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**Outline Stokes Vectors, Jones Calculus and Mueller Calculus**

Optics of Crystals: Birefringence Common polarization devices for the laboratory and for astronomical instruments Principles of Polarimetry: Modulation and Analysis. Absolute and Relative Polarimetry Principles of Polarimetry: Spatial modulation, Temporal modulation, Spectral modulation Principles of Polarimetry: Noise and errors Spurious sources of polarization

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**Stokes Vector, Jones Calculus, Mueller Calculus playing around with matrices**

A. López Ariste

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

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Jones Vector

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**Jones Vector: It is actually a complex vector with 3 free parameters**

It transforms under the Pauli matrices. It is a spinor

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**The Jones matrix of an optical device**

In group theory: SL(2,C)

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

Observables are made of quadratic forms of the wave function: J is a density matrix : The coherence matrix

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

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**The Stokes vector is the quadractic form of a spinor**

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

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

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**The Minkowski space I Partially polarized light Cone of**

(fully polarized) light Fully polarized light V Q

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**M is the Mueller matrix of the transformation**

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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

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**The cone of (fully polarized) light**

Lorentz boost = de/polarizer, attenuators, dichroism V Q

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**The cone of (fully polarized) light**

3-d rotation = retardance, optical rotation V Q

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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

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**Mueller Calculus: 3 basic operations**

Absorption of one component Retardance of one component respect to the other Rotation of the reference system

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**Mueller Calculus: 3 basic operations**

Absorption of one component

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**Mueller Calculus: 3 basic operations**

Absorption of one component Retardance of one component respect to the other

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**Mueller Calculus: 3 basic operations**

Absorption of one component Retardance of one component respect to the other Rotation of the reference system

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**Optics of Crystals: Birefringence**

A. López Ariste

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Chapter XIV, Born & Wolf

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!!

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Ellipsoïd

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Ellipsoïd

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**Three types of crystals**

A spherical wavefront

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**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

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**Three types of crystals**

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**The ellipsoïd of D in uniaxial crystals**

z s The ellipsoïd of D in uniaxial crystals De The two propagating waves are linearly polarized and orthogonal one to each other Do

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**Typical birefringences**

Quartz Calcite Rutile Lithium Niobate

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**Common polarization devices for the laboratory and for astronomical instruments**

A. López Ariste

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Linear Polarizer

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Retarder

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Savart Plate

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**Glan-Taylor Polarizer**

Glan-Taylor.jpg

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**Glan-Thompson Polarizing Beam-Splitter**

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**Rochon Polarizing Beamsplitter**

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Polaroid

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**Dunn Solar Tower. New Mexico**

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**Typical birefringences**

Quartz Calcite Rutile Lithium Niobate Zero-order waveplates Multiple-order waveplates

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Waveplates

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**Principles of Polarimetry Modulation Absolute and Relative Polarimetry**

A. López Ariste

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

Measure # 1 : I + Q Measure # 2 : I - Q Subtraction: 0.5 (M1 – M2 ) = Q Addition: (M1 + M2 ) = I How to switch from Measure # 1 to Measure # 2? MODULATION

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**Measure # 1 : I + Q Measure # 2 : I - Q**

Subtraction: 0.5 (M1 – M2 ) = Q Addition: (M1 + M2 ) = I Principle of Polarimetry Everything should be the same EXCEPT for the sign

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MODULATION

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MODULATION

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**O is the Modulation Matrix**

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**MODULATION Conceptually, it is the easiest thing**

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

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**MODULATION Del Toro Iniesta & Collados (2000)**

Asensio Ramos & Collados (2008) MODULATION

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**MODULATION Del Toro Iniesta & Collados (2000)**

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

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MODULATION

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**Design of a Polarimeter**

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

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**Absolute vs. Relative Polarimetry**

Efficiency in Q,U and V limited by efficiency in I What limits efficiency in I?

