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Transcript of Polarization
1
POLARIZATION
SOLO HERMELIN
Updated 27.05.06http://www.solohermelin.com
POLARIZATION
History
SOLO
TABLE OF CONTENT
Natural or Unpolarized Light
Monochromatic Planar Wave Equations
Linear Polarization or Plane-PolarizationCircular Polarization
Elliptically PolarizationMethods of Achieving Polarization
Polarization Ellipse
Degenerated States of Polarization Ellipse
The Stokes Polarization Parameters
Measuring the Stokes Parameters
Poincaré Sphere
The Mueller Matrices for Polarizing Components
The Jones Polarization Parameters
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POLARIZATIONSOLO
TABLE OF CONTENT (Continue – 1)
Faraday Effect
Electro – Optical Effects ( Pockels, Kerr)
References Optics Polarization
4
POLARIZATION
Erasmus Bartholinus, doctor of medicine and professor of mathematics at the University of Copenhagen, showed in 1669 that crystals of “Iceland spar” (which we now call calcite, CaCO3) produced two refracted rays from a single incident beam. One ray, the “ordinary ray”, followed Snell’s law, while the other, the “extraordinary ray”, was not always even in the plan of incidence.
SOLO
History
Erasmus Bartholinus1625-1698
http://www.polarization.com/history/history.html
5
POLARIZATIONSOLO
History
Étienne Louis Malus1775-1812
Etienne Louis Malus, military engineer and captain in the army ofNapoleon, published in 1809 the Malus Law of irradiance through aLinear polarizer: I(θ)=I(0) cos2θ. In 1810 he won the French AcademyPrize with the discovery that reflected and scattered light also possessed“sidedness” which he called “polarization”.
6
POLARIZATION
Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that tworays polarized at right angles to each other never interface.
SOLO
History (continue)
Dominique François Jean Arago1786-1853
Augustin Jean Fresnel
1788-1827
Arago relayed to Thomas Young in London the resultsof the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillationsin the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young
1773-1829
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7
POLARIZATIONSOLO
The Natural Light is emitted by excitation of material atoms. Each excited atomradiates a wave train for roughly 10-8 sec. The Natural light is the result of radiation of alarge collection of such atoms. New waves are emitted in a completely unpredictable fashionand the light is actually composed of a succession of short living polarization states.
Natural or Unpolarized Light
A light source consists of a very large number of randomly oriented atoms emitters.
We say that the Natural light is compose of a collection of monochromatic (polichromatic) unpolarized rays.
One mathematical description of a monochromatic unpolarized ray moving in z direction, at a certain location in space, is by a Electric Intensity phasor of constant amplitude and a random phase:
( ) ( )( ) ( )( ) yxtzktjtzktj yx eAeAtE 11
∧+−
∧+− += δωδω
zyx 111 ,,∧∧∧
are orthogonal unit vectors and δx (t) are δy (t) are randomly phase angles.
8
POLARIZATIONSOLO
Light is a transverse electromagnetic wave; i.e. the Electric and Magnetic Intensitiesare perpendicular to each other and oscillate perpendicular to the direction of propagation.
For the natural light the direction of the Electric Intensity vector changes randomlyfrom time to time. We say that the natural light is Unpolarized.
A Planar wave (in which the Electric Intensity propagates remaining in a plane – containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized.
If light is composed of two plane waves of equal amplitude but differing in phase by 90°then the light is aid to be Circular Polarized.
If light is composed of two plane waves of different amplitudes and/or the difference in phase is different than 0,90,180,270° then the light is said to be Elliptically Polarized.
E
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9
ELECTROMAGNETICSSOLO
To satisfy the Maxwell equations for a source free media we must have:
Monochromatic Planar Wave Equations
we haveUsing: 1ˆˆ&ˆˆ0 =⋅== kkknkkk εµω
=⋅∇=⋅∇
−=×∇=×∇
0
0
H
E
HjE
EjH
ωµωε
=⋅
=⋅
=×
−=×
0ˆ
0ˆ
ˆ
ˆ
0
0
00
00
Hk
Ek
HEk
EHk
εµ
µε
=⋅−
=⋅−
−=×−
=×−
⇒
⋅−
⋅−
⋅−⋅−
⋅−⋅−
−=∇ ⋅−⋅−
0
0
0
0
00
00
rkj
rkj
rkjrkj
rkjrkj
ekje
eHkj
eEkj
eHjeEkj
eEjeHkjrkjrkj
ωµ
ωε
( ) *2
*2
&2
ˆ
2
ˆHHwEEwwcn
kwwcn
kS meme
µε ====+=
Time Average Poynting Vector of the Planar Wave
( ) ( )rktjrktj eHHeEE
⋅−⋅− == ωω00 &
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POLARIZATIONSOLO
A Planar wave (in which the Electric Intensity propagates remaining in a plane – containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized.
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://www.enzim.hu/~szia/cddemo/edemo0.htm )Andras Szilagyi(
Linear Polarization or Plane-Polarization
( ) yyzktj
y eAE 1∧
+−= δω
Return to Table of Content
11
POLARIZATIONSOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
If light is composed of two plane waves of equal amplitude but differing in phase by 90°then the light is said to be Circular Polarized.
http://www.optics.arizona.edu/jcwyant/JoseDiaz/Polarization-Circular.htm
( ) ( ) yx xx zktjzktj eAeAE 11 2/∧
++−∧
+− += πδωδω
Circular Polarization
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POLARIZATIONSOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
If light is composed of two plane waves of different amplitudes and/or the difference in phase is different than 0,90,180,270° then the light is said to be Elliptically Polarized.
( ) ( ) yx yxzktj
y
zktj
x eAeAE 11∧
+−∧+− += δωδω
Elliptically Polarization
Return to Table of Content
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POLARIZATIONSOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Methods of Achieving Polarization
Polarization is based on one of four fundamental physical mechanisms:
1. Dichroism or selective absorbtion 3. Reflection
2. Scattering 4. Birefrigerence
To obtain Polarization we must have some asymmetry in the optical process.
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POLARIZATIONSOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Methods of Achieving Polarization (continue – 1)
Polarization by Dichroism
A dichroic material has different absorption properties for perpendicular incident planes. An example of a dichroic material is the tourmaline that is a class of borone silicate. The tourmaline has a unique optic axis, and any electronic field normal to it is strongly absorbed.
15
POLARIZATIONSOLO
Polaroids (Dichroism)
http://en.wikipedia.org/wiki/Edwin_H._Land
Edwin H. Land 1909-1991
In 1928 Edwin H. Land undergraduate at Harvard Collegeinvented the Polaroid J-sheet. It consists of many microscopic Crystals of iodoquinine sulphate embeded in a transparentNitrocellulose polymer film.
The sunglasses use polaroid material that uses dichroism to achieve absorption..
16
Methods of Achieving Polarization (continue – 1)
Wire-Grid Polarizer (Dichroism)
POLARIZATIONSOLO
Grid of parallel conducting wires with a spacing comparable to the wavelength of the electromagnetic wave.
The Electric Field vector parallel to the wires is attenuated because of the currents induced in the wires.
17
POLARIZATIONSOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Methods of Achieving Polarization (continue – 2)
Polarization by Scattering
18
POLARIZATIONSOLO
Methods of Achieving Polarization (continue – 3)
Polarization by Scattering
19
POLARIZATIONSOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Methods of Achieving Polarization (continue – 4)
Polarization by Reflection
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POLARIZATIONSOLO
Methods of Achieving Polarization (continue – 5)
Polarization by Reflection
The Pile-of-plate Polarizer
The problem encounter using the Brewster Effect is that the reflected beam although completely polarized is weak and the refracted beam is only partially polarized.
The solution is to use a pile-of-plates polarizer as in Figure.
This was invented by F.J. Arago in 1812.
Dominique François Jean Arago1786-1853
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POLARIZATIONSOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Polarization by Birefrigerence (continue – 4)
Polarization can be achieved with crystalline materials which have a different index ofrefraction in different planes. Such materials are said to be birefringent or doubly refracting.
Nicol Prism The Nicol Prism (1828) is made up from two prisms of calcite cemented with Canada balsam. The ordinary ray can be made to totally reflect off the prism boundary, leving only the extraordinary ray.
William Nicol(1768 ?– 1851) Scottish physicist
Methods of Achieving Polarization (continue – 6)
22
POLARIZATIONSOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Polarization by Birefrigerence (continue – 4)
Polarization can be achieved with crystalline materials which have a different index ofrefraction in different planes. Such materials are said to be birefringent or doubly refracting.
Wollaston Prism
William HydeWollaston1766-1828
Methods of Achieving Polarization (continue – 7)
23
POLARIZATIONSOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Polarization by Birefrigerence (continue – 4)
Polarization can be achieved with crystalline materials which have a different index ofrefraction in different planes. Such materials are said to be birefringent or doubly refracting.
Glan-Foucault Polarizer
Methods of Achieving Polarization (continue – 8)
24
POLARIZATION
Methods of Achieving Polarization (continue – 9)
25
POLARIZATION
Methods of Achieving Polarization (continue – 10)
26
POLARIZATIONSOLO
Polarizations Prisms Overview
http://www.unitedcrystals.com/POverview.html
Methods of Achieving Polarization (continue – 11)
27
POLARIZATIONSOLO
Polarizations Prisms Overview
http://www.unitedcrystals.com/POverview.html
Methods of Achieving Polarization (continue – 12)
Return to Table of Content
28
POLARIZATIONSOLO
The Polarization of a monochromatic planar wave is defined in terms of the behavior of the tip of the phasor vector as a function of timeE
( ) ( ) yx yxzktj
y
zktj
x eAeAE 11∧
+−∧+− += δωδω
Tacking the real part we obtain:( )x
x
x zktA
E δω +−= cos
( ) ( ) δδωδδωδδδωδ
sinsincoscoscos xxxyx
y
y zktzktzktA
E+−−+−=
−++−=
δδ sin1cos
2
−=
x
x
x
x
y
y
A
E
A
E
A
E
δδ 2
22
sin1cos
−=
−
x
x
x
x
y
y
A
E
A
E
A
E
δδ 2
22
sincos2 =
+−
y
y
y
y
x
x
x
x
A
E
A
E
A
E
A
E
1sinsin
cos2
sin
2
2
2
=
+−
δδ
δδ y
y
y
y
x
x
x
x
A
E
A
E
A
E
A
E ellipse
( )y
y
y zktA
Eδω +−= cos
Polarization Ellipse
29
POLARIZATIONSOLO
Let transform the ellipse equation to canonical form by using a linear transformationthrough an angle ψ (to be defined)
1sinsin
cos2
sin
2
2
2
=
+−
δδ
δδ y
y
y
y
x
x
x
x
A
E
A
E
A
E
A
E
−=
⇔
−
=
η
ξ
η
ξ
ψψψψ
ψψψψ
E
E
E
E
E
E
E
E
y
x
y
x
cossin
sincos
cossin
sincos
1sin
cossincos
sin
cossin
sin
sincos2
sin
sincos22
=
++
+
−−
−δ
ψψδ
δψψ
δψψ
δψψ ηξηξηξηξ
yyxx A
EE
A
EE
A
EE
A
EE
( ) 1sincossin
cos2
sin
cossin2
sin
cossin2
cossin
cos
sin
sin2
sin
cos
sin
sincos
sin
sin
sin
cos2
sin
sin
sin
cos
0
22
22222
22
2
22
22
22
2
22
22
=
−−+−+
+++
−+
ψ
ηξ
ηξ
ψψδ
δδψψ
δψψ
δδ
ψδ
ψδ
ψδ
ψδδ
ψδ
ψδ
ψδ
ψ
definingby
yxyx
yxyxyxyx
AAAAEE
AAAAE
AAAAE
Choose ψ such that the last term is zero, or
02coscos211
2sin22
=−
− ψδψ
yxxyAAAA ( ) δαδδψ
α
cos2tancos/1
/cos
22tan
/:tan
222
xy AA
xy
xy
yx
yx
AA
AA
AA
AA =
=−
=−
=
Polarization Ellipse (continue – 1)
30
POLARIZATIONSOLO
1
22
=
+
η
η
ξ
ξ
A
E
A
E
Define (see Figure)
δδ
ψδ
ψδ
ψδ
ψ
δδ
ψδ
ψδ
ψδ
ψ
χξ
η
cossin
cos
sin
sin2
sin
cos
sin
sin
cossin
sin
sin
cos2
sin
sin
sin
cos
:tan
22
2
22
2
22
2
22
2
2
2
yxyx
yxyx
AAAA
AAAA
A
A
++
−+=
=
( ) ( )( ) ( )
δψψαψψαδψψαψψα
δψψψψδψψψψ
χ
α
coscossintan2cossintan
coscossintan2sincostan
coscossin/2cossin/
coscossin/2sincos/tan
222
222tan/
