Polarimetry in Astrophysics and Cosmology

POLARIMETRY IN ASTROPHYSICS AND COSMOLOGY

Lingzhen Zeng

A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy.

Baltimore, Maryland

June, 2012

Astrophysicists are mostly limited to passively observing electromagnetic radiation from a distance, which generally shows some degree of polarization. Polariza- tion often carries a wealth of information on the physical state and geometry of the emitting object and intervening material. In the microwave part of the spectrum, polarization provides information about galactic magnetic fields and the physics of interstellar dust. The measurement of this polarized radiation is central to much modern astrophysical research. The first part of this thesis is about polarimetry in astrophysics. In Chapter 1, I review the basics of polarization and summarize the most important mechanisms that generate polarization in astrophysics. In Chapter 2, I describe the data analysis of polarization observation on M17 (a young, massive star formation region in the Galaxy) from Caltech Submillimeter Observatory (CSO) and show the physics that we learn about M17 from the polarimetry. Polarimetry also plays an important role in modern cosmology. Inflation theory predicts two types of polarization in the Cosmic Microwave Background (CMB) radiation, called E-modes and B-modes. Measurements to date of the E-mode signal are consistent with the predictions of anisotropic Thompson scattering, while the B-mode signal has yet to be detected. The B-mode power spectrum amplitude can be parameterized by the relative amplitude of the tensor to scalar modes r. For the simplest inflation models, the expected deviation from scale invariance (n = 0.963 0.012) is s ± coupled to gravitational waves with r 0.1. These considerations establish a strong ≈ 32 motivation to search for this remnant from when the universe was about 10− seconds

ii old. The second part of this thesis is about the Cosmology Large Angular Scale Sur- veyor (CLASS) experiment, that is designed to have an unprecedented ability to detect the B-mode polarization to the level of r 0.01. Chapter 3 is an introduction ≤ to cosmology, including the big bang theory, inﬂation, ΛCDM model and polarization of the CMB radiation. Chapter 4 is about CLASS, including science motivation, instrument optimization and lab testing.

Advisor: Prof. Charles L. Bennett Second reader: Prof. Tobias Marriage

iii Acknowledgements

The work described in this thesis would not have been possible without the support of many people. Foremost, I would like to express my sincere gratitude to my advisor Prof. Chuck Bennett for the continuous support of my Ph.D study and research, for his patience, motivation, enthusiasm, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D study. Besides my advisor, I would like to thank Dave Chuss. In many research projects, I have been aided for many years by him. Dave is patient and always ready to discuss whatever problems are on my mind. I would like to thank Prof. Giles Novak, who offered me much advice and insight on the millimeter/submillimeter polarimetry. I will miss the time when we worked together on Mauna Kea summit. I gratefully acknowledge Prof. Toby Marriage for his valuable advice in lab discussions, supervision on lab instrument development. I would also like to thank Toby for his great help in my job application. I would like to thank David Larson and Joseph Eimer. We worked together for many years and have so many useful discussions and collaborations. My sincere thanks also goes to Ed Wollack, John Vaillancourt, George Voellmer, Gary Hinshaw, John Karakla, Karwan Rostem, Tom Essinger-Hileman and Paul Mirel for offering me help and discussions on the various research projects. I thank my fellow graduate/undergraduate students in the research group at Johns Hopkins University: Dominik Gothe, Zhilei Xu, Aamir Ali, Dave Holtz, Connor Hen- ley and Tiffany Wei for the fun and proud of working together on the CLASS project. It is a pleasure to thank my friends at JHU for making my life fun: Jiming Shi,

iv Jianjun Jia, Jun Wu, Zhouhan Liang, Jian Su, Sunxiang Huang, Yuan Yuan, Longzhi Lin, Hao Chang, Di Yang, Xin Guo, Jie Chen, Xiulin Sun, Jianhua Yu, Xin Yu, Wen Wang, Hui Gao, Jinsheng Li, Jiarong Hong and Yuan Lu. I am grateful to many others for making my time at JHU enjoyable. Unfortunately, there are too many to name individually. I would also like to thank my undergraduate classmates: Huaze Ding, Xiao Hu and Jun Li for our longtime friendship. I wish all of you the best in the future. Last but not the least, I would like to thank my family: my parents Xiangxiong Zeng and Qiuying Li, for giving birth to me at the ﬁrst place and supporting me spiritually throughout my life, and my sister Lingfang Zeng and brother Lingyao Zeng, for their understanding and support in so many years.

v Contents

Abstract ii

Acknowledgements iv

List of Tables ix

List of Figures x

I Polarimetry in Astrophysics 1

1 Introduction to Polarization in Astrophysics 2 1.1 PlaneWave ...... 2 1.2 StokesParameters ...... 3 1.3 Poincar´eSphere...... 7 1.4 Polarization in Astrophysics ...... 8 1.4.1 Synchrotron Emission ...... 8 1.4.2 Thermal Dust Emission and Absorption ...... 10 1.4.3 Examples of Polarization from Absorption and Scattering . . . 13 1.4.4 Anomalous Dust Emission ...... 15

2 Submillimeter Polarimetry of M17 16 2.1 Introduction to Submillimter Polarimetry ...... 16 2.2 Polarimetry at Caltech Submillimeter Observatory ...... 17 2.3 SHARP Data Pipeline ...... 20 2.4 IntroductiontoM17 ...... 22 2.5 M17PolarimetryResults...... 23 2.5.1 GeneralResults...... 23 2.5.2 Polarization Spectrum ...... 27 2.5.3 Spatial Distribution of Magnetic Field and Polarization Spectrum 30 2.5.4 Conclusion ...... 38

vi II Polarimetry in Cosmology 39

3 Introduction to Polarization in Cosmology 40 3.1 The Big Bang Theory ...... 40 3.1.1 The Expanding Universe–Hubble’s Law ...... 40 3.1.2 Big Bang Nucleosynthesis (BBN) ...... 41 3.1.3 The Cosmic Microwave Background (CMB) Radiation . . . . 42 3.1.4 OtherEvidence ...... 42 3.2 CosmicInﬂation...... 43 3.2.1 TheStructureProblem...... 43 3.2.2 The Flatness Problem ...... 44 3.2.3 The Horizon Problem ...... 44 3.2.4 The Magnetic Monopole Problem ...... 44 3.3 ΛCDMCosmologicalModel ...... 45 3.3.1 Cosmological Principles and FLRW metric ...... 45 3.3.2 Einstein Field Equations and Friedmann Equation ...... 47 3.3.3 Best-ﬁtΛCDMModelParameters...... 49 3.4 The Cosmic Microwave Background Radiation ...... 54 3.4.1 TheCMBAnisotropy...... 56 3.4.2 The CMB Polarization ...... 57

4 The Cosmology Large Angular Scale Surveyor (CLASS) 62 4.1 Scientiﬁc Overview ...... 64 4.2 Sensitivity Calculation and Bandpass Optimization ...... 67 4.2.1 Sensitivity Calculation ...... 68 4.2.2 Bandpass Optimization ...... 71 4.3 The Variable-delay Polarization Modulator ...... 79 4.3.1 Polarization Transfer Function ...... 80 4.3.2 VPM Grid Optimization ...... 81 4.3.3 VPM Mirror Throw Optimization ...... 83 4.3.4 VPMEﬃciency...... 89 4.3.5 CurrentStatus ...... 91 4.4 CLASSOptics...... 92 4.5 Smooth-walledFeedhorn ...... 98 4.5.1 Smooth-walled Feedhorn Optimization ...... 98 4.5.2 Smooth-walledFeedhornforCLASS...... 102 4.6 CLASSDetectors ...... 111 4.6.1 FocalPlane ...... 111 4.6.2 TES Bolometers ...... 113 4.7 LabSetupforDetectorTesting ...... 116 4.7.1 Cryostat...... 116 4.7.2 Thermometry ...... 118

vii 4.7.3 CryostatPerformance ...... 118 4.7.4 DetectorReadout...... 119

A M17 Polarization Data 125 A.1 PolarziationSpectrum: 450umvs60um ...... 125 A.2 PolarziationSpectrum: 450umvs100um ...... 127 A.3 Polarziation Spectrum: 450 um vs 350 um at RA > 18h17m30s . . . . 129 A.4 Polarziation Spectrum: 450 um vs 350 um at RA < 18h17m30s . . . . 130 A.5 PolarizationVectors ...... 132

B Blackbody Radiation 136

C NEP of Photons in a Blackbody Radiation Field 138

D A Low Cross-Polarization Smooth-Walled Horn with Improved Band- width 140 D.1 Smooth-walled Feedhorn Optimization ...... 141 D.1.1 Beam Response Calculation ...... 141 D.1.2 Penalty Function ...... 142 D.1.3 Feedhorn Optimization ...... 143 D.2 FeedhornFabricationandMeasurement...... 145 D.3 Conclusion...... 148

E CLASS 40 GHz Feedhorn Proﬁle 153

F Lab Cryostat Thermometry Codes 159

Vita 189

viii List of Tables

2.1 SHARPInstrumentSpeciﬁcations ...... 19 2.2 M17PolarizationSpectrumData ...... 30

3.1 Best-ﬁtΛCDMModelParameters...... 50

4.1 CLASSScientiﬁcOverview...... 66 4.2 CLASSDetectorParameters...... 70 4.3 CLASS VPM Mirror Throw Optimization ...... 88 4.4 CLASSOpticsOverview ...... 94 4.5 CLASS40GHzFeedhornRequirements ...... 102 4.6 Feedhorn Proﬁle Approximation (in Millimeters) ...... 104 4.7 FeedhornPerformance ...... 105 4.8 BeamParameters...... 110 4.9 CryostatThermometryReadout...... 122

D.1 Spline Approximation to Optimized Proﬁle (in Millimeters) ...... 148 D.2 BeamParameters...... 150

ix List of Figures

1.1 A simple plane wave. The electric (E, in x-z plane) and magnetic field (B, in y-z plane) is perpendicular to each other and to the direction of propagation(z)...... 4 1.2 Polarization ellipse. It shows the (ξ, η) coordinates with respect to the (x, y) coordinates and the definitions of orientation angle ψ, ellipticity angle χ...... 5 1.3 Poincarésphere, defining the polarization in spherical coordinates. It also shows the relation between (Q, U, V ) and (Ip, χ, ψ) [1]...... 7 1.4 CMB foreground radiation in WMAP bands [2]. The synchrotron radiation dominates the low frequency range below 60 GHz. Radiation fromdustcontributesmostlyabove70GHz...... 9 1.5 Starlight polarization vectors in Galactic coordinates. The upper panel shows polarization vectors in local clouds. The polarization averaged over many clouds in the Galactic plane is shown in the lower panel. The magnetic field is parallel to the polarization angle...... 14

2.1 NEFD350 m measurements (points) from Jan 2003 compared to theoretical expectation (solid line) from equation 2.1 [3]. The performance 1/2 is about 1 Jy s for τ225 GHz =0.05...... 18 2.2 The polarization splitting optics of SHARP [4] for reconstituting the image with an offset between the two polarization components. Left: The expanding beam from the CSO focus is reflected by P1 (paraboloid), F1 (flat mirror), through the HWP (half wave plate), and reaches the XG (crossed grid), where the polarization radiation is separated into two orthogonal (horizontal and vertical) components. Right: View toward the CSO focus. The vertical and horizontal components undergo further reflections by a series of mirrors and grids, and are displaced laterally at the BC (beam combiner), before being directed toward SHARCII...... 19

x 2.3 Flow chart of “SharpInteg”. It starts by masking the raw data file with an “rgm” file. Then, it demodulates the chopping to calculate the chopped data. After applying the relative data gain factor between the horizontal and vertical array, it calculates the I, I-error, Q, Q-error, U and U-error components and saves them into a new file...... 20 2.4 Flow chart of “Sharpcombine”. It applies τ and telescope pointing correction, background subtraction (BS), instrument polarization (I.P.) subtraction and polarization angle rotation to sky coordinates (Rot) to each sub-map before it combines them into a large map and smooths it. 21 2.5 M17 is a premier example of a young, massive star formation region in the Galaxy. Left: A M17 image from my 80 mm aperture optical telescope. Right: A false color image from Spitzer GLIMPSE (red: 5.8 um; green: 4.5 um; blue: 3.6 um.) [5]...... 22 2.6 A M17 model from [6]. The system can be described as a central cluster + 0 of stars surrounded by successive layers of H ,H , and H2 gas, that expanding with different velocities to the outer side of the cloud. . . . 24 2.7 M17 polarization fraction vectors are plotted over the 450 um uncalibrated flux map. Thick vectors are detected with greater than or equal to 3σ level and thin vectors are between 2σ and 3σ level. The circle on the bottom right shows the SHARP beamsize. Some parts of the flux map is removed due to high noise levels. Offsets are from 18h17m32s, -16◦14′25′′ (B1950.0)...... 25 2.8 Histogram of M17 polarization fraction. This distribution includes all vectors at greater or equal to than 2σ level. All vectors greater than 10% are 2σ vectors...... 26 2.9 Histogram of M17 polarization angle. Polarization angles are measured from north to east. The resulting net magnetic field is almost parallel totheRAdirection...... 26 2.10 Magnetic field vectors from SHARP (red, 450 um), Stokes (green,100 um) [7] and optical observation [8] (purple) plotted on top of Spitzer GLIMPSE 8.00 um flux map. The magnetic vectors from SHARP and Stokes are perpendicular to their polarization angles, while those from optical polarization measurement are parallel to their polarization angles. All magnetic vectors (plotted with the same length) are used h m s to indicate the direction only. Offsets are from 18 17 32 , -16◦14′25′′ (B1950.0)...... 28 2.11 The common area (green shadow) for polarization spectrum analysis. h m s h m s It is between 18 17 30 and 18 17 37 in Ra (B1950), 16◦16′20′′ and − 16◦13′00′′ in Dec (B1950). The selected polarization vectors are at − 60 m (yellow), 100 m (green), 350 m (blue) and 450 m (red). Background is the 450 m flux map. Offsets are from 18h17m32s,- 16◦14′25′′ (B1950.0)...... 29

xi 2.12 Polarization spectrum of some popular interstellar molecular clouds [9]. The median polarization ratio are normalize by the value at 350 m. In contrast to the results from other clouds, our work shows that, the M17 has lower median polarization at 450 m than at 350 m. The polarization spectrum falls monotonically from 60 m to 450 m. . . 31 2.13 Magnetic vectors from SHARP plotted over the [21 cm]/[450 m] flux ratio map, showing that the shock front is passing through the cloud. The contour levels are 0.1, 0.3, 0.5, 0.7, 0.9 . The “X” axis is defined { } by fitting contour level = 0.1. The new “X-Y” coordinate system is about 66.3◦ with respect to the “Ra-Dec” coordinates. The shock is following the “-Y” direction. The “y=0” and “y=-50 arcsec” lines separate the cloud into “post-shocked” (y > 0), “shock front” (-50 < y < 0) and “pre-shocked” (y < -50) regions. The polarization directions and magnitudes in these regions are different (figure 2.14 and 2.15). The magnetic fields in the dense cloud (can also be seen in figure 2.10) at the top of the map survive the windswept. Offsets are from 18h17m32s, -16◦14′25′′ (B1950.0)...... 33 2.14 Correlation between polarization angle and the Y direction (zero at h m s 18 17 32 , 16◦14′25′′), showing a linear relationship. The “post- shocked” region− is at y > 0 and the “pre-shocked” region is at y < 50 arcsec...... − 34 2.15 Correlation between polarization fraction and Y direction (zero at h m s 18 17 32 , 16◦14′25′′), showing a “U” like shape. The polarization − fraction is higher at the “post-shocked” region at y > 0 and the “pre- shocked” region at y < 50arcsec...... 34 2.16 Magnetic field vectors (red)− and intensity contours of SHARP (green, levels = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 ) are over plotted on the 21 cm absorption-line{ contour and the ratio} of neutral HI (NHI) column density to the spin temperature Tspin distribution map in the 17.5-22 km/s velocity area from [6]. This velocity component is correlated with the “post-shocked” and part of “shock front” region. The NHI/Tspin density at the dense cloud region (see figure 2.13) is low...... 35 2.17 The [450 m]/[350 m] polarization ratio vectors over plotted on the [21 cm]/[450 m] flux ratio map with contour levels = 0.1, 0.3, 0.5, { 0.7, 0.9 . The blue (red) vectors represent P < (>) P . The } 450 350 length of the 2% bar at bottom left is equivalent to P450/P350 = 1.0. The directions of the vectors are parallel to their polarization angles. h m s Offsets are from 18 17 32 , -16◦14′25′′ (B1950.0)...... 36 2.18 The [450 m]/[350 m] polarization ratio vectors and 450 m intensity contours of SHARP (green, levels = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 ) over plotted on the Fig.1 from [7]. The blue{ vectors is found to be correlated} with the [OI] line, which is a tracer for the atomic gas...... 37

xii 3.1 Timeline of the universe. The CMB radiation from the last scattering surface (LSS) when the universe is about 380,000 years old with the temperatureofabout3,000K[10]...... 55 3.2 The internal linear combination map from WMAP [11], showing the allskyCMBtemperatureanisotropy...... 56 3.3 The angular power spectrum from WMAP [12], showing the detection of the ﬁrst three peaks. The ﬁrst peak is at ℓ 220, corresponding to ≈ an angular scale of about 1◦...... 58 3.4 Left: Quadrupole polarization from Thomson scattering of the CMB photons with free electrons. Right: The E and B mode patterns. The E-modes are curl-free components with no handedness. The B-modes are curl components with handedness...... 59 3.5 Plots of signal for TT (black), TE (red ), and EE ( green). The not- yet-detected BB (blue dots) signal is from a model with r = 0.3. The BB lensing signal is shown as a blue dashed line. The foreground model for synchrotron plus dust emission is shown as straight dashed lines [13]. 60

4.1 Two-dimensional joint marginalized constraint (68% and 95% CL) on scalar spectral index (ns) and tensor to scalar ratio (r), derived from the data combination of WMAP + BAO + H0 [14]. Three linear fits are from different simple inflation models...... 63 4.2 The background is the WMAP 7 year all sky Q band polarization map in Galactic coordinates showing the sky coverage of CLASS experiment. Observing from the Atacama Desert in Chile, CLASS covers 65.1% of the sky above 45◦ elevation. Excluding the Galactic mask area,∼ the visible sky left is 46.8% (bright region). The dark circle at ∼ the south pole is about 22◦ in radius. Figure courtesy of David Larson. 64 4.3 CLASS instrument overview for the 40 GHz band. The instrument consists a front-end variable-delay polarization modulator, catadioptric optic system and a field cryostat. The lenses are cooled to about 4 K and the smooth-walled feedhorn-coupled TES bolometer array operates at100mK.FigurecourtesyofJosephEimer...... 65 4.4 CLASS wavebands and sensitivity curve from [15]. Left: The frequency bands of CLASS are chosen to straddle the Galactic foreground spectral minimum and to minimize atmospheric effects (see section 4.2.2). Right: The CLASS sensitivity curve, shown by the dashed curve along the shaded boundary, is the 1σ limit for each l and assumes 3 years of observing with a conservative 50% efficiency for down-time (see section 4.2.1). CLASS has the sensitivity to definitively detect B-modes at the cosmologically interesting limit of r 0.01...... 68 ∼

xiii 4.5 Annual variation of the Precipitable Water Vapor (PWV) content at Chajnantor, based on 10 years of site testing. Conditions are worse during the winter from the end of December to early April. The expected median PWV for the rest of the year is around 1 mm, while conditions of PWV < 0.5 mm can be expected up to 25% of the time [16]...... 73 4.6 Atmospheric transmission and brightness temperature at CLASS site from 5 to 1000 GHz. ATM parameters: ground temperature = 275 K, ground pressure = 558 mb, PWV = 1.0 mm, elevation = 45◦, altidude = 5180 m. ATM version: atm2011 03 15.exe...... 74 4.7 Top: the CMB signal (equation 4.20) and Bottom: atmospheric noise source (equation 4.16) for the relative signal-to-noise ratio calculation (equation 4.21). The red, green and blue lines shows our optimized bandwidth for 40, 90 and 150 GHz band: (30.3 GHz - 40.3 GHz), (77.3 GHz-108.3GHz)and(126.8GHz-164.3GHz)...... 77 4.8 The 2-D plot of relative signal-to-noise ratio (equation 4.22) from 0 to 200 GHz showing our optimization results. The cross points of red, green and white lines are the locations of the local maxima. For the 40 GHz band, we only search for the maximum in the range of ν > 30 GHz. The coordinates are (30.3, 40.3), (77.3, 108.3) and (126.8, 164.3). 78 4.9 As shown in Poincarésphere, VPM modulates between Q and V , while the HWP mix Q and U. In the case of VPM, the residuals due to the spectral effects (shown in blue) are a function of measurable modulation parameters. Figure courtesy of David Chuss...... 79 4.10 VPM modulates polarization by introducing a controlled variable path difference between two orthogonal linear polarizations. Dots show the component with polarization angle parallel to the grid; Double arrow show that with angle perpendicular to the grid. By moving the mirror up and down, VPM introduces a path difference x(t) = 2d(t)cosθ between these two orthogonal polarization components...... 80 4.11 The wire grid performances for two different wavelengths from a sim- ulation [17]. In the limit of g/λ 1, a sinusoidal form for Stokes Q is in good agreement with an ideal≪ grid (equation 4.29). The VPM reflection phase delay differs from the free-space grid-mirror delay if theconditionsarechanged...... 82 4.12 The contour plot of relative signal-to-noise ratio for Stokes Q, calculated from equation 4.42 with cosine chopping mode. This plot is for the 40 GHz band (33 GHz to 43 GHz, λ0 = 7.89 mm). The maximum is at (0.19 λ0, 0.13 λ0) with the peak signal-to-noise ratio scaled to be 1.00. There are 4 other local maxima nearby: (0.19 λ0, 0.39 λ0), (0.44 λ0, 0.13 λ0), (0.44 λ0, 0.39 λ0) and (0.27 λ0, 0.26 λ0). Details are listed intable4.3...... 86

xiv 4.13 The contour plot of relative signal-to-noise ratio for Stokes Q, calculated from equation 4.42 with linear chopping mode. This plot is for the 40 GHz band (33 GHz to 43 GHz, λ0 = 7.89 mm). The maximum is at (0.46 λ0, 0.16 λ0) with the peak signal-to-noise ratio scaled to be 1.00. There are 2 other local maxima nearby: (0.63 λ0, 0.19 λ0) and (0.45 λ0, 0.42 λ0). Details are listed in table 4.3...... 87 4.14 VPM efficiency calculated from equation 4.55. The efficiency drops quickly from r = 1.0 to r = 5.0 and becomes almost flat after r > 10. The noise at large r is due to the rounding in the numerical calculations. 92 4.15 Photo of the prototype VPM grid. The wires are glued on an aluminium box frame with over 2 tons of stretch force. The diameter of the flattener ring is 50 cm. The wire diameter, 2a, is 63.5 m, with wire pitch, g = 200 m. 2a/g = 1/3.15 1/π. The flatness of the grid ≈ is better than 50 m. The total length of the wires is longer than 2 miles...... 93 4.16 Top: Drawings of CLASS 40 GHz optics. It consists of a front-end VPM, two mirrors, two lenses, a Lyot stop, a vacuum window and two infrared (IR) blocking filters. Bottom: Drawing and the ray trace of the cooled optics. Units are in mm. Figure courtesy of Joseph Eimer. 95 4.17 Ray trace of CLASS 40 GHz optics. Basic parameters: VPM diameter = 60.0 cm, effective focal length = 70.5 cm, f/2.0, focal plane diameter = 27.0 cm, Lyot stop diameter = 30.0 cm, FOV = 18.0◦, number of pixels=36. FigurecourtesyofJosephEimer...... 96 4.18 Point spread diagram of CLASS 40 GHz optics from Zeemax. Each diagram in this figure represents a separate direction on the sky. The circles show the first Airy disk at the corresponding location. This diagram shows that the optics is diffraction limited. Figure courtesy ofJosephEimer...... 97 4.19 Flow chart of smooth-walled feedhorn optimization. Optimization begins with a sin0.75 profile, the method from [18] is used to calculate the beam patterns. The feedhorn profile was found by this multi-step iterative solution with different thresholds in each step...... 101 4.20 CLASS 40 GHz feedhorn profile. The 10.00 mm long input waveguide has a radius of 3.334 mm, with fc = 26.349 GHz. The length of the feedhorn is 100.00 mm. The aperture is 35.828 mm. This is a monotonic profile that allows a progressive milling technique...... 103 4.21 CLASS feedhorn performance from 30 to 50 GHz. The dashed lines define the -30 dB line, and the waveband limit of 33 GHz and 43 GHz. The cut off frequency is fc = 26.349GHz...... 103 4.22 Beam patterns of the CLASS smooth-walled feedhorn within azimuth angles of 90◦,from33GHzto38GHz...... 106 ±

xv 4.23 Beam patterns of the CLASS smooth-walled feedhorn within azimuth angles of 90◦,from39GHzto44GHz...... 107 ± 4.24 Beam patterns of the CLASS smooth-walled feedhorn within azimuth angles of 15◦,from33GHzto38GHz...... 108 4.25 Beam patterns± of the CLASS smooth-walled feedhorn within azimuth angles of 15◦,from39GHzto44GHz...... 109 4.26 The averaged± cross-pol, return-loss and edge-taper plot for the tolerance calculation from 0 to 300 um. For each tolerance, these values were from the average of 120 calculations. (The plots are noisy at large tolerance, more calculation would be required to smooth the plots.) . 111 4.27 Section view of CLASS 40 GHz focal plane. It consists of a array of 36 smooth-walled feedhorns, waveguide adapter, detector mounting plate and clips. The focal plane will operate at a temperature of 100 mK. Figure courtesy of Thomas Essinger-Hileman...... 112 4.28 The feedhorn-couple TES bolometers set up [15] and prototype detector chip for the 40 GHz CLASS [19]. Left: The detector set up showing the feedhorn, detector housing, detector chip and backshort. Right: Photo of a 40 GHz prototype detector chip, showing the OMT, Magic Tees, filters and TES membranes...... 113 4.29 The electro-thermal circuit diagram of a TES bolometer (modified from [20]). Left: Each pix with a heat capacity of C at temperature T is connected by a thermal link G to a thermal source with a temperature of T . The total power to the pixel is P + P P . Right: TES bath γ J − G is biased by I = V /R , in the case of R R . For R R , B B B B ≫ SH ≫ SH the TES is bias by V = IBRSH, then fluctuations of R will result in fluctuation in current, which is read out by the inductor L and the superconducting quantum interference device (SQUID) amplifier. . . 114 4.30 Section view of model 104 Olympus ADR cryostat showing mechanical heat switch controller, vacuum valve, pulse tube (PT) head, 60 K plate, 4 K plate, adiabatic demagnetization refrigerator (ADR), high temp superconducting leads for 4 T magnet, thermal shielding, and vacuum jacket[21]...... 117 4.31 Left: The ADR and the He-4 refrigerator mounted on the 4 K plate of the HPD cryostat in the experimental cosmology lab at Johns Hopkins University. Photo courtesy of David Larson. Right: the rack-mounted devices for cryostat thermometry. From top to bottom, they are, a SRS SIM900 mainframe with 2 MUXs, a diode moniter and an AC bridge, a front panel, a NI GPIB to Ethernet adapter, a Lakeshore 370 AC resistance bridge and two Keithley 2440 current sources...... 119 4.32 Cryostat cool down curves. It takes about 24 hours for the cryostat to cool down to the state with stable temperature readouts. The typical values of the thermometers are listed in table 4.9...... 120

xvi 4.33 ADR cooling curves at 100 mK, showing the magnet current versus time of the ADR with the loads of from 2.0 to 10.0 W. Based on these curves, the FAA pill of the ADR have higher cooling capacities atlowerloads...... 121 4.34 The FLL block diagram for TES detector readout, showing the cold electronics inside the cryostat and the warm electronics (MCE) [22]. . 123 4.35 This photo shows the Multi-Channel Electronics (MCE) mounted on the wall the cryostat in the experimental cosmology lab at Johns Hop- kins University. The MCE is connected to a data-acquisition computer by a pair of ﬁber optic cables (the orange wires). Photo courtesy of DavidLarson...... 124

A.1 60 um polarization vectors from Stokes ([23], Yellow) and the 450 um result from SHARP (smoothed to 22′′ resolution, Red), center at h m s 18 17 32 ,-16◦14′25′′ (B1950.0)...... 126 A.2 100 um polarization vectors from Stokes ([23], Green) and the 450um h m s result from SHARP (smoothed to 35′′ resolution, Red), center at 18 17 32 ,- 16◦14′25′′ (B1950.0)...... 128 A.3 350 um polarization vectors from Hertz ([24]) and the 450 um result h m s from SHARP (smoothed to 20′′ resolution, Red), center at 18 17 32 ,- h m s 16◦14′25′′ (B1950.0). Blue: Hertz vectors at RA > 18 17 30 , Green: Hertz vectors at RA < 18h17m30s ...... 131

B.1 The Planck, Wien and Rayleigh-Jeans spectrum of a 2.725 K black body. The Wien limit is a good approximation at ν > 250 GHz and the Rayleigh-Jeans limit works well below 20 GHz...... 137

D.1 The initial, intermediate and final profiles are shown. All dimensions are given in units of the cuttoff wavelength of the input circular waveguide...... 145 D.2 (Top) The maximum cross-polar response across the band is shown for the three profiles in Figure D.1. Measurements of the maximum cross-polarization are superposed. (Bottom) The reflected power measurements for the final feed horn are shown plotted over the predicted reflected power for the initial, intermediate, and final feedhorn profiles. Frequency is given in units of the cutoff frequency of the input circular waveguide...... 146 D.3 A smooth-walled feedhorn operating between 33 and 45 GHz was constructed. The horn is 140 mm long with an aperture radius of 22 mm. The input circular waveguide radius is 3.334 mm...... 149

xvii D.4 The measured E-, H-, and diagonal-plane angular responses for the lower edge (33 GHz), center (39 GHz), and upper edge (45 GHz) of the optimization band are shown. The cross-polar patterns in the diagonal plane are shown in the bottom three panels for each of the threefrequencies...... 151 D.5 The maximum cross-polar response of the prototype feedhorn is compared to other implementations of smooth-walled feedhorns. The data presented have been normalized to the design center frequencies as speciﬁedbytherespectiveauthors...... 152

F.1 SRSreadoutprogramfrontpanel...... 160 F.2 PIDcontrolprogramfrontpanel...... 160 F.3 BlockdiagramoftheSRSreadoutprogram...... 161 F.4 Block diagramofthe PID control program. Part 1of3...... 162 F.5 Block diagramofthe PID control program. Part 2of3...... 163 F.6 Block diagramofthe PID control program. Part 3of3...... 164

xviii Part I

Polarimetry in Astrophysics

1 Chapter 1

Introduction to Polarization in Astrophysics

Astrophysicists are mostly limited to passively observing electromagnetic radiation from a distance. This radiation is most generally described by a speciﬁc intensity as a function of sky direction (θ, φ), frequency (ν) and polarization state. The polarization information is important for astronomy. Radiation from astronomical sources generally shows some degree of polarization. Although it is usually only a small fraction of the total radiation, the polarization component often carries a wealth of information on the physical state and geometry of the emitting object and intervening material. In the microwave part of the spectrum, polarization provides information about galactic magnetic ﬁelds and the physics of interstellar dust. The measurement of this polarized radiation is central to much modern astrophysical research.

