lezione8 2017.ppt - Modalità compatibilitàoberon.roma1.infn.it/lezioni/laboratorio_specialistico... · 2017. 11. 20. · CMB anisotropy Dark energy 68.3% Dark matter 26.8% Baryonic

1900 1920 1940 1960 1980 2000 2020 2040 2060

102

107

1012

1017

Langley's bolometerGolay Cell

Golay Cell

Boyle and Rodgers bolometer

F.J.Low's cryogenic bolometer

Composite bolometer

Composite bolometer at 0.3K

Spider web bolometer at 0.3KSpider web bolometer at 0.1K

1year

1day

1 hour

1 second

Development of thermal detectors for far IR and mm-waves

time

requ

ired

to m

ake

a m

easu

rem

ent (

seco

nds)

year

Photon noise limit for the CMB

Spider-Web Bolometers

Absorber

Thermistor

Built by JPL Signal wire

2 mm

•The absorber is micro machined as a web of metallized Si3N4 wires, 2 m thick, with 0.1 mm pitch.

•This is a good absorber for mm-wave photons and features a very low cross section for cosmic rays. Also, the heat capacity is reduced by a large factor with respect to the solid absorber.

•NEP ~ 2 10-17 W/Hz0.5 is achieved @0.3K

•150KCMB in 1 s

•Mauskopf et al. Appl.Opt. 36, 765-771, (1997)

1 10 100 1000 100000.01

0.1

1

10

150 GHz,10% BW, 2

150 GHz, 10% BW, 1 cm2sr 30 GHz, 10% BW, 2

erro

r per

pix

el (

K)

integration time (s)

The ultimate sensitivity plot !!

CMB BLIP

Quando si e’ limitati dal rumore intrinseco dell’ osservabile , l’ unico modo per migliorare la

misura e’ renderla piu’ efficiente in termini di durata.

Usando un mosaico di rivelatori di grande formato si aumenta la

“mapping speed”

• Con un mosaico di N rivelatori si puo’ aumentare il rapporto S/N di un fattore N1/2

• E in realta’ di piu’, a causa della presenza di striping e altri effetti non gaussiani nei segnali di ciascuno dei rivelatori, che possono essere rimossi sfruttando le correlazioni tra pixel adiacenti.• Comunque stiamo parlando di mosaici di grandi dimensioni, dell’ ordine di 10000 rivelatori

W-Detector arrays

A.Lee, Berkeley

CMB & Cosmology• In the primeval plasma a huge number of photons was in equilibrium with matter (109/b). Equilibrium was maintained by Thomson scattering between s and charged particles (mostly e‐).

• With the expansion of the universe, the photon/matter plasma cooled down, until H atoms could form (3000K, 380000y after the big bang).

• The interaction of photons with neutral matter becomenegligible, and they where released, free to propagate withoutfurther interactions with matter.

• At that epoch, they formed a 3000K blackbody, i.e. an overwhelming background of optical and IR light filling the universe.

• Those photons are still filling the universe today, after an exapansion of all distances (and wavelengths) by a factor 1100, and form a faint, cold background (2.735K blackbody), mostly atmm wavelengths: is the CosmicMicrowave Background (CMB).

• The CMB carries information about all the phases of the evolution of the Universe.

P. de Bernardis, Searching for B‐modes in the polarization of the CMB with QUBIC, Malargue, 19/09/2017

What you can do studying the CMB:Measurement of B‐mode polarization, Gaussianity, and absolute spectrum. Study the inflation process at ultra‐high energies (1019GeV)

Spectrum: proof of the hot big bang. Spectral distortios: probe epochs before recombination and new physics

Galaxy clusters studies and surveys via the Sunyaev‐Zeldovich effect: (>106 : all clusters with M>1014M within our horizon )

CMB lensing: Map the gravitational potential all the way to z=1100

Maps of the primary anisotropy and E‐mode polarization of the CMB: Study the oscillations of the primeval plasma, cosmological parameters.

2 3

1 4

5


What you can do studying the CMB:

That’s why since 1965 CMB measurements have been steadily improved ..

Measurement of B‐mode polarization, Gaussianity, and absolute spectrum. Study the inflation process at ultra‐high energies (1019GeV)

Spectrum: proof of the hot big bang. Spectral distortios: probe epochs before recombination and new physics

Galaxy clusters studies and surveys via the Sunyaev‐Zeldovich effect: (>106 : all clusters with M>1014M within our horizon )

CMB lensing: Map the gravitational potential all the way to z=1100

Maps of the primary anisotropy and E‐mode polarization of the CMB: Study the oscillations of the primeval plasma, cosmological parameters.

