A source that rotates its own light

The rotation measure is built on a picture with two parts that never touch. A source somewhere emits polarised light at an angle nobody knows; between it and the telescope lies a magnetised plasma that turns that angle by an amount proportional to the square of the wavelength; and so the angle, measured at several wavelengths and plotted against wavelength squared, lies on a straight line whose slope is the rotation measure and whose intercept is the unknown emission angle. The slope needs no source, which is the whole appeal: the unknown is pushed into the intercept and the wanted quantity is left in a place where it can be read.

The picture assumes the two parts are separate. For a pulsar seen through the interstellar medium, or a quasar seen through the halo of the Milky Way, they are: the emission comes from a region tiny compared to the path, and the rotating plasma lies entirely in front of it. But the sources that radio astronomers most want to measure the fields of — the lobes of radio galaxies, the discs of spiral galaxies, the diffuse halos of clusters — are not like that. They emit by synchrotron radiation, which needs relativistic electrons spiralling in a magnetic field, and the same field threads the thermal plasma that does the rotating. The emitter and the rotator are one body. And when they are, the straight line fails in a way that is instructive rather than merely inconvenient.

Light from the back of a source is turned further than light from the front #

Take the simplest version, which Burn worked out in 1966: a uniform slab that emits polarised light evenly through its depth and rotates it evenly too. Light emitted at the front surface reaches the telescope unrotated. Light emitted at the back surface crosses the whole slab and is rotated by the full Faraday depth — call it Φ, measured in the same radians per square metre as a rotation measure. Light from the middle is rotated by half of that. The telescope does not receive one polarisation vector; it receives the sum of a continuous fan of them, spread evenly across an angle of $2\Phi\lambda^2$.

At short wavelengths the fan is narrow and the vectors add almost as if they were parallel, so nothing is lost. As the wavelength grows the fan opens, and vectors pointing in different directions begin to cancel. When the fan spans a full turn — when $\Phi\lambda^2$ reaches π — every direction is represented equally and the sum is exactly zero. The source, which is emitting perfectly well polarised light from every point in it, looks entirely unpolarised.

The shape is the modulus of $\sin(\Phi\lambda^2)/(\Phi\lambda^2)$, and every part of the figure follows from it. The first zero sits at a wavelength of $\sqrt{\pi/\Phi}$: 56 cm for a slab of depth 10 radians per square metre, 28 cm for 40, 14.5 cm for 150. The depth needed to kill the polarisation at 21 cm, the wavelength of the neutral-hydrogen line and one of the commonest bands for polarimetry, is about 70 radians per square metre — a modest value, well inside what the disc of a spiral galaxy supplies. Beyond the first zero the polarisation comes back, but weakly, to a fifth of its intrinsic value at best and less at each successive return, because only the leftover fraction of an incomplete turn survives the cancellation.

The zeros are the part worth dwelling on. A foreground screen, however strong, never does this. It rotates every vector by the same amount, so their sum keeps its full length at every wavelength and only its direction turns. A polarised fraction that falls to nothing and returns is therefore a statement about the arrangement of the rotating plasma along the line of sight — that it is mixed into the emission rather than laid in front of it — and it is a statement no single-wavelength measurement could make, however precise.

The practical consequence ran through radio astronomy for decades. Surveys at 21 cm found that the discs of inclined spiral galaxies were strikingly weakly polarised, and that the polarisation that did appear came preferentially from one side of the disc. The first reading was that the fields were tangled — a reading that starlight polarised by aligned dust in the same discs, which reports a field’s direction without rotating anything, did not support, since it traced ordered spiral patterns. Much of it was Faraday depolarisation instead: the far side of the disc was depolarised by the near side, and the asymmetry was the sign of the large-scale field, which adds to the rotation on one side and subtracts on the other. A field ordered on the scale of the whole galaxy had been hidden by its own rotation.

An angle that turns at half the rate and gives no warning #

The polarised fraction is only half of what a polarimeter measures. The other half is the angle, and for the slab it does something that is harder to see and more dangerous.

The sum of a symmetric fan of vectors points along the middle of the fan. For the slab, the middle is the vector from the middle of the source, which has been rotated by half the total depth. So between zeros the angle of a Faraday-thick source rotates linearly in wavelength squared — exactly as a screen would make it — but at half the rate a screen of the same depth would give. At each zero the vector sum shrinks through nothing and emerges pointing the opposite way along the same line, which for a polarisation angle, defined only modulo 180°, is a jump of 90°.

