Outline
1.
2.
3.
4.
5.
6.
7.
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
 E x   Ax


 E   A e i
 y  y
 i ( kzt )
e


 E x   Ax


 E   A e i
 y  y
Jones Vector
 i ( kzt )
e


 E x   Ax


 E   A e i
 y  y
 i ( kzt )
e


Jones Vector:
It is actually a complex vector with 3 free parameters
It transforms under the Pauli matrices.
It is a spinor

 E x   a b  E x 
 Ex 
   
   C  
E 
 y   c d  E y 
 Ey 
C
a 
i 0,3
i
i
The Jones matrix of an optical device
C
a 
i 0,3
1
 0  
0
0
 2  
1
i
i
0

1
1 0 

 1  
 0  1
1

0
 0 i

 3  
  i 0
In group theory: SL(2,C)
 E x   Ax


 E   A e i
 y  y
 i ( kzt )
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:
 
J  EE
J is a density matrix : The coherence matrix
  E x E x*   E x E *y
J  
*
*

E
E


E
E
y
x
y
y



 
Like Jones matrices, J also belongs to the
SL(2,C) group, and can be decomposed in the
basis of the Pauli matrices.
J  I 0  Q  1  U  2  V  3
I
 
Q
U 
 
V 
 
Is the Stokes Vector
J  I 0  Q  1  U  2  V  3
I
 
 Q
I  
U
 
V 
 
The Stokes vector is
the quadractic form of
a spinor. It is a bispinor, or also a 4vector


I  Tr ( J )


I  Tr ( J )
I
 
Q
U 
 
V 
 
0 0
2
 1, 2,3   0
2
I  Q U V  0
2
2
2
2
4-vectors live in a Minkowsky
space with metric (+,-,-,-)
The Minkowski space
I
Cone of
(fully polarized)
light
I 2  Q2 U 2 V 2
Partially polarized light
Fully polarized
light
I 2  Q2 U 2 V 2
V
Q

 E x   a b  E x 
 Ex 
   






C
 E 
E 
E 
c
d
 y 
 y 
 y

  
 Ex 

 
J      E x
E y  CEE C  CJC 
 Ey 









I   Tr ( J  )  Tr (CJC  )  Tr (CC  ) I  MI
M is the Mueller matrix of the transformation
 
M  Tr (CC  )
 
M  Tr (CC  )
From group theory, the Mueller matrix belongs to a
group of transformations which is the square of SL(2,C)
SL(2, C )  SL(2, C )
Actually a subgroup of this general group called O+(3,1)
or Lorentz group
The cone of (fully polarized)
light
I
Lorentz boost = de/polarizer,
attenuators, dichroism
V
Q
The cone of (fully polarized)
light
I
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


 a 0  Ca  
   0   1 
C  
 0 0 2
1

a 1
M  
2 0

0

1 0 0

1 0 0
0 0 0

0 0 0 
Mueller Calculus:
3 basic operations
• Absorption of one component
• Retardance of one component respect
to the other
1 0 
i
i

C  

1

e


1

e
1
0
i 
0 e 




1

0
M 
0

0

0
0
1
0
0
cos 
0  sin 
0 

0 
sin  

cos  
Mueller Calculus:
3 basic operations
• Absorption of one component
• Retardance of one component respect
to the other
• Rotation of the reference system
 cos 
C  
  sin 
sin  
  cos  0  sin  3
cos  
0
1

 0 cos 2
M 
0  sin 2

0
0

0
sin 2
cos 2
0
0

0
0

1 
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
Three types of crystals
s
z
The ellipsoïd of D in
uniaxial crystals
De
Do
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
 0.5 0.5

 0.5 0.5
M 
0
0

 0
0

 0 . 5 0 .5

 0 . 5 0 .5
M  R( )
0
0

 0
0

0 0

0 0
0 0

0 0 
0 0
cos 2
sin 2
 1


0 0  1
cos 2 2
cos 2 sin 2
 cos 2
R ( )  0.5

0 0
sin 2 sin 2 cos 2
sin 2 2



 0
0 0
0
0

0

0
0

0 
Retarder
1

0
M 
0

0

1

0
M  R ( )
0

0

0
0
1
0
0
cos 
0  sin 
0
0
1
0
0
cos 
0  sin 
0 

0 
sin  

cos  
0 

0  1
R ( )  ?

sin 

cos  
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
  ne  n0 
d

Multiple-order waveplates
Waveplates
Principles of Polarimetry
Modulation
Absolute and Relative Polarimetry
A. López Ariste
Measure # 1 : I + Q
Measure # 2 : I - Q
Subtraction:
(M1 #–1M2
) = Q # 2?
How
to switch from0.5
Measure
to Measure
Addition:
MODULATION
0.5 (M1 + M2 ) =
I
Measure # 1 : I + Q
Measure # 2 : I - Q
Principle
Polarimetry
Subtraction:
0.5of(M1
– M2 ) = Q
Everything should be the same EXCEPT for the sign
Addition:
0.5 (M1 + M2 ) = I
MODULATION
M1 
c S
j 1, N
1
j
S1  I
S2  Q
j
S3  U