222
222
2
++−+
++−+
=
==xy EE
xyxy
xyxy
EEEE
EEEE
( ) ( ) ( )α
δψααψα
δψψααψψχχχ
2
2
2
222
2
2
tan1
cos2sintan2tan12cos
tan1
coscossintan4tan1sincos
tan1
tan12cos
++−=
++−−=
+−=
From δαψ cos2tan2tan =αψδ2tan
2tancos =
andααα
ααα
2
2
2 tan1
tan12cos&
tan1
tan22tan
+−=
−=
( )ψα
ψψψ
αα
ααψψααψ
χ
ψα
2cos
2cos
2cos
2sin2cos
tan1
tan1
tan12tan2tan
2sintan2tan12cos2cos
2cos/1
2
2cos
2
2
2
2
=
+
+−=
+
+−=
δαψ cos2tan2tan =
Polarization Ellipse (continue – 2)
31
POLARIZATIONSOLO
1
22
=
+
η
η
ξ
ξ
A
E
A
E
χαψ2cos
2cos2cos = δαψ cos2tan2tan =
χδαδα
χαψψψ
2cos
cos2sincos2tan
2cos
2cos2tan2cos2sin ===
Therefore
1cossin
cos
sin
sin2
sin
cos
sin
sincos
sin
sin
sin
cos2
sin
sin
sin
cos22
2
22
22
22
2
22
22 =
+++
−+ δ
δψ
δψ
δψ
δψδ
δψ
δψ
δψ
δψ
ηξyxyxyxyx
AAAAE
AAAAE
Also
δδ
ψδ
ψδ
ψδ
ψ
δδ
ψδ
ψδ
ψδ
ψ
η
ξ
cossin
cos
sin
sin2
sin
cos
sin
sin1
cossin
sin
sin
cos2
sin
sin
sin
cos1
22
2
22
2
2
22
2
22
2
2
yxyx
yxyx
AAAAA
AAAAA
++=
−+=
δηξ22222 sin
11111
+=+
yx AAAA
Polarization Ellipse (continue – 3)
To Stokes Parameters
32
POLARIZATIONSOLO
Let compute the area of the Polarization Ellipse
( )
( )yyyyy
xxxxx
AzktAEy
AzktAEx
δτδω
δτδω
τ
τ
+=
+−==
+=
+−==
coscos:
coscos:
( ) ( )
( ) ( ) δπτδδτδδ
τδτδτ
π
δ
π
sin2sinsin2
1
sincos
2
0
2
0
yxyxxyyx
xxyy
AAdAA
dAAdxyArea
=
++−−=
++−==
∫
∫∫
But the area of the Polarization Ellipse is1
22
=
+
η
η
ξ
ξ
A
E
A
Eηξπ AAArea =
Therefore δηξ sinyx AAAA =
Using
δηξ22222 sin
11111
+=+
yx AAAA δηξ
ηξ222
22
22
22
sin
1
yx
yx
AA
AA
AA
AA +=
+ 2222
yx AAAA +=+ ηξ
Energy Equation
Polarization Ellipse (continue – 4)
33
POLARIZATIONSOLO
δηξ sinyx AAAA =2222
yx AAAA +=+ ηξ Energy Equation
ξ
ηχA
A=:tanwe defined
Therefore
δαδδ
χχχ
ηξ
ηξ
ξ
η
ξ
η
sin2sinsin
1
2sin22
1
2
tan1
tan22sin
2222222=
+
=+
=+
=
+
=+
=
y
x
y
x
yx
yx
A
A
A
A
AA
AA
AA
AA
A
A
A
A
δαχ sin2sin2sin =
Polarization Ellipse (continue – 5)
δαψ cos2tan2tan =We also found that
ψχ
ψχψχ
αψχ
αψαχ
δδδ
2sin
2tan
2cos2cos2tan
2sin
2cos2tan
2sin
2tan/2tan
2sin/2sin
cos
sintan =
⋅⋅=
⋅===
ψχδ2sin
2tantan =
ψχα 2cos2cos2cos ⋅=
To StokesParameters
34
POLARIZATIONSOLO
Summary
Polarization Ellipse (continue – 6)
δηξ sinyx AAAA =2222
yx AAAA +=+ ηξ
==
δαχδαψ
sin2sin2sin
cos2tan2tan
=
⋅=
ψχδ
ψχα
2sin
2tantan
2cos2cos2cos
⇐
χψ
δα
⇐
δα
χψ
ξ
ηχA
A=:tan
x
y
A
A=:tanα
1sinsin
cos2
sin
2
2
2
=
+−
δδ
δδ y
y
y
y
x
x
x
x
A
E
A
E
A
E
A
E1
22
=
+
η
η
ξ
ξ
A
E
A
E
−=
η
ξ
ψψψψ
E
E
E
E
y
x
cossin
sincos
xy δδδ −=( ) ( ) yxyx yxzktj
y
zktj
xyx eAeAEEE 1111∧
+−∧+−
∧∧
+=+= δωδω
Return to Table of Content
35
POLARIZATIONSOLO
( ) ( ) yx yxzktj
yzktj
x eAeAE 11∧
+−∧
+− += δωδω
Linearly Horizontally Polarized (LHP):
( )x
x
x zktA
E δω +−= cos
Degenerated States of Polarization Ellipse
( ) 01 ==∧
+−y
zktjx AeAE xxδω
http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)
Linearly Vertically Polarized (LVP): ( )01 ==
∧+−
xzktj
y AeAE yyδω
( )y
y
y zktA
Eδω +−= cos
36
POLARIZATIONSOLO
Linear + 45° Polarized (L+45P)
Degenerated States of Polarization Ellipse
( )tzkj
yx eAAE yx ω−−∧∧
+= 11 ( )tzkj
yx eAAE yx ω−−∧∧
−= 11
Plane PolarizationMixed
0=−= xy δδδ πδδδ =−= xy
Linear - 45 ° Polarized (L-45P)
http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)
( ) ( ) yx yxzktj
yzktj
x eAeAE 11∧
+−∧
+− += δωδω
37
POLARIZATIONSOLO
( )xx zkt
A
E δω +−= cos ( )x
y zktA
Eδω +−−= sin
Degenerated States of Polarization Ellipse
Right Circular Polarization (RCP) AAA yxxy ===−= &2/πδδδ
1
22
=
+
A
E
A
E yx
http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)
( ) ( ) yx yxzktj
yzktj
x eAeAE 11∧
+−∧
+− += δωδω
( )xx zkt
A
E δω +−= cos ( )x
y zktA
Eδω +−= sin
Left Circular Polarization (LCP) AAA yxxy ===−= &2
3πδδδ
38
POLARIZATIONSOLO
( )xx zkt
A
E δω +−= cos ( )x
y zktA
Eδω +−= sin
Degenerated States of Polarization Ellipse
Left Circular Polarization (LCP) AAA yxxy ===−= &2
3πδδδ
1
22
=
+
A
E
A
E yx
http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)
Superposition of Two Circular Polarizations
( ) ( ) yx yxzktj
yzktj
x eAeAE 11∧
+−∧
+− += δωδω
Return to Table of Content
39
POLARIZATIONSOLOThe Stokes Polarization Parameters
George Gabriel Stokes 1819-1903
G.G. Stokes, “On the Composition and Resolution of Streams of Polarized Light from different Sources”,
Trans. Cambridge Phil. Soc., Vol.9, 1852, pp.399-416
( ) ( ) yx yyxx zktAzktAE 11 coscos∧∧
+−++−= δωδω
where
( ) ( ) ( ) ( )δδ 2
22
sin,
cos,,
2,
=
+−
y
y
y
y
x
x
x
x
A
tzE
A
tzE
A
tzE
A
tzEThe Polarization Ellipse is
All the information of polarization is contained in this equation.
In order to observe the quantities involved let take the time average <…> of the time dependent quantities in the Polarization Ellipse equation.
