POLARIMETRY IN ASTROPHYSICS AND COSMOLOGY Lingzhen ...

POLARIMETRY IN ASTROPHYSICS AND

COSMOLOGY

by

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

c© Lingzhen Zeng 2012

All rights reserved

Abstract

Astrophysicists are mostly limited to passively observing electromagnetic radia-

tion 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) radi-

ation, 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 param-

eterized by the relative amplitude of the tensor to scalar modes r. For the simplest

inflation models, the expected deviation from scale invariance (ns = 0.963± 0.012) is

coupled to gravitational waves with r ≈ 0.1. These considerations establish a strong

motivation to search for this remnant from when the universe was about 10−32 seconds

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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 polariza-

tion 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

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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 dis-

cussions, 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,

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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.

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Contents

Abstract ii

Acknowledgements iv

List of Tables ix

List of Figures x

I Polarimetry in Astrophysics 1

1 Introduction to Polarization in Astrophysics 21.1 Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Stokes Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Poincare Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Polarization in Astrophysics . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Synchrotron Emission . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Thermal Dust Emission and Absorption . . . . . . . . . . . . 101.4.3 Examples of Polarization from Absorption and Scattering . . . 131.4.4 Anomalous Dust Emission . . . . . . . . . . . . . . . . . . . . 15

2 Submillimeter Polarimetry of M17 162.1 Introduction to Submillimter Polarimetry . . . . . . . . . . . . . . . . 162.2 Polarimetry at Caltech Submillimeter Observatory . . . . . . . . . . . 172.3 SHARP Data Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Introduction to M17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 M17 Polarimetry Results . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.2 Polarization Spectrum . . . . . . . . . . . . . . . . . . . . . . 272.5.3 Spatial Distribution of Magnetic Field and Polarization Spectrum 302.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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II Polarimetry in Cosmology 39

3 Introduction to Polarization in Cosmology 403.1 The Big Bang Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.1 The Expanding Universe–Hubble’s Law . . . . . . . . . . . . . 403.1.2 Big Bang Nucleosynthesis (BBN) . . . . . . . . . . . . . . . . 413.1.3 The Cosmic Microwave Background (CMB) Radiation . . . . 423.1.4 Other Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Cosmic Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.1 The Structure Problem . . . . . . . . . . . . . . . . . . . . . . 433.2.2 The Flatness Problem . . . . . . . . . . . . . . . . . . . . . . 443.2.3 The Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . 443.2.4 The Magnetic Monopole Problem . . . . . . . . . . . . . . . . 44

3.3 ΛCDM Cosmological Model . . . . . . . . . . . . . . . . . . . . . . . 453.3.1 Cosmological Principles and FLRW metric . . . . . . . . . . . 453.3.2 Einstein Field Equations and Friedmann Equation . . . . . . . 473.3.3 Best-fit ΛCDM Model Parameters . . . . . . . . . . . . . . . . 49

3.4 The Cosmic Microwave Background Radiation . . . . . . . . . . . . . 543.4.1 The CMB Anisotropy . . . . . . . . . . . . . . . . . . . . . . . 563.4.2 The CMB Polarization . . . . . . . . . . . . . . . . . . . . . . 57

4 The Cosmology Large Angular Scale Surveyor (CLASS) 624.1 Scientific Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Sensitivity Calculation and Bandpass Optimization . . . . . . . . . . 67

4.2.1 Sensitivity Calculation . . . . . . . . . . . . . . . . . . . . . . 684.2.2 Bandpass Optimization . . . . . . . . . . . . . . . . . . . . . . 71

4.3 The Variable-delay Polarization Modulator . . . . . . . . . . . . . . . 794.3.1 Polarization Transfer Function . . . . . . . . . . . . . . . . . . 804.3.2 VPM Grid Optimization . . . . . . . . . . . . . . . . . . . . . 814.3.3 VPM Mirror Throw Optimization . . . . . . . . . . . . . . . . 834.3.4 VPM Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3.5 Current Status . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4 CLASS Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.5 Smooth-walled Feedhorn . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5.1 Smooth-walled Feedhorn Optimization . . . . . . . . . . . . . 984.5.2 Smooth-walled Feedhorn for CLASS . . . . . . . . . . . . . . . 102

4.6 CLASS Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.6.1 Focal Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.6.2 TES Bolometers . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.7 Lab Set up for Detector Testing . . . . . . . . . . . . . . . . . . . . . 1164.7.1 Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.7.2 Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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4.7.3 Cryostat Performance . . . . . . . . . . . . . . . . . . . . . . 1184.7.4 Detector Readout . . . . . . . . . . . . . . . . . . . . . . . . . 119

A M17 Polarization Data 125A.1 Polarziation Spectrum: 450 um vs 60 um . . . . . . . . . . . . . . . . 125A.2 Polarziation Spectrum: 450 um vs 100 um . . . . . . . . . . . . . . . 127A.3 Polarziation Spectrum: 450 um vs 350 um at RA > 18h17m30s . . . . 129A.4 Polarziation Spectrum: 450 um vs 350 um at RA < 18h17m30s . . . . 130A.5 Polarization Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 140D.1 Smooth-walled Feedhorn Optimization . . . . . . . . . . . . . . . . . 141

D.1.1 Beam Response Calculation . . . . . . . . . . . . . . . . . . . 141D.1.2 Penalty Function . . . . . . . . . . . . . . . . . . . . . . . . . 142D.1.3 Feedhorn Optimization . . . . . . . . . . . . . . . . . . . . . . 143

D.2 Feedhorn Fabrication and Measurement . . . . . . . . . . . . . . . . . 145D.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

E CLASS 40 GHz Feedhorn Profile 153

F Lab Cryostat Thermometry Codes 159

Vita 189

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List of Tables

2.1 SHARP Instrument Specifications . . . . . . . . . . . . . . . . . . . . 192.2 M17 Polarization Spectrum Data . . . . . . . . . . . . . . . . . . . . 30

3.1 Best-fit ΛCDM Model Parameters . . . . . . . . . . . . . . . . . . . . 50

4.1 CLASS Scientific Overview . . . . . . . . . . . . . . . . . . . . . . . . 664.2 CLASS Detector Parameters . . . . . . . . . . . . . . . . . . . . . . . 704.3 CLASS VPM Mirror Throw Optimization . . . . . . . . . . . . . . . 884.4 CLASS Optics Overview . . . . . . . . . . . . . . . . . . . . . . . . . 944.5 CLASS 40 GHz Feedhorn Requirements . . . . . . . . . . . . . . . . 1024.6 Feedhorn Profile Approximation (in Millimeters) . . . . . . . . . . . . 1044.7 Feedhorn Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.8 Beam Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.9 Cryostat Thermometry Readout . . . . . . . . . . . . . . . . . . . . . 122

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

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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 ofpropagation (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, ellipticityangle χ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Poincare sphere, defining the polarization in spherical coordinates. Italso shows the relation between (Q, U , V ) and (Ip, χ, ψ) [1]. . . . . . 7

1.4 CMB foreground radiation in WMAP bands [2]. The synchrotron ra-diation dominates the low frequency range below 60 GHz. Radiationfrom dust contributes mostly above 70 GHz. . . . . . . . . . . . . . . 9

1.5 Starlight polarization vectors in Galactic coordinates. The upper panelshows polarization vectors in local clouds. The polarization averagedover 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 µm measurements (points) from Jan 2003 compared to theo-retical expectation (solid line) from equation 2.1 [3]. The performanceis about 1 Jy s1/2 for τ225 GHz = 0.05. . . . . . . . . . . . . . . . . . . 18

2.2 The polarization splitting optics of SHARP [4] for reconstituting theimage 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 theXG (crossed grid), where the polarization radiation is separated intotwo orthogonal (horizontal and vertical) components. Right: View to-ward the CSO focus. The vertical and horizontal components undergofurther reflections by a series of mirrors and grids, and are displacedlaterally at the BC (beam combiner), before being directed towardSHARC II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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2.3 Flow chart of “SharpInteg”. It starts by masking the raw data filewith an “rgm” file. Then, it demodulates the chopping to calculate thechopped data. After applying the relative data gain factor between thehorizontal and vertical array, it calculates the I, I-error, Q, Q-error, Uand U-error components and saves them into a new file. . . . . . . . . 20

2.4 Flow chart of “Sharpcombine”. It applies τ and telescope pointingcorrection, background subtraction (BS), instrument polarization (I.P.)subtraction and polarization angle rotation to sky coordinates (Rot) toeach 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 regionin the Galaxy. Left: A M17 image from my 80 mm aperture opticaltelescope. Right: A false color image from Spitzer GLIMPSE (red: 5.8um; 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 clusterof stars surrounded by successive layers of H+, H0, and H2 gas, thatexpanding with different velocities to the outer side of the cloud. . . . 24

2.7 M17 polarization fraction vectors are plotted over the 450 um uncali-brated flux map. Thick vectors are detected with greater than or equalto 3σ level and thin vectors are between 2σ and 3σ level. The circle onthe bottom right shows the SHARP beamsize. Some parts of the fluxmap is removed due to high noise levels. Offsets are from 18h17m32s,-1614′25′′ (B1950.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Histogram of M17 polarization fraction. This distribution includes allvectors at greater or equal to than 2σ level. All vectors greater than10% are 2σ vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.9 Histogram of M17 polarization angle. Polarization angles are measuredfrom north to east. The resulting net magnetic field is almost parallelto the RA direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.10 Magnetic field vectors from SHARP (red, 450 um), Stokes (green,100um) [7] and optical observation [8] (purple) plotted on top of SpitzerGLIMPSE 8.00 um flux map. The magnetic vectors from SHARPand Stokes are perpendicular to their polarization angles, while thosefrom optical polarization measurement are parallel to their polarizationangles. All magnetic vectors (plotted with the same length) are usedto indicate the direction only. Offsets are from 18h17m32s, -1614′25′′

(B1950.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.11 The common area (green shadow) for polarization spectrum analysis.

