# A cosmic microwave background radiation polarimeter using superconducting magnetic bearings

код для вставкиСкачатьA C osm ic M icrow ave B a ck grou n d R a d ia tio n P o la rim eter U sin g S u p erco n d u ctin g M a g n etic B ea rin g s A TH ESIS S U B M IT T E D TO T H E FA CULTY OF T H E G R A D U A T E SCHOOL OF T H E U N IV E R S IT Y OF M IN N E SO T A B Y Tom otake M atsum ura IN PA RTIAL FU L FIL L M E N T OF T H E R E Q U IR E M E N T S FO R TH E D E G R E E OF D O C T O R OF P H IL O SO P H Y S eptem b er, 2006 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3234933 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignm ent can adversely affect reproduction. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3234933 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. © Tomotake Matsumura 2006 ALL RIGHTS RESERVED Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF MINNESOTA This is to certify th at I have examined this bound copy of a doctoral thesis by Tomotake Matsumura and have found th at it is complete and satisfactory in all respects and that any and all revisions required by the final examining committee have been made. Professor Shaul Hanany (Faculty Adviser) GRADUATE SCHOOL Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A C osm ic M icrow ave B ack grou n d R a d ia tio n P o la rim eter U sin g S u p erco n d u ctin g M a g n etic B ea rin g s by Tomotake Matsumura Under the supervision of Professor Shaul Hanany A B STR A C T We discuss the application of half-wave plate (HWP) polarimetry for measurements of the cosmic microwave background (CMB) polarization. In the first part of the thesis, we investigate the use a high-temperature superconducting magnetic bearing (HTS bearing) to support continuous rotation of the HWP at liquid helium (LHe) temperature. We have constructed a prototype HTS bearing and have carried out experiments to measure properties of the HTS bearing when it operates at LHe tem perature. We present the construction of a cryogenic induction motor, which can maintain the rotation frequency of the HTS bearing constant. In the second part of thesis, we discuss sources of systematic errors when using a HWP for a CMB polarization experiment. The systematic errors include instrumental-, cross-, de-polarization, and effects to a pointing. We develop a model to analyze the performance of a HWP and of an achromatic HWP, and quantify the systematic errors associated with HWP polarimetry. We also discuss designs of an anti-reflection coating by using trapezoidal-shape sub-wavelength grating structures to minimize reflection from the HWP and lens over broadband at cryogenic temperatures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A cknow ledgem ents First of all, I would like to thank my adviser, Prof. Shaul Hanany, for his endless support throughout the pursuit of my Ph.D. He has given me a number of opportuni ties in research with enormous patience. The research under his supervision awarded me with invaluable opportunities to experience his professional attitude, filled by his motivation, to push science forward. I would also like to thank Prof. Terry Jones for helpful discussions of polarization and Dr. John Hull for giving me ideas and input on SMBs. I would not have completed my thesis without former and current members of the observational cosmology lab. I enjoyed working with them and learned a lot from them through academic and social interactions. I would also like to thank Mr. Jon Kilgore and the physics departm ent machine shop machinists. On a personal level, I want to thank Tatsu-san, Chiho, Richard, Alan/Miki+1, Masaya, Hyuk-Jae, Justin, friends from physics grads and UMN, the guys from the Rambling Sturgeon, MKSA, Dunesday, and many more. W ithout them, my life in graduate school would have been nothing but going back and forth between the lab and home for 365 days x x years. Finally, I would like to thank to my family, the Matsumura, the Tsuchiya, the Kuwahata, and the Yamamoto, for their constant support and for being in Japan with good health while I am away from home for quite a long time. Special thanks to Akira Tsuchiya for giving me a telescope when I was little, even though I did not become an astronomer. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C ontents A bstract i A cknow ledgem ents ii List o f Tables vi List o f Figures ix 1 C osm ic M icrow ave Background R ad iation and its Polarization 2 1 1.1 In tro d u ctio n .................................................................................................... 1 1.2 HWP polarimetry and its experimental ch allen g es................................ 3 1.2.1 Rotation M e c h a n ism ......................................................................... 6 1.2.2 Systematic effects associate with HWP p o la rim e try ................... 10 Superconducting M agnetic B earings 13 2.1 M otivation........................................................................................................ 13 2.2 Theoretical background of an HTS b e a r in g ............................................. 15 2.2.1 Rotational loss .................................................................................. 15 2.2.2 S tiffn e s s ............................................................................................... 18 2.2.3 Damping property ............................................................................ 20 2.3 Hardware of prototype S M B ....................................................................... 21 2.4 Coefficient of f r ic tio n .................................................................................... 22 2.4.1 Tests, results, and interpretation .................................................. 22 2.4.2 Discussion and s u m m a ry .................................................................. 29 Stiffness and damping coefficient................................................................ 31 2.5 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5.1 In tro d u ctio n ............................................................................................ 31 2.5.2 Experimental s e tu p ............................................................................... 31 2.5.3 Resonant frequency...................................................... 2.6 2.7 3 2.5.4 Vibrational amplitudes and d a m p in g .............................................. 36 2.5.5 C o n clu sio n s............................................................................................ 39 Rotational frequency variation within oneperiod of rotation frequency 3.2 3.3 3.4 40 2.6.1 In tro d u ctio n ............................................................................................ 40 2.6.2 Dipole-dipole interaction m o d e l ........................................................ 41 2.6.3 Experimental S e tu p ............................................................................... 43 2.6.4 E xperim ents............................................................................................ 44 2.6.5 Results and D ata A n a ly s is ................................................................. 45 2.6.6 D iscussion............................................................................................... 45 2.6.7 C o n clu sio n s........................................................................................... 49 Electromagnetic drive mechanism:Induction m o t o r ................................... 51 2.7.1 In tro d u ctio n ........................................................................................... 51 2.7.2 Induction motor h a r d w a re ................................................................. 51 2.7.3 Induction motor m o d e l........................................................................ 52 2.7.4 Measurements and R e s u lts ................................................................. 55 2.7.5 Discussion and Conclusions 58 .............................................................. S y stem a tic effects in half-wave p late p olarim etry 3.1 3 61 In tro d u ctio n ....................................................................................................... 61 3.1.1 Stokes v e c t o r ........................................................................................ 62 3.1.2 List of systematic e ffe c ts ..................................................................... 63 Polarimeter model .......................................................................................... 65 3.2.1 Modulation efficiency........................................................................... 67 3.2.2 Phase vs. input polarization an g le.................................................... 67 3.2.3 Parameters used in the sim ulation.................................................... 68 Single HWP p o la rim e try ................................................................................ 69 3.3.1 Analytical d e riv a tio n ........................................................................... 69 3.3.2 Discussions 71 ........................................................................................... Achromatic half-wave plate ......................................................................... iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 3.5 3.6 3.7 3.4.1 R esu lts.................................................................................................. 77 3.4.2 Discussions 79 ........................................................................................ Spectrum of the incident radiation .......................................................... 84 3.5.1 M o d e l.................................................................................................. 85 3.5.2 R e su lts.................................................................................................. 86 3.5.3 D iscussion.................... 87 Reflection from multi-layered wave p la te s ........................................ . . . 93 3.6.1 In tro d u ctio n ........................................................................................ 93 3.6.2 A generalized transmission Mueller m a t r i x .................................. 94 3.6.3 Transmission coefficient ofmulti-layer birefriengent materials . 96 3.6.4 Results and discussions.................................................................... 104 Anti-reflection coating with SWG s tr u c tu r e s ......................................... 110 3.7.1 In tro d u ctio n ........................................................................................ 110 3.7.2 Subwavelength Grating S tru ctu res................................................... Ill 3.7.3 Model and D esign.............................................................................. 112 3.7.4 R esu lts.................................................................................................. 114 3.7.5 D iscussion........................................................................................... 121 3.8 Effects when a wave plate does not reside at an aperture stop .... 3.9 Oblique angle of incident radiation to the H W P .................................... 124 3.9.1 Double re fra c tio n .............................................................................. 124 3.10 Summary of systematic e ffe c ts.................................................................... 127 A A generalized transm ission Jones m atrix 122 129 B E xtraction o f th e polarization of C M B and dust from IVA curves in tw o bands 131 B .l In tro d u ctio n .................................................................................................... 131 B.2 P r o b le m .......................................................................................................... 132 B.3 S o lu tio n .......................................................................................................... 132 B.4 Conclusion....................................................................................................... 135 C Second order effective m edium th eory v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Tables 2 .1 Summary of measurements with the LNC. In measurement 9 the magnetic field of the m agnet has been altered to reduce inhomogeneity by shimming with high permeability shims (see te x t)...................................................................................................... 25 3 .1 The parameters used in the simulation throughout this chapter are shown. 68 3 .2 The maximum deviations of P out at Pin = 0.1 are shown. The indices of refraction . . . are used for sapphire [1]. The thickness of each wave plate is chosen such that the frequency of incident light, v w p = 300 GHz, satisfies AS = n ....................................... 3 .3 81 Top: The offset angles with four different spectra are shown. Bottom: The differ ence of the offset phase between different spectra. The number in a parenthesis is the difference in terms of the polarization angle a on the sky. A unit of the phase is in degrees...................................................................................................................................... 3 .4 86 The summary of the errors in the polarization angle and the degree of polarization when the incident radiation are two spectral components. In the 150 and 250 GHz bands, the errors in the polarization angle are for . In the 420 GHz, the error is for a T “st........................................................................................................................................ 3 .5 The table shows the instrumental- and cross-polarization induced by the effects of reflection at the two bands with the five-stack AH W P..................................................... 3 .6 109 The averaged reflectance and the difference of the averaged reflectance of the single sapphire wave plate are shown for the three bands............................................................ 3 .7 88 114 The averaged reflectance and the difference of the averaged reflectance of the TMM + sapphire are shown for the three bands............................................................................. 11 8 3 .8 The averaged reflectance of the rexolite for three bands is shown........................................ 119 3 .9 Instrum ental-polarization............................................................................................................ vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 3 .1 0 C r o ss-p o la r iz a tio n ...................................................................................................................... 127 3 .1 1 De-polarization .......................................................................................................................... 128 3 .1 2 The effects to a pointing.............................................................................................................. 128 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Figures 1.1 The expected l a determination of the E and B CMB polarization power spectra with the EBEX balloon-borne experiment [2] after 14 days of a long duration balloon flight (red points and error bars) and with the Planck Surveyor satellite after with 1 year of data (only B-mode is shown, blue dotted error bars). The solid lines are theoretical models for E and B power spectra in a standard ACDM cosmology with T / S = 0.1. The B-mode spectra due to the IG B and due to lensing are shown both separately and combined. The B-m ode power spectra of polarization from Galactic dust (short dash) and from synchrotron emission (long dash) at 150 GHz assume the recent results from the W M AP team [3] but have been scaled by the ratio of the dust and synchrotron RMS in the EBEX observing region to those in the W M AP data to provide a crude estim ate of the expected foregrounds in small, clean regions of the sky. Subtraction of residual dust and synchrotron foregrounds is possible using E B E X ’s multiple frequency capability. The power spectra of pixel noise (dash dot) show that EBEX could make a map of the B-m ode signal for I < 200 while Planck could only make a map for I < 7. . . 1 .2 A cross-sectional view of the EBEX optical system is shown........................................... 1 .3 A side view of a ring-shape PM (red) levitating above an array of YBCO HTS tiles (black). The HTSs were submerged in LN2, and the HTSs and the PM are lifted from the LN2 bath for taking a picture. The HTS bearing is orientated at 70 degrees with respect to the direction of earth gravity. The PM achieves a stable levitation in radial and vertical directions............................................................................. 2.1 Magnetic bearing apparatus during testing in a liquid helium cryostat. The size of the prototype SMB system is designed to implement in the M AXIPOL cryostat [4] ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 .2 Inhomogeneity of the magnetic field of the magnet as a function of distance from its surface for the bare magnet (stars) and for a configuration where the magnet is shimmed with high permeability steel (square). The inhom ogeneity is quantified using Equations 2.17 and 2.18. The insert zooms on the difference of 1.65 x 104 gauss between the shimmed and nonshimmed configurations. The straight line is an approximate fit to the nonshimmed data and was used to find interpolated values. 23 2 .3 COF as a function of magnetic field inhomogeneity [from Equation 2.17 ]. The data suggests a linear relation as predicted by theory....................................................... 2 .4 Coefficient of friction as a function of ambient pressure for two levitation distances. All data was measured at a temperature of 77 K ................................................................ 2 .5 24 26 Coefficient of friction as a function of frequency for temperatures of 16 (stars), 60 (open triangles), 70 (squares), 79 (circles), and 84 K (filled triangles) for a levitation distance of 6.0 mm. For clarity of presentation, the COF for temperatures of 70, 79, and 84 K, have been offset vertically up by 0.4, 0.8, and 1.2 ordinate units, respectively........................................................................................................................................ 2 .6 27 Coefficient of friction as a function of frequency for temperatures of 15 (stars), 50 (triangles), 62 (squares), and 79 K (circles), for a levitation distance of 7.2 mm. For clarity of presentation, the COF for temperatures of 62 and 79 K have been offset vertically up by 0.3 and 0.8 ordinate units, respectively....................................... 2 .7 Coefficient of friction is plotted as a function of temperature for a levitation distance of 6, 7.2, and 9 m m ........................................................................................................................ 2 .8 28 Coefficient of friction at 1 Hz as a function of 1 /J C , where Jc is the critical current in the H T S........................................................................................................................................ 2 .9 27 29 Schematic diagrams show experimental setups to measure the vertical (top) and horizontal (bottom ) vibration, respectively............................................................................ 32 2 .1 0 Vibration spectra for nonrotating levitated rotor in vertical and horizontal direc tions (top panel, heavy and light curves, respectively) compared with spectra of laboratory wall and cryostat shell (bottom panel).............................................................. 33 2 .1 1 Resonant frequency in vertical direction versus tem perature........................................... 34 2 .1 2 Resonant frequency in radial direction versus tem perature.............................................. 34 2 .1 3 Vibrational amplitude of the resonant frequencies versus temperature for a non spinning rotor.......................................................................................................... x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 2 .1 4 Vertical vibration amplitude versus tim e with period of shaking the cryostat for 10 sec. At a temperature of 16 K (top panel) and 64 K (bottom panel). Shaking occurs at tim es of 10 — 20, 70 — 80 and 130 — 140 s ........................................................... 2 .1 5 Time constant of vertical vibration damping versus tem perature................................. 2 .1 6 Time constant of horizontal vibration damping versus tem perature............................ 2 .1 7 A schematic of the dipole-dipole interaction that is used to model the torques that give rise to rotational speed variation...................................................................................... 2 .1 8 Cross-sectional view of the hardware. On the bottom right is an inset with a top view of the copper disk................................................................................................................. 2 .1 9 The rotation frequency as a function of tim e. The beginning of each rotation of the rotor is marked with a pair of high and low frequencies; see text. Repeatable sinusoidal variations of the rotational speed are evident within each period of rotation. 2 .2 0 Top: the rotational frequency, after removal of an offset and a gradient, as a func tion of rotational angle. The speed variations are synchronous with rotational position. There is a strong correlation between m axima and minima in rotational speed and m axim a and minima in the spatial magnetic field. Bottom: the axial component of the m agnetic field as a function of azimuthal angle. The measure ment is made at a radius where the axial component of the magnetic field is a maximum........................................................................................................................................... 2 .2 1 Same as the top panel of Fig. 2.20 but for rotation frequencies of about 5 Hz (top) and 10 Hz (b ottom )........................................................................................................................ 2 .2 2 Fractional speed variation as a function of frequency for environments (i), (ii) and (iii), top to bottom , respectively. The dashed lines are best fits to a constant and to Equation 2.29 and the solid line is a best fit to the sum of a constant and Equation 2.29. The fit constants (in %) and the values of a (in Hz2) from the solid lines are (1.1, 1.6 x 10- 2 ), (1.0, 1.1 x 10- 2 ), and (1.4, 1.1 x 10~2) for the top, middle, and bottom , respectively............................................................................................... 2 .2 3 The rotation frequency as a function of tim e. The RMS current changed from 32 mA to 53 mA at t = 470 sec. The rotation frequency increased from its initial state to the final state exponentially. The solid line is a fit to the data after t = 470 sec using the Equation 2.41......................................................................................... xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 .2 4 The deceleration as a function of rotation frequency during a free spin-down of the rotor..................................................................................................................................................... 55 2 .2 5 Top panel: stability of rotation frequency as a function of time. The inset shows a zoom on the data between 4 and 5 hours with an expanded scale for the vertical axis. B ottom panel: power spectral density of the data in the top panel after subtraction of a m ean.................................................................................................................... 56 2 .2 6 The temperature of the coil as a function of frequency of the applied current for RMS currents of 7.5 mA, 15 mA, 20 mA, and 60 m A ....................................................... 57 2 .2 7 The exponent as a function of the RMS current for frequencies of 12.5, 25, and 50 Hz. The continuous lines are quadratic fits to the data of each of the frequencies. 3 .1 A schematic diagram to show the HW P polarimeter m odel.............................................. 3 .2 The left panels show the IVA curves at a single frequency of 150 (solid), 200 (dash), 58 65 250 (d ot), 300 (dash-dot) GHz for a single HWP, three-stack AHWP, and five-stack AHWP. The right panels show the IVA curves that are averaged over the bandwidth 150 ± 30 GHz (solid) and 250 ± 30 GHz (dash) with a step size of 1 GHz. In all cases, the input polarization angle a lrl = 0, and the ellipticity /3,;n = 0 ....................... 3 .3 72 Pout is plotted as a function of Pin with various input polarization angles with the single HWP. The thickness of the wave plate is chosen such that the optimized frequency of the wave plate is 300 GHz.................................................................................. 3 .4 73 The m odulation efficiency with zero bandwidth is plotted as a function of frequency. The black curve is for the single HWP. The red curve is for the three-stack AHWP and the blue curve is for the five-stack AH W P.................................................................... 3 .5 A configuration assumed in our simulation is shown. The transmission axis of a linear polarizer is parallel to the x axis................................................................................... 3 .6 74 75 M odulation efficiency e(v, A v = 0 , a = 0,0) (top) and the phase § ( v , k . v = 0 , a = 0 ,9) (bottom ) for the three-stack (left) and five-stack (right) are plotted as a func tion of frequency with offset angles of 6 3 = (0 ,5 8 ,0 ) degrees and 9$ = (0 ,2 9 ,9 4 .5 ,2 9 ,2 ) de grees..................................................................................................................................................... xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 3 .7 Top: M odulation efficiency of the three-stack AHW P is plotted with an input polarization angle of 0 (solid line), 22.5 (dot), 45 (dash), 67.5 (dot-dash), and 90 (three-dot dash) degrees. Bottom: M odulation efficiency of the five-stack AHWP with same input polarization angles as the top panel........................................................ 3 .8 77 Top: P out as a function of P vn for the three-stack AHWP. Bottom: For the fivestack AHWP. Each curve in each panel corresponds to the input polarization angle of 0 (solid line), 22.5 (dot), 45 (dash), 67.5 (dot-dash), and 90 (three-dot dash) degrees................................................................................................................................................ 3 .9 78 The output phase is plotted as a function of the input polarization angle. Top left: Three-stack AHW P with a bandwidth of 150 ± 30 GHz (solid line) and 250 ± 30 GHz (dot). Top right: Three-stack AHW P with a bandwidth of 300 ± 0 (solid), 100 (dot), and 200 (dash) GHz. B ottom left: Five-stack AHWP with a bandwidth of 150 ± 30 GHz (solid) and 250 ± 30 GHz (dot). B ottom right: Fivestack AHW P with a bandwidth of 300 ± 0 (solid), 100 (dot), and 200 (dash) GHz. 79 3 .1 0 Left: The modulation efficiency of the three-stack AHW P is plotted as a function of the angle of the second plate 02 and the bandwidth A v around the center frequency of 1' w p = 300 GHz. The first and third plate angles are kept at 0 degrees with respect to the x axis. Right: The corresponding phase variation A (f> is plotted in units of degrees. A horizontal axis is the differential frequency Si/. In both plots, the input polarization angle is assumed to be a = 0 ........................................................ 80 3 .1 1 Same procedure as Figure 3.10 for five-stack AHWP. The first, third, and fifth plates are kept at {8 1 , 6 3 , 8 5 ) = (0,9 4 .5 ,2 ) degrees. The offset angles of the second and fourth wave plates are kept same and are varied from 0 to 90 degrees with a step of 1 degree................................................................................................................................ 81 3 .1 2 Left: The m odulation efficiency of the five-stack AHW P is plotted as a function of an offset angle of the third plate while the others are fixed at {6 1 , 8 2 , 6 4 , 8 5 ) = (0 ,2 9 ,2 9 ,2 ) degrees. Right: The m odulation efficiency of the five-stack AHW P is plotted as a function of an offset angle of the fifth plate while the others are fixed at (6»i, 8 2 , 8 3 , 8 4 ) = (0 ,2 9 ,9 4 .5 ,2 9 ) degrees............................................................................. 82 3 .1 3 The total intensity of two spectra is plotted. The higher intensity curve is the 2.73 K black body spectrum of the CMB. The lower intensity curve is the dust spectrum that assumes only one power law as shown in Equation 3 . 3 4 ..................................... xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 3 .1 4 The phase (left) and the output polarization (right) of the i-IVA curve are plotted as a function of the a f ^ B and at the 150 ± 30 GHz band. A unit of the phase contours is in degrees....................................................................................................... 