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

2. 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

4. Jones Vector

5. 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)

7. 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

9. The Stokes vector is the quadractic form of a spinor
The Stokes vector is the quadractic form of a spinor. It is a bi-spinor, or also a 4-vector

10. 4-vectors live in a Minkowsky space with metric (+,-,-,-)

11. The Minkowski space I Partially polarized light Cone of
(fully polarized) light
Fully polarized
light V Q

13. M is the Mueller matrix of the transformation

14. 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

15. The cone of (fully polarized) light
Lorentz boost = de/polarizer, attenuators, dichroism V Q

16. The cone of (fully polarized) light
3-d rotation = retardance, optical rotation V Q

17. 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

18. Mueller Calculus: 3 basic operations
Absorption of one component
Retardance of one component respect to the other
Rotation of the reference system

19. Mueller Calculus: 3 basic operations
Absorption of one component

20. Mueller Calculus: 3 basic operations
Absorption of one component
Retardance of one component respect to the other

22. Mueller Calculus: 3 basic operations
Absorption of one component
Retardance of one component respect to the other
Rotation of the reference system

24. Optics of Crystals: Birefringence
A. López Ariste

26. Chapter XIV, Born & Wolf

27. !!

31. Ellipsoïd

32. Ellipsoïd

33. Three types of crystals
A spherical wavefront

34. 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

37. Three types of crystals

40. The ellipsoïd of D in uniaxial crystalsz s
The ellipsoïd of D in uniaxial crystals De
The two propagating waves are linearly polarized and orthogonal one to each other Do

42. Typical birefringences
Quartz
Calcite
Rutile
Lithium Niobate

43. Common polarization devices for the laboratory and for astronomical instruments
A. López Ariste

44. Linear Polarizer

45. Retarder

47. Savart Plate

48. Glan-Taylor Polarizer
Glan-Taylor.jpg

49. Glan-Thompson Polarizing Beam-Splitter

50. Rochon Polarizing Beamsplitter

51. Polaroid

52. Dunn Solar Tower. New Mexico

53. Typical birefringences
Quartz
Calcite
Rutile
Lithium Niobate
Zero-order waveplates
Multiple-order waveplates

54. Waveplates

56. Principles of Polarimetry Modulation Absolute and Relative Polarimetry
A. López Ariste

57. How to switch from Measure # 1 to Measure # 2?
Measure # 1 : I + Q
Measure # 2 : I - Q
Subtraction: 0.5 (M1 – M2 ) = Q
Addition: (M1 + M2 ) = I
How to switch from Measure # 1 to Measure # 2?
MODULATION

58. Measure # 1 : I + Q Measure # 2 : I - Q
Subtraction: 0.5 (M1 – M2 ) = Q
Addition: (M1 + M2 ) = I
Principle of Polarimetry
Everything should be the same EXCEPT for the sign

59. MODULATION

60. MODULATION

61. O is the Modulation Matrix

62. MODULATION Conceptually, it is the easiest thing
Is it so instrumentally?
Is it efficient respect to photon collection, noise and errors?

63. MODULATION Del Toro Iniesta & Collados (2000)
Asensio Ramos & Collados (2008)
MODULATION

64. MODULATION Del Toro Iniesta & Collados (2000)
Asensio Ramos & Collados (2008)
Del Toro Iniesta & Collados (2000)
MODULATION

65. MODULATION

66. Design of a Polarimeter
Specify an efficient modulation scheme: The answer is constrained by our instrumental choices

67. Absolute vs. Relative Polarimetry
Efficiency in Q,U and V limited by efficiency in I
What limits efficiency in I?

68. Absolute vs. Relative Polarimetry
What limits efficiency in I?
Measure # 1 : I + Q
Measure # 2 : I - Q
Subtraction: 0.5 (M1 – M2 ) = Q
Addition: (M1 + M2 ) = I
Principle of Polarimetry
Everything should be the same EXCEPT for the sign

69. Absolute vs. Relative Polarimetry
What limits efficiency in I?
Measure # 1 : I + Q
Measure # 2 : I - Q
Subtraction: 0.5 (M1 – M2 ) = Q
Addition: (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

70. 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

71. 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

72. Absolute vs. Relative Polarimetry
Absolute error : 10-3 I
Relative error : 10-3 Q

73. Absolute vs. Relative Polarimetry
Li 6708
Absolute error : 10-3 I
Relative error : 10-3 Q

75. D2 D1 D2
Phase de 45 deg
Phase de 102 deg

76. 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

77. Principles of Polarimetry Spatial modulation, Temporal modulation, Spectral modulation
A. López Ariste

78. How to switch from Measure # 1 to Measure # 2?
Measure # 1 : I + Q
Measure # 2 : I - Q
Subtraction: 0.5 (M1 – M2 ) = Q
Addition: (M1 + M2 ) = I
How to switch from Measure # 1 to Measure # 2?
MODULATION

79. How to switch from Measure # 1 to Measure # n?

80. Analyser: Calcite beamsplitter

81. Analyser: Rotating Polariser

82. Analyser: Calcite beamsplitter
2 beams ≡2 images
Spatial modulation
Analyser: Rotating Polariser
2 angles ≡ 2 exposures
Temporal modulation

83. Modulator:
What about U and V?

84. Modulator:

85. Modulator:

86. Modulator: Rotating λ/4

87. The basic Polarimeter
Modulator
Analyzer

88. Examples QW1 QW2 Measure T1 0° 0 ° Q T2 22.5 ° U T3 -45 ° V T4 45 ° -V
2 Quarter-Waves + Calcite Beamsplitter
QW1
QW2
Measure T1 0°
0 ° Q T2
22.5 ° U T3
-45 ° V T4
45 ° -V ….

89. LCVR
Calcite

90. Examples Rotating Quarterwave plate + Calcite Beamsplitter
Photelastic Modulators (PEM) + Linear Polariser

91. Spectral Modulation
Chromatic waveplate:
Followed by an analyzer

92. See Video from Frans Snik (Univ. Leiden)
Spectral Modulation
Chromatic waveplate:
Followed by an analyzer
See Video from Frans Snik (Univ. Leiden)

93. Principles of Polarimetry Noise and errors
A. López Ariste

94. 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

95. 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)

96. Correcting some unknown errors Spatio-temporal modulation
Goal: to make the measurements symmetric respect to unknown errors in space and time
I+V
Detectin in different pixels
I-V
Exposure 1

97. Spatio-temporal modulation
Goal: to make the measurements symmetric respect to unknown errors in space and time
I+V
I-V
Detection at
different times
Detectin in different pixels
I-V
I+V
Exposure 1
Exposure 2

98. Spatio-temporal modulation
I+V
I-V
I-V
I+V
Exposure 1
Exposure 2

99. Spatio-temporal modulation
Let’s make it more general

100. Cross-Talk Is this true? This is our polarimeter
This is what comes from the outer universe
Is this true?

103. CrossTalk

105. 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

106. Dunn Solar Tower. New Mexico

108. 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