CHARACTERIZATION of the POLARIZATION and FREQUENCY SELECTIVE BOLOMETRIC DETECTOR ARCHITECTURE a Dissertation Submitted to the Gr

CHARACTERIZATION OF THE POLARIZATION AND FREQUENCY SELECTIVE BOLOMETRIC DETECTOR ARCHITECTURE

A Dissertation

Submitted to the Graduate School of Case Western Reserve University in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy in Physics

Jonathan Ryan Kyoung Ho Leong, B. S.

Thesis Advisor: John E. Ruhl

Graduate Program in Physics Cleveland, Ohio May 2009 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis/dissertation of

Jonathan Ryan Kyoung Ho Leong candidate for the Doctor of Philosophy degree∗.

John E. Ruhl(signed) (chair of the committee)

Daniel S. Akerib

Lawrence M. Krauss

R. Mohan Sankaran

November 24, 2008(date)

∗ We also certify that written approval has been obtained for any proprietary material contained therein. This document is in the public domain. To my mother

iii Contents

List of Tables ...... vi

List of Figures ...... vii

ACKNOWLEDGMENTS ...... xi

Chapter 1: INTRODUCTION ...... 1 1.1 Big Bang Cosmology ...... 1 1.2 CMB Temperature Anisotropies ...... 7 1.3 The Inﬂationary Paradigm ...... 17 1.4 CMB Polarization Anisotropies ...... 21

Chapter 2: PFSB MOTIVATION ...... 30 2.1 CMB Polarization Challenges ...... 30 2.2 PFSB Introduction ...... 37 2.2.1 Bolometers ...... 37 2.2.2 FSBs ...... 41 2.2.3 PFSBs ...... 43 2.3 PFSB Feed Architecture ...... 49

Chapter 3: CHARACTERIZATION FORMALISM ...... 57

Chapter 4: SIMULATION ...... 63 4.1 Overview ...... 64 4.2 Aperture Diﬀraction ...... 66 4.3 Waveguide Mode Decomposition ...... 68 4.4 Propagation & Termination ...... 71 4.5 Circular Waveguide Results ...... 73

Chapter 5: PFSB TEST CRYOSTAT ...... 77 5.1 Cryogenics ...... 77 5.2 SQUID Readout ...... 81

iv Chapter 6: CHARACTERIZATION TOOLS ...... 86 6.1 Spectra ...... 86 6.1.1 Fourier Transform Spectrometer ...... 87 6.1.2 Measurement Setup & Analysis ...... 93 6.2 Beam Maps ...... 98 6.2.1 Beam Mapper ...... 98 6.2.2 Measurement Setup & Analysis ...... 101

Chapter 7: PFSB OPTICS ...... 108 7.1 Power Loading Capabilities ...... 108 7.2 Optics ...... 114 7.2.1 On-axis System ...... 114 7.2.2 Beam Mapping Systems ...... 122 7.2.3 Square Guide System ...... 129

Chapter 8: RESULTS & DISCUSSION ...... 132 8.1 Spectra ...... 132 8.1.1 Circular Waveguide ...... 133 8.1.2 Square Waveguide ...... 136 8.2 Beam Maps ...... 137 8.2.1 Circular Waveguide ...... 138 8.2.2 Square Waveguide ...... 148

Chapter 9: PFSBS ON AN EXAMPLE TELESCOPE ...... 161

Chapter 10: CONCLUSION ...... 173

Appendix A: CHARACTERIZATION SYSTEMATIC CHECKS ...... 176 A.1 LEGS Linearity Tests ...... 176 A.1.1 Spectra ...... 177 A.1.2 Beam Maps ...... 181 A.2 Beam Mapper Source Characterization ...... 184 A.3 Fitted Parameter Errors ...... 185

Bibliography ...... 189

v List of Tables

2.1 PFSB Characteristics ...... 47 2.2 Simulated Winston Cone Parameters ...... 52

5.1 Cryogenic Storage Characteristics ...... 79 5.2 SQUID Readout Parameters ...... 83

7.1 On-axis System Parameters ...... 120 7.2 Parameters for the Beam Mapping Systems ...... 123 7.3 Small Aperture Stop Baﬄe Parameters ...... 127 7.4 Large Aperture Stop Baﬄe Parameters ...... 130

A.1 Square Guide Spectra R Ranges ...... 182 A.2 Large Aperture Beam Map Linearity Characterizations ...... 183 A.3 Square Guide Beam Map Linearity Characterizations ...... 185

vi List of Figures

1.1 Velocity-distance relation among extra-galactic nebulae ...... 2 1.2 BBN predictions and observations of the light element abundances 4 1.3 CMB decoupling ...... 5 1.4 Velocity-distance relation for SNe Ia ...... 8 1.5 Contents of the universe ...... 9 1.6 CMB surface of last scattering ...... 10 1.7 CMB blackbody spectrum as measured by FIRAS ...... 11 1.8 CMB temperature anisotropies ...... 12 1.9 Global curvature scenarios in the universe ...... 13 1.10 Acoustic oscillations in the photon-baryon ﬂuid prior to CMB decoupling ...... 14 1.11 CMB temperature power spectrum ...... 16 1.12 The inﬂationary universe ...... 20 1.13 CMB polarization generation ...... 22 1.14 Quadrupolar radiation patterns at the surface of last scattering . 23 1.15 Example E- and B-mode function patterns ...... 25 1.16 CMB power spectra ...... 27

2.1 CMB power spectra ...... 31 2.2 Temperature induced polarization via beam asymmetries . . . . . 33 2.3 CMB foreground power spectra ...... 35 2.4 CMB foreground spectra ...... 36 2.5 A schematic for a general bolometer ...... 38 2.6 A TES superconducting transition ...... 41 2.7 FSB surface absorber geometry ...... 42 2.8 Spectral absorption eﬃciency of two FSB stacks ...... 43

vii 2.9 FSB architecture ...... 44 2.10 A schematic of a PFSB ...... 45 2.11 Photographs of one of the PFSBs we tested ...... 46 2.12 ZEMAX simulation of the polarization properties of the Winston cone ...... 51 2.13 Winston cone exit radiance ...... 53 2.14 ZEMAX simulation of the polarization properties of back-to-back Winston cones ...... 54 2.15 Back-to-back Winston cones exit radiance ...... 55

3.1 Instrument action for a single detector on the incoming radiation ﬁeld ...... 57

4.1 PFSB waveguide simulation schematic ...... 64 4.2 Co-polar radiation pattern simulation of the large aperture system 74 4.3 Cross-polar radiation pattern simulation of the large aperture system 75 4.4 Rotation angle radiation pattern simulation of the large aperture system ...... 76

5.1 PFSB test cryostat ...... 78 5.2 SQUID readout for the PFSB test system ...... 82

6.1 The Michelson interferometer ...... 87 6.2 The Martin-Puplett interferometer conﬁgured as an FTS . . . . . 90 6.3 The PFSB test cryostat sitting on the FTS for oﬀ-axis spectra measurements ...... 94 6.4 Interferograms from the two output ports of the FTS ...... 95 6.5 Co- and cross-polar spectra as the PSDs of the interferograms in Figure 6.4 ...... 96 6.6 PFSB co- and cross-polar spectra where we have removed the source response from Figure 6.5 ...... 97 6.7 The 2-dimensional beam mapper ...... 99 6.8 A schematic of the source used on the beam mapper ...... 100 6.9 The PFSB test cryostat sitting on the beam mapper ...... 102 6.10 Reference conditioning for PFSB signal lock-in ...... 104 6.11 Demodulation of PFSB signals ...... 105

viii 6.12 Co-polar beam map example ...... 106

7.1 Power loading comparison between the PFSB design and characterization scenarios ...... 109 7.2 Load curves for two PFSB detectors ...... 112 7.3 On-axis PFSB test system ...... 115 7.4 On-axis aperture stop baffle ...... 116 7.5 On-axis aperture stop baffle simulation ...... 117 7.6 Absorbing backend to the PFSB waveguide ...... 124 7.7 Small aperture stop baffle ...... 125 7.8 Small aperture stop PFSB test system ...... 126 7.9 Large aperture stop PFSB test system ...... 128 7.10 Large aperture stop baffle ...... 129 7.11 Square waveguide PFSB test system inserts ...... 131

8.1 On-axis system spectra for the low- and high-G¯ PFSBs ...... 134 8.2 Small aperture system spectra for the high-G¯ PFSB ...... 135 8.3 Square guide system spectra for the high-G¯ PFSB ...... 137 8.4 Co-polar beam maps C(Ω) of the large aperture system ...... 139 8.5 Cross-polar beam maps X(Ω) of the large aperture system . . . . 140 8.6 Rotation angle beam maps ∆(Ω) of the large aperture system . . 141 8.7 Principal plane cuts through the beam maps of the large aperture system ...... 142 8.8 Diagonal plane cuts through the beam maps of the large aperture system ...... 143 8.9 First 12 modes of the circular PFSB waveguide ...... 145 8.10 Linear combinations of circular waveguide modes ...... 146 8.11 First 12 modes of the square PFSB waveguide ...... 149 8.12 Linear combinations of square waveguide modes ...... 150 8.13 Co-polar beam maps C(Ω) of the square guide system ...... 152 8.14 Cross-polar beam maps X(Ω) of the square guide system . . . . . 153 8.15 Rotation angle beam maps ∆(Ω) of the large aperture system . . 154 8.16 Principal plane cuts through the beam maps of the square guide system ...... 155

ix 8.17 Diagonal plane cuts through the beam maps of the square guide system ...... 156 8.18 Co-polar beam map simulation C(Ω) of a square waveguide excluding aperture stop diffraction effects ...... 157 8.19 Cross-polar beam map simulation X(Ω) of a square waveguide excluding aperture stop diffraction effects ...... 158 8.20 Rotation angle beam map simulation ∆(Ω) of a square waveguide excluding aperture stop diffraction effects ...... 159 8.21 Polarization efficiency beam map ρ(Ω) of the square guide system 160

9.1 Spider-like optics for predicting PFSB systematics on the sky . . . 162 9.2 Aperture illumination functions for a square waveguide PFSB in a Spider-like telescope ...... 164 9.3 Beams on the sky for a square waveguide PFSB in a Spider-like telescope ...... 166 9.4 Polarization response beams on the sky for an orthogonal PFSB diﬀerence pixel in a Spider-like telescope ...... 170 9.5 Beam window functions for PFSBs on example telescopes . . . . . 171

A.1 On-axis system linearity calibration ...... 178 A.2 Square guide system FTS linearity calibration ...... 180 A.3 Second square guide system FTS linearity calibration ...... 181 A.4 Second square guide system beam mapper linearity calibration . . 184 A.5 Beam mapper source box characterization ...... 186

x ACKNOWLEDGMENTS

There are so many of you whom I owe my deepest gratification for the completion of this thesis. My life is filled with wonderful experiences that I have shared with all of you. These are the lessons which I have based my growth upon, cul- minating in the work that lies herein. I apologize in advance for not being able to name all of you in the interest of space. I will make sure to mention you by group and pick out at least one representative person; the rest of you will know who you are. My life begins with my immediate family. My mother Lucretia Leong gave her all in raising her two boys who are indebted to her for that. Our father Albert Leong gave us financial support and provided an example of a good-hearted person who always looks out for the good of others. Thank you older brother Kevin Leong for being my youthful role model and looking after me in numerous ways. To my two half-sisters, Trisha and Shannon Leong, you have brought a special joy to my life from giving me the opportunity to be “older brother.” Grandma Lucille Lee, may you rest in piece. We all love you dearly. You have been my daytime mother and showed me what true aloha is. My extended ohana mostly consists of friends I have made while attending Moanalua High School in Honolulu, Hawaii. You have all provided me with a warm environment of love and aloha to return to on virtually every vacation I have ever taken. You make it very hard for me to leave Hawaii but fill me with the

xi nourishment needed to push forward in my studies. In particular, I would like to thank Jeremy Pobre whom I have befriended the longest since elementary school. He is faithful and loyal to the end. David “Da Bushla” Awaya for being “Mr. Aloha,” and Mariel Moriwake for our beautiful companionship throughout the years. Rick Nakahara my dearest friend, we were partners on so many occasions—I will keep you in my memories forever. At the start of my academic career I have the entire Caltech community to thank. What a special place to provide the foundation for learning to be tomor- row’s scientist. On the one hand my classmates and I, the Physics Communist Party (PCP) of Ricketts House, had a very intimate relationship that can only manifest itself from spending multiple 20 hour days a week torturing ourselves with physics problem sets. Yann De Graeve, thank you for rocking out with me when it was time to blow some steam, and Adam Scott for being my favorite part- ner in tackling those hard problems that would only show themselves through long collaborations. On the other side of my Caltech experience was the time spent working in the lab of my freshman adviser, Andrew Lange. This was my ﬁrst ex- posure to research and it taught me what it feels like to be a real “screw-turner.” For taking me under his wing and creating a very enthusiastic work environment at the forefront of our ﬁeld, Andrew is like a father to me. I am also thankful to his secretary Kathy Deniston for many motherly conversations. It is always nice to come back an catch up with all of you there. Luckily Caltech is such a central location for physics! During my year between undergraduate and graduate school, I have Alan Tokunaga to thank for giving me a job at the University of Hawaii’s Institute for Astronomy and Michael Liu for mentoring me. In addition to the two of you, all

xii of the staff there at the InfraRed Telescope Facility were so nourishing to me in a style that can only be found in Hawaii. This came at just the right time in my life and revitalized my deep love for physics and science in general. The difficult part of this section is to somehow succinctly thank the multitude of individuals who have played a significant role in my graduate education at Case Western Reserve University. After each prior graduation, I had always felt that the period prior was the one which had brought me the most growth in my life. After my bachelor’s degree I was sure that these growth spurts were sure to slow down. Upon nearing the completion of this work however, I realized that this thought was a vast misconception. Many people that I have discussed this with would agree that during one’s undergraduate years their life growth is actually put on hold so that they may learn how to learn. It is during graduate school where we learn how to live. Because of this, it is extremely hard for me to wrap up this portion of my life so that I may move on. I have had many very special friendships (often occurring in triplets) during these graduate student years which have helped to pull me through. First of all I would like to mention Jonathan Penoyar who was an old PCP friend that attended Case Western Reserve’s medical school. Through him I met his classmate Mohammad Hanizavareh. The three of us were partners in the exploration of Cleveland’s awesomeness. Mohammad, his now fiancéeChau Tran, and I formed another triplet. They are the couple in my life that I know most intimately. Also in the medical profession are my “Wacky Wednesdays” pizza friends Thomas Cowan and Roger Lin with whom I have had many philosophical and scientific conversations. My understanding of human nature has benefited considerably from these.

xiii Before mentioning my physics acquaintances, I would like to add three individuals. A special thanks goes out to Xiao She for the time we spent together. If I had to use one word to describe those times it would be colorful. Richard Pazol is someone who I have great respect for. I owe so much to him for being a mentor and confidant. Last of my non-physics friends, but certainly not least is Nathan Sears. Thank you for teaching me the ways of bicycling as well as many other things. Moving closer to people who were more directly applicable to my graduate work, I want to extend my thanks out to the entire physics department at Case Western Reserve. You provided a wonderful place for graduate studies and I could not have chosen better. First let me thank the entire administrative staff which includes Edith “Betty” Gaffney and Lori Morton. You have helped me with all of the logistics of my graduate education and also befriended me. It was nice to have so many people to chit-chat with during the day. Next, I would like to thank all of the faculty. Your aloha is really what made this a special time for me and resulted in my getting to know so many of you. In particular, “The People’s Chair” Daniel Akerib has been a great help in supporting the opinions and views of the graduate student body, making sure our word got heard. I would like to thank Robert Brown for advising me on a project in physics education. You are a truly innovative teacher. Finally, there is Lawrence Krauss. Your busy schedule prevented us from spending a large amount of time together, but the manner in which you interacted with me made it seem like so much. You have been an inspiration and a friend. Thank you. On the student side of my associations in graduate school there are even far too many groups to do justice to so I hope that you will all know who you are. I

xiv feel the need to mention my classmates. We all spent a very intimate first year together where we worked, ate, lived, and hung out together. Most notable is another triplet with Zachary Staniszewski and Georgi Dakovsky. Our intimacy extended far past the first year due to our common living arrangement for the next two years. Minhua Zhu has also been a real stand-up friend and classmate throughout all of graduate school. Finally coming upon the Ruhl Lab, I had a great time being a part of this small effort to study the Cosmic Microwave Background (CMB). You are all such an awesome group of people that it made working in the Ruhl Lab a unique and fun experience. This starts with Zachary, another classmate of ours Wenyang Lu, and I as a triplet of John Ruhl’s first students at Case Western Reserve. To be going through the early stages of professional life with two other classmates was a neat experience and it was nice that the choosing of thesis projects went so smoothly. Thanks to the three post-doctoral scholars that have occupied our lab during my time, Thomas Montroy, Kecheng Xiao, and Jonathan Goldstein. I have learned a lot from all of you. Tom in particular is responsible for teaching me the ways of a screw-turner, without which I would not have been able to accomplish anything. Jon G., I am grateful for the friendship that you also extended towards me and the various nick names. The lab got a little duller when you left. A very special thank you goes out to Richard Bihary. He is the machinist/me- chanical designer that we hired. I do not know what I would do without him. So much of my thesis work relied on his construction insight as well as his fast and still very functional machine work. Thank you, Rick, for also being a close friend. The two younger graduate students who have entered the laboratory have also played a large role in my life. Sean Bryan, thank you so very much for helping to

xv open up a new pathway in my life. James Sayre, thank you for helping to turn power supplies on and off during my last cryogenic run. Of course, in addition to this menial task that I imposed on you, I really enjoyed the time we spent together when you were shadowing me. You were a fun pupil. Finally, I would like to extend my gratitude towards all of the undergrads that have worked in the Ruhl Lab. The operation could not have run so smoothly without you. Specific to my thesis work, I have the entire Frequency Selective Bolometer (FSB) collaboration (see author list on Perera et al. [67]) to thank, whose work provides the foundation for my project. In particular, Grant Wilson of the Uni- versity of Massachusetts Amherst and Stephan Meyer of the University of Chicago for being very supportive and offering helpful advice. I am also indebted to the entire team at Goddard Space Flight Center who fabricated working detectors for me to test. Thank you to Mohan Sankaran for being my thesis committee member outside of the physics department. Our interaction was brief but made a very positive impact on my experience with the defense of this thesis. I would also like to mention one more member of the PCP, Roger O’Brient, who went to the University of California to carry out an extremely similar thesis project to mine [62]. My gratitude towards you for the numerous conversations we have had about our field of work and our specific projects. They have often helped me to understand my project and its motivations better. Last, but certainly not least, thank you to “Big” John Ruhl my advisor. Your direction was just the right balance which allowed me to grow independently yet still provided the guidance needed to keep me on the correct path. Most importantly, you provided me with a wonderful example of a working physicist to follow. I have learned much from watching and trying to imitate you. I am proud

xvi to be your student. Thank you all again for your contributions to this work. Without any one of you my life would not be so full of aloha, excitement, color, etc. All of that good stuﬀ that makes life worth living. I am happy to put this chapter of a long road of education behind me and move on towards my life as a professional scientist!

xvii Characterization of the Polarization and Frequency Selective Bolometric Detector Architecture

Abstract by Jonathan Ryan Kyoung Ho Leong

The Cosmic Microwave Background (CMB) has been a wonderful probe of fundamental physics and cosmology. In the future, we look towards using the polarization information encoded in the CMB for investigating the gravity waves generated by inﬂation. This is a daunting task as it requires orders of magnitude increases in sensitivity as well as close attention to systematic rejection and astrophysical foreground removal. We have characterized a novel detector architecture which is aimed at making these leaps towards gravity wave detection in the CMB. These detectors are called the Polarization and Frequency Selective Bolometers (PFSBs). They attempt to use all the available photon information incident on a single pixel by selecting out the two orthogonal polarizations and multiple frequency bands into separately stacked detectors in a smooth-walled waveguide. This approach is inherently multimoded and thus solves problems with downlink and readout throughput by catching more photons per detector at the higher frequencies where the number of detectors required is prohibitively large. We have found that the PFSB architecture requires the use of a square cross-section waveguide. A simulation we developed has illuminated the fact that the curved ﬁeld lines of the higher order modes can be eliminated by degeneracies which exist only

xviii for a square guide and not a circular one. In the square guide conﬁguration, the PFSBs show good band selection and polarization eﬃciency to a level of about 90% over the beam out to at least 20◦ from on-axis.

xix Chapter 1

INTRODUCTION

The purpose of this thesis is to further our understanding of the universe through the characterization of a novel detector architecture. We begin the discussion with the scientific background needed to place this new technology in a contextual setting. Section 1.1 covers big bang cosmology as the foundation for the vast majority of work done in cosmology over the past half century. Section 1.2 discusses one of our most valuable probes of the early universe, the Cosmic Mi- crowave Background (CMB) radiation, and its present impact. Adding to the big bang scenario, we then introduce inflation as a theoretical paradigm which solves many of its previous dilemmas. This introduction ends in Section 1.4 with a discussion of how future studies of the CMB may provide a stronger observational footing for the inflationary paradigm.

1.1 Big Bang Cosmology

We now live in a golden age of cosmology because the past century has produced notable advances in the understanding of the universe. The centerpiece of this knowledge is the big bang origin where the universe’s current epoch expanded from a very hot, dense ﬁreball. This model rests on three observational pillars. The ﬁrst is Edwin Hubble’s discovery [35] that the further away an astronomical

1 object is, the faster it recedes from us. See Figure 1.1 for the plot showing this that appeared in Hubble’s original paper. Combined with Einstein’s theory of general relativity [20], this leads us to postulate a universe in which spacetime itself is expanding. Extrapolating back in time, space shrinks, and assuming there is no mechanism for matter decimation as in steady state models, density increases and temperature rises. This single pillar alone strongly points towards the big bang model. However, the strength of the model relies on the other two.

Figure 1.1. Radial velocities in km/s, corrected for solar motion, are plotted against distances estimated from stars and mean luminosities of nebulae in a cluster. The black discs and full line represent the solution for solar motion using the nebulae individually, the circles and broken line represent the solution combining the nebulae into groups; the cross represents the mean velocity corresponding to the mean distance of 22 nebulae whose distance could not be estimated individually. The slopes of the lines shown here are ∼ 500 km/s·Mpc. We call this number the Hubble constant H0 where today our measurements result in km H0 ∼ 70 /s·Mpc. The discrepancy here is due to systematic errors in Hubble’s distance calibrations. Taken from Hubble [35].

2 The theoretical underpinnings of these pillars are based on applying our understanding of nuclear and atomic processes to the hot early universe. This leads us towards two implications: predictions for the light element abundances and the necessity of a relic radiation background. Both of these are examples of a concept called freeze-out, or alternatively, decoupling. In the early universe where energies and densities are high, particles routinely interact with each other. The particles and radiation involved exchange energy through both scattering and species conversion, keeping many forms of energy in thermal equilibrium. Due to dropping densities from the expansion and cooling of the universe as well as various processes in the interaction chain, some of these interaction rates decline significantly as the universe evolves. At some point the universe has expanded enough that it becomes extremely difficult for a given species to find interaction partners. For- mally, this occurs when the reaction rate for that species falls below the Hubble expansion rate. The comoving number density (one whose volume element grows with the expansion) for the species then becomes constant, or frozen. Thus the term “freeze-out”. We also refer to this as a “decoupling” because, as the interaction stops, the species falls out of thermal equilibrium with the rest of the cosmic soup. An evaluation of this decoupling for the nuclear processes that are involved in forming the light atomic nuclei leads to predictions of their freeze-out abundances. This calculation, originally studied by Alpher et al. [3], gives appreciable levels of D, 3He, 4He, and 7Li formed in the first three minutes after the big bang [21]. We normally refer to this phenomenon as Big Bang Nucleosynthesis (BBN). See Figure 1.2 for the BBN predictions and their dependency on the current density of baryonic matter (protons and neutrons). This density is given as a fraction of

3 Figure 1.2. BBN predictions and observations of the light element abundances as a function of the baryon density of the universe. As per the legend, we give the abundance of 4He in magenta at the top, the rapidly falling curve just below in red is the D abundance; the green curve in the middle of the plot is for 3He and the 7Li abundance is given in blue on the bottom. The observational constraints are given as the horizontal strips of similar colors to the prediction curves. A single value of the baryon density comes very close to fitting all of the data. Taken from Wright [91]. the critical energy density (the amount of energy needed for a flat universe). We normally refer to these fractions by Ωx where x is the type of energy. When the subscript is omitted we mean the total density of the universe. The figure also shows the observational evidence which makes BBN such a strong indicator of big bang cosmology. Note that a single value of the baryon density comes close to fitting all of the data. 7Li is the only discrepant species showing predictions which are a factor of two above the observations. However, these observations are made

4 Figure 1.3. CMB decoupling. In the early, hot universe shown on the left, photons are energetic enough to break up neutral hydrogen and helium. We show the nuclei as the large red (protons) and blue (neutrons) spheres and the electrons as the green small spheres. The photons constantly scatter off the electrons as depicted by the yellow glow. This creates a tight thermal equilibrium between matter and radiation. As we move towards the right in the figure, the universe cools and the photons no longer carry the ability to ionize. All the free electrons get taken up to form hydrogen and helium and the photons decouple from the matter, free-streaming (shown by the streaming yellow lines on the right) for lack of scattering partners. They subsequently get redshifted due to the expansion of the universe. Taken from The Universe Adventure Website [81]. by studying the compositions of the oldest stars, which actually destroy lithium. Another decoupling process that gives rise to the third and final pillar of standard big bang cosmology happens at a much later epoch then BBN. Figure 1.3 is a pictorial representation of these process. We consider the matter-radiation equilibrium created by Compton scattering of photons off of free electrons

e− + γ ↔ e− + γ . (1.1)

5 The constant scattering is depicted in the figure by the yellow colored (photons) background on the far left side. This equilibrium ceases due to a freeze-out process called recombination, p + e− ↔ H + γ , where the free electrons disappear and form neutral hydrogen and helium 300,000 years after the big bang as shown on the right of the figure. The Compton scattering (1.1) rates then drop below the Hubble expansion rate, decoupling the photons. They then free stream throughout the universe, shown by the streaming yellow over the black background. We call these photons, now redshifted by the expansion into the microwave region of the electromagnetic spectrum, the Cosmic Microwave Background (CMB) radiation. The CMB’s characteristic blackbody temperature was predicted by a series of authors in the late 1940s including George Gamow, Ralph Alpher, and Robert Herman as outlined by Gamow [24] and Kragh [45]. In 1965, Arno Penzias and Robert Wilson (two microwave engineers working at Bell Laboratories) report on the serendipitous discovery of the CMB [66]. This was much to the dismay of Robert Dicke, a prominent experimental physicist who made important contributions to radio communication techniques. Dicke, at the time, was leading a group intending to make such a discovery. His reaction to a phone call by Penzias and Wilson was, “Boys, we’ve been scooped.” It is this measurement of the CMB as an isotropic 3 K blackbody that later resulted in Penzias and Wilson being awarded a Nobel prize in physics. Combined with the predictions, it makes up the third pillar of the big bang scenario. The three, Hubble expansion, BBN, and the CMB, are all different observational probes of the same underlying model of a universe which expands, as described by Einstein’s general relativity, from a hot,

6 dense fireball. It is their concordance that provides the strength of the big bang scenario. Today, our knowledge of the universe has grown significantly from recent observations. Studies of Type Ia supernova (SNe Ia) which are standard candles have allowed us to extend Hubble’s diagram (Figure 1.1) by many orders of magnitude. See Figure 1.4 for the updated plot. The distances shown here probe a much larger fraction of the history of the universe which now shows an evolution to the expansion rate discovered by Hubble. As seen in the plot, we use this evolution to fit cosmological models with various forms of energy densities. Combining this data with measurements of anisotropies in the CMB, described in the next section, has enabled us to determine that we live in a geometrically flat universe

(Ω0 = 1 where the 0 subscript stands for the value today) with contents as shown in Figure 1.5. These results, like the standard big bang scenario discussed in the beginning of this section, are supported by a concordance of a large body of many types of observations. We will thus refer to this model (standard big bang with

Ω0 = 1 and mass budget given in Figure 1.5) as the concordance model. It is from a class of models called the ΛCDM cosmologies where Λ stands for dark energy and CDM stands for cold dark matter.

