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Primordial Inflation Polarization Explorer (PIPER). Halpern, Mark 2010

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The Primordial Inflation Polarization Explorer (PIPER) David T. Chuss,a Peter A.R. Ade,b Dominic J. Benford,a Charles L. Bennett,c Jessie L. Dotson,d Joseph R. Eimer,c Dale J. Fixsen,a Mark Halpern,e Gene Hilton,f James Hinderks,a Gary Hinshaw,a Kent Irwin,f Michael L. Jackson,a Muzariatu A. Jah,a Nikhil Jethava,a,g Christine Jhabvala,a Alan J. Kogut,a Luke Lowe,a Nuala McCullagh,c Timothy Miller,a Paul Mirel,a S. Harvey Moseley,a Samelys Rodriguez,a Karwan Rostem, a Elmer Sharp, a,g Johannes G. Staguhn,a,c Carole E. Tucker,b George M. Voellmer,a Edward J. Wollack,a and Lingzhen Zengc aNASA Goddard Space Flight Center, Observational Cosmology Laboratory, Code 665, Greenbelt, MD, USA; bCardiff University, UK cThe Johns Hopkins University, Department of Physics and Astronomy, Baltimore, MD, USA dNASA Ames Research Center, Moffett Field, CA, USA eThe University of British Columbia, Canada fNIST, Boulder, CO, USA gGlobal Systems Technology, Greenbelt, MD, USA ABSTRACT The Primordial Inflation Polarization Explorer (PIPER) is a balloon-borne instrument designed to search for the faint signature of inflation in the polarized component of the cosmic microwave background (CMB). Each flight will be configured for a single frequency, but in order to aid in the removal of the polarized foreground signal due to Galactic dust, the filters will be changed between flights. In this way, the CMB polarization at a total of four different frequencies (200, 270, 350, and 600 GHz) will be measured on large angular scales. PIPER consists of a pair of cryogenic telescopes, one for measuring each of Stokes Q and U in the instrument frame. Each telescope receives both linear orthogonal polarizations in two 32 × 40 element planar arrays that utilize Transition-Edge Sensors (TES). The first element in each telescope is a variable-delay polarization modulator (VPM) that fully modulates the linear Stokes parameter to which the telescope is sensitive. There are several advantages to this architecture. First, by modulating at the front of the optics, instrumental polarization is unmodulated and is therefore cleanly separated from source polarization. Second, by implementing this system with the appropriate symmetry, systematic effects can be further mitigated. In the PIPER design, many of the systematics are manifest in the unmeasured linear Stokes parameter for each telescope and thus can be separated from the desired signal. Finally, the modulation cycle never mixes the Q and U linear Stokes parameters, and thus residuals in the modulation do not twist the observed polarization vector. This is advantageous because measuring the angle of linear polarization is critical for separating the inflationary signal from other polarized components. Keywords: Cosmic Microwave Background, Polarimetry 1. INTRODUCTION The measurement of the polarization of the Cosmic Microwave Background offers a tool with which to probe the earliest epoch of the Universe’s history. Recent observations have significantly advanced our understanding of the history of the Universe and have hinted at new physics. Specifically, the flatness of the Universe, its unifor- mity, and the presence of a nearly scale-invariant spectrum of density perturbations in the early Universe have Further author information: (Send correspondence to D.T.C.) D.T.C.: E-mail: David.T.Chuss@nasa.gov, Telephone: 1 301 286 1858 Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy V, edited by Wayne S. Holland, Jonas Zmuidzinas, Proc. of SPIE Vol. 7741, 77411P © 2010 SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.857119 Proc. of SPIE Vol. 7741  77411P-1 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms supported the idea that the Universe underwent a brief period of