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
While considering a single interface between two layers (in the previous lecture) we had used a coordinate system ($$x_1,x_2,x_3$$). In the following we switch to a system ($$x, y, z$$) 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 ($$x, y, z$$) shown in the figure, the solution for the TE wave can be written as

E_y(x, z) = \tilde{E}_y(z)~e^{\pm i~k_x~x} $$ and the governing equation is

\left[ \mu(z)~\cfrac{d }{d z}\left(\mu(z)^{-1}~\cfrac{d }{d z}\right) + \omega^2~\epsilon(z)~\mu(z) - k_x^2\right]~\tilde{E}_y = 0 ~. $$ Recall that plane waves propagating in the $$z$$ direction have the form

\tilde{E}_y(z) = \tilde{E}_0~\exp[i(k_z~z - \omega~t)] ~. $$ Therefore, in region 1 (see Figure~1) for fixed $$\omega$$,
 * $$ \text{(1)} \qquad

\tilde{E}_{y1}(z) = \tilde{E}_0~\left[\exp(-i~k_{z1}~z) + R^{TE}~\exp(i~k_{z1}~z) \right] ~. $$ The first term of the left hand side of (1) represents the incoming wave while the second term on represents the reflected wave (hence the difference in signs of $$z$$). The quantity $$R^{TE}$$ is a reflection coefficient.

Similarly, in region 2 (see Figure 1),
 * $$ \text{(2)} \qquad

\tilde{E}_{y2}(z) = \tilde{E}_0~T^{TE}~\exp(-i~k_{z2}~z) $$ where $$T^{TE}$$ is a transmission coefficient.

Continuity at the interface requires that the following conditions be satisfied:
 * $$ \text{(3)} \qquad

\tilde{E}_{y1} = \tilde{E}_{y2} ~; \mu_1^{-1}~\cfrac{d \tilde{E}_{y1}}{d z} = \mu_2^{-1}~\cfrac{d \tilde{E}_{y2}}{d z} ~. $$

If we choose the coordinate system such that $$z = 0$$ at the interface, substitution of (1), (2) into (3) gives
 * $$ \text{(4)} \qquad

1 + R^{TE} = T^{TE} ~; \mu_1^{-1}~k_{z1}~(1 - R^{TE}) = \mu_2^{-1}~k_{z2}~T^{TE} ~. $$ Solving for $$R^{TE}$$ and $$T^{TE}$$ from equations (4) gives
 * $$ \text{(5)} \qquad

R^{TE} = \cfrac{\mu_2~k_{z1} - \mu_1~k_{z2}}{\mu_2~k_{z1} + \mu_1~k_{z2}}~; T^{TE} = \cfrac{2~\mu_2~k_{z1}}{\mu_2~k_{z1} + \mu_1~k_{z2}}~; $$ 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

k_z^2 = \omega^2~\mu~\epsilon - k_x^2 ~. $$ Therefore, if $$\omega^2~\mu_2~\epsilon_2 < k_{x2}^2$$, then $$k_{z2}$$ is purely imaginary. If $$k_{z1}$$ is real, then the first of equations (5) implies that the numerator and the denominator are complex conjugates. This means that

|R^{TE}| = 1 \qquad \implies \qquad \text{perfect reflection!} $$ If such a situation exists, the wave in region 2 is called  evanescent.

Multilayered Systems
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 $$z = -d_1$$ and that between regions 2 and 3 be located at $$z = -d_2$$. Then, using a change of coordinates $$z \leftarrow z + d_1$$, in region 1 (from equation 1) we have

\begin{align} \tilde{E}_{y1}(z) & = \tilde{E}_0~\left[\exp[-i~k_{z1}~(z+d_1)] + \tilde{R}_{12}~\exp[i~k_{z1}~(z+d_1)] \right] \\ & =     \tilde{E}_0~\exp(-i~k_{z1}~d_1)~\left[\exp(-i~k_{z1}~z) + \tilde{R}_{12}~\exp(i~k_{z1}~z)~\exp(2~i~k_{z1}~d_1) \right] \end{align} $$ or,

\tilde{E}_{y1}(z) = A_1~\left[\exp(-i~k_{z1}~z) + \tilde{R}_{12}~\exp(i~k_{z1}~z + 2~i~k_{z1}~d_1) \right] $$ where

A_1 = \tilde{E}_0~\exp(-i~k_{z1}~d_1) $$ and $$\tilde{R}_{12}$$ is the apparent reflection coefficient at the interface between regions 1 and 2 due to the slab.