1.2 CMB Temperature Anisotropies

The power of using the CMB as an observational tool for cosmology does not end with Penzias and Wilson’s discovery. The past 40 years have seen remarkable leaps in our understanding of the universe through studying the CMB. This thesis is directed towards taking the next step in probing the vast amount of information encoded in this radiation source.

7 Figure 1.4. Velocity-distance relation for binned SNe Ia data taken from Kowalski et al. [44]. The curves show a closed Universe (Ω = 2) in red, the critical density Universe (Ω = 1) in black, the empty Universe (Ω = 0) in green, the steady state model in blue, and the WMAP based concordance model with Ωm = 0.27 and ΩΛ = 0.73 in purple. This km model gives H0 = 71 /s·Mpc which has been used to scale the luminosity distances in the plot. The data show an accelerating Universe at low to moderate redshifts but a decelerating Universe at higher redshifts, consistent with a model having both a cosmological constant and a significant amount of dark matter. The dashed black curve shows an Einstein-de Sitter model with a constant co-moving dust density which can be ruled out. The dashed purple curve shows a closed ΛCDM model which is a good fit to the data. The dashed blue curve shows an evolving supernova model which is also a good fit. Taken from Wright [91].

Let us now discuss the CMB observable that has received the most attention in the recent past. These are minute anisotropies at a level of one part in 105

8 Figure 1.5. Contents of the universe shown as a pie chart. The part of the universe that has been observed directly only takes up the small sliver in light blue denoted as atoms. This is the material we and stars are made out of. The rest of the pie consists of elements we only postulate due to their gravitational eﬀects. The dark matter component is made up of a type of matter which feels the force of gravity and not much else. Because it does not feel electromagnetic forces it neither emits nor scatters light and is unseen. We also know that this matter is relatively slow moving and thus call it cold dark matter. Taking up the largest piece is a component we do not have any understanding for. This so called dark energy actually acts to create a repulsive gravitational force. It is required for the acceleration of the universe’s expansion described in the caption of Figure 1.4. Taken from the WMAP website [90]. of the average CMB temperature. In the photon decoupling described in the previous section (see Figure 1.3 for a summary), the CMB radiation emanates from everywhere, in all directions, for a brief period characterized by the timescale over which the decoupling takes place. For a later observer, the CMB photons seen will be sourced by a thin spherical shell of radius R ∼ ct where c is the speed of light and t is the time between the observation and decoupling. The sphere is centered on this observer as shown in Figure 1.6. We refer to this as the surface of last scattering.

9 Figure 1.6. CMB surface of last scattering. This is deﬁned by the thin shell in space which sources the CMB photons (shown as red squiggly lines with arrows) that we observe today. Since we live in a time which is ∼ 13 billion years from the big bang and decoupling, this surface is a sphere of radius 13 billion light years. We depict the decoupling event as the transition between the opaque plasma of the early universe (red) and the transparent neutral gas of the later universe (white). The outer edge of the red area marks the edge of the observable universe. From Ruhl and Meyer [72].

Tracing a CMB photon back in time to the surface of last scattering, it gets completely absorbed and thermalized by the photon-baryon plasma existing before decoupling. Thus, the CMB should be a perfect blackbody. This quality was measured with exquisite precision by FIRAS on the COBE satellite, [23] and [58], giving the best representation of a blackbody known to man (Figure 1.7). A neat consequence of the opacity of the CMB is that it is the oldest light in the universe that one can see. More importantly is the fact that the CMB, having free streamed towards an observer, retains information about the temperature of

10 Figure 1.7. FIRAS results for a measurement of the spectrum of the CMB. These results ﬁt the spectrum of a 2.725 K blackbody (shown on the top) to remarkable precision as shown by the residuals in the bottom plot. The colored lines in the bottom plot show various likely non-blackbody spectra that are allowed by the data. From Wright [91].

11 Figure 1.8. The detailed, all-sky picture of the infant universe from three years of WMAP [5] data. The image reveals 13.7 billion year old temperature fluctuations (shown as color differences) that correspond to the seeds that grew to become the galaxies. The signal from the our Galaxy was subtracted using the multi-frequency data. This image shows a temperature range of ±200 µK. From the WMAP website [89]. the region from which it last scattered. The temperature of the photon-baryon fluid is tightly linked to its density through two competing effects caused by the gravitational potential wells of over- dense regions. Mass infall into the potential wells causes an increase in pressure, particle velocities, and thus temperature. The second effect called the Sachs-Wolfe effect is that upon last scattering, a photon from the bottom of the well must climb out giving up some of its energy and thus decreasing its temperature. Through these two mechanisms the CMB acts as a probe of density perturbations over the surface of last scattering. This oldest image of the early universe has been of paramount importance to cosmology as the density perturbations contained within are to later seed the growth of structure in the universe. We show in Figure 1.8 the temperature anisotropies recorded by the WMAP satellite [5]. The most important contribution that these images have made to cosmology

12 Figure 1.9. Global curvature scenarios in the universe. On the top we show as closed universe which results from densities greater than the critical value. This scenario by the concave curvature as shown where the angles within a triangle sum to values greater than 180◦. In the middle, a below critically filled universe with convex curvature. The angles in a triangle here sum to less than 180◦. Finally, on the bottom we show a flat universe at the critical density where we have the familiar Euclidean geometry. From Wikipedia [87]. is in providing a way to directly probe the global curvature of the universe. In Einstein’s general theory of relativity, the curvature of spacetime is altered by the presence of any type of energy density. When applying the equations of general relativity to the global structure of the universe, we find that there are three basic forms of curvature as shown in Figure 1.9. We probe this curvature, precisely in the manner suggested in the figure, by measuring the angles interior to a large triangle in space. The CMB creates such a triangle for us, but to understand how this happens

13 Figure 1.10. Acoustic oscillations in the photon-baryon fluid prior to CMB decoupling. A gravitational potential well (shown here as the bowl shaped black line) created by a slight over-density in the fluid pulls mass (orange spheres) into it through gravitational infall. The springs shown here represent the photon pressure created by the tight coupling between matter and radiation provided by Compton scattering (1.1). As infall densities grow, so does the photon pressure which then serves to push out the infallen matter. The blue (hot) and red (cool) arrows drawn show what happens to the temperature of the fluid as matter falls in and gets pushed out of the well respectively. These temperature variations follow from the changing density during compression and subsequent rarefaction of the fluid. From the Wayne Hu Website [84]. we must first discuss more about the physics of the photon-baryon fluid before the time of decoupling. See Wayne Hu’s website [84] and listed papers therein for beautiful reviews of this as well as much of the physics behind the observable effects in the CMB. Gravitational infall pulls matter into potential wells set up by some initial spectrum of density perturbations. For this initial spectrum, we normally assume scale-invariance as is consistent with all our data. Considering the coupling between matter and radiation caused by Compton scattering (1.1), we see that there is a photon pressure that acts as a restoring force to the infall. The resultant tug-of-war on the matter (depicted in Figure 1.10) causes acoustic

14 oscillations in the plasma of the early universe. Given the finite sound speed in the fluid, density perturbations on different scales oscillate at different frequencies with the larger scales oscillating slower. At decoupling, the photons free stream out of the system carrying away the temperature information from this particular time slice of the universe’s history. Thus the CMB catches different modes in different phases of their oscillation. One special length scale in this scenario is the largest wavelength mode which has reached a peak of its compression-rarefaction cycle. Being the slowest, it had just enough time to carry out a half of one such cycle. Due to a damping term caused by photons random walking and thermalizing between the peaks and troughs of the smaller scale modes, this special mode has the largest amplitude of temperature, and thus density perturbations of all the modes undergoing acoustic oscillations. Armed with the time of decoupling and the sound speed which we can find via other techniques, we can calculate the physical size of the wavelength of this special mode. Adding to this a knowledge of the distance between us and the last scattering surface, also obtainable via other techniques, we see that this special scale gives us the desired triangle with all dimensions known. This triangle has one vertex at the observer and the other two at the ends of our “CMB ruler” on the surface of last scattering. Measuring the subtended angle of this ruler on the sky then gives us a measure of the global curvature of the universe. We can see the ruler in Figure 1.8 as the degree scale fluctuations over the entire map. In reality, we do not measure the angles directly on the map of the CMB anisotropies. Instead we carry out a reduction of this dataset by assuming an isotropic statistical nature to the CMB with near Gaussian variations. We expand the maps in spherical harmonics and find the angular scale where the power is

15 Figure 1.11. CMB temperature power spectrum corresponding to the map, shown in Figure 1.8, of the WMAP 3-year data (black). Data from other experiments are shown in color when their statistical weight dominates that of the WMAP data. The solid gray line is the best ﬁt cosmological model for all the data. Shown as a red strip is the cosmic variance errors due to the fact that we only have a single realization of the CMB. Notice that the combination of WMAP and ACBAR alone, suﬃce to describe the CMB power spectra out to l = 1500. From the Legacy Archive for Microwave Background Data Analysis (LAMBDA) Website [50]. highest. The spherical harmonic decomposition is

δT X (Ω) = a Y (Ω) (1.2) T lm lm where T is the temperature of the CMB, δT give the variations we measure, and

(Ω) ≡ (θ, φ) is the directional variable; Ylm are the normal spherical harmonics

16 and the alm are the complex expansion coeﬃcients. The above assumptions imply that most of the information is contained within the power spectrum deﬁned by

2 Cl ≡ h|alm| i (1.3) where the angle brackets denote an ensemble average over various realizations of the CMB. The power spectrum is not dependent on the spherical harmonic index m due to the isotropy. Figure 1.11 shows a plot of the spectrum corresponding to the map in Figure 1.8. As implied in the figure, multipole index l can serve as an inverse angle where l = 200 corresponds to nearly a degree. We see that this power spectrum exhibits peaks at different angular scales due to the CMB catching the acoustic oscillations at different phases, as mentioned above. Since we measure power here, the first peak is the compression corresponding to the standard CMB ruler, the second is the mode that had a chance to first compress then rarefy, the third is one which compressed, rarefied, and then compressed again, and so on. We obtain the desired curvature and other important cosmological parameters by creating a model that reproduces curves like the one shown in Figure 1.11, then fitting this family of theoretical curves to the data as shown. The latest results given in the analysis of 5-year WMAP data [43] is that the universe today is flat to within a couple percent. The other cosmological implications of the CMB temperature anisotropy dataset are numerous, however, they are beyond the scope of this short introduction.

1.3 The Inﬂationary Paradigm

In this section we will ﬁrst focus on a few of the diﬃculties encountered with the concordance model outlined in Section 1.1, then present the currently accepted

17 resolution. Three of the conflicts are known as the flatness, horizon, and monopole problems. We have just described the observed flatness of universe in the previous section. The number quoted there was that the universe today is flat to within a couple percent. This observation does not necessarily seem problematic until you consider the dynamics of the universe. Solving the equations of Einstein’s general relativity, it can be seen that as the universe slows in its expansion, any deviation from flatness grows. Using a cosmological model similar to the concordance one of Section 1.1, extrapolating back from today shows that flatness to a percent today requires flatness to within 10−15 back during the time of BBN [74]. This begs the question of why the universe exhibited such a high degree of flatness in its early times. Since no law of physics prefers one level of curvature to another, it seems to be a very remarkable coincidence that the universe has chosen such initial proximity to flatness. This coincidence became known as the flatness problem. It requires either a cosmological model that prefers flatness or a mechanism that produces flatness at early times. The next conflict called the horizon problem has to do with the high degree of isotropy (deviating by only a part in 105) seen in the CMB implying a near homogeneity of the universe at the time of decoupling. We find this to be discordant with the idea of horizons in the universe. A horizon from any given point forms the boundary between points that are and are not in causal contact with the point in question. It turns out that antipodal points on the CMB last scattering surface are far out of causal contact. How is it then that they seem to “know” about each others temperature to one part in 105? This is another remarkable coincidence of the concordance model that begs for some mechanism that can explain it. The final difficulty inherent in the standard big bang model is the lack of mag-

18 netic monopoles observed in the universe. This problem is based on unverified particle physics beyond the Standard Model and is thus not as compelling as the flatness and horizon problems. We mention it here because the solution presented for all of these was developed first as a solution to the monopole problem. When including Grand Unified Theories (GUTs—attempts to create a model which uni- fies all the forces except gravity) into the hot big bang scenario, we encounter a phase transition that occurs in the early universe at the point where the symmetry between the strong and electroweak forces is broken. Phase transitions are often accompanied by topological defects much like the domain walls in a ferromag- netic material when it is below its critical temperature. Depending on the type of phase transition one can obtain topological defects of varying dimension. Many GUTs predict phase transitions with point-like defects that behave as magnetic monopoles. It turns out that the number of these defects produced by GUT phase transitions is large enough that they would dominate the energy density of the universe for times later than 10−16 s [74]. This is obviously not the case and in fact not a single magnetic monopole has ever been discovered. This discrepancy is called the monopole problem. In 1981, Alan Guth introduced a way to solve all three problems [29] by pos- tulating a period of exponential expansion of the universe called inflation, illus- trated in Figure 1.12. Inflation solves the monopole problem by expanding the universe by many orders of magnitude after the GUT phase transition when all the magnetic monopoles were produced. This thins the density of monopoles in the universe to below the observational constraints. Inflation solves the flatness and horizon problems in much the same manner. Recalling that the geometry diverges from flatness in a decelerating universe, conversely, an accelerating phase

19 Figure 1.12. The inflationary universe. This plot, of the radius of the region that evolves to become the presently observable universe, shows the standard big bang model in red, and with an inflationary epoch of exponential expansion added in blue. The inflation occurs over a very small timescale following the GUT phase transition at around 10−36 s. It solves the flatness, horizon, and monopole problems by expanding the size of the universe over 50 orders of magnitude during this short period. From Wikimedia Commons [86]. like inflation drives the universe towards a flat geometry. Carrying this out over the huge range spanned by inflation shown in the figure, we can easily obtain the flatness needed upon entering the standard big bang scenario. Finally, we solve the horizon problem during the period before inflation. Note from the figure that the observable universe today evolved from a very small region of radius less than 10−50 m before inflation. Temperature homogeneity over this region was established during this epoch. Subsequently, after the exponential expansion this region grew to a size larger than the last scattering surface today, explaining the

20 observed isotropy of the CMB. In addition to solving the three problems mentioned above, inflation also provides an explanation for the creation of density perturbations in the early universe. Through the large expansion, quantum fluctuations in the energy density of the fields existing before inflation get blown up to super horizon scales. Upon reenter- ing the growing sound horizon later, these perturbations begin acoustic oscillations as measured by the CMB. Inflationary models also predict the spectrum of these perturbations to be scale-invariant conforming to the data. Inflation has been accepted as a necessary part of our current cosmological paradigm. It resolves these three otherwise difficult and troubling initial condition problems. While inflation is one of the key parts of our concordance model, we do not have direct evidence for its existence. Much like the hot big bang scenario postulated after Hubble’s discovery of the expanding universe, we look for other observational signatures to deliver strength to the inflationary paradigm.

1.4 CMB Polarization Anisotropies

Future measurements of the polarization of the CMB are the most promising probes of inflation and the subject of this thesis is to help develop the necessary technology required to make these incredibly difficult measurements. In addition to the density perturbations generated from the inflation of quantum fluctuations, we also get a scale-invariant spectrum of gravitational radiation. Similar to the way the CMB significantly strengthens the big bang scenario, an observation of this stochastic background of gravitational waves would be the smoking gun of inflation. Oddly enough, the CMB can also be used as a tool to probe this gravitational wave background. We find the gravity wave information encoded in

21 the polarization of the CMB, but before discussing how this comes about we must ﬁrst describe the mechanism for polarization generation.

Figure 1.13. CMB polarization generation by Thomson scattering of a quadrupolar radiation pattern. The blue and red crosses on the incoming fields represent the two components of the electric field coming from hot and cold spots as seen by the electron respectively. Note that the component which acts to oscillate the electron along the line of sight to the observer does not contribute to the radiation observed. Instead, one component of the observed radiation comes from the hot spot and the other from the cold spot leading to an amount of linear polarization to the field. Taken from Hu and White [33].

Figure 1.13 depicts an electron at the surface of last scattering in a situation where it is surrounded by hot and cold radiation spots arranged in a quadrupolar pattern. Each ﬁeld component of the radiation scattered towards the observer is sourced by orthogonal directions in the electron’s reference frame. For the

22 quadrupolar anisotropy shown here, these directions have opposing temperatures. This leads to an imbalance between the components of the observed ﬁeld, giving rise to a partial polarization.

(a) (b)

Figure 1.14. Quadrupolar radiation patterns at the surface of last scattering. We show here two possible sources. In (a), gravitational infall or radiation pressure outflow of plane wave density perturbations cause electrons sitting on a given plane to see differences in the radiation temperature coming from the neighboring planes verses directions within the plane. In (b), gravity waves stretch spacetime in the manner shown by the ovals drawn in the right side of the figure. This stretching causes differential redshifts, and thus temperature differences, in the radiation pattern viewed within a single plane of a gravity wave. From Hu and White [33].

Figure 1.14 shows two of the ways that these quadrupoles can be generated. In Figure 1.14(a) we show movement in the photon-baryon ﬂuid caused by gravitational infall or radiation pressure outﬂow from a plane wave density perturbation.

23 For an electron that is sitting in the middle of a compression of the fluid, it will see hotter radiation along the directions of the compression and cooler radiation along orthogonal directions. Figure 1.14(b) shows an entirely different mechanism. This time a plane gravitational wave is shown stretching spacetime in the manner depicted by the ovals drawn on the planes. The stretching induces a differential redshift between orthogonal directions in a single plane, sourcing the temperature variations. Thus we see that both density perturbations and gravitational waves around at the time of decoupling give rise to polarization of the CMB. For describing the complete set of CMB observables, including its temperature and polarization anisotropies, we turn to the Stokes parameters which describe any incoherent stream of photons:

2 2 I = h|Ex| + |Ey| i ,

2 2 Q = h|Ex| − |Ey| i , (1.4) ∗ ∗ U = hExEy + EyExi ,

∗ ∗ V = ihExEy − EyExi ,

where the angle brackets denote a time average and Ex and Ey are the complex phasor amplitudes of orthogonal components of the electric field. Stokes I gives a measure of the total intensity in the stream. Polarization is expressed through the other parameters where Q and U represent the linearly polarized part and V represents circular polarization. See Tinbergen [82] for an excellent description of polarized light. Remember that for a given band, we can relate CMB temperature to intensity by using the blackbody spectrum. Also, since we do not expect any circular polarization in the CMB, we find that mapping out the directional variations in (I, Q, U) of Equation (1.4) fully specifies the information encoded in

24 Figure 1.15. Example E- and B-mode function patterns around a local polarization intensity extremum. Black bars designate the polarization direction. The upper row depicts the even parity E-mode (negative on the left, positive on the right), and the lower row the odd parity B-mode (negative on the left, positive on the right). the CMB. We already mentioned how we reduce the temperature maps of the CMB to a power spectrum (1.3). For the polarization observables, polarization speciﬁc spherical harmonics must be utilized in analogy to Equation (1.2). We normally split these functions up into two categories following their parity properties and name them, in analogy to electromagnetism, E- and B-mode functions. Fig- ure 1.15 shows an example of the polarization patterns of these functions around a polarization intensity extremum. Notice that like electromagnetism, the E-mode functions are curl free. Adding the T -mode functions (the normal spherical harmonics of the temperature anisotropies) to these, we note that the three sets of functions (T -, E-, and B-mode functions), which serve as a basis for describing linear polarization maps on the sphere, arise from the three Stokes parameters (I, Q, U) used to describe a stream of linear polarized radiation.

25 Again using the assumption of near Gaussian statistical isotropy, we reduce the (I, Q, U) maps to a set of correlation functions

X Y ∗ hXY il ≡ h|almalm |i where X (and Y ) stands for any one of the three mode function types (T,E,B)

X Y and the alm (alm) are the corresponding expansion coeﬃcients for the three maps

δI δQ δU ( I , I , I ). For our purposes, we will mostly restrict our discussion to the polarization power spectra and drop the explicit l dependence. These are denoted by hEEi and hBBi which we will also refer to as E- and B-mode power spectra. The temperature power spectra hTT i was already discussed in Section 1.2 as the Cl’s of Equation (1.3). One interesting consequence of splitting the mode functions in this manner is that the only last scattering surface process capable of inducing a hBBi spectrum are the gravity waves described above. Thus the problem of probing gravitational waves at the surface of last scattering is reduced to measuring the CMB B-mode power spectrum. A sample plot of the three CMB power spectra is shown in Figure 1.16 for a cosmology which fits all of the data taken prior to 2003. Here T has been replaced with Θ and the explicit ensemble averaging is dropped. Because the CMB is sensitive to both density (scalar) and gravity (tensor) perturbations we normally characterize the relative amplitudes of these two types with a tensor- to-scalar ratio r ≡ T/S where T here is the tensor amplitude and S is the scalar amplitude. Shown in the plot are the temperature and E-mode spectra which both contain information about density and gravitational perturbations. However, the two effects on each are difficult to unravel. Near the bottom of the plot, bounding the gray band, we show two plausible B-mode spectra given the limits

26 Figure 1.16. Scalar CMB power spectra in temperature (ΘΘ) and E-mode polarization (EE) compared with B-mode polarization due to gravitational lensing and gravitational waves at the maximum allowable 2.6×1016 GeV (r = 0.3) [83] and minimum detectable 3.2×1015 GeV (r = 6 × 10−5) level [42]. The cold dark matter model with a cosmological constant (ΛCDM) shown has parameters given in Hu et al. [34], from which this figure is taken. on the tensor-to-scalar ratio r obtained from temperature measurements. Notice the vast difference in amplitude of these signals. We will comment more on the instrumental challenges posed by this in Chapter 2, but for now simply recall that the temperature anisotropies are only one part in 105 of the average background level. Also depicted in Figure 1.16 are two post-decoupling effects on the CMB photons. The first of these poses an ultimate limit to what we have access to in the gravitational wave B-mode spectra. As the CMB photons travel towards us, it is gravitationally lensed by the intervening Large Scale Structure (LSS) of the

27 universe. This does not really affect the temperature studies of the CMB because, although lensing distorts our maps, it does not have much effect on the temperature spectrum. However, referring back to Figure 1.15, we see that a lens can easily turn E-mode power into B-mode power by rotating the polarization. This lensing effect is shown in Figure 1.16 as the B-mode spectra due to gravitational lensing of the E-modes. A well measured E-mode spectrum and hTEi cross-correlation allows us to calculate these lensed B-modes which can then be cleaned out of the total spectrum. However, there is an ultimate limit to this process as described in Knox and Song [42]. We show the minimum detectible gravity wave B-mode signal as the bottom curve bounding the gray strip. The second late time effect on the CMB is due to the reionization of the universe which began during its star forming period. In the post-decoupling era, the lack of radiation pressure in the cosmic soup causes the growth of density perturbations through unchecked gravitational infall. As densities grow, the material inside potential wells heats up again and the universe eventually undergoes reionization. This leads to a portion of the CMB photons re-scattering, now containing information about the reionization epoch by the same mechanisms described above for the surface of last scattering. Luckily, due to the low density of the ionized material, its opacity is not high enough to obscure our view of the last scattering surface. In addition to this, because the reionization happens at a much later time in the history of the universe than decoupling (when the size of the universe has grown by a factor of 100), the angular scales that show up in the reionized CMB power are much larger. Reionization effects can be seen in Figure 1.16 as the low-l bumps in the hEEi and hBBi spectra. We are actually aided in our search for gravitational waves using the CMB by this reionization bump as the lensed

28 B-modes fall off at large angular scales. Here the gravity waves we are probing are those around during the time of reionization. The only caveat to the gravitational wave program for probing inflation is that symmetry breaking phase transitions in the early universe generate a spectrum of gravity waves that may be indistinguishable from the inflationary signal. This is worked out recently by Jones-Smith et al. [39]. However, these phase transition gravitational waves still come from a pre-decoupling era. Probing this energy density will then still result in furthering our observational reach in the universe. Thus, even if the gravity waves from inflation are buried by this phase transition signal, it is still a very worthy task to try to detect gravitational radiation via the CMB B-modes. Nonetheless, the inflationary paradigm is of such paramount importance to holding our picture of the universe together in a consistent manner, that setting it on stronger observational footing is identified as the top priority for upcoming experiments in NASA’s 2003 Structure and Evolution of the Universe Roadmap, Beyond Einstein: From the Big Bang to Black Holes [61]. Any chance at attaining this goal should be taken and the CMB B-modes happen to be the easiest way for probing the large-r region of the parameter space for the gravity waves from inflation. This thesis discusses the characterization of a novel detector architecture aimed at the task of observing the CMB B-modes.

29 Chapter 2

PFSB MOTIVATION

We present a novel detector architecture, called the Polarization and Frequency Selective Bolometer (PFSB), that is aimed at detecting the CMB B-modes. This technology could also ﬁnd application in furthering our understanding of galaxies and clusters via observations of polarized emission in the sub-mm regime. Sec- tion 2.1 covers the challenges that the instrument builders face in detecting the B-modes. In Section 2.2, we introduce the PFSBs and show how they potentially rise up to meet those challenges. We close this chapter in Section 2.3 with a discussion on the feed architecture we choose for coupling radiation to the PFSBs and our motivation for that choice.

