Results

Resulting plots of the balanced intensity differences of the two lowest modes were shown in Fig. 8.3. Finding the bound modes for each waveguide parameter choice, with $N \approx 5000$, typically took between 3 and 20 minutes of computing time on a desktop workstation^8.5. This was dominated by the sparse eigenvalue diagonalization time. The convergence time of the eigenvalues for the same $N$ could vary by an order of magnitude, and was largely dependent on how close to cut-off the guide was: closer to cut-off took longer. Little attempt was made to optimize this.

Future improvements to the method, which would increase the accuracy or the convergence rate $\gamma$, include using higher order elements (if done carefully, this could correctly represent physical $E$ and $H$ components at dielectric steps), and explicit modelling of the field singularities at guide corners using specialized elements.