Mitigation of the spectral dependent polarization angle response for achromatic half-wave plate
MMitigation of the spectral dependent polarization angle response for achromatichalf-wave plate
Tomotake Matsumura ∗ Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA)3-1-1 Yoshinodai, Chuo, Sagamihara, Kanagawa 252-5210, Japan.
Polarimetry using a half-wave plate (HWP) modulator provides the strong tools to avoid a de-tector 1 /f noise and instrument-originated spurious polarization systematic effects. While the Pan-charatnam achromatic HWP (AHWP) is commonly used for an application that needs a broadbandfrequency coverage, this technique introduces a frequency-dependent polarization angle rotation.In this paper we propose a new technique to mitigate this effect by introducing a second set ofan AHWP. One rotational and one stationary set of AHWPs achieve a broadband coverage ofmodulation efficiency without the frequency-dependent polarization angle rotation. We conductedmeasurements by using three layers of sapphire wave plates and demonstrated this technique atmillimeter wavelengths between 72 and 162 GHz. We also discuss a potential application in theCMB polarization experiment based on numerical simulations. I. INTRODUCTION
Measurements of the cosmic microwave backgroundradiation (CMB) have been playing an important roleto establish the ΛCDM cosmology. While vast amountof information is learned by the Planck satellite usingthe temperature anisotropy of the CMB [1], there is acommunity-wide effort to measure the polarization ofCMB to test the inflationary paradigm and to probethe evolution of the universe via interactions betweenthe CMB and gravitational potential from the large scalestructures [2]. These paradigms started to be disclosedby the recent results from the CMB polarization experi-ments, BICEP2, POLARBEAR, and SPTpol [3–5].Forthcoming CMB polarization experiments requirea polarimeter that is free from a detector 1 /f noiseand controls the instrument-originated polarization sys-tematic effects. A polarimeter using a half-wave plate(HWP) modulator provides two attractive features, i)avoiding the detector 1 /f by modulating and demod-ulating the signal frequency, and ii) eliminating thedetector differencing, which is a source of the instru-mentally induced spurious polarization effect, to re-construct the incident polarization state [6]. MAX-IPOL was the first CMB experiment that employedthe continuously rotating HWP. Currently a number ofCMB experiments, including ABS, EBEX, LiteBIRD,POLARBEAR-1, POLARBEAR-2, QUBIC, SPIDER,SWIPE are pursuing this technology [7–16].While the use of HWP becomes a popular polarime-try technique, a polarimetry using a single HWP limitsthe use of the electromagnetic frequency range, and thusthe observing detection bandwidth. The typical availablebandwidth with a single HWP is δν/ν ∼ . ∗ Corresponding author: [email protected]
FIG. 1. Schematic view of the AHWP polarimeter (top) andROC-AHWP polarimeter (bottom) using three wave plates( m = 3). of δν/ν ∼ π broader than δν/ν = 0 .
3. Whilethis is a very attractive option, one complication withthe Pancharatnam AHWP is that the amount of anglerotated by the AHWP becomes an electromagnetic fre-quency dependence. This effect can be rephrased as thepolarization sensitivity axis depends on the instrumentbandpass shape and the source spectrum. When the twoor more sources are mixed, the uncertainty of the po-larization sensitive angle is not only depending on thespectral shapes but also the relative polarized intensities.In this paper, we introduce the idea to mitigate thespectral dependence of the polarization angle. In sec-tion 2, we briefly review the AHWP polarimetery andintroduce the mitigation recipe. In section 3, we showthe experimental results as a demonstration of the idea.Finally in section 4 we discuss the actual implementa-tions for forthcoming CMB experiments. a r X i v : . [ a s t r o - ph . I M ] A p r II. MITIGATION RECIPEA. AHWP polarimetry
The detailed descriptions of the AHWP polarimetryand its formalism can be found in Matsumura et al. [18].Here we briefly review the AHWP polarimetry. Figure 1shows the configurations of the polarimeter we assumethroughout this paper.The expected signal can be formulated by using theMueller matrix as below (cid:126)S out = G x m (cid:89) i =1 [ R ( − ρ − θ i )Γ( δ ) R ( ρ + θ i )] (cid:126)S in , (1)where (cid:126)S in is the Stokes vector of the incident linearlypolarized signal. In this paper, we assume (cid:126)S in = ( I in , Q in , U in ,
