Submillimeter Polarimetry with PolKa, a Reflection-Type Modulator for the APEX Telescope

RPMs are simple yet efficient optical equivalents to birefringent materials (Fig. 1): In a parallel assembly of a grid and a mirror, mounted at an adjustable distance, the component of the incident radiation that is polarized parallel to the wires can excite an electromagnetic mode and is reflected off, while the component polarized perpendicular to the wires is transmitted and then reflected into the outgoing ray where it is superposed to the other ray, with a delay-induced phase difference. In a birefringent material, these two components correspond to the "fast" and the "slow" rays, respectively. If the device is tuned such that the phase difference between the two rays is 180°, the overall effect is that of a classical half-waveplate, i.e., transmitting both polarized and unpolarized radiation, but rotating the plane of polarization by the double position angle of the grid. When a linearly polarized incident signal is measured with a polarization-sensitive bolometer element (or through an analyzer grid, for elements insensitive to polarization), a rotation of the waveplate therefore modulates the received power. The amplitude and the phase of the modulation (which for a constant rotation speed is sinusoidal) then yields the linear polarization and its polarization angle, respectively.

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The Cassegrain focal plane of the APEX telescope is reimaged to the bolometer array of LABOCA (Siringo et al. 2009) with three concave mirrors, two flat mirrors, and a lens. The aperture ratio of the Cassegrain focus, F/D = 8.0, is reduced to F/D = 1.75 in the focus of LABOCA. We stress that the accomodation of a polarimeter was in the design specifications of the tertiary optics right from the beginning. In 2009, when PolKa was permanently installed, it simply replaced one of the two flat mirrors and a filterwheel with two analyzer grids added. The modulator of PolKa (Fig. 2) was designed and manufactured by the Fraunhofer Institute for Applied Optics and Precision Engineering (Fraunhofer IOF, Jena, Germany). The grid for the modulator was made in the division for submillimeter technologies at MPIfR, has an aperture of 246 mm and is made of 20 μm thick tungsten wires (Siringo et al. 2012). For the analyzer, two identical grids were contributed by RWTH University Aachen (Germany), each of 146 mm aperture and made of gold-coated tungsten wires. The two analyzer grids are mounted at different position angles in order to suppress systematic effects, an equipment feature to which we will come back in § 2.6. As shown in Figure 2 (right panel), the analyzer grid is used only in transmission. The smooth rotation of the grid-mirror unit is ensured by an air bearing. It consists of a rotor (pale orange in Fig. 2) and a stator (shown in orange). The compressed air enters through a channel machined into the stator and flows from the center of the bearing to the outside. The surface of the bearing is made of two hemispheres. This ensures that the bearing works in any orientation, which for a Cassegrain cabin is evidently an indispensable feature. We operate the air bearing at a pressure of 4 bar, which is appropriate given the altitude of the observatory (5105 m). The position angles are read by an encoder; in practice, a time stamp is written for each crossing of a well defined reference position. Interpolations between two successive reference crossings then provide the position angles for each record in the bolometer data stream, sampled at a speed of 1 kHz. The whole dataset (bolometer total power counts and the speed and position angle of the waveplate) are then downsampled to a frequency in the range from 25 to 50 Hz and written to raw data files (in multibeam FITS format; Muders et al. 2006), which are subsequently analyzed by the data processing software described in § 3.

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The measurement equation accounts for the multiple reflections by defining a suitable coordinate frame for the description of linear polarizations and therefore Stokes Q and U. The rotations and inversions of the coordinate system in the path of the beam are then described by successive similarity transformations of the coherency matrix (Born & Wolf 1999). This algebraic formulation of polarimetry owes its power to the capacity to represent both mixed polarization states (like Müller matrices) and phase information (like complex Jones vectors). For details, we refer to Appendix A. Here we present the result of the calculation, which is provided in full length in Appendix B. We define the coordinate system in the focal plane, located between the vertex of the telescope and its tertiary mirror, as shown in Figure 3. In the following, we will refer to this frame as "Cassegrain coordinate system" with Stokes parameters Q_c, U_c defined such that the polarization angle

is measured counterclockwise from the positive x-axis (IEEE definition), as viewed from the tertiary to the secondary mirror. The resulting measurement equation reads

with the Stokes parameters I, Q_c, U_c and V. The signs of the terms for Stokes V and U are opposite to the sign of the Stokes Q term because PolKa is a reflecting polarimeter. φ is the position angle of the grid but projected onto a plane normal to the incoming ray, measured counterclockwise from the plane of incidence. It is related to the position angle φ₀ of the wires of the grid (as read out by the encoder) by φ = arctan(cos α tan φ₀), where α is the angle of incidence defined in Figure 1. Because of the relatively small angle of incidence in our setup (), the difference between φ and φ₀ is at most , so that the sampling of position angles is not too distorted and remains reasonably regular. Φ is the phase difference created by the delay between the reflected and transmitted (thus, orthogonal) polarization. The angle Δθ accounts for the rotation of the (x, y) plane in Figure 3 that occurs when the beam is downfolded, and for the orientation of the analyzer grid. More precisely, the rotation angle Δθ is given by

where θ_PA is the rotation of the beam occurring between the RPM and the analyzer grid (measured with respect to its wires). θ_FP describes the beam rotation between the Cassegrain focal plane and PolKa, measured counterclockwise with respect to the x-axis in the focal plane (Fig. 3).

