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Microwave Cavities for Axion Dark Matter Detectors

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Title:
Microwave Cavities for Axion Dark Matter Detectors
Creator:
Stern, Ian Phillip
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (201 p.)

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Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
TANNER,DAVID B
Committee Co-Chair:
SULLIVAN,NEIL S
Committee Members:
SIKIVIE,PIERRE
STANTON,CHRISTOPHER JAY
CRANE,CARL D,III

Subjects

Subjects / Keywords:
admx -- amc -- anneal -- annealing -- antenna -- array -- avoided -- axion -- axions -- bead -- bead-pull -- breaking -- capp -- cavities -- cavity -- cdm -- coating -- coherent -- combine -- combining -- comsol -- configuration -- cp -- crossing -- crowding -- dark -- detection -- detector -- dfsz -- direct -- direct-detection -- electric -- electromagnetic -- electropolish -- factor -- flux -- form -- form-factor -- frequency -- frequency-spectrum -- frequncy-tuning -- gap -- haloscope -- haloscopes -- haystac -- height -- high -- high-frequency -- hole -- hybrid -- hybridization -- identification -- in-situ -- incoherent -- invariance -- ksvz -- longitudinal -- magnetic -- map -- mass -- matter -- mechanical -- method -- microwave -- misalignment -- mode -- mode-crossing -- mode-crowding -- mode-density -- mode-map -- modes -- multiple -- orthogonality -- particle -- partition -- partitioned -- penetration -- periodic -- periodic-array -- post -- power -- power-combining -- probe -- problem -- pull -- quality -- quality-factor -- radius -- rate -- ratio -- resonant -- rod -- rotation -- rotational -- scan -- scan-rate -- search -- spectrum -- strong -- superconducting -- symmetry -- symmetry-breaking -- te -- test -- testing -- thin-film -- tilt -- tm -- tm010 -- translate -- translational -- transverse -- transverse-electric -- transverse-magnetic -- tunable -- tunable-cavity -- tune -- tuning -- vane -- volume
Physics -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
Nearly all astrophysical and cosmological data point convincingly to a large component of cold dark matter (CDM) in the Universe. The axion particle, first theorized as a solution to the strong charge-conjugate/parity problem of quantum chromodynamics, has been established as a prominent CDM candidate. Cosmic observation and particle physics experiments have bracketed the unknown mass of such CDM axions between approximately microelectronvolts and millielectronvolts. The Axion Dark Matter eXperiment (ADMX) is a direct-detection CDM axion search which has set limits at the KSVZ coupling of the axion to two photons for axion masses between 1.9 and 3.7 microelectronvolts. But most of the viable search-space of axions has yet to be probed. It is now evident that the current cavity design for ADMX does not provide for successful scanning above ~4 microelectronvolts (~1 gigahertz), and new microwave cavity technologies need to be developed to enable searching for CDM axions of greater mass. A broad study into factors that affect high-frequency microwave cavity was conducted in an effort to expand the capabilities of current haloscope detectors. The study evaluated the effects of symmetry breaking on axion search potential and assessed state-of-the-art cavity technologies relevant to haloscope detectors. Periodic arrays of tuning rods or vanes were evaluated for increasing the search range of haloscopes, and a prototype haloscope cavity and an in-situ mode identification technique were developed and tested. The study revealed several significant findings. The cause for frequency gaps in search modes at mode crossing is mode mixing, which occurs due to longitudinal symmetry breaking. Symmetry breaking also results in increased mode crowding, which hinders mode identification and tracking, and further increases mode mixing. Arrays of tuning rods or vanes produce tunable modes, but result in increased sensitivity to symmetry breaking, requiring advanced mode identification techniques and additional studies. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2017.
Local:
Adviser: TANNER,DAVID B.
Local:
Co-adviser: SULLIVAN,NEIL S.
Statement of Responsibility:
by Ian Phillip Stern.

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MICROWAVE CAVITIES FOR AXION DARK MATTER DETECTORS By IAN P. STERN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2017

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2017 Ian P. Stern

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To Mom mom and Pop pop, who were unable to witness this moment in my life

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4 ACKNOWLEDGMENTS I wish to thank my advisor, Dr. David Tanner, and Dr. Pierre Sikivie and Dr. Neil Sullivan for years of guidance, assistance teaching, mentoring, and enabling this research to transpire I thank Dr. Christopher Stanton for being an outside observer and a constant check on the technical worth of my work. I also thank my external committee member, Dr. Carl Crane, for his years of support during my engineering studies and throughout my engine ering career leading up to my Ph.D. program; without you, I could have never made the successful leap into physics. I especially thank Martyna Levay for years of emotional support and dedication including uncoun table hours of proofreading, keeping me fo cused and on track helping me with planning and organizing, and putting up with more than she should have had to I thank my fellow graduate students Manish Amin, Alex Markoff Nilanjan Banik, and Chad Hopkins for their friendship and assistance with passing my classes. I would like to thank Dr. Gp Carosi and Dr. Gray Rybka for being role models as superior young scientists, and Dr. Lesley Rosenberg for his leader ship on ADMX. I acknowledge my ADMX collaborators Ch ristian Boutan, Jim Sloan, and all the others for assisting me through diffic ult technical challenges in my investigations I thank the staff at the University of Florida Department of Physics, especially Marc Link, Bill M alphurs Pete Axson, Greg Labbe, Clint Collins, Pam Marlin, and Janet Ge rmany, for years of support. I thank Byron Knight for his ongoing mentorship, wisdom, and encouragement. Finally, I wish to thank my mother for birthing me, feeding me, giving me a home, raising me, nurturing me, encouraging me, and always believing in me It is without doubt that I could have never accomplished anything without you.

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5 I acknowledge funding support by the U.S. Department of Defense through the National Defense Science and Engineering Graduate Fellowship Program, the National Aeronautics and Space Administration through the Florida Space Research Program, and the Armed Forces Communications and Electronics Association through the Intelligence Studies Scholarship award, as well as the Department of Energy through the Axion Dark Matter eXperime nt under Grant No. DE SC0010280

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6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 ABSTRACT ................................ ................................ ................................ ................... 14 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 16 2 BACKGROUND ................................ ................................ ................................ ...... 20 2.1 Axion Partic le ................................ ................................ ................................ 20 2.2 Dark Matter ................................ ................................ ................................ ... 22 2.3 CDM Axions ................................ ................................ ................................ .. 31 2.4 Axion Halo scope Detectors ................................ ................................ ........... 38 3 MICROWAVE CAVITIES FOR AXION HALOSCOPE DETECTORS ..................... 50 3.1 Microwave Cavity Theory ................................ ................................ .............. 50 3.2 Symmetry Breaking ................................ ................................ ....................... 61 3.3 Mode Identification Methods ................................ ................................ ......... 91 3.4 High Frequency Cavities ................................ ................................ ............. 100 3.5 Superconducting Hybrid Cavities ................................ ................................ 118 4 12 VANE CAVITY PROTOTYPE ................................ ................................ .......... 123 4.1 Design ................................ ................................ ................................ ......... 123 4.2 Fabrication ................................ ................................ ................................ .. 134 4.3 Test App aratus ................................ ................................ ............................ 140 4.4 Test Results ................................ ................................ ................................ 146 5 IN SITU MODE TEST ................................ ................................ ........................... 166 5.1 Test Methods ................................ ................................ .............................. 166 5.2 Test Results ................................ ................................ ................................ 172 6 CONCLUSION ................................ ................................ ................................ ...... 185 6.1 Future Wor k ................................ ................................ ................................ 185 6.2 Summary ................................ ................................ ................................ ..... 190 LIST OF REFERENCES ................................ ................................ ............................. 194

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7 BIOGRAPH ICAL SKETCH ................................ ................................ .......................... 201

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8 LIST OF TABLES Table page 3 1 Configuration of cavities with the highest scan rate s from the periodic array study ................................ ................................ ................................ ................. 115 4 1 Quality factors of the 12 vane prototype for di fferent channels during testing .. 164 5 1 Predicted combined power variables of modes shown in Fig. 5 2 .................... 170 5 2 Results of the empty cavity in situ mode test ................................ .................... 175 5 3 Results of the 2 rod cavity in situ mode test in the minimum TM 010 frequency configuration ................................ ................................ ................................ ..... 179 5 4 Results of the 2 rod cavity in situ mode test in the maximum TM 010 frequency configuration ................................ ................................ ................................ ..... 181 5 5 Results of the 2 rod cavity in situ mode test in the mid range TM 010 frequency configuration ................................ ................................ ................................ ..... 182 5 6 Results of the 12 v ane prototype in situ mode test ................................ ........... 184

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9 LIST OF FIGURES Figure page 2 1 Baryonic matter observed in the coma cluster ................................ .................... 23 2 2 Rotation curve of Galaxy M33 ................................ ................................ ............ 23 2 3 Rotation curve of nearby galaxies ................................ ................................ ...... 24 2 4 Gravitational lensing of the Abell 2218 galaxy cluster ................................ ........ 25 2 5 Observations from the Bullet Cluster ................................ ................................ .. 26 2 6 Limits on percent of baryons in the universe from nucleosynthesis .................... 27 2 7 Cosmic microwave backgrou nd measurements from Planck ............................. 28 2 8 Power spectrum of the cosmic microwave background measurements ............. 28 2 9 ....... 29 2 10 Visualization of current theories of dark matter ................................ ................... 30 2 11 Feynman Diagram of Axion photon Interaction ................................ .................. 34 2 12 Axion exclusion plot near the CDM axions mass range ................................ ...... 36 2 13 Local kinetic energy distribution of galactic axions ................................ ............. 39 2 14 Schematic of mechanical tuning of haloscope cavities ................................ ....... 41 2 15 Graphic representation of a haloscope detector using a solenoid magnet and a cylind rical microwave cavi ty, and an anticipated detection signal of CDM axions ................................ ................................ ................................ ............... 46 2 16 Schema tic of ADMX receiver chain and photograph of the inside of the ADMX microwave cavity ................................ ................................ ..................... 47 2 17 Axion exclusion plot showing published results of A DMX ................................ ... 48 3 1 Cross sectional images of the E z field in an empty cylindrical cavity .................. 53 3 2 Cross sectional images of the E z field in a cylindrical cavity with a coaxial conducting rod ................................ ................................ ................................ .... 56 3 3 Cross sectional images of the E z field in a cylindrical cavity with a coaxial dielectric rod ................................ ................................ ................................ ....... 57

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10 3 4 Antenna coupled to a cavity through a wall or endplate ................................ ..... 59 3 5 Cross sectional images of the E z field in a cylindrical cavity with an offset conducting rod ................................ ................................ ................................ .... 63 3 6 Example of the electric field of the TM 010 mode inside a tuned cylindrical cavity depicting the curvature of the field ................................ ............................ 63 3 7 Plot of the TM 010 frequency and form factor as a function of tuning rod displacement for a circular cylindrical cavity ................................ ....................... 65 3 8 Cross sectional images of the E z field for the TM 010 mode in a cylindrical cavity with multiple conducting rods ................................ ................................ ... 66 3 9 Cross sectional images of the E z field for the TM 110 modes in a cylindrical cavity with two conducting rods ................................ ................................ .......... 68 3 10 Cross sectional images of the E z field for higher order modes in a cylindrical cavity with two conducting rods ................................ ................................ .......... 68 3 11 Cross sectional images of the E z field for modes in a cylindrical cavity with two conducting rods ................................ ................................ ............................ 70 3 12 Cross sectional images of the E z field in a cylindrical cavity with an offset dielectric rod ................................ ................................ ................................ ....... 70 3 13 Plot of a TM 010 mode crossing during frequency tuning with a conducting rod for a circular cylindrical cavity ................................ ................................ ............. 72 3 14 Cross sectional images of | E | of the TM 010 mode in a coaxial cavity .................. 74 3 15 Plots of mode repulsion and search mode form factor for a circular cylindrical cavity with a discontinuous conducting tuning rod ................................ .............. 76 3 16 Plots showing the hole in the frequency search spectrum produced by longitudinal symmetry breaking ................................ ................................ .......... 80 3 17 Plot of hole in frequency search spectrum as a function of rod end gap size ..... 81 3 18 Cross sectional images of | E | with E vectors of the hybrid modes and the TE 215 mode at the point of maximum mode repulsion in a cylinder cavity .......... 83 3 19 Plot showing mode crowding due to increased mechanical gap at the rod ends ................................ ................................ ................................ ............... 85 3 20 Cross sectional images of | E | of the TM 010 mode in a tuned cavity with a rod tilt (angular misalignment) ................................ ................................ .................. 87

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11 3 21 Plots of mode repulsion and search mode form factor for a circular cylindrical cavity with a tilted conducting tuning rod ................................ ............................ 88 3 22 Plots of frequency and search mode form factor for a circular cylindrical cavity with a conducting tuning rod tilted 1.80 ................................ ................... 90 3 23 Example of frequency predictions from ADMX ................................ ................... 93 3 24 Example of a TM 010 mode map from ADMX ................................ ....................... 95 3 25 Example of a TM 020 mode map without predictions from ADMX ......................... 96 3 26 Sche matic of bead pull test and test results from a bead pull test ...................... 99 3 27 Plot of frequency range and form factor for a circular cylindrical cavity with a conducting tuning rod at varying diameters ................................ ...................... 102 3 28 Schematic of a 6 cavtiy packing ................................ ................................ ....... 105 3 29 Photograph of the inside of a 2 cell partitioned cavity ................................ ...... 106 3 30 Cross sectional images of the E z field of the TM 110 modes in a 8 cell partitioned cylindrical cavity ................................ ................................ .............. 107 3 31 Plot of effective volume verses TM 010 frequency for power combined cavities ................................ ................................ ................................ ......... 108 3 32 Plot of form factor verses frequency for the three lowest search modes of ADMX ................................ ................................ ................................ ............. 111 3 33 Geometry of example cavities from periodic array study ................................ .. 113 3 34 Cro ss sectional images of the E z field for the TM 010 mode in periodic arrays ... 114 3 35 Cross sectional images of the E z field for the TM 010 mode in a 19 rod periodic array cavity ................................ ................................ ................................ ....... 117 3 36 Schematic of perpendicular magnetic fields in superconducting hybrid cavities ................................ ................................ ................................ ............ 122 4 1 Schematic of a cross section through the x y plane of the 12 vane prototype .. 124 4 2 Plot of C 010 verse f 010 for the 12 vane cavity ................................ ..................... 125 4 3 Frequency predictions for the 12 vane prototype from a 2 dimensional analysis ................................ ................................ ................................ ............ 126 4 4 Cross sectional images of the E z field for the TM 010 mode of the 12 vane prototype design with bottom antenna positions overlaid ................................ 127

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12 4 5 CAD models of the 12 vane prototype ................................ .............................. 128 4 6 FEMs of the 12 vane prototype ................................ ................................ ........ 129 4 7 Cross sectional images of | E | of the TM 010 mode in 12 vane prototype as resolved by FEM simulation ................................ ................................ .............. 130 4 8 Cross sectional images of | E | of the TM 010 mode in the 12 vane prototype as resolved by the complex FEM simulation ................................ ......................... 131 4 9 Plot of the results from a convergence study for the complex 12 vane FEM model ................................ ................................ ................................ ........... 132 4 10 Frequency predictions for the 12 vane prototype from complex FEM simulations ................................ ................................ ................................ ........ 133 4 11 Select views from fabrication drawings of the 12 vane prototype ..................... 135 4 12 Plot of a generic electropolishing current density curve ................................ .... 136 4 13 Photograph of four vanes partly assembled ................................ ..................... 137 4 14 Photograph of prototype vanes assembled to the cross bracket and the bottom plate ................................ ................................ ................................ ...... 138 4 15 Photograph of prototype assembly with the top plate removed ........................ 139 4 16 Photograph of 12 vane prototype assembly ................................ ..................... 140 4 17 Schematic of 12 vane prototype test apparat us ................................ ............... 141 4 18 Photograph of the gearbox for the 12 vane prototype ................................ ...... 143 4 19 Photograph the 12 vane cavity cryostat insert ................................ .................. 145 4 20 Photograph the 12 vane cavity test apparatus ................................ ................. 146 4 21 Cross sectional images of the E z field of the TM 010 mode in the 4 vane cavity with bottom antenna positions overlaid ................................ ............................. 147 4 22 Power spectrum plots for the 4 vane cavity prototype ................................ ...... 148 4 23 Power spectrum plots for the 12 vane prototype at room temperature ............. 150 4 24 Mode maps for the 12 vane prototype ................................ .............................. 152 4 25 Mode map for the 12 vane prototype for channels S HE and S HF ....................... 153 4 26 Color plot mode maps for the 12 vane prototype ................................ ............. 154

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13 4 27 Plot showing the frequency step size of the 12 vane prototype ........................ 156 4 28 Power spectrum plots fo r the 12 vane prototype showing frequency discrepancies of a mode ................................ ................................ ................... 158 4 29 Partial measurement mode map in a LN 2 bath for the 12 v ane prototype ........ 159 4 30 Cryostat level detector reading during a L 4 He fill ................................ .............. 160 4 31 Maximum motor speed of the failing motor as a function of temperature ......... 161 4 32 Analytical predictions for the 12 vane prototype during L 4 He testing ................ 162 4 33 Partial measurement mode map in a L 4 He bath for the 12 vane prototype ...... 163 4 34 Measured Q L of the 12 vane prototype verses temperature ............................. 165 5 1 Cross sectional images of the E z field for the TM 010 mode in cylindrical cavities with various tuning configurations depicting parity symmetries ........... 168 5 2 Cross sectional images of the E z field of various modes in cylindrical cavities with two non symmetric, conducting tuning rods and two antenna locations .... 169 5 3 Schematic of in situ mode test ................................ ................................ .......... 171 5 4 Cross sectional images of the E z field in an empty cylindrical cavity depicting the location of the top plate in situ mode test antennas ................................ ... 173 5 5 Power spectrum plots for the empty cavity in situ mode test ............................ 176 5 6 Cross sectional images of the E z field for the TM 010 in a cylindrical cavity with two conducting tuning rods in the in situ test configurations depicting the location of the top plate antennas ................................ ................................ ..... 177 5 7 Power spectrum plots for the 2 rod cavity in situ mode test in the minimum TM 010 frequency configuration ................................ ................................ .......... 177 5 8 Cross sectional images of the E z field of the TM 020 in a cylindrical cavity with two conducting tuning rods in the in situ test configurations depicting t he location of the top plate antennas ................................ ................................ ..... 178 5 9 Power spectrum plot for the 4 vane cavity in situ mode test ............................ 183

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14 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MICROWAVE CAVITIES FOR AXION DARK MATTER DETECTORS By Ian P. Stern D e c e m b e r 2017 Chair: David B. Tanner Major: Physics Nearly all astrophysical and cosmological data point convincingly to a large component of cold dark matter (CDM) in the Univ erse. The axion particle, first theorized as a solution to the strong CP (charge conjugation times parity) problem of quantum chromodynamics, has been established as a prominent CDM candidate. Cosmic observation and particle physics experiments have bracke ted the unknown mass of such eXperiment (ADMX) is a direct detection CDM axion search which has set limits at the But most of the via ble search space of axions has yet to be probed. It is now evident that the current cavity design for ADMX does not provide for successful scanning above ~4 eV (~1 GHz), and new microwave cavity technologies need to be developed to enable searchin g for C DM axions of greater mass. A broad study into factors that affect high frequency microwave cavity was conducted in an effort to expand the capabilities of current haloscope detectors. The study evaluated the effects of symmetry breaking on axion search pot ential and assessed state of the art cavity technologies relevant to haloscope detectors. Periodic arrays of tuning rods or

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15 vanes were evaluated for increasing the search range of haloscopes, and a prototype haloscope cavity and an in situ mode identificat ion technique were developed and tested. The study revealed several significant findings. The cause for holes in the frequency search spectrum at mode crossing s is hybridization modes (mode mixing) which occurs due to longitudinal symmetry breaking. Sym metry breaking also results in increased mode crowding, which hinders mode identification and tracking, and further increases mode mixing. Arrays of tuning rods or vanes produce tunable modes, but result in increased sensitivity to symmetry breaking, requi ring advanced mode identification techniques and additional studies.

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16 CHAPTER 1 INTRODUCTIO N The detection of the axion is of significant interest to particle physics and to cosmology. The axion is a theoretical pseudoscalar particle [1 2] which was originally postulated to explain why P (parity) and CP (charge conjugation times parity) are conserved by the strong interactions [3] If the axion mass were in the range of ~10 6 10 3 eV, the axion would be a natural cold dark matter (CDM) candidate [4] With recent diminutions of the available par ameter space for the weakly interacting massive particle (WIMP) [5] the axion is now the most promising candidate for the constitution of cold dark matter and thi s has generated a proliferation of searches around the world. String theory further suggests the simultaneous presence of many ultra light axion like particles, possibly populating each decade of mass down to the Hubble scale of 10 33 eV [6] Axion detection has proven to be extremely challenging. CDM axions have a lifetime vastly greater than the age of the universe [5] have exceptionally weak interactions with matter and radiation, and were originally thought to be "invisible" to all detection techniques. However, in 1983 Sikivie [ 7] proposed a method by which these axions plausibly could be detected. He showed the decay rate of axions to photons can be greatly increased within a strong magnetic field, through the i nverse Primakoff [8] effect. The Lagrangian for the axion photon interaction is given by ( 1 1 ) where g is the axion photon coupling consta nt, proportional to the mass of the axion, a is the axion field, and E and B are the electric and magnetic fields of each photon

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17 respectively. The coupling allows the axion to decay to two photons (see Fig. 2 11) In a static external magnetic field, an axion may convert to a photon whose energy equals the total energy of the axion. The B in Eq. 1 1 is effectively changed to the static magnetic field, B 0 Thus, as the external magnetic field strength is increased, so is the conversion rate of the axion. This process is effective for both relativistic or non relativistic axions [9] A n axion haloscope detector uses a microwa ve cavity permeated by a strong magnetic field to convert axions to photons via the inverse Primakoff effect The axion photon conversion is resonantly enhanced when the frequency of a cavity mode is f mnp m a c 2 / h where c is the speed of light in vacuum and h and the mode sufficiently couples to the axion photon interaction (see Chapter 2.4) From Eq. 1 1 it can be shown that the coupling strength of a reso nant mode to the axion photon interaction d 3 x B 0 E mnp ( x ), where E mnp is the electric field of the mode. The indices m n and p identify the various modes. The power produced in the cavity for a particular mode is given by (1 2) where m a is the mass of the axion, a is the local mass density of the axion field, V is the volume of the cavity, and Q L is the loaded quality factor of the cavity (assumed to be less than the effective Q of the axion signal ) [10] C mnp is the normalized form factor describing the coupling of the axion to a specific mode. It is given by (1 3) where ( x ) is the permittivity within the cavity n ormalized to vacuum.

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18 The mass of CDM axions is constrained by cosmological [11 13] and astronomical [14] m a meV. Haloscope detectors conduct axion searches by incrementally tuning frequency of a coupling resonant mode within the microwave cavity, and scanning for a signal above the background noise. To date only a very small fraction of the viable mass range has been searched with sufficient sensitivity to detect the invisible axion. Current haloscope detectors a re gr eatly hindered by the frequency scan range of existing microwave cavity technology [15 16] (see Chapter 2.4 ) A study of microwave cavities for axion dark matter detectors was conducted in an effort to expand the capabilities of current haloscopes. Computer simulations were used to evaluate u tilizing periodic arrays of tuning rods and vanes for axion detector s and to assess the effects of symmetry breaking in cavities on the detection capabilities of detectors. The analysis showed that such periodic arrays could be used to increase and tune the search frequencies of m icrowave cavities used in axion detectors while maintain a large volume (see Chapter 3.4) The simulations further showed that symmetry breaking resulted in reduced sensitivity, mode mixing which caused gaps in the detector scan range, and mode crowding th at increased the likeliness of mode mixing and complicated mode detection and tracking (s ee Chapter 3.2) A cavity prototype was developed and tested to demonstrate the use of tuning vanes for axion detectors (see Chapter 4) The testing was conducted at temperatures of 300 K, 77 K, and 4.2 K. The results showed that tuning vanes adjusted the frequency of the modes the cavity. However, the prototype demonstrated the sensitivity of modes to symmetry breaking. Due to mechanical misalignments, fabrication def ects, and

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19 design fla ws, significant mode mixing, mode crowding and mode localization observed in the cavity prevented the desired search mode from being positively identified. To improve mode identification capabilities, an in situ mode test was develope d (see Chapter 5) The test combines power from two antenna ports to determine if the field at the antenna locations are in phase or out of phase. By comparing the phases of the fields at multiple locations and comparing the results to simulatio n predictio ns, different TM modes could be distinguished as well as certain TE modes. The test method was demonstrated successfully on an empty cylindrical cavity and a cylindrical cavity with two tuning rods. However, the method was not successfully demonstrated on the prototype multivane cavity.

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20 CHAPTER 2 BACKGROUND 2.1 Axion Particle The axion is a theoretical pseudoscalar particle which was originally postulated to explain, within the framework of the standard model of particle interactions, why P (parity) and CP (charge conjugation times parity) are conserved by the strong interactions. The Lagrangian for Quantum Chromodynamics (QCD) contains a CP symmetry violating term [17] (2 1) where is the gluon strength tensor and g is a colorless coupling. The parameter is a consequence of the non abelian nature of QCD, and is an observable parameter with a value between and [18 19] Without further information, would be expected to be of order 1. As the neutron is composed of quarks, the violation of CP symmetry in QCD should also be observed in the neutron. Indeed, the existence o f an electric dipole moment in the neutron would violate CP symmetry, and the strength of the moment could be used to determine the value of [20 22] To date, no experiments have detected a neutron electric dipole moment, with an upper limit being placed at 3.0 x 10 26 e cm (at 90% CL) [23 24] The resulting upper limit of is ~10 10 which is many orders of magnitude less than predict ed. This ex tremely small upper bound on is the so CP A solution to the strong CP problem was proposed by Peccei and Quinn [3] By int ro ducing a new U(1) symmetry to the early universe becomes a dynamical

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21 parameter by obtain ing field dependent term a / f a where a is a complex pseudo scalar fi eld introduced by the symmetry and f a is an energy scale ( temperature ) T he Peccei Quinn (PQ) symmetry was spontaneous broken when the universe cooled below f a causing < a > to settle to a minimum and to naturally relax to 0. Weinberg [1] and Wilczek [2] individually showed breaking the PQ symmetry results in the creation of a new pseudo Nambu Goldst one boson, dubbed the axion Though massless at the classical level, t he explicit breaking of the symmetry gives the axion a potential. Oscillations about the minimum of this potential result in the axion acquiring a mass. The mass of the axion arises from the explicit breaking of PQ symmetry by instanton effects and is related to f a by [5] (2 2) where the pi on mass and energy scale are m = 135 MeV and f respectively T he canonical value of z = m u / m d = 0.56 is used where m u and m d are the masses of the up and down quark s, respectively. Initially the PQ symmetry was believe d to break around the weak scale, f a ~ 100 keV [14] known as the PQWW ( Peccei Quinn Weinberg Wilczek ) model. However, beam dump experiments ruled out any such models [25] Current axion models allows f a to be orders of magnit ude higher than the weak scale, which predict a drastically reduced mass of the axion from the PQWW model Additionally the increase in energy scale also increases the difficulty detecting an axion (see Chapter 2.4) and such models are often called the accepted invisible axion models are the KSVZ ( Kim Shifman Vainshtein Zhakharov ) model [26 27] and the DFSZ (Dine Fischler Srednicki Zhitnitsky) model [28 29] The

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22 KSVZ model, which predates the DFSZ model by about a year, is sometimes referred to model adds a new, electrically neutral heavy quark that carries a U(1) PQ charge to the Standard Model. The DFSZ model requires at least two Higgs doublets, and ordinary quarks and leptons carry PQ charges. The KSVZ axion couples to the heavy quark but not to leptons, while the DFSZ axion couples to both quarks and leptons [5] 2.2 Dark Matter Significant astrophysical and cosmological evidence for the existence of dark matter exists I n 1933, red shift measurements from the Coma Clust er showed a large disparity between the measured velocity dispersion of the galaxies and the predicted values based on the computed luminous mass [30] Velocity mea surements were used to compute the lower limit on the average mass to light ratio [31] ( 2 3 ) where M and L are the mass and luminosity of the sun, respectively. Th e results were two orders of magnitude larger than expected and led to the claim that the Coma Cluster contained non luminous matter Figure 2 1 shows the observed luminous ma tter in the Coma Cluster. Measurements of the rotational veloci ties of spiral galaxies showed the velocity did not exhibit Keplerian motion beyond the cluster bulge, scaling to 1/ r 2 (the inverse of the radius squared). Instead, the velocities at large radii demonstrated rotational motion in an isometric potential, ma intaining a constant velocity regardless of radius [34] The results strongly suggested a large quantity of non observable matter. Figure 2 2 shows the rotation curved of galaxy M33 [35] demonstrating a near constant velocity beyond

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23 (a) (b) Figure 2 1 Baryonic matter observed in the coma cluster (a) Optical light [32] (b) X ray light [33] Velocity dispersion measurements of galaxies suggested significant non lumin ous mass was present. Figure 2 2 Rotation curve of G alaxy M 33 [36] The dashed white line shows Keplerian m otion assuming a gravitational potential based on light measurements The s olid yellow line shows rotation in a n isothermal halo. The data suggest that significantly more matter is present than can be observed by light emission the cluster bulge. The dotted lines show the predicted curve based on the luminous matter. Figure 2 3 shows the rotational curves for four additional galaxies demonstrating simil ar behavior.

