The finite element eigenmode (FEEM) solver calculates the modes or characteristic solutions of Maxwell’s equations over the cross-section of a long uniform structure like a waveguide or a fiber in the frequency domain. The solver determines the effective index, loss, and electromagnetic fields associated with each mode for a given frequency. This page provides an overview of the solver capabilities and the calculations that it performs.

## Solver physics

On a long uniform structure like a waveguide or fiber, one can take advantage of the fact that all characteristic solutions of Maxwell’s equations will be periodic in the direction in which the structure is uniform. To show how this simplifies the process of solving Maxwell’s equations, let us carry out such a simplification in Cartesian coordinates assuming the structure under consideration is long and uniform along the \(z\)-axis and contains only simple materials. In such a case, the characteristic solutions or modes will have a spatial dependency of the form \( e^{i\beta z} \), where \( \beta \) is a constant to be determined. In the frequency domain and in the absence of sources, Maxwell’s equations are given by

$$ i\omega\mu\overrightarrow{H}=\triangledown\times\overrightarrow{E} $$

$$ -i\omega\varepsilon\overrightarrow{E}=\triangledown\times\overrightarrow{H} $$

where \(\mu\) and \(\varepsilon\) are the permittivity and permeability, respectively, and \(\omega\) is the angular frequency. Assuming the structure is uniform along the \(z\)-axis, the electromagnetic fields will be periodic along the \(z\)-axis and have a spatial dependence of the form

$$ \overrightarrow{E}(x,y,z) = \overrightarrow{E}(x,y) e^{i\beta z} $$

$$ \overrightarrow{H}(x,y,z) = \overrightarrow{H}(x,y) e^{i\beta z} $$

Substituting these into Maxwell’s equations eliminates the \(z\)-axis dependence of the problem and results in a reduced 2D coupled eigenvalue problem of the form

$$-i\omega\varepsilon\left(\begin{array}{c}E_{x}\\E_{y}\\E_{z}\end{array}\right)-\left(\begin{array}{c}0&0&\partial_{y}\\0&0&-\partial_{x}\\-\partial_{y}&\partial_{x}&0\end{array}\right)\left(\begin{array}{c}H_{x}\\H_{y}\\H_{z}\end{array}\right)=i\beta\left(\begin{array}{c}0&-1&0\\1&0&0\\0&0&0\end{array}\right)\left(\begin{array}{c}H_{x}\\H_{y}\\H_{z}\end{array}\right) $$

$$ i\omega\mu\left(\begin{array}{c}H_{x}\\H_{y}\\H_{z}\end{array}\right)-\left(\begin{array}{c}0&0&\partial_{y}\\0&0&-\partial_{x}\\-\partial_{y}&\partial_{x}&0\end{array}\right)\left(\begin{array}{c}E_{x}\\E_{y}\\E_{z}\end{array}\right)=i\beta\left(\begin{array}{c}0&-1&0\\1&0&0\\0&0&0\end{array}\right)\left(\begin{array}{c}E_{x}\\E_{y}\\E_{z}\end{array}\right) $$

For a given value of \(\omega\), there will be a discrete set of eigenvalues corresponding to different values of \(\beta\). For each eigenvalue or propagation constant \(\beta\), there will be a corresponding eigenvector defining the electromagnetic field components \(E_{x},\ldots,H_{z}\). The FEEM solver will find a subset of these eigenvalues based on the specified number of trial modes. For each of the found eigenvalues, the FEEM solver reports the corresponding effective index

$$ n_\text{eff}=c_{0}\beta/\omega $$

where \(c_0\) is the speed of light in vacuum. In addition, the solver will extract the field components \(E\) and \(H\) from the corresponding eigenvector associated with each eigenvalue.

When the materials have losses (i.e. \(\varepsilon\) and \(\mu\) have imaginary parts), the effective index will have a complex part and the solution will contain an attenuation factor along the \(z\)-axis. This attenuation factor is reported as the loss or

$$ \frac{20\log_{10}(e)\omega}{c_{0}}\text{Im}(n_{\text{eff}}) $$

## Finite element method

To solve the 2D eigenvalue problem described above, the FEEM solver employs the finite element method with triangular vector elements of variable polynomial order. The triangulation is generated based on a maximum triangle edge size that is set based on the free space wavelength \(\left( \lambda=2\pi c_{0}/\omega \right)\). Mesh refinement controls are provided based on material properties and maximum triangle edge sizes for specific simulation domains. The finite element discretization leads to a sparse eigenvalue problem that is solved using standard linear algebra techniques.

## Use with a 2D simulation region

Because the FEEM solver reduces a 3D eigenvalue problem to 2D by assuming that the structure under consideration is uniform along a specified direction, it can only be used in conjunction with a 2D simulation region placed on a cross section of the structure. The direction normal to the simulation region indicates the direction in which the structure is assumed to be uniform. This direction can be set to \(x\), \(y\) or \(z\) as described previously. In each case, the appropriate 2D reduction of Maxwell’s equations is applied.

## Accuracy: mesh refinement vs. polynomial order

As with most finite element solvers, using smaller elements will usually lead to higher accuracy, however, that is not the only way to increase the accuracy of the FEEM solver. For structures without curved material boundaries, increasing the polynomial order of the triangular vector elements is usually a more effective mechanism to increase the solution accuracy. The polynomial order can be set in the solver options.

## Units and normalization

Unless otherwise specified, all quantities are provided in SI units. Note that some quantities, like the effective index, are rendered unitless by normalizing to frequency or wavelength. The amplitude of electric and magnetic fields of each mode are normalized so that the power carried by the mode is 1 W.

### References

- Jianming Jin, The finite element method in electromagnetics. Wiley, 2002.