Outline

1
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

3

• 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
6
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
8
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
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
I
Lorentz boost de/polarizer, attenuators,
dichroism
V
Q
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The cone of (fully polarized) light
I
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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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 0.009
• Calcite -0.172
• Rutile 0.287
• Lithium Niobate -0.085

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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 0.009
• Calcite -0.172
• Rutile 0.287
• Lithium Niobate -0.085

Zero-order waveplates
Multiple-order waveplates
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Waveplates
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Principles of PolarimetryModulation 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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MODULATION
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MODULATION
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)
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
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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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
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 ?/4
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The basic Polarimeter
Analyzer
Modulator
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Examples
2 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
.
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LCVR
Calcite
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Examples

1. Rotating Quarterwave plate Calcite Beamsplitter
2. Photelastic 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 quanti?ed 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 errorsSpatio-temporal
modulation
Goal to make the measurements symmetric respect
to unknown errors in space and time
IV
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
IV
I-V
Detection at different times
Detectin in different pixels
I-V
IV
Exposure 1
Exposure 2
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Spatio-temporal modulation
IV
I-V
I-V
IV
Exposure 1
Exposure 2
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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

• 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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