Please post a full description and your questions.yashabyadav said:I am trying to apply finite difference scheme for Beam propagation method by following this paper.
I was wondering if anyone can share their code if they have implemented this method. I can share my code which is not working as expected and can get some insights if possible.
The finite difference method (FDM) for the Beam Propagation Method (BPM) is a numerical technique used to solve the paraxial approximation of the Helmholtz equation. It involves discretizing the spatial domain into a grid and approximating the derivatives with finite differences, allowing the simulation of light propagation in optical waveguides and other photonic structures.
To set up the grid for FDM in BPM, you need to define the spatial domain and discretize it into a grid of points. This involves choosing the number of grid points (Nx, Ny) and the step sizes (dx, dy) in the x and y directions. The grid points should be fine enough to capture the variations in the field but not so fine as to make the computation excessively expensive.
Boundary conditions are crucial for the stability and accuracy of the FDM in BPM. Common boundary conditions include Dirichlet (fixed value), Neumann (fixed derivative), and periodic boundaries. For optical waveguides, absorbing boundary conditions like Perfectly Matched Layers (PML) are often used to simulate open boundaries and prevent reflections.
The stability of the finite difference method in BPM is influenced by the choice of step sizes (dx, dy) and the propagation step (dz). A common criterion is that the propagation step should be small enough to ensure numerical stability, often dictated by the Courant-Friedrichs-Lewy (CFL) condition. Ensuring that the refractive index contrast is not too high also helps maintain stability.
Sure! A simple example involves solving the scalar wave equation in a 2D waveguide. First, discretize the spatial domain and initialize the field. Then, use finite difference approximations to update the field at each grid point for each propagation step. Here's a pseudocode snippet:1. Define Nx, Ny, dx, dy, dz.2. Initialize the field E(x, y, 0).3. For each z-step: a. Compute the Laplacian using finite differences. b. Update the field using the BPM equation: E(x, y, z+dz) = E(x, y, z) + dz * Laplacian(E(x, y, z)).4. Apply boundary conditions at each step.This is a very basic outline, and actual implementation will depend on the specific problem and boundary conditions.