2 3

1 4

5


BOOMERanG (2000) the first resolved map of CMB anisotropy

8 minutes

13.7 billion years

Here, now

Here, now

Hot gas in the photosphere of the sun

Hot gas in the primeval universe, at recombination

redshift, redshift, redshift, redshift, redshift, redshift …

Looking at the primeval plasma at recombination is like looking at the photosphere of the sun:

COBE (1992) the most accurate spectrum of CMB brightness ever


visw

• The study of solar oscillations allows us to study the interior structure of the sun, well below the photosphere.

• The study of the structure of CMB anisotropy, due to oscillations of the primeval plasma, allows us to study the universe well behind (well before) the cosmic photosphere (the recombination epoch)


Looking at the primeval plasma at recombination is like looking at the photosphere of the sun:

PP. de Bernardis, Searching for B‐modes in the polarization of the CMB with QUBIC, Malargue, 19/09/2017

CMB anisotropyResults of the Planck satellite

Brightness fluctuations: + 180 ppmscale

15

A precision measurement: 100K rms

Over three decades in angle, with the same instruement(intercalibration!)

You see seven peaks by eye


CMB anisotropy

*,,2

)1(2

)1(mmaaCD

16

6 parameters model(‐CDM) best fit

HenceGeometry ()Kinematics (H) Composition (, DM , b)Origin of the structures (ns)


CMB anisotropy

Dark energy 68.3%

Dark matter26.8%

Baryonic matter4.9%

CMB polarization: why

P. de Bernardis, Searching for B-modes in the polarization of the CMB with QUBIC, Malargue, 19/09/2017

-

-

+

-

+ x

y

--

+

-

+

x

y

-x

y

-10ppm +10ppm

= e‐ at last scattering

• CMB photons are last scattered at recombination.

• It’s a Thomson scattering, and any quadrupole anisotropy in the incomingphotons produces a degreeof linear polarization in the scattered photons.

• Density perturbationsproduce a small degree of linear polarization (E‐modes)

Same flux as seen in the electronrest frame

Quadrupole anisotropydue to Doppler effect

redshift

blueshift

blueshift

redshift

+ +

+

+

- -

-

-

Velocity fieldat recombination

Expect E‐modes and a T‐E anti‐correlation

Hot, over‐denseregion

Cold, under‐dense region

Origin of E‐modesresulting

CMB polarizationfield


P. de Bernardis, Searching for B-modes in the polarization of the CMB with QUBIC, Malargue, 19/09/2017

E‐modes

Origin of B‐modes


• Cosmic Inflation in the very early universe stretches the geometry to flat, and produces:– Scalar (density) perturbations, responsible for

primary CMB anisotropy and velocity fields at recombination ‐> E‐modes (and also of current large‐scale structures in the universe)

– Tensor (gravitational waves) perturbations, with ultra‐long wavelengths, travelling across the universe.

• At recombination, tensor perturbations also induce a small degree of polarization in the CMB, with both gradient and curl symmetries.

• The latter is called the B‐mode. • Moreover, lensing of E‐modes by intervening

matter concentrations between recombination and us also produces B‐modes, important at small scales.

How to separate B‐modes and E‐modes


• From the measurements of the Stokes Parameters Q and U of the linear polarization field we can recover both irrotational and rotational alm by means of modified Legendre transforms:

nYiaaniUQ mm

Bm

Em

2

,)(

nYniUQnYniUQnWdi

a

nYniUQnYniUQnWda

mmBm

mmEm

22

22

)()(21

)()(21

E‐modes, produced by scalar and tensor perturbations

B‐modes, produced only by tensor perturbations

Polarization is a spin-2 quantity: spin-2 basis

Level of inflationary B‐modes


• Since scalar perturbations do not produce B‐modes, and lensing B‐modesare produced mainly at small scales, B‐modes at large scales are a signature of cosmic inflation

• The amplitude of this effect is very small, but depends on the energy‐scale of inflation. In fact the amplitude of tensor modes normalized to the scalar ones r is:

• and

• There are theoretical arguments to expect that the energy scale of inflation is close to the scale of GUT i.e. around 1016 GeV.

• If ones detects primordial B‐modes, can constrain the energy‐scale of inflation.

