Polarization light introduction

Polarization of Light:
from Basics to Instruments
(in less than 100 slides)
N. Manset
CFHT

N. Manset / CFHTPolarization of Light: Basics to 2
Introduction
• Part I: Different polarization states of light
• Part II: Stokes parameters, Mueller matrices
• Part III: Optical components for polarimetry
• Part IV: Polarimeters
• Part V: ESPaDOnS

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

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

Mathematical description of
the EM wave
Light wave that propagates in the z direction:
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx


εω
ω
+=
=

Graphical representation of the
EM wave (I)
One can go from:
to the equation of an ellipse (using trigonometric
identities, squaring, adding):
εε 2
0y
y
0x
x
2
0y
y
2
0x
x
sincos
E
E
E
E
2
E
E
E
E
=−








+





y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx


εω
ω
+=
=

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...

Vertically polarized light
If there is no amplitude in x (E0x = 0), there is
only one component, in y (vertical).
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx


εω
ω
+=
=
Part I: Polarization states, linear polarization

Polarization at 45º (I)
If there is no phase difference (=0) and
E0x = E0y, then Ex = Ey
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx


εω
ω
+=
=

Polarization at 45º (II)

Circular polarization (I)
If the phase difference is = 90º and E0x = E0y
then: Ex / E0x = cos Θ , Ey / E0y = sin Θ
and we get the equation of a circle:
1sincos
E
E
E
E 22
2
0y
y
2
0x
x
=Θ+Θ=








+





y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx


εω
ω
+=
=
Part I: Polarization states, circular polarization

Circular polarization (II)

Circular polarization (III)

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

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

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

A cool Applet
Electromagnetic Wave
Location: http://www.uno.edu/~jsulliva/java/EMWave.html

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

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
Part II: Stokes parameters

Stokes parameters (I)
The polarization ellipse is only valid at a given instant of time
(function of time):
εsinεcos
(t)E
(t)E
(t)E
(t)E
2
(t)E
(t)E
(t)E
(t)E 2
0y
y
0x
x
2
0y
y
2
0x
x
=−








+





To get the Stokes parameters, do a time average (integral over
time) and a little bit of algebra...

Stokes parameters (II)
described in terms of the electric field
( ) ( ) ( ) ( )2
0y0x
2
0y0x
22
0y
2
0x
22
0y
2
0x εsinEE2εcosEE2EEEE =−−−+
The 4 Stokes parameters
are:
εsinEE2V
εcosEE2U
EEQ
EEI
0y0x3
0y0x2
2
0y
2
0x1
2
0y
2
0x0
==
==
−==
+==
S
S
S
S

Stokes parameters (III)
described in geometrical terms














=












β
φβ
φβ
2sin
2sin2cos
2cos2cos
V
U
Q
I
2
2
2
2
a
a
a
a

Stokes vector
The Stokes parameters can be arranged in a Stokes vector:
( ) ( )
( ) ( )
( ) ( )











−
°−°
°−°
=














−
+
=












LCPIRCPI
135I45I
90I0I
intensity
εsinEE2
εcosEE2
EE
EE
V
U
Q
I
0y0x
0y0x
2
0y
2
0x
2
0y
2
0x
• Linear polarization
• Circular polarization
• Fully polarized light
• Partially polarized light
• Unpolarized light 0VUQ
VUQI
VUQI
0V0,U0,Q
0V0,U0,Q
2222
2222
===
++>
++=
≠==
=≠≠
Part II: Stokes parameters, Stokes vectors

Pictorial representation of the
Stokes parameters
Σ
∆

Stokes vectors for linearly
polarized light
LHP light












0
0
1
1
I0
LVP light +45º light -45º light












−
0
0
1
1
I0












0
1
0
1
I0












−
0
1
0
1
I0
Part II: Stokes parameters, examples

Stokes vectors for circularly
polarized light
RCP light












1
0
0
1
I0
LCP light












− 1
0
0
1
I0
Part II: Stokes parameters, examples

(Q,U) to (P,)
In the case of linear polarization (V=0):
I
UQ
P
22
+
= 





=Θ
Q
U
arctan
2
1
Θ= 2cosPQ Θ= 2sinPU

Mueller matrices
If light is represented by Stokes vectors, optical components are
then described with Mueller matrices:
[output light] = [Muller matrix] [input light]
