The danger is in the segment before the first zero. There, the slab’s angles lie on a perfect straight line. A three-wavelength fit returns a small scatter, a good reduced chi-squared, and a rotation measure of 20 radians per square metre for a source whose true Faraday depth is 40. Nothing in the angles signals the error. The line is straight because the physics makes it straight; it is simply the wrong line, and its slope measures the plasma to the middle of the source rather than through it.

What does signal the error is the polarised fraction, which is falling over the same segment — but only if the falling is noticed and attributed correctly. A fraction that decreases with wavelength is also what a turbulent foreground produces, and a turbulent foreground does not halve the slope. So the fraction alone does not decide it either. The two effects have to be separated, and the separation depends on how each one falls.

A turbulent foreground depolarises too, and never recovers #

The second way to lose polarisation is the one the essay on the slope named as its first failure: a foreground screen whose rotation measure is not uniform across the telescope’s beam. Different parts of the beam are rotated by different amounts, and if the spread in rotation measure is σ, the vectors fan out across an angle of about $2\sigma\lambda^2$. The loss has a different shape from Burn’s, because the rotations are scattered at random rather than spread evenly. For a Gaussian scatter the surviving fraction is $\exp(-2\sigma^2\lambda^4)$: a smooth fall with no zeros and no return.

The two shapes are distinguishable in principle and often not in practice. The Burn slab returns after its first zero; the turbulent screen does not. But the return is weak — a fifth of the intrinsic fraction at best — and it happens at a wavelength where both kinds of source are already faint in polarisation. A survey with three or four wavelengths, all on the short side of the first zero, sees the two curves falling together and cannot say which it is looking at. The half-height wavelengths make the problem concrete. A foreground dispersion of 15 radians per square metre halves the polarisation at 20 cm; a slab of depth 30 halves it at 25 cm; at 11 cm and 6 cm both keep more than nine tenths; at 21 cm the screen keeps four tenths and the slab seven. With the intrinsic fraction unknown, each model has one free parameter to set the curve and one to set its height, and a survey with four wavelengths short of the first zero fits either.

The angles can, if they are measured to the right precision. The turbulent screen does not change the mean rotation, only the spread about it, so its angle keeps the full slope of the mean rotation measure while its fraction falls. The slab halves its slope. A source whose rotation measure seems too small for the depolarisation it shows is a candidate for internal rotation. That test needs a model of the source’s structure before it can be applied, and here the method changes character: the separate fitting of a fraction and an angle gives way to something that uses both at once.

There are more ways to depolarise than these two. A slab whose emitting and rotating plasma are both turbulent depolarises in a mixture of both shapes; a beam that covers a gradient in rotation measure across the source produces yet another curve, a sinc in the gradient rather than in the depth; and every real source is some combination. The catalogue of functional forms, collected by Sokoloff and colleagues in 1998, runs to a dozen, and fitting each in turn to four wavelengths is an exercise in choosing a model the data cannot reject. What was needed was a way to ask the data itself how the rotating plasma is arranged along the line of sight.

Measuring in Faraday depth instead of in wavelength #

The way out, proposed by Brentjens and de Bruyn in 2005 and now the standard method, is to stop fitting and start transforming. It rests on noticing that the polarised signal a telescope measures at wavelength-squared $\lambda^2$ is a sum over the line of sight:

$$P(\lambda^2) = \int F(\phi)\, e^{2i\phi\lambda^2}\, d\phi$$

Here F(φ) is how much polarised emission comes from gas at Faraday depth φ — the rotation measure between that gas and the observer. The integral is over the same free electrons that delay and disperse a spacecraft’s radio signal, weighted this time by the field along the path. A foreground screen in front of a compact source is a spike in F at one depth. A Burn slab is a flat-topped box. And the equation is a Fourier transform, with Faraday depth and wavelength squared as conjugate variables — the same relation that ties time to frequency. So measure P across as many wavelengths as possible and invert the transform, and F comes back: a spectrum of polarised emission against Faraday depth, which says directly where along the line of sight the rotated emission originates.