exponential expansion, known as “inflation,” early in its history. Inflation is expected to produce a stochastic background of gravitational waves (tensor perturbations) whose amplitude depends only on the energy scale at which inflation occurred. The CMB temperature anisotropy is sensitive to a combination of scalar and tensor perturbations, but the CMB polarization anisotropy has a component that depends only on the gravitational wave background. Thus the polarization signal provides a way to directly measure the energy scale of inflation. This is a potential probe of physics some 12 orders of magnitude higher in energy than is achievable using terrestrial accelerators. The CMB anisotropy is weakly polarized due to anisotropic Thompson scattering at the epochs of recombi- nation and reionization. The polarization pattern that is induced by density perturbations is “curl-free”, due to the symmetry of the scalar perturbations. This curl-free pattern is referred to as “E-mode” polarization. The pattern induced by gravitational waves does not possess scalar symmetry, thus these perturbations can give rise to an additional divergence-free component, which is termed “B-mode” polarization.1,2 A detection of B-mode polarization thus provides a direct measure of the gravitational wave amplitude imparted by inflation. To date, the E-mode signal has been measured by several groups3–12 and is consistent with the predictions of anisotropic Thompson scattering of the primary temperature anisotropy. The B-mode signal has yet to be detected. The inflationary gravitational wave amplitude is parameterized by r, the “tensor-to-scalar” ratio. Models for inflation connect this parameter to the spectral tilt of the density perturbations, ns. Recent measurements13 have found that ns is measurably less than 1. For such a value, the simplest and arguably most compelling inflation models predict r > 0.01. Therefore, polarization measurements at this sensitivity provide a means to test specific inflationary models. The energy scale associated with such scales is consistent with that predicted for GUT scale physics. The Primordial Inflation Polarization Explorer (PIPER) is a balloon-borne experiment designed to search for the B-mode polarization signal in the Cosmic Microwave background as a test of inflation. PIPER will use a pair of liquid helium-cooled telescopes to observe a large fraction of the sky during turn-around flights from both the Northern and Southern Hemispheres. Each flight will be configured for a single frequency, and the frequency response of the instrument will be changed between campaigns. In Section 2, we summarize the predicted instrument performance. In Section 3 we briefly describe the optics. In Section 4, the polarization modulation for PIPER is described. The detectors are discussed in Section 5, and a discussion of systematics is included in Section 6. 2. INSTRUMENT SUMMARY The PIPER payload is shown in Figure 1A. PIPER consists of two telescopes that are cooled to 1.5 K via liquid helium. The telescopes are mirror images of each other and are aligned such that one measures Stokes Q and the other Stokes U simultaneously and on the same location on the sky. The polarization modulation is accomplished via variable-delay polarization modulators (VPMs). In each telescope, a modulator is placed at the front of the optical system so as to encode the modulation onto the signal before instrumental polarization is introduced by the optics. Both telescopes are located in a 3000 L bucket dewar that is filled with liquid helium before launch. Once at float altitude, the low atmospheric pressure reduces the temperature of the liquid to 1.5 K. The fore-optics, which are located in the same volume as the liquid cryogen, are cooled using a combination of evaporating liquid and superfluid helium pumps. There are no windows at the exit of the dewar; frost is prevented from