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We deliberately show the measurement equation (2) in full generality because it allows us to estimate the precision needed to tune the RPM, and to evaluate the bandwidth smearing that will be addressed below. We also keep Stokes V in equation (2), although in view of the aforesaid it can be expected to be small in most applications. Incidentally, it may still be produced by the spurious conversion of Stokes I into Stokes V arising when a grid is mounted in a divergent beam with its wires parallel to the plane of incidence (Chu et al. 1975; Thum et al. 2008). Obviously, this occurs in every other modulation cycle. However, even if this instrumental conversion was a first-order effect, its impact on the measured modulation would be of second order, owing to the half-waveplate tuning and to the relatively small phase error introduced by an imperfect tuning and/or the bandwidth effect discussed below.

We finally note here that a fast polarization modulation may also be generated by periodically varying the delay-induced phase difference Φ. As shown by Chuss et al. (2006) and Krejny et al. (2008), a sinusoidal modulation can then be achieved by adding a second polarization transforming reflector, rotated by -45° with respect to the first one (cf. Table 1). In such a design, known as a dual variable-delay polarization modulator (dual VPM), the system can be kept free from unwanted oscillations by using piezo-electric actuators (the translational polarization rotator introduced by Chuss et al. [2012b] is a further development). It seems fair to say that in both approaches the stability requirement makes the system more complex, either by adding a second VPM, or, as in our case, an air-bearing. A way to enable an even faster modulation without moving parts is the Faraday rotation modulator (Keating 2009) using a ferrite dielectric waveguide. This kind of magneto-optical devices has a promising performance in the polarimetry of the cosmic microwave background and of diffuse, large-scale Galactic dust emission at frequencies up to 150 GHz (Moyerman et al. 2013). However, to implement this technology for large detector arrays at higher frequencies is very challenging.

We now rewrite the measurement equation (2) for a vanishing Stokes V, but introduce efficiency factors η_tm, η_bp, and η_ts describing the optical transmission, and the bandpass- and time-smearing, respectively:

The efficiency factors will be quantified in the next section. A rotation of the analyzer grid by 90° obviously inverts the signs of Q_c and U_c. By adding the signals from measurements with orthogonal analyzer orientations it is in principle possible to separate Stokes I from Stokes Q and U. This requires a strict synchronization between the slew motion of the telescope and the waveplate rotation when observing in on-the-fly mode. In practice such a synchronization is difficult to achieve and its failure leads to a substantial loss of usable data. At best, such an approach may work for fast on-off observations of compact sources, an application not considered in this paper. For the extended sources we focus on here, scanning the sky in the on-the-fly mode and thereby obtaining a Nyquist-critical sampling, is the observing method of choice for which a fault-tolerant system (yet accurately tracing actual values) proves to be more efficient. We will show in § 3 that the Stokes parameters can be separated by a dedicated time series analysis. Notwithstanding, using two analyzer grids at different orientations may suppress a part of the instrumental polarization, which we will characterize below.

In order to obtain, for a given point of the mapped area, a stationary Stokes Q and U, we transform (Q_c, U_c) to the equatorial reference frame. From the transformation of the left-handed Cassegrain coordinates (x, y) to offsets (Δα cos δ,Δδ) in the right-handed system of the tangential plane,

with η the parallactic angle, we can readily derive the corresponding transformation of Stokes (Q_c,U_c) to (Q_eq,U_eq), leading in equations (2) and (4) to the substitution Q_c → Q_eq,U_c → -U_eq,Δθ → Δθ + η. In the IAU definition, which differs from the IEEE definition, the polarization angle is measured counterclockwise from north and is now given by

It should be kept in mind that Q_c and U_c consist of an intrinsic signal and a polarization of instrumental origin, arising mainly in the tertiary optics.

The fractional linear polarization pL used throughout the rest of this paper is given by

Stokes Q_eq and U_eq are fraught with systematic and random errors. The former will be removed in the correction for instrumental polarization (see § 2.6), but random errors still lead to a bias and therefore an overestimate of p_L (see, e.g., Wardle & Kronberg 1974). However, it can be shown, both with a Monte-Carlo simulation and analytically, that if the most likely values of Q_eq and U_eq are used (in general median values), no such corrections need to be done (for details see § 3). Therefore, p_L and τ_IAU are calculated only at the last data reduction stage.

The modulation efficiency can be decomposed into the three factors: η_bp, η_ts, and η_tm. From equation (4), one can see that only the third factor affects all Stokes parameters, while the other two factors only affect Stokes parameters Q_c and U_c.

