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.

Figure 1. A TE wave at an interface between two layers.

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.

Figure 2. Reflection and transmission in a three layer medium.

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






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




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.

Figure 3. Interpretation of series expansion of  .

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  .

Figure 4. A medium with   layers.

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




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

  1. This lecture closely follows the exposition in Chew95. For further details please consult that source.

References edit

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