Outline
description
Transcript of Outline
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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
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Stokes Vector, Jones Calculus,
Mueller Calculus
playing around with matrices
A. López Ariste
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)( tkzii
y
x
y
x eeA
A
E
E
Assumptions:
•A plane transverse electromagnetic wave•Quasi-monochromatic•Propagating in a well defined direction z
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)( tkzii
y
x
y
x eeA
A
E
E
Jones Vector
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)( 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
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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)
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)( 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
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**
**
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
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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
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V
U
Q
I
02222 VUQI
02
0
02
3,2,1
4-vectors live in a Minkowsky space with metric (+,-,-,-)
)(
JTrI
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The Minkowski space
I
VQ
Partially polarized light
Fully polarizedlight
Cone of (fully polarized) light
2222 VUQI
2222 VUQI
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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
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)( 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
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The cone of (fully polarized) light
I
VQ
Lorentz boost = de/polarizer, attenuators, dichroism
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The cone of (fully polarized) light
I
VQ
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
C
10200
0
aaC
0000
0000
0011
0011
2
aM
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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
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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
30 sincoscossin
sincos
C
1000
02cos2sin0
02sin2cos0
0001
M
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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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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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zs
De
Do
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 +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
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
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Retarder
cossin00
sincos00
0010
0001
M
?)(
cossin00
sincos00
0010
0001
)( 1
RRM
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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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dnne 0
Zero-order waveplates
Multiple-order waveplates
Typical birefringences
•Quartz +0.009
•Calcite -0.172
•Rutile +0.287
•Lithium Niobate -0.085
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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
Njj
njn
Njjj
ScM
ScM
,1
,1
11
VS
US
QS
IS
4
3
2
1
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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
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MODULATION
Njj
njn
Njjj
ScM
ScM
,1
,1
11
ii
i
cc
c
14,3,2
1 0
IOM
O is the Modulation Matrix
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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?
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MODULATION
IOM
MDMOI
1
nj
ijVUQIi D,1
2,,,
Del Toro Iniesta & Collados (2000)Asensio Ramos & Collados (2008)
nj
ijVUQIi Dn,1
2,,,
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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)
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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
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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
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?
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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
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
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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
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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?
VIM
VIM
UIM
UIM
QIM
QIM
6
5
4
3
2
1
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Analyser: Calcite beamsplitter
V
U
Q
I
M
0
0
QI
QI
0
0
QI
QI
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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
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Analyser: Rotating Polariser
Analyser: Calcite beamsplitter
2 beams ≡2 images Spatial modulation
2 angles ≡ 2 exposuresTemporal modulation
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Modulator:
V
U
Q
I
M Analyzer
0
0
QI
QI
What about U and V?
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Modulator:
V
U
Q
I
M Modulator
V
Q
U
I
V
U
Q
I
M Modulator
Q
U
V
I
![Page 85: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/85.jpg)
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
![Page 86: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/86.jpg)
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
![Page 87: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/87.jpg)
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
![Page 88: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/88.jpg)
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
….
![Page 89: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/89.jpg)
LCVR
Calcite
IwavelengthOM
)(
![Page 90: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/90.jpg)
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
![Page 91: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/91.jpg)
Spectral ModulationChromatic waveplate: )( f
V
U
Q
I
)(cos0)(sin0
0100
)(sin0)(cos0
0001
Followed by an analyzer )(cos1 QS
![Page 92: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/92.jpg)
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)
![Page 93: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/93.jpg)
Principles of Polarimetry Noise and errors
A. López Ariste
![Page 94: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/94.jpg)
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
![Page 95: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/95.jpg)
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)
![Page 96: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/96.jpg)
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
![Page 97: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/97.jpg)
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
![Page 98: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/98.jpg)
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
![Page 99: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/99.jpg)
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
![Page 100: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/100.jpg)
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?
![Page 101: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/101.jpg)
StarV
U
Q
I
StarV
U
Q
I
?Star
V
U
Q
I
![Page 102: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/102.jpg)
935.0323.000
323.0935.000
0099.0009.0
00009.099.0
M
![Page 103: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/103.jpg)
StarV
U
Q
I
StarV
U
Q
I
Star
Telescope
V
U
Q
I
M
CrossTalk
![Page 104: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/104.jpg)
935.0323.000
323.0935.000
0099.0009.0
00009.099.0
M
![Page 105: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/105.jpg)
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
![Page 106: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/106.jpg)
Dunn Solar Tower. New Mexico
![Page 107: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/107.jpg)
![Page 108: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/108.jpg)
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
![Page 109: Outline](https://reader036.fdocuments.us/reader036/viewer/2022062423/5681455c550346895db22ddc/html5/thumbnails/109.jpg)