2.1 CMB Polarization Challenges

The major challenges that face CMB B-mode observers in the future are vast increases in sensitivity, ﬁner control of instrument systematics, and astrophysical foreground rejection. In addition to these we must keep in mind the technical limitations of large format bolometric detector arrays. For proposed satellite missions such as EPIC [9], the size of the down-link data bandwidth is restrictive. Already for the 2000 detectors proposed there, a 34 m ground station antenna is required. Also, in future sub-mm focal planes where the detector counts reach

30 Figure 2.1. Reproduction of Figure 1.16. Scalar CMB power spectra in temperature (ΘΘ) and E-mode polarization (EE) compared with B-mode polarization due to gravitational lensing and gravitational waves at the maximum allowable 2.6×1016 GeV (r = 0.3) [83] and minimum detectable 3.2×1015 GeV (r = 6 × 10−5) level [42]. The cold dark matter model with a cosmological constant (ΛCDM) shown has parameters given in Hu et al. [34], from which this figure is taken. tens of thousands rather than a few thousand, reading out these focal planes are a major stretch for current systems like the ones on SPT [11] and SCUBA2 [4]. To illuminate the upcoming sensitivity challenges, we show the various CMB power spectra in Figure 2.1 which is simply a reprint of Figure 1.16. The current state-of-the-art set by QUaD [69] can detect the peaks of the E-mode spectra with a signal-to-noise of a few. As can be seen from the top of the shaded band in the figure, the primordial B-modes lay below the 0.2 µK level at least an order of magnitude in sensitivity away from the QUaD [69] result. We have almost two orders of magnitude to go before the CMB can wave the white flag for detecting

31 the primordial gravity waves due to the confusion limit shown as the bottom of the shaded region. Because the current detectors are background limited [92], in order to obtain these vast increases in sensitivities, future instrument builders must start using the brute force approach of catching more photons. The next challenge is a ﬁner control of instrument systematics. These are unwanted eﬀects that any real instrument induces on the signal it is intending to measure. A sample of potential sources and their impact on the various CMB power spectra are worked out in detail by Hu et al. [34], as well as Shimon et al. [77]. A good overview can be found in the Task Force on Cosmic Microwave Background Research (TFCR) [8]. For our purposes, since we are concerned with a single detector/feed system, we will only discuss two main sources:

• a loss of polarization eﬃciency, and

• creating spurious polarization through beam asymmetry.

Polarization measurements made with bolometers (as opposed to HEMTs) are normally carried out by differencing two orthogonally oriented polarized detectors [59]. A non-ideal polarized detector will always have some response to orthogonally polarized radiation which we will call the cross-polar response. This decrement in our ability to discriminate between two orthogonal polarization states results in a loss of sensitivity to polarized radiation when differencing the two detectors. The creation of spurious polarization signals mainly arises from differences in the beam patterns of the two detectors. Figure 2.2 shows an example of how beam ellipticity can play a major role here. When the beams from the two orthogonally oriented detectors sample different temperature structures on the sky, their difference (and thus the observed polarization) will be nonzero even if the underlying signal is unpolarized. This will result in converting temperature anisotropies into

32 Figure 2.2. CMB temperature anisotropies observed by ACBAR. The black ellipses show examples of asymmetric beam patterns emanating from two (solid and dashed) orthogonally oriented polarized detectors. The two beams sample different temperature anisotropies on the sky so, even when there is no underlying polarization signal, the difference between the two signals will be non-zero leading to an apparent polarization. This illustrates how beam asymmetry induces polarization from temperature anisotropies. Adapted from the ACBAR website [1]. polarization anisotropies (hΘΘi → hEEi, hBBi); a devastating effect due to the large difference in levels between the two (see Figure 2.1). Future detector/feed systems must attempt to minimize these effects. It is then important to characterize what systematics remain for the purposes of comparing technologies as well as the ability to take these effects into account in the analysis of CMB data. This work is mainly concerned with the characterization of the PFSB detector architecture and in understanding the contributions to the residual

33 systematics. The final major challenge that our field faces is the removal of polarized astrophysical foregrounds from our CMB observations. These foregrounds arise from galactic and extragalactic sources that we must look through in order to see the CMB emanating from the surface of last scattering. Figure 2.3 shows a forecast for the contributions of two types of foreground sources. One can see from this figure and our recent history, that temperature anisotropy and E-mode studies will not be hindered too much by the astrophysical foregrounds. However, any possible B-mode spectra, including the lensed E-modes, lays below the upper limit of the foreground forecasts. Fortunately, the emission spectra of these foregrounds differ from that of the CMB, as shown in Figure 2.4, and multi-frequency information can be exploited in their removal. Taking into account the three major challenges that face the CMB instrument builders, we can outline the properties of an ideal receiver to accompany any given telescope. The focal plane of a telescope is limited, in spatial extent, by aberrations [76]. If we are not to waste any of the unaberrated incident optical field, to achieve the highest sensitivity available to the telescope, we must

• tile the entire focal plane with detectors,

• detect both polarizations at each pixel,

• use all the important bandwidth (CMB and foreground sensitive) everywhere, and

• reach the photon noise limit in all bands.

In addition to meeting the sensitivity challenge, we must be very careful about controlling the systematic errors mentioned above as well as those outlined in the

34 Figure 2.3. Current estimates of the polarized Galactic foreground signals due to synchrotron emission from cosmic ray electrons and to thermal emission from interstellar dust grains. We show the foreground signals expected at a ﬁxed frequency, 94 GHz. Also shown are the limits on polarized synchrotron emission reported by the POLAR [63] and DASI [51] experiments. These observational limits have been extrapolated to 94 GHz, and plotted in orange with an estimated error band. The thermal dust signal, plotted in blue, was estimated by scaling the unpolarized dust model of Finkbeiner et al. [22], then applying a uniform 5% polarization. The large error band for dust emission reﬂects considerable uncertainty in both the polarization percentage and the coherence of polarization angle as a function of angular scale. It is largely accepted that we will be able to clean the foregrounds with multi-frequency instruments to a factor of ten below the levels shown here. This level of foreground removal is plotted as the dashed red line which, over a substantial range of l, lies below the B-mode signal for r > 0.01. Taken from Bock et al. [8].

35 Figure 2.4. The rms ﬂuctuations in the polarized CMB and foreground signals as a function of frequency. For each emission component, the band represents the rms signal expected from the large-scale emission (2 < l < 20), consistent with the models used in Figure 2.3. The orange band is the synchrotron emission, green is the dust emission, the upper dark band shows the hEEi portion of the CMB signal, and the lower dark band shows the hBBi portion of the CMB, assuming r = 0.01. Taken from Bock et al. [8].

TFCR report [8]. For the ability to do foreground subtraction, we must then split the important bandwidth into many individual bands for multi-frequency information. The TFCR report identiﬁed this frequency coverage to be from roughly 30–300 GHz. Using nominal 25% fractional bandwidths, this total range can be split up into 8 separate frequency bands centered at 30, 40, 60, 90, 120, 150, 200, and 300 GHz. One method to achieve the necessary polarization and frequency multiplexing is to use large oblique polarizers and dichroic splitters to create multiple focal planes. Due to optical limitations this solution reduces the focal plane area and thus sensitivity. It is also bulky and awkward when

36 considering the fact that the ultimate instrument will eventually be ﬂown on a satellite. It is thus important for us to design a compact way of carrying out this polarization and frequency multiplexing in an individual pixel on a single focal plane. This is what the PFSB endeavors to do.

2.2 PFSB Introduction

We will now present the PFSB solution to the challenges outlined in Sec- tion 2.1. This discussion will be split into three parts. First, in Section 2.2.1, we will discuss the general aspects of bolometers and how we use them. Section 2.2.2 will describe the technology upon which the PFSBs are based. We ﬁnally introduce the PFSB in Section 2.2.3 and discuss its admirable qualities with respect to the challenges above.

2.2.1 Bolometers

Bolometers, originally developed by Langley [47], have proven to be the most sensitive detectors for observations above 100 GHz. Low frequency measurements of the CMB have previously been dominated by High Electron Mobility Transistor (HEMT) based detection schemes. However, in a few years projected bolometer sensitivities [8] will reach or surpass those of the HEMT schemes in this range. Then bolometers will be able to span the full range of frequencies suggested by the TFCR. Figure 2.5 shows the major components of a bolometer. It works by receiving energy deposited onto a thermal mass, heating it up. The weak thermal link provides a connection to a bath which acts as a sink for the thermal energy. With steady power deposited on the thermal mass, this can be described succinctly by

37 Figure 2.5. A schematic for a general bolometer. Some form of energy (red) heats up a thermal mass shown in black. This mass is weakly linked (shown in red) to a bath (blue) allowing the energy to heat it up. The temperature of the mass is read out by a thermistor shown in green. the power balance equation ¯ P = G(T − T0) (2.1) where P is the total power being deposited, T is the temperature of the thermal ¯ mass, T0 is the bath temperature, and G, deﬁned by this equation, is the mean thermal conductivity of the link over the range [T0,T ]. The thermal link allows the temperature of the mass to reset on timescales comparable to τ = C/G where C

dP is the heat capacity of the mass and G ≡ dT is the dynamic thermal conductivity of the link. Finally, a thermistor, or temperature sensitive resistor, is used to measure the temperature of the mass, turning the energy deposition signal into an electrical one. In order to read out the resistance of the thermistor, we must electrically bias it in some way. Most commonly done is to bias with either a constant current or voltage, measuring the thermistor voltage drop or current respectively in order to

38 obtain the desired resistance. The choice between these two biasing schemes is set by the slope of the thermistor resistance as a function of temperature at the bias point and the type of reaction we would like to induce on the bolometer thermal mass upon energy deposition. Consider the power dissipated into the thermal mass by the thermistor electrical bias,

V 2 P (T ) = = I2R(T ) , (2.2) el R(T ) where V , I, and R are the thermistor voltage drop, current, and resistance respectively. The electrical power dissipation was written in its two forms to explicitly show its diﬀerent dependence on thermistor resistance, and therefore temperature, for a constant voltage or constant current bias respectively. Now considering the fact that the total power in Equation (2.1) is the sum of this electrical power and the incident power Q absorbed by the bolometer,

¯ P = Pel(T ) + Q = G(T − T0) , we see that the bias power induces an electro-thermal feedback in the bolometer. As the temperature changes in response to a change in the incident power Q we see a corresponding change in the electrical power Pel which itself is also responsible for thermal heating in the bolometer. The sign of this feedback is determined by the type of biasing scheme and the thermistor temperature dependence as shown by dP V 2 dR dR el = − = I2 . (2.3) dT R2 dT dT

Thus, given the sign of the thermistor’s resistance as a function of temperature, we can choose the type of feedback (positive verses negative) with a choice of the

39 bias type. For mm-wave detection, the thermal mass must be coupled to the radiant energy that is being measured. Two schemes are in use by the CMB community today. For the PFSBs and other absorber-coupled bolometers, some radiant power absorber, like a sheet of resistive metal matched to the impedance of free space, is deposited directly onto a transmissive thermal mass. To aid in absorption, a reflective layer, called a backshort, is placed at a distance λ/4 away so that the absorber always sits at a maximum in the wave propagation. In antenna-coupled bolometer schemes, a mm-wave antenna sends signals down a transmission line which is terminated by a shunt impedance on the thermal mass of the bolometer. These are much easier to fabricate as the thermally active region only needs to be large enough to support the shunt impedance and the thermistor (∼ 100 µm × 100 µm) rather than the entire microwave absorber (as small as 1 mm2). The PFSBs make use of the steep superconducting transition, shown in Fig- ure 2.6, to provide a highly sensitive thermistor. This is called a Transition-Edge Sensor (TES). When operating bolometers for CMB observations, we are interested in a stable bias point so that we can continuously track changes in the radiation field we are measuring. The figure shows that, for TES thermistors the slope of the R(T ) curve is positive, so from Equation (2.3), it is evident that we must voltage-bias the TES in order to obtain the required negative electro- thermal feedback for stability. We then read out the TES current as a measure of the TES resistance and thus the optical signal. See Section 5.2 for a description of the SQUID amplifier system used for this current readout. For a good review of infrared and mm-wave bolometer fundamentals see Richards [70]. Gildemeister [26] discusses the special case of the voltage-biased superconducting bolometers

40 Figure 2.6. A TES superconducting transition. This sketch, similar to the PFSB TES transitions, has a normal state resistance of Rn = 200 mΩ and a critical temperature of Tc = 500 mK. We see that the transition between the normal and superconducting states is very steep occurring over a range of only 2 mK. This large slope is utilized for extremely sensitive thermistors in bolometric detectors. With CMB observations, we invoke a bias which gives negative electro-thermal feedback, keeping us at a stable operating point in the transition. The temperature ﬂuctuations shown here can be caused by ﬂuctuations in the external power incident on the thermal mass of the bolometer. just described.

2.2.2 FSBs

The PFSBs are a simple modiﬁcation of a technology called the Frequency Selective Bolometer (FSB). The FSBs combine bolometric and band-pass ﬁltering techniques to create a detector which is only sensitive to a single frequency band of 20% fractional bandwidth. This is accomplished by converting the traditional

41 Figure 2.7. FSB surface absorber geometry. This resonant pattern absorbs over a restricted frequency band characterized by the three parameters shown here. l and w specify the length and width of the individual crosses respectively while g gives the grid spacing. sheet absorber of a mm-wave bolometer into an absorptive resonant pattern as shown in Figure 2.7. The absorption-aiding λ/4 backshort is a reﬂective layer patterned in the same manner. With this geometry, the FSBs absorb in-band radiation at high eﬃciency [67], as shown in Figure 2.8, while letting the out-of- band light pass through. This large out-of-band transmission property enables stacking of multiple frequency bands into a single waveguide creating a multicolor pixel. Figure 2.9 shows the entire FSB detector architecture for a 4-color pixel. Due to a long tail on the high frequency side of the absorption band shown in Figure 2.8, we place the higher frequency detectors upstream from the lower ones in the waveguide. This allows them to remove the high frequency photons from the stream before encountering the lower frequency detectors.

42 Figure 2.8. Spectral absorption efficiency of two FSB stacks. The difference between the two stacks is in the material used for leads on the thermistor which should not affect optical characteristics. In each stack, the 219 GHz detector is placed downstream of the 271 GHz detector in the light path. Overlaid with dashed and dotted curves are the corresponding models using a finite element method electromagnetic field simulator scaled to have the same peak absorption. Taken from Perera et al. [67].

2.2.3 PFSBs

PFSBs take the idea of resonant absorptive structures and add polarization selectivity by splitting the crosses in Figure 2.7 into bars separated onto two diﬀerent bolometers as shown in Figure 2.10. These bars restrict the allowable current densities in the absorbing material so that they can only be excited by a single linear polarization. As with the FSBs, stacking creates a multicolor, dual-polarization pixel. We show photographs of one of the PFSBs we tested in Figure 2.11. These detectors were fabricated at the Goddard Space Flight Center (GSFC) by a team headed by Robert Silverberg, including Tina Chen and Fred Finkbeiner. Shown in the ﬁgure is a fully assembled PFSB/backshort pair where the backshort is

43 Figure 2.9. Schematic of the SPEED [79] detector array assembly for one spatial pixel showing the structure of the FSB architecture. From Silverberg et al. [79].

glued to the front of the PFSB through λ/4 spacer blocks. The bars we see here are those of the reﬂective backshort with the PFSB absorber bars hidden behind. These are patterned onto a thin silicon-nitride membrane supported on all sides by the silicon frame shown. The gold ring shown marks the extents of the thermal mass of the PFSB. It is also patterned onto a thin silicon-nitride membrane and helps keep the active region isothermal as well as increases its heat capacity. The PFSB bars, which do the absorbing, lie inside this ring on the same membrane. In two corners of the thermal mass we place TESs to monitor the temperature. This entire mass is supported only on the four corners in the manner shown in Figure 2.11(b) where we can see the TES leads running down one of the support legs. The two detectors we ended up characterizing both have identical optical properties which are given in Table 2.1.

The advantage of the PFSB multiplexing architecture is that it does not waste photons, and thus sensitivity. PFSBs require a constant focal plane packing density as a function of frequency since the various bolometers sit in a single waveg-

44 Figure 2.10. A schematic of a PFSB. In the left figure, the PFSB resonant absorbers are shown face-on. The vertical resistive bars lie on one silicon-nitride membrane while the horizontal ones lie on a second membrane behind the first as shown on the right. Also on the right is the single resonant reflective backshort a λ/4 away from the bolometers. As in the FSBs, the bar geometries can be used to tune the frequency dependence of the optical properties of the device so that many of these can be stacked into a multi-color pixel. From Ruhl and Meyer [72]. uide. As the diffraction spot size for a given telescopic system decreases with increasing frequency, a single-moded frequency multiplexed system with a constant focal plane packing density would be using sub-optimal densities at the higher frequencies. However, the PFSB waveguide becomes multi-moded at higher frequencies (being single-moded only at the lowest frequency) propagating the light from the multiple diffraction spots down towards the bolometers. The throughput of a multi-moded system is given by

AΩ = mλ2 (2.4) where A is the area of the aperture, Ω is the accepted solid angle in steradians, and m is the number of modes. So instead of wasting the extra modes of the high-frequency radiation, the PFSBs combine them into a single bolometer. This

45 (a) (b)

Figure 2.11. Photographs of one of the PFSBs we tested. Shown in (a) is the entire PFSB/backshort pair. Here the backshort lies in front of the PFSB and consists of the reﬂective bars patterned on a thin silicon-nitride membrane held on all sides by a silicon frame. The thermal mass of the PFSB is contained within the gold ring shown and also consists of a thin silicon-nitride membrane. Its absorber bars are hidden behind the backshort’s in the ﬁgure. The PFSBs each have two TESs shown here on the top-left and bottom-right corners of the thermal mass. The entire structure is supported only on the four corners by thin silicon-nitride legs as shown in (b) which is an enlarged view of the bottom-right corner. In the top-right corner of (a), the backshort frame is cut away to expose the bonding pads for the TESs on the PFSB frame. multi-moded behavior is convenient because we achieve optimal sensitivity by multiplexing, without having to increase the number of detectors in the focal plane. Thus we stay clear of the technical limitations regarding large focal plane array readouts pointed out in the beginning of Section 2.1. To check this, lets us calculate the number of detectors needed in the 8 band, dual-polarization system

46 Table 2.1

PFSB Characteristics

Characteristic Value

target central absorption frequency 223 GHz

la 572 µm

wa 13.9 µm

ga 1020 µm

backshort spacing 336 µm

a See Figure 2.7 for deﬁnitions of FSB and PFSB absorber parameters. proposed at the end of that section. The multi-moded nature of the PFSBs actually allows control over the resolution of the telescope since the throughput of a given multi-moded pixel is given by Equation (2.4). We can see this by invoking conservation of throughput, which allows us to reinterpret A as the telescope aperture and Ω as the beam size. Since the telescope aperture is constant, increasing the number of modes increases the size of the beam. Given this freedom, let us choose a single moded system at 30 GHz. With the 2.5 m primary, 4◦ ﬁeld of view telescope system described in the TFCR report [8, Section 6.2], we get a resolution of 250 with 70 detectors at each frequency. The full 8 band, dual-polarization system will then only require 1200 detectors which is feasible for current readout and data down-link systems. One consequence of this multi-moded multiplexing scheme, illuminated by the above calculation, is that we lose resolution relative to the single-moded case.

47 This happens because the PFSB system is limited to the resolution of the lowest frequency in the system. Whereas we have calculated a 250 resolution here, a single-moded system at 150 GHz, historically the most sensitive CMB channel, has a resolution of 50. However from Figure 2.3, we see that the CMB B-modes peak at scales larger than 1◦. In comparison to the PFSBs, fully planar alternatives are currently getting a lot of attention for their scalable fabrication methods and compact geometry. However, the combination of polarization and frequency selectivity is difficult to engineer. Antenna-coupled architectures given in Goldin et al. [27] for comparison, only achieve one or the other. In order for these types of bolometers to obtain both types of multiplexing, very novel antenna designs must be employed. See O’Brient et al. [62] for ultra-wide band, log periodic, dual polarization antennas. A disadvantage of the log periodic antenna design is that the effective pixel area gets smaller with increasing frequency. Since close packing can only be achieved at the lowest frequency (largest effective pixel area), the highest efficiency obtainable for our system with 8 bands spanning 30–300 GHz is 26%. Doing away with frequency multiplexing entirely, as is done in the EPIC study [9], causes a huge loss of sensitivity. We see that the systems suggested there either contain multiple small, single-frequency telescopes, or split a single, larger throughput focal plane up into various single-frequency pixels. Either way, these instruments throw away all out of band radiation for each single frequency. Let us now summarize the power of the PFSB approach to detecting inflationary gravitational waves via the B-mode power spectrum of the CMB:

• compact polarization and frequency multiplexing in a single module that can be placed in the focal plane, compared to bulky and awkward dichroic

48 splitters and polarizers, makes use of the full sensitivity available to the telescope,

• multi-mode detectors provide for extra sensitivity without causing a readout or down link problem, and

• multi-frequency instruments allow for good rejection of astrophysical foregrounds.

In order to evaluate the true prospects of the PFSB architecture, we are left with figuring out a scheme for coupling radiation to it from a future telescope, investigating its contributions to instrument systematics, and finally, choosing a telescopic system for coupling to the sky. This work involves the choice of feed structure and the characterization of the PFSBs in that scenario. We leave the latter part of the evaluation to be done in future work. In our characterization we will take a look at band definition, depolarization, and beam asymmetry.

2.3 PFSB Feed Architecture

We now turn to motivating our strategy for coupling radiation from a telescope to the PFSBs. FSBs are fed by a pair of back-to-back Winston cones with a throughput of 4.5 mm2 sr. Winston cones [88] are non-imaging light concentrators originally developed for the solar power industry. In the ray optics approximation, they accept light over a limited angular range across their entire opening aperture, carrying out a maximal spatial concentration by dumping the light from the exit aperture over the full 2π steradians. See Figure 2.12 for the geometry of the Winston cone. Back-to-back Winston cones, as shown in Figure 2.14, are used by the FSBs to limit the angular extent of the rays which pass through the detectors

49 in order for the resonant structures to have similar dimensions when viewed by all rays. Because the PFSBs are polarization sensitive devices, we must use a feed that preserves polarization properties. When polarized light reflects off of metal surfaces, it undergoes changes in its polarization properties, which depend on the frequency-dependent real and imaginary parts of the complex dielectric constant of the metal [82]. These effects are minimal at normal and grazing incidence and with polarization parallel or perpendicular to the component of the surface normal perpendicular to the propagation direction. However, in the intermediary ranges of skew reflections with diagonal polarizations, it is imperative to take into account the material properties where the mirror can behave as both a polarizer and a waveplate. In completely rotationally symmetric on-axis imaging optical systems these effects can cancel, preserving the input polarization on output. By contrast, the Winston cone is non-imaging and there is no guarantee that it will combine the light in ways which lead to cancellation of the polarizing effects of skew reflections. Thus it is important for us to check the polarizing properties of Winston feeds. We have simulated the linear polarization effects of a single Winston cone using the ZEMAX Non-Sequential Components (NSC) software package [93]. ZEMAX- NSC is a ray tracing program where the user defines the spatial geometry of optical systems including sources and detector arrays. It launches rays from the sources by pulling from a distribution in angular and position space, then assigns a power to each ray depending on the total source power and the number of rays used in the analysis. These rays are then allowed to traverse the defined optical system. The NSC part denotes the fact that ZEMAX does not need to know what optical

50 element each ray will hit next in order to complete the calculation as do most traditional ray tracing programs. This feature is important for the non-imaging simulations we carry out in this thesis. The properties of the ray ensemble can be queried at various places in the system by placing a detector pixel in their path which sums the powers of the rays incident on it. This summing can be done in a coherent or incoherent manner. If a detector array is used, one can view either the positional or the directional variation of the rays which hit the array. When a ray reﬂects from a mirror surface in ZEMAX, the standard complex index of refraction chosen for that surface is

n = 0.7 − 7.0i . (2.5)

This is appropriate for aluminum at at a wavelength of about 1 µm [64]. Other values may be chosen but we stuck with the default value.

Figure 2.12. ZEMAX simulation of the polarization properties of the Winston cone. See Table 2.2 for dimensions. The source (red) on the left emits on-axis rays (blue) down the mouth of the Winston cone (black). After reﬂecting oﬀ the cone walls, the rays encounter a polarizer and then get recorded at a detector placed at the output of the Winston.

51 Table 2.2

Simulated Winston Cone Parameters

Throughput Large Aperture Small Aperture Length Acceptance Half Horn (mm2 sr) Diameter (mm) Diameter (mm) (mm) Angle (degrees)

Winston (Fig. 2.12) 2600 16 3.3 45 12 back-to-back Winston (Fig. 2.14) 4.5 10 0.48 57 5.5

In the simulation shown in Figure 2.12 we place in front of a Winston cone, whose dimensions and properties are given in Table 2.2, a completely polarized source with a spatial extent covering the entire entrance aperture. For simplicity, this source only emits on-axis rays (parallel to the rotational axis of the Win- ston cone). On the exit aperture of the Winston, we put a linear polarizer and a detector array just behind it. Two configurations of this system were run, each launching 104 rays. One has the exit polarizer parallel to the polarization of the source and the other uses a perpendicular orientation. The angular distribution of the rays hitting the detector is shown in Figure 2.13. We see from the figure that the Winston cone produces a large amount of cross-polarization. The power fractions show that almost all polarization information is lost on output, as expected given all the skew reflections with diagonally polarized light. The majority of the rays shown in Figure 2.13 come from single reflections off the Winston cone. Thus light coming out of the exit aperture traveling in the vertical or horizontal direction nominally reflects off the vertical or horizontal wall. Since the source polarization is along the vertical direction with respect

52 (a) vertical analyzer power fraction = 52% (b) horizontal analyzer power fraction = 40%

Figure 2.13. Winston cone exit radiance in angular space for a vertically polarized source. The plots show this function over the entire hemisphere projected onto the (u, v) plane where u = sin(θ) cos(φ) and v = sin(θ) sin(φ)(u, v ∈ [−1, 1]). The units of the color bar are arbitrary. In (a) the exit polarizer is oriented parallel to the source polarization where in (b) it is perpendicular. The power fractions stated in the captions are the fraction of the total source power which is incident on the detectors. Here we calculate the sums incoherently. In the figures we only include rays which actually reflect off of the cone (hence the dark circle at the center) while the power fractions include all rays. These do not add up to unity because of the straight through rays, which make up 4% of the power, and the losses in the cone defined by Equation (2.5). to the figures, this means that the radiance in these directions comes from skew reflections with polarizations aligned with the axes setup by the propagation direction and the surface normal. In these cases, we see that the polarizing effects of the reflection are minimal and most of the light passes through the parallel polarizer as shown in Figure 2.13(a), and gets blocked by the perpendicular polarizer (Figure 2.13(b)). However, in the directions diagonal to the polarization, we see the large polarizing effects from skew reflections of diagonally polarized light. This time, for the particular index of refraction chosen (2.5), the reflection almost rotates the polarization to the orthogonal state as can be seen by the relative lev-

53 els of the diagonal plane response from the perpendicular (Figure 2.13(b)) verses parallel (Figure 2.13(a)) polarizer conﬁgurations of the system.

Figure 2.14. ZEMAX simulation of the polarization properties of identical back-to-back Winston cones. See Table 2.2 for dimensions. The source (red) on the left emits on-axis rays (blue) down the mouth of the back-to-back Winstons (black). After reﬂecting oﬀ the horn walls, the rays encounter a polarizer and then get recorded at a detector placed at the output of the cone combination.

