Outline

1. Stokes Vectors, Jones Calculus and Mueller Calculus

2. Optics of Crystals: Birefringence3. Common polarization devices for the

laboratory and for astronomical instruments4. Principles of Polarimetry: Modulation and

Analysis. Absolute and Relative Polarimetry5. Principles of Polarimetry: Spatial modulation,

Temporal modulation, Spectral modulation6. Principles of Polarimetry: Noise and errors7. Spurious sources of polarization

Stokes Vector, Jones Calculus,

Mueller Calculus

playing around with matrices

A. López Ariste

)( tkzii

y

x

y

x eeA

A

E

E

Assumptions:

•A plane transverse electromagnetic wave•Quasi-monochromatic•Propagating in a well defined direction z

)( tkzii

y

x

y

x eeA

A

E

E

Jones Vector

)( tkzii

y

x

y

x eeA

A

E

E

Jones Vector:

It is actually a complex vector with 3 free parametersIt transforms under the Pauli matrices.It is a spinor

y

x

y

x

y

x

E

EC

E

E

dc

ba

E

E

3,0i

iiaC

3,0i

iiaC

10

010

10

011

01

102

0

03 i

i

The Jones matrix of an optical device

In group theory: SL(2,C)

)( tkzii

y

x

y

x eeA

A

E

E

From the quantum-mechanical point of view, the wave function cannot be measured directly.

Observables are made of quadratic forms of the wave function:

EEJ

J is a density matrix : The coherence matrix

**

**

yyxy

yxxx

EEEE

EEEEJ

3210 VUQIJ

Like Jones matrices, J also belongs to the SL(2,C) group, and can be decomposed in the basis of the Pauli matrices.

V

U

Q

I

Is the Stokes Vector

3210 VUQIJ

V

U

Q

I

I

The Stokes vector is the quadractic form of a spinor. It is a bi-spinor, or also a 4-vector

)(

JTrI

V

U

Q

I

02222 VUQI

02

0

02

3,2,1

4-vectors live in a Minkowsky space with metric (+,-,-,-)

)(

JTrI

The Minkowski space

I

VQ

Partially polarized light

Fully polarizedlight

Cone of (fully polarized) light

2222 VUQI

2222 VUQI

y

x

y

x

y

x

E

EC

E

E

dc

ba

E

E

CJCCEECEE

E

EJ yx

y

x

IMICCTrCJCTrJTrI

)()()(

M is the Mueller matrix of the transformation

)( CCTrM

)( CCTrM

From group theory, the Mueller matrix belongs to a group of transformations which is the square of SL(2,C)

Actually a subgroup of this general group called O+(3,1) or Lorentz group

),2(),2( CSLCSL

The cone of (fully polarized) light

I

VQ

Lorentz boost = de/polarizer, attenuators, dichroism

The cone of (fully polarized) light

I

VQ

3-d rotation = retardance, optical rotation

Mueller Calculus

• Any macroscopic optical device that transforms one input Stokes vector to an output Stokes vector can be written as a Mueller matrix

• Lorentz group is a group under matrix multiplication: A sequence of optical devices has as Mueller matrix the product of the individual matrices

Mueller Calculus: 3 basic operations

• Absorption of one component• Retardance of one component

respect to the other• Rotation of the reference system

Mueller Calculus: 3 basic operations

• Absorption of one component

C

10200

0

aaC

0000

0000

0011

0011

2

aM

Mueller Calculus: 3 basic operations

• Absorption of one component• Retardance of one component

respect to the other

10 110

01

ii

i eee

C

cossin00

sincos00

0010

0001

M

Mueller Calculus: 3 basic operations

• Absorption of one component• Retardance of one component

respect to the other• Rotation of the reference system

30 sincoscossin

sincos

C

1000

02cos2sin0

02sin2cos0

0001

M

Optics of Crystals: Birefringence

A. López Ariste

Chapter XIV, Born & Wolf

Ellipsoïd

Ellipsoïd

Three types of crystals

A spherical wavefront

Three types of crystals

Two apparent waves propagating at different speeds:•An ordinary wave, with a spherical wavefront propagating •at ordinary speed vo

