The pattern the sky is written in

Sunlight arrives unpolarised. The sky is not. Look at any patch of blue through a polarising filter and turn it, and the patch brightens and darkens — by a factor of several in some directions and not at all in others.

That is a pattern covering the whole hemisphere, fixed by nothing but the position of the sun, and it has been used as an instrument by insects for a long time and by people for two centuries. It follows from one fact about how a small scatterer radiates, and almost everything else about the sky’s polarisation is a departure from what that one fact predicts.

Why a scatterer polarises #

A molecule in the path of a light wave is driven by the wave’s electric field and re-radiates as a small oscillating dipole. Why the sky is blue and the sunset is not uses that picture to get the colour, from the fourth power of frequency in the radiated intensity. The same picture gives the polarisation, and it needs one further fact: a dipole does not radiate along its own axis.

Unpolarised sunlight driving a molecule sets it oscillating in both directions transverse to the sunbeam, equally. Now stand somewhere and look at that molecule. One of the two oscillation directions is transverse to the line of sight as well, and radiates towards the observer at full strength. The other is at an angle to the line of sight, and radiates in proportion to the sine of that angle.

At a right angle to the sunbeam the second oscillation points straight along the line of sight and radiates nothing at all in that direction. So only one of the two survives, and the light is completely polarised — with its electric field perpendicular to the plane containing the sun, the scatterer and the observer.

Straight towards the sun or straight away from it, both oscillations are equally transverse to the line of sight and both arrive equally. The light is unpolarised, exactly.

The pattern this forces #

Everything on the map follows from that curve plus a piece of geometry that needs no calculation.

The direction of vibration is perpendicular to the plane through the sun, the observer and the point being looked at. On the sky, that makes it tangent to the circle of constant angular distance from the sun. The pattern is therefore a set of concentric rings centred on the sun — and, since the sun and the point opposite it are both centres of the same family, the rings close around the antisolar point too.

The degree of polarisation depends only on angular distance from the sun, so it is constant around each of those rings and greatest on the ring at ninety degrees. That ring is a great circle, which means it runs from horizon to horizon through the zenith region rather than sitting in one part of the sky. With the sun low the band passes overhead; with the sun high it lies nearer the horizon all the way round.

Two consequences follow that are worth stating because they are commonly got wrong. The pattern does not depend on the time of year, the weather or the observer’s latitude except through the sun’s position — it is geometry. And the direction of vibration at a given point is not fixed in the sky; it rotates through the day as the sun moves, which is exactly what makes it a compass rather than a landmark.

What keeps it off a hundred per cent #

The ideal curve reaches one at ninety degrees and no real sky does. Two quite different things are responsible and the difference matters.

Air molecules are not spherical. A nitrogen molecule driven along one axis does not polarise purely along that axis, so its radiation is not that of an ideal dipole. The effect is small, fixed and unavoidable: it costs a few per cent everywhere, and it is the same on a mountain top as at sea level.

Light scattered twice has forgotten the geometry. A photon that has bounced off two molecules arrives from a direction that says nothing about where the sun is, carrying whatever polarisation the second event gave it. That contribution grows with the amount of air along the path, with the amount of dust in it, and with how bright the ground below is — since light reflected up and then scattered down is a large part of it.

So the maximum degree of polarisation is a measurement of the atmosphere rather than a constant. Eighty per cent is a very clear sky. Fifty is an ordinary one. Under thin cloud the pattern survives with the degree much reduced, which is precisely the condition in which an animal that can read it has an advantage over one that needs to see the sun.

The same rule, at other numbers #

A pattern fixed by geometry ought to change in a predictable way when the geometry changes, and the sun’s height is the only thing that can change it.

With the sun on the horizon, the ninety-degree band passes through the zenith, and the most strongly polarised light in the sky is directly overhead. With the sun overhead, the band lies on the horizon all the way round, and the zenith is at zero degrees from the sun and unpolarised. In between, the band is a circle tilted by whatever the sun’s elevation is — so through a day the band sweeps from the zenith down to the horizon and back, once each way.