For back-to-back Winston cones, the entrance and exit apertures are in a more symmetric situation and we should expect some of these effects to cancel after reflecting off the second Winston. We have carried out a similar simulation for identical back-to-back Winstons as shown in Figure 2.14. These Winston cones were chosen with the same geometry as the detector side Winston in the back-to- back pair that feeds the FSBs. We decided to go with an identical pair here in an attempt obtain the highest level of symmetry. See Table 2.2 for the dimensions and properties of this feed. Figure 2.15 shows the results of this simulation by plotting the detector illumination in position space. We have converted to viewing in position space because the back-to-back Winstons are intended to collimate

54 (a) vertical analyzer power fraction = 63% (b) horizontal analyzer power fraction = 12%

Figure 2.15. Back-to-back Winston cones exit radiance in position space for a vertically polarized source. The plots show this function over the entire exit aperture in the xy-plane with x, y ∈ [−8, 8]mm. The units of the color bar are arbitrary. In (a) the exit polarizer is oriented parallel to the source polarization where in (b) it is perpendicular. The power fractions stated in the captions are the fraction of the total source power which is incident on the detectors. Here we calculate the sums incoherently. In the figures we only include rays which actually reflect off of the cone while the power fractions include all rays. These do not add up to unity because of the straight through rays, which make up 1% of the power, and the losses in the cones defined by Equation (2.5). The losses shown here are far greater than the ones shown in Figure 2.13 because of the greater number of average reflections per ray in both the back-to-back scenario and for these particular cone dimensions. the beam and thus our angular distribution on exit is small. Notice that while we get a similar pattern to the single Winston cone, the difference in amplitude between the parallel (Figure 2.15(a)) and perpendicular (Figure 2.15(b)) polarizer configurations is increased. From the power fractions stated in the captions of the figures, we see that the back-to-back scenario still rotates a significant fraction of the power into the orthogonal polarization. Redoing this analysis with an index of refraction more suitable to copper at 220 GHz, n = 270 − 650i [65], shows that the qualitative results remain the same

55 but the power fractions change by as much as 10%. The single Winston cone gets better at preserving the polarization but the back-to-back Winston gets worse. It is important to keep in mind that ZEMAX simulations operate in the geo- metric optics limit (inﬁnite modes) and do not apply at all to the single-moded case. However, we are planning to operate the PFSBs in a many-moded regime, so we infer from these simulations the dangers of operating feed horns in this manner. Based on this ﬁnding we decided not to use Winston cones to feed our PFSBs. Instead we mount our detectors in the waveguide intended for polarization and frequency multiplexing and operate them without additional feeds. With this choice of light coupling architecture, the rest of this thesis concerns the characterization of the spectra and beam of the PFSB detectors as they emanate from the aperture of this waveguide.

56 Chapter 3

CHARACTERIZATION FORMALISM

The majority of the work leading to this thesis was the characterization of the PFSBs. We will now lay out the general formalism from which we will describe this characterization.

Figure 3.1. Instrument action for a single detector on the incoming radiation field (circular polarization ignored). Incoming light, varying over direction Ω and frequency ν, is modified by the telescope which illuminates its focal plane. The switch from Ω to Ω0 that occurs here is meant to denote the fact that the telescope does not simply alter the polarization state of the light. It often bends the light rays as well, mapping one direction to another. Some kind of feed structure exists in the focal plane which has its effects on the radiation as well as having a finite beam size that integrates out the angular dependence and sends a stream of power to a detector. The detector finally converts that power into an electrical signal. For a polarimeter, some form of polarization selectivity occurs in the feed/detector block. In some cases, two different detectors must be differenced to back out the intended Stokes parameter. Modified from Kaplan and Delabrouille [41].

57 An instrument or camera designed to map out the angular variations in a radiation field normally consists of a telescope and a detector system. The telescope serves to concentrate and focus the light onto a focal plane, thereby imaging the radiation field. We then place some detector technology in this focal plane to record that image. Typically, the light incident on the focal plane is binned into a number of finite-sized pixels with detector channels in them that first possibly condition the light, then measure the total incident power. Figure 3.1 shows a schematic of the instrument and the various components which effect the radiation field important for a single detector channel. The splitting of the instrument into various components shown here is not necessarily a unique one, but important for conceptualization. For a polarimeter, some form of polarization selectivity occurs in the feed/detector block where in certain cases, two different detectors must be differenced to back out the intended Stokes parameter. We will concentrate our efforts at measuring the systematic effects in this block of Figure 3.1. The results found here can be used with calculations of telescope effects to simulate the full instrument systematics. Our characterization will involve coupling fully polarized radiation to the PFSBs. In this scenario it is useful to use the language of the Jones calculus where we express a single frequency radiation stream as a two-element vector of orthogonal complex phasor components of the electric field

  E  x  E =   . Ey

See Tinbergen [82] for an excellent description of the Jones calculus. We describe

58 the instrument action on this ﬁeld as a 2 × 2 complex Jones matrix J such that

Eout = J · Ein (3.1)

where Ein and Eout are input and output radiation streams respectively. The Jones matrix is specified by 7 parameters and one irrelevant phase. Kaplan [40] gives elementary operations specified by 7 parameters that an instrument can perform on a radiation stream. Since we are not concerned with circular polarization, and the chain shown in figure 3.1 is terminated by a power detector, we are left with only these two elementary actions governed by three parameters:

  τ 0  k  Jpol =   , (3.2) 0 τ⊥   cos ∆ − sin ∆   Jrot =   (3.3) sin ∆ cos ∆

where Jpol is an imperfect polarizer whose co-polar direction is oriented along the x-axis with τk and τ⊥ giving the co-polar and cross-polar transmissions respectively, and Jrot is a polarization rotator by an angle ∆. Since the power detector signal is unaﬀected by any rotation immediately before it, we can write down the Jones matrix of a general linear polarimeter as a combination of a polarizer (3.2) following a rotator (3.3)

Jinst = Jpol · Jrot   τ cos ∆ −τ sin ∆  k k  =   . (3.4) τ⊥ sin ∆ τ⊥ cos ∆

59 Using this formalism, we can write out the post telescope chain of Figure 3.1. The rate of energy deposited onto the power detector is given by

Z T 0 ∗ 0 0 Ppd(ν) = hEfda(Ω , ν) · Efda(Ω , ν)idΩ where (Ω) ≡ (θ, φ) represents the directional coordinates, ν is the optical propagation frequency, and Efda is the radiation field after having been altered by the feed/detector architecture. Now from Equation (3.1) describing the Jones calculus, we can write the feed/detector architecture radiation field as the dot product between a characterization source radiation field Esrc and the feed/detector action

Jfda giving

Z 0 0 T 0 0 ∗ 0 Ppd(ν) = h[Jfda(Ω , ν) · Esrc(Ω , ν)] · [Jfda(Ω , ν) · Esrc(Ω , ν)] idΩ . (3.5)

As stated in the caption of Figure 3.1, the feed/detector architecture aﬀects the polarization state of the incoming radiation and integrates over the beam. It sends energy into the power detector which then records the total, frequency integrated, power given as

Z 0 0 s = Ppd(ν )dν ZZ 0 0 0 0 T 0 0 0 0 ∗ 0 0 = h[Jfda(Ω , ν ) · Esrc(Ω , ν )] · [Jfda(Ω , ν ) · Esrc(Ω , ν )] idΩ dν . (3.6)

In the case of the PFSBs Ppd is in the form of thermal energy and the power detector is the TES.

Jfda(Ω, ν), parametrized by Equation (3.4), contains all the characterization information contained in a single polarimeter channel. Ideally, in order to obtain

60 the three Jones characterization parameters, varying over the beam and spectra of the channel, we would use a polarized point source at a single frequency in which we were free to rotate the polarization:

  cos ψ 0 0   0 0 Esrc(Ω , ν ) =   δ(Ω − Ω)δ(ν − ν) (3.7) sin ψ where Ω is the angular position of the point source, ν is its frequency, ψ is the angle of the source polarization relative to some system of axes, and we have normalized the intensity. Substituting Equations (3.4) and (3.7) into Equation (3.6), it can be shown that

s(ψ;Ω, ν) = X(Ω, ν) + [C(Ω, ν) − X(Ω, ν)] cos2[ψ + ∆(Ω, ν)] . (3.8)

2 2 We have redefined the variables into power units, C ≡ τk and X ≡ τ⊥, so that C is the signal value when the source polarization is “aligned” (taking into account the rotator angle ∆) with the instrument polarizer and X is the signal when the source is oriented in the perpendicular direction. It is this signal dependence on the polarizer source angle s(ψ) which we will exploit to fit out the three instrument characterization parameters (C,X, ∆) as a function over beam and spectra. We will use the terms co-polar response, cross-polar response, and rotation angle respectively to denote these parameters. Obtaining them would serve as a complete characterization of a single linear polarimeter channel. In reality, we will only be able to easily measure a limited set of information about these functions. See Chapter 6 for a discussion of the tools we use to make these measurements. The difficulties in exploring the full angular and spectral dependence of the characterization functions will be clear there. For now, let us

61 simply give an overview of what we measure. The spectral dependence of the PFSBs are explored by placing a specialized, ﬁnite sized source somewhere in the beam. From this we gain access to

Z C(Ω¯, ν) ≡ C(Ω0, ν)dΩ0 , source extent Z (3.9) X(Ω¯, ν) ≡ X(Ω0, ν)dΩ0 source extent for a few places within the beam, where we have used Ω¯ to remind us that these are beam averaged quantities. We will not be presenting measurements of the spectral variations of ∆(Ω, ν). In our beam measurements we use a small, hot blackbody source having a wide-band continuous emission spectra. Thus we will not be able to unfold the spectral integral in Equation (3.6). Instead, we measure

Z C(Ω) ≡ C(Ω, ν0)ν02dν0 , Z X(Ω) ≡ X(Ω, ν0)ν02dν0 , (3.10) Z ∆(Ω) ≡ ∆(Ω, ν0)ν02dν0 where the ν02 appears in the integral from the Rayleigh-Jeans source spectra. This thesis comprises the measurements of the five functions given in Equations (3.9) and (3.10), for two different PFSB architectures (circular and square cross-section waveguides) and understanding the difference between the two scenarios through simulation.

62 Chapter 4

SIMULATION

We now discuss the method used to simulate the PFSB architecture. This simulation will give results that we can compare to the beam measurements of Equa- tion (3.10) described in the previous chapter. The spectral aspect of the PFSBs are very similar to that of the FSBs so we refer to Logan [53] for simulations of the FSB spectra. Our simulation illuminates the polarization discrimination advantages of using a square verses circular waveguide to mount the PFSBs. Fig- ure 4.1 shows a schematic of the main parts of this simulation. Section 4.1 gives a quick overview of the basic simulation elements. Section 4.2 presents a detailed discussion of the way we handle the diffraction caused by an aperture stop in the optical system used to measure PFSB radiation patterns. Moving on towards the waveguide, we split the discussion into two parts. Section 4.3 describes the decomposition of the PFSB waveguide aperture illumination into guide modes while Section 4.4 discusses the electric field propagation down the waveguide and its termination on the bolometer. We conclude this chapter in Section 4.5 showing the simulation results for the circular waveguide configuration of the PFSBs. We use this case as an example for describing the specific details of the two measurement comparison simulations that we carry out (circular and square cross-section waveguide). See Chapter 8 for the presentation of all of our simulation results.

63 Figure 4.1. PFSB waveguide simulation schematic (not to scale). Plane waves shown in red in the upper-right corner impinge upon an aperture stop depicted as the black-shaded boundary. From this aperture illumination Eap, we calculate the diffracted E-field which then illuminates the PFSB waveguide (black outlined cylinder). This field is decomposed into waveguide modes Emn, that are propagated down towards the PFSB absorber shown schematically as a green wire grid. The gray filled space on the bottom represents a reflecting stop that the modes bounce off, back towards the guide mouth. The total electric field is sampled at the position of the PFSB absorber and used to calculate deposited power.

4.1 Overview

As shown in Figure 4.1, the PFSB is mounted in a smooth-walled unbroken guide of either circular or square cross section. The guide is terminated with an

64 all-reflecting short (shown as solid gray in the figure), which serves the purpose of simulating the backshort behind the PFSB described in Section 2.2.3. For the PFSB absorber bars, we simply use incoherent E-field power monitors in the corresponding positions as described in more detail in Section 4.4. We draw these schematically in the figure as the green vertical bars just above the guide termination. Now in addition to the PFSB waveguide architecture, as is commonly done, we will be employing an aperture stop to cleanly limit the angular extent of the beam of our instrument. Because of the relatively small size of the aperture stop used in one of our systems, it is important to simulate its diffraction effects. This is accomplished by an infinitely thin reflecting/absorbing aperture placed above the PFSB waveguide as depicted by the hatched surface in Figure 4.1. We run the simulation in “receive mode” by allowing plane waves of known propagation direction and polarization to impinge upon this system. The choice to run this system in receive mode is driven by the incoherent nature of the way that bolometers thermally sense optical power. It would be very difficult to begin by positing random electric currents setup by the finite temperatures of the PFSB absorber, then attempting to propagate the fields these generate through the system. We carry out a much easier scenario of starting coherently via plane waves, propagating through the system in a completely coherent manner, and imposing an incoherent power sum on the bolometer at the end. All simulations are run with a propagation frequency of 220 GHz, near the central absorption frequency of the PFSBs we tested. We do not simulate the in- coherency caused by the PFSB’s finite bandwidth, because we found that changing the propagation frequency did not have a qualitative impact on the radiation patterns.

65 These radiation patterns are built up by running the above simulation for many plane wave propagation directions and polarizations. We obtain the functions of Equation (3.10) by using the polarization angle ψ dependence of the signal in Equation (3.8) to calculate the three characterization parameters (C,X, ∆) for a given plane wave direction Ω.

4.2 Aperture Diﬀraction

This section begins the detailed discussion of the simulation. We ﬁrst setup a system of axes by placing the origin at the center of the PFSB waveguide entrance aperture with the z-axis parallel to the waveguide axis of symmetry, pointed towards the plane wave source. The y-axis points along the PFSB absorber bars. We can now present the plane wave, which we normalize to unit amplitude because we only care about relative response throughout the beam:

i[k(Ω)·x−ωt] Epw(x;Ω, ψ) = Eˆ pw(Ω, ψ)e (4.1)

where Epw is the electric field, unit vector Eˆ pw specifies its polarization orientation, k is the wave vector and ω is the angular optical propagation frequency. As with all plane waves, Eˆ pw(Ω, ψ) · k(Ω) = 0 so ψ specifies the polarization angle in the plane perpendicular to the propagation direction. We show an explicit dependence of Epw on spatial position x, propagation direction Ω, and polarization orientation ψ, leaving the time dependence t implicit because we will soon convert to phasor notation. For diffraction of this plane wave by the aperture stop, we will use the general vector diffraction formula derived by Jackson [38] for an arbitrary blocking screen,

66 Z ikR 1 e 0 Ediﬀ (x) = ∇ × (n × E) da . (4.2) 2π apertures R

Here Ediff is the diffracted E-field, E is the total aperture E-field, and n is the unit normal for the aperture surfaces, pointing away from the source. The integration is taken over all apertures that may exist in a general blocking screen, and R is the distance between the observation point and the aperture integration point

R = |x − x0| .

When interpreting Equation (4.2), we must be careful to notice that ∇ acts on the observation coordinates x, whereas the aperture E-ﬁeld varies over the aperture coordinates x0. Thus the diﬀerentiation in ∇ only operates on the last part of the

ikR integrand e /R. Because of this fact, we can easily simplify Equation (4.2) by constricting its application to our current situation. These restrictions are that the apertures lie completely in the xy-plane, and that we can ignore the electric ﬁeld normal to the aperture (see Section 4.3 for the rationale behind ignoring this component of the ﬁeld). We obtain

z − z0 Z eikR E (x) = (ikR − 1)E da0 (4.3) tdiff t 3 2π apertures R where the “t” subscript denotes the fields transverse to the z- or waveguide symmetry axis. The location of the aperture screen along the z-axis is given by z0. The vector diffraction formula (4.2) is derived directly from Maxwell’s equations and is thus exact. We apply the standard approximation, where instead of using the full aperture E-field in Equation (4.3), we replace it with the sourced

67 E-ﬁeld from Equation (4.1)

E(xap) u Eap(xap;Ω, ψ) ≡ Epw(xap;Ω, ψ) (4.4)

where xap reminds us that we constrict the range here to within the aperture. This approximation ignores the diffracted contribution to the E-field in the aperture which mostly affects the area near the edges of the blocking screen. Using the combination of Equations (4.3) and (4.4), we can then calculate the diffracted

transverse E-ﬁeld in the PFSB waveguide aperture. Let us call this Etwg (xwg;Ω, ψ) where the “wg” subscript stands for the PFSB waveguide aperture.

4.3 Waveguide Mode Decomposition

For the propagation of electromagnetic ﬁelds down the waveguide, we rely on an impressive body of literature. This work was spurred by a massive research eﬀort to develop various radar systems during World War II at the Radiation Laboratory (Rad Lab) of the Massachusetts Institute of Technology (MIT). Their ground breaking advances, which encompasses much of the physics of electromagnetic communication, are nicely summarized in a series of publications collectively known as the MIT Radiation Laboratory Series. Two volumes worth mentioning here for their application to waveguides and microwave antennas are Marcuvitz [56] and Silver [78]. Good textbook treatments of waveguides are found in Jackson [38] and Kraus [46]. For future PFSB simulation work where the gaps in the waveguide may be accounted for, see Wexler [85] and Gesell [25] for a modal approach to applying electromagnetic boundary conditions to discontinuous waveguide sections. Returning to the problem at hand, we are armed with the sourced, transverse

68 electric ﬁeld in the PFSB waveguide aperture Etwg (xwg). Henceforth, we will drop the explicit dependence on the source direction and polarization, while remembering that it is always implicit. In order to continue the propagation, we must satisfy the boundary conditions imposed by the conducting walls of the waveguide. This can be done by decomposing the sourced E-ﬁeld into waveguide modes

Emn(x, y) where m and n are mode indicies and Emn is a transverse field. The z- component of the waveguide modes and their z-dependence are irrelevant for now. These waveguide modes automatically satisfy the boundary conditions and form a complete basis for the space of all such fields. For hollow waveguides of uniform cross section, the modes come in two flavors: one in which the z-component of the E-field vanishes called Transverse-Electric (TE) modes, and another where the z- component of the H-field vanishes called Transverse-Magnetic (TM) modes. See Marcuvitz [56, Chapter 2] for the mode functions of the guides we use. Following Jackson [38], we can write the decomposition of any transverse field in the guide over a single slice of constant z as

X (+) (−) Et = Cmn + Cmn Emn , mn (4.5) X (+) (−) Ht = Cmn − Cmn Hmn mn

(±) where the Cmn ’s are complex coefficients for waves traveling down the guide in the +z- (+) and −z-directions (−), and the Hmn are magnetic field waveguide modes. For fields that do not satisfy the boundary conditions such as the aperture

field Etwg (xwg) that we are interested in decomposing, the equalities are only approximate. Since the waveguide modes also come along with z-components and dependencies, specifying these coefficients would fully specify the field everywhere

69 within the waveguide. Using the orthonormality of the modal ﬁelds

Z Emn · Em0n0 da = δmm0nn0 , (4.6) Z 1 0 0 0 0 Hmn · Hm n da = 2 δmm nn Zmn where the integration is taken over the waveguide cross section and Zmn is the mode impedance, we can rewrite Equation (4.5) to be a solution for the modal coefficients given the transverse fields Et and Ht at a particular z-slice. Thus these four functions (two transverse components for both fields) across a cross section specify the fields in the entire waveguide. We use this formalism, making some significant approximations to simplify our calculations. The first, much like the diffraction approximation made earlier,

is to use the sourced, transverse diffracted plane wave E-field Etwg (xwg) as the transverse field in Equation (4.5). Again, this ignores edge effects caused by the start of the waveguide. Next, we will ignore another consequence of this edge discontinuity in the waveguide by assuming that the sourced field can only lead to forward-going waves down the guide. Remembering the axes system setup in

(+) Section 4.2, this leads to Cmn = 0. From now on, we then drop the designator, such

(−) that Cmn ≡ Cmn . Finally, we truncate the inﬁnite sum to only include propagating modes since evanescent ones will not travel far enough to reach the bolometer at the bottom of the guide. With these approximations we write Equation (4.5) as

X Etwg (xwg) = CmnEmn . prop modes

Now taking the dot product of both sides with the mode function Em0n0 , integrating

70 over the aperture, and invoking Equation (4.6) gives

Cmn = Emn · Etwg (xwg)da . (4.7)

These coeﬃcients are what we take to be the full speciﬁcation of the forward going waves down the guide.

4.4 Propagation & Termination

Since we have yet to apply the boundary conditions at the guide short, we will see that the sourced mode coefficients of Equation (4.7) actually fully specify the electric field within the waveguide. As stated in Silver [78, Chapter 7], because of the uniformity of the waveguide cross-section along its length, the z-dependence of the modal fields are given by e∓γmnz, where the (−) is for waves traveling in the +z-direction and the (+) is for those traveling in the −z-direction. The γmn above is the propagation constant for the mode and, for a non-dissipative guide, is given by

p 2 2 γmn = κmn − k

where κmn is the cutoﬀ wavenumber. See Marcuvitz [56, Chapter 2] for equations for the cutoﬀ wavelengths of modes in the guides we use. Notice that for k > κmn,

γmn is imaginary and the wave propagates; however, for k < κmn, γmn is real and we have an evanescent wave (thus the name cutoﬀ). We can now write down the full spatial dependence of the sourced modal ﬁelds as

γmnz Emni (x, y, z) = Emn(x, y)e (4.8)

71 where the “i” subscript denotes the incident field. To include the effects of the backshort, which we simulate as a guide short (Figure 4.1), we must add to the above, the wave reflected by the backshort. This wave will propagate in the opposite direction and have a different amplitude Amn and phase shift δ,

−γmnz+δ Emnr (x, y, z) = AmnEmn(x, y)e (4.9) where the “r” subscript stands for reﬂected. Putting Equations (4.8) and (4.9) together and applying the boundary conditions for a short at z = −zb gives the total transverse ﬁeld for each mode,

γmnz −γmn(z+2zb) Emn(x, y, z) = Emn(x, y) e − e (4.10) −γmnzb = 2Emn(x, y)e sinh[γmn(z + zb)] .

Finally, we carry out a coherent sum of the various propagating modes with the coefficients specified by Equation (4.7), resulting in the complete transverse electric field within the waveguide as promised,

X −γmnzb Et(x, y, z) = 2 CmnEmn(x, y)e sinh[γmn(z + zb)] . (4.11) prop modes

Equation (4.11) provides a prescription for propagating the diffracted plane wave aperture illumination down to the bolometer. These fields induce oscillating currents in the resistive absorber bars patterned on the PFSB membrane. On a uniform sheet absorber, these currents have an amplitude proportional to the transverse electric field and point in the same direction. However, the PFSB geometry restricts the current density and only allows the oscillations aligned with the bars. Through the resistive losses in the bars this electrical power is

72 converted into thermal energy which is read out by the TES. We simulate this situation by ﬁrst aligning the bars with the y-axis as stated in Section 4.2, then integrating the square of the y-component of the transverse electric ﬁeld over the area of the absorber bars

Z 2 P (Ω, ψ) ∝ [ˆy · Et(x, y, z0;Ω, ψ)] da (4.12)

absorber bars

where z0 is the position along the z-axis of the PFSB. We bring back the explicit dependence in the source conﬁguration variables (Ω, ψ) in order to address the power pattern P (Ω, ψ) on the left hand side. We are not concerned here with the constant of proportionality since we only care about the relative levels in the radiation patterns. This power pattern is what we spoke of in Section 4.1 that will allow us to back out the three characterization maps C(Ω), X(Ω), and ∆(Ω) of Equation (3.10).

4.5 Circular Waveguide Results

To give us the opportunity for discussing the speciﬁc details of how we carried out the simulation just described, we present here the results for PFSBs mounted in a circular waveguide. Figures 4.2–4.4 shows the three characterization maps (C,X, ∆) for our simulation of the circular guide system whose measured radiation pattern data we will be presenting in Chapter 8 (see Section 7.2.2 for a description of this “large aperture system”). The dimensions of the elements in our simulations match those of the systems we are trying to simulate. For this simulation see Tables 7.2 and 7.4. These plots were created from the simulation power pattern P (Ω, ψ) of Equa-

73 Figure 4.2. Co-polar radiation pattern C(Ω) simulation of the large aperture system where the PFSB is mounted in a circular waveguide. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right. We choose coordinates that are a projection of the hemisphere onto a flat surface with the extents of the plot at θ = 20◦. tion (4.12). We sample this pattern by evaluating the right hand side, for a discrete set of plane wave configurations. All the surface integrations needed are carried out on a grid with a spacing finer than the smallest structures in the integrand. These structures are limited by the fact that we carry out the simulation at a single frequency. The plane wave configurations sample the zenith angle θ with 7 points between 0.7◦ and 20◦. The azimuthal φ sampling is chosen to approximate equal area gridding on a spherical surface centered on the waveguide aperture. One constraint here is that we force points along the E- and H-planes so that plane-cut plots can be made. For a given antenna, the E-plane is the plane de-

74 Figure 4.3. Cross-polar radiation pattern X(Ω) simulation of the large aperture system where the PFSB is mounted in a circular waveguide. See Figure 4.2 for a description of the plot.

fined by the direction of the E-field along the antenna and the on-axis direction, while the H-plane is the one orthogonal to this with respect to the on-axis direction. In our PFSB simulation the E-plane is the yz-plane while the H-plane is the xz-plane. For each of the directions Ω just described (outlined in Figures 4.2–4.4 by the black diamonds), we choose three polarization orientations ψ = 0, 45, 90◦. Note that it is an arbitrary decision how to map the ψ = 0 directions over the sphere. Although many definitions exist in the literature we use Ludwig’s “definition 3” [55] for all our radiation patterns, which is the standard in our field. The results of the power pattern evaluations for the three polarizations ψ of a given direction Ω are used to calculate the three parameters (C,X, ∆). The values of these pa-

75 Figure 4.4. Rotation angle radiation pattern ∆(Ω) simulation of the large aperture system where the PFSB is mounted in a circular waveguide. See Figure 4.2 for a description of the plot. rameters are shown as the color in Figures 4.2–4.4 where the levels in the area between the sampled points are quadratic interpolates. We will withhold a detailed discussion of these results and the presentation of the square waveguide PFSB simulation till Chapter 8, where we can present both the measurements and simulations side-by-side. For now, let us comment on the alarming fact that the simulations show large oﬀ-axis cross-polar lobes centered on the E-plane. In some places in the beam there is a complete loss of polarization discrimination. We will provide an explanation for this phenomenon in Chapter 8 when we compare the circular and square waveguide architectures. Before doing so we must describe the instruments used to measure the patterns so that we can verify these results.