•An extraordinary wave with an elliptical wavefront, its speed •depends on direction with characteristic values vo and ve

Three types of crystals

zs

De

Do

The ellipsoïd of D in uniaxial crystals

The two propagating waves are linearly polarized and orthogonal one to each other

Typical birefringences

•Quartz +0.009

•Calcite -0.172

•Rutile +0.287

•Lithium Niobate -0.085

Common polarization devices for the laboratory and for

astronomical instruments

A. López Ariste

Linear Polarizer

0000

0000

005.05.0

005.05.0

M

0000

02sin2cos2sin2sin

02sin2cos2cos2cos

02sin2cos1

5.0)(

0000

0000

005.05.0

005.05.0

)( 2

21

RRM

Retarder

cossin00

sincos00

0010

0001

M

?)(

cossin00

sincos00

0010

0001

)( 1

RRM

Savart Plate

Glan-Taylor Polarizer

Glan-Taylor.jpg

Glan-Thompson Polarizing Beam-Splitter

Rochon Polarizing Beamsplitter

Polaroid

Dunn Solar Tower. New Mexico

dnne 0

Zero-order waveplates

Multiple-order waveplates

Typical birefringences

•Quartz +0.009

•Calcite -0.172

•Rutile +0.287

•Lithium Niobate -0.085

Waveplates

Principles of PolarimetryModulation

Absolute and Relative Polarimetry

A. López Ariste

Measure # 1 : I + Q

Measure # 2 : I - Q

Subtraction: 0.5 (M1 – M2 ) = Q

Addition: 0.5 (M1 + M2 ) = I

How to switch from Measure # 1 to Measure # 2?

MODULATION

Measure # 1 : I + Q

Measure # 2 : I - Q

Subtraction: 0.5 (M1 – M2 ) = Q

Addition: 0.5 (M1 + M2 ) = I

Principle of Polarimetry

Everything should be the same EXCEPT for the sign

MODULATION

Njj

njn

Njjj

ScM

ScM

,1

,1

11

VS

US

QS

IS

4

3

2

1

MODULATION

Njj

njn

Njjj

ScM

ScM

,1

,1

11

VS

US

QS

IS

4

3

2

1

ii

i

cc

c

14,3,2

1 0

MODULATION

Njj

njn

Njjj

ScM

ScM

,1

,1

11

ii

i

cc

c

14,3,2

1 0

IOM

O is the Modulation Matrix

MODULATION

VIM

VIM

UIM

UIM

QIM

QIM

6

5

4

3

2

1

1001

1001

0101

0101

0011

0011

O

Conceptually, it is the easiest thingIs it so instrumentally?

Is it efficient respect to photon collection, noise and errors?

MODULATION

IOM

MDMOI

1

nj

ijVUQIi D,1

2,,,

Del Toro Iniesta & Collados (2000)Asensio Ramos & Collados (2008)

nj

ijVUQIi Dn,1

2,,,

MODULATION

MDMOI

1

Del Toro Iniesta & Collados (2000)

nj

ijVUQIi Dn,1

2,,,

VUQii

I

,,

2 1

1

Del Toro Iniesta & Collados (2000)Asensio Ramos & Collados (2008)

MODULATION

VIM

VIM

UIM

UIM

QIM

QIM

6

5

4

3

2

1

1001

1001

0101

0101

0011

0011

O

3

1

1

,,

VUQ

I

Design of a Polarimeter

•Specify an efficient modulation scheme: The answer is constrained by our instrumental choices

Absolute vs. Relative Polarimetry

nj

ijVUQIi Dn,1

2,,,

VUQii

I

,,

2 1

1

Efficiency in Q,U and V limited by efficiency in I

What limits efficiency in I?

Absolute vs. Relative Polarimetry

What limits efficiency in I?

Measure # 1 : I + Q

Measure # 2 : I - Q

Subtraction: 0.5 (M1 – M2 ) = Q

Addition: 0.5 (M1 + M2 ) = I

Principle of Polarimetry

Everything should be the same EXCEPT for the sign

Absolute vs. Relative Polarimetry

What limits efficiency in I?

Measure # 1 : I + Q

Measure # 2 : I - Q

Subtraction: 0.5 (M1 – M2 ) = Q

Addition: 0.5 (M1 + M2 ) = I

Principle of Polarimetry

Everything should be the same EXCEPT for the sign

Usual photometry of present astronomical detectors is around 10-3

Absolute vs. Relative Polarimetry

What limits efficiency in I?

You cannot do polarimetry better than photometry

Usual photometry of present astronomical detectors is around 10-3

Absolute vs. Relative Polarimetry

What limits efficiency in I?

You cannot do ABSOLUTE polarimetry better than photometry

Usual photometry of present astronomical detectors is around 10-3

Absolute vs. Relative Polarimetry

I

Q

I

QIQIM

I

Q

I

QIQIM

11

11

2

1

QI

I

Q

I

I

QI

I

QI

I

QI

21

2

)()2(

)(1)(1

Absolute error : 10-3 IRelative error : 10-3 Q

Absolute vs. Relative Polarimetry

I

Q

I

QIQIM

I

Q

I

QIQIM

11

11

2

1

QI

I

Q

I

I

QI

I

QI

I

QI

21

2

)()2(

)(1)(1

Absolute error : 10-3 IRelative error : 10-3 Q

Li 6708

D2 D1

D2

Phase de 45 deg

Phase de 102 deg

Design of a Polarimeter

•Specify an efficient modulation scheme: The answer is constrained by our instrumental choices

•Define a measurement that depends on relative polarimetry, if a good sensitivity is required

Principles of Polarimetry Spatial modulation, Temporal

modulation, Spectral modulation

A. López Ariste

Measure # 1 : I + Q

Measure # 2 : I - Q

Subtraction: 0.5 (M1 – M2 ) = Q

Addition: 0.5 (M1 + M2 ) = I

How to switch from Measure # 1 to Measure # 2?

MODULATION

How to switch from Measure # 1 to Measure # n?

VIM

VIM

UIM

UIM

QIM

QIM

6

5

4

3

2

1

Analyser: Calcite beamsplitter

V

U

Q

I

M

0

0

QI

QI

0

0

QI

QI

Analyser: Rotating Polariser

0

2sin2cos2sin2sin

2sin2cos2cos2cos

2sin2cos

0000

02sin2cos2sin2sin

02sin2cos2cos2cos

02sin2cos1

2

2

2

2

UQI

UQI

UQI

V

U

Q

I

0

0

QI

QI

0

0

0

QI

QI

2

Analyser: Rotating Polariser

Analyser: Calcite beamsplitter

2 beams ≡2 images Spatial modulation

2 angles ≡ 2 exposuresTemporal modulation

Modulator:

V

U

Q

I

M Analyzer

0

0

QI

QI

What about U and V?

Modulator:

V

U

Q

I

M Modulator

V

Q

U

I

V

U

Q

I

M Modulator

Q

U

V

I

Modulator:

V

U

Q

I

M Modulator

V

Q

U

I

V

U

Q

I

M Modulator

Q

U

V

I

B

A

UI

UI

V

U

Q

I

MM ModAn

B

A

VI

VI

V

U

Q

I

MM ModAn

Modulator: Rotating λ/4

cossin

sincos

cos0sin0

0100

sin0cos0

0001

VQ

U

VQ

I

V

U

Q

I

Q

U

V

I

2

B

A

VI

VI

V

U

Q

I

MM ModAn

The basic Polarimeter

Modulator Analyzer

V

U

Q

I

3

2

1

S

S

S

I

3

2

1

1

S

S

SI

SI

3

2

1

1

S

S

SI

SI 1S

Examples2 Quarter-Waves + Calcite Beamsplitter

QW1 QW2 Measure

T1 0° 0 ° Q

T2 22.5 ° 22.5 ° U

T3 0 ° -45 ° V

T4 0 ° 45 ° -V

….