=












V
U
Q
I
V'
U'
Q'
I'
44434241
34333231
24232221
14131211
mmmm
mmmm
mmmm
mmmm
Part II: Stokes parameters, Mueller matrices

Mueller calculus (I)
Element 1 Element 2 Element 3
1M 2M 3M
I’ = M3 M2 M1 I

Mueller calculus (II)
Mueller matrix M’ of an optical component with
Mueller matrix M rotated by an angle :
M’ = R(- ) M R() with:












−
=
1000
02cos2sin0
02sin2cos0
0001
)R(
αα
αα
α

Jones formalism
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:








=
(t)E
(t)E
(t)J
y
x









=
2221
1211
jj
jj
J
BUT: Jones formalism can only deal with 100% polarization...
Part II: Stokes parameters, Jones formalism, not that important here...

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

Complex index of refraction
The index of refraction is actually a complex quantity:
iknm −=
• 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
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.
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

Wide-grid polarizers (II)
[dichroism]

Dichroic crystals
[dichroism]
Dichroic crystals absorb one
polarization state over the other
one.
Example: tourmaline.

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.
Part III: Optical components, polarizers – Polaroids, like in sunglasses!

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

Crystal polarizers (II)
[birefringence]
The 2 output beams are polarized (orthogonally).
isotropic
crystal
(sodium
chloride)
anisotropic
crystal
(calcite)

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, ...

Mueller matrices of polarizers
(I)
• (Ideal) linear polarizer at angle χ:












0000
0χ2sinχ2cosχ2sinχ2sin
0χ2cosχ2sinχ2cosχ2cos
0χ2sinχ2cos1
2
1
2
2

Mueller matrices of polarizers
(II)
Linear (±Q)
polarizer at 0º:












±
±
0000
0000
0011
0011
5.0
Linear (±U)
polarizer at 0º :












±
±
0000
0101
0000
0101
5.0
Circular (±V)
polarizer at 0º :












±
±
1001
0000
0000
1001
5.0

Mueller calculus with a
polarizer
Input light: unpolarized --- output light: polarized












=
























−
−
=












0
I-
0
I
5.0
0
0
0
I
0000
0101
0000
0101
5.0
V'
U'
Q'
I'
Total output intensity: 0.5 I

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.
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.

Half-Wave plate (II)

Quarter-Wave plate (I)
• Retardation of ¼ wave or 90º for one of the
polarizations
• Used to convert linear polarization to elliptical.

• Special case: incoming light polarized at 45º with respect to
the retarder’s axis
• Conversion from linear to circular polarization (vice versa)
Quarter-Wave plate (II)

Mueller matrix of retarders (I)
• Retarder of retardance τ and position angle ψ:
( ) ( )cosτ1
2
1
Handcosτ1
2
1
G:with
cosτcos2ψsinτsin2ψsinτ0
cos2ψsinτcos4ψHGsin4ψH0
sin2ψsinτsin4ψHcos4ψHG0
0001
−=+=












−
−
−+

Mueller matrix of retarders (II)
• Half-wave oriented at 0º
or 90º
• Half-wave oriented at
±45º












−
−
1000
0100
0010
0001
k












−
−
1000
0100
0010
0001
k

Mueller matrix of retarders
(III)
• Quarter-wave oriented at
0º
• Quarter-wave oriented at
±45º












− 0100
1000
0010
0001
k












± 0010
0100
1000
0001

k

Mueller calculus with a
retarder












=
























+
−
=












1
0
0
1
0
0
1
1
0010
0100
1000
0001
V'
U'
Q'
I'
kk
• Input light linear polarized (Q=1)
• Quarter-wave at +45º
• Output light circularly polarized (V=1)

(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.

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

Achromatic retarders (II)
=140-220º
not very
achromatic!
= 177-183º
much better!

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

Fresnel rhombs
• Quarter-wave and half-wave rhombs are
achieved with 2 or 4 reflections

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...

