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 +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
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 +0.009 Calcite -0.172 Rutile +0.287 Lithium Niobate -0.085 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: 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
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: 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 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: 0.5 (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° 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