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Review of Basic Polarization
Optics for LCDs
Module 4
1 1
V 

2  i
i   1 
1 1
  
 
1 0
2  i 
i  t  x 
i t




Ex (t )  Re  J x e   Re  Ax e

Module 4 Goals
• Polarization
• Jones Vectors
• Stokes Vectors
• Poincare Sphere
• Adiabadic Waveguiding
Polarization of Optical Waves
Objective: Model the polarization of light through an LCD.
Assumptions:
•
•
Linearity – this allows us to treat the
transmission of light independent of wavelength
(or color).
We can treat each angle of incidence independently.
Transmission is reduced to a linear superposition of the
transmission of monochromatic (single wavelength) plane
waves through LCD assembly.
Monochromatic Plane Wave (I)
A monochromatic plane wave propagating in isotropic and
homogenous medium:
E (t )  A(t )cos( t  k r )
A = constant amplitude vector
 = angular frequency
k = wave vector
n = index of refraction
c = speed of light
 = wavelength in vacuum
For transparent materials
k is related to frequency
k n

c
n
2

n  Re, n  n( )
Dispersion relation
Monochromatic Plane Wave (II)
• The E-field direction is always  to the direction of propagation
k E 0
• Complex notation for plane wave: (Real part represents actual E-field)
E  A exp i (t  k r )
• Consider propagation along Z-axis, E-field vector is in X-Y plane:
Ex  Ax cos(t  kz   x )
E y  Ay cos( t  kz   y )
Ax , Ay independent amplitudes
 x , y two independent phases    x , y  
Y-axis
EY
X-axis
Ex
Monochromatic Plane Wave (III)
• There is no loss of generality in this case.
• Finally, we define the relative phase as
   y x
    
• Now in a position to look at three specific cases.
1. Linear Polarization
2. Circular Polarization
3. Elliptical Polarization
Linear Polarization
• In this case, the E-field vector
follows a linear pattern in the
X-Y plane as either time or
position vary.
Y-axis
AY

• Occurs when    y   x  0
or    y   x  
•
Ax
X-axis
Important parameters:
1. Orientation
2. Handedness
3. Extent
tan  
Ay
Ax
Linear polarized or plane polarized are used interchangeably
Circular Polarization
• In this case, the E-field vector
follows a circular rotation in the
X-Y plane as either time or
position vary.
• Occurs when Ax  Ay
and    y   x  
•
Y-axis
AY
X-axis

2
Ax
Important parameters:
1. Orientation
2. Handedness
3. Extent
 

2
E x 2  E y 2  A2
(-) CCW rotation = RH, (+) CW rotation = LH
Circular Polarization
E x  A cos( t  kz )
E y  A cos( t  kz   / 2)
E x  E y  A2 cos2 ( t  kz )  A2 cos2 ( t  kz   / 2) 
 A2 cos2 ( t  kz )  A2 sin2 ( t  kz ) 
 A2 (cos2 ( t  kz )  sin2 ( t  kz ))  A2
Equation of a circle
Elliptic Polarization States
• This is the most general
representation of polarization.
The E-field vector follows an
elliptical rotation in the X-Y
plane as either time or position
vary.
• Occurs for all values of
  y x
•
Y-axis
AY
b
a

Ax
X-axis
Important parameters:
1. Orientation
2. Handedness
3. Extent of Ellipticity
b
e
a
2 Ax Ay
tan 2  2
cos 
2
Ax  AY
Elliptic Polarization States
E x  A cos( t  kz )
E y  A cos( t  kz   )
eliminate t
2
 Ex   Ey 
cos 
2


2
E
E

sin




 
x y

Ax Ay
 Ax   Ay 
2
2
 Ex   Ey  
 a   b  1

 

2
a2  Ax cos2   Ay sin2   2Ax Ay cos  cos  sin
2
2
b2  Ax sin2   Ay cos2   2Ax Ay cos  cos  sin
2
tan2 
2
2Ax Ay
Ax  Ay
2
2
cos 
Transformation:
x’
y’
a
b

Ax
X-axis
=3/4
=/2
=/4
=0
=/4
=/2
=3/4
=
=3/4
=/2
=/4
=0
=/4
=/2
=3/4
=
Review Complex Numbers
Im
 = 3 – 4i
-2+2i
 = ei = cos  + i.sin
 = e-i = cos (-) + i.sin (-) = cos  - i.sin
Remember the
identities:
ex ey = ex+y
ex / ey = ex-y
d/dz ez = ez
Re
3-4i
Complex Number Representation
Polarization can be described by an amplitude and phase angles
of the X-Y components of the electric field vector. This lends
itself to representation with complex numbers:
  e tan 
i
Ay
Ax
e
i (  y  x )
Im
linear
x  y  0

Ay
Ax
(cos 0  i sin(0)) 
Ay
Ax
Re
on x axis
circular
 

2
, Ay  Ax
  e i / 2  cos

2
 i sin

2
i
on y (imaginary axis)
Jones Vector Representation
Convenient way to uniquely describe polarization state of a
plane wave,using complex amplitudes as a column vector.
 Ax e i x
J 
 A e i y
 y



J is not a vector in real space, it is a mathematical abstraction in
complex space.
E x (t )  Re J x exp  i t    Re  Ax exp  i t   x  
Jones Vector
amplitude
phases
electric field
Polarization is uniquely specified
Jones Vector Representation (II)
If you are only interested in polarization state, it is most
convenient to normalize it.
J J 1
A linear polarized beam with electric field vector oscillating along
a given direction can be represented as:
 cos 
J 

sin



For orthogonal state,    
1

2
  sin 
J 

cos



Jones Vector Representation (III)
Normalize Jones Vector
 Ax e i  x
J 
 A e i y
 y




J *  ( Ax* e  i  x , Ay* e
 i y
)
J * J  Ax Ax*  Ay Ay* | Ax |2  | Ay |2  1
Take
 Ax e i  x
J 
 A e i y
 y
e  i x
| Ax |2  | Ay |2
 cos 

i 
 sin e 




 Ax e i  x

 A e i y
 y

 


1
| Ax |2  | Ay |2
 Ax 

i  
A
e
 y

Jones Vector Representation (IV)
The Jones matrix of rank 2, any pair of orthogonal Jones
vectors can be used as a basis for the mathematical space
spanned by all the Jones vectors.
When =0 for linear polarized light, the electric field oscillates
along coordinate system, the Jones Vectors are given by:
 cos   1 
x 
 
 sin   0 
For circular polarized light:
  sin   0 
y 
 
 cos   1 
 cos 
1  1
R    i / 2

 
e
sin

2


 i 
 cos 
1  1
L   i / 2

 
e
sin

2 i 


Mutually orthogonal condition R  L  0
Polarization Representation
Polarization Ellipse
Jones Vector
1
 
0
0
 
1
1
2
1
 
1
1  1 
 
2  1
(,)
 0, 0 
 0, / 2
 0, / 4
(,)
Stokes
 0, 0 
1
 
1
0
 
0
 / 2, 0
 1 



1


 0 


 0 
 / 4, 0
1
 
0
1
 
0
 , / 4   / 4,0
 1 


 0 
 1 


0


Polarization Representation
Polarization Ellipse Jones Vector
(,)
(,)
1  1 


2  i 
  / 2, / 4
 0, / 4
1 1
 
2 i
 / 2, / 4
 0,  / 4
1  1


5  2i 
1  2 


5  i 
1 2i


10  2  i 
Stokes
 1 


 0 
 0 

 1 



1
 
0
0

1

 

 1 


 3 / 5 
 0 

 4/5 




1 1 


/
2,
tan


2

 1 


 3/5 
 0 

 4 / 5 





1 1 

/
2,

tan
 / 2, tan 2 

2

1

1 1 
0,
tan


2


 
1 4
1 1 

tan
,

/
4

/
4,
tan

 