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24 Figure 2 3 Rotation c urve of n earby g alaxies [37] Data suggests the galaxies are rotating in an isothermal halo. Analyses of gravitational lensing have further reinforced the theory of dark matter. As light from distance galaxies passes by a massive gravitational potential, the light is bent by the potential. Observations in the near field depict curved light signatures around the mass sources. Lensing analyses compute the mass of the source. A vast number of w eak and strong lensing studie s of galaxy clusters have demonstrated a need for additional non luminous matter in the universe [38 40] In some instances, the observed lensing is visible without augmentation or computation. Figure 2 4 shows the gravitational lensing observed in the Abell 2218 galaxy cluster [41] The lensing is clearly visible on the left side of the figure as depicted by the curved, elongated galaxies The mass needed to produce such lensing is ten times greater than is perceived through light measurement. Recently discovered galaxy cluster mergers have revealed some of the strongest evidence for dark matter to date. The x ray gas from galaxy clusters that merged in the past scattered from collisions, forming a bulk luminous mass around the intersection point. Gravitational lensing studies show the majority mass passed through the intersection point without interaction, producing two separate mass bulge s [43 44] To

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25 Figure 2 4 Gravitational l ensing of the Abell 2218 galaxy c luster [42] Gravitational lensing of light is visible without augmentation. Weak lensing analyses predict ten times more mass is present than is observed. date, there is no scientific explanation for these observations besides the existence of dark matter. Figure 2 5 shows the observations from the Bullet Cluster merger ( 1E 0657 558 ) in the optical and x ray (shown in pink) spectru m s. The yellow lines are the mass contour lines resulting from a weak gravitational lensing analysis. The bulk of the x ray gas is located near the center of the figure. The mass peaks computed from the weak lensing is located at the center of the smallest yellow circles to the sides of the bulk gas, coinciding with the center of the optical cluster centers, suggesting most the mass passed through the intersection point without scattering [45] While the astronomical evidence for dark matter spans many length scales, the cosmological evidence reaches across several times scales. Observations of high redshift celestial objects provide the abundance of heavy e lements present moments after big bang nucleosynthesis. Measured on the ratios of d euterium ( 2 H), helium ( 3 He and 4 He), and lithium ( 7 Li) to hydrogen (H) place constraints on the percent of baryonic

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26 Figure 2 5 Observations f rom the Bullet Cluster [46] The yellow lines show the contour lines of the mass computed from weak gravitational lensing. The x ray gas is shown in pink. Weak lensing suggests the majority of the mass is collisi onless, thus non baryonic matter in the universe. Deuterium measurements provide strongest evidence for existence of non baryonic matter in the universe, as deuterium is extremely sensitive to the initi al baryon density of the universe and has no natural mechanism for creation in burning stars. Figure 2 6 show s the ratio of the measured abundance of 2 H (labeled D) 3 He, 4 He, and 7 Li to the measured abundance of H compared to the abundance predicted durin g Big Bang Nucleosynthesis The single lines represent the measurements and the double lines shows the predicted baryon percentage as a function of abundance The intersection of the lines is the baryon percentage of the universe predicted by each measurem ent. The measurements of 2 H, 3 He and 4 He estimate baryonic matter makes up only ~5 % of the total energy of the universe [47 48] leaving ~95 % of the energy (matter or otherwise) unknown.

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27 Figure 2 6 Limits on p ercent of b aryons in the u niverse from n ucleosynthesis [49] The four single line s shows the measured abundance of d euterium (D), h elium ( 3 He and He), and l ithium ( 7 Li ) divided by the measured abundance of hydrogen (H) The double lines show the predicted baryon percentage as a function of abundance of each substance The intersecti on of the lines show the limit on baryon percentage The measurements of D 3 He and He estimate baryons makes up ~5% of the universe. Measu rements of the cosmic microwave background (CMB) have provided additional evidence for non baryonic matter 380,000 years after the Big Bang. CMB measurements have shown the universe to be remarkably isotropic with relic quantum fluctuations and acoustic o scillation stretched out to cosmic scales. The power spectrum of the residual power from the fluctuations places constraints on the baryon density in the universe at ~ 0.02 3 [50] Figure 2 7 shows the measured temperature of CMB in the universe, depicting the maximum temperature spread of < 1mK. Figure 2 8 shows the power spectrum from the (top) Planck satellite observations with the (bottom) residual fluctuation nor malized.

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28 Figure 2 7 Cosmic m icrowave b ackground m easurements from Planck [51] The thermal spread of < 1 mK shows that the universe is almost perfectly isotropic on the cosmic scale. Figure 2 8 Power s pectrum of the c osmic m ic rowave b ackg round m easurements [51] The top shows the measured power spectrum with the best fit line. The bottom shows the residual of the measurements. The low power fluctua tions on the left side of the plot set limits on the baryon mass of the universe at ~2%. Further evidence for dark matter exists in the scale modeling for structure formation in the universe. Current the oretical modeling methods accurate predict the size scale of cosmic structures from observation only when non baryonic particles are included in the simulations To align the predictions with observation, the ratio of non

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29 baryonic matter to baryonic matter needs to be approximately 5 [52] The non baryonic matter must also be non interacting. Figure 2 9 shows the predicted scale of observations methods of cosm D M model (see below) and me asured scales. The plot shows that prediction well match observations over all scales. Figure 2 9 Predicted s cales of c osmic s tructure o bservations u DM m odel [53] Only when non baryonic matter is added to the model, do the simulations accurately predict the measurable scales of observation techniques Though the evidence for dark matter is plentiful, measurements have established modest bounds on the properties of dark matter. Dark matter must be non baryonic and collisionless to produce the a forementioned observations. Dark matter must be long lived as is necessary to produce the results o bserved from all cosmological time scale up to present day. Additionally, dark matter must have feeble electromagnetic couplings else signatures of dark matter would have been detected. The abundance of

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30 dark matter is determined by the redshift evolution of the universe pressure to density ratio to ~22% [54] Dark matter is generally considered to be nonrelativistic (cold) as a result of red shift based sky surveys [55] The property bound s contribute the dark cosmological model of the universe. Since the origination of the theory of dark matter, various candidates within and beyond the standard model have been proposed. However, the majority these candidates have been excluded through observation or experimentation [56] Various dark matter ca ndidates are still plausible, though the two most widely considered the strongest are weakly interacting massive particle s (WIMP) and the axion. Figure 2 10 shows a visualization of the current field of plausible dark matter candidates. Figure 2 10 Visualization of c urrent t heories of d ark m atter [57] Various candidates are viable, while WIMPs and CDM axions are generally considered the most plausible

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31 The WIMP consists of a group of theoretical particles, such as the n eutralino and the gravitino, that were produced t hermally in the early Universe. The most prominent WIMP theories viable today are predicted by supersymmetry Current experim ents have set strong limits on WIMPs [58] and supersymmetry [59] suggesting additional tuning of the theory ma y be required. Axions were first theorized to constitute dark matter in 1983 [4] The axion particle naturally possesses all the properties of dark matter from its simple extensi on of the standard model, and is a favored candidate due to is elegant sol ution to the strong CP problem. T he axion is predicted to form a Bose Einstein condensate, which potentially distinguishes it from other cold dark matter candidates through their pha se space structure, supported by astronomical observation [60] Recently string theory has enhanced the theory of CDM axions as well as other theoretical particles, known as (ALP) that spontaneously break a U(1) symmetry similarly to the PQ symmetry breaking of the axion [61] 2.3 CDM Axions As solutions to the strong CP problem, axions would need to be abundant at all time scales of the post inflationary universe. Axions produced through PQ symmetry breaking wo uld be non baryonic, nonrelativistic and collisionless without fine tuning The axion is dominated by gravitational forces and has an extremely weak coupling to the electromagnetic and weak force. Thus, the axion particle which arises from the Peccei Quin n solution to the strong CP problem is a natural dark matter candidate. The mass of the axions is not predicted by the PQ symmetry breaking theory, but cosmic observation and particle physics experiments have constrained the axion energy

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32 scale and thus t provided the lower bound of f a 10 9 GeV [14] The cosmic energy density argument places the upper limit. The cosmological energy density of axions increases as the mass of the axion decreases; if f a 10 12 GeV, the axion energy density would be to large, causing the early universe to collapse (overclosure) [11 13] From E q. 2 2 the allowable mass of the axion is found to be between approximately e V and meV. The cosmic number density of CDM axions in the allowable mass range is around 10 11 10 14 /cm 3 depending on the mass, though local densities due to caustics produced by the dark matter flow could be several orders of magnitude greater [62] The coupling of the axion to the standar d model particles and forces is so weak that initially it was believe d The various coupling s are model dependent, but in all axion models within the dark matter mass range, the coupling to stan da rd model particles and forces is 10 12 Gev 1 and more often is many orders of magnitude lower [63] The natural lifetime of a n axion is related to mass by (2 4 ) For CDM axions within the allowable mass range, the decay time to convert to a standard model particle is many orders of magnitude greater than the ag e of the universe, further suggesting that dark matter axions are undetectable I n 1983 Sikivie [7] suggested using the inverse Primakoff to significantly increase the effective coupling to the electromagnetic force leading to multiple axio n dark matter direct detection concepts The electromagnetic interaction of the axion is determined by the Lagrangian for the axion photon conversion

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33 (2 5 ) where a is the axion field, and E and B are the electric and magnetic fields of the two propagating photons, respectively (see Fig. 2 1 1). The axion photon coupling constant, g is proporti onal to the axion energy scale by (2 6 ) where is the fine structure constant and g is a model dependent constant of order 1 In the KSVZ and DFSZ models, g is equal to 0.97 and 0.36 respectively. Equation 2 6 leads directly to a model dependent axion photon coupling constant In the KSVZ model, the coupling is [26 27] (2 7) In the DFSZ mo del, the coupling is [28 29] . (2 8) The axion coupling to the photon in the DFSZ is weaker than in the KSVZ model, making detection of axions more challenging if the DFSZ model is accurate. The Primakoff effect, first theorized by Primakoff [8] in 1951, describes a coherent nuclear production mechanism for a single photon to meson conversion in the prese nce of a magnetic field. Sikivie [7] detailed how the same mechanism could be used to convert single photons to axions or single axion s to photons ( T he latter is m ost often referred to as the inverse Primakoff effect .) In the presence of a static magnetic field, the effective Lagrangian of the axion photon interaction becomes (2 9)

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34 where B 0 is the static magnetic field. Figure 2 11 shows the Feynman diagrams for the axion photon conversion (a) in vacuum and (b) in a static magnetic field from the Primakoff effect. (a) (b) Figure 2 11. Feynman Diagram of Axion photon Interaction. (a) Interaction in vacuum. (b) Interaction in a static mag netic field due to the Primakoff effect. The lifetime of the axion from conversion to photons is proportional to B 0 2 Conversion due to the inverse Primakoff effect in the presence of a significantly strong magnet produces a lifetime many orders of magnitu de lower than that of conversion in vacuum greatly increase the conversion rate This process is effective for relativistic or non relativistic axions. As axion dark matter is nonrelativistic, the frequency of the photon from the axion photon conversion is f m a c 2 / h where c is the speed of light in a vacuum and h is with a very small adjustment for the relative kinetic energy If CDM axions are virialized, 0 < v < c /1000, making the spread of energy for con versions ~10 6 m a c 2 [10] and has a Maxwellian distribution For non virialized axion flows, the energy spread could be significantly lower [64] Sikivie proposed two direct detection experiments utilizing the inverse Primakoff effect, a haloscope and a helioscope. A n axion haloscope uses a microwave cavity

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35 permeated by a stro ng magnetic field to resonant the conversion of CDM axions to photons producing an enhanced signal for detection (see Chapter 2.4) An axion helioscope converts axions produced in the sun The helio scope is composed of a long telescope permeated by a strong magnetic field; the telescope it pointed at the sun, converting relativistic solar axions (not CDM axions) released from the sun to photons inside the telescope. A photon detector at the back of t he telescope observes the signal produced from the conversion. The most sensitive haloscope and helioscope detectors presently are the Axion Dark Matter eXperiment (ADMX) [65] and the CERN Axion Solar Telescope (CAST) [66] respectively. Both experiments use cryogenic systems to reduce background noise and highly sensitive s ignal detectors to enhance sensitivity. The Haloscope At Yale Sensitive To Axion CDM (HAYSTAC) published results in 2016 that were not sensitive to KSVZ coupling [67] (not included in Fig. 2 12). No dark matter or solar axions have been directly detected to date. Figure 2 12 shows the axion exclusion plot around the mass range for dark matter axions. The y axis is the axion to photon coupling. The yellow bar bounds t he KSVZ (upper) and DFSZ (lower) coupling, depicting the linear relationship between photon coupling and mass of the axion for the two models (see Eq. 2 7 and Eq. 2 8). The green area shows the axion search space that has been excluded by haloscope detecto rs; ADMX is the leftmost green bar and the only exclusion area to include some KSVZ coupling region within the allowable mass range for CDM axions. The blue area shows the axion search space that has been excluded by the CAST helioscope. The

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36 Figure 2 12. A xion e xclusion p lot n ear the CDM a xions m ass r ange [68] The y axis is the axion to photon coupling. The yellow bar bounds the KSVZ (u pper) and DFSZ (lower) co upling The green blue, and pink area s show the axion search space that has been excluded by haloscopes, helioscopes, and LSW experiments respectively M ost of the search space of CDM axions is yet unexplored pink area shows axions excluded by light shi ning through wall (LSW) experiments (see below). The plot shows that most of the search space of CDM axions is yet unexplored. More recently, various axion detectors have been proposed, with a select few being developed. The LSW (light shining through wal l) experiment which uses the Primakoff effect and inverse Primikoff effect in series, is perhaps the most notable of these experiments. The LSW experi ments passes coherent light though a static magnetic field, converting a fraction of the photons to axion s which must conserve energy and momentum of the light. The axions would pass through a physical barrier (wall) into a second static magnetic field, where a fraction of the axions would convert back to a photon [69]

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37 LSW experiments are broad search es that detect any ALP, but a detection cannot determine the mass of the particle nor distinguish an axion from ALPs. The detector does not search for CDM axions, as the axions are generated in the laboratory. The Any Light Particle Search is a LSW detector that uses optical cavities to enhance sensitivity. The A ny L ight P article Search has set axion exclusion limits that are shown on Fig. 2 12, though they are less encompassing than the CAST limits. The Center for Axion and Precision Physics (CAPP) in South Korea is conducting research on axion dark matter detectors to search for axions of mass greater than ~20 ev. CAPP is developing novel microwave cavity concepts to include a toroidal cavity haloscope design. The proposed International Axion Observatory ( IAXO) experiment will be an improved solar axion search that is projected to be an order of magnitude more sensitive than CAST. Additional axion detection schemes have been proposed recently, though none have published limit yet. NMR based detectors detec t the spin precess ion caused by low mass (< 10 9 eV) axions ; the Cosmic Axion Spin Precession Experiment (CASPEr) is a proposed NMR based experiment [70] A detectio n method using a LC circuit was proposed by Sikivie et al. [71] The presence of a xions in a static magnetic field alter an oscillating electric current to flow along magnetic field lines. An LC circuit within a static magnetic field could be used to detect the oscillating current. The detector is believed to be sensitive to axions with mass approximately 10 7 10 9 eV. The University of Florida is developing a proto type detector Variants of haloscope d etectors using periodic dielectrics within waveguides and optical cavities are being developed at the University of Washington [72]

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38 antisymmetric mode within the waveguide or cavity, yielding higher order mode s with a nonzero form factor. The tuning is conducted by adjusting the cavity length and the s pacing between the dielectrics. 2.4 Axion Haloscope Detectors To date, t he only detection scheme that has been sensitive enough to detect CDM axions at KSVZ coupling is the haloscope As previously stated, the axion haloscope uses a microwave cavity resonator permeated by a high strength static magnetic field The magnetic field increases the conversion rate of axions to photons due the invers e Primakoff effect, proportional to | B 0 | 2 Increasing the strength of the magnetic field thus increases the power of the axion signal and, consequently, the sensitivity of a haloscope detector. Typ ically, a haloscope uses a superconducting solenoidal magnet to maximize the field strength. The sig nal of the axion conversion would have a bandwidth ( b a ) determined by the kinetic energy spread E at half of the peak probability The effective quality factor of the signal ( Q a ) would be the peak energy divided by the bandwidth, E E c 2 / v v where v and v are the peak velocity and velocity dispersion of the local axions From the predicted energy spread of the (virialized) axion conversion, Q a 10 6 Figure 2 13 shows the local k inetic energy distribution of galactic axions normalized such that the peak is one. The energy spread at half the peak is on order 10 6 m a c 2 The microwave cavity of the haloscope is of great importance. From Eq. 2 9, it can be shown that the cavities capacity to resonate the axion decay signal is dx 3 E mnp B 0 over the volume of the cavity, where E mnp is the electric field of each individual cavity mode identified by the indices m n and p For right circular

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39 Fi gure 2 13. Local k inetic energy distribution of galactic axions [73] The plot is normalized such that the peak has a value of one. The distribution is Maxwellian (virialized axions). The energy spread is ~10 6 m a c 2 For non virialized axions, the energy spread will be much less. cylindrical cavities, the indices correspond to the polar coordinates , and z respectively. established to quantify the capacity of a given cavity mode to resonantly enhance the signal from axion conversion, ( 2 10 ) where V is the volume of the cavity and is the permittivity within the cavity normalized to vacuum. The form factor can be interpreted as a coupling factor between a cavity mode and the axion photon interaction. C mnp has a value between zero, where no conversion signal is resonated, to one, where the signal is resonated h omogen e ously within the entire volume of the cavity. Note, due to bou ndary conditions of the electromagnetic field s a value of one is unobtainable. C 2 E

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40 The most common cavity configuration for a haloscope is a right circular cylindrical cavity oriented in the detector such that the center axis is normal to the dependent on the integral over the cavity volume, maximizing the volume of the cavity is desired, and utilizing a righ t circular cavity in a cylindrical solenoid obviously maximizes the available volume. From Eq. 2 10, only modes that have an electric field component parallel to the static magnetic field of the detector will yield an observable signal. In a right circular cylindrical cavity in a so lenoidal magnet, only transverse magnetic (TM) modes and hybrid modes (see Chapter 3.1) containing some longitudinal electric field will resonate the axion signal. Transverse electric (TE) and transverse electromagnetic (TEM) mod es will yield no detectable signal. In order for the axion signal to resonate a mode, the signal power spectrum must sufficiently overlap with the mode power spectrum. Thus, the frequency of the axion signal ( f m a c 2 / h ) must match the frequency of the mo de ( f mn p ) within f / Q min where Q min Q L (see Chapter 3.1) and the effective quality factor of the signal Q a Typically, Q L is significantly less than the Q a making Q min = Q L Because the mass of the axion is unknown, the frequency of searchable cavity modes must be incrementally adjusted, or tuned to s can for CDM axions within a finite mass range. As the frequency of the search mode is tuned, the measurement frequency of the detector is simultaneously changed to match. Cavity tuning has historically been done using one or two conducting or dielectric tuning rods that run approximately the length of the cavity that are mechanically rotated about a pivot point off center of the

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41 rods though other concepts have been proposed in recent years (see Chapter 3.4) A of a cavity that can be tuned across a frequency range that is within the allowable mass range of CDM axions and has a sufficient form factor, C mnp to yield a non negligible signal throughout the search range. The frequency step size, the change in frequency of the search mode from one scan to the next, must be n o larger than f mnp / Q L Most haloscope e xperiments will use step sizes f f m n p /5 Q L to maintain a more consistent sensitivity throughout the search range, reducing the integration time at each step but increasing the total number of steps across a range. F ig ure 2 14 shows a schematic of mechanical tuning using (a) one large tuning rod and (b) two smaller tuning rods. Generally c onducting rods will increase the search mode frequencies and dielectric rods will decrease the frequencies (see Chapter 3.1). (a) (b) Figure 2 14. Schema tic of mechanical tuning of haloscope cavi ties (a) One tuning rod. (b) Two tuning rods. A mechanical gap must exist between the top of the tuning rods and the endcaps of the cavity to enable motion. Generally conducting rods will increase the search mode frequencies and dielectric rods will decrease the frequencies.

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42 For an empty cylindrical cavity, only TM 0 n 0 will resonate the axion signal due to mode anti symmetry. However, an empty cavity cannot be tuned to scan across a fre quency range. Mechanical tuning requires longitudinal symmetry breaking within the cavity to maneuver the rods. The symmetry breaki ng causes modes to break pure TE /T M /TEM field co nfigurations (see Chapter 3.2). Though the search modes of such cavities are no longer pure TM 0n0 modes, the nomenclature TM 0 n 0 for cylindrical cavities is often used to indicate the mode the perturbed modes most closely resemble. In particular, the most common search mode for a haloscope is usually referred to as the TM 010 mode, though the mode is a pertur bated form of the pure mode and TM 010 like would be a more accurate description The TM 010 mode tends to be the most desired mode to use for axions searches as it yields the highest form factor, with higher values of n (e.g., TM 020 TM 030 etc.) providing reduced form factors with a value of ~ C 010 / 4 ( n 1) for a given cavity. C 010 refers to the form factor for the TM 010 ( like ) mode of the cavity at a given tuning configuration. Depending on the tuning rod(s) configuration, occasionally symmetry breaking of a mode with m > 0 yield a mode with a nonnegligible form factor (see Chapter 2.2). From the proportionalities noted above and those of the axion parameters the power transmitte d from the cavity is [10] (2 10) where a is the local density of CDM axions and g is the model dependent axion photon coupling constant given by Eq. 2 7 for the KSVZ model and Eq. 2 8 for the DFSZ model. Note, a / m a is the number density, n a of the axion. Q L is mode dependent, though the indices for the mode are conventionally not written. The

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43 sensitivity of the haloscope is maximized when the antenna coupling to the cavity is over coupled, such that the loaded quality factor is one third the unloaded cavity quality factor, Q L = Q /3 [15] However, imp lementing and maintaining that coupling throughout a frequency range scan is challenging; typically, the antenna is critically coupled to the cavity so that Q L = Q /2. The search scheme of an axion haloscope consists of tuning the search mode of the microwa ve cavity to an initial frequency, measuring the signal within the cavity for a set scan time, adjusting the search mode frequency by an amount less than f / Q L and repeating until the scan range of the search mode is exhausted. Because the axion signal add s coherently and noise adds incoherently, the longer the scan integrates measurements, the greater the sensitivity of the detector. Thus, the scan integration time is determined by the desired sensitivity to axion photon coupling and confidence of the sear ch detecting the axion if it exists at a scanned mass For example, a search at DFSZ coupling sensitivity and 95 % confidence requires considerably longer integration time per frequency step than a search with the same detector at KSVZ coupling sensitivity and 90% confidence (see Eq. 2 13). The confidence is related to a signal to noise ratio of an observation, which is related back to the power transmitted by the radiometry equation [74] , ( 2 11 ) where T n is the system noise temperature of the detector k B t is the scan integration time, and b a is the axion signal bandwidth. For a xion searches,

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44 typically s / n > 5 to obtain a confidence of >9 5 % The signal bandwidth can be expressed in terms of the axion mass (2 12) Solving Eq. 2 11 for t yields the integration time required for a given frequency (2 13) If the Q L is less than Q a then the number of signal bandwidths that can be scanned simultaneously is N = Q a / Q L Thus, the integration time for a single frequency step is ~ t / N The total integration t ) required to scan over a small frequency f ) is (2 14) Combining Eq. 2 10 and Eq. 2 14 and rearranging yie lds the haloscope scan rate [75] (2 15) Note the scan rate is proportional to g 4 so increasing the sensitivity of a haloscope detector from KSVZ coupling to DFSZ coupling would require an increase in scan time of ~50 if no changes were made to the detector Equation 2 15 demonstrates that an axion haloscope search is a tradeoff between sensitivity and time Because a typical haloscope search is conducted on order of a year, a higher scan rate is obviously desired As experiments have no control ove r axion parameters such as mass and local density the haloscope is designed to m aximize scan rate as much as possible. As mentioned earlier, the magnetic field strength is maximized to obtain the highest axion to photon conversion rate. The microwave cavi ty design and fabrication ideally maximizes volume, quality factor, and

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45 form factor, but, as discussed below, other search factors can constrain the cavity parameters (see also Chapter 3.4). The system noise temperature significantly affects the detectors sensitivity. From Eq. 2 15, a minimum temperature is desired. The temperature dictates the signal noise; the higher the system temperature is, the longer the required integration time will be for a given signal to noise ratio. Haloscope experiments make g reat efforts to reduce the system noise temperature by placing the detector within a cryostat. Recent experiments have utilized SQUID (superconducting quantum interference device) amplifiers to obtain near quantum limited electronic noise and dilution refr igerators to obtain physical te Figure 2 15 show s (a) a g raphic representation of a haloscope using a aa cavity within a solenoidal magnet and (b) the expected axion signal of a detection Axions converted to photons v ia the inverse Primakoff effect will resonate when the frequency of a mode with a sufficient form factor is nearly the same as the axion signal. The physical temperature of the cavity is minimized to reduce the noise. An antenna, usually critically coupled will extract the signal along with the physical temperature noise. The extracted signal is carried to a power meter often passing through amplifiers and gaining electrical noise The system noise is the total of the physical temperature and electrical n oise The power spectrum of the readings would yield a detection signal if the detector is sensitive enough. Under even the most optimistic circumstances, the axion signal within the haloscope cavity is extremely weak, requiring amplification and a well conceived receiver chain to enable detection. Each experiment has individual receiver chains that

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46 (a) (b) Figure 2 15. (a) Graphic representation of a haloscope detector using a solenoid magnet and a cylindrical microwave cavity, and ( b) an a nticipated detection signal of CDM axions [68] The signal can only be observed if the frequency of a mode with a nonnegligible form factor nearly matches the axion signal and the detector sensitivity is sufficient The signal to noise ratio shown is ~5 optimize thei r detector, but near ly all will have cryogenic and noncryogenic amplifiers to increase the signal strength filters to isolate the axion signal, mixers to make the signal more manageable, and circulators to prevent any reflected power from corrupting the m easurements. Detectors usually have power injection ports to take Q L measurements at different tuning configuration and a data storage system to conduct data analysis after a search is conducted. The largest and most sensitive microwave c avity axion search to date is ADMX Currently located at the University of Washington, the D epartment of Energy funded experiment searches for CDM axions with a haloscope detector using a 7.6 Tesla superconducting solenoid and a ~0.15 m 3 copper cylindrical microwave cavity. The TM 010 frequency of the cavity is tuned using two copper rods. The signal mea surement is extracted through a critically coupled antenna and amplified through a receiver chain using a SQUID ampl ifier and several HFET ( heterostructure field effect transi stors) The

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47 signal is passed through a crystal filter and mixed down to 35 KHz before being stored in medium resolution and high resolu tion bins for data analysis. A medium resolution analysis searches for axions signals with a Maxwellian velocity distribu tion [76] and a high resolution analysis searches for signals with a fine structure velocity spread [77] Figure 2 16 shows (a) the schematic of the receiver chain and (b) a photograph inside a cavity depicting the tuning rods. (a) (b) Figure 2 16. (a) Schematic of ADMX recei ver chain [78] and (b) p hotograph of the inside of the ADMX microwave cavity [68] The tuning rods are incrementa lly rotated about the offset pivot points to adjust the TM 010 frequency, provid i ng scanning across an axion mass range of ~2 4 eV. ADMX has successfully excluded CDM axions at the KSVZ coupling sensitivity in the mass range of 1.9 3.7 eV (460 890 MHz) [79 82] using a ~3 K system temperature detector. In 2016, ADMX installed a dilution refrigerator and i s currently operating the detector at ~300 mK system temperature over approximately the same

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48 mass range. The new data is projected to be at the DFSZ coupling sensitivity. However, the current detector is not sensitive above ~4 eV (~1 GHz) and a new halosc ope cavity is required to search at higher masses Figure 2 17 shows the exclusion limits published by ADMX. Figure 2 17. Axion exclusion plot showing published results of ADMX [68] The KSVZ Z coupling is labeled ADMX has excluded axions to KSVZ coupling in the mass range of 1.9 3.7 eV (460 890 MHz) [79 82] The following proportionalities illustrates the challenges of axion searches at higher mass The frequency of a search mode scales as the axion mass. The s earch mode frequencies scales inver sely with the cavity radius and the volume as the cube of the radius w hen the ratio of the length to the radius L / R is maintained. At temperatures below ~7 K, Q L scales as f 2/3 with constant L / R due to the anomalous skin effects (see Chapter 3.1) and t he power transmitted from the cavity will scale as ~ f 8/3 Thus, using the current state of the art for haloscope cavities, doubling the scan frequency of the

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49 detector by simply scaling the cavity will reduce the signal power by a factor of ~6.3 and increase the scan time by a factor of ~25 for the same confidence and coupling sensitivity. HAYSTAC (formerly ADMX HF) recently conducted a CDM axion search at a mass range of ~23.6 24.0 eV [67] The experiment only obtained a sensitivity of ~2.3 times KSVZ coupling despite using a dilution refrigerator and a near quantum limited superconducting amplif ier and searching a cross a modest mass range for about four months The results well demonstrate the difficulties of scanning for axions at frequencies above ~1 GHz as detail ed above. Researchers now know that ne w microwave cavity technologies need to be d eveloped to search for CDM axions with masses above ~4 eV (~1 GHz).