It is between 18h17m30s and 18h17m37s in Ra (B1950), −1616′20′′ and−1613′00′′ in Dec (B1950). The selected polarization vectors are at60 µm (yellow), 100 µm (green), 350 µm (blue) and 450 µm (red).Background is the 450 µm flux map. Offsets are from 18h17m32s, -1614′25′′ (B1950.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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2.12 Polarization spectrum of some popular interstellar molecular clouds [9].The median polarization ratio are normalize by the value at 350 µm.In contrast to the results from other clouds, our work shows that, theM17 has lower median polarization at 450 µm than at 350 µm. Thepolarization spectrum falls monotonically from 60 µm to 450 µm. . . 31

2.13 Magnetic vectors from SHARP plotted over the [21 cm]/[450 µm] fluxratio 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 definedby fitting contour level = 0.1. The new “X-Y” coordinate system isabout 66.3 with respect to the “Ra-Dec” coordinates. The shock isfollowing the “-Y” direction. The “y=0” and “y=-50 arcsec” lines sepa-rate the cloud into “post-shocked” (y > 0), “shock front” (-50 < y < 0)and “pre-shocked” (y < -50) regions. The polarization directions andmagnitudes in these regions are different (figure 2.14 and 2.15). Themagnetic fields in the dense cloud (can also be seen in figure 2.10) atthe top of the map survive the windswept. Offsets are from 18h17m32s,-1614′25′′ (B1950.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.14 Correlation between polarization angle and the Y direction (zero at18h17m32s, −1614′25′′), showing a linear relationship. The “post-shocked” region is at y > 0 and the “pre-shocked” region is at y < −50arcsec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.15 Correlation between polarization fraction and Y direction (zero at18h17m32s, −1614′25′′), showing a “U” like shape. The polarizationfraction is higher at the “post-shocked” region at y > 0 and the “pre-shocked” region at y < −50 arcsec. . . . . . . . . . . . . . . . . . . . 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 cmabsorption-line contour and the ratio of neutral HI (NHI) column den-sity to the spin temperature Tspin distribution map in the 17.5-22 km/svelocity area from [6]. This velocity component is correlated with the“post-shocked” and part of “shock front” region. The NHI/Tspin den-sity at the dense cloud region (see figure 2.13) is low. . . . . . . . . . 35

2.17 The [450 µm]/[350 µm] polarization ratio vectors over plotted on the[21 cm]/[450 µm] flux ratio map with contour levels = 0.1, 0.3, 0.5,0.7, 0.9. The blue (red) vectors represent P450 < (>) P350. Thelength 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.Offsets are from 18h17m32s, -1614′25′′ (B1950.0). . . . . . . . . . . . 36

2.18 The [450 µm]/[350 µm] polarization ratio vectors and 450 µm intensitycontours of SHARP (green, levels = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0) overplotted on the Fig.1 from [7]. The blue vectors is found to be correlatedwith the [OI] line, which is a tracer for the atomic gas. . . . . . . . . 37

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3.1 Timeline of the universe. The CMB radiation from the last scatteringsurface (LSS) when the universe is about 380,000 years old with thetemperature of about 3,000 K [10]. . . . . . . . . . . . . . . . . . . . 55

3.2 The internal linear combination map from WMAP [11], showing theall sky CMB temperature anisotropy. . . . . . . . . . . . . . . . . . . 56

3.3 The angular power spectrum from WMAP [12], showing the detectionof the first three peaks. The first peak is at ℓ ≈ 220, corresponding toan angular scale of about 1. . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Left: Quadrupole polarization from Thomson scattering of the CMBphotons with free electrons. Right: The E and B mode patterns. TheE-modes are curl-free components with no handedness. The B-modesare 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. TheBB lensing signal is shown as a blue dashed line. The foreground modelfor synchrotron plus dust emission is shown as straight dashed lines [13]. 60

4.1 Two-dimensional joint marginalized constraint (68% and 95% CL) onscalar spectral index (ns) and tensor to scalar ratio (r), derived fromthe data combination of WMAP + BAO + H0 [14]. Three linear fitsare from different simple inflation models. . . . . . . . . . . . . . . . 63

4.2 The background is the WMAP 7 year all sky Q band polarization mapin Galactic coordinates showing the sky coverage of CLASS experi-ment. Observing from the Atacama Desert in Chile, CLASS covers∼ 65.1% of the sky above 45 elevation. Excluding the Galactic maskarea, the visible sky left is ∼ 46.8% (bright region). The dark circle atthe south pole is about 22 in radius. Figure courtesy of David Larson. 64

4.3 CLASS instrument overview for the 40 GHz band. The instrumentconsists a front-end variable-delay polarization modulator, catadioptricoptic system and a field cryostat. The lenses are cooled to about 4 Kand the smooth-walled feedhorn-coupled TES bolometer array operatesat 100 mK. Figure courtesy of Joseph Eimer. . . . . . . . . . . . . . . 65

4.4 CLASS wavebands and sensitivity curve from [15]. Left: The frequencybands of CLASS are chosen to straddle the Galactic foreground spec-tral minimum and to minimize atmospheric effects (see section 4.2.2).Right: The CLASS sensitivity curve, shown by the dashed curve alongthe shaded boundary, is the 1σ limit for each l and assumes 3 yearsof observing with a conservative 50% efficiency for down-time (see sec-tion 4.2.1). CLASS has the sensitivity to definitively detect B-modesat the cosmologically interesting limit of r ∼ 0.01. . . . . . . . . . . . 68

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4.5 Annual variation of the Precipitable Water Vapor (PWV) content atChajnantor, based on 10 years of site testing. Conditions are worseduring the winter from the end of December to early April. The ex-pected median PWV for the rest of the year is around 1 mm, whileconditions 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 sitefrom 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 noisesource (equation 4.16) for the relative signal-to-noise ratio calculation(equation 4.21). The red, green and blue lines shows our optimizedbandwidth for 40, 90 and 150 GHz band: (30.3 GHz - 40.3 GHz), (77.3GHz - 108.3 GHz) and (126.8 GHz - 164.3 GHz). . . . . . . . . . . . 77

4.8 The 2-D plot of relative signal-to-noise ratio (equation 4.22) from 0 to200 GHz showing our optimization results. The cross points of red,green and white lines are the locations of the local maxima. For the40 GHz band, we only search for the maximum in the range of ν > 30GHz. The coordinates are (30.3, 40.3), (77.3, 108.3) and (126.8, 164.3). 78

4.9 As shown in Poincare sphere, VPM modulates between Q and V , whilethe HWP mix Q and U . In the case of VPM, the residuals due to thespectral effects (shown in blue) are a function of measurable modula-tion parameters. Figure courtesy of David Chuss. . . . . . . . . . . . 79

4.10 VPM modulates polarization by introducing a controlled variable pathdifference between two orthogonal linear polarizations. Dots show thecomponent with polarization angle parallel to the grid; Double arrowshow that with angle perpendicular to the grid. By moving the mir-ror up and down, VPM introduces a path difference x(t) = 2d(t)cosθbetween 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 Qis in good agreement with an ideal grid (equation 4.29). The VPMreflection phase delay differs from the free-space grid-mirror delay ifthe conditions are changed. . . . . . . . . . . . . . . . . . . . . . . . 82

4.12 The contour plot of relative signal-to-noise ratio for Stokes Q, calcu-lated from equation 4.42 with cosine chopping mode. This plot is forthe 40 GHz band (33 GHz to 43 GHz, λ0 = 7.89 mm). The maximumis at (0.19 λ0, 0.13 λ0) with the peak signal-to-noise ratio scaled to be1.00. There are 4 other local maxima nearby: (0.19 λ0, 0.39 λ0), (0.44λ0, 0.13 λ0), (0.44 λ0, 0.39 λ0) and (0.27 λ0, 0.26 λ0). Details are listedin table 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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4.13 The contour plot of relative signal-to-noise ratio for Stokes Q, calcu-lated from equation 4.42 with linear chopping mode. This plot is forthe 40 GHz band (33 GHz to 43 GHz, λ0 = 7.89 mm). The maximumis at (0.46 λ0, 0.16 λ0) with the peak signal-to-noise ratio scaled to be1.00. 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 dropsquickly 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 alu-minium box frame with over 2 tons of stretch force. The diameter ofthe flattener ring is 50 cm. The wire diameter, 2a, is 63.5 µm, withwire pitch, g = 200 µm. 2a/g = 1/3.15 ≈ 1/π. The flatness of the gridis better than 50 µm. The total length of the wires is longer than 2miles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.16 Top: Drawings of CLASS 40 GHz optics. It consists of a front-endVPM, two mirrors, two lenses, a Lyot stop, a vacuum window and twoinfrared (IR) blocking filters. Bottom: Drawing and the ray trace ofthe 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 ofpixels = 36. Figure courtesy of Joseph Eimer. . . . . . . . . . . . . . 96

4.18 Point spread diagram of CLASS 40 GHz optics from Zeemax. Eachdiagram in this figure represents a separate direction on the sky. Thecircles show the first Airy disk at the corresponding location. Thisdiagram shows that the optics is diffraction limited. Figure courtesyof Joseph Eimer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.19 Flow chart of smooth-walled feedhorn optimization. Optimization be-gins with a sin0.75 profile, the method from [18] is used to calculatethe beam patterns. The feedhorn profile was found by this multi-stepiterative solution with different thresholds in each step. . . . . . . . 101

4.20 CLASS 40 GHz feedhorn profile. The 10.00 mm long input waveguidehas a radius of 3.334 mm, with fc = 26.349 GHz. The length ofthe feedhorn is 100.00 mm. The aperture is 35.828 mm. This is amonotonic profile that allows a progressive milling technique. . . . . . 103

4.21 CLASS feedhorn performance from 30 to 50 GHz. The dashed linesdefine the -30 dB line, and the waveband limit of 33 GHz and 43 GHz.The cut off frequency is fc = 26.349 GHz. . . . . . . . . . . . . . . . 103

4.22 Beam patterns of the CLASS smooth-walled feedhorn within azimuthangles of ±90, from 33 GHz to 38 GHz. . . . . . . . . . . . . . . . . 106

xv




4.26 The averaged cross-pol, return-loss and edge-taper plot for the toler-ance calculation from 0 to 300 um. For each tolerance, these valueswere from the average of 120 calculations. (The plots are noisy at largetolerance, 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 36smooth-walled feedhorns, waveguide adapter, detector mounting plateand 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 de-tector chip for the 40 GHz CLASS [19]. Left: The detector set upshowing 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 isconnected by a thermal link G to a thermal source with a temperatureof Tbath. The total power to the pixel is Pγ + PJ − PG. Right: TESis biased by IB = VB/RB, in the case of RB ≫ RSH. For R ≫ RSH,the TES is bias by V = IBRSH, then fluctuations of R will result influctuation in current, which is read out by the inductor L and thesuperconducting quantum interference device (SQUID) amplifier. . . 114

4.30 Section view of model 104 Olympus ADR cryostat showing mechanicalheat switch controller, vacuum valve, pulse tube (PT) head, 60 K plate,4 K plate, adiabatic demagnetization refrigerator (ADR), high tempsuperconducting leads for 4 T magnet, thermal shielding, and vacuumjacket [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.31 Left: The ADR and the He-4 refrigerator mounted on the 4 K plate ofthe HPD cryostat in the experimental cosmology lab at Johns HopkinsUniversity. Photo courtesy of David Larson. Right: the rack-mounteddevices for cryostat thermometry. From top to bottom, they are, aSRS SIM900 mainframe with 2 MUXs, a diode moniter and an ACbridge, a front panel, a NI GPIB to Ethernet adapter, a Lakeshore 370AC resistance bridge and two Keithley 2440 current sources. . . . . . 119

4.32 Cryostat cool down curves. It takes about 24 hours for the cryostat tocool down to the state with stable temperature readouts. The typicalvalues of the thermometers are listed in table 4.9. . . . . . . . . . . . 120

xvi

4.33 ADR cooling curves at 100 mK, showing the magnet current versustime of the ADR with the loads of from 2.0 to 10.0 µW. Based onthese curves, the FAA pill of the ADR have higher cooling capacitiesat lower loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.34 The FLL block diagram for TES detector readout, showing the coldelectronics inside the cryostat and the warm electronics (MCE) [22]. . 123

