1. Lecture 9
Polarization of Light

2. Introduction Part I: Different polarization states of light
Part II: Stokes parameters, Mueller matrices
Part III: Optical components for polarimetry
Part IV: Polarimeters

3. Part I: Different polarization states of light
Light as an electromagnetic wave
Mathematical and graphical descriptions of polarization
Linear, circular, elliptical light
Polarized, unpolarized light

4. Light as an electromagnetic wave
Part I: Polarization states
Light as an electromagnetic wave
Light is a transverse wave,
an electromagnetic wave

5. Mathematical description of the EM wave
Part I: Polarization states
Mathematical description of the EM wave
Light wave that propagates in the z direction:

6. Graphical representation of the EM wave (I)
Part I: Polarization states
Graphical representation of the EM wave (I)
One can go from:
to the equation of an ellipse (using trigonometric identities, squaring, adding):

7. Graphical representation of the EM wave (II)
Part I: Polarization states
Graphical representation of the EM wave (II)
An ellipse can be represented by 4 quantities:
size of minor axis
size of major axis
orientation (angle)
sense (CW, CCW)
Light can be represented by 4 quantities...

8. Vertically polarized light
Part I: Polarization states, linear polarization
Vertically polarized light
If there is no amplitude in x (E0x = 0), there is only one component, in y (vertical).

9. Polarization at 45º (I) If there is no phase difference (=0) and
Part I: Polarization states, linear polarization
Polarization at 45º (I)
If there is no phase difference (=0) and
E0x = E0y, then Ex = Ey

10. Part I: Polarization states, linear polarization
Polarization at 45º (II)

11. Linear polarization E-field magnitude oscillates Direction fixed
Arbitrary polarization angle
superposition of x and y polarized waves
real numbers
Time
evolution
Example
45 ° linear polarization

12. Circular polarization (I)
Part I: Polarization states, circular polarization
Circular polarization (I)
If the phase difference is = 90º and E0x = E0y
then: Ex / E0x = cos  , Ey / E0y = sin 
and we get the equation of a circle:

13. Circular polarization (II)
Part I: Polarization states, circular polarization
Circular polarization (II)

14. Circular polarization (III)
Part I: Polarization states, circular polarization
Circular polarization (III)

15. Circular polarization (IV)
Part I: Polarization states, circular polarization... see it now?
Circular polarization (IV)

16. Circular polarization
E-field magnitude constant
Direction rotates
Complex superposition of x and y polarizations
x and y in quadrature
Time
evolution
Example:
right circular polarization

17. Elliptical polarization
Part I: Polarization states, elliptical polarization
Elliptical polarization
Linear + circular polarization = elliptical polarization

18. Elliptical polarization
General case
polarization partly linear and partly circular
E-field sweeps out ellipse
both magnitude and direction change with time
Superposition of L and R states
Superposition of L and R
Elliptical polarization

19. Polarization summary Decompose into x and y polarizations
Linear -- real superposition
Circular -- quadrature superposition
Linear polarizations
Circular polarizations Ey
E+45
i Ey Ex
ERight Ex Ey
ELeft
E-45
i Ey Ex Ex

20. Unpolarized light (natural light)
Part I: Polarization states, unpolarized light
Unpolarized light (natural light)

21. Producing linear polarization -- 1
Induce loss for one polarization direction
Examples: wire grid, polaroid filter

22. Producing linear polarization --
Separation in birefringent crystal -- ex: calcite
Ordinary wave -- behaves as expected
Extra-ordinary wave -- behaves differently
example -- E-field not perpendicular to propagation direction
Input light converted to two polarized beams

23. Producing linear polarization --
Brewster’s angle
Only one polarization reflected
Reflected light polarized
Works with most surfaces
Good way to calibrate polarizers
Refracted beam
creates dipoles in medium
Brewster angle:
dipole field zero
perpendicular to
reflection prop.
direction

24. Polarizing cubes
Uses fact that total internal reflection close to Brewster’s angle
reflection large for other polarization
Multi-layer coating enhances effect
reflections from multiple surfaces -- resonance
Brewster polarization reflection always zero

25. Waveplates Polarization converters
One linear polarization direction propagates faster
Half wave plate -- phase delay 180°
rotate linear polarization up to 90°
fast axis at 45° to input polarization direction
Quarter wave plate -- phase delay 90°
convert linear to circular polarization
R or L for fast axis +45 or -45 to input pol.
Rotate linear pol. by angle 2q
Create circular polarization
Retardation of
one polarization

26. Other circular polarizers
Use phase shift for total internal reflection
45° over broad range of angles
Two reflections give 90°
Converts linear to circular polarization

27. Part II: Stokes parameters and Mueller matrices
Stokes parameters, Stokes vector
Stokes parameters for linear and circular polarization
Stokes parameters and polarization P
Mueller matrices, Mueller calculus
Jones formalism

28. Part II: Stokes parameters
Stokes parameters (I) 1852: Sir George Gabriel Stokes took a very different approach and discovered that polarization can be described in terms of observables using an experimental definition
The polarization ellipse is only valid at a given instant of time (function of time):
To get the Stokes parameters, do a time average (integral over time) and a little bit of algebra...