87 3 .1 5 The phase (left) and the output polarization (right) of the i-IVA curve are plotted as a function of the bib and a bij bl ust at the 250 ± 30 GHz band. A unit of the phase contours is in degrees....................................................................................................... 88 3 .1 6 The phase (left) and the output polarization (right) of the i-IVA curve are plotted as a function of the afLMB and aP ^ st at the 420 ± 30 GHz band. A unit of the bib bi b phase contours is in degrees....................................................................................................... 3 .1 7 The variation of the phase A<fi — <j)max grees at a fixed 89 of the i-IVA curve over 0 < a ^ st < 180 de = 0 degrees is plotted. We use P Busl = 0.05. Three solid lines correspond to P ^ MB = 5 x 10~7, 1 x 10- 6 , and 5 x 10~6 (from left to right). The solid line assumes the total dust intensity in Equation 3.34. The dashed and dotted lines assume 10 % higher and 10 % lower of Idust(v), respectively. The right panel is a zoom up of the left panel.......................................................................................... 3 .1 8 The variation of the phase A(f> = <t>max—<Pmin of the i-IVA curve over 0 < 92 < 180 de grees at a fixed cnf^“st = 0 degrees is plotted. We use P ? ust = 0.05. Three solid lines correspond to P?nMB = 5 x 10~7, 1 x 10- 6 , and 5 x 10~6 (from left to right). The solid line assumes the total dust intensity in Equation 3.34. The dashed and dotted lines assume 10 % higher and 10 % lower of Idusti1'), respectively. The right panel is a zoom up of the left panel...................................................................................... 93 3 .1 9 Electric fields at boundaries. Each electric field E is a vector. Each layer has the two indices of refraction n 0 and n e. The relative angle between x and x[ is (pi and between x[ and x 2’ is <j> 2 ................................................................................................................ 97 3 .2 0 The transm ittances, Txx (black), Tyy (red), Txy (blue), and Tyx (green) are plot ted as functions of frequency for the single HWP (top), the three-stack AHW P (middle), and the five-stack AHW P (bottom )...................................................................... 105 3 .2 1 M odulation efficiency of a single HW P w ith zero-bandwidth is plotted as a function of frequency. The black curve takes into account of the effects of reflection and the red curve does n o t........................................................................................................................... x iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 3 .2 2 M odulation efficiency e ( v , A v = 0 , a = 0, 9) (top) and the phase (j>{v,Av = 0 ,a = 0 , 6 ) (bottom ) for the three-stack (left) and five-stack (right) are plotted as functions of frequency with offset angles of 6 3 = (0 ,5 8 ,0 ) degrees and 9$ = (0 ,2 9 ,9 4 .5 ,2 9 ,2 ) de grees. The red line assumes no reflection. The black line which has fringes is calcu lated by taking into account the effects of reflection between the vacuum and wave plate interface and between wave plate interfaces............................................................... 107 3 .2 3 A schematic diagram of a square pattern of the SWG structures are shown. . . 112 3 .2 4 A cross-sectional view of the pyramidal shape SWG structure is shown..................... 113 3 .2 5 The averaged reflection from the single sapphire wave plate at the 150 ± 30 GHz band is plotted as a function of the height h and the top-base width s. The left panel is for the case that the index of refraction of the sapphire is n a and the right panel is for n e, respectively. The reflectance is normalized to one................................. 115 3 .2 6 The reflectance for the ordinary (left) and extraordinary (right) axes from the single sapphire wave plate are plotted on the top panels. The bottom panels show the corresponding profile of the effective index of refraction for the ordinary (left) and extraordinary (right) axes............................................................................................................ 116 3 .2 7 The averaged reflection of TMM + sapphire at the 150 ± 30 GHz band is plotted as a function of the height h and the top-base width s. The left panel is for the case that the index of refraction of the sapphire is n 0 and the right panel is for n e, respectively........................................................................................................................................ 118 3 .2 8 The reflectance of TMM + sapphire for the ordinary (left) and extraordinary (right) axes are plotted on the top panels. The bottom panels show the corresponding profile of the effective index of refraction for the ordinary (left) and extraordinary (right) axes........................................................................................................................................ 119 3 .2 9 The averaged reflection of the rexolite at the 150 ± 30 GHz band is plotted as a function of the height h and the top-base width s .............................................................. 120 3 .3 0 The reflectance from the interface between the vacuum and the rexolite is plotted on the top panel. The bottom panel shows the corresponding profile of the effective index of refraction............................................................ 3 .3 1 120 Schematic diagrams of double refraction effects are shown............................................ 124 3 .3 2 The distance x is plotted as a function of the incident angle 9. Five curves that correspond to p = 0 (—), 30(- • ■), 45(----- ), 60(- • -), 90(-------) are over-plotted. XV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . . . 125 C hapter 1 Cosm ic M icrowave Background R adiation and its Polarization 1.1 In tro d u ctio n The cosmic microwave background (CMB) radiation has provided a tool to probe physics of the early universe since its discovery by Penzias and Wilson in 1965 [5]. To date, measurements of the CMB has established a standard cosmological paradigm. The current experimental efforts with state-of-the-art technology has just started to search for observational evidences of the universe further back in time as early as 10-43 seconds after the beginning of the universe, the big bang. The universe has expanded since the big bang. Until the universe was at the age of ~ 380,000 years old, photons were tightly coupled with free electrons due to Thomson scattering. As the universe expands, the tem perature of particles drops. Once the universe was cooled enough for an electron and a proton to form a hydrogen atom, a process called recombination, photons no longer interacted with electrons and propagated through the universe until they are observed by us as the 2.73 K CMB radiation today. In the past two decades, a number of im portant observational results came out from ground-base, balloon-borne, and satellite missions. Three im portant milestones among them can be highlighted: (i) a perfect black body spectrum at tem perature 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of 2.73 K th at was measured by COBE FIRAS [6]; (ii) tem perature anisotropy of the CMB th at was first detected by COBE DMR and characterized by WMAP and many other groups, including MAXIMA [7, 8, 9, 10, 11, 12]; and (iii) polarization anisotropy of the CMB th at was first discovered by DASI [13] and has just recently started to be characterized by several groups, including WMAP [14, 15, 16, 17, 3]. To date, the most recent results of the tem perature anisotropy are best summa rized in the WMAP three year data set [12]. The tem perature anisotropy is now well characterized and is fully consistent with a model: the acoustic oscillation of baryonphoton plasma in the background of the primordial density perturbation (PDP) [18]. If the model of the CMB tem perature anisotropy is correct, the CMB radiation is expected to be linearly polarized [19]. During the recombination, the CMB photons interact with electrons due to Thomson scattering. The differential cross section depends on the polarization as doT dQ, e2 \ e ■ e ' \ ( 1 .1) 4-7TC3 where e (or e') is a unit vector of the incident (or outgoing) polarization. When a uniform radiation field exists around the scattering center, no net polarization is created by the scattering. A local quadrupole intensity distribution of the incident radiation around the scattering center gives rise to a net linear polarization. This process results in a particular pattern of polarization on the sky that is correlated with the PDP, and thus is correlated with the tem perature anisotropy of the CMB. This vector field of the linear polarization on the sky has a curl-free pattern, and therefore is called as E-mode polarization due to the analogy to the electromagnetism. The E-mode polarization signal is fully predicted by cosmological parameters th at are determined by the tem perature anisotropy. The measurements of the E-mode improve the estimate of the cosmological parameters. This E-mode polarization signal was recently discovered [13] and has just started to be characterized [3]. The E-mode polarization is not only the pattern of the polarization if the infla tionary paradigm is true. The divergence-free pattern of the polarization anisotropy, called B-mode, is also imprinted in the CMB polarization anisotropy if the universe has gone through inflation. Inflation was first proposed by Guth to explain unsolved Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. problems in the standard cosmology, e.g. monopole problem, horizon problem, and flatness problem [20]. Although the postulation of inflation is consistent with current observational results, there is no direct observational evidence. A stochastic background of gravitational wave th at was left from inflation creates a distortion of metric in a quadrupole pattern due to the spin 2 nature of the gravity wave. As a result, this quadrupole pattern also left a signature in the CMB polarization. The polarization pattern from the inflationary gravitational-wave background (IGB) has E-mode and B-mode equally. Therefore, the detection of the B-mode polarization can open a new door to study physics of inflation observationally. The polarization anisotropy can be quantified by the power spectrum of the E- and B-mode signals. Figure 1.1 shows the E- and B-mode power spectra. The B-mode signals consist of three shallow bumps, reionization peak (I ~ 4), IGB peak (I ~ 80), and lensing peak (I ~ 1000). The B-mode has not been detected. Our primary goal is to measure the IGB B-mode. The signal strength of the IGB B-mode is quantified by a param eter called a tensor-to-scalar ratio r. The T / S ratio is related to the energy scale of inflation V as C 1/4 = 3.3 x 1016 r 1/4 GeV. (1.2) A current observational upper limit of the T / S ratio is r < 0.55 from WMAP threeyears data [3], 1.2 H W P p o la rim etry and its ex p erim en ta l chal len ges A polarimeter is a device to measure the state of polarized light. A common tech nique to measure linearly polarized radiation, particularly at infrared and optical wavelengths, is to use a rotating half-wave plate (HWP) together with a linear po larizer. Although HWP polarimetry has been used for years in many astrophysical observations, the implementation to a CMB polarization experiment is relatively new, and it was first employed by the balloon-borne CMB polarization experiment, MAX3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 S y n c h o tr o n PLANCK, Dust EBEX. it 1 a. t= <m \ cf 0.1 + \ Gravity iffavei 0.01 lensing 10 100 1000 F ig u r e 1.1: The expected 1a determination of the E and B CMB polarization power spectra with the E BEX balloon-borne experiment [2] after 14 days of a long duration balloon flight (red points and error bars) and with the Planck Surveyor satellite after with 1 year of data (only B-m ode is shown, blue dotted error bars). The solid lines are theoretical models for E and B power spectra in a standard ACDM cosmology with T / S = 0.1. The B-m ode spectra due to the IG B and due to lensing are shown both separately and combined. The B-mode power spectra of polarization from Galactic dust (short dash) and from synchrotron emission (long dash) at 150 GHz assume the recent results from the W M AP team [3] but have been scaled by the ratio of the dust and synchrotron RMS in the EBEX observing region to those in the W M AP data to provide a crude estim ate of the expected foregrounds in small, clean regions of the sky. Subtraction of residual dust and synchrotron foregrounds is possible using E B E X ’s multiple frequency capability. The power spectra of pixel noise (dash dot) show that EBEX could make a map of the B-m ode signal for I < 200 while Planck could only make a map for I < 7 . 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IPOL [21]. In this section, we describe how the HWP polarimetry works in general and list experimental challenges which the HWP polarimetry has particular to CMB observations. The polarimeter consists of a HWP, a linear polarizer, and a bolometer. A wave plate is a birefringent crystal th at is cut in a disk shape such that the ordinary and extraordinary axes lay in the plane of the disk. When linearly polarized light is transm itted through a wave plate with a propagation direction th at is normal to the surface of the wave plate, the phase difference A<$ between electric fields along the ordinary and extraordinary axes is = (1.3) c where d is a thickness of the wave plate, v is the electromagnetic frequency of the light, and c is the speed of light. The two indices of refraction in the ordinary and extraordinary axes are n 0 and n e, respectively. When AS = tt, the wave plate is called as a HWP. When linearly polarized light is incident on the rotating HWP at a frequency of /o, the transm itted polarization rotates at 2 x / 0. This rotating linearly polarized light passes through a fixed linear polarizer. The bolometer, which is not sensitive to the polarization, measures the modulation of intensity th at appears at 4 x / 0 when the signal is linearly polarized. We analyze this modulated intensity as a function of the HWP angle to reconstruct the state of incoming polarized light. The rotating HWP polarimeter has strong advantages to control systematic errors of an experiment. • The rotation frequency of the HWP can be chosen such th at the signal frequency resides above a 1 / / noise knee of the detector and readout system. • The facts th at the signal appears at known frequency 4 x / 0 and it is not the rotational synchronous frequency of the HWP provide us a strong tool to reject a spurious signal th at does not appears at 4 x / 0. • A single detector can fully reconstruct the state of incoming polarized light. Therefore, there is no need to cross-calibrate the multiple detectors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Secondary Mirror Wa\ Plat Primary Mirror Camei Lens Focal Planes Figure 1.2: Linear Polarizer A cross-sectional view of the EBEX optical system is shown. Although there are the advantages to use a HWP polarimeter to measure the CMB polarization, a number of technical issues need to be addressed to implement the HWP polarimeter into the CMB observation. We describe the technical challenges for the rest of this chapter. 1.2.1 R otation M echanism Figure 1.2 shows an example of the EBEX optical systems. EBEX is a long duration balloon-borne CMB polarization experiment th at is designed to detect the IGB and lensing B-mode signals as well as the polarization of the dust signal [2], Typically, an optical system designed for CMB observation has multiple mir rors or lenses to focus the incident radiation onto a focal plane. The CMB radiation is 2.7 K black body radiation. To measure this cold radiation through a number of optical elements, the tem perature of these elements need to be cooled down to mini mize the thermal emission from these elements. The HWP is also one of the elements th at needs to be cooled, and the ideal location to mount is at the liquid helium (LHe) 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tem perature stage. As a result, the HWP has to rotate at LHe temperature at a typical rotational frequency of 1 ~ 20 Hz. The continuous mechanical motion at LHe tem perature is known to be difficult to realize due to the absence of lubricants at this temperature. When a mechanical bearing is used to rotate the HWP at LHe temperature, stick-slip friction induces vibration. This vibration makes difficult to design a continu ous rotation mechanism at a vicinity of bolometric detectors because the microphonic vibration induces noise in the detector readout in two ways. First, the vibration of a spider web of the bolometer deposits heat on the absorber, and therefore this input energy appears as noise. Second, the vibration of signal wires induces the fluctua tion of impedance due to the inductive and capacitive couplings. As a resut, this fluctuation appears as noise in a detector readout. We consider two types of bolometers due to high sensitivity at electromagnetic frequencies of 100 —500 GHz wave band. One is a bolometer th at uses neutron trans m utation doped (NTD) Ge as a thermister. This thermister has a large impedance of ~ 1 MfL The other type is a transition edge sensor (TES) bolometer th at oper ates at the transition edge of superconductivity and its impedance is ~ 1 ST. The NTD bolometers are highly sensitive to microphonic noise. Typically astrophysical observations th at use NTD type bolometers rotate the HWP not continuously but step-by-step to minimize the microphonic noise. On the other hand, as of today, the relatively new technology of TES has no definitive answer whether or not the conventional mechanical bearing is workable with an acceptable noise level. The continuous rotation with a mechanical bearing is also a heat source at the cryogenic tem perature stage due to the friction of the bearing. When the observation continues for the time scale of weeks to years in a balloon- or space-borne platform, it is not suitable to have a high-power dissipative device at the cryogenic tem perature stage because of a limited available cryogen. To take advantages of HWP polarimetry, we need a reliable rotation mecha nism to support the continuous rotation of the HWP at cryogenic temperature. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F ig u r e 1.3: A side view of a ring-shape PM (red) levitating above an array of YBCO HTS tiles (black). The HTSs were submerged in LN2, and the HTSs and the PM are lifted from the LN2 bath for taking a picture. The HTS bearing is orientated at 70 degrees with respect to the direction of earth gravity. The PM achieves a stable levitation in radial and vertical directions. Superconducting m agn etic bearings Hanany et al. proposed to use a high-temperature superconducting (HTS) magnetic bearing (hereafter HTS bearing) as a bearing to support the rotation of a HWP [4]. As shown in Figure. 1.3, the HTS bearing consists of a ring-shape permanent magnet (PM) and an array of bulk HTS tiles. The PM is magnetized in the axial direction. Once the array of HTSs is placed underneath the PM with a separation of a few millimeters and is cooled below its critical tem perature Tc of the HTS (~ 95 K for YBCO) in the presence of the magnetic field of the PM, the HTSs shield the magnetic field due to the Meisner effect and also trap the magnetic field due to the property of a type II superconductor. This process is called field cooling (FC) as opposed to zerofield cooling (ZFC) th a t infers th at the HTS is cooled below its Tc without applying external magnetic field. As a result of FC, the PM achieves a stable levitation above the array of HTSs. Furthermore, as a consequence of azimuthal symmetry of the PM 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in geometry and magnetism about the axial direction, the PM does not only levitate but also rotate freely in azimuth without contact. The choice of the PM to be a rotor and the HTSs to be a stator, or vice versa, is arbitrary. A detailed description of the physics of levitation can be found in articles [22, 23, 24]. The levitated PM is stable in all degrees of freedom except in azimuthal di rection. It is stable enough to support its own weight as well as significant external force in all directions without any active control to maintain its position of levitation. Due to its contact-less rotation bearing, there is no stick-slip friction, and therefore the coefficient of friction (COF) of its rotation is about four order of magnitude lower than that of conventional mechanical bearings. This HTS magnetic bearing is appealing to use as a bearing to support the HWP at cryogenic tem perature with following reasons. • Low COF without stick-slip friction. • Passive stable levitation without any active motion control • No wear and tear for long term use • Minimum energy deposit the LHe tem perature stage during the rotation • No extra-effort to cool HTSs to achieve levitation If the noise associate with the rotation of the HWP is mainly due to micro phonic vibration, the HTS bearing should reduce the stick-slip friction, and thus the microphonic vibration. From the experiences with MAXIPOL [21], the conventional mechanical gear induces a noise level of 20 ~ 1000 times larger than the noise level of the NTD bolometer itself. There is no clear relationship between the COF and the noise level induced due to the microphonic noise. Nevertheless, if we assume that the noise decreases linearly with COF of the bearing, the noise induced by the HTS bearing will be negligible as compared to the noise level of the detector itself. Because the COF of the HTS bearing is small, the energy dissipated due to the friction at a LHe tem perature stage is much smaller than th at of a conventional me chanical bearing. W ith the same reason, the energy required to maintain its rotation is also smaller. Furthermore, the feature of no surface-to-surface contact does not 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. induce any wear and tear of the bearing. These three features of HTS bearings are particularly attractive to use for the time scale of weeks to years such as in sub-orbital and orbital missions of a CMB polarization experiment. The HWP needs to be cooled down to minimize the thermal emission. This indicates th at we do not need any extra effort to cool the HTS bearing to achieve its passively stable levitation. Although the HTS bearing has a number of attractive features, no group has studied its operation at the LHe temperature. A commonly used HTS for the HTS bearing is YBCO and its critical tem perature is ~ 95 K. Therefore, all the applications are often assumed to be used with a LN 2 bath to cool the HTSs. Properties of the HTS bearing may change depending on the operational temperature. We discuss the experiments th at we conducted to characterize the properties of the HTS bearing at LHe tem perature in Chapter 2. Furthermore, we describe the development of a cryogenic motor to maintain the rotation of the PM rotor at a fixed rotational frequency. 1.2.2 System atic effects associate w ith H W P polarim etry The rotation mechanism to support the HWP without inducing noise in bolometers is a crucial requirement toward the detection of the IGB B-mode signal, but this may not be a sufficient requirement to be imposed. As shown in Figure 1.1, the IGB B-mode signal is expected to be at least a factor of ~ 10 lower than E-mode signal. The required level of controlling systematic effects associate with HWP polarimetry is demanding, and therefore a detailed understanding of HWP polarimetry itself is essential. In this section, we highlight the systematic effects we anticipate. Detailed analyses to quantify these systematic effects are in Chapter 3. Broad-band coverage: Typical balloon-borne CMB experiments, including EBEX [2], cover electromagnetic frequencies of between 100 —500 GHz. Although ro tating HWP polarimetry is useful to reject spurious signals, a single HWP polarimeter is not suitable when the measurement of interest is in a wide frequency band because the chromatic nature of a single wave plate retarder forces the high modulation ef10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ficiency to reside in a narrow spectrum. An achromatic half-wave plate (AHWP) has been proposed to overcome this problem [25]. The use of the AHWP can induce cross and de-polarization. We need to understand how to extract the polarization of incident radiation when the AHWP is used, and all the possible systematic effects due to the AHWP need to be quantified. Incident radiation spectrum : A HWP is a wavelength sensitive device, and therefore the output signal of the HWP polarimeter depends on the incident radiation spectrum. We need to study how the HWP polarmeter behaves to the CMB spectrum and the dust spectrum. We quantify the difference of the extracted polarization states, input polarization angle and the degree of polarization, between the CMB spectrum and the dust spectrum. R eflection: Any interface between media th at have the two different indices of refraction causes reflection. The reflection reduces the total amount of power that is detected by a detector. The reflection also becomes a source of systematic errors. At a millimeter wave band, a wavelength is longer than the characteristic length scale of the surface roughness of the HWP surface. Therefore, two parallel surfaces create a resonant cavity and the reflection between these interfaces causes an constructive and destructive interference over frequency. As a result, we need to quantify the effects of reflection from the HWP and AHWP as a source of instrumental and cross polarization. Furthermore, we need to address the issues of how to minimize the reflection for broadband when the HWP and lens are at cryogenic temperature. A perture stop: An optical system is designed such th at the HWP resides at an aperture stop, and therefore the wave plate is uniformly illuminated by all the detectors. In reality, the HWP does not locate at the exact position of the aperture stop, and therefore any azimuthal inhomogeneity of the HWP properties appears as a modulated signal. We need to quantify this systematic effect. Oblique incident light to a H W P: The incident radiation to the HWP is not necessary normal to the surface. When the oblique incident light passes through the HWP, the refracted angle varies as a function of the rotation angle of the HWP. We need to quantify the effects to a pointing due to the oblique angle of incident. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Chapter 3, we present the results of analyzing the possible systematic effects when the HWP polarimeter is used. Tables 3.9, 3.10, 3.11, and 3.12 show the summary of the systematic effects. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 2 Superconducting M agnetic Bearings 2.1 M o tiv a tio n The critical tem perature of the YBCO bulk high-temperature superconductor (HTS) is ~ 95 K, and therefore YBCO becomes superconducting at liquid nitrogen (LN2) temperature. This feature of a high critical tem perature makes the use of YBCO attractive in a number of industrial applications, including a levitated transportation system, flywheel energy storage, fault current limiter, and trapped-field magnet [23, 26]. An HTS bearing is also one of the systems th at has many industrial applications [23], and therefore a number of studies have reported properties of the HTS bearing th at operates at LN2 temperature. No study has yet been conducted to characterize the properties of the HTS bearing at liquid helium (LHe) tem perature. In our application, we need to maintain the tem perature of the half-wave plate (HWP) at LHe tem perature to reduce the thermal emission from the HWP itself. A single detector receives 0.15 pW of radiative power from 2.73 K blackbody CMB radiation at 150 ± 30 GHz. If we want to maintain the thermal emission from a sapphire HWP below 10 % of the CMB radiative power at the same bandwidth, the HWP has to be cooled below 6 K (assuming the emissivity of sapphire wave plate as 1.6 %). Therefore the bearing to support the HWP has to 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be also at the LHe tem perature stage. To investigate the dynamical properties of the HTS bearing at LHe temper ature as well as to characterize system parameters for the use of HTS bearing in a CMB polarization experiment we have constructed a prototype HTS bearing. We also constructed an electromagnetic drive mechanism to maintain continuous rotation of the bearing. In particular, we conducted the following experiments. • Measurements of a coefficient of friction (COF) at LHe tem perature (Sec tion 2.4) [4] • Measurements of vibrational properties at LHe tem perature (Section 2.5) [27] • Characterization of the correlation between the magnetic inhomogeneity of the rotor permanent magnet (PM) and rotational frequency variation within one period of rotation (Section 2.6) [28] • Development of a cryogenic induction motor with HTS bearings to maintain a constant rotation frequency (Section 2.7) [29] In the following sections, we discuss the hardware, the measurements, and the construction of a motor. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2 T h eo retica l background o f an H T S b earing In this section, we describe the physics of an HTS bearing. 2.2.1 R otational loss Even though there is no physical contact between a rotor and stator, several physical mechanisms contribute to the friction of an HTS bearing, including hysteresis friction and eddy current friction. H ysteresis loss Time varying magnetic field in a type II superconductor creates hysteresis loss. Bean [30] describes th at the energy loss due to the hysteresis loss scales as A £ o c t ^ , (2.1) where A B is the peak-to-peak variation of the time varying magnetic field in a su perconductor, and Jc is the critical current of a superconductor. In a context of HTS bearings, A B arises from azimuthal inhomogeneity of the magnetic field about the axis of rotation due to imperfection in a fabrication process of a single ring-shape rotor. When a ring magnet consists of segmented magnets in az imuth, the joint between adjacent magnets also creates the azimuthal inhomogeneity of the magnetic field. As a rotor magnet rotates, this spatial magnetic field inhomo geneity becomes time varying magnetic field with respect to the stationary HTS tile. When the rotor wobbles during its rotation, the radial-, vertical-, or tilt-mode of the magnet vibration also creates time varying magnetic field in a superconductor. The magnetic field variation A B depends on the quality of the magnet and the relative distance between the rotor magnet and the array of HTS tiles. When the magnetization of a rotor magnet has a tem perature dependence, the tem perature of the rotor magnet also affects the magnitude of A B. The hysteresis loss is inversely proportional to critical current Jc. A super conductor has higher critical current as tem perature of the superconductor decreases. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore, the COF is expected to decreases as tem perature of the HTS tiles de creases. Zeiberger et al. reported th at the critical current of a bulk YBCO sample increases by factor of 20 ~ 30 from 77 K to 4 K when the externally applied magnetic field is in the range of 0 — 1 T [31]. The energy loss due to the hysteresis is related to the deceleration as AE ~At ~ td uj 1lr O,U.' a = A (2.2) where I m is the moment of inertia of a rotor, and u and a is angular speed and angular acceleration, respectively. Equation 2.2 shows th at the hysteresis loss appears to the deceleration as frequency independent term. E ddy current loss Any time varying magnetic field A B in an electrically conductive material induces an EMF and therefore eddycurrent. This process of eddy current loss eddy current dissipates asJoule heat. contributes as friction of a rotating magnet. This The energy loss in unit time due the eddy current [32, 33] is P = Fv a a ( A B ) 2uj2. (2.3) (2.4) Therefore, with the same argument as Equation 2.2, the eddy current loss contributes to the deceleration as an angular speed, u, dependent term. Sources of the time varying magnetic field are the same as the case we discuss for the hysteresis loss. One additional source of the time varying magnetic field is the trapped magnetic field in the HTS tiles. The stator HTS is not a continuous ring, but an array of HTS tiles. Therefore, the field th at is trapped in each tile has higher concentration of the flux at the center of the tile and the trapped magnetic field decays as approaching to the edge of the HTS tile. As the rotor magnet rotates above the array of HTS tiles, the spatially fixed magnetic field with respect to each HTS tile becomes time varying field 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in the rotor magnet and creates friction due to the eddy current interaction. The eddy current loss is proportional to the electrical conductivity of sur rounding metals. Equation 2.4 indicates th at it is best to construct hardware with non-electrically conductive material, such as G-10 and vespel. When the operational tem perature of the HTS bearing changes from LN2 tem perature to LHe temperature, the electrical conductivity of material changes corre spondingly. The electrical conductivity of metals tends to increase as the tempera ture of metal increases. For aluminum 6061, the resistance ratio, and OFHC copper has , is ~ 1.2, ~ 23 [34]. An OFHC copper is often used to maxi mize the thermal conductivity at cryogenic temperature. When the OFHC copper is used around the HTS bearing, the eddy current loss is expected to increase as the operational tem perature decreases. On the other hand, the eddy current loss due to the finite electrical conductivity of the rotating magnet may cause not only to increase the COF, but also to increase the tem perature of the levitating magnet. This is because the rotor is thermally isolated except through radiative heat exchange. When NdFeB is used as a rotor magnet, the electrical conductivity of NdFeB at room tem perature is about 5 order of magnitude lower than that of A1 at the same temperature. This effect adds the heat input to the levitating rotor magnet in addition to the absorption from the optical load. The eddy current loss is a frequency dependent loss, and therefore the contri bution to the COF may be small when the rotation frequency is low. Coefficient o f friction Hull et al. [35] proposed to quantify the coefficient of friction (COF) of an HTS bearing as the ratio of drag force FD to lift force FL as COF Fd Fl (2.5) I 01M g R D (2 .6 ) 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The drag force is F d = td / R d , where R D is the outer radius of the rotor and td is drag torque due to friction. The drag torque is calculated from the measured angular deceleration as td = I ma , where I m is the total moment of inertia of a rotor and a (a < 0) is the acceleration of a rotor magnet. The lift force is FL = M g , where M is the mass of a rotor and g is the deceleration of gravity. 2.2.2 Stiffness The stiffness of a levitating HTS bearing is quantified by a spring constant due to the analogy of a spring system. Hull shows the analytical relationship of the spring constant to the superconducting levitating system by using the frozen flux model [24], The derivation assumes a dipole magnet th at levitates above an infinite plane of a type II superconductor. When a dipole with the magnetization m is placed at the distance 2 above a type II superconductor with FC, the flux due to the Meisner effect and trapped flux at a pinning center can be treated as two images, diamagnetic image and frozen image. The diamagnetic image is a mirror image of the levitating dipole at the distance z from the interface of the superconductor. The frozen image in the superconductor appears at the same location as the diamagnetic image, but the direction of the magnetization differs by 180 degrees from the magnetization of the diamagnetic image. We treat these two images as external magnetic field sources and calculate the magnetic interaction between the external field and the real dipole above super conductor. We label these two images as B dm and B l rozen. If we assume th at the magnetization direction of the levitating dipole is normal to the plane of a supercon ductor as m = mz, the spring constant can be written as kz = _dF\ dz $ 2 B dia Q 2B f r o z e n + - a ? ~ > kx — ( 2 ' 7) dx pjl jdf r o z e n ~m~ ts r 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <2-8> If we substitute the explicit expression of the magnetic field B, such as for a dipole, it is easy to show th at the three spring constants relate as kx “I- h'y kz — 0. ( 2 . 10 ) This is analogue to Earnshaw’s theorem of a type II superconductor when the su perconductor is field-cooled. Equation 2.10 shows th at the spring constants, kx, ky, kz, can be all positive values simultaneously. Therefore, the field cooled levitating magnet is stable in all three directions. When a ring shape magnet is levitating above an array of HTS tiles, a levitating element has a symmetry in the x and y directions, and therefore kx = ky = kr. 2kr —kz = 0. (2 .11) When there is a spring constant to each degree of freedom, there is a corre sponding natural frequency as (2 .12) (2 .1 3 ) where M is the total mass of a rotor. When rotation frequency of the rotor coincides with a natural frequency, two frequencies resonate. As a consequence, the rotor wobbles unstably and the COF increases [23]. It is im portant to design the stiffness of the HTS bearing such th at the natural frequency is away from the rotation frequency. Equations 2.7, 2.8, 2.9 show th at the spring constant is proportional to m 2 and the magnetic field geometry. Therefore, a HTS bearing becomes stiffer when a stronger magnet is used. Also, a special configuration of a magnet is proposed to increase the second derivative of the magnetic field and therefore to increase the stiffness with a given magnetization (e.g. see [36]). These conclusions indicate that 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the stiffness of an HTS bearing does not depend on tem perature unless the magnetic properties of the levitating magnet depends on temperature. 2.2.3 D am ping property When a levitating magnet is forced to displace either in the radial or vertical directions from its equilibrium position instantaneously, the magnet oscillates. The oscillation decays as a function of time due to a damping property of the HTS bearing. The mechanism of the damping is the same as the energy loss in the COF, the hysteresis loss and the eddy current loss. The advantage of the damping property of the HTS bearing is th at the damping becomes stronger when the displacement of the rotor magnet from its equilibrium position becomes bigger. This is because the hysteresis loss and the eddy current loss are both as a function of magnetic field variation, (A B )3 and (AH)2, respectively. The damping property tends to be dominated by the hysteresis loss because the rotor magnet and the HTS tiles are in close proximity. When the operational tem perature changes from LN2 to LHe temperature, the primary concern is the reduction of the damping property due to the hysteresis loss in the superconductor. When we consider the COF due to the hysteresis loss, the COF decreases as the tem perature of the HTS decreases. On the other hand, the damping decreases as the tem perature of HTS decreases. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3 H ardw are o f p r o to ty p e SM B The magnet and the HWP are the rotor of a magnetic bearing th at is levitated above a ring of YBCO HTS materials. The sintered NdFeB magnet has an inside radius of 2.54 cm, an outside radius of 3.56 cm, thickness of 1.2 cm, and mass of 0.2 kg. It is magnetized in the axial direction and has a remnance of ~ 11 x 103 gauss and an energy product of 30 x 106 gauss-oersted. The moments of inertia of the HWP and magnet are 83 and 1910 gr-cm2, respectively. A HWP holder, made of Delrin and with a gear at its outer circumference, holds the m agnet/H W P combination together and is part of the rotor, see Figure 2.1. The magnet is held at an appropriate distance above a ring of HTS tiles which consists of 12 pieces of melt-textured YBCO [35]. The distance is a free parameter and is typically between 4 to 10 mm. Two clamps, each resembling a plier, hold the rotor in place during the cool-down of the system. A vacuum rotary feed-through th at is mounted outside of the cryostat rotates a shaft and a pair of cams is used to open and close the clamps. The HTS tiles and the clamps are mounted on a G-10 board. Rotation of the rotor is achieved by means of a half-gear th at is driven by a second vacuum feed-through. During cool-down the half-gear and the gear at the outer circumference of the rotor are engaged. Once the system has cooled to 4 K, the clamps are opened and the half-gear is turned. If the need arises to re-rotate the rotor, the half-gear can be slowly engaged with the rotor, and the process repeats. We tested all the mechanical components of the HTS bearing at liquid nitrogen, liquid helium and intermediate temperatures. Measurements between 4.2 and 77 K were conducted in a liquid He cryostat (LHC) in which the stator was mounted on a 0.9 cm thick copper cold-plate. The tem perature of one of the HTSs is monitored by calibrated silicon diode and carbon resistor thermometers attached to a point on its perimeter at mid-height. The tem perature of the HTS was controlled with heating resistors. The rotation rate was measured by an LED and a photodiode together with a reflective tape attached on the rotor. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.1: Magnetic bearing apparatus during testing in a liquid helium cryostat. The size of the prototype SMB system is designed to implement in the M AXIPOL cryostat [4], 2.4 C oefficient o f frictio n 2.4.1 Tests, results, and interpretation We carried out spin-down measurements between 3.5 and 0.3 Hz to determine the co efficient of friction of the bearing at various temperatures, levitation heights, ambient pressures, and as a function of the magnetic field structure of the magnet. Another set of spin-down measurements at 77 K were conducted in a liquid nitrogen cryostat (LNC) with the apparatus described by Hull et al. [35]. This apparatus was designed to minimize energy losses due to eddy currents. The mea surements with the LNC are summarized in Table 2.1. In all measurements the rotor consisted of the magnet without the HWP. We quantify the results of the spin-down measurements in terms of the coefficient of friction (COF) [35] given by COF = - C cl 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.14) CO to to 106 D CD CO CD < 0 6 4 2 8 10 D istance [mm] F ig u r e 2.2: Inhomogeneity of the m agnetic field o f the magnet as a function of distance from its surface for the bare magnet (stars) and for a configuration where the magnet is shimmed with high permeability steel (square). The inhomogeneity is quantified using Equations 2.17 and 2.18. The insert zooms on the difference of 1.65 x 104 gauss between the shimmed and nonshimmed configurations. The straight line is an approximate fit to the nonshimmed data and was used to find interpolated values. where C = M g R f) as shown in Equation 2.6. For our rotor magnet, C = 2.7 x 10 3 sec2 and a is the angular acceleration. The angular acceleration a can be a function of the frequency of rotation / and tem perature T, and we parameterize it as [33] a(T) = 2?r f t = - a „ ( T ) - 2 jtoi ( T ) / . (2.15) The coefficient do has been interpreted as the contribution of hysteresis to the angu lar acceleration and ai quantifies the contributions from eddy currents and ambient pressure [35]. If a\ is non-zero the solution of Equation 2.15 becomes 2 tt/ = ai + ( — + 2 7 r/0)e a i t , ai where /o is initial rotational frequency. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 2 .1 6 ) 4 CD o o 3 05 C zs u. 2 O O 10000 20000 30000 40000 AB3 [G auss3] Figure 2.3: COF as a function of magnetic field inhomogeneity [from Equation 2.17 ]. The data suggests a linear relation as predicted by theory. COF Versus D ista n ce at 77 K: H ysteresis Losses A comparison of lines 1, 4, 5 in Table 2.1 gives a comparison of the COF as a function of distance from the HTS with low ambient pressures, and negligible contribution from eddy currents. The COF decreases as the distance between the rotor and stator increases. This decrease is correlated well with a decrease in the inhomogeneity of the axial component of the magnetic field as the distance from the magnet increases [37]. We measured the strength of the magnetic field as a function of distance from the surface of the magnet. Measurements were taken at the inner, middle and outer radii of the magnet and every 10 degrees in azimuth. In these measurements the major contribution to the COF is expected to be hysteresis loss in the HTS. We use Equation 2.1 to estim ate the contribution of magnetic field inhomogeneity. We define ( A B ) 3 as (2.17) 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. measurement 1 2 3 4 5 6 7 8 9 distance (±0.2 mm) 8.6 8.6 8.6 9.5 10.7 10.7 10.7 10.7 10.5 Pressure (torr) 8 x 10~7 3 x 10“2 3 x 10-1 7 x 10~7 8 x 10-7 3 x 10"2 8 x 10~2 3 x 10” 1 8 x 10~7 COF do (rad/sec2) 1.4 x 10“3 3.7 x 10“° 1.4 x 10~3 1.4 x 10“3 1.8 x 10~6 6.6 x 10~4 1.3 x 10~6 4.9 x 10~4 4.9 x 10~4 4.9 x 10“ 4 4.9 x 10-4 6.5 x 10“ 7 2.4 x 10“ 4 ax (1/sec) 0 1.2 x 10~4 1.6 x 10~4 0 0 1.2 x 10“4 1.4 x 10“4 1.8 x 10~3 0 Table 2.1: Summary of measurements with the LNC. In measurement 9 the magnetic field of the magnet has been altered to reduce inhomogeneity by shimming with high permeability shims (see te x t). where runs over the inner, middle and outer radii and 5Bz(9,ri) = \BZjmax(8,ri,d) - B Ztmin(Q,rh d)\. (2.18) The variables cuj are weights th at are proportional to the ratios of elements of area as a function of radius. Figure 2.2 shows the quantity ( A B ) 3 as a function of d. The reduction in magnetic field inhomogeneity with distance is evident. By interpolating the data of A B 3 vs. distance (Figure 2.2) we plot in Figure 2.3 the COF as a function of ( A B ) 3. The number of data points is small, nevertheless the d ata is consistent with the linear relationship as C O F oc ( A B ) 3 that is predicted by theory in Equations 2.2 and 2.6 [38]. In order to reduce hysteresis losses and decrease the coefficient of friction we added shims in various locations on the surface of the magnet that pointed toward the HTS [38]. The shims were made of 25 —150 /i m thick high permeability steel leaf. Measurements of the magnetic field at a distance of 7 mm away from the magnet gave less magnetic field inhomogeneity, as shown in Figure 2.2. The COF was measured for a different shimming configuration (for which measurements of ( A B ) 3 are not available) and gave the lowest COF measured with the LNC, see measurement 9. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.00025 d=10.7mm d= 8.6mm 0.00020 „ 0.00015 ^ 0.00010 0.00005 0.00000 10-8 1 0 '7 10-5 1 0 '4 1 0 '3 1 0 '2 1 0 '1 10 ° Pressure [Torr] Figure 2.4: Coefficient of friction as a function of ambient pressure for two levitation distances. All data was measured at a temperature of 77 K. COF Versus P ressure Figure 2.4 shows the magnitude of the coefficient cq as a function of ambient pressure. The measurement is a compilation of data from two levitation distances. The data show a decrease of with ambient pressure and are in broad agreement with those of Weinberger et al. [39], which suggest th at oq becomes negligible at pressures below ~ 5 x 10~5 torr. COF Versus T em perature We used the LHC to measure the COF at levitation distances of 6 and 7.2 mm for various temperatures between 15 and 80 K and for frequencies between ~ 0.3 and 3.5 Hz. The magnet was released to levitate when the helium cold plate was at 50 K and we have no information about the tem perature of the magnet subsequently. Measurements of the COF as a function of rotation frequency are shown in Figures 2.5 and 2.6. At low tem peratures (~15 K) the data were consistent with a constant deceleration for most rotation frequencies with perhaps a slight increase in COF at frequencies below 1 Hz. At high temperatures, above about 60 K, the data 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.0 3 2.5 LO o 2.0 o & c3 1.5 O o 7 V V ^ 7 VV ! 1.0 'W 0.5 .......... V " 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Frquency [Hz] Figure 2.5: Coefficient of friction as a function of frequency for temperatures of 16 (stars), 60 (open triangles), 70 (squares), 79 (circles), and 84 K (filled triangles) for a levitation distance of 6.0 mm. For clarity of presentation, the COF for temperatures of 70, 79, and 84 K, have been offset vertically up by 0.4, 0.8, and 1.2 ordinate units, respectively. 5.0 ----- >----- 1----- '----- 1----- >----- 1----- ■ ----- 1----- ■----- 1----- >----- r ) ---- ,---- 1---- ,---- 1---- ,---- 1---- ,---- 1---- ,JL_j---- ,----L*—1----0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frquency [Hz] Figure 2.6: Coefficient of friction as a function of frequency for temperatures of 15 (stars), 50 (triangles), 62 (squares), and 79 K (circles), for a levitation distance of 7.2 mm. For clarity of presentation, the COF for temperatures of 62 and 79 K have been offset vertically up by 0.3 and 0.8 ordinate units, respectively. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25.0 6 and 7.6 mm _______ 9 mm 20.0 CD o o 15.0 w u_ 10.0 O O 5.0 0.0 10 20 30 40 50 60 70 80 90 Temperature [K] Figure 2.7: Coefficient of friction is plotted as a function of temperature for a levitation distance of 6, 7.2, and 9 mm. show an increasing COF with frequency. The data at a levitation distance of 6 mm and 60 K show a transition between a constant COF above 1 Hz to a decreasing COF below 1 Hz. Figure 2.7 shows the COF as a function of tem perature at a frequency of 1 Hz. A decrease in the COF by about a factor of ~ 3 is evident for both levitation heights. Hull et al. [38] argue th at the COF should be inversely proportional to the critical current Jc in the HTS. Using data from Zeisberger et al.1 [33] which give information about the critical current as a function of tem perature we plot the COF as a function of 1/ Jc in Figure 2.8. Since we do not have data about the critical current as a function of tem perature for our HTS samples, it is premature to conclude th at there is a disagreement between the data and the model. 1The data of Zeisberger et al. [31] is for 0, 1, and 2 Tesla and shows a weak dependence on magnetic field strength. In our case the m agnetic field is ~ 0.1 T and we use the 0 T data. 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 2.0 o o CO c3 LL o o 0.5 0.0 0 5 10 15 20 25 30 35 40 45 Jc' 1 [1/p.A-cm'2] Figure 2.8: Coefficient of friction at 1 Hz as a function of 1/JC, where Jc is the critical current in the HTS. 2.4.2 D iscussion and sum m ary We have presented a working prototype for an HTS bearing for use with polarimeters th at employ a HWP. The complete absence of stick-slip friction makes the magnetic bearing suitable for detector systems th at are sensitive to microphonic excitation. Our measurements indicate th at with a readily available shimmed magnet and HTS, at the range of gas pressures expected in a liquid He cryostat and with no particular efforts to reduce eddy current losses we can expect a COF of ~ 3 x 10-6 (factor 2 less than the measured COF at 15 K due to shimming), which would give rise to an angular deceleration of about 1.7 x 10~4 Hz/sec. If the rotor is set in motion and then slows down over a frequency range of 20 Hz, suitable for CMB observations between 21 and 1 Hz given the time constant of current bolometeric detectors, the bearing would need to be reset in motion once in 33 hours. Since re-setting the bearing in motion should take only a few minutes, the duty cycle of such a magnetic bearing system is close to 100%. The present design could be implemented in ground based or short duration balloon polarimeters. A COF of ~ 3 x 10-6 is still two orders of magnitude larger than the smallest 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COF reported for the 300 - 400 gr mass category [23], indicating that a more careful design of the bearing and the cryostat in which it operates can provide uninterrupted rotation for more than 4 months. Even just a factor of 10 improvement in the COF would give a continuous rotation for a month, more than adequate for a long duration balloon mission. W ith a slowly decelerating rotor the frequency where the polarization signal appears will also decrease. This could provide a powerful check on systematic errors, but could also complicate the analysis of the data. Rather than let the bearing slow down over a long period of time, it could be driven continuously with an induction or standard brushless motor [40]. If necessary a feed-back loop can be implemented to maintain constant speed. Because of the small COF, minimal power is required to drive such a motor and this approach appears most attractive for a future satellite polarimetry mission. Despite of the success of our prototype more testing needs to be done to characterize the bearing performance. In particular it is essential to measure bear ing oscillations, particularly at low tem peratures where hysteretic damping is lower compared to 77 K, and the stability of the bearing parameters, e.g., stiffness, and levitation distance, should be tested over a long period of time. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 Stiffness and d a m p in g coefficien t 2.5.1 Introduction The HTS bearing is a levitating system, and therefore the rotor PM has an unavoid able vibration. Any unwanted vibration becomes a source of systematic errors, and therefore it is im portant to understand the vibration properties of the HTS bearing. The vibrational properties of the levitating PM can be quantified by the ampli tude of the vibration, stiffness, and damping coefficient. If we quantify the stiffness as a spring constant, a corresponding natural frequency can be calculated with a given mass. The amplitude of oscillation and coefficient of friction of the PM rotor increase because the natural frequency and the rotation frequency resonate. Therefore, it is im portant to design the HTS bearing such th at the rotation frequency and the natural frequency do not coincide. Of particular concern is whether the damping of the bearing, which should be inversely proportional to Jc, will still be sufficient at the lower operating temperature. No measurement has conducted to characterize the damping properties of the HTS bearing at LHe temperature. The critical current of YBCO bulk is expected to decrease by a factor of 20 —30 from 77 K to 4 K [31]. In this section, we present measurements of the amplitude of the vibration, stiffness, and damping coefficient of the HTS bearings while it is levitating without rotation at tem perature range 4 —77 K. 2.5.2 E xperim ental setup The vibration of the rotor was measured with a reflected laser beam that was simul taneously analyzed by an interferometer and Doppler velocimeter. The outputs of these devices are input to a spectrum analyzer. Figure 2.9 shows the mounting for the mirrors th at allows measurement of either the vertical or horizontal component of vibration. The window of the cryostat is at a different height than the rotor and therefore minimizes heating of the bearing. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V ib ro m e te r M irror m o u n t M irror 1 C ry o s ta t sh ell m agnet HTS // / W / / // C old p la te , 4K V ib ro m e te r M irror 1 C ry o s ta t sh ell M irror m o u n t m agnet M irror 2 C old plate, 4K F ig u r e 2.9: Schematic diagrams show experim ental setups to measure the vertical (top) and horizontal (bottom ) vibration, respectively. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10. 000 — vertical - . 000 direction horizontal direction Q_ E < 0. 001 0 10 30 20 40 50 Frequency [Hz] 10.000 — c r y o s t a : s hel l — wal l .000 Cc 0.00 0 10 30 20 40 50 Frequency [Hz] Figure 2.10: Vibration spectra for nonrotating levitated rotor in vertical and horizontal directions (top panel, heavy and light curves, respectively) compared with spectra of laboratory wall and cryostat shell (bottom panel). 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a 23.4 40 50 60 Tem perature [K] F ig u r e 2 .1 1 : Resonant frequency in vertical direction versus temperature. 16.0 16.0 15.9 oc 0 15.9 0cr 0 15.8 -H- -H- *+— 0 oc 0 c ow 0 15.8 + + -H + +t- 15.7 DC 15.7 15.6 10 20 30 40 50 60 70 80 90 Tem perature [K] Figure 2 .1 2 : Resonant frequency in radial direction versus temperature. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.40 16Hz 21 Hz 23Hz 0.35 0.30 1 °-25 CD "g H — > 0.20 “q. E 0.15 < • • 0.10 + +++ 0.05 0.00 10 20 30 40 50 60 Temperature [K] Figure 2.13: Vibrational amplitude of the resonant frequencies versus tem perature for a nonspin ning rotor. 2.5.3 R esonant frequency Figure 2.10 shows the vibrational spectrum of the levitated but non-spinning rotor in the vertical and horizontal directions, together with vibrational spectra of the room wall and cryostat shell. Comparing data from the rotor with th at of the cryostat shell, the spectra show rotor resonances at 16 and 23 Hz in the vertical direction and possibly an additional resonance at 21 Hz in the radial direction. We hypothesize th at the 21 Hz vibration may be the tilt mode. Comparing the amplitudes of the spectra, the 16 Hz resonance appears as the radial mode, and the 23 Hz resonance appears as the vertical mode. As expected in Equation 2.11, the ratio of the frequencies is approximately the square root of two, which is consistent with the stiffness in the vertical direction being twice th at in the radial direction [41], We repeated these measurements over the tem perature range of 15 —60 K at intervals of 5 — 10 K. We observed no major differences in the overall shape of the spectra as a function of temperature. We define the resonant frequency as the frequency at which the maximum amplitude occurs. We then determine the frequency as a function of temperature. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.11 shows the radial resonant frequency vs. tem perature, and Figure 2.12 shows the vertical resonant frequency vs. temperature. For these data, the resolu tion of the spectrum analyzer is 0.05 Hz. To within 0.5 Hz, there is no dependence on tem perature for either resonant frequency. This is consistent with the stiffness of the bearing system responding diamagnetically to changes from the equilibrium position. However, the data may also suggest th at the frequency increases slightly as the tem perature decreases. If this suggestion is confirmed, it would be consistent with the increase in Jc as tem perature decreases, which has the tendency to decrease departures from a pure diamagnetic response. In the flux frozen model [24], a dia magnetic image and a frozen image appear when a magnet is field-cooled in a HTS. The magnetization of the diamagnetic image is M ~ H when the HTS becomes a superconducting state. When the tem perature of the HTS decreases, the critical current in the HTS increases. Correspondingly, the magnetization approaches to a perfect diamagnetic image as M —> H. Equation 2.7 indicates higher stiffness with higher magnetization. 2.5.4 V ibrational am plitudes and dam ping Figure 2.13 shows the amplitude of vibration as a function of tem perature for the three different resonant frequencies while the rotor is not spinning. Because the vibrations induced by the room are sub-dominant (see Figure 2.10), these results indicate that the vibrational amplitude is independent of temperature. We performed another series of tests in which the vibration amplitude was monitored while a shaker vibrated the cryostat for a fixed period of time and then was turned off. Figure 2.14 shows one set of results from this series. During the time that the shaker was turned off, the amplitude decayed. We fit an exponential envelope to these amplitudes to derive a time constant for the vibrational decay. The time constant versus tem perature is shown in Figure 2.15 for vertical motion. The time constant versus tem perature is shown in Figure 2.16 for horizontal motion. The data for the vertical motion are consistent with a decay time constant of 0.15 ± 0.05 s-1 th at is constant with tem perature. For the horizontal motion, the data suggest a decrease in the time constant as tem perature decreases. The reduction of the damping 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 300 2 00 E CD O E - ,0° < -200 -3 0 0 0 50 100 150 200 150 2 00 Time [ s e c ] 300 200 E a. J cO Z) 00 ^ -100 < -200 0 100 50 T im e [ s e c ] F ig u r e 2 .1 4 : Vertical vibration amplitude versus tim e with period of shaking the cryostat for 10 sec. At a tem perature of 16 K (top panel) and 64 K (bottom panel). Shaking occurs at tim es of 10 - 20, 70 - 80 and 130 - 140 s. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.30 o 0) C/5 ^ 025 0.20 a ■4—' | 0.15 « 0.10 (/) O +t- E h0.05 0.00 10 20 30 40 50 60 70 80 90 Tem perature [K] Figure 2.15: Time constant of vertical vibration damping versus temperature. 1.4 1 1 1 1 1 1 1.2 cT $ + 1 Bc 0.8 C 0.6 CO 00 n e 1 b- ...........1... + ■f + + - + -f- + 0 .. + + -f 0.2 0 10 _l_ -f- + + + - + -1 _ + + + f + + + 1 1 20 30 40 1 1 I 1 50 60 70 80 90 Temperature [K] Figure 2.16: Time constant of horizontal vibration damping versus temperature. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. coefficient is about a factor of 3 ~ 4. This is consistent with the measured reduction of COF as a function of temperature. The vertical time constant is significantly smaller than the horizontal time constant, which is consistent with a larger magnetic field gradient in the vertical direction. 2.5.5 C onclusions We have compared bearing behaviour in the tem perature range from 15 to 83 K. The resonant frequencies and damping are mostly independent of temperature. The COF decreases with temperature. This suggests th at many aspects of a bearing system can be developed at the relatively convenient tem perature of 77 K and are expected to perform the same or better at 20 K. 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.6 R o ta tio n a l freq u en cy v a riation w ith in on e p e riod o f ro ta tio n freq u en cy We give experimental evidence for the connection between magnetic field inhomo geneity of a permanent magnet and torque on the rotor in a high tem perature super conducting bearing. Spin-down measurements below 14 Hz are used to demonstrate a high degree of correlation between variations in the angular speed of the rotor within a single period of rotation with the measured spatial structure of the magnetic field of the rotor. At frequencies below ~1 Hz the fractional speed variation within a single period of rotation is inversely proportional to the square of the mean frequency of ro tation. We propose th at a dipole-dipole interaction gives rise to the torques th at lead to speed variations and we show th at this interaction explains the observed functional dependence on frequency. At frequencies above ~1 Hz the measured magnitude is about 1 % of the mean frequency of rotation, consistent with the noise level in the experiment. The results imply th at arcminute accuracy angular encoding of the rotor can be achieved with a single measurement of angle in each period. 2.6.1 Introduction We report on a new type of measurement which probes the relation between mag netic field inhomogeneity and torques in an SMB in a direct manner. Magnetic field inhomogeneity is not only a source of friction which decelerates the rotor, it also gives rise to torques which cause rotational speed variations. We measure these variations in the rotational speed of the rotor during a single period of rotation. We then cor relate the rotational location of speed variations with the measured structure of the magnetic field of the rotor. In various applications th at use SMBs it is im portant to record the angular position of the rotor [42]. For example, we have recently proposed to use an SMB in astrophysical polarimetry [4] where the angular position of the rotor is an essential element in the analysis of the data. In this and similar applications speed variations of the rotor may determine the accuracy of angular position extraction. Conversely, angular encoding can be simplified substantially if all angles can be extrapolated 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dipole m 1 i i Frozen im age (0,h) r D ipole m 2 Initial speed v0 0 1 (P A ) p Figure 2 .1 7 : A schematic of the dipole-dipole interaction that is used to m odel the torques that give rise to rotational speed variation. accurately using the mean angular speed and a single measurement of position during every cycle of rotation. Our measurements provide information about the level of rotational speed variations in an SMB system. This information can be used to set limits on the accuracy of determining the angular position of the rotor. 2.6.2 D ipole-dipole interaction m odel We model the interaction between the magnet (the rotor) and the magnetic field imprinted in the HTS as a dipole-dipole interaction as shown in Figure 2.17. We analyze the one dimensional motion of dipole fh\ under the potential created by dipole 777.2 along the p coordinate in a (p, z) plane. Dipole m 2 starts at p = —00 with initial velocity v0 and moves to +p. Dipole fhi represents the frozen flux in the HTS and dipole m,2 represents the azimuthal inhomogeneity of the axial component of the magnetic field of the rotor magnet. We assume th at the magnitudes of rh\ and 777.2 are equal to the magnetization th a t creates the magnetic field A B z, which is defined as A B z (z = h) = B z max - B z min, (2.19) where B z max (Bz min) is the maximum (minimum) of the axial component of the magnetic field along the circumference of the rotor at its levitation distance from the 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. HTS. The potential energy of m 2 is U = -m 2 • (2.20) where B(fhi) is the magnetic field due to rhi B{rhi) — y^^r[3(m i • f ) f - m j 47r r A = + h2. r (2-21) (2.22) Energy conservation at any coordinate p gives Ei(p = oo) 1 , , = E + U 9 1 -M r0 2 0 , , (2.23) U n m i7 Y i2 9 2 fi2 — p2 = -M u 2 , 1 ------ =-*2 4tr r5 ri p 0m l m 2 2h2 - p2 1 i v = u0 H -------- ?------ d 2L 2ttM r5 ugJ 2.24 v ' (2.25) v ' We now convert from translational motion to rotational motion and find the maximum fractional speed variation S. The maximum occurs when two dipoles are aligned with each other at p = 0. Then Equation 2.25 can be written as /(P = 0) = /o (l + ~h )K (2.26) Jo where „ _ ftr» im 2 2 1 2trM h3 {2nl)2' 1 J The distance / is the mid point of the inner and outer radii of the rotor magnet. The maximum fractional speed variation 5 is defined as i = / ( P = 0 ) - /0 JO + 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2 2 8 ) (2-29) Eddy current motor - ~..s Optical encoder Copper disk optical chopper Photodiode Permanent magnet I (NdFeB) High temperature superconductor (YBCO)^. ■ ■ ■ Top view of Cu disk and coils metallic cold plate F ig u r e 2 .1 8 : Cross-sectional view of the hardware. On the bottom right is an inset with a top view of the copper disk. In the limit, a / f ^ <C 1 which implies & = £;■ ZJ 0 (2-30) Therefore, qualitatively speaking, the fractional speed variation is expected to decay as oc l / / 2. This model does not consider any energy loss in the system. 2.6.3 E xperim ental Setup A sketch of the experimental setup is shown in Figure 2.18. A copper disk is mounted to the magnet holder with 6 aluminum rods. The copper disk has 60 slots th at are spaced by 6 degrees apart in azimuth along an equal radius. The slots are machined along radial lines and 59 of them have each a width of 2.9 degrees. One of the slots has a smaller opening angle and it provides the zero angle reference. The slots serve as an optical chopper of an optical encoder. The encoder consists of an LED and a photodiode th at are positioned above and below the disk at the radial location of the slots. The rotor is spun up by the stator of an eddy current motor, which consists of three C-shaped molybdenum permalloy cores th at are wound with 1500 turns of NbTi superconducting wire. Each coil is driven at a 120 degrees phase-shifted AC current of frequency 50 - 100 Hz. A detailed description of this induction motor is in 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.6 0.5 0.2 C 0 , I L U _ L _ _J 5 L_ i -t I 10 I I 1 i 1 15 I I I I I 20 Time [ s e c] Figure 2.19: The rotation frequency as a function of time. The beginning of each rotation of the rotor is marked with a pair of high and low frequencies; see text. Repeatable sinusoidal variations of the rotational speed are evident within each period of rotation. Section 2.7. 2.6.4 E xperim ents We conduct spin-down experiments from a frequency of 14 Hz down to 0.02 Hz. The spin-down measurements were carried out in (i) atmospheric pressure and with the HTS at 77 K, (ii) a pressure of ~ l x l 0 -6 torr and with the HTS at 77 K, and (iii) in a pressure of ~ l x l 0 ~ 6 torr and with the HTS at 4 K. We sample the voltage of the photodiode using an analog to digital converter with a sampling rate th at is a factor of 600 faster than the rotational frequency of the rotor. In addition to the spin-down measurements we also use a LakeShore Hall probe to measure the axial component of the magnetic field along the circumference of the magnet at the distance of 6.2 mm and at the radius where the axial component of the magnetic field is maximum. This is where the center of the array of YBCO tiles 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is located during the spin-down experiments. D ata was taken every 5 degrees and at room temperature. The zero angle for the rotational encoder and for the magnetic field measurement is the same. The bottom panel of Figure 2.20 shows the results of this measurement. 2.6.5 R esu lts and D ata A nalysis The raw d ata from the spin-down measurements consists of square waves th at are the readout of the optical encoder. The 60 periods of square waves represent a single rotation of the rotor. We subtract the mean voltage of the raw data, and determine the time interval between each of the 60 zero crossings. We then calculate the frequencies of rotation for each of the square waves from the known angular spacing of the slots and the zero crossings time intervals. Figure 2.19 shows the rotation frequency for '-'-'20 sec of data. The overall constant deceleration is accompanied by a periodic oscillation of the rotational speed. Once every 58 data points high and low rotational speed points correspond to the single narrow slot on the disk and mark the angular zero position. Thus, there are 6.5 revolutions in this particular data set. We work with d ata sets th at are short enough such th at the deceleration is constant and for the rest of this paper we focus on the variation of the rotational speed by removing an offset and a gradient from the frequency vs. time data. We bin the variation in rotational speed in angle. The binning assumes that the rotation frequency is constant within a 6 degree slot grid spacing. The top panel of Figure 2.20 shows a sample of the angle-binned variation in rotational speed during 6 consecutive periods of relatively slow rotation, about 0.4 Hz. Figure 2.21 shows the same data but for higher frequencies, about 5 Hz and 10 Hz. 2.6.6 D iscussion Speed V ariation and M agn etic Field In h om ogen eity The inhomogeneity of the axial component of the magnetic field along the azimuthal direction shows four peaks separated by ~90 degrees; see the bottom panel of Fig ure 2.20. Visual comparison of this data with the upper panel of Figure 2.20, which 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.02 F - 0 . 0 2 iE_ 0 ..................*.... ................................................ ......................... ..... ..... i 100 ....................... *.... *........... 200 3 300 400 300 400 Angle [ de gr e e ] 1640 e> 630 CD 620 610 a; c 1600 cn 5o 590 0 100 20 0 Angle [ de gr e e ] Figure 2.20: Top: the rotational frequency, after removal of an offset and a gradient, as a function of rotational angle. The speed variations are synchronous with rotational position. There is a strong correlation between m axim a and minima in rotational speed and m axim a and minima in the spatial m agnetic field. Bottom: the axial component of the m agnetic field as a function of azimuthal angle. The measurement is made at a radius where the axial component of the m agnetic field is a maximum. shows the variation in rotational speed as a function of angle at low frequencies, gives convincing evidence for the connection between magnetic field inhomogeneity and the torque th at gives rise to rotational speed variations. The variation in rotational speed of the rotor are associated with areas of weaker and stronger magnetic field. There are four periods for the variation in rotational speed and there are also 4 periods for the inhomogeneity of the magnetic field. These periods have the same angular phase. To check whether the prediction from the model is borne out by the experi ments we define the fractional change in rotational speed as £ ^ fp p fm a x To /'o fm in To * 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 01 \ 0.06 0.04 0.02 N X 0.00 0.02 - - 0.04 0.06 0.06 0.04 0.02 0.00 - 0.02 0.04 0.06 0 100 200 300 400 Angie [D e gree ] Figure 2.21: Same as the top panel of Fig. and 10 Hz (bottom ). 2.20 but for rotation frequencies of about 5 Hz (top) where f max and f rnrn are the maximum and minimum offset and gradient removed rotation frequencies during one period. The initial rotational frequency / 0 is defined as the rotation frequency at the angle at which the offset- and the gradient-removed rotational frequency is minimum in each cycle. Figure 2.22 shows 5 as a function of / 0 for the three environments in which the experiments were carried out. Below ~1 Hz the data shows the predicted l / / o 2 dependence. At frequencies above Hz 5 is close to constant at a level of ~1 %. Furthermore, for this frequency range the variations in rotational speed do not show four clear peaks with angle, but rather random or rotational synchronous sinusoidal structure as shown in Figure 2.21. We interpret the 1 % residual variation as due to radial vibrations of the copper disk. Measurements of radial vibrations in this system give an amplitude of ~0.25 mm, which would explains a 1 % level of noise in the readout of the optical encoder given the geometry of the disk and slots. The data in the bottom panel of Figure 2.22 show a slight increase in the 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fractional variation of the rotation frequency as the rotation frequency increases. This increase is due to the 16 Hz resonance frequency for vibration in the radial direction of this system [27]. We use Equation 2.29 to fit the data in Figure 2.22. We can apply Equa tion 2.27 to our geometry. We set the magnetization m i = m 2 = m and relate the strength of the magnetization m as 27th3 m = 27T/T E^ ( A 5 z j ) 2 ) (2.32) /F) \ i=1 where A B zi = B zmax — B zmin. In our geometry, A B zi = 40,37,16, and 28 Gauss (see bottom panel of Figure 2.20) and h = 5.5 mm. These values give a = 6 x 10 3 Hz2. This implies th at Equation 2.30 will be valid at frequency above 0.2 Hz. A ccuracy o f E n cod in g A ngular P osition The measurements of speed variation within a single period of rotation can be used to set an upper limit on the accuracy with which angular position can be determined in cases where only a single angular position measurement is taken every period, for example when only the zero position is measured. W ith our experiment the variation in angular speed appears to have a sinu soidal dependence PP j. — sin ojnt, • (2.33) where u n = 27m/0, (2.34) and n > 1 is an integer. It is straight forward to show th at in such a case the maximum uncertainty in the determination of the angular position of the rotor is (2.35) For 8 = 1% and n = 4 this gives A# = 4.3 arcminutes. This angle is the uncertainty of the angular position if we encode the angular position of the rotating magnet 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10.000 .000 0.010 0.001 10.000 1 .000 0.100 0.001 0.000 .0 0 0 0.100 0.010 0.001 0. 01 0.10 1 .0 0 10. 00 Frequency [Hz] Figure 2.22: Fractional speed variation as a function of frequency for environments (i), (ii) and (iii), top to bottom , respectively. The dashed lines are best fits to a constant and to Equation 2.29 and the solid line is a best fit to the sum of a constant and Equation 2.29. The fit constants (in %) and the values of a (in Hz2) from the solid lines are (1.1, 1.6 x 10- 2 ), (1.0, 1.1 x 10- 2 ), and (1.4, 1.1 x 10~2) for the top, middle, and bottom , respectively. only once in a cycle and then interpolate for all other angular positions. W hether the interpolation should be linear or quadratic depends on the magnitude of the deceleration of the rotor. If the variation in angular speed is random, such as appears for some of the data above 1 Hz, the worst error in angle estimation would correspond to the case where n — 1 (see Equation 2.35), which gives A 9 = 17 arcminutes. 2.6.T C onclusions We showed a high degree of spatial correlation between the structure of the magnetic field of the rotor and speed variations during a single period of rotation. We gave an analytical model demonstrating th at a dipole-dipole interaction, such as would be expected in this system, explains the dependence of the magnitude of speed variations 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. on frequency. Our experimental set-up gives a noise floor to the measurement of fractional speed variation of about 1 %. This noise floor dominates the measurements for frequencies above 1 Hz. Because the rotor rotates relatively smoothly and with fractional speed varia tions th at are only ~1 % it is possible to encode angular position with a single angle measurement in a period and maintain angular accuracy of less than 20 arcminutes. We showed th at the largest errors th at would be incurred are inversely proportional to the number of periods in the structure of the magnetic field inhomogeneity. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7 E lectro m a g n etic drive m ech an ism : In d u ctio n m otor In this application the polarimeter operates at the LHe tem perature and the HWP rotates with a rotation frequency of about 10 Hz. The typical duration of observa tions could be several hours for a ground-based observatory and years on a satellite platform. There is a need for a low heat dissipation and low noise motor th at will keep the HWP at a stable rotation speed, particularly for the longer observation times. 2.7.1 Introduction We have constructed an induction motor to drive the rotor of the SMB. The torque of this motor relies on the Lorentz force between the eddy current induced in a copper disk and an AC magnetic field th at is applied using coils. In this paper we report on the design of the motor and on measurements that characterize its operation. 2.7.2 Induction m otor hardware A sketch of the motor and the experimental setup used to test it is shown in Fig ure 2.18. The motor and the SMB are mounted on the cold plate of a liquid helium cryostat. The motor consists of three stator coils and a copper disk that is mounted to the rotor of the SMB. Each coil is made of a C-shaped molybdenum permalloy powder core th at is wound with 1500 turns of copper clad NbTi superconducting wire with diameter of 0.005 inches. The coils are driven with 120 degrees phase-shifted AC current th at is supplied by a commercially available linear amplifier [43] placed outside of the cryostat. They are located at a radius of 58 mm with respect to the center of the rotor and are tightly packed in azimuth, which results in a circumferential separation of about 20 mm. We found th at this configuration maximizes the torque to the rotor. We measure the rotation frequency using an optical encoder. The copper disk has 60 slots along its circumference and serves as an optical chopper for a cryogenic LED and a photodiode th at are positioned above and below the disk at the radial location of the slots. More details about this encoder are given by Matsumura 51 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. et al. [28]. The encoder output is sampled densely in time and analysis of this data gives both position and rotation speed of the rotor as a function of time. A thin film resistance tem perature sensor is mounted in proximity to one of the coils to measure its temperature. 2.7.3 Induction m otor m odel We use a formalism described by Richards et al. to model the induction motor [32]. Richards et al. analyzed the torque due to eddy currents th at is applied to an infinitely large conductive plane with the thickness d, when a sinusoidal magnetic field wave travels with a relative group velocity v with respect to the conductive plane. We adapt their formalism to describe our motor. The force induced by the sinusoidal magnetic field wave is F = T y 1 + (v/v0) (2.36) <cgs) (2 37) where = v° = 2h i (cgs)- ( 2 -3 8 ) The variable a is electrical conductivity of the conductive plane, R e/ / is magnetic field in the conductive plane, and c is speed of light. In the limit v <C vq, the induced force is proportional to v. If we apply the formalism in Richards et al. to our induction motor system, the equation of motion of the rotor can be written as (2.39) where r is the torque on the rotor, I m is its moment of inertia, and /o = uo/27rr, where r is the radial location on the rotor where the motor applies torque. In our analysis, we choose r for the outer radius of the conductive rotor-disk. The first term on the right hand side is called the torque due to ’slip’, th at is the difference between the frequency of the traveling wave /o and the instantaneous rotation frequency / . 52 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. There is no torque when the two friction terms on the right hand side balance the driving term coming from slip. The coefficient A can be expressed as A = A ( a , d ) I c2 , (2.40) where I c is the RMS of the current applied to the coils, a is the electrical conduc tivity of the conductive plane, and d is its skin depth. The friction terms describe torques th at originate from the SMB system and arise from eddy current losses, which are proportional to B and hysteresis losses, which are quantified by C. When using this model a correspondence needs to be made between the frequency of the alter nating current in the coils of the motor and the frequency of the traveling wave of the magnetic field. For our physical geometry and assuming coils that are equally spaced circumferentially around the entire disk this correspondence gives th at when the frequency of the alternating current is F the equivalent traveling wave frequency is /o = 0.16F. Because there are only three coils around a small part of the circum ference of the disk we expect this correspondence to be only approximate. A solution of Equation 2.39 gives m = (/.- - + ff (2.41) where /; is the initial frequency, f f is the final frequency of rotation, a A+ B Im ' Jl fpA-g A + B hAIl-g AH + B ' 1 ' ’ According to Equations 2.41 and 2.42 a steady state frequency of rotation / / is attained exponentially. The exponential time constant is determined by the RMS of the current, by the skin depth, by the electrical conductivity, and by friction. For sufficiently large currents in the coils Equation 2.42 gives f f = / 0, which for our geometry is f f = 0.16F 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 450 500 550 600 650 Time [ s e c ] F ig u r e 2 .2 3 : The rotation frequency as a function of time. The RMS current changed from 32 mA to 53 mA at t = 470 sec. The rotation frequency increased from its initial state to the final state exponentially. The solid line is a fit to the data after t = 470 sec using the Equation 2.41. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0. 025 0.020 0. 005 1 0.000 0 1 _L_! 2 3 4 5 Rot at i on f r e q u e n c y [Hz] Figure 2 .2 4 : The deceleration as a function of rotation frequency during a free spin-down of the rotor. 2.7.4 M easurem ents and R esults a. V alidity o f M otor M odel We measured the temporal evolution of the frequency of rotation by applying periodic changes in the RMS of the current supplied to the coils and then letting the rotor stabilize to a new final rotation speed. The current was changed in steps between 15 mA and 77 mA RMS and the experiment was repeated for alternating current frequencies of F = 12.5, 25, and 50 Hz. D ata from one such experiment is shown in Figure 2.23. Following Equation 2.41 we fit the data with exponentials and extracted a from the exponent of the fit. The agreement between the fit and the data shown in Figure 2.40 is representative of the quality of the fit in all cases. b. Source o f Friction To measure which of hysteresis or eddy currents is the major source of friction in our motor we let the rotor spin-down freely with the coils not energized. In Figure 2.24 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2. 5 2.0 1.5 2. 0 2 0 2.010 1. 0 2.000 o 1.990 1 0. 5 cr 0 .0 4 .0 0 2 8 hour s 4.2 4 Time [ hou r s ] 2. 8 ho u r s 17 min 4.4 4.6 4.8 5 .0 - 6 .7 min 10 s e c 10- ■5 F r e q u e n c y [Hz] Figure 2.25: Top panel: stability of rotation frequency as a function of tim e. The inset shows a zoom on the data between 4 and 5 hours with an expanded scale for the vertical axis. Bottom panel: power spectral density of the data in the top panel after subtraction of a mean. we plot the one sample of the data of deceleration as a function of rotation frequency / . According to Equation 2.39 we expect —2 ? r^ = a0 + 2-KCLif (2.43) A fit to this data gives the fit parameters ao and ai as 4.6 x 10~3 rad s~2 and 3.9 x 10-3 s_1, respectively. c. Long Term S tab ility o f R o ta tio n It is im portant to characterize the stability of the rotation over an extended period of time. The top panel of Figure 2.25 shows the rotation frequency with a fixed frequency F of 25 Hz and an RMS current of 39 mA. The rotation frequency stays approximately constant for over 8 hours of measurement. The bottom panel shows the power spectral density of the rotation frequency after subtraction of a mean over 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 : I i / 1 / ....... / / • / 11 1 ^ / / / o o 15 0) "tu a5 CL E CD h- / ! 10 • ; ' 9 - ' / / / / 4 ! / / / i /’ : - oV-/ j / ✓/ ,* / s I 7.5 15 20 60 I 300 400 ' ' - "" 4 U 4 100 ■' / s' s’ s’ 5 4 y / s’ / / / / / s' * / 6 / / / 7 — / / P / ✓ / / / / / / / 8 / /* / / 200 mA mA mA mA rms rms rms rms ■ • o I ♦ 500 600 Frequency [Hz] Figure 2.26: The tem perature of the coil as a function of frequency of the applied current for RMS currents of 7.5 mA, 15 mA, 20 mA, and 60 mA. 8 hours. The RMS variation is 5.4 x 10 3 Hz for time scales between 2 seconds and 8 hours. d. V ariation o f R o ta tio n Frequency w ith in a Single P eriod Using our data we extracted the variation of the rotation frequency within a single period of rotation. A description of the technique is given by M atsumura et al. [28]. The fractional speed variation, 5, is defined as A/,PP f (2.44) where A f pp is the peak to peak rotation frequency variation from its mean / within one cycle of rotation. Over the 8 hours of data shown in Figure 2.25 we find 5 = 3.4 ± 1%. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.20 12.5 Hz drive 25 Hz drive 50 Hz drive 0.15 'o <d V ). Bc 0.10 CD C o Q. X CD 0.05 A - - ■ 0.00 10 20 30 40 50 60 70 80 current to coil [mA rms] Figure 2.27: The exponent as a function of the RMS current for frequencies of 12.5, 25, and 50 Hz. The continuous lines are quadratic fits to the data of each of the frequencies. e. H eat D issip ation in th e Coils In Figure 2.26 we plot the tem perature of coil as a function of the frequency F. We have also found th at the tem perature of the coils remains below 5 K when the RMS current to the coil is below 60 mA and the frequency is below 25 Hz. 2.7.5 D iscussion and Conclusions There appears to be agreement between our model for the motor and the data. The temporal evolution of the rotation speed follows an exponential, as expected from Equation 2.41. Figure 2.27 shows the exponent a as a function of the RMS of the AC current for the three different frequencies / 0. The data shows a quadratic dependence on the current in agreement with Equations 2.40 and 2.42. When we apply sufficiently large currents the frequency / / does approach the expected value /o as predicted by Equation 2.42. The interpretation of the data in terms of the model is subject to some uncer58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tainty because of uncertainty in the tem perature of the rotor. For a fixed frequency /o the torque of the motor depends on the electrical conductivity of the disk, which is a function of its temperature. The change in the RMS of the driving current results in a change in the RMS of eddy currents in the disk and therefore a change in heat dissipation. Because the disk is levitated and has no conductive or convective ther mal path to the environment its heating or cooling time scale is long compared to the duration of accelerations or decelerations, and therefore the tem perature of the disk may not stabilized during our measurements. In our model, we assume v <C vq. This limit is true when the rotor starts, v ~ 0. The characteristic velocity vq depends on the electrical conductivity of the rotor disk, and therefore n0 changes with tem perature of the rotor disk. As a result, the limit »<<tio may be no longer valid depending on the tem perature of the rotor disk and the relative velocity v. Figure 2.27 shows the rotor speed increases exponentially as a function of time, and therefore the rotor behaves in the limit of v <C Vo. If we assume th at the motor is operating at the limit of v = Vq, we can estimate the upper limit of the rotor temperature. When v = v0, the electrical conductivity is ° = (2tt )2d r f (2'45') At the rotation frequency 2 Hz, the electrical conductivity of copper is 4-5*lp7 (12 cm)-1 . The corresponding tem perature of OFHC copper at 2 Hz of rotation frequency with given electrical conductivity is about 30 K. Our experiments show th at the coefficient of friction in our bearing system is dominated by losses due to eddy currents and not due to hysteresis. This indicates th at the presence of the motor increases the friction due to eddy currents because the same experimental setup without the motor gave friction which was dominated by losses from hysteresis [4], It is interesting to measure the fractional speed variation 5 because it informs the decision about the angular encoding of the SMB [28]. Previous measurements by our group [28] gave an upper limit of 1 % for 8 at rotation frequency above 1 Hz when the rotor was not driven. The 1 % limit was due to noise th at originated from the translational vibration of the rotor. We find an increase of 8 to a level of 3.4 % in the 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. presence of drive torque due to the motor. If we assume that the entire contribution to 5 comes purely from an actual change in rotation frequency, and is not induced by e.g. a translational motion of the rotor, we calculate that an angular encoding of the position of the rotor only once during a full period would give an upper limit on the uncertainty in the determination of angular position of only 1 degree. Such accuracy may be adequate for many polarimeters. The tem perature rise of the coil depends on both the RMS and frequency of the applied AC current. Possible heat inputs are Joule heating in the coils, hysteresis loss in the core and in the coils, and eddy currents in the core, in the copper cladding of the coils, and in metallic elements near the coils. The data of Figure 2.26 shows a quadratic nature of the increase in tem perature with respect to frequency F, which suggests that the source of heat are eddy currents. This conclusion relies on the assumption th at the thermal conductance between the source of heat and the thermometer stays approximately constant as a function of temperature. We have identified a range of operation parameters for the motor where the tem perature of the coil remains stable and far below the critical tem perature of the superconducting NbTi wire of the coils. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 3 S ystem atic effects in half-wave plate polarim etry 3.1 In tro d u ctio n We discuss systematic effects in half-wave plate (HWP) polarimetry. The HWP polarimetry has been used in optical and infrared astrophysics observations. The goal in this section is to identify systematic effects, if any, which prevents the use of a HWP polarimeter in a CMB polarization experiment. A general description of a HWP and HWP polarimeter can be found in several textbooks [44, 45, 46] and some of the effects discussed in this chapter were previously addressed for different applications or due to their own interest [47, 48, 49]. No extensive studies have been conducted specifically for the use in a CMB polarization experiment. We assume th at the polarimeter consists of a rotating HWP with a spatially fixed wire grid polarizer in front of a bolometric detector. The B-mode signal is polarized by the fractional polarization P ~ 10~8. Also, the distinction between Emode and B-mode requires the error in the determination of polarization angle by a < ~ 0.2 degrees [50]. To achieve precise measurements of weakly polarized signal, we need to study systematic effects th at are purely induced by the polarimeter itself, and therefore we need to quantify the effects th at cause instrumental-, cross-, de polarization, and effects due to pointing and beam size. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In this section, we define the Stokes vector to discuss polarization mathemati cally. Then, we give a list of the possible systematic effects th at cause instrumental-, cross-, de-polarization, and effects to a pointing. 3.1.1 Stokes vector In contrast to Jones formalism that treats radiation as an electromagnetic wave with a 2 x 2 Jones matrix, the Stokes vector describes polarization using intensity with a 4 x 4 Mueller matrix. We choose to describe the polarized light with a Mueller matrix because the detector th at is used by MAXIPOL and EBEX is a bolometer, which is a power detector. Throughout this chapter, all the results are based on computational work. The fully polarized electromagnetic wave at a given time and wavelength can be fully described by the horizontal and vertical amplitudes, E x and E y, and the phase difference (f>between orthogonal components of the electric fields. In addition to these three parameters, we need an extra parameter to describe the degree of polarization P. As a result, the four free parameters can describe the polarized light. A Stokes vector is defined as ’ I ' {El + ED Q u V {El - El) (2E xE y cos <j>) (2E xE y sin </>} The bracket indicates th at the each component is time-averaged assuming that the radiation is in steady state at the time scale of averaging. By introducing the geometrical parameters, the orientation of polarization a and an ellipticity /?, the above equation can be rewritten in terms of intensity as I ' Q u V Ip Ip cos 2aicos 2/3 Ip sin 2a; cos 2/3 Ip sin 2/3, ’ + In 0 0 ' 0, 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Ip and I u are polarized and unpolarized intensity, respectively. The first com ponent of the Stokes vector can be normalized as one, and therefore ' / i ’ Q P cos 2a: cos 2/3 u P s in 2a cos 2/3 V P sin 2/3, (3.3) where P = V Q 2 + U2 + V '2 (3.4) (3.5) Ip + Iu The quantity P is the degree of polarization, and P = 1 is fully polarized and P = 0 is unpolarized. The angle a is an polarization angle with respect to a reference coordinate. The angle /3 is ellipticity th at is defined as /3 = arctan (Ey/ E x). (3.6) Non-zero /3 indicates th at the light is elliptically polarized. The fourth component of the Stokes vector V is often taken to be zero through out this chapter because the CMB radiation is expected to be linearly polarized. 3.1.2 List of system atic effects Systematic effects in HWP polarimetry can be distinguished into several categories. We list all the possible effects for completeness, although some of the effects are not discussed in this thesis. • instrumental-polarization: I leaks to Q, U, V — harmonic leakage to 4x / peak due to the differential reflection (Section 3.6) — oblique incident angle of radiation to a HWPs 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - azimuth asymmetry of HWP properties when the HWP is not at an aper ture stop (Section 3.8) • cross-polarization: Q leaks into U, and vice versa - differential reflection at a HWP (Section 3.6) - offset phase when an achromatic HWP is used (Section 3.4) - effect due to the incident radiation spectrum (Section 3.5) - oblique incident radiation to a HWP - error in measurements of a HWP position angle due to a finite resolution of an encoder ([51]) • de-polarization: decreasing the degree of polarization of incident light P{n - finite frequency bandwidth (Section 3.3, 3.4) - oblique incident radiation to a HWP - effect due to the incident radiation spectrum (Section 3.5) • variation of a beam shape and a focus - the two indices of refraction varies the focus in vertical and horizontal direction at focal plane - effects of a pointing due to the double refraction of a birefringent material • reflection: reduction of total intensity I - any anti-reflection coating (Section 3.7) - a gap between AR coating thin film and a HW P/lens (Section 3.7) 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T ransm ission axis k \ ) \jU oc... = 0 a ou, i Single HWP Figure 3.2 3.1: = 0 $ offset Linear polarizer ~ 0 Detector A schematic diagram to show the HW P polarimeter model. P o la rim eter m o d el We consider a polarimeter th at consists of a rotating HWP at the frequency / 0, a linear polarizer, and a power detector (e.g. a bolometer), as shown in Figure 3 .1 . The rotation angle of the wave plate is labeled as p. Information about the incident polarization is contained in the intensity th at is detected by the detector as a function of p. To a good approximation, the detected intensity is sinusoidal as a function of p with a frequency of 4 x / 0 when there is a high signal-to-noise ratio. Our primary interest in this paper is to analyze the detected intensity as a function of p, which we call IVA (intensity vs. angle), with the purpose of reconstructing the incident polarization. We use Mueller matrices to describe the output signal in the approximation of normal incidence on the HWP. Consider an input Stokes vector Sin of radiation propagating along the z axis th at is incident on the polarimeter. The output Stokes vector is m Smt = G IJ[ZZ(—/> - Ot)TiR(p + 9i)}Sin(ain, Pin, Pin), (3 .7 ) 1=1 where S in — { l i n t Q in i U im (8-8) V in ) — (1, Pin COS l‘oiin COS 2/3jn, P;n sin 2Oiin COS 2/3jn, Pin COS 2/3jn), (3.9) where G is the Mueller m atrix of the linear polarizer, R is a rotation matrix, and T 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is the Mueller matrix of a retarder. The input Stokes vector is parameterized by the polarization angle and the degree of polarization Pin of the incident polarization. The index m is the number of the wave plate. A single HWP is m = 1 and an achromatic HWP (AHWP) is m > 2 (Section 3.4). The components of the Mueller matrices R are 1 0 0 0 0 cos 9 —sin 9 0 m 0 sin 9 0 cos 9 0 0 1 0 (3.10) The Mueller matrix of a linear polarizer in arbitrary transm ittance is expressed as ‘ -I 7 i + T 2 Tx —T 2 0 0 Ti - T 0 0 0 2 Ti + T 2 0 0 2^/2W 2 0 0 0 2 (3.11) VTYT2 where Ti is transm ittance in the x axis and T2 is in the y axis, which can take a value from 0 to 1. In our model, we choose the x axis as the transmission axis of the linear polarizer. Thus, the Mueller matrix of the polarizer becomes Gt — - 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 (3.12) The components of the retardar P are defined later because the components change whether the effect of reflection is taken into account or not. We define angles of rotation around the z axis according to the usual convention where angles increase in the counter-clockwise direction from the x axis, which is also assumed to be the transmission axis of the ideal grid, see Figure 3.1. The rotation angle of the HWP is given by p. The relative orientation of a plate i in the stack relative to the first plate is given by 9i. The ordinary axis of the HWP is aligned with 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the x axis when p = 0. Since the bolometers are only sensitive to the intensity of the radiation, we are only interested in calculating the first component of output Stokes vector as a function of the HWP angle. For the sake of simplicity, we assume th a t the intensity of the incident radiation is independent of frequency. We introduce a window function W(u) that describes the spectrum response of a detector and band-pass filters. Thus, the first element of the output Stokes vector within the bandwidth can be written as .fOO {lout) (Wj ®ini Pin> Pint Pi d") I Jo Vi fiin j Pin,9,p,d)du. (3.13) A plot of Equation 3.13 as a function of p is an integrated-IVA (i-IVA) curve. We use this curve to reconstruct the state of incident polarized light, Pin and cq„. The connection between the i-IVA curve and the state of incident polarized radiation is discussed in following subsections. 3.2.1 M odulation efficiency A useful figure of merit for the operation of a polarimeter is the ’modulation efficiency’. It is defined as e(W ,a,S,d) =^ . Mn (3.14) The quantity Pout is the measured degree of polarization defined as P, /tir n j \ _ {Iout)max {Iout)min outyW'} ^ 3 &) jj r (T \ 5 \*out/max “r \^out/min /Q . r\ where (I ) max and (I ) min are the maximum and minimum of the i-IVA for angles 0 < p < 90 [44], The modulation efficiency e is a measure of de-polarization introduced by the polarimeter. 3.2.2 P hase vs. input polarization angle In addition to the degree of polarization Pj„, the input polarization angle is another im portant variable to extract from the i-IVA curve. The input polarization angle 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Index of refraction of sapphire [1] Thickness of each wave plate, d Bandwidth of frequency, uc + A u Offset angles of three-stack AHWP, 63 Offset angles of five-stack AHWP, 05 Resolution of frequency Resolution of wave plate angle Table 3.1 : n 0 = 3.047, n e = 3.364 1.58 mm (-H- uwp = 300 GHz) 150 ± 30 GHz, 250 ± 30 GHz (0, 58, 0) degrees (0, 29, 94.5, 29, 2) degrees 0.5 GHz 0.1 degrees The parameters used in the simulation throughout this chapter are shown. relates to a phase of the IVA curve. We define this phase <fr as (lout) °c cos (4p - 4(f)). (3.16) When the single HWP is used, the relationship between a in and (f> is <j) = \ a in. When the AHWP is used, the phase is not only the function of a, but many other dependences, including the bandwidth and offset angles of wave plates. A detailed discussion of the phase of the AHWP is in Section 3.4. 3.2.3 Param eters used in th e sim ulation Table 3.1 shows the parameters that are used in the simulation in this chapter. When some other parameters are used, we state those parameters in each section. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 Single H W P p o la rim etry 3.3.1 A nalytical derivation We initially neglect the effects of reflection from the interface between vacuum and a wave plate. We discuss the effects of reflection in Section 3.6. When we consider a single HWP, it is straightforward to derive the analytical expression of the first component of the output Stokes vector. We use Gx, m = 1 and &i = 0 to calculate Equation 3.7. We use the Mueller matrix of the retardar T as r(A d) = 1 0 0 0 0 1 0 0 0 0 cos Ad - sin AS 0 0 sin AS cos AS where (3.17) v 27r—(ne —n 0)d. Ad The variable AS is the retardance of a single wave plate and is a function of the ordinary and extraordinary indices of refraction n 0 and n e, respectively, the thickness of the wave plate d, and the electromagnetic frequency of light v. Therefore the output intensity, Equation 3.7, is Iout(Pi A(5(z<'), Qtim fiim Pin) lout [lin + Qin (cos2 — + sin2 — cos 4p) + Uin sin2 — sin 4p - Vin sin Ad] = ^ = ^[1 + Pi„[(cos4psin2 ^ + sin 4p sin2 ^ + cos2 cos2cq„cos2(3in sin 2ain cos 2/3in —sin 2p sin Ad sin 2A/3in]]. A top-hat window function between f b top-hatiy) uq and zq is 1 U0 < U < Ui 0 otherwise. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.18) Equation 3.18 can be integrated analytically and the result is A +B +C ( I o u t ) - 2 l(l+ 2 ) A — B + C sin AS(v) cos AS(v). 2 A5{ v ) J + D A h ( z / ^ 3' 19^ where A = Pin cos 4pcos 2ain cos 2Pin P — C = Pin sin 4p sin 2a;jn cos 2(3in (3.22) D = Pin sin 2psin pin. (3.23) P in COS *20iin COS (3.20) 2P in (3.21) A plot of (lout) as a function of p is the i-IVA. For the sake of simplicity, we derive the modulation efficiency without taking into account the bandwidth. The maximum and minimum of the IVA curve appear at p = | and § + f , respectively. Therefore, the maximum and minimum intensity are c r IZ n = ^ [1 + Pin (sin2 ^ cos 2Pin + cos2 ^ = sin ain sin AS sin 2pin)\, 9 AS \ [1 t1 + P Pin ( - Ssin2 cos 2Pin + cos2 ^ in(~ in ^ — CC cos 2a in cos 2/5* cos 2ain cos 2pir —cos ain sin Ah sin 2/A (3.24) We can calculate Pout as jm ax Pout uul = Tmin — — j m a ------x Train I out ' 1out \ P in(2 sin2 (3.25)' I v cos 2P ^ - y/2 sin Ah sin 2pin sin (ain - f)) 1 + \Pin (2 cos2 ^ cos 2ain cos 2pin - a/2 sin Ah sin 2pin sin (a in + f ) 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When pin = 0 degrees, out Pin sin2 1 + Pin cos2 cos 2a i: (3.26) Although we define our modulation efficiency in Equation 3.14, the degree of input and output polarization do not relate linearly. Furthermore, Pout has a dependence not only on Pin , but also the input polarization angle a in. When Pin <C 1, the input polarization angle dependence disappears, and therefore the input and output degree of polarization relates linear as Pout = sin2 ^ - P in - (3.27) This relationship is also true when a in = 45 degrees regardless of the value Pin. In this limit, the modulation efficiency can be a single value per band as e = sin2 £ (3.28) Notice th at this analytical expression of the modulation efficiency is not averaged over frequency. Equation 3.18 can be also written as lout = 7) Pin sin2 -y- cos (4P - 2ain) + DC term. (3.29) When the single HWP is used, the phase 4>of the IVA relates to the input polarization angle ain as p = ^ a in. 3.3.2 (3.30) D iscussions The top panels of Figure 3.2 show examples of IVA curves. The left top shows the IVA curves with zero-bandwidth. The right top shows the two i-IVA curves th at are averaged over the bandwidths. Figure 3.3 shows the measured degree of polarization Pout as a function of the 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 >, 0.8 0.6 0.4 0.4 :e HWP -. 0.2 0.0 Av = 0 0.2 -single AHWP 0.0 hv = 60 GHz 0 20 40 W ave p la te 60 a n g le 80 0 100 [degrees ] 3 s t a c k AHWP -Av= 0 > -, 0.6 5 0.4 60 a n g le 80 100 [d e g re e s ] 0.2 0.0 20 40 W ave p la te 60 a n g le 80 0 100 [d eg rees ] 0.6 0.4 0.4 0.2 0.0 0.2 0.0 20 60 40 p la te a n g le 80 100 [d e g re e s ] 1.0 5 s t a c k AHWP >, 0.8 -Ar 0.6 W ave 20 W ave :5 s t a c k AHWP rAv= 0 0 40 1.0 :3 s t a c k AHWP 0.8 rAv = 60 GHz "m C 0.6 G 0.4 ~ 0.2 0.0 0 20 W ave p la te 40 p la te 60 a n g le 80 0 100 [d eg rees] = 60 GHz 20 W ave 40 p la te 60 a n g le 80 100 [d e g re e s ] F ig u r e 3.2: The left panels show the IVA curves at a single frequency of 150 (solid), 200 (dash), 250 (dot), 300 (dash-dot) GHz for a single HWP, three-stack AHW P, and five-stack AHWP. The right panels show the IVA curves that are averaged over the bandwidth 150 ± 30 GHz (solid) and 250 ± 30 GHz (dash) with a step size of 1 GHz. In all cases, the input polarization angle a rn = 0, and the ellipticity /?;„ = 0. incident degree of polarization Pin. In this plot, the slope corresponds to the mod ulation efficiency. The five curves in each plot correspond to the five different input polarization angles, and therefore the modulation efficiency changes depending on what input polarization angle the incident radiation has. Only the curve corresponds to ain = 45 degrees is linear in all the bands as shown in Equation 3.26. All the curves merge together into the curve with cqn = 45 degrees in the limit when Pj„ <C 1. The input polarization angle is often unknown at the time of an observation, and therefore the variation of the modulation efficiency can be a source of error in estimating the degree of polarization Pin. The maximum variation of Pout at Pin = 0.1 in 150, 250 and 300 GHz bands in Figure 3.3 are 0.05, 0.008 and 0.001, respectively. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 GHz 250 + 300 + / - 30 GHz 0.6 0.6 o Q- // 0.4 3 3 a.o 3 0.2 a. 0 .4 0 .4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.6 0 .4 Pin 0.8 1.0 0.0 0.0 0 .2 0.4 Pin 0 .6 0 .8 1.0 Pin Figure 3.3: Pout is plotted as a function of Pj„ with various input polarization angles with the single HWP. The thickness of the wave plate is chosen such that the optim ized frequency of the wave plate is 300 GHz. To measure the modulation efficiency in a laboratory, it is common to use a fully linearly polarized source and find the modulation efficiency based on the measured i-IVA and Equation 3.14. When the polarized source is oriented such that the input polarization angle is at 45 degrees with respect to the transmission axis of the linear polarizer th at resides in front of the detector, the measured modulation efficiency at Pin = 1 can be directly applicable to Pin <C 1. In this way, the modulation efficiency is defined as a single number per band. Figure 3.4 shows the modulation efficiency with zero bandwidth as a function of frequency. The black curve corresponds to the single HWP and the others are with AHWPs which we discuss in Section 3.4. The modulation efficiency of the single HWP can achieve above 0.9 over a bandwidth of A v / v c ~ 0.3 with a first harmonic peak. The thickness of the wave plate is chosen such th at the frequency th at is optimized for the wave plate is at vWP = 300 GHz. Strictly speaking, averaging Equation 3.28 over the bandwidth is not same as averaging the intensity and calculating the modulation efficiency as defined in Equation 3.14. The difference between two methods are ~ 1% when the bandwidth is chosen to be 150 ± 30 GHz. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.8 u..1 o 0.0 0 200 400 Frequency 600 [GHz] Figure 3.4: The m odulation efficiency with zero bandwidth is plotted as a function of frequency. The black curve is for the single HWP. The red curve is for the three-stack AHW P and the blue curve is for the five-stack AHWP. 3.4 A ch ro m a tic h alf-w ave p la te Single HWP polarimetry is not suitable when the measurement of interest is in a wide frequency band because the chromatic nature of a single wave plate retarder forces the high modulation efficiency to reside in a narrow spectrum. An achromatic half-wave plate (AHWP) has been proposed to overcome this problem. The AHWP is a stack of birefringent plates th a t are rotated relative to each other. W ith appropriate choices of the number of wave plates and the relative angles, it is possible to achieve close to 100 % modulation efficiency at a given frequency over a broadband of frequencies [25, 47, 52, 49, 46], Figure 3.4 shows the modulation efficiency with zero-bandwidth as a function of frequency. In contrast with the single HWP case, the AHWP can achieve high modulation efficiency over a broadband. We consider a stack of three (m = 3) and five (m = 5) wave plates as an AHWP. As with the single HWP, we consider a polarimeter th a t consists of a rotating AHWP at frequency / 0, a linear polarizer, and 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T ransm ission axis a a =0 t a ou, * 0 Linear polarizer AHWP Detector Figure 3.5: A configuration assumed in our simulation is shown. The transmission axis of a linear polarizer is parallel to the x axis. a bolometer as shown in Figure 3.5. The rotation angle of the AHWP is labeled as p. We assume th at the spectral response of the incident radiation and the detec tor/readout system is a top-hat function between vc —A v /2 and vc + A v /2 , where vc is the center frequency of the band and A v is the bandwidth. We use Equation 3.13 to compute the output intensity with the Mueller matrices T, R, and Gx th at are used in the calculation of the single HWP. The middle and bottom panels of Figure 3.2 show the IVA for a three- and five-stack AHWP. The thickness of each wave plate d is chosen such that At) = ir when vwp = 300 GHz. The input polarization angle is chosen to be a = 0. The left panels show the example IVA curves at a single frequency with A v = 0. We show the i-IVA in the right panels when finite bandwidth is taken into account. IVA curves with finite bandwidth are found by calculating the output intensities I out as a function of p for a set of discrete frequencies, then averaging the intensities over the bandwidth angle by angle. Several generic features are apparent. As the frequency varies from v = vWP, the modulation amplitude decreases. When only a single frequency is considered, the amplitude of modulation tends to decrease as the frequency of each curve deviates from vw p- The IVAs are normalized to the input intensity and linear input polar ization becomes somewhat elliptical when it passes through a wave plate optimized for a different frequency. The grid passes only one state of the polarization which results in non-zero output intensity even though the state of polarization is crossed with respect to the transmission axis of the grid. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.8 uc>. D o q u u 0.4 0.4 LL Ld 0.2 0.0 0 200 400 Frequency 0 600 [GHz] 200 Frequency 400 600 [GHz] 90 D a) D a) CD CO co o 60 Q_ 80 2 60 CL 50 0 50 200 0 600 Frequency [GHz] 200 Frequency 400 600 [GHz] F ig u r e 3.6: Modulation efficiency e(v, A v = 0, a = 0 , 9) (top) and the phase <f>(v, A v — 0, a = 0 , 9) (bottom ) for the three-stack (left) and five-stack (right) are plotted as a function of frequency with offset angles of 8 3 = (0,58,0) degrees and 05 = (0,29,94.5,29,2) degrees. Another noticeable feature is the difference in phase among the IVA curves. The middle and bottom left panels of Figure 3.2 show the IVA curves which have different phases depending on frequency. For the case of a single HWP, 4> = a/2. However, for an AHWP = <f>(a,vc, A v ,d ,6 ). The phase does have a frequency dependence, and therefore the IVA curves in the middle and bottom left of Figure 3.2 are not in phase even though they have the same input polarization angle. The detector measures all the intensities within the bandwidth. The resultant IVA curve th at accounts for the bandwidth is the average of multiple IVA curves th at have different amplitudes and phases. We analyze the i-IVA of Figure 3.2 to reconstruct the state of incident polarized light. It is the goal in this section to discuss quantitatively the loss of intensity and the phase for an AHWP, both of which are non-trivial functions of the parameters of the stack. Many of the results were calculated by two independent computer codes to check for consistency. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 100 200 Bandwidth 300 400 300 400 [GHz] S' 1-00 .1 0 .9 8 u a) 0 .9 6 J 0 .9 4 0 0 .9 2 1 0 .9 0 0 100 200 Bandwidth [GHz] Figure 3.7: Top: M odulation efficiency of the three-stack AHW P is plotted with an input polarization angle of 0 (solid line), 22.5 (dot), 45 (dash), 67.5 (dot-dash), and 90 (three-dot dash) degrees. Bottom: M odulation efficiency of the five-stack AHW P with same input polarization angles as the top panel. 