The first factor, η_bp accounts for the fact that an optimal tuning can only be achieved at a nominal frequency, while the bandpass of LABOCA, defined by a set of cold filters at the liquid nitrogen and helium-4 stages, extends over 60 GHz (Fig. 5 in Siringo et al. [2009]). Even if the grid and mirrors had a 100% efficiency, the modulation would suffer from a loss because the optimal half-wave phase shift (Φ = π) can only be achieved at a fixed frequency. Three micrometer screws allow to fine-tune the distance between the mirror and the grid, d(M,G). At the passband-weighted center frequency of 344 GHz (λ₀ = 870 μm) and for our angle of incidence, ,

The tuning distance is 217 μm to account for the finite thickness of the tungsten wires. The zero position of the micrometer screws has been confirmed by the disappearance of the Moiré pattern arising for a finite mirror-grid distance. At a wavelength offset Δλ from λ₀, the phase is shifted from its optimum value Φ = π to

Equation (2) is then used to calculate the modulation across the spectral bandpass. The latter may be modified by the atmospheric transmission, but for acceptable observing conditions, the loss of modulation efficiency does not strongly depend on the weather. For a water vapor column of 0.7 mm, 50° elevation, and 553 hPa ambient pressure, the resulting modulation efficiency is η_bp = 99.3%. Because ΔΦ < 0.1 rad, the bandpass smearing is a second order effect which explains why it affects the polarization efficiency only weakly.

The second factor, η_ts, results from the time smearing, i.e., the dilution of the modulation due to the elementary integration time step Δt, given by the sinc function

where ω = 2πf₀ is the angular speed of the waveplate rotation. In practice, we use f₀ = 1.56 Hz and for integration intervals of Δt = 20 or 40 ms the dilution factors are 97 and 90%, respectively.

The third factor, η_tm, can be calculated numerically from the optical properties of the polarizer (i.e., the grid labeled G in Fig. 1), namely, its reflectance, R_∥ and transmission, T_⊥ for the radiation power polarized parallel, respectively perpendicular, to the wire grids (Chuss et al. 2012a; further references therein). In the limiting case where λ > 2p, λ≫a, and a/p < 1/2π (where p is the spacing between successive wires and a their diameter) one can calculate R_∥ and T_⊥ from (Lamb 1898)

For PolKa, p = (63 ± 18) μm (the error has been determined from measurements under a microscope) and a = 20 μm. Therefore, the filling factor a/p does not obey the last of the three conditions above. Chambers et al. (1980) used a Green function method to semianalytically derive the transmission of wire grids. Using their results (Chambers et al. 1988) for our filling factor of a/p = 0.317 ± 0.091 and the spacing of p/λ = 0.072 ± 0.021 μm, we find R_⊥ = 0.1%(+0.4,-0.1)% for normal incidence; for our case of a slightly oblique incidence the resulting R_⊥ would decrease even more. We note (1) that the application of the approximative equation (11) yields a value for R_⊥ well within the errors due to the measured variance of p, and (2) that the impact of the latter is small due to the low ratio of p/λ (cf. Fig. 13 of Chambers et al. [1986]). Khazan (2002) showed with terahertz time-domain spectroscopy that the theoretical predictions from Green's function describe the actual measurements of Tungsten wire grids similar to ours fairly well. This discussion also strongly suggests that the polarization efficiency η_bpη_ts in equation (4) is dominated by the time-smearing factor η_ts.

We are therefore confident that η_tm is close to 0.99 (the waveplate mirror is of optical quality and therefore leads to no loss of efficiency). Its measurement by means of primary calibrators is notoriously difficult. When PolKa started its operation in 2011 December, the flatfielding was done on Mars without the modulator and analyzer grids. Folding the fluxes predicted by the model of Rudy et al. (1987) (and calculated with the online tool provided by Butler [2008]) into the bandpass of LABOCA yields a calibration factor of 4.74 Jy/μV. The model by Lellouch & Amri (2006) yields marginally (∼1% at 300 GHz) higher continuum fluxes. After insertion of the modulator grid calibration maps on Mars, taken without the analyzer grid, showed no significant flux loss: During the 2-week campaign, we measured daily the flux rise of Mars (the planet approached its 2012 March opposition). The uncertainty of the relative flux scale is typically 2%. Comparing the model with the measured fluxes yields a flux calibration factor of 4.63 Jy/μV. The accuracy is limited by the uncertainty of the model flux, estimated to ∼5%.—19 Uranus maps, made to characterize the instrumental polarization (see § 2.6), yield a calibration factor of 4.75 Jy/μV, using the Uranus model ESA-4 (Orton et al. 2014, estimated to be accurate to 2–3%) and assuming that the analyzer grid has a transmission of 100% in one polarization and no cross-polarization. The calibration factors agree within 5% with respect to the mean value, 4.7 Jy/μV. We stress that the observations of Uranus, unlike those of Mars, were done in polarimetry mode, which involves a more intricate data reduction (§ 3), while the flux scale is preserved. The good agreement between the calibrations done without grids, with the modulator grid, and then with the analyzer grids sets a lower limit of 95% to the transmission η_tm of PolKa.

Our calibration factor falls 27% below the value determined by Siringo et al. (2009). One reason for this discrepancy is the determination of the opacity correction. We obtain it from the precipitable water vapor (pwv) measured once per minute with the APEX 183 GHz radiometer and converted to a bandpass-weighted zenith opacity. A consistent and reproducible calibration is also important in view of the processing of polarization data obtained under varying weather conditions (see § 3). We use the am transmission model,⁵ predicting a zenith opacity of τ_ν = b_ν(p_amb)pwv + c_ν(p_amb), where the coefficients b_ν and c_ν parameterize the "wet" and "dry" atmosphere and depend on frequency and outside pressure (Guan et al. 2012).