1.1 Plane Wave

Polarization describes the orientation and phase coherence of the oscillations of electromagnetic waves. Speciﬁcally, the polarization of a wave is described by spec- ifying the orientation of the wave’s electric ﬁeld at a point in space. Polarization is most usefully illustrated using the concept of a plane wave, a monochromatic wave

2 having planar wave fronts that are inﬁnite in extent. Figure 1.1 shows a simple plane wave with its electric component parallel to the x axis. Generally, the electric ﬁeld of a plane wave can be written as:

E (r, t) = (E ,E ,E ) = (A cos(kz ωt + φ ),A cos(kz ωt + φ ), 0) (1.1) x y z x − x y − y where (Ax, Ay) and (φx, φy) are the amplitudes and phase offsets of the x and y component of the electric field; ω is the angular frequency; k is the wave number. In the x y plane, equation 1.1 can be simplified as: − E = A sin(ωt φ ) x x − x E = A sin(ωt φ ). (1.2) y y − y By defining φ = φ φ , equation 1.2 can be written into an elliptical form: x − y 2 2 Ex Ey ExEy 2 2 + 2 2 cosφ = sin φ. (1.3) Ax Ay − AxAy For different phase offsets, the polarization state varies. From equation 1.3, if φ = mπ (where m = 0, 1, 2, ...), then E /A E /A = 0 (linear polarization); if ± ± x x ± y y φ = (2m+ 1)π/2 and A = A , then E2 + E2 = A2 (circular polarization); if φ = mπ, x y x y x then it will be an elliptical polarization. In the latter cases (circular and elliptical polarization), the oscillations can rotate either towards the right (0 < φ < π) or towards the left ( π < φ < 0) in the direction of propagation. −

1.2 Stokes Parameters

The parameters A , A , φ above, used to describe polarization have diﬀerent { x y } units. In 1852, George G. Stokes deﬁned a set of 4 parameters (the Stokes parameters) as a mathematically convenient alternative. For the monochromatic plane wave described above, the Stokes parameters are:

2 2 I = Ax + Ay Q = A2 A2 x − y U = 2AxAycosφ

V = 2AxAysinφ (1.4)

3 y x B

Figure 1.1: A simple plane wave. The electric (E, in x-z plane) and magnetic ﬁeld (B, in y-z plane) is perpendicular to each other and to the direction of propagation (z).

where I is the intensity of the radiation; Q describes the horizontal and vertical

linear polarization components; U are the linear components with 45◦ angle and V represents the circular polarization components. Generally, the amplitude and phase offset of the radiation are time-dependent stochastic variables A (t),A (t), φ(t) and the observed radiation is a partially co- { x y } herent superposition of many waves. As a result, the Stokes parameters for a general radiation field are defined as averaged quantities over a period in time:

I = A2 (t) + A2(t) x y Q = A2 (t) A2(t) x − y U = 2 A (t)A (t)cosφ(t) x y V = 2 A (t)A (t)sinφ(t) (1.5) x y where angular brackets denote averaging over many wave cycles. Useful relation can be derived among the stokes parameters. For purely monochromatic (coherent) radiation I2 = Q2 + U 2 + V 2. (1.6)

4 y η E ξ

ψ x

Figure 1.2: Polarization ellipse. It shows the (ξ, η) coordinates with respect to the (x, y) coordinates and the deﬁnitions of orientation angle ψ, ellipticity angle χ.

For the partially-coherent radiation, the previous equation becomes an inequality

I2 Q2 + U 2 + V 2. (1.7) ≥ We can deﬁne a total polarization fraction (degree of polarization)

p = (Q2 + U 2 + V 2)1/2/I. (1.8)

Most sources of electromagnetic radiation contain a large number of emitters that are not necessarily correlated with each other either in phase or direction and emit over a limit bandwidth, in which case the light is said to be unpolarized (p = 0). If there is partial correlation between the emitters, the light is partially polarized (0 < p < 1). If the polarization is consistent across the bandwidth of detectors, partially polarized light can be described as a superposition of a completely unpolarized component, and a completely polarized one (p = 1). Another way to describe polarization is to use the polarization ellipse parameters, by giving the semi-major and semi-minor axes of the polarization ellipse, its orientation, and the sense of rotation (Figure 1.2). This method uses the orientation angle (ψ, the angle between the major semi-axis of the ellipse and the x-axis.) and ellipticity angle χ = arccot(ǫ), where ǫ is the ellipticity (the major-to-minor-axis ratio of

5 the ellipse).

We have a transform between (Eξ, Eη) and (Ex, Ey)

E = E cosψ E sinψ x ξ − η Ey = Eξsinψ + Eηcosψ (1.9) and

2 2 2 2 1/2 Ax = a0((cos χcos ψ + sin χsin ψ)

2 2 2 2 1/2 Ay = a0((cos χsin ψ + sin χcos ψ)

tanφx = tanχtanψ tanφ = tanχcotψ (1.10) y − where Eξ and Eη are the amplitudes of −→E along the semi-major and semi-minor axes, a0 is the average amplitude of −→E . From equation 1.2, equation 1.3, equation 1.9 and equation 1.10, we have

Eξ = a0cosχsinωt

Eη = a0sinχcosωt (1.11) and 2 2 Eξ Eη 2 2 + 2 2 = 1. (1.12) a0cos χ a0sin χ An ellipticity of zero (χ = π/2) or inﬁnity (χ = 0) corresponds to linear polarization and an ellipticity of 1 (χ = π/4) corresponds to circular polarization. The relation between Stokes parameters and polarization ellipse parameters is:

2 I = a0 2 Q = a0cos2χcos2ψ 2 U = a0cos2χsin2ψ 2 V = a0sin2χ (1.13) with the following inverse equations:

tan2ψ = U/Q sin2χ = V/(Q2 + U 2 + V 2)1/2. (1.14)

6 (V)

(U)

(Q)

Figure 1.3: Poincar´esphere, deﬁning the polarization in spherical coordinates. It also shows the relation between (Q, U, V ) and (Ip, χ, ψ) [1].

1.3 Poincar´eSphere

From equation 1.13, the polarization state can be described in spherical coordi- 2 nates, by replacing a0 with Ip:

Q = Ipcos2χcos2ψ

U = Ipcos2χsin2ψ

V = Ipsin2χ (1.15)

2 2 2 1/2 where Ip = (Q + U + V ) is the polarization intensity, 2χ and 2ψ are other two axes in the spherical coordinates. Equation 1.15 makes use of a convenient representation of the last three Stokes parameters as components in a three-dimensional vector space. The Poincar´esphere is the spherical surface occupied by polarization states having a constant polarization:

7 Q 1 S = U (1.16) Ip V    The Poincarésphere provides a convenient  way of representing polarization and representing how any given retarder (i.e. the VPM described in section 4.3) will change the polarization form. The north and south poles of the sphere represent left and right circular polarization (V ). The points on the equator correspond to linear polarization state (Q and U). Other points on the sphere represent elliptical polarizations. If an arbitrarily chosen point on the equator designates horizontal polarization, then the point which locates 180◦ opposite to it designates vertical polarization. A general point (Ip) on the surface of the Poincarésphere is specific in terms of the longitude (2ψ) and the latitude (2χ). The factor of 2 before ψ represents the fact that any polarization ellipse is indistinguishable from one rotated by 180◦, and the factor of 2 before χ indicates that an ellipse is indistinguishable from one with the semi-axis lengths swapped by a 90◦ rotation.

1.4 Polarization in Astrophysics

Many mechanisms generate polarized emission in astrophysics, including synchrotron emission, dust emission, absorption (extinction) and scattering, such as starlight polarization and free-free (bremsstrahlung) emission from cloud edges. Ad- ditional polarized components like the anomalous emission from dust have also been discovered.

1.4.1 Synchrotron Emission

Synchrotron emission arises from the acceleration of cosmic-ray electrons in magnetic ﬁelds. Based on the results of Cosmic Microwave Background (CMB) foreground studies [2] (Figure 1.4), synchrotron radiation dominates at frequencies below 60 GHz ( 5 mm). ≥

8 K K a Q V W

8 5% K, rms) 100 S CMB Anisotropy Synchrotronky (K p Free-free 7 2) 7% S ky Dust (K p0) 10

Antenna Temperature ( 1 20 40 60 80 100 200 Frequency (GHz)

Figure 1.4: CMB foreground radiation in WMAP bands [2]. The synchrotron radiation dominates the low frequency range below 60 GHz. Radiation from dust contributes mostly above 70 GHz.

If the energy spectrum of cosmic-ray electrons can be expressed as a power-law distribution: γ N(E) E− (1.17) ∝ where γ is the electron power-law index, then the synchrotron ﬂux density spectral index (α) and synchrotron emission spectral index (β) are related to γ, by:

γ 1 α = − − 2 γ + 3 β = (1.18) − 2 and we have the flux density S(ν) να and the brightness temperature T (ν) νβ. ∝ ∝ Assuming N(E) and a uniform magnetic field, the resulting emission is strongly polarized with fractional linear polarization: γ + 1 p = (1.19) syn γ + 7/3 and aligned perpendicularly to the magnetic field [25]. At microwave frequencies, the synchrotron emission spectral index observed is β 3 [26], so that synchrotron ≈ − 9 emission could have fractional polarization as high as psyn = 75% (equation 1.18 and equation 1.19), which is almost never observed. The main reason for that is the magnetic field distribution and the electron energy distribution are not uniform in the Galaxy. The line of sight and beam averaging effects reduce the observed polarization fraction by averaging over different regions in the Galaxy. At low frequencies (below a few GHz) Faraday rotation ( λ2) will also reduce the polarization fraction for a ∝ sufficiently wide passband.

1.4.2 Thermal Dust Emission and Absorption

The dominant source of Galactic emission at far-infrared (far-IR) and submillimeter (SMM) wavelengths (100 GHz - 6000 GHz) is thermal emission from interstellar dust grains at temperatures of 10 - 100 K. The spectrum of this radiation is generally modelled with one or more thermal components with diﬀerent temperatures by a frequency dependent emission:

n βi I(ν) = Aiν Bν(Ti) (1.20) i=1 Where ν is frequency, n is the total number of thermal components, Ai, νi and Ti are the coeﬃcient, spectral index and temperature of component i, Bν is the Planck blackbody function (equation B.1). Multiple temperatures and spectral indices are often needed to model the intensity spectrum at any single point on the sky. For example, The Galactic dust emission has been modelled by a two temperature component model of T1 = 9.5 K with β1 = 1.7 and T2 = 16 K with β2 = 2.7 [27].

Dust Grain Alignment

The radiation from the dust grains that have been aligned by interstellar magnetic ﬁelds is partially polarized. The alignment requires: (1) The small axis (symmetry axis) with the largest moment of inertia of the grain to be aligned with the spin axis; (2) The spin axis is then aligned with the local magnetic ﬁeld [28, 29, 30, 31, 32].

10 (1) Internal Dissipation Consider a dust grain with rotational energy of 1 E = (I Ω2 + I Ω2 + I Ω2) (1.21) rot 2 x x y y z z where Ix < Iy < Iz are the principal axes of inertia and Ωx, Ωy, Ωz are the angular velocities. Such a dust grain has an angular momentum,

2 2 2 2 2 2 1/2 J = (IxΩx + Iy Ωy + Iz Ωz) (1.22)

Suppose the total angular velocity Ω is not parallel to any of the principal axes, then periodic motions will be executed with respect to these axes, which will mechani- cally stress the grain by the alternating centrifugal forces. As a result, heat will be generated at the expense of Erot. Since J will not change (conservation of angular 2 2 momentum), this requires an increase in the time-average value of Ωz relative to Ωy 2 2 2 2 2 2 2 (or Ωy to Ωx). The dissipation will not stop until Ωx = Ωy = 0 and Ωz = J /Iz . The internal dissipation of the rotational energy in a free rotator forces the angular velocity Ω toward the axis with the largest moment of inertia Ωz [33].

(2) Barnett Dissipation In 1915, Barnett found the magnetization of an un- charged body when spun on its axis [34]. A paramagnetic or ferromagnetic body rotating freely will develop spontaneously a magnetic moment M parallel to the axis of rotation (Barnett Eﬀect): M = χΩ/γ (1.23) where Ω is the angular velocity, χ is the magnetic susceptibility and γ is the gyromag- netic ratio for the material. The Barnett eﬀect can be explained by considering that some of the angular momentum is transferred to the unpaired electrons thus aligning the magnetic moments. In the case of a dust grain, if the initial Ω is not parallel to any principal axis, it will precess in the grain coordinates. The magnetic moment will lag behind the precession, which will cause a dissipation (Barnett Dissipation)

of the rotational energy Erot. As a result, Ω will become parallel to Ωz and the local magnetic ﬁeld. There is a balance between the alignment of the symmetry and spin axis of dust grains with magnetic ﬁeld and the collisions between the grains and gas molecules.

11 In order for the dust grains to become aligned, the time scale of the alignment must be shorter than the time scale of the damping of collision. This condition is satisfied if the grains are rotating supra thermally, E kT . The torques produced by the rot ≫ formation and subsequent ejection of H2 molecules from grain surfaces could spin up the grain to the necessary speeds [35, 33]. Photons can also provide the necessary torques to spin up the grain [36, 37, 38, 39]. It has been suggested by observation that photons can produce a net torque on irregularly shaped grains because they present different cross sections to right- and left-hand circularly polarized photons [30]. Modern grain alignment theory favors radiative torques over H2 torques as the mechanism by which grains achieve high angular velocities and align with magnetic fields. The angular momentum of a grain may flip suddenly because of thermal fluctuations. One reason for this is that the H2 torques will change direction when the spin vector flips, causing the grain to spin- down [40, 41]. Due to these “thermal flipping” and “thermal trapping” effects, grains smaller than 1 m cannot reach supra thermal velocities [42]. However, this is not the case for radiative torques because the helicity of a grain does not depend on its orientation. While other alignment mechanisms may dominate in some select environments [43], the above mechanism is favored in conditions prevalent throughout most of the interstellar medium (ISM). The result of this mechanism is to align the grains with the longest axis perpendicular to the magnetic field. Since the grains will emit, and absorb, most efficiently along the long grain axis, polarization is observed perpendicular to the magnetic field in emission, but parallel to the field in absorption (extinction).

Polarization by Emission from Elongated Dust

The polarization of radiation emitted from dust grains is parallel to the long axis of the grain and perpendicular to the aligning magnetic ﬁeld. The lower limit on the column densities of the clouds that can be traced by emission polarimetry is set by the earth atmosphere absorption and instrument sensitivity. In some dense clouds, which the interstellar radiation cannot penetrate deeply into, the embedded stars can

12 still provide the necessary radiative torques to spin up the grains [44].

Polarization by Absorption from Elongated Dust

Polarization of starlight from ultraviolet to near-infrared (NIR) wavelengths is mostly due to selective extinction by grains that have been aligned by a local magnetic field [28]. The polarization will be parallel to the magnetic field, since starlight is preferentially absorbed along the long axis of the grain. Observations of starlight polarization have proven to be a useful tool for tracing the magnetic field structure in diffuse ISM regions [45, 46]. However, at high extinctions, photons are completely absorbed. Even at moderate extinctions, polarization by absorption is not a reliable tracer of the magnetic field due to the drop in grain alignment efficiency [47]. Po- larization by absorption cannot be used to reliably trace magnetic field structure in

regions where the extinction (AV ) is greater than 1.3 [48].

1.4.3 Examples of Polarization from Absorption and Scat- tering

Starlight Polarization

The polarization of starlight was ﬁrst observed by [49] and [50]. As concluded in the last section, starlight polarization is only measureable in regions of low extinction

(AV less than a few magnitudes for near-infrared observations), where near-visible photons can traverse the ISM. This makes it a feasible tool for inferring the Galactic magnetic ﬁeld. The extinction places a limit on the most distant stars for which polarization can be observed. At high Galactic latitude, most stars observed with polarization are within 1 kpc of the Sun. While at low latitude, this distance extends to as far as 2 kpc [45, 51]. Figure 1.5 shows an analysis [51] using the data from [45]. The low latitude stars have higher polarization fraction (p 1.7%) and extinctions (E(B V ) 0.5 ≈ − ≈ mag), while the high latitude stars have signiﬁcantly lower values (p 0.5% and ≈ E(B V ) 0.15 mag). There is a strong alignment of net starlight polarization − ≈

13 Figure 1.5: Starlight polarization vectors in Galactic coordinates. The upper panel shows polarization vectors in local clouds. The polarization averaged over many clouds in the Galactic plane is shown in the lower panel. The magnetic ﬁeld is parallel to the polarization angle. vectors with the Galactic plane (see the lower panel).

Free-free Emission from Cloud Edges

Free-free (Bremsstrahlung) emission is due to electron-electron scattering from ionized gas (with T 104 K) in the ISM. At frequencies higher than 10 GHz, the ≈ free-free thermal emission has a spectrum of T νβ, with β = -2.15 [2]. ∼ The free-free emission is intrinsically unpolarized because of the randomization of scattering directions. However, at the edges of bright free-free features (i.e. HII regions) a secondary polarization signature can occur as a result of anisotropic Thomson scattering [25, 52]. This could cause signiﬁcant polarization ( 10%) in the Galactic ≈ plane at high angular resolution. However, at high Galactic latitudes, and with a low resolution, the residual polarization is expected to be < 1%.

14 1.4.4 Anomalous Dust Emission

There are additional dust emission mechanisms that could produce a low level of polarized emission. Some studies at high Galactic latitude [53, 54, 55, 56] and individual Galactic clouds [57, 58], have observed unexpected emission in excess of that from the three components discussed above (synchrotron, thermal dust, and free- free emission). This emission has been termed “anomalous” for the reason that its provenance was not completely understood at this time. Some studies [57, 59, 60, 61] show that this emission is correlated with large-scale maps of far infrared emission from thermal dust. There are two main hypotheses for the anomalous emission. The first mechanism is the spinning dust model: small ( 1 nm), rapidly rotating dust grains emit electric ≈ dipole radiation at microwave frequencies [62, 63, 64]. The second is the vibrating magnetic dust model: large ( 100 nm), thermally vibrating grains undergoing ≥ fluctuations in their magnetization will emit magnetic dipole radiation at microwave frequencies [65]. The spinning dust model is favored by some observations [66, 67]. However, emission from vibrating magnetic dust should exist at some level, because large grains are known to exist from observed emission in the far infrared, and contain ferromagnetic material [68, 69]. This is important for polarization observations as magnetic dust is predicted to be better aligned to the magnetic fields than the spinning dust. The spinning dusts aligned by paramagnetic dissipation [28] emit polarized radiation. Theory predicts the polarization from spinning dust peaks at about 2 GHz ( 7%) and falls below 0.5% above 30 GHz [70]. Observations [71, 72] suggest that ≈ the spinning dust grains are inefficiently aligned and will produce little polarization at any frequency. There is evidence that the vibrating magnetic grains are well aligned with the magnetic field. Theory predicts a maximum polarization fraction to be 40% [65] with the polarization angle flipping within the 1 - 100 GHz range. The po- ∼ larization is perpendicular to the magnetic field at higher frequencies, but parallel to the field at lower frequencies.

15 Chapter 2

Submillimeter Polarimetry of M17

In this chapter, I present the data analysis process of 450 m polarization observations of the M17 molecular cloud from the Caltech Submillimeter Observatory (CSO) and discuss the physics of the cloud that we learn from the submillimeter polarimetry.

2.1 Introduction to Submillimter Polarimetry

Although it is possible to measure polarized thermal emission of aligned grains from mid-IR to millimeter wavelengths [73, 23], for a blackbody spectrum, the peak of the thermal emission spectrum of a typical molecular cloud (with a temperature of about 10 K [74]) falls in the submillimeter band (see appendix B). Thus, the submillimeter waveband is a very important window for studying the physics of these interstellar medium. Submillimeter polarimetry provides one of the best methods for mapping interstellar magnetic fields in star forming regions and other interstellar clouds [75]. Magnetic fields are believed to play an important role in the support and evolution of molecular clouds via the magnetic flux freezing effect [76]. The way in which polarization data traces the magnetic field is described in section 1.4.2. Basically, the magnetically aligned interstellar dust grains emit partially polarized thermal radiation. The direction of polarization gives the orientation of the interstellar magnetic field, as projected onto the plane of the sky (B ). ⊥

16 2.2 Polarimetry at Caltech Submillimeter Obser- vatory

The earliest detections of far-IR/submillimeter polarization in astronomical objects were obtained during the 1980s using single-pixel polarimeters from balloons [77] and aircraft [78]. In the 1990s, astronomers developed more powerful polarimeters with tens of pixels, such as Stokes [79] for the Kuiper Airborne Observatory (KAO), SCU-POL [80, 81] for the James Clerk Maxwell Telescope (JCMT) and Hertz [82] for the CSO. Since 2006, SHARP [83] has served as a new polarimeter for the CSO. The CSO is one of the world’s premier submillimeter telescopes on Mauna Kea. It consists of a 10.4 meter diameter dish with a root-mean-square (rms) surface error of about 20 m [84] and an active optics system [85]. The superconductor-insulator- superconductor (SIS) receivers [86] of the CSO are available from 180 to 720 GHz atmospheric windows with the performance close to the theoretical limit given by “Quantum Noise” [87]. Submillimeter High Angular Resolution Camera II (SHARC II) [88] is a background- limited “CCD-style” bolometer array with 12 32 semiconducting bolometric detec- × tors. As a facility camera for the CSO, SHARC II operates at 350 m and 450 m wavebands. In the best 25% of winter nights on Mauna Kea (with τ 0.05), 225 GHz ≈ SHARC II is expected to have a noise equivalent ﬂux density (NEFD 1) at 350 m of 1 Jy s1/2 or better (equation 2.1 [3] and ﬁgure 2.1).

NEFD = 1.0 exp(25.0 τ airmass 1.6) Jy s1/2. (2.1) 350 m × × 225 GHz × − SHARP [4] is a foreoptics module that converts the SHARC II camera into a sensitive dual-beam 12 12 pixel imaging polarimeter at wavelengths of 350 and × 450 m. It splits the incident radiation into two orthogonally polarized beams that are then reimaged onto 12 12 subarrays at opposite ends of the 32 12 array in × × SHARC II. The polarization signal is modulated by a warm rotating half wave plate (HWP) at front of the polarization-splitting optics.

1NEFD is deﬁned as the level of ﬂux density required to obtain a unity signal to noise ratio in 1

17 Figure 2.1: NEFD350 m measurements (points) from Jan 2003 compared to theoretical expectation (solid line) from equation 2.1 [3]. The performance is about 1 Jy s1/2 for τ225 GHz = 0.05.

Figure 2.2 shows the optics of SHARP. The submillimeter light beams from the focus of the CSO telescope enter SHARP from the left, and are relayed through an optical path including flat and curved mirrors and polarizing wire grids. The radiation then passes the M4 mirror and enters the SHARC II camera. The key idea of the design is to reconstitute the image with an offset between the two orthogonal linear polarization components. SHARC II can be easily converted back to photometric mode by removing mirror P1 and F5 in figure 2.2. The SHARP instrument specification is listed in table 2.1 [89]. With a resolution of about 5 arc seconds, high sensitivity and low systematic errors, SHARP is a powerful tool for submillimeter polarimetry. At present, SHARP and the submillimeter array (SMA) are the only two instruments with submillimeter polarimetric capabilities that are in service. The SMA is a interferometer consisting of 8 six-meter dishes focusing on high resolution on small scales. In addition, BLAST-pol, a successor to balloon-borne large-aperture submillimeter telescope (BLAST [90]), has had its first flight over Antarctica, and the data obtained at 250, 350 and 500 m are being reduced and analyzed. In the future, the second of integration with the detector. See secton 4.2.1 for the definitions of NEP, NET and NEQ.

18 Figure 2.2: The polarization splitting optics of SHARP [4] for reconstituting the image with an offset between the two polarization components. Left: The expanding beam from the CSO focus is reflected by P1 (paraboloid), F1 (flat mirror), through the HWP (half wave plate), and reaches the XG (crossed grid), where the polarization radiation is separated into two orthogonal (horizontal and vertical) components. Right: View toward the CSO focus. The vertical and horizontal components undergo further reflections by a series of mirrors and grids, and are displaced laterally at the BC (beam combiner), before being directed toward SHARC II.

Table 2.1: SHARP Instrument Speciﬁcations

λ0 (m) 350 450 Bandwidth (∆λ/λ0) 0.13 0.10 FOV (arc sec arc sec) 55 55 55 55 × × × Pixel Size (arc sec arc sec) 4.6 4.6 4.6 4.6 × × × Angular Resolution (arc sec) 9.0 11.0 FOV (arc sec arc sec) 55 55 55 55 × × × Point Source Flux for (σp = 1%) in 5 Hours (Jy) 3.6 2.0 Surface Brightness for (σp = 1%) in 5 Hours (Jy/pixel) 0.63 0.35 Max Separation of Main and Reference Beams (arc min) 5.0 5.0 Systematic Errors, σp (sys) < 0.2% < 0.2%

19 H/V rgm R Gain I De- ChoppedC A modulaon Data Q W U

Figure 2.3: Flow chart of “SharpInteg”. It starts by masking the raw data file with an “rgm” file. Then, it demodulates the chopping to calculate the chopped data. After applying the relative data gain factor between the horizontal and vertical array, it calculates the I, I-error, Q, Q-error, U and U-error components and saves them into a new file.

SCUBA-2 [91] instrument being commissioned at the JCMT also has a polarimeter, POL-2, and the ALMA interferometer should also have polarimetric capabilities at multiple submillimeter/millimeter wavelengths.

2.3 SHARP Data Pipeline

There are two data processing programs for SHARP pipeline: “SharpInteg” and “Sharpcombine”. “SharpInteg” is a program that takes a cycle of half wave plate measurement from SHARP and process it to for I, Q and U along with the corresponding errors. “Sharpcombine” is for map combining and smoothing. As shown in figure 2.3, “SharpInteg” first reads in the SHARC II raw data file and apply a pixel mask to it from a pixel mask file (“rgm” file). After that, the chopping is demodulated, and the data at different chop/nod positions is given a weight equals to the number of samples at that position. The chopped data is calculated by summing the weighted raw data within each sampling period. In the next step, the relative data

20 I I I I I B Q Q Q I.P. Q Q S Rot U U U S U U

I I I I I B I Q Q Q I.P. Q Q S Rot U U U S U U

I I I I I B Q Q Q I.P. Q Q S Rot U U U S U U

Figure 2.4: Flow chart of “Sharpcombine”. It applies τ and telescope pointing correction, background subtraction (BS), instrument polarization (I.P.) subtraction and polarization angle rotation to sky coordinates (Rot) to each sub-map before it combines them into a large map and smooths it. gain factor (f) between the horizontal (“H”) and vertical (“V”) array is calculated by taking all of the samples from a particular HWP position and fitting to the line of “V = a H + b” using numerical method. After this is done for all HWP positions, the median value is taken and f is set to the inverse value of the median. The “H” and “V” array samples are combined after calculating the f value. Finally, The I, I-error, Q, Q-error, U and U-error maps are calculated and saved to a FITS file. “Sharpcombine” starts with the output FITS files from “SharpInteg” containing the I, Q and U map. Each of them represents a small map to be combined to a large map. In the first step, it applies τ (atmospheric optical depth) and telescope pointing

21 Figure 2.5: M17 is a premier example of a young, massive star formation region in the Galaxy. Left: A M17 image from my 80 mm aperture optical telescope. Right: A false color image from Spitzer GLIMPSE (red: 5.8 um; green: 4.5 um; blue: 3.6 um.) [5].

corrections to the small maps. After that, it applies background subtraction (BS) to I, Q and U data, and instrument polarization (I.P.) subtraction to the Q and U data in each map. All the maps are rotated to the sky direction (Rot) before being combined to a big map. Finally, the I, Q and U maps are combined to a large map and smoothed by interpolation (see ﬁgure 2.4).

2.4 Introduction to M17

M17, the Omega Nebula, locating at the constellation Sagittarius with (l, b) = (15.05, -0.67), is a premier example of a young, massive star formation region in the Galaxy. It is one of the brightest IR and thermal radio sources in the sky. The distance of the M17 is measured to be 1.6 0.3 kpc [92]. It covers an area of about ± 11 arc min 9 arc min across the sky (ﬁgure 2.5). × A global shell structure geometric model of M17 is presented by [6]. In the in- ner part of the nebula, a bright, photoionized region with a hollow conical shape surrounds a central star cluster. This region is about 2 pc across and expanding westward into the outer molecular cloud. There is a large, unobscured optical HII region spreading into the low density medium at the eastern edge of the molecular

22 cloud. Gas photoexcited by the early OB stars is concentrated in the northern and southern bars. X-ray observations [93, 94] indicate that the region interior to the HII region is filled by hot (106 - 107 K) gas, which is flowing out to the east. [93] noted that this region is too young to have produced a supernova remnant and interpret the X-ray emission as hot gas filling a super bubble blown by the OB star winds. In the middle of the nebula, velocity studies show an ionized shell with a diameter of about 6 pc. On the western side of the outer part, all tracers of warm and hot gas are truncated by a wall of dense, cold molecular material which includes the dense cores known as “M17 Southwest” and “M17 North”, which exhibit many other tracers of current massive star formation. At this region, Only the most massive members of the young NGC6618 stellar cluster [95] exciting the nebula have been characterized, due to the comparatively high extinction. Figure 2.6 shows a simple M17 model. We can represent the system as a central + 0 cluster of stars surrounded by successive layers of H ,H , and H2 gas to the SW side and by a background sheet of ionized and neutral gas wrapping around to the NE.

