Far-infrared polarimetry from the Stratospheric Observatory for Infrared Astronomy
J. E. Vaillancourt, D. T. Chuss, R. M. Crutcher, J. L. Dotson, C. D. Dowell, D. A. Harper, R. H. Hildebrand, T. J. Jones, A. Lazarian, G. Novak, M. W. Werner
iin Proc. SPIE 6678: Infrared Spaceborne Remote Sensing & Instrumentation XV, 66780D, ed. M. Strojnik-Scholl (2007) Far-infrared polarimetry from the StratosphericObservatory for Infrared Astronomy
John E. Vaillancourt a , David T. Chuss b , Richard M. Crutcher c , Jessie L. Dotson d ,C. Darren Dowell a,e , D. Al Harper f , Roger H. Hildebrand f , Terry J. Jones g ,Alexandre Lazarian h , Giles Novak i , and Michael W. Werner ea Physics Department, California Institute of Technology, Pasadena, CA 91125, USA; b NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA; c Department of Astronomy, University of Illinois, Urbana, IL 61801, USA; d NASA Ames Research Center, Moffett Field, CA 94035, USA; e Jet Propulsion Laboratory, Pasadena, CA 91109, USA; f Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA; g Department of Astronomy, University of Minnesota, Minneapolis, MN 55455, USA; h Department of Astronomy, University of Wisconsin, Madison, WI 53706, USA; i Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA
ABSTRACT
Multi-wavelength imaging polarimetry at far-infrared wavelengths has proven to be an excellent tool for studyingthe physical properties of dust, molecular clouds, and magnetic fields in the interstellar medium. Although thesewavelengths are only observable from airborne or space-based platforms, no first-generation instrument for theStratospheric Observatory for Infrared Astronomy (SOFIA) is presently designed with polarimetric capabilities.We study several options for upgrading the High-resolution Airborne Wideband Camera (HAWC) to a sensitiveFIR polarimeter. HAWC is a 12 ×
32 pixel bolometer camera designed to cover the 53 – 215 µ m spectral rangein 4 colors, all at diffraction-limited resolution (5 – 21 arcsec). Upgrade options include: (1) an external set ofoptics which modulates the polarization state of the incoming radiation before entering the cryostat window;(2) internal polarizing optics; and (3) a replacement of the current detector array with two state-of-the-artsuperconducting bolometer arrays, an upgrade of the HAWC camera as well as polarimeter. We discuss a rangeof science studies which will be possible with these upgrades including magnetic fields in star-forming regionsand galaxies and the wavelength-dependence of polarization. Keywords: polarimetry, far-infrared astronomy, instrumentation, interstellar medium, magnetic fields, dust,airborne astronomy, SOFIA
1. INTRODUCTION
Mapping polarimetry at far-infrared/submillimeter wavelengths began on the Kuiper Airborne Observatory(KAO) more than 20 years ago. Over the course of the KAO’s lifetime polarimetric instruments evolved fromsingle pixel devices to larger arrays and eventually moved to ground-based observatories.
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Far-infrared (FIR)and submillimeter observations of dense molecular clouds, protostellar disks and envelopes, the Galactic center,and one external galaxy have shown that dust emission from these objects is polarized (at levels of ∼ The alignment of dust grains with the local magnetic field allows one to infer the directionof the magnetic field projected onto the plane of the sky. This allows studies of magnetic fields’ interactions withthe local interstellar medium and their role in star and galaxy formation as well as studies of dust and cloudproperties.
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As successor to the KAO, the Stratospheric Observatory for Infrared Astronomy (SOFIA) offers a newfrontier for observations in the infrared-to-submillimeter spectral regime. Unfortunately, no first-light SOFIA
Further author information: J.E.V.: E-mail: [email protected] a r X i v : . [ a s t r o - ph ] N ov Far-infrared polarimetry from SOFIA
Figure 1.
