The rings no ray can reach

How far the air lets a mountain be seen dealt with haze as a loss of contrast, and when the particle is the size of the wave computed scattering by a sphere from the Mie series across the full range of sizes, with a brief mention of a feature near 180 degrees that it called the strongest available argument that summing the series is not merely a technical convenience. This essay takes up that feature. It has a long history. In 1735, Pierre Bouguer’s expedition to measure the shape of the Earth in the Andes saw coloured rings around their own shadows on mist. They appeared in the accounts of climbers on the Brocken in Germany, along with the magnified shadow known as the Brocken spectre, and of balloonists. Passengers in aircraft see one around the plane’s shadow on any low bank of cloud with the Sun behind them. It is called a glory.

A glory looks as though it must be a sort of small rainbow, and the first attempts at explaining it treated it as one. It is sitting at a specific angle from the Sun’s direction, it has colours, and it is made by spherical drops of water. The angle the rainbow has to be explained one of those at 138 degrees by a stationary ray path. By that reasoning there ought to be a stationary ray returning near 180 degrees. There is not. The attempt to find one runs into a gap of about fourteen degrees that no ray in a water drop can cross.

Every ray a water drop returns #

A ray enters a sphere of refractive index $n$ at an impact parameter $b$ (measured as a fraction of the radius), with angle of incidence $i$ given by $\sin i = b$ and angle of refraction $r$ by $\sin r = b/n$. It refracts at the surface, reflects $k$ times inside, and refracts out. Its total deflection is

$$D = 2(i - r) + k(\pi - 2r).$$

With $k = 1$ in water, $D$ starts at 180° for the ray that goes straight through the centre and bounces back. As $b$ rises, the deflection falls, reaching a minimum of 137.9° near $b = 0.86$. That minimum is what gives the rainbow, since rays pile up at a stationary point, and it is the angle of the primary bow. Then it rises again as the ray goes off towards the rim, but it only gets as far as 165.6°, at $b = 1$, where the ray enters tangentially.

So at $b = 0$ the ray returns straight back, and its neighbours fall away quickly. This axial reflection is a real effect. It is what makes a drop retroreflect at all, and the fraction of incident light in that tiny bundle gives a small, smooth background. A nearby bundle spreads rapidly with angle. Two reflections, or more, give other curves, but none come back close to 180 degrees with any useful strength. So a water drop in the ray picture sends only a small amount of light in nearly the backward direction, and nothing special happens at that angle apart from a mild concentration.

Now change the index and watch what happens. For glass at $n = 1.5$, the rim ray is at 191°, past the backward direction; for $n = 1.9$, the curve flattens at 180° for small $b$ and goes further still. A sphere with $n = 2$ focuses parallel rays exactly on its back surface, and these return exactly toward the source, a neat retroreflector known in road paint: glass beads are used on road markings because they have a high enough index to come close to this. The corner that sends light home discussed another way to get light straight back, and the full moon that is too bright mentions a similar brightening on a rough surface.

Fourteen degrees that rays cannot make up #

A ray at the rim enters tangentially, so $i = 90°$, and refracts to the critical angle, $r = \theta_c$, with $\sin\theta_c = 1/n$. Putting that in the formula for $k = 1$ gives $D = 360° - 4\theta_c$. For this to reach 180°, $\theta_c$ must be 45°, which needs $n = \sqrt2 \approx 1.414$. Water’s index is 1.333, and its $\theta_c$ is 48.6°. So the shortfall is 14.4°.

This number has a crucial place in the story. The whole of the glory has to get from the end of the ray’s reach to the backward direction, across those 14.4°, with no ray to carry it. If ice or glass drops were at issue, a ray with a suitable $b$ could do it. For a water cloud, it cannot be done by rays at all, and since a water drop in a cloud has the wrong index by less than 7 per cent, a strong bright feature ought to be completely absent if rays were the whole story.

H. C. van de Hulst suggested in 1947 that the missing angle was traversed by a surface wave. A ray striking at the rim has a neighbouring set of paths, with slightly larger impact parameters, that would miss the drop by a small distance if they were rays. A wave is not so restricted. Its field near the surface extends a short distance beyond it, so that it can tunnel in, travel along the interface, tunnel out and go on. At each stage of the journey along the surface, it loses some energy, by radiating off tangentially. Yet those 14.4° are a short distance for a wave with a large drop. A wave can thus travel those fourteen degrees and still emerge with a substantial amplitude. Two such surface waves, from a whole ring of entry points round the drop’s edge, come out focused on the axis, in what is called the axial focusing effect. They arrive with a common phase for all azimuths, which is a very great concentration of energy, and that is what makes the glory bright.