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50 CHAPTER 3 MICROWAVE CAVITIES FOR AXION HALOSCOPE DETECTORS 3.1 Microwave Cavity Theory A resonant cavity is any arbitrarily shaped enclosure with electromagnetic reflecting bound ary conditions which are most commonly satisfied by conducting walls Resona nt cavity theory is derived as an extension of waveguide theory. A ssum ing a steady state solution that maintains an oscillating E and B field with a time component consisting of e i where = 2 , (3 1) (3 2) (3 3) (3 4) where and are the permittivity and the permeability inside the cavity, respectively. T he conducting boundary conditions are (3 5 ) (3 6 ) where n is the normal vector to the surface of the boundary. Combining Eq. 3 3 into Eq. 3 4 and vice versa yields the field separated equations (3 7 ) (3 8 )

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51 Under the most general conditions, the permittivity and the permeability inside the cavity are a function of location, x and can have any positive value. Expandi ng Eq. 3 7 and Eq. 3.8 yields (3 9 ) (3 10 ) If the cavity has constant permittivity and permeability, the right side of Eq. 3 9 and Eq. 3 10 go to zero, yielding the wave equations (3 11 ) (3 12 ) In arbitrary geometry, the waves flow in a direction dictated by the boundary conditions. Because the system is closed, standing waves are formed. If the boundary conditions are constant in the longitudinal direction (i.e., constant cross section) and the end s of the cavity (endcaps) a re normal to the longitudinal axis forming a right cylinder then the solution to the wave equations will be of the form (3 13 ) where L is the distance between the endcaps (cavity length), z is the distance (between 0 and L ) along t he longitudinal axis between the endcaps and p is a nonnegative integer. A and B are the electric or magnetic field in the cavity, depending on the solution. When the cavity maintains translational invariance or longitudinal symmetry, the solution can b e separated into transverse magnetic (TM) and transverse electric (TE) solutions. The wave equation can be reduced to the 2 dimension al wave problem

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52 (3 14 ) where is the magnitude of the electric or magnetic field in the z direction for the TM and TE solutions, respectively. By definition, B z and E z for the TM and TE solutions, respectively, are zero. The symbol t indicates the 2 dimensional vector differential operator in the transverse direction. For a right circular cylindrical cavity, the longitudinal direction is along the cylinder axis. The electric field in the z axis for the TM solutions is given by [83] (3 1 5 ) where J m is the m th order of the Bess el function of the first kind, and x mn is the n th root of J m ( x ) = 0. R is the radius of the cavity, is the distance from the center between 0 and R and is the angle between 0 and 2 relative to an arbitrary baseline on the cavity cross section. E 0 is a nonzero electric field used to indicate an arbitrary scalar value of the eigenfunction solution. The plus/minus in the exponential acts on and indicates a polarization of the modes. Modes with m > 0 have a 2 state degeneracy, with + and polarization. For rotationally invariant cavities, the orientation of the polarization is arbitrary. Equation 3 15 shows that the TM solution to the 2 dimension al wave equation Eq. 3 14 is a set of finite solutions, or modes, with integer values of m n and p The characteristics of the Bessel and cosign functions dictate that m and p are nonnegative while n must be positive. But none have an upper limit, indicating that the solution set is infinite. Each solution oscillates at a frequency (3 16)

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53 Note, the minimum TM frequency for an empty cavity is the TM 010 mode frequency denot ed as f 0 = 0.1145 GHz m/ R For TE modes, x mn is replaced with x mn the n th root of the J m ( x ) = 0. Figure 3 1 shows images of the E z field through a cross section of the cavity for the (a) TM 010 (b) TM 110 and (c) TM 020 The cross section is parallel to the cavity axis and the outer edge of the images are the conducting boundaries. Red indicates higher field and b lue indicates lower field, with zero field at the boundaries The blue diagonal line in (b) and the blue circle in the middle of (c) are nodes where the field is zero. On one side of the node E z is out of the page and on the other side E z is into the page. The orientation of the TM 110 mode is arbitrary. (a) (b) (c) Figure 3 1 Cross sectional images of the E z field in an empty cylindrical cavity (a) TM 0 10 (b) TM 110 (c) TM 020 The cross section is parallel to the cavity axis and the outer edge of the images are the conducting boundaries. Red indicates higher field and blue ind icates lower field with zero field at the outer boundaries The mechanism of selecting the orientation of the TM 110 mode is unknown. From Eq. 2 10, for empty circular cylindrical cavities, only TM 0 n 0 modes will have a nonzero form factor due to the anti symmetry of the other TM modes as the opposing fields will cancel each other out when integrated over the cavity volume, and the absence of a E z field in the TE modes. The TE solutions are similar in nature, containing

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54 an infinite set of modes with a B field and frequency indicated by some m n and p integer values. Equation 3 14 denotes two decoupled Sturm Liouville eigenvalue equation so the solution is a complete set of decoupled orthogonal modes (TM and TE) defined by (3 17 ) (3 18 ) where i and j indicate any TM or TE mode and ij is the Kronecker delta The orthogonality can be applied to degenerate modes, but a rotational transformation of /2 about the z axis must be applied to one of the modes. D ue to rotational invariance of the cavity the orientation of the degenerate mode is chosen arbitrarily or with no known mechanism (see Fig. 3 1(a)) No research has been published as to how the ori entation would be determined by nature. However, no theory could be tested, as any testing would require the presence of an antenna within the cavity breaking the symmetry and providing a basis of orientation. A microwave cavity used in a haloscope must h ave a tunable search mode, making an empty cylindrical cav ity unusable for the detector. Currently, t he most common method of generating a tunable mode that will resonate the axion photon conversion is to insert one or more tuning rods that are rotated off axis to translate through the cavity (see Fig. 2 14). The rod creates additional boundar ies within the cavity w hich changes the solution and the frequency of the modes. Because the cavity of a haloscope operates within a static magnetic field, a relative permeability of for the tuning rods As the Laplacian is a measure of scalar curvature of the mode Eq. 3 14 shows that conducting rods will

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55 typically result in an increase in frequency for TM The additional boundary conditions with zero tangential electric field require the TM mode to retu rn to zero at the boundary, causing the scalar curvature of the field to increase (see Fig. 3 6), unless the mode has a node at/ near the boundary condition already. T he TM 01 p modes will always experience an increase in frequency; higher order m and n modes will often experience an increase depending on wher e the rod is in location to the node (s) of the mode When the conducting rod runs the length of the cavity Eq. 3 14 is valid, and the solut ion is an infinite set of orthogonal modes that meet the criteria of Eq. 3 17 and Eq. 3 18. The addition of an interior boundary allows for an infinite set of transverse electromagnetic (TEM) modes to exist in addition to the TE and TM modes where (3 19 ) The E z field of the TM modes takes a form similar to Eq. 3 15, but with the single Bessel function ( J m ) r eplaced with a summation of Bessel functions of the first and second kind [84] Figure 3 2 shows images of the E z field through a cross section of the cavity with a coaxial conducting rod for the (a) TM 010 (b) TM 110 and (c) TM 020 The white area in the center is the location of the rod with no field solution The dark blue at the inner and outer boundaries indicates zero field The diameter of the rod is R /2. Dielectric rods, most often made of alumina ( Al 2 O 3 ) for haloscope cavities, typically results in a decrease in the TM modes, as the mod e tends to localize within the rod, resulting in an overall reduction in scalar curvature [15] However, the rod location relative to a node also contributes to the effect on modes with higher order than TM 01 p

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56 (a) (b) (c) Figure 3 2. Cross sectional images of the E z field in a cylindrical cavity with a coaxial conducting rod (a) TM 0 10 (b) TM 110 (c) TM 020 The cross section is para llel to the cavity axis. The white area in the center is the location of the conducting rod ( d = R /2) ; a field solution is not included in the images as there is zero field in side the conductor. The outer edge of the images and white center areas are the conducting boundaries. Re d indicates higher field and blue indicates lower fi eld with zero field at the conducting boundaries The mechanism of selecting the orientation of the TM 110 mode is unknown. From Eq. 3 9 and Eq. 3 10, the field equations of a cavity with dielectric rods a re [85] (3 20 ) (3 2 1 ) If the rods run the entire length of the cavity an d are parallel to the center axis, symmetry is preserved. TE 0 np and TM 0 np modes exist in the form of [86] (3 22 ) (3 23 ) In order to meet the boundary conditions, modes with m > 0 must break pure TE and TM configuration and incorporate some E z and B z field respectively [87 88] The new eigenfunction solutions are hybrid modes. The electric and magnetic fields of the

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57 hybr id modes are coupled, which basis defined as [89 90] (3 24 ) where E i is the complex conjugate of E i When the dielectr ic rod is off axis from the cavity, rotational invariance is broken, and all modes are hybrid to meet the boundary conditions [91] Figure 3 3 shows images of the E z field through a cross section of the cavity with a coaxial dielectric rod for the (a) TM 010 (b) TM 110 and (c) TM 020 The field is localized within the die lect ric outlined by a dotted line, when compa red to Fi g. 3 1. (a) (b) (c) Figure 3 3. Cross sectional images of the E z field in a cylindrical cavity with a coaxial dielectric rod. (a) TM 0 10 (b) TM 110 (c) TM 020 The cross section is parallel to the cavity axis and the outer edge of the images are the conducting boundaries. The dielectric rod is outline d by a dotted line and is the same diameter as the conducting rod in Fig. 3 2 ( d = R /2) Red indicates higher fi eld and blue indicates lower field with zero field at the outer boundaries Localization of the field with in the dielectric can be seen. The mechanism of selecting the orientation of the TM 110 mode is unknown. As energy is stored in a cavity as a resonance power is simultaneously lost through the boundaries as well as any lossy material that is inside the cavity. A frequency dependent measure of the energy storing efficiency of a resonating cavity is the quality factor, Q The quality factor of any resonat or is defined as

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58 ( 3 25 ) where U is the energy stored in the cavity and is the resonant frequency Though Q for a cavity is mode dependent, it is not common to use subscripts to annotate the mode. For a cylindrical cavity with conducting tuning rods, t he Q for a TM mode is given by [83] ( 3 26 ) where i is the skin depth of the fields in each cavity component identified by the subscript. The summation is over n number of tuning rods. The variables and are dimensionless number s of order unity, defined by ( 3 27 ) where is the 2 dimensional field solution to Eq. 3 14. A and S are the cross sectional area and circumference of the cavity, respectively. The prime differentiate s between the walls and the tuning rods : w ithout the prime, the line integral is around the boundary of the walls; with the prime the line integral is around the boundary of the n th tuning rod. For both, the surface integral is across the cavity. When all the boundaries of the cavity are homogen e ous (i.e., made of the same material), the sk in depth is constant throughout the ca vity. E quation 2 26 reduces to ( 3 28 ) The summation is over all the tuning rods plus the wall. For modes with p = 0, Eq. 3 28

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59 is multiplied by 2 and L is replaced by 2 L T he quality factor is highly dependent fabrication method of the cavity and tuning rods. To inject power into a resonating cavity, the cavity must be coupled to a power source; to extract power to a receiver chain or measurement device, the power sink must be coupled to the cavity. Coupling is achieved through an antenna in serted into the cavity or through a waveguide feed connecting to aperture in the cavity. An antenna can tak e the form of a probe or a loop. An infinitely thin probe antenna will only be excited by an electric field that runs parallel to the axis of the probe, while an infinitely thin loop antenna will only be exci ted by a magnetic field that is normal to the pl ane of the loop Figure 3 4 depicts (a) a probe antenna excited by an electric field and (b) a loop antenna excited by a magnetic field. The arrows indicate the e lectric field and the crosses indicate the magnetic field into the page. (a) (b) Figure 3 4. Antenna coup led to a cavity through a wall or endplate (a) Probe antenna. (b) Loop antenna. Blue arrows in (a) are electric field lines. Blue crosses in (b) are magnetic field line going into the page. The fields shown interact strongly with the ant ennas shown. The hatched area represents a dielectric used as a coaxial waveguide to transmit the signal to/from the cavity. The most common way to couple to a TM mode in a cylindrical cavity is via a probe antenna through an endplate. The amount of coupl ing is controlled by the penetration depth of the probe: the greater the depth, the stronger the coupling.

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60 Because the antenna is a source of power absorption, the Q of the cavity is reduced by the presence of the antenna. The quality factor of a cavity wi th an antenna is call a loaded quality factor, given by (3 29 ) where Q A is the quality factor of the antenna which is removing energy from the cavity Q A decreases with increased coupling and is approximately inversely proportional to y 4 where y is the insertion depth of the antenna When the power loss in the c avity equals the loss through the antenna, the antenna is critically coupled and Q L = Q /2 Because real antennas are not infinitely thin, the antenna will perturb the fields. Sp ecifically, a probe antenna will bend the electric field locally, which results in a weak interaction with the crossing field. TE and TEM modes are therefore commonly observed with a probe antenna inserted into the endplate of cylindrical cavities though usually at a much lower power than the TM modes. The frequenc y shift caused by the antenna can be estimated using perturbation theory for each mode [92] , (3 30 ) where U E and U B are the change in the stored electric and magnetic energy, respectively. T he effects on the field require numerical analysis to predict. Waveguide coupling is achieved by connecting to an aperture in the cavity. The apertur e can be designed to filter modes. T he aperture size also affects the coupling between the cavity and waveguide; a smaller aperture results in a reduced coupling. The Q of the aperture is ( 3 31 )

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61 where Z A and L A are the shunt impedance and inductance the of the aperture, respectively. The physical temperature of the cavity also a ffects the Q of the modes. As temperature of a conductor decreases, the mean free path of the electrons increases. Simultaneously, lower ing the temperature of a conducting cavity decreases the skin depth. A t temperature s below T ASE the mean free path of the electrons exceeds the cavity skin depth. The response of free electrons within a conductor to an externally applied electric field be comes nonlocal, such that the current at a location is dependent on the field elsewhere [93] T he long mean free path puts the cavity in the anomalous skin effects (ASE ) regime In ASE, the skin depth of the cavity is [94] ( 3 32 ) where m e and e are the mass and charge of the electron, respectively, v f is the Fermi velocity, and N is the electron number density. Equation 3 32 gives rise to a frequency dependence of Q ( 3 33 ) 3.2 Symmetry Breaking Symmetry breaking has been shown to have negative effects on the form factor of search modes in haloscope cavities [15 16] Two types of symmetry breaking exist: transverse and longitudinal. In a circular cylindrical cavity t ransverse symmetry b reaking is a loss of discrete or continuous rota tional invar iance. T ransverse symmetry can be broken by placing one or more tuning rods off axis or using noncircular tuning rods Longitudinal symmetry breaking is a loss of translational invariance within the

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62 boundaries of the cavity. Longitudinal symmetry can be broken using discon tinuous tuning rods that do not bridge the entire length of the cavity. T ilting the rod (s) or endcaps or warping the shape of the cavity wall break both transverse and longitudinal symmetry Transver se symmetry breaking from displacing a conducting rod c a uses localization of t he electric fields in TM modes, but the electric and magnetic field remain decoupled. Therefore, Eq. 3 14 is valid as is the orthogonality of modes defined by Eq. 3 17 and Eq. 3 18. The solution to the eigenfunction consists of a comb ination of several Bessel functions of the first and second kind [95] where the mode subscripts m n and p are preserved Figure 3 5 shows images of the E z field through a cross section of the cavity with an offset conducting rod for the (a) TM 010 (b) TM 110 and (c) TM 020 T he tuning rod is shown as a white circle; t he offset is x = R /4. The localization of the electric field is observed to the left of the rod One degeneracy of the TM 110 mode is shown; the second degenerate mode is less obvious by visual inspection. The orientation of the TM 110 mode s are now determined by the offset of the rod. The TM 02 0 mode is also less obvious ly identified visually than the TM 02 0 mode shown in Fig. 3 2. Localization of a mode from transverse symmetry breaking causes the frequency to change. ADMX and HAYSTAC take advantage of this phenomena to tune the search mode frequency of their detectors. In general, moving a tuning rod away from the center axis of the cavity will lower the frequency of the TM 010 because the curvature of the field is reduced. Figure 3 6 shows the electric field of the TM 010 mode inside a cylindrical cavity with the tuning rod (a) on axis and (b) off a xis. The field is localized in the larger

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63 (a) (b) (c) Figure 3 5. Cross sectional images of the E z field in a cylindrical cavity with an offset conducting rod. (a) TM 0 10 (b) TM 110 (c) TM 020 The cross section is para llel to the cavity axis. The white area in the center is the location of the conducting rod ( d = R /2) offset by R /4 ; a field solution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white center areas are the conducting boun daries. Red indicates higher field and blue indicates lower fi eld with zero field at the boundaries Localization of the field is seen to the left of the rod. (a) (b) Figure 3 6. E xample of the e lectric field of the TM 010 mode inside a tuned cylindrical cavity depicting the curvature of the field. (a) On center tuning rod. (b) Off center tuning rod. The cross section is cut through the x z plane and shows a portion of the cavity. The localization in (b) mirror s the localiz ation shown in Fig. 3 5(a). The curvature of (a) is greater than the curvature of (b), resulting in a higher TM 010 frequency in (a) than (b). The fields lines are for visualization and are not to scale cross sectional area when the rod is off axis. The widening of the field reduces the curvature. In the non localized area, the reduction of the field reduces the curvature. From Eq. 3 14, the reduced curvature of the field produces a reduced mode frequency. At higher order mode s translation of a tuning rod has even less intuitive effects and multiple rods further complicate the mode solutions. Small displacements of a

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64 single rod toward the center usually increase the frequency of higher TM 0 n 0 modes, but quite often the tuning rod position localizes the field in a way that causes a mode frequency to decrease as the tuning rods approach the center of the cavity. Predicting when this occurs typically require numerical analysis. Often when multiple rods are used, the next higher search mode above the TM 010 c hanges from one mode to another depending on the configuration, complicating the use of higher TM modes for axion searches. Breaking rotational invariance causes the form factor of the cavity to decrease due to transverse localization of the electric field Figure 3 7 shows a plot of the TM 010 mode (top) frequency and (bottom) form factor as a function of tuning rod displacement from a finite element model (FEM) of a circular cylindrical cavity The simulation used a n eigenfrequency solver conducted with a prepackaged program, COMSOL version 5.1 [96] The radius of the model cavity was R = 6.826 cm and the tuning rod diameter is 0.532 R The x axis is the displacement of the rod from the center axis ( x ) normalized by the cavity radius. The detector magnetic field is assumed to be homogen e ous and in the z direction for computing C Note TM 010 frequency for the empty cavity is f 0 = 1.68 GHz. The plot demonstrates moving the tuning rod off center reduces the frequency and breaking transverse symmetry decreases the form factor significantly. TM mode frequencies scale as the inverse of the radius when all other geometry (length, tuning rods, etc. ) is kept proportional and the mode subscript is p = 0 (see Eq. 3 16). The form factor is always independent of radius when all geometry is kept proportional.

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65 Figure 3 7. Plot of the TM 010 (top) frequency and (bottom) form factor as a function of tuning rod displacement for a circular cylindrical cavity Displacement is shown on the x axis as distance from the center axis ( x ) over the cavity radius ( R ). The plot shows f 010 and C 010 decrease as the tuning rod moves away from the axis. When multiple tuning rods are used in a microwave cavity, breaking discrete transverse symmetry can reduce the form factor of a search mode considerably Figure 3 8 shows images of the E z field through a cross section of two cavities with mu ltiple conducting tuning rod s shown as white circles A cavity with two rods is shown with (a) symmetry maintained and (b) symmetry broken, and a cavity with seven rods is show with (c) symmetry maintained and (d) symmetry broken. The local ization of the electric field when the symmetry is broken results in a form factor that is ~ 3 / 4 the form factor w hen the symmetry is maintained.

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66 (a) (b) ( c ) ( d ) Figure 3 8. Cross sectional images of the E z field for the TM 0 10 mode in a cylindrical cavity with multiple conducting rod s (a) Two rods maintaining discrete transverse symmetry. (b) Two rods with broken transverse symmetry. (c) Seven rods maintaining discrete transverse symmetry (d) Seven rods with broken transverse symmetry. The cross section is para llel to the cavity axis. The white areas are the location of the conducting rod s ; a field solution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero field at the boundaries Localization of the field s reduce s C to ~ 3 / 4 the form factor with symmetry maintained. Multiple tuning rods can produce unexpected changes in the TM 010 frequency when symmetry is broken. For both cases of symmetry breaking shown in Fig. 3 8, the breaking decreases f 010 though the one tuning rod is moved away from the axis in the cavity with two rod s, and one tuning rod is moved toward the axis in the cavity with s even rods. When multiple tuning rods are used, numerical analysis is almost always required to evaluate tuning and symmetry breaking effects

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67 Transverse symmetry breaking also causes degeneracy breaking. The eigenfunction solutions with subscript m > 0 con tain two indistinguishable degeneracies with plus/minus polarization about the axis of the circular cylinder. The modes have no preferred orientation, but the degeneracies are oriented 90 about the axis relative to each other to maintain orthogonality Br eaking rotational invariance localizes the fields of the degenerate modes in differing ways, causing the modes to have distinct solutions and preferred orientations. The two modes have different peak frequencies. Figure 3 9 shows images of the E z field through a cross section of a cylind rical cavity with two conducting tuning rod s (white circles) for the TM 110 modes The cavity has 180 rotational invariance. The modes are degenerate when the cavity maintains 90 rotational invariance, but in the configuration shown, the degeneracies are lifted. T hough the tuning rods are at the edge of the cavity where they have the smallest effect on the modes, t he frequencies of the modes vary considerably: (a) f 110 R = 16.30 GHz cm ; (b) f 110 R = 20.62 GHz cm Whe n the tuning rods are located closer to the center of the cavity, the degeneracy breaking is greater, and the modes are more distinct with a greater relative difference between frequencies. Note, the variable f nm 0 R for TM modes can be scaled to any cavity size by dividing by the cavity radius [16] With multiple tuning rods, degeneracy breaking can affect higher order modes in unexpected ways. Figure 3 10 shows images of the E z field through a cross section of a cylindrical cavities with two conducting tuning rod (white circles) for the (a) TM 2 10 (b ) unknown TM mode, and (c) TM 020 The solution set contains only one TM 210 mode even though there should be a polarized pair. A 90 rotation of the field within the cavity would result in no distinguishable change in the mode. The TM 020 mode is not obvious

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68 (a) (b) Figure 3 9 Cross sectional images of the E z field for the TM 1 10 modes in a cylindrical cavity with two conducting rod s The cross section is para llel to the cavity axis. The white areas are the location of the conducting rod s ; a field solution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero field at the bound aries The tuning rods break the degeneracy of the modes, resulting in two distinct modes at differing frequencies. An orientation relative to the rods is fixed for both modes. (a) (b) (c) Figure 3 10 Cross sectional images of the E z field for higher order mode s in a cylindrical cavity with two conducting rod s (a) TM 2 10 (b) Unknown mode (c) TM 020 The cross section is para llel to the cavity axis. The white areas are the location of the conducting rod; a field solution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red ind icates higher field and blue indicates lower field with zero field at the boundaries The unknown mode appears visually to be a degeneracy of the TM 020 mode, but is more likely the degeneracy of the TM 210 mode

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69 visually, as the shape of the field is consi derably different than that of the empty cavity (see Fig. 3 1(c)). The unknown mode visually appears to be a degeneracy of the TM 020 mode, though the mode does not have a polarization, as the subscript m = 0. The form factor of the unknown mode is 0.054. T he form factors of the other modes are C 2 1 0 = 0 and C 020 = 0.081. The subscript of the unknown mode could only be determined by finding a closed form solution to Eq. 3 14 and matchi ng electric field of one solution to the figure. Sometime localization of t he fields from transverse symmetry breaking produces a TM mode with m > 0 and C > 0 unambiguously Figure 3 11 shows the images of the E z field through a cross s ection of a cylindrical cavity with two conducting tuning rod (white circles) for the (a) TM 1 10 and (c) TM 020 The orientation of the rods breaks discrete and continuous rotational invariance. T he localization results in C 1 1 0 = 0 .080 and C 020 = 0.037, where the TM 110 is more sensitive to the axion for a ty pical haloscope ( s ee also Fig. 3 30 ) For cy lindrical cavities with offset dielectric rods, the solution requires an infinite series of Bessel functions of the first and second kind, with diminishing coefficients [91] To meet the boundary conditions, the electric and magnetic fields are coupled and all modes are hybrid modes. The mode subscripts m n and p are still preserved. The field is localized within/about the dielectric. The dielectric always decreases the frequency and form factor of TM 010 mode, but has varying effects on higher order modes, much like the conducting rods, requiring numerical analysi s to predict the effects. Figure 3 12 shows images of the E z field through a cross section of the cavity with an offset dielectric (alumina) rod for the (a) TM 010 (b) TM 110 and (c) TM 020 The

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70 (a) (b) Figure 3 1 1 Cross sectional images of the E z field for modes in a cylindrical cavity with two conducting rods. (a) TM 1 10 (b) TM 020 The cross section is para llel to the cavity axis. The white areas are the location of the conducting rod s ; a field solution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero field at the bound aries TM 11 0 mode has a higher form factor than the TM 020 mode, though modes with m > 0 are conventionally thought to have a form factor of zero. (a) (b) (c) Figure 3 12. Cross sectional images of the E z field in a cylindrical cavity with an offs et dielectric rod. (a) TM 0 10 (b) TM 110 (c) TM 020 The cross section is parallel to the cavity axis and the outer edge of the images are the conducting boundaries. The dielectric rod is outlined by a dotted line. The rod is the same diameter as the rod in Fig. 3 2 ( d = R/2) and is offset by R /4. Red indicates higher field and blue indicates lower field with zero field at the boundaries The field is localized inside the dielectric. offset is x = R /4. The electric field is localized inside the rod, outlined by a dotted line. One degeneracy of the TM 110 mode is shown; the second degenerate mode is less

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71 obvious by visual inspection. The orientation of the TM 110 is now fixed by the offset of the rod. Translation of a conducti ng or dielectric tuning rod will cha nge the frequency of all TM modes in the cavity, but will have no effect on the frequency of the TE or T E M modes. Rod insertion will alter the TE mode frequency some, but considerably less than the change it produces on the TM mode frequencies. As a result tuning a TM mode across a large frequency range will almost always cause the frequency to match that of a TE or TEM mode. A mode crossing occurs w hen the frequency of a search mode is the same as the frequency of a non search mode Figure 3 13 shows a p lot of the frequency tuning of the TM 010 mode with a conducting tuning rod through a mode crossing in a circular cylindrical cavity The frequency of the broken degeneracies of the TE 215 and the TM 0 1 1 are also shown. The results are from a FEM simulation conducted with COMSOL version 5.1 [96] T he radius of the model cavity i s R = 6.826 cm the length i s 5 R an d the tuning rod diameter is 0. 532 R The x axis is the displacement of the rod from the center axis ( x ) normalized by the cavity radius. The point where the TM 010 and TE 215 modes have the same frequency are mode crossing s It is important to note that mode repulsion, which is discussed below, is not observed Longitudinal symmetry breaking compromises translational invariance within a microwave cavity. Localization from longitudinal symmetry breaking has been predicted from the eigenfunction problem [97] as well as finite integration simulations [15] Longitudinal localization of fields in waveguides and cavities have been less studied than transverse localization due to the complex nature of the solution. However,

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72 Figure 3 13. Plot of a TM 010 mode crossing during frequency tuning with a conducting rod for a circular cylindrical cavity Displacement is shown on the x axis as distance from the center axis ( x ) over the cavity radius ( R ). The plot shows the TM 010 the (broken) degeneracies of the TE 215 and the TM 0 1 1 modes. The mode crossing occurs where the frequency of the TM 010 matches that of the TE 215 mode. longitudinal localization is more problematic than transverse localization for axion haloscope s [15] and requires in depth consideration for future searches, especial ly those at higher frequencies. Several types of longitudinal symmetry breaking exist in haloscope cavities. The most common type is a discontinuity between the t uning rods and the endcaps. Since the cavity must be tuned to different frequencies tuning rod s used to shift the frequency must be capable of moving within the cavity. If the rod ( s ) is inserted through an endcap a me chanical gap will exist between rod end and the opposing endcap. If the rod(s) translates inside the cavity, a mechanical gap needs to exist between the endcaps and both rod ends Dielectric rod insertion was analyzed by Hagmann et al. [15] quantifying the effects of longitudinal symmetry breaking in cavities with dielectric rods. As the primary focus of this study is conducting axion searches at higher frequencie s and dielectric TE 214 TM 010 TM 011