That prediction is checkable with a filter and five minutes, and it fails in one respect that is worth knowing. Near the horizon the path through the atmosphere is many times longer than overhead, so multiple scattering is much heavier there, and the band is measurably weaker where it meets the horizon than where it passes overhead. The geometry says the degree of polarisation depends only on the angle from the sun; the atmosphere says it also depends on how much air is in the way. The first is exact and the second is what is actually seen, and the discrepancy between them is again a measurement of the air rather than a failure of the geometry.

The points that should not exist #

If the polarisation depends only on the angle from the sun, it vanishes at two points and nowhere else. In a real sky it vanishes at several, and the extra ones were found by measurement long before there was a theory to accommodate them.

Arago found one above the antisolar point in 1809, using nothing but a crystal and his own eyes. Babinet found one above the sun in 1840 and Brewster another below it. All three sit fifteen to twenty degrees from their respective poles, and all three move — with the sun’s elevation, with the dustiness of the air, with how much light the ground is throwing back up.

The model that produces them is short: multiply scattered light near the sun and near the antisolar point is polarised in the opposite sense to the singly scattered light, so somewhere between them the two cancel. Getting the amount right needs a full multiple-scattering calculation and getting the existence right needs one number. The figure uses the one number and lands the crossings where they are observed.

The neutral points are still used as instruments. Their distance from the sun is a measure of atmospheric turbidity, which is a quantity otherwise awkward to obtain from the ground, and the measurement needs no calibration at all — only the ability to find a place in the sky where a filter stops making a difference.

There is a third neutral point in most skies and it belongs to the ground rather than the air. Light reflected upwards from a surface is itself polarised — that is the angle at which reflection picks a side — and when it is scattered back down it carries that polarisation with it. Over water the effect is large enough to move the Arago point measurably, and the direction it moves in depends on how wet the ground is. Anyone measuring turbidity from the neutral points has to know what they are standing on.

What a filter is actually doing #

A polarising filter passes the component of the field along its own axis, so on light that is a fraction $p$ polarised it passes between $(1-p)/2$ and $(1+p)/2$ of the intensity depending on which way it is turned. That is the direction of the shaking applied to a partially polarised beam, and the partial case is worked through in the light with no direction of shaking.

The figure explains something every photographer notices and few are told. A polariser is not a sky filter; it is a filter for one band of sky. On a long lens pointed at the right part of the sky it darkens the whole frame convincingly. On a wide lens it darkens a stripe across the middle of the frame and leaves pale sky on either side, because the frame is wider than the band — and no amount of turning the filter fixes that, since turning it moves the depth of the darkening and not the band.

Which band is darkened is decided by the sun, so the effect is strongest with the sun off to one side and absent with the sun behind the photographer. The rule of thumb — point a thumb at the sun and the strongest effect is where the extended forefinger points — is the ninety-degree ring drawn by hand.

Why the effect looks strongest through a windscreen #

The filter curves also explain a nuisance that has nothing to do with the sky. Light reflected from a road, a windscreen or a pane of glass is polarised horizontally, strongly so near Brewster’s angle, and a polariser turned to reject it is turned to a fixed direction — vertical — regardless of where the sun is.

The sky’s polarisation is not fixed that way. Its direction at any point runs round the sun, so a filter set vertically to kill glare is set to whatever angle the sky happens to present at each part of the frame. On a photograph of a lake with sky above it, the same filter setting can be doing its best work on the water and almost nothing on the sky, or the reverse, and the two cannot be optimised independently with one filter.

This is the practical difference between a polarisation that is fixed by a surface and one that is fixed by a direction in the sky. The first has one orientation everywhere; the second has a different orientation at every point, and a single flat filter samples that field one angle at a time.

Reading the sun off it #

Two readings of the vibration direction fix a great circle each, and two great circles cross at the sun. That is a compass which works on a patch of blue with no sun in it.

