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Outline 1.Stokes Vectors, Jones Calculus and Mueller Calculus 2.Optics of Crystals: Birefringence 3.Common polarization devices for the laboratory and for astronomical instruments 4.Principles of Polarimetry: Modulation and Analysis. Absolute and Relative Polarimetry 5.Principles of Polarimetry: Spatial modulation, Temporal modulation, Spectral modulation 6.Principles of Polarimetry: Noise and errors 7.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. 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 V Q Partially polarized light Fully polarized light Cone of (fully polarized) light

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

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

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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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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 v o An extraordinary wave with an elliptical wavefront, its speed depends on direction with characteristic values v o and v e

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Three types of crystals

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

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

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Waveplates

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

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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

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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

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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)

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

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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: 0.5 (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: 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

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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

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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

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Absolute vs. Relative Polarimetry Absolute error : I Relative error : Q

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Absolute vs. Relative Polarimetry Absolute error : I Relative error : Q Li 6708

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D2D1 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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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

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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 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: Rotating λ/4

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The basic Polarimeter Modulator Analyzer

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Examples 2 Quarter-Waves + Calcite Beamsplitter QW1QW2Measure T10°0°0 °Q T222.5 ° U T30 °-45 °V T40 °45 °-V ….

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LCVR Calcite

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Examples 1Rotating Quarterwave plate + Calcite Beamsplitter 2Photelastic Modulators (PEM) + Linear Polariser

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Spectral Modulation Chromatic waveplate: Followed by an analyzer

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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 quantied 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 Exposure 1 I+V I-V Detectin in different pixels

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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 Detectin in different pixels Detection at different times

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

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Spatio-temporal modulation Lets make it more general

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Cross-Talk 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 1.Avoid it: 2.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 3.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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