Part IV: Polarimeters
• Polaroid-type polarimeters
• Dual-beam polarimeters

Polaroid-type polarimeter
for linear polarimetry (I)
• Use a linear polarizer (polaroid) to measure
linear polarization ... [another cool applet]
Location: http://www.colorado.edu/physics/2000/applets/lens.html
• Polarization percentage and position angle:
)II(
II
II
P
max
minmax
minmax
=Θ=Θ
+
−
=
Part IV: Polarimeters, polaroid-type
Σ
∆

for linear polarimetry (II)
• Advantage: very simple to make
• Disadvantage: half of the light is cut out
• Other disadvantages: non-simultaneous
measurements, cross-talk...
• Move the polaroid to 2 positions, 0º and 45º
(to measure Q, then U)

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
• Same disadvantages as before

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
Part IV: Polarimeters, dual-beam type

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
Σ
∆

Switching beams by rotating the prism
rotate by
180º

Switching beams using a ½ wave plate
Rotated
by 45º

A real circular polarimeter
Semel, Donati, Rees (1993)
Quarter-wave plate, rotated at -45º and +45º
Analyser: double calcite crystal
Part IV: Polarimeters, example of circular polarimeter

free from gain (g) and atmospheric
transmission (α) variation effects
• First measurement with quarter-wave plate at -45º, signal
in the (r)ight and (l)eft beams:
• Second measurement with quarter-wave plate at +45º,
signal in the (r)ight and (l)eft beams:
• Measurements of the signals:
rl
SS 11 ,
rl
SS 22,
)()(
)()(
22222222
11111111
VIgSVIgS
VIgSVIgS
rrll
rrll
−=+=
−=+=
αα
αα

free from gain and atmospheric
transmission variation effects
• Build a ratio of measured signals which is free of gain and
variable atmospheric transmission effects:
1for
2
1
2
1
1
4
1
2
2
1
1
21211221
2112
1
2
2
1
<<





+≈
+−−
+
=





−=
V
I
V
I
V
F
VVVIVIII
VIVI
S
S
S
S
F r
r
l
l
average of the 2 measurements

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
– analyzer + half-wave plate
• Circular polarimetry:
– analyzer + quarter-wave plate
 2 positions minimum
 1 position minimum
Part IV: Polarimeters, summary

Part V: ESPaDOnS
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

ESPaDOnS: circular
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
Part V: ESPaDOnS, circular polarimetry mode

ESPaDOnS: circular
polarimetry of circular polarization
• half-wave
• 22.5º positions
• flips
polarization
• gain,
transmission
• quarter-
wave
• fixed
• circular
to linear
• analyzer
Σ
∆

ESPaDOnS: circular polarimetry of
(unwanted) linear polarization
• half-wave
• 22.5º
positions
• gain,
transmission
• quarter-
wave
• fixed
• linear to
elliptical
• analyzer
• circular part
goes through
not analyzed
and adds same
intensities to
both beams
• linear part is
analyzed!
• Add a
rotating
half-wave
to “spread
out” the
unwanted
signal

ESPaDOnS: linear polarimetry
• Half-Wave rhombs positioned at 22.5º
increments
• Quarter-Wave fixed
Part V: ESPaDOnS, linear polarimetry

ESPaDOnS: linear polarimetry
• 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
Part V: ESPaDOnS, linear polarimetry
Σ
∆

ESPaDOnS - Summary
• 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
Part V: ESPaDOnS, summary

Credits for pictures and movies
• Christoph Keller’s home page – his 5 lectures
http://www.noao.edu/noao/staff/keller/
• “Basic Polarisation techniques and devices”, Meadowlark Optics Inc.
http://www.meadowlark.com/
• 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
http://www.colorado.edu/physics/2000/polarization/polarizationII.html
• Reflection of wave http://www.physicsclassroom.com/mmedia/waves/fix.html
• ESPaDOnS web page and documents

References/Further reading
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
http://www.colorado.edu/physics/2000/polarization/polarizationI.html
• 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

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

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

Astronomy, advanced
• Introduction to Spectropolarimetry, J.C.
del Toro Iniesta – radiative transfer – ouch!
• Astrophysical Spectropolarimetry,
Trujillo-Bueno et al. (eds) – applications to
astronomy

Polarization light introduction

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Polarization light introduction

Similar to Polarization light introduction (20)

Recently uploaded

Recently uploaded (20)

Polarization light introduction