2.5 M17 Polarimetry Results

2.5.1 General Results

Our M17 map from the SHARP 450 m observation, is centered at 18h17m32.0s, h m s 16◦14′25.0′′ (B1950) or 18 20 25.2 , 16◦13′02.1′′ (J2000). It covers an area of − − about 4′25′′ 2′45′′ at the SW bar of M17 (figure 2.10). Taking the distance to M17 × to be about 1.6 kpc (section 2.4), our map coverage is equivalent to an area of 2.05 pc 1.28 pc. × M17 polarization vectors are plotted in figure 2.7 and a table of the vectors is listed in appendix A.5. As we can see in figure 2.7, for regions of high submillimeter flux, the average polarization fraction is lower than that in low flux regions. This is caused by the line of sight (LOS) effect: assuming the polarization angles at different distances along the los to be variable, the measured polarization fraction trends to

23 Figure 2.6: A M17 model from [6]. The system can be described as a central cluster + 0 of stars surrounded by successive layers of H ,H , and H2 gas, that expanding with diﬀerent velocities to the outer side of the cloud.

become diluted upon integration along the LOS. The magnetic ﬁeld projected onto the plane of the sky can be approximated by

rotating the polarization vectors by 90◦ (section 1.4.2). Our results for the magnetic ﬁeld direction are in good agreement with the those of previous observations at far-IR (Stokes, 60 and 100 m) [7, 23] and submillimeter (Hertz, 350 m) [24] wavebands, but with much higher resolution (ﬁgure 2.10). Figure 2.8 shows the distribution of polarization fraction of the measurements. The average polarization fraction is about 2.4%, a typical number for magnetically aligned molecular clouds. The mean polarization angle (from north to east) is about

5.0◦ (figure 2.9), which gives an average magnetic field almost parallel to the RA − direction. Figure 2.10 shows the magnetic field distribution from 100 um [7], 450 um (SHARP) and optical observations [8]. The 8.00 m Spitzer GLIMPSE flux map are mostly due to the polycyclic aromatic hydrocarbons (PAHs) molecular emission. The magnetic field follows the molecular cloud and the curvature of the HII region.

24 Figure 2.7: M17 polarization fraction vectors are plotted over the 450 um uncalibrated flux map. Thick vectors are detected with greater than or equal to 3σ level and thin vectors are between 2σ and 3σ level. The circle on the bottom right shows the SHARP beamsize. Some parts of the flux map is removed due to high noise levels. Offsets are h m s from 18 17 32 , -16◦14′25′′ (B1950.0).

25 Median = 1.90, Mean = 2.36, Std = 1.81

40 Number 30

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Polarization (%)

Figure 2.8: Histogram of M17 polarization fraction. This distribution includes all vectors at greater or equal to than 2σ level. All vectors greater than 10% are 2σ vectors.

Median = -4.00, Mean = -5.02, Std = 30.39 40

Number 15

90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80

Polarization Angle (degree)

Figure 2.9: Histogram of M17 polarization angle. Polarization angles are measured from north to east. The resulting net magnetic ﬁeld is almost parallel to the RA direction.

26 The center OB type stars heat the HII region and carve a hollow conical shape into the molecular cloud and separating it into two parts: the M17 SW and the M17 N. It is found that PAHs are destroyed over a short distance at the photodissociation region (PDR) around the edge of the HII bubble [5].

2.5.2 Polarization Spectrum

There are several instruments that contribute multiwavelength polarimetric data from far-IR (Stokes) to submillimeter (Hertz, SHARP, SCU-POL). If the source of the polarized emission is a single population of dust grains with similar polarization properties and temperature, then one expects the magnitude of the polarization (polarization fraction) to be nearly independent of wavelength higher than 50 m [96, 97]. The far-IR to submillimeter polarization spectrum of various molecular clouds have been studied by observations [97, 98, 99, 100] and simulations [101, 102]. The

polarization spectrum of M17 at 60 um (Stokes, 22′′ resolution), 100 um (Stokes, 35′′

resolution) and 350 um (Hertz, 20′′ resolution) had been studied by [97, 98] and their results are shown in ﬁgure 2.12. It has been found that the spectra are falling from far-IR to about 350 m and rising towards longer wavelengths. The analysis presented here incorporate the 450 m SHARP polarimetric data

(about 10′′ resolution). The polarization data points that are to be compared between two wavelengths are chosen based on the following criteria [97]: (1) The vectors are in the same region of the same cloud; (2) The difference between the polarization angle must be within 10◦; (3) The vectors are from the cloud envelope; (4) All vectors are greater or equal to 3σ level. Applying the above criterion, the surviving vectors at 60 m to 450 m are plotted in figure 2.11. They share a common area (marked by a green shadow) between h m s h m s 18 17 30 and 18 17 37 in Ra (B1950), 16◦16′20′′ and 16◦13′00′′ in Dec (B1950). − − A summary of the result is presented in table 2.2 and the details can be found in appendix A. The M17 polarization spectrum from 60 m to 450 m is plotted in figure 2.12. Our basic result is P450 < P350 < P100 < P60. In contrast to the

27 Figure 2.10: Magnetic field vectors from SHARP (red, 450 um), Stokes (green,100 um) [7] and optical observation [8] (purple) plotted on top of Spitzer GLIMPSE 8.00 um flux map. The magnetic vectors from SHARP and Stokes are perpendicular to their polarization angles, while those from optical polarization measurement are parallel to their polarization angles. All magnetic vectors (plotted with the same length) are h m s used to indicate the direction only. Offsets are from 18 17 32 , -16◦14′25′′ (B1950.0).

28 Figure 2.11: The common area (green shadow) for polarization spectrum analysis. It h m s h m s is between 18 17 30 and 18 17 37 in Ra (B1950), 16◦16′20′′ and 16◦13′00′′ in Dec (B1950). The selected polarization vectors are at 60−m (yellow), 100− m (green), 350 m (blue) and 450 m (red). Background is the 450 m ﬂux map. Oﬀsets are h m s from 18 17 32 , -16◦14′25′′ (B1950.0).

29 Table 2.2: M17 Polarization Spectrum Data Ratio Points Median Mean Std Note P450/P60 13 0.390 0.395 0.056 see appendix A.1 for details P450/P100 11 0.520 0.525 0.128 see appendix A.2 for details P450/P350 22 0.795 0.887 0.289 see appendix A.3 for details

results from other clouds, our work shows that, in the common area, the M17 has lower median polarization at 450 m than at 350 m. The polarization spectrum falls monotonically from 60 m to 450 m. There are many models to explain the rising (or falling) of the polarization spectrum from far-IR to submillimeter wavelength. Generally speaking, the radiation environment plays an important role in forming the polarization spectrum, since the grain alignment eﬃciency is dependent on radiative torques (section 1.4.2). In a weak radiation ﬁeld, the polarization spectrum normally has a positive slope (towards long wavelengths) [101]. That is what we observed from many clouds from 350 m to 450

m (ﬁgure 2.12). Our result of a negative slope (P450 < P350) from the east part of the cloud indicates the existence of a strong radiation ﬁeld from that direction.

2.5.3 Spatial Distribution of Magnetic Field and Polarization Spectrum

As already shown in ﬁgure 2.10, the center OB type stars in M17 heat the HII region and carve a hollow conical shape into the molecular cloud. Our analysis shows that this shock front is passing through our sampled region. Figure 2.13 shows 450 m magnetic vectors plotted over the [21 cm]/[450 m] ratio map (with the peak normalized to 1.0). The shock is following the “-Y” direction. We can separate the cloud into three regions: “post-shocked”, “shock front” and “pre-shocked” by the “y = 0” and “y = -50 arcsec” lines. The “post-shocked” region has consistent polarization angles with an average of

about 18◦; in the “shock front” region, where the magnetic ﬁeld is being distorted, the

4 W51 M17 OMC−1 NGC2024 3 DR21 OMC−3 This work

2 Median Polarization Ratio

40 100 200 400 1000 2000 Wavelength (um)

Figure 2.12: Polarization spectrum of some popular interstellar molecular clouds [9]. The median polarization ratio are normalize by the value at 350 m. In contrast to the results from other clouds, our work shows that, the M17 has lower median polarization at 450 m than at 350 m. The polarization spectrum falls monotonically from 60 m to 450 m.

31 polarization angle distribution spreads out in a large range from -40◦ to 60◦ and the polarization fractions become small; in the “pre-shocked” region, polarization angles become consistent again and the polarization fraction is much higher before being destroyed by the shock. The distributions described above are shown in figure 2.14 and 2.15, which show that the polarization angles and fractions are correlated with the “Y” axis in a linear and a “U” shape relationship. In the dense region of the cloud, the magnetic fields survive the windswept of the shock. There is an example in figure 2.13 (see the region marked by a blue box). The magnetic field in the dense cloud is different from other vectors in the “post-shocked” region. This cloud core can also be seen in figure 2.7, 2.10 and 2.11. Figure 2.16 shows the magnetic field vectors (red) and intensity contours of SHARP (green, levels = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 ) over plotted on the 21 cm absorption-line { } contour and the ratio of neutral HI (NHI) column density to the spin temperature

Tspin distribution map in the 17.5-22 km/s velocity area from [6]. The expanding HI region (see ﬁgure 2.6) is corresponding to the “post-shocked” and part of “shock

front” region. The NHI/Tspin density is low at the dense cloud region (see ﬁgure 2.13),

which is still dominated by the H2 molecular cloud. One conclusion from the above discussion on the polarization spectrum is at the

common region, P450 is smaller than P350. But in other parts of the cloud, we found that P450 is greater than P350 (figure A.3 and appendix A.4). In figure 2.17, the [450 m]/[350 m] polarization ratio vectors are over plotted on the [21 cm]/[M17 450um] flux ratio map. As we can see, most of the blue vectors are within the common region defined in figure 2.11. In spite of some red vectors distributed round the dense cloud region, the contour line with the [21 cm]/[450 m] = 0.1 separates the blue and red vectors into two regions. Figure 2.18 also agrees with this conclusion: the blue vectors are mostly related to the east “post-shocked” region, where the star radiation field is strong, while the red vectors are related to the molecular region with weak radiation fields.

32 Dec

y = 0 y = -50

Y Dense Cloud Ra

66.3o

Figure 2.13: Magnetic vectors from SHARP plotted over the [21 cm]/[450 m] flux ratio map, showing that the shock front is passing through the cloud. The contour levels are 0.1, 0.3, 0.5, 0.7, 0.9 . The “X” axis is defined by fitting contour level { } = 0.1. The new “X-Y” coordinate system is about 66.3◦ with respect to the “Ra- Dec” coordinates. The shock is following the “-Y” direction. The “y=0” and “y=-50 arcsec” lines separate the cloud into “post-shocked” (y > 0), “shock front” (-50 < y < 0) and “pre-shocked” (y < -50) regions. The polarization directions and magnitudes in these regions are different (figure 2.14 and 2.15). The magnetic fields in the dense cloud (can also be seen in figure 2.10) at the top of the map survive the windswept. h m s Offsets are from 18 17 32 , -16◦14′25′′ (B1950.0).

33 100 60um 80 100um 350um 60 450um

(degree) £

¡ ¡ 40

60 ¡

¡ 150 100 50 0 50 100 ¡ ¡ ¡

Y (arcsec) ¢

Figure 2.14: Correlation between polarization angle and the Y direction (zero at h m s 18 17 32 , 16◦14′25′′), showing a linear relationship. The “post-shocked” region is at y > 0 and− the “pre-shocked” region is at y < 50 arcsec. −

12 60um 100um 10 350um 450um

6 p ( % )

150 100 50 0 50 100 ¤ ¤ ¤

Y (arcsec) ¥

Figure 2.15: Correlation between polarization fraction and Y direction (zero at h m s 18 17 32 , 16◦14′25′′), showing a “U” like shape. The polarization fraction is higher at the “post-shocked”− region at y > 0 and the “pre-shocked” region at y < 50 arcsec. −

34 Figure 2.16: Magnetic ﬁeld vectors (red) and intensity contours of SHARP (green, levels = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 ) are over plotted on the 21 cm absorption-line contour and{ the ratio of neutral HI} (NHI) column density to the spin temperature Tspin distribution map in the 17.5-22 km/s velocity area from [6]. This velocity component is correlated with the “post-shocked” and part of “shock front” region. The NHI/Tspin density at the dense cloud region (see ﬁgure 2.13) is low.

35 Figure 2.17: The [450 m]/[350 m] polarization ratio vectors over plotted on the [21 cm]/[450 m] ﬂux ratio map with contour levels = 0.1, 0.3, 0.5, 0.7, 0.9 . The { } blue (red) vectors represent P450 < (>) P350. The length of the 2% bar at bottom left is equivalent to P450/P350 = 1.0. The directions of the vectors are parallel to their h m s polarization angles. Oﬀsets are from 18 17 32 , -16◦14′25′′ (B1950.0).

36 Figure 2.18: The [450 m]/[350 m] polarization ratio vectors and 450 m intensity contours of SHARP (green, levels = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 ) over plotted on the Fig.1 from [7]. The blue vectors is found{ to be correlated with} the [OI] line, which is a tracer for the atomic gas.

37 2.5.4 Conclusion

The combination of multi wavelength study and the polarimetric data from far-IR to submillimeter reveal the violent physical process in the M17 cloud. At large scale, the young OB type stars in the center of the cloud heat the HII region up to 106 - 107 K and create a high energy fountain towards the southeast direction. The HII wind push the HI and H2 regions outwards, creating a hollow conical shape into the cloud. The magnetic field is found following the curve of the HII region. On small scales, within our field of view, the shock is passing through the boundary between the HI and H2 region. There are significant differences between the dust alignment before and after the shock. The polarization properties and temperature of the dust population are also changed by the wind. Our study also shows that a high density molecular clump is being blasted by the out-going wind, while the magnetic field distribution in the area remains unchanged. It is the first time to observe the variance of polarization ratios across a molecular cloud. There are still no models that can explain this result. One conclusion we can make is that the grains in the cloud are not always aligned with the magnetic field perfectly, and any model trying to explain the polarization spectrum should take into account the variance of interstellar physical condition along the line of sight.

38 Part II

Polarimetry in Cosmology

39 Chapter 3

Introduction to Polarization in Cosmology

3.1 The Big Bang Theory

The Big Bang theory describes the evolution of our universe. It posits that our universe started from a hot and dense phase about 13.75 billion years [14] ago and the content of the universe kept evolving after that. In inflation theory, the universe was dominated by an energy field with a negative pressure, which drove an early period of accelerated expansion. It was then dominated by radiation, and later by matter. And now, it has again become dominated by a dark energy that is driving a slower accelerated expansion (equation 3.27). The term “Big Bang” was first coined by Fred Hoyle, when he was trying to belittle the credibility of the theory. However, The Big Bang has became the standard cosmological framework for understanding the universe and is supported by many lines of evidence.

3.1.1 The Expanding Universe–Hubble’s Law

In 1929, Edwin Hubble announced his discovery (now Hubble’s Law) [103], that describes the relation between radial velocity (V ) and distance (D) of extra-Galactic

40 “nebulae” (galaxies):

V = H0D (3.1)

where H0 is Hubble constant. Hubble’s law basically describes that the more distant the galaxy, the faster it is receding from us and galaxies are moving away from each other. It is the result that we expect for a uniformly expanding universe.

3.1.2 Big Bang Nucleosynthesis (BBN)

At the time of about 3 minutes after the Big Bang, the universe is hot and dense. There were no atomic elements, but rather a sea of neutrons, protons, electrons, positrons, photons and neutrinos. As the universe cooled, the following processes occurred:

+ + 2 n p + e− +ν ¯ n + p D + γ → e →1 2D + p+ 3 He + γ 2D +2 D 4 He + γ 1 →2 1 1 →2 2D +2 D 3 T + p+ 2D +3 T 4 He + n 1 1 →1 1 1 →2 ...... (3.2)

This process of light element formation in the early universe is called “Big Bang nucleosynthesis” (BBN). It lasted for only about 17 minutes. After that, the limited lifetime of free neutrons ended the process, while the temperature and density of the universe fell below that which is required for nuclear fusion. The brevity of BBN is important because it predicts that only light elements could be form, and the abundance of light elements are: Hydrogen(1H) 75%, Helium(4He) 25%, ≈ ≈ 2 3 7 10 Deuterium( H) 0.01%, Helium( H) 0.001%, Lithium( Li) 10− [104, 105]. ≈ ≈ ≈ This prediction is in agreement with observations. In 2011, pristine clouds of the primordial gas were found [106]. These clouds of gas was discovered by analysing the light from distant quasars. Absorption lines that can be used to measure the composition of the gas appeared in the spectrum where the

41 light was absorbed by the gas. The composition of the gas matches the predictions from BBN, providing the latest direct evidence in support of the modern cosmological explanation for the origins of elements in the universe.

3.1.3 The Cosmic Microwave Background (CMB) Radiation

The CMB radiation was predicted as radiation left over from an early stage in the development of the universe (section 3.4), and its discovery is considered a landmark success of the Big Bang theory, ruling out the competing Steady State Theory [107]. In 1964, Arno Penzias and Robert Wilson discovered the cosmic background radiation while conducting diagnostic observations using a microwave receiver [108]. Their discovery confirmed the CMB predictions from the Big Bang theory–an isotropic and consistent blackbody spectrum with a temperature of about 3 K. The Cosmic Background Explorer (COBE) satellite was launched in 1989. Its findings were consistent with the Big Bang’s predictions regarding the CMB. COBE found a precise blackbody spectrum (reflecting thermal equilibrium between matter and radiation in the early universe) with a temperature of 2.725 0.001 K and detected ± for the first time the fluctuations (anisotropy) in the CMB, at a level of about one part in 105 [109]. In early 2003, the first results of the Wilkinson Microwave Anisotropy Probe (WMAP) were released [11]. These results tested and refined a standard cosmological model with accurate values for the cosmological parameters. The WMAP results were also consistent with the inflation theory. A new generation space probe – the Planck satellite, was launched in 2009. It has goals similar to WMAP –to provide even more accurate measurements of the CMB anisotropy while further testing the model. There are also many other ground- and balloon-based experiments targeting various aspects of the CMB.

3.1.4 Other Evidence

Observations of the morphology and distribution of galaxies and quasars also provide strong evidence for the standard model of cosmology. Observations suggest

42 that the ﬁrst quasars and galaxies formed when the age of the universe was only about 0.5 billion years and distant galaxies (galaxies formed in the early universe) appear very diﬀerent from nearby galaxies (galaxies formed recently). These observations agree well with numerical simulations [110]. The age of universe determined to < 1% from the CMB is also in good agreement with other estimations, i.e. using the ages of the oldest stars.

3.2 Cosmic Inﬂation

Cosmic inflation was originally proposed by Alan Guth [111, 112], Alexei Starobin- sky [113], Andrei Linde [114], Andreas Albrecht and Paul Steinhardt [115]. It posits that there was a rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative pressure energy (ω < 1/3). Following − 43 36 the grand unification epoch (between 10− s and 10− s after the Big Bang), the 36 inflationary epoch comprises the first part of the electroweak epoch (between 10− s 12 36 32 and 10− s after the Big Bang). It lasted from 10− s to about 10− s. After that, the universe continued to expand at a rate that was much slower than inflation. While the detailed physics mechanism responsible for inflation is still unknown, inflation makes a number of predictions that have been confirmed by observations, such as CMB observations, galaxy surveys and 21 cm radiation observations. Inflation is thus now considered to be an extension of the Big Bang theory. It resolves several problems in the Big Bang cosmology.

3.2.1 The Structure Problem

Considerable structures in the universe, from stars to galaxies to clusters and super clusters of galaxies have been observed. How did these structures form? The Big Bang theory does not account for the needed fluctuations to produce the structure we see. Inflation gives a solution to this problem: Quantum fluctuations in the nearly-uniform density of the early universe expanded to cosmic scales during cosmic inflation. These fluctuations also would have left an imprint in the CMB radiation

43 in the form of temperature ﬂuctuations from point to point across the sky (the CMB anisotropy). The structures that we observe today grew from the gravitational pull of these ﬂuctuations.

3.2.2 The Flatness Problem

Observations show that the geometry of the current universe is nearly flat (section 3.3.3). However, under the nominal Big Bang theory, curvature grows with time. A universe as flat as we see it today would require an extreme fine-tuning of conditions in the past, which would be an unbelievable coincidence. Inflation provides a solution to this problem via the stretching of any initial curvature of the 3-dimensional universe to near flatness, resulting Ω 0. k ≈

3.2.3 The Horizon Problem

The uniformity of the CMB temperature (section 3.4) implies that the entire observable universe must have been in causal contact in the past. But now the distance between two regions with 2◦ apart in the sky are so far apart that, they ≥ could never have been in causal contact with each other, because the light travel time between them is greater than the age of the universe. This can be explained by inflation theory: Distant regions were actually much closer together prior to inflation than they would have been with only standard Big Bang expansion. Thus, such regions could have been in causal contact prior to inflation and could have attained a uniform temperature.

3.2.4 The Magnetic Monopole Problem

The Big Bang theory predicts that the early universe produced a very large number of heavy and stable magnetic monopoles. However, these magnetic monopoles have never been observed so far. The explanation from inﬂation is that, during in- ﬂation, the density of monopoles drops exponentially, so their abundance drops to undetectable levels.

44 3.3 ΛCDM Cosmological Model

The Λ Cold Dark Matter (ΛCDM) model is a model of the content of the universe that includes baryons, cold dark matter, photons, neutrinos and a cosmological constant Λ.

3.3.1 Cosmological Principles and FLRW metric

The cosmological principle is that, on suﬃciently large scales, the universe is homogeneous and isotropic. Homogeneity implies translational invariance and isotropy implies rotational invariance. These principles are distinct but closely related, because a universe that appears isotropic from any two locations must also be homogeneous. The isotropic principle is supported by the observations: (1) Radio galaxies are randomly distributed across the sky; (2) The large scale distribution of galaxies is isotropic in the range of greater than 200 Mpc; (3) The observed redshift distribution of distant galaxies is isotropic, which implies a uniform expansion of space in all directions; (4) The Cosmic Microwave Background (CMB) radiation is constant in all directions to within 1 part in 105 (section 3.1.3). A universe must be non-static if it follows the cosmological principle. In 1923, Alexander Friedmann derived a version of Einstein’s equations of general relativity describing the dynamics of a homogeneous and isotropic universe. After that, Georges Lemaˆıtre, Howard P. Robertson and Arthur G. Walker also derived the general relativity metric for the cosmological principle independently. It is named Friedmann - Lemaˆıtre - Robertson - Walker (FLRW) metric. In the four space-time dimensions, using Einstein notation, the invariant is:

2 ν ds = gν dx dx (3.3) where the and ν indices range from 0 to 3. The FLRW metric can be written as:

45 1 0 0 0 − a2  0 1 kr2 0 0  gν = − (3.4)  0 0 a2r2 0     2 2 2   0 0 0 a r sin θ   and its inverse is:  

1 0 0 0 − 1 kr2  0 − 0 0  gν = a2 (3.5) 1  0 0 2 2 0   a r   1   0 0 0 2 2 2   a r sin θ  where a = a(t) is the time-dependent cosmic scale factor and k is a constant representing the curvature of the space. The Riemann curvature tensor Rαβγ is defined by: dΓ dΓ R = αγ αβ + Γ Γσ Γ Γσ (3.6) αβγ dxβ − dxγ σβ γα − σγ βα where Γ are the Christoffel symbols: 1 ∂g ∂g ∂g Γα = gαν ν + λν λ . (3.7) λ 2 ∂xλ ∂x − ∂xν Also, the Ricci tensor Rν is defined as:

α Rν = Rνα. (3.8)

The diagonal elements of the Ricci tensor are: a¨ R = 3 00 − a aa¨ + 2˙a2 + 2k R = 11 1 kr2 − 2 2 R22 = r (aa¨ + aa˙ + 2k)

2 2 2 R22 = r (aa¨ + aa˙ + 2k)sin θ (3.9)

and the trace of the Ricci tensor is the scalar curvature R: aa¨ +a ˙ 2 + k R gνR = 6 . (3.10) ≡ ν a2

46 3.3.2 Einstein Field Equations and Friedmann Equation

The Einstein Field Equations (EFEs), that are used to determine the spacetime geometry resulting from the presence of mass-energy and momentum, can be written in the form of: 1 R g R + g Λ = 8πGT (3.11) ν − 2 ν ν ν where Λ is the cosmological constant, G is the gravitational constant and Tν is the energy-momentum tensor.

For perfect isotropic ﬂuid in equilibrium, Tν can be written as:

Tν = (p + ρ)uuν + pgν (3.12) where ρ is density, p is pressure and u is the four velocity. By plugging equation 3.8, equation 3.9, equation 3.10 and equation 3.12 into equation 3.11, we have:

a˙ 2 8πG k Λ H2 = ρ + (3.13) ≡ a 3 − a2 3 and a¨ 4πG Λ = (ρ + 3p) + (3.14) a − 3 3 where H is the Hubble parameter that gives the rate of expansion of the universe and k is the curvature constant that belongs to the set of -1, 0, +1 , standing for a { } negative, zero, positive curvature. { } Equation 3.13 is the Friedmann equation, which describes the expansion of a homogeneous and isotropic universe within the context of general relativity. The Hubble 1 1 constant is the current value of Hubble parameter H = H = 100h km s− Mpc− = 0 |t=t0 1 1 1 1 70.4 km s− Mpc− [14], where h = H0/(100 km s− Mpc− ) = 0.704. Equation 3.14 is the acceleration equation, describing the accelerated expansion rate of the universe. Λ can be absorbed into ρ and p, by replacing ρ + ρ ρ and p p p, where Λ → − Λ → ρΛ = Λ/(8πG) and pΛ = Λ/(8πG). Equation 3.13 and equation 3.14 can be simpliﬁed as:

a˙ 2 8πG k = ρ (3.15) a 3 − a2 47 and a¨ 4πG = (ρ + 3p). (3.16) a − 3 For a perfect isotropic fluid, from the first law of thermodynamics dE + P dV = 0, one can derive the fluid equation of the universe: a˙ ǫ˙ + 3 (ǫ + p) = 0, (3.17) a where ǫ = ρc2 is the energy density. We can define the equation of state as:

p = ωǫ (3.18) where the dimensionless number ω is the equation of state parameter. From equation 3.15, equation 3.16, equation 3.17 and equation 3.18, we have:

2 8πG 1 3ω k a˙ = ǫ a− − (3.19) 3 ω,0 − a2 ω and a¨ 4πG = (1 + 3ω)ǫ. (3.20) a − 3

where ǫω,0 is the energy density of the species with equation of state parameter ω. For a ﬂat universe, k = 0. From the above equations, we can derive:

3(ω+1) ρ a− (3.21) ∝ and 2 a t 3(ω+1) . (3.22) ∝ Diﬀerent species in the universe have diﬀerent equation of state parameters: (1) 4 1 For photons and other relativistic species, ω = 1/3, ρ a− and a t 2 ; (2) For ∝ ∝ 3 2 non-relativistic matter (cold dark matter and baryons), ω = 0, ρ a− and a t 3 ; ∝ ∝ (3) For dark energy, ω < 1/3 anda ¨ > 0. Dark energy accelerates the expansion − of the universe. (4) For Λ, ω = 1, ρ a0 = constant and a exp(H t). The − ∝ ∝ 0 cosmological constant represents a special kind of dark energy.

48 The total density of the universe, Ω0, is deﬁned as the ratio of the actual (observed)

density ρ0 to the critical density ρc,0 of the Friedmann universe: 3H2 ρ = 0 (3.23) c,0 8πG

ρ0 8πGρ0 Ω0 = = 2 . (3.24) ρc,0 3H0 If we introduce a new set of deﬁnition:

ǫr,0 ǫm,0 ǫΛ,0 Ωr,0 ,Ωm,0 ,ΩΛ,0 (3.25) ≡ ǫc,0 ≡ ǫc,0 ≡ ǫc,0 and

Ω0 = Ωr,0 + Ωm,0 + ΩΛ,0 Ω = 1 Ω (3.26) k − 0 2 where ǫc,0 = ρc,0c , indices r, m, Λ and k stand for radiation, matter, cosmological constant (dark energy) and curvature, respectively. Equation 3.15 can be written as: 2 H (t) Ωr,0 Ωm,0 Ωk 2 = 4 + 3 + ΩΛ,0 + 2 . (3.27) H0 a a a The last term on the right hand side is related to curvature. In a ﬂat universe,

Ω0 = 1, Ωk = 0. Equation 3.27 describes the evolution of the universe with a combination of diﬀerent species: After inﬂation (section 3.2), which was dominated 4 by a negative pressure, when a 1, the universe was dominated by radiation ( a− ), ≪ ∝ 3 and later by matter ( a− ). Now, at a = 1, it has again become dominated by a ∝ negative pressure energy (dark energy) that is driving an accelerated expansion.

3.3.3 Best-ﬁt ΛCDM Model Parameters

The ΛCDM model currently has six parameters: baryon density (Ωb), dark matter density (Ωc), dark energy density (ΩΛ), scalar spectral index of spatial fluctuation 2 (ns), curvature fluctuation amplitude (∆R) and reionization optical depth (τ). Other model values, including the Hubble constant and age of the universe, can be derived from these parameters. Table 3.1 lists the best-fit parameters of the ΛCDM model [14] based on data from Wilkinson Microwave Anisotropy Probe (WMAP), Baryon

Acoustic Oscillations (BAO) and Hubble constant (H0) measurements.