3.4.1 R esults We define the modulation efficiency of the AHWP as shown in Equation 3.14. Since the modulation efficiency e for a broad-band AHWP is calculated by averaging the IVA of a set of discrete frequencies, it is useful to examine both e and the phase <fi as a function of frequency. In Figure 3.6, e and (f) are plotted as a function of frequency with zero-bandwidth A v = 0. In both three- and five-stack AHWPs, e is close to 1 for a wide range of frequency near vwp and sharply drops away from vWp. The phase is constant around vWP and deviates away from <j>(vwp) as frequency deviates from vWpThe top panel of Figure 3.7 shows the modulation efficiency of the threestack AHWP as a function of bandwidth around vc = 300 GHz with various input polarization angles. The bottom panel of Figure 3.7 is for the five-stack AHWP. The modulation efficiency 0.99 is achieved with a bandwidth of 200 GHz by 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 0.8 3 s to c k AHWP 150 + / - 3 0 GHz 3 S tock AHWP 2 5 0 - r / - 30 GHz 0.8 0.6 Z3 o CL 0.0.2 0.0 0 .0 0 .2 0 .4 0 .6 0 .8 0 .0 1.0 0 .2 0 .4 0 .6 0 .8 1.0 Pin Pin 5 s ta c k AHWP 5 s ta c k AHWP 1 50 + / - 30 GHz 250 + / " 3 0 GHz F ig u r e 3.8: Top: P out as a function of P;n for the three-stack AHWP. Bottom: For the five-stack AHWP. Each curve in each panel corresponds to the input polarization angle of 0 (solid line), 22.5 (dot), 45 (dash), 67.5 (dot-dash), and 90 (three-dot dash) degrees. the three-stack AHWP and a bandwidth of 300 GHz by the five-stack AHWP. When e > 0.99, the modulation efficiency varies over the input polarization angles by less than 1.3 x 10~3 for both AHWPs. Figure 3.8 shows the degree of polarization Pout as a function of the degree of polarization of incident light Pin with two bands, 150 ± 30 GHz and 250 ± 30 GHz, with various input polarization angles. In this figure, the slope of each curve is the modulation efficiency e. The curves correspond to a = 45 degrees are linear in all the panels of Figure 3.8 with a reduced-^2 < 5 x 10- r . T he output phase </> vs. th e in pu t p olarization angle a Figure 3.9 shows the output phase ^ as a function of the input polarization angle a for the three- and five-stack AHWPs. The input radiation is Pin = 1. These results show that a and <f>have a linear relationship with a slope of 0.5 when the AHWP is used in various frequencies and bandwidths. On the other hand, the offset of the 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CO 0 <p 80 m m u su ■o. bO a; o 40 a 40 CL a 20 Out 0 0 20 40 60 80 0 0 100 20 40 60 80 100 I nput p o l a r i z a t i o n a n g l e [ d e g r e e s ] I nput p o l a r i z a t i o n a n g l e [ d e g r e e s ] 80 20 0 20 40 60 80 I nput p o l a r i z a t i o n a n g l e [dr 100 0 gree 20 40 Input p o l a r i z a t i o n 60 80 100 angle [ d eg re e s] Figure 3.9: The output phase is plotted as a function of the input polarization angle. Top left: Three-stack AHW P with a bandwidth of 150 ± 30 GHz (solid line) and 250 ± 30 GHz (dot). Top right: Three-stack AHW P with a bandwidth of 300 ± 0 (solid), 100 (dot), and 200 (dash) GHz. Bottom left: Five-stack AHW P with a bandwidth of 150 ± 30 GHz (solid) and 250 ± 30 GHz (dot). Bottom right: Five-stack AHW P with a bandwidth of 300 ± 0 (solid), 100 (dot), and 200 (dash) GHz. linear curve depends on vc and A v. 3.4.2 D iscussions M od u lation efficiency w ith various bandw idths and input polarization an gles Our calculation shows the three-stack AHWP achieves 0.99 modulation efficiency with A v l v c ~ 0.67 with offset angles 93 = (0,58,0) degrees. Similarly, the fivestack AHWP achieves 0.99 with A v / v c ~ 1 with offset angles 05 = (0,29,94.5,29,2) degrees. As previously shown, the modulation efficiency of a single HWP depends on the input polarization angle a. The AHWP shows the same feature. Figure 3.8 shows that the Pout — Pm relationship is close to linear as vc approaches to vwp- 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 100 200 300 Bandwidth [GHz] 400 0 50 100 Differentia! 150 Frequency 200 [ GHz] F ig u r e 3 .1 0 : Left: The modulation efficiency of the three-stack AHW P is plotted as a function of the angle of the second plate 62 and the bandwidth A v around the center frequency of v w p = 300 GHz. The first and third plate angles are kept at 0 degrees with respect to the x axis. Right: The corresponding phase variation Acp is plotted in units of degrees. A horizontal axis is the differential frequency Sv. In both plots, the input polarization angle is assumed to be a = 0. Table 3 .2 shows the the maximum variation of Pout at curves th at correspond to the input polarization angles = 0.1 among five 0, 2 2 .5 , 4 5 , 6 7 .5 , and 90 degrees. This residual becomes the maximum uncertainty in the measurements of the degree of polarization when Pin <0.1 e and (f>w ith various offset angles 9 The modulation efficiency depends on the choice of offset angles 9. We calculate the modulation efficiency as a function of bandwidth A v and offset angles 9 for three- and five-stack AHWPs as shown in the left panels of Figure 3 .1 0 , 3 .1 1 , and Figure 3 .1 2 . These plots show the optimal offset angles for high modulation efficiency over a broad band are 0, 2 9 , 9 4 .5 , 2 9 , 2 degrees. The phase also depends on the offset angle.To quantify the constancy of the 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 100 200 300 Bandwidth [GHz] 400 0 50 100 Differential 150 Frequency 200 [ GHz] Figure 3.11: Same procedure as Figure 3.10 for five-stack AHWP. The first, third, and fifth plates are kept at ( # i ,03,# s) = (0 ,9 4 .5 ,2 ) degrees. The offset angles of the second and fourth wave plates are kept same and are varied from 0 to 90 degrees with a step of 1 degree. phase as a function of frequency, we define the change of phases as Acf) = </>(u) - <f>(vwp). (3.31) The right panels of Figure 3.10, and 3.11 show A <j>as a function of the difference of frequency th at is defined as 5u — v — v w p While the offset angles of (0, 58, 0) degrees and (0, 29, 94.5 29, 2) degrees three-stack AHWP, A Pout five-stack AHWP, A Pout 150 ± 30 GHz 2 x 10“3 1 x 10“4 250 ± 30 GHz 5 x 10"5 5 x 10"5 Table 3.2: The maximum deviations of P oilt at P{n = 0.1 are shown. The indices of refraction are used for sapphire [1], The thickness of each wave plate is chosen such that the frequency of incident light, v w p = 300 GHz, satisfies AS = ir. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. , 0 _r> >,3:3 100 200 300 Bandwidth [GHz] 400 0 100 200 300 Bandwidth [GHz] 400 Figure 3 .1 2 : Left: The modulation efficiency of the five-stack AHW P is plotted as a function of an offset angle of the third plate while the others are fixed at ( 8 1 , 8 2 , #4, 8 5 ) = (0 ,2 9 ,2 9 ,2 ) degrees. Right: The m odulation efficiency of the five-stack AHW P is plotted as a function of an offset angle of the fifth plate while the others are fixed at (9\, 8 2 , 8 3 , #4) = (0 ,2 9 ,9 4 .5 ,2 9 ) degrees. achieve high modulation efficiency with broadest bandwidth in the three- and fivestack AHWPs, the corresponding phase variation A</> at the same offset angles and over the same bandwidth is less than 25 degrees in the three-stack AHWP and 20 degrees in the five stack-AHWP. When the five-stack AHWP is used, it is clear that the alignment of second, third, and fourth wave plates are much more demanding than th at of fifth plate to maintain high modulation efficiency. Offset phase (f>0 In single HWP polarimetry, the phase and the input polarization angle relate as 4>= a/2. Our calculation shows the slope of the </>— a relationship is 0.5 when the AHWP is used, but the offset phase </>0 th at is defined as 0 = 0.5a: + (j)0 depends on the bandwidth, wave plate offset angles, and the direction of rotation. Any error in the determination of the (ft — a relationship becomes a source of 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. systematic errors in cross-polarization. In principle, this relationship can be calcu lated with known information in advance as described in this paper. Nevertheless, it is best to measure this relationship if the experimental set up allows to do so. This relationship can be calibrated with a fully polarized source with known input polar ization angle with respect to the transmission axis of a linear polarizer. The amount of the cross polarization depends on how well one can measure this linear relationship with a fully assembled experimental setup. The fact that this relationship is linear makes easy to find the calibration between </>and a with high accuracy. In this way, we can calibrate the 4>—a relationship with a given incident radiation spectrum. The effects due to the incident radiation spectrum is discussed in Section 3.5. The rotational direction of the AHWP also affects the offset phase </>0. When the single HWP is used, the rotational direction does not m atter due to the rotational symmetry of the ordinary and extraordinary axes. On the other hand, the AHWP does not have the rotational symmetry due to the offset angles of each wave plate. The direction th at we assume in this paper is noted in Figure 3.5. We assume the receiver spectral response is a top-hat. However, the offset angle </>0 also depends on the receiver band shape. Therefore, it is best to determine the (f) — a relationship experimentally unless the spectral response of optical and detector systems is known to calculate the (f>— a relationship. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. q-16 \ X C\i I s in in < D c 0-24 0 400 200 600 F r e q u e n c y [ GHz ] F ig u r e 3 .1 3 : The total intensity of two spectra is plotted. The higher intensity curve is the 2.73 K black body spectrum of the CMB. The lower intensity curve is the dust spectrum that assumes only one power law as shown in Equation 3.34 3.5 S p ectru m o f th e in cid en t ra d ia tio n In this section, we consider the systematic effects of the HW P polarimeters when we take into account the incident radiation spectrum. We assume th at the spectral response of the incident radiation is a top-hat function in the previous sections. In a real experiment this is not a true assumption. We calculate the phase and the modulation efficiency from the i-IVA curve with the input Stokes vector th at includes the incident radiation spectrum. We also consider the case when the incident radiation is a combination of the CMB and dust. This effect results in a source of systematic errors of cross-polarization and of mis estimate of the polarized intensity. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5.1 M odel To take into account the spectral dependence of the incident radiation, we modify the input Stokes vector as Q u p ’ 1- p ’ ’ I = I{ v ){ 0 P cos 2a + 0 P sin 2a: 0 0 V (3.32) where I(v) is the spectral response of the incident radiation. In our analysis, we assume th at the degree of polarization P and the input polarization angle a are independent of frequency. To compute the phase </>, we consider four different spectra: CMB spectrum, dust spectrum, 300 K black body source in a laboratory, and a top-hat spectrum, as I c m b {u ) Idustty) habiy) — B { v , T Cm b (3.33) ) (3.34) = A u aoB {v ,T dust) = Itop-hatiy) = B(u,Tiab) (3.35) 1 (^i < v < v2), 0 (otherwise), (3.36) where TCMb = 2.73 K , d = 4 x 10~7, a 0 = 1-75, Tdust = 18 K, Tlab = 300 K, and B (v ,T )= 27rh 2 C vz QkgT — (3.37) J We assume a single power law for the dust spectrum. Equation 3.34 assumes the mean dust intensity for the EBEX sky area. We calculate the (j) — a relationship with different incident spectra for three bands: 150 ± 30 GHz, 250 ± 30 GHz, and 420 ± 30 GHz, by using Equation 3.7, 3.13, and 3.16 as described in Section 3.2. In addition, we simulate the case when the incident radiation has both the CMB and dust spectra. In a real balloon-borne observation th at probes the frequency range 100 —500 GHz, the incident radiation has a combination of these two spectra, 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 ± 30 GHz 250 ± 30 GHz 420 ± 30 GHz 150 ± 30 GHz 250 ± 30 GHz 420 ± 30 GHz Dust 56.69 51.16 54.50 CMB 0o = 57.86 51.12 53.85 Lab 57.33 51.14 54.49 CMB 0.53 -0.02 -0 .6 4 CMB — Dust A 0 = 1.17 (A ct = 2.34) -0 .0 4 (-0.08) -0.65 (-1.3) Top-flat 57.90 51.13 54.19 - Lab (1.06) (-0.04) (-1.28) Dust -0 .6 4 0.02 0.01 — Lab (-1.28) (0.04) (0.02) T a b le 3.3: Top: The offset angles with four different spectra are shown. Bottom: The difference of the offset phase between different spectra. The number in a parenthesis is the difference in terms of the polarization angle a on the sky. A unit of the phase is in degrees. at a given location on the sky. We neglect any other source of emission including the atmospheric emission and the emission from optical elements along the light path between the sky and detectors. When there are two sources of polarized emission, we need to reconstruct four unknown variables, P ? MB, 0^MB, Pt®wst, 4>?nSti from the i-IVA curve th at has a single phase and modulation amplitude. We modify the input Stokes vector as Q u 1 1 ’ I ' P g MB cos 2 = I c m b {v ) ^ + PfnMB s m 2 a g MB PiBust cos 2af™st iD ustiv) P ^ ust sin 2aT“st 0 0 V We neglect the effect of reflection and the incident radiation is assumed to be normal to the HWP surface. We assume P-^MB = 5 x 10 -6 and P ^ ust = 0 .0 5 [53]. The step size of the input polarization angle is 1 degree. 3.5.2 R esults We calculate the phase of the i-IVA curve with a given input polarization angle a when the various spectra are assumed. Table 3 .3 shows the offset phase angles 0o with four different spectra of the incident radiation and the difference of the offset 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CMB, 0 50 input polarization 100 150 angle 0 [degree] CMB, input 50 10 0 polarization 150 angle [degree] Figure 3.14: The phase (left) and the output polarization (right) of the i-IVA curve are plotted as a function of the cxfJaB and a f ^ st at the 150 ± 30 GHz band. A unit of the phase contours is in degrees. angles among three different spectra. We assume th at the incident polarization is fully polarized at a in = 0 degrees. We use the five-stack AHWP. Notice th at the phase and the input polarization angles are related as 4> oc \ a . Therefore, the corresponding shift in the input polarization angle becomes twice as large as the offset phase angle. The left panels of Figure 3.14, 3.15, and 3.16 show th at the phase of the IVA curve as a function of the input polarization angle of the CMB, afnMB, and dust, a f n Sti 250, and 420 GHz bands when the incident radiation has the CMB and dust spectra. The left panels of the same figures show the measured degree of polarization at the same bands. 3.5.3 D iscussion When an AHWP is used, the phase of the IVA curve depends on the electromag netic frequency. The frequency dependence comes from the spectrum response of an instrument as well as the spectrum of the incident radiation. The spectrum of the 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 1.0 e - 0 0 5 72 S.O e-006 54 6.0e-006 ■ 100 5 100 36 4 . 1 e —0 0 6 18 2.1 e - 0 0 6 1.0 e -0 0 7 CMB, 0 50 input polarization 100 0 150 angle [degree] C MB, input 50 100 polarization 150 angle [degree] Figure 3 .1 5 : The phase (left) and the output polarization (right) of the i-IVA curve are plotted as a function of the afnMB and a fnust at the 250 ± 30 GHz band. A unit of the phase contours is in degrees. A a [degrees] A P / P m ean 150 ± 30 GHz 5 (CMB) 0.1 250 ± 30 GHz 60 (CMB) 0.7 420 ± 30 GHz 5 (dust) 0.1 T a b le 3.4: The summary of the errors in the polarization angle and when the incident radiation are two spectral components. In the 150 and in the polarization angle are for a f :nMB ■ In the 420 GHz, the error is for the degree of polarization 250 GHz bands, the errors ayDUSt [ xin 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3 .1 6 : The phase (left) and the output polarization (right) of the i-IVA curve are plotted as a function of the a fnMB and a f ^ st at the 420 ± 30 GHz band. A unit of the phase contours is in degrees. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. incident radiation adds an extra weighting factor when the IVA is summed over the bandwidth. When only a single spectrum of the incident radiation is assumed, only the slope of the spectrum as a function of frequency affects the phase of the IVA curve. As shown in Table 3.3 , the largest difference in the phase appears between the CMB and dust spectra at 150 GHz. In this case, the difference in the polarization angle on the sky is ~ 0.5 degrees. As shown in Figure 3.13, the difference of the slopes of two spectra is largest in the 150 GHz band. When the incident radiation is a combination of the CMB and dust, there is no one-to-one relationship between the phase and the input polarization angle (0 vs. a ) and the modulation amplitude and the degree of polarization (Pout vs. Pin). The level of degeneracy depends on the ratio between QCMB and QDust (or ratio in Us). As shown in Figure 3.14, the phase of the 150 GHz band is dominated by afnMB- The weak contribution of the polarized dust signal gives an uncertainty of ~ 5 degrees in the determination of qVMjB based on the measured phase of the i-IVA curve. As the frequency goes higher, the contribution of the dust spectrum increases. The uncertainty in the determination of afnMB for the 250 GHz band is ~ 60 degrees. On the other hand, the phase for the 420 GHz band is mainly dominated by o t f ^ . The uncertainty in the determination of at the 420 GHz is ~ 5 degrees. The level of the degree of polarization Pout is 10-5 ~ 10~6 at all bands. The measured degree of polarization is defined as the ratio of the modulation amplitude of 4 x / curve to the DC level of the IVA curve. In the case th at the incident radiation has two spectra, the measured degree of polarization is jdC M B V C M B*in / t D Pout — i ' ------------------------ 7 7 ---------------------- VC MB T pDustX 1 Du st * in ) v±Ai/ —j -r ---------------r -------------------------------------------- • o n '1 ( d . d y j J-Dust) v ± A v This expression assumes th a t the input polarization angles of the CMB and dust are the same. As shown in this equation, even though P Bust = 0.05, the resultant output degree of polarization is suppressed by the total intensity in the denominator. As shown in Figure 3.13 the total intensity of the CMB is higher than that of the dust. On the other hand the degree of polarization of the CMB is much smaller than that of the dust. During a real observation, the sources of emission are not only the CMB 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and dust. The atmospheric emission and emission from the optical elements along the light path are not polarized add the extra contribution of a constant loading, and correspondingly the measured Pout decreases. To a good approximation and when the two incident polarization angles are the same, the degree of polarization that is extracted from the i-IVA curve is the highest. This is because the i-IVA curves only from the CMB and only from the dust are in phase when of ct>QVs ~ and therefore = o ^ “st within the approximation The difference between 0(pV5 and <j) \ a r e in Table 3.3. At a constant afnMB (or a £ “st), the degree of polarization in Figure 3.14, 3.15, and 3.16 varies by A P / P mean ~ 10 %, 70 %, and 10 % for the 150 GHz, 250 GHz, and 420 GHz bands respectively. Table 3.4 shows the summary of the errors in phase and the degree of polarization when the incident radiation are two spectral components. When there is only one band to measure the incident radiation with two spectra, this uncertainty is unavoidable. To eliminate these uncertainties, we need multiple bands to separate each spectral component. In Appendix, we describe the extraction of incident radiation with two spectra from IVA curves in two bands. The effects due to the incident radiation spectrum are not unique to the AHWP, but are intrinsic to HWP polarimetry. We have compared the magnitude of the effect between two cases, the single HWP with monochromatic incident radi ation and the five-stack AHWP with a finite bandwidth. W ithin the resolution in a and p of 1 degree, there is no difference in the phase variation between two cases. We define the phase variation as A(f) — (f*max rf'min- (3.40) At the 150 GHz and 250 GHz bands, the CMB signal dominates and therefore the phase of the i-IVA curve is mainly determined by afnMB. We fix the CMB polarization angle as afnMB = 0 degrees. We vary a ^ ust from 0 to 180 degrees and label the maximum phase of the i-IVA curve as and the minimum phase as For the 420 GHz band, the dust signal dominates over the CMB, and therefore we define the phase variation by fixing the dust polarization angle (a £ “s* = 0 degrees) and varying the CMB polarization angle. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 4 i: I 2 1 i 200 Figure 3.17: The variation of the phase A<j> = (pmax — <Pmin of the i-IVA curve over 0 < a f n St < 180 degrees at a fixed oV MB = 0 degrees is plotted. We use P ^ ust = 0.05. Three solid lines correspond to P ^ MB = 5 x 10~7. 1 x 10- 6 , and 5 x 10-6 (from left to right). The solid line assumes the total dust intensity in Equation 3.34. The dashed and dotted lines assume 10 % higher and 10 % lower of Idust(v), respectively. The right panel is a zoom up of the left panel. Figure 3.17 shows the phase variations A (f>of the i-IVA curve as a function of frequency at a fixed afnMB = 0 degrees. As the frequency goes higher, the contribution of the dust increases and the phase variation increases. Therefore the uncertainty of determining the phase, A(j>, increases. On the other hand, Figure 3.18 shows the phase variation at a fixed = 0 degrees. In this case, as the frequency goes higher, the contribution of the CMB decreases and the phase variation decreases. We use the single HWP with monochromatic incident radiation to produce these plots to shorten the computational time. The phase variation scales with the ratio {IBustP ^ ust) /{ I u MBPj0iMB)- There fore, when P ^ MB and P Bust increase by the same factor, the phase variation does not change at a given frequency. As examples, we assume following changes in I®ust and P ^ MB■ As shown in Figures 3.17 and 3.18, the ±10 % fluctuation of the dust spectrum does not contribute the m ajor change of the phase variation at a given frequency. On the other hand, the fluctuation of P ^ MB = 5 x 10~7 ~ 5 x 10~6 do affect the phase variation significantly. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 4 S £ 12 I G | 1 0 0 200 Froq u e r c y 400 6u0 000 4CC 500 600 FV c- q u e r c y [ 0 H z ] [G H z ] The variation of the phase A<j> = (pmax - <Pmm of the i-IVA curve over 0 < a ?nMB < ISO degrees at a fixed a f ^ st = 0 degrees is plotted. We use P Bust = 0.05. Three solid lines correspond to P ^ MB = 5 x 10~7, 1 x 10- 6 , and 5 x 10-6 (from left to right). The solid line assumes the total dust intensity in Equation 3.34. The dashed and dotted lines assume 10 % higher and 10 % lower of Idust{v), respectively. The right panel is a zoom up of the left panel. Figure 3.18: 3.6 R eflectio n from m u lti-la y ered w ave p la tes 3.6.1 Introduction This section describes the effects of reflection when the HWP and AHWP are used. The wavelength which we are interested in is in the range of 600 /rm - 2 mm. Therefore, the surface flatness of the wave plate material, such as sapphire, is easily achievable below ~ A/20 in this wave length. When the surface flatness is smaller than the wavelength of the incident radi ation by below ~ A/20, we have to take into account the effect of Fabry-Perot. The two parallel surfaces create a cavity and the reflection at the two interfaces causes the constructive and destructive interferences. This effect has not been an issue in HWP polarimetry at optical and infrared wavelengths due to their wavelength as compared to the length scale of the surface roughness. On the other hand, the millimeter wave polarimetry needs to take into account this effect. Clarke discusses the effects of reflection with the use of HWP and AHWP [48]. In our work, we calculate the modulation efficiency and phase with the effect of reflection and compare them with the case the effect of reflection is not taken into account. We also quantify the instrumental- and cross-polarization that 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are induced by this effect. We need two derivations to produce an IVA curve th at includes the effects of reflection. First, Equation 3.17 does not include the effect of reflection, and therefore we need to derive the Mueller matrix of retarder th at models the effects of reflection. We choose to use the Mueller matrix instead of Jones m atrix because it is straight forward to treat unpolarized radiation with the Mueller matrix. Second, we need to calculate the transmission coefficients at the interface between vacuum and a wave plate and between wave plates. These coefficients are the inputs to the new Mueller m atrix of the retarder. In this section, we limit our discussion to the radiation that incidents normal to the wave plate. 3.6.2 A generalized transm ission M ueller m atrix Let us consider the incoming and outgoing radiation of an optical element. We express the incident radiation as Ei = (Eixel(i>x, E iyel<l>y) , and substitute Ei into the Stokes vector as / Q.?.r TP 771 * TP 1 JP * \ ( E lxE*x EiyEjy R7. B7* W njiynjlx K7. W* ^ix-^iy TP E ixE*y \ ( u \ 1sin A 0 j Qi Ui iVi / \Eix\2 + \Eiy\2 \Eix\2 - \Eiy\2 2\Eix\\Eiy\cos A cf) TP* 2z\Eix| (3.41) where A <j>= 4>x — <py. We choose / , Q, U, V to be real numbers. We then define the Mueller matrix of arbitrary transm ittance as if ' Qf ( T ( t XX, t Xy , t y X, t y y ) uf \ vf /. \ Qi Ui (3.42) v> / J Notice that there are no imaginary numbers in the Stokes vector. We define the transmission coefficients, t x x , t x y , t y x , t yy Efx as t x x E ix T t x y E { y , 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.43) (3.44) E f y -- tyXE lX -)“ tyyEjy . We substitute Equations 3.43 and 3.44 into the output Stokes vector, /j1 L1* | C1 L'-t tifx&fx + ti fy ^ f y Sf = 77 77* 77 77* 77 77* 1 and regroup the terms such that t \3 = tU * f f* ~ l:yVxx lcyy yy _ i_ - Jn -cy(\ tbXXbi £'*XX ) ■tLx y t* bXy 4” ftLxV Xy t x x _!_ -{txxi 2i Z7 * t x y t Xy 4 " ^ y y ^ y y ) t d~ t*yx hc) \t l x x tb: (3.45) of the Mueller matrix T are ~ Tc\^(\ftLx t x x ttb:x x 4 " t y x t y x lr)i \ ti XXb: t tn 77 relate in the form of Equation 3.42. As and a result, the corresponding components = 17* 77 -Wz-Ws/ _ £11 77 - h fy&fy EfxEfy + tifyE17fx _ !_ ~ f f* by x byy — + ^yytyy)i (3.47) tyytyx)) (3.48) f f* — f bx y bXx Ly y Lyx)> fby xf byx * - ^I - t bx y bXy f* (3.46) t yt*byy}) ^ by (3.49) (3.50) hi = ^22 = l2 i \tbXX tL ; hs = 2 h i = J . [txyi ^31 = 2 {txx^i f* x - I -' t bx y byy f* ^-\- by t yf *bx y1) i 'bx (3.54) = 2 (txxti fbx *x ^33 = ~^(txxti tLy x t* bXy ^34 = ^41 = — — + E* — f* ( ^ x yt<bXX r — 1 f f* l y y l yx f f* ' bXXbXy _J_ f* — f f* y Ly x bX X bx y t Lx y byy f* — f f* r bx y by x 1_ Ly y Ly y ) i (3.51) f f* \ by X byy)') (3.52) f +* ) ' Ly x by y ) l (3.53) _L_ + bx y bXy ^ — -L ~fc* \ — tby y fbx* y' )J ) (3.55) _ |_ “ f f* } Ly y bxx)') (3.56) 1 2 ~Et x x i f* ~2 ((\ tfU bxXXby x t* bl X — tl x y tby *x — f f * _ |_ f +* l y X bxX ^ UX y byy .— __ _ — — f y tbxx)-> * 1 by f } Ly y Lx y ) l 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.57) (3.58) 2 (tlltyx tyx^xx ^43 — 2 * / \pxxt-yy fLyxL t *xy -\-' 1bxybyx t* ^44 n {txxtyy ^42 — tyxtXy ^-xytyy — tyytxy) , (3.59) tLyybxx)•> t* } (3.60) txytyx + tyytxx)- (3.61) To check the consistency, we calculate the Mueller matrix of a linear polarizer. If we assume a perfect grid that has a transmission axis along the x axis, G x. — ■f'Mueller (txx T]\/fn.pJ,lp.r(tXx 1 2 — IT.t?/ t Xy 1 ) ( 1 1 (i 0 N 1 1 0 0 0 0 0 0 0 0 °y 1° — tJy x — ^yy 0) (3.62) (3.63) We recover the Mueller m atrix of a linear polarizer. We can construct any Mueller m atrix as long as we know the transmission coefficients of that optical element. 