Transcript 2. Electro-optics of Nematic Liquid Crystals

Liquid Crystal Optics and
Electro-Optics
Chang-Kui Duan
2016/4/4
Chang-Kui Duan, Institute of
Modern Physics, CUPT
Introduction





Most studied & applied properties: light-scattering ability
externally applied field control or realign the anisotropic
liquid crystal axis, thereby controlling the effective
refractive index and phase shift
form the basis for various optical transmission, reflection,
switching, and modulation applications.
LCs are noted for their large birefringence and easy
susceptibility to external field perturbation.
basic principles and seek only some general
understanding by dealing with analytically or conceptually
solvable cases.
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LCD pixel

Schematic of a
typical liquid
crystal display
pixel consisting of
electronic driving
circuit, polarizers,
liquid crystal cell,
color filter, and
phase plate
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Content
0. Introduction
1. Electro-Optics of Anisotropic and Birefringent
Crystals
2. Electro-Optics of Nematic Liquid Crystals
3. Nematic Liquid Crystal Switches and Displays
4. Electro-Optical Effects in Other Phases of Liquid
Crystals
5. Nondisplay Applications of Liquid Crystals
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1. Electro-Optics of Anisotropic and
Birefringent Crystals
①
②
③
④
Anisotropic, Uniaxial, and Biaxial Optical Crystals
Index Ellipsoid in the Presence of an Electric
Field: Linear Electro-Optics Effect
Polarizers and Retardation Plate
Basic Electro-Optics Modulation
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Permittivity tensor （介电张量）
The polarization and dielectric displacement are
now given by
Pi   0  ij E j
Di  Pi   0 Ei   0 ( ij   ij ) E j
(i , j  x , y , z )
The elements of the
permittivity tensor
depend on the choice
of coordinate system
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Principal axes
A coordinate system can be found such that the tensor
is diagonal i.e.
2

D
n
 x
x
D     0
0
 y
D 
0
 z

0
n 2y
0
0   Ex 
 
0  Ey 
nz2   E z 
• This coordinate system define the principal axes and
principal planes associated to the crystal.
• The corresponding refractive indexes are known as
principal indexes.
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Biaxial, Uniaxial & isotropic crystal
Crystals with three different principal refractive
indexes are referred to as biaxial crystals
 Crystal with two different principal refractive
indexes are referred to as uniaxial crystals
 For uniaxial crystals, the refractive indexes are
nx=ny=no, and nz=ne where “o” stands for ordinary
axis and “e” for extraordinary axis.
 If no>ne the crystal is said to be a positive uniaxial
crystal

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Uniaxial crystal
such as nematic liquid crystal
n1 = n2 = no, ordinary ray; n3 = ne extraordinary ray
 index ellipsoid

x2 y2 z2
 2  2 1
2
n x n y nz
The ellipsoid in (x, y, z) intersect the axis at
x = ±nx; y = ±ny; z = ±nz
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For light propagate along direction k
The direction of D is in the plan
perpendicular to k.
Ordinary wave: Do perpendicular
to the z-k plane
no()  n0
Extraordinary wave:
De in the z-k plane but
perpendicular to k
 cos  sin 
1


2
2
ne ( )
n0
ne2
2
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k
Uniaxial crystals (cont’)
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Presence of an Electric Field:
Linear Electro-Optics Effect
In the presence of an applied field, the index
ellipsoid becomes:
(1/n2)i are dependent on the applied field E.
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Linear optical effect
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Examples
For a widely used electro-optics crystal such as
lithium niobate (LiNbO3), r33 = 30.8 (in units of 10-12
m/V), r13 = 8.6,r22 = 3.4, and r42 = 28, with ne = 2.29
and no = 2.20 (at 550 nm).
 For these values of electro-optics coefficients (10-11
m/V), an applied dc voltage of 10,000 V is needed
to create a phase shift of in a crystal of centimeter
length.
 liquid crystal electro-optics devices, the typical ac
voltage needed is around 1 V and the liquid crystal
thickness is on the order of a few microns