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**Absolute vs. Relative Polarimetry**

What limits efficiency in I? Measure # 1 : I + Q Measure # 2 : I - Q Subtraction: 0.5 (M1 – M2 ) = Q Addition: (M1 + M2 ) = I Principle of Polarimetry Everything should be the same EXCEPT for the sign

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**Absolute vs. Relative Polarimetry**

What limits efficiency in I? Measure # 1 : I + Q Measure # 2 : I - Q Subtraction: 0.5 (M1 – M2 ) = Q Addition: (M1 + M2 ) = I Usual photometry of present astronomical detectors is around 10-3 Principle of Polarimetry Everything should be the same EXCEPT for the sign

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**Absolute vs. Relative Polarimetry**

What limits efficiency in I? Usual photometry of present astronomical detectors is around 10-3 You cannot do polarimetry better than photometry

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**Absolute vs. Relative Polarimetry**

What limits efficiency in I? Usual photometry of present astronomical detectors is around 10-3 You cannot do ABSOLUTE polarimetry better than photometry

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**Absolute vs. Relative Polarimetry**

Absolute error : 10-3 I Relative error : 10-3 Q

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**Absolute vs. Relative Polarimetry**

Li 6708 Absolute error : 10-3 I Relative error : 10-3 Q

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D2 D1 D2 Phase de 45 deg Phase de 102 deg

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**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

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**Principles of Polarimetry Spatial modulation, Temporal modulation, Spectral modulation**

A. López Ariste

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

Measure # 1 : I + Q Measure # 2 : I - Q Subtraction: 0.5 (M1 – M2 ) = Q Addition: (M1 + M2 ) = I How to switch from Measure # 1 to Measure # 2? MODULATION

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

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**Analyser: Calcite beamsplitter**

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**Analyser: Rotating Polariser**

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**Analyser: Calcite beamsplitter**

2 beams ≡2 images Spatial modulation Analyser: Rotating Polariser 2 angles ≡ 2 exposures Temporal modulation

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Modulator: What about U and V?

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Modulator:

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Modulator:

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**Modulator: Rotating λ/4**

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The basic Polarimeter Modulator Analyzer

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**Examples QW1 QW2 Measure T1 0° 0 ° Q T2 22.5 ° U T3 -45 ° V T4 45 ° -V**

2 Quarter-Waves + Calcite Beamsplitter QW1 QW2 Measure T1 0° 0 ° Q T2 22.5 ° U T3 -45 ° V T4 45 ° -V ….

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LCVR Calcite

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**Examples Rotating Quarterwave plate + Calcite Beamsplitter**

Photelastic Modulators (PEM) + Linear Polariser

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Spectral Modulation Chromatic waveplate: Followed by an analyzer

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**See Video from Frans Snik (Univ. Leiden)**

Spectral Modulation Chromatic waveplate: Followed by an analyzer See Video from Frans Snik (Univ. Leiden)

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**Principles of Polarimetry Noise and errors**

A. López Ariste

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**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 quantiﬁed Parametrized by position of zero point for Q, U, V ABSOLUTE POLARIMETRY

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**Sensitivity vs. Accuracy**

SENSITIVITY: Smallest detectable polarization signal related to noise levels in Q/I, U/I, V/I. RELATIVE POLARIMETRY Gaussian Noise (e.g. Photon Noise, Camera Shot Noise)

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**Correcting some unknown errors Spatio-temporal modulation**

Goal: to make the measurements symmetric respect to unknown errors in space and time I+V Detectin in different pixels I-V Exposure 1

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**Spatio-temporal modulation**

Goal: to make the measurements symmetric respect to unknown errors in space and time I+V I-V Detection at different times Detectin in different pixels I-V I+V Exposure 1 Exposure 2

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**Spatio-temporal modulation**

I+V I-V I-V I+V Exposure 1 Exposure 2

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**Spatio-temporal modulation**

Let’s make it more general

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**Cross-Talk Is this true? This is our polarimeter**

This is what comes from the outer universe Is this true?

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CrossTalk

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**Solutions to Crosstalk**

Avoid it: Measure it Mirrors with spherical symmetry (M1,M2) introduce no polarization Cassegrain-focus are good places for polarimeters THEMIS, 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 wavelength It forces you to observe the full Stokes vector

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**Dunn Solar Tower. New Mexico**

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**Solutions to Crosstalk**

Compensate it Several 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

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