An accurate comparison of our calibration with that of other polarimeters is difficult. Different bolometer arrays have different spectral bandpasses, therefore flux measurements in sources with strong spectral indices and substantial contributions from spectral lines vary from instrument to instrument, e.g., for OMC1 (§ 4.3) where spectral lines in the SCUBA bandpass contribute up to 50% to the observed flux (Groesbeck 1995; Johnstone & Bally 1999). For Tau A (the Crab nebula, § 4.2), whose flux is dominated by synchrotron emission with a relatively flat spectral index (S ∝ ν^-0.3), our peak flux, applying the calibration factors derived from Mars and Uranus, differs, after correcting for the slightly different beam sizes, by only 3.6% from that of Green et al. (2004), measured with SCUBA at λ850 μm.

As stated in § 1, the most important quantity for the analysis of magnetic fields by means of polarimetry of dust or synchrotron emission is the polarization angle. Its accuracy depends on the instrumental conversion between Stokes I on the one hand and Stokes Q and U on the other hand, and between Stokes Q and U. The first conversion can be corrected by means of an unpolarized calibrator (we used Uranus; see § 2.6), while the second one arises in the tertiary optics and can be measured with an additional polarizer. In 2013 March, we performed a series of calibrations with a high-quality grid mounted in the focal plane of the telescope (Fig. 3), in reflection for vertical polarization as defined in the Cassegrain reference frame (i.e., with the wires along the y-axis of the Casssegrain coordinate system, perpendicular to the elevation axis of the telescope). The grid was mounted to better than 1° accuracy, and we can safely assume a fully polarized signal with Q_c/I = -1 and U_c/I = 0. We obtained and for the two analyzer grids mounted on the filterwheel and therefore an absolute polarization angle calibration with an accuracy well below the limitation by the sensitivity. The difference between these observed angles is confirmed by a direct measurement to within . Our absolute polarization angle calibration was further confirmed by observations of celestial sources that will be presented below together with other first-light observations. The statistical error of the polarization angle is given, for equal noise contributions from Stokes Q and U (which is the case here), by σ_ψ = σ_pL/2p_L. In astronomical polarimetry, cutoffs of 2 to 3σ_pL are commonly used, i.e., statistical polarization angle errors of up to and , respectively, are tolerated.

Thanks to its axisymmetric design, the level of instrumental polarization (hereafter we use the acronym IP) in a Cassegrain telescope is insignificant. As a matter of fact, the main contributions to IP arise from the tertiary optics in the receiver cabin (see also Thum et al. [2008] and further references therein), and ignoring them may lead to a severe misinterpretation of polarization data. In order to quantify and correct the IP, one ideally observes an unpolarized, unresolved source, e.g., a gas planet like Uranus. In principle, Mars and Mercury may also be useful. Their weak, radial polarization pattern cancels out if their disks remain unresolved (Mercury should be observed near full phase).

Figure 4 shows the IP measured in on-the-fly mode on Uranus in the Cassegrain coordinate system. As in the map-making of Stokes I, all available LABOCA pixels have been used, scanning the planet at different times. The resulting polarization is therefore a weighted average of the IP at different distances from the optical axis. We expect sign changes of the instrumental Stokes Q and U across the field-of-view; therefore, the IP in the averaged data partially cancels out. Because the sensitivity of the IP map is insufficient to be used for individual bolometer pixels, we cannot quantify to which extent this happens, but it seems fair to conclude that the IP in the final map, produced from the full dataset, is lower than for the individual pixels. The fractional linear IP toward the brightness peak of the Stokes I beam amounts to p_L = (0.10 ± 0.04)%. The spatial average within the 10 dB contour yields (0.33 ± 0.09)%. Using only the first analyzer grid yields (0.17 ± 0.06)% at the peak position and (0.47 ± 0.18)% within the -10 dB contour, and with the second analyzer we obtained (0.09 ± 0.04)% and (0.37 ± 0.36)%, respectively. These numbers demonstrate that the instrumental Stokes Q and U beams are wider than the Stokes I beam, leading to a larger IP at off-source positions. They also show that using an analyzer grid at different orientations can help to decrease the IP level. A further reduction occurs because the IP pattern, whose polarization vectors are fixed in the Cassegrain coordinate system, is smeared out on the sky thanks to the parallactic rotation. However, the above numbers substantiate that the removal of IP with a dedicated correction algorithm is essential. In Appendix C, we present such a procedure that accounts for the detailed coupling of instrumental Stokes Q and U beam patterns to the brightness distribution on the sky. This approach is far more sophisticated than the a posteriori application of a constant fractional IP to the final polarization maps. An IP map like that shown in Figure 4 can be used for our correction method, provided that in the sky plane the sampling of the source resembles as closely as possible that of the IP calibrator. Nothing can be said about the IP below the -10 dB contour due to the limitation of the dynamic range. To what extent this residual IP affects the accuracy of the measured polarization is difficult to say without deeper observations, each at several parallactic angles. It seems fair to say that only sensitive observations of weak polarizations (≲10 mJy) are concerned when a strong source (Stokes I ∼ 100 Jy) is located in the error beam.

