#### Transcript Polarization of Light - University of Hawaii

```Polarization of Light:
from Basics to Instruments
(in less than 100 slides)
N. Manset
CFHT
Introduction
•
•
•
•
•
Part I: Different polarization states of light
Part II: Stokes parameters, Mueller matrices
Part III: Optical components for polarimetry
Part IV: Polarimeters
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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
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Part I: Polarization states
Light as an electromagnetic
wave
Light is a transverse wave,
an electromagnetic wave
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Part I: Polarization states
Mathematical description of
the EM wave
Light wave that propagates in the z direction:


E x ( z, t )  E 0x cos(kz -  t) x


E y ( z, t )  E 0y cos(kz -  t   ) y
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Part I: Polarization states
Graphical representation of the
EM wave (I)
One can go from:


E x ( z, t )  E 0x cos(kz -  t) x


E y ( z, t )  E 0y cos(kz -  t   ) y
to the equation of an ellipse (using trigonometric
2
Ey
 Ex   Ey 
E
x
 2

  
cos  sin 2 


E0x E0y
 E0x   E0y 
2
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Part I: Polarization states
Graphical representation of the
EM wave (II)
An ellipse can be represented
by 4 quantities:
1. size of minor axis
2. size of major axis
3. orientation (angle)
4. sense (CW, CCW)
Light can be represented by 4 quantities...
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Part I: Polarization states, linear polarization
Vertically polarized light


E x ( z, t )  E 0x cos(kz -  t) x


E y ( z, t )  E 0y cos(kz -  t   ) y
If there is no amplitude in x (E0x = 0), there is
only one component, in y (vertical).
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Part I: Polarization states, linear polarization
Polarization at 45º (I)


E x ( z, t )  E 0x cos(kz -  t) x


E y ( z, t )  E 0y cos(kz -  t   ) y
If there is no phase difference (=0) and
E0x = E0y, then Ex = Ey
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Part I: Polarization states, linear polarization
Polarization at 45º (II)
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Part I: Polarization states, circular polarization
Circular polarization (I)


E x ( z, t )  E 0x cos(kz -  t) x


E y ( z, t )  E 0y cos(kz -  t   ) y
If the phase difference is = 90º and E0x = E0y
then: Ex / E0x = cos  , Ey / E0y = sin 
and we get the equation of a circle:
2
 Ex   Ey 
  cos2  sin 2  1

  
E 
E
 0x   0y 
2
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Part I: Polarization states, circular polarization
Circular polarization (II)
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Part I: Polarization states, circular polarization
Circular polarization (III)
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Part I: Polarization states, circular polarization... see it now?
Circular polarization (IV)
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Part I: Polarization states, elliptical polarization
Elliptical polarization
• Linear + circular polarization = elliptical polarization
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Part I: Polarization states, unpolarized light
Unpolarized light
(natural light)
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Part I: Polarization states
A cool Applet
Electromagnetic Wave
Location: http://www.uno.edu/~jsulliva/java/EMWave.html
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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
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Part II: Stokes parameters
Stokes parameters
A tiny itsy-bitsy little bit of history...
• 1669: Bartholinus discovers double refraction in calcite
• 17th – 19th centuries: Huygens, Malus, Brewster, Biot,
Fresnel and Arago, Nicol...
• 19th century: unsuccessful attempts to describe unpolarized
light in terms of amplitudes
• 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
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Part II: Stokes parameters
Stokes parameters (I)
The polarization ellipse is only valid at a given instant of time
(function of time):
2
E y (t)
 E x (t)   E y (t) 
E
(t)
x
 2

  
cos εsin 2 ε


E0x (t) E0y (t)
 E0x (t)  E0y (t)
2
To get the Stokes parameters, do a time average (integral over
time) and a little bit of algebra...
