This topic compares the analytical solutions and results simulated with MODE for surface plasmon modes on the air/gold interface at a wavelength of 632.8nm.

## Simulation setup

The figure above shows the structure and physical constants of air/gold interface. The gold layer is many wavelengths thick, with a refractive index of n_{m} = 0.238+3.385i at a wavelength of 632.8nm. A 1D FDE solver region is used.

## Analysis

The analytical solutions for the effective index and propagation loss for the surface plasmon modes are given by the following expressions:

$$ N_{s p}=\frac{k_{s p}}{k_{o}} $$

$$ L_{s p}=\frac{40 \pi}{\lambda \log (10)} \frac{\operatorname{imag}\left(N_{s p}\right)}{1000} $$

where

$$ k_{s p}=\sqrt{\frac{k_{0}^{2} \varepsilon_{i} \varepsilon_{m}}{\left(\varepsilon_{i}+\varepsilon_{m}\right)}} $$

$$ \varepsilon_{i}=1, \varepsilon_{m}=n_{m}^{2} $$

These will be used in the evaluation of the MODE results.

## Results

The script surface_plasmon.lsf finds the TM1 mode and plots the effective index and propagation loss as well as the corresponding % errors as a function of the number of grid points. The script also analyzes the improvements in performance of using a graded mesh instead of a uniform mesh.

Effective index calculated for air/gold interface at a wavelength of 632.8nm when using a graded mesh. The blue curves denote the MODE calculations, and the horizontal green lines show the analytic results. The x axis shows the number of grid points used in the calculation.

Propagation loss calculated for air/gold interface at a wavelength of 632.8nm when using a graded mesh. The blue curves denote the MODE calculations, and the horizontal green lines show the analytic results. The x axis shows the number of grid points used in the calculation.

The above figures show the error in percentage of the effective index and propagation loss values from the MODE calculation for fundamental surface plasmon mode of air/gold interface at a wavelength of 632.8 nm. The x-axis is the number of grid points in the one-dimensional calculation region using a graded mesh.

The above figure compares the error amplitude over number of grid points between using a uniform mesh and a graded mesh where a mesh override region is used to force a finer mesh in the region around the interface between the two materials.

The results show that it is more efficient to use a graded mesh where the mesh is finer near the interface of the materials since the surface plasmon mode has a sharp peak at the interface, and the graded mesh allows for higher resolution of the mode profile without requiring as many total grid points.