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Before we present the further data reduction methods, a few words about speed considerations seem appropriate here. The lowest frequencies of the atmospheric fluctuations in Figure 5 can be suppressed by choosing an adequate mapping speed. In the logarithmic spiral mode, a single subscan takes 36 s. Atmospheric fluctuations on longer timescales are therefore avoided. To what extent faster fluctuations degrade the image quality depends both on the source structure and on how it is sampled in the on-the-fly observing mode, scanning the source with a linear or spiral stroke pattern. The latter is used with a constant angular speed of 90°/s, i.e., a filamentary source, repeatedly appearing in the time series of a scan, has Fourier components at a frequency of 0.25 Hz and its harmonics. The removal of atmospheric fluctations is therefore mandatory and will be discussed in § 3.3. On the other hand, for a speed of 200''/s (the largest speed occurring in the spiral stroke pattern), the main lobe of the diffraction pattern of a point-like source has a width of 8.8 Hz (FWHM) in frequency domain, which implies that the largest power is at frequencies above those of the fluctuations.

These considerations hold for Stokes I. Assuming a constant fractional polarization across the source, its modulation appears in frequency domain as a scaled version of the Stokes I spectrum but now centered at the modulation frequency f_M, i.e., well above the atmospheric fluctuations, which explains why they affect the polarization far less than Stokes I. The consequences for the demodulation will be treated in § 3.4.

A closer inspection of the time series shows a spurious signal on top of the expected output (Fig. 6a). The spectral power density (Fig. 5) confirms that it is a harmonic total power beating, starting at the fundamental frequency f₀ of the mechanical modulator rotation and visible in all the harmonics up to the Nyquist frequency. The spurious signal is not due to a gain variation and therefore independent from the total power received from the sky. Its origin can be manifold. It is common wisdom that a modulator housed in the cold part of the optics provides intrinsically more stable signals, but instability is not an issue here (the beating is a deterministic signal). The resonances that can occur in reflecting polarizers (Houde et al. 2001; Krejny et al. 2008) are also unlikely to be of concern here (the characteristics of the grid as described in § 2.4 are optimal).

Imaging the beating across the full array and as a function of time, i.e., record by record, reveals a bar-like, asymmetric total power distribution, rotating about the center of the array with the frequency f₀. The strength of this rotating feature is modulated with the frequency 2f₀ of the second harmonic. In the time series of the total power beating the combination of these spatial and temporal variations leads, for a given bolometer channel, to the observed profile. We also note here that the sampling frequency of 25 Hz is not an exact multiple of f₀ and that the spikes of the beating are not fully resolved in time; in the Fourier spectrum this leads to a further redistribution of spectral power among the harmonics. This discussion may suggest that the beating is due to the differential emissivity of the grid and the mirror of the modulator, so that the higher emissivity of the mirror leads to a net polarization perpendicular to the wires of the modulator grid. However, we cannot corroborate such a conclusion. The mirror is made from an aluminum alloy whose conductivity falls short of that of pure aluminum but can be exptected to be of the order of σ = 2.5 × 10⁷ S/m, while that of tungsten is 1.8 × 10⁷ S/m. From the Hagen–Rubens law for the spectral emissivity, _ν = 4(πν₀/σ)^1/2, and Kirchhoff's law, we can then estimate the expected emission. Assuming for the modulator a typical temperature of 283 K, we obtain a polarized signal with a Rayleigh–Jeans temperature of ∼200 mK, accounting for the filling factor of the grid. The strength of the total power beating is comparable to that of our Orion KL data, i.e., a ∼100 Jy source observed with an atmospheric transmission of 55–72%. With an aperture efficiency of 0.6 (Güsten et al. 2006), this converts to an antenna temperature of 1.4–1.8 K which is an order of magnitude above our estimate of the beating expected from a differential emissivity of mirror and grid. More dedicated measurements will be needed to clarify the origin of the beating and to further improve the system.

In practice, the total power beating dominates the signal but is strictly harmonic. In the raw data from a single scan and pixel, the modulation of the signal due to its polarization is still negligible; with the aforementioned noise-equivalent flux density, the noise in a 40 ms dump amounts to 275 mJy. In the Fourier transform of the time series the beating can be easily distinguished from the spectrum of white noise (which has a constant amplitude but a random phase) and is removed from the time series pixel-wise and scan by scan (Fig. 6b), by eliminating the narrow spikes in the Fourier spectrum of the adequately apodized time series, and interpolating between the real and imaginary parts next to them. In practice, this correction leads to an insignificant distortion of the observed polarization, because the latter is only piecewise periodic. Its spectral power is distributed across a much wider frequency range determined by the window function (unity when a pixel crosses the source, and zero elsewhere), especially if a filamentary structure is observed perpendicular to its long axis. Another reason for the widened Fourier spectrum of the modulation centered at 4f₀ is that the polarization vectors on the sky, while the source is scanned, either rotate with, or counter-rotate against, the motion of the waveplate. Tests with simulated sources have shown that the removal of the beating does not significantly affect the measurement of the intrinsic linear polarization. One of these test sources, a 6^' wide model consisting of three Gaussians, whose power is as weak as 0.5% of that of the simulated beating (100 Jy), is shown in Figure 7. After removal of the beating and of the simulated atmospheric total power fluctuations, the intrinsic polarization of p_L = 10% (modeled as the projection of a dipole field onto the plane of the sky) could be restored. Toward zones with a strong curvature of the field lines the 20'' wide beam (fwhm) leads to depolarization, while an excess of the fractional polarization is observed where the Stokes I emission is weak; this is due to the difficulties in conserving the largest spatial scales in the reconstruction of the data. In summary, the removal of the total power beating by the software filter described in this section shows that mounting a polarimeter in the warm part of the optics, making it more prone to resonances of this kind, can be compensated for by an adequate data reduction algorithm.