4.35 This photo shows the Multi-Channel Electronics (MCE) mounted onthe wall the cryostat in the experimental cosmology lab at Johns Hop-kins University. The MCE is connected to a data-acquisition computerby a pair of fiber optic cables (the orange wires). Photo courtesy ofDavid Larson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.1 60 um polarization vectors from Stokes ([23], Yellow) and the 450um result from SHARP (smoothed to 22′′ resolution, Red), center at18h17m32s,-1614′25′′ (B1950.0). . . . . . . . . . . . . . . . . . . . . . 126

A.2 100 um polarization vectors from Stokes ([23], Green) and the 450umresult from SHARP (smoothed to 35′′ resolution, Red), center at 18h17m32s,-1614′25′′ (B1950.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

A.3 350 um polarization vectors from Hertz ([24]) and the 450 um resultfrom SHARP (smoothed to 20′′ resolution, Red), center at 18h17m32s,-1614′25′′ (B1950.0). Blue: Hertz vectors at RA > 18h17m30s, Green:Hertz vectors at RA < 18h17m30s . . . . . . . . . . . . . . . . . . . . 131

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

D.1 The initial, intermediate and final profiles are shown. All dimensionsare given in units of the cuttoff wavelength of the input circular waveg-uide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

D.2 (Top) The maximum cross-polar response across the band is shownfor the three profiles in Figure D.1. Measurements of the maximumcross-polarization are superposed. (Bottom) The reflected power mea-surements for the final feed horn are shown plotted over the predictedreflected power for the initial, intermediate, and final feedhorn profiles.Frequency is given in units of the cutoff frequency of the input circularwaveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

D.3 A smooth-walled feedhorn operating between 33 and 45 GHz was con-structed. 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 thelower edge (33 GHz), center (39 GHz), and upper edge (45 GHz) ofthe optimization band are shown. The cross-polar patterns in thediagonal plane are shown in the bottom three panels for each of thethree frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

D.5 The maximum cross-polar response of the prototype feedhorn is com-pared to other implementations of smooth-walled feedhorns. The datapresented have been normalized to the design center frequencies asspecified by the respective authors. . . . . . . . . . . . . . . . . . . . 152

F.1 SRS readout program front panel. . . . . . . . . . . . . . . . . . . . . 160F.2 PID control program front panel. . . . . . . . . . . . . . . . . . . . . 160F.3 Block diagram of the SRS readout program. . . . . . . . . . . . . . . 161F.4 Block diagram of the PID control program. Part 1 of 3. . . . . . . . . 162F.5 Block diagram of the PID control program. Part 2 of 3. . . . . . . . . 163F.6 Block diagram of the PID control program. Part 3 of 3. . . . . . . . . 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) = (Ex, Ey, Ez) = (Axcos(kz − ωt+ φx), Aycos(kz − ωt+ φy), 0) (1.1)

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:

Ex = Axsin(ωt− φx)

Ey = Aysin(ωt− φy). (1.2)

By defining φ = φx − φy, equation 1.2 can be written into an elliptical form:

E2x

A2x

+E2

y

A2y

− 2ExEy

AxAycosφ = sin2φ. (1.3)

For different phase offsets, the polarization state varies. From equation 1.3, if

φ = mπ (where m = 0,±1,±2, ...), then Ex/Ax ± Ey/Ay = 0 (linear polarization); if

φ = (2m+1)π/2 and Ax = Ay, then E2x+E

2y = A2

x (circular polarization); if φ 6= mπ,

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 Ax, Ay, φ above, used to describe polarization have different

units. In 1852, George G. Stokes defined a set of 4 parameters (the Stokes parame-

ters) as a mathematically convenient alternative. For the monochromatic plane wave

described above, the Stokes parameters are:

I = A2x + A2

y

Q = A2x −A2

y

U = 2AxAycosφ

V = 2AxAysinφ (1.4)

3

y

x

z

E

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 Ax(t), Ay(t), φ(t) and the observed radiation is a partially co-

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 = 〈A2x(t) + A2

y(t)〉

Q = 〈A2x(t)− A2

y(t)〉

U = 2〈Ax(t)Ay(t)cosφ(t)〉

V = 2〈Ax(t)Ay(t)sinφ(t)〉 (1.5)

where angular brackets denote averaging over many wave cycles.

Useful relation can be derived among the stokes parameters. For purely monochro-

matic (coherent) radiation

I2 = Q2 + U2 + V 2. (1.6)

4

ψ x

y ξ η

E

χ

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 + U2 + V 2. (1.7)

We can define a total polarization fraction (degree of polarization)

p = (Q2 + U2 + 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 orienta-

tion, 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 ellip-

ticity 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)

Ex = Eξcosψ − Eηsinψ

Ey = Eξsinψ + Eηcosψ (1.9)

and

Ax = a0((cos2χcos2ψ + sin2χsin2ψ)1/2

Ay = a0((cos2χsin2ψ + sin2χcos2ψ)1/2

tanφx = tanχtanψ

tanφy = −tanχcotψ (1.10)

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)

andE2

ξ

a20cos2χ

+E2

η

a20sin2χ

= 1. (1.12)

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:

I = a20

Q = a20cos2χcos2ψ

U = a20cos2χsin2ψ

V = a20sin2χ (1.13)

with the following inverse equations:

tan2ψ = U/Q

sin2χ = V/(Q2 + U2 + V 2)1/2. (1.14)

6

(V)

(U)

(Q)

Figure 1.3: Poincare sphere, defining the polarization in spherical coordinates. It alsoshows the relation between (Q, U , V ) and (Ip, χ, ψ) [1].

1.3 Poincare Sphere

From equation 1.13, the polarization state can be described in spherical coordi-

nates, by replacing a20 with Ip:

Q = Ipcos2χcos2ψ

U = Ipcos2χsin2ψ

V = Ipsin2χ (1.15)

where Ip = (Q2 + U2 + V 2)1/2 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 Poincare sphere

is the spherical surface occupied by polarization states having a constant polarization:

7

S =1

Ip

Q

U

V

(1.16)

The Poincare sphere 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 Poincare sphere 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 syn-

chrotron 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 mag-

netic 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

CMB Anisotropy

Frequency (GHz)

Ante

nna

Tem

pera

ture

(K,

rms)

1

10

100

10020 6040 80 200

85%Sky (Kp2)

Synchrotron

Free-free Dust

Synchrotron

77%Sky (Kp0)

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

If the energy spectrum of cosmic-ray electrons can be expressed as a power-law

distribution:

N(E) ∝ E−γ (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:

α = −γ − 1

2

β = −γ + 3

2(1.18)

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:

psyn =γ + 1

γ + 7/3(1.19)

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 submillime-

ter (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 gener-

ally modelled with one or more thermal components with different temperatures by

a frequency dependent emission:

I(ν) =n

∑

i=1

AiνβiBν(Ti) (1.20)

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

Erot =1

2(IxΩ

2x + IyΩ

2y + IzΩ

2z) (1.21)

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,

J = (I2xΩ2x + I2yΩ

2y + I2zΩ

2z)

1/2 (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

momentum), this requires an increase in the time-average value of Ω2z relative to Ω2

y

(or Ω2y to Ω2

x). The dissipation will not stop until Ω2x = Ω2

y = 0 and Ω2z = J2/I2z .

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, Erot ≫ kT . The torques produced by the

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 µm 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 ab-

sorb, 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 mag-

netic 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 panelshows polarization vectors in local clouds. The polarization averaged over manyclouds in the Galactic plane is shown in the lower panel. The magnetic field isparallel 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 re-

gions) 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 vibrat-

ing 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, emis-

sion 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 radi-

ation. 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 µm polarization observa-

tions 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 interstel-

lar 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 sec-

tion 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 ob-

jects 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 µm [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 µm and 450 µm

wavebands. In the best 25% of winter nights on Mauna Kea (with τ225 GHz ≈ 0.05),

SHARC II is expected to have a noise equivalent flux density (NEFD 1) at 350 µm

of 1 Jy s1/2 or better (equation 2.1 [3] and figure 2.1).

NEFD350 µm = 1.0× exp(25.0× τ225 GHz × airmass− 1.6) Jy s1/2. (2.1)

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 µm. 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 µm measurements (points) from Jan 2003 compared to theoreti-cal 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 instru-

ments 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 submil-

limeter telescope (BLAST [90]), has had its first flight over Antarctica, and the data

obtained at 250, 350 and 500 µm 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 imagewith an offset between the two polarization components. Left: The expanding beamfrom 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 radiationis separated into two orthogonal (horizontal and vertical) components. Right: Viewtoward the CSO focus. The vertical and horizontal components undergo furtherreflections 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 (µm) 350 450Bandwidth (∆λ/λ0) 0.13 0.10

FOV (arc sec × arc sec) 55 × 55 55 × 55Pixel Size (arc sec × arc sec) 4.6 × 4.6 4.6 × 4.6Angular Resolution (arc sec) 9.0 11.0FOV (arc sec × arc sec) 55 × 55 55 × 55

Point Source Flux for (σp = 1%) in 5 Hours (Jy) 3.6 2.0Surface Brightness for (σp = 1%) in 5 Hours (Jy/pixel) 0.63 0.35Max Separation of Main and Reference Beams (arc min) 5.0 5.0

Systematic Errors, σp (sys) < 0.2% < 0.2%

19

I

Q

U

R

A

W

rgm H/V

Gain

De-

modula!on

Chopped

Data

C

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. Afterapplying the relative data gain factor between the horizontal and vertical array, itcalculates the I, I-error, Q, Q-error, U and U-error components and saves them intoa 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 mea-

surement 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

Q

U

I

Q

U

I

Q

U

B

S

I

Q

U

I

Q

U

I.P.

S

I

Q

U Rot

I

Q

U

I

Q

U

I

Q

U

B

S

I

Q

U

I

Q

U

I.P.

S

I

Q

U Rot

I

Q

U

I

Q

U

I

Q

U

B

S

I

Q

U

I

Q

U

I.P.

S

I

Q

U Rot

I

Q

U

I

Q

U

III

Q

U

QQ

UU

Figure 2.4: Flow chart of “Sharpcombine”. It applies τ and telescope pointing cor-rection, background subtraction (BS), instrument polarization (I.P.) subtraction andpolarization angle rotation to sky coordinates (Rot) to each sub-map before it com-bines 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 inthe 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: 3.6um.) [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

cluster of stars surrounded by successive layers of H+, H0, 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 µm observation, is centered at 18h17m32.0s,

−1614′25.0′′ (B1950) or 18h20m25.2s, −1613′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 clusterof stars surrounded by successive layers of H+, H0, and H2 gas, that expanding withdifferent 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 µm) [7, 23] and submillimeter (Hertz, 350 µm) [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 µm 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 uncalibratedflux map. Thick vectors are detected with greater than or equal to 3σ level and thinvectors are between 2σ and 3σ level. The circle on the bottom right shows the SHARPbeamsize. Some parts of the flux map is removed due to high noise levels. Offsets arefrom 18h17m32s, -1614′25′′ (B1950.0).