3.6.3 Transm ission coefficient of m ulti-layer birefriengent ma terials We solve the Maxwell’s equation with multi-layer birefringent materials to calculate transmission coefficients of the HWP and the AHWP. We assume that the crystal axis of a birefringent material resides such th at the ordinary and extraordinary axes create a plane parallel to the surface of the wave plate. We limit our discussion to the radiation th at is normally incident to a wave plate surface. As Figure 3.19 shows, we label the incident radiation as Ei = (E^°\ E ^ ) . The subscripts x and y indicate the spatially fixed coordinate system in a vacuum layer. The labels x\ and y[ indicate a coordinate system th at is fixed to the first wave plate. We choose the x\ axis to be the ordinary axis of the first wave plate. At the first boundary, we can equate the boundary conditions for E and H as follows, E x\ {V ] = E ix\ {0) +' E {0] rx\= e tx\ \ 1], +' E rx\ '{1),1 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. po) po) ix i coordinate (x.y) iy E rx (0) i -E^ r (0) y relative angle 0j Interface I / r (1) /T (1) 4I /T '(1) p O ) fI rxi' l L txl ' ’ r 'ty !' Interface 2 ^ £ (1) IEL ry\ (l) £ (1) £ (1) 4I r 'txx'>n 't y \ f(2) tXi f ’ ( 2) ty \ coordinate (x’hy"}) f| relative angle 0 2 I 4 coordinate ( x ^ y ’z) F ig u r e 3 .1 9 : Electric fields at boundaries. Each electric field E is a vector. Each layer has the two indices of refraction n 0 and n e. The relative angle between x and x\ is 0 i and between x\ and x 2’ is 0 2- E (0,l) = E (0) y\ iy\ E (0) = E (l) ry \ £,(1) ty \ ry[ ’ (3.64) # ( 0 , 1) Xl rr(Od) *Vi Wi ry i ry\ (3.65) The (x,y) and coordinates are related with the relative angle 0 i as E r( cos 0 i —sin 0 i sin 4>i cos (f>i = Ex Equations 3.64 and 3.66 yield E (?’1} xi Em E-^) cos 0i —E $ sin 0 X+ E^J cos 0i —E ^ sin 0i 4 1) p'(t) ^tx\ tx\ > — E P°) -^ sin 0! + E «/^ cos 0! + E ^ sin 0 X+ E ry ^ cos 0i Vi ryi 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.66) (3.67) The electric field and the magnetic field are related through Faraday’s law as = ~ (3.68) The speed of light of the electric field along the ordinary and extraordinary axes vary due to the two indices of refraction. Therefore, Faraday’s law for incident light is, 27t - Euyt — 2tt Ap An — Enx' (3.69) for transm itted light, 2tt 27r r Etiyi = o j B f i x t Etix> coBtiy'., (3.70) and for reflected light, 2tt t E r\y> Ae 27T 7 E r\xf u)Brix1, ujBr\yi. (3.71) Notice th at the sign of reflected light is flipped due to the opposite direction of prop agation. We apply the same coordinate transformation to the boundary conditions of the magnetic field and rewrite them in terms of the electric field as H $ ' l) = 1 " H ^ ’1] = 2/1 - y - E f V cos fa - y ~ e £ > sin fa + j ~ e ^ ^e\ /*oi Wi cos fa - ( - - ^ ) £ g ) sin fa Wi C E {1) I C E l{1) ~ Ae2 1 ^e2 1 ’ - j ~ E $ sin fa + y ~e ^ cos fa + y s i n Aei y A01 Aei = 02 1 02 0 : + ( - ^ - ) ^ ° ) cos 0i A01 (3-72) 1 where C = — , U jl o 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.73) where v is the electromagnetic frequency of incident radiation in vacuum and jj,0 is a permeability of vacuum. Throughout this calculation, we assume that the perme ability of the wave plate material is equal to th at of vacuum. The boundary conditions at the second interface are E j>2 $ ' 2) = E f J2 + E ^ ,2= E f J 2, E f (3'74> = H ti>2 ^ ’2) = # 1^2 $ + # >$ 2 = # $ 2, E f * $ + < = ■ » £ ’• = (3.75) The ordinary axis of the second wave plate coincides the x 2 axis. When the single HWP is considered, the coordinate {x'2, y^) has to coincide with the coordinate (x,y). We label the relative angle between the x[ and x 2 axes as faThe outgoing electric fields from the first interface and the incoming electric fields at the second interface are related by the phase lag of the propagation in the wave plate material as E i{1) = x\ E t x^\ e ^ 01, ’ (3.76) Also the reflected light from the second boundary and the back incident light at the first boundary are related as E r{1) x\ = E rl(x1\)ei5°i E r yi x = d r (yA1 * ' - (3.77) The phase lag along the ordinary and extraordinary axes are ij S0 = 2,K—n 0d, c Se = 27r—n ed. 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.78) The boundary conditions of the electric field at the second boundary are written with Equations 3.66, 3.74, 3.76 and 3.77 as £ f!’2) = E t$x \e l5°i cos 02 - E^ )e lSei sin (f>2 + E'r^)elS°l cos 02 - E'r^ e %5ei sin 02 f (2 ) E§f> V2 = E tx$ e ~iS°i sin 0 2 + E ^ e ~ l5^ cos 02 + E'^)eiS^ sin 0 2 + E ^V ryi e^i cos 02 J p i . 2) (3.79) *2/2' Also the boundary condition of the magnetic field with Equations 3.66 and 3.74 with Maxwell’s equations, Equations 3.69, 3.70, and 3.71, and 3.76, 3.77, become H xx2 ' ,2) 1 cos 02 - sin 02 'V *ei 1 + J - E l y ' / 5ei C0Sfa - ( ~ J - ) E rxl Sin<^2 -IL K V2 e ® ty'^ - - ^ - E l l ) e ~ iSei sin 0 2 + t ~ E ^ J e"^ 01 cos 02 Aei A01 1 +^ } e ^ ^ei 1 sin 0 2 + ( - S L ) E ' ry e i5°i cos0 '01 E Em K 111' (3.80) As a result, the boundary condition from the first interface is written in a matrix form as ( E ^ ’1) ^ E( °’1) V = Mi H i 0.0 Hm J EU \ p i 1) ty'i p ' 0-) rxj l y 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.81) Also, the boundary condition from the second interface is written as ( E {)'2) \ ( x2 E {)’2) < A (3.82) M, Vi K [S e \ ,/(i) :ry[ Therefore, the above two equations yield ( E(°’^ \ fi?(o.i) v Mi Mo ( E xC ] ^ E {}'2) Vi (3.83) h % 2) if(o.i) 4 ? where ( Mx = el5°i cos (f>2 M o1 = —ej5ei sin ^>2 e-j<501 cos - \ _ e-2<5ei g^n 1 0 1 0 \ 0 1 c 0 1 c 0 c \ A(J; Ae i 0 e^0! sin (f>2 ei5ei CQS ^ 2 e -i<501 s j n e_l5ei cos 4>2 0 c A0l (3.84) Ae i 0 _^2±ei501 gjn 02 r^Lel<5°i sjn ^2 ,^£l.pi5ei C O S (j>2 c ^c2Xe-iSo -e “'°ix sin (f)2 b^Q -^ei COg (j)2 _ ^ _ ei5ei sjn 02 —^ e _l5°i cos (f>2 ^ h e ~i6ei sin </>2 y (3.85) Equation 3.83 shows th at the electric and magnetic fields of the incident radiation and the outgoing radiation in the (x'2, y'2) coordinate relate directly by a single 4x4 matrix mi = (3.86) This indicates th at Equation 3.83 can be extended to multi-layer birefringent mate- 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rials as ( E ^ ~ hN) \ ( E i0-1) ^ ^(0.1) y N p ( i V - l , J V ) v’ n h (N-1,N) (3.87) X1V rr(N-l,N) Hi o.i) y v’ n N yN — m (3.88) ( n - i ,n ) h XN rr{N-l,N) V’n We want to express the outgoing electric fields {E\x^ , E \ v^), ( E i ^ , E i ^ ) in terms of the incident radiation ( E ^ , E ^ ) . The boundary at the iVth interface gives p (2) E (N-l , N) E J{ ~ hN) = e S ] , “JV VN C XiV Xe 'e?f *2/jv ’ Vn -(N-l,N) _ v’n ~ ~ p ---- - e {n) ''Xoon tXN' (3.89) Therefore, Equation 3.88 can be rewritten by using Equation 3.89 as f Ei0,1} \ £ 7 (0 ,1 ) y Hi°V (3.90) = A J The boundary of the first interface gives ( Ei.0’1) \ fi(OA) v 4 0) B H i0'1) (3.91) Ei°) rx \\ H ry (0) J/ 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore, E i01 4 0) £(°) rx £f(°) ry fp(N) h tx'N Fm B ~ XA h ty'N J where c ( mn + ^f-m u mu ^eN m + - f - m 2i m 22 *eN m 23 , C _ msz A AoN m 21 \ m s i + vA— m u oN m 32 77141 mi2 + T ^ - m A°'~N 13 c °N A B -1 (3.92) (3.93) N 44 Y~miz eN cos (f>i sin 4>i — sin <j>i cos 4>i cos 4>i sin (p i — sin ( p i cos (p i Ae^ sin </>i COS</»i sin ( p i ^ cos (p i COS ( p i ll£l sin (p i c Ae^ cos <4 ~ (3.94) -% h sin ( p i The element of the m atrix m is expressed as r r i i j . l A as a^-, the electric field of the If we define the element of the m atrix B reflected and transm itted light are expressed as E (0) +I1r1'X E 1-( 0 ) i ^7T V-LJf ' xx-^ix xy-^iy ^rx ~ £(°) ry — r1 y x - ^ i x E {N) f E™ ty'N + + ' r1 y y ^ i y 1 (3.95) (3.96) and 4- p f 0 ) bx x J-J i x + 7? (° ) ', x y J- /i y > E (0) l'yx-LE^ix 4~r~- 7byy^iy (3.97) (3.98) where 7 XT. " Q 22 Q 31 ~ Q 32 q 21 ----- ^22^11 —012&21 QllQ32 ~ Q3lQl2 ^22011 —O12 O21 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.99) (3.100) rVx = (241^22 <^42«21 --------------------, /o m i\ (3.101) O22 O11 — a l2&21 (242 ( 2 l r.yy l ~ & 41&12 (222a ll —<212*221 5 (3.102) and 4* = --------— -------- , (3-103) -----------— -------- , 022011 —012(221 (3.104) tyx = ----------- — -------- , (222(211 —(212(221 (3.105) tyy = (3.1.06) txy = --------— -------- ■ When a single HWP is considered, (j>1 = —(f>2- When the multiple wave plates are considered, it is im portant to make sure th at the electric fields are expressed by the same coordinate system in the 0th and N th media 3.6.4 R esu lts and discussions Figure 3.20 shows the transmittances (T^ = tijt*■) as a function of frequency for a single HWP, and three-stack and five-stack AHWPs. The transm ittance curve has a fringe pattern because of the constructive and destructive interferences between the two parallel interfaces. The thickness of each wave plate is fixed at d = 1.58 mm, and therefore the overall thickness of the entire AHWP becomes thicker. The periodicities of the fringes are different among the single HWP, the three-stack AHWP, and the five-stack AHWP. In the cases of the AHWPs, the main fringes results from the interferences between the front and back surface of the entire AHWP. This fringe has the fastest periodicity because the distance between the front and back surfaces of the wave plate is farthest. The fringes due to the interferences between adjacent wave plates are also found to be in 3 x and 5x the periodicities in the three-stack and the five-stack AHWPs, respectively. The transm ittance of the single HWP does not have any cross-transmittance Txy nor Tyx. On the other hand, the transm ittance of the AHWP has non-zero cross-transmittance. This is because the optic axes of two 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. $L> 0.8 I 0.6 ! 0.4 c P 0.2 i— 100 300 200 400 500 400 500 400 500 Fr equency [GHz] <8 0.8 o J 0.6 I 0.4 c o q: 0.2 0.0 100 300 200 Frequency [GHz] ] q 0 0.8 £ 0.6 1 0.4 0.2 0.0 100 300 200 Fr equency [GHz] Figure 3.20: The transm ittances, Txx (black), Tyv (red), Txy (blue), and Tyx (green) are plotted as functions of frequency for the single HW P (top ), the three-stack AHW P (m iddle), and the five-stack AHW P (bottom ). adjacent HWPs are neither parallel nor perpendicular to each other. This non-zero cross-transmission explains why there is a phase offset 4o when the AHWP is used. In all cases, the cross-transmittance Txy = Tyx. To calculate the IVA curve th at includes the effects of reflection, we use the Mueller m atrix of a generalized retarder as, Sout G4-R( p)T^[uener(txixi, t x'y', ty'x', ty'y1')-fi'(p)Sin- (3.107) The x' and y' axes of the (x1, y') coordinate system coincide with the ordinary and extraordinary axes of the first wave plate. Once the output Stokes vector is calculated by Equation 3.107, we use Equa105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0. 8 (J c o o 0 .6 (D O O 0.4 TO 3 0.2 0.0 100 200 400 300 Frequency 500 [GHz] F ig u r e 3.21: M odulation efficiency of a single HW P with zero-bandwidth is plotted as a function of frequency. The black curve takes into account of the effects of reflection and the red curve does not. tion 3.13 to average over the bandwidth. The transm ittance, which has a frequency dependence, effectively adds an extra window function when the intensity is averaged over the bandwidth. Therefore, non-zero reflectance affects the modulation amplitude and the phase of the resultant i-IVA curve because of the same reason as described in Section 3.4. To calculate the measured degree of polarization Pout, we do not use Equa tion 3.15. This is because the i-IVA curve th at takes into account the effects of reflection is expected to have multiple mode of modulation. Therefore, it is not clear how to define appropriate maximum and minimum of (I out)• Instead, we use the fit as hut = C\ + C 2 cos 2p -1- (73 sin 2p + C4 cos 4p + C5 sin4p. 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.108) C c 0 .6 a) 0.6 CD 'o U- 'o T 0 .4 0.4 L lJ L lJ 0.2 c.o 0.0 0 2 00 400 0 600 2 00 400 600 F req ue nc y [GHz] F re q u en cy [GHz] 90 90 in T on 0) oU 0on) CD CO a 60 o 60 CL Q_ 50 50 0 2 00 400 0 600 200 400 600 Fre qu e n cy [GHz] Frequency [GHz] Figure 3.22: M odulation efficiency e{y, A v = 0 , a = 0,6 ) (top) and the phase A v = 0 ,a = 0 , 9) (bottom ) for the three-stack (left) and five-stack (right) are plotted as functions of frequency with offset angles of 9 3 = (0 ,5 8 ,0 ) degrees and 9 5 = (0 ,2 9 ,9 4 .5 ,2 9 ,2 ) degrees. The red line assumes no reflection. The black line which has fringes is calculated by taking into account the effects of reflection between the vacuum and wave plate interface and between wave plate interfaces. We re-define the measured degree of polarization Pout as J c i + Cl Pout = -• The modulation efficiency is calculated as the ratio of Pout to (3.109) The definition of the phase is the same as Equation 3.16 and it is calculated to be 1 C <b = - arctan —p. 4 C4 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.110) The phase is folded such th at 0 < (ft < 90. Figure 3.21 shows the modulation efficiency of a single HWP with zero-bandwidth as a function of frequency. In contrast to the case where the effects of reflectance are neglected, the modulation efficiency also contains the fringe pattern. Figure 3.22 shows the modulation efficiency and the phases for the threeand five-stack AHWPs. All are calculated with zero-bandwidth and are plotted as functions of frequency. Similar to the single HWP, the black curves of the modulation efficiency and the phase have the same overall features as the red curves th at do not take into account the effects of reflection. The fringes th at appear in the black curves are mainly due to interference at the two interfaces between vacuum and the wave plate. Although the magnitude of effect is secondary, there are also fringes th at result from reflections at the interfaces between adjacent wave plates. The reflection from a single interface between wave plates is at a maximum when the ordinary and extraordinary axes are orthogonal to each other, and the reflectance is 2.4 x 10~3 when a sapphire wave plate is assumed at 150 GHz. We anticipate the following systematic effects from reflection. First, the dif ferential transmission due to the two different indices of refraction in a birefringent material induces a modulation at the frequency of 2 x / 0 and its harmonics even though the incident radiation is unpolarized. The leakage of the differential transmission into the frequency 4 x / 0 is instrumental polarization. Second, we are particularly interested in the effects of reflection when the AHWP is used. Any reflection from two parallel interfaces creates an interference fringe pattern over the bandwidth of an experiment. This fringe pattern gives a weighting factor when the IVA is integrated over the bandwidth. Therefore, the fringe pattern affects the offset phase and the modulation efficiency of the resultant i-IVA curve when the bandwidth is taken into account. As shown in Table 3.5, we calculate the instrumental and cross polarization induced by the five-stack AHWP without taking into account any AR coatings on interfaces, as a worst case estimate. We choose the offset angles of (0, 29, 94.5, 29, 2) degrees. The state of the incident polarization, a in = 0 degrees and Pin = 1, is used to calculate the cross polarization. The cross polarization is defined as the difference of the phases A = \4> r — 4> n r \-, where 4> r is the phase of the i-IVA curve that takes 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Instrumental polarization Cross polarization [degrees] 150 ± 30 GHz 3 x 10-7 0.046 250 ± 30 GHz 5 x 10~8 0.006 Table 3.5: The table shows the instrumental- and cross-polarization induced by the effects of reflection at the two bands with the five-stack AHWP. into account the effects of reflection and 4>^r is not. When the spectral bandwidth of the experiment is large compared to the interference fringe pattern from the reflection at all the interfaces of the AHWP, these effects are averaged over the bandwidth, see Figure 3 .2 2 . Therefore, these effects may come out to be negligible depending on the bandwidth of the experiment. Furthermore, an appropriate AR coating th at minimizes the amplitude of the fringe pattern and the cross-transmission (i.e. differential reflection) in Figure decrease the values in Table 3 .5 . 10 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 .2 0 should 3.7 A n ti-reflectio n co a tin g w ith S W G stru ctu res 3.7.1 Introduction Any interface between media th at have different indices of refraction causes reflection. When a HWP is used in the optical system of an experiment, the reflection at the interface between the vacuum and medium reduces the total power of incident radi ation that is detected by a detector. Furthermore, the reflection become a source of systematic errors as discussed in Section 3.6. In this section, we address a technique to reduce the reflection over a broadband when the HWP resides at cryogenic tem perature. The entire discussion in this section is applicable not only to the HWP, but also to lenses, which also requires the means to reduce the reflection for a broadband at cryogenic temperature. A common technique to minimize the reflection is to use a thin film coating at the interface between the vacuum and the wave plate. Detailed conceptual and mathematical descriptions of an anti-reflection (AR) coating are found in a number of text books, including Hecht and Fowles [44, 54]. A single layer of the AR coating minimizes the reflection only at a single frequency and its harmonics with carefully chosen index of refraction and thickness of the thin film material. The typical band width of the reflectance below 1 % with a single layer is A v f v ~ 0.2, and therefore it is not appropriate when the bandwidth of the experiment is 120 - 450 GHz. To increase the frequency coverage, the standard technique is to vary the index of refraction from the index of the vacuum to the index of the wave plate material continuously. The technique of using the gradient index of refraction can be achievable either by using a material th at has gradient in the index of refraction or by using a number of thin-film layers th at have different indices of refraction and stacking them in layers. The first option is commonly used for the optical fiber application at an optical wavelength, but not available in millimeter wavelength. The second option is limited by the availability of material th at has the appropriate index of refraction. Even though the m aterial is available, the layers of thin films need to stay as a stack of layers when it is used at cryogenic tem perature. Typically, a glue is used to hold the films together. However, after a few thermal cycles, the glued thin films tend to peel off due to the differential thermal contraction. 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To achieve the broadband cryogenic AR coating at millimeter wavelength, we investigate a design of the AR coating using a sub-wavelength grating (SWG) structure. 3.7.2 Subwavelength G rating Structures The AR coating with SWG structures employs patterned structures on the surface of the wave plate in place of layers of thin-film. The structure size needs to be on the order of the wavelength of incident radiation to avoid unwanted scattering from the structures. This SWG replaces the use of glues because the structure can be directly patterned on the substrate surface. Furthermore, the shape of the structure determines the effective index of refraction of the structured layer, and therefore no need to search a material th at matches to the appropriate index of refraction to minimize the reflection. The diagram (1) in Figure 3.23 shows a side view of an square array of SWG structures. These structures are equivalent to a single layer th at has the effective index of refraction neff , 1 < n ef f < n s, as shown in the diagram (2). The structure is characterized by the grid spacing g, width w, and height d. The effective index of refraction is determined by the volume fraction of the vacuum to the substrate in a given layer. The diagram (3) shows the top view. The structure is patterned in two dimensions because the AR coating needs to be effective to two orthogonal states of polarization. The diagram (1) of Figure 3.24 shows the structure that is tapered from the substrate side to the vacuum side. This shape achieves to vary the index of refraction continuously as the incident radiation transm its into the substrate, and therefore it is suitable for the broadband AR coating. The AR coating with the SWG structure is already demonstrated in the op tical and infrared wavelengths [55, 56, 57, 58]. The application to the HWP in the millimeter wavelength (120 —450 GHz), however, has just begun. Ill Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Side view n =1 (vacuum ) h ns (substrate) T op view W (3) m4 isp km F ig u r e 3 .2 3 : 3.7.3 S8S - ' ■ mm ■mam A schematic diagram of a square pattern of the SWG structures are shown. M odel and D esign We design the pyramidal shape for various materials to minimize the reflection. To model the shape in the diagram (1) of Figure 3.24, we slice the pyramidal shape in layers as shown in the diagram (3). The stepped pyramidal shape in the diagram (3) can be treated as a stack of thin-film layers on the substrate as shown in the diagram (4). Once the structure is modeled as a multi-layer thin-film stack, we calculate the transmission and reflection coefficients by using the m atrix method th at is described in Hecht [44]. In our calculation, we choose to slice the smooth pyramidal shape in 30 steps to model as a stack of thin-film layers. In addition to the dimensions, w ,g ,h, we introduce the length s to denote the width of the top-base when the trapezoidal shape is considered. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X 9 (1) (2 ) m m w, (3) k (4) k Figure 3.24: k k k k A cross-sectional view of the pyramidal shape SWG structure is shown. There has been studies to search the optimal changes of the indices of refrac tion as a function a distance along the z axis to increase the bandwidth th at has low reflectance [59]. Although Figure 3.24 shows a geometrically linear slope in the pyra midal shape, in principle the slope does not have to be straight. The choice of this functional form, n = n(z), determines the slope of the pyramidal structure. Although it is best to find the optimal n = n ( z ) and to design the corresponding slope of the pyramid, the typical structure size is ~ 500 fxm and this structure size is rather too small for a mechanical machining and is too large for an etching or deposition type processes. Therefore, there is less capability to control the slope. Throughout our de113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sapphire R0 Re AR 150 ± 30 GHz 0.011 0.022 0.011 250 ± 30 GHz 0.026 0.020 0.006 420 ± 30 GHz 0.012 0.014 0.002 Table 3.6: The averaged reflectance and the difference of the averaged reflectance of the single sapphire wave plate are shown for the three bands. signing, we fix the slope of the pyramidal structure as geometrically linear to simplify the fabrication process even though the corresponding functional form, n = n(z), may not be optimum. We use the approximated second order effective medium theory (EMT) to calculate the effective index of refraction with a given volume fraction [60]. The second order EMT is valid at the limit when the wavelength of the incident radiation A satisfies the condition of the grid spacing g as ?A < — T Tig ~’ Tly (3-m ) where n v and n s are the indices of refraction of two sides. Typically, nv is the index of refraction of the vacuum and n s is the index of refraction of a substrate, i.e. lens or wave plate material. To achieve a smooth transition of the index of refraction, it is sensible to choose the width w at the bottom of the pyramid shape to be w = g. Therefore, the only remaining dimension to specify is the height h. A detailed mathematical description of the EMT is in Appendix. 3.7.4 R esults Sapphire We calculate the reflectance from a single sapphire wave plate when the trapezoidal structure is patterned directly on the sapphire surface. We assume that the thickness of the wave plate is 1.69 mm. This thickness does not include the height of the pyramidal structure. The trapezoidal SWG structures are patterned on both faces of the single sapphire HWP. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 40 t o p —b a s e 60 80 dim ension 100 20 [um] 40 top-base 60 80 dim ension 100 [ urn] Figure 3.25: The averaged reflection from the single sapphire wave plate at the 150 ± 30 GHz band is plotted as a function of the height h and the top-base width s. The left panel is for the case that the index of refraction of the sapphire is n 0 and the right panel is for n e, respectively. The reflectance is normalized to one. 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.10 0.08 0.08 c 0 .0 6 c 0.06 0.04 0.02 0.00 0 100 20 0 300 400 100 500 0 100 20 0 300 400 20 0 30 0 400 500 F r e q u e n c y [GHz] Fr e q u e n c y [GHz] 500 600 0 100 200 30 0 40 0 500 600 he i g h t [urn] h e i g h t [urn] Figure 3.26: The reflectance for the ordinary (left) and extraordinary (right) axes from the single sapphire wave plate are plotted on the top panels. The bottom panels show the corresponding profile of the effective index of refraction for the ordinary (left) and extraordinary (right) axes. 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.25 shows the averaged reflectance over the frequency 150 ± 30 GHz as a function of the top-base width s and the height of the trapezoid h. We calculate the reflectance by assuming th at the index of refraction of sapphire is either n 0 or n e. Based on Figure 3.25, we choose the geometry to minimize the differential reflection between the ordinary and extraordinary axes as s 0 = 53 firm, h 0 = 504 gm, g0 = 164 g m se = 64 g m , he = 504 gm, ge = 152 gm. The dimensions s 0 and g0 are parallel to the ordinary axis of the sapphire and se and ge are parallel to the extraordinary axis. Therefore, the base of the trapezoid becomes a rectangular shape. The height of the trapezoid has to be same for both, h 0 = he. The top two panels of Figure 3.30 shows the reflectance as a function of fre quency at the ordinary axis and the extraordinary axis with the geometry we chose. The bottom two panels of Figure 3.30 shows the profile of the index of refraction for the ordinary and extraordinary axes at a frequency of 150 GHz with a given geometry. Table 3.6 shows the averaged reflectance and differential reflectance for three bands. T M M on Sapphire TMM is a dielectric material th at has the index of refraction of 3.13 at 10 GHz [61]. TMM is easier to machine than sapphire, and therefore it is an option to glue TMM on the sapphire wave plate surface and machine the SWG structure on TMM instead of directly machining structures on the sapphire. We calculate the reflectance from structured TMMs th at are glued on the both surfaces of the single sapphire wave plate. We assume th at the index of refraction of TMM at v = 120 ~ 450 GHz is same as th at at 10 GHz. Figure 3.27 shows the averaged reflectance over the frequency 150 ± 30 GHz as a function of the top-base width s and the height of the trapezoid h. Based on this figure, we choose the following dimension for the structure as s = 65 gm, h = 468 gm, g = 156 gm. 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.112) Figure 3.27: The averaged reflection of TMM + sapphire at the 150 ± 30 GHz band is plotted as a function of the height h and the top-base width s. The left panel is for the case that the index o f refraction of the sapphire is n 0 and the right panel is for n e, respectively. TMM + sapphire R0 Re AR 150 ± 30 GHz 0.011 0.016 0.005 250 ± 30 GHz 0.030 0.046 0.016 420 ± 30 GHz 0.021 0.027 0.006 Table 3.7: The averaged reflectance and the difference of the averaged reflectance of the TMM + sapphire are shown for the three bands. The top panel of Figure 3.28 shows the reflectance as a function of frequency with the geometry we chose. The bottom panel of Figure 3.28 shows the index profile at a frequency of 150 GHz with a given geometry. Table 3.7 shows the averaged reflectance and the differential reflectance for three bands. R exo lite The rexolite is a material th at is a candidate to be used as a lens material. Therefore, the AR coating is assumed to be applied on a curved lens surface. We calculate the reflectance at a single interface between the vacuum and the rexolite not to take into 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.10 0.06 0.03 a; a; I I § 0.06 0) I a. 0.04 0.02 0 .0 0 i h 0 .0 2 0 .0 0 100 0 L/J. o 100 200 300 400 F r e q u e n c y [GHz] r r e q u e n c y [GHz] 3. 5 3.D ■§ s: 1 3 n 1 3.Q £ 2.5 £ 2.0 .0 « t- 1.5 c* 0 0 o 2 .5 o o o S 2 .0 •o .0 « 1.5 *— 1.0 0 o o O. 0 O, °O o 1.0 100 200 300 400 h e :g h t [ur n] 500 0 600 100 200 300 400 h e i g h t [u r n] 500 600 F ig u r e 3.28: The reflectance of TMM + sapphire for the ordinary (left) and extraordinary (right) axes are plotted on the top panels. The bottom panels show the corresponding profile of the effective index of refraction for the ordinary (left) and extraordinary (right) axes. Rexolite R Table 3.8 : 150 ± 30 GHz 0.004 250 ± 30 GHz 0.004 420 ± 30 GHz 0.002 The averaged reflectance of the rexolite for three bands is shown. account the interference effect with two parallel surfaces. Figure 3.29 shows the averaged reflectance over the frequency 150 ± 30 GHz as a function of the top-base width and the height of the trapezoid. We find the reflectance is 0.5 % with following geometry, 106 fim, h = 540 gm, g = 254 g m (3.113) We chose this dimensions based on the reflectance at the 150 GHz band. The top panel of Figure 3.30 shows the reflectance as a function of frequency with the geometry we chose. Table 3.8 shows the averaged reflectance at three bands. The bottom panel of Figure 3.30 shows the index profile at a frequency of 150 GHz with given geometry. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 40 top 60 b o s e c i m e n s i o n [ u m] SO 100 F ig u r e 3 .2 9 : The averaged reflection of the rexolite at the 150 ± 30 GHz band is plotted as a function of the height h and the top-base width s. 0.05 o ^.04 B 0.03 o ^ 0.02 - 0.01 0.00 0 10 0 100 200 300 f r e q u e n c y [GHz] 200 300 h [micron] 400 400 500 500 Figure 3 .3 0 : The reflectance from the interface between the vacuum and the rexolite is plotted on the top panel. The bottom panel shows the corresponding profile of the effective index of refraction. 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.T.5 D iscussion The SWG AR coating can reduce the reflectance and the differential reflectance below a few percents for the 150, 250, and 420 GHz bands simultaneously. Although we did not put an effort to make the pyramid base of TMM as a rectangular shape to minimize the differential reflection, we achieve the differential reflection below 1 % for the 150 and 420 GHz bands. If we optimize the shape of the square base to a rectangular, we expect to reduce the differential reflection furthermore. Also, the choice of the structure is made purely by the reflectance at 150 GHz, but the structure size can be optimized for three bands simultaneously. 