Waves in composites and metamaterials/Continuum limit and propagator matrix

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 that we have been dealing with the TE equation [1]




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




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

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






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


Similarly, igoring powers   and higher, we get




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




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 Aj edit

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




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




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




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

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:




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




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






Note that, for a point   that is different from  ,




Therefore we can write (30) in the form






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

  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.