The inversion is never complete, because no telescope measures P at every wavelength. It measures over a band — and for negative $\lambda^2$, which does not exist, it measures nothing at all. So what comes back is the true Faraday spectrum convolved with a spread function set entirely by which wavelengths were sampled. This is the same structure as aperture synthesis in radio imaging, where an image comes back convolved with a beam fixed by which baselines exist, and it has the same two consequences: a resolution, set by the width of the band in $\lambda^2$, and a largest recoverable scale, set by its shortest wavelength.

The resolution is $2\sqrt{3}$ divided by the band’s width in $\lambda^2$. For a band from 1 to 2 GHz — the classic L band, used by most of the large radio arrays — $\lambda^2$ runs from 0.022 to 0.090 square metres, and the resolution in Faraday depth is about 52 radians per square metre. Two features closer together than that blur into one.

The example is modest and representative. A slab 40 radians per square metre deep, sitting beside a screen at 60, is not resolved in L band: the two blend into a single asymmetric hump, the strongest peak slides from 60 to 54, and a program that reports the peak as “the” rotation measure reports a number describing neither component. The shoulder on the hump’s near side is the only sign that two things are present, and at the signal-to-noise of a typical survey source it is below the noise. What a transform does, it does honestly — the blending is the resolution, stated in advance, not a failure of the method — but a spectrum is only as fine as the band that made it.

A band that sees the edges of a slab and not its body #

The second limit is stranger, and the figure it produces is the one most often misread.

A slab of emission extended in Faraday depth needs short wavelengths to be seen. At a long wavelength its whole fan of rotated vectors cancels — that is the Burn zero — so the only wavelengths at which a broad slab is visible at all are those short enough that $\Phi\lambda^2$ stays below π. A band that does not reach those wavelengths cannot see a slab deeper than about π divided by its shortest $\lambda^2$: the largest recoverable scale. For L band that is about 140 radians per square metre.

A slab 100 radians per square metre deep is broader than L band’s resolution and smaller than its largest scale, and so ought to come back as a flat-topped plateau. It does not. Its body is recovered at well under half its true level, and what survives with any clarity is a pair of features near its two edges — the places where the box has a step, which is where a Fourier transform with a missing low end puts its power. A reader of this spectrum would report a thin component near zero, a weaker one near −110, and the screen at 60: three thin components, two of which do not exist, and no slab.

This is the misreading that matters. A spectrum with several peaks is the most natural thing in the world to interpret as several screens at several distances — a cloud here, a shell there — and much of the literature on Faraday complexity is devoted to deciding which of those peaks are real. A box seen through a band that cannot reach its full depth looks like two screens. The method is linear, so this is not a numerical artefact that care can remove. It is the correct convolution of the correct spectrum with a spread function that has a hole at short spacings, and it will be there however long the source is observed.

The low-frequency band sees fine structure and misses anything broad #

The new generation of low-frequency arrays observes at metre wavelengths — the band drawn here runs from 120 to 170 MHz, $\lambda^2$ from 3.1 to 6.2 square metres. Their reason for doing so is the resolution, and it is extraordinary: $\lambda^2$ spans three square metres, so two features are resolved when they are about one radian per square metre apart. That is about fifty times finer than L band, fine enough to separate the Faraday depth of a nearby cloud from the one behind it, and fine enough to detect the rotation of the tenuous gas outside the disc of the galaxy.

The same arithmetic that makes the resolution fine makes the largest scale tiny. The shortest wavelength in the band is about 1.8 metres, so the largest structure it can see is π/3.1 — one radian per square metre. Anything broader cancels at every wavelength the band contains. A slab of Faraday depth 100, spread through the disc of a galaxy, is not blurred or attenuated at these frequencies; it is absent, and its absence is indistinguishable from there being no extended emission at all. The only emission a low-frequency survey detects in polarisation is emission that is Faraday-thin — screens, or slabs so shallow they are nearly screens — and so the catalogues it produces describe a population selected by that property. The selection is invisible from inside the catalogue.

This explains a result that looked like a puzzle when the first low-frequency polarisation surveys came in: diffuse polarised emission from the galaxy detected at a small fraction of what was expected, often with sharp structure in Faraday depth and no broad component. The sharp structure is real. The missing broad component is the band’s largest scale, which is about the depth of a single cloud.