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.8 E ffects w h en a w ave p la te d o es n o t resid e at an ap ertu re sto p In this section, we study the effects when the wave plate does not reside at the exact location of the aperture stop. It is ideal to place a wave plate at an aperture stop in the optical system because all the detectors illuminate the wave plate uniformly, and therefore any azimuthal asymmetry of the wave plate properties is averaged over within the beam. We consider the azimuthal asymmetry of the temperature, emissivity, and thickness of the wave plate. Throughout this calculation, we assume the EBEX optical system [2]. Tem perature We assume the total optical load 0.380 pW on the detector at the 150 GHz band. This total optical load includes the power from the CMB, the atmospheric emission, and the thermal emission from optical elements. We also assume that the mean tem perature of the wave plate is 6 K and the amplitude of the quadrupole variation of the wave plate tem perature is 5 % of 6 K. When the wave plate is displaced from the aperture stop by 1 cm, 11 % of the beam is not uniformly illuminated by all the detectors. We assume the emissivity of the sapphire wave plate as 1.6 %. The resultant induced 4 x / modulation due to the quadrupole tem perature variation appears as instrum ental polarization. The magnitude of the instrumental polarization is P = 3 x 10“ 4. E m issivity We use the same parameters th at are used to estimate the effects due to the tempera ture variation. Instead of varying the tem perature, we fix the tem perature at 6 K and assume the quadrupole variation of the emissivity by 5 % of 1.6 %. The instrumental polarization due to this emissivity variation is 1 x 10-4 . 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thickness When there is a thickness variation, the transmission property and the retardance of the wave plate change. The change in the transmission property results in instrumen tal polarization. The change in the retardance results in the change in the modulation efficiency. The transmission property of the wave plate changes when the thickness varies due to the interference between two parallel surfaces. The change in the transm it tance due to the 20 f in i peak-to-peak quadrupole variation results in the instrumental polarization by 1.25 x 10~4. To assess the effect of the retardance variation as an order of magnitude esti mate, we calculate the retardance of the single HWP based on Equation 1.3 with the amplitude of 10 fim quadrupole thickness variation. We weigh the retardance by the area where the wave plate is uniformly illuminated and the area where it is not. The resultant variation of the modulation efficiency is Ae = 1 x 10-3 . 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. { p - 90 degrees) { p = 0 degrees) isotropic medium x' birefringent layer isotropic medium je-wave ; e-wave o-wave o-wave Ax Figure 3.9 3 .3 1 : Schematic diagrams of double refraction effects are shown. O blique an gle o f in cid en t ra d ia tio n to th e H W P When the oblique angle of incident radiation transm its through a birefringent mate rial, the beam splits into two beams. In this section, we discuss how this effect affects to observations. 3.9.1 D ouble refraction When unpolarized light incidents to a birefringent material with non-zero incident angle, the refracted wave splits into two waves in the birefringent material. Figure 3 .3 1 shows a schematic diagram of the double refraction. The refracted angles can be calculated by the Snell’s law as 90 = sin-1 (— sinflj) c (3 .1 1 4 ) 9P = sin-1 (— sin#*), (3 .1 1 5 ) where u0 = — nQ (3 .1 1 6 ) 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 300 200 D 00 s 100 u 40 20 60 Incident Angle Ih eta 80 100 [degrees] Figure 3.32: The distance x is plotted as a function of the incident angle 6 . Five curves that correspond to p = 0 (—),3 0 (- • -),4 5 (----- ), 60(- • -),9 0 (------- ) are over-plotted. up ue = U PC (3.117) \Jc2 + (u2 —u2) sin2 p sin2 9i —. Tle (3.118) In contrast to the speed of ordinary wave u0, the speed of extraordinary wave up depends on the incident angle 9i and the angle p between the ordinary axis and the x' axis that defines the plane-of-incident with the z' axis. We calculate the distance A x between the ordinary and extraordinary waves when two waves transm itted from the birefringent m aterial as a function of the in cident angle 9. Figure 3.32 shows A x as a function of 9i for a single HWP with a thickness of 1.58 mm x 5 wave plates = 7.9 mm. If we assume th a t this distance A x directly corresponds to the displacement on the focal plane, we can associate the displacement of a ray on the focal plane to the displacement of a beam on the sky. The displacement at 0* = 30 degrees corresponds 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to 10 arcsec on the sky. We assume the plate scale 1.3 arcmin/mm. This indicates th at the ordinary and extraordinary beams do not point at exactly same spot on the sky by 10 arcsec. The refracted angle of the ordinary wave 0o is always stationary with respect to the instrument frame, the (x1, z') coordinate. On the other hand, the refracted angle of the extraordinary wave 9P moves as the HWP rotates and this is indicated by the spead of curves in Figure 3.32. At the angle 0i = 30 degrees, the maximum variation of the distance A x is A x(p = 90) —Ax(p = 0) = 3 /mi. W ith the same assumptions, this displacement corresponds to the 0.2 arcsec on the sky. As a summary, two beams has an offset by 10 arcsec and the beam which corresponds to the extraordinary wave moves by 0.2 arcsec with 2x the rotation frequency of the HWP. At p = 0 degrees, the ordinary wave is a p-wave (parallel to the plane-ofincident) and the extraordinary wave is an s-wave. On the other hand, at p = 90 degrees, the ordinary wave is the s-wave and the extraordinary wave is the pwave. While the HWP rotates, the wave vector of the ordinary wave stays same and the wave vector of the extraordinary wave stays nearly same within 3 pm. But the polarization states of the ordinary wave and extraordinary wave switch at 2x the rotation frequency of the HWP. Correspondingly, the beams for the ordinary and extraordinary waves collect the two polarization states at 2x the rotation frequency of the HWP alternately. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.10 S u m m ary o f sy ste m a tic effects The systematic effects addressed in this chapter is summarized in Table 3.9, 3.10, 3.11, and 3.12. S ystem atic effects - Differential transmission leakage to 4 x / If the wave plate is not at aperture stop - 0.3 K of quadrupole tem perature variation - 5 % of quadrupole emissivity variation - 10 /urn of quadrupole thickness variation Table 3.9: M agnitude P = 3 x 10~7 (150 ± 30 GHz) w/o AR coating (5-stack AHWP) P = 3 x 10-4 P = 1 x 10-4 P = 1 x 10“4 Instrumental-polarization S ystem atic effects - Phase offset with frequency dependence - Difference between w / and w /o reflection - AHWP phase offset between CMB and dust - Incident radiation spectrum mixing between CMB and dust induces Table 3.10: M agnitude <fi = 0.5a: + </>o(zq A d , 9, d) A (f) = 0 .0 5 degs (1 5 0 ± 3 0 GHz) Ac/) = 0 .0 0 6 degs (2 5 0 ± 3 0 GHz) A <f>= 1 A<f> ~ Ac/> ~ A <j) ~ deg 5 (1 5 0 ± 3 0 degs (1 5 0 ± 3 0 GHz) degs (2 5 0 ± 3 0 GHz) degs (4 2 0 ± 3 0 GHz) 60 5 GHz) Cross-polarization 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S ystem atic effects - Broadband modulation efficiency - Multiplicative factor - Thickness variation - Incident radiation spectrum mixing between CMB and dust induces the errors in the estimate of Pjn M agnitud e e > 99 % with A v / v ~ 0.6 (3-stack AHWP) A v / v ~ 1 (5-stack AHWP) Pout = ePin(Pin < 0.1) with A P out = 1 x 10~4(Pj„ = 0.1) Ae < 1 x 10~3 (single HWP) A P / P ~ 10 % (at 150 GHz) A P / P - 70 % (at 250 GHz) A P / P - 10 % (at 420 GHz) Table 3.11: De-polarization S y stem atic effects Beam wobbling due to the oblique angle of incident ( 9i — 30 degrees at edge field) M agnitude 10 arcsec offset between o- and e-beams e-beam moves 0.2 arcsec at 2 x / p-state and s-state switch between o- and e-beams at 2 x / Table 3.12: The effects to a pointing. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p en d ix A A generalized transm ission Jones m atrix Transmission coefficients th at are derived in Section 3.6 can also be used by Jones matrices as t NX j E tN y where _ QJx]lJ ^ p } rJ'J (txlx^ t X'y', tylxl, tytyl)Rj(p) J ^ | ], (A.l) R in y y R j is a 2 x 2 rotation m atrix th at is the same as Equation 3.66. The Jones m atrix G jx is a linear polarizer and T j is the retarder defined as Gj* = \ 0 Q I> (A-2) and T j ( t X'X', t Xly', t y ' X', t y l y l ) ~ [ ^ ^ ty ' x ' . t y 'y ' 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A -3) 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ppen d ix B E xtraction o f th e polarization of C M B and dust from IVA curves in two bands B .l In tro d u ctio n When the radiation is linearly polarized, we need three parameters to describe the polarized radiation: intensity / , the degree of polarization P, and polarization angle a. This is equivalent to say th at we need / , Q, and U of Stokes vector to describe the linearly polarized light. When the incident radiation is a combination of two sources with different spectrum (CMB and dust), there are 6 parameters to fully describe the incident radiation. When a half-wave plate (HWP) polarimeter is used to measure the polarization of the incident polarized radiation, the output of the signal is modulated intensity as a function of the HWP angle. This intensity vs. HW P angle (hereafter IVA) is described by three parameters, a DC offset, amplitude of modulation, and phase. EBEX and other CMB polarization anisotropy experiments are generally not designed to measure the absolute intensity of incident radiation, and therefore we only measure 2 quantities per band (amplitude and phase for a HWP polarimeter). In this memo, we show th at two IVAs at different frequency are enough to 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. reconstruct the polarization state from two sources. We also discuss how the spectral dependence of the iVA phase affects to the extraction of the incident radiation when an achromatic HWP is used. B .2 P ro b lem • unknow n param eters: 12 - CMB( l a , Pci, oici) and dust (I d i , P d i ,®-d i ) at 150 GHz - CMB(Ic 2, Pc 2, 0^02) and dust (Id 2, Pd 2,a D2) at 420 GHz Note that the subscripts 1 and 2 refer 150 GHz and 420 GHz bands, respectively. Also the subscripts C and D refer the CMB and dust. • assum ptions: - P and a of the CMB and dust are independent of electromagnetic fre quency v (12 —>- 8) - total intensity of the CMB is known to a level we need (8 —>6) - total intensity of the Dust is known to a level we need (6 —» 4) • As a result, the number of unknown parameters is 4. If we measure the IVA at two bands, we obtain 4 known parameters (2xamplitudes + 2 xphases). Q uestion: Howto extract 4 unknown parameters from two measured IVAcurves? How doesthe uncertainty in the total intensity of the dust spectrum affect to the extraction? B .3 S o lu tio n When incident radiation has one spectrum, the measured IVA at a given frequency can be written as 1= + ^I„eP cos ( 4 p - 4(f)). 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.l) When the finite band width is taken into account, the measured IVA is I = ^(I)u c,Ar' + \ ( I )^cA^cA uPcos(4p-A (t)). (B.2) The phase 0 of the measured IVA relates to the input polarization angle as cf>= - a + 4>o(uc, Au, spectral shape). (B-3) 2 The offset phase is due to an AHWP polarimeter. When a single HWP polarimeter is considered, the offset phase is zero. When the incident radiation has two spectra, the resultant IVA becomes Itotal = Ic + Id + I atm + hpt = \lc + tc P c cos (4p - 40c) +\ID\tIDPD + cos (4p ~~ 4<^d ) = C + A cos (4p —40). (B-4) (B.5) (B '6) (B.7) where - ( 1 ° + I d + Iatm + hpt) C = A = ^ y (/c ^ ^ o s l0 ^ T 7 ^ P ^ o s # ^ p T 7 ic ^ s in A 0 ^ k /^ P ^ in A 0 ^ (B-8) (B.9) w = 1 I CPC sin 40c + I d P d sin 40c - a rc ta n ——------ —------ ——------- ——. 4 I CPC cos 40c + I d P d c o s 40d (Jd.10) The variable e is modulation efficiency. In this memo, we assume th at the modulation efficiency is independent of electromagnetic frequency. When there are two bands for measurements, there are two measured IVAs h = Ci + Ax cos (4p —4 ^ i) (B -ll) I2 = C 2 + A 2 cos (4p —40>2)■ (B.12) 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore, we can reconstruct the 4 unknown parameters (P c ,^ D ,o c ,c to ) from 4 known parameters by using Equations B.8, B.9, and B.10 as I d i I d = 2Ci — Ici = 2 (B.13) C'-2 —I c 2 (B-14) qc = Pc cos 4<f>c 2 (B.15) 2 I uc = = = d {I D2 2 — I \I d c c i I d c (I d 2 \I d c c 2Id 1 — I — Id \— cos 4 ^ 2) (B.16) Ao sin 4 ^ —I d i — sin 4 ^ 2 ) Cl (B.18) (-2 (B.19) (Ic2 \I c c A i 2 d A \ Cl 2 PD sin 4(f)D 2 I 4t/>! (B.17) 2 — I 2Id \ — I COS Cl 2 Pc cos 4(f)c 2 I UD I Pc sin 4(pc 2 I qu c i Ic i— cos 4 ^ 2 ) ^2 (B.20) (B.21) (■IC2 \I d A i 2 Cl Ici— s in 4 ^ 2 )- (B.22) ( B . 22) £2 (B.23) Therefore, 4>c = Pc = 4>d = Pd = 1 , Uc - arctan — 4 qc \/qc + uc 1 , Ud - arctan — 4 qo J q b + ud (B.24) (B.25) (B.26) (B.27) Notice th at q c ,u c ,q D ,u D are not expressed by the polarization angles but by the phases. To associate the phase to the input polarization angle, we can use the rela tionship in Equation B.3. This relationship depends on the incident spectrum. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B .4 C on clu sion We show th at the combination of two polarized radiation with different spectra can be reconstructed by using the HWP polarimeter with two bands. Even though <fi — a relationship varies depending on the incident spectrum with the use of the AHWP po larimeter, this source of errors can be removed by knowing the two (f>—a relationships for the CMB and dust independently in advance. Therefore, under the assumptions we made, the source of cross polarization depends only on the preflight calibration of 4>— a relationship for a given frequency spectrum. We assume th at the total intensity of dust spectrum is known to the level we need. But in reality this may not be true. The uncertainty of the dust spectrum induces errors in the preflight calibration of <f>— a relationship and in the extraction of 4>c and 4>d - The former error is due to the AHWP polarimetry and the latter error is due to the foreground removal. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ppendix C Second order effective m edium theory Brauer et al. [60] shows the connections relationship between the effective index of refraction and the area fraction of two media for the two dimensional squire pattern by using the approximated second order effective medium theory. They also show the consistency between the approximated 2nd order EM T and the rigorous coupledwave analysis (RCWA) [62] th at does not assume any approximation t o provide a same relationship. We employed the EMT, because the RCWA is computationally intensive method even though it does not use the approximation. The approximated 2nd order EMT and the RCWA are same at the limit of TA < — n v +[—n s . (C .l) where nv and n s are the indices of refraction of two sides of media. Typically, nv is the index of refraction of vacuum and the n s is the index of refraction of the substrate. The variable d is the grid spacing and the A is the wavelength of the incident radiation. The incident radiation is assumed to be normal. (2) The effective index of refraction due to the square pattern n 2£, can be estimated by averaged 2nd order ID EMT as n ?D = + 2^2n + 2 ” 2§)> 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.2) where n 4d V4S (n = (1 - / 2)ni + f n 2 (C.3) ~ (1 —/)el + / e±^ (C-4) = (!- /) /£i + /A||2) (C-5) = ./I ) , (C.6) where 42) = ' H 1 + 6 z e± <2) = + <c '7) e I Co J°) <J e0 (0) 1!C— )2) ’ i £2 <c ' 8 ) where 4°^ — (1 —/ ) ei + / e2 1 / cj°^ “ (1 —/ ) A i + / / e2- (C-9) (C.10) The variables ei and e2 are the dielectric constants for the first medium (vacuum) and the second medium (substrate), respectively. The parameter / and r are defined as i - i <c -n > r = j, (C.12) where w is the width of square as shown in Figure 3.23. Notice th at the effective index of refraction depends on the wavelength of the incident radiation. Throughout our discussion, the index of refraction of the substrate is fixed at 120 GHz ~ 450 GHz. In reality, the index of refraction is a function of frequency. Additionally, the SWG AR coating has a variation of the index of refraction at a given geometry of the structure. 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B ibliography [1] E. V. Loewenstein. Optical Constants for Far Infrared Materials. I. Crystalline Solids. Applied Optics, 12(2):398-406, February 1973. [2] P. Oxley, P. Ade, C. Baccigalupi, P. deBernardis, H.-M. Cho, M. J. Devlin, S. Hanany, B. R. Johnson, T. Jones, A. T. Lee, T. Matsumura, A. D. Miller, M. Milligan, T. Renbarger, H. G. Spieler, R. Stompoer, G. S. Tucker, and M. Zaldarriaga. The EBEX Experiment. In W. L. Barnes and J. J. Butler, editors, Earth Observing Systems IX: Infrared Spacebourne Remote Sensing, volume 5543 of Proceedings of SPIE, pages 320-331, 2004. astro-ph/0501111. [3] L. Page, G. Hinshaw, E. Komatsu, M. R. Nolta, D. N. Spergel, C. L. Bennett, C. Barnes, R. Bean, O. Dore, M. Halpern, R. S. Hill, N. Jarosik, A. Kogut, M. Limon, S. S. Meyers, N. Odegard, H. V. Peiris, G. S. Tucker, L. Verde, J. L. Weiland, E. Wollack, and E. L. Wright. Three Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Polarization Analysis. A p J submitted, March 2006. astro-ph/0603450. [4] S. Hanany, T. Matsumura, B. Johnson, T. Jones, J. R. Hull, and K. B. Ma. A cosmic microwave background radiation polarimeter using superconducting bearings. IEEE Transactions on Applied Superconductivity, 13:2128-2133, june 2003. [5] A. A. Penzias and R. W. Wilson. A Measurement of Excess Antenna Tempera ture at 4080 Mc/s. Ap. J., 142:419-421, July 1965. [6] J. C. Mather, E. S. Cheng, D. A. Cottingham, R. E. Eplee, D. J. Fixsen, T. Hewagama, R. B. Isaacman, K. A. Jensen, S. S. Meyer, P. D. Noerdlinger, 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S. M. Read, L. P. Rosen, R. A. Shafer, E. L. Wright, C. L. Bennett, N. W. Boggess, M. G. Hauser, T. Kelsall, S. H. Moseley, R. F. Silverberg, G. F. Smoot, R. Weiss, and D. T. Wilkinson. Measurement of the cosmic microwave back ground spectrum by the COBE FIRAS instrument. Ap. J., 420:439-444, January 1994. [7] G. F. Smoot, C. L. Bennet, A. Kogut, E. L. Wright, J. Aymon, N. W. Boggess, E. S. Cheng, G. De Amici, S. Gulkis, M. G. Hauser, G. Hinshaw, P. D. Jackson, M. Janssen, E. Kaita, T. Kelsall, P. Keegstra, C. Lineweaver, K. Lowenstein, P. Lubin, J. Mather, S. S. Meyer, S. H. Moseley, T. Murdock, L. Rokke, R. F. Silverberg, L. Tenorio, R. Weiss, and D. T. Wilkinson. Structure in the COBE Differential Microwave Radiometer First-Year Maps. Ap. J., 396:L1-L5, 1992. [8] S. Hanany, P. Ade, A. Balbi, J. Bock, J. Borrill, A. Boscaleri, P. de Bernardis, P. G. Ferreira, V. V. Hristov, A. H. Jaffe, A. E. Lange, A. T. Lee, P. D. Mauskopf, C. B. Netterfield, S. Oh, E. Pascale, B. Rabii, P. L. Richards, G. F. Smoot, R. Stompor, C. D. W inant, and J. H. P. Wu. Maxima-1: A measurement of the cosmic microwave background anisotropy on angular scales of 10’-5°. Ap. J. Lett., 545:L5-L9, December 2000. astro-ph/0005123. [9] P. de Bernardis, P. A. R. Ade, J. J. Bock, J. R. Bond, J. Borrill, A. Boscaleri, K. Coble, B. P. Crill, G. De Gasperis, P. C. Farese, P. G. Ferreira, K. Ganga, M. Giacometti, E. Hivon, V. V. Hristov, A. Iacoangeli, A. H. Jaffe, A. E. Lange, L. Martinis, S. Masi, P. V. Mason, P. D. Mauskopf, A. Melchiorri, L. Miglio, T. Montroy, C. B. Netterfield, E. Pascale, F. Piacentini, D. Pogosyan, S. Prunet, S. Rao, G. Romeo, J. E. Ruhl, F. Scaramuzzi, D. Sforna, and N. Vittorio. A Flat Universe from High-Resolution Maps of the Cosmic Microwave Background Radiation. Nature, 404:955-959, April 2000. astro-ph/0004404. [10] N. Halverson et al. Ap. J., 568:38, 2001. [11] C. L. Kuo, P. A. R. Ade, J. J. Bock, C. Cantalupo, M. D. Daub, J. Goldstein, W. L. Holzapfel, A. E. Lange, M. Lueker, M. Newcomb, J. B. Peterson, J. Ruhl, M. C. Runyan, and E. Torbet. High-Resolution Observations of the Cosmic Mi140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. crowave Background Power Spectrum with ACBAR. Ap. J., 600:32-51, January 2004. astro-ph/0202289. [12] G. Hinshaw, M. R. Nolta, C. L. Bennett, R. Bean, 0 . Dore, M. R. Greason, M. Halpern, S. Hill, N. Jarosik, A. Kogut, E. Komatsu, M. Limon, N. Odegard, S. S. Meyer, L. Page, H. V. Peiris, D. N. Spergel, G. S. Tucker, L. Verde, J. L. Weiland, E. Wollack, and E. L. Wright. Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis. A p J subitted, March 2006. astro-ph/0603451. [13] J. M. Kovac, E. M. Leitch, C. Pryke, J. E. Carlstrom, N. W. Halverson, and W. L. Holzapfel. Detection of polarization in the cosmic microwave background using DASI. Nature, 420:772, December 2002. astro-ph/0209478. [14] E. M. Leitch, J. M. Kovac, N. W. Halverson, J. E. Carlstrom, C. Pryke, and Smith. M. W. E. Dasi three-year cosmic microwave background polarization results. Astrophys. J., 624:10-20, 2005. astro-ph/0409357. [15] T. Montroy, P. A. R. Ade, J. J. Bock, J. R. Bond, J. Borrill, A. Boscaleri, P. Cabella, C. R. Contaldi, B. P. Crill, P. de Bernardis, G. de Gasperis, A. de Oliveira-Costa, G. de Troia, G. di Stefano, E. Hivon, A. H. Jaffe, T. S. Kisner, W. C. Jones, A. E. Lange, S. Masi, P. D. Mauskopf, C. MacTavish, A. Melchiorri, P. Natoli, C. B. Netterfield, E. Pascale, F. Piacentini, D. Pogosyan, G. Po lenta, S. Prunet, S. Ricciardi, G. Romeo, J. E. Ruhl, M. Santini, M. Tegmark, M. Veneziani, and N. Vittorio. A Measurement of the CMB jEE^ Spectrum from the 2003 Flight of BOOMERANG. Submitted to Astrophysical Journal. astro-ph/0507514. [16] D. Barkats, C. Bishoff, P. Farese, L. Fitzpatrick, T. Gaier, J. O. Gunderson, M. M. Hedman, L. Hyatt, J. J. McMahon, D. Samtleben, S. T. Staggs, K Vanderlinde, and B. Winstein. First Measurements of the Polarization of the Cos mic Microwave Background Radiation at Small Angular Scales from CAPMAP. Ap. J., 619:L127-L130, 2005. astro-ph/0409380. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [17] J. L. Sievers, Achermann C., J. R. Bond, L Bronfman, R. Bustos, C. R. Contaldi, C. Dickinson, P.G. Ferreira, M. E. Jones, A. M. Lewis, B. S. Mason, J. May, S. T. Myers, S. Padin, T. J. Pearson, M. Pospieszalski, A. C. S. Readhead, R. Reeves, A. C. Taylor, and S. Torres. Implications of the Cosmic Background Imager Polarization Data. Submitted to Ap. J., astro-ph/0509203, 2005. [18] W. Hu, N. Sugiyama, and J. Silk. The physics of microwave background anisotropies. Nature, 386:37, 1997. astro-ph/9604166. [19] W. Hu and M. White. A CMB polarization primer. New Astronomy, 2:323-344, 1997. astro-ph/9706147. [20] A. H. Guth. Inflationary universe: A possible solution to the horizon and flatness problems. Phys. Rev. D., 23:347-356, January 1981. [21] B. R. Johnson, M. E. Abroe, P. Ade, J. Bock, J. Borrill, J. S. Collins, P. Fer reira, S. Hanany, A. H. Jaffe, T. Jones, A. T. Lee, L. Levinson, T. Matsumura, B. Rabii, T. Renbarger, P. L. Richards, G. F. Smoot, R. Stompor, H. T. Tran, and C. D. W inant. MAXIPOL: a balloon-borne experiment for measuring the polarization anisotropy of the cosmic microwave background radiation. New A s tronomy Review, 47:1067-1075, December 2003. astro-ph/0308259. [22] F. C. Moon. Superconducting Levitation. Wiley and Sons, New York, 1994. [23] J. R. Hull. Topical Review: Superconducting bearings. Superconductor Science Technology, 13:1, February 2000. [24] Hull, John R. Effect of permanent-magnet irregularities in levitation force mea surements. Supercond. Sci. Technol., 13:854-856, 2000. [25] S. Pancharatnam. Achromatic combinations of birefringent plates. Raman Re search Inst. Bangalore, Memoir, 71:137-144, 1955. [26] Hull, John R. and Murakami, M. Applications of bulk high-temperature Su perconductors. IEEE Transactions on Applied Superconductivity, 92:1705-1718, 2004. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [27] J. R. Hull, S. Hanany, T. Matsumura, B. Johnson, and T. Jones. Characteriza tion of a high-temperature superconducting bearing for use in a cosmic microwave background polarimeter. Supercond. Sci. Technol., page submitted, 2004. [28] T. Matsumura, S. Hanany, B. Johnson, T. Jones, J. R. Hull, and P. Oxley. Magnetic field inhomogeneity and torque in high tem perature superconducting magnetic bearings. IEEE Trans. Appl. Super., Volume: 15, Issue: 2:2316-2319, 2005. [29] T. Matsumura, S. Hanany, J. R. Hull, B. Johnson, T. Jones, and P. K. Oxley. Development of a cryogenic induction motor for use with a superconducting magnetic bearing. Physica C Superconductivity, 426:746-751, October 2005. [30] Bean, C. P. Reviews of Moder Physics, page 31, 1962. [31] M. Zeisberger, W. Gawalek, T. Habisreuther, B. Jung, D. Lizkendorf, and T. Straber. In Applied Superconductivity Conference, 1998. [32] Richards, P. L. and Tinkham, M. J. Appl. Phys., 43:2680, 1972. [33] Zeisberger, M. and Gawalek, W. Losses in magnetic bearings. Mat. Scie. Eng., page 193197, 1998. [34] http://cryogenics.nist.gov/. [35] Hull, John R. and Mulcahy, T. M and Uherka, K. L. and Abboud, R. G. Low rotational drag in high-temperature superconducting bearings. IEEE trans. Appl. Supercon., 5:626-629, 1995. [36] G. Sotelo, A. Ferreira, and de Andrade R. Halbach Array Superconducting Magnetic Bearing for a Flywheel Energy Storage System. IEEE Trans. Appl. Supercond., 15:2253-2256, 2005. [37] Bean, C. P. Phys. Rev. Lett., 8:250, 1962. [38] J. R. Hull, J. F. Labataille, T. M. Mulcahy, and J. A. Lockwood. Reduced hys teresis loss in superconducting bearings. Applied Superconductivity, 41:1, 1996. 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [39] Weinberger, B. R. and Lynds, L. and Hull, J. R. and Balachandran, U. Low friction in high tem perature superconductor bearings. Appl. Phys. Lett., 59:1132 - 1134,1991. [40] Y. Zhang, Y. Postrekhin, K. B. Ma, and W. K. Chu. Reaction wheel with HTS bearings for mini-satellite attitude control. Supercond. Sci. Technol., 15:823-825, May 2002. [41] John R. Hull and Ahmet. Cansiz. Vertical and lateral forces between a perma nent magnet and a high-temprature superconductor. Journal of Applied Physics, 86:6396, 1999. [42] J. R. Hull and T. M. Mulcahy. Gravimeter using high-temperature supercon ducting bearing. IEEE Transactions on Applied Superconductivity, 9:390, 1999. [43] Trust Automation Inc. TA 310 linear amplifier. [44] E. Hecht. Optics. Addison-Wesley, 1998. [45] W. A. Shurcliff. Polarized light. Harvard U. P., Cambridge, 1966, 1966. [46] J. Tinbergen. Astronomical Polarimetry. Cambridge University Press, Cam bridge, UK, 1996. [47] A. M. Title. Improvement of birefringent filters. Applied Optics, 14:229-237, 1975. [48] D. Clarke. Interference effects in compound and achromatic wave plates. J. Opt. A: Pure Appl. Opt., 6:1041-1046, 2004. [49] S. Hanany, H. Hubmayr, B. R. Johnson, T. Matsumura, P. Oxley, and Thibodeau M. Millimeter-wave achromatic half-wave plate. Applied Optics, 44:4666-4670, August 2005. [50] h ttp://w ww.nsf.gov/m ps/ast/tfcr_final_report.pdf. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [51] Joseph E. Wells. Angular encoder for a superconducting magnetic bearing in a cosmological application. M aster’s thesis, University of Minnesota, Twin Cities, 2005. [52] A. M. Title and W. J. Rosenberg. Achromatic retardation plates. In Polarizers and Applications, SPIE Proceedings Vol. 307. Edited by Giorgio B. Trapani. Bellingham, WA: Society for Photo-Optical Instrumentation Engineers, 1981., p. 120, January 1981. [53] T. J. Jones. Polarimetry in the visible and infrared: application to CMB polarimetry. New Astronomy Review, 47:1123-1126, December 2003. [54] Grant R. Fowles. Introduction to Modern Optics. Dover, 1975, 1975. [55] Enger, Rolf C. and Case, Steven K. Optical elements with ultrahigh spatialfrequency surface corrugations. Applied Optics, 22, 1983. [56] Ono, Yuzo and Kimura, Yasuo and Ohta, Yoshinori and Nishida, Nobuo. Antire flection effect in ultrahigh spatial-frequency holographic relief gratings. Applied Optics, 26, 1987. [57] Y. Kanamori, M. Sasaki, and K. Hane. Broadband antireflection gratings fabri cated upon silicon substrates. Optics Letters, 24(20), 1999. [58] K. Kintaka, J. Nishii, A. Mizutani, H. Kikuta, and H. Nakano. Antireflection microstructures fabricated upon fluorine-doped sio2 films. Optics Letters, 26(21), November 2001. [59] Dobrowolski, J. A. and Poitras, D. and Penghui, M. and Vakil, H. and Acree, M. Toward perfect antireflection coatings: numerical investigation. Appl. Optics, 41:3075-3083, 2002. [60] R. Brauer and O. Bryngdahl. ’’design of antireflection gratings with approximate and rigorous methods” . Applied Optics, 33(34), December 1994. [61] http://w w w .rogerscorporation.com /. 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [62] T. K. Gaylord and M. G. Moharam. Analysis and application of optical diffrac tion by gratings. In The Cosmological Model, volume 73 of Proceedings of the IEEE, page 894, 1985. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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