0) (2)= I in (1 , P in cos 2 α in , P in sin 2 α in , , (3)where I in is the incident intensity, P in is the incidentdegree of polarization, α in is the incident polarizationangle with respect to the detector polarization sensitivityaxis, the x -axis. The output Stokes vector is (cid:126)S out . TheMueller matrices, Γ, R , G x are for a retarder, rotationand wire grid. The detailed matrix elements are shownin Appendix. The angle θ i is the offset wave plate angleabout the z -axis for an i th plate with respect to the x -axis, see Figure 1. The total number of wave plates thatconsist of one set of AHWP is m . The angle ρ is the waveplate angle. The retardance δ is δ = 2 π νd | n e − n o | c , (4)where c is the speed of light, and n o and n e are theordinary and extra-ordinary indices of the refraction ofthe wave plate, respectively. As a part of the constructionparameters, we determine the thickness of the HWP, d c ,as d c = 12 cν c | n e − n o | , (5)where ν c is the central frequency of the detection band.Without taking into account the effect of reflectionfrom a wave plate, the intensity as a function of a waveplate angle (intensity vs angle = IVA), I out , for a singleHWP can be analytically expressed as I out = 12 { I in + (cid:15) [ Q in cos 4 ρ + U in sin 4 ρ ] } = 12 [ I in + (cid:15) (cid:113) Q + U cos (4 ρ − α in )] , (6)where (cid:15) is modulation efficiency defined as (cid:15) ≡ I p out I p in , where I p = (cid:112) Q + U . (7) We also define the phase of IVA and relate to the incidentpolarization angle as φ = 12 α in . (8)When we apply Equation (6) to the IVA of AHWP, weintroduce an extra phase, φ ν , as I out = 12 [ I in + (cid:15) (cid:113) Q + U cos (4 ρ − α in − φ ν )] (9)in order to take into account the phase variation as afunction of frequency, and thus the phase of the modula-tion is expressed as φ = 12 α in + φ ν . (10)The extra term, φ ν , has a spectral dependence, i.e. thespectra of a source and instrument. Thus, a polarimeterusing AHWP has the intrinsic source of uncertainty in itspolarization angle unless one knows the source spectrumto the required precision. The quantitative discussionabout this effect in the context of a CMB polarizationexperiment can be found in Matsumura et al. and Baoet al.[18, 19]. B. Introducing the rotational-offset-cancelingAHWP
The idea to mitigate the frequency dependence of thepolarization angle with the use of rotating AHWP is toplace a second set of the AHWP that is moving withrespect to the first set of the AHWP. The simplest con-figuration for the second set is to prepare an identicalstationary AHWP as shown in the bottom of Figure 1.The first set is placed to modulate the incident po-larization angle. The second set still maintains thehigh modulation efficiency and also rotate the offset ofthe incident polarization angle back. Hereafter, we callthis configuration the rotational-offset-canceling AHWP(ROC-AHWP). The above configuration can be writtendown by using the Mueller matrices as (cid:126)S out = G x m (cid:89) i =1 [ R ( − ρ − θ i )Γ( δ ) R ( ρ + θ i )] × m (cid:89) i =1 [ R ( − ρ − θ i )Γ( δ ) R ( ρ + θ i )] (cid:126)S in , (11)The detected intensity is the first element of the outputStokes vector. The rotational angles of the two AHWPs, ρ and ρ , need to be not identical. The simplest con-figuration is to set ρ = 0 and let ρ to rotate. Wereconstruct the phase, φ ν , from IVA and this now doesnot have the spectral dependence while maintaining highmodulation efficiency, (cid:15) . FIG. 2. Schematic view of the measurement setup.
III. MEASUREMENTS
We conducted an experiment to demonstrate the con-cept of the ROC-AHWP.
A. Sample preparation
We prepare the 50 mm diameter of an A-cut singlecrystal sapphire as a wave plate. We construct the twosets of the stacked three wave plates ( m = 3) with theoffset angles of (0, 55, 0) degrees. The relative offsetangles between the wave plates are aligned within 0.5degrees. The thickness of each sapphire sample is 3 . ± .
005 mm. The ordinary and extra-ordinary indices ofthe sapphire at the room temperature are assumed to be3.07 and 3.40 [8]. The surfaces of the first and the thirdsapphire layers are anti-reflection (AR) coated with twolayers of dielectric materials. The first AR layer on thesapphire is Stycast2850FT with a thickness of 0 .