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Part II: Stokes parameters
Stokes parameters (II)
described in terms of the electric field
E
2
0x
E
  E
2 2
0y
2
0x
E
  2E
2 2
0y
The 4 Stokes parameters
are:
E0ycos ε   2 E0x E0ysin ε 
2
0x
2
2
S0  I  E0x
 E0y
2
2
S1  Q  E0x
 E0y
S2  U  2 E0x E0ycos ε
S3  V  2 E0x E0ysin ε
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2
Part II: Stokes parameters
Stokes parameters (III)
described in geometrical terms

a2
I 

   2
 Q   a cos2  cos2 

 U   a 2 cos2  sin 2 

  
 a 2 sin 2 

V
  

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Part II: Stokes parameters, Stokes vectors
Stokes vector
The Stokes parameters can be arranged in a Stokes vector:
2
2

 
E

E
I
intensity 
 
0x
0y
 
  

2
2
 Q   E0x  E0y   I0  I90 
 U    2 E E cos ε    I45  I135 
 
   0x 0y


 2 E E sin ε 
V
   0x 0y
  IRCP   ILCP
• Linear polarization
• Circular polarization
• Fully polarized light
• Partially polarized light
• Unpolarized light
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Q  0, U  0, V  0
Q  0, U  0, V  0
I2  Q2  U 2  V2
I2  Q2  U 2  V 2
QUV0
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Part II: Stokes parameters
Pictorial representation of the
Stokes parameters


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Part II: Stokes parameters, examples
Stokes vectors for linearly
polarized light
LHP light
LVP light
+45º light
-45º light
1
 
1
I0  
0
 
 0
1
 
  1
I0  
0
 
0
1
 
 0
I0  
1
 
 0
1
 
0
I0  
1
 
0
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Part II: Stokes parameters, examples
Stokes vectors for circularly
polarized light
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RCP light
LCP light
1
 
 0
I0  
0
 
1
1
 
0
I0  
0
 
 1
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Part II: Stokes parameters
(Q,U) to (P,)
In the case of linear polarization (V=0):
Q2  U 2
P
I
Q  P cos 2
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1
U
  arctan 
2
Q
U  P sin 2
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Part II: Stokes parameters, Mueller matrices
Mueller matrices
If light is represented by Stokes vectors, optical components are
then described with Mueller matrices:
[output light] = [Muller matrix] [input light]
 I'   m11 m12
  
 Q'   m21 m22
 U'    m
m32
31
  
 V'   m41 m42
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m13
m23
m33
m43
m14  I 
 
m24  Q 
m34  U 
 
m44  V 
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Part II: Stokes parameters, Mueller matrices
Mueller calculus (I)
Element 1
Element 2
Element 3
M1
M2
M3
I’ = M3 M2 M1 I
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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:
0
0
1

 0 cos2 sin 2
R( )  
0  sin 2 cos2

0
0
0
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0

0
0

1
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Part II: Stokes parameters, Jones formalism, not that important here...
Jones formalism
Stokes vectors and Mueller matrices cannot describe
interference effects. If the phase information is important (radioastronomy, masers...), one has to use the Jones formalism, with
complex vectors and Jones matrices:
• Jones vectors to describe the • Jones matrices to represent
polarization of light:
optical components:

 j11 j12 

 E x (t )


J  
J (t )   
 E (t )
 j21 j22 
 y 
BUT: Jones formalism can only deal with 100% polarization...
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Part III: Optical components
for polarimetry
• Complex index of refraction
• Polarizers
• Retarders
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Part III: Optical components
Complex index of refraction
The index of refraction is actually a complex quantity:
m  n  ik
• real part
• imaginary part
• optical path length,
refraction: speed of light
depends on media
• absorption, attenuation,
extinction: depends on
media
• birefringence: speed of
light also depends on P
• dichroism/diattenuation:
also depends on P
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Part III: Optical components, polarizers
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.
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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
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Part III: Optical components, polarizers
Wide-grid polarizers (II)
[dichroism]
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Part III: Optical components, polarizers
Dichroic crystals
[dichroism]
Dichroic crystals absorb one
polarization state over the other
one.
Example: tourmaline.
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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.
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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
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Part III: Optical components, polarizers
Crystal polarizers (II)
[birefringence]
isotropic
crystal
(sodium
chloride)
anisotropic
crystal
(calcite)
The 2 output beams are polarized (orthogonally).
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Part III: Optical components, polarizers
Crystal polarizers (IV)
[birefringence]
• Crystal polarizers used as:
• Beam displacers,
• Beam splitters,
• Polarizers,
• Analyzers, ...
• Examples: Nicol prism, GlanThomson polarizer, Glan or GlanFoucault prism, Wollaston prism,
Thin-film polarizer, ...
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Part III: Optical components, polarizers
Mueller matrices of polarizers
(I)
• (Ideal) linear polarizer at angle :
cos2χ
sin 2χ
 1

cos2 2χ
sin 2χ cos2χ
1  cos2χ
sin 2 2χ
2  sin 2χ sin 2χ cos2χ

0
0
 0
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Polarization of Light: Basics to Instruments
0

0
0

0
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Part III: Optical components, polarizers
Mueller matrices of polarizers
(II)
Linear (±Q)
polarizer at 0º:
 1 1

1 1
0 .5 
0
0

0
 0
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0
0
0
0
0

0
0

0
Linear (±U)
polarizer at 0º :
1

 0
0 .5 
1

 0
0 1
0 0
0 1
0 0
0

0
0

0
Circular (±V)
polarizer at 0º :
1

 0
0 .5 
0

1
Polarization of Light: Basics to Instruments
0
0
0
0
0  1

0 0
0 0

0 1
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Part III: Optical components, polarizers
Mueller calculus with a
polarizer
Input light: unpolarized --- output light: polarized
 I' 
1
 

 Q' 
0
 U'   0.5   1
 

 V' 
0
0 1
0 0
0 1
0 0
0  I 
I
 
 
0  0 
0
 0.5  



0 0
-I
 
 
0  0 
0
Total output intensity: 0.5 I
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Part III: Optical components, retarders
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.
