Waves in composites and metamaterials/Continuum limit and propagator matrix

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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 | edit source]

Recall from the previous lecture that we have been dealing with the TE equation [1]

where

For a a multilayered medium with layers, we found that in the -th layer

where is a generalized reflection coefficient. This coefficient can be obtained from a recursion relation of the form

where

is the Fresnel reflection coefficient for TE waves. Equation (3) may also be written as

We will now proceed to determine the generalized reflection coefficient in the continuum limit.

Continuum Limit[edit | edit source]

Consider a medium where the layer thickness if . Denote the reflection coefficient and the generalized reflection coefficient at the interface as and , respectively. Also denote the phase velocity just below the interface as and the permeability as . \footnote {This implies that we are measuring the phase velocity and the permeability at the center of the layer. However, this is not strictly necessary and we could alternatively measure these quantities at .}

Then, equation (2) can be written as

where

with

Expanding in Taylor series about and ignoring higher order terms, we get

Similarly, igoring powers and higher, we get

and

Plugging the expansions (7) and (8) into (5) gives

Substituting (6) into (9) and dropping terms containing and higher gives

If we assume that is small such that the denominator can be expanded in a series, we get

After expanding and ingoring terms containing , we get

or,

We thus get an equation that gives a continuous representation of the generalized reflection coefficient . Equation (13) can be solved numerically using the Runge-Kutta method.

For example, in the stuation shown in Figure 1, the generalized reflectivity coefficient at the point is while that at point is 0. If we wish to determine the value of at a point inside the smoothly varying layer, then one possibility is to assume that and is constant for and compute the value of in the usual manner.

Figure 1. Reflectivity in a smoothly graded layered material.

There can also be a situation where there are a few isolated strong discontinuities inside the graded layer as shown in Figure 2. If there is a discontinuity at , we can use the discrete solution with layer thickness 0 at the discontinuity.

Figure 2. Reflectivity in a smoothly graded material with a strong discontinuity.

Then, from (2), at the discontinuity

Also, from (4)

Hence we can find the generalized reflection coefficients at isolated discontinuities within the material.

Determining the coefficients [edit | edit source]

Recall equation (1):

So far we have determine the value of is this equation. But how do we determine the coefficients in multilayered media?

Let us start with the coefficients for a single layer that we determined in the previous lecture. We had

where

We can rewrite (16) as

Using the same arguments as before, we can generalize (17) to a medium with layers. Thus, for the -th layer, we have

Define

Then we can write (18) as

The second of equations (19) gives us a recurrence relation that can be used to compute the other s. Thus, we can write

We can introduce a generalized transmission coefficient

Then,

So the downgoing wave amplitude in region at is times the downgoing amplitude in region 1 (). Due to the products involved in the above relation, a continuum extension of this formula is not straightforward.

State equations and Propagator matrix[edit | edit source]

The propagator matrix relates the fields at two points in a multilayered medium. This matrix is also known as the transition matrix or the transfer matrix.

Let us examine the propagation matrix for a TM wave. Recall the governing equation for a TM wave:

where

Define . Also,

Therefore, (21) can be written as

To reduce (22) to a first order differential equation, introduce the quantity

Clearly, has to be continuous across the interface for the differential equation (22) to be satisfied.

Plugging (23) into (22) gives

Therefore, (23) and (24) for a system of differential equations which can be written as

Define

Then, equations (25) can be written as

If is constant, particular solutions to (26) can be sought of the form

Plugging (27) into (25) leads to the eigenvalue problem

Solutions exist only if

Therefore, the general solution of (26) is

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

Equation (28) can be written more compactly in the form

or,

where

Note that, for a point that is different from ,

Also,

Therefore we can write (30) in the form

or,

where

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

In a multilayered system (see Figure~3), since the vector is discontinuous, we can show that

where depends on and depends on .

Figure 3. Multilayered medium.

Footnotes[edit | edit source]

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

References[edit | edit source]

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