collecting on the optics by the outflow of evaporating helium. These techniques were first demonstrated by the ARCADE experiment.14 The reimaging optics, filters, and detectors are located inside a vacuum vessel. An adiabatic demagnetization refrigerator (ADR) cools the detectors to ∼100 mK. Cooling the optics to 1.5 mK lowers the background photon noise on the detectors significantly as compared to the case with warm fore-optics, as shown in Figure 1B. The decreased noise in combination with the ability to simultaneously detect large numbers of modes from the CMB enables PIPER to reach low statistical limits on r in a series of turn-around flights. Proc. of SPIE Vol. 7741  77411P-2 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms VPMs Bucket Dewar Optics to Measure Stokes Q (cooled to1.5 K) Warm VPM motors Vacuum Vessel (Reimaging Optics, Filters, Detectors, ADR) Optics to Measure Stokes U (cooled to1.5 K) A B Figure 1. (A) A schematic diagram of the PIPER payload is shown. The fore-optics are cooled to 1.5 K via superfluid pumps and the evaporating cryogen. The reimaging optics, filters, and detectors are located inside a vacuum vessel. Polarization modulation is accomplished via variable-delay polarization modulators (VPMs). (B) The calculated background photon noise contributions for PIPER are compared to that of an equivalent system with warm mirrors. For PIPER, the dominant photon noise contribution is from the residual atmosphere at float altitude. Approximate PIPER bands are shown (shaded bands). By flying each frequency of PIPER in both the Southern and Northern Hemisphere, PIPER is able to cover a large fraction of the sky. This enables PIPER to be sensitive to the “reionization bump” of the B-mode spectrum that is present at large angular scales (low l). This signal is expected to be free of the gravitation lensing foreground signal that is likely to affect polarimetric observations at finer angular scales. Figure 2 shows both the sky coverage and the statistical sensitivity of PIPER. These calculations are based on 2 flights at each of 4 different frequencies. Table 1 gives a summary of various specified and predicted instrument properties for the four bands of PIPER. Table 1. Instrument Summary Property Band 1 Band 2 Band 3 Band 4 Frequency (GHz) 200 270 350 600 Wavelength (μm) 1500 1100 850 500 Bandwidth δν/ν 0.30 0.30 0.08 0.07 Beam Width (arc-min) 21 15 14 14 Optical Efficiency 0.30 0.30 0.30 0.15 Detector Absorption 0.90 0.90 0.70 0.50 Bolometer (Phonon) NEP (W Hz−0.5) 3.8× 10−18 3.8× 10−18 3.8× 10−18 3.8× 10−18 Total NEP (W Hz−0.5) 4.7× 10−18 5.9× 10−18 5.1× 10−18 7.1× 10−18 Detector Noise (mJy √ s) 160 147 466 877 Detector NET (μK √ s) 80 80 377 6600 Detector NEQ (μK √ s) 113 113 534 9300 Number of Detectors 5120 5120 5120 5120 Instrument NEQ (μK √ s) 1.6 1.6 7.5 130 3. OPTICS The optics for PIPER are described in detail in a companion paper,15 so we will only briefly summarize them here. Figure 3 shows the basic layout for a single telescope. Each telescope consists of three distinct sections. The first section is the VPM which will be discussed in Section 4. The second section is the fore-optics. This Proc. of SPIE Vol. 7741  77411P-3 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms Proc. of SPIE Vol. 7741  77411P-4 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms Proc. of SPIE Vol. 7741  77411P-5 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms simultaneously. The capacitance sensors have been tested cryogenically and are accurate to within a fraction of a micron over throws up to 2 mm. The nominal throw for the long wavelength of PIPER is ∼ 0.5 mm. Wire     Grid Input Signal Output Signal Mirror  z  Wire GridMirror Motor Drive Shaft Rotary Flexures Counterweight Cam Figure 4. (LEFT) The VPM uses a wire grid positioned in front of and parallel to a moving mirror. The phase delay introduced between the two orthogonal polarizations is controlled by modulating the grid-mirror separation. (RIGHT) A VPM CAD model for PIPER is shown. The PIPER VPM will be actuated by a warm motor connected via a cam to a shaft. A rotary flexure couples the motion of the shaft to a flexure that maintains the parallelism between the grid and mirror over the stroke of the modulation. 5. DETECTORS PIPER utilizes the Backshort-Under-Grid (BUG) architecture20 for its bolometer arrays. These detectors are described in greater detail in a companion paper,21 but a brief description is included here for completeness. Each array contains 32 × 40 planar absorbing elements each thermally coupled to a transition-edge sensor (TES). Four such arrays are included in PIPER, each used to detect a single linear polarization in each of the two telescopes. A BUG array (a 32×40 test device is shown in Fig. 5) is a two-dimensional structure that is sized to mate to the two-dimensional multiplexers developed by NIST for SCUBA-2.22 Prototype BUG arrays have been demonstrated in the GISMO instrument on the IRAM telescope.23 The BUG array consists of two pieces. The first is the grid of suspended membranes that make up the sensing part of the array. The electrical signals for the TES bolometers are routed to the back of the grid structure using “wrap-around vias,” or electrical paths that are deposited around each wall. A second grid that consists of an array of reflecting elements is nested into the back of the first grid in order to provide a backshort for each detector. In the case of PIPER, the backshort distance was chosen to maximize detector absorption over the three lowest frequencies which are those that contain the CMB signal. Because the pixels are close to the size of the wavelength for the PIPER bands, few modes exist within the detector volume. Because of this, we used full electromagnetic models to calculate the desired detector-backshort spacing in order to accurately account for the boundary conditions. 6. SYSTEMATIC ERROR CONTROL Because B-mode measurement involves the detection of a small polarized signal in the presence of much larger unpolarized sources, systematic error control is critical. PIPER is designed to avoid such systematics where possible and to mitigate those it cannot avoid. 6.1 T −→ B The mixing of unpolarized flux into a false polarized signal, commonly referred to as “instrumental polarization,” is especially to control given that the unpolarized flux is likely to be > 108 larger than the B-mode signal. Scattering, off-axis reflections, and transmission through dielectrics can all induce polarization on an initially unpolarized signal. PIPER’s VPMs are placed at the front of the optical system so that the polarization modulation is encoded on the CMB before any of these effects can occur within the instrument. Proc. of SPIE Vol. 7741  77411P-6 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms Figure 5. PIPER will utilize four 32×40 BUG arrays. It is conceivable to get some polarization at the VPM due to the small asymmetry between the boundary conditions for the two polarization components; however, the modulation architecture mitigates this in the following way. This class of systematic is manifest as a differential decrease in the linear polarization basis that defines the Stokes U -basis. The analyzer grid is oriented at a 45◦ angle with respect to this bases, and so detection is done in the Stokes Q-basis. In practice, a true sky polarization will induce a signal with opposite signs in the two oppositely-polarized detector arrays within a single telescope. The systematic signal will have the same sign in the two detectors. Mitigation of this class of systematic is dependent upon gain stability between the two detectors. 