other polarimetersoberon.roma1.infn.it/.../pdf_2006/lezione_09.pdfother polarimeters • wave plates...
TRANSCRIPT
Other polarimeters• Wave plates can be used to build
polarimeters sensitive to linear or to circular polarization.
• Half-wave plate (φ=180o)
• Quarter-wave plate (φ=90o)
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−=
01001000
00100001
QWP
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
=
1000010000100001
HWP
Polarimeter with HWP
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
0000000000110011
HP
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
=
1000010000100001
HWP
θ
source Polarizer
Intensity detector
HWP
( ) ( ){ } SRHWPRPDW H ×××−××= θθ
( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−=
100000000001
22
22
cssc
R θ
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
VUQI
S ( )0001=D ( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
=
100000000001
22
22
cssc
R θ
Polarimeter with HWP
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
10000000000001
100000000001
1000010000100001
2
2
22
22
cc
cssc
θ
source Polarizer
Intensity detector
HWP
( ) ( ){ } SRHWPRPDW H ×××−××= θθ
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
10000000000001
10000000000001
100000000001
22
22
2
2
22
22
cc
cc
cssc
Polarimeter with HWP
θ
source Polarizer
Intensity detector
HWP
( ) ( ){ } SRHWPRPDW H ×××−××= θθ
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
00000000001001
10000000000001
0000000000110011
22
22
22
22 c
c
cc
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
++
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
00
00000000001001
22
22
22
22
QcIQcI
VUQI
cc
( ) ( )222
122
22
21
00
0001 QcIQcIQcI
W +=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
++
= ( )off
QIW4
2cos221
=
+= θ
Polarimeter with QWP
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
0000000000110011
HP
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−=
01001000
00100001
QWP
θ
source Polarizer
Intensity detector
QWP
( ) ( ){ } SRQWPRPDW H ×××−××= θθ
( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−=
100000000001
22
22
cssc
R θ
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
VUQI
S ( )0001=D ( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
=
100000000001
22
22
cssc
R θ
Polarimeter with QWP
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−001000
000001
100000000001
01001000
00100001
22
22
22
22
cs
sccssc
θ
source Polarizer
Intensity detector
QWP
( ) ( ){ } SRHWPRPDW H ×××−××= θθ
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
000000
0001
001000
000001
100000000001
22222
22222
22
22
22
22
csscsscc
cs
sccssc
Polarimeter with QWP
θ
source Polarizer
Intensity detector
QWP
( ) ( ){ } SRHWPRPDW H ×××−××= θθ
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−−−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
00000000
11
000000
0001
0000000000110011
22222
22222
22222
22222 sscc
sscc
csscsscc
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−++−++
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
00
00000000
11
22222
22222
22222
22222
VsUscQcIVsUscQcI
VUQI
ssccsscc
( )θθθθ 2sin2sin2cos2cos 221 VUQIW +++=
of4 of2of4
Modulation techniques for polarization experiments
All these techniques suffer for the need of long integration time: One needs to point to the same sky pixel during many cycles ofthe analyzer. It is thus difficult to produce extended maps of the CMB polarization.
A two-bolometers polarimeter ?
We can map the two orthogonal components of the linear polarization with two separate bolometers, and combine the two signals to retreive the Stokes parameters Q and U.