LCVR

Calcite

IwavelengthOM

)(

Examples1 Rotating Quarterwave plate + Calcite Beamsplitter2 Photelastic Modulators (PEM) + Linear Polariser

tVttUtQS 2sin2sin2cos2cos21

0

1 QS

2

0 2

11

VSS

4

0

43

2 43

11

2

4

11

USSSS

Spectral ModulationChromatic waveplate: )( f

V

U

Q

I

)(cos0)(sin0

0100

)(sin0)(cos0

0001

Followed by an analyzer )(cos1 QS

Spectral ModulationChromatic waveplate: )( f

V

U

Q

I

)(cos0)(sin0

0100

)(sin0)(cos0

0001

Followed by an analyzer )(cos1 QS

See Video from Frans Snik (Univ. Leiden)

Principles of Polarimetry Noise and errors

A. López Ariste

Sensitivity vs. Accuracy

SENSITIVITY: Smallest detectable polarization signal

related to noise levels in Q/I, U/I, V/I.RELATIVE POLARIMETRY

ACCURACY: The magnitude of detected polarization signal That can be quantifiedParametrized by position of zero point for Q, U, VABSOLUTE POLARIMETRY

Sensitivity vs. Accuracy

SENSITIVITY: Smallest detectable polarization signal

related to noise levels in Q/I, U/I, V/I.RELATIVE POLARIMETRY

MDMOI

1

nj

ijVUQIi Dn,1

2,,,

Gaussian Noise (e.g. Photon Noise, Camera Shot Noise)

Correcting some unknown errorsSpatio-temporal modulation

Goal: to make the measurements symmetric respect to unknown errors in space and time

Exposure 1

I+V

I-V

Det

ecti

n in

dif

fere

nt p

ixel

s

Spatio-temporal modulation

Goal: to make the measurements symmetric respect to unknown errors in space and time

Exposure 1

I+V

I-V

Exposure 2

I-V

I+V

Det

ecti

n in

dif

fere

nt p

ixel

s

Detection at different times

Spatio-temporal modulation

2

2

2

1

1 41

I

Vo

I

V

VI

VI

VI

VI

Exposure 1

I+V

I-V

Exposure 2

I-V

I+V

:IV

Spatio-temporal modulation

2

2

2

1

1 41

I

Vo

I

V

VI

VI

VI

VI

Let’s make it more general

:IV

2

002

2

1

1

I

Io

I

IK

IO

IO

IO

IO

Cross-Talk

B

A

SI

SI

V

U

Q

I

MM ModAn1

1

This is our polarimeter This is what comes from the

outer universe

Is this true?

StarV

U

Q

I

StarV

U

Q

I

?Star

V

U

Q

I

935.0323.000

323.0935.000

0099.0009.0

00009.099.0

M

StarV

U

Q

I

StarV

U

Q

I

Star

Telescope

V

U

Q

I

M

CrossTalk

935.0323.000

323.0935.000

0099.0009.0

00009.099.0

M

Solutions to Crosstalk

1. Avoid it:

2. Measure it

Mirrors with spherical symmetry (M1,M2) introduce no polarizationCassegrain-focus are good places for polarimetersTHEMIS, CFHT-Espadons, AAT-Sempol,TBL-Narval,HARPS-Pol,…

Given find its inverse and apply it to the measurements

It may be dependent on time and wavelengthIt forces you to observe the full Stokes vector

TelescopeM

Dunn Solar Tower. New Mexico

Solutions to Crosstalk

3. Compensate itSeveral procedures:• Introduce elements that compensate the

instrumental polarization• Measure the Stokes vector that carries the

information• Project the Stokes vector into the

Eigenvector of the matrix