Similarly, for region 2, we have

\tilde{E}_{y2}(z) = A_2~\left[\exp(-i~k_{z2}~z) + \tilde{R}_{23}~\exp(i~k_{z2}~z + 2~i~k_{z2}~d_2) \right] $$ and $$\tilde{R}_{23}$$ 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

\tilde{R}_{23} = R_{23} = \cfrac{\mu_3~k_{z2} - \mu_2~k_{z3}}{\mu_3~k_{z2} + \mu_2~k_{z3}}~; $$

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

\tilde{E}_{y3}(z) = A_3~\exp(-i~k_{z3}~z) ~. $$

At this stage we don't know what $$\tilde{R}_{12}$$ 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 $$z = -d_1$$,
 * $$ \text{(6)} \qquad

A_2~\exp(i~k_{z2}~d_1) = T_{12}~A_1~\exp(i~k_{z1}~d_1) + R_{21}~A_2~R_{23}~\exp(-i~k_{z2}~d_1 + 2~i~k_{z2}~d_2) $$ where $$T_{12}$$ is the transmission coefficient between regions 1 and 2 and $$R_{21} = - R_{12}$$ 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 $$z = -d_1$$ we have

A_1~\tilde{R}_{12}~\exp(-i~k_{z1}~d_1 + 2~i~k_{z1}~d_1) = R_{12}~A_1~\exp(i~k_{z1}~d_1) + T_{21}~A_2~R_{23}~\exp(-i~k_{z2}~d_1 + 2~i~k_{z2}~d_2) $$ or,
 * $$ \text{(7)} \qquad

A_1~\tilde{R}_{12}~\exp(i~k_{z1}~d_1) = R_{12}~A_1~\exp(i~k_{z1}~d_1) + T_{21}~A_2~R_{23}~\exp[-i~k_{z2}~d_1 + 2~i~k_{z2}~d_2) ~. $$

Eliminating $$A_2$$ from (6) gives
 * $$ \text{(8)} \qquad

A_2 = \cfrac{T_{12}~A_1~\exp[i~(k_{z1} - k_{z2})~d_1]} {1 - R_{21}~R_{23}~\exp[2~i~k_{z2}~(d_2-d_1)]} ~. $$ Plugging (8) into (7), we get
 * $$ \text{(9)} \qquad

\tilde{R}_{12} = R_{12} + \cfrac{T_{12}~T_{21}~R_{23}~\exp[2~i~k_{z2}~(d_2 - d_1)]} {1 - R_{21}~R_{23}~\exp[2~i~k_{z2}~(d_2-d_1)]} $$ which gives us an expression for the generalized reflection coefficient $$\tilde{R}_{12}$$.

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

0 \le R_{21}~R_{23}~e^{2~i~k_{z2}~(d_2-d_1)} < 1 ~. $$ Then we can expand (9) in series form to get
 * $$ \text{(10)} \qquad

\tilde{R}_{12} = R_{12} + T_{12}~T_{21}~R_{23}~e^{2ik_{z2}(d_2 - d_1)} \left[ 1 + R_{21}~R_{23}~e^{2ik_{z2}(d_2-d_1)} + R_{21}^2~R_{23}^2~e^{4ik_{z2}(d_2-d_1)} + \dots \right] ~. $$ 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 $$N$$ layers (see Figure 4 for a schematic of the situation). If one additional layer is added, then we only need to replace $$R_{23}$$ in equation (9) with $$\tilde{R}_{23}$$.

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

\tilde{E}_{yj}(z) = A_j~\left[\exp(-i~k_{zj}~z) + \tilde{R}_{j, j+1}~\exp(i~k_{zj}~z + 2~i~k_{zj}~d_j) \right] ~. $$ For the last layer,

\tilde{R}_{N, N+1} = 0 ~. $$ For all other layers we get a recursion relation
 * $$ \text{(11)} \qquad

\tilde{R}_{j, j+1} = R_{j, j+1} + \cfrac{T_{j, j+1}~T_{j+1, j}~\tilde{R}_{j+1, j+2}~ \exp[2~i~k_{z, j+1}~(d_{j+1} - d_j)]} {1 - R_{j+1, j}~\tilde{R}_{j+1, j+2}~ \exp[2~i~k_{z, j+1}~(d_{j+1}-d_j)]} ~. $$ Recall from equation (4) that
 * $$ \text{(12)} \qquad

T_{ij} = 1 + R_{ij} \qquad \text{and} \qquad R_{ij} = - R_{ji} ~. $$ Using equation (12), equation (11) simplifies to
 * $$ \text{(13)} \qquad

{ \tilde{R}_{j, j+1} = \cfrac{R_{j, j+1} + \tilde{R}_{j+1, j+2}~ \exp[2~i~k_{z, j+1}~(d_{j+1} - d_j)]} {1 + R_{j, j+1}~\tilde{R}_{j+1, j+2}~ \exp[2~i~k_{z, j+1}~(d_{j+1}-d_j)]} ~. } $$ where

R_{j, j+1} = \cfrac{\mu_{j+1}~k_{z, j} - \mu_j~k_{z, j+1}} {\mu_{j+1}~k_{z, j} + \mu_j~k_{z, j+1}} $$ 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.