Fourth-order variational mode solving for anisotropic planar structures

A variational method that gives 4th-order accuracy by only using linear basis functions within the computational domain is proposed for the mode solving of anisotropic planar stratified waveguides with diagonal permitivity tensor. A non-uniform mesh is used to get rid of the necessity of incorporating interface corrections and higher-order basis functions, while Richardson-like extrapolation is used to obtain 4th-order accuracy. The scheme was derived for both TE and TM mode analysis and completed with infinite elements as transparent boundary conditions. The use of a simple extrapolation technique to further refine the results by using two consecutive calculated results is also demonstrated.