49 Table 3.1: Best-ﬁt ΛCDM Model Parameters Basic parameter Value Description Ω 0.0456 0.0016 Baryon density b ± Ωc 0.227 0.014 Cold dark matter density ±+0.015 ΩΛ 0.728 0.016 Dark energy density − ns 0.963 0.012 Scalar spectral index 2 1 +0.088± 9 ∆R(k0 = 0.002Mpc− ) (2.441 0.092) 10− Curvature ﬂuctuation amplitude τ 0.087− 0.×014 Reionization optical depth ± Extended parameter Value Description +1.3 1 1 H0 70.4 1.4 Hubble constant (km s− Mpc− ) − t0 13.75 0.11 Age of the universe (Gyr) r < 0.24(95%± CL) Tensor-to-scalar ratio +0.0056 Ω0 1.0023 0.0054 Total density −+0.0054 Ωk 0.0023 0.0056 Curvature density − −+0.68 z 1090.89 0.69 Redshift at decoupling ∗ +3205− t 377730 3200 Age at decoupling (yr) z ∗ 10.4 −1.2 Redshift of reionization reion ±

The universe is nearly ﬂat

If the density of the universe, ρ0, is greater than critical density, ρc,0, then Ω0 > 1.0,

Ωk < 0, the geometry of space is closed. In this space, initially parallel photon paths

converge and return back to their starting point; If ρ0 < ρc,0,Ωk > 0, then the geometry of space is open and negatively curved like the surface of a saddle; From +0.0056 +0.0054 table 3.1, we have Ω0 = 1.0023 0.0054,Ωk = 0.0023 0.0056. These measurements − − − show that the geometry of the universe is within measurement error of a flat space. If flat or negatively curved, it is infinite in extent, unless the cosmological principle does not hold on scales much greater than the horizon scale. A solution to the flatness problem is given by the inflation theory (section 3.2). It proposes a period of extremely rapid (a factor of 1026 in scale in only a small ∼ fraction of a second) expansion of the universe prior to the more gradual Big Bang expansion. Inflation stretches the geometry of the universe towards flatness.

50 Relativistic species in the universe

The main relativistic species are the Cosmic Microwave Background (CMB) photons (see section 3.4) and neutrinos. The energy density of photon can be calculated by Bose-Einstein distribution:

d3p ρ = g f(x, p)E(p), (3.28) (2π)3 where, ρ is the energy density, g is the degeneracy, f(x, p) is the distribution function, and E(p) = (p2 + m2)1/2 is the energy at a given stage p. For photons, g = 2, f(x, p) = 1/(eE(p)/Tγ 1) and E(p) = (p2 + m2)1/2 = p, thus − (note that d3p = 4πp2dp):

d3p p ργ = 2 (2π)3 ep/Tγ 1 3− 8π ∞ p = dp (2π)3 ep/Tγ 1 0 − π2 = T 4 (3.29) 15 γ So,

2 2 ργ,0 π 4 1 π 4 8πG Ωγ,0 = = Tγ,0 = Tγ,0 2 ρc,0 15 ρc,0 15 3H0 2 π 4 2 7 5 = T h− 6.808 10− = 4.98 10− (3.30) 15 γ,0 × × ×

where Tγ,0 = 2.725 K is the temperature of the CMB measured today and h = 0.704. Cosmic neutrinos have not been directly observed, because they are weakly interacting particles. We can compute the relative energy density of neutrinos by relating the temperature of neutrinos to the temperature of photons in CMB radiation, since neutrinos were once in equilibrium with the rest of the cosmic plasma. Theory pre- 5 dicts T 0.71 T = 1.945 K and Ω 0.68 Ω = 3.40 10− . WMAP ν,0 ≈ × γ,0 ν,0 ≈ × γ,0 × has found that a cosmic neutrino background is also needed as a part of the standard model of cosmology.

51 Matter in the universe

Observations have indicated the presence of dark matter in the universe, including the rotation curves of galaxies, gravitational lensing of background objects by galaxy clusters, the temperature distribution of hot gas in galaxies and clusters of galaxies, and the CMB measurement, which can distinguish the dim baryon “dark matter” from the non-baryonic dark matter. There are three types of hypothetical dark matter: cold, warm and hot dark matter. Cold dark matter is the dark matter composed of particles with typical speeds much below the speed of light (generally < 0.1c). Warm dark matter are particles traveling at relativistic speeds, but less than ultra-relativistic speeds (typically between 0.1c and 0.95c). Hot dark matter are particles that travel at ultra-relativistic velocities (> 0.95c). Cold dark matter is currently the area of greatest interest for dark matter research, as hot and warm dark matter do not seem to be viable for galaxy and galaxy cluster formation. Most of the matter in the universe is cold dark matter. The measurements of cosmic abundances of light elements suggest that the baryon density is only a small fraction of the critical density. From table 3.1, we have Ωb = 0.0456 and Ωc = 0.227. The cold dark matter constitute about 83% of the matter in the universe. The visible universe (baryon) only contributes about 17% of the total mass and 4.56% of the total mass-energy. Weakly Interacting Massive Particles (WIMPs) are candidates for cold dark matter. These particles interact only through the weak force and gravity. WIMPs do not interact via electromagnetism, so they cannot be observed directly. They do not react with atomic nuclei because they do not interact with the strong force either. There are many experiments currently running (or planned), aiming to search for WIMPs [116, 117, 118, 119, 120]. WIMPs could also be produced in the laboratory. Experiments with the Large Hadron Collider (LHC) may be able to detect WIMPs produced in collisions of the proton beams. Although WIMPs are a more popular dark matter candidate, there are also experiments searching for other particle candidates. It is also possible that dark matter

52 consists of very heavy hidden sector particles that only interact with ordinary matter via gravity.

Dark energy in the universe

Dark energy is a hypothetical energy that permeates the entire space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain that the universe is expanding at an accelerating rate [121]. In the ΛCDM model, dark energy needs to account for 72.8% of the total mass-energy of the universe (table 3.1) to reconcile the measured flat geometry of space with the total density equals the critical density. A direct signal of dark energy in a flat universe is from the late-time Integrated Sachs-Wolfe effect (ISW) [122, 123, 124]. Dark energy is thought to be homogeneous, and is not known to interact through any of the fundamental forces other than gravity. With ω < 1/3, dark energy has negative pressure as − gravitational repulsion to accelerate the expension of the universe. There are two proposed forms for dark energy: the cosmological constant and scalar fields having a time-dependant energy density. The cosmological constant may includes the contribution from scalar fields that are constant in time. It may be difficult to distinguish scalar fields from the cosmological constant because the time variation of the fields could be extremely small and the value of ω could be very close to -1. The simplest explanation for dark energy is the cosmological constant and the simplest explanation for the cosmological constant is vacuum energy. That is, a volume of space has some intrinsic, fundamental energy. There are many ways to predict and estimate this energy, including quantum field theory and string theory. The cosmological constant remains a subject of theoretical and empirical interest. The explanation of this small but positive value is still an outstanding challenge.

53 3.4 The Cosmic Microwave Background Radiation

Figure 3.1 shows the timelime of CMB radiation formation. The ΛCDM model includes the abrupt appearance of expanding space-time containing radiation at temperatures of around 1019 K. The universe was intensely hot, remarkably smooth and essentially homogeneous. However, small fluctuations in density originating as quantum fluctuations, began to appear and grow. Inflation stretched the curvature of the universe to be nearly (but not exactly) flat and expanded these quantum fluctuations in the density of the early universe to the cosmic scale. At redshift (z > 1100), when the temperature was still above 3000 K, photons were tightly coupled to free electrons through Thomson scattering. As the universe cooled down to about 3000 K, clumps of matter (baryons) began to condense and within them protons captured electrons +3205 and became atoms (recombination). Radiation decoupled from matter at 377730 3200 − years (table 3.1) after the Big Bang. The last scattering of the CMB photons was at redshift of about 1100, at which point the universe was almost exclusively composed of hydrogen, helium, dark matter, photons and neutrinos. This is the period when the CMB radiation was last scattered. After that, the CMB photons were free streaming. The color temperature of the CMB photons has continued to diminish ever since the Last Scattering Surface (LSS) and now down to about 2.7 K. Their temperature will continue dropping as the universe expands. The radiation from the sky we measure today comes from the surface of last scattering (figure 3.1). Most of the radiation energy in the universe is in the CMB radiation, making up a fraction of 5 roughly 5 10− (equation 3.30) of the total density of the universe. × Precise measurements of cosmic background radiation are critical to cosmology. The CMB has a thermal black body spectrum at a temperature of 2.725 K. In the Planck spectrum, it peaks at the microwave range frequency of about 160.2 GHz, corresponding to a 1.873 mm wavelength (see appendix B for details).

54 Figure 3.1: Timeline of the universe. The CMB radiation from the last scattering surface (LSS) when the universe is about 380,000 years old with the temperature of about 3,000 K [10].

55 Figure 3.2: The internal linear combination map from WMAP [11], showing the all sky CMB temperature anisotropy.

3.4.1 The CMB Anisotropy

While it is nearly perfectly homogeneous, the CMB radiation does have temperature anisotropy at the level of one part in 105 (figure 3.2). The CMB anisotropy was firstly measured by COBE (section 3.1.3). There are two sorts of CMB anisotropy: primary anisotropy, due to effects that occurred at the last scattering surface and earlier; and secondary anisotropy, due to effects such as interactions of the CMB radiation with hot gas or gravitational potentials, which occurred between the last scattering surface and the observer. To characterize the statistical properties of the CMB temperature T (ˆn) on the celestial sphere, we can expand it in a spherical harmonics basis Ylm as:

T (ˆn) = almYlm(ˆn) (3.31) l,m then the angular power spectrum for our actual sky will be 1 Csky = a 2. (3.32) l 2l + 1 | lm| m 56 The structure of the CMB anisotropy power spectrum is mainly determined by three effects: initial fluctuations (presumably from inflation), acoustic oscillations and diffusion damping (collisionless damping). In the early universe photon-baryon plasma, the pressure of the photons tended to weaken the anisotropy, while the gravitational attraction from the baryons tended to strengthen it. These two effects competed to create acoustic oscillations that generated characteristic peak structures in the CMB power spectrum. The peaks roughly correspond to the resonances in which the photons decouple when a particular mode was at its peak amplitude. The WMAP satellite improved the sensitivity and resolution of the measurements and detected the first three peaks in the angular power spectrum (figure 3.3). These peaks contain important physical signatures about the universe: The angular scale of the first peak determines the curvature of the universe. The amplitude ratio of the first and second peak determines the baryon density. The amplitude of all three peaks is related to the dark matter density. The locations of the peaks also give important information about the nature of the geometry. More power spectrum peaks at higher multipole moment have been measured by ACBAR [125], ACT [126, 127] and SPT [128]. Collisionless damping was caused by two effects: the increasing mean free path of the photons as the primordial plasma rarefied when the universe expanded and the finite depth of the LSS, which caused the mean free path to increase rapidly during decoupling, while some Thomson scattering was still occurring.

3.4.2 The CMB Polarization

CMB polarization arose from the Thomson scattering of the CMB photons at the LSS. As shown in ﬁgure 3.4, when an electromagnetic wave is incident on a free electron (from x or y direction), the scattered wave is polarized perpendicular to the incidence direction (z direction). If the incident radiation is isotropic or has only a dipole variation, the scattered radiation would have no net polarization (xp1 = yp1). However, if the incident radiation from perpendicular directions (x and y) have different intensities, a net linear polarization would result (xp1 = yp1). Such anisotropy

57 Figure 3.3: The angular power spectrum from WMAP [12], showing the detection of the first three peaks. The first peak is at ℓ 220, corresponding to an angular scale ≈ of about 1◦. have a quadrupole pattern. At the LSS, there was temperature inhomogeneity. So the scattered radiation is polarized. There were three different perturbations in the early universe plasma: scalar, vector and tensor perturbation. The scalar perturbation was the energy density fluctuations in the plasma that caused velocity distributions. The fluid velocity from hot to colder regions caused blueshift of the photons, resulting in quadrupole anisotropy. The vector perturbation was the vorticity in the plasma that caused Doppler shifts. However, vorticity would be damped by inflation and is expected to be negligible. The tensor perturbation is from the inflationary gravitational waves that stretched and squeezed space in orthogonal directions (+, ), which also stretches the wavelength × of radiation. The above perturbations result in two types of polarization in the CMB radiation, called E-modes and B-modes (in analogy to electromagnetism) [129]. The E-modes, curl-free components with no handedness, are due to both the scalar and tensor perturbations. The B-modes, curl components, are due to only tensor perturbations

58 (hot radiaon) X

E > 0 E < 0

xp1

xp2 yp2 e

Y yp1 (cold radiaon) xp1 yp1 B > 0 B < 0

Z (polarizaon)

Figure 3.4: Left: Quadrupole polarization from Thomson scattering of the CMB photons with free electrons. Right: The E and B mode patterns. The E-modes are curl-free components with no handedness. The B-modes are curl components with handedness. because of their handedness (ﬁgure 3.4). The amplitudes of tensor and scalar ratio is parametrized by the tensor-to-scalar ratio (r), which is related to the energy scale of inﬂation. Similar to the temperature anisotropy, the CMB polarization at each point on the sky can be characterized by combining its Q and U Stokes parameters (section 1.2) in terms of spin-2 spherical harmonics:

Q(ˆn) iU(ˆn) = a 2,lm 2Ylm(ˆn) (3.33) ± ∓ ∓ lm then we decompose them into E- and B-like components:

a 2,lm = Elm iBlm. (3.34) ± ±

These Elm and Blm parameters can be estimated from polarization maps as for the temperature anisotropy spectrum. Additionally, the cross-correlation between the temperature and the polarization can be taken. Figure 3.5 shows the TE, EE, and BB power spectra measured by WMAP [13].

59 Figure 3.5: Plots of signal for TT (black), TE (red ), and EE ( green). The not- yet-detected BB (blue dots) signal is from a model with r = 0.3. The BB lensing signal is shown as a blue dashed line. The foreground model for synchrotron plus dust emission is shown as straight dashed lines [13].

60 The E-modes polarization had been measured over a range of angular scales [130, 131]. The B-modes, which have not been detected, are expected to be extraordinarily faint. To set meaningful limits on inflationary models, any experiment designed to detect the inflationary B-mode signal should have a polarization sensitivity near 30 8 3 nK. That is 10− of the CMB blackbody temperature and 10− of the primordial CMB temperature anisotropy. The B-mode observation provides the only known way to measure the energy scale of inflation since inflation produced these gravitational waves whose amplitude depends only on the energy scale at which inflation occurred. Detection of B-modes would also be the first ever direct detection of gravitational waves.

61 Chapter 4

The Cosmology Large Angular Scale Surveyor (CLASS)

In chapter 3, we discussed the origin and evolution of the universe. A fundamental question is, “Did inflation really happen?” Inflation posits that the universe grew from quantum fluctuations of the vacuum driven by negative pressure energy to expand exponentially to astronomical scales. The simplest (and therefore most compelling) versions of inflation produce a stochastic background of gravitational waves whose amplitude depends only on the energy scale at which inflation occurred. The gravitational waves imprint a polarization pattern on the CMB (B-mode polarization), that then provides a direct way to measure the energy scale of inflation (section 3.4.2). Measurements to date of the E-mode signal are consistent with the predictions of anisotropic Thompson scattering [130, 131], while the B-mode signal has yet to be detected. The B-mode power spectrum amplitude can be parameterized by the relative amplitude of the tensor to scalar modes (the tensor-to-scalar ratio), given by

2 ∆h(k0) r 2 , (4.1) ≡ ∆ (k0) ℜ 2 2 where, ∆ (k) and ∆h(k) denote the dimensionless scalar and tensor power spectra, ℜ ℜ denotes the intrinsic curvature perturbation, h denotes the amplitude of gravitational

waves, and k0 is some pivot wavenumber [132, 133].

62 Figure 4.1: Two-dimensional joint marginalized constraint (68% and 95% CL) on scalar spectral index (ns) and tensor to scalar ratio (r), derived from the data combination of WMAP + BAO + H0 [14]. Three linear fits are from different simple inflation models.

If inflation produced the structure we see today, and it is associated with the energy scale ( 1016 GeV) of grand unified theories (GUTs) [134], then r 0.01 (for ∼ ≥ the simplest models). The current upper limit, inferred from WMAP + BAO + H0 is r < 0.24 (Table 3.1). The WMAP + BAO + H0 data also show 3σ deviation from a scale-invariant (scalar spectra index, ns = 1.0) scalar perturbation spectrum, with ns = 0.963 0.012 [14]. For the simplest inflation models (see figure 4.1), this expected ± deviation from scale invariance is coupled to gravitational waves with r 0.10. These ≈ considerations establish a strong motivation to search for this remnant from when the 32 universe was about 10− seconds old. The Cosmology Large Angular Scale Surveyor (CLASS) is an experiment with an unprecedented ability to detect the B-mode polarization to the level of r 0.01. It ≤ consists of 4 ground-based wide-field polarimeters, operating at 40, 90 and 150 GHz. CLASS will measure the large angular scale CMB polarization signature by observing

65% of the sky above 45◦ elevation from the Atacama Desert, 5180 meters above ∼ sea level (ﬁgure 4.2).

63 Figure 4.2: The background is the WMAP 7 year all sky Q band polarization map in Galactic coordinates showing the sky coverage of CLASS experiment. Observing from the Atacama Desert in Chile, CLASS covers 65.1% of the sky above 45◦ elevation. ∼ Excluding the Galactic mask area, the visible sky left is 46.8% (bright region). The ∼ dark circle at the south pole is about 22◦ in radius. Figure courtesy of David Larson.

As shown in figure 4.3, each CLASS telescope has a large front-end polarization modulator, called Variable-delay Polarization Modulator (VPM) (section 4.3), that rapidly modulates the polarization sensitivity for each observed sky pixel with no reliance on spatial scanning. Each of CLASS’s telescopes is a diffraction limited catadioptric system (section 4.4). The optics are fast (f/2.0) and have a large field of view (FOV), low cross-polarization and high Strehl ratio across the FOV. On the focal planes (section 4.5), the smooth-walled feedhorn arrays couple the radiation from the optics to transition edge sensor (TES) bolometer detectors. The focal planes and detectors are cooled to about 100 mK.

4.1 Scientiﬁc Overview

Table 4.1 shows the CLASS scientiﬁc overview. It lists the main challenges of B- mode detection and the solutions to them that CLASS provides. The B-mode signal

64 Figure 4.3: CLASS instrument overview for the 40 GHz band. The instrument consists a front-end variable-delay polarization modulator, catadioptric optic system and a ﬁeld cryostat. The lenses are cooled to about 4 K and the smooth-walled feedhorn- coupled TES bolometer array operates at 100 mK. Figure courtesy of Joseph Eimer.

65 Table 4.1: CLASS Scientific Overview B mode challenge Requirement CLASS solution B-mode signal Systematic control Polarization modulator is small is essential at front of optics Noise is Large number 4 telescopes dominated of detectors 4 focal planes by with 396 pixels atmosphere low noise 792 TESs Foregrounds are Polarization must 40 GHz: 36 detector pairs polarized: Synchrotron, be measured at 90 GHz: 300 detector pairs polarized dust emission multiple frequencies 150 GHz: 60 detector pairs At small angular Separate lensing Focus on large scales, gravitational B-modes from angular scales lensing converts E B inflationary B-modes ( 65% sky coverage) → ∼ is predicted to be extremely weak ( 30 nK) and hides behind the 2.725 K CMB ∼ monopole signal, which requires the experiment to be designed to minimize systematic measurement errors. The critical front-end VPM can minimize the systematic error by separating the instrumental effects from sky signals. It modulates the very large angular scale polarization rapidly (> 3 Hz) to remove 1/f noise. Because the polarization signal is spatially correlated, relying on the usual approach (scanning to remove 1/f noise) will convert unpolarized structure into false polarization signals. The CLASS objective is to avoid spatial scanning to remove 1/f noise. This instrument modulates polarization at the front-end with a small motion over a large aperture. It combines an unprecedented sensitivity to the inflationary B-mode signal with powerful systematic error suppression. Since CLASS is a ground-based experiment, the signal is dominated by photon noise from the atmosphere and the instrument (see section 4.2.2 for details). Sensi- tivity requirements demand a combination of substantial observing time with a large numbers of detectors, each operating well below the background limit. CLASS has 792 TES bolometers operating at 100 mK in 4 different focal planes. These are novel integrated focal planes that combine the clean beam properties of smooth-walled feedhorns with planar microwave filters and sensitive TES bolometers, which have been

66 demonstrated in astronomical instruments [135]. To characterize the Galactic foreground contamination from synchrotron and polarized dust emission (section 1.4), CLASS observes in three frequency bands (40, 90 and 150 GHz), accessible from the ground, as seen in figure 4.4. The data from these bands will be used to characterize the foreground signals for subsequent removal. Free-free emission is unpolarized so it does not affect the CMB polarization measurement. Gravitational lensing can convert E-mode polarization to B-mode polarization in small angular scales, but on large angular scales inflation is the only known extra- galactic source of B-mode polarization. Thus, observations of the large angular scale CMB polarization signals provide a clean way to directly verify inflation and measure the energy scale of inflation [136]. By targeting the large scale “reionization bump” of the B-mode signal at l 10, where the B-mode signal emerges most clearly from the ≤ gravitational lensing foreground, CLASS avoids gravitational lensing contamination (see section 4.2). In summary, the CLASS experiment has the following design criteria for B-mode searching: (1) Improves instrument sensitivity by using a larger number of background limited detectors; (2) Achieves excellent systematic control by placing the polarization modulator at front of the optics; (3) Observes in multi-waveband for foreground removal. (4) Focuses on large angular scale to avoid gravitational lensing contamination.

4.2 Sensitivity Calculation and Bandpass Optimiza- tion

Figure 4.4 shows the results of CLASS waveband optimization and sensitivity calculation. CLASS observing near the frequency of minimum Galactic foregrounds, achieves maximum sensitivity to the level of r 0.01 at the “reionization bump” ∼ and avoids the lensing contamination that dominates at small scales. Additional experiments, such as PIPER [137], SPIDER [138], and EBEX [139] will take various

67 apigterm notu adit f1Hrzi qiaett afascn of second a half Nyquist-Shannon to equivalent the is on Hertz Base 1 signal- of [140]. unity bandwidth a bandwidth output gives output an that Hertz theorem, power sampling 1 signal a the as in defined ratio is to-noise It astronomy. in used mally NEQ and NET NEP, of Definition (see NEFD as such NEQ: parameters, or instrument-specific NET In other 2.2), that NEP, section to as time. converted quoted be integration normally also total is sensitivity can and detector detectors waveband, millimeter the to of far-IR sensitivity and number efficiency, Calculation Sensitivity 4.2.1 CLASS. to complementary are that the approaches at B-modes detect definitively to sensitivity of the limit has interesting CLASS cosmologically 1 4.2.1). the section is boundary, (see shaded minimum the sensitivity along spectral CLASS curve The l foreground dashed Right: the Galactic f by 4.2.2). the The shown section (see curve, straddle Left: effects to atmospheric minimize [15]. chosen to from and are curve CLASS sensitivity of and wavebands bands CLASS 4.4: Figure $QWHQQD n sue er fosrigwt osraie5%ecec o down-time for efficiency 50% conservative a with observing of years 3 assumes and $WPRVSKHULF 7HPSHUDWXUH µ. 7UDQVPLVVLRQ os qiaetPwr(E)i esr ftesniiiyo eetrnor- detector a of sensitivity the of measure a is (NEP) Power Equivalent Noise size, beam instrument coverage, sky the on based is calculation sensitivity The V\QFK (PRGHV

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∼ %% 2 O O)&O π[µK ] 0 .01. 68 WMAP U U U Bump Reionization BICEP DASI Multipole Moment Moment Multipole BOOMERanG O σ

Grav. Lensing ii o each for limit requency QUaD CAPMAP CBI 1/2 integration time. NEP is a detector-specific parameter. It has the unit of WHz− . NET is Noise Equivalent Temperature. It is defined as the signal (in temperature units) from a source needed to produce a signal-to-noise ratio value of unity in a 1.0 second integration [141]. It is an instrument-specific parameter and quoted in units of Ks1/2. To measure polarization signal, we have an equivalent definition to the NET, that is Noise Equivalent Q Stokes parameter (NEQ). It is defined as the polarized signal from a linearly polarized source aligned with the detector orientation that is required to produce a signal-to-noise ratio value of unity in a 1.0 second integration. It is also quoted in units of Ks1/2. The above definitions can be quoted for a single detector or a pair of detectors following the relations:

NEPs = √2NEPp

NETs = √2NETp

NEQs = √2NEQp (4.2)

where the indices “s” means a single detector and “p” means a detector pair. The conversion between NET and NEP is [142]: NEP NET = (4.3) √2ηdηtAΩ∆ν∂Bν /∂T

where, ηd and ηt are the detector and instrument eﬃciencies, AΩ describes the optics,

∆ν is the bandwidth and ∂Bν /∂T is the derivative of the source emission (the CMB) with respect to temperature (Tcmb). The factor of √2 is from the conversion between Hz and second. We also have [141]:

NEQs = √2NEQp = 2NETs = 2√2NETp (4.4)

Table 4.2 shows CLASS detector sensitivities in NEQp.

69 Table 4.2: CLASS Detector Parameters Channel 40GHz 90GHz 150GHz Number of pixels (detector pair) (Np) 36 300 60 Number of detectors (Ns) 72 600 120 Beamsize (◦) 1.50 0.67 0.40 1/2 NETp of detector (Ks ) 68 60 93 1/2 NEQp of detector (Ks ) 135 120 186

Beamsize and Window Function

1 For a Gaussian beam, the beamsize is normally deﬁned as the full width at half maximum (FWHM). Then, the standard deviation of the beamsize can be written as: FWHM σbeam = . (4.5) √8ln2 Under the ﬂat-sky approximation, the solid angle of the beam will be:

2 Ωbeam = 2πσbeam (4.6) because the integral over a Gaussian plane with unit height gives: x2 + y2 exp dxdy = 2πσ2. (4.7) − 2σ2 The window function is a function that contains information about the beamsize and chopping angle of the experiment [143]. For a Gaussian beam, in multipole-space it can be written as: ω = exp[ ℓ(ℓ + 1)σ2 ]. (4.8) ℓ − beam

Noise Power Spectra

2 The noise of the Q or U measurement is given by:

2 2 2 NEQp NEQs σQ = 2 = 2 (4.9) (ηdηt) Nptpix (ηdηt) Nstpix where tpix is the integration time for each pixel (beam).

1This section is mostly from [142] 2This section is mostly from [142]

70 For an experiment with fsky coverage, Ωbeam beam solid angle, tobs total observa-

tion time and ηobs observation eﬃciency (e.g., including precipitable water eﬀects), assuming the experiment scans uniformly across the sky, then 4πf η t Ω t = η t / sky = obs obs beam . (4.10) pix obs obs Ω 4πf beam sky For Gaussian white noise on the sky, where the Q and U measurements in each beam-sized pixel are uncorrelated with each other and with the Q and U values in every other pixel, the noise power in E and B modes is:

BB EE 2 Nℓ = Nℓ = ΩbeamσQ. (4.11)

BB Then the expected error in the Cℓ measurement is:

2 N BB ∆CBB = CBB + ℓ (4.12) ℓ (2ℓ + 1)f ℓ ω sky ℓ By substituting equation 4.5 - equation 4.11 in to equation 4.12, we have:

2 BB 2 BB 4πfskyNEQp 2 ∆Cℓ = Cℓ + 2 exp[ℓ(ℓ + 1)σbeam] (4.13) (2ℓ + 1)fsky { ηobs(ηdηt) tobsNp }

2 To calculate the CLASS sensitivity (see ﬁgure 4.4), we used the NEQp,Np and

σbeam values listed in table 4.2 and assumed tobs = 3 years, ηobs = 50%, ηdηt = 0.80, f = 65% and CBB is from the current upper limit of r 0.2. sky ℓ ≈

4.2.2 Bandpass Optimization

The scientiﬁc goal of the CLASS project is to detect the B-mode polarization of the CMB. To calculate the instrument signal-to-noise ratio, we should use the B- mode polarization as our signal. However, the B-mode has not yet been detected. By assuming the B-mode signal is a tiny fraction of (and is proportional to) the CMB monopole (black body radiation with T = 2.725 K), in a given bandwidth, we can calculate the relative signal-to-noise ratio by using the CMB monopole spectrum as our signal. Observing from the ground, the dominant noise of CLASS is from atmospheric emission. In the signal-to-noise ratio calculation, we should also take the

71 eﬃciency of the VPM into account, since it depends on the bandwidth. The bandwidth optimization is based on maximizing the total signal-to-noise ratio integrated over each of these bandwidths.

Atmosphere Model

The Atmospheric Transmission at Microwaves (ATM) model [144] was used to calculate the transmission of the atmosphere at the CLASS site - Chajnantor Plateau, Chile. The ATM model was improved from many widely used older models such as the Microwave Propagation Model (MPM) [145]. It has been developed to perform radiative transfer calculations trough the terrestrial atmosphere. ATM treats the clear sky case to evaluate absorption/emissivity, but also polarization and scattering effects. It is currently used by several millimeter/subllimeter wave telescopes such as the Atacama Large Millimeter Array (ALMA) to evaluate atmospheric transmission and phase dispersion. Validation of this model has been undertaken with a series of observational experiments using a Fourier Transform Spectrometer (FTS) installed at the Caltech Submillimeter Observatory (CSO) [146]. CLASS will be deployed at Chajnantor Plateau, close to the Atacama Pathfinder Experiment (APEX) telescope. According to the site testing result from the APEX (figure 4.5), it is reasonable to have the annual Precipitable Water Vapor (PWV) as 1.0 mm. Figure 4.6 shows the ATM model of the atmosperic transmission and brightness temperature from 5 to 1000 GHz at 45◦ elevation with PWV = 1.0 mm. There are 3 main atmosphere windows below 200 GHz, centered at about 40, 90 and 150 GHz. As shown in figure 1.4, the 40 GHz band can be used to characterize the synchrotron foreground radiation and the 150 GHz band data can be used to do polarized dust foreground removal. CLASS has two polarimeters operating at 90 GHz band, near the minimum of the foreground contamination.

Signal to Noise Ratio

The NEP of a bolometer can be described by [147]:

72 Figure 4.5: Annual variation of the Precipitable Water Vapor (PWV) content at Chajnantor, based on 10 years of site testing. Conditions are worse during the winter from the end of December to early April. The expected median PWV for the rest of the year is around 1 mm, while conditions of PWV < 0.5 mm can be expected up to 25% of the time [16].