• The current upper limits on B‐modes at large scales give T/S<0.1 (at 2)

GeV1006.101.0 16

4/14/1

Vr

GeV102

1.02

)1(16

4/1

maxVKcB

inflaton potential

The signal is extremely weak• Nobody really knows how to detect B‐modes.• Whatever smart, ambitious experiment we design to detect

the B‐modes:– needs to be extremely sensitive – use large arrays of detectors to boost

the speed of the sky survey– needs an extremely careful control of systematic effects – use a clean

optical configuration and polarization modulation techniques– needs careful control of local polarized emission (foregrounds) – use

multibandmeasurements– will need independent experiments, with orthogonal systematics,

confirming the result.

• There is still a long way to go.• QUBIC is an original instrument, devoted to sensitive and

accurate CMB polarization measurements with an uniqueexperimental setup.


CMB polarization: state of the art

CMB-S4 science book, 2016

???

A lot of work, but no detection of inflationary B‐modes (large scales, low level !)

Inflation B‐modesP. de Bernardis, Searching for B‐modes in the polarization of the CMB with QUBIC, Malargue, 19/09/2017

Dotted lines = primordial B‐modes (to be detected)Continuous lines = primordial B‐modes + lensing B‐modes

Detected signal is Interstellar Dust + Lensing [Planck+BICEP2]

Focussing on B‐modes:


The current effort of a number of experiments is to obtain deeper and cleaner measurements, to detect the primordial part of the signal (dotted lines)


Focussing on B‐modes:

How to measure CMB polarization: 1) polarization modulation

• i.e. how to modulate an extremely weak polarized signal in an overwhelming, structured unpolarized background.

• OPTION 1: take a mm‐wave photometer array, add a polarizer in front of the detectors, and rotate the entire photometer around its optical axis. – This was made with Planck, BICEP, etc.– The main disadvantage is that if the beam is slightly elliptical (as usually is), unpolarized

sources offset from the beam center will be modulated exactly as linearly polarized signals. Intensity to polarization leakage has to be corrected for.

• OPTION 2: take a mm‐wave photometer array and convert it into a Stokes polarimeter, i.e. add a Half‐Wave‐Plate and a polarizer, and spin the HWP to modulate polarization (without modulating intensity). – This is the option used by several new experiments, including e.g. SPIDER, LSPE and QUBIC


SRHRPDW VV )()(

SKY

polarizedsignal

x

y

principalaxis

HWPPOLARIZERDETECTORS

HWP rotation: continuous or stepping

i

i

i

i

VUQI

S

Polarized sky brightness:

Power on detectors:

t )24sin()24cos(21 sssV UQIW

How to measure CMB polarization: 2) beam forming

• CMB polarimeter arrays work at mm‐waves, where the beam shape is set by diffraction effects in the optics, in addition to detector properties.

• Three classes of instruments: imagers or interferometers (coherent or bolometric).

• Coherent interferometers are too complex to make, for a large array of detectors (we need large arrays, with thousands of pixels to achieve the required sensitivity). So we will consider only bolometric (Fizeau) interferometers in the following.

• Direct imager: telescope followed by an array of detectors in its focal plane. All recent CMB polarization instruments (but one) use this configuration.

• Fizeau interferometer: Array of apertures (horns) whose signals are combined so that each aperture illuminates the entire detector array. QUBIC uses this configuration, with significant advantages.


Distant source(boresight)

Focal plane withImage of the source(only center pixel illuminated)

Lens (telescope)

Regular array of apertures

Distant source(boresight)Imager Fizeau

Interferometer

focal len

gth

focal len

gth

Focal plane with uniform illumination from all the apertures (all pixels illuminated): Vector sum of the fields from all the apertures in phase.

Lens(beam combiner)

Each aperture is a source. In this case, since the source in the sky is in the boresight, they are all in‐phase.

S. Masi, CSN2‐INFN 2016/09/14

Distant source(off‐axis)

Focal plane withImage of the source(only side pixel illuminated)

Lens (telescope)

Regular array of apertures

Each aperture is a source, in this case they have different phases, depending on how‐much off‐axis the source is.

Lens(beam combiner)

Distant source(off‐axis)Imager Fizeau

Interferometer

focal len

gth

focal len

gth

Phase shift

xuie2

Focal plane with illumination from all the apertures. The vector sum of the fields from all the apertures is a system of fringes, depending on how‐much off‐axis the source is.