The HAWC cryostat showing the locations of several proposed polarimetric upgrades: dual-polarization detectorarrays, wave-plates (HWP), and the variable-delay modulator. instruments have existing capability for accurate polarimetry. Both a far-infrared ( Hale ) and mid-infrared polarimeter have been proposed as future SOFIA instruments. Here we discuss three options for upgrading theHigh-resolution Airborne Wideband Camera (HAWC) to a sensitive FIR polarimeter.HAWC is a first-light instrument for SOFIA. It consists of a 12 ×
32 pixel bolometer camera designed to coverthe 53 – 215 µ m spectral range in 4 colors, all at diffraction-limited resolution (5 (cid:48)(cid:48) – 21 (cid:48)(cid:48) ). A complete upgradeof the HAWC camera into a polarimeter (which we call Hale ) will include replacing the existing 384 pixel arrayof silicon bolometers with two superconducting bolometer arrays. Each such array would consist of over 5000pixels and detect orthogonal polarization components selected by wire grids close to the detectors. A cold (4 K)crystal half-wave plate (HWP) upstream from the grids modulates the incoming radiation. In this way we createa sensitive, diffraction-limited, 4-color polarimeter. Additionally, the dual-polarization measurement and thedetector upgrade yield a photometer with a much larger field-of-view than the original HAWC instrument.We consider two other options (called HAWC-pol) as “pathfinder” programs working towards the morecomplete
Hale instrument. These options are attractive as it is feasible that they may be incorporated intoHAWC soon after its first light on SOFIA. In both HAWC-pol options we retain the existing 12 ×
32 siliconbolometer array and measure only a single component of polarization. Option (1) places one or more rotatingHWPs at the cold HAWC pupil stop, followed by a polarizing grid. In option (2) we place only a polarizing gridat the pupil and modulate the polarization outside the cryostat using a variable-delay polarization modulator, orpolarization “switch”. Details of the upgrade components are discussed in Sect. 3; their locations in the HAWCcryostat are shown in Fig. 1. The technical specifications for both
Hale and HAWC-pol are shown in Table 1. . E. Vaillancourt et al. Table 1.
HAWC-pol and
Hale specifications.
Parameter Band 1 Band 2 Band 3 Band 4Central Wavelength ( µ m) 53 89 155 216Bandwidth FWHM (∆ λ/λ ) 0.16 0.19 0.22 0.21Pixel Size (arcsec) 2.3 3.5 6.0 8.0Resolution FWHM (arcsec) 5.4 9.0 16 22Field of view, HAWC-pol (arcmin) 0 . × . . × . . × . . × . Hale (arcmin) 2 . × . . × . . × . . × . a (fW Hz − / ) 0.43 0.30 0.24 0.14NEFD b (Jy s / ) 0.87 0.59 0.60 0.44Polarization uncertainty, c HAWC-pol (%) 0.26 0.18 0.18 0.13Polarization uncertainty, c Hale (%) 0.18 0.13 0.13 0.09Position angle uncertainty ( P = 3%), c HAWC-pol (degrees) 2.5 1.7 1.7 1.3Position angle uncertainty ( P = 3%), c Hale (degrees) 1.7 1.2 1.2 0.9 a Noise Equivalent Power: background limited, per pixel b Noise Equivalent Flux Density for HAWC camera: background limited, per beam, chopped c In 5 hours for 5 Jy source; chopped; assumes 100% polarization efficiency and 100% observing efficiency.
2. SCIENTIFIC GOALS2.1. Turbulent Star Formation
The paradigm for star formation, once seen as depending on slow diffusion of magnetic fields out of cloud cores,has shifted to a violent one, the dynamics of which are governed by magnetic compressible turbulence.
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A unique view of the turbulent field is provided by polarimetry of emission from magnetically aligned grains.Problems of interpreting the statistics of 2D polarization maps in terms of the underlying statistics of the 3Dmagnetic fields are becoming tractable with modern theoretical work on radiative transfer, grain alignment, andturbulence, even in the case of inhomogeneous clouds containing dense clumps and clusters of embedded stars. The problem of field bending due to large-scale non-turbulent effects is being addressed by choosing referencesystems along the local projected direction of the field rather than the mean field for the whole cloud.When maps of polarized emission can be made with SOFIA’s resolution, dynamic range, and spectral coverage,it will be feasible, not only to test predictions of turbulent vs. static star formation, but also to determinecharacteristics of the magnetic fields such as the ratio of fluctuating to mean components, the energy spectrum,and the distribution of magnetic field strengths.