Herch Moysés Nussenzveig, in a long sequence of papers from the 1960s onward, made this exact by expressing the Mie series in terms of complex angular momenta. Terms of the series with angular momentum greater than the drop’s size parameter, in which a ray would miss the drop, are found to matter strongly, since they represent light that tunnels through the classically forbidden barrier into the drop. The wall that is not quite a wall describes that same process in quantum mechanics, and the same mathematics applies in both cases: partial waves with an angular momentum above the classical limit see a centrifugal barrier, and a wave can pass through such a barrier. The glory, in a real sense, is a light-tunnelling effect. It occurs with a very weak resonance that keeps a wave near the surface, as the mode that will not turn a corner discusses in connection with whispering-gallery modes.

The rings the full solution finds #

A sphere’s scattering has an exact solution, due to Gustav Mie in 1908. The incident plane wave is expanded in spherical partial waves, each of which is matched at the surface, giving two amplitudes for each integer $n$. About $x + 4x^{1/3}$ terms are needed, where $x = 2\pi r/\lambda$ is the size parameter. Everything about the drop’s interaction with light is contained in this one sum, including rays, rainbows, diffraction and surface waves. It does not need to be told which of these it contains.

For a single drop, the result is a sharp and noisy pattern, because each drop has high-order resonances, narrow peaks in response at particular sizes. Averaging over a modest spread of sizes, as nature does, removes that noise. The figure uses 25 sizes spread over ten per cent about 10 µm. What emerges is a series of rings, equally spaced by about one and a half degrees and increasingly bright, as they get closer to the backward direction. The first and brightest is 1.5° away, with the backward point itself sitting in a slight dip for this polarisation average.

These rings are a real prediction of the calculation, and the sum knows nothing about a surface wave. They come from interference between the surface waves emerging on opposite sides of the drop, and they are thus a two-beam pattern, with the sources separated by the drop’s own diameter. The spacing is close to $\lambda/2r$ radians. Their position in the sum is unambiguous, and a person who computes this series for a size parameter of 114 obtains them without having to understand any of the reasoning above. A comparable computation before electronic computers was extremely long, and H. C. Bryant and N. Jarmie were among the first to see the rings in full numerical calculation in the 1970s.

Smaller drops, wider rings #

Since it is an interference pattern, its scale depends inversely on the drop’s size: 20 µm drops have rings about half as wide as 10 µm drops. A person seeing a glory is therefore reading off the mean droplet size in the cloud, and satellites have used that fact. Instruments such as POLDER, flying on the PARASOL satellite, have retrieved cloud droplet sizes from the shape of the glory seen in polarised reflected sunlight.

Sharp rings need a narrow range of sizes. That is why glories favour fresh, thin layers of cloud, where droplets started growing together and have reached a similar size. Large spreads of size wash out everything except a general brightening, and very old clouds with drizzle do not give a strong glory. Colours follow from the wavelength dependence: blue rings are smaller than red at a given size, so the rings are bluish on the inside and reddish on the outside. For a white light source, the second and third rings overlap, so the colours fade after a few orders.

An interesting fact about the shape of the shadow follows. An observer in an aircraft looking down sees the glory centred on the antisolar point, which is where the shadow of the observer’s own eye falls. On a plane, the rings are centred on the window where the observer sits and not on the aircraft’s centre. A pair of climbers side by side each sees a glory round their own head only, which impressed people greatly for centuries and appeared in the religious imagery of mountain regions. This has nothing to do with the physics of scattering, only with a geometrical point about direction: the rings surround the antisolar direction from every observer’s eye, and the distance to the cloud plays no part.

A drop that returns more than its surface can #

A useful comparison is with what a surface can do by itself. A water surface reflects 2 per cent at normal incidence, so a sphere’s external reflection returns about 2 per cent of the geometric cross-section per unit solid angle in the appropriate normalisation. The Mie efficiency near the backward direction is an average of about 1.6, or 78 times that. The back-scattered light is not a small correction to reflection. Radar and lidar depend on this enhanced backscatter from cloud drops, and the way these sensors estimate a cloud’s water content needs an accurate value for it.

It is equally striking that this does not go away with increasing size. There is no value of size parameter in the plotted range at which this approaches the ray-optics value. Glass, at $n = 1.5$, is similar. Since a typical cloud droplet has a size parameter of about 100, it is well past where optics texts would call ray optics “accurate”. Nearly everything about the drop at that size is given well by rays: the total scattering cross-section is close to the geometric-optics value of two, and the rainbow’s angle and most of the angular distribution are correct. Nevertheless, in this one direction, rays give the wrong answer entirely, because the effect is not a small refinement but a separate channel that they lack altogether.

Geometric optics treats a wave as if its wavelength were zero, and that limit is an asymptotic and not a uniform one. There may be directions and regions where a term of relative order $x^{-1/3}$ is more significant than the leading term, because the leading term vanishes there. The size parameter would have to reach $x^{1/3} \gg 1$, or $x \gg 1000$, for these corrections to become small in general, and in this case an axial focus raises the importance of the surface term further still. Where rays stop being enough dealt with the same failure in the simplest case, Poisson’s spot, in which the wave picture gives a bright point where rays have none. The glory is the same kind of phenomenon at a far more demanding level, because of the geometry of its axial focus.

Other places where the fourteen degrees matter #

The arithmetic of the shortfall also explains why such a thing does not show up in other spheres. For ice crystals, geometry fails altogether, as they are not spheres, and they produce a different family of phenomena, such as those in the ring at twenty-two degrees. Cirrus clouds, as a rule, give no glory. The presence of a glory is thus a good clue to the phase of a cloud top, liquid or ice, and satellite retrievals use this as well: a glory in an instrument’s view means liquid water in spherical droplets.

The planet Venus showed one too. In 2011 the Venus Express orbiter imaged what looks like a glory in the planet’s upper clouds, at the antisolar point, and this has been used to measure droplet sizes near 1.2 µm and to constrain their composition, since the refractive index of Venus’s sulphuric acid droplets, near 1.44, is close to $\sqrt2$ and gives an index-dependent glory of a different form. The observed ring pattern suggested that the droplets’ composition may differ from pure sulphuric acid. The light that comes back bluer when it is warm gives another example where a small spectral feature in reflected light says something about a planet’s materials.

The same mathematics applies in a domain nobody would associate with clouds. Nuclear physicists studying the scattering of particles from atoms and nuclei find a backward glory too, in the same partial-wave expansions. In atom-atom scattering, an attractive potential combined with a repulsive core can give a deflection function passing through 180 degrees. The classical trajectories there really do reach it, and the backward peak is called by the same name. The analogy is close enough for theorists of optics and of nuclear physics to use the same terms and much of the same mathematics, including Regge poles.

Perfect spheres at one wavelength #

The drops here are perfect spheres of pure water at one refractive index, and the wavelength is fixed. The refractive index of water changes slightly across the visible range, and real cloud drops carry dissolved material and are slightly flattened if they are large. Because the glory depends on an exact axial focus, it is very sensitive to non-sphericity, and drops larger than a few tens of micrometres, distorted by their fall, will not give a good glory. Multiple scattering in a thick cloud washes it out to a degree, and the visible glory comes mostly from a thin top layer. Averaging here assumes a simple spread of sizes with a Gaussian-like weight, and a real cloud’s distribution may differ. An observer’s perceived colour also depends on the full spectrum, while this calculation has been done at a single wavelength of green light.

The intensities plotted are also the result of averaging over polarisation. Near the backward direction, the two polarisations behave very differently, and the structure at the very centre depends on that. In addition, the Mie series gives an exact number for a sphere, but the interpretation in terms of surface waves is not unique: it rests on the rearrangement that Nussenzveig developed, in which the series is transformed into a contour integral. Other equally exact accounts have been offered, and the correct physical picture remains under some discussion. What can be said firmly is that ray paths alone do not suffice.

Still open: how much the surface wave does #

The claim that the glory depends on a surface wave has a clean mathematical statement and a controversial quantitative one. Philip Laven computed in detail in 2005 what happens when the series is decomposed by contributions, and he argued that what dominates comes from the interference of several different terms, including high-order internal reflections with their associated surface waves, and that no single simple picture is accurate at all sizes. He noted that surface-wave contributions depend sensitively on the phase of the drop’s resonances, and when a whole size distribution is averaged, the result is not what the simplest two-wave account would imply. There are also experiments on single levitated drops, in which a drop is held still by a laser or an electric field and its scattered pattern is measured at known size. Those give a very clean test, and they are technically demanding.

A separate and practical question concerns retrievals, since remote sensing relies on what a real cloud’s mix of droplet sizes does to the rings and on how well a simple functional form can represent that distribution. How accurately a glory’s fine detail can determine what a cloud is made of, compared with other sensors, remains a current subject in cloud science, and it matters for climate models that depend on knowing the sizes of droplets.

In a drop of water, a ray entering at the rim turns through 165.6° after one internal reflection, 14.4° short of straight back, and water’s index does not allow any ray to do better. Yet the exact solution returns about 78 times what the surface reflects, in rings about 1.5° apart for a 10 µm drop. That light crosses the 14.4° along the surface, as a wave that a ray does not have. A drop with size parameter over 100 is large by every other test, and for this one direction rays have nothing to say.