73 Atmospheric Transmission Atmospheric Brightness Temperature 1 300

0.9 250 0.8

0.7 200 0.6

0.5 150

0.4 Transmission 100 0.3 Brightness Temperature (K) 0.2 50 0.1

0 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Frequency (GHz) Frequency (GHz)

Figure 4.6: Atmospheric transmission and brightness temperature at CLASS site from 5 to 1000 GHz. ATM parameters: ground temperature = 275 K, ground pressure = 558 mb, PWV = 1.0 mm, elevation = 45◦, altidude = 5180 m. ATM version: atm2011 03 15.exe.

2 2 2 2 NEP = NEPJohnson + NEPthermal + NEPphoton 2 2 2 + NEPload + NEPampliﬁer + NEPexcess. (4.14)

2 Since detectors for the CLASS experiment are background limited, NEPphoton from atmosphere and instrument dominates over other sources of noise. In a black-body 2 radiation ﬁeld, NEPphoton can be written as [147] (see appendix C for details):

AΩ (k T )5 x4 αǫf NEP2 = 4 B s 1 + (αǫf)dx (4.15) photon c2 h3 ex 1 ex 1 − − Where AΩ describes the optics, f is the transmissivity of the optics, Ts is the temperature of the source, ǫ is the emissivity of the source, α is the detector absorptivity,

α, ǫ, and f are evaluated at ν, and x = hν/(kBTs). The atmosphere is treated as a blackbody source with varying emissivity:

2hν3/c2 Iν = ǫ (4.16) ehν/(kB Ts) 1 −

where Iν is the intensity of the atmosphere emission.

74 The number of photons in a given state x is: αǫf n(ν) = n(x) = . (4.17) ex 1 − From equation 4.16 and equation 4.17 , we have:

3 2 2 x 2hν /c αfc n = αfIν/(e 1)/ = Iν. (4.18) − ehν/(kB Ts) 1 2hν3 − Equation 4.15 can be written in n as: AΩh2 (k T )5 NEP2 = 4 B s n(1 + n)ν4dν photon c2 h3 h2 = 4 AΩn(1 + n)ν4dν = 4h2 n(1 + n)ν2dν (4.19) c2 where AΩ = c2/ν2 for diﬀraction-limited optics. The CMB signal has a blackbody spectrum. Since NEP is the standard deviation in power from a 0.5 second integration, our signal is 2hν3/c2 hν S = (0.5 s) AΩαfξ dν = αfξ dν (4.20) ehν/(kB Tcmb) 1 ehν/(kBTcmb) 1 − − where ξ is the transmission of the atmosphere. From equation 4.19 and equation 4.20, we have: ν S S 2 αfξhν/(ehν/(kBTcmb) 1)dν = = ν1 − (4.21) N NEP (4h2 ν2 n(1 + n)ν2dν)1/2 ν1 for the waveband between ν1 and ν2. This equation teats the atmosphere with a single temperature model and does not include the loading from the instrument.

Optimization Results

The VPM efficiency is strongly depend on its operational bandwidth (see section 4.3.4 for details). Theoretically, we should absorb the VPM efficiency into the coefficient f in equation 4.21 to calculate the total signal-to-noise ratio, but this would result in complicated calculations. As an approximation, we multiply equation 4.21 by equation 4.55 to get the total relative signal-to-noise ratio: ν 2 αfξhν/(ehν/(kB Tcmb) 1)dν SNR η(ν /ν ) ν1 − (4.22) ∝ 2 1 (4h2 ν2 n(1 + n)ν2dν)1/2 ν1 75 The ATM model provides the transmission (ξν) and brightness temperature (Tν) of the atmosphere (figure 4.6). One can calculate the emission intensity Iν from Tν using the Planck function:

2hν3 1 Iν = (4.23) c2 ehν/(kB Tν ) 1 − then, from equation 4.18, we have:

αfc2 αf n = Iν = . (4.24) 2hν3 ehν/(kB Tν ) 1 − We can calculate the relative signal-to-noise ratio numerically by substituting equation 4.55 and equation 4.24 into equation 4.22. The bandwidths were optimized by maximizing the signal-to-noise ratio value. Figure 4.7 shows the CMB signal transmitted through the atmosphere (equation 4.20) and atmosphere emission intensity (equation 4.16) for the relative signal- to-noise ratio calculation (equation 4.21). Figure 4.8 shows the result of the optimization. The 2-D plot of relative signal-to-noise ratio over 0 to 200 GHz shows 3 local maxima in the bandwidth from 0 to 200 GHz. They are located at (30.3 GHz, 40.3 GHz), (77.3 GHz, 108.3 GHz) and (126.8 GHz, 164.3 GHz). For the 40 GHz band, we search for the maximum in the range of ν > 30 GHz. In this model, we assumed αf = constant across the bandwidths, and found that within the tolerance of the optimization (0.02 GHz), in the range of 0.5 αf 1.0, ≤ ≤ the optimization result does not depend on the αf value. We also found that the optimization of equation 4.21 and equation 4.22 gave the same result. The VPM efficiency does not affect the optimization. The above are the results from relative signal-to-noise ratio optimization only. As we can see in figure 4.8, the plot does not show strong gradients around the peaks. The nearby points can also provide a similar signal-to-noise ratio level. We should take into account other instrument effects, such as the bandwidth limit of a feedhorn. The 40 GHz bandwidth is set to be 33.0 GHz to 43.0 GHz. The 90 and 150 GHz bands have not been fixed yet.

76 −18 x 10 CMB x Atmosphere Transmission 4

3.5 ]

−2 3 sr −2 2.5 m −1 2

1.5

Intensity [W Hz 1

0.5

0 0 20 40 60 80 100 120 140 160 180 200 Frequency [GHz]

−15 x 10 Atmosphere Emission Intensity

2.5 ] −2 2 sr −2 m −1 1.5

1 Intensity [W Hz 0.5

0 0 20 40 60 80 100 120 140 160 180 200 Frequency [GHz]

Figure 4.7: Top: the CMB signal (equation 4.20) and Bottom: atmospheric noise source (equation 4.16) for the relative signal-to-noise ratio calculation (equation 4.21). The red, green and blue lines shows our optimized bandwidth for 40, 90 and 150 GHz band: (30.3 GHz - 40.3 GHz), (77.3 GHz - 108.3 GHz) and (126.8 GHz - 164.3 GHz).

77 Relative Signal to Noise Ratio

3 20

40 2.5 60 2 80

100 1.5 120 Frequency [GHz] 140 1

160 0.5 180

200 0 20 40 60 80 100 120 140 160 180 200 Frequency [GHz]

Figure 4.8: The 2-D plot of relative signal-to-noise ratio (equation 4.22) from 0 to 200 GHz showing our optimization results. The cross points of red, green and white lines are the locations of the local maxima. For the 40 GHz band, we only search for the maximum in the range of ν > 30 GHz. The coordinates are (30.3, 40.3), (77.3, 108.3) and (126.8, 164.3).

78 V (circular) V (circular)

U U

Q Q HWP VPM

Figure 4.9: As shown in Poincar´esphere, VPM modulates between Q and V , while the HWP mix Q and U. In the case of VPM, the residuals due to the spectral eﬀects (shown in blue) are a function of measurable modulation parameters. Figure courtesy of David Chuss.

4.3 The Variable-delay Polarization Modulator

VPM is the the first element of CLASS instrument. It modulates the sky polarization signal by introducing a controlled variable path difference between two orthogonal linear polarizations of the incident radiation. Compared to the conventional spinning Half Wave Plate (HWP), the advantages of the VPM can be summarized as follows: [148] (1) The VPM is used in reflection, eliminating the effects from the dielectrics (e.g., nonuniformity, birefringence, ...); (2) The VPM modulation employs small motions, making it easier to achieve rapid modulation; (3) The VPM has more flexibility in size than the HWP. This allows larger apertures that enable front-end modulators for low frequency systems; (4) The modulation symmetry of the VPM allows spectro-polarimetry; (5) The VPM does not convert between Stokes Q and U, as opposed to the HWP (figure 4.9).

79 Grid d Mirror

Figure 4.10: VPM modulates polarization by introducing a controlled variable path diﬀerence between two orthogonal linear polarizations. Dots show the component with polarization angle parallel to the grid; Double arrow show that with angle perpendicular to the grid. By moving the mirror up and down, VPM introduces a path diﬀerence x(t) = 2d(t)cosθ between these two orthogonal polarization components.

4.3.1 Polarization Transfer Function

As shown in figure 4.10, the VPM is made of a polarizing wire grid (the wires only run one direction in this grid) and a movable parallel mirror behind it. The polarized radiation from sky can be decomposed into two orthogonal components: The component with polarization angle parallel to the grid will be reflected by the wire grid, the component with angle perpendicular to the grid will pass through the grid and get reflected by the mirror. For an ideal VPM, the optical path difference between these two components is [149]:

x = 2dcosθ, (4.25)

where d is the grid-mirror separation, θ is the incident angle. An ideal VPM is equivalent to a birefringent plate with its birefringent axis ori- ented at an angle α with a delay φ followed by a reﬂection. The Mueller matrix for a VPM system can be written as [150]:

80 1 0 0 0 0 cos22α + cosφsin22α sin2αcos2α(1 cosφ) sin2αsinφ  Mvpm(α, φ) = − − 0 sin2αcos2α(1 cosφ) sin22α cosφcos22α cos2αsinφ  − − − −    0 sin2αsinφ cos2αsinφ cosφ   −   (4.26) where, in the long wavelength limit,

φ = kx = 2kdcosθ (4.27) is the phase delay, k is wave number, α is the angle of the grid with respect to the orientation of detectors.

By setting α = 45◦, we have:

I I 1 0 0 0 I Q Q 0 cosφ 0 sinφ  Q = Mvpm = . (4.28) U U 0 0 1 0  U      −            V  V  0 sinφ 0 cosφ V   det  sky  −   sky         Then, for a detector sensitive to Stokes Q, the signal at the detector will be:

Qdet = Qskycosφ + Vskysinφ (4.29)

Equation 4.29 is the polarization transfer function describing the way that a VPM modulates the incident polarized signal. The key to understanding this polarization transfer function is to determine how the phase delay, φ, is related to the grid-mirror separation, d, since the latter is the quantity that can be directly measured in an instrumental setup.

4.3.2 VPM Grid Optimization

To optimize the performance of a wire grid, analytical approximations suggest a desire to achieve λ a and 2a/g 1/π, where λ is the wavelength, a is the ≫ ≈ 81 oe a endvlpdt iuaet efrac faVMgi narneof range a in grid VPM a of performance to simulate line 0 to transmission A developed been component. has perpendicular the model for transmissive high enable time odlfnto ftepaedly oee,teVMrflcinpaei dependent is geometry. phase grid reflection the VPM sinu- of the a details diameter however, remains the wire delay; upon response the phase polarization the As the of delay. wavelength, function a phase soidal of VPM fraction the finite for a approximation becomes good a with is illumination wave 4.27 plane tion For limit. geometric different with models eeto o ohprle n epniua oaiaincmoet.Giswith Grids components. polarization perpendicular 2 and parallel both for reflection aiso h wire, the of the from radius differs delay changed. phase are reflection conditions VPM the The if delay 4.29). grid-mirror (equation free-space grid ideal fr an of wavelengths different limit with two the for In performances [17]. grid tion wire The 4.11: Figure .02 a/g iue41 hw h oaiaintase ucinfrasnl rqec ftwo of frequency single a for function transfer polarization the shows 4.11 Figure < ≈ 7 2 1 / < a/g /π lo iheog eetv o h aallcmoetada h mean the at and component parallel the for reflective enough high allow 1 0[17]. .00 g stecne-ocne iepth agr2 Larger pitch. wire center-to-center the is g/λ ≪ ,asnsia omfrSoe si odagreement good in is Q Stokes for form sinusoidal a 1, 82 a/g ed ohigher to leads λ masimula- a om ≫ a equa- , 4.3.3 VPM Mirror Throw Optimization

In the above discussion, a VPM grid with λ a and 2a/g 1/π is a reasonable ≫ ≈ approximation to an ideal VPM. This section will focus on the optimization of the mirror throw for an ideal VPM base on maximizing the signal-to-noise ratio of its output.

From equation 4.29, for a given bandpass kl to kh, the average output signal of an ideal VPM as a function of x is:

1 kh S(x) = [I(k) + Q(k)cos(kx) + V (k)sin(kx)]dk (4.30) 2(k k ) h − l kl where, I(k), Q(k), V (k) are the stokes parameters of the incident signal from the sky, kl and kh are the wave numbers of the lower and higher limit of the waveband. In CLASS wavebands, the atmospheric transmission is high and roughly constant (ﬁgure 4.6), thus I(k), Q(k) and V (k) can be ﬁt by a black body spectrum ( ∝ AΩBν (T )):

k I0k I(k) = I0 = ehν/(kB Tcmb) 1 eAk 1 − − k Q0k Q(k) = Q0 = ehν/(kB Tcmb) 1 eAk 1 − − k V0k V (k) = V0 = (4.31) ehν/(kB Tcmb) 1 eAk 1 − − where, I0, Q0, V0 are constants and A = hc/(2πkBTcmb). Then, equation 4.30 can be written as:

1 kh I k Q k V k S(x) = 0 + 0 cos(kx) + 0 sin(kx) dk 2(k k ) eAk 1 eAk 1 eAk 1 h − l kl − − − = S I + S (x) Q + S (x) V (4.32) I × 0 Q × 0 V × 0

83 where,

1 kh k S = dk = constant I 2(k k ) eAk 1 h − l kl − 1 kh kcos(kx) S = dk Q 2(k k ) eAk 1 h − l kl − 1 kh ksin(kx) S = dk. (4.33) V 2(k k ) eAk 1 h − l kl − “Cosine” and “Linear” are two candidate VPM mirror chopping modes. The chopping can be approximated by N discrete steps as:

p0 + ∆p (cos(i/(N 1) π) + 1)/2 Cosine mode d(i) = × − ∗ (4.34) p0 + ∆p (i/(N 1)) Linear mode × − where p0 is the starting position of the mirror, ∆p is the peak-to-peak mirror throw and i is an integer in the range of (0, 1, ..., N 1) and the optical path diﬀerence is: − x(i) = 2d(i)cosθ (4.35)

Then, equation 4.32 can be written in matrix format:

AX = s (4.36) where

SI(0) SQ(0) SV (0)

 SI(1) SQ(1) SV (1)  A =  ......  , (4.37)     SI (N 2) SQ(N 2) SV (N 2)  − − −    SI (N 1) SQ(N 1) SV (N 1)  − − −    I0

X = Q0 (4.38)

V0     

84 and S(0)  S(1)  s =  ...  . (4.39)     S(N 2)  −    S(N 1)  −  All elements in A can be calculated by substituting equation 4.35 and equation 4.34 into equation 4.33. For a given signal matrix s, We can solve equation 4.36 for X and its covariance matrix CovX:

T 1 T X = (A A)− A s (4.40)

T 1 CovX = (A A)− (4.41)

Theoretically, both X and CovX should be diagonal matrices, since Q and V noise are uncorrelated. Then, the relative signal-to-noise ratio will be:

Q SQ(i) SNQ = | | (4.42) σ ∝ √CovX Q 22

V SV (i) SNV = | | (4.43) σ ∝ √CovX V 33 The CLASS VPM mirror throw is optimized by maximizing the relative signal-to-

noise ratio for Stokes Q. The parameters in our calculation are as following: p0 and

∆p are in the range of 0.01 λ0 to 1.00 λ0, with the step size of 0.01 λ0 and N = 100. Based on these setting, the resolution is 79 m for the 40 GHz band, 32 m for 90 GHz and 21 m for the 150 GHz band. Figure 4.12 and ﬁgure 4.13 show the optimization plots of “cosine” and “linear” chopping modes for the 40 GHz band. In the plots, the relative signal-to-noise ratios are normalized to the peak values. The details are listed in table 4.3. For the “cosine”

mode, (0.19 λ0, 0.39 λ0) is preferred, while the “linear” mode prefers (0.46 λ0, 0.16 λ0). The peak relative signal-to-noise ratio of the “cosine” mode is 6.4 105 and 5.4 105 × × for the “linear” mode. The “cosine” chopping mode oﬀers a higher signal-to-noise ratio than that from the “linear” mode.

85 1

0.9 0.9

0.8 0.8

0.7 0.7 ) 0 λ 0.6 0.6

0.5 0.5

0.4 0.4 Mirror Start Position ( 0.3 0.3

0.2 0.2

0.1 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Peak to Peak Throw (λ ) 0

Figure 4.12: The contour plot of relative signal-to-noise ratio for Stokes Q, calculated from equation 4.42 with cosine chopping mode. This plot is for the 40 GHz band (33 GHz to 43 GHz, λ0 = 7.89 mm). The maximum is at (0.19 λ0, 0.13 λ0) with the peak signal-to-noise ratio scaled to be 1.00. There are 4 other local maxima nearby: (0.19 λ0, 0.39 λ0), (0.44 λ0, 0.13 λ0), (0.44 λ0, 0.39 λ0) and (0.27 λ0, 0.26 λ0). Details are listed in table 4.3.

86 1

0.9 0.9

0.8 0.8

0.7 0.7 ) 0 λ 0.6 0.6

0.5 0.5

0.4 0.4 Mirror Start Position ( 0.3 0.3

0.2 0.2

0.1 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Peak to Peak Throw (λ ) 0

Figure 4.13: The contour plot of relative signal-to-noise ratio for Stokes Q, calculated from equation 4.42 with linear chopping mode. This plot is for the 40 GHz band (33 GHz to 43 GHz, λ0 = 7.89 mm). The maximum is at (0.46 λ0, 0.16 λ0) with the peak signal-to-noise ratio scaled to be 1.00. There are 2 other local maxima nearby: (0.63 λ0, 0.19 λ0) and (0.45 λ0, 0.42 λ0). Details are listed in table 4.3.

87 Table 4.3: CLASS VPM Mirror Throw Optimization

Cosine chopping mode: Waveband λ0 Maxima Relative SNR Throw Star-Pos Throw Star-Pos (GHz) (mm) (#) (-) (λ0) (λ0) (mm) (mm) 1 1.000 0.19 0.13 1.500 1.026 33.0 2 0.942 0.19 0.39 1.500 3.077 - 7.890 3 0.912 0.44 0.13 3.472 1.026 43.0 4 0.861 0.44 0.39 3.472 3.077 5 0.849 0.27 0.26 2.130 2.051 1 1.000 0.19 0.13 0.614 0.420 77.3 2 0.907 0.19 0.39 0.614 1.260 - 3.231 3 0.892 0.44 0.13 1.422 0.420 108.3 4 0.838 0.27 0.25 0.872 0.808 5 0.811 0.44 0.39 1.422 1.260 1 1.000 0.19 0.13 0.391 0.268 126.8 2 0.944 0.19 0.40 0.391 0.824 - 2.060 3 0.912 0.45 0.13 0.927 0.268 164.3 4 0.862 0.45 0.39 0.927 0.803 5 0.849 0.28 0.26 0.577 0.536 Linear chopping mode: Waveband λ0 Maxima Relative SNR Throw Star-Pos Throw Star-Pos (GHz) (mm) (#) (-) (λ0) (λ0) (mm) (mm) 33.0 1 1.000 0.46 0.16 3.629 1.262 - 7.890 2 0.932 0.63 0.19 4.971 1.500 43.0 3 0.884 0.45 0.42 3.551 3.314 77.3 1 1.000 0.45 0.16 1.454 0.517 - 3.231 2 0.908 0.59 0.19 1.906 0.614 108.3 3 0.811 0.46 0.42 1.486 1.357 126.8 1 1.000 0.46 0.16 0.948 0.330 - 2.060 2 0.933 0.62 0.20 1.277 0.412 164.3 3 0.889 0.46 0.42 0.948 0.865

88 4.3.4 VPM Eﬃciency

This section is about VPM eﬃciency as a function of bandpass. To simplify the calculation, we assume that the I, Q and V signal are constants in a given bandpass and the VPM is operating at the best optimized chopping position with max signal to noise ratio. From section 4.3, the output power of the VPM can be written as:

Qcosφ + V sinφ + I (4.44)

where Q and V are the Stokes parameters, I is the total intensity, and φ is the phase delay in the VPM: φ = kx = 2kdcos(θ) (4.45) where k is the wave number, x is the optical path diﬀerence, d is the grid-mirror separation and θ = 20◦ is the angle of the incident radiation with the normal of the VPM. As an approximation, we assume the intensity and all the Stokes parameters, are constants independent of frequency over the region of the passband. The Time- Ordered Data (TOD) from the experiment will be a function of time integrated over the waveband: 1 φh TOD(t) = (Qcosφ + V sinφ + I)dφ (4.46) φ φ h − l φl where, the indices l and h mean the lower and higher limit of the bandwidth: φl = klx = 2kldcos(θ) and φh = khx = 2khdcos(θ) . In order to ignore the eﬀects on photon noise from changing the passband size, we normalize the TOD by the width of the passband. Instead of referring to the low and high values of the phase delay, we switch coordinates to the geometric mean and ratios:

φ φ φ = x k k 0 ≡ l h l h r φ /φ ≡ h l φl = φ0/√r

φh = φ0√r (4.47)

89 where r > 1.0 is deﬁned as the ratio of the upper to the lower frequency of the bandwidth in this section (NOT the tensor-to-scalar ratio). Equation 4.46 can be re-written as: 1 φ0√r TOD(t) = (Qcosφ + V sinφ + I)dφ φ √r φ /√r 0 − 0 φ0/√r = Q TOD (t) + V TOD (t) + I (4.48) × Q × V where √r TOD (t) = [(sin(φ √r) sin(φ /√r)] Q φ (r 1) 0 − 0 0 − √r TOD (t) = [(cos(φ /√r) cos(φ √r)]. (4.49) V φ (r 1) 0 − 0 0 − The shape of TODQ(t) and TODV (t) is dependent only on r. That is, one can change φ0, and then plot TODQ(t), but changing φ0 will only expand or contract the horizontal direction, it won’t change the number of peaks within one period. As a consequence, the VPM eﬃciency depends only on r, not on φ0 = x√klkh. In order to solve for the Q, V , and I values, we can perform a linear least squares analysis. Assuming white noise and that the error bar on each TOD measurement is the same, then, equation 4.48 can be written as:

Q Q TOD (t), TOD (t), 1 V  = A V  = TOD (4.50) Q V × ×  I   I      then the least-squares solution is:    

Qˆ T 1 T Vˆ  = (A A)− A TOD (4.51)  Iˆ     Assuming a sinsoidal oscillation of φ0(t) = x(t)√klkh with time:

φ (t) = x k k + (x k k x k k )(cos(t) + 1)/2 (4.52) 0 min l h max l h − min l h where φ0 ranges from the ﬁrst local minimum in TODQ to the next local maximum. We take the integral over a half period of that oscillation to get:

90 π T 2 (A A)QQ = TODQ(t)dt 0 π T T (A A)QV = (A A)VQ = TODQ(t)TODV (t)dt 0 π T T (A A)QI = (A A)IQ = TODQ(t)dt 0 π T 2 (A A)VV = TODV (t)dt 0 π T T (A A)VI = (A A)IV = TODV (t)dt 0 π T (A A)II = dt = π. (4.53) 0 We then invert the AT A matrix to get the covariance matrix. To the extent that AT A is a diagonal matrix, we have: 1 σQ = . (4.54) π 2 0 TODQ(t)dt From equation 4.49 and equation 4.54, we have σQ = σQ(r). We can deﬁne the VPM eﬃciency as a function of the bandwidth (r):

η(r) = SNR(r)/SNR(r 1.0) = σ (r 1.0)/σ (r) (4.55) → Q → Q that is the signal-to-noise ratio of the bandwidth (r) normalized by the signal-to- noise ratio of a delta bandwidth (r 1.0). Figure 4.14 shows the eﬃciency plot, → from r = 1.0 to r = 50.0.

4.3.5 Current Status

The prototype VPM grid was built [151] (ﬁgure 4.15) and measured [152]. It is made of over 2 miles long, 63.5 m (0.0025”) diameter, gold plated tungsten wires. The wires were glued on an aluminium frame with a total of 2 tons stretching force. This prototype grid has 2a/g = 1/3.15 1/π, with 200 m wire pitch. It has a 50 cm ≈ diameter clear aperture, with ﬂatness within 50 m. The mechanical wire resonant frequency is higher than 128 Hz.

91 VPM Efficiency 100

70 (%) η 60

30 5 10 15 20 25 30 35 40 45 50 r

Figure 4.14: VPM efficiency calculated from equation 4.55. The efficiency drops quickly from r = 1.0 to r = 5.0 and becomes almost flat after r > 10. The noise at large r is due to the rounding in the numerical calculations.

With a/λ 0.004, this wire grid can be considered as a perfect VPM grid for 0 ≈ the 40 GHz waveband. It should also have good performance for the 90 GHz band. However, for the 150 GHz band, there will be significant difference between the VPM grid reflection phase delay and the free-space grid-mirror delay (equation 4.27). CLASS needs large aperture VPMs as the front-end modulators, i.e. the 40 GHz band requires a 60 cm diameter VPM. Operating at room temperature, gold plated tungsten wires are preferred because it provides adequate electrical conductivity and high yield strength. The VPM mirror can be control by a Proportional-Integral- Derivative (PID) controller. Since the required mirror throw is short, a linear piezo motor may be a good choice to drive the mirror.

4.4 CLASS Optics

Table 4.4 show the parameters of CLASS optics. The CLASS 40 GHz optics design was completed [153]. The geometric parameters are: VPM diameter = 60.0 cm, eﬀective focal length = 70.5 cm, f/2.0, focal plane diameter = 27.0 cm, Lyot stop

92 Figure 4.15: Photo of the prototype VPM grid. The wires are glued on an aluminium box frame with over 2 tons of stretch force. The diameter of the ﬂattener ring is 50 cm. The wire diameter, 2a, is 63.5 m, with wire pitch, g = 200 m. 2a/g = 1/3.15 1/π. ≈ The ﬂatness of the grid is better than 50 m. The total length of the wires is longer than 2 miles.

93 Table 4.4: CLASS Optics Overview Waveband 40 GHz 90 GHz 150 GHz

FOV (◦) 18.0 7.0 3.5 Beamsize (◦) 1.50 0.67 0.40 Strehl Ratio > 0.995 > 0.990 > 0.990 f 2.0 2.0 2.0 # of detector pairs 36 150 60

diameter = 30.0 cm, FOV = 18.0◦, number of pixels = 36. The 90 GHz and 150 GHz optics will share a similar design as the 40 GHz band. Figure 4.16 and figure 4.17 shows the drawings of CLASS 40 GHz optics and its ray trace. It is a diffraction limited catadioptric system with fast speed, large FOV, low cross-polarization and high Strehl ratio across the entire focal plane. It consists of a front-end VPM, two mirrors, a vacuum window, a Lyot stop, two lenses and two infrared (IR) blocking filters. The VPM and mirrors operate at room temperature, while the lenses, filters and the focal plane are cooled by a cryostat. The Point Spread Function (PSF) diagrams on the focal plane are shown in figure 4.18. The size of point spread is much smaller than the size first Airy disk (shown as circles), showing a diffraction limited optics. There are many methods to build the optical components by using different ma- terials. The following are the tentative methods for CLASS: The cold lenses will be made of high density polyethylene (HDPE) plastic, which is commonly used in millimetre wavebands. The lenses can be anti-reflective (AR) coated by direct bond- ing of dielectric layers with the right thickness and index of refraction [154]. The IR blocking filter can be made of polytetrafluoroethylene (PTFE). These filters have high transmission for the wavelength longer than 60 m [155], while absorb most of the radiation with higher frequency. The 5 inch thick vacuum window can be constructed by sandwiching 5 layers of the 1 inch thick Zotefoam (HD30).

lcigﬁtr.Bto:Daigadterytaeo h oldotc.Uisare Units optics. front cooled a the Eimer. of (IR) of Joseph infrared trace consists of two ray It courtesy and the Figure window and vacuum mm. optics. Drawing a in GHz stop, Bottom: Lyot 40 ﬁlters. a CLASS lenses, blocking of two mirrors, Drawings two VPM, Top: 4.16: Figure 1146.2

1820

2200

95 500 40° 127

941 600

1026.5

54.1° -end Figure 4.17: Ray trace of CLASS 40 GHz optics. Basic parameters: VPM diameter = 60.0 cm, eﬀective focal length = 70.5 cm, f/2.0, focal plane diameter = 27.0 cm, Lyot stop diameter = 30.0 cm, FOV = 18.0◦, number of pixels = 36. Figure courtesy of Joseph Eimer.

96 Figure 4.18: Point spread diagram of CLASS 40 GHz optics from Zeemax. Each diagram in this figure represents a separate direction on the sky. The circles show the first Airy disk at the corresponding location. This diagram shows that the optics is diffraction limited. Figure courtesy of Joseph Eimer.

97 4.5 Smooth-walled Feedhorn

CLASS requires feedhorns having symmetric beam patterns and low reflected power over a large bandwidth. Conventional corrugated feedhorns can produce beam patterns with low sidelobe levels, low cross polarization and low reflected power. How- ever, corrugated feedhorns are difficult to manufacture and require high machining precision. The cost of large arrays with hundreds of feedhorns is high, especially for high frequency bands (i.e., CLASS 90 GHz and 150 GHz band). As an alternative, smooth-walled feedhorns with monotonic profile are much more straightforward to build. They can provide performance comparable to that of the corrugated feedhorns. Smooth-walled feedhorns do not require high fabrication precision and are cost effective to build. A smooth-walled feed that has a 30% operational bandwidth over which the cross-polarization response is better than -30 dB and reflected power is better than -28 dB was designed, built and measured [156]. This smooth-walled feedhorn, however has relatively low aperture efficiency and high sidelobe levels due to its big aperture-to-length ratio. By reducing the aperture-to-length ratio, a feedhorn with both cross polarization and return loss lower than -30 dB across 30% bandwidth was designed [157]. It provides a sidelobe level lower than -25 dB, and an aperture efficiency of about 60%.

4.5.1 Smooth-walled Feedhorn Optimization

Input Waveguide

At the waveguide end of the of the horn, a short section of input circular waveguide is included. The waveguide radius provides a homogeneous interface to a rectangular waveguide by maintaining a uniform cutoﬀ frequency across the discontinuity [158]. The cutoﬀ frequency can be written as:

′ fc = c/(2ao) = p11c/(2πaguide) (4.56) where ao is the rectangular waveguide broadwall width, aguide is the circular waveguide ′ radius, and p 1.841 is the eigenvalue for TE mode, and c is the speed of light. 11 ≈ 11 The cutoﬀ wavelength is λc = c/fc.