Recent years have been marked by substantial progress in the understanding of grain alignment. Research onradiative torques and grain dynamics have made it possible to explain the otherwise perplexing resultsof FIR polarization from starless cores and to make quantitative predictions of grain alignment along lines ofsight through dense clouds.The anisotropic radiation responsible for aligning grains is ubiquitous in astrophysics. Thus polarized emissionfrom dust grains can trace magnetic fields in various astrophysical environments. (While radiative torquesprovide the aligning mechanism, the alignment occurs with respect to the local magnetic field.) Features in themeasured polarization spectra of molecular clouds (Fig. 2) have been attributed to varying dust temperaturesand polarizing efficiencies along the line-of-sight.
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The alignment of grains, their temperature, and theiremission properties depend on the radiation field to which they are exposed. As a result, multi-wavelengthpolarimetry can be used for detailed studies of magnetic fields even within unresolved objects (e.g. Sect. 2.3;Fig. 2).A polarimeter on SOFIA will make it possible to pursue these studies with more passbands, more pixels, andimproved spatial resolution.
Far-infrared polarimetry from SOFIA
Figure 2.
Left: Measured polarization spectra of several molecular cloud envelopes, each cloud is normalized at350 µ m. Right: Predicted spectral energy distribution (bottom) and polarization spectrum (top) for protoplane-tary disk emission. Note that the peak in the polarization occurs at 45 µ m, corresponding approximately to the shortestHAWC passband. By the time a young stellar object reaches the Class II stage, a.k.a. T-Tauri star (TTS) stage, the star has acquiredapproximately its final mass, but a residual disk still contains diffuse matter that will presumably be incorporatedinto planets. According to some current theories of planet formation, the first step is for the grains in such aprotoplanetary disk to coagulate, forming particles with sizes of order one millimeter or larger. Observationalevidence of such “large grains” has been obtained via modeling of the submillimeter-millimeter spectral energydistributions of TTS disks. If these models are correct, then the optical depths for absorption and scatteringwill approach or exceed unity across much of the FIR-to-millimeter spectrum. In this case, the polarization ofthe disk thermal emission will likely contain contributions from selective absorption and scattering. Althoughthis complicates the interpretation, it also presents us with a new opportunity: Because the cross-sections forabsorption and scattering, and thus the polarization characteristics, depend strongly on grain size for sizescomparable to the wavelength, we may be able to confirm the existence of millimeter-sized grains and furtherconstrain the grain size via polarimetry of the disk emission.Polarimetry of protoplanetary disk emission has been sparse and the modeling is at an early stage, but withthe advent of SOFIA, ALMA ∗ , and EVLA it will become possible to carry out sensitive polarimetry across theFIR-to-millimeter wavebands for a sample of bright TTS disks. A recent model shows that the polarizationcharacteristics depend strongly on grain size (Fig. 2). Initial polarimetric observations of TTS disk emission,obtained at the James Clerk Maxwell Telescope and at the Caltech Submillimeter Observatory (M. Krejny,priv. comm.) suggest polarization levels of one to several percent. Four-color polarimetry with Hale or HAWC-pol will place constraints on grain size and provide information on the magnetic fields (the latter of which mayinfluence disk evolution ). ∗ . E. Vaillancourt et al. The role played by magnetic fields in star formation has remained uncertain. At one extreme of the theories ofstar formation is that magnetic fields control the formation and evolution of the molecular clouds from which starsform, including the formation of cores and their gravitational collapse to form protostars. The other extreme isthat magnetic fields are unimportant, with molecular clouds forming at the intersection of turbulent supersonicflows in the interstellar medium. In spite of considerable effort to observe magnetic fields in molecular clouds inorder to resolve this uncertainty, the issue remains controversial. SOFIA with
Hale or HAWC-pol will provide acrucial link between large- and small-scale data on interstellar magnetic fields that will be essential to understandthe role of magnetic fields in star formation.On the largest scales, observations of stellar polarization