Mn 
c S
j 1, N
n
j
j
S4  V
MODULATION
M1 
c S
j 1, N
1
j
S1  I
S2  Q
j
S3  U

Mn 
c S
j 1, N
n
j
j
S4  V
c 0
i
1
c
i
2 , 3, 4
c
i
1
MODULATION
M1 
c S
j 1, N

Mn 
1
j


M  OI
j
O is the Modulation Matrix
c S
j 1, N
n
j
j
c 0
i
1
c
i
2 , 3, 4
c
i
1
MODULATION
M1  I  Q
M2  I Q
M3  I U
M 4  I U
M5  I V
M 6  I V
1 1 0 0 


1  1 0 0 
1 0 1 0 

O
1 0  1 0 
1 0 0 1 


 1 0 0  1


Conceptually, it is the easiest thing
Is it so instrumentally?
Is it efficient respect to photon collection, noise and errors?
Del Toro Iniesta & Collados (2000)
Asensio Ramos & Collados (2008)
MODULATION


M  OI



1
I  O M  DM
 i  I ,Q ,U ,V  
D
j 1, n
2
ij
 i  I ,Q ,U ,V  n  Dij2
j 1, n
Del Toro Iniesta & Collados (2000)
Asensio Ramos & Collados (2008)
MODULATION



1
I  O M  DM
 i  I ,Q ,U ,V  n  Dij2
j 1, n
I 1

i Q ,U ,V
2
i
1
MODULATION
M1  I  Q
M2  I Q
M3  I U
M 4  I U
M5  I V
M 6  I V
1 1 0 0 


1  1 0 0 
1 0 1 0 

O
1 0  1 0 
1 0 0 1 


 1 0 0  1


I 1
 Q ,U ,V
1

3
Design of a Polarimeter
•Specify an efficient modulation scheme: The answer is constrained
by our instrumental choices
Absolute vs. Relative Polarimetry
 i  I ,Q ,U ,V  n  Dij2
j 1, n
I 1

i Q ,U ,V
2
i
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
Principle
of Polarimetry
Subtraction:
0.5 (M1
– M2 ) = Q
Everything
should0.5
be the
same
EXCEPT
Addition:
(M1
+ M2
) = Ifor the sign
Absolute vs. Relative Polarimetry
What limits efficiency in I?
Measure # 1 : I + Q
Measure # 2 : I - Q
Usual photometry of
present astronomical detectors is around 10-3
Principle
of Polarimetry
Subtraction:
0.5 (M1
– M2 ) = Q
Everything
should0.5
be the
same
EXCEPT
Addition:
(M1
+ M2
) = Ifor 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
 Q  Q
M 1  I  Q  I 1     1  
I 
I

 Q  Q
M 2  I  Q  I 1    1  
I 
I


Q 
 Q
 
I 1    ( I  )1 
I

 ( I  ) 
Q
   (2 I  )

( I  )


  Q 





   2  
    2  Q
I  1   
I




I 

Absolute error : 10-3 I
Relative error : 10-3 Q
Absolute vs. Relative Polarimetry
 Q  Q
M 1  I  Q  I 1     1  
Li 6708I   I 
 Q  Q
M 2  I  Q  I 1    1  
I 
I


Q 
 Q
 
I 1    ( I  )1 
I

 ( I  ) 
Q
   (2 I  )

( I  )


  Q 





   2  
    2  Q
I  1   
I




I 

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
Measure # 1 : I + Q
Measure # 2 : I - Q
Subtraction:
(M1 #–1M2
) = Q # 2?
How
to switch from0.5
Measure
to Measure
Addition:
MODULATION
0.5 (M1 + M2 ) =
I
M1  I  Q
M2  I Q
M3  I U
M 4  I U
M5  I V
M 6  I V
How to switch from Measure # 1 to Measure # n?
Analyser: Calcite beamsplitter
I
 
Q
M  
U
 
V 
 
 I  Q


 I  Q
 0 


 0 


 I Q


 I Q
 0 


 0 


Analyser: Rotating Polariser
 1

 cos 2
 sin 2

 0

cos 2
sin 2
cos 2 2
cos 2 sin 2
sin 2 cos 2
sin 2 2
0
0
 0
I

I




 Q

 Q
0 

0 
0  I  
I  Q cos 2  U sin 2

  

