Download Optics of Liquid Crystals

Optics of LC displays
1
Chap.2
Polarization of optical waves
2
• A monochromatic plane wave is
 
 
E  A cos(t  k  r )
For mathematical simplicity
 

 i (t  kr )

E  A exp[ i (t  k  r )]  Ae
Only the real part represents the actual electric field.
• The polarization state of a light beam is
specified by its electric field.
– Consider a light advances along z axis, then its
electric field vectors must lie in xy plane.
Ex  Ax cos(t  kz   x )
E y  Ay cos(t  kz   y )
3
Linear polarization state
• Consider the time evolution of the electric
filed vector at the origin z=0.
E x  Ax cos(t   x )
E y  Ay cos(t   y )
Define relative phase as
   y x, -    
Linear polarized, or plane polarized if
   y   x  0, or 
Ey
Ex

Ay
Ax
,
or 
Ay
Ax
The vibration of the electric field vector is confined in
a specific plane.
4
• Now examine the space evolution of the
electric field vector at a fixed point in time
(say t=0)
Ex  Ax cos(kz   x )
E y  Ay cos(kz   y )
Linear polarized, or plane polarized if
   y   x  0, or 
Linear polarization is most widely used in optics.
5
Circular polarization state
1
   y x   
2
Ax  Ay
• A beam of light is said to be circularly
polarized if the electric field vector
undergoes uniform rotation in the xy plane.
6
The notation is different from Hecht.
7
2.3 Jones vector representation
• The plane wave is expressed in terms of its complex
amplitude as a column vector [ignore time domain]
 Ax e i x 
J 
i y 
A
e
 y 
• To obtain the real x component of the electric field,
we must perform the operation.

E x (t )  Re J x e
it
  ReA e
x
i (t  x )

8
• If we are interested only in the polarization state of
wave, we will use the normalized Jones vector that
satisfy the condition.
J  J*  1
• Thus, a linearly polarized light with electric field
oscillating along a given direction is
y
cos 
 sin  



x
9
10
右
左
Hecht representation
11
Yeh representation
4. Jones Matrix Method
• A powerful technique.
• In LCDs, the orientation of the director depends
on the applied voltage and the boundary condition.
• The polarization state of the input light beam can
be converted to any other polarization state by
means of a suitable retardation plate, or LC plate.
• Assume there is no reflection of light from either
surface of the plate
12
• Two coherent P-states in the principal coordinate
system, some how caused to have a phase lag.
光程差   d( n e  n o ),
相位差   k 0  
z = Slow axis
Optic axis
E //

E //
E
(8.31)
2
d( n e  n o ),
0
(8.32)
ne = n3

y
E
E
no
LL
x = Fast axis
A retarder plate. The optic axis is parallel to the plate face. The o- and e-waves travel
in the same direction but at different speeds.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
13
Slow axis (LC director, C axis, ne axis)
Fast axis
Lab axis
• Incident linear polarized light described by Jones
vector
V 
V  
Vy 
x
• Vx , Vy are complex numbers, x and y axes are fixed lab. axes.
• Ψis the angle between the s-f coordinate and the x-y coordinate,
with the z axis as the axis of the coordinate rotation.
14
• Decompose the light into the linear combination of the
fast and slow normal modes in the LC cell. This is done
by coordinate transformation.
Vs   cos
V   
 f   sin 
sin   Vx 
Vx 
V   R( ) V 

cos   y 
 y
• After entering the medium, due to the difference in
phase velocity, one component is retarded relative to
the other.
• The retardation changes the polarization state of the
emerging beam.
15
The polarization state of a light
can be converted to any other
polarization state by suitable
LC plate.
16
• The polarization state of the emerging beam
in the medium s-f coordinate system is
 V
0  s 
2
d  V 
in f
e   f 
2
1
2   i ( n s  n f ) d
1

2
i ( n s  n f ) d
e
 
e 2

0

2
in s d

 V's  e 
V'   
 f
 0
 V
 s
0

2
1
i ( n s  n f ) d  V 

 f
e2

17
• Define the phase retardation

2

(ns  n f )d
which is the relative phase change due to propagation, not
the absolute change.
• The mean absolute phase change is defined as
1
2
  ( ns  n f )
d
2