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Polarizers and Retardation Plate
Typical electro-optic modulation scheme with polarizer–
analyzer sandwiching an electro-optics crystals and a
retardation plate.
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Linear and circular polarizers
Linear polarizers are usually made of anisotropic
absorbing materials in which the absorption along
a crystalline axis is much stronger than the
orthogonal axis
 Circular polarizers are usually made by putting in
tandem(串连) a linear polarizer and a birefringent
retardation (相位延迟) plate, with the polarization
vector bisecting the so-called fast and slow axes
of the retardation plate

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Polarization of output light
Various states of polarization resulting from the
addition of two orthogonal components of a polarized
light with a relative phase shift.
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Basic Electro-Optics Modulation
For A is oriented at
45° with respect to
the crystalline axes


At the exit plane of the crystal of length l
crystal   x   y  2
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( n y  nx )l
0
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Basic Electro-Optics Modulation
 =  crystal + phase
shift by retardation
plate.
E x (l )  A exp( t  k x l )
E y (l )  A exp( t  k x l  crystal )
By summing the components of Ex and Ey on the
transmission axis of the output polarizer (along y)


 

2


I  E y (d )  E x i   E y j  . j  Ai sin   
  2 
2
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2
2. Electro-optics of Nematic Liquid
Crystals
In general, the distortions on the electronic wave
function of liquid crystal molecules caused by an
applied field do not cause appreciable change to
its contribution to the refractive indices
 However, the orientation of the molecules can be
dramatically altered by the applied field
 principal mechanism used in liquid-crystal-based
electro-optical devices.

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Dual-Frequency Liquid Crystals
transparent conductor ITO to allow the
application of an electric field across the cell
AC instead of DC:
Avoid current flow,
degration
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ne  no
along E
ne  no
away E
Mixing and doping
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Dual-frequency liquid crystal

Since the dielectric anisotropy is frequency
dependent (cf. Fig. 3.5), one could create a
mixture of liquid crystals with different dielectric
dispersions such that the resulting so-called dualfrequency liquid crystal (DFLC) possesses an
effective positive anisotropy at one frequency of
the applied ac electric field, but possesses a
negative anisotropy at another ac frequency.
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Freedericksz Transition Revisited
Geometry for observing
(a) the S (splay)
deformation, (b) the B
(bend) deformation, and
(c) the T (twist)
deformation.
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Case 1: One-elastic-constant
approximation.
Standard variation method:
d 2
 E2
K 2 
sin  cos  0
dz
4
1/ 2
2

d 
E
 c 
cos(2 ) 
dz 
8 K



d
E
  c 
cos(2 ) 
0
2
8 K


 E2
c
cos(2 m )
8 K
m
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作业：
Reminder:
(2/d)=m
d/dz | z=2/d =0!
1/ 2
d
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Solution
For relatively small reorientation angles
only if E > EF
 4 K 
VF  EF d   

  
1/ 2
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Case 2: Freedericksz transition voltage
including elastic anisotropies.
 4 K11 
VF  EF d   

  
1/ 2
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Case 3: Freedericksz transition voltage
including elastic conductivity.
The maximum reorientation angle m is described by
 4 K11 
VF   
 ， =  K 33  K11  K11
  
1/ 2
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Field-Induced Refractive Index Change
and Phase Shift
Director axis reorientation
profile in the cell at
various applied voltage
above the Freedericksz
transition.
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Phase shift for light passing through
Approximation:
Twisted configuration with maximum angle 900
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current liquid crystal display devices:
twisted configuration.