2
0  Q   I cos 2  Q cos 2  U cos 2 sin 2 




0 U
I sin 2  Q sin 2 cos 2  U sin 2 2 
  






0  V  
0



2
 I Q 


 I  Q
 0 


 0 


Analyser: Calcite beamsplitter
2 beams ≡2 images
Spatial modulation
Analyser: Rotating Polariser
2 angles ≡ 2 exposures
Temporal modulation
Modulator:
I
 
Q
M Analyzer   
U
 
V 
 
I

I




 Q

 Q
0 

0 
What about U and V?
Modulator:
I I
   
 Q  U 
M Modulator     
U
Q
   
V  V 
   
I I
   
 Q  V 
M Modulator     
U
U
   
V   Q 
   
Modulator:
I I
   
 Q  U 
M Modulator     
U
Q
   
V  V 
   
I I
   
 Q  V 
M Modulator     
U
U
   
V   Q 
   
 I  I U 
  

Q  I U 
M An M Mod    
U
A 
  

V   B 
  

 I  I V 
  

Q  I V 
M An M Mod    
U
A 
  

V   B 
  

Modulator: Rotating λ/4
0
1

 0 cos 
0
0

 0  sin 

0  I  
I

  

0 sin   Q   Q cos   V sin  





1
0
U
U
  

0 cos   V    Q sin   V cos  
0
I
 

V 
   
U
2
 
Q
 
 I  I
  
Q  I
M An M Mod    
U
  
V  
  
V 

V 
A 

B 
The basic Polarimeter
Modulator
I
 
Q
U 
 
V 
 
 I 
 
 S1 
S 
 2
S 
 3
Analyzer
I

I




 S1 

 S1 
S2 

S 3 
I

I




 S1 

 S1 
S2 

S 3 
S1
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
….
LCVR
Calcite


M  O( wavelength) I
Examples
1
2
Rotating Quarterwave plate + Calcite Beamsplitter
Photelastic Modulators (PEM) + Linear Polariser
S1  Q cos 2t  U cos 2t sin 2t  V sin 2t
2

S
Q
1

0

2
 S   S
1
1
0

S
0
3
 2
4
1

1
4
2

4
 S   S
2
V
1

 S
3
4
1
U
Spectral Modulation
Chromatic waveplate:
0
1

 0 cos  ( )
0
0

 0  sin  ( )

Followed by an analyzer
  f ( )
 I 
 
0 sin  ( )  Q 
U 
1
0
 
0 cos  ( )  V 
0
0
S1  Q cos  ( )
Spectral Modulation
Chromatic waveplate:
0
1

 0 cos  ( )
0
0

 0  sin  ( )

Followed by an analyzer
  f ( )
 I 
 
0 sin  ( )  Q 
U 
1
0
 
0 cos  ( )  V 
0
0
S1  Q cos  ( )
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



1
I  O M  DM
Gaussian Noise (e.g. Photon Noise, Camera Shot Noise)
 i  I ,Q ,U ,V  n  D
j 1, n
2
ij
Correcting some unknown errors
Spatio-temporal modulation
Detectin in different pixels
Goal: to make the measurements symmetric respect to
unknown errors in space and time
I+V
I-V
Exposure 1
Spatio-temporal modulation
Detectin in different pixels
Goal: to make the measurements symmetric respect to
unknown errors in space and time
I+V
I-V
Detection at
different times
I-V
Exposure 1
I+V
Exposure 2
Spatio-temporal modulation
I  V 1 I  V 2
I  V 1 I  V 2
V  I :
V
 1  4
I
 V 
  o 
 I
2
I+V
I-V
I-V
I+V
Exposure 1
Exposure 2
Spatio-temporal modulation
V  I :
I  V 1 I  V 2
I  V 1 I  V 2
V
 1  4
I
 V 
  o 
 I
Let’s make it more general
 

I
O I O2 I
    K 
O I O2 I
 I0

1

1

 I
  o
 I
  0




2
2
Cross-Talk
 I  I
  
Q  I
M An M Mod    
U
  
V  
  
This is our
polarimeter
 S1 

 S1 
A 

B 
This is what comes from the
outer universe
Is this true?
I
 
Q
U 
 
V 
  Star
I
 
Q
U 
 
V 
  Star
?
I
 
Q
U 
 
V 
  Star
0
0 
 0.99  0.009


0
0 
  0.009 0.99
M 
0
0
 0.935 0.323 


 0
0
 0.323  0.935

I
 
Q
U 
 
V 
  Star
CrossTalk
I
 
Q
M Telescope  
U
 
V 
  Star
I
 
Q
U 
 
V 
  Star
0
0 
 0.99  0.009


0
0 
  0.009 0.99
M 
0
0
 0.935 0.323 


 0
0
 0.323  0.935

Solutions to Crosstalk
1. Avoid 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,…
2. Measure it
Given M
find its inverse and apply it to the measurements
Telescope
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
3. 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