If we want to know the optical field how to distribute in the medium rather not time evolution such as the waveguide, then we can calculate the Helmholtz equation to get the effective eigenmodes in the medium. Here, we only consider the 1-dimensional constant and gradient refractive index as our examples.
The Helmholtz equation is derived from the Maxwell equations, and assume that the relative permittivity is constant.
First, we modify the Helmholtz equation with the finite difference method, and then rearrange this modified equation into a matrix form. Next, the Helmholtz equation becomes an eigen problem. Finally, the effective eigen-mode must satify that the effective refractive index is between the shedding layer (n0) and the core medium (n1).
Code6_HelmholtzEq.m
Code6_HelmholtzEq.pptx
Case 1: The constant refactive index in 1-dim.
Case 2: The gradient refactive index in 1-dim.
back to Content