Fit the rotating polarization vector

Two coordinates describe linear polarization

Fix two perpendicular axes on the receiver. Q distinguishes the first axis from the second; U distinguishes axes rotated by forty-five degrees. Their shared normalization is the linearly polarized intensity L. A polarization angle repeats after a half-turn, because reversing an oscillating electric-field arrow describes the same line. That is why the trigonometric formulas use twice the angle.

A complex number stores two coordinates: its real part is Q and its imaginary part is U. The symbol i turns a vector through a quarter-turn when multiplied. Multiplication by a complex exponential rotates the vector without changing its length. This notation packages two ordinary sine-and-cosine equations; it adds no new physical observable.

The Lab fixes an instrument reference direction with an intrinsic angle of twenty degrees and intrinsic linear fraction of eighty percent. It does not assign an absolute position angle on the sky. A calibrated sky convention would measure angle from north through east.

Q=L\cos(2\chi),\qquad U=L\sin(2\chi),\qquad P=Q+iU=L e^{2i\chi},\qquad \chi=\tfrac12\operatorname{atan2}(U,Q)

Q, U and L share intensity units. The two-argument arctangent chooses the correct quadrant; the resulting angle remains equivalent modulo pi.

Reading: Brentjens & de Bruyn (2005), Faraday Rotation Measure Synthesis, arXiv v2 · Condon & Ransom, Essential Radio Astronomy, Appendix D

Add a signed magnetic field

A magnetic field changes the two circularly polarized propagation modes in opposite ways. Their relative phase rotates the direction of linear polarization. Far above the plasma and gyro frequencies, the accumulated angle is proportional to wavelength squared. We define a positive line-of-sight field to point from the source toward the observer.

RM is the rotation measure in radians per square meter. Its path integral weights the field by the free-electron density. DM weights only the electrons. If one uniform screen supplies both quantities, their ratio estimates the field. Field reversals inside a complicated path can cancel in RM even when DM is large.

For an isolated screen at redshift z, observed wavelengths are longer than screen-frame wavelengths by one plus z. Writing the same angle in observed wavelengths gives an observed RM smaller than the rest-frame value by the square of one plus z. Separate the Milky Way, ionosphere and host contributions before interpreting a host field.

\chi(\lambda^2)=\chi_0+\mathrm{RM}\lambda^2,\qquad \frac{\mathrm{RM}}{\mathrm{rad\,m^{-2}}}=0.812\int\frac{n_e}{\mathrm{cm^{-3}}}\frac{B_\parallel}{\mu\mathrm G}\frac{dl}{\mathrm{pc}}

A positive slope follows from the chosen field direction. Wavelength is the observer-frame vacuum wavelength c divided by frequency.

\frac{\langle B_\parallel\rangle_{n_e}}{\mu\mathrm G}=\frac{\mathrm{RM}/(\mathrm{rad\,m^{-2}})}{0.812\,\mathrm{DM}/(\mathrm{pc\,cm^{-3}})},\qquad \mathrm{RM}_{\rm obs}=\frac{\mathrm{RM}_{\rm rest}}{(1+z)^2}

The ratio requires the same physical electron population in both integrals. The redshift relation applies to a localized screen.

Reading: Condon & Ransom, Essential Radio Astronomy, Appendix D · Brentjens & de Bruyn (2005), Faraday Rotation Measure Synthesis, arXiv v2 · Pandhi et al. (2024), CHIME/FRB polarimetry, arXiv v2

Average the vector inside each channel

A detector channel spans a range of frequencies. Its Q and U are averages over that range, not measurements at a perfectly sharp wavelength. If the polarization turns substantially within one channel, differently directed vectors cancel. Intensity can remain high while the measured linear polarization becomes small.

Integrate the complex vector against the channel response before fitting it. For a rectangular channel in frequency, wavelength squared changes nonlinearly across the interval. The familiar sinc factor is exact for a rectangle in wavelength squared, a different response. The Lab integrates frequency channels numerically with progressively finer subdivisions as the vector turns faster.

The frequency centers are twelve and a half megahertz apart in this teaching detector, but a default channel is only about 0.391 megahertz wide. The gaps and the response widths have different effects. The gaps produce aliases in the RM search; the width causes cancellation within each channel.

r_j(\phi)=\frac1{\Delta\nu_j}\int_{\nu_j-\Delta\nu_j/2}^{\nu_j+\Delta\nu_j/2}\exp\!\left(2i\phi\frac{c^2}{\nu^2}\right)d\nu,\qquad P_j=A\,r_j(\mathrm{RM})+\epsilon_j

The response r is dimensionless. Phi is a trial rotation measure, A the common intrinsic complex polarized fluence, and epsilon the recorded noise.

