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]
[
μ
(
z
)
d
d
z
(
μ
(
z
)
−
1
d
d
z
)
+
ω
2
ϵ
(
z
)
μ
(
z
)
−
k
x
2
]
E
~
y
=
0
{\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}
where
E
y
(
x
,
z
)
=
E
~
y
(
z
)
e
±
i
k
x
x
.
{\displaystyle E_{y}(x,z)={\tilde {E}}_{y}(z)~e^{\pm i~k_{x}~x}~.}
For a a multilayered medium with
N
{\displaystyle N}
layers, we found that in the
j
{\displaystyle j}
-th
layer
(1)
E
~
y
j
(
z
)
=
A
j
[
exp
(
−
i
k
z
j
z
)
+
R
~
j
,
j
+
1
exp
(
i
k
z
j
z
+
2
i
k
z
j
d
j
)
]
{\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]}
where
R
~
j
,
j
+
1
{\displaystyle {\tilde {R}}_{j,j+1}}
is a generalized reflection coefficient. This
coefficient can be obtained from a recursion relation of the form
(2)
R
~
j
,
j
+
1
=
R
j
,
j
+
1
+
R
~
j
+
1
,
j
+
2
exp
[
2
i
k
z
,
j
+
1
(
d
j
+
1
−
d
j
)
]
1
+
R
j
,
j
+
1
R
~
j
+
1
,
j
+
2
exp
[
2
i
k
z
,
j
+
1
(
d
j
+
1
−
d
j
)
]
{\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})]}}}
where
(3)
R
j
,
j
+
1
=
μ
j
+
1
k
z
,
j
−
μ
j
k
z
,
j
+
1
μ
j
+
1
k
z
,
j
+
μ
j
k
z
,
j
+
1
{\displaystyle {\text{(3)}}\qquad 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 TE waves. Equation (3)
may also be written as
(4)
R
j
,
j
+
1
=
k
z
,
j
μ
j
−
k
z
,
j
+
1
μ
j
+
1
k
z
,
j
μ
j
+
k
z
,
j
+
1
μ
j
+
1
.
{\displaystyle {\text{(4)}}\qquad R_{j,j+1}={\cfrac {{\cfrac {k_{z,j}}{\mu _{j}}}-{\cfrac {k_{z,j+1}}{\mu _{j+1}}}}{{\cfrac {k_{z,j}}{\mu _{j}}}+{\cfrac {k_{z,j+1}}{\mu _{j+1}}}}}~.}
We will now proceed to determine the generalized reflection coefficient
in the continuum limit.
Consider a medium where the layer thickness if
Δ
{\displaystyle \Delta }
. Denote the
reflection coefficient and the generalized reflection coefficient at the
interface
z
=
−
d
j
{\displaystyle z=-d_{j}}
as
R
(
z
)
{\displaystyle R(z)}
and
R
~
(
z
)
{\displaystyle {\tilde {R}}(z)}
, respectively. Also
denote the phase velocity
k
z
,
j
+
1
{\displaystyle k_{z,j+1}}
just below the interface as
k
z
(
z
−
Δ
/
2
)
{\displaystyle k_{z}(z-\Delta /2)}
and the permeability
μ
j
+
1
{\displaystyle \mu _{j+1}}
as
μ
(
z
−
Δ
/
2
)
{\displaystyle \mu (z-\Delta /2)}
.