The modulation is obtained by scanning the sky at constant rate, so that the polarization signals are detected at a frequency far from the 1/f knee of the noise and far from the effect of instrumental drifts (similar to the anisotropy measurements with B98)f = v l/π : with l=300-1000 and v=1o/s f = 1.7-5.5 Hz
scan
I1 I2Q = I1(t1) – I2(t2)
Terminology• Co-polar response of the polarimeter C:
response of the polarimeter to incoming radiation 100% polarized along the principal axis of the polarimeter. It is the product of the detector responsivity times the integral of the Co-polar beam response. Units: V/W or V/K
• Cross-polar response of the polarimeter X:response of the polarimeter to incoming radiation 100% polarized and orthogonal tothe principal axis of the polarimeter. It isthe product of the detector responsivity times the integral of the cross-polar beam response. Units: V/W or V/K
• Polarization efficiency = 1 – X/C
I
I
S=CI={(Σ+Δ)/2}I
S=XI ={(Σ−Δ)/2}I
Incomingdiffuseradiation
Polarimeter
Principalaxis
Principalaxis
Frequency content of detected signal• The detected signal is the sum of several contributions. • First order approximation, for bolometer #1:
S1 = C1 x ( TCMB + O + D + ΔB/2 + ΔTU /2 + ΔP1) + N1+ X1x (.. +ΔP2)
Co-polarresponse CMB &
Offset0 Hz
InstrumentDrifts0-0.01 Hz
Cross-polarresponseCMB
Anisotropy,UnpolarizedComponentAlong dir. 10.5-5.5 Hz
CMBPolarizedComponentAlong dir.11.7-5.5 Hz
Noise0-20Hz
These dominantcomponents can be removed using a high-pass filter
These components are detected
ΔBKGIn thef range of interestunpolarized
High-pass filtered signals
S1 = C1 x (ΔB /2 + ΔTU /2 + ΔP1) + X1 x (ΔB /2 + ΔTU /2 + ΔP2) + N1
S2 = C2 x (ΔB /2 + ΔTU /2 + ΔP2) + X2 x (ΔB /2 + ΔTU /2 + ΔP1) + N2
S1 = (C1 + X1) x (ΔB + ΔTU)/2 + C1 x ΔP1 + X1 x ΔP2 + N1S2 = (C2 + X2) x (ΔB + ΔTU)/2 + C2 x ΔP2 + X2 x ΔP1 + N2
Let’s assume C1C2 >> X1X2
(example: with a polarization efficiency = 90% we have already 0.81 >> 0.01 )and do the math: solve for ΔP1 and ΔP2 and compute the Stokes parameter:Q = ΔP1 - ΔP2 = …..
“nuisance” signals “interesting” signals noise
Q = ΔP1 - ΔP2 = = (S1/C1)(1+X2/C2) – (S2/C2)(1+X1/C1)+ (ΔB + ΔTU) x {X1/C1 – X2/C2}/2 + (N2/C2)(1+X1/C1) – (N1/C1)(1+X1/C1)
• The signal term is simply S1- S2 with the two signals corrected for different co-polar and cross-polar responses, including different responsivities, efficiencies, etc.
• The common mode signal (ΔB + ΔTU) is reduced by a factor {X1/C1 – X2/C2}/2 with respect to the polarization signal. This means that it disappears if the two detectors have identical polarization efficiencies (notice that perfect responsivity matching of the two bolometers is not important for this term). Otherwise this term will become comparable to what we want to measure if{X1/C1 – X2/C2} ΔB /2 =ΔP and {X1/C1 – X2/C2} ΔTU/2 = ΔP
• If we are able to reduce the cross-polar response of the two polarimeters below 5% of the co-polar one, then ΔB and ΔTU can be up to 40 times larger than ΔP ; if, in addition, the cross-polar and co-polar responses are matched within 5%, then ΔB and ΔTU can be upto 8000 times larger than ΔP
Noise terms
Common mode term
Difference Signal
• Sources of cross-polarization are asimmetries in the optical system, the filters, the feed horns, and imperfection of the analyzers used in the detectors.
• The best system is a completely azimuthally simmetricone: a Cassegrain telescope is perfect.
• However, this perfection degrades immediately as soon as we consider off-axis locations of the detector in the focal plane.
• The same is true for off-axis telescopes, even if they satisfy the Mizuguchi-Dragone condition. (IEEE Trans.AP-30, 331, 1982; IEEE Trans.AP-22, 472, 1974 )
• In BOOMERanG we have an off-axis telescope (close to Dragone) and we have analyzed in detail its polarization properties on a wide region of the focal plane.
How much is the Cross Polarization X ?
X and the BOOMERanG telescope• The telescope is an off-axis gregorian
with a 1.3m diameter aluminum primary enclosed in a low emissivity cavity.
• Secondary and tertiary (Lyot stop) are inside the cryostat, and reimagethe primary focal plane inside thedewar, improving sidelobes rejectionand significantly reducing the straylight on the detectors.
