This video is taken from the EME 100 course on Lumerical University.

## Transcript

The Eigenmode Expansion, or EME, method solves Maxwell's equations in the frequency domain.

The method works by first slicing the device along the main propagation axis, which is

the x-axis, and solving for the modes, also called eigenmodes, that are supported in each

slice, or cell.

Only one cell is needed in regions with uniform cross section such as the input and output

waveguides here, since the set of supported modes will not change over a uniform region.

More cells are needed over regions where the cross section of the device is smoothly varying,

such as over the tapered region.

The EME method uses modal decomposition, which is the idea that fields at a particular position

can be represented as a linear combination of supported modes of the cross section of

the structure.

If we have an infinite number of modes, then we will have a full basis set, and we will

be able to represent the fields perfectly using a linear combination of modes.

However, since we can't practically use an infinite number of modes, we need to choose

the number of modes to use, and we'll discuss how to figure out the number of modes to use

later on in the Convergence Testing section of the course.

As we mentioned in the My First Simulation section, running an EME simulation involves

2 steps: Finding modes, and Propagating.

During the Finding modes step which happens when you click on the Run button, the Finite

Difference Eigenmode, or FDE, method is used to calculate a set of supported modes in each

cell.

The cross section of the structure that is used will be taken from the center of each

cell.

The FDE method uses a rectangular mesh to discretize the cross section of the structure,

and solves Maxwell's equations in the form of a matrix eigenvalue problem to return the

effective index and the mode profiles of the supported modes.

The modes from the FDE solver are always orthogonal, which means that you can't construct one mode

using a linear combination of other modes.

The FDE solver algorithm and settings are covered in detail in Lumerical University's

FDE 100 course, and taking the FDE 100 course is recommended.

For more information, see the related links below this video.