24 mmand the second layer on Stycast2850FT is the teflon sheetwith a thickness of 0 . B. Experimental setup
Figure 2 shows the experimental setup to measure themodulation efficiency and the phase of IVA. We use themillimeter-wave generator and the × × α in = 0 degrees as the polarization vector referencedwith respect to the x -axis. The linearly polarized sig-nal is measured by a diode detector that is sensitive tolinearly polarized light. The signal is chopped at 13 Hzand the detected modulated signal is demodulated by thelock-in amplifier and read by data acquisition system.The incident beam from the source is collimated by aspherical lens made by Rexolite and the beam is further FIG. 3. Typical IVA data and the fit using Equation (12).FIG. 4. The modulation efficiency (top) and the phase (bot-tom) as a function of the incident electromagnetic frequencyfor the three-layer AHWP (left) and ROC-AHWP (right).The blue points are measured data and the red lines are nota fit but predicted curves. The incident polarization angle is α in = 0 degrees. collimated by the 2.5 cm diameter aperture. The AHWP(sample x -axis within 5 degrees. We calibrate theglobal offset angle about the z -axis between the sampleand the x -axis by using the wire grid. IV. RESULTS
Figure 3 shows the typical IVA data. The IVA is fittedwith the following model, I m = A + A cos (2 ρ + 2 φ ) + A cos (4 ρ + 4 φ ) . (12)We define the normalized modulation efficiency, ˜ (cid:15) , andthe phase, φ , of the IVA as˜ (cid:15) ≡ A A , φ ≡ φ (13)The normalized efficiency for a single HWP can be ana-lytically solved as ˜ (cid:15) = P out P in . (14)We use the normalized modulation efficiency instead ofthe modulation deficiency defined in Equation (7). Thisis because the source intensity, I p in , is not well knownover the spectral range while we can ensure to achieve P in = 1 by using the combination of the linearly polarizedsource and the wire grid.Figure 4 shows the modulation efficiency and the phaseas a function of the electromagnetic frequency for theAHWP and the ROC-AHWP. The phase of the AHWPshows the spectral dependence. On the other hand, thephase of the ROC-AHWP has a flat response. The mod-ulation efficiency of AHWP and ROC-AHWP is close to1 over broadband, δν/ν ∼ . × − over 1 sec of the integration timeand the temperature of the source is stationary within0.1 degrees. The thermal drift is estimated by repeatingthe measurements multiple times over a few hour sep-arations. The systematic error due to the reflection isestimated by the deviation of IVA from Equation (12).The red lines are the predicted curves without takeninto account the effect of reflection from the wave platesurfaces. The predicted curves have thickness due to theuncertainties in the thickness of each wave plate and theoffset angles between the wave plates. The data pointsand the predicted curves are well agreed within 1- σ errorbar.Figure 5 shows the phase as a function of the incidentpolarization angle. The phase at each incident angle isaveraged between 72 and 108 GHz. The error bar on eachdata point is dominated by the uncertainty of the inci-dent polarization angle. The uncertainty of the wire gridorientation is ± . FIG. 5. The band averaged phase as a function of the incidentpolarization angle. The solid line is a linear fit that is wrappedaround between 0 and 90 degrees.
V. DISCUSSIONS
We discuss the detailed feature that may become thepotential systematic effect with the use of ROC-AHWP.We also discuss the design parameters for two upcomingCMB experiments that potentially use the AHWP basepolarimetry.