18
• The birefringence of a typical retardation is
small, that is
ns  n f  ns , n f
For LCs, ns~1.74, nf ~1.52
ns-nf ~ 0.22
the absolute change >> the relative phase retardation
19
• The polarization state of the emerging beam can
be rewritten as

i

V 's  i e 2


e
V ' 

 f
 0
 V 
0 s
 

i  V
e 2  f 
• The polarization state of the emerging beam in the
x-y coordinate is by transforming back from the s-f
coordinate system
V ' x  cos
V '   
 y   sin 
 sin   V 's 
V ' 

cos   f 
20
• Finally, the polarization state of the emerging
light in the lab axes is
V ' x 
Vx 
V '   R( )W0 R( ) V   WV
 y
 y
R() is the coordinate rotation matrix and W0 is the Jones
matrix for the retardation plate (LC plate) in the s-f coordinate
 cos
R( )  
 sin 
sin  
cos 
 i 2
W0  e i e

 0

0

i 
e 2
The phase factor e-i can be neglected if interference effects
due to multiple reflection are not important, or not observable.
21
• The Jones matrix of a retardation plate is
characterized by its phase retardation  and its
azimuth angle , and is represented by the product
of three matrices
W  R( )W0 R( )
22
• Coordinate transformation is included.
23
• The Jones matrix of a linear polarizer orientated
with transmission axis parallel to the lab x axis
P0  e
i '
1 0
0 0 


‘ is the absolute phase
• Neglecting the absolute phase ‘, the matrix
representations of the polarizer’s transmitting axis
parallel to the x or y axes, is
1 0
Px  

0
0


0 0 
Py  

0
1


24
• The transmission axis of a polarizer has an angle
 with respect to lab x axis is
P  R( ) P0 R( )
25
26
Example: a half wave retardation plate

2
(ns  n f )d  

0 
• Incident light, vertically polarized, V   
• Azimuthal angle =45°
• Jones matrix for the half wave plate
1
1 1  1  i 0 1  1 1  0 i 
W
1 1   0 i 
 1 1   i 0
2

 2
 

• The result is
 i 
1
V '  WV     i  
0
0 
W  R( )W0 R( )
horizontal polarized light
27
Regardless of the azimuth angle
28
Example: a quarter wave retardation plate

2
( ns  n f ) d   / 2

• Incident light, vertically polarized
0 
V  
1
• Azimuthal angle =45°
• Jones matrix for the quarter wave plate
1 1  1 e i / 4
W
1 1  
2
 0
0  1  1 1 1  1  i 
 1 1 
 i 1 
i / 4 
e  2
2



• The result is
1  i   i 1 Left handed circularly
V '  WV 
1

2 
2 i  polarized light
29
30
Intensity transmission spectrum
• Incident light
• Incident light
  Ex 
E 
Ey 
 
2
2
I  E  E  Ex  Ey
• Emerging light
  E 'x 
E'   
E' y 
• Emerging light
E' x  E' y
2
T
Ex  Ey
2
2
2
31
A birefringent plate between
parallel polarizers
y-axis
y-axis
x-axis
V
LC plate
V’
V '2
T 2
V
32

2

(ns  n f )d
• The corresponding Jones matrix is
W  R( )W0 R( )
cos

 sin 

i

 sin   e 2


cos  
 0
 cos
0 

i   sin 
e 2 


sin    cos 2


cos   i sin 
2


 i sin 
2
 
cos 
2 
• The incident light be unpolarized, thus when it passes
through the first polarizer

1 0 
E
1
2 
Assume the intensity of the incident
light is unity and only half of the
intensity passes through the polarizer
33
• The emerging beam
analyzer
0
E'  
0
1

2


0  cos 2


1  i sin 
2

 0 


cos 2 
Birefringent plate

 i sin  1 0
2
  2 1
cos 
2 
After pass the 1st polarizer
1
1
2 
2   ( ne  no ) d 
I  cos
 cos 

2
2 2



The transmitted light is y polarized, because the light
goes through a polarizer.
34
A birefringent plate between
crossed polarizers
1
E'  
0
i

2


0  cos 2


0  i sin 
2

 
sin 2 
 0 



 i sin  1 0
2
  2 1
cos 
2 
1 2  1 2   (ne  no )d 
I  sin
 sin 

2
2 2



The transmitted light is x polarized
35
Parallel Aligned Cells
x
y
36
A birefringent plate between a
pair of polarizers Homework
• The analyzer forms an angle  with x axis



cos

i
sin
 cos 
 1 0 
cos sin   
2
2
E'  