52, 53 and Planck polarization maps at 5 (cid:48) resolutionenable mapping of the Galactic to molecular cloud scale structure of the interstellar magnetic field. HAWC-polwill make it possible to study structure of cores relative to large-scale magnetic fields. Some predicted phenomenathat require the correlation of the large- and small-scale maps of magnetic field structures include collapse ofmass along field lines to form cores flattened along field lines, hourglass morphology fields in cores with the corefields connecting to the larger scale molecular cloud fields, magnetic braking of cores that will twist the fields asangular momentum is transferred outward from cores to envelopes, and bipolar outflows from protostars withmagnetic fields being parallel to the outflows.On the smallest scales ( < few arcseconds), ALMA will image protostars, protoplanetary disks, and theinner parts of protostellar outflows. In spite of the much larger telescope collecting area of ALMA, the shorterwavelengths and much higher bandwidths observed by HAWC mean that HAWC-pol will have comparablesensitivity to ALMA for polarization mapping of extended emission. Theoretical models predict the morphologyof the connection of magnetic fields in these very small scale phenomena to the larger scale fields in the coresfrom which protostars form. Combining ALMA and HAWC-pol maps will probe the full spatial structure ofthese phenomena and allow testing of the star formation theoretical results. Our proximity to the Galactic Center provides an opportunity to study the physics of galactic nuclei in greatdetail. There is evidence that magnetic fields play an important role in the dynamics of the Galactic center regionas evidenced by the prominent Radio Arc and other non-thermal filaments
56, 57 that exist in this region. Thesefilaments are thought to be the result of relativistic electrons spiraling along lines of magnetic flux, and thoughthe brightest are oriented perpendicular to the plane, more recently discovered filaments have been found withother orientations. Submillimeter polarimetry has indicated that the large-scale magnetic field in the moleculargas is generally parallel to the plane in stark contrast to the brightest radio filaments. However, on smallerscales the field has been found to be more complex (Fig. 3). Key outstanding questions concerning the magnetic fields in the Galactic center are: (1) What is the geometryof the field, (2) what is the strength of the field, and (3) how do the radio filaments form? The multi-frequencycapability of a HAWC polarimeter will allow us to separately measure the field in different components alongthe line-of-sight. This is particularly important in the Galactic center, as the region encompasses a wide rangeof temperatures. In addition, the high angular resolution may enable one to probe the interaction sites betweenfilaments and associated molecular clouds. Such interactions may provide a mechanism for electron accelerationwhich is required for the formation of the filaments. This has the potential for providing estimates of the localmagnetic field strength and constraining models for filament formation.External galaxies give us the opportunity to study the magnetic field geometry of an entire system viewedfrom the outside . Edge on spiral galaxies in particular will allow us to sample the integrated interstellar mediumalong a number of lines of sight, weighted of course by the distribution of dust, stars, and heating radiation as afunction of galactocentric radius. For many galaxies such as NGC 891, NGC 4565, etc., the edge-on disk is onlyabout one beam wide at 60 µ m and spans many beam widths across the sky. Comparison of very active galaxiessuch as M82, moderately active disks such as in NGC 891 and more quiescent disks such as in NGC 4565 willallow Hale to find and distinguish large-scale structures in the magnetic field deep within these galaxies. Thesefeatures could include massive blowouts, for example, with a projected magnetic field geometry vertical to thegalactic plane, undetectable with optical and near-infrared polarimetry.
Far-infrared polarimetry from SOFIA
Figure 3.
Polarization and radio map of the Galactic center. Inferred magnetic field vectors from FIR/submillimeterpolarimetry are superposed on a 20 cm continuum VLA image. The 100 µ m vectors appear to trace the thermal archedfilaments while the magnetic field in the molecular cloud associated with G0.18-0.04 (350 µ m polarimetry) is perpendicularto the field traced by the non-thermal filaments of the Radio Arc.
3. INSTRUMENT DESIGN3.1. Polarization Modulation
Atmospheric constraints on suborbital FIR/submillimeter polarimetry have been described in the literature.
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The two main effects are: 1) the temporal change of the atmospheric transmission, and 2) “sky noise”, whichis the spatially and temporally variable emission from the atmosphere which is not completely eliminated bytechniques such as chopping. Both effects can be mitigated by a polarimeter which observes the field of viewin two polarizations simultaneously. However, it is often not practical to implement the dual-polarizationapproach in an existing instrument design. An alternate solution is to perform polarization modulation quickly,so as to “freeze” the atmosphere.