3
2

 
 1 


 0 
 3/5 

 4 / 5 



Jones Matrix Limitations
Jones is powerful for studying the propagation of plane waves
with arbitrary states of polarization through an arbitrary
sequence of birefringent elements and polarizers.
Limitations:
• Applies to normal incidence or paraxial rays only
• Neglects Fresnel refraction and surface reflections
• Deficient polarizer modeling
• Only models polarized light
Other Methods:
• 4x4 Method – exact solutions
(models refraction and multiple reflections)
• 2x2 Extended Jones Matrix Method
(relaxes multiple reflections for greater simplicity)
Partially Polarized & Unpolarized Light
We discussed monochromatic/polarization thus far.
If light is not absolutely monochromatic, the amplitude and
relative phase  between x and y components can vary
with time, and the electric field vector will first vibrate in one
ellipse and then in another.  The polarization state of a
polychromatic wave is constantly changing.
If polarization state changes faster than speed of observation,
the light is partially polarized or unpolarized.
Optics – light of oscillation frequencies 1014s-1
Whereas polarization may change 10-8s (depending on source)
Partially Polarized & Unpolarized Light
Consider quasi monochromatic waves (D<<)

Light can still be described as: E  A(t )exp i  t  k  r 
Provided the constancy condition of A is relaxed.
 denotes center frequency
A denotes complex amplitude

Because (D<< ), changes in A(t) are small in a time interval
1/D (slowly varying).
If the time constant of the detector td>1/D, A(t) can change
originally in a time interval td.
Partially Polarized & Unpolarized Light
To describe this type of polarization state, must consider time
averaged quantities.
S0 = <<Ax2+Ay2>>
S1 = <<Ax2-Ay2>>
S2 = 2<<AxAy cos>>
S3 = 2<<AxAy sin>>
Ax, Ay, and  are time dependent
<< >> denotes averages over time interval td that is the
characteristic time constant of the detection process.
These are STOKES parameters.
Stokes Parameters
Note: All four Stokes Parameters have the same dimension of
intensity.
They satisfy the relation:
S0  S  S2  S3
2
2
1
2
2
the equality sign holds only for polarized light.
Stokes Parameters
Example: Unpolarized light
No preference between Ax and Ay
(Ax=Ay),  random
S0 = <<Ax2+Ay2>>=2<<Ax2>>
S1= <<Ax2-Ay2>>=0
S2,3=2<<AxAy cos>>=2<<AxAy sin>>=0
since  is a
random function
of time
if S0 is normalized to 1, the Stokes vector parameter is
for unpolarized light.
Example: Horizontal Polarized Light
1


0


0

0



Ay=0, Ax=1
S0=<<Ax2>>=1
S1=<<Ax2>>=1
S2,3=2<<AxAy cos>>=2<<AxAy sin>>=0
1


1


0

0



Stokes Parameters
Example: Vertically polarized light
Ay=1, Ax=0
S0 = <<Ax2+Ay2>>=<<Ay2>>=1
S1 = <<Ax2-Ay2>>=<<-Ay2>>=-1
S2,3 = 2<<AxAy cos>>=2<<AxAy sin>>=0
Example: Right handed circular polarized light
S0 = <<Ax2+Ay2>> = 2<<Ax2>>
S1 = <<Ax2-Ay2>> = 0
S2 = 2<<AxAy cos1/2>> = 0
S3 = 2<<AxAy sin1/2>> = -1







1 

0 
0 

1 

 1 



1


 0 

 0 



(=-1/2) Ax=Ay
Stokes Parameters
Example: Left handed circular polarized light
(=1/2) Ax=Ay
1


0


0

1



S0 = <<Ax2+Ay2>> = 2<<Ax2>>
S1 = <<Ax2-Ay2>> = 0
S2 = 2<<AxAy cos1/2>> = 0
S3 = 2<<AxAy sin1/2>> = 1
Degree of polarization:



S1  S2  S3
2
2
2
S0
Unpolarized S12 = S22 = S32 = 0
Polarized S12+S22+S32 = 1
useful for describing partially polarized light

1
2
0   1
Poincare’ Sphere:
Linear Polarization
States
Poincare’ Sphere:
Elliptic Polarization
States
Polarization Conversion:
Polarization Conversion:
Y-axis
f
s

X-axis
Z-axis

Jones Matrix Method (I)
f
Y-axis

s

X-axis
Z-axis
Vx 
• The polarization state in V  


V
 y 
a fixed lab axis X and Y:
• Decomposed into fast and slow
coordinate transform:
Vs 
 cos


V
  sin
 f 
Vx 
sin  Vx 
  R   


V
cos  Vy 
y


(notation: fast (f) and slow (s)
component of the polarization state)
rotation matrix
• If ns and nf are the refractive indices associated with the propagation of slow and fast components, the emerging beam has
the polarization state:

2 

exp

in
d
s

Where d is the
V   



s

thickness and  is 
V   
the wavelength
 f  
0


 V
 s 


2    Vf 

exp  inf
d 



0
Jones Matrix Method (II)
• For a “simple” retardation film, the following phase changes occur:
2
1
2
   ns  nf 
d

 ns  nf  d
2


(relative phase retardation)
(mean absolute phase change)
• Rewriting previous retardation equation:


2 

exp

in
d
0
 V 
 s  
V   


s
 s  
 
V   
2    Vf 

 f  
0
exp  inf
d 






  ns  nf ns  nf  2 
exp

i

d
0


 
  
2
2




 Vs


  ns  nf ns  nf  2    Vf

0
exp  i 

d  


2   
  2

i / 2
0  Vs 
 i  e
e 
 
i / 2  
e   Vf 
 0




Jones Matrix Method (III)
•The Jones vector of the polarization state of the emerging
beam in the X-Y coordinate system is given by transforming
back to the S-F coordinate system.
V  
 x    cos

V  
 sin
y


 sin  Vs 


cos   V  
 f 
Jones Matrix Method (IV)
• By combining equations, the transformation due to the retarder
V  
plate is:
Vx 
 x   R    W0R   


V  
 y 
Vy 
where W0 is the Jones matrix for the retarder plate and R(Y) is the
coordinate rotation matrix.
 cos
R    
  sin
sin 

cos 
i  / 2

e
W0  e i 
 0
0 
i / 2 
e 
(The absolute phase can often be neglected if multiple reflections can be ignored)
 A retardation plate is characterized by its phase retardation 
and its azimuth angle , and is represented by:
W  R   W0R  
Examples
Polarizer with transmission
axis oriented  to X-axis
 i   1
P0  e

0
Polarization State
Y-axis
0

0
E

X-axis
Jones Vector
 cos  


 sin  
’ is due to finite optical thickness
of polarizer.
1 1
 
2  i 
If polarizer is rotated by  about Z
P  R    P0 R  
1 1
 
2 i
ignoring ’ polarizers transmitting
light with electric field vectors  to
x and y are:
0 0
1 0
Px  
 Py  

0
1
0
0




b
b
a

a


 a cos   ib sin  


 a sin   ib cos  
 a cos   ib sin  


 a sin   ib cos  
Examples
¼ Wave Plate
 and the thickness t  

4 Dn
2
and   450 and incident beam is
vertically polarized:
Polarization State
Y-axis
E
X-axis

Incident
Jones Vector
1
2
1
 
i
W
1 1

2  i
1

1
Emerging Jones Vector
1
 
0
 cos  


 sin  
1 1
W 
2 1
  i 

exp

0

1   4 
 1 1


1 
 i   2  1
0
exp   

 4 

i 

1
V 
V 
1 1

2  i
1 1

2  i
V 
i   1 
1 1

 
 
1 0
2  i 
i   cos  
1  cos   i sin  





1   sin  
2  sin   i cos  
1 1

2  i
i  1  1 
    
1 i  0
Birefringent Plates
45
45
Parallel polarizers


cos

 0 0
2
E'  

 0 1   i sin 

2



i sin 

2  1  0   1  cos 
2
 
  2 1
2 

cos 
 0 
2 
1
 1
  (ne  no )d 
I  cos 2    cos 2 

2

2 2


Cross polarizers


cos

 1 0
2
E'  

 0 0   i sin 

2


 
2  1  0   i  sin 
2
 
  2 1
2 

cos 
 0 
2 
i sin
1
 1
  ( ne  no ) d 
I  sin 2    sin 2 

2

2 2


Wave Plates
y
c-axis
c-axis
c-axis
In
general:
c-axis
450

x
Jones Matrices
 ei  / 2
W 
 0
0 

ei  / 2 
 ei  / 2
0 
W 
i / 2 
e
 0



cos

2
W 
 i sin 


2

i sin 
2

 
cos 
2 
Remember:

2
 ne  no  d

 ei  / 2
0 
W  R  Y  
R Y
i  / 2 
e
 0

 cos Y  sin Y   ei  / 2 0   cos Y sin Y 
W 



ei  / 2    sin Y cos Y 
 sin Y cos Y   0
Polarizers
Jones Matrices
y
transmission
axis
x
transmission
axis
transmission
axis
450
In
general:
transmission
axis

1
W 
0
0

0
0
W 
0
0

1
 1/ 2 1/ 2 
W 

 1/ 2 1/ 2 
Remember:
 1 0
W  R  Y  
R Y
0
0


 cos Y  sin Y  1 0  cos Y sin Y 
W 



 sin Y cos Y  0 0   sin Y cos Y 
 cos 2 Y
cos Y sin Y 


2
sin
Y
cos
Y
sin
Y


General Matrix For LCD
e – component || director
o – component
director
 sin X

V    cos X  i
2X
 e 
V   
sin X

 o  

X

X  2   
2
2
sin X
X
 sin X
cos X  i
2X


 Ve

 Vo






 Twist angle
 Phase retardation
Adiabatic Waveguiding
• Consider light polarized parallel to the slow axis of a twisted
LC twisted structure:
Ve   1 
  
Vo   0 
 E mode 
• Then, the output polarization will be:
 sin X

V    cos X  i
2X
 e 
V   
sin X

 o  

X






90° Twist
with X 

  
2
2
2
Adiabatic Waveguiding
• Notice that for TN displays since << (twist angle much
smaller than retardation ):

2Dnd


2  0.23  20  m 
0.5  m
 18.4
• Then the output
polarization reduces to:
 sin X

cos
X

i
V  

2X
e

 
V  
sin X


 o 


X


 ei / 2 


0





which means that the electric field vector “follows” the nematic
director as beam propagates through medium – it rotates –
90º Twisted Nematic
(Normal Black)
• Consider twisted structure between a pair of parallel polarizers
and consider e-mode operation.
V  
 e 
V  
 o 
1 0

2 0
 sin X

cos
X

i
0
2X

1 
sin X



X
sin X
X
 sin X
cos X  i
2X


 1

0


0   1
 
0   1
e-mode input
• The transmission after the second polarizer:
2 
2 
sin

1

u
1


T 
2
1 u 2
u

2
 Dnd
2 





2 

Transmission of Normal Black
0.5
u
T (%)
0.4
first
minimum
0.3
u 3
0.2
2d Dn

second
minimum
15
35
0.1
third
minimum
0
0
2
4
6
u
8
10
12
14
Normal White Mode (I)
• Consider twisted structure between a pair of parallel polarizers
and consider e-mode operation.
V  
 e 
V  
 o 
1 1

2 0
 sin X

cos
X

i
0
2X

0
sin X



X
sin X
X
 sin X
cos X  i
2X


 1

0


0   1
 
0   1
e-mode input
• The transmission after the second polarizer:
2 
2 
sin

1

u
1 1


T  
2 2
1 u 2
u

2
 Dnd
2 





2 

Normal White Mode (II)
0.5
35
0.4
T (%)
u 3
15
0.3
0.2
u
2d Dn

0.1
0
0
2
4
6
u
8
10
12
14
n
n
Y-axis
n
E
z 0
X-axis
E
z  d /10
n
E
E
z  4d /10
z  3d /10
n
n
E
n
z  7d /10
E
z  9d /10
zd
z  5d /10
n
E
E
n
z  2d /10
(n)
n
E
z  6d /10
E
E
z  8d /10
Phase Retardation at Oblique
Incidence: Complicating Matters
z
n 1
B D
F
o
C
e
d

A
n 1
Summary of Optics
Vital to understanding LCD’s and their viewing angle solutions:
• Linear, circular, elliptical polarization
• Jones Vector
• Stokes Parameters
• Jones Matrixes
• Adiabatic Waveguiding
• Extended Jones and 4x4 Methods