76 Chapter 5

PFSB TEST CRYOSTAT

We now describe the test cryostat facility (shown in Figure 5.1) which houses the detectors, their readout, and part of the optical chain used in our measurements. We begin in Section 5.1 by describing the cryogenic system that cools most of these elements, then describe the readout system in Section 5.2. A description of the various optical systems tested in this cryostat is postponed till Chapter 7, just before presenting the results of our measurements in Chapter 8.

5.1 Cryogenics

The PFSB detectors, readout, and sub-mK refrigerator are all based oﬀ a 4 K cold plate which is passively cooled by a liquid helium (LHe) reservoir as shown in Figure 5.1. Once the air inside of the cryostat is pumped out, the dominant heat load becomes radiative. We buﬀer the 300 K radiation from the vacuum jacket with an intermediate 77 K stage passively cooled by liquid nitrogen (LN2). The

77 K stage consists of a tank which holds the LN2, a 77 K cold plate that allows mounting of devices at this temperature, and a radiation shield to complete the enclosure. There is a radiation shield over the 4 K cold plate to block the 77 K radiation from the sub-mK stages described below. For added radiative conduction stopping power, the outsides of the 4 K and 77 K stages are wrapped with many layers of super-insulation. These are very

77 Figure 5.1. PFSB test cryostat. The detector is mounted on the red colored piece in the middle of the PFSB stack at about the same height as the ultracold table. It is cooled to 300 mK by the active sorption pump refrigerator strapped to the ultra- and inter-cold tables as shown. We read out the detectors with SQUID amplifiers housed in the box mounted to the outside of the 4 K radiation shield. All of this is based off a 4 K cold plate, passively cooled by LHe. This stage is shielded from room temperature radiation by a 77 K, LN2 cooled enclosure. In operation the cryostat is oriented upside down from what is shown here because the cryogen fill tubes exit the vacuum space at the bottom of the figure.

78 thin reﬂective sheets of aluminized mylar that are crinkled to prevent good thermal contact between sheets. See Table 5.1 for a summary of the cryogenic storage characteristics of this dewar, manufactured by Precision Cryogenic Systems, Inc. [68].

Table 5.1

Cryogenic Storage Characteristics

Characteristic Value

LHe storage volume 20 L

LN2 storage volume 40 L LHe hold timea 2.7 days

a LN2 hold time 1.5 days

a Hold times are quoted for the square guide system described in Section 7.2.3.

The various stages are all structurally mounted off each other by the G10 cylinders shown in the figure. The re-entrant design increases the path length that heat must traverse by conduction from one stage to the next. The fill tubes to the cryogen storage tanks are made out of stainless steel with a bellows section to allow for flexibility and added thermal path. The purpose of the concentric design of the storage tanks are to increase the 4 K shielded work space, and to allow for a long, cold, optical system which can

79 exit the cryostat on either end. As seen from Figure 5.1, we do not make use of this space. Our optical systems are relatively small and look out through the end of the dewar opposite the fill tubes. This is accomplished by cutting windows in both of the radiation shields and the vacuum jacket allowing a path for radiation all the way through to the 4 K work area. On these windows we mount various filters that will be described in Section 7.2. The sub-mK temperature stage on which the detectors and some of the optical components sit, is cooled by an active 3He/4He sorption refrigerator manufactured by Chase Research [12]. This is a closed cycle unit which utilizes cooled charcoal sorption pumps to lower the vapor pressure above LHe baths. In the final state of the cycle, the refrigerator contains two pumped baths of liquid 3He, one which can reach 230 mK, and an intermediate stage attaining 350 mK in order to buffer the heat load from 4 K. Bhatia et al., [6] and [7], give an extensive description of these types of refrigerators where the latter describes a refrigerator with similar helium charge and bath sizes to ours. We use a cycling procedure resembling the one in Bhatia et al. [6]. Since the heads or baths of the refrigerator are not mechanically stiff, we thermally attach them to copper tables with copper straps. These are structurally mounted off the 4 K cold plate with low thermal conductivity Vespel legs, as shown in Figure 5.1. The ultracold table shown in the figure is so named because it is strapped to the head providing the coldest temperature (the “ultrahead”) in the entire system. The intercold table is strapped to the other cold head (the “interhead”) of the refrigerator. We then proceed to mount the devices for testing and the some of the optics to the ultracold table is as shown in Figure5.1. The wires that lead up to the this table are also tied down to the intercold table in

80 order to reduce the heat load on the ultracold table. In Chapter 7 we will be describing the various optical systems that were mounted to the ultracold table. Through radiative loading, these systems deliver thermal power to the refrigerator heads at diﬀerent rates resulting in diﬀerent cooling abilities. Although it takes about three hours to cycle the refrigerator in all cases, the hold times and base temperatures that we achieve show wide variation. In the systems that do not have a large absorptive ultracold surface area (the on-axis and small aperture systems of Sections 7.2.1 and 7.2.2 respectively) we are able to reach an average base temperature of 265 mK, holding for 22 hours. However, for the other systems (the large aperture and square guide systems of Sections 7.2.2 and 7.2.3 respectively), due to the extra loading we only reach an average ultracold temperature of 320 mK, holding for 6 hours.

5.2 SQUID Readout

As described in Section 2.2.1, we bias the PFSBs with a constant voltage and read out the TES current to measure its resistance. A schematic of our entire readout chain is shown in Figure 5.2. See Table 5.2 for the resistance and inductance values of the various components shown in the figure. This system is modified from the one used on the South Pole Telescope (SPT) SZ receiver [71]. It uses a constant amplitude AC voltage-bias, sourced by the oscillator shown in the figure, to modulate the signal above the 1/f knee of the SQUIDs as well as to provide a frequency-domain multiplexing scheme [48] for reading out the SPT bolometers. We do not make use of the multiplexing capabilities of this SQUID system. Instead we simply choose to bias our bolometer at 20 kHz, and use a separate SQUID to read out each of our two bolometers.

81 Figure 5.2. SQUID readout for the PFSB test system. The TES is denoted by the variable resistor labeled R with current I. The various components of the system are sectioned off to show where they reside in the test cryostat. All of the warm electronics are outside of the vacuum jacket and the electronics on the 4 K stage are mounted on a G10 card in the SQUID readout box (see Figure 5.1). The TES AC voltage-bias comes in from a function generator outside the cryostat as shown in the upper-right corner. We read the TES current by coupling to SQUID amplifiers via the input inductors on the bottom-left. The SQUIDs are biased with a constant current through lines that are not shown here. To extract the signal from the SQUID output we must demodulate using a lock-in amplifier. A point of entry for sending currents directly to the SQUID input inductors, which we use to measure the system gain, is given by line labeled “Carrier Nulling/Flux Bias Input”. Modified from Dobbs [16].

82 Table 5.2

SQUID Readout Parameters

Parameter Value

normal state TES resistance R 200 mΩ

Lw 520 nH

Lin 80 pH

Rs 10 mΩ

Rb 200 Ω

Rf 3.3 kΩ

Rcn 3.3 kΩ

This system sets up an approximate voltage-bias across the TES by sending an approximate current-bias, created by a large bias resistor Rb, into the parallel pair made up of the very small shunt resistor Rs and the TES resistance R. For the voltage-biasing to work well, we need Rb Rs and Rs R. Included in the

figure is an unavoidable wire inductance Lw and the input inductance Lin needed to supply the SQUID amplifiers described below with a magnetic flux signal. To not spoil the voltage-biasing, we need the sum of these, L ≡ Lw + Lin, to obey

2πfbL R where fb is the bias frequency. In the SPT system, this constraint does not exist and more inductance is actually added because an impedance nulling capacitor is used in series to create a resonant circuit. On resonance, this leg behaves as if it only contained the TES resistance. This resonance is important

83 for the multiplexing scheme mentioned above where many of these TES legs with different resonances are strung together in parallel. Even though we do not use the multiplexing capabilities of the system we are prevented from using a DC bias because the bias lines going through the SQUID controller electronics are isolated with transformers. We measure the current flowing through the TES with SQUID amplifiers. The DC SQUIDs we use measure magnetic fields by sensing the quantum interference between super-currents tunneling through two parallel Josephson junctions. When current-biased, they output an oscillating voltage as a function of the applied flux. Because of this non-linearity, SQUIDs are normally operated in a very strong feedback regime. See Clarke [13] for a more detailed overview of DC SQUIDs. We use them as current sensors by coupling the SQUID to a small input inductor which turns the current into magnetic flux. As shown in Figure 5.2, our system uses a SQUID array which provides extra amplification of the desired current [36]. These SQUIDs are fabricated by the National Institute of Standards and

Technology (NIST) Quantum Sensors Group. The resistor Rf provides the SQUID feedback by converting the output voltage into a current which is sent back to the SQUIDs via the input inductors. This system provides other points of access for sourcing current to the input inductors. We show these inputs schematically as a single one in Figure 5.2, through the voltage → current conversion resistor Rcn. One of the purposes for these inputs is to provide a carrier nulling signal which subtracts the carrier out of the currents that are sent to the SQUIDs. This provides room for the SQUID dynamic range to be used for the TES modulations rather than the carrier. However, in all the data presented in chapter 8, because of a large excess in the signal-to-noise

84 ratio for our measurements, the carrier nulling was not necessary. We do make use of this access to the SQUID input inductors for the purposes of calibrating the SQUID outputs. This is accomplished by sourcing a known current and recording the output. Finally, we use a lock-in ampliﬁer, as shown in Figure 5.2, to lock into the carrier wave extracting the TES signal. This summarizes the readout chain for a single bolometer in our test cryostat. We have four such channels but only use them one at a time for all our measurements. See Chapter 7 for a description of the optical systems housed within the test facility just described.

85 Chapter 6

CHARACTERIZATION TOOLS

We now move to a description of the tools used to perform the characterization procedure, outlined in Chapter 3, on the PFSB detector architecture. As mentioned at the end of that chapter, we will not be reporting on the full spectral and beam dependence of the characterization functions C(Ω, ν), X(Ω, ν), and ∆(Ω, ν). Instead, Section 6.1 will describe the method used for exploring the spectral dependence of these functions, at a few places in the beam, as described by Equation (3.9). The description of the radiation pattern measurements, presented in Equation (3.10), will be carried out in Section 6.2. Both methods described in this chapter can be viewed as placing a conﬁgurable source in the beam of the test receiver.

6.1 Spectra

We will split the discussion of the spectra measurements into two parts. Sec- tion 6.1.1 will introduce the instrument which serves as our spectral source, while a description the measurement setup and data processing chains used is given in Section 6.1.2.

86 Figure 6.1. The Michelson interferometer. Light enters the instrument from the input port denoted by the source on the left, and divides into two arms upon encountering the beam splitter. The end of each arm contains a mirror to reﬂect the light back towards the splitter which then recombines, illuminating the detector at output port on the bottom. In a spectrometer, the movable mirror moves at a constant speed, scaling optical frequencies to much lower-frequency time variations of the intensity on the detector. See the text for a full description of this phenomenon.

6.1.1 Fourier Transform Spectrometer

In mm-wave optics, we normally take spectra using a device called the Fourier Transform Spectrometer (FTS). Our FTS is a polarized Martin-Puplett interferometer. However, let us ﬁrst restrict our discussion to the simpler Michelson spectrometer, shown in Figure 6.1, because its operation is virtually identical to the Martin-Puplett. First, consider monochromatic light of frequency ν sourced into the interferometer. When the two arms have equal optical path lengths (∆L = 0 where ∆L is the optical path length diﬀerence between the two arms), the beams from the

87 two arms constructively interfere giving the maximum signal that can be seen by the detector at the output port. Varying the optical path length of one arm by a half wavelength, ∆L = λ/2, will result in a minimum signal since the beams will now destructively interfere. This modulation behavior continues over integer wavelength intervals of ∆L. We write down the behavior of the detector signal as a function of optical path length diﬀerence

2π∆L s ∝ 1 + cos . (6.1) λ

Now if we were to move one of the mirrors with a velocity v our optical path diﬀerence would vary as ∆L = 2vt and Equation (6.1) would become

2v s(t) ∝ 1 + cos 2π ν t (6.2) c where c is the speed of light. Thus the Michelson spectrometer performs a frequency scaling, converting an optical frequency ν to low frequency variations in intensity, which can be read out by a power detector. The signal from this detector modulates at 2v ν ≡ ν . (6.3) det c

Now consider a ﬂat broadband source at the input port of the Michelson (we normally use Rayleigh-Jeans thermal sources which instead has a ∝ ν2 spectrum). At the output port, we place in front of the detector a test piece whose power transmission coeﬃcient spectra τ(ν) we would like to measure. Assuming the detector simply measures the total power incident on it, using Equation (6.2) we

88 obtain a signal

Z 2v s(t) ∝ τ(ν) 1 + cos 2π ν t dν c Z c ∝ τ ν cos(2πν t)dν + constant . (6.4) 2v det det det

We see from Equation (6.4) that the detector timestream on the output of the FTS is simply given (up to an irrelevant multiplicative constant and dc offset) by the cosine transform of the spectrum τ(ν) we are interested in measuring, frequency scaled by the factor given in Equation (6.3). We get the cosine transform here because all the different frequencies travelling through the system result in maximum intensity at the equal optical path length configuration of the Michelson. The timestream thus has an interference fringe-like quality which we normally refer to as an interferogram (see Figure 6.4 for example timestreams). The point in the timestream of maximum signal is called the white light fringe. In order to retrieve the desired spectrum, we simply cosine transform the timestream and correct for the frequency scaling Equation (6.3). One of the difficulties in using a Michelson as a spectrometer is the require- ment that the beam splitter have relatively flat and broadband transmission and reflection properties. Instead of developing such a beam splitter in the millimeter region of the spectrum, we circumvent this problem by replacing the beam splitter with a polarizer. In order to get the correct behavior, the Michelson must be modified in other ways which are shown in Figure 6.2. This configuration is called the Martin-Puplett interferometer. See Lesurf [52] and Martin [57] for excellent descriptions of this type of FTS. Here the light enters into the interferometer from two different input ports that are split using a polarizer. One delivers horizontally

89 Figure 6.2. The Martin-Puplett interferometer configured as an FTS. The light paths are shown as the repetitive symbols representing the various polarization states that make their way around the instrument. ◦ represents vertically polarized light; | stands for horizontal; stands for polarization aligned with the 45◦ polarizer and gives the orthogonal polarization; × represents some mixture of both polarizations; gives the specific mixture of unpolarized light. The arrows show the direction of propagation. See the text for an explanation of how the Martin-Puplett produces spectra. polarized light and the other vertically polarized as shown in the figure. Splitting of the input light occurs with a polarizer oriented 45◦ to the horizontal. This splits both the input beams (carried on orthogonal polarization states) down the two arms of the interferometer. The most significant modification comes from the

90 fact that the light that leaves the splitter polarizer by either reflecting or trans- mitting must do the opposite on the way back. This is so that the interference beams can exit away from the input port in a direction on the opposite side of the splitter polarizer. Using roof mirrors instead of flat mirrors accomplish this task by exchanging between the two 45◦ polarization states upon exit from the second reflection. See Lesurf [52] for a detailed description of the roof mirrors. The resultant recombined beams exiting from the 45◦ polarizer contain a mixture of the two orthogonal “45◦” polarizations. This mixture holds the interferogram information as a relative phase offset between the two states. Accessing these interferograms requires examining either the vertical or horizontal polarization states, since looking at one of the 45◦ states only inspects light that is sourced from a single arm of the interferometer. To accomplish this task, we split the mixture using a final polarizer oriented along one of the vertical-horizontal axes, which creates two output ports (signified by the detectors in Figure 6.2). In the equal-arm length configuration of the interferometer, the vertical polarization output port is only sourced by the cold source in the figure while the other is sourced by the hot source. Thus the white light fringe is a dip in power in one output port and a peak in the other (see Figure 6.4). For configurations other than the white light one, the two ports contain a mixture of the light from the two sources that works out to be a comparison between the their intensities. When we use the FTS to characterize the spectra of a mm-wave receiver, we do not decouple the band defining elements from the detector system. For the PFSB architecture, this decoupling is impossible since band definition occurs on the detector. Instead of placing a test piece somewhere in the beam of the FTS, we simply place our receiver on one of the output ports and ignore the light coming

91 out of the other one. In this picture we can view the FTS as a special source of ﬁnite angular size that we place in the beam of the receiver we are trying to characterize. The angular size is set by a focusing optic placed on the receiver side of the exit polarizer in our FTS. There is another optic between the input polarizer and the splitter polarizer that provides the FTS with collimated light from the two sources. These optics have a focal ratio of f/4 and thus our FTS

1 ◦ source has an angular size characterized by a cone with a 7 8 half angle. All FTS spectra that we take are therefore beam integrated over the area of this cone. It is for this reason that Equation (3.9) contains only beam averaged spectra. We mention only being able to take measurements at a few places within the beam because of the restrictive size of the FTS (40 × 30 × 10). Building a structure that moves the FTS around the beam of a test receiver at a constant distance would be a difficult undertaking. Obtaining the three characterization spectra at any one place in the beam would require us to rotate the exit polarization of the FTS. This can be accomplished by rotating the entire instrument with respect to the receiver being tested. However, again taking into account the large size of our FTS, as well as the difficulty of maintaining constant beam alignment throughout this rotation (especially for off-axis orientations—see Figure 6.3), we decided to use the exit polarizer for this task. Since our FTS is contained within a box with walls coated by Eccosorb AN-72 [17] to absorb stray light, one minor modification carried out to facilitate this was to move the exit polarizer to a place outside the FTS. This puts it between the focusing optic and the instrument we wish to characterize. In this configuration, we can easily switch between the two output ports by rotating the exit polarizer by 90◦. We are then essentially rotating the polarization orientation

92 of the FTS “source” without having to rotate the FTS. This gives us easy access to comparable co- and x-polar spectra, C(Ω¯, ν) and X(Ω¯, ν) respectively. One might argue that since we are switching between two different output ports our interferograms will be characteristically different (one with a white light fringe of maximum power and the other with minimum power); however, this is only a phase difference and the power spectra will have equivalent transfer functions. Recalling that we lose the interference between the two arms when viewing an exit polarization 45◦ to one of the output ports, we see that this rotation of the FTS polarization is restricted to only these two angles. This is the reason why we are not able to access the rotation angle ∆(Ω¯, ν); a small price to pay for the confidence of constant beam alignment.

6.1.2 Measurement Setup & Analysis

We now describe the setup and data processing details pertaining to our spectra measurements. As just mentioned, we place our cryostat at one of the output ports of the FTS shown in Figure 6.2, maintaining the ability to switch output ports by a 90◦ rotation of the final polarizer. Figure 6.3 depicts the PFSB test cryostat in this configuration for an off-axis measurement. The part of the instrument stack shown here that is relevant to the signal coming off the PFSBs, is the lock-in amplifier terminating the SQUID readout electronics shown in Figure 5.2. Its reference signal comes from the oscillator that provides the bias signal. As mentioned in Section 5.2, all measurements are made with a 20 kHz TES bias. On the final low-pass filter of the lock-in, we use a time constant of 10 ms with a

24 dB/octave roll-oﬀ. We acquire the signal from this lock-in directly which, in this case, is the raw

93 Figure 6.3. The PFSB test cryostat sitting on the FTS for oﬀ-axis spectra measurements. The output opening of the FTS enclosure is just below the exit polarizer. The beam exiting the FTS through this opening is coming to a focus 1000 above the top plate of the FTS enclosure. We hold the cryostat such that the PFSB aperture is approximately at this point with the wooden blocks shown. As described in Section 6.1.1, we have moved the exit polarizer to the position shown for easy rotation enabling output port switching. The cold source input port is oﬀ the bottom of the picture on the right hand side where we have an emissive LN2 cooled piece of Eccosorb HR-10. interferogram as described in Section 6.1.1. The interferograms corresponding to the two output ports of the FTS are shown in Figure 6.4. This is on-axis data

94 Figure 6.4. Interferograms from the two output ports of the FTS. These are taken with the low-G¯ PFSB in the on-axis system described in Section 7.2.1. The top interferogram is from the output port which has its polarization aligned with the PFSB leading to the co-polar spectra, and the bottom is the orthogonally polarized output port which gives the cross-polar spectra. The units for the PFSB output are arbitrary, but identical, so it is safe to compare the levels from the two. The full interferograms are much longer than shown (about 850 s) as indicated by the time axis. for the low-G¯ PFSB in the on-axis system that will be described in Section 7.2.1. The PFSB absorption orientation is approximately aligned with one of the output ports so these are co- and cross-polar interferograms. While we expect diﬀer- ences, note that as discussed in Section 6.1.1, the white light fringes go in the opposite direction for the two output ports. Once we have these interferograms, we calculate the Power Spectral Density (PSD) as shown in Figure 6.5. Because of the excess in signal-to-noise for our spectra measurements, we use this simple approach that is less sensitive compared to the cosine transform as outlined in

95 Figure 6.5. Co-polar spectra indicated by a red solid line, and cross-polar spectra shown by a green dashed line, as the PSDs of the interferograms in Figure 6.4. The spike at 170 GHz in the cross-polar spectrum is a cryogenic artifact.

Section 6.1.1. The frequency axis has already been scaled by the FTS frequency scaling relation, Equation (6.3). These spectra still have the source response folded into them. We divide out a ν2 spectra because the entirety of the band lies on the Rayleigh-Jeans side of the blackbody curves for the two source temperatures of 300 and 77 K. The resultant spectra are shown in Figure 6.6. It is these source response removed spectra that we will present as the absorption spectra of the various PFSB test systems in Section 8.1. We will be presenting spectra measurements for both on- and oﬀ-axis positions in the beam of the PFSBs. The on-axis conﬁguration of the cryostat has the PFSBs looking straight into the FTS output ports mounted in much the same manner as shown in Figure 6.3, but with an equivalent height of wooden spacer

96 Figure 6.6. PFSB co-polar spectra shown in red, and cross-polar spectra in dashed green. We have removed the ν2 source response from Figure 6.5. Again, the spike at 170 GHz in the cross-polar spectra is a cryogenic artifact. The noise walls on the left-hand side of the plot are not optical, but are due to the 1/f noise in the readout being amplified by the source removal. blocks on each side. For the off-axis spectra, we select two different off-axis points in the beam from which to take measurements. The easiest way is to choose points in either the E-plane or the H-plane. To get the cryostat to look at the FTS output ports off-axis, we tilt it along one of these planes in the manner shown in Figure 6.3. Depending on the specific measurement, our two off-axis spectra points are E- and H-plane 11 or 12◦ points.

97 6.2 Beam Maps

As was just done for the spectra measurements, we will split the discussion of the beam measurements into two parts. Section 6.2.1 will introduce the instrument which serves as our beam mapping source, while a description the measurement setup and data processing chains used is given in Section 6.2.2.

6.2.1 Beam Mapper

We explore the directional dependence of the three characterization functions C(Ω, ν), X(Ω, ν), and ∆(Ω, ν) by using a 2-dimensional beam mapper shown in Figure 6.7. As mentioned at the end of Chapter 3, this instrument can only measure the spectra integrated radiation patterns given in Equation (3.10). The beam mapper is a two axis system which allows us to vary the angular position of a source placed within the beam of an antenna feed system, while keeping the source at a constant distance and pointed directly at the feed. As shown in the figure, arm 1 is attached to the structure holding the feed through a rotational bearing whose axis goes through the center of the feed entrance aperture. Arm 2 is mounted to arm 1 through another bearing whose axis also goes through the center of the entrance aperture but in an orthogonal direction. We place the source on arm 2. Its position is varied by torquing the two arms with two linear drivers. Arm 1’s driver is mounted to the structure which holds the feed while the driver for arm 2 is mounted to arm 1. Encoders are used to read out the position to an acquisition system. Figure 6.8 shows a schematic of the source that we place on the beam mapper. The radiation originates from a hot filament which is placed at the focus of a parabolic reflector that collimates the light. We place a chopper directly in

98 Figure 6.7. The 2-dimensional beam mapper is an instrument used to map out the radiation pattern of an antenna system. The green downward facing horn in the ﬁgure is where the antenna feed would go. It is placed at the center of a sphere swept out by the double axis arm system shown where the axes of rotation are indicated by the blue bearing rods. A set of these rods mount one arm to the other, while another set mounts the entire system to the feed. The second arm holds a source box shown in red which illuminates the feed with conditioned radiation from a variable direction. front of this source to modulate the signal. Before the source radiation leaves the enclosure, we set its polarization state by passing it through a rotatable polarizer. This polarizer is composed of thin conductive metal lines patterned onto a thin

99 Figure 6.8. A schematic of the source used on the beam mapper. The radiation emanates from a hot IR source at the focus of a parabolic reflector (red). It is chopped with a 300 K load shown in black to modulate the signal. A rotatable polarizer (green) polarizes the light before it exits the enclosure (black). Finally, a baffle (black) is used for extra collimation. Heat is prevented from hurting the fragile membrane polarizer by sectioning off the box with a Kapton film window shown in orange. We attach Eccosorb HR-10 to most of the walls to kill reflections. membrane and is very fragile. To protect it from the heat produced by the source we block the convective transfer by separating the source box into two compart- ments using a thin Kapton film window. Finally, we further collimate the light at the exit of the enclosure using a cylindrical baffle made out of aluminum coated with Eccosorb HR-10 [19]. In addition to this stray light baffle, we also coat most of the surfaces inside the aluminum enclosure box with Eccosorb HR-10. See Ap- pendix A for a measurement where we verify that unpolarized light emanates from

100 the hot filament-Kapton film combination. Once the source box is mounted to the secondary arm of the beam mapper we balance both arms by releasing the linear drivers, allowing them to swing freely. Counterweights are attached in various places to minimize the torque on the arms in any configuration. This is the purpose of the rod extenders coming out of the secondary arm as shown in figure 6.7. When running the beam mapper, we coat all surfaces “seen” by both the antenna feed and the source box with Eccosorb HR-10, except the direct path between the two. This reduces the sensitivity of the test receiver to radiation from reflections and other unwanted sources.