25

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

0

10

20

30

40

50

60

70

80

Num

ber

Median = 1.90, Mean = 2.36, Std = 1.81

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

908070605040302010 0 10 20 30 40 50 60 70 80Polarization Angle (degree)

0

5

10

15

20

25

30

35

40

Num

ber

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

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

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 polariza-

tion properties and temperature, then one expects the magnitude of the polarization

(polarization fraction) to be nearly independent of wavelength higher than 50 µm

[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 µm and rising towards longer wavelengths.

The analysis presented here incorporate the 450 µm 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 µm to 450 µm are plotted

in figure 2.11. They share a common area (marked by a green shadow) between

18h17m30s and 18h17m37s in Ra (B1950), −1616′20′′ and −1613′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 µm to 450 µm 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 umflux map. The magnetic vectors from SHARP and Stokes are perpendicular to theirpolarization angles, while those from optical polarization measurement are parallelto their polarization angles. All magnetic vectors (plotted with the same length) areused to indicate the direction only. Offsets are from 18h17m32s, -1614′25′′ (B1950.0).

28

Figure 2.11: The common area (green shadow) for polarization spectrum analysis. Itis between 18h17m30s and 18h17m37s in Ra (B1950), −1616′20′′ and −1613′00′′ inDec (B1950). The selected polarization vectors are at 60 µm (yellow), 100 µm (green),350 µm (blue) and 450 µm (red). Background is the 450 µm flux map. Offsets arefrom 18h17m32s, -1614′25′′ (B1950.0).

29

Table 2.2: M17 Polarization Spectrum Data

Ratio Points Median Mean Std NoteP450/P60 13 0.390 0.395 0.056 see appendix A.1 for detailsP450/P100 11 0.520 0.525 0.128 see appendix A.2 for detailsP450/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 µm than at 350 µm. The polarization spectrum

falls monotonically from 60 µm to 450 µm.

There are many models to explain the rising (or falling) of the polarization spec-

trum 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 µm to 450

µm (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

µm magnetic vectors plotted over the [21 cm]/[450 µm] 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

30

40 100 200 400 1000 2000

1

2

3

4

Wavelength (um)

Med

ian

Pol

ariz

atio

n R

atio

W51M17OMC−1NGC2024DR21OMC−3This work

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

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

µm]/[350 µm] 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 µm] = 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

Y

X

Dec

Ra

66.3o

y = 0 y = -50

Dense Cloud

Figure 2.13: Magnetic vectors from SHARP plotted over the [21 cm]/[450 µm] fluxratio map, showing that the shock front is passing through the cloud. The contourlevels 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=-50arcsec” lines separate the cloud into “post-shocked” (y > 0), “shock front” (-50 < y <0) and “pre-shocked” (y < -50) regions. The polarization directions and magnitudesin these regions are different (figure 2.14 and 2.15). The magnetic fields in the densecloud (can also be seen in figure 2.10) at the top of the map survive the windswept.Offsets are from 18h17m32s, -1614′25′′ (B1950.0).

33

150 100 50 0 50 100Y (arcsec)

80

60

40

20

0

20

40

60

80

100

(deg

ree)

60um100um350um450um

Figure 2.14: Correlation between polarization angle and the Y direction (zero at18h17m32s, −1614′25′′), showing a linear relationship. The “post-shocked” region isat y > 0 and the “pre-shocked” region is at y < −50 arcsec.

150 100 50 0 50 100Y (arcsec)

0

2

4

6

8

10

12

p (%

)

60um100um350um450um

Figure 2.15: Correlation between polarization fraction and Y direction (zero at18h17m32s, −1614′25′′), showing a “U” like shape. The polarization fraction is higherat 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-linecontour and the ratio of neutral HI (NHI) column density to the spin temperatureTspin distribution map in the 17.5-22 km/s velocity area from [6]. This velocitycomponent 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 µm]/[350 µm] polarization ratio vectors over plotted on the[21 cm]/[450 µm] flux ratio map with contour levels = 0.1, 0.3, 0.5, 0.7, 0.9. Theblue (red) vectors represent P450 < (>) P350. The length of the 2% bar at bottom leftis equivalent to P450/P350 = 1.0. The directions of the vectors are parallel to theirpolarization angles. Offsets are from 18h17m32s, -1614′25′′ (B1950.0).

36

Figure 2.18: The [450 µm]/[350 µm] polarization ratio vectors and 450 µm intensitycontours of SHARP (green, levels = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0) over plotted on theFig.1 from [7]. The blue vectors is found to be correlated with the [OI] line, which isa 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:

n→ p+ + e− + νe n+ p+ →21 D+ γ

21D + p+ →3

2 He + γ 21D +2

1 D →42 He + γ

21D+2

1 D →31 T+ p+ 2

1D +31 T →4

2 He + n

... ... (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%,

Deuterium(2H) ≈ 0.01%, Helium(3H) ≈ 0.001%, Lithium(7Li) ≈ 10−10 [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 radia-

tion 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

the grand unification epoch (between 10−43 s and 10−36 s after the Big Bang), the

inflationary epoch comprises the first part of the electroweak epoch (between 10−36 s

and 10−12 s after the Big Bang). It lasted from 10−36 s to about 10−32 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 (sec-

tion 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 solu-

tion to this problem via the stretching of any initial curvature of the 3-dimensional

universe to near flatness, resulting Ωk ≈ 0.

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 num-

ber 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 uni-

verse 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 ho-

mogeneous 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

Lemaıtre, Howard P. Robertson and Arthur G. Walker also derived the general rela-

tivity metric for the cosmological principle independently. It is named Friedmann -

Lemaıtre - Robertson - Walker (FLRW) metric.

In the four space-time dimensions, using Einstein notation, the invariant is:

ds2 = gµνdxµdxν (3.3)

where the µ and ν indices range from 0 to 3.

The FLRW metric can be written as:

45

gµν =

−1 0 0 0

0 a2

1−kr20 0

0 0 a2r2 0

0 0 0 a2r2sin2θ

(3.4)

and its inverse is:

gµν =

−1 0 0 0

0 1−kr2

a20 0

0 0 1a2r2

0

0 0 0 1a2r2sin2θ

(3.5)

where a = a(t) is the time-dependent cosmic scale factor and k is a constant repre-

senting the curvature of the space.

The Riemann curvature tensor Rµαβγ is defined by:

Rµαβγ =

dΓµαγ

dxβ−dΓµ

αβ

dxγ+ Γµ

σβΓσγα − Γµ

σγΓσβα (3.6)

where Γ are the Christoffel symbols:

Γαλµ =

1

2gαν

(

∂gµν∂xλ

+∂gλν∂xµ

− ∂gµλ∂xν

)

. (3.7)

Also, the Ricci tensor Rµν is defined as:

Rµν = Rαµνα. (3.8)

The diagonal elements of the Ricci tensor are:

R00 = −3a

a

R11 =aa+ 2a2 + 2k

1− kr2

R22 = r2(aa+ aa2 + 2k)

R22 = r2(aa+ aa2 + 2k)sin2θ (3.9)

and the trace of the Ricci tensor is the scalar curvature R:

R ≡ gµνRµν = 6aa+ a2 + k

a2. (3.10)

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:

Rµν −1

2gµνR + gµνΛ = 8πGTµν (3.11)

where Λ is the cosmological constant, G is the gravitational constant and Tµν is the

energy-momentum tensor.

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

Tµν = (p+ ρ)uµuν + pgµν (3.12)

where ρ is density, p is pressure and uµ 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:

H2 ≡(

a

a

)2

=8πG

3ρ− k

a2+

Λ

3(3.13)

anda

a= −4πG

3(ρ+ 3p) +

Λ

3(3.14)

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 ho-

mogeneous and isotropic universe within the context of general relativity. The Hubble

constant is the current value of Hubble parameterH0 = H|t=t0 = 100h km s−1 Mpc−1 =

70.4 km s−1 Mpc−1 [14], where h = H0/(100 km s−1 Mpc−1) = 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+ ρΛ → ρ and p− pΛ → p, where

ρΛ = Λ/(8πG) and pΛ = Λ/(8πG). Equation 3.13 and equation 3.14 can be simplified

as:

(

a

a

)2

=8πG

3ρ− k

a2(3.15)

47

anda

a= −4πG

3(ρ+ 3p). (3.16)

For a perfect isotropic fluid, from the first law of thermodynamics dE+PdV = 0,

one can derive the fluid equation of the universe:

ǫ+ 3a

a(ǫ+ p) = 0, (3.17)

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:

a2 =8πG

3

∑

ω

ǫω,0a−1−3ω − k

a2(3.19)

anda

a= −4πG

3(1 + 3ω)ǫ. (3.20)

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:

ρ ∝ a−3(ω+1) (3.21)

and

a ∝ t2

3(ω+1) . (3.22)

Different species in the universe have different equation of state parameters: (1)

For photons and other relativistic species, ω = 1/3, ρ ∝ a−4 and a ∝ t12 ; (2) For

non-relativistic matter (cold dark matter and baryons), ω = 0, ρ ∝ a−3 and a ∝ t23 ;

(3) For dark energy, ω < −1/3 and a > 0. Dark energy accelerates the expansion

of the universe. (4) For Λ, ω = −1, ρ ∝ a0 = constant and a ∝ exp(H0t). The

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:

ρc,0 =3H2

0

8πG(3.23)

Ω0 =ρ0ρc,0

=8πGρ03H2

0

. (3.24)

If we introduce a new set of definition:

Ωr,0 ≡ǫr,0ǫc,0

, Ωm,0 ≡ǫm,0

ǫc,0, ΩΛ,0 ≡

ǫΛ,0ǫc,0

(3.25)

and

Ω0 = Ωr,0 + Ωm,0 + ΩΛ,0

Ωk = 1− Ω0 (3.26)

where ǫc,0 = ρc,0c2, indices r, m, Λ and k stand for radiation, matter, cosmological

constant (dark energy) and curvature, respectively. Equation 3.15 can be written as:

H2(t)

H20

=Ωr,0

a4+

Ωm,0

a3+ ΩΛ,0 +

Ωk

a2. (3.27)

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

by a negative pressure, when a≪ 1, the universe was dominated by radiation (∝ a−4),

and later by matter (∝ a−3). 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

(ns), curvature fluctuation amplitude (∆2R) 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 DescriptionΩb 0.0456± 0.0016 Baryon densityΩc 0.227± 0.014 Cold dark matter densityΩΛ 0.728+0.015

−0.016 Dark energy densityns 0.963± 0.012 Scalar spectral index

∆2R(k0 = 0.002Mpc−1) (2.441+0.088

−0.092)× 10−9 Curvature fluctuation amplitudeτ 0.087± 0.014 Reionization optical depth

Extended parameter Value DescriptionH0 70.4+1.3

−1.4 Hubble constant (km s−1 Mpc−1)t0 13.75± 0.11 Age of the universe (Gyr)r < 0.24(95% CL) Tensor-to-scalar ratioΩ0 1.0023+0.0056

