Waves in composites and metamaterials/Waves in layered media and point sources

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.

Previous Lecture

edit

Recall from the previous lecture the governing equation for a TM wave. [1]

 

where

 

and

 

Also recall that we introduced a state vector   which is continuous across boundaries. The state vector is defined as

 

where

 

Then,

 

The general solution of (2) is

 

where   and   are the eigenvectors corresponding to the eigenvalues   and  , respectively.

We also found that the solution of the differential equation (2) can be expressed in terms of a propagator matrix   such that

 

where

 

where the columns of   are the eigenvectors of   and

 

The matrix   is called the propagator matrix or the transition matrix that related the fields at   and  .

Also, in a multilayered system (see Figure 1)

 

where   depends on   and   depends on  .

 
Figure 1. Multilayered medium.

A special case

edit

Let us now consider the particular case shown in Figure 2. The medium consists of three layers. Regions 1 and 3 are isotropic while the sandwiched region 2 is multilayered. The interface between regions 1 and 2 is located at   while the interface between regions 2 and 3 is located at  . Let the propagator matrix of region 2 be  . We want to find the reflection coefficient and the transmission coefficient of the system.

 
Figure 2. A multilayered medium sandwiched between two layers.

In reqion 1, the state vector is given by (see equation (3))

 

Define

 

where   is like a scalar reflection coefficient. Then

 

Proceeding as we did in the previous lecture, let us define matrices

 

Then equation (5) can be written as

 

In region 3, there is only a transmitted wave. Therefore, the state vector is given by

 

Define

 

where   is a factor that acts like a scalar transmission coefficient. Then, equation (7) can be written in matrix form as

 

where

 

Since we have the propagator matrix for region 2,  , we can use it to connect regions 1 and 3. The continuity of the state vector across the interfaces implies that

 

Also, using equation (4) we have

 

Therefore, using equations (9), we can write (10) as

 

From (6), at  , we have

 

Also, from (8), at  , we have

 

Plugging into (11) gives

 

or

 

Equation (14) can then be solved to find the reflection and transmission coefficients   and  .

Extension to waves in anisotropic layered media

edit

In a layer medium where each of the layers is isotropic, the TE and TM waves are uncoupled at the interface. However, this is not true when each of the layers is anisotropic and we have to consider the full Maxwell's equations. The state vector approach can still be used for anisotropic media by choosing the variables such that they are continuous across interfaces.

Let us start with Maxwell's equations

 

Recall that continuity of the fields requires that the tangential components of   and   be continuous across material interfaces. Therefore, an appropriate state vector for anisotropic media is

 

where   and   are the tangential components of   and   (i.e., the components on the surface normal to the   direction).

Let us decompose the vector fields into a sum of the normal and tangential components:

 

The gradient operator can also be split along the same lines, i.e.,

 

where   are the unit vectors in the   directions, respectively. Let us express the tensors   and   in matrix form (with respect to the basis  ) as

 

where   are   matrices,   are   matrices,   are   matrices, and   are   matrices, i.e., scalars.

Using the splits of the various quantities and the gradient operator, we can show ({\Red work this out}) that   can be expressed in terms of   as

 

After some further manipulations, the Maxwell equations may be expressed in matrix form as (see ~Chew95 for details)

 

where   is a   matrix. If you compare equation (15) with (2), you will see that the form of the equations is the same, except that instead of a   matrix   for the isotropic case, we now have a   matrix.

Plane wave expansion of sources in a homogeneous medium

edit

It is often useful to expand sources in terms of plane waves so that the results of the previous lectures may be used directly on the basis of superposition. In this section we look at the expansion of point sources in terms of plane waves (for a homogeneous medium).

Let us look at the two-dimensional scalar wave equation (which can be used for acoustics, TE and TM waves, antiplane elasticity, etc.) In the presence of a point source, the wave equation has the form

 

Assume that   has a small positive imaginary part (it is a slightly lossy material), i.e.,

 

Expressed in cylindrical coordinates, equation (16) becomes (since the equation is symmetric about the origin)

 

The solution of (17) is

 

where   is a Hankel function of the first kind. \footnote{ Recall that a Hankel function of the first kind is defined as

 

where   is a Bessel function of the first kind and   is a Bessel function of the second kind. }

We can also solve (16) using Fourier transforms. Let us assume that the function   has a Fourier transform, i.e.,

 

Also note that

 

Plugging (19) and (20) into (16) gives

 

or,

 

Since the above equation holds for all values of  , the Fourier components must agree, i.e.,

 

Defining

 

we get

 

Note that now the equation has a source only at  . Away from the source (i.e.,  ), the right hand side of (21) is zero, and the solution corresponds to the homogeneous part of the equation. Therefore,

 

This solution must be matched with the singularity at  . This can be achieved by requiring that the solution have discontinuous second derivatives at  . We then have ({\Red full explanation is needed here}), considering only waves that are damping away from the source rather that those growing exponentially,

 

Plugging (23) into (19) gives

 

Equation (24) is a plane wave solution for the wave equation with a point source. So the point source has been converted into a sum of propagating plane waves and some evanescent terms.

Note that the denominator in (24) contains  . Hence, when   the solution blows up. Hence there are branch points at these locations as shown in Figure 3. In a lossless medium,   and these points appear as pole on the real   axis. The integral in equation (24) can then be computed using the residual theorem. The region between the two poles is where waves are allowed to propagate in the   direction while the region outside the poles is where these waves are evanescent.

 
Figure 3. Poles and integration path for plane wave solutions corresponding to a point source.

If we now compare the solutions (18) and (24), we have

 

which provides a definition for the Hankel function.

Also, differentiating (16) with respect to   and  , we get

 

Note that the products   and   correspond to dipole sources in the   and   directions, respectively.

Define

 

Therefore, from (24), we have

 

These are the plane wave expansions of dipoles in the   and   directions respectively. Taking higher derivatives gives results from quadrupoles and other multipoles.

Footnotes

edit
  1. This lecture closely follows the work of Chew~Chew95. Please refer to that text for further details.

References

edit

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