6.2.2 Measurement Setup & Analysis

For our PFSB beam measurements we place the cryostat on the optics table of our beam mapper which holds the PFSB waveguide aperture at the position of the antenna feed shown in Figure 6.7. Figure 6.9 shows a photograph of this setup. In all of our beam maps, the parabolic source reﬂector aperture is 25 mm in diameter. It is kept 840 mm from the PFSB aperture (no larger than 10 mm in diameter)

2 2 which puts it in the far field since D /λ = (10 mm) /1.37 mm = 73 mm 840 mm. Here we have used the 220 GHz PFSB band center for the propagation frequency. In characterizing the radiation pattern emanating from the PFSB architecture, we need to chart out the polarization efficiency curve, Equation (3.8), at various points in the beam to fit out the three characterization parameters (C,X, ∆) and map out their dependence on beam position. Here, we take a step, sit, and chop approach to operating the beam mapper. Step means to change the configuration of the beam mapper through its three degrees of freedom—two arm rotations alter the angular position of the source, and the source box polarizer rotation sets the

101 Figure 6.9. The PFSB test cryostat sitting on the beam mapper. It is supported by the optics table which hangs off of the large beam going across the top of the figure. Arm 1 is mounted to the optics table through the shaft going through encoder 1. Arm 2 is mounted to arm 1 as shown. Arm 1 rotates from the torque applied to a bolt riding a screw which is turned by the motor shown. The motor which rotates arm 2 is hung off arm 1 to the right of the figure and works by the same mechanism. As described in Section 6.2.1 we attach a conditioned source pointed at the cryostat to arm 2. The entire system is balanced using counterweights to minimize the required torque. We stack the readout electronics on the top of the beam mapper for proximity to minimize radio-frequency pickup. source polarization. Sit and chop refer to the type of data we take at each point of the beam mapper configuration space. This data simply consists of the PFSB looking out at a small stationary polarized load which is chopped between 300

102 and 900 K. The rest of the beam sees unpolarized 300 K radiation. We use this discretized approach to simplify our analysis so that the bolometer response time constants are irrelevant. This chopping method also allows us to identify a zero in beam response as a null modulation of the signal. All of our beam maps will contain the same configuration space points. The directional (Ω) sampling is identical to the one used in our simulations described in Section 4.5. For charting out the polarization efficiency curve, however, we use 6 polarizer angles ψ between 0 and 150◦ with respect to the Ludwig definition 3 co-polar direction. We do this identically for each Ω point. This time we need 6 samples instead of three because our measurements are not in the ultra-high signal- to-noise ratio regime that exists for our simulations. Thus to obtain (C,X, ∆), we fit Equation (3.8) to these 6 points for each direction Ω. The PFSB readout and acquisition is done in exactly the same manner as for the FTS measurements described in Section 6.1.2. In addition to this signal, we send a reference/chopper sensing signal into the data acquisition system for the purposes of software demodulation. This decision to demodulate in software rather than hardware was driven by the slow 4 Hz chopping speed that we use. In a hardware lock-in, we would have to use a long time constant with a correspond- ingly long wait for the measured value to settle after each step in our beam map measurements. The software demodulation procedure is outlined in Figures 6.10–6.11. First we remove the means of the PFSB output as well as the reference signal. Then a pre-processing is carried out on the reference signal as shown in Figure 6.10. This is done to create a reference which mimics the shape of the PFSB output, increasing the signal to noise on the lock. We accomplish this by first shifting the reference to

103 Figure 6.10. Reference conditioning for PFSB signal lock-in. The top plot shows a sample PFSB output from the lock-in of Figure 5.2. Just beneath, is the corresponding chopper sensing signal. Both have had their means removed. We first shift the chopper signal as shown in the third plot, then apply a low-pass filter with a time constant near to the value of the thermal bolometer response time mentioned in Section 2.2. The resultant timestream is an attempt to closely mimic the shape of the PFSB output on the top. We then normalize this signal to have a peak-to-peak amplitude of unity so that we are demodulating (Figure 6.11) with a standard reference. The final conditioned reference signal is showed on the bottom plot.

104 account for a difference in chopper sensing position verses source aperture blocking position. As shown in the third plot of Figure 6.10, the resultant timestream exhibits ringing because this shift is carried out in the frequency domain. In addition to the shift, we apply a low-pass filter, again in the frequency domain, to the shifted reference. This filter has a time constant commensurate with the bolometer reaction time discussed in Section 2.2. Finally, we standardize this reference by normalizing it to have a peak-to-peak amplitude of unity.

Figure 6.11. Demodulation of PFSB signals. We obtain this timestream by multiplying the PFSB output and the conditioned reference signals (top and bottom timestreams of Figure 6.10), as shown in the black curve which is periodic at twice the frequencies of the input signals. The demodulation occurs when we basically ﬁlter this timestream taking its mean which is the level shown in red.

105 Figure 6.12. Co-polar beam map of the large aperture PFSB test system to be described in Section 7.2.2. Shown with diamonds are the places in the beam where the data samples are taken. The color, mapped by the color-bar on the right, is the interpolated PFSB output normalized to the peak. We choose coordinates that are a projection of the hemisphere onto a ﬂat surface, with the extents of the plot at θ = 20◦.

Once this conditioned reference is in hand, we can easily demodulate the PFSB output by multiplying it by the reference as shown in Figure 6.11. We then find the mean of the resultant timestream. It is this value (shown in red in the plot) that we take to be the PFSB response to a given beam mapper configuration. Figure 6.12 is an example beam map plot with the data sampling shown in diamonds. We start the beam map near on-axis and complete the revolutions of the polarizer before moving the beam mapper arms into the next configuration. The data portions that are used for the fitting of the characterization parameters (C, X, δ) are thus all adjacent. The mapping continues to move outwards in

106 sections of constant polar angle θ, sampling the azimuthal angle φ in a counter clockwise fashion. The PFSB output is denoted in color where the values between the data samples are quadratic interpolates (as was done for the simulation maps shown in Section 4.5). Section 8.2 will present the maps for all three characterization parameters derived from any given beam measurement. We will normalize the co-polar and cross-polar beam maps to the peak value of the co-polar map. This concludes our discussion of the tools we use to characterize the PFSB architecture. We are only left with a description of the various optical systems used within the test cryostat. Then in Chapter 8, we will ﬁnally be able to present the results of these characterization measurements side-by-side with our simulation results. The interaction between the two is what paved the road towards the square waveguide.

107 Chapter 7

PFSB OPTICS

This chapter describes the various cold optical systems we use in our measurements of the PFSB architecture. The presentation of these systems occurs in Section 7.2. However, in Section 7.1 we must ﬁrst discuss a diﬃculty encountered with the power loading on our detectors as part of the motivation for these systems.

7.1 Power Loading Capabilities

All other things being equal, the optical power impinging on a detector with a throughput AΩ, from a blackbody source of temperature Tsrc in the Rayleigh- Jeans limit is

PRJ ∝ AΩ Tsrc .

With the choice of coupling architecture described in Section 2.3, we have significantly increased the throughput of the system feeding the PFSBs with respect to that of the FSBs. We make this comparison to the FSBs because the PFSBs we tested were unfortunately designed to handle loading conditions equivalent to the FSBs. For our circular waveguide systems we used a guide with a diameter of ∼ 1 cm. A naive calculation shows that operating this waveguide bare gives a throughput of 250 mm2 sr, while the FSBs sit behind 4.5 mm2 sr back-to-back

108 Winston cones. In addition to this, the FSBs are meant to look at the sky from a ground based telescope whose loading is dominated by the 20 K in-band emission of the atmosphere. The measurements we perform involve room temperature (300 K) sources. Thus, without taking any precautions, we would be loading our detectors with ∼ 800 times the design optical power. Figure 7.1 is a succinct representation of this.

Figure 7.1. Power loading comparison between the PFSB design (left) and characterization (right) scenarios. The design throughput is limited by the back-to-back Winston cones in front of the waveguide; the no-Winston case has 55 times more throughput. In addition this diﬀerence, the source temperatures we will be pointing our receiver at for characterization are much hotter (300 K) than the 20 K atmosphere envisioned when designing the PFSBs, a factor of 15. There is a factor of 800 diﬀerence in the power loading between the two.

109 This diﬀerence in power loading becomes a problem when we consider the fact that the PFSBs use a TES thermistor (see Section 2.2.1) whose R(T ) curve (where R is the thermistor resistance) is shown in Figure 2.6. Note that the transition region is the only sensitive portion of the curve. Thus we can rewrite Equation 2.2.1 as ¯ P = Pel + Q = G(Tc − T0)

where Tc is the critical temperature of the TES superconductor. We normally interpret this equation as a statement that, given the radiant power absorbed on the bolometer, we adjust the electrical bias to bring the TES into its transition. A problem arises when ¯ Q > G(Tc − T0) (7.1) where the radiant power alone drives the TES into the unresponsive normal state. We refer to this situation as overloading the detector. As shown in Equation 7.1, what sets the amount of power a detector can handle is the size of the thermal link to the bath G¯. Normally a G¯ is chosen such that the optical power alone brings the TES near to transition because the detector response (power → current gain) is inversely proportional to the electrical power applied [26]. Since the PFSBs were designed for identical loading conditions to the FSBs they have a G¯ half as large due to single polarization absorption being half that of dual-polarization. We thus see that loading the PFSBs during characterization with 800 times the design power is sure to overload them. Due to fabrication diﬃculties, only a small amount of functional PFSBs were produced during the fabrication funding period. Detector casualties then left us with only two to characterize. After having worked out the feed strategy and testing procedure, no further design iterations with an increased G¯ were possible.

110 Use of a Neutral Density Filter (NDF) is often employed in this situation to lower the power loading, placing the detector into the active region of the thermistor R(T ) curve. However, for our purposes an NDF is unacceptable because no NDF technology completely preserves polarization and beam properties. We circumvent part of the loading problem by cutting the throughput with various cold aperture stops described in Section 7.2. However, our work would not be possible without the use of these two serendipitous discoveries:

• a fabrication error resulting in a PFSB G¯ 10 times larger than designed and

• the LEGS transition (see below).

Before discussing these discoveries lets give a quick overview of the tool that enabled them—the load curve. This is a characterization procedure commonly applied to two terminal electronic components. It often consists of applying a known current to the device and measuring the resultant voltage across the terminals. Because of the stability reasons mentioned in Section 2.2.1, for the PFSBs and other TES devices used for CMB observations, we instead apply a known voltage across the terminals and measure the resultant current. More generally, we may view the load curve for any bolometer as tracking the resistance of the thermistor as a function of the applied electrical power deposited on the thermal mass. Figure 7.2 shows the load curves of our two PFSBs. The fabrication “error” mentioned above occurred early on in the development of the FSBs when investigating diﬀerent materials for the leads of the of the TES running down the legs providing the structural support for the thermal mass of the detectors (see Figure 2.11(b)). Fortunately, one of the materials investigated by the GSFC team resulted in a much larger G¯ then the target value. Figure 7.2 compares the load curves of this high-G¯ PFSB to the nominal- or low-G¯ one. We

111 Figure 7.2. Load curves for the two PFSB detectors we characterize. Plotted here is the bolometer resistance R as a function of electrical power Pel deposited on the thermal mass of the PFSB. The blue curve with the multiple transitions is from the low-G¯ PFSB and the red is from the high-G¯ PFSB. The optical system used in these measurements was the on-axis system described in Section 7.2.1 and we have placed a 77 K blackbody in the beam. There is approximately 7 pW of optical power absorbed by the PFSB in this scenario. The high frequency wiggles in the curves are an artifact of the ﬁnite reaction time of the PFSBs and the way the applied electrical power was changed in intervals. Also, the transition to the purely superconducting state of zero resistance appears to be somewhat gradual because of errors in the measured series inductance in the TES leg of the readout circuit (Figure 5.2). see from the ﬁgure that it takes 10 times the power for the anomalous PFSB to reach the same transition temperature as the target one thus the conductivity is 10 times larger. Unfortunately we were left with only one of these high-G¯ bolometers, the other being one of the target value bolometers. The other aid to our power loading problem is the second region of sensitivity

112 shown in the very high power regime of the low-G¯ PFSB load curve in Figure 7.2. The high-G¯ PFSB also exhibits this phenomenon but the plot range is not large enough to contain its depiction. We see from this ﬁgure that the TES superconducting transition happens at an applied electrical power of Pel = 20 pW. At 100 pW the bolometer becomes active again showing a more complicated dependence on Pel. This may be explained by considering the following: in all bolometers the electrical leads of the thermistor must travel down the legs which form the structural supports for the thermal mass. The parallel combination of these form the thermal link to the bath. Figure 2.11(b) shows the leg geometry of the PFSBs. For the TES bolometers these leads need to be superconducting so as not to dominate the resistance of the TES. Given this scenario, the second region of bolometer sensitivity shown in the ﬁgure is most likely due to the bolometer thermal mass getting so hot that it starts to transition part of the lead material running down the legs of the bolometer. This transition is much wider than a normal superconducting one because the superconducting material, instead of being isothermal for any given applied power to the thermal mass, runs along a thermal gradient. Thus as we apply more power to the mass we transition a longer length of the lead material. Being that the bolometer does not care whether the applied power is electrical or optical, we can use this transitioning of the lead material as a high optical power sensor on the PFSBs. Thus we have dubbed this the Large Extra Glow Sensor (LEGS). In reference to Figure 7.2, we see that the TES transition occurs at 30 pW total power while the LEGS remains sensitive to ∼ 350 pW. Thus the LEGS buys us a factor of ∼ 12 in power loading. This has been a tremendous asset in our work as we would not have been able to accomplish the characterization of

113 the PFSB architecture using these devices without it. Because of the complicated shape of the LEGS transition, we were careful to check its linearity for most of the measurements we present (see Appendix A). The combination of the high-G¯ PFSB and the LEGS provides us with a factor of ∼ 120 in power loading capabilities. Because we can never measure the PFSB beams over the entire hemisphere, the rest of the power loading problem (the need to handle 800 times the design loading) is solved by the cold aperture stops described in the next section.

7.2 Optics

Keeping in mind the power loading considerations discussed in the previous section, we can now discuss the cold optical systems needed to carry out the required characterization measurements of the PFSBs. Our results involve four such optical configurations of the receiver. These will be discussed in the next three sections. Section 7.2.1 will describe a configuration where we stop a large fraction of the throughput down so that we only have access to the on-axis portion of the beam. This is the only configuration in which we were able to get two PFSBs operating at once. In Section 7.2.2 we open up the angular aperture to two different, much larger sizes so that we can carry out beam maps. Finally, Section 7.2.3 discusses a configuration in which the PFSB waveguide is square rather than circular.

7.2.1 On-axis System

To enable both of the PFSBs we had at our disposal to be active (the low-G¯ in its LEGS transition and the high-G¯ in its TES transition), we reduced the

114 Figure 7.3. On-axis PFSB test system. The high-G¯ PFSB and the low-G¯ PFSB are oriented perpendicular to one another. The vaned baffle shown significantly reduces the throughput of the system which allows both PFSBs to be active. The waveguide is terminated in a Winston cone feeding an unpolarized detector for diagnostic reasons. All of this is mounted to the ultracold (∼ 260 mK) table, cooled by the active sorption pump refrigerator shown in Figure 5.1. The warmer filters at the top are mounted to the two radiation shields, while the Zotefoam window covers an opening in the vacuum jacket. Where more than one filter is represented by a single element in the figure, the placement order is reflected by the labels.

115 optical loading with a cold baffle which significantly vignettes the throughput of the waveguide. A model of this system is presented in Figure 7.3. As seen in the figure, the 6 vaned baffle prevents the portion of the beam that reflects off the “blackened” surfaces from escaping. The entire baffle is cooled by the sorption pump refrigerator described in Section 5.1 to the same temperature as the PFSB bath (∼ 260 mK).

Figure 7.4. On-axis aperture stop baffle. We show the absorptive Eccosorb coatings in dark blue which sit in copper molds (yellow). The individual sections of the baffle are separated for illustrative purposes. Notice that the entire baffle consists of only three distinct types of pieces: an aperture stop, multiple knife-edge vane mid-sections, and a PFSB waveguide mating section.

116 We machined this baffle from cast Eccosorb CR-112 [18] in a copper mold. See Figure 7.4 for a close up of the baffle. As in the prescription given in Lee et al. [49], the beveled side of the knife-edge vanes are oriented towards the incoming light on the source side of the baffle, but in the opposite orientation on the detector side. When mixing the cast Eccosorb, small (0.2–0.3 µm) glass particles known as Cab-o-Sil [10] were added to prevent the iron filings from settling towards the bottom during the casting process. The iron filings are lossy at microwave frequencies and provide the absorptive power of the Eccosorb. This mixture was done in accordance to Hemmati et al. [32] where one can also find transmission and reflection measurements. We use a 1 mm thickness for our Eccosorb coatings.

Figure 7.5. On-axis aperture stop baffle simulation. Here we use two rays, colored blue, emanating from the green source at the bottom-left hand side of the figure. At the top-right, we see the detector in white at the position of the aperture stop. The copper part of the baffle is colored in red while the Eccosorb is gray. The emissivity of the Eccosorb is = 0.7.

117 The baffle was simulated using the ZEMAX NSC optical design package (see Section 2.3). For the exposed copper surfaces we use the standard mirror reflection properties given by (2.5). The Eccosorbed surfaces were defined as ideal coatings in ZEMAX. These materials are defined only by their intensity transmission and reflection coefficients. Within this coating category we will constrain ourselves to purely emissive materials with a vanishing transmission coefficient. Thus

= 1 − R (7.2) where is the emissivity of the material, R is its power reflection coefficient. We use an emissivity = 0.7 for Eccosorb CR-112. This simulation was done in broadcast mode where we place a source with a uniform beam over the entire hemisphere covering the PFSB waveguide aperture. In order to encompass the most extreme case of near perfect reflection off this aperture we put a mirror surface directly behind the source. A detector is placed over the aperture stop to measure the power exiting the baffle. Figure 7.5 shows an example ray trace which uses two rays emanating from the source. Notice that the rays get trapped by the vanes and must reflect many times before emerging from the aperture stop of the baffle. In carrying out the full simulation we use 5 × 104 rays then split the rays that strike the detector into two categories: those that have reflected off of the baffle at least once and those which have not. With this distinction we are able to characterize the baffle with two parameters:

• the fraction of the PFSB waveguide throughput passed in an ideal case where

the baﬄe works perfectly ≡ ηAΩ, and

• the fraction of the PFSB waveguide throughput passed which has reﬂected

118 oﬀ the baﬄe (the unwanted throughput) ≡ AΩ.

Table 7.1 presents the geometry of the simulation as well as the results and also gives other system parameters which we will describe next. We see from the table that our on-axis baffle cuts the throughput by a factor of 320 with less than 1% of the remaining due to the imperfect absorption of the Eccosorb. Recalling the polarizing properties of reflections mentioned in Section 2.3, it is this 1% which exits the baffle after reflecting off of it that we hope to keep to a minimum. Recall from the previous section that we need to cut the design throughput by at least a factor of 800. Since the on-axis baffle cuts the optical throughput by a factor of 320 we are able to use the LEGS on the low-G¯ PFSB, which buys an extra factor of 12, and the normal transition on the high-G¯ PFSB which gives a factor of 10.

The baffle feeds the PFSB stack, which contains two separate PFSB/backshort pairs, mounted with their polarization orientations orthogonal to each other. The characteristics of the waveguide that the PFSBs sit in are given in Table 7.1. Ending the optical chain is a Winston horn fed BOOMERanG 1998 flight [15] (B98) 400 GHz bolometer. Table 7.1 gives the properties of this Winston cone; because its acceptance angle and throughput is much larger than the resultant baffled PFSB waveguide’s, this cone should accept all the radiation incident upon it. Since we only used this backend as a diagnostic tool, we do not present measurements made with the B98 bolometer; it is mentioned here only for its possible effects on the complete optical chain. Filters are used to block unwanted high-frequency radiation, and a Zotefoam window blocks the air from spoiling the vacuum. These are shown in Figure 7.3. The 9, 14, and 18 icm (icm stands for cm−1 and gives the cut-off frequency of the filter in inverse wavelength units) low-pass filters are metal mesh filters fabricated

119 Table 7.1

On-axis System Parameters

Parameter Value

aperture stop diameter 8.1 mm

center-to-center distance between waveguide end and aperture stop 76 mm

vane diameter 13 mm

vane chamber diameter 23 mm

ηAΩ 0.31%

AΩ 0.0028%

PFSB waveguide diameter 9.5 mm

PFSB waveguide AΩ 220 mm2 sr

total length of PFSB waveguide 59 mm

distance between waveguide end and high-G¯ PFSB 19 mm

distance between waveguide end and low-G¯ PFSB 38 mm

B98 Winston AΩ 12 mm2 sr

B98 Winston large aperture diameter 11 mm by Ade et al. [2]. We place the 9 icm low-pass filter in the beam at 260 mK to simulate the effects of having a 270 GHz PFSB upstream from the 220 GHz ones we are testing. As noted in Section 2.2.2, this is important because of the high- frequency leak inherent in the FSBs (see Figure 2.8). The rest of the filters are a combination of the schemes used by the BOOMERanG

120 2002 flight [59] (B2K) and the FSB characterization receiver [67] to eliminate the transmission of high-frequency radiation. The 14 icm and 18 icm low-passes are used at 4 K and 77 K respectively and are important for staggering resonant ripples and harmonic transmission leaks in the stop bands of these filters. We also place a IR heat-blocking filter, which is just a thin (1 µm) reflective layer, over these two filters on the two radiation shields. These blockers serve to reflect the IR loading coming from the various hotter stages within the cryostat enabling longer cryogenic hold times. The Yoshinaga/black-poly filter at 260 mK is a low-pass with a stop band that does not contain transmission leaks and is thus used to ensure we are not allowing very high frequencies into our PFSB waveguide. These filters were used in the B2K receiver [59] which has a measured polarization efficiency greater than 98% for some of the channels. We can trust that these filters can at most produce errors on the 2% level. The two other filters in the system, Fluoro-Gold (see Halpern et al. [30] for far- IR transmission properties) and GORE-TEX, are both low-pass filters with stop bands that do not have transmission leaks and were simply installed for extra filtering when we were suspicious of high-frequency leaks internal to the cryostat. Because these two come from the FSB characterization optics, we do not have measurements of their cold polarization properties. Finally, the last element in the optical chain is the vacuum window. We use a replica of the ACBAR [73] window which is made from a 1.200 thick piece of Zotefoam. BICEP [92] also used a similar, but thicker window with good polarization fidelity. Results of on-axis testing using the setup described in this section are given in Section 8.1.1.

121 7.2.2 Beam Mapping Systems

In order to measure the beam of the PFSB system, we need a wider aperture and larger throughput. This increases the power absorbed by the PFSBs to the point where even the LEGS is overloaded on the low-G¯ PFSB. For the remaining systems, we can only characterize the high-G¯ detector by making use of its LEGS transition. As stated in Section 7.1, this combination brings us up to a factor of 120 out of 800 in power loading capabilities. We will need an absorbing baffle to take up the remaining factor of 6.7, suppressing the throughput to 15% of the bare PFSB waveguide level. We will present measurements for two beam mapping systems where we accomplish this task by opening the beam up to an approximate 20◦ half angle cone. Before describing the baffles used in these two systems we present the changes from the on-axis system that are common to both. We remove the low-G¯ PFSB and replace it with the high-G¯ one. We then put the 9 icm low-pass filter in the previous location of the high-G¯ PFSB. In the beam mapping systems, suppressing the high-frequency tail shown in the FSB absorption (Section 2.2.2) becomes extremely important because of the band integration in Equation (3.10). See Ta- ble 7.2 for the geometry of this new waveguide configuration.

For the backends of these systems, we removed the B98 Winston cone and bolometer and replaced it with an absorbing cavity shown in Figure 7.6. This is important for these beam mapping systems because of the larger throughput. The baﬄes we will soon describe reduce the bare PFSB waveguide throughput by a factor of 6.7, giving 33 mm2 sr. From Table 7.1 we see that the B98 Winston cone has a throughput of only 12 mm2 sr. Thus, leaving it in the optical chain would result in reﬂecting over half of the light back for a second pass through the

122 Table 7.2

Parameters for the Beam Mapping Systems

Parameter Value

PFSB waveguide diameter 9.5 mm

PFSB waveguide AΩ 220 mm2 sr

total length of PFSB waveguide 59 mm

distance between waveguide end and 9 icm low-pass 17 mm

distance between waveguide end and high-G¯ PFSB 38 mm

absorbing back waveguide extension length 30 mm

absorbing back chamber diameter 23 mm

absorbing back chamber length 44 mm

PFSBs. The new waveguide termination begins as a lossy extension to the PFSB waveguide created by the same 1 mm thick coating of Eccosorb CR-112 in a thick copper rod. This waveguide opens up to the larger lossy cavity which feeds into a small tile of tessellating THz Radar Absorbing Material (RAM) manufactured by Tera- hertz [80]. The THz RAM is a polypropylene based absorber which is tessellated with a pyramidal structure on its surface to reduce reflections. For angles of incidence up to 60◦ the reflected power is less than −20 dB of the incident power [75]. In this case, the full range of angles of incidence is unimportant because the aperture stop baffle limits the incoming angles to less than 20◦. The filtering scheme used here (shown in Figure 7.8) is a reduction of the one

123 Figure 7.6. Absorbing backend to the PFSB waveguide. The bulk of the absorbing power of this backend is provided by the tessellating THz RAM at the bottom which reflects less than −20 dB of the incident power. This is fed by a lossy extension of the PFSB waveguide at the top, which opens up into a much larger chamber. used in the on-axis system. This time we only use filter and window materials that have been used in CMB polarization experiments (B2K and BICEP) and are known to have little if any effect on the polarization. See Section 7.2.1 for descriptions of these and the motivation for their use. The Zotefoam window serves as a low-pass filter with a stop band that does not contain transmission leaks. This acts to prevent high-frequency radiation outside the dewar from coming inside. We can then exclude all the other low-pass filters meant to cover the stop band transmission leaks of the 9, 14, and 18 icm filters since at this point we actually welcome some level of internal high-frequency leaks to ensure that the high-G¯

124 Figure 7.7. Small aperture stop baffle. We show the absorptive Eccosorb coatings in dark blue which sit in copper molds (yellow). The individual sections of the baffle have been separated for illustrative purposes. We simply use some of the same pieces from the on-axis baffle shown in Figure 7.4 with a minor modification to one of them.

PFSB gets into the LEGS transition. Now that we have described the elements of the beam mapping systems that are common to both, let us discuss the two throughput reducing baffles. The first system uses a cold aperture stop and baffle that are similar to those used in the on-axis system; a detailed view of the baffle is in Figure 7.7 while Figure 7.8 shows the whole system which we will refer to as the small aperture stop system. We simulate the optical properties of the small aperture stop baffle in the exact manner as for the on-axis baffle described in the last section. Table 7.3 gives the results of this simulation as well as the baffle geometry. The simulation shows that this baffle cuts the total PFSB waveguide throughput by a factor of 6.4. Less than 4% of the remaining throughput has reflected off of a baffle wall. Remember that it is this portion of the beam which we are trying to minimize due to the polarizing properties of reflections (Section 2.3). This system is used for the off-axis spectra measurements of the circular waveguide architecture whose results are presented in Section 8.1.1.

125 Figure 7.8. Small aperture stop PFSB test system. The vaned baffle shown is a modification of the on-axis baffle (see Section 7.2.1) that gives a much larger throughput for beam maps. We terminate the waveguide with an absorbing stop shown at the bottom. All of this is mounted to the ultracold table, cooled by the active sorption pump refrigerator shown in Figure 5.1. The warmer filters at the top are mounted to the two radiation shields, while the Zotefoam window covers an opening in the vacuum jacket.

