The ring at twenty-two degrees

A halo is a ring of light round the sun or the moon, at a radius of about twenty-two degrees — roughly a handspan at arm’s length. It appears when the sky is covered by thin cirrus, it has a sharp inner edge, and the inside of the ring is noticeably darker than the sky just outside it.

Every part of that description follows from one property of a prism: its deviation has a minimum, and a minimum is flat.

The crystal and its two prisms #

Ice crystallises hexagonally, and a cirrus crystal is a hexagonal plate or column. The angles between its faces are therefore not arbitrary: alternate side faces meet at 60°, and a side face meets an end face at 90°.

Everything here is built from refraction at a single face. The ray bends towards the normal on entering the ice and away from it on leaving, and the two bends do not cancel because the two faces are not parallel — which is the whole reason a prism deviates and a window pane does not.

A ray that enters one side face and leaves an alternate one has passed through a 60° prism. One that enters a side face and leaves an end face has passed through a 90° prism. Those are the only two prism angles a hexagonal crystal offers to a ray that enters and leaves without reflecting, and they produce the two halos: the common one at 22° and the rare one at 46°.

Tracing a ray through either is Snell’s law applied twice and nothing else. The entry bends the ray towards the normal, the geometry of the apex changes which normal the second face presents, and the exit bends it away again. Because the faces are not parallel the two bends do not cancel, and the total is the deviation.

There is a range of incidence over which nothing gets through at all. If the internal angle at the second face exceeds the critical angle the ray is totally internally reflected and takes some other path entirely. That cutoff is what ends both curves in the opening figure, and the width of the window it leaves is the reason one halo is common and the other is not.

Why a minimum makes a ring #

The deviation is not a single number. For a 60° ice prism it runs from 21.84° upwards over a window of incidence 76° wide, so a crystal in a random orientation sends light almost anywhere in a forty-degree range.

What makes a ring is that the deviation is stationary at its minimum. Near a stationary point a function changes only in second order, so a wide band of orientations all deliver light to within a fraction of a degree of the same angle, while orientations away from the minimum spread their light thinly over the rest of the range.

The histogram makes the consequence explicit. There is a cliff at 21.84° with nothing inside it — no orientation of the crystal can bend a ray by less, so the region between the sun and the ring receives no refracted light at all and appears darker than the sky beyond. Outside, the intensity falls away gradually. A halo therefore has a sharp inner rim and a soft outer one, which is the shape of this distribution and is what anybody who has looked at one has seen.

This is the same argument that makes a rainbow, in the same words, applied to a different stationary point. It is also the same argument that makes a bright caustic wherever rays are focused by a stationary path length — the general statement being that light piles up where the mapping from source to angle has a fold.

The passage at the minimum #

There is a fact about the minimum that is worth extracting rather than assuming, because it explains why the number is so easy to compute.

At minimum deviation the passage is symmetric: the ray enters and leaves at the same angle, and inside the crystal it runs parallel to the base. That was not built into the drawing. The incidence was found by searching the traced curve for its smallest value, and the exit angle is then whatever Snell’s law gives at the second face; it comes out equal to eight decimal places.

The reason is a reversibility argument and it is short. A ray sent backwards along its own path has the same deviation, so every asymmetric passage has a partner with the same deviation on the other side of symmetry. A smooth function taking equal values at two arrangements has a stationary point between them, and by symmetry that point is the symmetric passage.

Once the symmetry is granted, the minimum deviation follows in one line: $D_{\min} = 2\arcsin\!\left(n\sin\frac{A}{2}\right) - A$. For $A = 60°$ and $n = 1.31$ that is 21.84°, and for $A = 90°$ it is 45.73°. Both are drawn from the traced rays and agree with the closed form to eight decimals, which is the check that the trace and the formula are about the same prism.