29. Stokes parameters (II) described in terms of the electric field
Part II: Stokes parameters
Stokes parameters (II) described in terms of the electric field
The 4 Stokes parameters are:

30. Stokes parameters (III) described in geometrical terms
Part II: Stokes parameters
Stokes parameters (III) described in geometrical terms

31. Part II: Stokes parameters, Stokes vectors
The Stokes parameters can be arranged in a Stokes vector:
Linear polarization
Circular polarization
Fully polarized light
Partially polarized light
Unpolarized light

32. Pictorial representation of the Stokes parameters
Part II: Stokes parameters
Pictorial representation of the Stokes parameters

33. Stokes vectors for linearly polarized light
Part II: Stokes parameters, examples
Stokes vectors for linearly polarized light
LHP light
LVP light
+45º light
-45º light

34. Stokes vectors for circularly polarized light
Part II: Stokes parameters, examples
Stokes vectors for circularly polarized light
RCP light
LCP light

35. Part II: Stokes parameters, Mueller matrices
If light is represented by Stokes vectors, optical components are then described with Mueller matrices:
[output light] = [Muller matrix] [input light]

36. Mueller calculus (I) I’ = M3 M2 M1 I Element 1 Element 2 Element 3
Part II: Stokes parameters, Mueller matrices
Mueller calculus (I)
Element Element Element 3
I’ = M3 M2 M1 I

37. Part II: Stokes parameters, Mueller matrices
Mueller calculus (II)
Mueller matrix M’ of an optical component with Mueller matrix M rotated by an angle :
M’ = R(- ) M R() with:

38. Part II: Stokes parameters, Jones formalism, not that important here...
Stokes vectors and Mueller matrices cannot describe interference effects. If the phase information is important (radio-astronomy, masers...), one has to use the Jones formalism, with complex vectors and Jones matrices:
Jones vectors to describe the polarization of light:
Jones matrices to represent optical components:
BUT: Jones formalism can only deal with 100% polarization...

39. Part III: Optical components for polarimetry
Complex index of refraction
Polarizers
Retarders

40. Complex index of refraction
Part III: Optical components
Complex index of refraction
The index of refraction is actually a complex quantity:
real part
optical path length, refraction: speed of light depends on media
birefringence: speed of light also depends on P
imaginary part
absorption, attenuation, extinction: depends on media
dichroism/diattenuation: also depends on P

41. Part III: Optical components, polarizers
Polarizers absorb one component of the polarization but not the other.
The input is natural light, the output is polarized light (linear, circular, elliptical). They work by dichroism, birefringence, reflection, or scattering.

42. Wire-grid polarizers (I) [dichroism]
Part III: Optical components, polarizers
Wire-grid polarizers (I) [dichroism]
Mainly used in the IR and longer wavelengths
Grid of parallel conducting wires with a spacing comparable to the wavelength of observation
Electric field vector parallel to the wires is attenuated because of currents induced in the wires

43. Wide-grid polarizers (II) [dichroism]
Part III: Optical components, polarizers
Wide-grid polarizers (II) [dichroism]

44. Dichroic crystals [dichroism]
Part III: Optical components, polarizers
Dichroic crystals [dichroism]
Dichroic crystals absorb one polarization state over the other one.
Example: tourmaline.

45. Polaroids [dichroism]
Part III: Optical components, polarizers – Polaroids, like in sunglasses!
Polaroids [dichroism]
Made by heating and stretching a sheet of PVA laminated to a supporting sheet of cellulose acetate treated with iodine solution (H-type polaroid). Invented in 1928.

46. Crystal polarizers (I) [birefringence]
Part III: Optical components, polarizers
Crystal polarizers (I) [birefringence]
Optically anisotropic crystals
Mechanical model:
the crystal is anisotropic, which means that the electrons are bound with different ‘springs’ depending on the orientation
different ‘spring constants’ gives different propagation speeds, therefore different indices of refraction, therefore 2 output beams

47. Crystal polarizers (II) [birefringence]
Part III: Optical components, polarizers
Crystal polarizers (II) [birefringence]
isotropic
crystal
(sodium
chloride)
anisotropic
(calcite)
The 2 output beams are polarized (orthogonally).

48. Crystal polarizers (IV) [birefringence]
Part III: Optical components, polarizers
Crystal polarizers (IV) [birefringence]
Crystal polarizers used as:
Beam displacers,
Beam splitters,
Polarizers,
Analyzers, ...
Examples: Nicol prism, Glan-Thomson polarizer, Glan or Glan-Foucault prism, Wollaston prism, Thin-film polarizer, ...