S. Masi, CSN2‐INFN 2016/09/14

31

0.5 mm PWV2 mm PWV

40 km

240K=2%

240K=0.1%

How to measure CMB polarization: 3) site


2 options:• Ground based, low PWV, stable

site, long integration time: South Pole, Andes

• Stratospheric Balloon, no PWV, limited integration time

• QUBIC will work from the Andes, for at least 2 years

Lo sviluppo di mosaici di grande formato di rivelatori sensibili alla

polarizzazione CMB :

• Puo’ dare ottimi risultati per misure dall’ Antartide e da pallone (tempo scala 4 anni)

• Puo’ permetterci di candidarci come fornitori dei rivelatori per la missione “deep space” post-Planck dedicata alla polarizzazione del fondo a microonde (tempo scala 10-15 anni)– NASA - Inflation Probe– ESA - Cosmic Vision– ASI - BPOL

Planck

Bolom.Array

From theEBEXproposal

Large Bolometer Arrays

30’

150 GHz 220 GHz 340 GHz 540 GHz

• >400 TES bolometers for the OLIMPO balloon telescope devoted to SZ and CMB anisotropy (Silvia Masi, Roma)

Altri impieghi di questi mosaici:• Astrofisica e Astroparticelle:

– Studio di nubi molecolari nel sub-mm (origine della vita, SAFIR)

– Spettroscopia ottica risolta temporalmente di pulsars (astrofisica relativistica, relativita’ generale, fisica dei plasmi magnetizzati e della materia ad altissme densita’)

– Studio del mezzo intergalattico nell’ UV (condizioni di ionizzazione, formazione delle strutture in fase non lineare)

– Spettroscopia X a immagini (astrofisica delle alte energie, Constellation X, ZEUS)

– Fisica dei Raggi Cosmici• Medicina

– camere X sensibili, camere FIR per mappe T)• Controllo Ambientale, controlli non invasivi

Mosaici di rivelatori fotolitografati: Due tecnologie possibili

• TES– Mosaici di Transition Edge Sensors, bolometri con

termometro a superconduttore.– Diversi gruppi hanno gia’ sviluppato la tecnologia del

sensore (NIST, Berkeley, Genova, Cardiff, SRON..)– Nessuno ha un multiplexer funzionante (Array di

SQUID), che comunque sara’ costosissimo.• KID

– Mosaici di Induttanze superconduttrici sensibili alla temperatura (Kinetic Inductance Detectors) o direttamente alle microonde.

– Un solo gruppo (Caltech) sta sviluppando la tecnologia

– Readout in principio semplice, con tecnologie RF da 1GHz (wireless, telefonini)

Kinetic Inductance Detectors (KIDs) for microwaves

• A KID is a strip of superconductor.• The incident radiation with

creates quasiparticles by breaking Cooper pairs. • The increased number of quasiparticles changes the

surface inductance of the superconductor because block the Cooper pairs from occupying some of the electron states (exclusion principle).

• This modifies the effective pairs energy and reduces the density of pairs.

• The change of L is sensed by making the strip part of a microwave resonant circuit, and sensing the change in the phase of a microwave signal transmitted through the resonator.

2h Superconducting gap energy

ckT5.32

data fromDay et al. 2003

120 mK

260 mK

T << TcDensity of states for quasiparticles

hNqp

sss LjRZ

ssqp LRN ,

Mazin (Caltech)

GHz RF (…+fN-1+fN+fN+1+…)

CMB CMB CMB

Pixel N-1fN-1

Pixel NfN

Pixel N+1fN+1

0.3K - 0.1K

RF mux• A KID has high transmission at f away from resonance.

This fact can naturally be used for multiplexing many detectors, tuned at different resonances f, all loading the same transmission line.

• Using excitation in the GHz range:

• high quality wireless components are available

• thousands of detectors can be multiplexed, with a single coax and a single HEMT

More Advantages:• A KID is simple to fabricate (no junctions, nor complex

processes) in e.g. Aluminum (Tc=1K, Top=0.1K), sapphire, silicon: all rugged materials suitable for space environment.

• At Top<<Tc the sensitivity scales as exp(-/kTop)• There is flexibility in the selection of

– Microwave circuit layout (striplines, microstrip lines, coplanar waveguides etc.)

– Superconducting material selection: Aluminum, Tantalum, Niobium, Titanium, Zirconium

– Substrates selection (Ge, Si, Sapphire etc.)

Responsivity• Is the phase change per injected quasiparticle . • It can be computed using the Mattis-Bardeen theory.

