Review%20of%20Basic%20Polarization%20Optics%20for%20LCDs%20Module%204

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Review of Basic PolarizationOptics for
LCDsModule 4
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Module 4 Goals

• Polarization
• Jones Vectors
• Stokes Vectors
• Poincare Sphere
• Adiabadic Waveguiding

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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.
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Monochromatic Plane Wave (I)
A monochromatic plane wave propagating in
isotropic and homogenous medium
A constant amplitude vector
? angular frequency
is related to frequency
k wave vector
For transparent materials
Dispersion relation
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Monochromatic Plane Wave (II)

• The E-field direction is always ? to the
  direction of propagation

• Complex notation for plane wave (Real part
  represents actual E-field)

• Consider propagation along Z-axis, E-field
  vector is in X-Y plane

Y-axis
EY
X-axis
Ex
independent amplitudes
two independent phases
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Monochromatic Plane Wave (III)

• There is no loss of generality in this case.

• Finally, we define the relative phase as

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Linear Polarization

• In this case, the E-field vectorfollows a
  linear pattern in the X-Y plane as either time
  orposition vary.

Y-axis
AY
X-axis
Ax

• Important parameters
• Orientation
• Handedness
• Extent

Linear polarized or plane polarized are used
interchangeably
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Circular Polarization

• In this case, the E-field vectorfollows a
  circular rotation in the X-Y plane as either
  time orposition vary.

Y-axis
AY
X-axis
Ax

• Important parameters
• Orientation
• Handedness
• Extent

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Circular Polarization
Equation of a circle
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Elliptic Polarization States

• This is the most generalrepresentation of
  polarization. The E-field vector follows an
  elliptical rotation in the X-Y plane as either
  time or position vary.

Y-axis
AY
a
b

• Occurs for all values of

X-axis
Ax

• Important parameters
• Orientation
• Handedness
• Extent of Ellipticity

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Elliptic Polarization States
Transformation
eliminate ?t
x
y
a
b
X-axis
Ax
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d3p/4
dp/2
dp/4
d0
dp/4
dp/2
d3p/4
dp
d3p/4
dp/2
dp/4
d0
dp/4
dp/2
d3p/4
dp
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Review Complex Numbers
Im

• 3 4i
• ei? cos ? i.sin?
• e-i? cos (-?) i.sin (-?) cos ? - i.sin?

-22i
Re
3-4i
Remember the identities
ex ey exy ex / ey ex-y d/dz ez ez
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Complex Number Representation
Polarization can be described by an amplitude and
phase angles of the X-Y components of the
electric field vector. This lendsitself to
representation with complex numbers
Im
Re
on x axis
on y (imaginary axis)
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Jones Vector Representation
Convenient way to uniquely describe polarization
state of aplane wave,using complex amplitudes as
a column vector.
Polarization is uniquely specified
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Jones Vector Representation (II)
If you are only interested in polarization state,
it is most convenient to normalize it.
A linear polarized beam with electric field
vector oscillating along a given direction can be
represented as
For orthogonal state,
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Jones Vector Representation (III)
Normalize Jones Vector
Take
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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 y0 for linear polarized light, the electric
field oscillates along coordinate system, the
Jones Vectors are given by
For circular polarized light
Mutually orthogonal condition
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Polarization Representation
Polarization Ellipse
Jones Vector
(d,y)
(f,q)
Stokes
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Polarization Representation
Polarization Ellipse
Jones Vector
(d,y)
(f,q)
Stokes
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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)

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Partially Polarized Unpolarized Light
We discussed monochromatic/polarization thus far.
If light is not absolutely monochromatic, the
amplitude and relative phase d 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.
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Partially Polarized Unpolarized Light
Consider quasi monochromatic waves (D?ltlt?)
Light can still be described as
Provided the constancy condition of A is
relaxed.
? denotes center frequency
A denotes complex amplitude
Because (D?ltlt ?), changes in A(t) are small in a
time interval 1/Dw (slowly varying).
If the time constant of the detector tdgt1/Dw,
A(t) can change originally in a time interval
td.
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Partially Polarized Unpolarized Light
To describe this type of polarization state, must
consider time averaged quantities.
S0 ltltAx2Ay2gtgt
S1 ltltAx2-Ay2gtgt
S2 2ltltAxAy cosdgtgt
S3 2ltltAxAy sindgtgt
Ax, Ay, and d are time dependent
ltlt gtgt denotes averages over time interval td
that is the characteristic time constant of the
detection process.
These are STOKES parameters.
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Stokes Parameters
Note All four Stokes Parameters have the same
dimension of intensity.
They satisfy the relation
the equality sign holds only for polarized light.
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Stokes Parameters
Example Unpolarized light
No preference between Ax and Ay (AxAy), d
random
S0 ltltAx2Ay2gtgt2ltltAx2gtgt
S1 ltltAx2-Ay2gtgt0
since d is a random function of time
S2,32ltltAxAy cosdgtgt2ltltAxAy sindgtgt0
if S0 is normalized to 1, the Stokes vector
parameter is for unpolarized light.
Example Horizontal Polarized Light Ay0,
Ax1
S0ltltAx2gtgt1
S1ltltAx2gtgt1
S2,32ltltAxAy cosdgtgt2ltltAxAy sindgtgt0
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Stokes Parameters
Example Vertically polarized light Ay1,
Ax0
S0 ltltAx2Ay2gtgtltltAy2gtgt1
S1 ltltAx2-Ay2gtgtltlt-Ay2gtgt-1
S2,3 2ltltAxAy cosdgtgt2ltltAxAy sindgtgt0
Example Right handed circular polarized light
(d-1/2p) AxAy
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Stokes Parameters
Example Left handed circular polarized light
(d1/2p) AxAy
Degree of polarization
Unpolarized S12 S22 S32 0
Polarized S12S22S32 1
useful for describing partially polarized light
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Jones Matrix Method (I)
Y-axis
f

