**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*: