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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 - PowerPoint PPT PresentationTRANSCRIPT
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
),2(),2( CSLCSL
The cone of (fully polarized) light
I
VQ
Lorentz boost = de/polarizer, attenuators, dichroism
The cone of (fully polarized) light
I
VQ
3-d rotation = retardance, optical rotation
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
C
10200
0
aaC
0000
0000
0011
0011
2
aM
Mueller Calculus: 3 basic operations
• Absorption of one component• Retardance of one component
respect to the other
10 110
01
ii
i eee
C
cossin00
sincos00
0010
0001
M
Mueller Calculus: 3 basic operations
• Absorption of one component• Retardance of one component
respect to the other• Rotation of the reference system
30 sincoscossin
sincos
C
1000
02cos2sin0
02sin2cos0
0001
M
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
zs
De
Do
The ellipsoïd of D in uniaxial crystals
The two propagating waves are linearly polarized and orthogonal one to each other
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
0000
0000
005.05.0
005.05.0
M
0000
02sin2cos2sin2sin
02sin2cos2cos2cos
02sin2cos1
5.0)(
0000
0000
005.05.0
005.05.0
)( 2
21
RRM
Retarder
cossin00
sincos00
0010
0001
M
?)(
cossin00
sincos00
0010
0001
)( 1
RRM
Savart Plate
Glan-Taylor Polarizer
Glan-Taylor.jpg
Glan-Thompson Polarizing Beam-Splitter
Rochon Polarizing Beamsplitter
Polaroid
Dunn Solar Tower. New Mexico
dnne 0
Zero-order waveplates
Multiple-order waveplates
Typical birefringences
•Quartz +0.009
•Calcite -0.172
•Rutile +0.287
•Lithium Niobate -0.085
Waveplates
Principles of PolarimetryModulation
Absolute and Relative Polarimetry
A. López Ariste
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
Njj
njn
Njjj
ScM
ScM
,1
,1
11
VS
US
QS
IS
4
3
2
1
MODULATION
Njj
njn
Njjj
ScM
ScM
,1
,1
11
VS
US
QS
IS
4
3
2
1
ii
i
cc
c
14,3,2
1 0
MODULATION
Njj
njn
Njjj
ScM
ScM
,1
,1
11
ii
i
cc
c
14,3,2
1 0
IOM
O is the Modulation Matrix
MODULATION
VIM
VIM
UIM
UIM
QIM
QIM
6
5
4
3
2
1
1001
1001
0101
0101
0011
0011
O
Conceptually, it is the easiest thingIs it so instrumentally?
Is it efficient respect to photon collection, noise and errors?
MODULATION
IOM
MDMOI
1
nj
ijVUQIi D,1
2,,,
Del Toro Iniesta & Collados (2000)Asensio Ramos & Collados (2008)
nj
ijVUQIi Dn,1
2,,,
MODULATION
MDMOI
1
Del Toro Iniesta & Collados (2000)
nj
ijVUQIi Dn,1
2,,,
VUQii
I
,,
2 1
1
Del Toro Iniesta & Collados (2000)Asensio Ramos & Collados (2008)
MODULATION
VIM
VIM
UIM
UIM
QIM
QIM
6
5
4
3
2
1
1001
1001
0101
0101
0011
0011
O
3
1
1
,,
VUQ
I
Design of a Polarimeter
•Specify an efficient modulation scheme: The answer is constrained by our instrumental choices
Absolute vs. Relative Polarimetry
nj
ijVUQIi Dn,1
2,,,
VUQii
I
,,
2 1
1
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
Principle of Polarimetry
Everything should be the same EXCEPT for the sign
Usual photometry of present astronomical detectors is around 10-3
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
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
Absolute vs. Relative Polarimetry
I
Q
I
QIQIM
I
Q
I
QIQIM
11
11
2
1
QI
I
Q
I
I
QI
I
QI
I
QI
21
2
)()2(
)(1)(1
Absolute error : 10-3 IRelative error : 10-3 Q
Absolute vs. Relative Polarimetry
I
Q
I
QIQIM
I
Q
I
QIQIM
11
11
2
1
QI
I
Q
I
I
QI
I
QI
I
QI
21
2
)()2(
)(1)(1
Absolute error : 10-3 IRelative error : 10-3 Q
Li 6708
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
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?
VIM
VIM
UIM
UIM
QIM
QIM
6
5
4
3
2
1
Analyser: Calcite beamsplitter
V
U
Q
I
M
0
0
QI
QI
0
0
QI
QI
Analyser: Rotating Polariser
0
2sin2cos2sin2sin
2sin2cos2cos2cos
2sin2cos
0000
02sin2cos2sin2sin
02sin2cos2cos2cos
02sin2cos1
2
2
2
2
UQI
UQI
UQI
V
U
Q
I
0
0
QI
QI
0
0
0
QI
QI
2
Analyser: Rotating Polariser
Analyser: Calcite beamsplitter
2 beams ≡2 images Spatial modulation
2 angles ≡ 2 exposuresTemporal modulation
Modulator:
V
U
Q
I
M Analyzer
0
0
QI
QI
What about U and V?
Modulator:
V
U
Q
I
M Modulator
V
Q
U
I
V
U
Q
I
M Modulator
Q
U
V
I
Modulator:
V
U
Q
I
M Modulator
V
Q
U
I
V
U
Q
I
M Modulator
Q
U
V
I
B
A
UI
UI
V
U
Q
I
MM ModAn
B
A
VI
VI
V
U
Q
I
MM ModAn
Modulator: Rotating λ/4
cossin
sincos
cos0sin0
0100
sin0cos0
0001
VQ
U
VQ
I
V
U
Q
I
Q
U
V
I
2
B
A
VI
VI
V
U
Q
I
MM ModAn
The basic Polarimeter
Modulator Analyzer
V
U
Q
I
3
2
1
S
S
S
I
3
2
1
1
S
S
SI
SI
3
2
1
1
S
S
SI
SI 1S
Examples2 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
….
LCVR
Calcite
IwavelengthOM
)(
Examples1 Rotating Quarterwave plate + Calcite Beamsplitter2 Photelastic Modulators (PEM) + Linear Polariser
tVttUtQS 2sin2sin2cos2cos21
0
1 QS
2
0 2
11
VSS
4
0
43
2 43
11
2
4
11
USSSS
Spectral ModulationChromatic waveplate: )( f
V
U
Q
I
)(cos0)(sin0
0100
)(sin0)(cos0
0001
Followed by an analyzer )(cos1 QS
Spectral ModulationChromatic waveplate: )( f
V
U
Q
I
)(cos0)(sin0
0100
)(sin0)(cos0
0001
Followed by an analyzer )(cos1 QS
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 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
information• Project the Stokes vector into the
Eigenvector of the matrix