6.2 ΔT −→ B Asymmetry between the vertical and horizontal polarization sensitivity of an instrument can cause unpolarized map features (such as the primary CMB anisotropy or foregrounds) to manifest themselves as a polarization. The front-end VPM mitigates this effect by switching the beam’s polarization sensitivity without altering the beam shape. In addition, the primary modulation is done in polarization rather than scanning, and so this provides a nearly diagonal covariance matrix. 6.3 E −→ B Mixing Stokes Q and U lead to E, B mixing. Because, the VPM modulates the polarization from a single linear Stokes parameter to circular polarization, there is no Q-U mixing inherent in the modulation cycle. To illustrate this, it is useful to compare the VPM to the half-wave plate (HWP) that modulates between Q and U . For example, in the presence of cross-polarization, the spatial distribution of polarization within the beam can cause some degree of Q-U mixing. The left side of Figure 6 shows a simulated patch of sky in Stokes Q, U , and V . The contours show the co- and cross-polar beam patterns, assuming a -25 dB peak cross-polarization. On the right hand side of this figure, raw signals for a HWP and VPM are shown along with the error signals. For an E-mode RMS amplitude of 1.3 μK, we find that the HWP generates a modulated error of 147 nK. The VPM generates an offset of 3 nK. The details for the cross-polar dependent mixing are given in the Appendix. 6.4 Foregrounds PIPER’s four frequencies are all in the spectral regime in which dust is the dominant foreground with polarized synchrotron being negligible. The multiple frequencies will allow the characterization of the dust foreground for removal. Ultimately, it is likely that the limiting factor in CMB polarization measurements will be determined by the degree of complexity of the foreground signal. 7. SUMMARY PIPER is a balloon-borne polarimeter for searching for the polarized signature from inflation. It observes with two cryogenic co-pointed telescopes, each of which has a front end VPM for polarization modulation and detects both orthogonal linear polarization components in order to both maximize sensitivity and to monitor and remove systematic effects. PIPER will fly the first of four frequency configurations in 2013. Proc. of SPIE Vol. 7741  77411P-7 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms Proc. of SPIE Vol. 7741  77411P-8 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms B., Taylor, A. N., Thompson, K. L., Tucker, C., Turner, A. H., Wu, E. Y. S., and Zemcov, M., “QUaD: A High-Resolution Cosmic Microwave Background Polarimeter,” The Astrophysical Journal 692, 1221–1246 (Feb. 2009). [10] Pryke, C., Ade, P., Bock, J., Bowden, M., Brown, M. L., Cahill, G., Castro, P. G., Church, S., Culverhouse, T., Friedman, R., Ganga, K., Gear, W. K., Gupta, S., Hinderks, J., Kovac, J., Lange, A. E., Leitch, E., Melhuish, S. J., Memari, Y., Murphy, J. A., Orlando, A., Schwarz, R., O’Sullivan, C., Piccirillo, L., Rajguru, N., Rusholme, B., Taylor, A. N., Thompson, K. L., Turner, A. H., Wu, E. Y. S., and Zemcov, M., “Second and Third Season QUaD Cosmic Microwave Background Temperature and Polarization Power Spectra,” The Astrophysical Journal 692, 1247–1270 (Feb. 2009). [11] Chiang, H. C., Ade, P. A. R., Barkats, D., Battle, J. O., Bierman, E. M., Bock, J. J., Dowell, C. D., Duband, L., Hivon, E. F., Holzapfel, W. L., Hristov, V. V., Jones, W. C., Keating, B. G., Kovac, J. M., Kuo, C. L., Lange, A. E., Leitch, E. M., Mason, P. V., Matsumura, T., Nguyen, H. T., Ponthieu, N., Pryke, C., Richter, S., Rocha, G., Sheehy, C., Takahashi, Y. D., Tolan, J. E., and Yoon, K. W., “Measurement of Cosmic Microwave Background Polarization Power Spectra from Two Years of BICEP Data,” The Astrophysical Journal 711, 1123–1140 (Mar. 2010). [12] Jarosik, N., Bennett, C. L., Dunkley, J., Gold, B., Greason, M. R., Halpern, M., Hill, R. S., Hinshaw, G., Kogut, A., Komatsu, E., Larson, D., Limon, M., Meyer, S. S., Nolta, M. R., Odegard, N., Page, L., Smith, K. M., Spergel, D. N., Tucker, G. S., Weiland, J. L., Wollack, E., and