This example shows how to use the FEEM to calculate the characteristic modes of a graded-index fiber. A spatially varying refractive index profile is generated using the (n, k) material attribute tool. The field distributions and effective indices of the lower order modes are calculated and compared with analytical results.

## Simulation setup

The above figure shows the refractive index profile of the graded-index fiber, which is defined through an (n,k) material attribute under the FEEM solver. The spatially varying refractive index profile is loaded into the (n,k) material attribute using a rectilinear data set named "index_profile" that is generated by the script nk_material_graded_index_fiber.lsf. The script generates the refractive index profile data using the following equation:

$$ n(x,y) = n_o \sqrt{1-\frac{x^2+y^2}{h^2}} $$

For this example, a peak value of the refractive index of \(n_o=2\) and a core size parameter value of \(h = 40 \mu m\) are used. The wavelength of interest is \(\lambda=1\mu\text{m}\).

## Results

### Analytical Solution

The analytical solution for the effective index of the TE modes is given by

$$ N_{mn} = n_o \sqrt{1- \frac{2\alpha}{h k_o}}, \quad \alpha = 1+(m+n), \quad k_o = \frac{2 \pi n_o}{\lambda} $$

Note that higher values of \(\alpha\) lead to increasing degeneracies. Furthermore, the TE and TM modes for this fiber are also degenerate, so each of the effective indices from the above expression is essentially duplicated in the full list of modes reported by the FEEM solver. In the next section, these analytical results are used to validate the FEEM calculations.

### FEEM results

The modal fields for the fiber mode TE_{01} at \(\lambda=1\mu\text{m}\) are shown below. This mode can be found by running the FEEM mode calculation in graded_index_fiber.ldev. Note that in the FEEM settings the number of edges per wavelength is set to \(0.8\), which results in the mesh shown in the top figure. This mesh is sufficiently fine since the modal fields are expected to have a slow spatial variation, as confirmed in the figure below. The TE_{01} mode can be identified in the mode list by its calculated effective index, close to the analytical value \(N_{01} = 1.99203\), and by its \(E_z\) component, which is negligible compared to the dominant \(E_x\) and \(E_y\) components.

Electric field intensity

|Ex|

|Ey|

Modal field profile for TE_{01} mode, with effective index N_{01} = 1.99203. The Ez field component (not shown) is negligible for this mode.

The script graded_index_fiber_FEEM.lsf performs a more detailed analysis by calculating the effective indices for the first twelve modes analytically using the formula above and numerically using the FEEM solver. The calculation is done for different interpolation settings and mesh refinements by modifying the "polynomial order" and "edges per wavelength" settings under the Mesh tab in the FEEM solver properties. The script plots the list of effective indices calculated by both methods and the maximum relative error with respect to the analytical result (see below). These results clearly show that for higher polynomial order the convergence to the analytical results is faster, and the relative error can be minimized to values below \(0.001\%\). Also, note how the mode degeneracy increases as we move down the mode list, and how it can be obtained more accurately for finer meshes and higher polynomial orders for interpolation.

Effective indices of the first twelve modes found for the graded-index fiber. We compare the results for different polynomial orders and number of edges per wavelength with the analytical solution.

Maximum relative error of FEEM calculation for the first twelve modes of the graded-index fiber, compared with analytic result.

### Related references

A. W. Snyder and J. D. Love, "Optical Waveguide Theory". London: Kluwer Academic Publishers (1983).