That expression run backwards is also how a prism spectrometer measures a refractive index: rotate the prism to minimum deviation, measure the angle, and read off $n$. It is one of the oldest precision measurements in optics and it needs no knowledge of where the prism is pointing, because the minimum locates itself.

Which edge is red #

The refractive index of ice falls with wavelength, from 1.3197 in the violet to 1.3090 in the red. Blue is bent more.

So the minimum deviation is smallest for red, and red therefore sits at the inner edge of the ring, which is the sharp edge where the light piles up. Outside it the colours overlap and wash into white within a couple of degrees. That is why a halo shows a distinct red inner rim and then gives up: the whole spread for the 60° prism is 0.81°, against a rainbow’s two degrees.

The direction is worth holding onto because it is the reverse of a rainbow’s. A rainbow’s red is on the outside, because there the stationary point is a maximum of the deviation and the caustic sits on the other side of the light. Same dispersion, same substance in the sense that both are water, opposite arrangement — and the difference is entirely which way the stationary point turns.

Anybody who has both a halo and a rainbow to look at has therefore a direct test of the two mechanisms available with no instrument at all.

Why the second halo is rare #

The 46° halo exists, is drawn in the same figures, and is seen perhaps once for every twenty times the 22° one is. The same calculation says why.

Ice against air has its critical angle at 49.8°, and past it nothing is transmitted at all. That angle is what closes the window of incidence over which a prism transmits: rays arriving too steeply never leave the second face. The two halos differ in rarity for exactly that reason — one prism angle leaves a wide window of incidence open and the other leaves a narrow one, and the narrow one needs more crystals to fill it.

A ray through the 90° prism has a much narrower window of incidence that gets through: 32° against the 60° prism’s 76°. Fewer orientations therefore contribute, so the ring is fainter in proportion for the same population of crystals.

There is a second effect the figure cannot show. The minimum deviation of a 90° prism is more sensitive to the crystal being tilted out of the plane of the drawing, so a real population smears the 46° ring over a wider angle than the 22° one, spreading the same smaller amount of light further. Between the two effects the outer ring is a marginal object requiring good crystals and a dark sky.

It also requires crystals with clean end faces, which are the faces most likely to be rounded or roughened. Column crystals with intact ends are common enough for the ring to appear, and less common than crystals with intact side faces.

The other things ice does #

Because the crystals are hexagonal and not spherical, orientation matters — and when a population is not randomly oriented, quite different shapes appear.

A raindrop is a sphere, so its orientation cannot matter and the only variable is where on it the ray strikes. That single degree of freedom is what makes a rainbow one bow with one radius. An ice crystal has an orientation as well as an impact parameter, and that extra freedom is why ice makes so many more shapes than water does — arcs, sun pillars, sun dogs and circles, each one a different subset of orientations selected by how the crystals happen to fall.

Flat plate crystals falling through still air settle with their faces horizontal, like leaves. Light from a low sun then passes through 60° prisms whose axes are vertical, which puts bright patches at 22° to the left and right of the sun and nowhere else — the sundogs, which are the same minimum deviation with the orientation restricted to a plane.

Column crystals settling with their long axes horizontal, and randomly oriented about them, produce the circumzenithal arc, an upside-down rainbow high above the sun that is far more strongly coloured than the halo because the light passes through a 90° prism at a near-symmetric passage.

None of those exists for water drops, because a sphere has no orientation. The whole zoo of halo phenomena — and there are dozens of named ones — exists because ice has a shape and a preferred way of falling, and it is a good example of an entire family of effects that follows from a symmetry being broken.

What a halo is worth measuring for #

A ring in the sky is a measurement instrument that costs nothing, and several things can be read off one.

Everything a halo says about the substance it is made of comes from dispersion. The angle depends on the refractive index, and the width of the coloured rim depends on how that index varies with wavelength — so measuring the ring’s radius measures $n$, and measuring the rim’s width measures $dn/d\lambda$. Ice disperses weakly, which is why a halo’s colours are faint where a rainbow’s are not.