( ) ( ) ( ) ( )δδ 2
2
2
2
2
sin,
cos,,
2,
=+−y
y
yx
yx
x
x
A
tzE
AA
tzEtzE
A
tzE
( ) ( ) ( ) ( ) yxjidttzEtzET
tzEtzET
jiT
ji ,,,,1
:,,0
lim == ∫→∞
( ) ( ) ( ) ( ) ( ) 22222 sin2,4cos,,8,4 δδ yxyxyxyxxy AAtzEAtzEtzEAAtzEA =+−
40
POLARIZATIONSOLOThe Stokes Polarization Parameters (continue – 1)
We obtain
( ) ( ) ( ) ( ) ( ) 22222 sin2,4cos,,8,4 δδ yxyxyxyxxy AAtzEAtzEtzEAAtzEA =+−
( ) ( ) ( ) ( )
( ) ( ) δδδωδδ
δωδω
δ
cos2
22coscos1
2
coscos1
,,
0
0
lim
lim
yxT
xyyxT
yx
T
yxyxT
zx
AAdtzkt
T
AA
dtzktzktAAT
tzEtzE
=
++−−−=
+−+−=
∫
∫
→∞
→∞
( ) ( ) ( )[ ]2
2cos11
2cos
1,
2
0
2
0
222
limlim xT
xT
xT
xxT
x
Adtzkt
T
AdtzktA
TtzE =+−−=+−= ∫∫
→∞→∞δωδω
( ) ( )2
cos1
,2
0
222
limy
T
yyT
y
AdtzktA
TtzE =+−= ∫
→∞δω
( ) ( ) 222222 sin22cos22 δδ yxyxyxyx AAAAAAAA =+−
By adding and subtracting on the left side of this equation we obtain
( ) ( ) ( ) ( ) 22222222 sin2cos2 δδ yxyxyxyx AAAAAAAA =−−−+
442 yx AA +
41
POLARIZATIONSOLOThe Stokes Polarization Parameters (continue – 2)
The Stokes Parameters are defined as
( ) ( ) ( ) ( ) 22222222 sin2cos2 δδ yxyxyxyx AAAAAAAA ++−=+
δδ
sin2
cos2
3
2
221
220
yx
yx
yx
yx
AAVS
AAUS
AAQS
AAIS
==
==
−==
+==
2
3
2
2
2
1
2
0 SSSS ++=
The Stokes Vector is defined as
−
+
=
=
=
δδ
sin2
cos2
22
22
3
2
1
0
yx
yx
yx
yx
AA
AA
AA
AA
V
U
Q
I
S
S
S
S
S
The Stokes Parameters are observable using aproper experiment.
42
POLARIZATIONSOLOThe Stokes Polarization Parameters (continue – 3)
Consider a quasi-monochromatic wave of mean frequency ω propagating in z direction
( ) ( )( ) ( ) ( )( ) ( ) ( ) yxyx ztEztEetAetAE yx
ttzkj
y
ttzkj
xxy 1111 ,,
∧∧∧+−−
∧+−− +=+= δωδω
For monochromatic waves Ax, Ay, δx, δy, ω are constant.
Quasi-monochromatic Light
For quasi-monochromatic waves Ax, Ay, δx, δy , ω are slowly changing with time.
We have
( ) ( ) ( ) ( ) yxjidttzEtzET
tzEtzET
jiT
ji ,,,,1
:,,0
lim == ∫ ∗
→∞
∗
( ) ( ) 2
0
21:,, lim x
T
xT
xx AdtAT
tzEtzE == ∫→∞
∗
( ) ( ) 2
0
21:,, lim y
T
yT
yy AdtAT
tzEtzE == ∫→∞
∗
( ) ( ) ( )yxyxj
yx
Tj
y
j
xT
yx eAAdteAeAT
tzEtzE δδδδ −−
→∞
∗ == ∫0
1:,, lim
( ) ( ) ( )yxxy j
yx
Tj
x
j
yT
xy eAAdteAeAT
tzEtzE δδδδ −−−
→∞
∗ == ∫0
1:,, lim
43
POLARIZATIONSOLOThe Stokes Polarization Parameters (continue – 3)
( ) ( )( ) ( ) ( )( ) ( ) ( ) yxyx ztEztEetAetAE yx
ttzkj
y
ttzkj
xxy 1111 ,,
∧∧∧+−−
∧+−− +=+= δωδω
Quasi-monochromatic Light
The Stokes Parameters are defined as
( )( )
( )( ) ∗∗
∗∗
∗∗
∗∗
−=−=
+=−=
−=−=
+=+=
xyyxxyyx
xyyxxyyx
yyxxyx
yyxxyx
EEEEjAAS
EEEEAAS
EEEEAAS
EEEEAAS
δδ
δδ
sin2
cos2
3
2
22
1
22
0
( ) ( ) 2
0
21:,, lim x
T
xT
xx AdtAT
tzEtzE == ∫→∞
∗
( ) ( ) 2
0
21:,, lim y
T
yT
yy AdtAT
tzEtzE == ∫→∞
∗
( ) ( ) ( )yxyxj
yx
Tj
y
j
xT
yx eAAdteAeAT
tzEtzE δδδδ −−
→∞
∗ == ∫0
1:,, lim
( ) ( ) ( )yxxy j
yx
Tj
x
j
yT
xy eAAdteAeAT
tzEtzE δδδδ −−−
→∞
∗ == ∫0
1:,, lim
44
POLARIZATIONSOLOThe Stokes Polarization Parameters (continue – 4)
Stokes Vector for Different Polarization Types
LHP
=
0
0
1
1
0IS
2
0
0
x
y
AI
A
=
=
LVP
2
0
0
y
x
AI
A
=
=
−
=
0
0
1
1
0IS
=
0
1
0
1
0IS
L+45P
2
0 2
0
x
yx
AI
AA
=
==δ
−=
0
1
0
1
0IS
L-45P
2
0 2 x
yx
AI
AA
=
== πδ
=
1
0
0
1
0IS
RCP
2
0 2
2
x
yx
AI
AA
=
=
= πδ
−
=
1
0
0
1
0IS
2
0 2
2
3
x
yx
AI
AA
=
=
= πδ
LCP
δδ
sin2
cos2
3
2
22
1
22
0
yx
yx
yx
yx
AAS
AAS
AAS
AAS
==
−=
+=
2
3
2
2
2
1
2
0 SSSS ++=
( ) ( ) yxyx yxzktj
y
zktj
xyx eAeAEEE 1111∧
+−∧+−
∧∧
+=+= δωδω
45
POLARIZATIONSOLOThe Stokes Polarization Parameters (continue – 5)
The Stokes Parameters are defined as
δδ
sin2
cos2
3
2
22
1
22
0
yx
yx
yx
yx
AAS
AAS
AAS
AAS
==
−=
+=
2
3
2
2
2
1
2
0 SSSS ++=
δαχ sin2sin2sin =
αψχ 2cos2cos2cos =⋅
δαψχ cos2sin2sin2cos ⋅=⋅
We found
22
22
2
2
222 tan1
tan12cos&
tan1
tan2sin
yx
yx
yx
yx
AA
AA
AA
AA
+−
=+−=
+=
+=
ααα
ααα
x
y
A
A=:tanα
0
122
22
2cos2cosS
S
AA
AA
yx
yx =+−
=⋅ ψχ
0
222
coscos2sin2sin2cos
S
S
AA
AA
yx
yx =+⋅
=⋅=⋅δ
δαψχ
0
322
sinsin2sin2sin
S
S
AA
AA
yx
yx =+⋅
==δ
δαχ
=
=
−
−
0
31
1
21
sin2
1
tan2
1
S
S
S
S
χ
ψ
Return to Table of Content
46
POLARIZATIONSOLO
Consider a quasi-monochromatic wave of mean frequency ω propagating in z directioncomposed of a Unpolarized component AUP with random phases δrx and δry and aPolarized component Ax, δx , Ay, δy
( ) ( ) ( )tzkjj
yP
j
UP
j
xP
j
UPyx eeAeAeAeAEEE yxyx yyrxxr ωδδδδ −−∧∧∧∧
+++=+= 1111
Measuring the Stokes Parameters
Pass the beam through a waveplate that induces a wave retardation of φ and a polarizer with a transmission axis at an angle β relative to x axis
( )( ) ( )( ) ( )tzkjj
yP
j
UP
j
xP
j
UP eeAeAeAeAE yx yyrxxr ωϕδδϕδδ −−∧
−∧+
+++= 11
2/2/'
( )( ) ( )( )[ ] ( )tzkjj
yP
j
UP
j
xP
j
UP eeAeAeAeAE yyrxxr ωϕδδϕδδ ββ −−−+ +++= sincos" 2/2/
The waveplate that induces a wave retardation of φ between the phases of x and ycomponents of the polarized light but will not affect the random phase of the unpolarized light.
