Eigenmode expansion (EME) is a computational electrodynamics modelling technique.
[2] Eigenmode expansion is a linear frequency-domain method.
It offers very strong benefits compared with FDTD, FEM and the beam propagation method for the modelling of optical waveguides,[3] and it is a popular tool for the modelling linear effects in fiber optics and silicon photonics devices.
Eigenmode expansion is a technique to simulate electromagnetic propagation which relies on the decomposition of the electromagnetic fields into a basis set of local eigenmodes that exists in the cross section of the device.
The eigenmodes are found by solving Maxwell's equations in each local cross-section.
The guided and radiation modes together form a complete basis set.
Many problems can be resolved by considering only a modest number of modes, making EME a very powerful method.
It uses the scattering matrix (S-matrix) technique to join different sections of the waveguide or to model nonuniform structures.
For structures that vary continuously along the z-direction, a form of z-discretisation is required.
Advanced algorithms have been developed for the modelling of optical tapers.
In a structure where the optical refractive index does not vary in the z direction, the solutions of Maxwell's equations take the form of a plane wave: We assume here a single wavelength and time dependence of the form
are the eigenfunction and eigenvalues of Maxwell's equations for conditions with simple harmonic z-dependence.
We can express any solution of Maxwell's equations in terms of a superposition of the forward and backward propagating modes:
When there is a change in the structure along the z-direction, the coupling between the different input and output modes can be obtained in the form of a scattering matrix.
The scattering matrix of a discrete step can be obtained rigorously by applying the boundary conditions of Maxwell's equations at the interface; this requires calculation of the modes on both sides of the interface and their overlaps.