Scattering efficiency, all the way from small to large

At its defaults it draws scattering efficiency, all the way from small to large. How strongly a sphere of index 1.333 scatters, as a multiple of its own geometric cross-section, against the size parameter — the circumference divided by the wavelength. The horizontal axis is logarithmic and covers three and a half decades. On the left the curve is Rayleigh's, rising as the fourth power of size and drawn dashed for comparison; the two are indistinguishable up to a size parameter of about a half and differ by a sixth at one, after which the fourth power runs away and the series does not. The efficiency then climbs to a first and largest maximum of 3.98 at x = 6.49, where the light that went through the sphere emerges 0.69 of a wavelength behind the light that went round it and the two interfere constructively in the forward direction. Past that it oscillates with diminishing amplitude toward two — not one — so a large sphere removes twice as much light from a beam as it geometrically blocks, which is the extinction paradox; the curve here has reached 2.12 by the right-hand edge. Nothing in the figure is a fitted or drawn shape: every point is the Mie series summed at that size.

mie-size is one function in lib/figures/optics.js — rays, lenses, mirrors and what light does to a surface. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

How strongly a sphere of index 1.333 scatters, as a multiple of its own geometric cross-section, against the size parameter — the circumference divided by the wavelength. The horizontal axis is logarithmic and covers three and a half decades. On the left the curve is Rayleigh's, rising as the fourth power of size and drawn dashed for comparison; the two are indistinguishable up to a size parameter of about a half and differ by a sixth at one, after which the fourth power runs away and the series does not. The efficiency then climbs to a first and largest maximum of 3.98 at x = 6.49, where the light that went through the sphere emerges 0.69 of a wavelength behind the light that went round it and the two interfere constructively in the forward direction. Past that it oscillates with diminishing amplitude toward two — not one — so a large sphere removes twice as much light from a beam as it geometrically blocks, which is the extinction paradox; the curve here has reached 2.12 by the right-hand edge. Nothing in the figure is a fitted or drawn shape: every point is the Mie series summed at that size.

The light scattered back by water drops of about 10 µm radius, at 550 nm, from the exact Mie series with around 134 terms for each drop, averaged over a ten per cent spread of sizes so that the fine ripple each single drop has is washed out. Instead of the smooth, weak background of the ray picture, the intensity rises sharply toward 180° through a series of rings about one and a half degrees apart, each brighter than the last, to a first ring 1.5° from straight back. These are what an observer sees round the shadow of an aircraft on a cloud, or round a shadow on a mist at a mountain top: the glory. It is present in the same mathematics that gives the rainbow, and it takes the whole series, with its surface-wave terms, to find.

The angle through which a ray is turned by a sphere when it enters, reflects once inside and comes out, against how far from the centre line it entered. In water the deflection falls from 180° on the axis to a minimum of 137.9°, the rainbow's angle, then rises to 165.6° at the rim. Apart from a ray exactly on the axis, nothing with one internal reflection is sent straight back, and only a vanishingly narrow bundle near the axis comes close. At an index of 1.5 the rim ray is at about 193°, and at the square root of 2 it would reach exactly 180°. Water falls short by about 14°. A ray optics of water drops therefore predicts no brightening toward the antisolar point, and that is what a glory contradicts.

The shortfall in a rim ray's total deflection from 180°, against refractive index, 4 arcsin(1/n) − 180°. A ray grazing a sphere enters at the critical angle, reflects once and leaves at the critical angle, turning through 360° less four critical angles. For water that comes to 14.4° short of straight back, and only above an index of the square root of 2 can a ray turn past it. So no geometrical path connects the sunlight to the point opposite the Sun in a water drop. Light can cross the gap only by travelling along the surface for those fourteen degrees, as a wave clinging to the boundary and shedding energy tangentially, which is not a ray at all. That route was suggested by H. C. van de Hulst in 1947 and worked out in detail by Nussenzveig much later.

The back-scattered intensity from drops of 5, 10 and 20 µm radius at 550 nm, each averaged over a ten per cent spread of sizes and scaled to its own largest value. The rings' angular spacing is roughly inversely proportional to the radius: smaller drops give wider rings, as any diffraction pattern does. An aircraft passenger who looks at a glory is looking at a measurement of the sizes of the cloud's droplets, and it has been used that way from satellites. A wide spread of droplet sizes washes the rings out, which is why glories appear only in clouds with a narrow distribution of drop sizes, and why sharp ones are found mostly at the tops of the fresh layers where droplets have grown from a similar start.

The efficiency of a sphere at scattering light straight back, averaged over a small spread of sizes to remove the sharp resonances, against the sphere's size parameter, with the line a drop's surface reflection alone would give. Reflection from the front of a water drop is 2 per cent; the full solution for water averages 2, about 78 times that. There is simply far more light going straight back than the ray picture provides, and it comes from waves carried round the surface of the drop and shed along the axis. The efficiency stays high all the way to the largest drop computed, so this is not a matter of the drop being small enough for rays to fail. The limit is approached so slowly that rays never catch up over any size that actually occurs in clouds.

physicscheck asserts something about mie-size that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.