Tilting and unwinding of the director axis of a 90°
twisted nematic liquid crystal cell under the action
of an applied field.
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3. NEMATIC LIQUID CRYSTAL
SWITCHES AND DISPLAYS
To obtain higher resolution, faster response, wider field
of view, larger display area, and more functions in
each display pixel.
 Two types: transmissive and reflective
 make use of the polarizing and birefringent properties
 conjunction with polarizers and phase (retardation)
plates
 broadband (from near UV to far infrared)
birefringence, and transparency
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A twisted nematic liquid-crystal switch.
(a) When
the electric
field is absent, the
LC cell acts as a
polarization rotator;
the light is transmitted.
(b) When the electric
field is present, the
cell’s rotatory power
is suspended and
the light is blocked.
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Liquid Crystal Switch: On-Axis
Consideration for Twist,
Planar, and Homeotropic Aligned Cells
normally black (NB) mode: two parallel polarizers
 normally white (NW) mode: two orthogonal
polarizers

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Off-Axis Transmission, Viewing Angle,
and Birefringence Compensation
Has to be considered for display application
 transmission function T is now a function of many
variables
Example: NB mode
 For on-axis light, the initial transmission is 0.
When the voltage is on, the transmission is at a
maximum for the on-axis light
 for the off-axis light, the e and o waves will pick up
an extra phase shift because of the extra optical
path length

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Birefringent compensation film

to place a birefringent film (of opposite anisotropy
to that of the liquid crystal) adjacent to the LC film
limiting case of = 0
compensation film should have birefringence of
opposite sign to that of the liquid crystal
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Sophisticated treatment
For arbitrary angle  or director axis angular
and spatial distributions, and more complicated
cell structure, the phase shift, and therefore the
transmission of light through the cell and other
accompanying polarization selective elements, is
not amenable to simple analytical treatment. More
sophisticated Jones matrix methods or numerical
technique such as the finite difference time domain
(FDTD) numerical methods discussed in the next
chapter are needed to solve such a complex
propaga- tion problem.
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Liquid Crystal Display Electronics
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Optical modulation of LCD
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4. Electro-optical Effects in Other
Phases of Liquid Crystals
nematics are the most extensively used
 other phases (smectic, cholesteric, etc.) and
‘‘mixed systems’’ capable of field-induced
reorientation have also been employed for electrooptical studies and applications
 ferroelectric liquid crystals, generally switch faster
than nematic cells

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Surface Stabilized FLC
Ferroelectric liquid crystal under an applied field,
Typical values:
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Approximation

Under the assumption that e is appreciable, the
first term can be neglected:
solution
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An practical case
: tilt angle
phase retardation :
 = 2dn/
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Soft-Mode FLCs
• SMFLCs use
changes in the tilt 
while  remains
constant. capable
of continuous
intensity change
• SMFLCs employ
smectic-A* phase
• experimental setup
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5. NONDISPLAY APPLICATIONS OF
LIQUID CRYSTALS
extremely broad spectral range (from near UV to
far infrared and into the microwave regime).
 fluid nature and compatibility with most
optoelectronic materials
 a whole host of tunable lens, filters, switches, and
beam/image processing devices have emerged.
 good candidates for biochemical sensing
applications due to organic nature
 light emitting diodes and electroluminescence
devices

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LC Spatial Light Modulator
A typical optically
addressed liquid
crystal spatial light
modulator
(OALCSLM)
operating in the
reflective mode
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Tunable Photonic Crystals with LC
Infiltrated Nanostructures
Photonic crystals in 1-, 2- and 3D forms made of
various optoelectronic materials
 photonic crystals can function as tunable filters,
switches, and lasing devices
 optical holography offers a quick one-step process
for the fabrication of photonic crystals (limited)

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Tunable Frequency Selective Planar
Structures
Transmission
Unit cell of an all-dielectric
polarization independent
FSS for operation in the
visible region as a stopband filter.
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Covered with LC
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Liquid Crystals for Molecular Sensing
and Detection
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Beam Steering, Routing, and Optical
Switching and Laser Hardened Optics
Although most optical elements involve low level
light, liquid crystals are actually excellent laserhardened materials capable of handling very
intense pulsed lasers or high power continuous
wave cw lasers.
 Intensity 1010 W/cm2
 liquid crystals also do not suffer any
structural/chemical damages.

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