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The overlap of the near-field beam patterns of the individual pixels leads to a strong correlation between the signals received by a pair of pixels, because they "see" the same atmospheric fluctuations ("1/f noise"). For each data record the median signal across the array provides a good measure for the dominating contribution of the atmosphere to the received total power. However, the gains of the individual bolometer pixels, as determined from observations of Mars, Uranus, or Neptune, are not appropriate because, unlike these primary calibrators, the atmospheric emission fills the entire forward-beam power pattern of each pixel. The strong correlation among the signals allows us to recalculate and to apply the gains for the reception of the atmospheric total power. Then for each record the median signal across the array is calculated and removed from the time series, and the resulting signal is scaled to the correct far-field gain for each pixel.

While the 1/f noise is efficiently suppressed by means of this correlated-noise filter, the power at the highest frequencies (f > 10 Hz) originates from the readout electronics and is suppressed by means of a wavelet filter. We refer the reader interested in wavelet filtering to Appendix D.

Both the correlated-noise and wavelet filter may not be used for the modulated (i.e., polarized) part of the signal. Depending on the polarization structure of the observed source, the modulation leads to a partial correlation of the signals and would be distorted by the removal of correlated noise. Fortunately, such a step is not necessary here: As demonstrated in Figure 5, the atmospheric fluctuations hardly leak into the modulation of the polarized fraction of the total power. We note, however, that although it is not modulated, Stokes I contaminates the polarization, due to the scanning motion. This happens, e.g., for a point source scanned at maximum speed, leading to an 8.8 Hz wide (fwhm) Fourier spectrum (the most extreme case demonstrated in § 3.1). The correction for this contribution in the demodulation of the polarization will be described in § 3.4.

For the modulated, i.e., polarized fraction of the signal a frequency filter of the form

is applied to the Fourier transform of suitably apodized segments of data (typically individual scans), with f_low = 3 Hz and f_high = 10 Hz.

Depending on the polarization structure of the source, the phase and amplitude of the modulation changes rapidly when the telescope sweeps across the sky. Therefore, the demodulation of the data and the map-making must be performed by the same data reduction step, to retrieve Stokes Q_eq and U_eq. The map-making, i.e., the gridding of Stokes I to a regular grid, follows the usual procedure to construct a regularly sampled image from a critically, but irregularly sampled stream of data: For each pixel of the output image, only data within a cutoff radius around this pixel will be considered, and the flux assigned to this pixel is an average of this data, weighted with a convolution kernel (here a Gaussian is used). For Stokes Q_eq and U_eq, the approach is different due to the modulation: first, the set of data located within the cutoff radius is binned into discrete waveplate position angle intervals, then this binned data is demodulated. In the logarithmic spiral mode (Siringo et al. 2009), LABOCA scans the sky at a varying speed (at constant angular velocity). Therefore the sampling of waveplate angles is no longer strictly synchronized with the spatial sampling of the area to be mapped, and the demodulation scheme needs to be generalized⁶. We will show now that this generalization is straightforward.

The measurement process can be represented via equation (2) as a time series, whose elements correspond to discrete time steps and therefore different waveplate angles and celestial positions. We write this time series as a vector . Since the polarization angle changes when the telescope sweeps the source, for the demodulation process we have to consider a subset of the data, , whose distance from a given pixel of the output map is within the cutoff radius of the convolution kernel used in the map-making. As already mentioned, may still contain a residual of the unmodulated signal, i.e., Stokes I. This residual is removed from by a baseline subtraction, and we are left with the modulated, i.e., polarized, signal fraction only, . The aim of generalized synchronous demodulation is to construct weight vectors w _Q, w _U, such that Stokes Q and U are obtained through the scalar products

In the following, we assume that PolKa is at its nominal λ/2 tuning, and denote the corresponding sine and cosine time series as vectors with elements

where φ_j = φ(t_j). The sensitivity of Stokes Q and U can then be derived from the radiometric noise σ_rms in the time series with

Because the sampling is in general irregular, we cannot expect that (which was the prerequisite for the demodulation scheme in Siringo et al. 2004). However, generalized synchronous demodulation offers many possibilities for the construction of the weight vectors w _Q, w _U, and also for other than sinusoidal modulations. In our case an obvious choice consists of using cos 4φ and sin 4φ as basis functions,

and to determine the coefficients μ, ν, ξ and ρ such that equation (13) is fulfilled. This yields

Thanks to the Cauchy–Schwarz inequality the denominator of these coefficients is positive, and zero only in the case that and are linearly dependent, i.e., for tan 4φ_j = 1 or -1 for all φ_j. The separation of Stokes Q from Stokes U in a technique applying a sinusoidal modulation is then impossible, but this situation is unlikely to occur, not least due to the parallactic rotation.

