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.

How many times more strongly a sphere scatters 450 nm light than 650 nm light, against its radius, with the radius on a logarithmic axis running from a couple of nanometres to twenty micrometres. On the left the ratio sits at 4.35, which is the fourth power of the wavelength ratio and the whole reason the daytime sky is blue. It does not stay there. By a radius of 245 nm the preference has halved, and by a micrometre it has essentially gone: a particle comparable with the wavelength scatters every visible colour within a few per cent of equally, which is why a cloud is white, why fog is white, why milk is white and why the exhaust of a cold diesel is white while the smoke of a cigarette — whose particles are ten times smaller — is blue. Nothing about the material changed between one end of this axis and the other; only the size did.

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 scattered intensity against scattering angle for 4 particle sizes, each normalised to its own forward value and plotted logarithmically, so what is being compared is shape rather than strength. The smallest is Rayleigh's dumbbell: as much light goes backwards as forwards, with a minimum at ninety degrees, which is why the sky is bright in every direction away from the sun. As the particle grows the pattern tips forward, and by a size parameter of twenty the forward lobe is 1.5e+3 times the intensity at right angles and the backward hemisphere has essentially emptied. The asymmetry parameter — the mean cosine of the scattering angle — runs 0.002 at x = 0.1, 0.185 at x = 1, 0.844 at x = 5, 0.772 at x = 20. That forward tipping is why a cloud is bright when the sun is behind it and merely grey when the sun is behind the observer, and why fog defeats headlights: the light comes back not because it is reflected but because it is scattered many times, each one a small deflection forward.

The degree of linear polarisation of light scattered through exactly ninety degrees, against the size of the scatterer. A Rayleigh scatterer polarises completely at that angle — the value is one on the left of the figure — because a dipole cannot radiate along its own axis, so one of the two incident polarisations contributes nothing in that direction. The full series shows the value collapsing as the particle grows past a size parameter of about 2.09, where it has fallen to a half, and oscillating about zero thereafter — a large droplet's ninety-degree scattering is not polarised at all, and can even be polarised the other way round. It is the cleanest test there is of what is doing the scattering: a polarising filter turned against the blue sky ninety degrees from the sun darkens it markedly, and turned against a white cloud in the same part of the sky does almost nothing.

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.

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.