A. Incident angle dependent modulation efficiency
When one designs the AHWP, there is a tradeoff be-tween the high modulation efficiency and the bandwidthby tuning the offset angles. In one application, if onedoes not necessary need the modulation efficiency of 1 inthe detection bandwidth, one can gain wider bandwidthwith modulation efficiency less than 1.As for the ROC-AHWP design, one needs a carefultreatment in this tradeoff. When the retardance is notequal to π after the first set of the AHWP, a partialcircularly polarized light is incident to the second AHWP.In such a case, the outgoing light is further circularlypolarized. Therefore, the second AHWP enhances to thelower modulation efficiency.If the retardance is δ (cid:54) = π , the polarized light after thefirst AHWP gains some amount of V (cid:54) = 0. This non-zero V polarized light incidents to the second AHWP withsome polarization angle with respect to the axis of thesecond AHWP. In such a case, the degradation of themodulation efficiency depends on what the initial polar-ization angle is with respect to the first AHWP. If onedesigns the ROC-AHWP for an application that needs noincident angle dependence in the modulation efficiency,one has to carefully choose the construction parametersin order to maximize the modulation efficiency for allthe incident polarization angles with some sacrifice of thebandwidth. Experiment d c [mm] ( ν c [GHz]) Band [GHz] Bandwidth [%] HWP offset angles [degs]POLARBEAR-2 3 3.74 (126.6) 95, 150 30 (0, 56, 0)LiteBIRD 9 2.52 (187.5) 60, 78, 100 23 (0, 18.5, 217.5, 73.9, 141.5,140, 195, 280 30 73.9, 217.5, 18.5, 202.7)TABLE I. The design parameters for ROC-AHWP for POLARBEAR-2 and LiteBIRD. The thickness is for a single plate andthe variable, ν c , corresponds to the frequency that has a retardant of π . B. ROC-AHWP design for POLARBEAR-2 andLiteBIRD
We describe the candidate ROC-AHWP design for up-coming CMB polarization experiments, POLARBEAR-2and LiteBIRD. Our design assumes the design parame-ters described in Table I. We optimize the offset anglesin such that the modulation efficiency is as close as ∼ > . > . −
320 GHz. This is one of the candidate offset anglesthat cover the detection bandwidth of LiteBIRD. Table IIshows the band averaged modulation efficiency and theRMS fluctuation of the phase within the band.
C. Implementations
This ROC-AHWP inevitably doubles the total thick-ness of the AHWP in the optical path and the surfacethat potentially reflects the incident radiation. In theapplication that needs high transmittance, one has tochoose a low-loss birefringent material and proper anti-reflection coating in the required bandwidth.When the two parallel optical elements are placed like
FIG. 6. The simulation results of the modulation efficiencyand the phase for the three layer AHWP and ROC-AHWP.The parameters are tuned for POLARBEAR-2 specification.The dashed vertical lines indicate the 30 % bandwidth of95 GHz and 150 GHz. The different color corresponds tothe different incident angle, α in , in 18 degree step.FIG. 7. The simulation results of the modulation efficiencyand the phase for the nine layer AHWP and ROC-AHWP.The parameters are tuned for LiteBIRD specification. Thedashed vertical lines indicate the 23 % bandwidth for 60, 78,and 100 GHz bands and 30 % bandwidth of 140, 195 and280 GHz. The different color corresponds to the differentincident angle, α in , in 18 degree step. Band [GHz] (cid:15) φ rms [degs]60 0 . +0 . − . . × −
78 0 . +0 . − . . × −
100 0 . +0 . − . . . +0 . − . × −
195 1 . +0 . − . × −