98 Beam Calculation

The details of the feedhorn beam calculation technique can be found in [18] and [156]. Basically, this method matches boundary conditions across adjacent concentric cylindrical waveguide sections to determine the mode content at the aperture end of the feedhorn based on a TE11 excitation at the circular waveguide end. The beam pattern in the E- and H- planes is calculated directly from the modal content. The full beam patterns can be calculated from the E- and H-plane proﬁles, since the horn in this calculation is known to be a BOR1 antenna [159] from the symmetry of the calculation. The smooth-walled feedhorn is approximated by a proﬁle that consists of discrete cylindrical sections, each of constant radius. For this approximation to be valid, the section length ∆l should be less than λc/20. It is also possible in principle to dynamically set the length of each section to optimize the approximation to the local curvature of the horn to increase the speed of the optimization.

Penalty Function

Generally, the smooth-walled feeds have good return loss performance [156]. It is not necessary to include it in the penalty function. The penalty function is constructed to depend on cross polarization and the edge taper at a give angle deﬁned by the optics. For the 40 GHz feedhorn, the bandwidth is from 33 GHz to 43 GHz, that is

1.25 fc to 1.63 fc (∆f/f0 = 26.3 %). The penalty function to minimize is

N M 2 2 χ = αj∆j(fi) , (4.57) i=1 j=1 where i is the sum is over a discrete set of (N) frequencies in the optimization frequency band, and j sums over the number (M) of discrete parameters one wishes to take into account for the optimization. The weights αj can be adjusted. In this work, uniform weights (αj = 1) have been implemented. The CLASS feedhorn is optimized by minimizing this penalty function including only the cross polarization and edge taper (M = 2 in equation 4.57). Other parameters such as beam shape could also be

99 employed for diﬀerent optimization requirements. The explicit forms used for ∆1(f) and ∆2(f) are

XP (f) XP0 if XP (f) > XP0, ∆1(f) = − 0 if XP (f) XP0, ≤ ∆ (f) = ET (f) ET (4.58) 2 − 0 where XP (f) is the maximum of the cross-polarization XP (f) = Max[XP (f, θ)], ET (f) is the edge taper at a given angle at frequency f. Our target beam pattern was for the D-plane to be -10 dB at the azimuth angle of 14◦. Respectively, XP0 and

ET0 are the threshold cross polarization and edge taper level. If the cross polarization or edge taper at a sampling frequency is less than or equal to its critical value, then it does not contribute to the penalty function. Otherwise, its squared diﬀerence is added to the penalty function.

Feedhorn Optimization

As shown in Figure 4.19, the feedhorn was optimized in a multi-step process that employs a modified version of Powell’s method [160] at each step. Powell’s method is a rapidly-converging method for finding the minima of a multi-variables function without explicit analytical expression for its partial derivatives. In this method, every variable of the function is free to float during the optimization. Generically, this algorithm can produce an arbitrary profile. To produce a feed that is easily machinable, we impose a restriction that the optimization is limited to the subset of profiles for which the radius increases monotonically along the length of the horn. Without this constraint, the serpentine profiles explored in [161] are accessible. Given enough degrees of freedom, this method can recover the corrugated horn solution. An initial input is required for the modified Powell method. A profile that is constructed by a sin0.75 converter section and a flare section that matches the expansion of a Gaussian beam [162] is used for the initial profile. The feed radius, r, can be written analytically as a function of the distance along the length of the horn z, as:

100 Feedhorn beam calculaon method from James

Opmizaon Opmizaon Opmizaon Inial input: step 1: step 2: step 3: Sin0.75+Gaussian Final profile 5 points natural 10 points natural 20 points natural addon profile spline profile spline profile spline profile

XP0=25dB XP0=30dB XP0=34dB ET0=10 dB ET0=10 dB ET0=10 dB

Figure 4.19: Flow chart of smooth-walled feedhorn optimization. Optimization begins with a sin0.75 profile, the method from [18] is used to calculate the beam patterns. The feedhorn profile was found by this multi-step iterative solution with different thresholds in each step.

0.75 aguide + acsin (πz/L) if 0 z L/2, r(z) = ≤ ≤ (4.59) 2 1/2 aguide + ac 1 + [C(z L/2)] if L/2 < z L, { − } ≤ where C = (2/L)[((a a )/a )2 1]1/2 (4.60) f − guide c − ac is the radius at the end of the converter section, af is the final radius of the flare section and L is the total length of the feedhorn. The initial profile of the CLASS feedhorn has aguide = 0.293λc, ac = 0.650λc, af = 1.582λc and L = 8.789λc. This profile is then approximated by natural spline of a set of 5 points. In the first step,

XP0 and ET0 are set to -25 dB and -10 dB. The minimum of the penalty function is found by the modified Powell method in this 5-dimensional space. The output profile from the first step is the initial input to the next optimization step. In the following optimization steps, the number of points in the natural spline is increased to be 10 and 20. The modified Powell’s method optimizes the profile in 10-dimensional and 20-dimensional spaces. Based on the result from the first step,

XP0 is set to be -30 dB and -34 dB for the 10-dimensional and 20-dimensional space.

ET0 remains unchanged in these steps. In a previous work [156], a 560-points spline was used in the ﬁnal optimization

101 Table 4.5: CLASS 40 GHz Feedhorn Requirements Item Requirement Note

Waveguide 3.334 mm fc = 26.349 GHz, λc = 11.378 mm, WR 22.4 Bandwidth 33 - 43 GHz 1.25 fc - 1.63 fc , ∆f/f0 = 26.3 % Aperture 36.00 mm feedhorn wall thickness = 1.00 mm Length 100.00 mm D = 36.00 mm requires L 75 mm ≥ Cross pol 30 dB within 15◦ azimuth angle, across the bandwidth ≤ − Return loss 30 dB across the bandwidth ≤ − Edge taper 10 dB at 14◦, at center frequency (38 GHz) ≈ −

step. The 20-point spline provided a sufficient number of degrees of freedom to achieve the desire result since only small improvements are realized by doubling the number of points from 10 to 20 and starting with the 5-point spline did produce the general features of the final horn, and significantly reduces the time required by the slower 10-point and 20-point algorithm.

4.5.2 Smooth-walled Feedhorn for CLASS

Table 4.5 lists the requirements for optimizing the CLASS 40 GHz band feedhorn. The feedhorn is optimized in the bandwidth of 33 GHz to 43 GHz (section 4.2.2). The

input waveguide has a radius of 3.334 mm, with fc = 26.349 GHz, λc = 11.378 mm (equation 4.56). The packing pattern of the feedhorn array and the size of the focal plane set a limit of 38.00 mm on the outer diameter of each feedhorn. The optical

design specifies about a -10 dB edge taper at a 14◦ angle on the VPM. The beam at the angle greater than that will be terminated. The cross polarization should be lower than -30 dB within this angle and across the entire bandwidth and the return loss should be always lower than -30 dB. Figure 4.20 shows the feedhorn profile. A 20-point approximation of this profile is listed in table 4.6. A 500 point table can be found in appendix E. The final profile

has aguide = 0.293λc, ac = 0.853λc, af = 1.574λc and L = 8.789λc. Table 4.7 and ﬁgure 4.21 shows the cross polarization, return loss and edge from 30 to 50 GHz. The cross polarization at the trough near the center frequency is about

102 Smooth−walled Feedhorn Profile 20

0 Radius (mm) −10

−20 −10 0 10 20 30 40 50 60 70 80 90 100 Length (mm)

Figure 4.20: CLASS 40 GHz feedhorn proﬁle. The 10.00 mm long input waveguide has a radius of 3.334 mm, with fc = 26.349 GHz. The length of the feedhorn is 100.00 mm. The aperture is 35.828 mm. This is a monotonic proﬁle that allows a progressive milling technique.

Cross−Pol, Return−Loss and Edge−Taper

−10

Cross−Pol −20 Return−Loss Edge−Taper

−30

−40 Power (dB) Power

−50

−60

−70 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 Freqency (fc)

Figure 4.21: CLASS feedhorn performance from 30 to 50 GHz. The dashed lines deﬁne the -30 dB line, and the waveband limit of 33 GHz and 43 GHz. The cut oﬀ frequency is fc = 26.349 GHz.

103 Table 4.6: Feedhorn Proﬁle Approximation (in Millimeters)

Step Length (z) Radius (r)

0 0.00 3.334 1 5.00 4.555 2 10.00 5.749 3 15.00 6.882 4 20.00 7.906 5 25.00 8.755 6 30.00 9.330 7 35.00 9.578 8 40.00 9.672 9 45.00 9.776 10 50.00 9.811 11 55.00 9.836 12 60.00 9.866 13 65.00 11.042 14 70.00 12.919 15 75.00 15.010 16 80.00 16.718 17 85.00 17.577 18 90.00 17.838 19 95.00 17.892 20 100.00 17.914

-40 dB and about -30 dB at the edges of the bandwidth. The cross polarization is below -30 dB in entire Q band (33 - 45 GHz), which is better than the requirement (table 4.5). The return loss is about -30 dB at the low frequency edge and drops below -40 dB at high frequencies. The edge taper is about -10.8 dB at 38 GHz, and rise slowly towards the edges of the bandwidth. For a feedhorn with a perfect beam pattern and zero cross polarization, the edge taper at a given angle should decrease as the frequency increases (a wavelength eﬀect). The edge taper in our penalty function is deﬁned as the power of the diagonal plane (the average of the E- and H- plane) at

14◦. From the low to high frequency edges, the power levels of the E- and H-plane flip (see figure 4.24 and figure 4.25, at 33 GHz, E-plane < H-plane, while E-plane

104 Table 4.7: Feedhorn Performance

f f λ Cross Pol Return Loss Edge Taper

[GHz] [fc] [mm] [dB within 15◦] [dB] [dB at 14◦]

30 1.139 9.993 -22.77 -21.80 -7.65 31 1.177 9.671 -25.00 -24.11 -7.95 32 1.214 8.369 -26.71 -27.95 -8.32 33 1.252 9.085 -30.01 -30.09 -8.69 34 1.290 8.817 -31.26 -32.89 -9.20 35 1.328 8.565 -33.74 -37.01 -9.63 36 1.366 8.328 -34.30 -36.99 -10.05 37 1.404 8.102 -39.62 -45.20 -10.45 38 1.442 7.889 -39.00 -39.86 -10.80 39 1.480 7.687 -36.51 -59.07 -11.45 40 1.518 7.495 -35.75 -41.55 -11.65 41 1.556 7.312 -34.83 -64.50 -11.79 42 1.594 7.138 -32.49 -41.61 -11.88 43 1.632 6.972 -32.17 -49.95 -11.68 44 1.670 6.813 -30.21 -42.31 -11.57 45 1.708 6.662 -29.11 -43.99 -11.42 46 1.746 6.517 -27.85 -44.88 -11.15 47 1.784 6.379 -26.40 -41.39 -11.13 48 1.822 6.246 -25.57 -51.60 -10.84 49 1.860 6.118 -24.11 -40.68 -10.99 50 1.898 5.996 -23.57 -51.10 -10.75

> H-plane at 44 GHz), while their average value (diagonal plane) remains the same, ending up with higher edge tapers at frequency edges (an average effect). From 30 to 42 GHz, where the wavelength effect dominates, the edge taper drops. At higher frequency, where the average effect dominates, the edge taper increases with a slow rate.

Figure 4.22 and ﬁgure 4.23 show the beam pattern within a 90◦ angle from 33 ± to 44 GHz. The sidelobe level of this feedhorn is about -15 dB. The FWHM at the

center frequency is about 14.6◦. The penalty function only takes the beam within 14◦ into account. Figure 4.24 and ﬁgure 4.25 show the beam pattern zoomed in to within

15◦.

105 33GHz HWHM=8.37deg 34GHz HWHM=8.12deg 0 E Plane 0 H Plane D Plane -20 X Pol -20

Power (dB) -40 Power (dB) -40

-60 -60 -90 -70 -50 -30 -10 10 30 50 70 90 -90 -70 -50 -30 -10 10 30 50 70 90 Angle (degrees) Angle (degrees) 35GHz HWHM=7.93deg 36GHz HWHM=7.72deg 0 0

-20 -20

Power (dB) -40 Power (dB) -40

-60 -60 -90 -70 -50 -30 -10 10 30 50 70 90 -90 -70 -50 -30 -10 10 30 50 70 90 Angle (degrees) Angle (degrees) 37GHz HWHM=7.55deg 38GHz HWHM=7.33deg 0 0

-20 -20

Power (dB) -40 Power (dB) -40

-60 -60 -90 -70 -50 -30 -10 10 30 50 70 90 -90 -70 -50 -30 -10 10 30 50 70 90 Angle (degrees) Angle (degrees)

Figure 4.22: Beam patterns of the CLASS smooth-walled feedhorn within azimuth angles of 90◦, from 33 GHz to 38 GHz. ±

106 39GHz HWHM=7.10deg 40GHz HWHM=6.92deg 0 E Plane 0 H Plane D Plane -20 X Pol -20

Power (dB) -40 Power (dB) -40

-60 -60 -90 -70 -50 -30 -10 10 30 50 70 90 -90 -70 -50 -30 -10 10 30 50 70 90 Angle (degrees) Angle (degrees) 41GHz HWHM=6.73deg 42GHz HWHM=6.55deg 0 0

-20 -20

Power (dB) -40 Power (dB) -40

-60 -60 -90 -70 -50 -30 -10 10 30 50 70 90 -90 -70 -50 -30 -10 10 30 50 70 90 Angle (degrees) Angle (degrees) 43GHz HWHM=6.42deg 44GHz HWHM=6.26deg 0 0

-20 -20

Power (dB) -40 Power (dB) -40

-60 -60 -90 -70 -50 -30 -10 10 30 50 70 90 -90 -70 -50 -30 -10 10 30 50 70 90 Angle (degrees) Angle (degrees)

Figure 4.23: Beam patterns of the CLASS smooth-walled feedhorn within azimuth angles of 90◦, from 39 GHz to 44 GHz. ±

107 33GHz HWHM=8.37deg 34GHz HWHM=8.12deg 0 0

E Plane H Plane -20 -20 D Plane X Pol

Power (dB) -40 Power (dB) -40

-60 -60 -15 -12 -9 -6 -3 0 3 6 9 12 15 -15 -12 -9 -6 -3 0 3 6 9 12 15 Angle (degrees) Angle (degrees) 35GHz HWHM=7.93deg 36GHz HWHM=7.72deg 0 0

-20 -20

Power (dB) -40 Power (dB) -40

-60 -60 -15 -12 -9 -6 -3 0 3 6 9 12 15 -15 -12 -9 -6 -3 0 3 6 9 12 15 Angle (degrees) Angle (degrees) 37GHz HWHM=7.55deg 38GHz HWHM=7.33deg 0 0

-20 -20

Power (dB) -40 Power (dB) -40

-60 -60 -15 -12 -9 -6 -3 0 3 6 9 12 15 -15 -12 -9 -6 -3 0 3 6 9 12 15 Angle (degrees) Angle (degrees)

Figure 4.24: Beam patterns of the CLASS smooth-walled feedhorn within azimuth angles of 15◦, from 33 GHz to 38 GHz. ±

108 39GHz HWHM=7.10deg 40GHz HWHM=6.92deg 0 0

E Plane -20 H Plane -20 D Plane X Pol

Power (dB) -40 Power (dB) -40

-60 -60 -15 -12 -9 -6 -3 0 3 6 9 12 15 -15 -12 -9 -6 -3 0 3 6 9 12 15 Angle (degrees) Angle (degrees) 41GHz HWHM=6.73deg 42GHz HWHM=6.55deg 0 0

-20 -20

Power (dB) -40 Power (dB) -40

-60 -60 -15 -12 -9 -6 -3 0 3 6 9 12 15 -15 -12 -9 -6 -3 0 3 6 9 12 15 Angle (degrees) Angle (degrees) 43GHz HWHM=6.42deg 44GHz HWHM=6.26deg 0 0

-20 -20

Power (dB) -40 Power (dB) -40

-60 -60 -15 -12 -9 -6 -3 0 3 6 9 12 15 -15 -12 -9 -6 -3 0 3 6 9 12 15 Angle (degrees) Angle (degrees)

Figure 4.25: Beam patterns of the CLASS smooth-walled feedhorn within azimuth angles of 15◦, from 39 GHz to 44 GHz. ±

109 Table 4.8: Beam Parameters

f λ Beam Solid Antenna Aperture Main Beam [-] [-] Angle Gain Eﬃciency Eﬃciency

[GHz] [mm] [Sr] [dBi] [-] [within 14◦]

30 9.993 0.131 19.81 0.741 0.914 31 9.671 0.126 20.00 0.725 0.927 32 9.369 0.121 20.18 0.709 0.936 33 9.085 0.115 20.41 0.702 0.946 34 8.817 0.109 20.63 0.697 0.953 35 8.565 0.105 20.80 0.683 0.958 36 8.328 0.100 20.99 0.675 0.966 37 8.102 0.095 21.22 0.674 0.974 38 7.889 0.092 21.36 0.660 0.977 39 7.687 0.087 21.61 0.664 0.979 40 7.495 0.083 21.79 0.658 0.979 41 7.312 0.081 21.91 0.643 0.977 42 7.138 0.078 22.09 0.639 0.975 43 6.972 0.076 22.18 0.622 0.973 44 6.813 0.074 22.33 0.615 0.971 45 6.662 0.072 22.44 0.603 0.970 46 6.517 0.070 22.54 0.590 0.968 47 6.379 0.068 22.66 0.582 0.968 48 6.246 0.067 22.71 0.565 0.966 49 6.118 0.065 22.85 0.559 0.966 50 5.996 0.065 22.87 0.541 0.967

The beam parameters are shown in table 4.8. The antenna gain is at the level

of about 21 dBi, aperture efficiency is about 67% and the main beam (within 14◦) efficiency is above 95%. A tolerance study was conducted based on the averaged values of cross polarization, return loss and edge taper across the waveband. In the tolerance study, the tolerance (the x axis of figure 4.26) is the amplitude of the Gaussian random noise applied to the radius of feedhorn profile. For each tolerance between 0 and 300 m, these average values were calculated 120 times and then averaged. By taking the average value of cross polarization = -30 dB as the criterion, then this Q band feed-

110 $YHUDJHG&URVV3RO5HWXUHQ/RVVDQG(GJH7DSHU

&URVV3RO 5HWXUQ/RVV (GJH7DSHU

3RZHU G%

7ROHUDQFH XP

Figure 4.26: The averaged cross-pol, return-loss and edge-taper plot for the tolerance calculation from 0 to 300 um. For each tolerance, these values were from the average of 120 calculations. (The plots are noisy at large tolerance, more calculation would be required to smooth the plots.)

horn has a fabrication tolerance at 1σ level of about 250 m (0.010 inch). With a machining tolerance of 25 m (0.001 inch), this design can be scaled up to 400 GHz.

4.6 CLASS Detectors

4.6.1 Focal Plane

The focal planes of CLASS (ﬁgure 4.27) consist of feedhorn-coupled, TES bolometers. The smooth-walled feedhorn provides well-controlled symmetric angular beam pattern through the optics (see section 4.5). A symmetric planar orthomode trans- ducer (OMT) with the horizontal (H) and vertical (V ) probe antennas is utilized to couple the orthogonal linear polarizations at feedhorn throat into independent

111 Figure 4.27: Section view of CLASS 40 GHz focal plane. It consists of a array of 36 smooth-walled feedhorns, waveguide adapter, detector mounting plate and clips. The focal plane will operate at a temperature of 100 mK. Figure courtesy of Thomas Essinger-Hileman. superconducting microstrip transmission lines. Band-defining filters limit the spectral range for each of the H and V polarizations. The signals terminate in resistors thermally coupled to TESs that are capable of providing background-limited performance. A λ/4 backshort is positioned behind the OMT to maximize the power from the waveguide to the microstrip lines (figure 4.28). The CLASS 40 and 90 GHz detectors are designed by the Goddard Space Flight Center (GSFC) and the 150 GHz detectors are design by National Institute of Stan- dards and Technology (NIST). In the GSFC design, broadband hybrid couplers (Magic Tees) combine the signals from opposite antennas and outputs the difference between these signals. For the 150 GHz channel, the signals from opposite antennas

112 Figure 4.28: The feedhorn-couple TES bolometers set up [15] and prototype detector chip for the 40 GHz CLASS [19]. Left: The detector set up showing the feedhorn, detector housing, detector chip and backshort. Right: Photo of a 40 GHz prototype detector chip, showing the OMT, Magic Tees, ﬁlters and TES membranes.

share a thermal coupling to the same TES.

4.6.2 TES Bolometers

A TES consists of a superconducting ﬁlm operated in the narrow temperature region between the normal and superconducting state, where the electrical resistance varies between zero and its normal value. In this state, the device has a ﬁnite electrical resistance, R, that is less than the resistance in the fully non-superconducting state,

Rn. Energy (Pγ ) coupled to the detector increases its temperature, pushing it further into the non-superconducting state and thereby increasing its electrical resistance. This increase in resistance can be used to detect very small changes in temperature, and hence in energy [20]. Figure 4.29 shows the electro-thermal circuit diagram of a TES bolometer. In this simplest model, the bolometer has a heat capacity C at temperature T , which is linked to the thermal bath with temperature Tbath (T > Tbath) by a thermal conductance G.

The bolometer is heated up by the absorbed radiation Pγ and the Joule power PJ ,

and power PG is conducted away through the weak link:

113

Figure 4.29: The electro-thermal circuit diagram of a TES bolometer (modified from [20]). Left: Each pix with a heat capacity of C at temperature T is connected by a thermal link G to a thermal source with a temperature of Tbath. The total power to the pixel is P + P P . Right: TES is biased by I = V /R , in the case γ J − G B B B of RB RSH. For R RSH, the TES is bias by V = IBRSH, then fluctuations of R will≫ result in fluctuation≫ in current, which is read out by the inductor L and the superconducting quantum interference device (SQUID) amplifier.

114 dT C = P + P P (4.61) dt γ J − G where 2 PJ = I R (4.62)

is the power dissipated when bias the TES with current I, and

T PG = G(T )dT. (4.63) Tbath The electrical circuit in the right panel of ﬁgure 4.29 can be written as: dI L + IR = (I I)R (4.64) dt B − SH where I is the current running through TES. Operating at equilibrium (dT/dt = 0 and dI/dt = 0), the saturation power of a TES bolometer can be written as I R 2 P = P I2 R = P B SH R (4.65) sat G − min n G − R + R n SH n

where, Rn and Imin are the resistance and the current running through TES at normal state. In the limit of a voltage bias (R R ), and a narrow transition, so that n ≫ SH P V 2/R is approximately constant at equilibrium, equation 4.65 reduces to G ≈ o P = (1 R /R )P (4.66) sat − o n G

where, Ro is the resistance of the TES at equilibrium (operating point). The total NEP of a TES can be written as:

NEP = (NEP2 + NEP2 )1/2 (4k G T 2 + 2P hν)1/2 (4.67) det γ ≈ B o o γ where NEPdet is the NEP due to phonons noise in the detector, NEPγ is due to fuctuations in the radiation load from sky background. Go and To are thermal conductance and temperature of the TES at the equilibrium point. CLASS detectors are background limited, that is NEPdet is smaller than NEPγ from the background radiation.

115 A TES bolometer loses all sensitivity when the signal power exceeds Psat.To maximize the TES performance, the thermal conductance, Go, must be chosen to be large enough that any important signal does not saturate the bolometer. Increasing

Go so that the highest signal power does not saturate, however, degrades the NEP for even the lowest measured signal power. The requirement of CLASS 40 GHz detectors 17 is : P = 3.5 pW, G = 116 pW/K, T = 0.150 K and NEP = 1.2 10− sat o o det × 1/2 WHz− .

4.7 Lab Set up for Detector Testing

4.7.1 Cryostat

The cryostat for CLASS detector testing in the experimental cosmology lab at Johns Hopkins University is a model 104 Olympus pulse tube (PT) cryostat manufactured by the High Precision Devices (HPD) Inc. A Helium-4 (He-4) refrigerator and an adiabatic demagnetization refrigerator (ADR) are mounted on the 4 K plate of the cryostat. The He-4 refrigerator is launched from the 4 K plate with a base temperature of 2.7 K, while the ADR can be launched from 4 K plate or the He-4 head with a base temperature of about 660 mK. Figure 4.30 shows the section view of the lab cryostat. It is a two-stage PT cryostat, with cooling power of 40 W at 45 K (1st stage, 60 K plate) and 1.5 W at 4.2 K (2nd stage, 4 K plate). On the 4 K plate, there is a two stage ADR system with a gadolinium gallium garnet (GGG) crystal and a ferric ammonium alum (FAA) salt pill. Each pill has its own ultra low thermal conducting support structure isolating it from the 4 K ﬂange and the intermediate stage. The two stages operate at a typical temperature of 1 K (GGG) and 100 mK (FAA) with the cooling capacity of 1.2 J ∼ ∼ (GGG) and 118 mJ (FAA). A He-4 refrigerator with 80 J cooling capacity at 660 ∼ mK can be mounted on the 4 K plate optionally (left panel of ﬁgure 4.31).

116 Figure 4.30: Section view of model 104 Olympus ADR cryostat showing mechanical heat switch controller, vacuum valve, pulse tube (PT) head, 60 K plate, 4 K plate, adiabatic demagnetization refrigerator (ADR), high temp superconducting leads for 4 T magnet, thermal shielding, and vacuum jacket [21].

117 4.7.2 Thermometry

The thermometry of the lab cryostat includes a general thermometer readout system and an ADR Proportional-Integral-Derivative (PID) control system. The general sensor readout system consists of a Stanford Research Systems (SRS) SIM900 mainframe with GPIB port, two SIM925 octal four-wire multiplexers (MUXs), a SIM922 diode temperature monitor and a SIM921 AC resistance bridge. MUX 1 is for reading out silicon diodes and MUX 2 is for reading out ruthenium oxide (RuOx) and other resistance temperature detectors (RTDs). The ADR PID control system includes a Lakeshore model 370 AC resistance bridge, a calibrated GR-50-AA germanium resistance temperature (GRT) sensor (mounted inside the cryostat) and two Keithley model 2440 5 A sourcemeters. The above devices communicate with a cryostat computer through a National Instruments (NI) model GPIB-ENET/100 GPIB to Ethernet adapter (right panel of figure 4.31). The mechanical heat switch of the ADR is controlled via a NI USB-6009 Data Acquisition (DAQ) device. The thermometry is control by LabVIEW programs (see appendix F for details). For general thermometer readout, the program loops over the two octal MUXs to produce real time plots. In the mean time, it also displays and saves all data with time stamps. The readout process is in series. Due to the delay in each readout caused by the response time of the diode monitor and the AC resistance bridge, it take about 120 seconds to finish a single loop for reading out all channels of two MUXs. To regulate the ADR temperature, the PID program reads the temperature from the GRT sensor through the Lakeshore AC resistance bridge, calculates the output to the Keithley current source by the PID algorithm, and controls the current in the ADR magnet. This program also produces real time plots of the temperature and error between the temperature and the set point. The temperature and the magnet current are saved with time stamps in a file.

4.7.3 Cryostat Performance

Figure 4.32 shows the cool down curves of the cryostat from warm temperature ( 300 K) to cool temperature (equilibrium temperature). It takes about 24 hours ∼

118 Figure 4.31: Left: The ADR and the He-4 refrigerator mounted on the 4 K plate of the HPD cryostat in the experimental cosmology lab at Johns Hopkins University. Photo courtesy of David Larson. Right: the rack-mounted devices for cryostat thermometry. From top to bottom, they are, a SRS SIM900 mainframe with 2 MUXs, a diode moniter and an AC bridge, a front panel, a NI GPIB to Ethernet adapter, a Lakeshore 370 AC resistance bridge and two Keithley 2440 current sources.

for the cryostat to cool down to the state with stable temperature readouts. The typical values of the thermometers are listed in table 4.9. The 60 K plate can reach the temperature of 50.0 K and the 4 K plate can get as lower as 2.7 K. Launching from the 4 K plate with a 2.7 K base temperature, the ADR can last for about 210 hour at 100 mK with no load (FAA pill). Thus, the intrinsic thermal load of the FAA pill is around 0.15 W. Figure 4.33 shows the cooling curves (the magnet current versus time) of the ADR with the loads of from 2.0 to 10.0 W. Based on these curves, the FAA pill of the ADR have higher cooling capacities at lower loads. With a 2 W load, the cooling capacity is about 130 mJ, which is close to the theoretical number of 118 mJ. While with a 10 W load, the capacity drops to about 50 mJ.

4.7.4 Detector Readout

The signal from the TES bolometers is read out by two stages of the superconducting quantum interference device (SQUID) amplifier. The output of the SQUID amplifier is a voltage that is approximately a sinusoidal function of the magnetic flux

119 &U\RVWDW&RRO'RZQFXUYHV .SXOVHWXEH .SXOVHWXEH .SODWH .SODWH $'5PDJQHW

7HPSHUDWXUH .

7LPH +U

Figure 4.32: Cryostat cool down curves. It takes about 24 hours for the cryostat to cool down to the state with stable temperature readouts. The typical values of the thermometers are listed in table 4.9.

120 $'5FRROLQJFXUYHV#P. X: X: X: X: X:

&XUUHQW $

7LPH +U

Figure 4.33: ADR cooling curves at 100 mK, showing the magnet current versus time of the ADR with the loads of from 2.0 to 10.0 W. Based on these curves, the FAA pill of the ADR have higher cooling capacities at lower loads.