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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.
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Part III: Optical components, retarders
Half-Wave plate (II)
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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.
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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)
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Part III: Optical components, retarders
Mueller matrix of retarders (I)
• Retarder of retardance  and position angle :
0
0
1

H sin4ψ
 0 G  H cos4ψ
0
H sin4ψ
G  H cos4ψ

 sinτ cos2ψ
 0 sinτ sin2ψ
1
with : G  1  cosτ  and H 
2
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

 sinτ sin2ψ 
sinτ cos2ψ 

cosτ

1
1  cosτ 
2
Polarization of Light: Basics to Instruments
0
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Part III: Optical components, retarders
Mueller matrix of retarders (II)
• Half-wave oriented at 0º
or 90º
1

0
k
0

0
0 0
0

1 0
0
0 1 0 

0 0  1
N. Manset / CFHT
• Half-wave oriented at
±45º
1 0

0 1
k
0 0

0 0
0 0

0 0
1 0

0  1
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Part III: Optical components, retarders
Mueller matrix of retarders
(III)
• Quarter-wave oriented at
0º
1

0
k
0

0
0 0
1 0
0 0
0 1
N. Manset / CFHT
0

0
1

0
• Quarter-wave oriented at
±45º
1 0

0 0
k
0 0

0 1
0 0

0  1
1 0

0 0
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Part III: Optical components, retarders
Mueller calculus with a
retarder
• Input light linear polarized (Q=1)
• Quarter-wave at +45º
• Output light circularly polarized (V=1)
 I' 
1 0
 

 Q' 
0 0

k
 U' 
0 0
 

 V' 
0 1
N. Manset / CFHT
0 0  1 
1
 
 
0  1  1 
 0

k
 0
1 0  0 
 
 
0 0  0 
1
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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.
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Part III: Optical components, retarders
Achromatic retarders (I)
• Retardation depends on wavelength
• Achromatic retarders: made of 2 different materials with
opposite variations of index of refraction as a function of wavelength
• Pancharatnam achromatic retarders: made of 3
identical plates rotated w/r one another
• Superachromatic retarders: 3 pairs of quartz and MgF2
plates
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Part III: Optical components, retarders
Achromatic retarders (II)
=140-220º
not very
achromatic!
= 177-183º
much better!
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Part III: Optical components, retarders
Retardation on total internal
reflection
• Total internal
reflection
produces
retardation (phase
shift)
• In this case, retardation is very achromatic
since it only depends on the refractive index
• Application: Fresnel rhombs
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Part III: Optical components, retarders
Fresnel rhombs
• Quarter-wave and half-wave rhombs are
achieved with 2 or 4 reflections
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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...
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Part IV: Polarimeters
• Polaroid-type polarimeters
• Dual-beam polarimeters
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Part IV: Polarimeters, polaroid-type
Polaroid-type polarimeter
for linear polarimetry (I)
• Use a linear polarizer (polaroid) to measure
linear polarization ... [another cool applet]
• Polarization percentage and position angle:
I max  I min
P
I max  I min
  ( I  I max )
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Polarization of Light: Basics to Instruments


61
Part IV: Polarimeters, polaroid-type
Polaroid-type polarimeter
for linear polarimetry (II)
• Move the polaroid to 2 positions, 0º and 45º
(to measure Q, then U)
• Advantage: very simple to make
• Disadvantage: half of the light is cut out
measurements, cross-talk...
