This document describes OpenLCDFDM, an open-source finite-difference LCD simulator. It summarizes the key components of LCD display structure and models used in the simulator, including the birefringence of liquid crystals, calculation of liquid crystal orientation under electric fields using the Oseen-Frank model, solving the Laplace equation in anisotropic media, and using the extended Jones matrix method for optical calculations. The document also outlines the program structure and demonstrates simulation results for different LCD modes. Future work is proposed to improve the modeling capabilities.

1. OpenLCDFDM: an finite-difference LCD
simulator
Zong-han, Xie
icbm0926@gmail.com
June 7, 2015

2. License of this slide
This slide made by Zong-han, Xie (icbm0926@gmail.com) is
licensed under Creative Commons Attribution-NonCommercial 4.0
International License.

3. LCD on consumer products

4. LCD display structure
Liquid crystal layer is served as an optical switch, it determines
how much light from light source (backlight in LCD display) can
propagate through.

5. Polarization

6. Birefringence

7. Birefringence
A material with two refractive index (no and ne) means there are
two refracted ray.
nair sin(θ1) = no sin(θo) (1)
nair sin(θ1) = ne sin(θe)

8. Birefringence
A material with two refractive index means that when light
propagates through this material it sees two light speed.
The other important feature for birefringence is that it causes
phase retardation and the polarization of light is changed.

9. Birefringence
If the light incidents with an angle relative to the optical axis, the
refractive index for extraordinary light is ne(θ).
ne(θ) =
cos2(θ)
no
2
+
sin2
(θ)
ne
2
(2)

10. Electric controlled LC orientation
Due to the nonisotroipic dielectric property of liquid crystal, when
external electric field is applied, the LC molecule will start to align
with the electric field.

11. Calculation models
In order to model such device, one has to model the transition
liquid crystal orientation under electric field and calculate phase
retardation under given LC distribution.
Model LC distribution:
Oseen-frank free energy for liquid crystal
Laplace equation in nonisotroipic media
Optical calculation: Extended Jones matrix method

12. OpenLCDFDM 1D structure

13. Calculate LC Orientation
Using Oseen-frank elastic free energy density to model the elastic
property of the liquid crystal layer. The LC molecule distribution is
acquired by minimizing the total energy density which includes
elastic free energy density and electric field energy density. The
total energy density has the following form [2].
f (x) =
1
2
K11( · n)2
+
1
2
K22(n · × n)2
+
1
2
K33(n × × n)2
+q0K22(n · × n) −
1
2
(D · E)
where D = ←→(x) · E and ←→(x) is a 3X3 dielectric tensor. n(x) is
a unit vector for local liquid crystal orientation.

14. Calculate LC Orientation

15. Calculate LC orientation
To calculate minimize free energy, Euler-Lagrange equation is
applied. OpenLCDFDM uses finite difference method to solve
Euler-Lagrange equation of the total energy density to get liquid
crysal distribution of minimized free energy.
The Euler-Lagrange equation of the free energy is solved through
iterative method.
∂ni
∂t
=
1
γ
−
δf
δni
, i = x, y, z (3)
γ is the rotational viscosity.

16. Laplace equation in nonisotroipic and inhomogeneous
media
Laplace equation:
· ←→(x) · φ(x) = 0
←→(x) is the local dielectric tensor decided by the local oritation of
LC molecule.
←→(x) =


⊥+ sin2
θ(x) cos2 φ(x) 2 sin2θ(x) sin 2φ(x) 2 sin 2θ(x) cos φ(x)
2 sin2
θ(x) sin 2φ(x) ⊥+ sin2
θ(x) sin2
φ(x) 2 sin 2θ(x) sin φ(x)
2 sin 2θ(x) cos φ(x) 2 sin 2θ(x) sin φ(x) ⊥+ cos2 θ(x)


where θ(x) and φ(x) is the local orientation of liquid crystal
molecule, = − ⊥.

17. Extended Jones matrix method for uniaixal media
Under the assumption of ne no, the e-ray and o-ray propagate
on the same direction. With this assumption, Jones matrix can be
modified into extended Jones matrix method.
Jout =
t s 0
0 t p
e · s o · s
e · p o · p
eike z 0
0 eiko z
s · e p · e
s · o p · o
ts 0
0 tp
Jin
t s, t p t s and t p are transmission rate for s wave and p wave
acquired through Fresnel’s euations.

18. Extended Jones matrix method for isotropic mediaand
calculation for transmission rate
For isotropic media, the Jones matrix only needs to calculate
transmission rate for s wave and p wave.
Jout =
ts 0
0 tp
Jin
In above formulas, Jin and Jout are the Jones vectors of light.
Transmision rate for incoherent light source:
M = M1M2M3... =
m11 m12
m21 m22
T = 0.5 ∗ (m2
11 + m2
12 + m2
21 + m2
22)
For multi-wavelength calculation:
T =
T(λ)P(λ)V (λ)dλ
P(λ)V (λ)dλ

19. OpenLCDFDM program structures

20. Setup parameters for FDM solver

21. Setup parameters for optical calculation

22. Calculate and get results

23. Demo: V-T curve of TN mode

24. Demo: Normally white vs. normally black

25. Demo: Conoscopy

26. Demo: TN viewing angle problem

27. Future works
Calculate Stokes values to track change of polarization.
Implement Berreman 4X4 method
Add bixial material support for extended Jones matrix method
Bind OpenLCDFDM with FDTD simulation module(ex.
meep-FDTD)
Move toward 2D and 3D simulation

28. References
Optics of Liquid Crystal Display by Pochi Yeh and Claire Gu.
ISBN: 0470181761
Fundamentals of Liquid Crystal Devices by Shin-Tson Wu and
Deng-Ke Yang. ISBN: 978-0-470-03202-2

29. Errata and supplements
The word ”nonisotropic” in this slide should be ”anisotropic”.
In page Calculate LC orientation , the graph to describe k constants is
referenced from [1].