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73 rods lower the frequency of search modes, longitudinal symmetry breaking with dielectric rods are not reevaluated. See Zaki et al. [90] for modeling discontinuities of dielectric loaded waveguides The mode for a cylindrical cavity with discontinuities at conducting rod end s can be solved using mode matching techniques [98 99] where each translationally invariant section contains an infinite se t of mode solutions consisting of reflective and transmissive parts from the ends of their respective section. Ends that correspond to the endcaps have only a reflective part. Each mode solution has an associated coefficient. The infinite set of solutions for each section are equated together at the transition locations along the z axis, and t he coefficients of the series are solved simultaneously to obtain a complete set of eigenfuntions. In practice, only a finite number of coefficients are solved to obta in a sufficiently accurate solution to the desired modes. The subscripts m n and p of the modes are not maintained, but the nomenclature of TM mnp and TE mnp are often used in microwave cavities, as the modes are commo nly perturbed versions of the TE /T M mnp solutions. For TM modes, the mechanical gap between a rod end and an endcap acts as a capacitor. Electric currents oscillate in the longitudinal direction along the surface of the rod. At each oscillation, a n electric potential is built up at the rod discontinuity, as the charge is attracted to the endplate, forming a capacitive effect [100] The potential results in localizing the electr ic field Figure 3 14 shows images of | E | for the TM 010 mode through a cross section in the x z plane of the coaxial cavity with (a) translational invariance maintained and (b) translational invariance broken by discontinuities (mechanical gaps) of the ce nter rod

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74 (white area) The endcaps are located at the top and bottom of the image and the cavity wall bounds the left and right side. The diameter of the conducting rod is d = R /2, the cavity length is L = 5 R and the mechanical gap at both rod ends for (b) is g = 0.02 L The cavity with longitudinal symmetry maintains a constant E field (all E z ) along the z axis; the cavity with the gap has a varying field along the z axis. The red area above and below the cent er rod in (b) illustrate the electric field caused by the electric potential buildup (i.e., capacitive effect). Visual inspection reveals t he mode in (a) meets the form of Eq. 3 22 while the mode in (b) does not (a) (b) Figure 3 14 Cross sectional images of | E | of the TM 0 10 mode in a coaxial cavity. (a) Longitudinal symmetry maintained. (b) Longitudinal symmetry broken by discontinuities at both rod ends The cross section is through the x z plane. The white area is the center conductor; t he outer edge of the image and the white area are the conducting boundaries. Red indicates higher field and blue indicates lower field The interface between the endplate and the center conductor in (a) is not distinguished. The localization of the electric f ield due to the capacitive effect is shown in red above a nd below the center rod in (b). When the frequencies of two modes are reasonably close, the longitudinal symm etry breaking will often cause the modes to mix. The mixing breaks the mode purity, formin g a pair of hybrid modes. As a tuning rod is translated or inserted in a

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75 manner that would otherwise bring the frequencies of the two modes together, the mode mixing increases, further breaking the mode purity and causing the two modes to take on similar f orm. As this happen s, the two modes do not come to the same frequency but instead maintain some inequality through what would be a mode crossing in a longitudinal ly symmetric cavity This phenomenon is mode repulsion [15 101] Figure 3 15 shows plots of (top) mode frequencies and (bottom) form factor verses tuning r od displacement for a mechanical gap at both ends of (a) g = 0.001 L (b) g = 0.003 L and (c) g = 0.005 L from a FEM of a circular cylindrical cavity The simulation was conducted using COMSOL version 5.1 [96] The radius of the model cavity i s R = 6. 826 cm the length i s 5 R and the tuning rod diameter is 0. 532 R The x axis is the displacement of the rod from the center axis ( x ) normalized by the cavity radius. The detector magnetic field is assumed to be homogen e ous and in the z direction for computing C The frequency plots show the frequency of the TM 010 the TM 011 and the degenerate TE 215 modes, as labeled. The modes are not true TE/TM mnp modes, but perturbed forms of the modes. The nomenclature is most commonly used, though TE/TM nmp like better describes the true nature of the modes. Data shown in the plots are partial to reduce computation time; the plot scales are the same for compariso n purposes. The plots depict several important characteristics of longitudinal symmetry breaking. First, mode repulsion is observed in all plots. As the TM 010 and TE 215 mode are brought closer in frequency due to the translation of the tuning rod away fro m the axis, one of the broken degeneracies of the TE 215 begins to change frequency with

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76 (a) (b) (c) Figure 3 15. Plots of (top) mode repulsion and (bottom) search mode form factor for a circular cylindrical cavity with a discontinuous conducting tuni ng rod (a) Mechanical gap of 0.001 L (b) Mechanical gap of 0.003 L (c) Mechanical gap of 0.0 05 L Displacement is shown on the x axis as distance from the center axis ( x ) over the cavity radius ( R ). The frequency plot shows the TM 010 the (broken) degeneracies of the TE 215 and the TM 0 1 1 modes, as well as the TE/TM hybrid modes through the mode re pulsion phase. The form factor plot shows the form factor of the search mode ( C 010 ) through the tuning ; all other form factors are approximately zero tuning rod displacement The change is away from the TM 010 frequency leading to the term mode repulsion; the TM 010 repels the TE 215 in the frequency spectrum, keeping TE 214 TM 010 TM 011 TM 010 TE 214 TE 214 TM 010 TM 011 TM 010 TE 214 TE 214 TM 010 TM 011 TE 214 g / L = 0.00 1 g / L = 0.00 3 g / L = 0.00 5

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77 some difference between them and avoiding a mode crossing. When this occurs, the rate of change of the TM 010 frequency with a change in rod positio n decreases. This is depicted in the plot by the curving of the TM 010 frequency trace near the TE 215 trace This phenomenon is due to the two modes mixing. When the mode frequencies are close enough, the electric and magnetic fields in the two modes couple slightly within the localizations caused by longitudinal symmetry breaking. The two modes become hybrid modes, breaking TE/TM purity. As the tuning rod is displaced to bring the mode frequencies closer, the mix ing increases, furt her breaking TE/TM purity more. At the point of maximum repulsion, the two modes take on similar form (see Fig. 3 16). As the tuning rod is further translated away from the axis the mixing of the modes decreases However, the mode that i s the TM 010 on the left side of the plot now transforms into a TE 215 like mode, and the mode is the TE 215 on the left side of the plot transforms into a TM 010 like mode. When the tuning rod is translated to a point where the frequencies of the two modes are sufficiently different, the mode return to pure TE/TM modes but with the once TM 010 mode transforming fully into a TE 215 mode and the once TE 215 transforming fully into TM 010 Though the hybrid modes do not cross (i.e., match frequencies) within the mode repulsion phase of the tuning, it is conve ntion to classify the mode repulsion as a mode crossing. For purposes of clarity, the mode repulsion phase will be identified as such to distinguish this phenomenon from a mode crossing without mode mixing as depicted in Fig. 3 13. The plots show that any longitudinal symmetry breaking from a rod end gap will cause mode mixing, and the larger symmetry breaking (i.e, gap size ) the greater the

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78 repulsion and thus mixing (see Fig. 3 17) Further, only one of the two TE degeneracies mix with the TM mode. This is true for all TE TM mode mixing; o nly one TE mode and one TM mode will mix. If degeneracies exist, one degeneracy will maintain mode purity. The mechanism for selecting which broken degenerate mode mixes is not fully understood but is related to the br oken degeneracies and the TM mode configuration s at the localizations In the case s shown in the Fig. 3 15 the degenerate TE 215 mode that does not mi x also does not change frequency. As the mode is a pure TE mode throughout the repulsion phase, it will n ot be significantly detectable with a probe antenna. It is important to note that transverse symmetry breaking does not cause mode repulsion; mode repulsion is a specific result of longitudinal symmetry breaking only. Further, the quality factor s of the mo de s do not alter the mode repulsion, and increasing Q will not reduce the negative effects of the phenomenon discussed below. The mechanical gap decreases the frequency of TE 215 modes and slightly increases the frequency of the TM modes in the plots The gap increas es the effective length of the cavity which decreases the frequency of any mode with p > 0 (see Eq. 3 16 ). The gap decreases the effective radius of the cavity which increases the frequency of all modes. Since the TM modes are perturbed fo rm of modes with p = 0, only the radius change has an effect, and their frequencies are slightly increased by the gaps. The TE modes are a perturbed form of a p = 4 mode, so the length change effect is approximately four time greater than the radius change effect, resulting in a decrease in the frequency larger than the change observed in the TM modes. The changes in frequencies cause the repulsion phase to move to the right in the plots as the

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79 mechanical gap is increased. Different modes will have differen t frequency responses to the gaps, depending on the p value of the mode. Another important characteristic of longitudinal symmetry breaking in haloscope cavities is the discrete transition of the search mode. The form factor plots show the highest form fac tor of the modes in the corresponding frequency plot. The form factor of the TM 110 mode and the non mixing TE 215 mode are approximately zero in the plots. The form factor of the TM 010 mode when the mode is not mixed is ~0.5 0.6 and the form factor of other degenerate TE 215 mode is about zero. However, in the mode repulsion phase, due to the two modes mix ing the form factor of the TE 215 mode increase and the form factor of the TM 010 mode decrease as the tuning rod is moved away from the center axis. At the point of maximum repulsion, the form factors of the two modes are equal. As the tuning rod is moved further from the axis, the form factor of the mode that is the TE 215 mode in the left side is the largest, and this mode becomes the search mode. To accomp lish this change, the measurement frequency must be instantly change d from one mode frequency to the other and continue scanning with the new mode caus ing a hole in the frequency search spectrum (see Fig. 3 1 6 ) [15] The form factor of the hybrid modes at the point of maximum repulsion is C = ~0.3 for all gap sizes evaluated, suggesting the gap size has almost no effect on the minimum form factor due to mode mixing. The transition of the search mode from one hybrid mode to the other causes the V shape in the form factor plots. As the mechanical gap size increase, the width of the V increases, resulting in a reduced form factor over a larger frequency span Thus, a

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80 larger mechanical gap typically leads to a less effective microwave cavity for axion searches. Further, reviewing the frequency and form factor plots simultaneously reveals a larger mechanical gap size causes a larger hole in the frequency search spectrum. Figure 3 16 is a replot of Fig. 3 15(c) identifying the hole in the frequency search spectrum caused by the mechanical gaps at the rod ends, g / L = 0.005. The difference in frequency between the two hybrid modes when their form factors are identi cal is the hole size f The scale of the plot has been changed from Fig. 3 15 to emphasize the hole. Figure 3 16. Plots showing the hole in the frequency search spectrum produce d by longitudinal symmetry breaking. The plot shows the (top) frequency and (bottom) search mode form factor through the mode repulsion phase for a circular cylindrical cavity with a discontinuous conducti ng tuning rod g / L = 0.005. The plot is the same as Fig. 3 15(c) with the scale adjusted to emphasize the h ole. The dotted line indicates the rod displacement at the point of maximum repulsion. T he hole in the spectrum is f The degradation of C as well as the mode repulsion is a result of the mode mixing. Mode crossings without mode mixing does not degrade th e sensitivity of a f

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81 haloscope detector like mode repulsion, though it can make mode identification more challenging if the crossing mode is detected by the instrumentation (see Chapter 3.3). When longitudinal symmetry is maintained, mode repulsion does not occur; mode tuning that does not break longitudinal symmetry thus will never experience a loss in C at mode crossings (see Chapter 3.4). In reality, a n axion search will stop scanning prior to the point of maximum repulsion, tune the cavity beyond the poi nt, and continue scanning on the new mode. This procedure is done because the simulation cannot perfectly predict the form factor of the modes, so additional precautions must be made to ensure a scan is not conducted with a mode that has a negligible form factor inadvertently. Thus, the actual f Figure 3 15 demonstrates that f Figure 3 f normalized by the mode frequency verses gap size for the cavities modeled in Fig. 3 15. f / f g / L for the cavity modeled. Figure 3 17 Plot of hole in frequency search spectrum as a function of rod end gap size Gap size i s normalized by the length of the cavity and the frequency hole is normalized by the search mode frequency. Fo r a single tuning rod of diameter ~ R /2 f / f g / L at small gaps.

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82 Figure 3 1 8 shows images of | E | and the E vectors for the (left) lower frequency hybrid, (middle) non mixing degenerate TE mode, a nd (right) higher frequency hybrid mode in a cylindrical cavity at the point of maximum repulsion. The cross section s are through the (top) x z plane and (bottom) y z plane of the cavity The mechanical gap of the ends of the tuning rod (white area) is g = 0.005 L The geometry of the cavity is identical to that of Fig. 3 15. The black arrows show the E vector s ; the location of the vector origin is the tail of the arrow. The mode configuration in the x z plane of the lower and upper hybrid (TM 010 like ) modes, Fig. 3 18(a) and Fig. 3 18(c), are notably similar. The left side of both images appear to be opposing hybrid modes while the right side appears to be mirrored TE like modes. The mode configuration in the x z plane of the hybrid modes, Fig. 3 18(d ) and Fig. 3 18(f), also have similar configurations, both appearing TM like, though the E z field in the lower frequency mode is observably stronger. The contrasting configurations of the modes are like that of broken degeneracies. Vectors that are into th e page are not seen, so ~half of the vectors for the hybrid modes in the x z plane are not visible. The mode configuration of the TE 215 demonstrates mode purity in both planes. The vectors are projected onto a plane, so the in/out of plane magnitude is not depicted. The vectors in Fig. 3 18(b) are in or out of the plane, so appear as dots or are not shown in the image. The mode configure does not significantly change with translation of the tuning rod. Currently no common nomenclature of a hybrid mode in haloscope cavities exists. Most often the search mode is to call the TM 0n0 though it is neither a pure TM

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83 (a) (b) (c) ( d) ( e ) (f) F igure 3 1 8 Cross sectional images of | E | with E vectors of the hybrid mode s and the TE 215 mode at the point of maximum mode repulsion in a cylinder cavity. (a) and (d) Lower frequency TM 0 10 like (b) and (e) TM 215 (c) and (f) Higher frequency TM 0 10 like. (a) (c) Cross sections through the x z plane. (d) (f) Cross sections through the y z plane. The white area is the center conductor; t he outer edge of the image and the white area are the conducting boundaries Red indicates higher field and bl ue indicates lower field The electric field vector is depicted by the black arrows originating from the tail of the arrow. The hybrid (TM 010 like) modes are not pure TE /T M modes. mode nor are the subscript m n and p of any meaning mathematically. Sometimes the modes are given a suffix (as done herein) the non hybrid modes. During the mode repulsion phase, mode mixing makes mode

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84 identification extremely difficult as the hybridization make different par ts of the mode appears as different mode types as seen in Fig. 3 18. Perhaps for this reason, hybrid modes in waveguides are sometimes indicated by H mnp B ut when m n and p subscripts lose meaning, such as in the mode repulsion phase even this method i s insufficient. Hybrid mode in waveguides where complex mixing occurs are sometime designated H i where i simply indicates the number of the hybrid mode counting by frequency from the lowest frequency mode; H 1 indicates the lowest frequency hybrid mode. I n general, there is no universally agreed upon nomenclature for hybrid modes in waveguides or cavities. Degeneracy breaking of modes due to longitudinal symmetry breaking causes additional modes to exist within a given frequency range of a microwave cavit y. In addition, r eentrant modes appear in the mechanical gaps and evanescent modes will form The result is modes will cluster closer together and increase in mode density about a search mode, causing mode crowding. Debate still exists as to whether the mo des are formed from the symmetry breaking, or if the modes existed at much higher frequencies prior to the breaking and were greatly reduced due to the breaking. The mode crowding interferes with identifying and tracking a search mode (see Chapter 3.3) Ad ditionally, mode crowding increases the number of mode repulsion type crossings across a frequency scan further degrading the capability of a detector. Figure 3 19 shows the nearest 15 modes to the TM 010 mode plus the TM 010 from a FEM simulation of a cylindrical cavity with the tuning rod along the center axis for various mechanical rod end gaps The radius of the cavity is R = 6.826 cm, the length is 5 R and the tuning rod diameter is 0.532 R

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85 Figure 3 19 Plot showing m ode c rowding d ue to increased mechanical g ap at the rod ends At each gap size with data, 16 modes are shown; the TM 010 mode is in bold red. The tuning rod i s lo cated along the center axis. Gap size is normalized by the length of the cavity. As the gap is increased, mode density increases. At each mechanical gap with data a total of 16 modes are shown (vertically) though some modes are to o close in frequency to distinguish visually. Each gap size represents a single eigenfunction solution of 16 modes about, and incl uding, the TM 010 mode (bold red) T he simulation was conducted using COMSOL [96] version 5.1. Because exactly 16 modes are solv ed for in each run, mode crowding is revealed as the frequency span across the modes shrinking Th e mode density is clearly greater on the right side of the plot than the left The mode crowding is caused by reentrant and /or evanescent modes existing near the TM 010 mode, when they were not f ound at smaller gaps, and degeneracy breaking of nearby modes. In general, mode crowding will be observed at almost any frequency when a mechanical gap is introduced; the TM 010 mode is only used as a reference point in the plot. T wo distinct clustering of modes is seen ; the mechanism for the separation in to two clusters is unknown

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86 A second significant symmetry breaking in tuned microwave cavities is Tilting of the tuning rod is a more complex symmetry breaking, as it breaks both transverse and longitudinal symmetry simultaneously. In theory, the mathematical method used to find the eigenfunction solutions for modes in tuned cavities with mechanical ga ps could be used to find the solutions for cavities with a tilted tuning rod, but the summation would need to be replaced with an integration. However, no such mathematic formulation has been established to date. Instead, eigenfunction solutions are found with a finite element (or similar) type computer simulation. Mode localization due to rod tilt is dependent on the displacement of the rod. Figure 3 20 shows images of | E | for the TM 010 mode through a cross section in the x z plane of a cylindrical cavity with longitudinal symmetry broken by a 1.0 of the tuning rod (white area) with (a) no displacement from the center axis and (b ) a rod displacement of R /100 in the x direction The endcaps are located at the top and bottom of the image and the cavity wall bounds the left and right side. When the rod is not displaced from center of the cavity, symmetry breaking is done by the tilt alone and the mode takes on a discrete antisymmetric form in the longitudinal direction. A small displacement of the rod breaks the anti parity and the localization moves to one section of the cavity, greatly diminishing the form factor. Visual inspection reveals the modes do not meet the form of Eq. 3 22. Symmetry breaking from tuning rod tilt manifests mode repulsion and the asso ciated reduction in form factor as seen with mechanical gaps. Figure 3 21 shows plots of (top) mode frequencies and (bottom) form factor verses tuning rod displacement

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87 (a) (b) Figure 3 20. Cross sectional images of | E | of the TM 0 10 mode in a tuned cavity with a rod tilt (angular misalignment). (a) No transverse displacement of the rod (b) Transverse displacement of the rod of x = R /100 The cross section is through the x z plane. The white area is the center conductor; t he outer edge of the image and the white area are the conducting boundaries. Red indicates higher field and blue indicates lower field The interface between the endplate and the center conductor is not distinguished. for a tuning rod tilt of (a) = 0.25, (b) = 0.50, and (c) = 1.00 from a FEM of a for a circular cylindrical cavity The simulation was conducted using COMSOL version 5.1 [96] The radius of the model ca vity is R = 6.826 cm, the length is 5 R and the tuning rod diameter is 0.532 R The x axis is the displacement of the rod from the center axis ( x ) normalized by the cavity radius. The detector magnetic field is assumed to be homogen e ous and in the z direction for computing C The plots demonstrate many of the same characteristics of Fig. 3 15. Any longitudinal symmetry breaking from a rod tilt will cause mode mixing and mode repulsions. In all cases shown, the mode hybridization chara cteristics observed with the mechanical gap are observed with the rod tilt. However, the mixing is less severe than that experienced with mechanical gaps. In particular, the mode repulsion is not even visible in Fig. 3 22(a). Only through observing the for m factor of the modes does the

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88 (a) (b) (c) Figure 3 21 Plots of (top) mode repulsion and (bottom) search mode form factor for a circular cylindrical cavity with a tilted conducting tuni ng rod (a) Tilt of 0. 25 (b) Tilt of 0.50 (c) Tilt of 1.00 Displacement is shown on the x axis as distance from the center axis ( x ) over the cavity radiu s ( R ). The frequency plot shows the TM 010 the (broken) degeneracies of the TE 215 and the TM 0 1 1 modes, as well as the TE/TM hybrid modes through the mode repulsion phase. The form factor plot shows the form factor of the search mode ( C 010 ) through the tun ing; all other form factors are approximately zero. mode mixing and thus repulsion become apparent. As a result, the hole in the frequency search spectrum and the width of the form factor degradation is considerably smaller. = 0.25 = 0.50 = 1.00 TE 214 TM 010 TM 011 TM 010 TE 214 TE 214 TM 010 TM 011 TM 010 TE 214 TE 214 TM 010 TM 01 0 TE 214

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89 Contrary to the mode repulsion effects, the reduction in form factor is more extreme from the conducting rod tilt, which could be predicted from the mode configurations seen in Fig. 3 20. As the tilt is increased, the form factor across the frequency range decrease considerably until C for a tilt of = 1.00 even when the TM 010 is not in the mode repulsion phase. At the point of maximum repulsion, the form factor is ~0.2 0.3, similar to Fig. 3 15. At a tilt above ~1. 00, mode configuration s and interactions become complex. Figure 3 22 shows plots of (top) mode frequencies and (bottom) form factor verses tuning rod displacement for a tuning rod tilt of = 1.80 from a FEM of a circular cylindrical cavity The simulation was conducted using COMSOL version 5.1 [96] and the cavity parameters are the same as Fig. 3 21. The TM 010 and TE 215 mode enter mode repulsion as soon as the tuning rod is moved from the center axis. At x / R TM 010 mode is no longer mixing with the TE 215 mode. The TM 010 continues to decrease as the tuning rod is moved away from the center axis, maintain a form factor of ~0.3. The next higher mode, labeled TM 010 like, initially increases with tuning rod displac ement and has a form factor of about zero. The form factor of the mode increase as the rod moves away from the center axis, until obtaining a C approximately the same as the TM 010 mode at x / R of the TM 010 like continues to increase until x / R form factor decrease with rod displacement. 010 the mode configuration looks like a separate TM 010 mode, leading the TM 010 like label. The e xample shown is an extreme case, but demonstrates the complex nature of cavity

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90 Figure 3 2 2 Plots of (top) frequency and (bottom) search mode form factor for a circular cylindrical cavity with a conducting tuni ng rod tilted 1. 8 0 Displacement is shown on the x axis as distance from the center axis ( x ) over the cavity radius ( R ). The frequency plot shows the TM 010 the (broken) degeneracies of the TE 215 and a second TM 010 like mode as well as the TE/TM hybrid modes through the mode repulsion phase. The form factor plot shows the form factor s of all modes with C > 0.2 The mode interaction is complex. modes when longitudinal and transverse symmetries are broke, as well as highlights the complica tion with nomenclature for such modes. Bending of a tuning rod produces similar effe cts to tilt. The localization is either near the center of the cavity or near the two ends, depending on the tuning rod bend in relation to the rod displacement. Longitudi nal symmetry breaking can also be a result of cavity fabrication or assembly errors. But such symmetry breaking produce insignificant changes to the frequency or form factor of search modes except in extreme cases. Angular misalignments of an endcaps relat ive to the center axis would need to be many degrees to produce any noticeable effect. Bowing of the cavity walls would need a / L TE 214 TM 010 like TM 010 TM 010 TM 010 like

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91 0.1 to produce any measurable difference where a is the bow height (maximum distance from the theoretical location to the ac tual location. 3.3 Mode Identification Methods Axion searches with a haloscope detector must be conducted with modes that are sufficiently sensitive to the particle conversion (i.e., have a large enough C ) to achieve a possible detection. At each tuning r od configuration, an infinite number of modes exist while only a very small number of modes will yield any statistical probability of a detection within a feasible time frame. In theory, a threshold for a minimum form factor of a search mode could be deter mined by the cavity parameters (see Eq. 2 10), the frequency range of the mode(s) to be scanned, the desired model sensitivity (see Fig. 2 17), and the available search time In practice, the TM 010 mode has been exclusively used to conduct axion searches, as it nearly always has the highest form factor of any mode. Higher modes have been investigated, but instead of establishing a threshold, the detector sensitivity is established from the available modes and time, and the sensitivity is evaluated for scientific worth. Until recently, no experiment had established sufficient sensitivity with a mode other than the TM 010 to detect the KSVZ axion. T he form factor of a mode is not directly measurable; th ere is no current measurement technique that could evaluate the electric field in the entirety of the cavity without significantly disrupting/destroying the field. Absent any preplanning, t he signal received from a desired search mode is indistinguishab le from signals from any other mode Thus, mode identification is critical to the success of an axion detector. Further, the search must match the scan frequency of the receiver chain with the frequency of the search mode throughout the cavity tuning range. M ode identification must be

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92 performed repeatedly and rapidly at all times, and mode tracking, staying on the search mode throughout the search, is necessary. The most common technique used for mode identification is mode mapping. Mode mapping consists of t wo parts: analyt ical predictions and cavity measurements. The analytical predictions of mode frequency verses tuning configuration are calculated typically using computer simulations, throughout the tuning range of the cavity. The predictions will be with in a frequency window, usually spanning the lowest frequency of any TM mode to some upper limit established by the search objectives. The predicted form factor and quality factor of each mode is also calculated. The frequency of the modes is plotted again st the tuning configuration parameters of the cavity, forming an analytical mode map. The tuning parameters would be, for example, the angle of rotation against some baseline for rods that rotate or the depth of insertion for rods that are inserted. A 2 di mensional map exists for ea ch tuning parameter, yielding n +1 dimensional mapping where n is the number of degrees of freedom. For a cavity with only one tuning rod, the mapping is a single plot; for a cavity with two tuning rods, a 3 dimensional map will r esult. The display of a 3 dimensional (or higher) map requires slicing the map into 2 dimensional plots by holding all but one parameter constant. The predictions can also be displayed with a c olor axis as the form factor, enabling easier identification of search modes, particularly at mode repulsion frequencies Figure 3 23 shows an example of analytical predictions from ADMX [100] The predicted frequency of several modes is shown verses the angular rotation of one tuning rod ( 2 ) while the other tuning rod is held fixed ( 1 ) Only part of the full tuning

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93 Figure 3 2 3 Example of frequency predictions from ADMX [102] The red circles show the predicted frequency while the blue line identifies the predicted f 010 as determined by C 010 of the analytical results. One tuning rod is rotated while the other is held fixed. Results are from a 2 dimensional analysis. range is shown. The red circles show the predicted frequency while the blue line identifies the predicted f 010 as determined by C 010 of the analytical results. Several mode crossings are depicted but no mode repulsions are shown, as the simulation assumed longitudinal symmetry. The cavity measurement mode map is composed from the power spectrum of the cavity at various configurations across the scanning range. The spectrum is obtained from sweeping an injection signal, usually through a weakly coupled port, across a frequency range that bounds the search frequency range of the cavity, and measuring the power t hrough the receiver chain (an S 12 measurement). The measurements are visually displayed in two forms. One form is a color plot with the data shown with the x

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94 axis as one tuning parameter, the y axis as the frequency, and the color axis as the measured pow er. The second form uses a power threshold, plotting only the frequencies of the peak power that exceed the threshold. M easurement data maps without the predictions are also referred to as a mode map s ; maps with analytical predictions should be identified clearly. The measured data is compared to the analytical predictions to identify the modes of the cavity. The identification method is often performed manually using visual techniques, as misalignments in the cavity make it nearly impossible to bound error in the predictions prior to the mode mapping. Figure 3 24 shows an example of a mode map from ADMX showing both the analytical predictions and the measured data [100] using the threshold method. The analytica l data is identical to Fig. 3 23 The red dots are the analytical predictions and the blue asterisks are the measured data. Mode repulsion is visible at two of the mode crossings of the TM 010 around 725 MHz and 745 MHz. Though the 2 dimensional simulation did not predict the mode repulsion, the predictions are accurate enough to identify the TM 010 However, a 3 dimensional model is needed to de termine where the mode transfer should occur in the mode repulsion phase. Due to prediction errors, the form factor at the mode repulsion is not well bound, requiring a broadening of the frequency hole in search spectrum to avoid scanning with an insuffici ent form factor. A few false positive mode measurements are also observed as single scattered points. Mode mapping has been successfully used in haloscope searches historically. However, as searches move into higher frequencies, mode maps become less effec tive. At higher order search modes, mode density increases greatly. Using a power threshold

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95 Figure 3 24 Example of a TM 010 mode map from ADMX [102] The red circles show the predicted frequency and the blue asterisks are the measured data One tuning rod is rotated while the other is held fixed. The predictions are accurate enough to identify the TM 010 type map is more useful for mode identification, but can result in data overcrowding, such that modes cannot be sufficiently distinguished from each other for tracking. Additionally, threshold maps often identify false positives or miss the search mode depending on where the threshold is set further com plicating mode identification. A colo r plot relieves some of the issues, but is usually too co ngested to be of much practical use. Figure 3 25 shows a 2 dimensional mode map without analytical predictions of ADMX at the next higher order search mode abov e the TM 010 The mode was identified as TM 020 [100] though the subscript is n ot fully accurate (see Fig. 3 32 ) The map is very crowded and depicting a search mod e, one that yields a viable form factor, over TM 010

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96 Figure 3 25 Example of a TM 02 0 mode map without predictions from ADMX [102] The red dots are frequency peaks of measured data above a specified power threshold One tuning rod is rotated while the other is held fixed. The map is too congested to accurate ly identify the search mode throughout tuning. modes with C map even more complicated and small misalignments or symmetry breakings cause the modes to move too far from the analytical predic tions to identify the desired search modes. Searching higher order mode s is only one of several methods of conducting higher frequency axion searches. Other methods include power combining multiple smaller cavities or using a greater number of tuning rod s (see Chapter 3.4). However, these methods have distinct issues that hinder the effectiveness of mode maps. Smaller cavities are more sensitive to symmetry breaking an d mechanical misalignments, causing a greater error in the predictions espe cially aroun d mode repulsions. L arge

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97 number s of tuning rods tend to drive the search mode into more mode dense frequencies, resulting in mode maps like Fig. 3 25 For these reasons, additional mode identification techniques need to be developed for successful axion se a r ch at higher frequencies. The simplest method of mode identification is rod insertion. Inserting a conducting rod a short distance inside the cavity will always increase the TM 010 mode and will typically increase the frequency of high er order search modes, though not always (see Chapter 3.2). Rod insertion can help identify the search mode by comparing shifts in the S 12 measurements of the mode frequency to the pred icted shifts from a preexisting analysis. Both the magnitude and the direc tion of the shift can be used for identification. The method is particularly useful when only two candidate modes exist. In theory, the rod insertion could be conducted using the receiver antenna of the haloscope antenna. Because the antenna needs to be c oupled at different frequencies, a haloscope cavity usually uses an adjustable probe antenna. The antenna could be inserted further to force the frequencies of the modes to shift. However, the insertion of the antenna also changes the coupling and Q L A ve ry precise analysis would be required to accurately predict the frequency shifts from the insertion. Characterizing the method prior to a search would be necessary. Alternatively, a separate rod with associated drive system could be used. Some haloscope ca vities use such a system as part of their tuning method. A more sophisticated and robust technique of mode identification is a bead pull (or bead perturbation) test. A small dielectric or conducting bead is placed inside the

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98 cavity shifting the frequencie s of the modes slightly depending on the local field at the bead. The bead is translated within the cavity usually by pulling on a nonconducting cord attached to the bead and multiple S 12 measurements of the mode frequency verses bead location are taken to map the frequency shift along a straight line. For a small lossless dielectric bead, the frequency shift of any mode can be predicted by [103] (3 34 ) w here V b and b are the volume and relative permittivity of the bead, respectively, and E is the field at the center of the bead Mapping the frequency shift of a mode against the location of the bead revea ls the relative strength of the field. When the bead i s traversed in the z direction through the length of the ca vity, the measurements produce a local profile of the mode s The f the value of the subscript p of the mode (or the subscript for the closest TE/ TM mnp mode ) Using a conducting bead can provide a mapping of the magnetic field. T he measurement can also identify misalignments or other internal issues. For example, a tuning rod f to vary linearly with height, where an aligned rod f T f Using the bead pull test in multiple transverse locations yields a field map of the cavity that can be compared to predictions to validate the analysis and bound errors in form factor. The bead pull test is the closest c heck an axion haloscope has to a direct measurement of the form factor. HAYSTAC [104] is currently conducting bead pull tests to enhance their microw ave cavity development program.

