OpenLCDFDM: an finite-difference LCDsimulator

Zong-han, Xieicbm0926@gmail.com

June 7, 2015

License of this slide

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

LCD on consumer products

LCD display structure

Liquid crystal layer is served as an optical switch, it determineshow much light from light source (backlight in LCD display) can

propagate through.

Polarization

Birefringence

Birefringence

A material with two refractive index (no and ne) means there aretwo refracted ray.

nair sin(θ1) = no sin(θo) (1)

nair sin(θ1) = ne sin(θe)

Birefringence

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

Birefringence

If the light incidents with an angle relative to the optical axis, therefractive index for extraordinary light is ne(θ).

ne(θ) =

√cos2(θ)

no2+

sin2(θ)

ne2(2)

Electric controlled LC orientation

Due to the nonisotroipic dielectric property of liquid crystal, whenexternal electric field is applied, the LC molecule will start to alignwith the electric field.

Calculation models

In order to model such device, one has to model the transitionliquid crystal orientation under electric field and calculate phaseretardation under given LC distribution.

I Model LC distribution:I Oseen-frank free energy for liquid crystalI Laplace equation in nonisotroipic media

I Optical calculation: Extended Jones matrix method

OpenLCDFDM 1D structure

Calculate LC Orientation

Using Oseen-frank elastic free energy density to model the elasticproperty of the liquid crystal layer. The LC molecule distribution isacquired by minimizing the total energy density which includeselastic free energy density and electric field energy density. Thetotal energy density has the following form [2].

f (~x) =1

2K11(∇ · ~n)2 +

1

2K22(~n · ∇ × ~n)2 +

1

2K33(~n ×∇× ~n)2

+q0K22(~n · ∇ × ~n)− 1

2(~D · ~E )

where ~D =←→ε (~x) · ~E and ←→ε (~x) is a 3X3 dielectric tensor. ~n(~x) isa unit vector for local liquid crystal orientation.

Calculate LC Orientation

Calculate LC orientation

To calculate minimize free energy, Euler-Lagrange equation isapplied. OpenLCDFDM uses finite difference method to solveEuler-Lagrange equation of the total energy density to get liquidcrysal distribution of minimized free energy.The Euler-Lagrange equation of the free energy is solved throughiterative method.

∂ni∂t

=1

γ

(− δfδni

), i = x , y , z (3)

γ is the rotational viscosity.

Laplace equation in nonisotroipic and inhomogeneousmedia

Laplace equation:

∇ ·←→ε (~x) · ∇φ(~x) = 0

←→ε (~x) is the local dielectric tensor decided by the local oritation ofLC molecule.

←→ε (~x) =

ε⊥+ M ε sin2 θ(~x) cos2 φ(~x) Mε2 sin2θ(~x) sin 2φ(~x) Mε

2 sin 2θ(~x) cosφ(~x)Mε2 sin2 θ(~x) sin 2φ(~x) ε⊥+ M ε sin2 θ(~x) sin2 φ(~x) Mε

2 sin 2θ(~x) sinφ(~x)Mε2 sin 2θ(~x) cosφ(~x) Mε

2 sin 2θ(~x) sinφ(~x) ε⊥+ M ε cos2 θ(~x)

where θ(~x) and φ(~x) is the local orientation of liquid crystalmolecule, M ε = ε‖ − ε⊥.

Extended Jones matrix method for uniaixal media

Under the assumption of ne u no , the e-ray and o-ray propagateon the same direction. With this assumption, Jones matrix can bemodified into extended Jones matrix method.

~Jout =

[t ′s 00 t ′p

] [~e ·~s ~o ·~s~e · ~p ~o · ~p

] [e ikeMz 0

0 e ikoMz

] [~s · ~e ~p · ~e~s · ~o ~p · ~o

] [ts 00 tp

]~Jin

t ′s , t ′p t ′s and t ′p are transmission rate for s wave and p waveacquired through Fresnel’s euations.

Extended Jones matrix method for isotropic mediaandcalculation for transmission rate

For isotropic media, the Jones matrix only needs to calculatetransmission rate for s wave and p wave.

~Jout =

[ts 00 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 ∗ (m211 + m2

12 + m221 + m2

22)

For multi-wavelength calculation:

T =

∫T (λ)P(λ)V (λ)dλ∫

P(λ)V (λ)dλ

OpenLCDFDM program structures

Setup parameters for FDM solver

Setup parameters for optical calculation

Calculate and get results

Demo: V-T curve of TN mode

Demo: Normally white vs. normally black

Demo: Conoscopy

Demo: TN viewing angle problem

Future works

I Calculate Stokes values to track change of polarization.

I Implement Berreman 4X4 method

I Add bixial material support for extended Jones matrix method

I Bind OpenLCDFDM with FDTD simulation module(ex.meep-FDTD)

I Move toward 2D and 3D simulation

References

Optics of Liquid Crystal Display by Pochi Yeh and Claire Gu.ISBN: 0470181761

Fundamentals of Liquid Crystal Devices by Shin-Tson Wu andDeng-Ke Yang. ISBN: 978-0-470-03202-2

Errata and supplements

I The word ”nonisotropic” in this slide should be ”anisotropic”.

I In page Calculate LC orientation , the graph to describe k constants isreferenced from [1].