# Waves in composites and metamaterials/TE waves in multilayered media

**The content of these notes is based on the lectures by Prof. Graeme W. Milton (University of Utah) given in a course on metamaterials in Spring 2007.**

## Review edit

While considering a single interface between two layers (in the
previous lecture) we had used a coordinate system ( ).
^{[1]}
In the following we switch to a system ( ) to make our notation
a bit less confusing for multilayered media. In the following we assume
that the material properties of each layer in a multilayered material
are piecewise constant.

Consider the TE wave shown in Figure 1.

Then, from the previous lecture and using the new notation ( ) shown in the figure, the solution for the TE wave can be written as

and the governing equation is

Recall that plane waves propagating in the direction have the form

Therefore, in region 1 (see Figure~1) for fixed ,

The first term of the left hand side of (1) represents the incoming wave while the second term on represents the reflected wave (hence the difference in signs of ). The quantity is a reflection coefficient.

Similarly, in region 2 (see Figure 1),

where is a transmission coefficient.

Continuity at the interface requires that the following conditions be satisfied:

If we choose the coordinate system such that at the interface, substitution of (1), (2) into (3) gives

Solving for and from equations (4) gives

Note that these quantities are the Fresnel coefficients of the bilayer and that the reflection and transmission coefficients may be complex.

Recall from the previous lecture that

Therefore, if , then is purely imaginary. If is real, then the first of equations (5) implies that the numerator and the denominator are complex conjugates. This means that

If such a situation exists, the wave in region 2 is called ** evanescent**.

## Multilayered Systems edit

Let us first examine the problem of reflection and transmission in a three layer medium (see Figure 2). Our goal is to find the effective reflection and transmission coefficients in this medium. Once we know these coefficients, we can choose the materials in the layers to achieve a desired reflectivity or transmissivity.

Let the interface between regions 1 and 2 be located at and that between regions 2 and 3 be located at . Then, using a change of coordinates , in region 1 (from equation 1) we have

or,

where

and is the apparent reflection coefficient at the interface between regions 1 and 2 due to the slab.

Similarly, for region 2, we have

and is the apparent reflection coefficient for a downgoing wave at the interface between regions 2 and 3 due to the slab. However, since the wave is transmitted in region 3 and there are no further reflections, we have

Since the wave is only transmitted in region 3, we have

At this stage we don't know what is. To find this quantity, note that the downgoing wave in region 2 equals the sum of the transmission of the downgoing wave in region 1 and a reflection of the upgoing wave in region 2 (see Figure 2). Hence, at the top interface ,

where is the transmission coefficient between regions 1 and 2 and is the reflection coefficient of waves from region 2 incident upon region 1.

Also, the upgoing wave in region 1 is the sum of the reflection of the incoming wave in region 1 and the transmission at interface 2-1 of the reflected wave at interface 2-3. Hence, at we have

or,

Eliminating from (6) gives

Plugging (8) into (7), we get

which gives us an expression for the generalized reflection coefficient .

We have considered only two internal reflections so far. How about further reflections? It turns out that equation (9) can be interpreted to include all possible internal reflections. To see this, let us assume that

Then we can expand (9) in series form to get

This equation can be interpreted as shown in Figure 3. However, sometimes the series may not converge at it is preferable to use (9) for computations.

We can now generalize the above results to a medium with layers (see Figure 4 for a schematic of the situation). If one additional layer is added, then we only need to replace in equation (9) with .

Therefore, in general, the wave in the -th region takes the form

For the last layer,

For all other layers we get a recursion relation

Recall from equation (4) that

Using equation (12), equation (11) simplifies to

where

is the Fresnel reflection coefficient for transverse electric waves.

In the next lecture we will take the continuum limit of these equations and derive equations for the effective reflection coefficient of a smoothly graded multilayered medium with a few isolated jumps.

## Footnotes edit

## References edit

- W. C. Chew.
*Waves and fields in inhomogeneous media*. IEEE Press, New York, 1995.