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99 Figure 3 2 6 shows (a) a schematic of a bead pull test setup and (b) results [105] from a test on an empty cylindrical cavity with a dielectric bead The red block depicts the bead and the grey line depicts the cord used to translate the be ad. The blue arrows indicate the E field of the TM 010 (not to scale). The black arrows indicate a force: the force down maintains tension on the cord while the force up translates the bead some z after each S 12 measurement. (a) (b) Figure 3 2 6 ( a) Schematic of bead pull test a nd (b) test results from a bead pull test [68] The red bar depicts the bead and the blue arrows are the E field for the TM 010 mode (not to scale) The grey cord is pulled taunt and translated a finite distance after each S 12 measurement. f for several locations inside an empty cylindrical cavity next to the predicted field lines for the cavity mode that matches the results. The x axis is the displacement of the bead from an arbitrary baseline. The node for the TM 011 is to the left of the plot. f of about half the cavity for the (top) TM 010 and (bottom) TM 011 modes. The frequency shift for the TM 010 mode is approximately constant; the shift for the TM 110 mode decreases from the edge to the cente r revealing one node at

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100 the center ( p = 1). The predicted field for the modes from a computer simulation is shown to the right of the data. The z axis of the cavities is horizontal in (b). All preexisting mode identification techniques beyond the mode map require perturbing the field in some manner. While it may be feasible to perturb the field when conducting an axion search, recent development testing revealed the need for an in situ method of enhancing mode identification. Chapter 5 d etails a new approa ch that uses the phase of the fields to distinguish modes 3.4 High Frequency Cavities ADMX has successfully excluded KSVZ The experiment is currently searching for axions at DFSZ sensitivity up to about the same mass limit. Extending the search up to 1 GHz while maintaining similar sensitivity is viable, but searching at frequencies beyond 1 GHz requires new cavity technologi es. For axion searches, the term frequency refers to a haloscope cavity that is capable of successfully search ing for axion s to at least the KSVZ sensitivity above 1 GHz within a feasible time frame Since the limits set on the WIMP [58] and on the existence of supersymmetry [59] in recent years, interest in ax ion search es has increased significantly. In particular, several scientific organizations from around the world have joined the effort to expand current haloscope detector capabilities above 1 GHz. The Axion Dark Matter eXperiment High Frequency (ADMX HF), a National Science Foundation funded axion search now name d HAYSTAC, began in 2011 as a path finder experiment for high frequency haloscope cavities. The Republic of Korea formed CAPP, an Institute of Basic Science, around 2014 to search for axions at fre quencies above 1 GHz. ADMX

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101 development program for high frequency cavities, while incorporating additional research facilities into the organization, to include Fermilab and Pacific Northwest National Laboratories Other research institutes have also joined the efforts, such as the University of Western Australia. The original research on high frequency cavit ies was conducted by Hagmann et al. [15] in 1990. The work evaluated reducing the cavity radius, conducting searches with higher ord er TM modes, both with and without dielectric materials and using higher number of conducting tuning rods. While the groundwork for the research was established nearly 30 years ago, only in the past ~5 years has any significant work been conducted to advance the current science. A few additional concepts have been conceived since the original study such as utilizing multiple frequency locked cavities of smaller radius, but limited development was conducted until recently. The simplest method to increase the frequency of a microwave cavity is to increase the size of a conducing tuning rod. HAYSTAC [24] utilized one tuning rod with a large diameter instead of two smaller diameter rods, like ADMX, to drive up the TM 010 frequencies. However, t he form factor of t he cavity is reduced as the tuning rod diameter is increased. Figure 3 2(a) shows that a larger tuning rod results in less volume with E z field, as no field exists in the rod, which reduces C for all search modes Figure 3 2 7 shows the TM 010 frequency rang e times the cavity radi us, and the associated form factor of a cavity with one conducting tuning rod of varying diameter. The data was obtained from a 2 dimensional COMSOL simulation [96] The tuning rod diameter is normalized by the cavity radius. The frequency is multiplied by the cavity radius for scaling. The scan range of a cavity would be between the minimum and

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102 Figure 3 2 7 Plot of (top) frequency range and (bottom) form factor for a circular cylindrical cavity with a conducting tuning rod at varying diameters. Rod diameter ( d ) divided by the cavity radius ( R ) is shown on the x axis. The frequency is multiplied by the cavity radius. The frequency plot shows the maximum and minimum frequencies for the TM 010 mode. The form factor plot shows C 010 for the associated maximum/minimum frequency. As the tuning rod diameter increases, the f 010 increases while C 010 decreases. maximum f 0 10 vertically, for a given rod diameter. To tune, the rod is moved from on center until displaced such that the rod touches the wall. As the frequencies increase with the rod diameter, the scan range increases and the C 010 drops. The form factors shown a re not the lowest form factors, as the minimum form factor occurs between the minimum and maximum frequency configurations (not shown). A modestly more complex method to increase the search frequency is to reduce the radius of a cavity. Perturbed forms TM mn 0 modes scale s as ~ 1/ R To maintain the same frequency range proportionality the tun ing rods would need to be scaled the same as the radius (i.e., d / R is constant)

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103 The length of the cavity dictates the TE modes, w ith a longer cavity lowering each TE m ode frequency The length of the cavity should be minimized in general to reduce mode crossings, thus driving the TE modes above the search mode. However, reducing the length of course reduces the volume of the cavity and therefore the sensitivity of the h aloscope. In simple cavities with one or two tuning rods, length of ~5 R volume. For this reason, the aspect ratio of L / R haloscope c avities. A t higher frequencies and with more complex cavities, more consideration needs to be given to mode crossings then previously. Maintaining all other haloscope parameters and assuming anomalous skin effects (ASE) reducing the cavity geometry yields a scan rate scaling of ~ f 010 14/3 [16] Thus, cutting a cavity radius in half along with the length and tuning geometry, will double the minimum and maximum frequencies of the search range but will take 25 times longer to obtain the same axion sensitivity. The primary reason for th e sensitivity loss is the reduction in volume; the scan rate scales as V 2 (see Eq. 2 15) and the volume scales as R 3 if the aspect ratio is not changed. One way to regain some of the sensitivity of th e haloscope is to use a stronger magnet and/or operate at colder temperatures. The scan rate scales as B 0 4 and 1/ T n 2 Since the cavity is physically smaller, the bore of the magnet can be smaller. Solenoidal magnets with smaller bores can be made at higher strengths To recoup the sensitivity reduction from reducing R the magnetic strength would need scale as R 7 /6 Similarly, smaller cavities have less mass, and thus can be cooled to a lower temperature more easily. To recover the sensitivity loss from reducing R T n would need

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104 to scale as R 7/3 B ut the receiver chain is at best case, quantum limited. The quantum limited noise goes as ~ 1/ R so recovering the sensitivity loss from reducing the cavity size could never be fully recovered by reducing the temperature alone. All axion searches are, in some way, investigating using smaller cavities while increasing the magnet strength and reducing the operating temperature of the detector. HAYSTAC recently published results [67] utilizing this technique to scan for axions at 2.3 g KSVZ (i.e., less than half the sensitivity needed to detect KSVZ axions) despite scanning at a rate ~10 times slower than ADMX. Note, the scan rate scales as g 4 ; HAYSTAC would need to scan 28 times longer to obtain KSVZ sensitivity, or about 300 times slower than ADMX (an AD MX cavity search is ~1/2 a year). The results demonstrate the need for further research and development of microwave cavity technology to search for axions at frequencies above 1 GHz. Another way to improve on t he sensitivity of a reduce sized cavity is t o combine the power from seve ral cavities Each cavity c ould be run as independent searches and the power spectra from each search would be added analytically. Assuming the power of each cavity is roughly the same at a given frequency, the combined power w ould be [106] ( 3 35 ) where N is the number of cavities and the summation is over all cavities. A haloscope detector has a finite volume for which to fit the cavities dictated predominately by the bore size of the magnet N will instantaneously increase as R is decreased to a size where one or more cavities can just fit within the volume, so N ( R ) is

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105 a step function. Volume between the cavities is unus ed. If the aspect ratio is maintained, volume in the longitudinal direction would also be unused except for specific cases. The effective volume, V eff = N V of all the cavities therefore discon tinuous, decreasing as R 3 for a constant N until a discrete increase in N causes an instantaneous increase in effective volume (see Fig. 3 31 ) Figure 3 2 8 shows a schematic of a 6 cavity packing (a) top view and (b) side view inside the available volume depicting the loss in volume (white area) between and below the cavities The cavities i n (b) are pushed to the top to emphas ize the unused volume. The aspect ratio of the cavities and the available volume is L / R = 5. S tudies us ing different packing ratios have been conducted [106] (a) (b) Figure 3 2 8 Schematic of a 6 cavtiy packing [78] (a) Cross section through the x y plane (b) Cross section through the x z plane The inside of the cavity is in blue. The white area within the available volume (outer black lines) is unuse d volume. (a) and (b) are not at the same scale.

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106 To utilize the available volume fully, partitioned cavities have been proposed. Partitioned cavities divide a large cavity that fills the available volume with solid dividers. Each cell would constitute a se parate cavity. T his technique increases the effective volume of the cavities, but maintain an aspect ratio of ~5 is considerably more challenging, resulting in more mode crossings typically. Figure 3 2 9 shows a photograph of the inside of a 2 cell partitio ned cavity prototype. Limited testing has been conducted to advance the technology [106] Figure 3 2 9 Photograph of the inside of a 2 cell partitioned cavity [68] The cavity wall is cooper plated stainless steel and the divider is copper plate. All surfaces are electropolished. The electrical conduction between the divider and the walls is maintained with silver paint. Figure 3 30 shows an image of the E z field of the TM 010 modes through a cross section of a n 8 cell partitioned cavity with one tuning rod in each cell The white circle in each cell is the tuning rod. The black lines indicate the cavity walls and the partitions. Red indicates higher field and blue indicates lower field with zero field at the conducting boundaries. The green dots are E vector arrow coming out of the page. The cells are at the same frequency.

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107 Figure 3 30 Cross sectional images of the E z field of the TM 110 modes in a 8 cell partitioned cylindrical cavity The cross section is para llel to the cavity axis. Each cell has a single conducting rod. The white areas are the conducting rods; field inside the conductor is zero. The black lines show the outer cavity boundary and the cell dividers. Red indicates higher field and blue indicates lower field with zero field at the conducting boundaries The green dots are the E vectors out of the page Frequency gains of ~2.5 can be achieved with an 8 cell partitioned cavity, but the TM 010 mode experiences many mode crossings unless the length of the cavity is ~2 R Several cavities must be stacked to maximize volume reducing the sensitivity gain, as V eff scales as N and increasing complexity. Additionally, t he proportional tuning range is considerably less that of the ADMX detector. Above ~8 partitions, each cell becomes very narrow and the tuning range becomes limited, minimiz ing the effectiveness of the concept. Equation 3 35 assumes the power is combined from the differe nt cavities or cells incoherently. At high numbers of cavities, the sensitivity gain from using multiple cavities tappers off. Alternatively, i f the power is combined coherently, the total power is P i assuming the individual

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108 powers are approximately the same. The effective volume is the sum of the volume of all the cavities, V eff = NV Figure 3 31 shows a plot of the effective volume of multiple cavities verses TM 010 frequency when the power is combined coherently and incoherently. The effective volume is normalized by the available volume ( V max ) and the frequency is normalized by the TM 010 frequen cy of a n equivalent cavity that fills the available volume ( f V max ). The spikes in the plot represent discontinuities where N incrementally increases at a given R The plot shows coherent power combining produces a higher gain in sensitivity especially at higher frequencies (i.e., higher N ) Because scan rate is proportional to V eff 2 incoherently combined cavities would need to scan N times as long as the same coherently combined cavities t o obtain the same sensitivity. Figure 3 31 Plot of effective volume verses TM 010 frequency for power combined cavities. Results are shown for power combined coherently (red solid line) and incoherently (blue dashed line). The volume is normalized by the available volume The frequency is normalized by the TM 010 freq uency of a single cavity that fills the available volume Combining power coherently is much more effective at increasing the detector sensitivity. Coherent power combining is accomplished by frequency locking the cavities (cells) together and combining t he analog signal measurements in phase. Frequency

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109 locking is accomplished with a feed back system that controls the tuning rod(s) and/ or an a uxili ary tuning device to match the search mode frequency of all the cavities. The power produced from axion conver sion within haloscope cavities within close proximity will always be in phase because the de Broglie wavelength of the axion is tens of meters or greater. The combined power from two cavities close in mode frequency neglecting losses through the combine r, is [107] (3 3 6 ) where is the difference between the mode frequencies and f is the average frequency of the two modes. P i is the power extracted from the two cavities The quali ty factor is assumed to be approximate the same for both cavities. When < f / Q Eq. 3 36 simplifies to (3 3 7 ) where (3 3 8 ) If a frequency matching criterion is set at < f / 10 Q L Eq. 3 37 is expanded for N cavities as (3 3 9 ) where the sum is over N Power is lost from phase shifting of the signal between the cavities and the combiner if the cable lengths from the cavi ti es/cells to the combiner are not the same. P ower is lost ADMX

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110 is currently developing frequency locked cavities and in phase power combining receivers for high frequency axion searches. A separate method of search ing for axions at higher frequencies is to scan higher order mode s. Though the TM 010 has the highest form factor by a considerable amount, with a large volume cavity, a strong magn etic field, and a very low temperature, KSVZ sensitivit y could be achieved with higher order TM modes within a viable time frame. ADMX has recently instrumented their detector with a receiver ch annel dedicated to scann ing higher order mode s. N o results have been released to date. The next higher search mode above the TM 010 mode yields a form factor of ~0.05 0.25 depending on the tuning rod(s) configuration. The mode is often referred to as the TM 020 mode, though the desired search mode is quite commonly another mode (see Fig. 3 11). Figure 3 32 shows a scatter plot of form factor verses frequency for the three lowest TM mn 0 modes of ADMX. The results are from a 2 dimensional eigenfrequency solver [96] The values shown are for various rod configurations (not specified), but are not all inclusive. The frequency on the x axis is multiplied by the cavity radius (0.21 m) for scalability. The TM 010 and both broken degeneracies of the TM 110 modes are shown. The TM 110 mode(s) is the next higher search mode above the TM 010 for the ADMX haloscope, though the experiment calls the mode TM 020 (A) [100] The true TM 02 0 mode is the third highest search mode for ADMX, having a frequency of f R f 020 ADMX) and a form factor of ~0.05 0.10 [1 08] At first glance, scanning higher order TM modes with adequate form factors might appear to the be simplest method to search for high frequency axions. To scan

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111 Figure 3 32 Plot of f orm factor verses frequency for the three lowest search modes of ADMX The frequency on the x axis is multiplied by the cavity radius for scaling; for ADMX, R = 0.21 m. The TM 010 and broken degeneracies of the TM 110 modes are shown. Due to transverse symmetry breaking, the TM 110 mode is a viable search mode, and has a higher form factor than the TM 020 mode which is higher in frequency (not shown). ADMX refers to the mode as TM 020(A) [100] higher order mode s the only hardware change is the receiver chain electronics to account for the hi gher frequency signal. But the same changes would be required of a higher frequency search regardless of method. No ne w cavity technology would seemingly be required. Howev er, as demonstrated by Fig. 3 25 mode density around higher order mode is very high. Mode identification and tracking becomes extreme ly challenging. Characterizing the searc h mode prior to a search and having an in situ technique of verifying the mode is a necessity. Mode identification techniques to support searching higher order mode s are being developed (see Chapter 3.3 and Chapter 5.0). HAYSTAC is investigation using a di electric cylindrical ring to spoil parts of higher order mode s to improve the form factor, but the technique will not mitigate the mode identification problem

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112 The most effective method of elevating the search modes of a cavity for axion detection is to use additional conducting tuning rods [15] As Fig. 3 6 alludes to, increasing the number of rods will increase the curvature of the field in the TM 010 mode s, driving up the frequency. Higher order mode s will also increase, though field localization will result in a lesser increase than TM 010 proportionally. The complex solutions of higher order mode s require analytical computation to evaluate the frequencie s and form factors Further, additional tuning rods increase m ode density. TE mode frequenc ies are not significantly altered driving search modes into h ighly crowded frequency ranges. In collaboration with HAYSTAC a cavity study was conducted to evaluate using h igher n umber of tuning rods to elevate the TM 010 frequency of the cavity [16] The study initially evaluated linear translation and off axis rotation of th e rods. However, linear translation was deeming too difficult to implement and the initial results were not notably better than the initial results of the off axis rotation concept. So, t he study focused primarily on off axis rotating rods. In addition to circular cross section rods, the study included wedge shaped rods, or vanes. The study was conducted using a 2 dimensional FEM and an eigenfrequency solver in COMSOL version 4.2 [96] The study analyzed 99 tuning configurations, consisting of 4, 6, 7, and 8 tuning rods, and 4 and 6 tuning vanes. For each configuration, a finite number of rods/vanes were stationary and a f inite number rotated about the center axis. The configuration s are commonly referred to as periodic array s of tuning rods or vanes The TM 010 tuning range for the various configurations were computed as well as C 010 and Q throughout the range.

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113 Fig ure 3 33 shows the geometry of example (a) 6 rod an d (b) 4 vane cavities The diameter of the rods is d and circle radii S and R are the location of rotating and stationary rods, respectively, measured from the center of the cavity. The vanes have an angle of and rounded ends The circle radii I and O are the location of center of the inner and outer vane end rounds respectively, measured from the center of the cavity. The outer circle are the cavity walls. The solid hatched rods/vanes depict the maximum TM 010 frequency configuration. The arrows show the direction of the rotating rods/vanes, and the dashed hatched rod s/vanes depict configuration of the rotating rods/vanes in the minimum TM 010 frequency configuration. Figure 3 33 Geometry of example cavities from periodic array study. (a) 6 rod. (b) 4 vane. Three rods move in (a) and two vanes move in (b). The solid hatched lines show the maximum frequency configuration for the TM 010 mode. The dotted hashed lines show the rotating rods/vanes in the minimum frequency configuration. The arrows show the direction of the moving rods/vanes. Figure 3 34 shows images of the E z field of the TM 010 mode through a cross section of a (top) 6 rod and (bottom) 4 vane cavities. The (left) minimum frequency and (right) maximum frequency configurations are shown. The white area is the location of d R S I O

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114 (a) (b ) (c) (d ) Figure 3 34 Cross sectional images of the E z field for the TM 0 10 mode in periodic arrays (a) 6 rod minimum frequency configuration (b) 6 rod maximum frequency configuration. (c) 4 vane minimum frequency configuration. (d) 4 vane maximum frequency configuration The cross section is para llel to the cavity axis. The white areas are the location of the conducting rod s/vanes ; a field solution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero field at the boundaries the rods/vanes with no field solution. Red indicates higher field and blue indicates lower field, with zero field at the boundaries. The s tudy revealed that periodic arrays are extremely sensitive to transverse localizations. Small changes in the configuration can drive the predominance of the field to a ve ry small area(s) of the cavity cross section, causing C 010 to be reduced greatly The study highlighted the importance of discrete transverse symmetry in haloscope cavities and demonstrated the need for A well distributed E z field, like those shown in

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115 Fig. 3 34 As a result, many of the 99 configurations analyzed proved to be unusabl e, as large sections of their tuning range had too low of a form factor. The study identified 13 configurations tha t yields the highest scan rates Table 3 1 shows the tuning rod/vane configurations of the 13 cavities normalized by the cavi ty radius for scaling. The number of tuning rods/vanes is N and the subscripts R and S indicate s rotating and stationary rod s/vanes Table 3 1. Con figuration of cavities with the highest scan rates from the periodic array study. Rods Design N N R N S R / R S / R d / R A 4 2 2 0.413 0.413 0.350 B 4 2 2 0.425 0.425 0.300 C 4 2 2 0.454 0.454 0.350 D 6 3 3 0.375 0.625 0.250 E 6 3 3 0.375 0.750 0.375 F 6 3 3 0.400 0.675 0.275 G 6 3 3 0.400 0.800 0.300 H 7 3 4 0.450 0.450 0.200 J 7 3 4 0.525 0.525 0.175 K 8 4 4 0.550 0.550 0.200 V ane s Design N N R N S I / R O / R ( ) L 4 2 2 0.250 0.800 27.0 M 4 2 2 0.300 0.700 27.0 N 6 3 3 0.300 0.600 9.0 The results of the study showed that periodic array cavities could be used to conduct high frequency axion searches. The configurations shown in Table 3 1 obtained a frequency range of 0.2 GHz m f 010 R 0.3 GHz m (~1.0 1.5 GHz for an ADMX size cavity) and form factors of 0.3 C 010 0.7 S can rates approximately equivalent to that of four coh erently power combined cavities that fit in the same allowable space were also achieved demonstrating the method detection potential

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116 Higher frequencies could be attained with more conducting rods/vanes, but at a cost of great er sensitivit y to localization. See Stern et al. [16] for detailed results of the study. The study did not evaluate the effects of longitudinal symmetry breaking or mode crossings. At the time of the study, the significant effects of longitudinal localization were not fully realized; only after prototype development was the extreme sensitivity of periodic array discovered (see Chapter 4.0). Mode crossing was expected to be an unavoid able problem. The increase of mode crossing s due to mode crowding from longitudinal symmetry breaking was also not realized until prototype testing. As a result of the study, 6 rod and 12 vane prototype cavities were developed by HAYSTAC and ADMX, respec tively. The 6 rod prototype testing was unsuccessful, causing the experiment to utilize the single tuning rod cavity described above. The 12 vane prototype was tested in cryogenic temperatures to advance the scientific understanding of longitudinal symmetr y breaking and identify technology needs for expanding the capabilities haloscope detectors (see Chapter 4.0). A 19 rod perio dic array was also evaluated. Higher frequency ranges were obtained as expected, but localization of the field caused the form fac tor to drop to an unusable value during the frequency tuning. Figure 3 35 shows images of the E z field of the TM 010 mode through a cross section of a 19 rod cavity concept. The (a ) minimum frequency and ( b ) maximum frequency configurations are shown. T he m inimum frequency configuration demonstrates how transverse localization can greatly hinder the form factor where C 010 = 0.26 The lowest form factor observed throughout the tuning range was < 0.1 (not shown).

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117 (a) (b) Figure 3 35 Cross sectional images of the E z field for the TM 0 10 mode in a 19 rod periodic array cavity. (a) Minimum frequency configuration. (b) Maximum frequency configuration. The cross section is para llel to the cavity axis. The white circles are the conducting ro ds; a field solution is not included as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero field at the boundaries A for m factor of C 010 < 0.1 was computed during a significant part of the tuning range. ADMX is researching dielectric coatings for tuning. A variable dielectric layer acro ss the dielectric changes the dielectric constant, which alters the field s and frequency of search modes. Because the coating runs the entire length of the cavity, longitudinal symmetry is maintained. No mode repulsion would be observed at mode crossings, eliminating holes in the frequency search spectrum However, in order to obtain an even change in property across the entire coating, only a modest dielectric layer can be uses. Thus, very small tuning can be obtained with the concept under the best circumstances, and the method could likely only be used for fine tuning of a cavity. Superconducting thin film hybrid cavities are being developed by HAYSTAC and ADMX. The cavity surfaces normal to the magnetic field would utilize supercond ucting films while the surfaces perpendicular to the field would maintain copper or aluminum

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118 boundaries. If superconductivity of the longitudinal surfaces is maintained, a gain of ~10 can be achieved in Q 3.5 Superconducting Hybrid Cavities One potential technique of recapturing some of the loss in sensitivity from utilizing small volume cavities is to significantly increase the quality factor of the cavity. The scan rate is proportion to Q L (see Eq. 2 15 ) up to Q L = Q a believed to be ~10 6 or greater Ho lding the aspect ratio constant increasing Q L as R 6 / N will maintain the same detection sensitivity. Superconducting hybrid cavities may enable such a gain Superconductivity, first discovered in 1911 [109] demonstrated that select metals below a critical temperature T c allow curr ent to flow with zero resistance. Superconductivity originates from the pair correlation of electrons through electr on phonon interaction, a mechanism of the Bardeen Cooper Shriffer (BCS) theory [110] with each pair having opposite momentum and spin. Two types of superc onductors exist. Type I superconductors expel magnetic flux up to a critical field, H c 1 T and above H c the superconductivity is destroyed. Type II superconductors typically have a much higher H c and allow partial flux penetrations below H c Each flux p enetration results in a local current vortex, which incrementally increases the overall lossiness of the material [111] M icrowave cavities with superconducting boundaries such as those used in particle accelerators, are well developed and have demonstrated Q > 10 10 However, such cavities use t ype I superconductors which are not suitable for use in a strong magnetic field. Placing a t ype I cavity in a haloscope detector would fully destroy the superconductivity, result ing in the energ y dissipating through the base material which is

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119 more lossy than copper or aluminum Even a cavity using a t ype II superconductor would experience many vortexes which would result in the material being more lossy. R ecent experiments, which directly confirmed the pair breaking theory of films [112] have le d to the prospect of using type II superconducting thin film coatings on haloscope cavit ies [113] T he cy linder walls and tuning rod s/vanes surfaces are coated with the film, while the end caps remain uncoated conductors In order to avoid flux penetration, the thickness of the coating needs to be less than the lattice constant of the flux lattice, (3 40) where 0 is the magnetic flux quantum This configura tion is often referred to as a hybrid cavity. The ideal hybrid cavity in a haloscope would maintain perfectly superconducting vertical surfaces while operating in a magnetic field with no rad ial field strength, resulting in no magnetic flux penetrating the superconducting surfaces. From Eq. 3 26 the unloaded quality factor of the hybrid cavity is (3 41 ) where SC is the skin depth in the superconducting surfaces and the summation is over all superconducting surfaces Equation 3 41 can be recast in the form Q H = [1/ Q end + 1/ Q SC ] 1 where (3 42 ) ( 3 43 )

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120 When no magnetic field penetrates the superconducting surfaces, SC ends and Q H Q ends Note, Q H is no longer dependent on the R or the rods/vanes configuration and size The gain in Q H is ( 3 44 ) where eff i ( ) For an empty cavity, eff = 1 ; for a cavity with two tuning rods, such as ADMX, eff eff will increase with more surface area, and thus more tuning rods/vanes though the mode configuration also contributes significantly (see Eq. 3 27) A superconducting hybrid cavity similar to the ADMX cavity w ould see a gain in Q of ~10 From 2 dimensional analyses, a periodic array cavity using superconducting film s could obtain gains as high as ~5 0. The results are intuitive. Each additional tuning rod adds significant surface for which power is lost through resi stive heating. Superconducting currents have no resistance so no heating occurs and, thus, no loss in power The result s assume longitudinal symmetry. T h e interaction of the capacitive effect with the superconducting films has yet to be investigated and could prove problematic T he rods/vanes must be superconducting or Q reverts to Eq. 3 41 with the absence of the walls in the summation. The gain in Q would not be significant as the influence of the rods/vanes on eff is considerably greater than the that of the walls. So, the capacitive effect is unavoidable. Misalignments of the cavity relative to the magnetic field, and radial components of the magnetic field may result in flux penetrations, which will diminish Q H F irst order

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121 approximation s of the effects can be computed by relating Q H to the optical conductivity of the surface The classical skin depth is (3 45 ) where is the conductivity at the boundary. Maxwell Garnett theory estimates the effective conductivity of superconductors at frequencies of axion searches as (3 4 6 ) where n and SC are the classical (non superconducting) and superconducting conductivity of the thin film, and = B / is the volume fraction of the vortice s, with B the magnetic field strength perpendicular to the surface. Experimental data has estimated 10.9 T for niobium titanium nitride (Ni 0.5 T 0.5 N) [114] a promising candidate for thin film materials of hybrids For an angular misalignment , of the cavity relative to the field, the effective perpendicular field is estimated (3 4 7 ) Equations 3 45 through 3 47 are solved backwards to resolve the effective skin depth of a misaligned hybrid cavity. The skin depth is entered into Eq. 3 43 to find Q H for the cavity. When SC = ends Q H becomes same Q Because the classical conductivity o f a type II superconducting material is lower than that of conventional cavity conductors (e.g., copper and aluminum) a sizable misalignment can cause Q H to be lower than Q For fields that are inhomogeneous, Eq. 3 47 can be used to estimate the effective perpendicular field where is now the angle of the field relative to the cavity at the endcap interfaces Figure 3 36 shows a schematic of a hybrid cavity (a) with an

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122 (a) (b) Figure 3 36 Schematic of perpendicular magnetic field s in superconducting hybrid cavities. (a) Cavity tilted relative to the field. (b) Inhomogeneous magnetic field. The magnetic field is depicted as green arrows. angular misalignment relative to the magnetic field and (b) in an inhomogeneous magnetic field. The magnetic fields are depicted by green arrows.