The polarizer will transmit only the component along the transmission axis
47
POLARIZATIONSOLOMeasuring the Stokes Parameters (continue – 1)
( )( ) ( )( )[ ] ( )tzkjj
yP
j
UP
j
xP
j
UP eeAeAeAeAE yyrxxr ωϕδδϕδδ ββ −−−+ +++= sincos" 2/2/
( ) ( )( ) ( )( )[ ] ( )( ) ( )( )[ ] zz yyrxxryyrxxrj
yP
j
UP
j
xP
j
UP
j
yP
j
UP
j
xP
j
UP eAeAeAeAeAeAeAeAcnkEEcnkS 11 sincossincos"", 2/2/2/2/∧−−−+−−−+
∧∗ +++⋅+++=⋅= ββββϕβ ϕδδϕδδϕδδϕδδ
( )( ) ( )( )[ ] ( )( ) ( )( )[ ]( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) βββ
βββ
βββ
βββ
ββββ
ϕδδϕδδϕδδ
δϕδδϕδδδ
ϕδδϕδδδϕδ
δδδϕδδϕδ
ϕδδϕδδϕδδϕδδ
22
0
2/
0
2/
2
0
2/2
0
2/
0
2
0
2/22
0
2/
00
2/22
0
2/2
2/2/2/2/
sincossin
sin2/cossin
cossincos
cossincos2/
sincossincos
++
++
++
++
++
++
++
+=
+++⋅+++
−−−−−−−
−−−+−
−−+−−+
−−−−+
−−−+−−−+
yP
jj
yPUP
j
yPxP
jj
yPUP
jj
yPUPUP
jj
UPxP
jj
UP
j
xPyP
jj
xPUPxP
jj
xPUP
jj
UPyP
jj
UP
jj
xPUPUP
j
yP
j
UP
j
xP
j
UP
j
yP
j
UP
j
xP
j
UP
AeeAAeAAeeAA
eeAAAeeAAeeA
eAAeeAAAeeAA
eeAAeeAeeAAA
eAeAeAeAeAeAeAeA
yxpxyyxr
yyyrxyrrr
xyxyrxx
xryxryxrx
yyrxxryyrxxr
The time average Poynting vector is
48
POLARIZATIONSOLO
( ) ( )( )[ ] zyP
jj
yPxPxPUP AeeAAAAcnkS 122222 sincossincos2/∧
−−− ++++= ββββ ϕδϕδ
2
2sincossin
2
2cos1sin
2
2cos1cos
2
2
βββ
ββ
ββ
=
−=
+=
( ) ( ) ( ) ( )[ ]{ }( ) ( ) ( ) ( )[ ]( ) ( )[ ]
[ ]βϕβϕβ
βϕδβϕδβ
βϕβϕβ
βϕϕϕϕβ
δδδδ
δδ
2sinsin2sincos2cos2
2sinsinsin22sincossin22cos2
2sinsin2sincos2cos2
2sinsincossincos2cos2
********
22222
22222
22222
xyyxxyyxyyxxyyxx
yPxPyPxPyPxPyPxPUP
jj
yPxP
jj
yPxPyPxPyPxPUP
jj
yPxPyPxPyPxPUP
EEEEjEEEEEEEEEEEEcnk
AAjAAAAAAAcnk
eeAAjeeAAAAAAAcnk
jejeAAAAAAAcnk
S
−+++−++=
++−+++=
−−++−+++=
++−+−+++=
−−
−
Measuring the Stokes Parameters (continue – 2)
( )( ) ( )( )[ ] ( )( ) ( )( )[ ] zz yyrxxryyrxxrj
yP
j
UP
j
xP
j
UP
j
yP
j
UP
j
xP
j
UP eAeAeAeAeAeAeAeAcnkEEcnkS 11 sincossincos 2/2/2/2/∧−−−+−−−+
∧∗ +++⋅+++=⋅= ββββ ϕδδϕδδϕδδϕδδ
[ ]βϕβϕβ 2sinsin2sincos2cos2 3210 SjSSScnk
S +++=
( )( )
( )( )xyyPxPxyyx
xyyPxPxyyx
yPxPyyxx
yPxPUPyyxx
AAEEEEjS
AAEEEES
AAEEEES
AAAEEEES
δδ
δδ
−=−=
−=+=
−=−=
++=+=
∗∗
∗∗
∗∗
∗∗
sin2
cos2
3
2
22
1
222
0
49
POLARIZATIONSOLOMeasuring the Stokes Parameters (continue – 3)
( ) [ ]βϕβϕβϕβ 2sinsin2sincos2cos2
, 3210 SjSSScnk
S +++=
( )( )
( )( )xyyPxPxyyx
xyyPxPxyyx
yPxPyyxx
yPxPUPyyxx
AAEEEEjS
AAEEEES
AAEEEES
AAAEEEES
δδ
δδ
−=−=
−=+=
−=−=
++=+=
∗∗
∗∗
∗∗
∗∗
sin2
cos2
3
2
22
1
222
0
The Stokes Parameters are measured by first removing the waveplate φ = 0
( ) [ ]ββϕβ 2sin2cos2
0, 210 SSScnk
S ++==
Now the polarizer is sequentially rotate to β = 0, π/4 and π/2
( ) [ ]1020,0 SS
cnkS +=== ϕβ
( ) [ ]2020,4/ SS
cnkS +=== ϕπβ
( ) [ ]1020,2/ SS
cnkS −=== ϕπβ
For the final measurement we add the waveplate with φ = π/2 and polarizer at β = π/4
( ) [ ]3022/,4/ SS
cnkS −=== πϕπβ
( ) ( )[ ]0,2/0,02
0 ==+=== ϕπβϕβ SScnk
S
( ) ( )[ ]0,2/0,02
1 ==−=== ϕπβϕβ SScnk
S
( ) 02 0,4/22
SScnk
S −=== ϕπβ
( )2/,4/22
03 πϕπβ ==−= Scnk
SS
50
POLARIZATIONSOLOMeasuring the Stokes Parameters (continue – 4)
( ) [ ]βϕβϕβϕβ 2sinsin2sincos2cos2
, 3210 SjSSScnk
S +++=
( )( )
( )( )xyyPxPxyyx
xyyPxPxyyx
yPxPyyxx
yPxPUPyyxx
AAEEEEjS
AAEEEES
AAEEEES
AAAEEEES
δδ
δδ
−=−=
−=+=
−=−=
++=+=
∗∗
∗∗
∗∗
∗∗
sin2
cos2
3
2
22
1
222
0
The Stokes Parameters can measure the degree of polarization of a beam.
We can see that a beam is unpolarized iff: 00 321
2
0 ===>=+= ∗∗ SSSAEEEES UPyyxx
The Degree of Polarization is defined as: 100
2
3
2
2
2
1
222
22
≤≤++
=++
+= P
S
SSS
AAA
AAP
yPxPUP
yPxP
If the beam is completely polarized then AUP = 0: 02
3
2
2
2
1
2
0 >++= SSSS
Return to Table of Content
51
POLARIZATIONSOLO
H. Poincaré, “Théorie Mathématique de la Lumiere”,Gauthiers-Villars, Paris, 1892, Vol.2, Ch.12
Poincaré Sphere
Jules Henri Poincaré1854-1912
We found
03
02
01
/2sin
/2sin2cos
/2cos2cos
SS
SS
SS
==⋅=⋅
χψχψχ
We see that the normalized Stokes Parameters can be represented as a unit vector on a sphere (called Poincaré Sphere) using the Polarized Ellipse parameters χ and ψ.
52
POLARIZATIONSOLOPoincaré Sphere
J.D. Kraus, “Electromagnetics”, 4th Ed., McGraw Hill, 1992, p.607
Return to Table of Content
53
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
In 1948 Hans Mueller, then a professor of physics at MIT devised a matrix methodthat deals with the relation of Stokes vector at the output of an optical device to theStokes vector at the input of the optical device.
=
3
2
1
0
33323130
23222120
13121110
03020100
3
2
1
0
'
'
'
'
S
S
S
S
mmmm
mmmm
mmmm
mmmm
S
S
S
S
inputoutput SMS
=
or
Hans Mueller, “The Foundation of Optics”, J. Opt. Soc. Am., 37, pg. 110 (1947) 38, pg. 661(1948)
54
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Suppose that we have a few optical elements through which the ray passes .
( ) inputinputoutput SMSMMMSMMSMS
==== 12312323
or
123 MMMM =
55
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Polarizer
10'
10'
≤≤=≤≤=
yyyy
xxxx
pEpE
pEpE
The Polarizer is described by two orthogonal transmission axes that are characterized, respectively, by transmission factors px and py (0 ≤ px,py ≤ 1).
( )∗∗
∗∗
∗∗
∗∗
−=
+=
−=
+=
xyyx
xyyx
yyxx
yyxx
EEEEjS
EEEES
EEEES
EEEES
3
2
1
0
( )∗∗
∗∗
∗∗
∗∗
−=
+=
−=
+=
'''''
'''''
'''''
'''''
3
2
1
0
xyyx
xyyx
yyxx
yyxx
EEEEjS
EEEES
EEEES
EEEES
+−
−+
=
3
2
1
0
2222
2222
3
2
1
0
2000
0200
00
00
2
1
'
'
'
'
S
S
S
S
pp
pp
pppp
pppp
S
S
S
S
yx
yx
yxyx
yxyx
( )
+−
−+
=
yx
yx
yxyx
yxyx
yxPOL
pp
pp
pppp
pppp
ppM
2000
0200
00
00
2
1,
2222
2222
56
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Polarizer
( )22
1
:
/tan:
yx
xy
ppp
pp
+=
= −βLet define
=
3
2
1
0
2
2
22
22
3
2
1
0
2sin000
02sin00
002cos
002cos
2
1
'
'
'
'
S
S
S
S
p
p
pp
pp
S
S
S
S
ββ
ββ
( )
=
ββ
ββ
β
2sin000
02sin00
0012cos
002cos1
2,
2ppM POL
ββ
sin
cos
pp
pp
y
x
==
57
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Polarizer (continue -1)
0'
10'
=≤≤=
y
xxxx
E
pEpE
Ideal Linear Polarizer py = 0 and 0 ≤ px ≤ 1.
( )
( )
+=
=
0
0
1
1
2
0000
0000
0011
0011
2
'
'
'
'
10
2
3
2
1
0
||
2
3
2
1
0
SSp
S
S
S
S
p
S
S
S
S
x
M
x
POL
Regardless the state of polarization of the input beam the output beam isLinear Horizontally Polarized (LHP)
10'
0'
≤≤==
yyyy
x
pEpE
E
Ideal Linear Polarizer px = 0 and 0 ≤ py ≤ 1.
( )
( )
−
−=
−
−
=
⊥
0
0
1
1
2
0000
0000
0011
0011
2
'
'
'
'
10
2
3
2
1
0
2
3
2
1
0
SSp
S
S
S
S
p
S
S
S
S
y
M
y
POL
Regardless the state of polarization of the input beam the output beam isLinear Vertically Polarized (LVP)
58
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Polarizer (continue -2)
Crossed Polarizers
( ) ( )
=
−
−
=
⊥
0
0
0
0
22
0000
0000
0011
0011
2
0000
0000
0011
0011
2
'
'
'
'22
3
2
1
0
2
||
2
3
2
1
0
yx
M
y
M
xpp
S
S
S
S
pp
S
S
S
S
POLPOL
Regardless the state of polarization of the input beam the output beam iscompletely blocked
Crossed Polarizers are a combination of a Linear Polarizer with the transmission axes normal to the x-axis, followed by a Linear Polarizer with the transmission axes parallel to the x-axis (or vice-versa).
59
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Polarizer (continue -3)
Eigenvalues and Eigenvectors of the Linear Polarizer Mueller Matrix
Let find the Input Stokes Vectors that are unaffected by the Polarizer; i.e.:
=
+−
−+
=
3
2
1
0
3
2
1
0
2222
2222
3
2
1
0
2000
0200
00
00
2
1
'
'
'
'
S
S
S
S
S
S
S
S
pp
pp
pppp
pppp
S
S
S
S
POLM
yx
yx
yxyx
yxyx
λ
( ) 0
22000
02200
002
002
2
1,
2222
2222
44 =
−−
−+−
−−+
=−
λλ
λ
λ
λ
yx
yx
yxyx
yxyx
xyxPOL
pp
pp
pppp
pppp
IppM
==
0
0
1
1
& 1
2
1 Spx
λ
−
==
0
0
1
1
& 2
2
2 Sp y
λ
1st eigenvalue-eigenvector
2nd eigenvalue-eigenvector
We can see that only LHP and LVP are unaffected by the Linear Polarizer.