• Where is the fraction of the line inductance which is kinetic

• No is the density of single spin electron states at the Fermi energy of the metal (No=1.72x1010 mm3/eV for Al)

• V is the volume of the superconductor

hNqp VNQ

dNd

oqp

total

kin

LL

o

qp

s

s

NN

ZZ

2

Day et al., 2003, Nature, 425 , 817

Mazin (Caltech)

Mazin (Caltech)

Polarization Basics• The equations

represent a pair of plane waves: the two components of the electrical field of an EM wave propagating in the z direction, not necessarily monochromatic.

• The amplitudes Eox,y(t) and phases x,y(t) fluctuate slowly with respect to the rapid oscillation of the carrier cos(t).

)](cos[)()()](cos[)()(

tttEtEtttEtE

yoyy

xoxx

z

Ex

Ey E

Polarization Basics

If we eliminate the term cos(t) between the two equations, and define (t)= y(t)- x(t), we find the polarization ellipse (valid in general at a given time), which is the locus of points described by the optical field as it propagates:

)](cos[)()()](cos[)()(

tttEtEtttEtE

yoyy

xoxx

)(sin)(cos)()()()(2

)()(

)()( 2

2

2

2

2

tttEtEtEtE

tEtE

tEtE

oyoy

yx

oy

y

ox

x

fast fast fast

slow slow slowslow slow

t

)()()( tEtEtE yx

z

Ex

Ey E

Polarization Basics• For purely monochromatic waves, amplitudes and

phases must be constant with time:

And the polarization ellipse is also constant:

]cos[)(]cos[)(

yoyy

xoxx

tEtEtEtE

22

2

2

2

sincos)()(2)()(

oyox

yx

oy

y

ox

x

EEtEtE

EtE

EtE

fast fast fast

t

)()()( tEtEtE yx

z

Ex

Ey E

Polarization Basics

• In general a beam of light is “elliptically polarized”.

• The polarization ellipse degenerates to special forms for special values of the amplitudes and of the phases.

• Linear polarized waves: when the ellipse collapses to a line, i.e. when =0,. The direction of the E vector remains constant.

• Circularily polarized waves: when the ellipse reduces to a circle, i.e. when = and Eox=Eoy=Eo.

E

x

y

E

x

y

E

x

y

Polarization Basics• The polarization ellipse is specified by the amplitude parameters Eox,Eoy,.

• But it can be expressed equivalently by the elliptical parameters:

• Orientation angle :

• Ellipticity angle :

• For linearly polarized light =0.

a

b

x

y’x’

y

22

cos22tan

oyox

oyox

EEEE

ab

tan

22

sin22sin

oyox

oyox

EEEE

Stokes Parameters• Our detectors are too slow to follow the time

evolution of the EM field. What we can measure are time averages, over periods much longer than 2/.

• Due to the periodicity of the EM waves, it is enough to compute time averages over a single period of oscillation. These are represented by the symbol <…>.

• So we take the time average of the polarization ellipse:

22

2

2

2

sincos)()(2)()(

oyox

yx

oy

y

ox

x

EEtEtE

EtE

EtE

Stokes Parameters• Multiplying by 4Eox

2Eoy2 we find

• Since Ex(t) and Ey(t) are sine waves, we can compute their time averages and substitute above:

• Since we want to express this in terms of intensities, we can add and subtract Eox

4+Eoy4:

2

2222

)sin2(cos)()(28

)(4)(4

oyoxyxoyox

yoxxoy

EEtEtEEE

tEEtEE

22

2222

)sin2()cos2(

22

oyoxoyox

oyoxoxoy

EEEE

EEEE

Stokes Parameters

• We find

• We define the Stokes Parameters:

• so that our equation reduces to

22

222222

)sin2()cos2(

)()(

oyoxoyox

oyoxoxoy

EEEE

EEEE

sin2

cos2

3

2

221

22

oyox

oyox

oyox

oxoyo

EES

EES

EES

EES

23

22

21

2 SSSS o

Stokes Parameters• If light is not purely monochromatic,

the amplitudes and phases fluctuate with time.

