Twisted Nematic Liquid Crystal Display Simulator

Transmittance

T(V)

99.8% Max transmission

Electric Field

E_z

0.00 V/µm

V / V_th = 0.00

Mauguin Waveguide Factor

1.00 η(V)

Waveguide active (90° rotation)

Exit Polarisation Angle

90.0° vs Analyser (90°)

Parallel to analyser (Pass)

Optical Layers Y-Axis (Beam) ↑

Top Analyser (90° / Z)

90° Grooved Top Plate

Nematic Directors (n)

0° Grooved Bottom Plate

Bottom Polariser (0° / X)

Backlight Illuminator

Polarisation Vector (±E)

Applied Voltage (

V

) 0.00 V

Controls the electric field $E_y = V/d$ across transparent ITO electrodes.

0 V (OFF) V_th ≈ 1.62 V 10 V (Full ON)

Nematic Cell Parameters

Cell Thickness (

d

) 5.0 µm

Birefringence (

Δn = n_e - n_o

) 0.120

Light Wavelength (

λ

) 550 nm (Green)

Surface Anchoring Condition

Visualization Options

Show Light Wave Vectors (

\mathbf{E}_\text{opt}

)

Show Grooves on Alignment Plates

De-emphasize LC Molecules (Ghost Mode)

Light Pulse Propagation Speed 1.0x

Twisted Nematic (TN) Architecture

A Twisted Nematic liquid crystal cell comprises rod-like calamitic molecules confined between two glass substrates treated with uniaxially rubbed polyimide alignment layers whose rubbing directions are mutually orthogonal (0° and 90°). The cell is positioned between two crossed polarizers whose transmission axes match the respective rubbing directions.

1. Field-OFF State (Bright / Transmissive)

In the absence of an external electric field ( $V = 0$ ), elastic twist energy minimization forces the molecular director $\mathbf{n}(z)$ to execute a continuous 90° helical twist along the cell normal:

ϕ (z) = π 2 (z d), θ (z) = 0

Incoming light polarized along the 0° (x-axis) enters the medium. Provided the Mauguin waveguide condition is satisfied:

2 d Δn ≫ λ \text{or} u = π d Δn λ ≫ 1

the light wave remains an eigenmode of the local birefringent medium. Its polarization plane rotates adiabatically through 90°, exiting aligned with the top analyzer at 90°, resulting in maximum transmission (bright state).

2. Field-ON State (Dark / Extinction)

When a potential difference $V$ is applied across the transparent Indium Tin Oxide (ITO) electrodes, an electric field $\mathbf{E} = E_y \hat{\mathbf{y}}$ forms along the light propagation axis. Because the nematic liquid crystal possesses positive dielectric anisotropy ( $Δε = \varepsilon_\parallel - \varepsilon_\perp > 0$ ), the electric field exerts an electrostatic torque:

\mathbf{\Gamma}_e = ε0 Δε ( \mathbf{n} · \mathbf{E} ) ( \mathbf{n} × \mathbf{E} )

Above the Freedericksz threshold voltage ( $V_\text{th} \approx 1.62\,\text{V}$ ), directors in the bulk tilt homeotropically toward $\theta(z) \to 90^\circ$ (parallel to the beam). Normal-incidence light experiences zero in-plane birefringence ( $\Delta n_\text{eff} \approx 0$ ).

Without in-plane birefringence, the waveguide condition collapses. The light does not twist and remains polarized at 0° (x-axis) across the entire cell. Upon striking the crossed top analyzer (fixed at 90° / z-axis), it is extinguished according to Malus' law:

I = I0 \cos2(90^\circ - ψ\text{exit}) = I0 \cos2(90^\circ) = 0

3. Gooch–Tarry Optical Transmission

The analytical transmission $T$ of a 90° twisted nematic cell between crossed polarizers is governed by the Gooch–Tarry formula:

T = 1 - \sin2(√1 + u2 \frac{π}{2}) 1 + u2

Transmission reaches unity whenever the Mauguin parameter $u = 2d\Delta n/\lambda$ corresponds to the Gooch–Tarry minima: $u = \sqrt{4m^2 - 1}$ for integers $m = 1, 2, 3 \dots$ .