60
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Waveplate
y
j
y
x
j
x
EeE
EeE2/
2/
'
'ϕ
ϕ
−==
The Waveplate is a polarizing element that introduced a phase shift φ between the orthogonal components of an optical beam. A Waveplate is a phase-shifter but is also called a retarder or a compensator.
( )∗∗
∗∗
∗∗
∗∗
−=
+=
−=
+=
xyyx
xyyx
yyxx
yyxx
EEEEjS
EEEES
EEEES
EEEES
3
2
1
0
( )∗∗
∗∗
∗∗
∗∗
−=
+=
−=
+=
'''''
'''''
'''''
'''''
3
2
1
0
xyyx
xyyx
yyxx
yyxx
EEEEjS
EEEES
EEEES
EEEES
−=
3
2
1
0
3
2
1
0
cossin00
sincos00
0010
0001
'
'
'
'
S
S
S
S
S
S
S
S
ϕϕϕϕ
( )
−=
ϕϕϕϕ
ϕ
cossin00
sincos00
0010
0001
WPM
61
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Waveplate (continue – 1)Quarter-Waveplate φ = π/2
The Quarter-Waveplate transforms Linearly Polarized (L+45P or L-45P) toRight or Left Circularly Polarized (RCP or LCP) or vice-versa.
( )
−==
0100
1000
0010
0001
2/πϕWPM
( )
RCPPLPL
WPM
=
−=
=
++
1
0
0
1
0
1
0
1
0100
1000
0010
0001
0
1
0
1
2/
4545
πϕ ( )
LCPPLPL
WPM
−
=
−
−=
−=
−−
1
0
0
1
0
1
0
1
0100
1000
0010
0001
0
1
0
1
2/
4545
πϕ
( )
PLRCPRCP
WPM
45
0
1
0
1
1
0
0
1
0100
1000
0010
0001
1
0
0
1
2/
−
−=
−=
= πϕ ( )
LCP
PLLCPLCP
WPM
45
0
1
0
1
1
0
0
1
0100
1000
0010
0001
1
0
0
1
2/
+
=
−
−=
−
= πϕ
62
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Waveplate (continue – 2)Half-Waveplate φ = π
( )
−−
==
1000
0100
0010
0001
πϕWPM
−−
=
−−
=
3
2
1
0
3
2
1
0
3
2
1
0
1000
0100
0010
0001
'
'
'
'
S
S
S
S
S
S
S
S
S
S
S
S
χ
ψχ
ψχ
2sin
2sin2cos
2cos2cos
0
3
0
2
0
1
=
⋅=
⋅=
S
S
S
S
S
S
=
=
−
−
0
31
1
21
sin2
1
tan2
1
S
S
S
S
χ
ψ
2'
2'
πχχ
ψπψ
−=
−=
The Half-Waveplate reverses the orientation and ellipticity of the polarization ellipse (polarization state).
63
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Waveplate (continue – 3)
The phases of two adjacent Half-Waveplates add.
( ) ( )( ) ( )
+++−+
=
−
−=
3
2
1
0
2121
2121
3
2
1
0
22
22
11
11
3
2
1
0
cossin00
sincos00
0010
0001
cossin00
sincos00
0010
0001
cossin00
sincos00
0010
0001
'
'
'
'
S
S
S
S
S
S
S
S
S
S
S
S
ϕϕϕϕϕϕϕϕ
ϕϕϕϕ
ϕϕϕϕ
Eigenvalues and Eigenvectors of the Waveplate Mueller Matrix
( ) 0
cossin00
sincos00
0010
0001
4 =
−−−
−−
=−
λϕϕϕλϕ
λλ
λϕ IMWP
LHP
S
==
0
0
1
1
&1 11
λ
LVP
S
−
=−=
0
0
1
1
&1 22
λ
1st eigenvalue-eigenvector
2nd eigenvalue-eigenvector
We can see that only LHP and LVP are unaffected by the Waveplate Polarizer.
64
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
The Mueller Matrix of a Rotator (Coordinate Rotation)
( )( ) θθθγ
θθθγcossinsin'
sincoscos'
yxy
yxx
EEEE
EEEE
+−=−=+=−=
Assume that the polarizing device rotates its orthogonal axes along the ray propagation direction by an angle θ. The orthogonal axes are defined as (‘).
( )∗∗
∗∗
∗∗
∗∗
−=
+=
−=
+=
xyyx
xyyx
yyxx
yyxx
EEEEjS
EEEES
EEEES
EEEES
3
2
1
0
( )∗∗
∗∗
∗∗
∗∗
−=
+=
−=
+=
'''''
'''''
'''''
'''''
3
2
1
0
xyyx
xyyx
yyxx
yyxx
EEEEjS
EEEES
EEEES
EEEES
−=
3
2
1
0
3
2
1
0
1000
02cos2sin0
02sin2cos0
0001
'
'
'
'
S
S
S
S
S
S
S
S
θθθθ ( )
−=
1000
02cos2sin0
02sin2cos0
0001
θθθθ
θROTM
'1'1
1111
''
sincos
yx
yxyx
yx
yx
EE
EEEE
∧∧
∧∧∧∧
+=
+=+= γγ
65
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
Assume that the polarizing device rotates its orthogonal axes along the ray propagation direction by an angle θ. The Mueller Rotation Matrix from x,y to x’,y’ is
( )
−=
1000
02cos2sin0
02sin2cos0
0001
θθθθ
θROTM
The Mueller Polarizator Matrix is
( )
=
ββ
ββ
β
2sin000
02sin00
0012cos
002cos1
2,
2ppM POL
( ) SMS ROT
θ='
( ) ( ) ( ) SMpMSpMS ROTPOLPOL
θββ ,'," ==
The Mueller Rotation Matrix from x’,y’ to x,y is
( )
−
=−
1000
02cos2sin0
02sin2cos0
0001
θθθθ
θROTM ( ) ( ) ( ) ( ) SMpMMSMS ROTPOLROTROT
θβθθ ,"'" −=−=
( ) ( ) ( ) ( )θβθθβ ROTPOLROTPOL MpMMpM ,,, −=
The Mueller Matrix of a Rotated Polarizer
66
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
( ) ( )( )
+−
−+=
βθβθθθβθβ
θθβθβθθβθβθβ
θβ
2sin000
02cos2sin2sin2cos2sin2sin12sin2cos
02cos2sin2sin12sin2sin2cos2cos2cos
02sin2cos2cos2cos1
2,,
22
222ppM POL
The Mueller Matrix for Rotated Polarizer is
( ) ( ) ( ) ( )
−
−
=
−=
1000
02cos2sin0
02sin2cos0
0001
2sin000
02sin00
0012cos
002cos1
1000
02cos2sin0
02sin2cos0
0001
2
,,,
2
θθθθ
ββ
ββ
θθθθ
θβθθβ
p
MpMMpM ROTPOLROTPOL
The Mueller Matrix of a Rotated Polarizer (continue – 1)
67
POLARIZATIONSOLOThe Mueller Matrices for Polarizing Components
( ) ( )( )
−+−
−−+=
ϕϕβϕβϕβϕββϕββϕβϕββϕββ
ϕβ
cossin2cossin2sin0
sin2coscos2cos2sincos12cos2sin0
sin2sincos12cos2sincos2sin2cos0
0001
,22
22
WPM
The Mueller Matrix for Rotated Waveplate is
( ) ( ) ( ) ( )
−
−
−
=
−=
1000
02cos2sin0
02sin2cos0
0001
cossin00
sincos00
0010
0001
1000
02cos2sin0
02sin2cos0
0001
,,
θθθθ
ϕϕϕϕθθ
θθ
θϕθθβ ROTWPROTPOL MMMpM
The Mueller Matrix of a Rotated Waveplate
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68
POLARIZATIONSOLOThe Jones Polarization Parameters
R. Clark Jones, “A New Calculus for the Treatment of Optical Systems”,J. Opt. Soc. Am., Vol.31, July 1941, pp.500-503J. Opt. Soc. Am., Vol.32, Aug. 1942, pp.486-493J. Opt. Soc. Am., Vol.37, Feb. 1947, pp.107-110J. Opt. Soc. Am., Vol.37, Feb. 1947, pp.110-112J. Opt. Soc. Am., Vol.38, Aug. 1948, pp.671-585 R. Clark Jones
1916-2004
Mueller matrices deal with the intensity of the beam. If the phase information is important we must use Jones formalism. Jones calculus was developed in the same time with the Mueller calculus by R. Clark Jones who introduced Jones vectors and Jones matrices:
Jones vectors describe the polarization of light:
Jones matrices describe the optical component:
+=
y
x
yxE
E
EEJ
22
1
[ ]
=
2221
1211
jj
jjJ
Jones calculus deals only with polarized light.