The lunar continuum radiation from radio to far-infrared wavelengths is dominated by the thermal emission from the regolith covering the surface, originating from a frequency dependent depth of 10 m at 3 GHz (the typical thickness of the regolith layer; Keihm & Langseth [1975]) to a few centimeters at 37 GHz (Fa & Jin 2007). The Fresnel coefficients for the last refraction between the regolith and the vacuum are different for polarizations perpendicular and parallel to the plane of incidence. The resulting radial linear polarization is a useful calibration source for polarimeters. As part of the commissioning of PolKa, the Moon was observed on 2011 December 8, at phase 94.1% shortly before full moon, scanning the disk in zig-zag mode twice, using different position angles for the filterwheel. The expected radial polarization pattern has been reproduced, and the fractional linear polarization amounts to up to ∼2% (Fig. 8). The terminator is not sharp, due to the delayed heating of the subsurface layers. The coupling of the polarized sidelobes due to the telescope's error beam pattern, irrelevant for the observations of more compact structures, leads to a slight deviation from a purely radial pattern, owing to the phase effect.

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Tau A, the Crab nebula, is a plerion-type supernova remnant. The polarization of its radio emission, discovered in 1957 independently by Mayer et al. (1957) and Kuz'min & Udal'Tsov (1959), has confirmed synchrotron radiation as the underlying mechanism. The distribution of the linear polarization across the center of the nebula, close to the pulsar which powers the synchrotron emission, is fairly smooth and the polarization angle does not vary significantly from the radio emission over visible light (Forman & Visvanathan 1971) to X-rays (Weisskopf et al. 1978). For more recent work we refer to Hester (2008). While only upper limits have been reported for the circular polarization (at λ3 mm, < 0.2%, Wiesemeyer et al. 2011, further references therein), the linear polarization at λ3 mm amounts to up to 30% (Aumont et al. 2010). Our polarization map of Tau A is shown in Figure 9; the results are summarized in Table 2 with and, for comparison, without the correction for instrumental polarization. The map was processed from 36 on the fly maps of 150 s each, corresponding to an on-source observing time of 1.5 hr, at a typical atmospheric transmission of 70%. The sensitivity is ∼20 mJy/beam. The correction for instrumental polarization (see Appendix C) changes the linear polarization by a few percent and the polarization angle by up to a few degrees. The overall agreement with other polarimetry campaigns (at λ3 mm with XPOL at the 30 m telescope [Aumont et al. 2010], and at λ850 μm with SCUPOL at the JCMT [Matthews et al. 2009]) is reasonably good toward the synchrotron emission peak. The largest discrepancies occur toward the pulsar position; for the fractional polarization they are significant, but not for the polarization angles. To date it is impossible to say whether this difference is intrinsic or due to a bias introduced by the measurements and their analysis. Depolarization can be ruled out as an explanation, since SCUPOL and XPOL measure the same fractional polarization, despite their different beams (20'' and 27'' fwhm, respectively). We note that the Pulsar position is 0'.5 north of the brightness peak. As mentioned in § 2.6, the instrumental polarization may matter; the three polarimeters account for it in different ways: Aumont et al. (2010) applied jackknife tests to ascertain the robustness of their results, while SCUPOL applies an approximative pixel-wise correction (Greaves et al. 2003). For PolKa we refer to Appendix C.

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Here we eventually examine whether Faraday rotation or a dust contribution to the continuum emission can rotate the polarization vector. The large-scale rotation measure toward Tau A is ∼-21 rad m^-2 (Bietenholz & Kronberg 1991) and mostly external to the nebula, while in unresolved filaments of thermal gas it rises up to 300 rad m^-2. Even in the latter medium, the differential Faraday rotation between λ3 mm and λ850 μm is not measurable. A dust emission component of polarized flux P_dust and polarization angle ψ_dust that adds to the synchrotron emission of polarized flux P_sync and polarization angle ψ_sync leads to a rotation Δψ of the polarization vector, to first order in P_dust/P_sync ≪ 1, by

The largest rotation of the polarization angle occurs for dust emission polarized at 45° from the synchrotron emission. Then the λ3 mm and λ870 μm polarization angle difference of 15° would require the dust emission to be polarized with 0.52 P_sync which is certainly not conceivable: Green et al. (2004) find only a small amount (≲0.07 M_⊙, ∼1.5% of the nebula mass; Bietenholz et al. 2001) of silicate or graphite dust at a temperature of about 50 K.

Our polarization map of Tau A is well consistent with a 32 GHz polarization map from the Effelsberg telescope, with a similar spatial resolution (26'' fwhm; Reich et al. 1998). A further comparison with a 5 GHz VLA map (Bietenholz et al. 2001) with 1'.4 resolution shows that the magnetic field orientation observed with PolKa is in the plane of the X-ray torus (Weisskopf et al. 2000). South-east of the torus, i.e., along the southern lobe of the jet, the polarization angles are similar and suggest here a magnetic field that is toroidal with respect to the jet axis. As a matter of fact, the kinked jet was successfully modeled by Mignone et al. (2013) assuming such a magnetic field configuration, naturally leading to the observed polarization signature. In the outer part the magnetic field structure is more complicated. Remarkably, the distribution of polarization angles (Fig. 10) shows two peaks which correspond to components that are roughly orthogonal. The peak at 160° corresponds to the emission near the torus, while the other peak represents the body of the nebula. Our findings favor a scenario in which the torus is magnetically confined because plasmas with crossed magnetic fields cannot penetrate each other (Hester 2008). Furthermore, a histogram (Fig. 10) of polarization angles in the inner part of the synchrotron nebula, measured in the optical (HST/ACS; Moran et al. 2013), peaks at a polarization angle of 150°, which is within 10° from the peak in the corresponding distribution measured with PolKa (the histogram of the optical data is wider because the subarcsecond resolution of the HST/ACS data traces the polarization of individual filaments). We take these correspondences as genuine pieces of evidence that PolKa reproduces the sky-plane magnetic field component of Tau A correctly, and confirms that the observed structures are magnetically controlled.