• On the edge of the focal plane (1.5o
off-axis in azimuth, 0. 5o off-axis in elevation) we have (phys.optics)
X1/C1=1.2 10-3 X2/C2=3.2 10-3
• Most of the cross polar response comes from the asimmetry of theprimary illumination which derives from the limited size of the tertiary
R
TS
P
21
• ΔB are fast fluctuation of the background. In order to be really dangerous, theyshould be synchronous with the sky. Otherwise, they just add noise. Difficult to quantify how much they are. The sky-synchronous component has to be small.
• Instead of writing more or less optimistic estimates of ΔB , we have used real data from our B98 flight, where we had several pixels, not sensitive to the polarization, but sensitive to all possible background fluctuations. From Netterfield et al.:
How much is ΔB ?
D = location of detectors
4o on the sky
B150A B150B
B150A1
B150A2
B150B1
B150B2
• In B98 six 150 GHz bolometers (not sensitive to polarization)scanned the same sky pixel at different times.
• All the bolometers had independent readout channels, so we can have an idea from real world data of the effects of noise, drifts, responsivity changes one gets measuring the difference of maps obtained by independent bolometers.
• The sensitivity (using 4 channels) would have been just enough to detect the first peak of the polarization spectrum at l= 410, and the first peak of the polarization-temperature cross-spectrum at l = 330 with high S/N.
0 100 200 300 400 500 600 700
-100
-50
0
50
100
PS {(A+A1)/2 - (A2+B2)/2} (B98, 9 days, 1.8% of sky) <EE> for best fit model to anisotropy data <TE> for best fit model to anisotropy data
l(l+1
)cl /
2π
(μK
2 )multipole
• We conclude that even without any improvement with respect to the B98 setup, it would be possible to constrain significantly the polarization of the CMB with two independent bolometers sensitive to orthogonal polarization components.
• There are many ways to improve !
• The first one is to optimize the observation strategy. In the “discovery mode” we want to map a sky region smaller than B98, accumulating much more integration time for each pixel:– B98 : few seconds per pixel, 100000 pixels, 7’x7’ each– B2K: >100 seconds per pixel, 7000 pixels, 7’x7’ each
A factor 5-6 improvement in S/N (sampling variance not dominant)
PSB: Polarization Sensitive Bolometers (JPL+Caltech)
• 150 GHz• Two wire-grid-like
absorbers with matched NTD thermistors
• Rotated 90°• Very close each
other (60 μm) inside the same groove of a corrugated circular feedhorn
MetalizedSi3N4 wires
T sensorB. JonesA. Lange
MetalizedSi3N4 wires
T sensorB. JonesA. Lange
Minimize Xpolby concentratingthe E field in thecenter
ε ~ 1.3%
ε ~ 5.5%
HFSS simulations
B.Jones et al. Astro-ph/0209132
Polarization-sensitive bolometersJPL-Caltech
3 μm thickwire grids,Separated by60 μm, in the same groove of a circular corrugatedwaveguide
Planck-HFItestbed
B.Jones et al. Astro-ph/0209132
Universita’ di Roma, La Sapienza:P. de Bernardis, S. Masi, F. Piacentini,
A. Iacoangeli, A. Melchiorri, G. Polenta, G. de Troia, S. Ricciardi, F. NatiCase Western Reserve University:J. Ruhl, T. Kisner, E. Torbet, T. MontroyCaltech/JPL: A. Lange, J. Bock, W. Jones, V. HristovUniversity of Toronto: B. Netterfield, C. MacTavish, E. PascaleCardiff University: P. Ade, P. Mauskopf IROE: A. Boscaleri
ING: G. Romeo, G. di StefanoCSU, Dominguez Hills: B. Crill IPAC: K. Ganga, E. HivonCITA: D. Bond, C. Contaldi LBNL, UC Berkeley: J. BorrillImperial College: A. Jaffe Institut d’ Astrophysique: S. PrunetUniversity of Alberta: D. Pogosyan U. Penn.: M. Tegmark, A. de Oliveira-CostaUniversita’ di Roma, Tor Vergata: N. Vittorio, G. de Gasperis, P. Natoli, A. Balbi, P. Cabella
Boomerang 2003
Polarization Sensitive Bolometers
PSB Pair
Corrugated cilindrical feedhorns
The Back to Back input feeds
2-Color Photometer
30’
30’
30’ 30’azimuth
elev
atio
n
Receiver Specifications
Polarization Efficiency
PSB’s have an efficiency of 90-95%Photometer channels have an efficiency > 97%
PSB pair Two 245 GHz detectors 45° apart
S(θ) = γ(1 – β sin2(θ-θ0))
β is the polarization efficiency
Measuring Polarization with BOOMERANG
Each Detector is sensitive to one linear polarization.