121 Table 4.9: Cryostat Thermometry Readout Thermometer Make Model Warm readout Cool readout Diode [-] [-] [K] [V] [K] [V] Magnet Lakeshore DT670 294.6 0.5720 2.701 1.619 4 K plate Lakeshore DT670 293.8 0.5739 2.722 1.619 60 K plate Lakeshore DT670 293.9 0.5738 50.28 1.073 60 K PT Lakeshore DT670 294.1 0.5732 35.94 1.097 4 K PT Lakeshore DT670 295.1 0.5709 2.745 1.618 He4 HS Unknow Unknow 300.0 0.5684 2.310 1.142 He4 Charcoal Lakeshore DT670 293.4 0.5749 2.937 1.614 RTD [-] [-] [K] [kΩ] [K] [kΩ] 50 mK Ruox Scientiﬁc Inst RO600 265.8 1.003 2.821 1.551 1 K Ruox Scientiﬁc Inst RO600 281.8 1.002 2.828 1.550 Magnet RTD AMI Unknow 293.0 0.1047 2.641 2.452 R1 Lakeshore RX-202A 115.0 2.000 2.918 3.146 R2 Lakeshore RX-202A 109.1 2.011 2.912 3.147 R3 Lakeshore RX-102A 216.1 1.003 2.725 1.548

φ through the SQUID junction loop [163]:

V = (R/2)(I2 (2I cos(πφ/φ ))2)1/2, (4.68) − c 0 where R is the resistance of the Josephson junctions, I is the SQUID bias current, Ic is the Josephson junction critical current, and φ0 is the magnetic flux quantum: φ0 15 = h/2e = 2.07 10− Wb. × As shown in the left panel of figure 4.34 (Cold Electronics), fluctuations in the TES current generate fluctuations in the flux of SQUID 1 (SQ 1), which is coupled to SQ 2 and then the SQUID Series Array (SSA). A flux-locked loop (FLL) is used to keep the system response linear, that is, the resulting change in SQUID voltage is a linear function of flux φ. The voltage output of the SSA (SSA SIG) is input to a differencing amplifier of which the other input is wired to a fixed voltage (SA OFFSET). Then, the output of this amplifier drives a feedback coil (Lfb), for coupling magnetic flux back to SQ 1. With the FLL, a fluctuation in current from the TES results in a

ﬂuctuation on the feedback coil, Lfb, which cancels the ﬂux from the input. To read out numbers of TES bolometers on a focal plane, the In-focal-plane SQUID multiplexers (MUXs) have been developed [164]. The TESs and SQUIDs

122 Cold Electronics Warm Electronics (MCE) Ibias TES Bias

Rshunt 14b n SQ1_FB D/A ÷2 Data Mode 2 LPF RTES I SA_BIAS ÷2m L fb Data Mode 1 15 k Lin PID SSA_SIG SQ1 14b + (110 ) A=195 A/D Data Mode 0 - SQ2 - ADC_OFFSET 0.1 Data Mode 3

SA_OFFSET

(110 )

SA_FB

SQ2_BIAS (SB)

SQ2_FB

SQ1_BIAS (RS)

Figure 4.34: The FLL block diagram for TES detector readout, showing the cold electronics inside the cryostat and the warm electronics (MCE) [22].

are operated at cold temperature in the cryostat (left panel of figure 4.34). The Multi-Channel Electronics (MCE), which controls the bias setting and the FFL feedback control (warm electronics) is mounted on the wall of the cryostat with magnetic shielding (right panel of figure 4.34). The MCE is provided by the University of British Columbia (UBC). Figure 4.35 shows a photo of the MCE at Johns Hopkins University. The MCE controls the SQUID amplifiers and MUXs, and reads signals from the TES array. Each box of the MCE is in turn connected by fiber optic cables to data-acquisition computers running real-time Linux and data-acquisition software (DAS).

123 Figure 4.35: This photo shows the Multi-Channel Electronics (MCE) mounted on the wall the cryostat in the experimental cosmology lab at Johns Hopkins University. The MCE is connected to a data-acquisition computer by a pair of ﬁber optic cables (the orange wires). Photo courtesy of David Larson.

124 Appendix A

M17 Polarization Data

A.1 Polarziation Spectrum: 450 um vs 60 um

1 1 2 2 3 ∆α ∆δ P450 σp P.A. σP.A. P60 σp P.A. σP.A. P450/P60 80.0 44.0 2.2 0.5 18.5 5.9 6.7 0.5 21.7 2.2 0.33 70.1 -3.5 1.5 0.3 21.9 5.2 4.5 0.3 23.0 1.6 0.33 62.7 17.9 1.8 0.2 17.6 3.8 4.5 0.2 22.8 1.4 0.40 60.3 -51.0 2.5 0.4 38.2 4.2 4.8 0.5 33.6 2.8 0.52 57.8 39.2 2.0 0.2 15.4 3.4 5.3 0.2 20.6 1.3 0.38 50.4 60.6 2.4 0.3 23.4 3.0 5.7 0.3 26.0 1.7 0.42 42.9 13.1 1.1 0.2 7.3 4.3 2.8 0.2 14.3 1.8 0.39 38.0 -58.1 2.1 0.2 40.6 2.7 5.2 0.4 43.5 2.1 0.40 33.0 -36.8 1.4 0.2 33.0 3.0 4.7 0.3 28.8 1.7 0.30 30.6 53.5 1.6 0.2 19.7 2.9 3.8 0.2 15.8 1.9 0.42 28.1 -15.4 1.4 0.1 12.7 2.6 3.7 0.2 17.9 1.6 0.38 23.1 77.2 1.6 0.2 20.4 3.2 3.4 0.5 21.3 3.7 0.47 13.2 -41.5 1.5 0.1 26.4 2.2 3.8 0.3 34.4 2.0 0.39

1Oﬀsets in arcseconds from 18h17m32s,-16◦14′50′′ (B1950.0). 2Position angle of E vector east from north. 3Median = 0.390, mean = 0.395 and std = 0.056.

125 Figure A.1: 60 um polarization vectors from Stokes ([23], Yellow) and the 450 um h m s result from SHARP (smoothed to 22′′ resolution, Red), center at 18 17 32 ,-16◦14′25′′ (B1950.0).

126 A.2 Polarziation Spectrum: 450 um vs 100 um

1 1 2 2 3 ∆α ∆δ P450 σp P.A. σP.A. P100 σp P.A. σP.A. P450/P100 119.9 -160.4 3.0 0.5 36.9 4.7 3.9 0.3 27.5 2.3 0.77 110.0 -124.8 1.4 0.2 30.7 4.4 4.4 0.2 23.0 1.6 0.32 100.1 -89.1 1.6 0.2 16.1 3.0 3.8 0.3 20.4 2.2 0.42 92.7 -205.5 1.5 0.3 50.5 6.1 3.5 0.3 51.9 2.9 0.43 90.2 -51.1 2.2 0.2 22.0 2.4 4.2 0.3 26.3 2.0 0.52 85.3 -169.9 2.0 0.2 41.4 2.7 3.5 0.2 35.5 1.9 0.57 80.3 -17.9 1.7 0.2 28.6 3.6 3.4 0.4 34.7 3.1 0.50 63.0 -98.6 1.2 0.1 5.7 1.9 2.3 0.2 10.1 1.8 0.52 55.6 -63.0 1.3 0.1 9.8 1.9 2.8 0.2 13.2 2.4 0.46 45.7 -25.0 1.5 0.1 17.2 1.8 2.0 0.1 25.5 1.9 0.75 38.3 -143.8 1.4 0.1 11.4 1.4 2.7 0.3 21.3 3.0 0.52

1Oﬀsets in arcseconds from 18h17m30s,-16◦13′03′′ (B1950.0). 2Position angle of E vector east from north. 3Median = 0.520, mean = 0.525 and std = 0.128.

127 Figure A.2: 100 um polarization vectors from Stokes ([23], Green) and the 450um h m s result from SHARP (smoothed to 35′′ resolution, Red), center at 18 17 32 ,-16◦14′25′′ (B1950.0).

128 A.3 Polarziation Spectrum: 450 um vs 350 um at RA > 18h17m30s

1 1 2 2 3 ∆α ∆δ P450 σp P.A. σP.A. P350 σp P.A. σP.A. P450/P350 71.7 -54.6 1.6 0.3 35.4 5.4 2.0 0.2 27.8 3.5 0.80 61.8 -19.0 1.1 0.2 20.4 4.9 1.5 0.1 13.1 2.6 0.73 59.4 -76.0 2.1 0.3 40.2 4.2 2.3 0.3 35.1 3.7 0.91 56.9 -2.4 1.4 0.2 5.9 3.7 2.3 0.1 4.5 1.7 0.61 54.4 71.2 1.8 0.4 30.1 6.5 2.3 0.4 24.5 5.1 0.78 51.9 14.2 1.6 0.2 9.7 3.3 2.0 0.2 7.3 2.9 0.80 44.5 -23.8 1.1 0.2 15.3 4.1 2.0 0.1 5.9 1.2 0.55 42.0 49.9 1.7 0.2 22.0 3.8 1.3 0.2 15.0 3.9 1.31 39.6 -7.1 1.3 0.2 2.4 3.3 2.0 0.1 3.0 1.0 0.65 37.1 66.5 1.7 0.3 21.9 4.3 1.2 0.2 17.2 4.6 1.42 34.6 9.5 1.3 0.1 2.8 3.2 2.2 0.1 5.0 1.0 0.59 32.1 -45.1 1.4 0.1 12.0 2.3 1.9 0.1 6.7 0.9 0.74 29.7 26.1 1.3 0.1 17.7 3.2 1.8 0.1 7.9 1.3 0.72 27.2 -28.5 1.8 0.1 -0.1 1.9 2.2 0.1 180.0 0.6 0.82 24.7 -85.5 2.2 0.2 35.2 2.4 1.7 0.2 30.9 3.4 1.29 22.2 -11.9 1.5 0.1 1.3 2.0 2.1 0.1 178.7 0.7 0.71 19.8 61.7 1.7 0.2 19.3 2.7 1.0 0.1 14.6 3.4 1.70 17.3 4.7 1.4 0.1 0.5 2.2 2.0 0.0 177.7 0.6 0.70 14.8 -49.9 1.7 0.1 -0.9 1.5 1.8 0.1 178.6 0.8 0.94 12.4 21.4 1.1 0.1 4.6 3.1 1.4 0.1 177.3 1.1 0.79 9.9 -33.3 2.2 0.1 -7.8 1.1 2.1 0.1 177.1 0.6 1.05 4.9 -16.6 1.8 0.1 -9.7 1.1 2.0 0.0 173.2 0.5 0.90

1Oﬀsets in arcseconds from 18h17m31.4s,-16◦14′25′′ (B1950.0). 2Position angle of E vector east from north. 3Median = 0.795, mean = 0.887 and std = 0.289.

129 A.4 Polarziation Spectrum: 450 um vs 350 um at RA < 18h17m30s

1 1 2 2 3 ∆α ∆δ P450 σp P.A. σP.A. P350 σp P.A. σP.A. P450/P350 2.5 -73.6 2.0 0.2 0.4 2.9 0.6 0.1 2.3 3.3 3.33 2.5 57.0 1.2 0.1 11.3 2.8 0.5 0.1 6.6 2.9 2.40 -0.0 -0.0 1.4 0.1 -15.1 1.4 1.6 0.0 167.3 0.6 0.87 -2.5 -57.0 1.6 0.2 -8.5 2.7 1.1 0.1 173.4 1.7 1.45 -5.0 16.6 1.3 0.1 -22.2 1.8 1.1 0.0 162.1 0.8 1.18 -7.4 -38.0 2.0 0.1 -7.4 1.4 1.2 0.1 174.5 1.2 1.67 -9.9 33.2 1.1 0.1 -18.6 2.3 0.7 0.0 156.9 1.6 1.57 -12.4 -21.4 2.1 0.1 -13.5 1.0 1.7 0.0 171.4 0.6 1.24 -14.9 49.9 0.8 0.1 -6.5 2.7 0.4 0.0 174.3 3.7 2.00 -17.3 -4.8 1.5 0.1 -19.8 1.3 1.4 0.0 167.1 0.7 1.07 -24.8 -45.1 0.8 0.1 -20.1 4.5 0.8 0.1 153.8 2.1 1.00 -29.7 -26.1 1.8 0.1 -13.2 1.8 1.1 0.1 159.0 1.8 1.64 -34.6 -9.5 1.9 0.1 -12.9 1.2 1.4 0.1 165.7 1.3 1.36 -42.1 80.7 0.7 0.1 -8.5 4.0 0.6 0.0 179.1 2.0 1.17 -44.5 -104.5 3.4 0.3 -46.1 2.6 2.1 0.2 141.6 2.3 1.62 -47.0 -30.9 1.5 0.1 -27.3 2.7 1.4 0.1 143.0 2.0 1.07 -49.5 -87.9 3.8 0.4 -43.5 2.4 2.5 0.1 140.7 1.4 1.52 -56.9 2.4 1.2 0.1 -16.6 1.3 0.9 0.1 164.5 2.5 1.33 -59.4 76.0 1.5 0.2 -26.3 3.4 0.5 0.1 151.3 3.1 3.00 -61.9 -109.3 4.8 0.4 -47.7 2.2 2.8 0.3 136.5 3.4 1.71 -66.8 -92.6 4.5 0.3 -45.5 1.9 3.7 0.2 136.9 1.3 1.22 -66.8 -19.0 1.6 0.1 -29.3 2.5 1.5 0.1 147.7 1.9 1.07 -71.7 54.6 0.7 0.1 -55.4 5.0 0.2 0.1 123.4 10.8 3.50 -76.7 71.2 1.4 0.2 -49.6 4.9 0.9 0.1 128.6 1.8 1.56 -89.1 -80.8 6.3 0.9 -49.8 3.3 3.3 0.3 135.8 2.9 1.91 -94.0 66.5 1.7 0.4 -52.9 6.9 0.8 0.2 128.3 7.6 2.12 -94.0 -64.1 4.4 1.1 -50.4 6.5 3.6 0.3 139.3 2.3 1.22 -94.0 9.5 1.3 0.3 -39.6 8.1 1.9 0.5 134.1 7.1 0.68

1Oﬀsets in arcseconds from 18h17m31.4s,-16◦14′25′′ (B1950.0). 2Position angle of E vector east from north. 3Median = 1.485, mean = 1.624 and std = 0.688.

130 Figure A.3: 350 um polarization vectors from Hertz ([24]) and the 450 um result from h m s SHARP (smoothed to 20′′ resolution, Red), center at 18 17 32 ,-16◦14′25′′ (B1950.0). Blue: Hertz vectors at RA > 18h17m30s, Green: Hertz vectors at RA < 18h17m30s

131 A.5 Polarization Vectors

1 1 2 ∆α ∆δ P σp P.A. σP.A. ∆α ∆δ P σp P.A. σP.A. 70.2 -114.0 5.8 2.5 51.7 12.1 -18.9 -19.0 2.5 0.2 -18.4 2.7 70.2 -95.0 5.0 2.1 36.4 11.9 -18.9 -9.5 1.6 0.2 -23.6 3.9 70.1 -57.0 3.1 1.5 32.0 12.8 -18.9 -0.0 1.4 0.3 -34.6 5.9 70.1 -38.0 3.4 1.7 5.6 13.7 -18.9 9.5 1.4 0.3 -29.1 5.5 70.1 76.0 10.1 5.0 37.3 13.3 -18.9 19.0 1.6 0.2 -26.9 3.8 60.3 -47.5 2.6 0.8 36.1 8.8 -18.9 28.5 1.3 0.3 -19.8 7.6 60.2 -19.0 2.5 0.8 14.7 8.4 -18.9 38.0 1.2 0.3 -20.4 7.8 60.2 -9.5 2.2 0.9 28.8 11.7 -18.9 47.5 1.0 0.3 -19.2 8.9 60.2 -0.0 2.6 1.1 21.8 11.4 -18.9 57.0 1.0 0.3 -2.1 9.4 60.2 9.5 2.8 0.9 3.4 8.3 -18.9 66.5 1.9 0.4 -3.9 5.4 60.2 19.0 4.1 2.0 25.7 12.4 -18.9 76.0 1.1 0.2 2.8 6.3 60.2 28.5 5.0 1.3 27.5 6.7 -18.9 85.5 1.6 0.2 2.0 3.2 60.2 38.0 3.6 1.6 17.3 12.6 -18.9 95.0 1.2 0.3 10.3 7.2 50.4 -95.0 1.9 0.7 43.9 10.8 -18.9 104.5 1.7 0.8 30.2 12.8 50.4 -85.5 2.7 0.7 48.8 7.2 -28.8 -47.5 1.0 0.5 -34.8 13.3 50.4 -76.0 3.8 1.1 33.0 8.3 -28.8 -28.5 2.0 0.3 -8.6 4.9 50.4 -19.0 1.9 0.9 31.0 13.9 -28.8 -19.0 2.9 0.8 -3.0 8.0 50.4 -0.0 2.1 0.8 6.1 10.3 -28.8 -9.5 1.3 0.3 -22.3 6.0 50.4 9.5 1.6 0.7 -8.6 12.0 -28.8 -0.0 1.2 0.2 -26.0 5.8 50.4 28.5 3.1 0.8 17.5 7.4 -28.8 9.5 1.4 0.3 -31.2 6.1 50.3 47.5 2.8 1.0 33.1 9.5 -28.8 19.0 1.0 0.3 -33.3 7.7 40.5 -85.5 2.4 1.0 23.1 11.4 -28.8 28.5 0.8 0.3 -39.5 9.4 40.5 -66.5 3.7 1.3 42.9 9.3 -28.8 38.0 1.0 0.4 -12.5 13.2 40.5 -57.0 1.9 0.8 32.9 11.8 -28.8 47.5 0.7 0.2 -7.9 9.3 40.5 -47.5 2.0 0.7 49.5 9.5 -28.8 57.0 1.0 0.2 15.6 5.7 40.5 -28.5 2.1 0.6 36.4 7.9 -28.8 66.5 1.0 0.2 0.8 4.9 40.5 -0.0 1.7 0.7 6.8 10.9 -28.8 76.0 1.1 0.2 -15.0 4.8 40.5 9.5 2.4 0.6 0.8 7.1 -28.8 85.5 1.4 0.2 5.7 5.1

132 40.5 19.0 2.5 0.8 20.9 8.8 -28.8 95.0 1.0 0.2 12.3 6.6 40.5 38.0 3.7 1.1 16.5 6.9 -28.8 104.5 1.1 0.5 40.3 12.7 30.6 -104.5 1.8 0.9 42.6 14.3 -38.7 -104.5 1.8 0.9 -38.8 13.8 30.6 -95.0 2.2 0.7 33.5 8.1 -38.7 -76.0 8.8 3.3 -71.6 10.1 30.6 -85.5 1.6 0.7 40.6 13.0 -38.7 -28.5 1.7 0.5 -3.3 8.7 30.6 -76.0 3.7 1.0 46.3 6.8 -38.7 -19.0 3.1 0.4 -7.6 3.6 30.6 -66.5 2.1 1.0 45.4 13.9 -38.7 -9.5 2.6 0.4 -6.3 3.8 30.6 -57.0 1.6 0.7 7.0 11.8 -38.7 -0.0 1.4 0.2 -10.7 4.0 30.6 -38.0 2.0 0.6 11.0 8.9 -38.7 9.5 1.1 0.2 -32.1 4.7 30.6 -28.5 2.1 0.8 12.9 11.2 -38.7 19.0 1.3 0.3 -37.8 7.6 30.6 -0.0 1.2 0.6 17.2 13.8 -38.7 28.5 1.0 0.2 -41.7 6.9 30.6 9.5 2.1 0.6 -7.7 7.7 -38.7 38.0 1.1 0.3 -43.9 6.6 30.6 28.5 2.0 0.9 15.7 13.2 -38.7 76.0 1.3 0.4 18.4 8.8 20.7 -104.5 1.9 0.7 51.7 10.5 -38.7 95.0 1.2 0.2 11.8 5.8 20.7 -95.0 1.8 0.7 47.9 11.8 -48.6 -104.5 3.1 1.4 89.6 12.8 20.7 -85.5 2.8 0.5 36.6 5.4 -48.6 -95.0 5.2 1.2 -37.6 6.2 20.7 -76.0 3.5 0.7 36.0 5.7 -48.6 -76.0 4.4 2.0 -75.1 12.3 20.7 -66.5 2.1 0.7 41.0 9.0 -48.6 -57.0 3.9 1.3 -72.4 9.5 20.7 -57.0 0.8 0.4 30.5 14.0 -48.6 -38.0 1.1 0.5 -42.3 12.9 20.7 -47.5 2.1 0.5 23.5 6.7 -48.6 -28.5 1.3 0.6 -19.8 13.6 20.7 -38.0 1.5 0.5 11.1 9.5 -48.6 -19.0 2.9 0.7 -6.2 6.3 20.7 -28.5 2.4 0.5 -0.4 6.1 -48.6 -9.5 1.5 0.4 -10.4 7.2 20.7 -9.5 1.5 0.5 6.9 9.4 -48.6 -0.0 1.7 0.2 -17.2 3.2 20.7 -0.0 2.3 0.5 4.1 5.6 -48.6 9.5 1.3 0.1 -28.0 2.5 20.7 28.5 2.1 1.0 28.3 13.1 -48.6 19.0 0.9 0.2 -21.5 5.6 20.7 47.5 1.6 0.7 23.5 13.0 -48.6 28.5 0.5 0.1 -32.0 7.1 20.7 66.5 2.4 0.9 26.6 10.5 -48.6 38.0 0.8 0.1 -43.7 5.2 20.7 76.0 2.8 1.2 42.8 11.6 -48.6 47.5 0.5 0.2 -35.5 10.4 20.7 85.5 3.0 1.3 25.2 11.9 -48.6 66.5 1.0 0.2 3.6 6.2 10.8 -85.5 3.2 1.3 0.1 10.1 -48.6 76.0 0.7 0.2 -31.6 10.3

133 10.8 -66.5 2.9 0.7 24.8 6.7 -48.6 95.0 1.2 0.4 30.1 10.2 10.8 -57.0 1.7 0.4 15.1 6.6 -48.6 104.5 1.3 0.6 29.0 13.6 10.8 -47.5 1.2 0.3 0.6 6.8 -58.5 -114.0 6.2 3.0 -64.0 12.5 10.8 -38.0 1.5 0.3 -1.1 6.1 -58.5 -104.5 3.0 0.9 -56.9 8.5 10.8 -28.5 2.0 0.5 -7.8 6.8 -58.5 -95.0 3.3 0.8 -36.6 6.9 10.8 -19.0 2.2 0.6 2.1 8.2 -58.5 -85.5 6.4 2.3 -36.3 8.8 10.8 -9.5 2.5 0.6 0.2 6.7 -58.5 -57.0 3.2 1.4 -70.9 11.9 10.8 -0.0 1.6 0.5 3.6 7.9 -58.5 -28.5 2.2 0.6 -42.3 7.0 10.8 9.5 1.0 0.4 22.6 11.9 -58.5 -19.0 2.7 0.6 -16.1 6.3 10.8 19.0 1.0 0.5 15.3 13.3 -58.5 -9.5 2.9 0.5 -7.1 4.8 10.8 28.5 1.8 0.6 20.7 8.7 -58.5 -0.0 1.4 0.3 -7.7 5.5 10.8 38.0 2.6 0.6 19.6 6.6 -58.5 9.5 1.1 0.2 -3.9 4.0 10.8 47.5 1.6 0.6 26.5 10.0 -58.5 28.5 0.5 0.2 -30.0 10.5 10.8 66.5 2.3 0.9 9.6 11.9 -58.5 38.0 0.9 0.2 -40.6 6.4 10.8 76.0 2.8 1.4 19.5 14.1 -58.5 57.0 1.3 0.3 -30.2 8.0 10.8 114.0 5.0 2.0 24.1 11.3 -58.5 66.5 3.0 0.4 -4.3 3.9 0.9 -104.5 1.7 0.7 33.2 11.5 -58.5 76.0 1.2 0.4 -7.7 9.9 0.9 -85.5 1.8 0.7 24.1 11.1 -68.4 -114.0 11.1 4.2 -52.5 7.8 0.9 -76.0 2.1 0.8 5.0 10.8 -68.4 -104.5 6.2 1.1 -46.9 4.3 0.9 -66.5 2.2 0.5 -3.7 7.1 -68.4 -95.0 3.4 0.7 -41.1 5.4 0.9 -57.0 2.4 0.4 -7.6 5.2 -68.4 -85.5 3.9 0.9 -54.6 6.0 0.9 -47.5 2.2 0.3 -9.5 4.1 -68.4 -38.0 2.2 1.0 -64.1 12.2 0.9 -38.0 2.8 0.3 -4.7 2.9 -68.4 -28.5 2.5 0.5 -42.1 5.9 0.9 -28.5 2.6 0.3 -3.2 3.6 -68.4 -19.0 2.1 0.5 -28.6 6.2 0.9 -19.0 1.4 0.4 -3.1 8.0 -68.4 -9.5 2.0 0.4 -23.2 5.9 0.9 -9.5 1.6 0.3 -4.2 6.2 -68.4 -0.0 1.6 0.3 -18.6 4.6 0.9 -0.0 1.8 0.4 -17.7 6.5 -68.4 9.5 0.7 0.1 -20.5 5.0 0.9 9.5 1.1 0.4 -2.1 11.1 -68.4 19.0 0.9 0.1 -18.3 4.3 0.9 19.0 1.6 0.8 -4.1 13.1 -68.4 28.5 0.5 0.1 -2.1 6.9 0.9 38.0 1.4 0.4 17.0 9.3 -68.4 47.5 1.0 0.3 -75.8 7.5

134 0.9 47.5 1.8 0.4 5.4 6.9 -68.4 57.0 1.2 0.4 -65.4 10.4 0.9 57.0 1.3 0.4 16.6 9.0 -68.4 66.5 2.1 0.7 -31.4 9.0 0.9 66.5 1.5 0.6 18.3 10.8 -68.4 76.0 2.2 0.8 -1.6 10.3 0.9 123.5 16.6 8.0 25.1 12.0 -78.3 -95.0 7.4 1.5 -41.7 4.6 -9.0 -66.5 1.7 0.6 -8.6 11.2 -78.3 -85.5 4.2 1.1 -42.0 6.6 -9.0 -57.0 2.1 0.6 -14.0 8.7 -78.3 -76.0 5.2 1.3 -54.0 6.3 -9.0 -47.5 2.1 0.7 -3.4 9.8 -78.3 -38.0 5.1 1.3 -69.2 7.0 -9.0 -38.0 2.5 0.4 -8.5 4.5 -78.3 -28.5 3.3 1.3 -67.2 11.3 -9.0 -28.5 2.4 0.3 -8.3 3.1 -78.3 -19.0 1.2 0.6 -43.4 14.3 -9.0 -19.0 2.2 0.2 -18.9 2.9 -78.3 -9.5 1.6 0.5 -48.6 9.0 -9.0 -9.5 2.0 0.3 -8.8 4.4 -78.3 9.5 1.9 0.4 -18.0 5.4 -9.0 -0.0 1.4 0.3 -10.1 5.7 -78.3 28.5 1.0 0.2 -9.2 5.9 -9.0 9.5 1.7 0.3 -20.7 5.5 -78.3 66.5 3.0 0.8 -29.3 7.8 -9.0 19.0 1.3 0.3 -15.0 7.6 -88.2 -85.5 5.1 2.2 -55.5 8.9 -9.0 38.0 1.9 0.6 -6.4 8.9 -88.2 -76.0 6.5 2.4 -50.2 7.0 -9.0 57.0 1.2 0.5 14.2 11.8 -88.2 -66.5 5.9 2.3 -40.5 9.6 -9.0 66.5 1.3 0.5 13.4 9.8 -88.2 -28.5 2.7 1.1 -37.4 10.9 -9.0 76.0 1.2 0.3 -32.0 8.1 -88.2 -19.0 2.3 1.1 -16.1 13.1 -9.0 85.5 1.1 0.3 -18.6 6.6 -88.2 -9.5 3.5 0.8 -24.2 6.7 -9.0 95.0 1.7 0.5 17.6 9.2 -98.1 -57.0 4.3 1.5 -50.4 9.2 -18.9 -47.5 1.5 0.5 11.1 9.6 -98.1 -9.5 4.3 1.4 -15.2 9.1 -18.9 -38.0 1.9 0.4 1.9 5.6 -98.1 -0.0 2.6 0.7 -45.7 8.1 -18.9 -28.5 1.4 0.3 -8.3 5.8 -98.1 47.5 1.2 0.4 -47.4 9.3

1Oﬀsets in arcseconds from 18h17m32s,-16◦14′25′′ (B1950.0). 2Position angle of E vector east from north.

135 Appendix B

Blackbody Radiation

All matter emits electromagnetic radiation when it has a temperature above absolute zero. A black body is an idealized physical body that absorbs all incident radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum that depends on the body’s temperature. The thermal radiation from a black body is called black body radiation. Planck’s law describes the radiation from a black body with a temperature of T:

2hν3 1 B (T ) = (B.1) ν c2 ehν/(kT ) 1 − Where h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant 1 2 1 1 and Bν(T ) is in the unit of Js− m− sr− Hz− . In the Wien limit, where hν kT , the Plank’s spectrum is approximately: ≫ 3 2hν hν/(kT ) B (T ) e− (B.2) ν ≈ c2 At low frequency range, where hν kT (Rayleigh-Jeans limit), it is approxi- ≪ mately: B (T ) 2kT ν2/c2 (B.3) ν ≈ Figure B.1 shows the Plank spectrum, Wien limit and Rayleigh-Jeans limit of a black body with a temperature of 2.725K. The black body radiation spectrum peaks

136 − %ODFN%RG\5DGLDWLRQ 7 . [ 3ODQFN :LHQ

5D\OHLJK−-HDQV

− +]

− VU

− P

− -V

ν %

ν *+]

Figure B.1: The Planck, Wien and Rayleigh-Jeans spectrum of a 2.725 K black body. The Wien limit is a good approximation at ν > 250 GHz and the Rayleigh-Jeans limit works well below 20 GHz.

at ∂Bν /∂ν = 0: T ν 2.82kT/h = 58.7 GHz. (B.4) peak ≈ × K

The CMB has a thermal black body spectrum at a temperature of Tcmb = 2.725 K. In the Planck spectrum, it peaks at the microwave range frequency of about 160.2 GHz, corresponding to a wavelength of 1.873 mm.

