# Outline Stokes Vectors, Jones Calculus and Mueller Calculus

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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

Stokes Vector, Jones Calculus, Mueller Calculus playing around with matrices
A. López Ariste

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

Jones Vector

Jones Vector: It is actually a complex vector with 3 free parameters
It transforms under the Pauli matrices. It is a spinor

The Jones matrix of an optical device
In group theory: SL(2,C)

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

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

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

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

The Minkowski space I Partially polarized light Cone of
(fully polarized) light Fully polarized light V Q

M is the Mueller matrix of the transformation

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

The cone of (fully polarized) light
Lorentz boost = de/polarizer, attenuators, dichroism V Q

The cone of (fully polarized) light
3-d rotation = retardance, optical rotation V Q

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

Mueller Calculus: 3 basic operations
Absorption of one component Retardance of one component respect to the other

Mueller Calculus: 3 basic operations
Absorption of one component Retardance of one component respect to the other Rotation of the reference system

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

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

Typical birefringences
Quartz Calcite Rutile Lithium Niobate

Common polarization devices for the laboratory and for astronomical instruments
A. López Ariste

Linear Polarizer

Retarder

Savart Plate

Glan-Taylor Polarizer
Glan-Taylor.jpg

Glan-Thompson Polarizing Beam-Splitter

Rochon Polarizing Beamsplitter

Polaroid

Dunn Solar Tower. New Mexico

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

Waveplates

Principles of Polarimetry Modulation Absolute and Relative Polarimetry
A. López Ariste

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

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

MODULATION

MODULATION

O is the Modulation Matrix

MODULATION Conceptually, it is the easiest thing
Is it so instrumentally? Is it efficient respect to photon collection, noise and errors?

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

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

MODULATION

Design of a Polarimeter
Specify an efficient modulation scheme: The answer is constrained by our instrumental choices

Absolute vs. Relative Polarimetry
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: (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: (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

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

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

Absolute vs. Relative Polarimetry
Absolute error : 10-3 I Relative error : 10-3 Q

Absolute vs. Relative Polarimetry
Li 6708 Absolute error : 10-3 I Relative error : 10-3 Q

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

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

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

Analyser: Calcite beamsplitter

Analyser: Rotating Polariser

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

Modulator: What about U and V?

Modulator:

Modulator:

Modulator: Rotating λ/4

The basic Polarimeter Modulator Analyzer

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 ° Q T2 22.5 ° U T3 -45 ° V T4 45 ° -V ….

LCVR Calcite

Examples Rotating Quarterwave plate + Calcite Beamsplitter
Photelastic Modulators (PEM) + Linear Polariser

Spectral Modulation Chromatic waveplate: Followed by an analyzer

See Video from Frans Snik (Univ. Leiden)
Spectral Modulation Chromatic waveplate: Followed by an analyzer 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 quantiﬁed Parametrized by position of zero point for Q, U, V ABSOLUTE POLARIMETRY

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)

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

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

Spatio-temporal modulation
I+V I-V I-V I+V Exposure 1 Exposure 2

Spatio-temporal modulation
Let’s make it more general

Cross-Talk Is this true? This is our polarimeter
This is what comes from the outer universe Is this true?

CrossTalk

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

Dunn Solar Tower. New Mexico

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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