The radius fixes the refractive index. Measuring a halo’s radius to a tenth of a degree fixes $n$ to about two parts in a thousand, which is enough to say that the cloud is water ice rather than any other substance likely to be up there. Carbon dioxide ice, for instance, has an index near 1.40 and would put the ring at 27°; there are Martian images showing exactly that.

The presence of particular arcs fixes the crystal habit. Sundogs require plates; the circumzenithal arc requires plates with intact end faces; a Parry arc requires columns with a preferred rotation about their long axes. A display containing several named arcs is a strong constraint on the size and shape distribution of the crystals producing it, and satellite retrievals of cirrus properties are checked against ground-based halo observations for that reason.

And the sharpness fixes how well aligned the crystals are. A random population gives the histogram drawn above; a population with even a weak preferred orientation gives a ring of uneven brightness, and the pattern of that unevenness says which direction is preferred.

None of that needs anything but a camera and a way of measuring angles, which is why halo observation is one of the few areas of atmospheric physics where amateur data are used directly.

The same shape, one field over #

A stationary point turning a broad distribution into a sharp line is not an optical idea, and recognising it elsewhere is most of what makes it worth naming.

The same mechanism appears with a phase in place of a direction. Sum contributions from a continuum of paths and where the phase changes quickly they cancel; near a stationary point a whole band of them adds. A halo is that argument with deviation angle in place of phase — the crystals whose deviation is stationary against orientation are the ones whose light piles up, and every other orientation spreads its light thinly across the sky.

In the sum over paths the same argument decides which trajectory a particle appears to follow: contributions from paths where the action changes quickly cancel in pairs, and only the band near the stationary path survives. The halo’s version is the same statement with a direction in place of a phase — orientations away from the minimum spread their light and effectively cancel, and the band near the minimum survives.

The same structure appears in a spectrum’s band edge, where the density of states piles up because a band’s energy is stationary in wavevector; in the caustics on the bottom of a swimming pool, where a rippled surface focuses light along the folds of a map; and in the rainbow, which is this essay’s own neighbour. The mathematics of these — catastrophe theory — classifies which shapes of stationary point are stable against perturbation, and a fold is the simplest and by far the most common.

That is a good reason not to treat a halo as meteorological trivia. It is a fold catastrophe with an angular radius that can be measured to a tenth of a degree by anybody with a camera, and there are not many places where a piece of mathematics is that directly visible.

The halos at other angles #

The essay has said that a hexagonal crystal offers exactly two prism angles, sixty and ninety degrees, and therefore two halos. That is right for the ordinary prism and plate habits, and there are rarer displays containing rings at nine, eighteen, twenty, twenty-three, twenty-four and thirty-five degrees.

They come from a different crystal. Ice sometimes grows with pyramidal end faces inclined to the long axis rather than perpendicular to it, and such a crystal presents a whole set of additional face pairs — with angles between them of about twenty-eight, fifty-two, fifty-six, sixty-two and eighty degrees, among others.

Every one of those is a prism, and every prism has a minimum deviation given by the one-line formula in this essay. Putting each apex angle into it with ice’s refractive index returns a radius, and the radii returned are the ones observed. The 28° prism gives the nine-degree halo, the 52.4° gives the eighteen, the 56° gives the twenty, the 62° gives the twenty-three, the 63.8° gives the twenty-four, and the 80.2° gives the thirty-five.

The test is a good one because nothing was adjusted. The face angles of a pyramidal ice crystal are fixed by the lattice, the refractive index is measured in a laboratory, and the halo radii are measured in the sky — three independent numbers, related by an expression with no free parameters, agreeing to a fraction of a degree.

The sequence also runs the other way historically. Rings at those unusual radii were photographed and catalogued before anybody had sampled the crystals responsible, and the existence of ice with pyramidal faces was inferred from the ring radii by inverting the formula. Aircraft sampling of the clouds in question confirmed it later.