−0.0054 Total densityΩk −0.0023+0.0054

−0.0056 Curvature densityz∗ 1090.89+0.68

−0.69 Redshift at decouplingt∗ 377730+3205

−3200 Age at decoupling (yr)zreion 10.4± 1.2 Redshift of reionization

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

table 3.1, we have Ω0 = 1.0023+0.0056−0.0054, Ωk = −0.0023+0.0054

−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) pho-

tons (see section 3.4) and neutrinos. The energy density of photon can be calculated

by Bose-Einstein distribution:

ρ = g

∫

d3p

(2π)3f(~x, ~p)E(p), (3.28)

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):

ργ = 2

∫

d3p

(2π)3p

ep/Tγ − 1

=8π

(2π)3

∫ ∞

0

p3

ep/Tγ − 1dp

=π2

15T 4γ (3.29)

So,

Ωγ,0 =ργ,0ρc,0

=π2

15T 4γ,0

1

ρc,0=π2

15T 4γ,0

8πG

3H20

=π2

15T 4γ,0h

−2 × 6.808× 10−7 = 4.98× 10−5 (3.30)

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 inter-

acting 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-

dicts Tν,0 ≈ 0.71 × Tγ,0 = 1.945 K and Ων,0 ≈ 0.68 × Ωγ,0 = 3.40 × 10−5. WMAP

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 mat-

ter. 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 be-

tween 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 mat-

ter. 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 ex-

periments 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 tem-

peratures of around 1019 K. The universe was intensely hot, remarkably smooth and

essentially homogeneous. However, small fluctuations in density originating as quan-

tum 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

and became atoms (recombination). Radiation decoupled from matter at 377730+3205−3200

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 tempera-

ture 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

roughly 5× 10−5 (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 scatteringsurface (LSS) when the universe is about 380,000 years old with the temperature ofabout 3,000 K [10].

55

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

3.4.1 The CMB Anisotropy

While it is nearly perfectly homogeneous, the CMB radiation does have tempera-

ture 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 (n) on the

celestial sphere, we can expand it in a spherical harmonics basis Ylm as:

T (n) =∑

l,m

almYlm(n) (3.31)

then the angular power spectrum for our actual sky will be

Cskyl =

1

2l + 1

∑

m

|alm|2. (3.32)

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 dif-

ferent intensities, a net linear polarization would result (xp1 6= yp1). Such anisotropy

57

Figure 3.3: The angular power spectrum from WMAP [12], showing the detection ofthe first three peaks. The first peak is at ℓ ≈ 220, corresponding to an angular scaleof 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 fluc-

tuations 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

Y

(cold radiaon)

Z

(polarizaon)

xp1

xp2 yp2

yp1

yp1

xp1

e-

E > 0 E < 0

B > 0 B < 0

Figure 3.4: Left: Quadrupole polarization from Thomson scattering of the CMBphotons with free electrons. Right: The E and B mode patterns. The E-modes arecurl-free components with no handedness. The B-modes are curl components withhandedness.

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(n)± iU(n) =∑

lm

a∓2,lm ∓2Ylm(n) (3.33)

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 lensingsignal is shown as a blue dashed line. The foreground model for synchrotron plusdust 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

nK. That is 10−8 of the CMB blackbody temperature and 10−3 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 gravita-

tional 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

r ≡ ∆2h(k0)

∆2ℜ(k0)

, (4.1)

where, ∆2ℜ(k) and ∆2

h(k) denote the dimensionless scalar and tensor power spectra, ℜdenotes 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) onscalar spectral index (ns) and tensor to scalar ratio (r), derived from the data com-bination of WMAP + BAO + H0 [14]. Three linear fits are from different simpleinflation 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

universe was about 10−32 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 inGalactic coordinates showing the sky coverage of CLASS experiment. Observing fromthe 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). Thedark 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 con-sists a front-end variable-delay polarization modulator, catadioptric optic system anda 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 modulatoris small is essential at front of opticsNoise is Large number 4 telescopes

dominated of detectors 4 focal planesby with 396 pixels

atmosphere low noise 792 TESsForegrounds are Polarization must 40 GHz: 36 detector pairs

polarized: Synchrotron, be measured at 90 GHz: 300 detector pairspolarized dust emission multiple frequencies 150 GHz: 60 detector pairs

At small angular Separate lensing Focus on largescales, 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 system-

atic 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 sig-

nals. 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 feed-

horns 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 po-

larized 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 re-

moval. 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 back-

ground 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

µ

dn

aB z

HG

04

dn

aB z

HG

09

dn

aB z

HG

05

1

PIPER Bands

Multipole Moment

)π

[µK

2]

Grav. Lensing

Reionization

Bump

BOOMERanG

DASI

WMAP

QUaD

CBI

CAPMAP

BICEP

Figure 4.4: CLASS wavebands and sensitivity curve from [15]. Left: The frequencybands of CLASS are chosen to straddle the Galactic foreground spectral minimumand to minimize atmospheric effects (see section 4.2.2). Right: The CLASS sensitivitycurve, shown by the dashed curve along the shaded boundary, is the 1σ limit for eachl 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 thecosmologically interesting limit of r ∼ 0.01.

approaches that are complementary to CLASS.

4.2.1 Sensitivity Calculation

The sensitivity calculation is based on the sky coverage, instrument beam size,

efficiency, number and sensitivity of the detectors and total integration time. In

far-IR to millimeter waveband, detector sensitivity is normally quoted as NEP, that

can also be converted to other instrument-specific parameters, such as NEFD (see

section 2.2), NET or NEQ:

Definition of NEP, NET and NEQ

Noise Equivalent Power (NEP) is a measure of the sensitivity of a detector nor-

mally used in astronomy. It is defined as the signal power that gives a unity signal-

to-noise ratio in a 1 Hertz output bandwidth [140]. Base on the Nyquist-Shannon

sampling theorem, an output bandwidth of 1 Hertz is equivalent to half a second of

68

integration time. NEP is a detector-specific parameter. It has the unit of WHz−1/2.

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 µKs1/2.

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 µKs1/2.

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]:

NET =NEP√

2ηdηtAΩ∆ν∂Bν/∂T(4.3)

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 40 GHz 90 GHz 150 GHzNumber of pixels (detector pair) (Np) 36 300 60Number of detectors (Ns) 72 600 120Beamsize () 1.50 0.67 0.40NETp of detector (µKs1/2) 68 60 93NEQp of detector (µKs1/2) 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:

σbeam =FWHM√

8ln2. (4.5)

Under the flat-sky approximation, the solid angle of the beam will be:

Ωbeam = 2πσ2beam (4.6)

because the integral over a Gaussian plane with unit height gives:

∫ ∫

exp

(

−x2 + y2

2σ2

)

dxdy = 2πσ2. (4.7)

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:

ωℓ = exp[−ℓ(ℓ + 1)σ2beam]. (4.8)

Noise Power Spectra

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

σ2Q =

NEQ2p

(ηdηt)2Nptpix=

NEQ2s

(ηdηt)2Nstpix(4.9)

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

tpix = ηobstobs/

(

4πfskyΩbeam

)

=ηobstobsΩbeam

4πfsky. (4.10)

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:

NBBℓ = NEE

ℓ = Ωbeamσ2Q. (4.11)

Then the expected error in the CBBℓ measurement is:

∆CBBℓ =

√

2

(2ℓ+ 1)fsky

(

CBBℓ +

NBBℓ

ωℓ

)

(4.12)

By substituting equation 4.5 - equation 4.11 in to equation 4.12, we have:

∆CBBℓ =

√

2

(2ℓ+ 1)fskyCBB

ℓ +4πfskyNEQ

2p

ηobs(ηdηt)2tobsNp

exp[ℓ(ℓ+ 1)σ2beam] (4.13)

To calculate the CLASS sensitivity (see figure 4.4), we used the NEQ2p, Np and

σbeam values listed in table 4.2 and assumed tobs = 3 years, ηobs = 50%, ηdηt = 0.80,

fsky = 65% and CBBℓ is from the current upper limit of r ≈ 0.2.

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 at-

mospheric 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 band-

width 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 atChajnantor, based on 10 years of site testing. Conditions are worse during the winterfrom the end of December to early April. The expected median PWV for the rest ofthe year is around 1 mm, while conditions of PWV < 0.5 mm can be expected up to25% of the time [16].

73

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Atmospheric Transmission

Frequency (GHz)

Tran

smis

sion

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300Atmospheric Brightness Temperature

Frequency (GHz)

Brig

htne

ss T

empe

ratu

re (K

)

Figure 4.6: Atmospheric transmission and brightness temperature at CLASS site from5 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.

NEP2 = NEP2Johnson +NEP2

thermal +NEP2photon

+NEP2load +NEP2

amplifier +NEP2excess. (4.14)

Since detectors for the CLASS experiment are background limited, NEP2photon from

atmosphere and instrument dominates over other sources of noise. In a black-body

radiation field, NEP2photon can be written as [147] (see appendix C for details):

NEP2photon = 4

AΩ

c2(kBTs)

5

h3

∫

x4

ex − 1

(

1 +αǫf

ex − 1

)

(αǫf)dx (4.15)

Where AΩ describes the optics, f is the transmissivity of the optics, Ts is the tem-

perature 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:

Iν =

(

2hν3/c2

ehν/(kBTs) − 1

)

ǫ (4.16)

where Iν is the intensity of the atmosphere emission.

74

The number of photons in a given state x is:

n(ν) = n(x) =αǫf

ex − 1. (4.17)

From equation 4.16 and equation 4.17 , we have:

n = αfIν/(ex − 1)/

(

2hν3/c2

ehν/(kBTs) − 1

)

=αfc2

2hν3Iν . (4.18)

Equation 4.15 can be written in n as:

NEP2photon = 4

AΩh2

c2(kBTs)

5

h3

∫

n(1 + n)ν4dν

= 4h2

c2

∫

AΩn(1 + n)ν4dν = 4h2∫

n(1 + n)ν2dν (4.19)

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

S = (0.5 s)

∫

AΩαfξ2hν3/c2

ehν/(kBTcmb) − 1dν =

∫

αfξhν

ehν/(kBTcmb) − 1dν (4.20)

where ξ is the transmission of the atmosphere.

From equation 4.19 and equation 4.20, we have:

S

N=

S

NEP=

∫ ν2ν1αfξhν/(ehν/(kBTcmb) − 1)dν

(4h2∫ ν2ν1n(1 + n)ν2dν)1/2

(4.21)

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 sec-

tion 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:

SNR ∝ η(ν2/ν1)

∫ ν2ν1αfξhν/(ehν/(kBTcmb) − 1)dν

(4h2∫ ν2ν1n(1 + n)ν2dν)1/2

(4.22)

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:

Iν =2hν3

c21

ehν/(kBTν) − 1(4.23)

then, from equation 4.18, we have:

n =αfc2

2hν3Iν =

αf

ehν/(kBTν) − 1. (4.24)

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 (equa-

tion 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 optimiza-

tion. 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

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

3.5

4 x 10−18

Frequency [GHz]

Inte

nsity

[W

Hz−1

m−2

sr−2

]

CMB x Atmosphere Transmission

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

x 10−15

Frequency [GHz]

Inte

nsity

[W

Hz−1

m−2

sr−2

]

Atmosphere Emission Intensity

Figure 4.7: Top: the CMB signal (equation 4.20) and Bottom: atmospheric noisesource (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 GHzband: (30.3 GHz - 40.3 GHz), (77.3 GHz - 108.3 GHz) and (126.8 GHz - 164.3 GHz).