• It can be shown that, in general,

• The = sign is valid for fully polarized light, while the > sign is valid for partially polarized or unpolarizedlight. P=degree of polarization:

• The intensity is related to So:• The orientation of the polarization

ellipse is related to S1 and S2:• The ellipticity of the polarization

ellipse is related to S3:

23

22

21

2 SSSSo

1

222

cos22tan

SS

EEEE

oyox

oyox

ooyox

oyox

SS

EEEE 3

22

sin22sin

sin2

cos2

3

2

221

22

oyox

oyox

oyox

oxoyo

EES

EES

EES

EES

22oxoyo EES

10

23

22

21

PS

SSSII

Pototal

pol

Stokes Parameters• Note that, for linear polarized

light (=0), both parameters S1and S2 represent the difference in intensity carried by two orthogonal components:

• S1 is the difference in intensity between the components along axis x and y

• S2 is the difference in intensity between the components along two axis x’ and y’ rotated 45o

with respect to x and y.

x

y

x

yy’

x’

221 yx EES

2'

2'''''2

2

1

2

1

2

1

2

122 yxyxyxyx EEEEEEEES

Ex

Ey

Ex’

Ey’ 45o

Stokes Parameters: examples• Unpolarized light:

=random<Eox

2>=<Eoy2>=Io

• Linearly polarized light:– Horizontal(Eoy=0) Vertical (Eox=0) +45o (Eoy= Eoy; = 0) o

• Circular polarized light:– Left Right

sin2

cos2

3

2

221

22

oyox

oyox

oyox

oxoyo

EES

EES

EES

EES

0001

2 oIS

0011

oIS

001

1

oIS

0101

oIS

02sin2cos

1

oIS

1001

oIS

1001

oIS

Stokes Parameters• The waves can be represented as complex functions:

• This helps in the time-averaging process needed to compute the Stokes Parameters. They can be rewritten as follows (Stokes vector):

sin2cos2

)(

22

22

**

**

**

**

3

2

1

oyox

oyox

oyox

oyox

xyyx

xyyx

yyxx

yyxxo

EEEE

EEEE

EEEEiEEEEEEEEEEEE

SSSS

)](exp[]cos[)()](exp[]cos[)(

yoyyoyy

xoxxoxx

tiEtEtEtiEtEtE

Stokes Parameters• The Stokes vector can also be expressed in terms of

So, , . • From

• And from

• Using we find S1, so we have:

2sin2sin2cos2cos2cos

1

3

2

1o

o

S

SSSS

2tan can write wecos2

2tan 1222 SSEE

EE

oyox

oyox

2sin can write wesin2

2sin 322 ooyox

oyox SSEE

EE

23

22

21

2 SSSS o Poincare’

Classical measurement of the Stokes Parameters

• The measurement of the 4 Stokes Parameters needs two optical components:– A retarder (wave plate): it is a phase-shifting element, whose

effect is to advance the phase of the x component by and to retard the phase of the y component by - . So the field emerging from the retarder is E’x= Ex ei and E’y= Ey e-i

– A polarizer. The optical field can pass only along one axis, the transmission axis. So the total field emerging from the polarizer is E”=E’xcos+E’ysin, where E’ is the incident field and is the angle of the transmission axis.

• So the beam arriving on the detector is E”=Ex ei cos+Eye-i sin

source retarder polarizer

detector


• E”=Ex ei cos+Eye-i sin• The detector measures its intensity, i.e. I= E”E”*• So we get

• Which can be rewritten using the half-angle formulas:


detector

cossincossin

sincos),(**

2*2*

iyx

iyx

yyxx

eEEeEE

EEEEI

2sinsin2sincos

2cos),(

****

****

21

xyyxxyyx

yyxxyyxx

EEEEiEEEE

EEEEEEEEI

22sin

22cos12

22cos12 cossin sin cos

2sinsin2sincos2cos),( 32121 SSSSI o


• This is the formula derived in 1852 by Sir George Gabriel Stokes.

• The first three parameters can be measured by removing the retarder (=0) and measuring the intensity with three orientations of the polarizer =0o,45o,90o:

• The fourth parameter can be measured by inserting a 90o retarder (quarter wave plate):


detector 2sinsin2sincos2cos),( 3212

1 SSSSI o

321

121

221

121

)90,45()0,90()0,45()0,0(

SSISSISSISSI

ooo

ooo

ooo

ooo

)0,90()0,0()90,45(2)0,90()0,0()0,45(2

)0,90()0,0()0,90()0,0(

3

2

1

oooooo

oooooo

oooo

ooooo

IIISIIIS

IISIIS


• The great advantage of the Stokes Parameters is that they are observable. The polarization ellipse is not (too fast).

• Moreover, the Stokes parameters can be used to describe unpolarized light: light which is not affected by the rotation of a polarized or by the presence of a retarder. Stokes was the first one to describe mathematically unpolarized and partially polarized light.

