ExamplesMeep Examples
Polarization Grating
Simulate liquid crystal polarization gratings with twisted nematic layers
Polarization Grating Simulation
This example simulates polarization gratings based on twisted nematic liquid crystals, computing diffraction efficiency for both uniaxial and bilayer configurations.
Overview
Polarization gratings are used for:
- Beam steering: Non-mechanical deflection
- Holography: Polarization-based recording
- Displays: High-efficiency light modulation
- Spectroscopy: Polarization-sensitive measurements
Parameters
| Parameter | Value | Description |
|---|---|---|
resolution | 50 | Pixels per μm |
n_0 | 1.55 | Ordinary index |
delta_n | 0.159 | Birefringence |
wvl | 0.54 μm | Wavelength (green) |
gp | 6.5 | Grating period |
Physical Setup
Two configurations are compared:
Uniaxial Grating
- Single twisted layer
- Phase delay: Δnd/λ
- 0th and ±1st orders
Bilayer Twisted Grating
- Two oppositely-twisted layers
- Enhanced first-order diffraction
- Reduced zeroth-order
The liquid crystal director rotates spatially, creating a periodic phase modulation that diffracts light into multiple orders.
Anisotropic Material
The permittivity tensor is position-dependent:
def lc_mat(p):
# Rotation matrix for director angle φ(p)
Rx = rotation_matrix(phi(p))
# Rotate diagonal permittivity tensor
lc_epsilon = Rx @ epsilon_diag @ Rx.T
return mp.Medium(
epsilon_diag=diag_elements,
epsilon_offdiag=offdiag_elements
)Analytic Comparison
For the uniaxial grating:
- η₀ = cos²(πΔnd/λ)
- η₁ = sin²(πΔnd/λ)
These match the FDTD results for small angles.
Results
The simulation produces:
- Diffraction efficiency vs phase delay
- Comparison with analytic formulas
- Angular distribution of diffracted orders
Related Examples
- Binary Grating Analysis - Amplitude gratings
- Binary Grating Phase Map - Phase gratings
- Faraday Rotation - Polarization effects