On the Convergence and Accuracy of Numerical Mode Computations of Surface Plasmon Waveguides

Two-dimensional surface plasmon-polariton waveguides must generally be analysed using numerical methods. However, accurate analysis is challenging due to large permittivity contrasts and to strong localisation of mode fields, especially near corners where they tend to diverge. These difficulties impact the convergence of numerical methods, yet understanding convergence is essential if accuracy is to be claimed. The convergence and accuracy of two vectorial numerical methods commonly used, the method of lines and the finite element method, were assessed by computing the propagation constant of modes supported by the metal slab, the metal stripe and the 90 degrees metal corner. A discretisation strategy that yields smooth monotonic convergence is demonstrated for both methods. More accurate results are then extrapolated from the convergence histories and anticipated errors (relative to extrapolated results) are computed. Both methods yield similar anticipated errors for a comparable discretisation spacing, with the finite element method yielding slightly lower errors but the method of lines requiring less computational effort. Convergence was slower for highly confined modes, particularly those having fields localised near corners. Convergence to within an anticipated error of +/-2% was readily achieved with both methods, except for the attenuation of modes that are highly confined to corners which remained in error by 10 to 20%. However, the percentage difference between the extrapolated results computed from both methods ranged from 3.75 to 0.012%. The convergence and accuracy of the methods of lines for curves was also investigated, and it was found that the absorbing boundary condition used along the radiating side of the curve introduces errors that can further limit the accuracy of the computations.