• It is evident from Stokes formula that, for unpolarized light, S1=S2=S3=0, while So>0.

• The fully polarized light had • The intermediate state is partially polarized light, where


detector 2sinsin2sincos2cos),( 3212

1 SSSSI o

23

22

21

2 SSSSo

23

22

21

2 SSSSo

Partially polarized light• The Stokes parameters of a combination of independent

waves are the sums of the respective Stokes parameters of the separate waves.

• If we combine a fully polarized wave with an independent, unpolarized one, we find partially polarized light.

• This expression will be useful in the following.

10 23

22

21

P

SSSS

II

Pototal

pol

3

2

1

3

2

1

000

)1(

SSSS

P

S

P

SSSS

S

ooo

Polarization-active optical components

• When a beam of light interacts with matter its polarization state is almost always changed.

• It can be changed by– changing the amplitudes– changing the phases– changing the directions

of the orthogonal field components.• Their effect can be described by means of the

Mueller matrices: M is a 4x4 matrix such that the emerging Stokes vector is S’=M S .

Polarizer (Diattenuator)

RotatorWave-plate (Retarder)

1) Polarizer or Diattenuator• It attenuates the orthogonal

components of an optical beam unequally:

• Using the definitions of S and S’

• And inserting the expressions for E’we get

yyy

xxx

EpEEpE

'

'

)( '*''*'

'*''*'

'*''*'

'*''*'

'3

'2

'1

'

xyyx

xyyx

yyxx

yyxxo

EEEEiEEEEEEEEEEEE

SSSS

)( **

**

**

**

3

2

1

xyyx

xyyx

yyxx

yyxxo

EEEEiEEEEEEEEEEEE

SSSS

3

2

12222

2222

'3

'2

'1

'

200002000000

21

SSSS

pppp

pppppppp

SSSS o

yx

yx

yxyx

yxyxo

Special cases

• If the diattenuator is simply an attenuator, i.e. if px=py=p we have a neutral density filter:

• If the Polarizer is ideal and horizontal, i.e. if py=0 we have

• If the Polarizer is ideal and vertical, i.e. if px=0 we have

3

2

12

'3

'2

'1

'

1000010000100001

SSSS

p

SSSS oo

3

2

12

'3

'2

'1

'

0000000000110011

2SSSS

p

SSSS o

x

o

3

2

12222

2222

'3

'2

'1

'

200002000000

21

SSSS

pppp

pppppppp

SSSS o

yx

yx

yxyx

yxyxo

3

2

12

'3

'2

'1

'

0000000000110011

2SSSS

p

SSSS o

y

o

Polarizer:• The characteristics of

the polarizer px and pycan be rewritten in terms of new parameters p and :

• With these parameters the Mueller matrix of a polarizer is:

• An ideal polarizer converts any incoming beam into a linearly polarized beam:

yx

yx

yxyx

yxyx

P

pppp

pppppppp

M

200002000000

21 2222

2222

sincos

pppp

defy

defx

2sin00002sin000012cos002cos1

2

2pMP

001

1

21

0000000000110011

21

1

3

2

1

'3

'2

'1

'

SS

SSSS

SSSS

o

oo

2) Retarder• It introduces a phase shift between

the orthogonal components of an optical beam :



)()()()(

2/'

2/'

tEetEtEetE

yi

y

xi

x

)( '*''*'

'*''*'

'*''*'

'*''*'

'3

'2

'1

'

xyyx

xyyx

yyxx

yyxxo

EEEEiEEEEEEEEEEEE

SSSS

)( **

**

**

**

3

2

1

xyyx

xyyx

yyxx

yyxxo

EEEEiEEEEEEEEEEEE

SSSS

3

2

1

'3

'2

'1

'

cossin00sincos00

00100001

SSSS

SSSS oo

Special cases

• If the retarder is a quarter-wave plate (=90o):

• Such a retarder converts a +45o linearly polarized beam into a right/left circularly polarized beam:

• If the retarder is a half-wave plate (=180o):

• This reverses the ellipticity and orientation of the incomin polarization state.