1
2


cos

sin

sin

2
 

  i sin
cos 
2
2 



 i cos sin  sin  cos cos 
2
2

 sin  
2


2
1
1 2
2
2 
2 
I  cos  sin
 sin  cos
2
2 2
2

2

(ne  no )d
37
Optical properties of a twisted
nematic liquid crystal (TN-LC)
• The orientation of the director is a function of
position. Thus, the director is twisted.
• Subdivide the medium into a large number of thin
plates. Each of the thin plate is approximated by a
homogeneous medium.
• Assume the twisting is linear and the azimuth
angle is
 ( z )  z
 is a constant
38
Twisted Nematic Liquid Crystal Displays:
Normally White (e-mode)
39
• Γis the phase retardation of the plate when it is untwisted. The
director is parallel to the surface plate
2
d is cell thickness

(ne  no )d

• The total twist angle is
   ( d )  d
• Divide the cell into N equally thin plate. Each plate has a phase
retardation of =/N. The plate are orientated at a azimuth angle
, 2, 3, ……(N-1), N, with =/N.
• The overall Jones matrix for these N plates is given by
N
N
m 1
m 1
M  WNWN 1.....W3W2W1   Wm   R(m )W0 R(m )
R is the coordinate rotation matrix and Wm is the Jones matrix
for the mth plate, W0 is
 i 1 

2N
e

W0 

 0
0 
1
i

e 2N 
40
• Using =/N and R(1)R(2)=R(1+2), the overall Jones
matrix can be written as
 

M  R( ) W0 R( )
N 

N
• Finally, the overall Jones matrix becomes
  i / 2 N

 cos N e
M  R( ) 
 i / 2 N
 sin e
N

sin
cos

N

N
e
 i / 2 N
e i / 2 N





N
41
• Further simplified and in the limit when N ∞
R ()
cos 
M 
 sin 
 sin X

 sin   cos X  i 2 X

sin X
cos   

X

sin X



X
 sin X 

cos X  i
2 X 
 2
where X    ( )
2
• M is the exact expression for the Jones matrix of a linearly
twisted nematic LC plate.
• V: initial polarization, V’: polarization state after exiting the
plate
2
 V’=MV, and T=(V’/V)2=M2,
42
• It is often useful to examine the polarization states
in the local principal coordinate system.
• In the local principal system, the e component is
along the direction of the director, and the o
component is perpendicular to the director. The
results can be written as
 sin X

V 'e  cos X  i 2 X
V '   
sin X
 o 

X

sin X


 Ve 
X
 sin X  Vo 

cos X  i
2 X 
43
Adiabatic following
(waveguiding in TN_LCD)
44
• The polarization of the input light is parallel
to c axis (the director) of LCs in the
entrance plane E-mode operation
• In the principal coordinate system, the
Jones vector of the input beam is
Ve  1
V   0
 o  
45
• In the principle coordinate system, the
Jones vector of the output beam is
 sin X
sin X



 1
V 'e  cos X  i 2 X
X
 
V '   
sin
X

sin
X
 o 
 0 

cos X  i
X
2 X 

 sin X 

cos X  i 2 X 


(4-3-15)
sin X



X



X   2  ( )2
2
46
• In TN_LC, if φ << Γ
– Ex: LC cell of E7 with 20mm, twist angle /2.
Dn=0.23.
2

(ne  no )d
– / Γ =1/37 at =500nm.

• The 2nd term (o-component) in eq. (4.3-15)
is near zero, thus, the output Jones vectors
is approximated as
i
 

V 'e  e 2

V '   
 o   0 
47
• The electric field vector of the beam remain
parallel to the local director as the beam
propagates in the twisted nematic LC medium.
– The same in the O-mode operation.
– Adiabatic following or waveguiding phenomenon.
• Adiabatic following occurs if the incident
beam is polarized either parallel or
perpendicular to the director at the entrance
plane, and the twist rate is small.
• Strictly, the polarization state in the medium is
elliptical as shown in the figure.
48
49
90º twisted nematic liquid
crystal-1
• 90º twisted nematic liquid crystal sandwiched
between a pair of parallel polarizers.
– So called normally black(NB) configuration.
• We consider the e-mode operation, thus the
input Jones vector is
Ve  1
V   0
 o  
50
• The transmission axis of the second polarizer is
perpendicular to the local director at the exit
plane.
• According to eq(4-3-15), the output intensity is

2
2
2
sin
X
sin

1

u
T  2

2
2
X
1 u


X   2  ( )2
2
Where ψ is the twist angle of the cell, u is the so-called
Mauguin parameter
 2
u
 (ne  no )d
2 
( 