Stokes Parameters
I = Ex2 + Ey
2 Q = Ex2 - Ey
2 U = 2ExEy
A pair of orthogonal detectors measures:
Sx = γx{ (1 – βx/2) I + (βx/2) Q}
Sy = γy{ (1 – βy/2) I - (βy/2) Q}
If γx = γy and βx = βy, we have
I = Sx + Sy
2 γ (1 – β/2)
Q = Sx - Sy
γ β
Instrument Calibration
• Absolute Calibration with respect to CMB anisotropies
- Primary method: cross-calibration with WMAP and B98
- Secondary methods: CMB dipole and RCW 38
• Beams
- Primary: Quasar in Deep Scan Region
- Secondary: Pre-flight using tethered source 1 km from Telescope
• Polarization
- No measured polarized astrophysical sources at our frequencies
- Primary polarization calibration done with a polarized far-field simulator
Theory
ΔT
QScan and
MapMaking
Cross-Polarization effects
X=10%
X=1%
Observation
Theory
ΔT
Q
X=1%
X=10%+ 10% accuratecorrection
Scan andMap
Making
Cross-Polarization effects
Observation
Calibration• We need 1% calibration.• This can be done, in the lab by means of a special full beam
calibrator {measures Σ, Δ (or C and X) and the principal axis direction for each bolometer}
• In flight, thanks to WMAP (which is calibrated to better than 1% !).
• If we correlate the two bolometers separately with the unpolarized WMAP maps, we can estimate the responsivity to better than 1%.
0.95 +/- 0.010.95 +/-0.03Sum
0.95 +/- 0.020.95 +/- 0.03B150B2
0.98 +/- 0.020.98 +/- 0.03B150A2
0.92 +/- 0.030.91 +/- 0.03B150A1
0.96 +/- 0.020.95 +/- 0.03B150A
C(l) basedPixel basedChannel
E.Hivon on B98
Pre-flight Beam Mapping
1 km
Dirigible
Kevlar String
Thermal Source 1 km
Collimated Polarized Source
Used to measure the polarization efficiency and the polarization angles of the full integrated telescope.
Pre-flight calibrations• The calibration of a polarimeter at ground is
more difficult than the usual photometer calibrations.
• In particular is very important to study the co and cross polarization response (beam and integral) of the polarimeter
• We have developed a polarized, sine-modulated source filling the beam of the instrument to carry out a through polarization characterization of all the detectors.
• There are two wire grid polarizers (P1 and P2), and a 77K blackbody source with a diaphragm in the focus of the 1.3 m off-axis paraboloid, producing a 10’ beam.
• Rotating P2 at constant speed we modulate the signal (sine wave).
• Rotating P1 (in steps) we change the illuminator from co-polar to cross-polar (and all intermediate directions).
P1P2
To polarimete
r
77K
M
S
1.3m
The Calibrator
PayloadSCAN
Modulation
•Additional Pointing Sensors
Tracking Star Camera Pointed Sun Sensor
with 16 bit abs. encoders
BOOM03 Flight
11.7 days of good data
Launched:January 6, 2003
From:
McMurdo Station,
Antarctica
Measurements OKfor 11.6 days
BOOMERanGlanded nearDome Fuji (h=3700m)after 14 daysof flight.The data have beenrecovered. The payloadis still there.