77

Frequency [GHz]

Freq

uenc

y [G

Hz]

Relative Signal to Noise Ratio

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200 0

0.5

1

1.5

2

2.5

3

Figure 4.8: The 2-D plot of relative signal-to-noise ratio (equation 4.22) from 0 to200 GHz showing our optimization results. The cross points of red, green and whitelines are the locations of the local maxima. For the 40 GHz band, we only search forthe 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)

U

Q

HWP VPM

V (circular)

U

Q

Figure 4.9: As shown in Poincare sphere, VPM modulates between Q and V , whilethe 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 courtesyof David Chuss.

4.3 The Variable-delay Polarization Modulator

VPM is the the first element of CLASS instrument. It modulates the sky polar-

ization signal by introducing a controlled variable path difference between two or-

thogonal 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

d

Mirror

Grid

Figure 4.10: VPM modulates polarization by introducing a controlled variable pathdifference between two orthogonal linear polarizations. Dots show the componentwith polarization angle parallel to the grid; Double arrow show that with angle per-pendicular to the grid. By moving the mirror up and down, VPM introduces a pathdifference 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

Mvpm(α, φ) =

1 0 0 0

0 cos22α+ cosφsin22α −sin2αcos2α(1− cosφ) sin2αsinφ

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

Q

U

V

det

= Mvpm

I

Q

U

V

sky

=

1 0 0 0

0 cosφ 0 sinφ

0 0 −1 0

0 sinφ 0 −cosφ

I

Q

U

V

sky

. (4.28)

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

Figure 4.11: The wire grid performances for two different wavelengths from a simula-tion [17]. In the limit of g/λ≪ 1, a sinusoidal form for Stokes Q is in good agreementwith an ideal grid (equation 4.29). The VPM reflection phase delay differs from thefree-space grid-mirror delay if the conditions are changed.

radius of the wire, g is the center-to-center wire pitch. Larger 2a/g leads to higher

reflection for both parallel and perpendicular polarization components. Grids with

2a/g ≈ 1/π allow high enough reflective for the parallel component and at the mean

time enable high transmissive for the perpendicular component. A transmission line

model has been developed to simulate to performance of a VPM grid in a range of

0.02 < 2a/g < 1.00 [17].

Figure 4.11 shows the polarization transfer function for a single frequency of two

models with different geometric limit. For plane wave illumination with λ≫ a, equa-

tion 4.27 is a good approximation for the VPM phase delay. As the wire diameter

becomes a finite fraction of a wavelength, the polarization response remains a sinu-

soidal function of the phase delay; however, the VPM reflection phase is dependent

upon the details of the grid geometry.

82

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:

S(x) =1

2(kh − kl)

∫ kh

kl

[I(k) +Q(k)cos(kx) + V (k)sin(kx)]dk (4.30)

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 )):

I(k) = I0k

ehν/(kBTcmb) − 1=

I0k

eAk − 1

Q(k) = Q0k


Q0k

eAk − 1

V (k) = V0k


V0k

eAk − 1(4.31)

where, I0, Q0, V0 are constants and A = hc/(2πkBTcmb).

Then, equation 4.30 can be written as:

S(x) =1

2(kh − kl)

∫ kh

kl

[

I0k

eAk − 1+

Q0k

eAk − 1cos(kx) +

V0k

eAk − 1sin(kx)

]

dk

= SI × I0 + SQ(x)×Q0 + SV (x)× V0 (4.32)

83

where,

SI =1

2(kh − kl)

∫ kh

kl

k

eAk − 1dk = constant

SQ =1

2(kh − kl)

∫ kh

kl

kcos(kx)

eAk − 1dk

SV =1

2(kh − kl)

∫ kh

kl

ksin(kx)

eAk − 1dk. (4.33)

“Cosine” and “Linear” are two candidate VPM mirror chopping modes. The

chopping can be approximated by N discrete steps as:

d(i) =

p0 +∆p× (cos(i/(N − 1) ∗ π) + 1)/2 Cosine mode

p0 +∆p× (i/(N − 1)) Linear mode(4.34)

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

A =

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

SI(1) SQ(1) SV (1)

... ... ...

SI(N − 2) SQ(N − 2) SV (N − 2)

SI(N − 1) SQ(N − 1) SV (N − 1)

, (4.37)

X =

I0

Q0

V0

(4.38)

84

and

s =

S(0)

S(1)

...

S(N − 2)

S(N − 1)

. (4.39)

All elements in A can be calculated by substituting equation 4.35 and equa-

tion 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:

X = (ATA)−1AT s (4.40)

CovX = (ATA)−1 (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:

SNQ =Q

σQ∝

∑ |SQ(i)|√CovX22

(4.42)

SNV =V

σV∝

∑

|SV (i)|√CovX33

(4.43)

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 µm for the 40 GHz band, 32 µm for 90

GHz and 21 µm 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

Peak to Peak Throw (λ0)

Mirr

or S

tart

Pos

ition

(λ0)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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

86

Peak to Peak Throw (λ0)

Mirr

or S

tart

Pos

ition

(λ0)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4.13: The contour plot of relative signal-to-noise ratio for Stokes Q, calculatedfrom equation 4.42 with linear chopping mode. This plot is for the 40 GHz band (33GHz to 43 GHz, λ0 = 7.89 mm). The maximum is at (0.46 λ0, 0.16 λ0) with the peaksignal-to-noise ratio scaled to be 1.00. There are 2 other local maxima nearby: (0.63λ0, 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)

7.890

1 1.000 0.19 0.13 1.500 1.02633.0 2 0.942 0.19 0.39 1.500 3.077- 3 0.912 0.44 0.13 3.472 1.026

43.0 4 0.861 0.44 0.39 3.472 3.0775 0.849 0.27 0.26 2.130 2.051

3.231

1 1.000 0.19 0.13 0.614 0.42077.3 2 0.907 0.19 0.39 0.614 1.260- 3 0.892 0.44 0.13 1.422 0.420

108.3 4 0.838 0.27 0.25 0.872 0.8085 0.811 0.44 0.39 1.422 1.260

2.060

1 1.000 0.19 0.13 0.391 0.268126.8 2 0.944 0.19 0.40 0.391 0.824- 3 0.912 0.45 0.13 0.927 0.268

164.3 4 0.862 0.45 0.39 0.927 0.8035 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)

33.07.890

1 1.000 0.46 0.16 3.629 1.262- 2 0.932 0.63 0.19 4.971 1.500

43.0 3 0.884 0.45 0.42 3.551 3.314

77.33.231

1 1.000 0.45 0.16 1.454 0.517- 2 0.908 0.59 0.19 1.906 0.614

108.3 3 0.811 0.46 0.42 1.486 1.357

126.82.060

1 1.000 0.46 0.16 0.948 0.330- 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:

TOD(t) =1

φh − φl

∫ φh

φl

(Qcosφ+ V sinφ+ I)dφ (4.46)

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:

φ0 ≡√

φlφh = x√

klkh

r ≡ φh/φl

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:

TOD(t) =1

φ0

√r − φ0/

√r

∫ φ0√r

φ0/√r

(Qcosφ+ V sinφ+ I)dφ

= Q× TODQ(t) + V × TODV (t) + I (4.48)

where

TODQ(t) =

√r

φ0(r − 1)[(sin(φ0

√r)− sin(φ0/

√r)]

TODV (t) =

√r

φ0(r − 1)[(cos(φ0/

√r)− cos(φ0

√r)]. (4.49)

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:

(

TODQ(t), TODV (t), 1)

×

Q

V

I

= A×

Q

V

I

= TOD (4.50)

then the least-squares solution is:

Q

V

I

= (ATA)−1AT ·TOD (4.51)

Assuming a sinsoidal oscillation of φ0(t) = x(t)√klkh with time:

φ0(t) = xmin

√

klkh + (xmax

√

klkh − xmin

√

klkh)(cos(t) + 1)/2 (4.52)

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

(ATA)QQ =

∫ π

0

TOD2Q(t)dt

(ATA)QV = (ATA)V Q =

∫ π

0

TODQ(t)TODV (t)dt

(ATA)QI = (ATA)IQ =

∫ π

0

TODQ(t)dt

(ATA)V V =

∫ π

0

TOD2V (t)dt

(ATA)V I = (ATA)IV =

∫ π

0

TODV (t)dt

(ATA)II =

∫ π

0

dt = π. (4.53)

We then invert the ATA matrix to get the covariance matrix. To the extent that

ATA is a diagonal matrix, we have:

σQ =1

√

∫ π

0TOD2

Q(t)dt. (4.54)

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) = σQ(r → 1.0)/σQ(r) (4.55)

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 µm (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 µm wire pitch. It has a 50 cm

diameter clear aperture, with flatness within 50 µm. The mechanical wire resonant

frequency is higher than 128 Hz.

91

5 10 15 20 25 30 35 40 45 5030

40

50

60

70

80

90

100VPM Efficiency

r

η (%

)

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

With a/λ0 ≈ 0.004, this wire grid can be considered as a perfect VPM grid for

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 aluminiumbox frame with over 2 tons of stretch force. The diameter of the flattener ring is 50 cm.The wire diameter, 2a, is 63.5 µm, with wire pitch, g = 200 µm. 2a/g = 1/3.15 ≈ 1/π.The flatness of the grid is better than 50 µm. The total length of the wires is longerthan 2 miles.

93

Table 4.4: CLASS Optics Overview

Waveband 40 GHz 90 GHz 150 GHz

FOV () 18.0 7.0 3.5Beamsize () 1.50 0.67 0.40Strehl 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 fig-

ure 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 µm [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).

94

600

500

127

1026.5

1146.2

1820

2200

40°

54.1°

941

Figure 4.16: Top: Drawings of CLASS 40 GHz optics. It consists of a front-endVPM, 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 arein mm. Figure courtesy of Joseph Eimer.

95

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 courtesyof Joseph Eimer.

96

Figure 4.18: Point spread diagram of CLASS 40 GHz optics from Zeemax. Eachdiagram in this figure represents a separate direction on the sky. The circles showthe first Airy disk at the corresponding location. This diagram shows that the opticsis 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 preci-

sion 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 re-

flected power is better than -28 dB was designed, built and measured [156]. This

smooth-walled feedhorn, however has relatively low aperture efficiency and high side-

lobe 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) = p′

11c/(2πaguide) (4.56)

where ao is the rectangular waveguide broadwall width, aguide is the circular waveguide

radius, and p′

11 ≈ 1.841 is the eigenvalue for TE11 mode, and c is the speed of light.

