1. Introduction
Let’s apply the material from Making Friends with Electromagnetic Boundary Conditions to calculate a transfer matrix. We will consider a TE wave travelling between materials with parameters and
:
There are a few things to note:
- This is a TE wave, so the electric field
is transverse to the plane of incidence. In this case we chose to let
point into the page, which then determines the direction of the magnetic field since
form a right-handed coordinate system.
- The total wavevector
is made up of a component
in the
-direction and
in the
direction.
- The field on each side is made up of a forward-propagating wave (
) and a backward propagating wave (
). On the left we have
and similarly for the right.
We want to represent the electric field in the first medium, but do we want them to represent
or
? To decide, let’s take a look at the boundary conditions:
The electric field is perpendicular to the boundary, so . On the other hand since
is perpendicular to
, you can convince yourself that the other boundary condition for the electric field gives
. Thus the electric field is continuous across the boundary, so we want
to represent the electric field. In this case if
are the fields just before the boundary and
the fields just after, we have
Note that we have chosen coordinates such that the exponentials in Eq. (1) are one right at the boundary.
Aside: If we had a TM wave we would choose to represent
rather than
, since
is what would be continuous across the boundary and we would again have
. Note that this is just a convenience; you could use
if you really wanted to, and using
the boundary condition would then become
.
We need one more relation between the , and we have two two boundary conditions we haven’t used so far:
These are in terms of the magnetic field, which we want to relate to the electric field. This comes from Maxwell’s equation:
What would happen if we took the equation? Since
is being dotted with
, we see that the
-component of
is continuous across the boundary. This tells us that
is continuous, which is:
since only is nonzero. Meanwhile the
equation gives us a relation for the
-component of
, and
The wave is propagating in the direction, so this is what we want.
Using , Eq. (7) tells us that
Looking at \cref{eq:TotalEField} the left hand side is
while the right hand side will be
Thus this boundary condition is:
The last thing is to write in terms of the parameters of the material. We know that
, and
. Thus
Eq. (11) thus gives us our second boundary equation:
This is in terms of the frequency of the wave , the parameters
of the material, as well as
, which is a parameter telling us the angle of the input wave. For normal incidence we have
.
2. References
[1] The LaTeX was written using the excellent tool LaTeX to WordPress: