A polarisation measured faster than the air changes

A photometer throws away the direction of the light it counts, and a polarimeter recovers it by counting twice: once through an analyser that passes light vibrating one way, once through one that passes light vibrating at right angles. The Stokes parameter $Q$ is the normalised difference,

$$q = \frac{Q}{I} = \frac{I_\parallel - I_\perp}{I_\parallel + I_\perp},$$

and $U$ is the same thing at forty-five degrees. For most stars $q$ and $u$ are fractions of a per cent; for the polarisation of a planet’s reflected light seen against its star, or of starlight scattered by a dust disc, they are parts per million.

Measuring a difference that small between two numbers that are each a whole star’s light is a problem of a familiar shape: the precision depends far less on how well each number is measured than on whether they are measured under the same conditions. The instrument’s own polarisation is one source of difference between the two conditions. This essay is about the other, which is the air.

Two parameters, and why the errors add a bias #

A polarisation has a size and a direction, and the two Stokes parameters $q$ and $u$ carry both: the degree of polarisation is the length of the vector $(q, u)$ and the angle of polarisation on the sky is half its direction in that plane. Every measurement of each comes with an error, and the errors of a careful instrument are independent and equal in the two parameters, so a measurement is a point scattered around the true one in the $(q, u)$ plane.

The figure carries a warning that matters more as the polarisation gets smaller. A length is never negative, so noise in $q$ and $u$ always lengthens the measured vector on average; a star with no polarisation at all, measured with errors of a part in ten thousand, appears polarised at a part in ten thousand. When the polarisation sought is comparable to the errors — which is the regime of every measurement at the frontier — the degree must be corrected for this bias, and the correction depends on knowing the errors well. A systematic error that is not included in the error estimate is not only an error in the result: it biases the degree upward, and it does so in a way that looks like signal.

That is why the false polarisations of this essay matter even when they are random from one measurement to the next. Random false signals average down over many measurements, but they inflate the scatter in the $(q, u)$ plane, and if the inflated scatter is not recognised, the bias correction is too small and a polarisation is reported that is not there.

A difference taken at two moments #

The simplest polarimeter has one detector and a rotating analyser, or a fixed analyser with a modulator in front of it that switches which polarisation the analyser passes. It measures $I_\parallel$, then $I_\perp$, then $I_\parallel$ again, and forms the difference from each pair. If the star’s brightness were constant, the pairs would differ only by the polarisation.

The star’s brightness is not constant. Seen through the turbulent atmosphere, it twinkles — scintillation from turbulence high in the air focusing and defocusing the wavefront — and on a large telescope the twinkling of a bright star is about a per cent rms, with most of its power at frequencies of tens to hundreds of hertz. Guiding errors and the image wandering across a slit add slower fluctuations of their own. A brightness change between the two halves of a pair is indistinguishable from polarisation: if the star brightens by a per cent while the polarimeter switches from one state to the other, the measured $q$ is half a per cent.

The false signal is set by how much the brightness changes in half a modulation period. If the brightness’s fluctuations have a spectrum $S(f)$, the variance of the change over an interval $\Delta t$ is

$$\langle (I(t+\Delta t) - I(t))^2 \rangle = 2\int S(f)\,\left[1 - \cos 2\pi f\Delta t\right]df,$$

which is small only when $\Delta t$ is short compared with the period of every frequency that carries significant power. Switching ten times a second leaves the whole flicker in; switching a thousand times a second removes the slow part and leaves the fast part, because the twinkling’s spectrum falls off above its turnover only as a modest power of the frequency. To bring the false polarisation to a part in ten thousand, the first figure says, the switching must run at kilohertz to tens of kilohertz.

Switching at tens of kilohertz #

No mechanical device rotates an analyser at tens of kilohertz. The instruments that do it use a photoelastic modulator — a block of glass squeezed periodically by a piezoelectric transducer at its own resonant frequency, so that it becomes birefringent and unbirefringent tens of thousands of times a second, swinging the polarisation that reaches the analyser back and forth. The detector must keep up: either a photomultiplier or avalanche photodiode whose counts are separated into bins synchronised with the modulator, or a CCD whose charge is shifted back and forth under a mask in step with it, accumulating the two states in alternate rows.

The second scheme, built in Zurich for solar polarimetry, became the model for imaging polarimeters at tens of kilohertz. A detector that shifts its charge in step with a modulator effectively takes two images interleaved in time at microsecond intervals, so that whatever the air does is common to both, and then reads them out after many thousands of cycles. The same principle at slightly lower speeds, with ferroelectric liquid-crystal modulators switching at hundreds of hertz to a kilohertz and fast photon-counting detectors, reached parts per million on the brightest stars.

Two beams at once #

There is another way to remove the air’s flicker, and it costs no speed: measure both polarisations simultaneously. A Wollaston prism splits the light into two beams polarised at right angles and sends them to two parts of a detector. Each exposure records $I_\parallel$ and $I_\perp$ through the same air at the same moment, and the normalised difference is untouched by scintillation.

It is touched by something else. The two beams fall on different pixels, pass through slightly different optics and are recorded with different efficiencies, and their ratio of gains is never exactly one. A gain mismatch of a per cent produces a false $q$ of half a per cent in every exposure — the same size as the scintillation error the beam-splitter was introduced to remove.

The remedy is to exchange the beams. A half-wave plate in front of the beam-splitter, rotated by forty-five degrees between two exposures, swaps which polarisation goes to which beam. In the first exposure beam A carries $I_\parallel$ and beam B carries $I_\perp$; in the second, A carries $I_\perp$ and B carries $I_\parallel$. The ratio of the two beams in each exposure contains the gain ratio once and the polarisation once; dividing the first exposure’s ratio by the second’s cancels the gain ratio and doubles the polarisation. The transparency of the air, which may differ between the two exposures, cancels too, because it multiplies both beams in each exposure equally.

The method, the double ratio or double difference, is what most dual-beam polarimeters on large telescopes do. It needs two exposures per Stokes parameter, and in practice four, at four angles of the half-wave plate, so that residual errors that depend on the plate’s angle also average out. What it cannot cancel is any change between the exposures that affects one beam and not the other: a gain that drifts, a flat field that shifts, a ghost that moves. Those are second-order — a drift in the mismatch rather than the mismatch itself — and the figure shows the size: a per cent of gain mismatch drifting by a thousandth of itself leaves a few parts in a million.

The Sun, where the image itself moves #

Solar polarimetry faces the same problem in a sharper form. Its target is not a point but an image of the solar surface, covered in granulation — bright cells and dark lanes a few hundred kilometres across, changing in minutes — and the polarisation it seeks is the faint circular and linear signal of magnetic fields in spectral lines, often a fraction of a per cent of the line’s depth.

The air does not only make the Sun’s image flicker; it makes the image move. Every few milliseconds the image shifts by a fraction of an arcsecond, and a pixel that was looking at a bright granule finds itself looking at a dark lane. If that happens between the two states of a modulation cycle, the difference in brightness is not a per cent but tens of per cent, and the false polarisation follows. Solar polarimeters therefore combine fast modulation with dual beams and with adaptive optics that hold the image still, and they have been among the drivers of fast detectors and modulators: the kilohertz modulation schemes were developed for the Sun before they were used on stars.

The same structure appears in space, where there is no air but the spacecraft jitters. A solar spectropolarimeter on a satellite, freed from seeing, reached precisions that ground instruments could not, and its residual errors came from the spacecraft’s pointing — the same image motion, from a different source.

What each scheme costs in photons #

Every scheme has a price, and it is paid in light.

A single-beam polarimeter throws away half the light at its analyser in each state. A fast modulator with a photoelastic block does not spend equal time in the two pure states — its retardance swings sinusoidally — and its efficiency at isolating each Stokes parameter is below one, typically about seventy per cent, so the noise on the measured polarisation is larger than the photon count alone would give. A dual-beam polarimeter keeps both polarisations, but it needs at least two exposures per parameter for the beam exchange, and four for the most robust schemes, so each exposure must be correspondingly shorter.

The trade is between the efficiency with which photons are converted into polarisation measurements and the size of the systematic floor. For faint sources, where photons are scarce and the floor is far below the noise, efficiency wins, and simple schemes are preferred. For bright sources, where photons are plentiful and the floor is everything, the most robust scheme wins even if it wastes half the light. The last figure is the decision chart: where the photon-noise curve meets an instrument’s floor is the brightness at which its designers should stop caring about efficiency.

The surprising part: two instruments, one idea #

Fast modulation and beam exchange look like opposite strategies. One measures the two states at nearly the same time on the same detector; the other measures them at exactly the same time on different detectors and then repeats with the detectors swapped. They are the same idea seen from two sides.

Each converts an absolute measurement into a differential one. A single-beam, fast-modulated polarimeter compares two states on the same pixels, so pixel gains cancel; it needs speed so that the air does not change between them. A dual-beam, exchanged polarimeter compares two states at the same moment, so the air cancels; it needs the exchange so that pixel gains cancel too. Each removes one source of difference by symmetry and the other by repetition, and the most precise instruments combine both — two beams, fast modulation, and periodic exchange — so that every systematic error has at least one symmetry working against it.

The same logic governs adaptive optics, which must correct the wavefront faster than the air distorts it, and differential photometry, which measures a target against comparison stars through the same air at the same time. Every precise measurement through the atmosphere is in the end a race against the air or a comparison made inside the same patch of it.

Where photons stop helping #

For faint sources, photon noise sets the precision, and the systematic errors in this essay hardly matter. For bright stars the situation is reversed. A star bright enough to deliver a hundred million photons a second reaches a precision of a part in ten thousand in a second, and its measurement from then on is limited by whatever systematic floor the instrument has. Halving the floor is worth a quadrupling of the observing time; reducing it tenfold is worth a hundredfold.

That is why precise polarimetry is an instrument-design problem rather than a telescope-size problem. A larger telescope collects photons faster, which matters only up to the floor; below it, a modest telescope with a better modulation scheme beats a giant one with a worse. Some of the most precise stellar polarimetry has been done on telescopes of a few metres, with instruments whose design was entirely about systematic errors, reaching parts per million on bright stars — precision that made the polarisation of starlight reflected by a hot planet a measurable quantity in principle, although the detections claimed at that level have not survived re-examination.

Checking the zero #

A differential scheme removes errors that are different between the two states; it cannot tell whether the zero point of the whole measurement is right. For that, a polarimeter observes stars known to be unpolarised — nearby stars with little interstellar dust in front of them and no circumstellar material — and stars known to be polarised by a well-measured amount. The first set the zero, the second the scale and the angle.

The standards are themselves measurements, and at the frontier they run out. A nearby star is unpolarised to the precision with which it has been measured, which for the best-studied is a few parts in a million; below that, its own magnetic activity, the slight asymmetry of its rapidly rotating surface, and the dust grains of the local interstellar medium all polarise its light by amounts nobody has catalogued. An instrument whose floor is below its standards’ precision cannot verify its own zero point, and its results become a statement about the difference between targets rather than about any one of them.

That is a familiar limit in a new place. Every instrument that measures something to a precision better than any reference eventually has to be checked against itself — by observing the same target at different orientations of the telescope, by rotating the whole polarimeter, by measuring in different seasons — and its results are trusted in proportion to how many independent ways it has found to fail and has not.

What the figures assume #

The flicker spectrum is a single shape. A real star’s brightness through the air has a spectrum set by the turbulence profile, the wind speed at each height, the telescope’s aperture and the guiding; it is flat at low frequency and falls as some power of the frequency above a turnover that ranges from a few hertz to a few hundred. The required modulation frequencies drawn here move with the turnover and the slope.

The flicker is the only error in the single-beam figures. Real fast modulators have their own systematic errors — the modulator’s retardance varies with temperature and wavelength, the demodulation timing must be exact, and the detector’s response to rapidly alternating light must be linear — and their floors are set by those once the air is frozen.

The gain drift is a single number. The exchanged-beam residual depends on how the gains of the two beams change between exposures, which depends on the detector’s temperature, the flat field’s stability and the illumination’s position on the detector. A thousandth is plausible for a well-controlled instrument and optimistic for some.

Instrumental polarisation is not included. Everything here is about the difference between two measurements; the polarisation the telescope itself adds to the light before it reaches the polarimeter is a separate, additive error, corrected by observing unpolarised standard stars, and it often sets the true floor for absolute measurements.

Still open: the floor below a part in a million #

The polarisation that would be most valuable to measure is the smallest. Starlight reflected by an exoplanet is polarised by scattering in the planet’s atmosphere, and the polarisation of the combined light of star and planet varies with the planet’s orbit at a level of parts in ten million for the most favourable known systems. Measuring it from the ground would reveal the scattering particles in the atmosphere — clouds, hazes, the Rayleigh scattering of a clear sky — without separating the planet’s light from the star’s at all.

Reaching that level needs a systematic floor ten times lower than the best instruments have achieved. The candidates for the dominant remaining error are the polarisation of the telescope itself, which drifts as the telescope tracks and is calibrated only to the precision of the standard stars; the residual non-common effects between beams; and the polarisation of the star itself, which for an active star varies with its own magnetic spots and faculae. Which of these sets the floor, and whether it can be pushed below a part in ten million, is what the next generation of polarimeters is designed to find out.