2
)
The transmission is near 0 for LCs with ψ<<
51
90º twisted nematic liquid
crystal-2
• 90º twisted nematic liquid crystal sandwiched
between a pair of parallel polarizers.
– So called normally black(NB) configuration.
• We consider the o-mode operation, thus the
input Jones vector is
Ve  0
V   1
 o  
52
• The output polarization state in terms of the Jones
vectors in the local principal coordinate is
 sin X
sin X



 0 
V 'e  cos X  i 2 X
X
 
V '   
sin
X

sin
X
 o 
 1

cos X  i
X
2 X 

sin X



0


X

 sin X 
cos X  i

2 X 

Again, the transmission is near 0 for LCs with ψ<<
sin X

0
X
53
Transmission properties of a
general TN-LCD
b

54
The input and the output polarization, as determines
by the orientation angles of the polarizer
transmission axes, are
Vx  cos  ent 
V   

 y   sin  ent 
V ' x  cos  exit 
V '   

 y   sin  exit 
The transmission of the system, according to Jones
matrix is
T  V '* MV
2
V and V’ are the input and output Jones vectors
55
• After some steps
T  cos 2 (   exit   ent )
 sin 2 X sin 2(   exit ) sin 2 ent


sin 2 X sin 2(   exit   ent )
2X
2
sin
X
2

cos 2(   exit ) cos 2 ent
2
X
• In various forms, define
   ent
b   exit  
56
• The transmission become
T  cos 2 (  b )  sin 2 X sin 2b sin 2

2
sin
X
2

sin 2 X sin 2(  b )  
cos 2 cos 2b
2
2X
X
• Also, it can be written as
T  cos 2 (  b )


 cos X cos 2b cos 2 [ tan X  tan 2 ][ tan X  tan 2b ]
X
X
2
This expression is particular useful in the design of TNLCDs or STN-LCDs
57
4.4 phase retardation at oblique incidence
• In general, two independent modes of propagation
given a direction of propagation in an anisotropic
medium.
– Ordinary and extraordinary (mutual orthogonal).
– O-mode is independent of the propagation direction, but emode depends on the incident direction.
– Thus, phase retardation depends on the incident direction.
• Consider an a plate, c-axis (director) parallel to the
surface plate. homogeneous LC cell.
– For normally incident, the phase retardation is:   2 (n

– A general expression is   (kez  koz )d
– Such general expression can be derived as follows.
e
 no )d
58
59
1. Wave approach
• Given an arbitrary incident plane wave, both
ordinary and extraordinary waves are generated in
the medium. The electric field amplitude can be
written as
E  Ee exp[ i(x  by  kez z )]  Eo exp[ i(x  by  koz z )]
– Tangential components (, b) are the same due to
continuity condition at the boundary.
– Thus, the phase difference between the two modes is
  (kez  koz )d
60
2. Ray approach
• Referring to figure 4.10, the phase retardation is
  kne ' AB  k BD  kno AC
• Using the continuity condition (snell’s law) and
simple trigonometry.
sin   ne ' sin e  no sin o
• We obtain
  k[ne ' cose  no coso ]d
Which is exactly
  (kez  koz )d
61
4.5 conoscopy
• A conoscopy is a polarizing microscope
designed to examine the transmission
property of a sample plate at various
directions of incidence simultaneously.
• For uniaxial media, a conoscope can be
used to determine the orientation of the caxis (director).
62
• Employing a convergent or
divergent light as the source,
which consists of many
plane waves over a cone of
solid angle.
• A lens converts each
component of the wave into
a point at the focal plane.
• The transmission
sandwiched between
polarizers depends on the
sample orientations and the
angle of incidence.
63
4.5.1 a plate of uniaxial crystals
For small incident angle <<1
  (ke z  ko z )d
 sin 2 
 0 1 
2
2
n
o

 no no  ne

 
cos 2  
ne
 ne

0  2 (ne  no )d / 
Γ0 : phase retardation at normal incidence.
For small
birefringence,
64
• If the phase retardation is
2m (full wave), the
emerge beam at the exit
face have the same
polarization as that at the
entrance face.
• This beam will be blocked
by the second polarizer
and leading to a dark
fringe at the focal plane.
65
4.5.2 c plate of uniaxial crystals
• At normal incidence, the wave
propagating along the c-axis. Both
components of the wave are
propagating at exactly the same
phase velocity, the polarization
state remain unchanged, the
transmission is zero.
• At a general incident angle, the
phase retardation is introduced
between the two modes. Since the
symmetry of the structure, the
phase retardation is a function of ,
forming a series of dark and bright
rings.
Provided <<1
66
67