Scan Strategy
4.7 secPolarized Foregrounds390 Galactic Plane3.3 sec<TE> and <TT>1130Shallow CMB
60 sec<EE>115Deep CMB
Time per 7’ pixelGoalSize (sq deg)Region
Optimal maps obtained with IGLS, the Rome pipeline (Natoli et al. 2001)
Optimal maps obtained with IGLS, the Rome pipeline (Natoli et al. 2001)
Optimal maps obtained with IGLS, the Rome pipeline (Natoli et al. 2001)
Optimal maps obtained with IGLS, the Rome pipeline (Natoli et al. 2001)
ell bins ( Δ ell = 75) < 10% correlated
Boomerang 2002: ~ 200 square degrees
<TT>
<TE>
<EE>
Bill Jones
Boomerang 2002: ~900 square degrees
ell bins ( Δ ell = 75) < 10% correlated
<TT>
<TE>
<EE>
Bill Jones
MAP 2 year data
<TT>
<TE>
<EE>
Bill Jones
Survey Strategy
4.7 secPolarized Foregrounds390 Galactic Plane3.3 sec<TE> and <TT>1130Shallow CMB
60 sec<EE>115Deep CMB
Time per 7’ pixelGoalSize (sq deg)Region
IRAS 3000 GHz BOOM 150 GHz
B2K - Galactic Plane SurveyRCW38IRAS08576
IRAS 3000 GHz BOOM 245 GHz
B2K - Galactic Plane Survey
IRAS 3000 GHz BOOM 340 GHz
B2K - Galactic Plane Survey
IRAS 3000 GHz BOOM 340 GHz
B2K - Galactic Plane Survey
What is this ?
IRAS 3000 GHz WMAP Ka 33 GHz
IRAS 3000 GHz WMAP Q 41 GHz
IRAS 3000 GHz WMAP V 60 GHz
IRAS 3000 GHz WMAP W 94 GHz
IRAS 3000 GHz CO map
COLD DUST associated to a molecular cloud
Scans on RCW38 a bscan
a+b
a-b
Polarization not visible in single scans
Scans on RCW38
a
b
scan
a+b
a-b
Polarization not visible in single scans
Scans on RCW38
a+b
a-b
ab
scan
Polarization not visible in single scans
Scans on RCW38
a+b
a-b
a
bscan
Polarization not visible in single scans
BOOM 145 GHz T BOOM 145 GHz P
22P UQ +=polarization is visible when coadding,in the brightest sources at the few % level.
fainter
brighter
fainter
brighter
Optimal maps obtained with IGLS, the Rome pipeline (Natoli et al. 2001)
Optimal maps obtained with IGLS, the Rome pipeline (Natoli et al. 2001)
Optimal maps obtained with IGLS, the Rome pipeline (Natoli et al. 2001)
Is there any signal in the Q,U maps ?
( ) ( )[ ]
( ) ( ) ( ) ( )22
1
;;
exp2
1)(
kijij
kksky
kskyij
i
i
i
jkskyijik
skyiji
ksky
NRCUQT
XUQT
k
XCXC
L
+=⎪⎩
⎪⎨
⎧=
⎪⎩
⎪⎨
⎧=
−∏= −
θσσ
σσπ
σ
• Likelihood to measure the observed signals in the i=1..N pixels. For gaussian correlated signal and noise:
NormalizedSky covariance
Pixel Noisecovariance
Is there any signal in the Q,U maps ?
( )
⎪⎩
⎪⎨
⎧=
⎪⎩
⎪⎨
⎧=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−
+∏=
i
i
i
skyi
i
skyi
iksky
UQT
XUQT
k
XL
;
21exp
21)( 22
2
22 σσσσπσ
• For a rough estimate, we can neglect the correlations:
• In the deep region we have N=5137 pixels (7’ side).• Average noise = (20,28,28 μK) per pixel (T,Q,U).• Expected CMB rms: (100, 2, 2 μK) for (T,Q,U).
0.0 0.5 1.0 1.5 2.0
1
10
100 Q U
L(σ sk
y)/L(
0)
σsky/σth