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

χ2 =

N∑

i=1

M∑

j=1

[

αj∆j(fi)2]

, (4.57)

where i is the sum is over a discrete set of (N) frequencies in the optimization fre-

quency 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

∆1(f) =

XP (f)−XP0 if XP (f) > XP0,

0 if XP (f) ≤ XP0,

∆2(f) = ET (f)− ET0 (4.58)

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 con-

structed 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

Inial input:

Sin0.75+Gaussian

add-on profile

XP0=-25dB

ET0=-10 dB

Feedhorn beam calculaon method from James

Opmizaon

step 1:

5 points natural

spline profile

Opmizaon

step 2:

10 points natural

spline profile

Opmizaon

step 3:

20 points natural

spline profile

Final profile

XP0=-30dB

ET0=-10 dB

XP0=-34dB

ET0=-10 dB

Figure 4.19: Flow chart of smooth-walled feedhorn optimization. Optimization beginswith 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 differentthresholds in each step.

r(z) =

aguide + acsin0.75(πz/L) if 0 ≤ z ≤ L/2,

aguide + ac1 + [C(z − L/2)]21/2 if L/2 < z ≤ L,(4.59)

where

C = (2/L)[((af − aguide)/ac)2 − 1]1/2 (4.60)

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 22.4Bandwidth 33 - 43 GHz 1.25 fc - 1.63 fc , ∆f/f0 = 26.3 %Aperture 36.00 mm feedhorn wall thickness = 1.00 mmLength 100.00 mm D = 36.00 mm requires L ≥ 75 mm

Cross pol ≤ −30 dB within 15 azimuth angle, across the bandwidthReturn loss ≤ −30 dB across the bandwidthEdge 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

−10 0 10 20 30 40 50 60 70 80 90 100−20

−10

0

10

20Smooth−walled Feedhorn Profile

Length (mm)

Rad

ius

(mm

)

Figure 4.20: CLASS 40 GHz feedhorn profile. The 10.00 mm long input waveguidehas a radius of 3.334 mm, with fc = 26.349 GHz. The length of the feedhorn is100.00 mm. The aperture is 35.828 mm. This is a monotonic profile that allows aprogressive milling technique.

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 1.9−70

−60

−50

−40

−30

−20

−10

Cross−Pol, Return−Loss and Edge−Taper

Freqency (fc)

Pow

er (d

B)

Cross−PolReturn−LossEdge−Taper

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

103

Table 4.6: Feedhorn Profile Approximation (in Millimeters)

Step Length (z) Radius (r)

0 0.00 3.3341 5.00 4.5552 10.00 5.7493 15.00 6.8824 20.00 7.9065 25.00 8.7556 30.00 9.3307 35.00 9.5788 40.00 9.6729 45.00 9.77610 50.00 9.81111 55.00 9.83612 60.00 9.86613 65.00 11.04214 70.00 12.91915 75.00 15.01016 80.00 16.71817 85.00 17.57718 90.00 17.83819 95.00 17.89220 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 -7.6531 1.177 9.671 -25.00 -24.11 -7.9532 1.214 8.369 -26.71 -27.95 -8.3233 1.252 9.085 -30.01 -30.09 -8.6934 1.290 8.817 -31.26 -32.89 -9.2035 1.328 8.565 -33.74 -37.01 -9.6336 1.366 8.328 -34.30 -36.99 -10.0537 1.404 8.102 -39.62 -45.20 -10.4538 1.442 7.889 -39.00 -39.86 -10.8039 1.480 7.687 -36.51 -59.07 -11.4540 1.518 7.495 -35.75 -41.55 -11.6541 1.556 7.312 -34.83 -64.50 -11.7942 1.594 7.138 -32.49 -41.61 -11.8843 1.632 6.972 -32.17 -49.95 -11.6844 1.670 6.813 -30.21 -42.31 -11.5745 1.708 6.662 -29.11 -43.99 -11.4246 1.746 6.517 -27.85 -44.88 -11.1547 1.784 6.379 -26.40 -41.39 -11.1348 1.822 6.246 -25.57 -51.60 -10.8449 1.860 6.118 -24.11 -40.68 -10.9950 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

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

33GHz HWHM=8.37deg

E PlaneH PlaneD PlaneX Pol

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

34GHz HWHM=8.12deg

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

35GHz HWHM=7.93deg

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

36GHz HWHM=7.72deg

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

37GHz HWHM=7.55deg

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

38GHz HWHM=7.33deg

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

106

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Pow

er (

dB)

39GHz HWHM=7.10deg


-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Pow

er (

dB)

40GHz HWHM=6.92deg

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Pow

er (

dB)

41GHz HWHM=6.73deg

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Pow

er (

dB)

42GHz HWHM=6.55deg

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Pow

er (

dB)

43GHz HWHM=6.42deg

-90 -70 -50 -30 -10 10 30 50 70 90-60

-40

-20

0

Angle (degrees)

Pow

er (

dB)

44GHz HWHM=6.26deg


107

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

33GHz HWHM=8.37deg


-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

34GHz HWHM=8.12deg

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

35GHz HWHM=7.93deg

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

36GHz HWHM=7.72deg

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

37GHz HWHM=7.55deg

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Po

we

r (d

B)

38GHz HWHM=7.33deg


108

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Pow

er

(dB

)

39GHz HWHM=7.10deg


-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Pow

er

(dB

)

40GHz HWHM=6.92deg

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Pow

er

(dB

)

41GHz HWHM=6.73deg

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Pow

er

(dB

)

42GHz HWHM=6.55deg

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Pow

er

(dB

)

43GHz HWHM=6.42deg

-15 -12 -9 -6 -3 0 3 6 9 12 15-60

-40

-20

0

Angle (degrees)

Pow

er

(dB

)

44GHz HWHM=6.26deg


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 0.91431 9.671 0.126 20.00 0.725 0.92732 9.369 0.121 20.18 0.709 0.93633 9.085 0.115 20.41 0.702 0.94634 8.817 0.109 20.63 0.697 0.95335 8.565 0.105 20.80 0.683 0.95836 8.328 0.100 20.99 0.675 0.96637 8.102 0.095 21.22 0.674 0.97438 7.889 0.092 21.36 0.660 0.97739 7.687 0.087 21.61 0.664 0.97940 7.495 0.083 21.79 0.658 0.97941 7.312 0.081 21.91 0.643 0.97742 7.138 0.078 22.09 0.639 0.97543 6.972 0.076 22.18 0.622 0.97344 6.813 0.074 22.33 0.615 0.97145 6.662 0.072 22.44 0.603 0.97046 6.517 0.070 22.54 0.590 0.96847 6.379 0.068 22.66 0.582 0.96848 6.246 0.067 22.71 0.565 0.96649 6.118 0.065 22.85 0.559 0.96650 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 polariza-

tion, 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 µm,

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

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

horn has a fabrication tolerance at 1σ level of about 250 µm (0.010 inch). With a

machining tolerance of 25 µm (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 bolome-

ters. 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 of36 smooth-walled feedhorns, waveguide adapter, detector mounting plate and clips.The focal plane will operate at a temperature of 100 mK. Figure courtesy of ThomasEssinger-Hileman.

superconducting microstrip transmission lines. Band-defining filters limit the spec-

tral 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 perfor-

mance. 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 detectorchip 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 prototypedetector 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 bya thermal link G to a thermal source with a temperature of Tbath. The total powerto the pixel is Pγ + PJ − PG. Right: TES is biased by IB = VB/RB, in the caseof RB ≫ RSH. For R ≫ RSH, the TES is bias by V = IBRSH, then fluctuations ofR will result in fluctuation in current, which is read out by the inductor L and thesuperconducting quantum interference device (SQUID) amplifier.

114

CdT

dt= Pγ + PJ − PG (4.61)

where

PJ = I2R (4.62)

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

PG =

∫ T

Tbath

G(T )dT. (4.63)

The electrical circuit in the right panel of figure 4.29 can be written as:

LdI

dt+ IR = (IB − I)RSH (4.64)

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

Psat = PG − I2minRn = PG −(

IBRSH

RSH +Rn

)2

Rn (4.65)

where, Rn and Imin are the resistance and the current running through TES at normal

state. In the limit of a voltage bias (Rn ≫ RSH), and a narrow transition, so that

PG ≈ V 2/Ro is approximately constant at equilibrium, equation 4.65 reduces to

Psat = (1−Ro/Rn)PG (4.66)

where, Ro is the resistance of the TES at equilibrium (operating point).

The total NEP of a TES can be written as:

NEP = (NEP2det +NEP2

γ)1/2 ≈ (4kBGoT

2o + 2Pγhν)

1/2 (4.67)

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 con-

ductance 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

is : Psat = 3.5 pW, Go = 116 pW/K, To = 0.150 K and NEPdet = 1.2 × 10−17

WHz−1/2.

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 manu-

factured 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 mechanicalheat switch controller, vacuum valve, pulse tube (PT) head, 60 K plate, 4 K plate,adiabatic demagnetization refrigerator (ADR), high temp superconducting leads for4 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 theHPD cryostat in the experimental cosmology lab at Johns Hopkins University. Photocourtesy 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 diodemoniter and an AC bridge, a front panel, a NI GPIB to Ethernet adapter, a Lakeshore370 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 µW. 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 µW.

Based on these curves, the FAA pill of the ADR have higher cooling capacities at

lower loads. With a 2 µW load, the cooling capacity is about 130 mJ, which is close

to the theoretical number of 118 mJ. While with a 10 µW 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 supercon-

ducting 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

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

120

Figure 4.33: ADR cooling curves at 100 mK, showing the magnet current versus timeof the ADR with the loads of from 2.0 to 10.0 µW. Based on these curves, the FAApill 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 1.6194 K plate Lakeshore DT670 293.8 0.5739 2.722 1.61960 K plate Lakeshore DT670 293.9 0.5738 50.28 1.07360 K PT Lakeshore DT670 294.1 0.5732 35.94 1.0974 K PT Lakeshore DT670 295.1 0.5709 2.745 1.618He4 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 1.5511 K Ruox Scientific Inst RO600 281.8 1.002 2.828 1.550

Magnet RTD AMI Unknow 293.0 0.1047 2.641 2.452R1 Lakeshore RX-202A 115.0 2.000 2.918 3.146R2 Lakeshore RX-202A 109.1 2.011 2.912 3.147R3 Lakeshore RX-102A 216.1 1.003 2.725 1.548

φ through the SQUID junction loop [163]:

V = (R/2)(I2 − (2Iccos(πφ/φ0))2)1/2, (4.68)

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

= h/2e = 2.07× 10−15 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

SQ1

SQ2

SA_BIAS

SA_OFFSET

A/D

D/A

PID

SA_FB

SQ2_BIAS (SB)

SQ2_FB

SQ1_BIAS (RS)

0.1

15 k

TES Bias

ADC_OFFSET

SQ1_FB

(110 )

(110 )

-

14b

14b

SSA_SIG

+

-

LPF

Cold Electronics Warm Electronics (MCE)