69
POLARIZATIONSOLO
Stokes and Jones Vector for Different Polarization Types
LHP
=
=0
1
0
0
1
1
0 JIS
2
0
0
x
y
AI
A
=
=
LVP
2
0
0
y
x
AI
A
=
=
=
−
=1
0
0
0
1
1
0 JIS
=
=1
1
2
1
0
1
0
1
0 JIS
L+45P
2
0 2
0
x
yx
AI
AA
=
==δ
−
=
−=
1
1
2
1
0
1
0
1
0 JIS
L-45P
2
0 2 x
yx
AI
AA
=
== πδ
=
=i
JIS1
2
1
1
0
0
1
0
RCP
2
0 2
2
x
yx
AI
AA
=
=
= πδ
−
=
−
=i
JIS1
2
1
1
0
0
1
0
2
0 2
2
3
x
yx
AI
AA
=
=
= πδ
LCP
δδ
sin2
cos2
3
2
22
1
22
0
yx
yx
yx
yx
AAS
AAS
AAS
AAS
==
−=
+=
2
3
2
2
2
1
2
0 SSSS ++=
( ) ( ) yxyx yxzktj
y
zktj
xyx eAeAEEE 1111∧
+−∧+−
∧∧
+=+= δωδω
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70
POLARIZATIONSOLO
Faraday Effect (Hecht p.261) Michael Faraday (England) 1845 described the rotation of the plane of polarized light that passed through glass in a magnetic field.
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71
POLARIZATIONSOLO
Pockels Effect (Hecht p.263, Chuang p.509, Meyer-Arent p.318)
Vnr
nretardatio
Ern
n
n
Ln
Ln
yx
z
yy
xx
3
0''
32
''
''
2
:
21
2
2
λπ
λπ
λπ
=Φ−Φ=Φ
⇒
=∆−=
∆
=Φ
=Φ
Pockels Effect 1893
Electro – Optical Effects The electro – optical effects are called Pockels or Kerr where the refractive indexchanges linearly or quadraticly, respectively.
==
=
∆
Kerrk
PockelskEKn
nk
2
1102
Frederich Carl AlwinPockels
(1865-1913)
http://www.physi.uni-heidelberg.de/~schmiedm/Vorlesung/LasPhys02/LectureNotes/OpticsCrystals.pdf#search='Pockels%20Effect'
http://en.wikipedia.org/wiki/Pockels_effect
72
POLARIZATIONSOLO
Electro – Optical Effects
Kerr Effect (Hecht p.263, Chuang p.509, Meyer-Arent p.318)
The electro – optical effects are Kerr or Pockels
2
0 EKn λ=∆2
2
02
d
VKλπ=Φ∆
In the Kerr Electro-optic Effect 1875 is the electric field that causes the substance tobecome birefringent.
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73
POLARIZATIONSOLO
Plane Polarized Wave in an Absorbing Medium
Circular Polarized Wave in an Absorbing Medium
74
POLARIZATIONSOLO
Plane Polarized Wave in an Refracting Medium
Circular Polarized Wave in an Refracting Medium
75
POLARIZATIONSOLO
Plane Polarized Wave in a Medium Showing Circular Dichroism
Plane Polarized Wave in a Medium Showing Circular Birefrigens
76
POLARIZATIONSOLO
Plane Polarized Wave in a Medium Showing Both Circular Dichroism and Circular Birefrigens
77
OPTICSSOLO
http://microscopy.fsu.edu/
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78
OPTICSSOLO
References Optics Polarization
A. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons, 1984
M. Born, E. Wolf, “Principles of Optics”, Pergamon Press,6th Ed., 1980
E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979, Ch.8
C.C. Davis, “Lasers and Electro-Optics”, Cambridge University Press, 1996
G.R. Fowles, “Introduction to Modern Optics”,2nd Ed., Dover, 1975, Ch.2
M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986
http://en.wikipedia.org/wiki/Polarization
W.C.Elmore, M.A. Heald, “Physics of Waves”, Dover Publications, 1969
E. Collett, “Polarization Light in Fiber Optics”, PolaWave Group, 2003
W. Swindell, Ed., “Polarization Light”, Benchmark Papers in Optics, V.1, Dowden, Hutchinson & Ross, Inc., 1975
http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)
79
ELECTROMAGNETICSSOLO
References Electromagnetics
J.D. Jackson, “Classical Electrodynamics”, 3rd Ed., John Wiley & Sons, 1999
R. S. Elliott, “Electromagnetics”, McGraw-Hill, 1966
J.A. Stratton, “Electromagnetic Theory”, McGraw-Hill, 1941
W.K.H. Panofsky, M. Phillips, “Classical Electricity and Magnetism”, Addison-Wesley, 1962
F.T. Ulaby, R.K. More, A.K. Fung, “Microwave Remote Sensors Active and Passive”, Addison-Wesley, 1981
A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”,John Wiley & Sons, 1988
80
ELECTROMAGNETICSSOLO
References
1. W.K.H. Panofsky & M. Phillips, “Classical Electricity and Magnetism”,
2. J.D. Jackson, “Classical Electrodynamics”,
3. R.S. Elliott, “Electromagnetics”,
4. A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”,
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January 4, 2015 81
SOLO POLARIZATION
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 –2013
Stanford University1983 – 1986 PhD AA