rainbow angle two rays with different impact parameters emerge in the same direction, one from either side of the minimum — and that pairing is the whole mechanism. Two paths arriving together with different path lengths interfere, the difference between them varies with angle, and the result is the fringe pattern. Below the rainbow angle the two rays do not exist, which is why there is nothing there to interfere and nothing to see.

The wavefront is a cubic #

The wave calculation is easier than it sounds, because near a caustic every optical system looks the same.

Write the phase across the emerging wavefront as a function of position. At an ordinary focus it is quadratic; at a caustic the quadratic term vanishes — that is what a caustic is — and the leading behaviour is cubic. Everything else is detail.

The far-field amplitude is then the integral of $e^{i(\text{cubic})}$ over the aperture, which is the Airy function by definition:

$$\int e^{i(u^3/3 + \zeta u)}\,du = 2\pi\,\mathrm{Ai}(\zeta)$$

so the intensity is $\mathrm{Ai}^2$ of a scaled angular offset. That gives three predictions the ray picture does not make. The maximum is finite and sits inside the geometric angle, at $\zeta = -1.0188$. There are further maxima behind it, at −3.2482, −4.8201 and so on — the supernumerary arcs. And on the other side, where geometry forbids any light, the Airy function decays exponentially rather than stopping.

The same three features appear at the edge of a shadow: a bright fringe displaced into the lit side, oscillations behind it, and light leaking into the geometric shadow. That is not an analogy — a shadow edge and a rainbow are both places where a ray calculation predicts a discontinuity, and the wave answer replaces the discontinuity with the same family of oscillations in both cases. The difference is only in the shape of the wavefront being integrated.

The scale, which is where the drop size enters #

The Airy variable carries all the physics of the particular caustic:

$$\zeta = -(\theta - \theta_c)\left(\frac{2k^2}{D''}\right)^{1/3}$$

with $D''$ the curvature of the deviation function in impact parameter — computed here by tracing the ray, not quoted — and $k = 2\pi/\lambda$. Since $D''$ scales as $1/a^2$ for a drop of radius $a$, the angular scale of the whole pattern goes as $(\lambda/a)^{2/3}$.

That two-thirds power is the signature of a fold, and it is neither of the exponents an optics course usually supplies. A two-slit fringe spacing goes as $\lambda/d$; a focal spot goes as $\lambda/D$; a caustic goes as $\lambda^{2/3}$, which is intermediate, and it comes from the cubic.

Why they are not always visible #

The fringes are always there and are usually invisible, and the reason is the source rather than the drops.

The sun is half a degree across. Every point of it makes its own rainbow, displaced by its own angle, and what reaches the eye is the sum. Where the fringes are closer together than half a degree, one point of the sun puts a maximum where another puts a minimum, and the pattern washes out.

So supernumeraries are a report about drop size. They appear in mist and light drizzle, where the drops are a tenth of a millimetre, and not in a thunderstorm, where they are millimetres. They are seen more often at the top of a bow than at its sides, because the largest drops have fallen furthest and the small ones are still high up. And a bow lit by a star rather than by the sun would show them at every drop size.

This is the sense in which a caustic is an instrument. The angle of the bow measures the refractive index; the fringes below it measure the drops.

The pattern from a circular aperture is named after the same person for the same reason, and it is worth keeping the two apart. Both are what happens when a ray calculation predicts an abrupt edge and a wave calculation is done instead — but the aperture case integrates over a flat wavefront and gives a Bessel function, and this one integrates over a cubic wavefront and gives the Airy function proper. Same author, same motivation, different integral.

What the two rays are doing #

It is worth putting the interference into words, because the standard account — “the two rays interfere” — is right and leaves out why the pattern is not a simple cosine.

Near the caustic, two impact parameters give the same emergent direction: one on either side of the minimum of the deviation curve. Their optical paths differ, and — this being an interference of two paths from one source — the pattern is decided by that difference in the same way two slits decide theirs. The difference grows with the angular distance from the caustic — so far from the caustic the two beams are far apart in phase and produce many closely spaced fringes, and close to it they merge.

That merging is what a two-beam calculation cannot handle. As the caustic is approached the two rays’ amplitudes both diverge and their phase difference goes to zero, and the correct answer requires treating them as one integral rather than as two beams. The Airy function is what that integral produces, and the two-beam cosine is its asymptotic form far from the caustic — where, satisfyingly, the fringe positions Young computed in 1804 come out right after all.

Airy’s calculation, and why it took so long #

The history is worth a paragraph because the delay is informative.

Descartes had the angle in 1637 and Newton had the colours by 1704, both from rays. The supernumerary arcs were known and unexplained: Newton’s theory could not produce them, and their spacing changing with the weather made them look like an atmospheric accident rather than a property of light.

Young identified them as interference in 1804 — between the two rays that leave a drop in the same direction — which was correct and gave the wrong intensities, because a two-beam interference calculation has equal-amplitude beams and the real ones are not equal near a caustic.