A=195

÷2n

÷2m

Ibias

I

Lin

Lfb

Rshunt

RTES

Data Mode 2

Data Mode 0

Data Mode 1

Data Mode 3

Figure 4.34: The FLL block diagram for TES detector readout, showing the coldelectronics 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 feed-

back 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 onthe 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

∆α1 ∆δ1 P450 σp P.A.2 σP.A. P60 σp P.A.2 σP.A. P450/P603

80.0 44.0 2.2 0.5 18.5 5.9 6.7 0.5 21.7 2.2 0.3370.1 -3.5 1.5 0.3 21.9 5.2 4.5 0.3 23.0 1.6 0.3362.7 17.9 1.8 0.2 17.6 3.8 4.5 0.2 22.8 1.4 0.4060.3 -51.0 2.5 0.4 38.2 4.2 4.8 0.5 33.6 2.8 0.5257.8 39.2 2.0 0.2 15.4 3.4 5.3 0.2 20.6 1.3 0.3850.4 60.6 2.4 0.3 23.4 3.0 5.7 0.3 26.0 1.7 0.4242.9 13.1 1.1 0.2 7.3 4.3 2.8 0.2 14.3 1.8 0.3938.0 -58.1 2.1 0.2 40.6 2.7 5.2 0.4 43.5 2.1 0.4033.0 -36.8 1.4 0.2 33.0 3.0 4.7 0.3 28.8 1.7 0.3030.6 53.5 1.6 0.2 19.7 2.9 3.8 0.2 15.8 1.9 0.4228.1 -15.4 1.4 0.1 12.7 2.6 3.7 0.2 17.9 1.6 0.3823.1 77.2 1.6 0.2 20.4 3.2 3.4 0.5 21.3 3.7 0.4713.2 -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,-1614′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 umresult from SHARP (smoothed to 22′′ resolution, Red), center at 18h17m32s,-1614′25′′

(B1950.0).

126

A.2 Polarziation Spectrum: 450 um vs 100 um


119.9 -160.4 3.0 0.5 36.9 4.7 3.9 0.3 27.5 2.3 0.77110.0 -124.8 1.4 0.2 30.7 4.4 4.4 0.2 23.0 1.6 0.32100.1 -89.1 1.6 0.2 16.1 3.0 3.8 0.3 20.4 2.2 0.4292.7 -205.5 1.5 0.3 50.5 6.1 3.5 0.3 51.9 2.9 0.4390.2 -51.1 2.2 0.2 22.0 2.4 4.2 0.3 26.3 2.0 0.5285.3 -169.9 2.0 0.2 41.4 2.7 3.5 0.2 35.5 1.9 0.5780.3 -17.9 1.7 0.2 28.6 3.6 3.4 0.4 34.7 3.1 0.5063.0 -98.6 1.2 0.1 5.7 1.9 2.3 0.2 10.1 1.8 0.5255.6 -63.0 1.3 0.1 9.8 1.9 2.8 0.2 13.2 2.4 0.4645.7 -25.0 1.5 0.1 17.2 1.8 2.0 0.1 25.5 1.9 0.7538.3 -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,-1613′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 450umresult from SHARP (smoothed to 35′′ resolution, Red), center at 18h17m32s,-1614′25′′

(B1950.0).

128

A.3 Polarziation Spectrum: 450 um vs 350 um at

RA > 18h17

m30

s


71.7 -54.6 1.6 0.3 35.4 5.4 2.0 0.2 27.8 3.5 0.8061.8 -19.0 1.1 0.2 20.4 4.9 1.5 0.1 13.1 2.6 0.7359.4 -76.0 2.1 0.3 40.2 4.2 2.3 0.3 35.1 3.7 0.9156.9 -2.4 1.4 0.2 5.9 3.7 2.3 0.1 4.5 1.7 0.6154.4 71.2 1.8 0.4 30.1 6.5 2.3 0.4 24.5 5.1 0.7851.9 14.2 1.6 0.2 9.7 3.3 2.0 0.2 7.3 2.9 0.8044.5 -23.8 1.1 0.2 15.3 4.1 2.0 0.1 5.9 1.2 0.5542.0 49.9 1.7 0.2 22.0 3.8 1.3 0.2 15.0 3.9 1.3139.6 -7.1 1.3 0.2 2.4 3.3 2.0 0.1 3.0 1.0 0.6537.1 66.5 1.7 0.3 21.9 4.3 1.2 0.2 17.2 4.6 1.4234.6 9.5 1.3 0.1 2.8 3.2 2.2 0.1 5.0 1.0 0.5932.1 -45.1 1.4 0.1 12.0 2.3 1.9 0.1 6.7 0.9 0.7429.7 26.1 1.3 0.1 17.7 3.2 1.8 0.1 7.9 1.3 0.7227.2 -28.5 1.8 0.1 -0.1 1.9 2.2 0.1 180.0 0.6 0.8224.7 -85.5 2.2 0.2 35.2 2.4 1.7 0.2 30.9 3.4 1.2922.2 -11.9 1.5 0.1 1.3 2.0 2.1 0.1 178.7 0.7 0.7119.8 61.7 1.7 0.2 19.3 2.7 1.0 0.1 14.6 3.4 1.7017.3 4.7 1.4 0.1 0.5 2.2 2.0 0.0 177.7 0.6 0.7014.8 -49.9 1.7 0.1 -0.9 1.5 1.8 0.1 178.6 0.8 0.9412.4 21.4 1.1 0.1 4.6 3.1 1.4 0.1 177.3 1.1 0.799.9 -33.3 2.2 0.1 -7.8 1.1 2.1 0.1 177.1 0.6 1.054.9 -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,-1614′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 < 18h17

m30

s


2.5 -73.6 2.0 0.2 0.4 2.9 0.6 0.1 2.3 3.3 3.332.5 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,-1614′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 fromSHARP (smoothed to 20′′ resolution, Red), center at 18h17m32s,-1614′25′′ (B1950.0).Blue: Hertz vectors at RA > 18h17m30s, Green: Hertz vectors at RA < 18h17m30s

131

A.5 Polarization Vectors

∆α1 ∆δ1 P σp P.A.2 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,-1614′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 ab-

solute 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 char-

acteristic, 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:

Bν(T ) =2hν3

c21

ehν/(kT ) − 1(B.1)

Where h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant

and Bν(T ) is in the unit of Js−1m−2sr−1Hz−1.

In the Wien limit, where hν ≫ kT , the Plank’s spectrum is approximately:

Bν(T ) ≈2hν3

c2e−hν/(kT ) (B.2)

At low frequency range, where hν ≪ kT (Rayleigh-Jeans limit), it is approxi-

mately:

Bν(T ) ≈ 2kTν2/c2 (B.3)

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

−

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-Jeanslimit works well below 20 GHz.

at ∂Bν/∂ν = 0:

νpeak ≈ 2.82kT/h = 58.7× T

KGHz. (B.4)

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

n(ν) = n(x) =αǫf

ex − 1(C.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

= t2AΩ

c2(kBTs)

5

h3

∫

x4

ex − 1(1 +

αǫf

ex − 1)(αǫf)dx (C.3)

1This chapter is mostly from [142]

138

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

NEP2 =2σ2

t

=4AΩ

c2(kBTs)

5

h3

∫

x4

ex − 1(1 +

αǫf

ex − 1)(αǫf)dx (C.4)

NEP has the unit of W2Hz−1/2.

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 dis-

persion 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 tech-

niques. 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 ≤ λc/20 were found to be sufficient by trial and error, where λc is the cutoff

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

χ2 =

N∑

i=1

M∑

j=1

(

αj∆j(fi)2)

, (D.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 (|S11|2) with uniform

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

∆1(f) =

XP (f)−XP0 if XP (f) > XP0,

0 if XP (f) ≤ XP0,(D.2)

∆2(f) =

RP (f)− RP0 if RP (f) > RP0,

0 if RP (f) ≤ RP0,(D.3)

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 re-

flection 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 ∼ 4λc, but was allowed

to vary slightly to achieve the desired beam size. A single discontinuity exists be-

tween 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:

r(z) =

0.293 + 0.703 sin0.75(0.255z) 0 ≤ z ≤ 6.15,

0.293 + 0.7031 + [0.282(z − 6.15)]2 12 6.15 < z ≤ 12.30,

(D.4)

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

−2 0 2 4 6 8 10 12−3

−2

−1

0

1

2

3

Rad

ius/

λ c

Length/λc

Initial ProfileIntermediate ProfileFinal Profile

Figure D.1: The initial, intermediate and final profiles are shown. All dimensions aregiven in units of the cuttoff wavelength of the input circular waveguide.

is reproduced with a low-spatial frequency error of ∼ 0.015λc. This effect has a

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 fre-

quency 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

−34

−30

−26

−22

−18

Ret

urn

Loss

(dB

)

1 1.2 1.4 1.6 1.8 2

−55

−45

−35

−25

−15

Max

imum

Cro

ss-P

ol (d

B)

Final ProfileIntermediate Solution

Measurements

Initial Guess

f/fc

Figure D.2: (Top) The maximum cross-polar response across the band is shown forthe three profiles in Figure D.1. Measurements of the maximum cross-polarizationare superposed. (Bottom) The reflected power measurements for the final feed hornare shown plotted over the predicted reflected power for the initial, intermediate, andfinal feedhorn profiles. Frequency is given in units of the cutoff frequency of the inputcircular 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 rect-

angular 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 s11 ∼ 1.841 is the eigenvalue

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 deter-

mine 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 3.331 7.0 5.772 14.0 7.913 21.0 9.904 28.0 10.865 35.0 11.136 42.0 11.277 49.0 11.668 56.0 11.909 63.0 11.9610 70.0 12.2411 77.0 12.4412 84.0 12.7613 91.0 13.7014 98.0 15.4015 105.0 17.0116 112.0 17.7117 119.0 20.0518 126.0 21.7519 133.0 21.9120 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 con-structed. The horn is 140 mm long with an aperture radius of 22 mm. The inputcircular 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 0.092534 8.82 21.1 0.098435 8.57 21.4 0.090436 8.33 21.3 0.092937 8.11 21.3 0.093038 7.89 21.9 0.081539 7.69 22.0 0.078840 7.50 22.7 0.067641 7.32 22.9 0.064342 7.14 23.5 0.055643 6.98 23.7 0.054044 6.82 24.2 0.047945 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

-50

-40

-30

-20

-10

0

Pow

er (d

B)

33 GHz

39 GHz

45 GHz

-50

-40

-30

-20

-10

0

Pow

er (d

B)

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

-60

-50

-40

-30

-20

-10

0

Pow

er (d

B)

-80 -60 -40 -20 0 20 40 60 80

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

Azimuth Angle(degrees)

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

E-plane

H-plane

D-plane

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

151

0.85 0.9 0.95 1 1.05 1.1 1.15−40

−35

−30

−25

Normalized Frequency

Cro

ss−P

olar

izat

ion(

dB)

This WorkYassin 2007Granet 2004Neilson 2002Pickett 1984Potter 1963

Figure D.5: The maximum cross-polar response of the prototype feedhorn is comparedto other implementations of smooth-walled feedhorns. The data presented have beennormalized 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


163


164

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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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