The purpose of the Finding modes step is to obtain a set of basis modes for describing
the EM fields within each cell.
Next, it's necessary to determine how each mode in a cell couples into the modes of neighboring
This information is stored in scattering matrices.
The physical principle used to calculate the S matrices is that the tangential E and H
fields are continuous at the interface between cells, so for each incident mode on the interface,
the sum of the incident and reflected tangential fields equals the transmitted tangential fields
in the next cell.
By modal expansion we can expand the transmitted and reflected modes in terms of the set of
modes on either side of the interface where the fields are weighted by the reflection
and transmission coefficients.
If we multiply both sides of this equation by the H fields, we can replace the E and
H fields by overlap integrals which have known values that are calculated during the Finding
modes step of the simulation.
Since we have an equation of this form for each incident mode on the interface, we end
up with a system of linear equations which can be solved by matrix inversion to obtain
the reflection and transmission coefficients for each mode incident on the interface.
A more detailed explanation of the mathematics involved can be found in the reference listed
During the Finding modes step of the simulation, in addition to calculating the modes in each
cell, the amplitude of the modal fields are also normalized so that the real part of E
cross H integrated over the mode is 1.
For lossless waveguides this means the mode will be carry 1 Watt of power.
Overlap integrals are calculated between the set of modes in each cell with the set of
modes in their adjacent cells, and they take this form.
As mentioned on the previous slide, the overlap integrals facilitate the calculation of the
transmission and reflection coefficients for each incident mode on the interface.
During the Propagate step of the simulation, the scattering matrices, or S matrices, which
contain the transmission and reflection coefficients are formulated.
There will be an S matrix for each cell interface, and the S matrix will contain the reflection
and transmission coefficients for each mode incident on the cell interface from the cells
on either side.
If M modes are supported on one side of the cell interface and P modes are supported in
the cell on the other side of the interface, then the S matrix for that interface will
have size M plus P by M plus P. The EME method finds the best solution for
the S matrix that minimizes any discontinuities in the tangential E and H fields at the interface
for the finite number of modes that are used.
As a result, the standard EME method does not have any notion of energy conservation
built into it.
The energy conservation is an additional constraint that can be added to ensure that energy is
This may result in greater field discontinuities at the interfaces,
so it's not possible to get everything (energy conservation and continuous tangential fields)
because we have to work with a finite set of modes.
The settings of the solver that are used for specifying the energy conservation will be
discussed later on in the Solver Region section of the course.
The S matrix for each cell interface is just one of the results calculated during the Propagate
step, and the next unit will discuss the other results.