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.
Recall from the previous lecture that we have been dealing with the
TE equation [1]
![{\displaystyle \left[\mu (z)~{\cfrac {d}{dz}}\left(\mu (z)^{-1}~{\cfrac {d}{dz}}\right)+\omega ^{2}~\epsilon (z)~\mu (z)-k_{x}^{2}\right]~{\tilde {E}}_{y}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32a1cecc4cb6581d22ac4cb9327c804315b13b8f)
where

For a a multilayered medium with
layers, we found that in the
-th
layer
![{\displaystyle {\text{(1)}}\qquad {\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dffdb5ba30b3db50368736fd345ccf28881d19c)
where
is a generalized reflection coefficient. This
coefficient can be obtained from a recursion relation of the form
![{\displaystyle {\text{(2)}}\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})]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04a55081b7f12ec8a704a10313a337cfdf7d8660)
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.
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
![{\displaystyle {\text{(5)}}\qquad {\tilde {R}}(z)={\cfrac {R(z)+{\tilde {R}}(z-\Delta )~\exp \left[2~i~k_{z}\left(z-\Delta /2\right)~\Delta \right]}{1+R(z)~{\tilde {R}}(z-\Delta )~\exp \left[2~i~k_{z}\left(z-\Delta /2\right)~\Delta \right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8bfe1acb30f872aa9c4b0b7e03cf53460ca4a0)
where

with

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

Similarly, igoring powers
and higher, we get
![{\displaystyle {\text{(7)}}\qquad \exp \left[2~i~k_{z}\left(z-\Delta /2\right)~\Delta \right]\approx 1+2~\Delta ~i~k_{z}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5342e6990b42da36083b606bb26138143f82ab8)
and

Plugging the expansions (7) and
(8) into (5) gives
![{\displaystyle {\text{(9)}}\qquad {\tilde {R}}(z)\approx {\cfrac {R(z)+\left[{\tilde {R}}(z)-\Delta ~{\tilde {R}}^{'}(z)\right]~\left[1+2~\Delta ~i~k_{z}(z)\right]}{1+R(z)~\left[{\tilde {R}}(z)-\Delta ~{\tilde {R}}^{'}(z)\right]~\left[1+2~\Delta ~i~k_{z}(z)\right]}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/343303c90afd758991e36d1d5032f9565440037f)
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
![{\displaystyle {\text{(11)}}\qquad {\tilde {R}}(z)\approx \left[{\cfrac {\Delta ~{\tilde {k}}^{'}(z)}{2~{\tilde {k}}(z)}}+{\tilde {R}}(z)-\Delta ~{\tilde {R}}^{'}(z)+2~\Delta ~i~k_{z}(z)~{\tilde {R}}(z)\right]\left[1-{\cfrac {\Delta ~{\tilde {k}}^{'}(z)}{2~{\tilde {k}}(z)}}~{\tilde {R}}(z)+O(\Delta ^{2})\right]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149eca84b7006b22e26175b2c025b238c7e7118c)
After expanding and ingoring terms containing
, we get
![{\displaystyle {\text{(12)}}\qquad {\tilde {R}}(z)\approx {\cfrac {\Delta ~{\tilde {k}}^{'}(z)}{2~{\tilde {k}}(z)}}+{\tilde {R}}(z)-\Delta ~{\tilde {R}}^{'}(z)+2~\Delta ~i~k_{z}(z)~{\tilde {R}}(z)-{\cfrac {\Delta ~{\tilde {k}}^{'}(z)}{2~{\tilde {k}}(z)}}~[{\tilde {R}}(z)]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7548d6d2536d8afe12f859af4efafe1df7bd4513)
or,
![{\displaystyle {\text{(13)}}\qquad {{\tilde {R}}^{'}(z)=2~i~k_{z}(z)~{\tilde {R}}(z)+{\cfrac {{\tilde {k}}^{'}(z)}{2~{\tilde {k}}(z)}}\left\{1-[{\tilde {R}}(z)]^{2}\right\}~.}\qquad {\text{Ricotti equation}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5079034465f79d9bccb8ca0c20a5b706905f9ab1)
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):
![{\displaystyle {\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]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb44332055530d822cc6cf6d3d7e4049b6c7278)
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
![{\displaystyle {\text{(16)}}\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})]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db39b2c6ef5ffa223dc36140ce0810fe91b6fe6a)
where

We can rewrite (16) as
![{\displaystyle {\text{(17)}}\qquad A_{2}~\exp(i~k_{z2}~d1)={\cfrac {T_{12}~A_{1}~\exp(i~k_{z1}~d_{1})}{1-R_{21}~R_{23}~\exp[2~i~k_{z2}~(d_{2}-d_{1})]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/746fbd3e461b2b43ffa465645325b7619cbcf606)
Using the same arguments as before, we can generalize (17) to
a medium with
layers. Thus, for the
-th layer, we have
![{\displaystyle {\text{(18)}}\qquad A_{j}~\exp(i~k_{z,j}~d_{j-1})={\cfrac {T_{j-1,j}~A_{j-1}~\exp(i~k_{z,j-1}~d_{j-1})}{1-R_{j,j-1}~{\tilde {R}}_{j,j+1}~\exp[2~i~k_{z,j}~(d_{j}-d_{j-1})]}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d31a0bdc7f4394335f894307bbd8275f4477164e)
Define
![{\displaystyle S_{j-1,j}:={\cfrac {T_{j-1,j}}{1-R_{j,j-1}~{\tilde {R}}_{j,j+1}~\exp[2~i~k_{z,j}~(d_{j}-d_{j-1})]}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b21defe7604d26faedd267d094bccba6f5ccc409)
Then we can write (18) as
![{\displaystyle {\text{(19)}}\qquad {\begin{aligned}A_{j}~\exp(i~k_{z,j}~d_{j-1})&=S_{j-1,j}~A_{j-1}~\exp(i~k_{z,j-1}~d_{j-1})\\&=S_{j-1,j}~A_{j-1}~\exp(i~k_{z,j-1}~d_{j-2})~\exp[i~k_{z,j-1}~(d_{j-1}-d_{j-2})]~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72944e46696a2358bec9bc5a40596b478530e3e4)
The second of equations (19) gives us a recurrence relation
that can be used to compute the other
s. Thus, we can write
![{\displaystyle {\text{(20)}}\qquad {\begin{aligned}A_{j}~\exp(i~k_{z,j}~d_{j-1})&=A_{j-1}~\exp(i~k_{z,j-1}~d_{j-2})~S_{j-1,j}~\exp[i~k_{z,j-1}~(d_{j-1}-d_{j-2})]\\&=A_{j-2}~\exp(i~k_{z,j-2}~d_{j-3})~S_{j-2,j-1}~\exp[i~k_{z,j-2}~(d_{j-2}-d_{j-3})]~S_{j-1,j}~\exp[i~k_{z,j-1}~(d_{j-1}-d_{j-2})]\\&=\dots \\&=A_{1}~\exp(i~k_{z1}~d_{1})~\prod _{m=1}^{j-1}S_{m,m+1}~\exp[i~k_{z,m}~(d_{m}-d_{m-1})]~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e299bc78f78b758539ab379c8111faca02b0ace9)
We can introduce a generalized transmission coefficient
![{\displaystyle {\tilde {T}}_{1N}:=\prod _{m=1}^{N-1}S_{m,m+1}~\exp[i~k_{z,m}~(d_{m}-d_{m-1})]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0f60568614ef0fffcb5f2245debc43bd7585c4)
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:
![{\displaystyle {\text{(21)}}\qquad \left[\epsilon (z)~{\cfrac {d}{dz}}\left(\epsilon (z)^{-1}~{\cfrac {d}{dz}}\right)+\omega ^{2}~\epsilon (z)~\mu (z)-k_{x}^{2}\right]~{\tilde {H}}_{y}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/435d8f18592dc58252e6502d29893f78b417198d)
where

Define
. Also,

Therefore, (21) can be written as
![{\displaystyle {\text{(22)}}\qquad \left[\epsilon (z)~{\cfrac {d}{dz}}\left(\epsilon (z)^{-1}~{\cfrac {d}{dz}}\right)+k_{z}^{2}\right]~\varphi =0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a712b0c37805e7dcdbc51dc137d5e0daff7c0339)
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.
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- ↑ This lecture closely follows the work of
Chew~Chew95. Please refer to that text for further details.
W. C. Chew. Waves and fields in inhomogeneous media. IEEE Press, New York, 1995.