Airy solved it properly in 1838 by writing the diffraction integral over the cubic wavefront and inventing the function now named after him to evaluate it. That the special function had to be created for the problem is the reason the delay was a hundred and thirty years rather than ten.

A single slit’s pattern comes from the same kind of integral performed over a flat wavefront rather than a cubic one, and which special function appears is decided by the shape of the wavefront and by nothing else. That is why Airy’s calculation took so long to be done: the cubic case has no elementary answer, and he had to define the function that solves it before he could compute the spacing.

The secondary bow, and the dark band between #

The same analysis applied to two internal reflections gives a second bow at 51° with its own caustic, its own Airy pattern and its own supernumeraries — fainter, wider, and with the colours reversed because the deviation function’s minimum has become a maximum.

Between the two bows is Alexander’s band, named for Alexander of Aphrodisias who described it around 200 AD. It is darker than the sky on either side, and the reason is exactly the ray argument this essay is about: no ray of either order emerges into that range of angles at all. The primary bow sends light only outside 42° and the secondary only inside 51°, so the band between receives neither.

The wave treatment says the band is not perfectly dark: the exponential tail of the Airy function reaches into it from both sides. That is a small correction and a real one, and it is the same tail that puts light into a geometric shadow.

Where the same fold appears #

At every shadow edge and every focus, since a fold is the generic singularity of a ray map and cannot be removed by perturbing the optics. That structural stability is the content of catastrophe theory, and the fold and the cusp are the only two singularities that survive in two dimensions.

In a spherical mirror, whose caustic is a nephroid and whose failure to focus is the aberration a sphere cannot avoid. The bright cusp in a coffee cup is that caustic, and it too is finite and fringed on close inspection.

In gravitational lensing, where a lens with no focal length has caustics of exactly this kind and they produce the enormous magnifications that make individual stars visible at cosmological distances — and where the divergence is regulated by the source’s size rather than by diffraction.

The caustic of a spherical mirror is the same fold traced from the exact law of reflection, and everything in this essay applies to it. The brightness along that surface is finite rather than infinite, the fringes sit on its bright side, and the spacing depends on wavelength — which is why the bright cusp in a coffee cup is not quite the sharp line geometry predicts, and why it is faintly coloured at the edge.

And in the twinkling of stars, where atmospheric turbulence produces a moving pattern of caustics on the ground. The bright lines on the bottom of a swimming pool are the same thing made by surface waves, and their sharpness is limited by exactly the argument in this essay.

The white bow, and why it is white #

The two-thirds scaling has a limit at the small-drop end that is worth following, because it produces a different-looking phenomenon out of the same arithmetic.

Take the drops down from a tenth of a millimetre to ten micrometres — fog rather than drizzle. The angular width of the Airy pattern goes as $(\lambda/a)^{2/3}$, so shrinking the drops by a factor of ten widens everything by a factor of about 4.6, and the main peak that was a few tenths of a degree wide becomes several degrees wide.

That width is now comparable with something else in the problem. The rainbow angle itself depends on wavelength — that is where the colours come from — and it moves by about two degrees between red and violet. Once the Airy pattern of a single colour is wider than the separation between colours, the colours overlap and the bow is no longer coloured.

What is seen instead is a fogbow: a broad, pale, almost white arc, centred a degree or two inside where the rainbow would be, often with one or two faint bluish and reddish fringes on its inner edge. Every feature of that description follows from the Airy function. The whiteness is the overlap. The displacement inward is the same $\zeta = -1.0188$ shift that moves the peak inside the geometric angle, now large enough to see. And the faint inner fringes are the supernumeraries, spread so far apart that only one or two fit before the pattern fades.

Fogbows are seen from hilltops above a valley of fog, from ships, and — as a complete circle — from an aircraft looking down at cloud. They are the same optics as a rainbow with one number changed.

The same widening explains something more ordinary. Cloud droplets are of that size, and a cloud is white rather than coloured because each of its droplets smears the colours together in exactly this way. The progression from a sharp coloured rainbow through a pale fogbow to a white cloud is one continuous function of drop size, and the whole of it is $(\lambda/a)^{2/3}$.

The same function at a quantum turning point #

The Airy function was invented for this problem and it turns up in a place with no optics in it at all, for a reason that is exactly the reason it turns up here.

Consider a particle in a potential well, with energy $E$, approaching the point where $V(x) = E$. Classically it stops there and turns round. The classical description says the particle spends more time near the turning point than anywhere else, so the probability of finding it there diverges — as the inverse square root of the distance from the turning point, which is precisely the exponent the ray calculation gives at a caustic.

That is not a coincidence of exponents. A classical turning point is a caustic: it is where the family of classical trajectories folds back on itself, so that two branches of the motion meet and the density of paths diverges. The ray approximation in optics and the classical approximation in mechanics are the same approximation, and both fail in the same place for the same reason.