Wright, E. L., “Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results,” ArXiv e-prints (Jan. 2010). [13] Komatsu, E., Smith, K. M., Dunkley, J., Bennett, C. L., Gold, B., Hinshaw, G., Jarosik, N., Larson, D., Nolta, M. R., Page, L., Spergel, D. N., Halpern, M., Hill, R. S., Kogut, A., Limon, M., Meyer, S. S., Odegard, N., Tucker, G. S., Weiland, J. L., Wollack, E., and Wright, E. L., “Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation,” ArXiv e-prints (Jan. 2010). [14] Singal, J., Fixsen, D. J., Kogut, A., Levin, S., Limon, M., Lubin, P., Mirel, P., Seiffert, M., Villela, T., Wollack, E., and Wuensche, C. A., “The ARCADE 2 Instrument,” ArXiv e-prints (Jan. 2009). [15] Eimer, J. R., Bennett, C. L., Benford, D. J., Chuss, D. T., Kogut, A. J., Mirel, P., Voellmer, G. M., Wollack, E. J., Ade, P. A. R., and Tucker, C. E., “The optical design of the piper experiment,” Proc. SPIE 7733 (July 2010). [16] Chuss, D. T., Wollack, E. J., Moseley, S. H., and Novak, G., “Interferometric polarization control,” Applied Optics 45, 5107 (2006). [17] Krejny, M., Chuss, D., D’Aubigny, C. D., Golish, D., Houde, M., Hui, H., Kulesa, C., Loewenstein, R. F., Moseley, S. H., Novak, G., Voellmer, G., Walker, C., and Wollack, E., “The Hertz/VPM polarimeter: design and first light observations,” Applied Optics 47, 4429 (Aug. 2008). [18] Ade, P. A. R., Chuss, D. T., Hanany, S., Haynes, V., Keating, B. G., Kogut, A., Ruhl, J. E., Pisano, G., Savini, G., and Wollack, E. J., “Polarization modulators for CMBPol,” Journal of Physics Conference Series 155 (Mar. 2009). [19] Voellmer, G. M., Chuss, D. T., Jackson, M., Krejny, M., Moseley, S. H., Novak, G., and Wollack, E. J., “A kinematic flexure-based mechanism for precise parallel motion for the Hertz variable-delay polarization modulator (VPM),” Proc. SPIE 6273 (July 2006). 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Terms of Use:  http://spiedl.org/terms [23] Staguhn, J. G., Benford, D. J., Allen, C. A., Maher, S. F., Sharp, E. H., Ames, T. J., Arendt, R. G., Chuss, D. T., Dwek, E., Fixsen, D. J., Miller, T. M., Moseley, S. H., Navarro, S., Sievers, A., and Wollack, E. J., “Instrument performance of GISMO, a 2 millimeter TES bolometer camera used at the IRAM 30 m Telescope,” Proc. SPIE 7020 (Aug. 2008). [24] Withington, S. and Saklatvala, G., “Characterizing the behaviour of partially coherent detectors through spatio-temporal modes,” Journal of Optics A: Pure and Applied Optics 9, 626–633 (July 2007). [25] Saklatvala, G., Withington, S., and Hobson, M. P., “Coupled-mode theory for infrared and submillimeter wave detectors,” Journal of the Optical Society of America A 24, 764–775 (Mar. 2007). [26] Ludwig, A., “The Definition of Cross Polarization,” IEEE Transactions on Antennas and Propagation AP- 21, 116–119 (January 1973). APPENDIX: CROSS-POLAR COUPLING IN HWPS AND VPMS The power absorbed by a polarization-sensitive detector can be expressed in the following way:24,25 〈p(t)〉 = ∫ ∞ 0 φ(ω)dω ∫ S dΩ̂1 ∫ S dΩ̂2G † (Ω̂1, Ω̂2, ω) · ·E(Ω̂1, Ω̂2, ω) (2) Here, the ·· refers to complete tensor contraction (the trace of the matrix product), and φ(ω) describes the passband of the system. In addition, G † (Ω̂1, Ω̂2, ω) = 〈G∗(Ω̂1, ω)G(Ω̂2, ω)〉 is the detector response expressed as a dyadic, and E † (Ω̂1, Ω̂2, ω) = 〈E∗(Ω̂1, ω)E(Ω̂2, ω)〉 is the cross spectral dyadic of the source. Here, the beam pattern G(Ω̂, ω) is the system response or gain in the direction Ω̂ at frequency ω, and E(Ω̂, ω) is the electric field of the source in the direction Ω̂ at frequency ω. The angle brackets represent a time average. This expression is extremely general since it decomposes both the beam and the sky in terms of their second- order statistical properties. This also makes for a convenient representation for a polarization analysis. In our case, the sky is partially coherent at each point (partially-polarized), but there are