Reading: Schnitzeler & Lee (2014), Rotation measure synthesis revisited, arXiv v1 · Brentjens & de Bruyn (2005), Faraday Rotation Measure Synthesis, arXiv v2

Solve the fit without unwrapping an angle

First integrate each recorded Q and U series over the saved time window. The pulse shape is common to the polarization components, and the window covers almost the entire scattering tail. For each trial RM, calculate every channel response and fit one common complex amplitude. Equal noise variances give equal channel weights in this model.

Minimizing the sum of squared complex residuals is equivalent to fitting the real and imaginary coordinates together. Expand each squared residual into a data-only term, a cross term and a term proportional to amplitude squared. Setting both amplitude derivatives to zero gives the weighted projection below. Substituting it back means the best RM maximizes the squared projection divided by the response norm.

A secondary peak means that another rotation aligns some sampled vectors. Inspect the whole search curve before quoting a single maximum. A narrow formal peak does not remove ambiguity from a sparsely sampled band. Real data also require noise weights, instrumental leakage and cross-hand calibration.

S(A,\phi)=\sum_j|P_j-A r_j(\phi)|^2,\quad \widehat A(\phi)=\frac{\sum_jP_jr_j^*(\phi)}{\sum_j|r_j(\phi)|^2},\quad T(\phi)=\frac{|\sum_jP_jr_j^*(\phi)|^2}{\sum_j|r_j(\phi)|^2}

The star means complex conjugation: keep the real coordinate and reverse the imaginary coordinate. The displayed matched amplitude is the square root of T.

Reading: Schnitzeler & Lee (2014), Rotation measure synthesis revisited, arXiv v1 · Brentjens & de Bruyn (2005), Faraday Rotation Measure Synthesis, arXiv v2

Worked example and saved detector results

Use the default column of twenty parsecs per cubic centimeter and a field of two microgauss toward the observer. Multiplying by 0.812 predicts an RM of 32.48 radians per square meter. Between four hundred and eight hundred megahertz, wavelength squared changes by about 0.4213 square meters. The angle therefore turns by 13.68 radians, over four half-turns. Joining principal-value angles with a straight line would lose those turns.

The saved synthetic run uses seed 606, thirty-two channels, 1,024 samples and independent noise with a standard deviation of 0.004 in each Stokes sample. Its default fit gives 32.5 radians per square meter and a recovered intrinsic linear fraction of 79.74 percent. The search step is one quarter of a radian per square meter. Those digits describe this injection test and grid, not an uncertainty on a celestial measurement.

In the coarse-channel example the response-aware fit recovers an intrinsic fraction near eighty percent even though individual observed Q/U amplitudes shrink. The fit divides out a specified response. If almost all channels depolarize, noise dominates that correction and the recovered amplitude becomes unstable.

Synthetic detector recovery, seed 606; generated by freeze-synthetic.ts. RM unit: radians per square meter.
Preset	Injected RM	Recovered RM	Recovered linear fraction
Default	$32.48$	$32.50$	$79.74\%$
Empty path	$0.00$	$0.00$	$79.98\%$
Long tail	$32.48$	$32.50$	$79.41\%$
Reversed field	$-32.48$	$-32.50$	$79.81\%$
Coarse channels	$162.40$	$162.25$	$79.65\%$

Two angles are not enough to choose a branch

Suppose two channels give a polarization-angle difference. Adding any integer number of half-turns produces the same displayed pair. Each choice implies a different RM. More well-spaced channels reject many branches, but their coverage still determines the width and sidelobes of the response.

This is the entry to RM synthesis in the frontier guide. A single rotating vector is one Faraday-thin component. Several emitting and rotating regions require a distribution in Faraday depth, and limited wavelength coverage blurs that distribution.

\mathrm{RM}_n=\frac{\Delta\chi+n\pi}{\lambda_2^2-\lambda_1^2},\qquad n\in\mathbb Z

Every integer n fits the same two principal-value angles. The data need additional frequencies or prior information to select a branch.

Reading: Brentjens & de Bruyn (2005), Faraday Rotation Measure Synthesis, arXiv v2 · Schnitzeler & Lee (2014), Rotation measure synthesis revisited, arXiv v1

Fit the rotating polarization vector

Two coordinates describe linear polarization

Add a signed magnetic field

Average the vector inside each channel

Solve the fit without unwrapping an angle

Worked example and saved detector results

Two angles are not enough to choose a branch

Exercises and explanations

Try the prediction

Recover a signed field and test an averaging loss

Sources and calculation scope