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123 CHAPTER 4 12 VANE CAVITY PROTOTYPE 4.1 Design The results of a study of 2 dimensional periodic arr ay cavities showed that using multiple tuning rod s /vane s in microwave cavities has high potential for use in high frequency axion searches. Based on the findings of the study, various period ic array configuration s using rods and vanes were investigated to evaluate the search range and sensitivity throughout the range. Though most of the concepts yielded moderate sensitivity and/or range a 12 vane configuration demonstrated significantly greater sensitivity across a considerable range The 12 vane design used 4 stationary vanes and 8 vanes that rotated abo ut the center axis, with 4 ro tating clockwise and 4 counterclockwise symmetrically In the minimum f 010 frequency configuration, four sets of three vanes are group ed together, forming four cells of equal size As the vanes move, the vanes in each set fan out, forming 12 cells. The cells are of two different sizes with four cells of one size and eight cells of another, until the maximum f 010 frequency configuration, where the 12 cells are equal in size. Figure 4 1 shows a schematic of a cross section of the 12 vane cavity in the (a) minimum f 010 frequency configuration and (b) maximum f 010 frequency configuration. The white area represents the vanes. The stationary vanes are on the x axis and y axis as oriented in the figure. The grey arrows show the direction of rotation of the moving vanes between the configurations. The inner and outer tip of the vanes have sharp edges rather than rounds for ease of fabrication.

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124 (a) (b) Figure 4 1. Schematic of a cross section through the x y plane of the 12 vane prototype. (a) Minimum f 010 frequency configuration. (b) Maximum f 010 frequency configuration. The resonant ar ea of the cavity is in blue; the white areas are the vanes. The gray arrows indicate the direction of the moving vanes between con figurations The 12 vane configuration was optimized for tuning range, while keeping a lower bound on C > 0.3, using a 2 dimensional FEM eigenfrequency solver in COMSOL version 4.2 [96] At the time of the simulations the significant effects of longitudinal symmetry breaking on periodic array s were not realized. The analysis yielded a cavity with a search range of 0.2 7 GHz f 010 R 46 GHz m (1.1 1.8 GHz for a n ADMX C 010 0.59. The profile angel of the vanes was = 5 (see Fig. 3 33) and a tip to tip length of the vane profile was 0.138 R Figure 4 2 shows C 010 verses f 010 R as resolved from the 2 dimension al simulations. The results indicated that the 12 vane design used in the ADMX haloscope would yield similar sensitivity as the current ADMX det ector with equivalent scan times but at about double the frequency coverage Higher modes were not extensively ev aluated, but the simulations showed C 020 in the range of 0.1 0.2. A radius of R = 6.826 cm was selected for the prototype to fit the available cryogenic equipment; the vane geometry was scaled appropriately (see Fig. 4 11 )

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125 Figure 4 2. Plot of C 010 verse f 010 for the 12 vane cavity. The frequency is multiplied by the cavity radius for scaling. The form factor is comparable to ADMX with a frequency range that is about twice as high. yield ing a predicted TM 010 frequency range of 3.9 6.7 GHz An aspect ratio of L / R = 4 was used to reduce mass and cost of the cavity. The analytical mode predictions were initially done with a 2 dimension al simulation. Figure 4 3 shows the analytical predictions from the 2 dimensional model. The TM 010 mode frequency is sho wn as red triangles. The angle of vane rotation is with the minimum f 010 position = 0 and the maximum f 010 position = 25. The analysis predicted only 14 modes crossings, which is higher than ADMX, but not so great as to expect the configuration t o be unworkable. However, the 2 dimensional model does not evaluate modes with subscripts p > 0, as those modes require 3 dimensional boundary conditions. So, a greater number of mode crossing was anticipated. Successful testing of the prototype was depend ent on verifying an evenly dispersed E z field throughout the cavity to ensure a high form factor; localization in one or two cells would significantly degrade C 010 The cavity was fitted with eight antenna probes, six on the bottom plate and two on the top plate. The antennas on the bottom

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126 Figure 4 3. Frequency predictions for the 12 vane prototype from a 2 dimensional analysis. The blue circles show all predicted mode frequencies while the red triangles show the predicted f 010 The vane rotation is shown on the x axis from the minimum f 010 position ( = 0) to the maximum f 010 position ( = 25). The simulation predicted 14 mode crossings. plate are pos itioned to measure the field in 4 cells ; the antennas on the top plate are in line wi th two of the antenna on the bottom plate to compare the field at opposite ends of the cavity Figure 4 4 shows images of the E z field through a cross section of the 12 vane cavity for the TM 010 in the (a) minimum and (b) maximum frequency configuration s The location s of the bottom antennas are overlaid as small black circles labeled A F The top antennas are in the same location as antennas B and E and are identified as antennas G and H, respectively To enable mechanical motion of the rotating vanes, each set of four rotating vanes were physically connected with a cross bracket at the top and bottom of the cavity, for a total o f four cross brackets. The cross brackets have a hole in the middle to accommodate a drive shaft or an idler shaft (for stability). A coaxial drive shaft is inserted through the top plate and connects to the holes in each of the top cross brackets to drive the sets of 4 vanes in opposite directions. A mechanical gap of

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127 (a) (b) Figure 4 4 Cross sectional images of the E z field for the TM 0 10 mode of the 12 vane prototype design with bottom antenna positions overlaid (a) Minimum frequency configuration. (b) Maximum frequency configuration. The cross section is para llel to the cavity axis. The black circles indicate the location of the bottom antennas, labeled A F. The white wedges are the conducting vanes; a field solution is not included as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero field at the boundaries 0.13 cm between each mov ing piece and endcaps was maintained to allow sufficient clearance for the parts to move. The stationary vanes were rig idly connected to both endcaps. Figure 4 5 shows a computer aided desig n (CAD) model of the cavity prototype (a) complete and (b) with th e top plate and half of the cavity wall removed. Antennas G and H are seen on the top plate of (a). The vanes are in the maximum f 010 frequency configuration in (b). The idler shaft is just visible in (b) at the bottom center of the cavity. Additional holes and assembly components (e.g., screws) are also visible; the holes/hardware are for assembly or inte rfacing to the test equipment. During the initial testing of the complete cavity (see Chapter 4.2), significantly more modes were detected than were p redicted. As a result, 3 dimensional FEM simulations using COMSOL version 4.3b [96] were employed to identify the p > 0

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128 (a) (b) Figure 4 5. CAD models of the 12 vane prototype (a) Complete model. (b) Model with top plate and half of the wall removed. The maximum f 010 frequency configuration is shown in (b). modes. Two models were d eveloped. The simpl ified model consisted of the outer boun daries and the vanes. The stationary vanes ran the entire length of the ca vity while the rotating vanes maintained a mechanical 0.13 cm gap at either end. The more complex model included basic geometry of the cross bracket s with the 0.13 cm gaps maintained between all moving components like the cavity design Figure 4 6 shows the geometry of the (a) simplified and (b) complex 3 dimensional FEMs. The top plate and half of the wall is removed. The simplified 3 dimensional model was initially used to evaluate mode crossings and form factor discrepancies from the 2 dimensional model. The first results from the simplified model showed a significantly greater mode density than the 2 dimensional mode l. Upon inspecting the details, an unexpected mode solution was observed. The electric field was localized in the center of the x y cross section of the model at both ends. While some small localization was expected, at the time of the simulations a

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1 29 (a) (b) Figure 4 6 FEM s of the 12 vane prototype. (a) Simplified. (b) Complex. The top plate and half of the wall is removed. The maximum f 010 frequency configuration is shown. mechanical gap of g / L ect on the mode. The ADMX cavity has maintained a mechanical gap of g / L = 0.007 though all the Figure 4 7 shows image s of | E | of the TM 0 10 mode in a cross section as resolved by the 3 dimensional FEM simulations using (a) the simplified model and (b) the simplified model with no mechanical gaps (all vanes run the full length of the cavity). The model is in the maximum f 010 frequency configuration. T he cross section is through the z axis and is 45 to x z planes, symmetrically cutting between the vanes. For visualization, the cut would go through antennas A, C, and E in Fig. 4 4(b), thus avoiding all vanes. A large localization of the field can be seen at either end of (b). The localization was greater t han expected and was not exclusively within the mechanical gaps, as was demonstrated in previous studies [15] Careful review of the image shows a small field

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130 (a) (b) Figure 4 7 Cross sectional images of | E | of the TM 0 10 mode in 12 vane prototype as resolved by FEM simulation (a) Simplified model (b) Simplified model with mechanical gaps removed The cross section is through the z axis and at a 45 angle to the x z plane The cross section does not cut though any vanes. Red indicates higher field and blue indicates lower field The first observations of localization induced by the capacitive effect is seen at the top and bottom of ( a ). between the c enter localizations and the main field running the length of the cavity, depicted by four light blue areas. Further study of the results exposed the small fields to be fringing fields coming off the nearest mechanical gaps. Inside each of the gaps was a ve ry high electric field, uncovering the previously unknown capacitive effect. The large field in the center of the cavity near the endcaps resulted from a large charge buildup at the ends of the vanes on their inner tip. The findings lead to an investigatio n of the capacitive effect which in turn developed into a more comprehensive evaluation of symmetry breaking in microwave cavities (see Chapter 3.2). Due to discovering the capacitive effect, the complex 3 dimensional model was created to further gage how the phenomenon affects the cavity The center component of the cross bracket s and each vane in the complex model act as capacitors. Figure 4 8 shows images of | E | of the TM 0 1 0 as resolved by the complex 3 di mensional simulations

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131 (a) (b) Figure 4 8. Cross sectional images of | E | of the TM 0 10 mode in the 12 vane prototype as resolved by the complex FEM simulation. (a) Through the x y plane (b) Through the z axis and at a 45 angle to the x z plane The white area s are the conducting vanes in (a) and the center of the cross bracket s in (b); a field solution is not included as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher fie ld and blue indicates lower field (a) and (b) are not the same scale. The capacitive effect is visible in both images. through a cross section in the (a) x y plane and (b) through the same plane as Fig. 4 7. The model is in the maximum f 010 frequency conf iguration. The x y plane is cut though the center of the upper gap in (b). Localization due to the capacitive effect can be seen in both images. In (a), each vane and the center section is a capacitor, and the cross section shows localization between every rotating vane and the endcap as well as between the center of the cross bracket and the endcap. A blue ring can be seen depicting a node along a constant indicative of the subscript m = 1 This was the first prediction of mode hybridization

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132 caused by symmetry breaking. In (b), the capacitive effect can be seen in the small mechanical gaps at the top and bottom of the cavity. The model took ~ 24 hours to solve one vane configuration with only 20 modes, where the 2 dimensional model took ~1 minute to solve per configuration with 100 modes each. In an effort to reduce comput ation time a convergence study was conducted. The resolution of the FEM me sh, measured in degrees of freedom (DOF) of the numerical problem was varied to evaluate the computed f 010 and C 010 T he finer the mesh, the more DOF and the longer the computation time. Figure 4 9 shows the results of the convergence study. The plot show s that ~ 2M DOF are required to obtain convergence on frequency and ~1M DOF are need for convergence of C The simulation time was reduced to ~12 hours, but still very high for extensive modeling. Figure 4 9. Plot of the results from a convergence study for the complex 12 vane FEM model. The number degrees of freedom (DOM) in millions is on the x axis. The TM 010 frequency ( red ) and form factor (blue) are on the y axes A coa rse mode predict ion was constructed to compare to the 2 dimensional model. The simulation computed an extremely high number of modes around the TM 010 far more than were measured (see Chapter 4.4) Figure 4 10 shows the coa rse mode predict ion computed by the complex model. The frequency of 20 modes near the f 010

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133 Figure 4 10. Frequ ency predictions for the 12 vane prototype from complex FEM simulations The blue dots show the predicted mode frequencies at each interval while the red line traces the f 010 The vane rotation is shown on the x axis. The modes are indistinguishable. were calculated at 0.5 incremental steps. Due to geometry interference in the model when the cross brackets are closed, only ~20 of rotation could be simulated. The TM 010 is identified by a red line. The two holes in the data are configurations where the mode l failed. The modes are too close together to distinguish one from the other. Many of these modes are evanescent modes which would not be detected in the cavity prototype while others are too low in power or have insufficient E z field to be detected by t he probe antenna. Regardless, the modes are computed by the simulations with no way of removing them, causing computational time to increase further for mode mapping Additionally identifying modes that would not be detected from the analytical results is extremely challenging and would require a significant exploration beyond the scope of this study Thus the scientific and technical value of additional simulations of the 12 vane prototype were deemed not worthy of the time requirem ents presently Addit ional

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134 simulations on less complex structures need to be evaluated before such a complex structure as the 12 vane cavity should be considered for axion detection (see Chapter 6.1). Because the prototype testing demonstrated the complications of identifying and tracking modes in a periodic array cavity (see Chapter 4.4) and the issues caused in fabrication made simulating the cavity with sufficient accuracy not feasible (se e Chapter 4.2), a complete mode predict mapping across the entire tuning range from a 3 dimensional FEM simulation was never constructed. Mode predictions from the complex model were computed o ver limited range to support cryogenic testing (see Fig. 4.32 ). 4.2 Fabrication The 12 vane cavity prototype utilized a stainless steel tube that was electroplated [115] with o xygen free high thermal conductivity (OFHC) copper as the cavity walls. The endcaps and vanes were made from pure OFHC copper. To all ow for repeated access inside the cavity, a standard bolted interface instead of a knife edge seal [107] was chosen to connect the endcaps to the cavity wall at a c ost to Q The decision removed the possibility of using the prototype for axion searches in the future, but enhanced the prospect for science discovery. Figure 4 11 shows views from the drawings of (a) the cavity wall and (b) a vane. A cross section of the side view of the cavity is shown, oriented with the z axis horizontal. The profile of the vane is shown. The dimensions are shown in inches, which was used for fabrication in the U.S. The flanges on the cavity wall are used to connect the endcaps. The not es on the cavity wall are for electroplating and attaching the flange. The threaded holes in the vanes are used to connect to the endcaps for stationary vanes or the cross brackets for the rotating vanes.

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135 (a) (b) Figure 4 11. Select views from fabrication drawings of the 12 vane prototype. ( a) Cross sectional side view of the cavity wall. (b) Top view of a vane. Dimensions are in inches. The views are not to the same scale. Several fabrication errors led to issues during testing. The boundary surfaces were intended to be electropolished to reduce surface resistance [116] However, due to fu nding limits, the electroplating was made m thick, which is insufficient to electropolish as the entire coating could to be removed in the process The vanes were wet sanded with 1000 grit sandpaper to prepare for electropolishing. The electrop olishing was conducted with Electro Glo #200 [117] solution, but inadequate controls were used to maintain the proper surface current across the anode for polishing OFHC c opper polishing is highly sensitive to the current which is depende nt on the surface resistance. But the surface resistance changes as the surface of the copper is polished [118] requiring a feedback system to obtain a proper finish. T he temperature and flow rate of the solution affect the surface resistance [119] which were not controlled during polishing. Lastly, the wedge shape of the vanes made is more difficult to obtain an even surface current and to remove bubbles that form, which cause pitting. As a r esult, the current was often too high, causing pitting or too low, causing etching or dulling. The endcaps were not electropolished as a result.

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136 Figure 4 12 shows a plot of a generic electropolishing current density curve The for OFHC copper i s narrower than typical metals and the zone moves due to surface polishing and/or heating more rapidly than typical metals. Figure 4 13 shows a photograph of four vanes assembled to two cross brackets Some pitting is just visible on the outer tip of the vanes as small dots Figure 4 12 Plot of a generic e lectropolishing current density curve Th e voltage applied is on the x axis and the surface current density of the anode is on the y is between points C and D. The zone is narrow er for OFHC copper than for most metals. ADMX electroplates stainless steel tubes/rods with OFHC copper to fabricate their cavity walls and tuning rods. The experiment does not electropolish the cavity. Rather, the components of the cavity are heated individually at 40 0C for 12 hours and cooled at room temperature [120] The annealing causes the copper to soften, but the temperature is too low to significantly alter the stainless steel stiffness ; the stainless steel maintains the shape of the electroplated parts. The process causes new grains to grow in the copper, destroying defective grains and yielding larger average g rain sizes [121] The result is a longer mean free path in the copper and a lower surface impedance at the cavity boundaries.

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137 Figure 4 13 Photograph of four vane s partly assembled. The vanes are attached to two cross brackets at each end forming one 4 vane set. The tips of the vanes show some pitting. The annealing process was performed on the 12 vane cavity prototype components. All parts were places in an autoclave for annealing simultaneously. To conserve volume in the autoclave, the vanes were places upright within the center of the cavity wall, with no supporti ng structure. Because the vanes are solid OFHC copper, the components softened at the annealing temperature and buckled under their weight. When the pieces cooled to room temperature, the vanes were noticeably curved along the longitudinal axis. Additional ly, the process further dulled the finish of several vanes causing severe discolor ation. The deformed vanes were used for the prototype testing, which led to a deeper understanding of longitudinal symmetry breaking in periodic array cavities though not intentionally Figure 4 14 shows a photograph of the vanes assembled together with the cross brackets and attached to the bottom plate. The assembly is lying sideways.

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138 Figure 4 1 4 Photograph of prototype vanes assembled to the cross bracket and the bot tom plate The assembly is turned sideways from the operational orientation. The vanes are attached to two cross brackets on the left, with nominal 0.013 cm mechanical gaps. Significant discoloration from the annealing is visible on seve ral vanes. The vane s are curved along the z axis of the prototype ( horizontally in the picture) Discoloration from the annealing process is visible. The vanes are also curved, though not as obviously from the photograph. The mechanical gaps between the moving vanes can als o be seen on the left side of the picture. The annealing process improved the quality factor of the empty cavity. Prior to annealing, Q = 1 1 8 00 for the TM 010 mode; after annealing, Q = 15, 1 00 The theoretical quality factor is Q = 34,300. However, the Q of the cavity with the vanes was not measured prior to annealing due to disassembly issues with the design. The effect of annealing on The stationary vanes were fastened to the bottom plate. The rotating vanes wer e attached to the crossing brackets and assembled to the bottom plate via an idler shaft. An alignment pin was used to initially align the vanes in the maximum f 010 frequency configuration The cavity wall and top plate was integrated and the drive shaft o f the gearbox (see Chapter 4.3) was installed to hold the vanes location Figure 4 15 shows a

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139 Figure 4 1 5 Photograph of prototype assembly with the top plate removed. The vanes are in the initial alignment configuration. photograph of the cavity prior to installation of the top plate and gearbox, displaying the vanes in the initial alignment configuration. A ntennas were produced by exposing 1.5 mm of the center conductor of a rigid copper coax cable. All antennas were installed such that the cut in the cable was flush with the cavity boundary, similar to Fig. 3 4(a). The antenna coupling was intentionally made weak to minimize the perturbation effects to the cavity modes The losses in the antennas were a ssumed to be the same. The vanes were re align ed by taking power meas urement in the various cells and adjusting the vanes location to best fit the predicted power values for the maximum f 010 frequency configuration As each vane subassembly moved as unit, o nly relative alignment could be achieved The final alignment of the vanes was made by rotating all the vanes together clockwise and counterclockwise until the cluster contacted the

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140 stationary vanes, and centering the subassembly between the two bounds. Du e to physical interference of the cross brackets, the tuni ng rods could only be rotated ~ 2 1 Figure 4 16 shows a photograph of the prototype fully assembled, except the bottom antennas. Stands are attached to the cavity for testing and part of the gearbo x is visible on top Figure 4 1 6 Photograph of 12 vane prototype assembly. The bottom antennas have been removed. Four posts are used as stands for testing. Part of the gearbox is visible at the top. 4.3 Test Apparatus The 12 vane cavity prototype was te sted in room temperature, within a liquid nitrogen (LN 2 ) bath and within a liquid helium four (L 4 He) bath The test apparatus was composed of a cryostat that contained the prototype, a drive system and controls, and measurement instrumentation and a data recording system. Figure 4 17 shows a schematic of the apparatus The schematic includes the mechanical systems such as

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141 Figure 4 17. Schematic of 12 vane prototype test apparatus. A computer sy stem controls the dr ive system and signal generator and stores the instrumentation measurements. The data is transmitted over a GPIB while the drive system is controlled analogically Only three antennas and associated switches are shows; the cavity has a total of eight antennas that the signal generator an d power meter can couple to. Components are not to scale. the motors and a gearbox, and the electrical components such as a power meter and signal generator. The tuning rods are represented by an adjustable LC circuit within a circle to symbolize the frequency adjusting capability. The cryostat is indicated by a labeled box. The temperature in side the box is at the test temperature while everything outside the box is at room temperature. The double lines at the top represent a g eneral purpose i nterface b us (GPIB). Only three antennas and two three way switches are shown to simplify the figure; the experiment had eight antennas and two eight way switch es (see below) The cavity prototype was attached at the bottom of a cryostat insert which was seal ed i nside a vacuum dewar. The insert was designed for L 4 He bath testing and

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142 consisted of a top plate, six radiation baffles an aluminum interface plate to attach the cavity and support structure Five resistors were characterized at L 4 He ( 4.2 K ) LN 2 (77 K) and room (300 K) temperatures The resistors were attached at differing heights of the insert to serve as a liquid level detector. The resistances were measured with voltmete rs, and th e data was saved to the computer. The liquid/gas filling and evacuatio n of the cryostat was done manually. The drive system was controlled by a computer using an analog input/output (I/O) device which applies voltages across a motor controller to operate two DC stepper motors [122] The motors were mounted to the top of the cryostat and rotated two stainless steel shafts that passed through the top plate The metal shafts were attached with flex couplers to G 10 shafts [123] that passed through all the radiation shields of the cryostat. Teflon bushing were used to keep the G 10 shafts in place. The stepper motors had a single step size of 1.8. The frequency step size of a haloscope cavity needs to be < f / 5 Q L so a gear reduction is required to meet the tuning requirements The reduction required for cavity with a single tuning variable is estimated ( 4 1) where s is the step size of the motor, max is the maximum rotation of the vanes, f max and f min are the maximum and minimum frequencies of the search mode, respectively. A minimum gear reduction of ~1: 10, 00 was needed for the testing assuming ano malous skin effects (ASE) To enable finer adjustability a gear reduction of 1 : 19,600 was used. The gearbox was divided into two sections, each composed of two worm gear pairs [124] in series for total of four worms/gears. The gears had 140 teeth each The gearbox had two

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143 separate input shafts, but the two output shafts were coaxial, a llowing one gearbox to drive both 4 vane sets simultaneously. Anti backlash gears were used to minimize the dead space when reversing direction with the motors. Figure 4 18 shows a photograph of the gear box. The gears are made of bronze to wear faster than the worms. The ou tput shaft of the upper section is nestled in the out put shaft of the bottom section; during operation, the shaft s would rotate in opposite direction at the same rate. Figure 4 1 8 Photograph of the gearbox for the 12 vane prototype The gearbox was divided into two sections, each with two worm gear pairs in series for a 1:19,600 gear ratio. The output shafts are coaxial. The G 10 shafts were attached to the gearbox input shafts with flex couplers to account for misalignments The gearbox was attached to the top plate of the cavity with three flex rods to accommodate dissimilar thermal expansions at cryogenic temperatures The coaxial output shafts penetrate the prototype through a center bearing (see Fig. 4 5(a)). The testing required 8 cable lines from each antenna to transfer the input and output signals, since the switch es w ere outside the cryostat. To

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144 reduce the thermal load, s emi rigid coax cables with beryllium copper center conductors [125] were installed. Figure 4 19 shows the cryostat insert with the 12 v ane cavity prototype at the bottom. The top plate of the insert is fixed to a frame; the dewar sits inside the pit beneath t he insert (not visible) and was lifted b y a hoist and bolted to the top plate for cryogenic testing (see Fig. 4 20) The six radiation baffles sit below the top plate. Three resistor rest on the baffle s and one lies on the interface plate above the protot ype. The gearbox is seen between the interface plate and the cavity. The cavity is attached below the interface plate with three rods. Eight flexible cables [126] c onnect the semi rigid cables to the cavity antennas The testing required taking S 12 measurements across a relatively wide frequency spectrum though the tuning range of the cavity. A signal generator [127] produced a narrow bandwidth signal at a frequency controlled by the computer system. A power meter [128] sensitive to 0. 1 nW measured the average power across the bandwidth and transmit the data digitally to the computer. The process would be repeated over a frequency range to obtain a power spectrum at a given cavity configuration. The signal generator and power meter coul d be connected to any of the eight antennas to obtain any S 12 measurement combination. The switching of the connections was done manually. Antennas lines that were not being use as input or output ports were terminated at the cryostat top plate. The test configuration did not accommodate injecting signal and taking measurements from the same port (S 11 ). The motor drive and instrumentation was controlled using visual code written in LabVIEW version 2016 [129] The code also id entified modes using a peak search

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145 Figure 4 19. Photograph the 12 vane cavity cryostat insert. The top plate of the insert is mounted to a frame. All electrical and mechanical feedthrough s are in the top plate. The dewar is below the cryostat ( not visib le ) program with a minimum power threshold and recorded the measurements from the power meter and voltmeter, as well as convert ed the resistance measurements to an approximate liquid level height The liquid height was use d to determine the test temperat ur e and plan cryogenic operations. Figure 4 20 shows the test apparatus with the insert sealed inside the orange dewar. The monitor of the control computer is in the center of the picture and the computer sits behind it. Four voltmeters are on the left sid e of the screen. The signal generator is on top of the blue frame facing to the right; the power meter is hidden

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146 Figure 4 20. Photograph the 12 vane cavity test apparatus. The cryostat is on the right of the picture and the controlling computer is in th e center. Four voltmeters (on the left) and the signal generator (at the top) can also be seen. behind the signal generator. The motor controllers and I/O device are on the backside of the blue frame, not visible in the photo. 4.4 Test Results The cavity prototype was initially tested at room temperature with only the four stationary vanes installed. The vane s run the length of the cavity, so no mechanical gap is present. Figure 4 21 shows an imag e of the predicted E z field for the TM 010 mode through a cr oss section of the 4 vane cavity with the bottom antenna locations overlaid The cross section is parallel to the cavity axis. The white areas are the vanes with no field solution. The outer edge of the images and the white area s are the conducting boundar ies. Red indicates higher field and blue indicates lower field, with zero field at the boundaries. Antenna ports G and H are on the top plate in line with antennas B and E, respectively.