280 0 . +0 . − . . × − TABLE II. The design parameters for ROC-AHWP forPOLARBEAR-2 and LiteBIRD. The thickness is for a sin-gle plate and the variable, ν c , corresponds frequency that hasa retardant of π . The quoted error in modulation efficiencyis from the incident angle dependence. the ROC-AHWP, the standing wave may cause undesiredfringe pattern in the spectrum of the detection band. Thesimplest solution is to apply a proper AR coating. As aresult, the requirement to the broadband AR coating maybecome more demanding with the ROC-AHWP.The modulation efficiency and phase response ataround (cid:15) ∼ φ ∼ > − K − between 4 and 100 K) [20]. Therefore,the stationary sapphire with a proper AR coating canserve not only as a part of the second set of the AHWPin the ROC-AHWP but also serve as a thermal filter thatabsorbs infrared radiation at cryogenic temperature. Forthis use, the correct choice of the AR coating materialwith features, absorptive at IR and transparent in mil-limeter wave (such as a thin layer of Stycast), shouldachieve a functionality as a thermal filter. The similarconcept is proposed using alumina with the Stycast ARcoatings as an IR filter at the cryogenic temperature [21].In a HWP based polarimetry using a continuous rota-tion, another possible operational mode is to rotate thetwo AHWPs of the ROC-AHWP in opposite directions.In Equation (11), one can set ρ = ωt and ρ = − ωt ,where ω is the angular velocity of the AHWP and t istime. In this mode, the total angular momentum ofthe ROC-AHWP can be canceled. When the angularmomentum due to the rotation of the AHWP can be asource of disturbance in a system, such as a satellite or balloon payload attitude control system, one can achievethe continuously rotating HWP based polarimetry withzero-momentum. In this case, one should aware that thepolarization signal appears at 8 ωt instead of 4 ωt , i.e. thenominal frequency that the polarization signal appearswith a single AHWP. Therefore, this double spinningROC-AHWP in the opposite directions can add an extrafeature of increasing the signal band. This is a partic-ularly interesting option when a physical rotational fre-quency of the AHWP is limited.While we demonstrated this concept of the ROC-AHWP at millimeter wavelengths experimentally, theprinciple of the ROC-AHWP can be applicable to anywavelengths. Thus, this ROC-AHWP can be a viablecandidate technology for any applications that use broad-band polarimetry, including solar physics, expoplanetsearch, aerosol characterization, and biomedical applica-tions. VI. CONCLUSIONS
We introduced the recipe to mitigate the spectral de-pendent phase response of the AHWP polarimeter by in-troducing the second set of the AHWP that rotates or isstationary with respect to the first set of the AHWP. Weexperimentally show that the ROC-AHWP configurationwith m = 3 achieves ∆ φ ν < φ ν ∼
10 degrees between 72and 162 GHz. We also computationally show the poten-tial ROC-AHWP design for POLARBEAR2 with m = 3and for LiteBIRD with m = 9. Although we demon-strate this ROC-AHWP at the millimeter wavelength,the ROC-AHWP concept is applicable to any broadbandpolarimeter application. ACKNOWLEDGEMENT
We would like to thank to Masashi Hazumi for usefuldiscussions. This work was supported by MEXT KAK-ENHI (Grant Nos. 24740182 and 24111715).
APPENDIXElements of Mueller matrix
The elements of the Mueller matrix that are used inthis paper are listed below.Γ( δ ) = δ − sin δ δ cos δ (15) R ( θ ) = θ − sin θ
00 sin θ cos θ
00 0 0 1 (16) G x = 12 . (17)For more general expression, please see Appendix inShurcliff [22]. [1] Planck Collaboration,“Planck 2013 results. I. Overviewof products and scientific results,” Astronomy & Astro-physics manuscript no. Planck Mission 2013.[2] Kamionkowski M and Kosowsky A 1999 Ann. Rev. Nucl.Part. Sci. 49 77.[3] BICEP2 collaboration, “BICEP2 I: Detection Of B-modePolarization at Degree Angular Scales,” arXiv:1403.3985.[4] POLARBEAR collaboration, “A Measurement of theCosmic Microwave Background B-Mode PolarizationPower Spectrum at Sub-Degree Scales with POLAR-BEAR”, arXiv:1403.2369.[5] D. Hanson et al. (SPTpol Collaboration), “Detectionof B-Mode Polarization in the Cosmic Microwave Back-ground with Data from the South Pole Telescope,” Phys.Rev. Lett. 111, 141301, 30 September 2013.[6] W. Hu, M. Hedman, M. Zaldarriaga, “Bench-mark Parameters for CMB Polarization Experiments,”Phys.Rev.D67:043004,2003[7] Wu et al., “MAXIPOL: Data Analysis and Results”, TheAstrophysical Journal, 665:55-66, 2007 August 10[8] B. R. Johnson, “MAXIPOL: a bolometric, balloon-borneexperiment for measuring the polarization anisotropy ofthe cosmic microwave background radiation,” Ph.D. dis-sertation (University of Minnesota, Twin Cities, Min-neapolis, 2004).