The second beam mapping baffle is physically much larger but cuts the throughput by nearly the same amount. This was done to reduce off-axis vignetting for points in the PFSB aperture near the edge. Figure 7.9 shows the configuration of

126 Table 7.3

Small Aperture Stop Baﬄe Parameters

Parameter Value

aperture stop diameter 13 mm

center-to-center distance between waveguide end and aperture stop 16 mm

vane diameter 13 mm

vane chamber diameter 23 mm

ηAΩ 15%

AΩ 0.60% this large aperture stop system, with a close-up of the baffle in Figure 7.10. Notice that the baffle for this aperture stop is significantly different than the previous baffles. It is made out of a single, square cross-section box that traps spurious reflections with the same type of knife-edge aperture used in previous baffles. This aperture is cut out of the top plate of the box. The much larger baffle gave us space to use the foam Eccosorb HR-10 [19], to coat the sides. This Eccosorb has very good properties holding the reflectance to less than 1% for angles of incidence less than 60◦, and raising only to about 3% at 75◦ [54]. We simulate the baffle’s optical properties in a similar manner to the last two baffles where we use an emissivity of = 0.9 for the HR-10 coated walls (as opposed to the = 0.7 used for the CR-112 coated walls). This choice of emissivity is very conservative since the Eccosorb reflectivity is much lower than 10% over most of the angular incidence range. Table 7.4 presents the results of this simu-

127 Figure 7.9. Large aperture stop PFSB test system. This system is identical to that shown in Figure 7.8 but with the larger baffle installed. The baffle shown here gives nearly the same throughput as the small aperture stop baffle in Figure 7.8, but is much larger to allow for a large aperture which vignettes less of the off-axis rays within the ±20◦ acceptance cone. lation as well as the baffle geometry. The simulation shows that this baffle cuts the total PFSB waveguide throughput by a factor of 7.9, and less than 5% of the

128 Figure 7.10. Large aperture stop baffle. We show the absorptive cast Eccosorb coatings in dark blue which sit in copper molds (yellow). The lighter blue pieces on the sides of the box are the foam Eccosorb HR-10, which is used for its superior reflection properties. These are attached to the walls with the cast Eccosorb. The individual sections of the baffle have been separated for illustrative purposes. remaining throughput has reflected off of a baffle wall. See Section 8.2.1 for the beam maps of the circular waveguide mounted PFSBs taken with this system.

7.2.3 Square Guide System

The final system that we will present measurements for, is the square guide system, which is so named because the high-G¯ PFSB sits in a square cross-section waveguide as opposed to a circular one. It is this configuration of the PFSB architecture which we find (in Section 8.2.2) to have reasonable levels of cross- polar response. See Figure 7.11 for a depiction of the only change between this test system and the one shown in Figure 7.9: the insertion of the square waveguide into the circular one. The square waveguide was fabricated by machining copper square waveguide inserts to slip into the circular guide. The inserts were made

129 Table 7.4

Large Aperture Stop Baﬄe Parameters

Parameter Value

aperture stop diameter 38 mm

center-to-center distance between waveguide end and aperture stop 54 mm

baﬄe chamber side length 83 mm

ηAΩ 12%

AΩ 0.64% using a broach to cut a square hole (0.2500 × 0.2500) in short sections of copper rod, so that the copper is continuous around the perimeter of the waveguide cross- section. We epoxy the inserts into the original circular waveguide sections using MS-907 epoxy [60]. In addition to this change, we also removed the IR heat blocker on the 77 K radiation shield window, which was damaged between the circular guide testing and this run. Measurement results obtained for this square waveguide architecture are found in Sections 8.1.2 and 8.2.2. We have ﬁnally completed the descriptions of all the various systems and simulations used to characterize the PFSB detector architecture. It is now time to present the results of our work.

130 Figure 7.11. Square waveguide PFSB test system inserts. The rest of the system is identical to the one shown in Figure 7.9 except tat we removed the IR blocker at 77 K.

131 Chapter 8

RESULTS & DISCUSSION

We now begin a summary of the results of our characterization work. The two fundamentally different architectures compared here are circular and square cross-section waveguides for housing the PFSBs. Recall that Chapters 5 and 7 describes the various systems used to make these measurements, and Chapter 6 discusses the characterization tools used, the details of how the measurements were performed, and the signal processing chains which lead to the creation of the plots we display. In Section 8.1 we will present the PFSB spectra and in Section 8.2 we will show the radiation patterns. To facilitate a discussion of the differences between the two waveguide geometries, we will draw upon the results of our simulation work described in Chapter 4. In order to provide an overview of the benefits and the disadvantages of both configurations, we will make extensive use of the intermediate data products of those simulations.

8.1 Spectra

In this section, we describe the spectra measurements of the PFSB architecture and are primarily concerned with testing that

• the spectra show good band deﬁnition and

132 • in-band, the detectors do a good job of discriminating between the co- and cross-polar directions.

The discussion is divided into two sections: Section 8.1.1 describes the results for the PFSB mounted in a circular waveguide and Section 8.1.2 shows the corresponding results for the square waveguide.

8.1.1 Circular Waveguide

We begin by presenting the only data we collected with a configuration of the system designed to keep both PFSB detectors active. This requires the use of the LEGS transition on the low-G¯ PFSB. In Appendix A.1.1 we show that this measurement contains no more than 2% rms deviations from linearity. The optical setup used here is the on-axis system described in Section 7.2.1, which has the detectors sitting in a circular waveguide. Figure 8.1 shows the co- and cross-polar spectra taken with both bolometers. As expected, the co-polar pass-bands are centered on 220 GHz with approximate 20% fractional bandwidths. This system achieves cross-polar suppression to less than 7% of the co-polar peak across the entire band. The detailed aspects of these spectra (e.g. the various wiggles in the passband) are unimportant here. As mentioned above, we are concerned only with the band definition and polarization discrimination. In addition to these properties, for this system we see that the shape and relative level between the co- and cross-polar spectra are not altered much by the presence of another detector upstream. Also, from a load curve analysis we find that the total power absorbed by the detectors are not more than 10% from each other with the detector further downstream actually absorbing more. This

133 Figure 8.1. On-axis system spectra for the low- and high-G¯ PFSBs. The green lines denote data taken with the low-G¯ PFSB, while the red lines are with the high-G¯ one. Recall from Figure 7.3 that the light encounters the high-G¯ detector before the low-G¯ one in the waveguide, and that both are oriented orthogonally to each other. The solid lines show the co-polar spectra and dashed is for cross-polar. The two sets of spectra are normalized to the maximum co-polar in-band transmission for each detector. Here we have already removed the source’s spectral dependence as described in Section 6.1.2. We ignore the large low-frequency response as this is due to the 1/f noise being amplified by the source removal. The spikes at 170 GHz in the cross-polar spectra are cryogenic artifacts. is a meaningful value to compare between the two detectors, as the absorbing materials and thicknesses are identical. From this test, we can say that, at least on-axis, the PFSBs mounted in a circular waveguide behave as designed. In opening up the aperture, we see that one of the above qualities change significantly off-axis. See Figure 8.2 for spectra taken at three different points in the beam of the small aperture system (Section 7.2.2). These points are on-

134 Figure 8.2. Small aperture system spectra for the high-G¯ PFSB. The black lines denote spectra taken on-axis, green lines are for 12◦ off-axis in the H-plane, and red for 12◦ off-axis in the E-plane. We use solid lines for the co-polar spectra and dashed for cross-polar. We see here a loss of polarization discrimination at off-axis points in the beam. axis and 12◦ off-axis in both the E- and H-planes. Recall that this system still mounts the PFSB in a circular waveguide. Although the LEGS transition on the high-G¯ PFSB was used, we failed to take linearity characterization data. We see that the co-polar spectrum maintains good band definition across the beam (20% bandwidth centered on 220 GHz). However, the ability to discriminate between two orthogonal polarizations varies widely over position in the beam. The on-axis cross-polar spectra, shown by the dotted lines (black is the co- polar while purple is the cross-polar), is still well suppressed, peaking at a value not much more than 10% of the co-polar peak. This level has risen with respect to the on-axis system measurement shown in Figure 8.1 because we are sampling

135 1 ◦ a larger part of the beam in this system. We utilize here the full 7 8 half angle source size of the FTS; whereas, in the on-axis system the beam is very well collimated and only views a small portion of this. What is disconcerting about these spectra is that we seem to lose polarization discrimination oﬀ-axis. In the E-plane, we almost lose this ability entirely as seen by the solid lines in the plot where red shows the co-polar spectra and aqua depicts the cross-polar. We will discuss this problem further in Section 8.2 where we use our simulation work to provide insight into why this happens.

8.1.2 Square Waveguide

We now focus on the square guide system of Section 7.2.3 where the PFSB is mounted in a square cross-section waveguide. This system utilizes the LEGS transition on the high-G¯ PFSB so we carry out a linearity test in Section A.1.1. We show there that the level of non-linearity in these measurements is below a 5% rms. Figure 8.3 shows the results of the square waveguide PFSB spectra, taken at nearly the same three points in the beam used above for the small aperture system. Here, oﬀ-axis points are only 11◦ (rather than 12◦) from on-axis. The spectra shown here exhibit good qualities in both measures that concern us. Again, we see the 20% bandwidth centered on 220 GHz with strong suppression out-of- band. In addition to this, unlike the situation with the circular waveguide, we obtain good polarization discrimination with the cross-polar spectra all kept to less than 7% of the peak co-polar level.

136 Figure 8.3. Square guide system spectra for the high-G¯ PFSB. The black lines denote spectra taken on-axis, green lines are for 11◦ oﬀ-axis in the H-plane, and red is for 11◦ oﬀ-axis in the E-plane. We use solid lines for the co-polar spectra and dashed for cross-polar. This data shows a high degree of polarization discrimination at all points in the beam.

8.2 Beam Maps

The main goal of measuring the absorption spectra of the PFSBs is to check the band deﬁnition. When mounting the PFSB in a circular guide (versus the square one), we see a problem with the polarization discrimination. However, with this data set, we do not gain much insight into the causes of this phenomenon. The radiation pattern data and simulations provide an excellent tool for exploring these issues as well as an invaluable characterization measure for the PFSB architecture. In Section 8.2.1 we begin by discussing the results (both measurement and simulation) for the circular waveguide architecture. Section 8.2.2 carries the discussion over to the square waveguide architecture, and provides an explanation for the

137 increase in polarization discrimination found in Section 8.1.2.

8.2.1 Circular Waveguide

We present here the beam map results for the large aperture system described in Section 7.2.2, where the PFSB is mounted in a circular waveguide. Results from the measurement and simulation are shown in tandem to facilitate comparisons. Figure 8.4 displays the co-polar maps and the cross-polar maps are shown in Figure 8.5. The rotation angle maps are given in Figure 8.6. In addition to the 2-dimensional views of the beam maps, we also plot plane cuts so that a better quantitative comparison can be made between the data and simulation. The cuts through the E- and H-planes are shown in Figure 8.7; cuts through diagonal planes in Figure 8.8. Since we make use of the LEGS transition on the high-G¯ PFSB, we check its linearity in Section A.1.2. This shows the non-linearity to be below the 2% rms level. These measurements show the same loss of polarization discrimination off-axis that was seen in the spectra of Figure 8.2. We need not compare these in great detail with the spectrum plots as they were taken with different optical systems; the large and small aperture systems respectively. In comparing the measurements to the simulation, we see in Figures 8.4–8.6 that the simulation achieves an overall qualitative agreement to the data. The main differences are that the simulation has an extra side-lobe structure in the E-plane of the co-polar map, and the cross-polar lobes are more intense. This is more evident in the plane-cut plots of Figures 8.7 and 8.8 where we notice that the simulation agrees rather well with the data in the H-plane. Although it appears that something is preferentially cutting power from the measurement

138 (a) measurement

(b) simulation Figure 8.4. Co-polar beam maps C(Ω) of the large aperture system with the high-G¯ PFSB in a circular waveguide. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right.

139 (a) measurement

(b) simulation Figure 8.5. Cross-polar beam maps X(Ω) of the large aperture system with the high-G¯ PFSB in a circular waveguide. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right.

140 (a) measurement

(b) simulation Figure 8.6. Rotation angle beam maps ∆(Ω) of the large aperture system with the high-G¯ PFSB in a circular waveguide. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right.

141 (a) E-plane (u = 0 in Figures 8.4–8.6)

(b) H-plane (v = 0 in Figures 8.4–8.6)

Figure 8.7. Principal plane cuts through the beam maps of the large aperture system with the high-G¯ PFSB in a circular waveguide. The symbols denote the data sampling where we use red crosses and X’s for the data, and blue squares and diamonds for the simulation. Solid lines are drawn through the co-polar points and dashed lines go through the cross-polar ones.

142 (a) 45◦-plane (v = u in Figures 8.4–8.6)

(b) 135◦-plane (v = −u in Figures 8.4–8.6)

Figure 8.8. Diagonal plane cuts through the beam maps of the large aperture system with the high-G¯ PFSB in a circular waveguide. The symbols denote the data sampling where we use red crosses and X’s for the data, and blue squares and diamonds for the simulation. Solid lines are drawn through the co-polar points and dashed lines go through the cross-polar ones.

143 near the E-plane, we do not address these details any further because they are not fundamental for the goals of this work. Given the qualitative agreement between the measurements and the simulation, we proceed to examine the details of the simulation to better understand what is happening. Figure 8.9 plots a collection of electric field modes in the circular PFSB waveguide. These are the basis functions that we use to decompose any PFSB aperture illumination in order to satisfy the waveguide boundary conditions as described in Chapter 4. It is these functions that we propagate separately down the guide and carry out a coherent sum at the position of the PFSB absorbers. Equations (4.8) and (4.4) show that this propagation is harmonic with a wavelength in the guide that is dependent on the cutoff wavenumber shown in the plot. Notice that, except for the first two, the circular guide modes do not exhibit a large degree of uniformity in polarization orientation over the guide cross section. This results in an inability to separate the two orthogonal polarizations using the absorber pattern on the PFSBs. Fortunately, we are not limited to single modes, but some linear combination of them. Given the large number of modes (239) that propagate down this waveguide at our operating frequency of 220 GHz, we should be able to decompose any aperture illumination resulting from the diffracted plane waves used in our simulation. These illuminations will have a constant polarization orientation across the entire aperture, and we should be able to separate the orthogonal components there. To demonstrate this fact, let us take some simple linear combinations of the modes in Figure 8.9. These are shown in Figure 8.10. We see that it is easy to greatly reduce the non-uniformities in the polarization orientation over the cross section. The particular combinations chosen here are ones that arise from either an

144 Figure 8.9. First 12 modes of the circular PFSB waveguide. These are designated with a mode type (TE, TE2, TM, TM2—where the “2” specifies a degenerate set of modes required by rotational symmetry given by Marcuvitz [56]) and mode indices (m, n). The cutoff wavenumber is also given to show degeneracies. In each plot, the circular waveguide boundary is drawn in as is the PFSB absorber bars (red). The E-field is depicted as arrows on a grid (with the tails on the grid points) whose length is proportional to the magnitude of the field. 145 (a)

(b)

Figure 8.10. Linear combinations of circular waveguide modes. See Figure 8.9 for a description of the individual plots. We show here how uniformity of polarization orientation can be obtained by a suitable choice of linear combinations. Both of these were chosen so that the resultant polarization orientations are mostly parallel with the PFSB absorber bars, maximizing the response. In (a) we show a combination that might arise from an oﬀ-axis E-plane illumination, whereas (b) shows one from an H-plane illumination of about the same angle from on-axis. Notice that for the circular guide shown here, the combinations needed have diﬀerent guide wavelengths and thus uniformity is lost further down the waveguide.

146 E- or H-plane illumination with a polarization aligned with the PFSB absorber. Taking a closer look at the pairs of modes we needed to sum in order to obtain the uniform polarization, we notice that they have close but not identical cutoff wavenumbers—and thus propagation constants. The fact that they are close is sensible because it is important that the patterns over the cross section be nearly identical for the uniformity to arise. However, their non-identity is what matters for PFSB absorption. Since the propagation constants give the wavelength of the mode in the guide, we see that modes with unequal propagation constants will develop a relative phase shift as we move along the guide. This phase shift grows with respect to the propagation length and the difference in propagation constants. The result is that it would be an unlikely scenario for the uniformity of polarization to last all the way to the PFSB absorber. This is especially true for modes far from cutoff that have guide wavelengths comparable to the wavelength of light in the system. For 220 GHz the wave-number is 7.3 icm, thus all of the lower-order modes that were shown in Figure 8.9 satisfy this property. Searching for degenerate modes that can combine into a propagating uniform polarization field is a daunting task. Looking closely at Figure 8.9, we see that the degeneracy we get for free (via the second set of modes mentioned in Figure 8.9) are all simply rotations of each other. This set exists only to regain the circular symmetry that is broken by the mode patterns. It is therefore useless for forming polarization uniformity because of the different patterns. In addition to the free degeneracy, there is an occasional accidental degeneracy between a rotationally symmetric TE mode and one of the pairs of TM modes. An example of this is shown by the modes with κmn = 1.28 icm in Figure 8.9. For these cases, we find that the patterns are different and do not lend themselves well towards

147 polarization uniformity via combination. Thus it is the different guide wavelengths of the modes that sum to a uniformly polarized pattern that is the cause of the loss in polarization selectivity seen in the spectra and beam maps of the PFSB in a circular waveguide. Before discussing the square waveguide, we note that for on-axis waves the illumination is flat, not just in the polarization orientation, but also in electric field direction. Thus, near on-axis waves mostly couple to the fundamental as the other modes have symmetry properties that cause the mode coupling integral (Equation 4.7) to approximately vanish. The fundamental mode, as shown in Figure 8.9, has a uniform polarization direction, which explains why the circular guide PFSB shows good properties on-axis.

8.2.2 Square Waveguide

Regarding the square waveguide, we continue here with the discussion of waveguide modes and table the presentation of the measurement and simulation results until we can discuss the case for setting the PFSBs in this conﬁguration. We plot the electric ﬁeld modes in Figure 8.11. Note the addition of more pairs of uniform polarization modes (TE20 and TE02 as well as TE10 and TE01). These happen for all TE modes where m or n = 0. Similarly to the TE0n and TM0n modes of the circular waveguide, these modes follow the symmetry of the guide boundaries. However, with the square guide, the geometry follows symmetries more closely linked to our goal of separating out two orthogonal polarization states. What is initially alarming about the square waveguide modes is that, similar to the circular guide, there is no lack of modes with a non-uniform polarization.

148 Figure 8.11. First 12 modes of the square PFSB waveguide. These are designated at the top of each plot with a mode type (TE or TM) and mode indices (m, n). The cutoff wavenumber is also given to show degeneracies. In each plot, the square waveguide boundary is at the plot extents and the PFSB absorber bars appear in red. The E-field is depicted as arrows on a grid (with the tails on the grid points) whose length is proportional to the magnitude of the field.

149 (a)

(b)

Figure 8.12. Linear combinations of square waveguide modes. See Figure 8.11 for a description of the individual plots. We show here how uniformity of polarization orientation can be obtained by a suitable choice of linear combinations. Both of these were chosen because the resultant polarization orientations are parallel with the PFSB absorber bars, maximizing the response. In (a) we show a combination that might arise from an oﬀ-axis E-plane illumination. Since TE20 itself can handle an H-plane illumination, in (b) we show a sum that would propagate a plane wave incident from one of the diagonal cut planes. The important diﬀerence between these sums and the ones shown in Figure 8.10 for the circular guide, are that they are between degenerate modes with identical guide wavelengths. The polarization uniformity shown here is thus preserved as the wave propagates down the guide.

150 We take a closer look here, as previously done, by forming linear combinations of modes. These are shown in Figure 8.12. The most notable property of these linear combinations is that they are between degenerate modes. This ensures that the polarization uniformity afforded by these combinations remains intact all the way down the guide. Thus, a PFSB in a square waveguide should have no problem preserving the two orthogonal polarizations. As a testament to this statement, we now present the measurement data and simulation results for the square guide system described in Section 7.2.3. Again, we use the LEGS transition on the high-G¯ PFSB. The linearity calibration in Section A.1.2 was inconclusive but hinted at the non-linearities here being kept to within the 5% rms level. Figure 8.13 displays the co-polar maps and the cross-polar maps are shown in Figure 8.14. The rotation angle maps are given in Figure 8.15. In addition to the 2-dimensional views of the beam maps, we also plot plane cuts so that a better quantitative comparison can be made between the data and simulation. The cuts through the E- and H-planes are shown in Figure 8.16. We show cuts through diagonal planes in Figure 8.17. The first thing to notice is the very low level of cross-polar response in both the data and simulation shown in Figure 8.14. Thus, as we have just predicted, we see that the square waveguide has enabled the PFSB to do a much better job at discriminating between the two orthogonal polarizations. In addition to this, we see from Figure 8.15 that this system also results in very little rotation angle variation across the beam. This will become important later when we discuss differencing detectors for polarization signals. Again, we see that there is good qualitative agreement between the simulation and the data. One factor worth noting here is that the rings in the co-polar maps (data and

151 (a) measurement

(b) simulation Figure 8.13. Co-polar beam maps C(Ω) of the square guide system with the high-G¯ PFSB in a square waveguide. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right.

152 (a) measurement

(b) simulation Figure 8.14. Cross-polar beam maps X(Ω) of the square guide system with the high-G¯ PFSB in a square waveguide. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right. To within the errors of the calculation, the simulation (b) is identically zero.

153 (a) measurement

(b) simulation Figure 8.15. Rotation angle beam maps ∆(Ω) of the large aperture system with the high-G¯ PFSB in a square waveguide. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right. These plots are not identically zero where the simulation residuals can be entirely explained by the relative orientations of Ludwig’s deﬁnition 3 co-polar direction and the PFSB absorber. It requires no physical rotation mechanism.

154 (a) E-plane (u = 0 in Figures 8.13–8.15)

(b) H-plane (v = 0 in Figures 8.13–8.15)

Figure 8.16. Principal plane cuts through the beam maps of the square guide system with the high-G¯ PFSB in a square waveguide. The symbols denote the data sampling where we use red crosses and X’s for the data, and blue squares and diamonds for the simulation. Solid lines are drawn through the co-polar points and dashed lines go through the cross-polar ones. As stated in Figure 8.14, the simulation cross-polar pattern is identically zero.

155 (a) 45◦-plane (v = u in Figures 8.13–8.15)

(b) 135◦-plane (v = −u in Figures 8.13–8.15)

Figure 8.17. Diagonal plane cuts through the beam maps of the square guide system with the high-G¯ PFSB in a square waveguide. The symbols denote the data sampling where we use red crosses and X’s for the data, and blue squares and diamonds for the simulation. Solid lines are drawn through the co-polar points and dashed lines go through the cross-polar ones. As stated in Figure 8.14, the simulation cross-polar pattern is identically zero.

156 Figure 8.18. Co-polar beam map simulation C(Ω) of a square waveguide excluding aperture stop diffraction effects. This uses a slightly larger waveguide with a 0.3300 side length. The black diamonds denote the data sampling while the color shows interpolated values as indicated by the bar on the right. simulation) are a product of diffraction at the cold aperture stop. This can be seen by presenting simulation maps where the aperture diffraction part is left out. Figure 8.18 shows the co-polar map. We carry out this simulation with a larger square waveguide side length of 0.3300; otherwise, all other parameters are identical to the square guide system simulation. Note that unlike the square guide system maps of Figure 8.13 which show a ring shaped peak, this map peaks on-axis. Due to the lack of an aperture stop, we have beam information over the entire hemisphere as shown in the figure. We show in Figure 8.19, the vanishing of the cross-polar response over the entire beam. Thus, the utilization of more of the PFSB beam should not degrade the ability to discriminate between polarization

157 Figure 8.19. Cross-polar beam map simulation X(Ω) of a square waveguide excluding aperture stop diffraction effects. This uses a slightly larger waveguide with a 0.3300 side length. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right. states. See Figure 8.20 for the rotation angle map which completes the depiction of this simulation. Now that we have a working PFSB architecture, let us present the radiation pattern data in a slightly different way which highlights its abilities to discriminate between the orthogonal polarizations. In Figure 8.21 we plot the polarization efficiency ρ, as defined by Montroy [59, Section 4.11], where

C − X ρ = . (8.1) C

Note that the eﬃciency never drops below 75% while maintaining a fairly constant

158 Figure 8.20. Rotation angle beam map simulation ∆(Ω) of a square waveguide excluding aperture stop diﬀraction eﬀects. This uses a slightly larger waveguide with a 0.3300 side length. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right.

90% efficiency over most of the beam. Taking the beam average over this plot we obtain ρavg = 89%. This concludes the discussion of the results of our characterization work. We have found that PFSBs exhibit far better cross-polar rejection when situated in a square verses circular cross-section waveguide. For the square waveguide architecture, the plots shown in part (a) of Figures 8.13–8.15 are the maps of the three characterization parameters (C,X, ∆), defined in Equation (3.10). The rotation angle ∆ map shows variations mostly below 3◦. These are the maps that future instrument builders will find useful when trying to evaluate the PFSB as a viable option for detectors in their focal planes. In the next chapter we carry out such

159 Figure 8.21. Polarization eﬃciency beam map ρ(Ω) of the square guide system with the high-G¯ PFSB in a square waveguide. The black diamonds denote the data sampling while the color shows interpolated values, as indicated by the bar on the right. The beam averaged ◦ eﬃciency over the entire beam (20 half-angle) is ρavg = 89%. an evaluation for a very simple example telescope.

160 Chapter 9

PFSBS ON AN EXAMPLE TELESCOPE

Given the working square waveguide PFSB architecture described in Sec- tion 7.2.3 with both measurement and simulation results presented in Section 8.2.2, we use these characterization results in a simple mock telescope to see the level of systematics this detector system induces for the beams on the sky. We will be taking a look at the efficiency of an orthogonally differenced detector system to detect polarized radiation as well as the intensity leakage into such a channel. The telescopic system we use for this calculation is similar to the one used in Spider [14] and BICEP [92]. The important optics for this discussion can be condensed into a simple f/2.2 lens primary with a 0.26 m aperture stop, feeding a single pixel focal plane. This is depicted in Figure 9.1. At the center of the focal plane, as our single pixel, we use an architecture that consists of two orthogonally oriented PFSBs mounted in a single square waveguide. For the beams emanating out of this waveguide, we use the measured maps presented in Section 8.2.2 for both of the detectors and simply rotate the end result on the sky 90◦ with respect to each other. We use the scalar Fraunhofer diffraction formula for a non-uniformly illuminated aperture to calculate beams on the sky. We reproduce a version of this formula here, taken from Hecht and Zajac [31, Chapter 11], but modified so that the observation point lies on a spherical surface with coordinates (θ, φ), rather

161 Figure 9.1. Spider-like [14] optics for predicting PFSB systematics on the sky. The telescope optics can be simplified, for this on-axis configuration, to a single f/2.2 lens refractor with a 0.26 m aperture stop at the primary. We place our PFSB square waveguide system at the center of the focal plane. For the discussion, we place two orthogonally oriented PFSBs in this waveguide that have beams identical to the ones measured in Section 8.2.2. than a flat one,

Z E(θ, φ) = A(x, y)eik sin θ(x cos φ+y sin φ)da (9.1)

aperture where E is the diffracted scalar field over the spherical observation screen, A is the scalar aperture illumination pattern over a flat aperture with coordinates (x, y), and k is the propagation wavevector. In obtaining this form, we assume that the spherical observation screen is centered on the center of the aperture and has a radius much larger than the aperture extents. This assumption is, of course, very reasonable for the CMB surface of last scattering and our 0.26 m primary aperture.