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Both the dual-polarization and rapid-polarization-modulation techniqueswork because atmospheric emission is intrinsically unpolarized.For HAWC-pol, we are considering two approaches to rapid polarization modulation – a variable-delay po-larization modulator (VPM), and a continuously-spinning half-wave plate (HWP). Both devices are consideredin more detail in the following sections. However, several aspects of the design are common.In our observing plan, the chopping secondary mirror of SOFIA is the fastest modulation, and the VPM orHWP is a slower modulation. Secondary chopping is ideally done at a suitably high frequency where the noiseof the difference is dominated by white noise from the photon arrival statistics. The typical atmospheric noisespectrum for SOFIA FIR observations is not yet known, but the mechanical capabilities support chopping at 5– 20 Hz, which we expect will be fast enough in most conditions.Although the atmospheric sky noise is intrinsically unpolarized, it will become partially polarized by non-normal reflections in the optics, in particular the dichroic tertiary mirror. Past experience with this type ofmirror indicates that dichroic mirror polarization will be ∼ ∼ × reduced. Since the polarization modulation is at a lower frequency, thefrequency spectrum of the atmospheric noise matters. If the noise spectrum is proportional to 1 /f (in amplitudeunits), then the polarization modulation needs to be at a frequency at least as large as ∼ . × the requirement . E. Vaillancourt et al. Polarizing GridInput Port Output PortMovable Mirror
DetectorsDFMLensWindowFM FiltersVPM Grid
Figure 4.
Left: A schematic for the Variable-delay Polarization Modulator (VPM). A controlled, variable delay (shadedline) can be introduced between two orthogonal linear polarizations. Right: The current HAWC fore-optics can bemodified for the polarimeter. The optics to the left of the window are at the ambient temperature; these include theVPM and the Focusing Mirror (FM). The optics to the right of the window are all ∼ for the chopping. For HAWC-pol, we plan on polarization modulation at ∼ ∼ ∼ In this section, we describe an approach to HAWC polarimetry using a uniformly rotating birefringent quartzHWP. Cryogenic crystal quartz has well characterized o-ray and e-ray indices of refraction, and thin wave-plates should have loss <
5% over most of the FIR.
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Although waveplates can be stacked to achieve near-achromaticity over an octave of wavelength, the HAWC filter bands cover two octaves. We expect that at leasttwo HWPs will be required and perhaps four.We have identified the HAWC pupil wheel as an ideal location to locate polarimeter HWPs. An existingmechanism provides infrequent selection of up to eight pupil apertures for diagnostic purposes. In practice, nomore than four of these apertures are needed for the camera, leaving four positions available for HWPs. Thepupil wheel is located at an image of the SOFIA primary mirror, which is an optimum location for a HWP. The pupil image has a diameter of 38 mm, a manageable size for quartz elements. A wire-grid polarizer will belocated behind each HWP to provide polarization detection. The polarimeter is removed from the optical pathby rotating the pupil wheel to select the camera-mode pupil aperture.For 1 Hz polarization modulation, the HWP must rotate at 15 rpm. Even at this low speed, induced vibrationand microphonic response of the high-impedance bolometer detectors is a concern. We will avoid ball bearingsand gear trains in the design and will instead use jewel bearings and three rollers to suspend each half-wave plateand a motor which directly torques permanent magnets attached to the waveplate. We expect dissipation in thebearings to be under 1 mW, which can be carried away through the pupil wheel. An optical encoder will monitorthe HWP position. Implementing this polarimeter solution is straightforward and will require reconstruction ofthe pupil wheel, the plate which holds it, and the Geneva mechanism which defines the eight positions.
In the case of the Variable-delay Polarization Modulator (VPM), polarization is modulated by allowing a con-trolled phase delay to be introduced between two orthogonal linear polarizations. The basic principle is shownin Fig. 4, and consists of a grid/mirror pair. Incident partially-polarized radiation comes in from the left. One
Far-infrared polarimetry from SOFIA
Table 2.
Measuring Polarization with a VPM
VPM Rotation Angle ∆ φ Polarization State(degrees) (radians)0 2 π I + Q π I − Q
45 2 π I + U π I − U component of polarization is reflected from the front of a grid. The other passes through the grid to a parallelmirror and is reflected back through the grid where the two beams are recombined. The component transmittedby the grid experiences an additional path length (which is shaded in the figure). As the grid-mirror separation ismodulated, the polarization measured by polarization-sensitive detectors changes in a predictable way. Variousinstruments have utilized devices with this architecture and have referred to them by various names.
