A robust two-dimensional (2D) formulation for the electrical characterization of nematic liquid crystals (N-LCs) under low-frequency (LF) AC biasing conditions is proposed. The finite-difference (FD) method is first implemented to solve Poisson's equation in the domain of interest in order to obtain the governing LF electric field, which affects the local dielectric properties of the anisotropic material. Then, the nonlinear Euler-Lagrange partial differential equation (PDE), governing the orientation of the directors, is solved using one of three FD schemes with relaxation proposed in this paper. Once the N-LC layer is characterized, the average refractive index as a function of the x-coordinate is calculated assuming a normally incident monochromatic laser beam. The results are compared with published data in the literature obtained using a finite element method (FEM). Solution of the PDE governing the orientation of the directors in a non-uniform 2D electric field is obtained using either strong anchoring or soft anchoring. An investigation of the effects of boundary conditions on the average refractive index is presented.
- Finite difference method
- Maxwell equations
- Modal analysis
- Mode-matching technique
- Nonlinear differential equations