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The first detection of the polarization of the dust emission from the Orion Molecular Cloud I (OMC1) was made at λ270 μm with the Kuiper Airborne Observatory at a spatial resolution of 90'' (Hildebrand et al. 1984; further references to earlier attempts therein). Schleuning (1998) observed OMC1 at far-infrared/submillimeter wavelengths (λ100 μm and λ350 μm, respectively) across an 8^' × 8^' large field. He confirmed the relatively weak linear polarization and its position angle measured by Hildebrand et al. (1984) toward the Kleinman-Low nebula (Orion KL) which is thought to be powered by an explosive event (Zapata et al. 2011), while the neighboring high mass star-forming region Orion-South and the envelope of Orion KL exhibit a stronger polarization. Schleuning (1998) explains the depolarization in Orion KL by the rising dust opacity toward the far-infrared, while the low polarization in the Orion Bar, a photon-dominated region seen edge-on (Tielens & Hollenbach 1985), is attributed to a magnetic field pointing to the observer. Vaillancourt et al. (2008) studied the polarization spectrum of OMC-1 and conclude that a polarization minimum occurs between λ100 μm and λ350 μm while Houde et al. (2004), investigating the large-scale structure of the magnetic field in Orion A, confirmed the relatively smooth polarization angle structure toward OMC1 and interprete the weak polarization levels in the Orion bar with the lower dust temperature in that region.

Our polarization map of OMC1 is shown in Figure 11 and confirms the hourglass-like structure of the magnetic field found by the aforementioned studies. The map is obtained from a total of 54 on-the-fly scans (i.e., a total of 2.25 hr on source), and observed with a typical atmospheric transmission of 55 to 72%. The sensitivity across the map is ∼30 mJy/beam. Our results agree with those from SCUPOL (Matthews et al. 2009), except for the fractional linear polarization that we detect toward OMC1-South which is as weak as in the BN/KL region where it agrees with SCUPOL (pL = 0.7%). The polarization angles in both regions are ∼30°. We note that in general the correction for instrumental polarization improves, across the map, the agreement between our results and those of SCUPOL. The different levels of line contamination in SCUPOL and PolKa, contributing substantially to Stokes I but barely to p_L, may explain part of the differences. The polarization in the filament extending northwards from Orion BN/KL is parallel to its long axis (i.e., the projected magnetic field perpendicular to it), which is consistent with previously reported results. We also note that our polarization angles agree, within the errors, with those measured by Hildebrand et al. (1984) despite the different wavelengths and spatial resolutions. We therefore confirm a fairly smooth magnetic field structure where only at smaller spatial scales (< 20'', i.e., 0.04 pc) depolarization occurs toward the cores of Orion KL and South. This is presumably due to either a more complex magnetic field structure there or a mixture of dust grains whose emission traces the same volume but have different properties such as temperature and size distribution, as pointed out by Vaillancourt & Matthews (2012). Their histogram of differential polarization angles at λ350/850 μm indeed peaks in the 0°–10° interval.

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As for the Orion Bar, a prototypical photon-dominated region, we find a fractional polarization that is significantly higher, by a factor of 3–4, compared to the BN/KL and South peaks. It traces a magnetic field of which the sky-plane projection is actually along the Bar, in disagreement with the suggestion by Schleuning (1998). It is uncertain whether this result confirms the conjecture of Houde et al. (2004) that the dust temperature is lower in the Bar than in BN/KL. From their CO excitation modeling, Peng et al. (2012) infer gas temperatures in excess of 350 K and 250 K toward BN/KL and the Bar, respectively, corroborating the H₂ excitation study of Shaw et al. (2009). An in-depth discussion of these findings is beyond the scope of the work at hand and will be followed up in a forthcoming study.

We owe the APEX observatory staff a debt of gratitude. We thank Dr. S. Risse from the Fraunhofer IOF (Jena, Germany) for very fruitful discussions concerning the design of PolKa. The analyzer grids were donated by Manfred Tonutti (RWTH Aachen University, Germany). H.W. acknowledges helpful comments from C. Thum and insightful discussions with D. Muders, S. Heyminck, A. Lobanov and M. Houde. The referee helped to improve the paper further. T. Hezareh's research was funded by the Alexander von Humboldt foundation. The data reduction software used the GILDAS and CFITSIO libraries (www.iram.fr/IRAMFR/GILDAS and heasarc.gsfc.nasa.gov/docs/software/). The opacity coefficients were obtained from the am model (S. Paine, SMA technical memo #152, 2012) via the kalibrate module of the KOSMA software, Department of Physics, Cologne University.