no correlations from point-to- point. Thus, we only have to consider the terms of the dyadic in which Ω̂ ≡ Ω̂1 = Ω̂2, reducing the expression for the cross-spectral dyadic of the sky to E(Ω̂, ω) = I(Ω̂, ω)σ0 + Q(Ω̂, ω)σ1 + U(Ω̂, ω)σ2 + V (Ω̂,ω)σ3. (3) Here, σ0 is the identity matrix, σi are the Pauli matrices, and (I,Q, U, V ) are the Stokes parameters. In contrast to the dyadic representing the sky, the dyadic representing the detector response can be highly correlated. For example, the beam pattern of a single-mode feedhorn is highly correlated over the entirety of a beam. Multi-mode systems can be represented by incoherent sums of coherent modes and will be only partially correlated. The gain can be expressed in the co- and cross-polarization basis. G(θ, φ) = C(θ, φ)êC + R(θ, φ)êR (4) where R(θ, φ) and C(θ, φ) are the co- and cross-polarization beam patterns, respectively.26 Therefore, the correlation dyadic is Gij(θ, φ) = êi ·G(Ω̂) · êj = ( |C(θ, φ)|2 C∗(θ, φ)R(θ, φ) C(θ, φ)R∗(θ, φ) |R(θ, φ)|2 ) . (5) The contracted dyadics in the integrand of Equation 2 can be written analytically as ∫ S dΩ̂1 ∫ S dΩ̂2G † (Ω̂1, Ω̂2, ω) · ·E(Ω̂1, Ω̂2, ω) = ∫ S d(cos θ)dφ{I(θ, φ)[C2 + R2] + Q(θ, φ)[C2 −R2] + (6) U(θ, φ)[2(R∗C)] + V (θ, φ)[2(R∗C)]}. Proc. of SPIE Vol. 7741  77411P-10 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms Here we have suppressed the explicit dependence of the Stokes parameters on frequency and the co- and cross- polarization dependence on θ and φ. The action of a modulator can be represented by transforming the beam correlation dyadic using a represen- tative Jones matrix. In general, this Jones matrix is frequency-dependent. It is unitary at each frequency for an ideal modulator; non-ideal effects can be introduced by relaxing the unitary condition. The total power from the detector can then be represented by 〈p(t)〉 = ∫ ∞ 0 φ(ω)dω ∫ S dΩ̂1 ∫ S dΩ̂2J † modG † (Ω̂1, Ω̂2, ω)Jmod · ·E(Ω̂1, Ω̂2, ω) (7) Note that the Jones matrix representing the modulator will depend on frequency, the internal modulation parameters (such as phase or rotation angle), and possibly Ω̂. Equation 7 provides a method for coding the polarization-modulated response of a detector to an arbitrary sky. As an example, we will look at the cases for ideal single-frequency half-wave plate (HWP) and a variable-delay polarization modulator (VPM). The Jones matrix for an ideal HWP is (see e.g.16) JHWP (α) = i ( − cos 2α − sin 2α − sin 2α cos 2α ) . (8) where α is the rotation angle of the waveplate. Using Equations 8, 3, and 5, we get the following expression for the integrand ∫ S dΩ̂1 ∫ S dΩ̂2J † HWPG † (Ω̂1, Ω̂2, ω)JHWP · ·E(Ω̂1, Ω̂2, ω) = ∫ S d(cos θ)dφ{I(θ, φ)[C2 + R2] + Q(θ, φ)[(C2 −R2) cos 4α + 2(C∗R) sin 4α] + U(θ, φ)[(C2 −R2) sin 4α + 2(C∗R) cos 4α] + V (θ, φ)[2(R∗C)]} (9) The Jones matrix for an ideal VPM oriented at 45◦ with respect to the polarization axis of the detector is16 JV PM (δ) = ( cos δ/2 −i sin δ/2 i sin δ/2 − cos δ/2 ) . (10) where δ is the phase delay between the two orthogonal linear polarizations. Using Equations 10, 3, and 5, we get the following expression for the integrand ∫ S dΩ̂1 ∫ S dΩ̂2J † V PMG † (Ω̂1, Ω̂2, ω)JV PM · ·E(Ω̂1, Ω̂2, ω) = ∫ S d(cos θ)dφ{I(θ, φ)[C2 + R2] + Q(θ, φ)[(C2 −R2) cos δ + 2(C∗R) sin δ] + U(θ, φ)[2(C∗R)] + V (θ, φ)[(C2 −R2) sin δ − 2(R∗C) cos δ]} (11) For the HWP, the term proportional to the product of the cross- and co-polarization causes leakage between Q and U . For the VPM, the leakage terms are between Q and V rather than Q and U . For a VPM measuring Q, U is unmodulated. The VPM does mix Q and V , as expected from its path on the Poincaré sphere. However, it is expected that V is close to zero for the astrophysical case in question. Proc. of SPIE Vol. 7741  77411P-11 Downloaded from SPIE Digital Library on 13 Sep 2011 to 137.82.117.28. Terms of Use:  http://spiedl.org/terms

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