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147 Figure 4 21. Cross sectional images of the E z field of the TM 0 10 mode in the 4 vane cavity with bottom antenna positions overlaid The cross section is para llel to the cavity axis. The white areas are the vanes; a field solution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero field at the conducting boundaries S 12 measurements of the 4 vane cavity were made with a spectrum analyzer for various antenn a input/output configuration. Figure 4 22 shows the power spectrum measurements of the 4 vane cavity for (a) S GA (b) S G B (c) S G C, and ( d ) S G D The subscripts for the S 12 measurements use the antenna location nomenclature to indicate the input (1) and out put (2) antennas, respectively. The power is in dBm. The green vertical lines show the predicted TM eigenfrequencies from a 3 dimensional COMSOL [96] simulation. T he predicted frequencies for the TM 010 and TM 013 are labeled. The measurements provided several important findings that were gateways to understanding symmetry breaking effects. The most notable observation is all the modes detected could be identified as a TM mode from simulations, supporting the later discovery that mechanical gaps cause mode repulsions. Mode repulsions are seen only when TE/TEM modes are detected by the probe antenna, which is a result of mode mixing. Because no TE/TEM modes appeared to be detected by the probe antenna, no

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148 (a) (b) ( c ) ( d ) Figure 4 22. Power spectrum plot s for the 4 vane cavity prototype. (a) S GA (b) S GB (c) S GC (d) S GD The predicted TM mode frequencies are shown as green vertical lines in all plots. The TM 010 and TM 013 mode predictions are labeled. The results helped explain vario us symmetry breaking phenomena. mode mixing appears to be occurring The conclusion of no mode mixing match es the predictions of simulations (see Fig. 3 13). M ost of the closely to the pre dicted TM modes. However, all the TM 01 p frequencies were lower than the predictions by ~15 MHz. The shift can be seen in the plots at the labeled modes. The lowest detected mode in the plots is the TM 010 which is 15 MHz less than the prediction. The mode to the left of the 013 013 mode, which is ~ 13 MHz less than predicted. Similar discrepanc ies are se en for the TM 011 and TM 012 modes. The frequency shifts are due to a symmetry breaking that caused transverse localization, which always reduces the TM 01p mode frequencies possibly due to the TM 010 TM 01 3 TM 010 TM 01 3 TM 010 TM 01 3 TM 010 TM 01 3

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149 curvature of some vanes Modes with m > 0 orient ed so that a node passes through the symmetry breaking greatly reducing the localization effects (see Fig. 3 5 ) on their frequency. Degeneracy breaking was also observed for modes with m > 0 in several of the plots, further suggesting a break in the discrete r otational invariance The power measured for the broken degenerac ies was different in each cell, and t he power measured for the TM 01 p mode in the S GB channel was greater than the other channels The discrepancies f urther demonstrate localization of the mo des caused by symmetry breaking Note, the B and G antennas are in the same cell, so power was lost transferring the signal from the injection cell to the other cells. However, the basic fingerprint of the mode s is similar r egardless of which antenna pair was used. This pattern was repeated in other measurement configurations, suggesting that the input and output ports of S 12 measurements are significant in periodic array cavities. After final assembly and alignment the 12 vane cavity prototype was tested at room temperature. The initial test runs immediately showed that a much higher number of modes was detected by a weakly coupled probe antenna than were predicted. Figure 4 23 shows the power spectrum for the 12 vane prototype in the maximum TM 010 frequency configuration (see Fig. 4 4(b)) for channels (a) S GB and (b) S GE The predicted TM 010 frequency from a 2 dimensional simulation is depicted as a vertical green line. The x axis is power in mW on a log scale. The 2 dimensional simulation predicte d only two TM modes in the frequency range shown in the plots. The measurement showed ~20 30 modes. Though a few more modes with p > 0 were expected to be observed than predicted, the discrepancy was significantly more than anticipated. Also, the number of modes in the different cells

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150 ( a ) ( b ) Figure 4 23. Power spec trum plots for the 12 vane prototype at room temperature (a) S G B (b) S G E The predicted TM 010 frequency is shown as a green vertical line Power is shown on a log scale in mW. were not the same and very few modes were seen in both channels. No mode appeared more or less likely to be the TM 010 Several variables were eliminated as causes for the discrepancy. The injection and receiver chain s were investigated for standing waves. A small number of standi ng waves were identified, but were determined not a significant contributor as they were not within the frequency range or did not have high enough energies to alter the power spectrum in the cavity. The non operating antenna s were removed and holes were plugged to identify power leaks or local localized modes, but the experiments showed no nota ble change in the spectrum. Terminations were removed from non operating antennas w hich yield ed inconsistencies in measurements, justifying t he use of terminations on unused antennas. Two 3 dimensional simulations were develop ed to identify the unknown modes (see Chapter 4.1). However, the complex simulations demonstrated that ~100 modes existed in the frequency range of the measurements and small adjustments to the model moved modes up and down in frequency significantly. The curvature of the vanes was never measured, as the expectation of their im pact was not significant prior to the

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151 testing. Because the exact geometry of the as built prototype was not known, properly modeling the cavity to identify the modes was impossible. Even bounding the symmetry breaking would yield no valuable data, as the f requency range of a single mode within the bounds would overlap many other modes, making them indistinguishable from S 12 measurements alone. Simulations were also used to predict which modes were more likely to couple to a probe antenna with no success. Th e coupling of the non TM modes to a probe antenna was discovered to be caused by modes mixing with TM modes. But the mixing of the modes was highly sensitive to the symmetry brea king. Without exact geometry input s for the simulation, predicting which TE/TE M modes mixed with TM modes was not possible providing no value in mode identification. The finding, however, was significant and contributed to a deeper understanding of the effects of symmetry breaking (see Chapter 3.2). M easurement m ode map s were deve loped using the threshold method Figure 4 24 shows mode maps for the entire tunable range of the 12 vane prototype for channels (a) S H E and (b) S H F The maps are typical of the channels. The tuning is from the minimum TM 010 frequency configuration dictated by the interference of the cross brackets to the maximum TM 010 frequency configuration The maps show the S HE channel is more mode dense than the S HF channel, showing how localizations cause varying results in different cells. It is not clear if the couple to the antennas sufficiently in the cells where they were not observed. Note, antennas E and H are in the same cell, suggesting that power was a factor in the

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152 ( a ) ( b ) Figure 4 24. M ode maps for the 12 vane prototype. (a) S HE (b) S HF The vane rotation is on the x axis in degrees, where 0 is the minimum TM 010 frequency configuration. The threshold method is used to identify modes. Analytical predictions are not shown. measurements. The results again show ed that the location of the input and output antennas are not negligible in exciting and detecting a mode, and would not be trivial in an axion search. Tuning of several modes is seen in the maps ; this is the first known demonstration of tuning a microwave cavity using vanes. The lowest frequency mode which tunes (~4.2 6.3 GHz) is believed to be the TM 010 but being the lowes t frequency does not guarantee the high est form factor (see Chapter 3.2) W ithout add itional confirmation, the TM 010 is not sufficiently known to ensure adequate sensitivity for axion searches. T he lowest frequency mode that is tuned does not align well with the 2 dimensional or 3 dimensional predictions, and has sever al very large gaps in the map, especially in the S HE channel. Figure 4 25 shows the mode maps from Fig. 4 24 together, zoomed in on the lower left corner. The 2 dimension al TM 010 frequency predictions are shown in green. The lowest tuned mode lines up well between the channel s, but in much of the tuning

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153 Figure 4 25. Mode map for the 12 vane prototype for channels S HE and S HF The vane rotation is on the x axis in degrees, where 0 is the minimum TM 010 frequency configuration. 2 dimensional a nalytical predictions are shown as a green line range shown, only one of the channels detects the mode. The mode does not align well with the predictions. Many mode crossings are seen, as are some mode repulsions. Because the threshold method of identifying mode peaks can miss modes when th e peak is below the threshold, color plot mode maps were constructed to determine if the lower frequency tuned modes are more observable throughout the tuning range. Figure 4 26 shows the color plot mode map for the 12 vane cavity prototype for chan nels S GA and S GB Light yellow is higher power and dark blue is lower. As seen before, the cell with the injection port has recorded higher power than the other cell. Mode repulsions are more visible than in Fig. 4 24, as seen by the vertical lines that di srupt the tuning modes. However, no additional information was presented in the color plots that aided mode identification. In particular, holes in the maps did not appear to be just below the threshold, but seem to be not present in the measurement at all The lowest mode appears to completely vanish above 4 in channel S GB and has

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154 ( a ) ( b ) Figure 4 26. Color plot m ode maps for the 12 vane prototype. (a) S GA (b) S GB The vane rotation is on the x axis in degrees, where 0 is the minimum TM 010 frequency configuration. Analytical predictions are not shown. many large holes in S GA If this mode were the TM 010 it would be unusable for axion searches as is. Perturbation techniques for mode identification (see Chapter 3.3) were implemented to deter mine the TM 010 mode. A copper rod was inserted in ~ L /4 increment into various cells to determine which modes increased in frequency. The technique was not able to distinguish modes very well. Specifically, vane configuration s where mode repulsions did not occur in one cell often occurred in another, so it was not possible to separate TM modes from mixed modes and the process resulted in data that could n ot be correlated between cells. A coa rse bead pull test was also conducted. An alumina bead about 1 mm in diameter was translated in the cavity z axis in increments of ~ L /10. The results were too coarse and inconsistent to yield meaningful results. A more sophisticated experiment was considered, but was decided not worthy of the time investment. The apparatus for the experiment was not local and the set up is extremely involved. Due to the extensive symmetry breaking causing a lack of equivalency in the cells, a fine bead pull

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155 experiment seemed no more probably to identify the TM 010 yielding likely zero fruit for considerable time. Mode correlation was also attempted. By assuming a mode is the TM 010 higher order TM modes could be estimated based on Eq. 3 16. The mode calculations are compared to the measurement at various tuning configurations to determine if any combination provides a superior match in modes. The process did not positively identify the TM 010 Because of the failed attempt to identify the TM 010 mode conclusively, and the growing awareness of the need for a nonperturbative in situ mode identif ication technique, the in situ mode test was developed (see Chapter 5) However, though the process was proven with cavities where no longitudinal symmetry is broken, it was not able to positively confirm the TM 010 mode. After all mode identification methods were exhausted, the TM 010 was not identified with sufficient confidence need for an axion dark matter search. Because a significant information about advanced tuning of microwave cavities had been obtained, the planned testing was co ntinued to glean any further knowledge from the prototype. The frequency step size was measured over a short tuning range. Figure 4 27 shows the frequency tuning for the smallest vane rotation of the apparatus for channels S HE and S HF The rod was rotated ~0.002 around 14.25. The mode shown i s the third lowest tuning mode. T he mode and the form factor of the mode are not known The mode and vane angle w ere selected because the tuning was identifiable and no mode crossing was obse rved. The average frequency step size for the prototype at the f K Hz, below the required f /5 Q L 27 K Hz using the theoretical Q

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156 Figure 4 27. Plot showing the frequency step size of the 12 vane prototype. Channels S HE and S HF are shown. The vane rotation is on the x axis in degrees, where 0 is the minimum TM 010 frequency configuration. The average minimum step size is ~9 K Hz. Q L of the measurements was ~1200. value in ASE. ( Q for the prototype was much less than the theoretic al value, discussed below.) The step size plot clearly demonstrates the frequency mismatch between cells. The mode in the two channels are believed to be the same, but their peak frequencies are ~70KHz different due to symmetry breaking between the cells. The measurements in both channels were taken simultaneously. The frequency mismatch would cause a reduction in power generated by the axio n conversion, on the order of Eq 3 37. Effectively, the cells can be considered individual cavities with the area tha t joins them as short waveguides that couple the cavities. When the area is zero, the cavities are complet ely decoupled, allowing the modes to be completely distinct in each cavity. When a hole is introduced between cavities, similar modes begin to couple, forcing their frequencies closer together even when geometrical difference exist. When the partition between the cells is removed, a single empty cavity remains (i.e., complete coupling).

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157 The 12 vane prototype demonstrated partial coupling, but consistent ly measurements of what appear to be the same TM mode had slightly different frequencies in the cells. The discrepancies are exacerbated due to mode repulsions. TE/TEM modes in the cells were also not at the same frequency, so mode repulsions would occur at differing frequencies in the cells. When a slightly higher frequency TM mode begins a mode repulsion at a vane angle lower than the ot her cells, the mode repulsion will drive the TM mode frequency even higher relative to the other cells making the variation greater. Figure 4 28 shows the power spectrum of the 12 vane prototype for the third lowest tuning mode across a small frequency ra nge. The vanes are at 0 . C hannels S GA and S GB are shown in red and blue, respectively. The curved lines are the power spectrum and the vertical lines identify the peak power of each channel, coordinated by color. Difference between the peaks is df z. If such a discrepancy was observed within a high Q cavity, it would be of order f / Q L The measurements were taken simultaneously. The mode is at a mode repulsion in the cell with antenna A, but not in the cell with antenna B. A TE mode is to th e right of the plot in both channels (not shown), but it is only mixing with the S GA channel, pushing the peak frequencies farther apart. This has the effect of further widening the hole in the frequency search spectrum of a detector than would exist with transverse symmetry maintained. However, this phenomenon could only be detected with multiple antenna ports, which suggests that multiple antenna ports are necessary for testing periodic array cavities. Multiple antennas are also necessary for the in situ mode test (see Chapter 5).

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158 Figure 4 28. Power spectr um plots for the 12 vane prototype showing frequency discrepancies of a mode C hannels S G A and S G B are in red and blue, respectively Power is shown on a log scale in mW. The different between the frequency peaks is ~300 KHz. The 12 vane prototype testing was conducted in a LN 2 bath to verify the apparatus for L 4 He testing. Frequency shifts in the modes due to the change in the dielectric constant was used to ensuring the cavity was submerged in th e bath and validate the level detector. At LN 2 temperatures (77 K), one of the drive systems locked. Modifications to the cryostat insert were made to minimize friction, but did not relieve the problem. The rotation speed of the motors was reduced to incre ase their torque. This foreshadowed issues that would occur during L 4 He testing. Tuning was verified at LN 2 temperatures and Q L was measured (see below). A small range of tuning at ~14 was verified for the third lowest tunable mode because tuning was clearly identifiable. Figure 4 29 shows the mode tuning for channels (a) S HD and (b) S HE Other channels showed the tuning but less clearly. The mode is labeled TM 3 to indicate the third lowest (assumed) TM mode; the subscripts have no relation to the conventional m n and p subscripts.

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159 ( a ) ( b ) Figure 4 29. Partial measurement mode map in a LN 2 bath for the 12 vane prototype. (a) S HD (b) S HE The vane rotation is on the x axis in degrees, where 0 is the minimum TM 010 frequency configuration. The third lowest tuning mode is labeled TM 3 Analytical predictions are not shown. The frequencies are lowered by the refraction index of LN 2 which is 1.199 at the boiling point [131] A mode repulsion is visible in channel S HD a round 14.15 (4.73 GHz) and in S HE around 14.05 (4.73 GHz) Note, as the mode repulsions are at differing vane angles the frequency mismatch of the mode between the cells will be enlarged as in Fig. 4 28 in the repulsions Because antenna E and H are in the same cell, the power of the modes was larger in S HE than S HD resulting in more modes exceeding the threshold. Adjusting the threshold for modes for different channe ls was not performed. A djusting the threshold for periodic array cavities depending on the antenna location, cell coupling, and symmetry breaking will be necessary for axion searches. The LN 2 tests were repeated in L 4 He to confirm tuning near axion search temperature s and compare Q L measurements across temperatures. The level detector was used to confirm the temperature of the cavity. Figure 4 30 shows the resistance measurements of the four resistors on the insert during a L 4 He fill. The resistors are numbered by height from top to bottom of the insert at ~18 cm intervals. T he cavity sat TM 3 TM 3

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160 Figure 4 30. Cryostat level detector reading during a L 4 He fill. The resistors are labeled by location from top to bottom. The time scale is arbitrary. The cavi ty sat below R 4 When the resistance is constant, the liqu id level is above the resistor. ~40 cm below R 4 The fifth resistor sat at the bottom of the dewar and was not recorded during testing. When the measured resistance is constant, the resistor is subm erged in the liquid. Resistance of R 1 was not recorded until R 4 was submerged. The resistances were also compared to characterization data to verify the values were as predicted. At L 4 He temperatures ( 4.2 K ), the same motor that locked up during the LN 2 t esting again locked up. However, reducing the motor speed did not allow the motor to drive the vanes and the testing was conducted with only one 4 vane set rotating during the tuning measurements. After cryogenic tes ting, the motor was verified operational at full speed at room temperature and at an estimated 240 K, based on resistor readings and characterization Figure 4 31 shows the maximum motor speed verses temperature for the motor that locked. The maximum speed capability is 60 RPM. The results show a linear relationship with temperature suggesting a misalignment due to thermal mismatch or some semi viscous fluid ( e. g. grease) solidifying As the drive system was inspected multiple times for wear and any signs of high friction were removed, the gear box is most

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161 Figure 4 31. Maximum motor speed of the failing motor as a function of temperature. The temperature at ~240 K was estimated based on resistor measurements likely the cause for the lockup. No grease was used intentionally in the gearbox, bu t was used for vacuum sealing the cryostat and may have gotten into a gearbox accidentally. Additional testing of the gearbox has not been conducted to confirm the issue, as it was deemed unnecessary at the present time. Analytical predictions for the L 4 H e testing were computed using the complex FEM simulation [96] (see Chapter 5.1) for the tuning range used in the cryogenic testing. However, t he predictions were b ased on all rotating vanes moving and do not correlate to the test results. As the model ing time is considerably long (see Chapter 4.1) time did not permit computing analytical predictions with only one vane set moving. Figure 4 32 shows the analytical predicts from the complex model The TM 010 mode is identified as red triangles. A few mode crossings and mode repulsions are seen in some of the tuned modes, but the simulation step size was too coarse to clearly discern crossings from repulsions. The frequency is reduced by 1.02 6 to account for t he refraction index of L 4 He [132]

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162 Figure 4 32. Analytical predictions for the 12 vane prototype during L 4 He tes ting. The vane rotation is on the x axis in degre es, where 0 is the minimum TM 010 frequency configuration. The predictions are based on the complex model and rotate both set of vanes as designed. The tuning of the cavity ( df / ) measured during the LN 2 testing was ~2/3 that of the model predictions. T o account for only one vane set rotating, measurements were made with the unmovable vane set fixed at 2 = 14.0 and 2 = 14.25 while the other set tuned the cavity. Thus, the tuning observed during the L 4 He te sting was ~1/2 that of the LN 2 tests, but the tuning was not observed across the entire vane rotation in most channels Figure 4 33 shows the mode tuning for channels (top) S H E and (bottom) S H B with 2 fixed at ( left ) 14.0 and ( right ) 14.25. The vane rotation on the x axis is for the vane driven by the motor ( 1 ) The third lowest tuning mode is labeled TM 3 The plots are not to the same scale as changing the non moving vane set will shift the tuning range of the mode s Additionally, relocating 2 was done manually so some repeatability errors occurred, causing select test runs to not align well with others. Tuning was observed in all cells and with all channels measured, but some channels observed TM 3 noticeably less across the vane rotation. For example, TM 3 is

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163 ( a ) ( b ) ( c ) ( d ) Figure 4 33. Partial measurement mode map in a L 4 He bath for the 12 vane prototype (a) S HE with 2 = 14.0. (b) S HE with 2 = 14.25. (c) S HB with 2 = 14.0. (d) S HE with 2 = 14.25. The vane rotation for only one 4 vane set is on the x axis in degrees, where 0 is the minimum TM 010 frequency configuration. The third lowest tuning mode is labeled TM 3 The plots are not to the same scale. Analytical predictions are not shown. not observed in (a) at 1 < 14.15. The reason for the disappearance of the mode in one cell is not precisely known, though the root cause is breaking discrete rotational invariance M ode repulsion s are seen in (b) (c), and (d) Tuning below 1 14. 2 was very limited in all channels TM 3 T M 3 2 = 14.0 2 = 14.25 TM 3 TM 3 2 = 14.0 2 = 14.25

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164 The quality factors of the cavity were measured at various configuration during the tests. Table 4 1 shows the measured quality factor at the three test temperatures for the third lowest tunable mode The vane sets were both at = 14.25 during the measurements The measurements are Q L with weakly coupled input and output antenna s so the value will be close to the unloaded Q The measurement for S HF at 4.2 K is blank because the mode was not obs erved in the configuration during L 4 He testing. Table 4 1. Quality factors of the 12 vane prototype for different channels during testing. Channel Q L (300 K) Q L (77 K) Q L ( 4.2 K ) S G A 920 3131 4302 S G B 968 3719 4073 S HC 1665 2676 4278 S HD 1439 3085 3968 S HE 1161 2594 3453 S HF 1373 2829 N/A The theoretical Q for room temperature and 4.2 K is ~15,000 and ~83,000, respectively. The increase in Q at low temperature is due to ASE (see Chapter 3.1). The table shows that the measured Q is more than an order of magnitude lower than predicted Figure 4 34 shows the measured values for channels S GB S HC and S HE from the 12 vane prototype testing. A gain of ~3 4 in Q L was achieved at 4.2 K revealing the cavity boundaries entered the ASE regime despite ex tensive issues during fabrication The prototype testing demonstrated the first cryogenic tuning of a periodic array cavity using vanes, though the results were modest. Tuning was less than predicted and mode crowding made identifying the TM 010 mode impos sible with current technology. Significant findings from simulations were supported or confirmed by the results of the tests, such as longitudinal symmetry breaking is the mechanism for mode mixing and frequency decoupling between cavity cells is enhanced by variances in mode repulsion frequencies between cells.

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165 Figure 4 34. Measured Q L of the 12 vane prototype verses temperature. Channels S GB S HC and S HE are shown. The vanes are at = 14.25. The cavity boundaries entered the ASE regime.

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166 CHAPTER 5 IN SITU MODE TEST 5.1 Test Methods Based on the results of the 12 vane prototype testi ng at room temperature and high frequency axion searches conducted by HAYSTAC and ADMX, an in situ mode identification method was developed The method compares the po wer measurement s from combining the sig nal in two antenna lines to measurements from the lines individually. The comparison provides information about the local phase and relative power of modes at th e antenna locations. R epeating the process with multiple antennas pairs a t various location s provide s a mapping of mode s with in the cavity at a given tuning configuration P ass fail criteria based on the analytical predictions are used to determine the desired search mode. Probe antennas respond primar ily to the electric field parallel to the probe axis (see Fig. 3 4) In a haloscope cavity, the probe must be oriented parallel to the externally applied magnetic field, because the electric field in the cavity must be oriented in the same direction (see Eq. 2 10 ), typically defined as the z axis. For cylindrical haloscope cavities, probe antennas are mounted to the endcaps. At every location on the endcaps, E z is either in phase, 180 out of phase, or nonexistent (for TE/TEM modes) due to the boundary conditions (see Chapter 3.1) The phase of the field is dependent on the mode configuration. For the TM 010 mode, the signal extracted by weakly coupled antennas at any point on the same endcap will be effectively in phase, while weakly coupled antenna s on opposing e ndcaps will be effectively 180 out of phase. S light discrepancies will occur due to the differences in

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167 coupling. The use of weakly coupled antennas will typically keep the discrepancies below the error bars of the measurement as well a s minimize perturba tions For higher order TM modes in cylindrical cavities, the phase will be dictated by the mode order. For example, modes with even values of subscript p will yield 180 out of phase signals extracted by two inline antennas on opposite endcaps, while mode s with odd values will yield in phase signals from the same antennas. The m and n subscripts will dictate the phase of signal extracted from the same endplate, but in a more complicated way than the p subscript. In general, the number of nodes between two antennas determines the phase, with an odd number producing i n phase signals and an even number producing 180 out of phase signal. However, with tuning rods, the always hold. Simulat ions are then requ ired to determine the phase of the signal s By predicting the phase of signals at various antenna ports, patterns can be determined that can be used to discern a search mode from other modes. In particular, the TM 010 mode often maintains known parity symmetries that can be exploited for mode detection. Figure 5 1 shows images of the E z field of the TM 010 mode through a cross section of various tuned cav ity geometries and illustrates the symmetries The symmetries provide ideal mode configu rations the in situ measurements to identify the mode. Antennas on the same endcap at points of symmetry would extract identical signals providing confirmation of the mode Signals extracted from symmetry points of a TM 010 mode on the same endcap would combine coherently if the cable lengths between the antenna and combiner are identical. The combined power would be double the power from a single antenna. Using

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168 (a) (b) ( c) (d ) Figure 5 1 Cross sectional images of the E z field for the TM 0 10 mode in cylindrical cavit ies with various tuning configurations depicting parity symmetries (a) Two conducting rods. (b) Four conducting vanes (c) Seven conducting rods (d) 12 conducing vanes The cro ss section is para llel to the cavity axis. The white areas are the location of the conducting rod s/vanes ; a field solution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the condu cting boundaries Red indicates higher field and blue indicates lower field with zero field at the boundaries The dotted black lines depict the symmetries antenna locations that are not on symmetry points would produce a combine d power given by Eq. 3 39. The TM 010 mode does not need parity symmetry for this relation, and Eq. 3 39 is valid for combining any two TM 010 signals coherently from the same endcap. Analytical predictions can estimate the power ratio of differing locations, are also used as part of the mode evaluation. Higher order modes can be assessed in a similar way. When signals are combined coherently, Eq. 3 39 dictates the power output. When signals are combine d

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169 180 out of phase, the power combines as ( 5 1 ) w here P i is the power extracted in each antenna Figure 5 2 shows images of the E z field through a cross section of a cylindrical cavity with two conducting tuning rods in a non symmetric configuration for the (a) TM 010 (b) TM 110 (c) TM 210 and (d) TM 020 Two arbitrary antenna locations are depicted as black circles and labeled A and B. (a) (b) ( c) (d ) Figure 5 2 Cross sectional images of the E z field of various modes in cylindrical cavities with two non symmetric, conducting tuning rods and two antenna locations (a) TM 010 (b) TM 1 10 (c) TM 2 10 (d) TM 02 0 The cross section is para llel to the cavity axis. The antennas are depicted as black circles. The white areas are the location of the conducting rods; a field solution is not inclu ded in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero field at the boundaries

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170 Ratios of P A / P B can be estimated from the simulations as can the combining of the powers, assuming phase is maintained. Table 5 1 shows the predicted power ratios, predicted relative phase, and predicted combined power of the modes in Fig. 5 1 The combined power is normalized b y the maximum power from the individual antennas (max( P A P B )) The power is assumed to come from antennas on the same endcap and maintain phase through combining. The TM 010 and TM 210 signals would combine coherently while the TM 110 and TM 020 would combine destructively. The power ratios of the TM 110 and TM 020 are distinct enough, that measurements could be used to distinguish them, but the power ratios of the TM 010 and TM 210 are probably too close to discern the two modes conclusively. Table 5 1. Predicted combined power variables of modes shown in Fig. 5 2. Mode (Fig.) Relative Phase ( ) ~ P A / P B ~ P AB / max( P A P B ) TM 0 10 (a) 0 0.5 1.5 TM 1 10 (b) 180 2.5 0.2 TM 2 10 (c) 0 0.3 1.2 TM 02 0 (d) 180 1.0 0.0 TE and TEM modes would not excite a weakly coupled probe antenna sufficiently to produce a signal that could be mistaken for a TM mode, but mixed (hybrid) modes could. The conf igurations of hybrid modes are much more challenging to predict but, w ith a properly built cavity, accurate modelin g should provide sufficient estimates to separate hybrid modes from TM modes using power combining and power ratio comparisons. However, testing is yet to confirm the method with mixed modes (see Chapter 5.2). As observed during the 12 vane prototype testi ng, symmetry breaking can cause what appears to be one mode to resonate at differing frequencies in various locations in the cavity (see Chapter 4.4). The frequency shift will cause coherently combined powe r

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171 to be reduced given by Eq. 3 37. Comparing peak frequency measurement s from the two antennas can predict the power loss. An alternative or augmenting method is to measure the phase directly. When the cavity geometry is reasonably simple, this approach may suffice without the combiners. But when the con figuration is more complex or the mode density is high, power combining will likely be necessa ry for positive mode identification. The configuration of the in situ mode test is reasonably simple. Two weakly coupled antennas measure the peak power of a mode individually and combined. The individual measurements must go through the power combiner to account for insertion losses. Figure 5 3 shows a schematic of the test. The switches are arranged to measure the power from each antenna and the combined power. T he power ratio and the normalized combine power are compared to predictions and pass fail criteria are used to systematical determine the desired search mode. Figure 5 3. Schematic of in situ mode test. The antennas are weakly coupled. The configuration shown is for measuring P A ; changing the switch configuration enables measuring P A P B or P A B The power meter can be replaced by a network or spectrum analyzer. A phase sensitive detector can also be added in parallel to the receiver chain.

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172 Antenna pairs on the same endplate are used to determine the subscripts m and n as demonstrated in Table 5 1. Multiple pairs will typically be needed to positively identify a search mode. The location of the antennas needs to be preplanned using analytica l predictions to maximize mode identification throughout the cavity tuning range Util izing antenna pairs that are in line on opposite endcaps will reveal the parity (not to be confused with parity symmetry) of subscript p as even or odd. Ideally, the tes t is conducted prior to use in a haloscope to calibrate the pass f ail algorithm for in situ use. The p recision of the measurement can be used to gauge the error bars for the criteria. The calibration will also verify the antenna configuration are adequate and provide evidence of positive mode identification. 5.2 Test Results The in situ mode test was assessed to varying degrees with four cavity configurations: an empty circular cylindrical cavity a cylindrical cavity with two tuning rods, a cylindrical c avity with four tuning vanes, and the 12 vane prototype (see Chapter 4) The two rod and empty cavit ies maintained longitudinal symmetry, while the 4 vane and 12 vane cavities did not. The experiments were used to prove the concept as well as gauge the applicability and technical challenges of implementation in haloscope detectors. The four cavity configurations utilized the same boundary conditions. A stainless steel tube of r adius R = 6.826 cm and length of L = 5 R was electroplated with OFHC copper [115] The endplates rods, and vanes were made of solid OFHC copper. The components for the vane cavities were annealed (see Chapter 4.2). The endplates for the empty cavity were electropolished.