Bees do this. The upper part of a bee’s eye carries receptors tuned to two perpendicular directions, and comparing their outputs is the same measurement as turning a filter and watching the brightness. A patch of blue the size of a coin at arm’s length is enough, which means the navigation survives cloud that hides the sun entirely — a real advantage in a temperate climate, and the reason the ability is worth having.

Whether Viking navigators used a birefringent crystal for the same purpose is argued about and not settled. The physics would work: the crystal that answers twice describes a stone that splits an image into two whose relative brightness depends on the polarisation, which is exactly the comparison a bee’s eye makes. What is missing is a stone found aboard a ship rather than an argument that one would have worked.

The geometry in the figure is the reason such a compass is usable at all. Two sightings close together give almost nothing, because their circles cross at a shallow angle and the intersection is smeared along both. Two at a right angle give the reading error back unchanged. An animal or a navigator taking readings from opposite sides of the sky is doing the right thing for a reason that can be computed.

The measurement, in one afternoon #

Nothing here requires equipment beyond a filter and a way of measuring brightness, which makes the whole pattern unusually easy to verify.

Point a camera at a patch of sky, record the exposure with the filter at four settings forty-five degrees apart, and the four numbers give the degree of polarisation and the direction of vibration at that point — four measurements for three unknowns, with one left over as a check that the light really is linearly polarised and not partly circular. Repeat across the sky and the map above comes back, in an afternoon, with the sun’s position recoverable from the data alone.

The one experimental care needed is that the sky’s brightness varies enormously across the hemisphere for reasons that have nothing to do with polarisation — it is much brighter near the sun and near the horizon — so the four exposures at each point have to be compared with each other and never with the four at a different point. The quantity being measured is a ratio at a point, which is the usual way an awkward absolute measurement is turned into an easy relative one.

Where the model stops #

Only two scatterings are represented, and one of them by a constant. The neutral points come out of a phenomenological background rather than a calculation, and the calculation is a genuinely hard one — radiative transfer in a scattering atmosphere with a reflecting ground, solved numerically.

Aerosols are absent. Dust and water droplets are large compared with a wavelength and scatter quite differently, mostly forward and much less polarised, which is why a hazy sky is both whiter and less polarised. When the particle is the size of the wave is where that regime is worked out, and it is what dominates a polluted sky.

The ground is ignored. Light reflected from below and scattered back down is a significant part of the multiply scattered contribution, and it depends on what is underneath — snow, sea and forest give measurably different neutral point positions.

And the sky is treated as thin. For a low sun the path through the atmosphere is long enough that single scattering stops being the dominant term at all, which is why the pattern is weakest and least regular at sunrise and sunset — precisely when the colour effects are strongest.

What the pictures cannot show #

The map draws the direction of vibration and cannot draw what a viewer sees, which is nothing: the pattern is invisible to human eyes except through a filter or as the faint smudge of Haidinger’s brush. A map of a quantity nobody perceives is a diagram of an instrument reading, and it should be read that way.

The map also flattens a hemisphere onto a disc, which preserves angular distance from the zenith and distorts everything else. The ninety-degree circle drawn on it is a true great circle on the sky and looks like an off-centre oval on the page, and the strokes near the rim are more crowded than they are overhead. No projection avoids this — something has to be given up, and what was kept here is that equal angles from the zenith are equal distances from the centre, so that the ninety-degree locus is a real locus rather than a distorted one.

Where the ladder goes next #

The polarisation ladder began with the direction of the shaking and the filter that only asks about it, went through the angle at which reflection picks a side, the crystal that answers twice, and the rotation a return trip doubles, and most recently reached partially polarised light in the light with no direction of shaking. This rung takes that apparatus outside and finds a pattern covering the sky that a single scattering event, repeated everywhere, has drawn.

The rung after it is the pattern’s circular component: multiple scattering and oriented particles leave a small handedness in the sky’s light, measurable at the level of a part in a thousand and different over ice, over sea and over vegetation. The habit worth carrying is the one this rung is built on: when a local rule holds everywhere, look for the global pattern it forces, because the pattern is usually simpler than the rule and is often the thing that can be measured.