[9] A. Kusaka et al., “Modulation of CMB polarization witha warm rapidly-rotating half-wave plate on the AtacamaB-Mode Search (ABS) instrument”, Rev. Sci. Instrum.,85, 024501 (2014).[10] B. Reichborn-Kjennerud et al., “EBEX: a balloon-borneCMB polarization experiment”, Proc. SPIE 7741, Mil-limeter, Submillimeter, and Far-Infrared Detectors andInstrumentation for Astronomy V, 77411C (July 15, 2010); doi:10.1117/12.857138.[11] T. Matsumura et al., “Mission design of LiteBIRD”,Journal of Low Temperature Physics, February 2014.[12] H. Nishino et al., “POLARBEAR CMB Polarization Ex-periment”, Proceedings of the 12th Asia Pacific PhysicsConference JPS Conf. Proc. 1, 013107 (2014).[13] T. Matsumura et al., “Cosmic Microwave Background B-mode Polarization Experiment POLARBEAR-2,” Pro-ceedings of the 12th Asia Pacific Physics Conference JPSConf. Proc. 1, 013108 (2014).[14] A. Ghribi et al., “Latest Progress on the QUBIC Instru-ment”, Journal of Low Temperature Physics, February2014.[15] S. Bryan et al., “Modeling and characterization of theSPIDER half-wave plate”, Proc. SPIE 7741, Millimeter,Submillimeter, and Far-Infrared Detectors and Instru-mentation for Astronomy V, 77412B (15 July 2010).[16] P. de Bernardis et al., “SWIPE: a bolometric polarimeterfor the Large-Scale Polarization Explorer”, Proceedingsof the SPIE Astronomical Telescopes + Instrumentation2012 Conference - Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for AstronomyVI - Amsterdam 1-6 July 2012, paper [1] Planck Collaboration,“Planck 2013 results. I. Overviewof products and scientific results,” Astronomy & Astro-physics manuscript no. Planck Mission 2013.[2] Kamionkowski M and Kosowsky A 1999 Ann. Rev. Nucl.Part. Sci. 49 77.[3] BICEP2 collaboration, “BICEP2 I: Detection Of B-modePolarization at Degree Angular Scales,” arXiv:1403.3985.[4] POLARBEAR collaboration, “A Measurement of theCosmic Microwave Background B-Mode PolarizationPower Spectrum at Sub-Degree Scales with POLAR-BEAR”, arXiv:1403.2369.[5] D. Hanson et al. (SPTpol Collaboration), “Detectionof B-Mode Polarization in the Cosmic Microwave Back-ground with Data from the South Pole Telescope,” Phys.Rev. Lett. 111, 141301, 30 September 2013.[6] W. Hu, M. Hedman, M. Zaldarriaga, “Bench-mark Parameters for CMB Polarization Experiments,”Phys.Rev.D67:043004,2003[7] Wu et al., “MAXIPOL: Data Analysis and Results”, TheAstrophysical Journal, 665:55-66, 2007 August 10[8] B. R. Johnson, “MAXIPOL: a bolometric, balloon-borneexperiment for measuring the polarization anisotropy ofthe cosmic microwave background radiation,” Ph.D. dis-sertation (University of Minnesota, Twin Cities, Min-neapolis, 2004).[9] A. Kusaka et al., “Modulation of CMB polarization witha warm rapidly-rotating half-wave plate on the AtacamaB-Mode Search (ABS) instrument”, Rev. Sci. Instrum.,85, 024501 (2014).[10] B. Reichborn-Kjennerud et al., “EBEX: a balloon-borneCMB polarization experiment”, Proc. SPIE 7741, Mil-limeter, Submillimeter, and Far-Infrared Detectors andInstrumentation for Astronomy V, 77411C (July 15, 2010); doi:10.1117/12.857138.[11] T. Matsumura et al., “Mission design of LiteBIRD”,Journal of Low Temperature Physics, February 2014.[12] H. Nishino et al., “POLARBEAR CMB Polarization Ex-periment”, Proceedings of the 12th Asia Pacific PhysicsConference JPS Conf. Proc. 1, 013107 (2014).[13] T. Matsumura et al., “Cosmic Microwave Background B-mode Polarization Experiment POLARBEAR-2,” Pro-ceedings of the 12th Asia Pacific Physics Conference JPSConf. Proc. 1, 013108 (2014).[14] A. Ghribi et al., “Latest Progress on the QUBIC Instru-ment”, Journal of Low Temperature Physics, February2014.[15] S. Bryan et al., “Modeling and characterization of theSPIDER half-wave plate”, Proc. SPIE 7741, Millimeter,Submillimeter, and Far-Infrared Detectors and Instru-mentation for Astronomy V, 77412B (15 July 2010).[16] P. de Bernardis et al., “SWIPE: a bolometric polarimeterfor the Large-Scale Polarization Explorer”, Proceedingsof the SPIE Astronomical Telescopes + Instrumentation2012 Conference - Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for AstronomyVI - Amsterdam 1-6 July 2012, paper