1

Other polarimeters• Wave plates can be used to build

polarimeters sensitive to linear or to circularpolarization.

• 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

2

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 techniquesfor 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 linearpolarization with two separate bolometers, and combine the twosignals 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 withB98)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 incomingradiation 100% polarized along the principal axis of the polarimeter. It is the product of the detector responsivity timesthe 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 incomingradiation 100% polarized and orthogonal tothe principal axis of the polarimeter. It isthe product of the detector responsivitytimes the integral of the cross-polar beamresponse. 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 beremoved using a high-pass filter

These components are detected

ΔBKGIn thef rangeof 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

3

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 fordifferent co-polar and cross-polar responses, including differentresponsivities, 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 meansthat it disappears if the two detectors have identical polarizationefficiencies (notice that perfect responsivity matching of the twobolometers is not important for this term). Otherwise this term willbecome 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 twopolarimeters below 5% of the co-polar one, then ΔB and ΔTU can beup 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 up to 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 soonas we consider off-axis locations of the detector in the focal plane.

• The same is true for off-axis telescopes, even if theysatisfy 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 toDragone) and we have analyzed in detail its polarizationproperties 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 aluminumprimary enclosed in a low emissivitycavity.

• Secondary and tertiary (Lyot stop) are inside the cryostat, and reimagethe primary focal plane inside the dewar, 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 responsecomes from the asimmetry of the primary illumination which derivesfrom 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 toquantify 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 realdata from our B98 flight, where we had several pixels, not sensitive to the polarization, but sensitive to all possible background fluctuations. FromNetterfield 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 hadindependent readout channels, so we can have an idea fromreal world data of the effects of noise, drifts, responsivitychanges one gets measuring the difference of maps obtained byindependent 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 withrespect to the B98 setup, it would be possible toconstrain significantly the polarization of the CMB withtwo independent bolometers sensitive to orthogonalpolarization 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 regionsmaller than B98, accumulating much more integrationtime 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)

4

PSB: Polarization Sensitive Bolometers(JPL+Caltech)

• 150 GHz• Two wire-grid-like

absorbers withmatched NTD thermistors

• Rotated 90°• Very close each

other (60 μm) inside the samegroove of a corrugated circularfeedhorn

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 grooveof a circularcorrugatedwaveguide

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. MauskopfIROE: A. Boscaleri

ING: G. Romeo, G. di StefanoCSU, Dominguez Hills: B. CrillIPAC: K. Ganga, E. HivonCITA: D. Bond, C. ContaldiLBNL, UC Berkeley: J. BorrillImperial College: A. Jaffe Institut d’ Astrophysique: S. PrunetUniversity of Alberta: D. PogosyanU. Penn.: M. Tegmark, A. de Oliveira-CostaUniversita’ di Roma, Tor Vergata: N. Vittorio, G. de Gasperis, P. Natoli, A. Balbi, P. Cabella

Boomerang 2003

5

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

6

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 principalaxis direction for each bolometer}

• In flight, thanks to WMAP (which is calibrated to betterthan 1% !).

• If we correlate the two bolometers separately with the unpolarized WMAP maps, we can estimate the responsivityto 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

7

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 photometercalibrations.

• In particular is very important to study the co and cross polarization response (beamand 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-axisparaboloid, 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

8

•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

9

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

10

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 ?

11

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

12

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

13

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 signalin 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 correlatedsignal and noise:

NormalizedSky covariance

Pixel Noisecovariance

Is there any signalin 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

14