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Part IV: Polarimeters, polaroid-type
Polaroid-type polarimeter
for circular polarimetry
• Polaroids are not sensitive to circular
polarization, so convert circular polarization
to linear first, by using a quarter-wave plate
• Polarimeter now uses a quarter-wave plate
and a polaroid
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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
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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
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Part IV: Polarimeters, dual-beam type
Dual-beam polarimeters
Switching beams by rotating the prism
rotate by
180º
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Part IV: Polarimeters, dual-beam type
Dual-beam polarimeters
Switching beams using a ½ wave plate
Rotated
by 45º
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Part IV: Polarimeters, example of circular polarimeter
A real circular polarimeter
Semel, Donati, Rees (1993)
Quarter-wave plate, rotated at -45º and +45º
Analyser: double calcite crystal
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Part IV: Polarimeters, example of circular polarimeter
A real circular polarimeter
free from gain (g) and atmospheric
transmission () variation effects
• First measurement with quarter-wave plate at -45º, signal
l
r
S
,
S
in the (r)ight and (l)eft beams: 1
1
• Second measurement with quarter-wave plate at +45º,
l
r
S
,
S
signal in the (r)ight and (l)eft beams: 2
2
• Measurements of the signals:
S1l  g l1 ( I1  V1 ) S1r  g r1 ( I1  V1 )
S  g 2 ( I 2  V2 ) S  g  2 ( I 2  V2 )
l
2
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l
r
2
r
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Part IV: Polarimeters, example of circular polarimeter
A real circular polarimeter
free from gain and atmospheric
transmission variation effects
• Build a ratio of measured signals which is free of gain and
variable atmospheric transmission effects:
 1
1  S1l S2r
I 2V1  I1V2
F   l r  1 
4  S2 S1
 2 I1 I 2  I 2V1  I1V2  V1V2
1  V1 V2 
F     forV  1
2  I1 I 2 
average of the 2 measurements
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Part IV: Polarimeters, summary
Polarimeters - Summary
• 2 types:
– polaroid-type: easy to make but ½ light is lost, and affected
by variable atmospheric transmission
– dual-beam type: no light lost but affected by gain
differences and variable transmission problems
• Linear polarimetry:
– analyzer, rotatable
 2 positions minimum
– analyzer + half-wave plate
• Circular polarimetry:
– analyzer + quarter-wave plate
N. Manset / CFHT
 1 position minimum
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Optical components of the
polarimeter part :
• Wollaston prism: analyses the
polarization and separates the 2
(linear!) orthogonal polarization
states
• Retarders, 3 Fresnel rhombs:
– Two half-wave plates to switch the
beams around
– Quarter-wave plate to do circular
polarimetry
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Part V: ESPaDOnS, circular polarimetry mode
polarimetry
• Fixed quarter-wave rhomb
• Rotating bottom half-wave, at 22.5º
increments
• Top half-wave rotates continuously at about
1Hz to average out linear polarization when
measuring circular polarization
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Part V: ESPaDOnS, circular polarimetry mode
polarimetry of circular polarization


• analyzer
N. Manset / CFHT
• half-wave
• 22.5º positions
• flips
polarization
• gain,
transmission
• quarterwave
• fixed
• circular
to linear
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Part V: ESPaDOnS, circular polarimetry mode
(unwanted) linear polarization
• analyzer
• circular part
• half-wave
goes through
not analyzed
intensities to
both beams
• 22.5º
positions
• linear part is
analyzed!
N. Manset / CFHT
• gain,
transmission
• quarterwave
• fixed
• linear to
elliptical
Polarization of Light: Basics to Instruments
rotating
half-wave
out” the
unwanted
signal
76
• Half-Wave rhombs positioned at 22.5º
increments
• Quarter-Wave fixed
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• Half-Wave rhombs positioned as 22.5º
increments


– First position gives Q
– Second position gives U
– Switch beams for gain and atmosphere effects
• Quarter-Wave fixed
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• ESPaDOnS can do linear and circular
polarimetry (quarter-wave plate)
• Beams are switched around to do the
measurements, compensate for gain and
atmospheric effects
• Fesnel rhombs are very achromatic
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Credits for pictures and movies
http://www.noao.edu/noao/staff/keller/
• “Basic Polarisation techniques and devices”, Meadowlark Optics Inc.
• Optics, E. Hecht and Astronomical Polarimetry, J. Tinbergen
• Planets, Stars and Nebulae Studied With Photopolarimetry, T.
Gehrels
• Circular polarization movie
http://www.optics.arizona.edu/jcwyant/JoseDiaz/Polarization-Circular.htm
• Unpolarized light movie
• Reflection of wave http://www.physicsclassroom.com/mmedia/waves/fix.html
• ESPaDOnS web page and documents
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On the Web
• Very short and quick introduction, no equation
http://www.cfht.hawaii.edu/~manset/PolarIntro_eng.html
• Easy fun page with Applets, on polarizing filters
• Polarization short course
http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/u12l1e.html
• “Instrumentation for Astrophysical
Spectropolarimetry”, a series of 5 lectures given at the
IAC Winter School on Astrophysical Spectropolarimetry,
November 2000 –
http://www.noao.edu/noao/staff/keller/lectures/index.html
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Polarization basics
• Polarized Light, D. Goldstein – excellent book,
easy read, gives a lot of insight, highly
recommended
• Undergraduate textbooks, either will do:
– Optics, E. Hecht
– Waves, F. S. Crawford, Berkeley Physics Course vol. 3
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Astronomy, easy/intermediate
• Astronomical Polarimetry, J. Tinbergen –
instrumentation-oriented
• La polarisation de la lumière et l'observation
astronomique, J.-L. Leroy – astronomy-oriented
• Planets, Stars and Nebulae Studied With
Photopolarimetry, T. Gehrels – old but classic
• 3 papers by K. Serkowski – instrumentation-oriented
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