The bidirectional eigenmode expansion (EME) solver is ideal for simulating light propagation over long distances. The computational cost of the method scales exceptionally well with the device length, making it much more efficient for the design and optimization of long tapers and periodic devices compared to FDTD-based methods.

## Solver physics

The EME method is a frequency domain method for solving Maxwell's equations. The algorithm is fully vectorial and bi-directional, and offers a good alternative to FDTD-based methods for long propagation distances. The methodology involves 2 key steps:

1. The modal decomposition of electromagnetic fields into a basis set of eigenmodes. These modes are computed by dividing the geometry into multiple cells and then solving for the modes at the interface between adjacent cells. Scattering matrices for each section are then formulated by matching the tangential E and H fields at the cell boundaries. This is the most time consuming portion of the EME calculation. In this step, FDE solver is used.

2. The simulation is now in analysis mode, and the solution to each section can be propagated bi-directionally to calculate the S matrix of the entire device. The internal fields can also be reconstructed, if desired. This step can be carried out very quickly.

Once in analysis mode, the user can change the propagation distance of each section arbitrarily without having to repeat step 1. This is why the EME method is very efficient for scanning the lengths of devices, as demonstrated in the Spot Size Converter Example.

The EME method has several advantages over other propagation methods:

- Beam propagation methods (BPM): unlike BPM, which relies on a slowly varying envelope approximation, the EME method makes no such approximations and is a rigorous technique. The accuracy of BPM also becomes compromised for propagation at large angles, or in components with high refractive-index contrast, making it unsuitable for photonic components manufactured from silicon or other high index contrast material systems.
- Finite-difference time-domain (FDTD) methods: the EME method scales exceptionally well with propagation distance and is an ideal method for simulating long structures whereas FDTD-based methods, while rigorous, exhibit significant increases in simulation times as the length of the device increases.

## Meshing

### Transverse mesh (y,z)

The EME solver uses a rectangular, Cartesian style mesh, like the one shown in the figure to the right. It is important to understand that the fundamental simulation quantities (material properties and geometrical information, electric and magnetic fields) are calculated at each mesh point. Obviously, using a smaller mesh allows for a more accurate representation of the device, but at a substantial cost. As the mesh becomes smaller, the simulation time and memory requirements will increase. MODE provides a number of features, including the conformal mesh algorithm, that allow you to obtain accurate results, even when using a relatively coarse mesh.

By default, the simulation will use a uniform mesh in the transverse directions. You simply set the number of mesh points along each axis. In some cases, it is necessary to add additional meshing constraints. Usually, this involves forcing the mesh to be smaller near complex structures where the fields are changing very rapidly.

Note: In FDTD-based simulations, it's important to use a smaller mesh in high index materials, and to maintain a minimum number of mesh points per wavelength. This constraint does not exist for the EME solver. |

### Longitudinal mesh (x)

The mesh along the propagation direction is defined by adding cells to the EME simulation region. An eigenmode simulation is carried out at the center x location of each cell to calculated all the modes supported at that cross section. More cells will allow for a more accurate representation of the geometry in the longitudinal direction, but at the cost of increasing the simulation time and memory.

Cell groups can be used to specify a non-uniform longitudinal mesh. For example, in the MMI structure to the right, only 1 cell is required for the uniform sections, while more cells are required for the tapered sections.

## Continuously Varying Cross-sectional Subcell (CVCS) method

The traditional EME method represents continuously varying structures, such as tapers, with a staircase approximation to resolve geometrical or material variations along the direction of propagation. This leads to non-physical reflections and calculation inaccuracies. The typical workaround involves increasing the number of expansion interfaces, which results in increased computational costs, both in terms of time and memory. Lumerical’s CVCS method avoids this staircasing effect and, through extensive benchmarking versus 3D finite-difference time-domain (FDTD) simulations for a wide variety of waveguide structures, offers excellent accuracy at a fraction of the time for longer structures.

## Units and normalization

Unless otherwise specified, all quantities are returned in SI units. Please see Units and Normalization for more information.