Odd-radius displays are rare because the pyramidal habit is rare, and because a display containing several of them at once requires a population of crystals all grown the same way. The best of them appear in diamond dust — ground-level ice crystals at polar stations — where the population is uniform enough that a dozen named arcs can be photographed in a single frame, which is why the finest halo photographs in the literature were taken at the South Pole.

The three centuries it took to get the crystal right #

The rainbow was explained in 1637 and the halo in 1686, and the half-century gap between two problems that now look equally elementary is worth accounting for.

Descartes wrote Les Météores in part because a spectacular halo display had been seen over Rome in 1629 and reported to him. He solved the rainbow completely: traced rays through a spherical drop, found the stationary deviation, and got the angle. He did not solve the halo, and proposed instead that it came from rings of ice in the air.

What made the rainbow tractable is that a drop is a sphere. A sphere has no orientation, so the only variable is where on it a ray strikes, and the problem is one-dimensional. Everything Descartes needed was Snell’s law and a search over one parameter.

A halo needs something a seventeenth-century natural philosopher did not have: the knowledge that ice crystallises in a hexagonal form, and therefore that its faces meet at sixty and ninety degrees. That is crystallography, and crystallography did not exist. Huygens and Newton both attempted the halo and neither got it.

Mariotte proposed the correct mechanism in 1686 — refraction through a sixty-degree ice prism — and it was a guess about a crystal shape as much as a piece of optics.

The systematic theory arrived in 1847, from Auguste Bravais, who worked out the whole family: the two halos, the sundogs, the parhelic circle and the circumzenithal arc, each from a hexagonal crystal in a stated orientation. Bravais is better known for the fourteen lattices that carry his name, and the coincidence is not one. Solving the halo requires knowing which face angles a crystal can present, and the man who classified how crystals may be arranged was the man in a position to compute what they do to sunlight.

Which is the honest summary of what a halo is. It is a measurement of a crystal’s face angles and refractive index, made on a specimen ten kilometres up that nobody has touched, by an instrument consisting of the sun and an eye.

What the pictures cannot show #

The orientations are taken uniformly in the angle of incidence, which is the approximation this treatment makes. The correct weighting is uniform on the sphere with each face weighted by how much of it the sun sees, and it sharpens the inner edge further without moving it.

What wave optics puts in place of a geometrical caustic’s infinity is an Airy function: a finite peak just inside the caustic, oscillations behind it, and an exponential tail beyond. The same correction applies at a halo’s inner edge and is negligible there, because the crystals are enormous compared with a wavelength — a hundred micrometres against half a micrometre — so the geometrical answer is the answer to a part in ten thousand.

Everything is geometrical optics. At a caustic the ray treatment predicts an infinite intensity and the wave treatment replaces it with a finite Airy-function peak; for ice crystals a hundred micrometres across the correction is far too small to see, which is why a halo has no supernumerary fringes where a rainbow does.

Nothing here is polarised. The Fresnel coefficients at each face depend on polarisation, so a halo is partially polarised — weakly, and measurably.

And the crystals are assumed perfect. Real cirrus contains a great many irregular particles that contribute a featureless glare, and the visibility of a halo is largely a question of what fraction of the population has intact hexagonal faces. That fraction is what a halo display is really a measurement of.

The ladder from here #

Later rungs on this anchor: the sundog and its colour separation, which is minimum deviation with the orientation constrained; the circumzenithal and circumhorizontal arcs, which use the 90° prism at near-symmetric passage and are the most saturated colours the sky produces; the parhelic circle, which is reflection rather than refraction and is therefore white; and the rarer arcs whose existence pins down the crystal habit in a way no other ground-based measurement does.

The neighbouring ladders are the angle the rainbow has to be, which is the same argument at a maximum instead of a minimum, the fringes below the rainbow, where the wave correction to a caustic is large enough to see, and the path that does not change, which is the general statement that light accumulates where a path is stationary.