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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Outline 1.Stokes Vectors, Jones Calculus and Mueller Calculus 2.Optics of Crystals: Birefringence 3.Common polarization devices for the laboratory and.
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Outline
1. Stokes Vectors, Jones Calculus and Mueller Calculus
2. Optics of Crystals: Birefringence3. Common polarization devices for the
laboratory and for astronomical instruments4. Principles of Polarimetry: Modulation and
Analysis. Absolute and Relative Polarimetry5. Principles of Polarimetry: Spatial modulation,
Temporal modulation, Spectral modulation6. Principles of Polarimetry: Noise and errors7. Spurious sources of polarization
Stokes Vector, Jones Calculus,
Mueller Calculus
playing around with matrices
A. López Ariste
)( tkzii
y
x
y
x eeA
A
E
E
Assumptions:
•A plane transverse electromagnetic wave•Quasi-monochromatic•Propagating in a well defined direction z
)( tkzii
y
x
y
x eeA
A
E
E
Jones Vector
)( tkzii
y
x
y
x eeA
A
E
E
Jones Vector:
It is actually a complex vector with 3 free parametersIt transforms under the Pauli matrices.It is a spinor
y
x
y
x
y
x
E
EC
E
E
dc
ba
E
E
3,0i
iiaC
3,0i
iiaC
10
010
10
011
01
102
0
03 i
i
The Jones matrix of an optical device
In group theory: SL(2,C)
)( tkzii
y
x
y
x eeA
A
E
E
From the quantum-mechanical point of view, the wave function cannot be measured directly.
Observables are made of quadratic forms of the wave function:
EEJ
J is a density matrix : The coherence matrix
**
**
yyxy
yxxx
EEEE
EEEEJ
3210 VUQIJ
Like Jones matrices, J also belongs to the SL(2,C) group, and can be decomposed in the basis of the Pauli matrices.
V
U
Q
I
Is the Stokes Vector
3210 VUQIJ
V
U
Q
I
I
The Stokes vector is the quadractic form of a spinor. It is a bi-spinor, or also a 4-vector
)(
JTrI
V
U
Q
I
02222 VUQI
02
0
02
3,2,1
4-vectors live in a Minkowsky space with metric (+,-,-,-)
)(
JTrI
The Minkowski space
I
VQ
Partially polarized light
Fully polarizedlight
Cone of (fully polarized) light
2222 VUQI
2222 VUQI
y
x
y
x
y
x
E
EC
E
E
dc
ba
E
E
CJCCEECEE
E
EJ yx
y
x
IMICCTrCJCTrJTrI
)()()(
M is the Mueller matrix of the transformation
)( CCTrM
)( CCTrM
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
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 quantifiedParametrized by position of zero point for Q, U, VABSOLUTE POLARIMETRY
Sensitivity vs. Accuracy
SENSITIVITY: Smallest detectable polarization signal
related to noise levels in Q/I, U/I, V/I.RELATIVE POLARIMETRY
MDMOI
1
nj
ijVUQIi Dn,1
2,,,
Gaussian Noise (e.g. Photon Noise, Camera Shot Noise)
Correcting some unknown errorsSpatio-temporal modulation
Goal: to make the measurements symmetric respect to unknown errors in space and time
Exposure 1
I+V
I-V
Det
ecti
n in
dif
fere
nt p
ixel
s
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
Det
ecti
n in
dif
fere
nt p
ixel
s
Detection at different times
Spatio-temporal modulation
2
2
2
1
1 41
I
Vo
I
V
VI
VI
VI
VI
Exposure 1
I+V
I-V
Exposure 2
I-V
I+V
:IV
Spatio-temporal modulation
2
2
2
1
1 41
I
Vo
I
V
VI
VI
VI
VI
Let’s make it more general
:IV
2
002
2
1
1
I
Io
I
IK
IO
IO
IO
IO
Cross-Talk
B
A
SI
SI
V
U
Q
I
MM ModAn1
1
This is our polarimeter This is what comes from the
outer universe
Is this true?
StarV
U
Q
I
StarV
U
Q
I
?Star
V
U
Q
I
935.0323.000
323.0935.000
0099.0009.0
00009.099.0
M
StarV
U
Q
I
StarV
U
Q
I
Star
Telescope
V
U
Q
I
M
CrossTalk
935.0323.000
323.0935.000
0099.0009.0
00009.099.0
M
Solutions to Crosstalk
1. Avoid it:
2. Measure it
Mirrors with spherical symmetry (M1,M2) introduce no polarizationCassegrain-focus are good places for polarimetersTHEMIS, 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 wavelengthIt forces you to observe the full Stokes vector
TelescopeM
Dunn Solar Tower. New Mexico
Solutions to Crosstalk
3. Compensate itSeveral procedures:• Introduce elements that compensate the
instrumental polarization• Measure the Stokes vector that carries the