137 Appendix C

NEP of Photons in a Blackbody Radiation Field

1 The variance in the number n of photons in a given state x = hν/(kBTs), is σ2 = n(n + 1), where αǫf n(ν) = n(x) = (C.1) ex 1 − where f is the transmissivity of the optics, Ts, ǫ are the temperature and emissivity of the source, α is the detector absorptivity. Then, the variance in energy is n(n+1)h2ν2. The number of states traveling toward the detector in a volume of ctA, is (2ν2/c3)ctAΩdν, where 2ν2/c3 is the number of states per unit volume per solid angle per frequency [25], Ω is the solid angle of the beam, A is the eﬀective area, and t is the integration time. The total number of states in the waveband is: (2ν2/c3)ctAΩdν (C.2) Then the total variance of energy is:

σ2 = (2ν2/c3)ctAΩn(n + 1)h2ν2dν 2AΩ (k T )5 x4 αǫf = t B s (1 + )(αǫf)dx (C.3) c2 h3 ex 1 ex 1 − − 1This chapter is mostly from [142]

138 NEP is deﬁned as the error in power in a half-second integration (section 4.2.1):

2σ2 NEP2 = t 4AΩ (k T )5 x4 αǫf = B s (1 + )(αǫf)dx (C.4) c2 h3 ex 1 ex 1 − − 2 1/2 NEP has the unit of W Hz− .

139 Appendix D

A Low Cross-Polarization Smooth-Walled Horn with Improved Bandwidth

1 Many precision microwave applications, including those associated with radio astronomy, require feedhorns with symmetric E- and H-plane beam patterns that possess low sidelobes and cross-polarization control. A common approach to achiev- ing these goals is a “scalar” feed, which has a beam response that is independent of azimuthal angle. Corrugated feeds [165] approximate this idealization by providing the appropriate boundary conditions for the HE11 hybrid mode at the feed aperture. Alternatively, an approximation to a scalar feed can be obtained with a multimode feed design. One such “dual-mode” horn is the Potter horn [166]. In this implemen- tation, an appropriate admixture of TM11 is generated from the initial TE11 mode using a concentric step discontinuity in the waveguide. The two modes are then phased to achieve the proper ﬁeld distribution at the feed aperture using a length of waveguide. The length of the phasing section limits the bandwidth due to the dispersion between the modes. Lier [167] has reviewed the cross-polarization properties of dual-mode horn antennas for selected geometries. Other authors have produced variations on this basic design concept [168, 169]. Improvements in the bandwidth

1This appendex is from [156]

140 have been realized by decreasing the phase difference between the two modes by 2π [170, 171]. To increase the bandwidth, it is possible to add multiple concentric step continu- ities with the appropriate modal phasing [172, 173]. A variation on this technique is to use several distinct linear tapers to generate the proper modal content and phasing [174, 175]. Operational bandwidths of 15-20% have been reported using such techniques. A related class of devices is realized by allowing the feedhorn profile to vary smoothly rather than in discrete steps. Examples of such smooth-walled feedhorns with 15% fractional bandwidths exist in the literature [162, 176]. ∼ In this work, we describe the design and optimization of a smooth-walled feed that has a 30% operational bandwidth, over which the cross-polarization response is better than -30 dB. The optimization technique is described, and the performance of the feed is compared with other published dual-mode feedhorns. The feedhorn described here has a monotonic profile that allows it to be manufactured by progressively milling the profile using a set of custom tools.

D.1 Smooth-walled Feedhorn Optimization

The performance of a feedhorn can be characterized by angle- and frequency- dependent quantities that include beam width, sidelobe response and cross-polarization. Quantities such as reflection coefficient and polarization isolation that only depend on frequency are also important considerations. All of these functions are dependent upon the shape of the feed profile. In the optimization approach described, a weighted penalty function is used to explore and optimize the relationship between the feed profile and the electromagnetic response.

D.1.1 Beam Response Calculation

The smooth-walled horn was approximated by a proﬁle that consists of discrete waveguide sections, each of constant radius. With this approach, it was important to verify that each section is thin enough that the model is a valid approximation of

141 the continuous profile. For profiles relevant to our design parameters, section lengths of ∆l λ /20 were found to be sufficient by trial and error, where λ is the cutoff ≤ c c wavelength of the input waveguide section. It is possible in principle to dynamically set the length of each section to optimize the approximation to the local curvature of the horn. This would increase the speed of the optimization; however, for simplicity, this detail was not implemented in our study. For each trial feedhorn the angular response was calculated directly from the modal content at the feed aperture. This in turn was calculated as follows. The throat of the feedhorn was assumed to be excited by the circular waveguide TE11 mode. The modal content of each successive section was then determined by matching the boundary conditions at each interface using the method of James [18]. The cylindrical symmetry of the feed limits the possible propagating modes to those with the same azimuthal functional form as TE11 [177]. This azimuthal-dependence extends to the resulting beam patterns, allowing the full beam pattern to be calculated from the E- and H- plane angular response. Ludwig’s third definition [178] is employed in calculation and measurement of cross-polar response. We note that an additional consequence of the feedhorn symmetry is that to the extent that the E- and H-planes are equal in both phase and amplitude, the cross-polarization is zero [159]. Changes in curvature in the feed profile can excite higher order modes (e.g., TE12 and TM12), the presence of which can potentially degrade the cross-polarization response of the horn.

D.1.2 Penalty Function

We constructed a penalty function to optimize the antenna proﬁle. The penalty function with normalized weights, αj, is written as

N M 2 2 χ = αj∆j(fi) , (D.1) i=1 j=1 where i sums over a discrete set of (N) frequencies in the optimization frequency band, and j sums over the number (M) of discrete parameters one wishes to take into account for the optimization. In the parameter space considered, this function

142 was minimized over the frequency range 1.25fc < f < 1.71fc (∆f/f0=0.3) to ﬁnd the desired solution. Results reported here were obtained by restricting this penalty function to include only the cross-polarization and reﬂection ( S 2) with uniform | 11| weights (M = 2). Additional parameters were explored; however, they were found to be subdominant in producing the target result. These functions were evaluated at 13

equally-spaced frequency points in Equation D.1. The explicit forms used for ∆1(f)

and ∆2(f) are

XP (f) XP0 if XP (f) > XP0, ∆1(f) = − (D.2) 0 if XP (f) XP0, ≤

RP (f) RP0 if RP (f) > RP0, ∆2(f) = − (D.3) 0 if RP (f) RP0, ≤ where XP (f) and RP (f) are the maximum of the cross-polarization XP (f) =

Max[XP (f, θ)] and reflected power at frequency f, respectively. XP0 and RP0 are the threshold cross-polarization and reflection. If either the cross-polarization or reflection at a sampling frequency were less than its critical value, it was omitted from the penalty function. Otherwise, its squared difference was included in the sum in Equation D.1.

D.1.3 Feedhorn Optimization

The feedhorn was optimized in a two-stage process that employed a variant of Powell’s method [160]. Generically, this algorithm can produce an arbitrary profile. To produce a feed that is easily machinable, we restricted the optimization to the subset of profiles for which the radius increases monotonically along the length of the horn. Without this constraint, this method was observed to explore solutions with corrugated features and the serpentine profiles explored in [161]. The aperture diameter of the feedhorn was initially set to 4λ , but was allowed ∼ c to vary slightly to achieve the desired beam size. A single discontinuity exists between the circular waveguide and the feed throat. The remainder of the horn profile adiabatically transitions to the feed aperture. The total length of the feedhorn from

143 the aperture to the single mode waveguide was fixed at 12.3 λc during optimization. This length is somewhat arbitrary, but chosen to produce a stationary phase center and a diffraction-limited beam in a practical volume. The approach of [162] was followed as an initial input to the Powell method. Specifically, the feed radius, r, is written analytically as a function of the distance along the length of the horn, z, as:

0.293 + 0.703 sin0.75(0.255z) 0 z 6.15, r(z) = ≤ ≤ (D.4) 2 1 0.293 + 0.703 1 + [0.282(z 6.15)] 2 6.15 < z 12.30, { − } ≤

where parameters are given in units of λc. This profile was then approximated by natural spline of a set of 20 points equally-spaced along the feed length. Throughout the optimization, we explicitly imposed the condition that radius of each section be greater than or equal to that of the previous section. This sampling choice effectively limits the allowed change in curvature along the feed profile. In the first stage of optimization, both XP0 and RP0 were set to -30 dB. The minimum of the penalty function was found by the modified Powell method in this 20-dimension space. In the second stage of the optimization, the number of points explicitly varied along the profile was increased to 560. The modified Powell method was used to

optimize the proﬁle in this 560-dimensional space. In this stage, both of XP0 and

RP0 were decreased to -34 dB. In principle, it is possible to use either of these techniques alone to find our solution. There are enough degrees of freedom in the 20-point spline to do so and the 560-point technique should be able to recover the solution regardless of the starting point. We found, however, that the 20-point spline did not converge readily to the final profile given the initial conditions above, but rather converged to a broad local minimum. In addition to finding the general features of the desired performance, this first stage of optimization provided a significant reduction in the use of computing resources compared to the slower 560-point parameter search. Figure D.1 shows the initial, intermediate, and final feedhorn profiles. It is possible to approximate the final profile with a 20-point spline. The final profile of the feed

144 3

1 c λ 0 Radius/ −1

Initial Profile −2 Intermediate Profile Final Profile −3 −2 0 2 4 6 8 10 12 Length/λ c

Figure D.1: The initial, intermediate and final profiles are shown. All dimensions are given in units of the cuttoff wavelength of the input circular waveguide. is reproduced with a low-spatial frequency error of 0.015λ . This effect has a ∼ c negligible influence on the modeled performance. This suggests that the optimization procedure could be done completely using a spline with fewer than 20 points if the location of the spline points were dynamically varied. Future optimization algorithms could be made more efficient by implementing this approach. Figure D.2 shows the improvement in cross-polarization for the two stages of optimization. The reflection is also shown for the initial profile, the intermediate optimization, and the final feedhorn profile.

D.2 Feedhorn Fabrication and Measurement

A feed (Figure D.3) that operates in circular waveguide with a TE11 cutoff frequency of fc=26.36 GHz was fabricated to test the proposed design. The structure was optimized between 33 and 45 GHz. The prototype feed was manufactured via electroforming in order to validate the design using a process that allows the feed structure to be measured and compared to the design profile. The final design profile is well-approximated by splining the radius (r) as a function of length (z) provided in Table D.1. The full 560-point profile is available upon request.

145 −18

−22

−26

−30

−34 Maximum Cross-Pol (dB)

−15 Initial Guess Intermediate Solution Final Profile −25 Measurements

−35

−45 Return Loss (dB)

−55

1 1.2 1.4 1.6 1.8 2 f/f c

Figure D.2: (Top) The maximum cross-polar response across the band is shown for the three profiles in Figure D.1. Measurements of the maximum cross-polarization are superposed. (Bottom) The reflected power measurements for the final feed horn are shown plotted over the predicted reflected power for the initial, intermediate, and final feedhorn profiles. Frequency is given in units of the cutoff frequency of the input circular waveguide.

146 The feedhorn was measured in the Goddard Electromagnetic Anechoic Chamber (GEMAC). The receivers and microwave sources used in the measurement provide a > 50 dB dynamic range from the peak response over 2π steradians with an ∼ absolute accuracy of < 0.5 dB. A five section constant cutoff transition from rectangular waveguide (WR 22.4, fc = 26.36 GHz) to circular waveguide [179] was used to mate the feedhorn to the rectangular waveguide of the antenna range infrastruc- ture. The constant cutoff condition was maintained in the transition by ensuring acircle = abroadwalls11/π where acircle is the radius of the circular guide, abroadwall is the width of the broadwall of the rectangular guide, and s 1.841 is the eigenvalue 11 ∼ for the TE11 mode [158]. The alignment of the circular waveguide feed interface was maintained to avoid degradation of the cross-polar antenna response. Pinning of this interface as specified in [180] or similar is recommended. Beam plots and parameters at the extrema and the middle of the optimization frequency range are shown in Figure D.4 and Table D.2. The cross-polarization response as a function of frequency of this device is compared to other published implementations of multi-mode scalar feeds (Fig. D.5). As is common for applications requiring the beam symmetry provided by a scalar horn, the aperture efficiency is low. In addition, we note that the phase center for this horn is near the aperture and is stable in frequency. An HP8510C network analzyer was used to measure the reflected power (see Fig. D.2) with a through-reflect-line calibration in circular waveguide. If desired, the match at the lower band edge can be improved by using a transition to a larger diameter guide. The measured observations are in agreement with theory. Imperfections in the profile may occur during manufacturing due to chattering of the tooling or similar physical processes. We performed a tolerance study to determine the effect of such high-spatial frequency errors in the feed radius. Negligible degradation in performance was observed for Gaussian errors in the radius up to

0.002 λc. The feed’s monotonic proﬁle is compatible with machining by progressive plunge milling in which successively more accurate tools are used to realize the feed proﬁle. This technique has been used for individual feeds and is potentially useful for fabricating large arrays of feedhorns. Examples include fabrication of multimode

147 Table D.1: Spline Approximation to Optimized Proﬁle (in Millimeters) Section Length (z) Radius (r) 0 0.0 3.33 1 7.0 5.77 2 14.0 7.91 3 21.0 9.90 4 28.0 10.86 5 35.0 11.13 6 42.0 11.27 7 49.0 11.66 8 56.0 11.90 9 63.0 11.96 10 70.0 12.24 11 77.0 12.44 12 84.0 12.76 13 91.0 13.70 14 98.0 15.40 15 105.0 17.01 16 112.0 17.71 17 119.0 20.05 18 126.0 21.75 19 133.0 21.91 20 140.0 21.92

Winston concentrators [181, 182], direct-machined smooth-walled conical feed horns for the South Pole Telescope [183], and the exploration of this technique for dual-mode feedhorns [174].

D.3 Conclusion

An optimization technique for a smooth-walled scalar feedhorn has been presented. Using this ﬂexible approach, we have demonstrated a design having a 30% bandwidth with cross-polar response below -30 dB. The design was tested in the range 33-45 GHz and found to be in agreement with theory. The design’s monotonic proﬁle and tolerance insensitivity enable the manufacturing of such feeds by direct machining.

148 Figure D.3: A smooth-walled feedhorn operating between 33 and 45 GHz was constructed. The horn is 140 mm long with an aperture radius of 22 mm. The input circular waveguide radius is 3.334 mm.

149 Table D.2: Beam Parameters Frequency Wavelength Antenna Gain Beam Solid Angle [GHz] [mm] [dBi] [Sr] 33 9.09 21.3 0.0925 34 8.82 21.1 0.0984 35 8.57 21.4 0.0904 36 8.33 21.3 0.0929 37 8.11 21.3 0.0930 38 7.89 21.9 0.0815 39 7.69 22.0 0.0788 40 7.50 22.7 0.0676 41 7.32 22.9 0.0643 42 7.14 23.5 0.0556 43 6.98 23.7 0.0540 44 6.82 24.2 0.0479 45 6.67 24.2 0.0473

This approach is useful in applications where a large number of feeds are desired in a planar array format.

150 33 GHz 39 GHz 45 GHz

-10 E-plane -20

-30

-40 Power (dB)

-50

-10 H-plane -20

-30

-40 Power (dB)

-50

-10 D-plane -20

-30

-40 Power (dB)

-50

-60 -80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80 Azimuth Angle(degrees) Azimuth Angle(degrees) Azimuth Angle(degrees)

Measured Co-Pol Predicted Co-Pol Measured Cross-Pol Predicted Cross-Pol

Figure D.4: The measured E-, H-, and diagonal-plane angular responses for the lower edge (33 GHz), center (39 GHz), and upper edge (45 GHz) of the optimization band are shown. The cross-polar patterns in the diagonal plane are shown in the bottom three panels for each of the three frequencies.

151 −25

−30

−35 This Work Yassin 2007

Cross−Polarization(dB) Granet 2004 Neilson 2002 Pickett 1984 Potter 1963 −40 0.85 0.9 0.95 1 1.05 1.1 1.15 Normalized Frequency

Figure D.5: The maximum cross-polar response of the prototype feedhorn is compared to other implementations of smooth-walled feedhorns. The data presented have been normalized to the design center frequencies as speciﬁed by the respective authors.

152 Appendix E

CLASS 40 GHz Feedhorn Proﬁle

Step Length Radius Step Length Radius Step Length Radius - mm mm - mm mm - mm mm 0 0.000 3.334 167 33.400 9.529 334 66.800 11.710 1 0.200 3.383 168 33.600 9.537 335 67.000 11.785 2 0.400 3.432 169 33.800 9.544 336 67.200 11.860 3 0.600 3.481 170 34.000 9.551 337 67.400 11.935 4 0.800 3.531 171 34.200 9.557 338 67.600 12.012 5 1.000 3.580 172 34.400 9.563 339 67.800 12.089 6 1.200 3.629 173 34.600 9.569 340 68.000 12.167 7 1.400 3.678 174 34.800 9.575 341 68.200 12.246 8 1.600 3.727 175 35.000 9.580 342 68.400 12.325 9 1.800 3.776 176 35.200 9.585 343 68.600 12.404 10 2.000 3.825 177 35.400 9.590 344 68.800 12.485 11 2.200 3.874 178 35.600 9.595 345 69.000 12.566 12 2.400 3.923 179 35.800 9.599 346 69.200 12.647 13 2.600 3.972 180 36.000 9.604 347 69.400 12.729 14 2.800 4.021 181 36.200 9.608 348 69.600 12.811 15 3.000 4.070 182 36.400 9.612 349 69.800 12.894 16 3.200 4.119 183 36.600 9.616 350 70.000 12.977

153 17 3.400 4.168 184 36.800 9.619 351 70.200 13.061 18 3.600 4.217 185 37.000 9.623 352 70.400 13.145 19 3.800 4.265 186 37.200 9.626 353 70.600 13.229 20 4.000 4.314 187 37.400 9.630 354 70.800 13.313 21 4.200 4.363 188 37.600 9.633 355 71.000 13.398 22 4.400 4.412 189 37.800 9.636 356 71.200 13.483 23 4.600 4.460 190 38.000 9.640 357 71.400 13.568 24 4.800 4.509 191 38.200 9.643 358 71.600 13.652 25 5.000 4.558 192 38.400 9.646 359 71.800 13.737 26 5.200 4.606 193 38.600 9.649 360 72.000 13.822 27 5.400 4.655 194 38.800 9.653 361 72.200 13.907 28 5.600 4.703 195 39.000 9.656 362 72.400 13.992 29 5.800 4.752 196 39.200 9.659 363 72.600 14.077 30 6.000 4.800 197 39.400 9.663 364 72.800 14.162 31 6.200 4.848 198 39.600 9.666 365 73.000 14.246 32 6.400 4.897 199 39.800 9.670 366 73.200 14.330 33 6.600 4.945 200 40.000 9.674 367 73.400 14.414 34 6.800 4.993 201 40.200 9.678 368 73.600 14.498 35 7.000 5.041 202 40.400 9.682 369 73.800 14.581 36 7.200 5.089 203 40.600 9.686 370 74.000 14.664 37 7.400 5.137 204 40.800 9.690 371 74.200 14.746 38 7.600 5.185 205 41.000 9.694 372 74.400 14.828 39 7.800 5.233 206 41.200 9.698 373 74.600 14.909 40 8.000 5.280 207 41.400 9.703 374 74.800 14.990 41 8.200 5.328 208 41.600 9.707 375 75.000 15.070 42 8.400 5.376 209 41.800 9.712 376 75.200 15.150 43 8.600 5.423 210 42.000 9.716 377 75.400 15.229 44 8.800 5.471 211 42.200 9.721 378 75.600 15.307 45 9.000 5.518 212 42.400 9.725 379 75.800 15.384 46 9.200 5.565 213 42.600 9.729 380 76.000 15.461

154 47 9.400 5.612 214 42.800 9.734 381 76.200 15.536 48 9.600 5.660 215 43.000 9.738 382 76.400 15.611 49 9.800 5.707 216 43.200 9.743 383 76.600 15.685 50 10.000 5.754 217 43.400 9.747 384 76.800 15.758 51 10.200 5.800 218 43.600 9.751 385 77.000 15.830 52 10.400 5.847 219 43.800 9.755 386 77.200 15.901 53 10.600 5.894 220 44.000 9.759 387 77.400 15.970 54 10.800 5.940 221 44.200 9.763 388 77.600 16.039 55 11.000 5.987 222 44.400 9.767 389 77.800 16.107 56 11.200 6.033 223 44.600 9.770 390 78.000 16.173 57 11.400 6.079 224 44.800 9.774 391 78.200 16.238 58 11.600 6.126 225 45.000 9.777 392 78.400 16.302 59 11.800 6.172 226 45.200 9.780 393 78.600 16.364 60 12.000 6.218 227 45.400 9.783 394 78.800 16.425 61 12.200 6.263 228 45.600 9.786 395 79.000 16.485 62 12.400 6.309 229 45.800 9.788 396 79.200 16.543 63 12.600 6.354 230 46.000 9.791 397 79.400 16.599 64 12.800 6.400 231 46.200 9.793 398 79.600 16.654 65 13.000 6.445 232 46.400 9.794 399 79.800 16.708 66 13.200 6.490 233 46.600 9.796 400 80.000 16.760 67 13.400 6.535 234 46.800 9.797 401 80.200 16.810 68 13.600 6.580 235 47.000 9.798 402 80.400 16.859 69 13.800 6.625 236 47.200 9.799 403 80.600 16.906 70 14.000 6.669 237 47.400 9.799 404 80.800 16.951 71 14.200 6.713 238 47.600 9.799 405 81.000 16.995 72 14.400 6.758 239 47.800 9.800 406 81.200 17.037 73 14.600 6.802 240 48.000 9.801 407 81.400 17.078 74 14.800 6.845 241 48.200 9.802 408 81.600 17.118 75 15.000 6.889 242 48.400 9.803 409 81.800 17.156 76 15.200 6.933 243 48.600 9.804 410 82.000 17.192

155 77 15.400 6.976 244 48.800 9.805 411 82.200 17.228 78 15.600 7.019 245 49.000 9.806 412 82.400 17.262 79 15.800 7.062 246 49.200 9.807 413 82.600 17.294 80 16.000 7.105 247 49.400 9.808 414 82.800 17.325 81 16.200 7.147 248 49.600 9.809 415 83.000 17.355 82 16.400 7.189 249 49.800 9.810 416 83.200 17.384 83 16.600 7.232 250 50.000 9.811 417 83.400 17.412 84 16.800 7.274 251 50.200 9.812 418 83.600 17.438 85 17.000 7.315 252 50.400 9.813 419 83.800 17.464 86 17.200 7.357 253 50.600 9.814 420 84.000 17.488 87 17.400 7.398 254 50.800 9.815 421 84.200 17.511 88 17.600 7.439 255 51.000 9.816 422 84.400 17.533 89 17.800 7.480 256 51.200 9.817 423 84.600 17.554 90 18.000 7.521 257 51.400 9.818 424 84.800 17.574 91 18.200 7.561 258 51.600 9.819 425 85.000 17.593 92 18.400 7.601 259 51.800 9.820 426 85.200 17.612 93 18.600 7.641 260 52.000 9.821 427 85.400 17.629 94 18.800 7.681 261 52.200 9.822 428 85.600 17.645 95 19.000 7.720 262 52.400 9.823 429 85.800 17.661 96 19.200 7.759 263 52.600 9.824 430 86.000 17.676 97 19.400 7.798 264 52.800 9.825 431 86.200 17.689 98 19.600 7.837 265 53.000 9.826 432 86.400 17.703 99 19.800 7.875 266 53.200 9.827 433 86.600 17.715 100 20.000 7.913 267 53.400 9.828 434 86.800 17.727 101 20.200 7.951 268 53.600 9.829 435 87.000 17.738 102 20.400 7.989 269 53.800 9.830 436 87.200 17.748 103 20.600 8.026 270 54.000 9.831 437 87.400 17.758 104 20.800 8.063 271 54.200 9.832 438 87.600 17.767 105 21.000 8.100 272 54.400 9.833 439 87.800 17.776 106 21.200 8.136 273 54.600 9.834 440 88.000 17.784

156 107 21.400 8.172 274 54.800 9.835 441 88.200 17.792 108 21.600 8.208 275 55.000 9.836 442 88.400 17.799 109 21.800 8.244 276 55.200 9.837 443 88.600 17.805 110 22.000 8.279 277 55.400 9.838 444 88.800 17.812 111 22.200 8.314 278 55.600 9.839 445 89.000 17.817 112 22.400 8.348 279 55.800 9.840 446 89.200 17.823 113 22.600 8.382 280 56.000 9.841 447 89.400 17.828 114 22.800 8.416 281 56.200 9.842 448 89.600 17.833 115 23.000 8.450 282 56.400 9.843 449 89.800 17.838 116 23.200 8.483 283 56.600 9.844 450 90.000 17.842 117 23.400 8.515 284 56.800 9.845 451 90.200 17.846 118 23.600 8.547 285 57.000 9.846 452 90.400 17.850 119 23.800 8.579 286 57.200 9.847 453 90.600 17.854 120 24.000 8.611 287 57.400 9.848 454 90.800 17.857 121 24.200 8.642 288 57.600 9.849 455 91.000 17.861 122 24.400 8.673 289 57.800 9.850 456 91.200 17.864 123 24.600 8.703 290 58.000 9.851 457 91.400 17.867 124 24.800 8.733 291 58.200 9.852 458 91.600 17.869 125 25.000 8.762 292 58.400 9.853 459 91.800 17.872 126 25.200 8.791 293 58.600 9.854 460 92.000 17.874 127 25.400 8.819 294 58.800 9.855 461 92.200 17.877 128 25.600 8.847 295 59.000 9.856 462 92.400 17.879 129 25.800 8.875 296 59.200 9.857 463 92.600 17.881 130 26.000 8.902 297 59.400 9.858 464 92.800 17.882 131 26.200 8.929 298 59.600 9.859 465 93.000 17.884 132 26.400 8.955 299 59.800 9.860 466 93.200 17.885 133 26.600 8.981 300 60.000 9.876 467 93.400 17.887 134 26.800 9.006 301 60.200 9.905 468 93.600 17.888 135 27.000 9.030 302 60.400 9.936 469 93.800 17.889 136 27.200 9.054 303 60.600 9.969 470 94.000 17.890

157 137 27.400 9.078 304 60.800 10.004 471 94.200 17.890 138 27.600 9.101 305 61.000 10.041 472 94.400 17.891 139 27.800 9.124 306 61.200 10.079 473 94.600 17.892 140 28.000 9.146 307 61.400 10.119 474 94.800 17.892 141 28.200 9.167 308 61.600 10.160 475 95.000 17.892 142 28.400 9.188 309 61.800 10.203 476 95.200 17.892 143 28.600 9.209 310 62.000 10.248 477 95.400 17.892 144 28.800 9.228 311 62.200 10.294 478 95.600 17.893 145 29.000 9.248 312 62.400 10.341 479 95.800 17.894 146 29.200 9.266 313 62.600 10.390 480 96.000 17.895 147 29.400 9.284 314 62.800 10.441 481 96.200 17.896 148 29.600 9.302 315 63.000 10.493 482 96.400 17.897 149 29.800 9.319 316 63.200 10.546 483 96.600 17.898 150 30.000 9.335 317 63.400 10.601 484 96.800 17.899 151 30.200 9.351 318 63.600 10.657 485 97.000 17.900 152 30.400 9.366 319 63.800 10.714 486 97.200 17.901 153 30.600 9.380 320 64.000 10.773 487 97.400 17.902 154 30.800 9.394 321 64.200 10.833 488 97.600 17.903 155 31.000 9.407 322 64.400 10.894 489 97.800 17.904 156 31.200 9.420 323 64.600 10.956 490 98.000 17.905 157 31.400 9.432 324 64.800 11.019 491 98.200 17.906 158 31.600 9.444 325 65.000 11.084 492 98.400 17.907 159 31.800 9.455 326 65.200 11.150 493 98.600 17.908 160 32.000 9.466 327 65.400 11.216 494 98.800 17.909 161 32.200 9.477 328 65.600 11.284 495 99.000 17.910 162 32.400 9.486 329 65.800 11.353 496 99.200 17.911 163 32.600 9.496 330 66.000 11.422 497 99.400 17.912 164 32.800 9.505 331 66.200 11.493 498 99.600 17.913 165 33.000 9.513 332 66.400 11.565 499 99.800 17.914 166 33.200 9.522 333 66.600 11.637 500 100.000 17.914

158 Appendix F

Lab Cryostat Thermometry Codes

Figure F.1 shows the front panel of the SRS readout code. The configuration panel shows the hardware settings: The GPIB address of the SIM900 mainframe is 2. MUX 1 is installed in slot 2 of the mainframe; Diode monitor is in slot 1; MUX 2 is in slot 8 and AC bridge is in slot 6. The readout code loops over all available channels of MUX1 and MUX2 and produce real time plots. It also displays and saves all data with time stamps. The block diagram of the readout code is shown in figure F.3. Figure F.2 shows the front panel of the PID temperature controller. It consists of six sub panels: the Lakeshore AC resistance bridge, the Keitheley current sources, the Mechanical heat switch, the PID control, the file writing and a magnet current display panel. In the Lakeshore AC bridge panel, the GPIB address is 12. There are different excitation options available, because different thermometers require different excitations in different temperature ranges. The germanium resistance temperature (GRT) sensors require voltage excitation mode, while the ruthenium oxide (RuOx) sensors require current excitation. There are two Keithley current sources, with GPIB addresses of 22 and 24. Each can ramp up to 5 A. The ADR magnet allows a max current of 9.0 A, and a max ramp rate of 0.01 A/s (0.005 A/s is recommended). A basic ADR operation procedure is: (1)Close the heat switch and ramp both Keithley 1 and 2 up to 4.5 A; (2)Open the heat switch and ramp Keithley 2 down to 0; (3)Turn on the PID control. In the control panel, the temperature set point and PID gains can be changed. Block diagram is shown in figure F.4, F.5 and F.6.

159 Figure F.1: SRS readout program front panel.

Figure F.2: PID control program front panel.

160 Figure F.3: Block diagram of the SRS readout program.

161 Figure F.4: Block diagram of the PID control program. Part 1 of 3. 162 Figure F.5: Block diagram of the PID control program. Part 2 of 3. 163 Figure F.6: Block diagram of the PID control program. Part 3 of 3.

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188 Vita

Lingzhen Zeng was born in Laibin, Guangxi province, China, on 02 October 1982, the son of Xiangxiong Zeng and Qiuying Li. After completing his work at No.1 High School of Laibin, he went on to the University of Science and Technology of China (USTC) in Hefei, Anhui province, China, where he studied astronomy and received his Bachelor degree in July 2005. After that, he entered the department of physics and astronomy at Johns Hopkins University (JHU) in Baltimore, Maryland as a graduate student.

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