No single band does both #

The two limits pull in opposite directions, and they come from the same number. A band’s resolution improves as its $\lambda^2$ span grows, which favours long wavelengths; its largest scale grows as its shortest $\lambda^2$ shrinks, which favours short ones. A band placed at long wavelengths gets the first and loses the second.

The diagram is the reason polarimetry has become a multi-band discipline. L band resolves 51 radians per square metre and sees up to 140; S band, 2 to 4 GHz, resolves 206 and sees 559; C band, 4 to 8 GHz, resolves 822 and sees 2,237. The low-frequency band resolves 1.1 and sees 1.0 — it sits on the diagonal, which means it can recover almost nothing that is resolved at all. Every band is a trade between the two corners, and the corner that matters for a disc galaxy or a cluster halo — fine enough to separate its components, broad enough to see its extended emission — is reached only by combining bands with wide gaps between them. The combination has its own difficulty, because the gap between bands is a gap in $\lambda^2$, and a gap in the sampled $\lambda^2$ produces sidelobes in the spread function, exactly as a gap in the baselines of an interferometer does.

Two responses to all of this are now standard. The first is deconvolution in Faraday depth — an iterative algorithm that finds the brightest peak, subtracts a scaled copy of the spread function, and repeats. It inherits both the strengths and the assumptions of the algorithm on which radio imaging has leaned for half a century, and it inherits its bias: it assumes a sky made of points, so it represents a slab as a row of spikes. The second is to abandon the transform and fit parametric models — a screen, a slab, a turbulent screen, a sum of them — directly to the Stokes Q and U values across the band, choosing between them by a likelihood penalised for the number of parameters. Fitting uses the physics that the transform ignores; the transform makes no assumption that the physics might have got wrong. Neither is enough alone, and surveys now commonly report both.

What depolarisation is a measurement of #

The two ways of losing polarisation this essay has followed are not nuisances layered on top of a measurement. They are measurements.

A Burn zero at a known wavelength fixes the Faraday depth of the emitting region, and with the synchrotron intensity, which depends on the field and the relativistic electron density, and the thermal electron density from another tracer, it constrains the field inside the source rather than in front of it. That is the only way to reach the field of a radio lobe, where most of the magnetic energy of a radio galaxy is stored, and it is how the fields of galaxy clusters were first measured. The halo gas that holds most of a cluster’s baryons turns out to carry fields of a few microgauss, tangled on scales of a few to tens of kiloparsecs, which is known because background sources seen through it lose their polarisation in the way a turbulent foreground predicts, and the loss scales with the path through the cluster.

A turbulent depolarisation fixes the spread of rotation measure within a beam, and with a model of how many turbulent cells a beam covers, that spread is a measurement of the size of the cells. The same quantity sets the timing noise that the interstellar medium imposes on pulsars used as clocks, measured by an entirely different instrument along entirely different lines of sight, and the two are consistent with a single turbulent spectrum for the interstellar medium across a range of scales no one method reaches.

And a Faraday spectrum, when it is recovered well, is one of the very few observations in astronomy that place emission along the line of sight without a distance indicator. The polarisation of a pulsar, a quasar or a galaxy disc is sorted by how much magnetised plasma lies in front of it — a depth coordinate made of field and electrons rather than of parsecs, but a depth coordinate nonetheless. That is the reason the method was worth the difficulty. It turns the thing the straight-line fit was blind to, the arrangement of plasma along the path, into the thing it measures.

Still open: whether a Faraday spectrum can be read without a model #

Every recovered Faraday spectrum is a convolution, and no band covers the negative $\lambda^2$ that would make the inversion complete, so every interpretation of a complex spectrum adds information the data do not contain. Deconvolution adds the prior that the sky is made of points; parametric fitting adds the prior that it is made of screens and slabs. The two are known to disagree on real sources, and in exactly the cases that matter most — Faraday-thick emission near the largest recoverable scale — they disagree about how many components there are.

The square kilometre arrays under construction will observe from about 50 MHz to above 15 GHz, which on the diagram above covers both corners at once for the first time. Whether a spectrum from a band that wide can be inverted with a prior weak enough that two independent methods return the same answer is not yet known. Until then, a published rotation measure for an extended source carries an unstated model, and the question of whether two measurements of the same galaxy disagree because the galaxy is complex, or because the two surveys assumed different things about it, often has no answer in the papers that report them.