49. Part III: Optical components, retarders
In retarders, one polarization gets ‘retarded’, or delayed, with respect to the other one. There is a final phase difference between the 2 components of the polarization. Therefore, the polarization is changed.
Most retarders are based on birefringent materials (quartz, mica, polymers) that have different indices of refraction depending on the polarization of the incoming light.

50. Part III: Optical components, retarders
Half-Wave plate (I)
Retardation of ½ wave or 180º for one of the polarizations.
Used to flip the linear polarization or change the handedness of circular polarization.

51. Part III: Optical components, retarders
Half-Wave plate (II)

52. Quarter-Wave plate (I)
Part III: Optical components, retarders
Quarter-Wave plate (I)
Retardation of ¼ wave or 90º for one of the polarizations
Used to convert linear polarization to elliptical.

53. Quarter-Wave plate (II)
Part III: Optical components, retarders
Quarter-Wave plate (II)
Special case: incoming light polarized at 45º with respect to the retarder’s axis
Conversion from linear to circular polarization (vice versa)

54. (Back to polarizers, briefly) Circular polarizers
Part III: Optical components, polarizers
(Back to polarizers, briefly) Circular polarizers
Input light: unpolarized Output light: circularly polarized
Made of a linear polarizer glued to a quarter-wave plate oriented at 45º with respect to one another.

55. Part III: Optical components, retarders
Other retarders
Soleil-Babinet: variable retardation to better than 0.01 waves
Nematic liquid crystals... Liquid crystal variable retarders... Ferroelectric liquid crystals... Piezo-elastic modulators... Pockels and Kerr cells...

56. Part IV: Polarimeters Polaroid-type polarimeters
Dual-beam polarimeters

57. Polaroid-type polarimeter for linear polarimetry (I)
Part IV: Polarimeters, polaroid-type
Polaroid-type polarimeter for linear polarimetry (I)
Use a linear polarizer (polaroid) to measure linear polarization .
Polarization percentage and position angle:

58. Dual-beam polarimeters Principle
Part IV: Polarimeters, dual-beam type
Dual-beam polarimeters Principle
Instead of cutting out one polarization and keeping the other one (polaroid), split the 2 polarization states and keep them both
Use a Wollaston prism as an analyzer
Disadvantages: need 2 detectors (PMTs, APDs) or an array; end up with 2 ‘pixels’ with different gain
Solution: rotate the Wollaston or keep it fixed and use a half-wave plate to switch the 2 beams

59. Dual-beam polarimeters Switching beams
Part IV: Polarimeters, dual-beam type
Dual-beam polarimeters Switching beams
Unpolarized light: two beams have identical intensities whatever the prism’s position if the 2 pixels have the same gain
To compensate different gains, switch the 2 beams and average the 2 measurements

60. Dual-beam polarimeters Switching beams by rotating the prism
Part IV: Polarimeters, dual-beam type
Dual-beam polarimeters Switching beams by rotating the prism
rotate by 180º

61. Dual-beam polarimeters Switching beams using a ½ wave plate
Part IV: Polarimeters, dual-beam type
Dual-beam polarimeters Switching beams using a ½ wave plate
Rotated by 45º

62. Faraday effect Magnetic field induces polarization rotation
Orients electron spins in medium
Angular momenta of electrons and photons interact
R and L have different propagation delays
Useful for magnetometers
example: rubidium vapor

63. Kerr effect Electro optic effect
Polarization in direction of applied field changes propagation speed
Input linear polarization
In direction of applied field -- phase modulator
Perpendicular to applied field -- nothing
45° to applied field -- variable waveplate
output polarizer gives intensity modulator

64. Pockels effect Similar to Kerr effect
Apply electric field along propagation direction
Crystal with no center of symmetry -- also piezoelectric
Delay linear in applied field -- Kerr effect quadratic

65. Liquid crystals
Electric field changes average orientation of molecules
Delay depends on polarization direction
Phase modulator or variable waveplate
Intensity modulator needs polarizers
Used for displays -- ex: computer monitors

66. Isolators -- 1 Polarizer and quarter waveplate
Double pass through quarter wave plate
same as half wave plate
rotate polarization by up to 45°
Polarizer blocks reflected light
Quarter wave
Polarizer
Reflecting element

67. Isolators -- 2 Faraday effect non-reciprocal
Opposite for different propagation directions
Put passive polarization rotator and Faraday rotator in series
One direction -- no effect
Opposite direction -- rotate polarization 90°
Polarizer blocks reflections
Used for fiber optics, laser diodes
performance good dB
Passive
rotator
Reflecting
element
Faraday
crystal
Polarizer
Ex: optically
active crystal
magnet

68. Mathematical description of polarization
Stokes vectors
Elements give:
1/2 total intensity
horizontal linear
+45° linear
right circular
Jones vectors
Elements give
E-field x-component
E-field y-component
Only applicable to polarized light
Unpolarized state
only I0 is non-zero