The repair is also the same. Expand the potential about the turning point; to leading order it is linear, and Schrödinger’s equation with a linear potential is Airy’s equation. The wavefunction near any ordinary turning point is $\mathrm{Ai}(\zeta)$, with $\zeta$ a scaled distance built from the potential’s slope and from $\hbar$ — the place where the discarded quantity re-enters, exactly as the wavelength did above.

The three features then read across without alteration. The oscillations on the classically allowed side are the standing wave in the well. The exponential tail on the forbidden side is the wavefunction leaking under the barrier, which is tunnelling — and it is the same tail that faintly illuminates Alexander’s dark band. And the displacement of the first maximum inside the turning point is the reason a quantum well’s effective width is slightly larger than its classical one.

The displacement has a well-known consequence. Matching the Airy solution to the oscillating region shows that the wave picks up a quarter-cycle of phase at each turning point, and a bound state must accumulate a whole number of cycles in a round trip. Two turning points at a quarter cycle each is half a cycle, which is where the one-half in the quantisation condition $\oint p\,\mathrm{d}q = (n + \tfrac12)h$ comes from.

That half is usually presented as a correction to be remembered. It is the Airy function’s asymptotic phase, and it is the same number that displaces the peak of a rainbow inside the angle Descartes computed.

The general lesson about divergences #

A physical quantity that comes out infinite is always a statement about the calculation, and this one is a clean case worth generalising from.

The divergence appeared because the ray approximation ignores a length — the wavelength — and any approximation that has thrown a length away will produce answers with no scale in them, including infinite ones. Restoring the length restores the scale, and the resulting pattern has a width, a displacement and a fringe spacing all built from it.

The pattern is common. The intensity at a focus is infinite in geometric optics and finite at $\lambda/D$; the field at a point charge is infinite in classical electromagnetism and regulated by quantum mechanics at the Compton wavelength; the energy in a blackbody’s short-wavelength modes is infinite classically and regulated by Planck’s constant. In every case the cure is the same shape: the divergent theory is the limit in which some quantity is taken to zero, and putting it back gives a finite answer whose size is a power of that quantity.

The exponent is what differs, and it is worth predicting rather than fitting. Here it is two-thirds, and it follows from the deviation function’s first two derivatives vanishing at the caustic — one because it is stationary, one because that is what makes it a fold.

Reading a photograph #

The practical use of all this is that a photograph of a bow can be measured, and the measurement gives the drops.

Three quantities are available. The angular position of the main peak relative to the geometric bow gives the scale directly, since the displacement is 1.0188 in the scaled variable. The spacing between the first and second supernumerary gives the same scale a second time, independently. And the contrast of the fringes gives a check, because it depends on the spread of drop sizes as well as on their mean — a shower with a wide size distribution smears its own fringes even when the sun is a point.

Doing this on a good photograph gives drop radii to within about ten per cent, which is comparable with what a disdrometer on the ground achieves and covers a volume of sky no instrument could sample. The technique has a literature, and its limitation is that the bow reports the drops along the line of sight at the moment of the exposure — which is a strength for meteorology and a nuisance for calibration.

The pattern here is one this site meets repeatedly. A quantity that a ray calculation treats as a nuisance — the width of a feature that should have been a line — turns out to carry the information, because the width is where the wavelength entered.

What the pictures cannot show #

The cubic is the leading term only. Far from the caustic the higher terms matter and the Airy form is wrong; the figures are drawn over the range where it holds, which is a few fringes. That range is also where all the observable structure is.

The rainbow calculation is monochromatic and scalar. Real bows are coloured, and the different wavelengths’ patterns overlap — which is why supernumeraries appear pink and green rather than as bright and dark bands, since the maxima of different colours fall in different places. Polarisation is also ignored; the rainbow is about 96 per cent polarised, which follows from the internal reflection happening near the angle at which reflection picks a side.

Drops are not spheres. Above about a millimetre falling drops flatten, and the flattening changes the deviation function differently for rays in the vertical and horizontal planes. That is why the top of a large-drop bow can look different from its sides beyond the drop-size effect described here.

And the Airy theory itself is superseded. The exact answer for a sphere is the Mie series, and comparing the two shows Airy’s approximation is good for drops above about 50 micrometres and poor below, where the drop is only a few wavelengths across and there is no ray picture to perturb.

The ladder from here #

Later rungs on this anchor: the cusp caustic and the Pearcey function, which is the two-parameter case; catastrophe optics as an organising scheme for diffraction patterns; the Mie solution and the range over which Airy’s approximation holds; and the higher-order bows, whose caustics are folds of the same kind at different angles.

The neighbouring ladders are the rainbow’s angle, which is the ray calculation this essay repairs, where rays stop being enough, which is the same failure in a simpler geometry, and the mirror that cannot focus, whose caustic is the other standard example.