3

2

1

'3

'2

'1

'

cossin00sincos0000100001

SSSS

SSSS oo

3

2

1

'3

'2

'1

'

01001000

00100001

SSSS

SSSS oo

01

01

01001000

00100001

1001

3

2

1

'3

'2

'1

'

1000010000100001

SSSS

SSSS oo

3) Rotator• Here



cos)(sin)()(sin)(cos)()(

'

'

tEtEtEtEtEtE

yxy

yxx

)( '*''*'

'*''*'

'*''*'

'*''*'

'3

'2

'1

'

xyyx

xyyx

yyxx

yyxxo

EEEEiEEEEEEEEEEEE

SSSS

)( **

**

**

**

3

2

1

xyyx

xyyx

yyxx

yyxxo

EEEEiEEEEEEEEEEEE

SSSS

3

2

1

'3

'2

'1

'

100002cos2sin002sin2cos00001

SSSS

SSSS oo

Rotated Optical Components• We have assumed that the optical axis of the components we

have considered were aligned to the coordinate system.• If they are not (as often happens), we have to

1. rotate the incident beam from the original coordinate system to the one aligned with the component: S’ = MR ()Sin

2. Multiply S’ by the Mueller matrix MC of the optical component S”= MCS’

3. Rotate the output beam back into the original coordinate system: Sout= MR (-) S’’

• So we have:Sout = MR (-) MC MR () SinWhere is the rotation on the optical component C.

100000000001

100002cos2sin002sin2cos00001

)(22

22

cssc

MdefR

Rotated Polarizer

100000000001

0000000000

100000000001

21)(

22

22

22

22

cssc

XXcs

scMP

• Here

• so

• and

XX

pppp

pppppppp

MMdef

yx

yx

yxyx

yxyx

PC

0000000000

21

200002000000

21)0()0(

2222

2222

XXcsXcssXcsXscc

sc

MP

0000)(0)(0

21)( 2

222222

2222

222

22

2cos2sin

2

2

2

22

22

cs

ppXpppp

yx

yx

yx

Linear Polarimeter

• A polarimeter is a device able to detect polarized light and measure its polarization characteristics.

• The simplest polarimeter we can imagine is a linear polarimeter, which can be built with a rotating polarizer in front of an intensity detector.

• An intensity detector is represented by a Stokes vector D=(1,0,0,0). The power detected by the detector from an optical beam with Stokes vector S is simply w=DS=So

• If we put a polarizer in front of the detector, the polarizer is called analyzer, and the power detected will be w() =DMP()S

source polarizer

Intensity detector

Linear Polarimeter

source Polarizer (analyzer)

Intensity detector

0000)(0)(0

21)0,0,0,1()(

3

2

122

22222

2222

222

22

SSSS

XXcsXcssXcsXscc

sc

SDMw

o

P

2sin2cos 2121 SSSw o

This polarimeter is not sensitive to circular polarization (no S3).It is sensitive to linear polarization (S1 and S2) and to unpolarized light (So). If the polarizer is ideal:

2sin2cos0X ; 1 ; 1

2121 SSSw o

Linear Polarimeter

source Rotating analyzer

Intensity detector

• If we are interested to the linear polarized component only, we can rotate continuously the polarizer: =t and look only for the AC signal at frequency 2.

• This allows to reject the unpolarized component, even if it is dominant, and to remove all the noise components at frequencies different than 2 (synchronous demodulation).

tStSSw o 2sin2cos 2121

)(2sin2cos)()()( 2121 tNtStSSRtNtRwtV o

constantsignal (DC)

modulatedsignal (AC)

noise(AC)detector

responsivity

Linear Polarimeter

source Rotating analyzer

x

Rw+N

A(Rw+N)

C

RAC

Ref(2)

A[Rw(2)+N()]Demodulated signal

Log P()

Log

P( ) d

2

-

-=1/T

1/RC

signalnoise

<…>T

Detector

How do we separate S1 and S2

• Neglecting the stochastic effect of noise (we integrate enough that N becomes negligible) and of the constant term (which we remove with the AC decoupling)

• We measure V and we want to estimate S1 and S2. We can use two reference signals, out of phase by T/8 and synchronously demodulate with them:

tStSRtRwtV 2sin2cos)()( 2121

)(2sin2cos)()()( 2121 tNtStSSRtNtRwtV o

How do we separate S1 and S2

TT

TR

T

T

TT

TR

T

T

tdttStdttStdttVY

tdttStdttStdttVX

02

012

1

0

1

02

012

1

0

1

2cos2sin2cos2cos2cos)(

2sin2sin2sin2cos2sin)(

181

281

SRYSRX

• So the double linear polarimeter is

insensitive to So and it is easy to calibrate.

• Is this a troubleless instrument ? No !• It is inefficient (factor 1/8 from

modulation and demodulation)• It can be microphonic.• And, as all polarimeters, needs a

telescope.

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