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A cosmic microwave background radiation polarimeter using superconducting magnetic bearings

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A 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
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UMI Number: 3234933
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© Tomotake Matsumura 2006
ALL RIGHTS RESERVED
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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
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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.
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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
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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
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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
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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
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137
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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
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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
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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
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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
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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
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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
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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
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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
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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
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. . .
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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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(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
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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
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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
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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
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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
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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
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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
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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
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(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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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- 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
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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
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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
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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
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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
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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
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(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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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,
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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(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
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(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
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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
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(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
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(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
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(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
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(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
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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
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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
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(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
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$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
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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
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(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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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(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
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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
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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
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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
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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
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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
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{ 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
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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
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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
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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
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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
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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
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(A -3)
130
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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
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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
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(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
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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
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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
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136
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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
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(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
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