This topic compares the analytical solutions and results simulated with MODE for a exponential index profile slab waveguide at wavelength 633nm.

## Simulation setup

The figure above shows the dimensions and permittivity as a function of depth for the exponential index profile slab waveguide. The permittivity of the slab decays exponentially from a value of 6.4 at the air/core interface surface into the substrate according to ε(y) = 6.1 + 0.3exp(-y/2.5), where y represents the depth below the surface measured in microns. Calculations performed with MODE are compared to the analytic response at a wavelength of 633nm. This structure can be constructed using rectangular structures of air(etch) on top of a dielectric defined with an index of n(y) = sqrt(ε(y)) = sqrt(6.1 + 0.3*exp((y-yspan/2)/2.5)). Note that y=0 at the center of the core, not at the origin (see Equation Interpreter).

## Results

The script exp_wg.lsf finds the effective indices of the TE1 and TM1 modes corresponding to the wavelength 633nm and plots the results and the corresponding % errors as a function of the number of grid points. The analytical solution for the effective index of the slab waveguide can be calculated with the Matlab script exp_wg.m, and it is used to verify the MODE results.

Set the flag "use_matlab" to be 1 in order to execute the analytical calculation and plot the results using Matlab (Matlab Integration must be enabled). Otherwise, download the provided text files with the calculated results for the analytical solution, which will be read and used to generate the plots with Lumerical's plotting functions.

(Left) Effective index values for TE (red) and TM (blue) modes as calculated via MODE (o) and via analytic relations (lines). The x-axis shows the convergence of MODE. (Right) The same figure generated using only built-in MODE functions (no Matlab interface).

(Left) Error amplitude of MODE calculation. The x-axis shows the number of grid points in the one-dimensional calculation region. (Right) The same figure generated using only built-in MODE functions (no Matlab interface).