Tests of accuracy and convergence

For the rectangular guide, the fractional error $\epsilon$ in the propagation constant $k_z$ was less than $1\%$, and the accuracy of the electric field strengths in the trapping region $\approx 3\%$, when we used $N \sim 5000$ degrees of freedom. This was sufficiently accurate for the present work.

We estimated the accuracy of the method by solving a cylindrical guide in an identical fashion with the same $N$ and a very similar non-uniform grid, for which there are known field solutions [183]. This system is quite similar to the rectangular guide in question, although it does not possess sharp corners. However, representing the circular cross-section by a pixellated approximation on a square grid (which cannot represent surfaces other than horizontal and vertical) was taken to be a stringent test of the method's ability to handle corners sensibly.

We performed a sweep in radius of the guide from below cut-off to where there are several bound modes: Fig. 8.4 shows the propagation constant agrees with the analytics to within 1%, even close to cut-off. The convergence with $N$ was measured for the rectangular guide case, and found to be $\epsilon \sim N^{-\gamma}$ with $0.55 < \gamma < 0.7$. This is less than optimal for first-order elements (which have a maximum possible convergence of $\gamma = 1$), and is believed to be due to an inability of the bilinear functions to represent physical in-plane $E$ and $H$ components at dielectric steps, or the weak field singularities which can physically occur at any exterior dielectric sharp edges (regardless of whether acute or obtuse).