Thenomenclature “VPM” used here emphasizes the functionality as a polarization modulator and that the degreeof freedom employed is that of a variable phase delay between linear polarizations.The polarization modulation for HAWC-pol must be flexible enough to accommodate all 4 HAWC passbands.As long as the grid is sufficiently fine for the 53 µ m band, we only need to change the grid-mirror separations tooptimize the VPM for polarization modulation across the different bands.By making two changes to the current HAWC configuration, we can use a single VPM to measure Stokes Qand U for the four HAWC passbands: 1) Replace the warm folding flat mirror with a VPM that is mounted on arotatable stage, and 2) Add two analyzer grids to the open slots on the aperture (pupil) wheel that have a relativeorientation of 45 ◦ . Figure 4 also shows the warm and cold optics for HAWC with the proposed modifications fora VPM-based polarimeter.The polarimeter will have two modes, one to measure Stokes Q in the instrument coordinate system andthe other to measure Stokes U. These will be differentiated by use of the two different analyzer grids that areoriented at a 45 ◦ with respect to one another. The VPM is placed on a rotator such that the relative angle ofthe VPM wires with respect to whichever analyzer grid is used is always 45 ◦ . In this way the measured signalon the detectors in each of the modes is S Q = I + Q cos(∆ φ ) + V sin(∆ φ ) , (1) S U = I + U cos(∆ φ ) + V sin(∆ φ ) , (2)where ∆ φ is the phase delay introduced by the VPM. This can be scaled to any HAWC frequency by changingthe amplitude of the grid-mirror separation. By using half- and full-wave delays for a given band, we can usethe VPM as a polarization “switch” to rapidly change the polarization state to which the detectors are sensitivefrom I + Q → I − Q or I + U → I − U depending upon the mode of operation. Table 2 illustrates the polarizationmeasurement strategy. To enable revolutionary FIR polarimetry,
Hale will require simultaneous detection of two orthogonal linearpolarizations. To accomplish this, we plan to implement two focal planes with 5,120 detectors each, usingGoddard Space Flight Center’s Backshort-Under-Grid (BUG) technology. As in the SCUBA-2 design, foursuch 32 ×
40 subarrays will be tiled to produce each 5,120 element array.The BUG architecture is a two-piece, keyed assembly in which a two-dimensional array of backshorts aremounted under a matching array of suspended membranes. (It should be noted here that for
Hale , the term“backshort” is a misnomer because in order to enable good performance over all four passbands, we will actuallyemploy a “back termination.”). Transition-Edge Sensors (TES) are fabricated on the membranes and the leadsare “wrapped around” the frame to the back of the device at which point electrical connections to the multiplexercan be made via bump-bonding. The pixel size in this architecture is 1.135 mm so that an array of BUGs can . E. Vaillancourt et al. A B
Figure 5. a) The Backshort-Under-Grid (BUGs) architecture is shown schematically. The absorber/detector membranesare suspended in a frame of silicon. Electrical and thermal connections to the frame are made over thin silicon legs. Thebackshort/termination assembly fits into keyed structures in the detector grid frame. Electrical connections are made tothe SQUID multiplexer by wrap-around vias that are made along the grid frame. b) A prototype 8 × ×
40 arrays. be mounted to the SCUBA-2 32 ×
40 two-dimensional time-domain SQUID multiplexer developed by NIST. Figure 5a shows a schematic of the BUG architecture.The suspended silicon membranes on the front piece of the BUG assembly will be ion-implanted to achievea surface impedance of 157 Ω per square. This process has been demonstrated for the Atacama CosmologyTelescope (ACT) detectors. The back short assembly, which for single-band implementations is made reflective,will be coated with an absorbing film to prevent unwanted resonances. This absorbing strategy is similar to thatused on HAWC, and will result in a 50 % efficiency across all four passbands. As in the ACT detectors, a TESconsisting of a molybdenum-gold bilayer with noise-reducing normal metal bars will be fabricated at the edge ofeach pixel.NASA/Goddard has developed an 8 × × ACKNOWLEDGMENTS
We would like to thank Fabian Heitsch, Jungyeon Cho, Diego Falceta-Gon¸calves, Megan Krejny, Jesse Wirth,Harvey Moseley, and Leslie Looney for useful discussions regarding the material in this manuscript. This work hasbeen partially supported by NSF grants 0505124 to the University of Chicago, AST-0540882 to the CaliforniaInstitute of Technology, and AST-0505230 to Northwestern University. A.L. acknowledges support from theNSF Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas and grant AST-0507164.R.M.C. acknowledges partial support from NSF grant AST-0606822.
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