# MODE - Finite Difference Eigenmode (FDE) solver introduction

The Finite Difference Eigenmode (FDE) solver calculates the spatial profile and frequency dependence of modes by solving Maxwell's equations on a cross-sectional mesh of the waveguide. The solver calculates the mode field profiles, effective index, and loss. Integrated frequency sweep makes it easy to calculate group delay, dispersion, etc. The solver can also treat bent waveguides.

## Solver physics

In the z-normal eigenmode solver simulation example shown in the figure above, we have the vector fields:

$$E(x,y)e^{i(-\omega t+\beta z)}$$

$$H(x,y)e^{i(-\omega t+\beta z)}$$

where ω is the angular frequency and β is the propagation constant. The modal effective index is then defined as \(n_{eff}=\frac{c\beta}{\omega}\).

The Eigensolver find these modes by solving Maxwell's equations on a cross-sectional mesh of the waveguide. The finite difference algorithm is the current method used for meshing the waveguide geometry, and has the ability to accommodate arbitrary waveguide structure. Once the structure is meshed, Maxwell's equations are then formulated into a matrix eigenvalue problem and solved using sparse matrix techniques to obtain the effective index and mode profiles of the waveguide modes. This method is based on Zhu and Brown1, with proprietary modifications and extensions.

The fields are normalized such that the maximum electric field intensity |E|^2 is 1.

The FDE mode solver is also capable of simulating bent waveguides. For more information, see the Bent waveguide solver page.

Note: The FDE solves an eigenvalue problem where beta2 (beta square) is the eigenvalue (see the reference below) and in some cases, such as evanescent modes or waveguides made from lossy material, beta2 is a negative or complex number. The choice of root for beta2 determines if we are returning the forward or backward propagating modes. By default, the root chosen is the one with a positive value of the real part of beta which, in most cases, corresponds to the forward propagating mode. However, we know that a waveguide will not create gain if the material has no gain. To ensure that the correct forward propagating modes are reported, the FDE may flip the sign of the default root to ensure that the mode has loss (and a negative phase velocity) which is physical. Detailed settings can be found in Advanced options.

### Helical waveguide mode solver

The FDE mode solver is also capable of simulating helical waveguides. For more information, see the Helical waveguide solver page.

## Meshing

The MODE Eigenmode Solver uses a rectangular, Cartesian style mesh, like the one shown in the following screenshot. It's important to understand that of 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. 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 Eigenmode solver.

### References

- Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853

### Selected Application Examples

- PN Depletion Phase Shifter
- Travelling Wave Mach-Zehnder Modulator
- Waveguide (FDE)
- Photonic crystal fiber
- Plasmonic waveguide
- Evanescent Waveguide Coupler
- Arrow Slab Waveguide
- SMF-28 Fibre

### Video resources

- MODE overview (YouTube.com)