162 We will be converting Equation (9.1) into a vector formula by taking the scalar fields to be the components of the electric field vectors in the aperture and on the observation screen. The only complication here is that the directions of the E-field are then set by the axes in the flat illumination aperture which do not fit precisely in the spherical system. However, the angular extents of the resulting beams are less than a degree in diameter, so to a good approximation, the observation screen is a flat surface. For the aperture illumination field, we will use the square root of the interpolated co- and cross-polar maps shown by the colors in Figures 8.13(a) and 8.14(a). Figure 9.2 shows these patterns already truncated by the aperture. We can do this because the function of the lens in the system is to flatten the wavefronts emanating from the PFSB aperture. Recall that we plotted the spherical data in the maps by projecting onto a flat surface (the u, v transformation). Thus, we simply truncate the interpolated data at a radius corresponding to f/2.2 and map these onto the 0.26 m primary. This makes use of various approximations. First, our beam maps do not contain information about the phase of the electric field. However, the telescope aperture is in the far field, and therefore the phase front is very nearly spherical. The next approximation uses the fact that the rotation angle maps shown in Fig- ure 8.15(a) nearly vanish over the entire measurement. We will set this rotation angle to be identically zero and assume that the co- and cross-polar maps arise from fields aligned with the axes defined by definition 3 in Ludwig [55]. Finally,

◦ since the angles contained within an f/2.2 system are less than 13 , the co- and cross-polar unit vectors in Ludwig’s third deﬁnition are virtually aligned with the yˆ and xˆ unit vectors respectively. This allows us to take the co-polar map as rep-

163 (a) co-polar

(b) cross-polar

Figure 9.2. Aperture illumination functions for a square waveguide PFSB in a Spider-like [14] telescope. The patterns shown here are the square roots of the measurements shown in Figures 8.13(a) and 8.14(a). We have also truncated and scaled the patterns as appropriate for the Spider optics. For a PFSB oriented along they ¯ direction, the co-polar plot will be used for the Ey aperture illumination and the cross-polar plot for Ex and vice versa for a PFSB in thex ¯ direction.

164 resenting the square of the E-field amplitude in the y-direction and the cross-polar for the x-direction in Equation (9.1). Now applying the diffraction formula (9.1) to the illuminations shown in Fig- ure 9.2 and squaring again to get back power gives us the co- and cross-polar beam maps on the sky as shown in Figure 9.3, which are very symmetric. This symmetry minimizes the contamination of the polarization measurements by coupling to intensity variations as depicted in Figure 2.2. Let us denote the pattern in Figure 9.3(a) by Csky(θ, φ) and that of Figure 9.3(b) by Xsky(θ, φ). To understand the significance for CMB observations, we need to create a polarization sensitive pixel by differencing two orthogonally oriented detectors.

The beams Csky(θ, φ) and Xsky(θ, φ) are the sky responses for a single PFSB. We add to these the fact that we set the rotation angle to zero giving ∆sky(θ, φ) = 0. To get the corresponding characteristics for the second detector, we rotate the beam patterns by 90◦ and add the same amount to the rotation angle. In summary, for the two detectors, we have the following polarization characteristics,

C1(θ, φ) = Csky(θ, φ)

X1(θ, φ) = Xsky(θ, φ)

∆1(θ, φ) = 0 (9.2) ◦ C2(θ, φ) = Csky(θ, φ + 90 )

◦ X2(θ, φ) = Xsky(θ, φ + 90 )

◦ ∆2(θ, φ) = 90

where (C1,X1, ∆1) are the characterization parameters for one detector through the entire system and (C2,X2, ∆2) are for the other. Recall that the characterization parameters given above are derived from the

165 (a) co-polar (Csky)

(b) cross-polar (Xsky)

Figure 9.3. Beams on the sky for a square waveguide PFSB in a Spider-like [14] telescope. These beams were calculated using Equation (9.1) and inserting the patterns from Figure 9.2 for the aperture illumination. We then square the results to get units of power. Finally, the plots here are normalized to the peak of the co-polar beam.

166 Jones formalism for 100% polarized radiation described in Chapter 3. To discuss the characterization relevant for partially polarized sources like the CMB, we need to introduce the Mueller calculus. Again, see Tinbergen [82] for a more detailed description. We represent a radiation stream as a four-element vector of Stokes parameters (1.4)   I      Q    S =   .  U      V

Analogous to the Jones calculus, the eﬀect of the instrument on the radiation stream is given by a 4 × 4 Mueller matrix M such that

Sout = M · Sin (9.3)

where Sin and Sout are the input and output radiation streams respectively. Us- ing this formalism we can write down the complete instrument chain shown in Figure 3.1, the analogs to Equations (3.5) and (3.6),

Z ˆ T 0 0 0 Ppd(ν) = S0 · Minst(Ω , ν) · Sﬀ (Ω , ν)dΩ (9.4) Z 0 0 s = Ppd(ν )dν ZZ ˆ T 0 0 0 0 0 0 = S0 · Minst(Ω , ν ) · Sﬀ (Ω , ν )dΩ dν (9.5)

where this time Minst is the full instrumental action containing telescopic and feed/detector effects; Sff is the far field radiation stream and S0 is the unit col- umn vector (1, 0, 0, 0) showing that the power detector only measures intensity represented by Ppd, resulting in a signal s.

167 As with the Jones calculus, the instrument characterization is entirely contained within the matrix elements of Minst where from Equation (9.5) we see that only the ﬁrst row of elements

ˆ T S0 · Minst ≡ ( PI PQ PU PV ) (9.6) are of importance. The CMB community traditionally is only concerned with the ﬁrst three elements of Equation (9.6) since it is assumed that the CMB contains no circular polarization. These three elements are commensurate with our three Jones characterization parameters (C,X, ∆). We can calculate the Mueller parameters by using the Jones to Mueller matrix conversion formulas found in Kaplan [40]. This gives

1 P = (C + X) I 2 1 P = (C − X) cos 2∆ (9.7) Q 2 1 P = − (C − X) sin 2∆ . U 2

Now using the parameters from Equation (9.2), Equations (9.7), (9.6), and (9.5) allow us to calculate the diﬀerenced detector signal response for any stream of photons coming from a given direction on the sky. To do this we use a point source far ﬁeld radiation pattern,

  I   0   0 Sﬀ (Ω ) =  Q  δ(Ω − Ω)     U which keeps the beam variations intact through the integration. Also, remember

168 that the beam measurements were done with a Rayleigh-Jeans source and the frequency integration has already been carried out. Putting this all together gives

1 1 s(θ, φ) = [(C1 + X1) − (C2 + X2)]I + [(C1 − X1) + (C2 − X2)]Q 2 2 (9.8)

≡ PI (θ, φ)I + PQ(θ, φ)Q + PU (θ, φ)U

where the Px functions are the Mueller characterization parameters for the polarized pixel (two diﬀerenced PFSBs in a single waveguide) specifying how various types of radiation couple into the signal. We see that PU is identically zero because of our assumptions about the rotation angle. Figure 9.4 plots PQ and

PI for the PFSB in a square waveguide looking through a Spider-like telescope. Most importantly, we see that the maximum I response is suppressed by a factor of 140 to the maximum Q response. Thus we have a pixel which measures the polarization signal allowing only a little more than 0.7% percent intensity/tem- perture leakage. The plots of Figure 9.4 are normalized from the fact that Csky and Xsky are normalized to the maximum Csky. Thus, the maximum level of 92% shown in PQ is the total eﬃciency for the two detectors. We can convert the plots to more meaningful quantities by calculating the window functions in multipole moment l-space. This is done by calculating the two-dimensional Fourier transform of the maps shown in Figure 9.4 then assigning an approximate l value to each pixel in k-space according to the prescription found in Goldstein [28, Equation 6.2.16]. We then carry out an azimuthal average by binning the pixels in l-space which produces the desired window functions. These are shown as the purple (PQ window function) and red (PI window function) curves in the top panel of Figure 9.5. Combining these two by normalizing the

169 (a) Q response (PQ)

(b) I response (PI )

Figure 9.4. Polarization response beams on the sky for an orthogonal PFSB diﬀerence pixel in a Spider-like[14] telescope. These beams were calculated using Equations (9.8) and (9.2). Note that the scale on the I response in (b) is magniﬁed by a factor of 1000. The response to U polarization is zero by construction.

170 Figure 9.5. Beam window functions for PFSBs on example telescopes. The top plot shows the beam window functions where purple denotes those for the Q-coupling PU while the red shows those for I-coupling PI . The solid lines denote the results for PFSBs set in the Spider-like telescope described in this chapter while the dashed shows the results for a similar system with an SPT-sized primary (8 m). Notice that for the SPT-sized system the Q-coupling window function is nearly unity for the entire l range plotted. These are normalized to the maximum of the Q-coupling window function for each telescopic system. In the bottom panel we show various CMB power spectra similar to the ones in Figure 2.1. Purple is for the hTT i spectra, blue denotes the hEEi spectra, and the B-modes due to gravitational waves and lensed hEEi power are shown in green and aqua respectively. In red we plot the contamination to polarization measurements due to temperature leakage and beam size as a consequence of the window functions shown at the top. Again, the solid line denotes the Spider-like system while dashed denotes the SPT-sized system. In the Spider-like system the contamination curve is mostly the result of numerical error for l > 1200 where we see a spike. This is due to the fact that both window functions fall off there (see top panel). 171 PI window function with the PQ one, multiplying by the temperature anisotropy power spectrum then gives us a contamination due to temperature leakage and finite beam size for a given system. This is plotted as the red curves in the bottom panel of the figure. Comparing this contamination to the B-mode spectra in the figure shows that, for PFSBs on a Spider-like telescope, we can already start to make low signal- to-noise ratio measurements of B-modes below the current tensor-to-scalar ratio limits over a very limited l range. For a similar telescope with a primary aperture the size of the SPT primary, temperature leakage due to beam asymmetry gets pushed to much higher l. PFSBs on this kind of system show very good properties as shown in Figure 9.5. Recalling that the estimated foreground cleaning limits lie near the 10 nK level, we see that this system should be sufficient for any future CMB mission covering up to l = 50. In order to do better than these limits we either need to employ polarization modulation via a half-wave plate or characterize the beam very well and use temperature measurements to subtract the residual leakage. This exercise of setting the PFSBs in an example telescope shows that when housed in an square waveguide, this architecture shows a high polarization efficiency and symmetric enough beams to measure B-modes below l = 50 on an SPT-sized telescope. In a Spider-like telescope, PFSBs alone are not good enough to make significant advances towards detecting gravity waves with the CMB. We would need to supplement a mission like this with either the wave plate or beam measurement method as described above.

172 Chapter 10

CONCLUSION

We have completed a characterization of the PFSB detector architecture. It has been found that the waveguide we mount these detectors in plays a crucial role in the detector’s ability to separate out two orthogonal polarizations. This separation task is aided by choosing a waveguide symmetry that better follows orthogonal separation, such as a square guide rather than a circular one. For the PFSBs mounted in a square guide, refer to part (a) of Figures 8.13–8.15 which show maps of the three characterization parameters (C,X, ∆) defined in Equation (3.10). We carried out these measurements over a 20◦ half-angle beam. To summarize, we note that the polarization efficiency (8.1) averaged over the entire measurement beam is 89%. Also, the rotation angle ∆ shows variations mostly below the 3◦ level. Finally we show, with Figure 8.3, that these detectors exhibit good band definition and rejection of out-of-band radiation. With these qualities, we have argued in Chapter 9, that the PFSBs are a de- sirable technology for future CMB B-mode studies. There we show that a simple telescopic design can utilize PFSBs to give a polarization coupling efficiency of 92% with only 0.7% of this coming from temperature leakage into the polarization signal. This level of temperature rejection is already good enough for the foreseeable future of low-l B-mode investigations when using large telescopes.

173 However, there is still some work to be carried out before PFSBs can be con- sidered a viable focal plane technology. Most importantly of all, it is important to test a fully-functional PFSB pixel with dual-polarizations and multiple frequencies stacked in a single waveguide. Because of a lack of testable detectors, this thesis has not even investigated the simplest of these tasks which is to look into the polarization properties of a dual-polarization pixel. It would also be useful to carry out the characterization measurements found in this work for a system with a much larger aperture stop. This way diffraction effects can be minimized and the real primary aperture illumination can be sampled. With this more accurate data one can gain a better picture of the sky polarization systematics induced by any telescopic system with PFSBs at its focal plane by following the analysis done in Chapter 9. Before closing, we comment on the far better suppression of cross-polar response in our simulation shown by part (b) of Figures 8.13 and 8.14. In addition to this, the rotation angle ∆ shown in Figure 8.15(b) shows mostly the fact that Ludwig’s definition 3 [55] of setting up co-polar directions on the sphere are not aligned with the PFSB absorption direction. The deviations from this are less than 40. We have a hunch that our inability to reproduce these properties in our measurements has to do with the fact that the breaks in the waveguide, needed for insertion of the PFSBs, are so large (∼ 3 mm). New fabrication methods are being developed by Kent Irwin of the NIST Quan- tum Devices Group [37] that will be able to shrink the breaks in the guide down to a size comparable to the propagation wavelengths. This may enable the PFSBs to obtain the much cleaner polarization properties seen in the simulation. To conclude this discussion, first recall that the PFSB architecture allows for

174 polarization and frequency multiplexing that makes full usage of the available telescope sensitivity. With this in mind, the results of this work show that the PFSBs have answers to all of the challenges outlined in Section 2.1: increased sensitivity, good suppression of instrument systematics, and astrophysical foreground rejection aﬀorded by multi-frequency observation. For these reasons, we are excited about the potential role of the PFSB detector architecture in future CMB missions!

175 Appendix A

CHARACTERIZATION SYSTEMATIC CHECKS

We now present some systematic checks that we carried out on our characterization instruments. These tests verify that the tools we use are behaving as expected. Section A.1 covers the various tests of the linearity of the LEGS on our detectors. In Section A.2 we confirm that the radiation emanating from the hot IR source and Kapton film combination in our beam mapper source box (Sec- tion 6.2.1) is unpolarized. Section A.3 closes this appendix with a discussion of how these non-linearities are translated into errors in the characterization parameters (C,X, ∆) that we find from the data.

A.1 LEGS Linearity Tests

When using the LEGS transition for a given measurement it is critical to check its linearity over a power loading range that is comparable to that covered in the measurement in question. To accomplish this task we make use of the polarization eﬃciency curve (3.8) which we reproduce here as

s(ψ) = X + (C − X) cos2(ψ + δ) where s is the optical signal incident on the detector, ψ is the polarization orientation of the light incident on the system, and (C, X, δ) are characterization

176 parameters that are unimportant for the purposes of checking the linearity of the LEGS or any other sensor. This relation gives us a way of modulating the optical power incident on the bolometer in an understandable manner. We then use it to compare to the electrical signal read oﬀ of the detector. Our parametriza- tion of the non-linearity exhibited by the LEGS is based on this comparison. Section A.1.1 will described the LEGS characterizations relevant to our PFSB spectra measurements while Section A.1.2 covers those relevant to our radiation pattern measurements.

A.1.1 Spectra

For the spectra taken with the on-axis system described in Section 7.2.1, we use the LEGS on the low-G¯ PFSB. Let us now describe the linearity testing of this device. Due to the small throughput of this system, this linearity is characterized over the entire range possible for LN2 sources. We place a source directly below the cryostat that is very similar to the beam mapper source box shown in Figure 6.8. The difference here is that this linearity characterization source fills the entire beam by being very near to the cryostat and replacing the small hot IR source with a large piece of LN2 cooled Eccosorb HR-10. We must use a larger chopper to ensure full beam chopping and we also exclude the source box enclosure to fit the new source and chopper. Because of this full beam chopping, any measurement comparing 300 K to 77 K sources with the on-axis system will be subject to non- linearities below the level found in this characterization. We take data with this chopped load between 300 K and 77 K at many different polarizer positions. Figure A.1 shows the results of this test. We send the output of the SQUID readout lock-in shown in Figure 5.2 straight into a data acquisition

177 Figure A.1. On-axis system linearity calibration. The diamonds in the top plot show the amplitudes of the first harmonics in the chopped signals for various polarizer positions. We use arbitrary units here because, ultimately, we are only concerned with a percent deviation from linearity. The solid red line is a fit to Equation (3.8). On the bottom, we show the fit residuals as a percentage of the mean of the data on top. The solid red lines here bracket off the ±rms of these residuals which is printed in red on the upper-left hand corner. system in the same manner as is done with the FTS measurements described in Section 6.1.2. This lock-in demodulates the AC bias but leaves the chopped signals intact. The points plotted here in the top plot are the first harmonic amplitudes of these signals. We get this by taking the PSD of the timestream for a single polarizer position and selecting the peak value near to the first harmonic. The data is fitted to the polarization efficiency curve, Equation (3.8), as outlined above. To obtain a measure of the non-linearities of the LEGS, we calculate the fit residuals (bottom plot) as a percentage of the mean of the data in the top plot.

178 The rms value of these residuals is then quoted as the degree of non-linearity. In our on-axis system spectra measurements we can safely say that the non-linearities are below the 2.1% rms level. For our off-axis spectra measurement systems we use the LEGS on the high- G¯ PFSB. However, for the small aperture system used to represent the circular waveguide PFSB architecture we failed to get linearity measurements. For the square guide system (Section 7.2.3), we carry out the linearity measurement in a slightly different manner than described above. Were we to chop the entire beam between 300 K and 77 K sources, we would see a high level of non-linearity. This is because of the extremely large throughput of the beam mapping systems. For the FTS measurements, we only modulate a very small portion of the beam; therefore we only need to characterize the linearity over this restricted range. To do this, we set the cryostat on the FTS and set the moving roof mirror in the FTS near the equal arm length position. This allows the maximum amount of optical power to traverse the FTS because it lets all frequencies through. We then modulate the signal by placing a chopper in front of the cold (77 K) source of the FTS. The polarization efficiency modulation, Equation (3.8), is then accomplished by rotating the exit polarizer between the FTS and the cryostat. Carrying out the linearity characterization in this manner restricts its throughput to something very similar to the throughputs involved in the FTS measurements. The data acquisition and analysis of this linearity characterization are identical to the process described above for the on-axis system. We take one set of linearity data prior to carrying out the spectra measurements with the cryostat in the on-axis configuration pointed straight into the FTS. The results are shown in Figure A.2. As mentioned in the figure caption, the LEGS resistance during

179 Figure A.2. Square guide system FTS linearity calibration. See Figure A.1 for a description of the plot elements. The LEGS resistance values spanned by this measurement is 457.1 mΩ < R < 459.1 mΩ. this test varied over 457.1 mΩ < R < 459.1 mΩ. At the end of the spectra measurements we carried out another test to get a wider LEGS resistance sampling. In this characterization we oriented the cryostat in the 11◦ oﬀ-axis E-plane con- ﬁguration. Figure A.3 shows the results of this measurement where we now span 458.2 mΩ < R < 460.4 mΩ in LEGS resistance. Since both of these linearity characterizations do not span the entire range of optical power modulation capable for the square guide system for 77 K loads, we must quote the LEGS resistance ranges encountered in our spectra measurements of this system. These are found in Table A.1. We can see from the table that the measurement ranges are almost entirely contained within the ranges spanned by our two linearity characterizations. Taking the larger of the two we summarize

180 Figure A.3. Second square guide system FTS linearity calibration. See Figure A.1 for a description of the plot elements. The LEGS resistance values spanned by this measurement was 458.2 mΩ < R < 460.4 mΩ. the results of these test as showing rms deviations of no more than 4.8% in our square guide system spectra measurements.

A.1.2 Beam Maps

For the linearity characterization of our beam map measurements we take the same data set as done for the spectra measurements above. This time we place the cryostat on the beam mapper, as shown in Figure 6.9, and simply use the source box on the beam mapper with the arms in the on-axis conﬁguration as our linearity characterization source. We also carry out the signal sensing a little diﬀerently than in Section A.1.1. Here we use the lock-in algorithm described in Section 6.2.2 to demodulate the chopped signal so that we remain consistent with

181 Table A.1

Square Guide Spectra R Ranges

Conﬁguration Minimum R (mΩ) Maximum R (mΩ)

on-axis co-polar 458.8 459.7

on-axis cross-polar 457.8 457.9

E-plane co-polar 458.9 460.5

E-plane cross-polar 458.2 458.4

H-plane co-polar 458.6 460.1

H-plane cross-polar 457.1 457.2 the way that the beam mapper data is sensed. The beam measurements for the large aperture system described in Section 7.2.2 (representing the circular waveguide PFSB architecture) span a LEGS resistance range of 398.3 mΩ < R < 398.9 mΩ. Table A.2 summarizes the results of the linearity characterizations of this system. We do not have a single linearity measurement that spans the entire range given above. However, all of the characterizations cover a very similar extent and we have four of them at diﬀerent places all showing a very low level non-linearity. Thus for our circular waveguide PFSB beam maps we quote at most a 2.2% rms deviation from linearity.

For the square guide system (Section 7.2.3), our beam measurements span a larger range: 479.4 mΩ < R < 480.4 mΩ. Table A.3 summarizes the results of our linearity characterizations of this system. In this group, we have one measurement that is in the above range but about half the extent. Because we do not

182 Table A.2

Large Aperture Beam Map Linearity Characterizations

Minimum Maximum rms linearity Characterization R (mΩ) R (mΩ) error (%)

1 398.0 398.5 1.9

2 398.9 398.4 1.9

3 399.9 393.4 1.9

4 394.9 395.4 2.2 have linearity data over the full TES resistance range of our beam measurements of this system, we must look a little closer at this restricted characterization data set. An inspection of Figure A.4 shows that the rms linearity error is dominated by noise in the measurement. A comparison to the spectra linearity characterizations shown in Figures A.1 and A.3 depicts a signiﬁcant diﬀerence between the non-linearity exhibited here, and the obviously systematic non-linearities exhibited in these previous characterizations. In the current case, it is not so clear that doubling the resistance range of any given measurement, beyond that which occurred with this second linearity characterization, would give rise to an increase in non-linear behavior. Although these tests were somewhat inconclusive, based on the above argument we estimate that the non-linearities exhibited in our square guide beam maps do not reach above the 5% rms level.

183 Figure A.4. Second square guide system beam mapper linearity calibration. The diamonds in the top plot show the outputs of the software lock-in method described in Section 6.2.2 for various polarizer positions. We use arbitrary units here because we are only concerned with a percent deviation from linearity for the end results. The solid red line is a fit to Equation (3.8). On the bottom we show the fit residuals as a percentage of the mean of the data on top. The solid red lines here bracket off the ±rms of these residuals which is printed in red on the bottom left hand corner. The TES resistance values spanned by this measurement was 479.6 mΩ < R < 480.0 mΩ.

A.2 Beam Mapper Source Characterization

This section discusses a measurement aimed at confirming that the IR source and Kapton film combination inside the beam mapper source box (Figure 6.8) are unpolarized. We take data sets equivalent to the linearity characterizations discussed above (i.e. the data set shown in Figure A.4) with three different orientations of the PFSB test cryostat. The three orientations (data is shown in Figure A.2) are the orientation which the beam maps are taken, and rotations

184 Table A.3

Square Guide Beam Map Linearity Characterizations

Minimum Maximum rms linearity Characterization R (mΩ) R (mΩ) error (%)

1 458.6 459.1 1.9

2 479.6 480.0 2.2

3 480.6 481.1 1.9 of the cryostat by roughly 45◦ and 90◦ about the waveguide symmetry axis. In the figure, we have shifted the polarizer angles by an amount determined by a fit of each individual data set to Equation (3.8). In this manner, we get a good alignment of all the peaks in the data. The values found by these fits are within 3◦ of the rotations specified above. This is well below our measurement errors of this angle. We then carry out another fit of Equation (3.8) to all of the data together. From the unified fit residuals, we see that there is a small systematic difference in the three data sets; however, the rms of the entire data set is below the 2% level confirming that the source box outputs highly unpolarized light before encountering the polarizer.

A.3 Fitted Parameter Errors

To get some feel for how percent rms deviations from the polarization eﬃciency curve (3.8) translates into errors in the parameters (C,X, ∆) that we ﬁt out from the beam map data presented in Chapter 8, we simulate two types of deviations. Both of these can be thought of as arising either from non-linearities in the LEGS

185 Figure A.5. Beam mapper source box characterization. This data is taken with the small aperture system of Section 7.2.2. The symbols in the top plot show the outputs of the software lock-in method (Section 6.2.2) for various polarizer and cryostat positions. Purple squares correspond to the cryostat being in an orientation identical to that of the beam mapping measurements, blue triangles denote the cryostat rotated by roughly 45◦ about the waveguide symmetry axis, and green diamonds represent a roughly 90◦ rotation. The polarizer angles of this data set have been shifted to account for the cryostat orientation. We use arbitrary units here because we are only concerned with a percent deviation from the unpolarized situation for our end results. The solid red line is a fit of all of the data to Equation (3.8). On the bottom we show the fit residuals as a percentage of the mean of the data on top. The solid red lines here bracket off the ±rms of these residuals which is printed in red on the bottom. The rms deviation shown here characterizes the source variation as a function of cryostat orientation and its low value implies a low level of source polarization. as characterized in Section A.1, or from a small partial polarization to the source on the beam mapper as explored in Section A.2. The fiducial curve that we use for this investigation has a cross-polar level at

186 5.4% of the co-polar level and a rotation angle that we vary. This is similar to the red curve shown in the top of Figure A.4. We sample this curve at the same six polarizer angles that are used in the beam mapping data (see Section 6.2.2). The rms deviations described below are then added to these sampled data points and we fit the polarization efficiency curve to the resultant data. We then compare the parameters extracted from this fit to the fiducial curve parameters and quote the deviations as the errors in our beam map measurements arising from LEGS non-linearity and beam mapper source polarization. The first deviation we investigate is a systematic one of the nature found in Figures A.1 and A.3. Notice that the residuals found here modulate at a frequency twice that of the radiometric signal. This is characteristic of the simplest form of gain compression with only first order deviations from linearity. As observed, we implement this by using a cosine deviation in the procedure outlined above, with a period of 2π/4. The amplitude we chose results in an rms of 5% which is just slightly larger than the largest observed deviation from linearity shown in Figure A.2. Comparing the fitted parameters to the fiducial curve parameters then shows that the errors due to this systematic deviation are negligible (not more than 0.0001% in all parameters for a range of rotation angles of the fiducial curve) probably due to the deviation arising at a different frequency than the extracted signal. We also try a random deviation which is highly unlikely for the types of errors we are trying to simulate but should cover the range of possibilities. This is pulled from a Gaussian random distribution with a 5% rms. After trying many instances, the fitted parameters show errors in the co- and cross-polar levels of less than 3% of the co-polar level and not more than 2◦ in the rotation angle.

187 Now we apply this to our square waveguide results shown in Figure 8.21 where we see that the smallest level of cross-polar response we observe comes on-axis at 5%. Thus we see that the 3% errors that we may attribute to non-linearities in the LEGS and a small amount of partial polarization of the beam mapper source can not explain the non-vanishing cross-polar level we are seeing.

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