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173 The empty cavity was used for the initial demonstration of the method Four weakly coupled antennas were used in the testing. Two antennas were mo unted 6.5 cm ( 0.95 R ) apart on the top plate in the x z plane of the cavity identified as A and B The other two antennas were mounted on the bottom in line with the first antennas with the letter same letter (without the pr ime) designation. Figure 5 4 is a reproduction of Fig. 3 1 with the top plate antennas overlaid. The orientation of the TM 110 mode is assumed because the opposite polarize d degeneracy rotated 90, will have the node in line with the antennas, due to small perturbations from the probes. (a) (b) (c) Figure 5 4. Cross sectional images of the E z field in an empty cylindrical cavity depicting the location of the top plate in situ mode test antennas. (a) TM 0 10 (b) TM 110 (c) TM 020 The cross section is parallel to the cavity axis and the outer edge of the images are the conducting boundaries. The antennas are depicted as black circles. Red indicates higher field and blue indicates lower field with zero field at the outer boundaries The number of node s between the antennas in (a) is zero, in (b) is one, and in (c) is two. From this, we can conclude that the power from antennas A and B will add coherently for the TM 010 and TM 020 and destructively for the TM 110 when phase is maintained Extending this over all TM modes in an empty cavity shows that coherent combining indicates an even value for the subscript m and destructive co mbining indicates an odd value. As stated above, t he subscript p will be odd when the powers

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174 from inline antennas combine coherently and even when they combine destructively (see Chapter 5.1) The initial test only evaluated the ratio of the combined power to the independent power measurements and did not use analytical predictions to compare P A / P B The measurements include S 12 of the top plate antennas (A and B) combined and independent, and S 12 of one top plate (A) and one bottom plate combined and independent for a total of six measurements. The power measurements we re swept from 1.5 GHz to 4 GHz to capture the lowest 15 TM modes of the cavity. was used as the input antenna for the A, B, and AB power measur ements, and antenna power measurements. The results were used to predict the parity of subscripts m and p When P AB > max( P A P B ), m is even, and P AB < max( P A P B ), m is odd. Similarly, w hen P A > max( P A P ), p is odd, and P A < max( P A P ), p is even. A few modes were observed during the measurements to have a maximum power c onsiderably lower in power than other modes. Those modes were removed from the algorithm and assumed to be TE modes. Table 5 2 shows t he results of the test compared to the predicted frequencies of the TM modes. The modes are ordered in ascending frequency. The predicted frequencies are from Eq. 3 16. The measured frequencies are from the modes not eliminated from the algorithm The righ t two columns show the parity of the subscripts as determined by the algorithm. The frequency of the measured modes closely matched the predicted frequencies of the TM modes. T he algorithm matched the parity of the m

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175 Table 5 2 Results of the empty cavity in situ mode test. Mode Predicted f mnp (GHz) Measured f (GHz) Algorithm m Algorithm p TM 010 TM 011 TM 012 TM 013 TM 110 TM 111 TM 014 TM 112 TM 113 TM 015 TM 114 TM 210 TM 211 TM 212 TM 115 1.6773 1.7645 2.0035 2.3483 2.6726 2.7282 2.7596 2.8885 3.1375 3.2119 3.4561 3.5821 3.6237 3.7459 3.8269 1.6783 1.7651 2.0060 2.3532 2.6752 2.7304 2.7671 2.8920 3.1430 3.2220 3.4638 3.5861 3.6276 3.7028 3.8372 even even even even odd odd even odd odd even odd even even even odd even odd even odd even odd even even odd odd even even odd even odd and p subscripts of every TM mode. As the test matched the subscripts for all predicted TM modes, the assumption that the modes with low power were TE modes is justified. F igure 5 5 shows the S 12 measurement s using antennas A and B as outputs The blue line shows the S the purple line shows S B and the red line shows S { A B} where {AB} indicates the combined signal from antennas A and B. The blue squares, purple diamonds, and red triangles depict P A P B and P A B for each mode, respectively. Select modes are identified for visualization. The two low power modes around 3 GHz are TE modes that were removed from the algorithm. The data can be traced back to the results shown in Table 5 2 for subscript m The S 12 meas urement using antennas A yielding the results for subscript p Equations 3 39 and 5 1 could predict P AB to within ~20%, which was deemed a cceptable for initial testing. The in situ mode test was conducted on a cylindrical cavity with two tuning rods. The rods ran the length of the cavity and attached to both endplates, maintaining

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176 Figure 5 5 Power spectrum plots for the empty cavity in si tu mode test. Power is shown in W on a log scale. S 12 measurements are shown using antenna indicates the output is the combined signal from antennas A and B. The blue square, purple diamond, and red triangle show the peak powers from antennas A, B, and {AB}, respectively, for each mode. longitudinal symmetry. d = 0.28 R The rods were repositioned manually. Three weakly coupled probe antennas were used on each endcap for a total of six antennas. Two an tennas were attached in line with the cavity axis on either endcap. The remaining four antennas were mount ed in the y z plane, with two on each endplate spaced 1.15 R apart. Figure 5 6 shows images of the E z field for the TM 010 mode through a cross section of the cavity in the three configurations tested with the top plate antenna location s overlaid The configurations are the (a) minimum TM 010 frequency (b) mid range TM 010 frequency, and (c) maximum frequency TM 010 The antennas are indicated by black circle s and labeled A, B, and C A prime is used to indicate the antennas inline on the bottom plate. The test used several antenna combinations to identify the TM 020 mode in the minimum and maximum TM 010 frequency configurations. Figur e 5 7 shows the S B A TM 012 TM 013 TM 112 TM 113 TM 114 TM 210

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177 (a) (b) (c) Figure 5 6. Cross sectional images of the E z field for the TM 010 in a cylindrical cavity with two conducting tuning rods in the in situ test configurations depicting the location of the top plate antennas. (a) Mini mum TM 0 10 frequency configuration (b) M id range TM 0 10 frequency configuration (c) Maximum TM 0 10 frequency configuration The cross section is para llel to the cavity axis. The antennas are depicted as black circles. The white areas are the location of the conducting rods; a field solution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero fi eld at the boundaries spectrum in the minimum TM 010 frequency configuration at around the predicted frequency range of the TM 020 mode ( f 020 = 4.269 GHz ) T he three candidate modes are identified as TM i from minimum to maximum frequency Figure 5 7. Power spectrum plots for the 2 rod cavity in si tu mode test in the minimum TM 010 frequency configuration Power is shown in W on a log scale. S BA measure ments are shown. TM 2 TM 1 TM 3

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178 For each configuration, three output antenna pairs were used to con duct the mode identification. The input antenna was varied based on the output pairs. The measurement sets were: S BA S BC S B{AC} ; S A S S } ; S A B S S A { } The first two sets were used to match the mode profile and the third set determines the parity of p The measurements were also made on the TM 010 mode to adjust the predictions for systematic errors The power ratio of the individual antennas and the power ratio of the combined power to maximum individual power were used to identify the TM 020 Figure 5 8 shows images of the E z field for the TM 0 2 0 mode through a cross section of the cavity in the (a) minimum and (b) maximum TM 010 configurations with the top plate antenna locations overlaid. The antennas are indicated by a black circle and labeled A, B, and C. (a) (b) Figure 5 8. Cross sectional images of the E z field of the TM 02 0 in a cylindrical cavity with two conducting tuning rods in the in situ test configurations depicting the location of the top plate antennas. (a) Minimum TM 0 10 frequency configuration. ( b ) Maximum TM 0 10 frequency configuration The cross section is para llel to the cavity axis. The antennas are depicted as black circles. The white areas are the location of the conducting rods; a field s olution is not included in the images as there is zero field inside the conductor. The outer edge of the images and white areas are the conducting boundaries. Red indicates higher field and blue indicates lower field with zero field at the boundaries

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179 Tabl e 5 3 shows the test results for the minimum TM 010 frequency configuration The measurement ratios are shown for the TM 010 and three candidate modes. Because the predicted power ratios for the TM 020 match the values for the TM 010 the measured values of the TM 010 are used as predicts for the TM 020 to account for systematic error. Table 5 3 Results of the 2 rod cavity in situ mode test in the minimum TM 010 frequency configuration. Mode P A / P C P A C /max( P A P C ) Predicted P A / P C Predicted P A C /max( P A P C ) Pass/Fail TM 010 0.59 2 .41 1.0 2.0 N/A TM 1 4.03 2.49 0.59 2.4 F TM 2 0.52 2.77 0.59 2.4 P TM 3 0.61 2.34 0.59 2.4 P P A / P B P AB /max( P A P B ) Predicted P A / P B Predicted P AB /max( P A P B ) TM 010 0.34 2.20 0.6 1.5 N/A TM 1 0.90 4.74 0.34 2.2 F TM 2 0.22 2.43 0.34 2.2 P TM 3 0.21 1.99 0.34 2.2 P P B / P B P B B /max( P B P B ) Predicted P B / P B Predicted P B B /max( P B P B ) TM 010 0.29 0.14 1.0 0.0 N/A TM 1 0.34 3.39 0.29 0.14 F TM 2 0.41 0.15 0.29 0.14 P TM 3 0.36 2.04 0.29 0.14 F TM 1 failed the criteria in the first two data sets, because the P A /P C and P A /P B measurements were more than triple the predicted values (i.e., the values measured for TM 010 ). TM 1 and TM 3 failed the criteria in the third set, because the P B B /max( P B P B ) measurements were more than triple the predicted values. TM 3 pass the criteria in all data sets as the measured ratio values were within a factor of three (above or below) of the predictions. Thus, the test conclude TM 2 was the TM 020 mode. TM 2 had a meas ured frequency of 4.265 GHz, almost exactly matching the analytical prediction. Based on the results, TM 3 was likely the TM 021 which has a predicted frequency of 4.304 GHz; the

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180 measured frequency for TM 3 was 4.301 GHz. TM 1 had odd values for p but the pa rity for m and n could not be established. The pass/fail criterion was made broad to account for the error bars on the prediction values and large systematic error observed in the test ing For example, the ratio of P A / P C for the TM 010 mode should be 1.0 but was measured at 0.59 The discrepancy could be due to inconsistencies in the combiner ports, attenuation in a receiver line, or longitudinal symmetry breaking in the cavity. An investigation into the source was not conducted, as the test consistently predicted the TM 020 when the TM 010 measurements were used for the predicted values The measured combined power for all the configurations was considerably higher than predicted by Eq. 3 39 and Eq. 5 1. This behavior was observed across all the in situ mode testing of the 2 rod cavity. The discrepancy may have been due a loss in power at the terminations but the cause of the difference was not investigated. Table 5 4 shows the test results for the maximum TM 010 frequency configuration. The configuration had four candidate modes for the TM 020 The measurement ratios are shown for the TM 010 and the candidates The measured values of the TM 010 are used as predictors for the TM 020 to account for systematic error like Table 5 3. A factor of t hree was used for the pass/fail criteria as well. The test conclude TM 2 was the TM 020 mode. TM 2 had a measured frequency of 4.009 GHz, almost exactly matching the analytical prediction of 4.012 GHz. The criterion for P /max(P B ,P ) was opened to a facto r of 10 to account for the very low prediction value, but a factor of three would have yielded the same conclusion. Unlike the results in Table 5 3, where the third data set would have been sufficient to identify

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181 Table 5 4 Results of the 2 rod cavity in si tu mode test in the maximum TM 010 frequency configuration. Mode P A / P C P A C /max( P A P C ) Predicted P A / P C Predicted P A C /max( P A P C ) Pass/Fail TM 010 0.53 3 .87 1.0 2.0 N/A TM 1 0.18 4.63 0.53 3.9 F TM 2 0.51 3.05 0.53 3.9 P TM 3 0.44 3.11 0.53 3.9 P TM 4 0.54 3.42 0.53 3.9 P P A / P B P AB /max( P A P B ) Predicted P A / P B Predicted P AB /max( P A P B ) TM 010 13.64 1.55 4 .0 1.6 N/A TM 1 3.17 0.17 14 1.6 F TM 2 2 2.54 1.70 14 1.6 P TM 3 0.10 0.46 14 1.6 F TM 4 15.98 1.71 14 1.6 P P B / P B P B B /max( P B P B ) Predicted P B / P B Predicted P B B /max( P B P B ) TM 010 1.35 0.01 1.0 0.0 N/A TM 1 1.66 0.10 1.4 0.01 P TM 2 1.10 0.02 1.4 0.01 P TM 3 1.28 0.01 1.4 0.01 P TM 4 1.51 3.53 1.4 0.01 F the TM 020 mode, the second and third data set s are needed in Table 5 4. In general, the greater the number of candidates, the more data sets will be needed to detect a search mode. Based on the results, TM 4 was likely the TM 021 which has a predicted frequency that exactly matches the measured value of 4.049 GHz TM 1 had an even value for p but the parity for m and n could not be established. TM 3 was discovered to be a TE mode that was not filtered out by the power threshold. The test method cannot be used to identify a TE mode with probe antennas. The in situ mode test positively identified the TM 020 mode in both 2 rod configurations tested. An additional check was conducted to verify the TM 010 mode in the mid range frequency configuration (see Fig. 5 6(b)). Table 5 5 shows the results.

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182 Table 5 5 Results of the 2 rod cavity in situ mode test in the mid range TM 010 frequency configuration. Output Antennas P 1 / P 2 P 12 /max( P 1 P 2 ) Predicted P 1 / P 2 Predicted P 12 /max( P 1 P 2 ) Pass/Fail A,C 0.19 1.68 0. 5 1.5 P A,B 0.44 2.51 0. 6 1.6 P 1.12 0.01 1.0 0.0 P The measurements passed the pass/fail criteria of the algorithm identifying the mode as the TM 010 The measured frequency was 2 4 54 GHz and the predicted frequency is 2.458 GHz. The combined power was still considerably higher than predicted, particularly the measured value for the S B { AB } channel. The source of the discrepancy was not detected. The in situ mode test was conducted on the 4 vane cavity prototype (see Chapter 4.4) to verify that the powers combined as expected. Figure 5 9 shows t he power spectrum of the 4 vane cavity for the S GE S GF and S G{EF} channels (see Fig. 4 21) The power is in dBm. Select modes are labeled. The red line indicates the combined power P EF ; the peak of the red line is higher than the other channels only for the TM 01 p modes. Thus, o nly the TM 01 p mode signals combined constructively due to discrete transverse symmetry as expected Other antenna combinations were also tested yielding the same positive results. Because the vanes were known to have some bend, lon gitudinal symmetry breaking existed in the cavity. The degeneracy breaking of the TM 011 mode (second from the left) is distinctly depicted as two peaks conjoined, which is a direct result of longitudinal symmetry breaking. The 4 vane in situ mode test demo nstrated the mode identification method works with modest longitudinal symmetry breaking. But the extent

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183 Figure 5 9. Power spectrum plot for the 4 vane cavity in situ mode test. Power is shown in dBm S 12 measurements are shown using antenna G as the input antenna. S indicates the output is the combined signal from antennas A and B. The blue square, purple diamond, and red triangle show the peak powers from antennas A, B, and {AB}, respectively, for each mode. of allowable symmetry breaking is not known, since the test did not work on the 12 vane cavity prototype. The mode test was conducted multiple times on th e 1 2 vane prototype in an attempt to positively identify the TM 010 mode. The test was conducted at room temperature in the maximum TM 010 frequency confi gurations, in conjunction with c ourse bead pull and rod insertion tests (see Chapter 4.4). The in situ test unfortunately eliminated all candidate modes, suggesting the TM 010 mode was in some hybrid state in the test configuration The test was also run in LN 2 and L 4 He on the third lowest TM mode with no success. The test used five antenna pair to measure the p ower ratios of TM 3 with the vane angles set at = 14.25 (see Fig. 4 29 and Fig. 4 33) The measurements were compared to predictions for the TM 010 Equation 3 37 was also used to estimate the combined power with limited success. Table 5 6 shows the results from the LN 2 testing. The antenn a labels are depicted on Fig. 4 4. Input lists the input antenna used and TM 010 TM 1 1 3 TM 0 1 3 TM 111 TM 01 2 TM 110 TM 213

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184 Table 5 6 Results of the 12 vane proto type in situ mode test. Input Output P 1 / P 2 Predicted P 1 / P 2 P 12 /max( P 1 P 2 ) Predicted P 12 /max( P 1 P 2 ) Computed P 12 /max( P 1 P 2 ) G A,B 2.2 1. 0 0.4 1.0 0.2 A B,G 1.7 1.0 2.2 0.0 1.6 H C,D 12.7 1.0 1.5 1.0 0.8 H E,F 49.7 1.0 1.0 1.0 0.6 F E,H 0.6 1.0 1.7 0.0 1.6 output list the output antennas that constitute the three channels use in each line of the table, with the first antenna list corresponding to subscript 1 and the second antenna corresponding to subscript 2. The results suggest TM 3 is not the TM 010 mode. Specifically, the S A{BG} and S F{EH} measurements show ed coherent adding, signifying p has an odd value. The high values for P 1 / P 2 indicate significant symmetry breaking. The cell with antennas D and F has an extremely week field at the mode frequency The computed P 12 was le ss than the measured values in every channel similar to the results seen in the other tests, suggesting an error in the test set up. The error was not investigated further. The in situ mode tests showed the meth od ha s potential for identifying search modes in haloscope cavities. The test successfully identified parity of subscripts, could depict desired modes when no longitudinal symmetry was broken, and could assist in identifying the TM 010 mode with modest symm etry breaking. However, the method was not able to identify the TM 010 mode no r determine the subscripts of other TM mode s in the 12 vane prototype showing the method may have limited use in highly complex cavities without further development. The incorpor ation of a phase sensitive detector could significantly improve the reliability of the testing.

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185 CHAPTER 6 CONCLUSION 6.1 Future Work The interest in searching for axion dark matter at masses above ~4 eV (~1 GHz) is growing tremendously in the science community. Every notable CDM axion experiment is currently researching techniques to conduct KS VZ sensitive searches at higher masses The advancement of tunable microwave cavities is a significant part of that effor t. The findings of the research presented herein furthers the progress of high frequency axion detectors, but considerable more work must be done to el evate the search frequencies through the allowable mass coupling phase space Continuing the development of test methods and analysis techniques discussed above will be pivotal to successfully observing axion dark matter. ADMX, HAYSTAC, and CAPP have a multitude of high frequency cavity prototypes that are in various stages of fabrication and testing. Many of the lessons learned from the 12 vane cavity prototype yield ed insight into potential i ssues and best practices for next generation cavities. Additional analysis and testing of longitudinal sy mmetry breaking needs to be performed to better predict how th e more complex cavities will be effected. The correlation of the mechanical gap size (i.e., capacitive effect) and rod/vane misalignment to the frequency spectrum hole need s to be verified A test cavity would provide an adjustable gap and /or tilt. Mode m aps for varying gaps and tilt should be compared to the analytical predictions. The results will confirm the finding that longitudinal symmetry is the mechanism for mode repulsions. Further analysis would

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186 look as surface currents, charge densities, and fie ld lines in greater detail to better understand how their interaction alter the mode solutions. Other symmetry breaking constructs should also be studied. The capacitive effect experience with tuning rod gaps will also be observed when the tuning rod pass es through a slot in the endplates. However, the orientation of the resulting field lines is now transverse. How the change effects mode repulsions and mode mixing is not known. A deeper understanding of how the effects compare to tuning rod gaps in terms of holes in the frequency spectrum and losses in form factor is needed Additional a nalysis and testing of more nuanced discoveries must also to be conducted. The obs ervation of mode decoupling should be modeled to verify cause is mode decoupling and provide predictability A eigenfrequency simulation would provide such results. CAPP is investigating a multicell tuned cavity similar in fashion to the 12 vane prototype. S ymmetry breaking between the cells is proposed to cause mode decoupling, resul ting in each cell resonating at slightly differing frequencies. Verification of the mechanism with testing and correlation to modeling is necessary for the success of the design Because the form factor is not measured, but is a critical value in the sensi tivity of the detector a high confidence in simulations is critical to detect CDM axions. Without justifying the simulation predictions of longitudinal symmetry breaking in complex cavities the computed C will require high error bars. The location of th e input and output antennas have been shown to be nonnegligible, especially in periodic array cavities and for conducting in situ mode testing. Additional studies need to focus solely on antenna placement and how losses between cells will affect axion sign al absorption by the output antenna. A periodic array

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187 prototype with multiple antenna ports could be built to observe the effects of varying input and output antenna on S 12 measurements. Additionally, the perturbation effects of the output antenna on the m ode configurations and form factors are not well known, and need further study. The location of the antenna will affect the penetration depth needed t o critically couple to the mode. The relationship between mode hybridization and longitudinal symmetry br eaking is a n area of study independent from axion searches. While hybridization in waveguides with in homogeneous dielectric constants has been well explored little is known about mode mixing in homogeneous conditions. Currently, no mode mixing has been ob served between a TM mode and a non degenerate TE mode. But insufficient evidence exists to postulate a theorem. Understanding what cause s some mod es to mix and others not to mix could uncover new discoveries in waveguide and resonant cavities as well as g reatly enhance haloscope cavity development. The mechanism for mode mixing in cavities with homogeneous permittivity should be explored using v arious numerical analysis techniques to ensure consistent conclusions Analytical expansion of the effects into particle accelerators, reentrant cavities, and microstrip resonators could open unknown potential. Select FEM simulations have suggested that mode hybridization breaks the orthogon ality condition of Eq. 3 17. During mode repulsions, the orthogonality condi tion appears to be violated in proportion to the strength of the repulsion. The orthogonality condition is maintained for the non mixing broken degenerate TE mode. However, the results are not well vetted and seem to violate boundary conditions Further st udy to

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188 determine if this is a real phenomenon or a construct of nu merical simulation is required. Longitudinal symmetry breaking introduces evanescent modes to the observed frequency range. It is unknown if evanescent mode mix with other modes or are detec table with probe antennas. Further understanding of how these modes interact with conventional and reentrant modes and effect haloscope measurements is needed. F indings of simulations should be confirmed with testing. Cavities with intentional symmetry bre aking could be used to demonstrate mode hybridization. Such tests should be accompanied by bead pull tests to clearly depict mode mixing. The effects of changing boundary conditions (e.g., mechanical gaps, rod tilt) sho uld be quantified in terms of symmetr y breaking and mixing parameters. Mode identification methods are being advanced by many current haloscope experiment s In particular, HAYSTAC has made significant progress in utilizing bead pull tests to characterize microwave cavities. Successfully imple menting bead pull testing will be critical in CDM axion detection as search fr equencies continue to increase, and axion collaboration s are beginning to incorporate the process into the cavity production flow. Rod i nsertion should also be reviewed as a tool for mode identification, especially for detectors with a separate vernier tuning device such as HAYSTAC. The in situ mode test will be necessary to conduct search es with higher order modes. The validation testing showed the method has potential for mode identification during data taking runs However, significant mode hybridization may plague the

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189 method and issues need to be flushed out with additional testing. Conducting the test with varying symmetry breaking will expose the limits of the method. Custom combiners should be used to minimize insertion losses when only one antenna signal is measured. The cause for the inconsisten cies between predicted and measured combined power during testing (see Chapter 5.2) must be determined. Establish ing the proper te rmination for the combiner ports is critical to verifying the power readings match predictions and can b e repeatedly used for mode confirmation. The method should be tested with a network analyzer and phase sensitive detector to improve reliability Combin ing the bead pull characterization with in situ mode testing prior to data taking, will yield a powerful tool in ensuring the proper mode is tracked. The antennal location testing and in situ mode testing could be performed in conjunction, as the test se t ups are essentially identical and the results are interconnected Changing the location of the antennas alter the effectiveness of the mode test and the detection sensitivity. Period ic array cavities are particularly sensitive to symmetry breaking, both discrete transverse and longitudinal. Such cavities require a comprehensive understanding of how antenna locations translate into more repeatable mode identification and tracking. Finally, the superconducting thin fil m hybrid cavities have significant potential to increase the sensitivity of a detector. Most of the power loss in the cavity is through the surfaces that are oriented parallel to B 0 Eliminat ing those losses could boost Q L by an order of magnitude or more. The gain s are greatest in periodic array cavities which use many tuning rods and thus have more longitudinal surfaces to dissipate energy With the possible reduction in form factor and cell coupling due to even modest symmetry

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190 breaking, superconducting hybrids may be the only way to recover the losses in sensitivity. Unfortunately superconducting hybrids have proven t o be tremendously challenging. Various t hin film coating techniques have been tested by HAYSTC, but have yield only modest progress in th e last ~5 years. The collaboration has a homogeneous ~8 T magnet with a radial field of < 0.005 T developed specifically for testing superconducting hybrid cavities. However, no testing has been performed to date due to a lack of a test cavity. Recent ana lyses have shown that a substrate with thickness comparable to the wavelength is required to maintain high reflectivity at the superconducting boundaries. But limited simulations have been conducted. The effe cts of flux penetrations on Q of hybrids was onl y briefly examined in this study, and a much more comprehensive investigation is warranted. But, until coating techniques are successful, testing the findings of su ch analyses will be impossible. 6.2 Summary The detection of axion dark matter is of great interest to the physics community sensitivity, and is currently probing to the DFSZ model coupling at about the same mass range, no search has successfully achieved KSVZ sensitivity above 1 GHz. In order to observe axion dark matter at such masses, detector technologies must continue to advance. Developing microwave cavities with sufficient tuning range and detection parameters is the most critical component. The findings of the study detailed within continue the march toward axion detection The investigation uncovered intricacies of symmetry breaking not previously

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191 known and revealed lesser realized challenges of complex microwave cavities such as periodic a r rays. Several discoveries provide critic al information to researchers as haloscope cavities evolve toward higher frequencies. Most notably, uncovering that the mechanism for mode repulsion is longitudinal symmetry breaking enables cavity developments a poi nt of entry for mitigating the problem. Longitudinal symmetry breaking was previously known to cause mode localization and degrade form factor [15] and mode repul sion has been documented before [100 101 107] But the precise mechanism for mode repulsion had not been determined prior The results of the study strongly suggest that mode repulsion is due to mode mixing, which is directly caused by longitudinal symmetry breaking within the cavity. Cavities with no or little symmetry breaking demonstrated no mode mixing while cavities with considerable symmetry breaking were riddled with mode repulsions and hybrid modes. The finding was supported by simulations, power spectrum measurements, mode maps, and in situ mode testing. Further, simulations showed a clear correlation between the amount of symmetry breaking and the strength of the mode repulsion, quantified by the hole size in the frequency sear ch spectrum. Longitudinal symmetry breaking was also discovered to be a source of m ode crowding Numerical simulations uncovered the finding, which was substantiated by experimentation. The revelation that, during a mode repulsion, only one broken degenerate TE mode mixes with a TM mode while the other remains pure was also new, leaving open the question of why to future studies

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192 While the 12 vane prototype testing failed to demonstrate conclusively the concept s feasibility for haloscope detectors, the testing yielded significant insight which supported the simulation results and revealed key aspects of periodic array cavities. The experiment demonstrated the first mode tuning with conducting vanes in room and cryogenic temperatures. The testing als o displayed cell decoupling of modes, for which no pr evious documentation was found. The extreme sensitivity of periodic arrays to longitudinal symmetry breaking could only have been realized through prototype testing. The unexpectedly large mode density i n measurements and simulations showed the tremendous impact the capacitance effect has on mode crowding with evanescent, reentrant, and broken degenerate mode s filling the previously empty frequency space. And only th r ough measurements did the need for mo re robust mode detection methods come to light. The demonstra tion of the in situ mode test i s also significant for high frequency haloscope detectors. ADMX had already experience d trouble tracking higher order TM modes for axion searches. The need for mod e identification techniques during data taking is only going to grow as frequency ranges move up. T he in situ mode test conceived and proven in this study provides a noninvasive method of mode identification. The test demonstrated the capability of depicti ng a search mode out of select candidates and partially identifying multiple modes in various cavity configurations. Improving the instrumentation and components of the test could produce a robust tool for high frequency axion searches. Additionally, this research provided further guidance into tunable high frequency cavities. Establishing quantifiable differences between cavity concepts that could be

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193 used to establish a normalized figure of merit will help steer development programs in the field. The first order approximation of the effects of flux penetrations on superconducting thin film hybrid cavities can open the door to more complex simulations and can be validated with experimentation in the foreseeable future. While the original intent of the invest igation, to prove the feasibility of periodic array cavit ies for haloscope cavities, was not fully achieved, significant progress towards next generation axion dark matter detectors at higher frequencies was obtained. The findings herein will continue the community march toward s observation of the axion particle.

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201 BIOGRAPHICAL SKE TCH Ian P. Stern was awarded a US army commission at the New Mexico Military Institute at age 19. After earning a B.S. degree in aerospace engineering from Embry Riddle Aeronautical University, he served three years on active duty as an infantry officer. H e earned a M.S. degree in mechanical engineering from the University of Florida while serving as an Officer Candidate School instructor in the Florida National Guard. Prior to his Ph.D. pursuit, he worked in the space industry, developing deploy able system s for satellites with Harris Corp. and L ockheed Martin Space System Co.