• The polarization state in a fixed lab axis X and
  Y

s
y
X-axis
y

• Decomposed into fast and slow
• coordinate transform

Z-axis
(notation fast (f) and slow (s) component of the
polarization state)
rotation matrix

• If ns and nf are the refractive indices
  associated with the pro-pagation of slow and fast
  components, the emerging beam has the
  polarization state

Where d is the thickness and l isthe wavelength
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Jones Matrix Method (II)

• For a simple retardation film, the following
  phase changes occur

(relative phase retardation)
(mean absolute phase change)

• Rewriting previous retardation equation

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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.

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Jones Matrix Method (IV)

• By combining equations, the transformation due
  to the retarder
• plate is

where W0 is the Jones matrix for the retarder
plate and R(Y) is the coordinate rotation matrix.
(The absolute phase can often be neglected if
multiple reflections can be ignored)

• A retardation plate is characterized by its
  phase retardation G
• and its azimuth angle y, and is represented by

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Examples
Polarizer with transmission axis oriented ?? to
X-axis
Jones Vector
Polarization State
E
Y-axis
f
X-axis
f is due to finite optical thickness of
polarizer.
If polarizer is rotated by y about Z
ignoring f polarizers transmitting light with
electric field vectors ?? to x and y are
a
b
a
b
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Examples
¼ Wave Plate
IncidentJones Vector
Polarization State
Emerging Jones Vector
Y-axis
E
X-axis
f
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Jones Matrices
Wave Plates
y
x
c-axis
Remember
c-axis
c-axis
450
Ingeneral
y
c-axis
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Jones Matrices
Polarizers
y
transmissionaxis
x
transmissionaxis
Remember
transmissionaxis
450
transmissionaxis
y
Ingeneral
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Birefringent Plates
45?
45?
Parallel polarizers
Cross polarizers
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Poincares Representatives Method
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Poincare Sphere
Linear Polarization States
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Poincare Sphere
Elliptic Polarization States
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Polarization Conversion
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Polarization Conversion
Y-axis
s
f
y
X-axis
Z-axis
?
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Some Examples

• TN LCD Formulations

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General Matrix For LCD
e component director o component
director

• Twist angle
• ? Phase retardation

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Adiabatic Waveguiding

• Consider light polarized parallel to the slow
  axis of a twisted LC twisted structure

• Then, the output polarization will be

90 Twist
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Adiabatic Waveguiding

• Notice that for TN displays since fltltG (twist
  angle muchsmaller than retardation G)
• Then the outputpolarization reduces to

which means that the electric field vector
follows the nematicdirector as beam propagates
through medium it rotates
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90º Twisted Nematic (Normal Black)

• Consider twisted structure between a pair of
  parallel polarizers
• and consider e-mode operation.

e-mode input

• The transmission after the second polarizer

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Transmission of Normal Black
first minimum
second minimum
third minimum
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Normal White Mode (I)

• Consider twisted structure between a pair of
  parallel polarizers
• and consider e-mode operation.

e-mode input

• The transmission after the second polarizer

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Normal White Mode (II)
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n
n
n
Y-axis
E
X-axis
E
E
(n)
n
n
E
E
E
n
n
n
E
E
E
n
n
E
E
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Phase Retardation at Oblique Incidence
Complicating Matters
z
D
F
B
C
qo
qe
d
A
q
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Summary of Optics
Vital to understanding LCDs and their viewing
angle solutions

• Linear, circular, elliptical polarization

• Jones Vector

• Stokes Parameters

• Jones Matrixes

• Adiabatic Waveguiding

• Extended Jones and 4x4 Methods