\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
z
−
Δ
{\displaystyle z-\Delta }
.}
Then, equation (2) can be written as
(5)
R
~
(
z
)
=
R
(
z
)
+
R
~
(
z
−
Δ
)
exp
[
2
i
k
z
(
z
−
Δ
/
2
)
Δ
]
1
+
R
(
z
)
R
~
(
z
−
Δ
)
exp
[
2
i
k
z
(
z
−
Δ
/
2
)
Δ
]
{\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]}}}
where
R
(
z
)
=
k
z
(
z
+
Δ
/
2
)
μ
(
z
+
Δ
/
2
)
−
k
z
(
z
−
Δ
/
2
)
μ
(
z
−
Δ
/
2
)
k
z
(
z
+
Δ
/
2
)
μ
(
z
+
Δ
/
2
)
+
k
z
(
z
−
Δ
/
2
)
μ
(
z
−
Δ
/
2
)
=
k
~
(
z
+
Δ
/
2
)
−
k
~
(
z
−
Δ
/
2
)
k
~
(
z
+
Δ
/
2
)
+
k
~
(
z
−
Δ
/
2
)
{\displaystyle R(z)={\cfrac {{\cfrac {k_{z}(z+\Delta /2)}{\mu (z+\Delta /2)}}-{\cfrac {k_{z}(z-\Delta /2)}{\mu (z-\Delta /2)}}}{{\cfrac {k_{z}(z+\Delta /2)}{\mu (z+\Delta /2)}}+{\cfrac {k_{z}(z-\Delta /2)}{\mu (z-\Delta /2)}}}}={\cfrac {{\tilde {k}}(z+\Delta /2)-{\tilde {k}}(z-\Delta /2)}{{\tilde {k}}(z+\Delta /2)+{\tilde {k}}(z-\Delta /2)}}}
with
k
~
:=
k
z
μ
.
{\displaystyle {\tilde {k}}:={\cfrac {k_{z}}{\mu }}~.}
Expanding in Taylor series about
z
{\displaystyle z}
and ignoring higher order terms, we get
(6)
R
(
z
)
≈
k
~
(
z
)
+
Δ
/
2
k
~
′
(
z
)
−
k
~
(
z
)
+
Δ
/
2
k
~
′
(
z
)
k
~
(
z
)
+
Δ
/
2
k
~
′
(
z
)
+
k
~
(
z
)
−
Δ
/
2
k
~
′
(
z
)
=
Δ
k
~
′
(
z
)
2
k
~
(
z
)
.
{\displaystyle {\text{(6)}}\qquad R(z)\approx {\cfrac {{\tilde {k}}(z)+\Delta /2~{\tilde {k}}^{'}(z)-{\tilde {k}}(z)+\Delta /2~{\tilde {k}}^{'}(z)}{{\tilde {k}}(z)+\Delta /2~{\tilde {k}}^{'}(z)+{\tilde {k}}(z)-\Delta /2~{\tilde {k}}^{'}(z)}}={\cfrac {\Delta ~{\tilde {k}}^{'}(z)}{2~{\tilde {k}}(z)}}~.}
Similarly, igoring powers
Δ
2
{\displaystyle \Delta ^{2}}
and higher, we get
(7)
exp
[
2
i
k
z
(
z
−
Δ
/
2
)
Δ
]
≈
1
+
2
Δ
i
k
z
(
z
)
{\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)}
and
(8)
R
~
(
z
−
Δ
)
≈
R
~
(
z
)
−
Δ
R
~
′
(
z
)
.
{\displaystyle {\text{(8)}}\qquad {\tilde {R}}(z-\Delta )\approx {\tilde {R}}(z)-\Delta ~{\tilde {R}}^{'}(z)~.}
Plugging the expansions (7) and
(8) into (5) gives
(9)
R
~
(
z
)
≈
R
(
z
)
+
[
R
~
(
z
)
−
Δ
R
~
′
(
z
)
]
[
1
+
2
Δ
i
k
z
(
z
)
]
1
+
R
(
z
)
[
R
~
(
z
)
−
Δ
R
~
′
(
z
)
]
[
1
+
2
Δ
i
k
z
(
z
)
]
.
{\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]}}~.}
Substituting (6) into (9) and dropping terms
containing
Δ
2
{\displaystyle \Delta ^{2}}
and higher gives
(10)
R
~
(
z
)
≈
Δ
k
~
′
(
z
)
2
k
~
(
z
)
+
R
~
(
z
)
−
Δ
R
~
′
(
z
)
+
2
Δ
i
k
z
(
z
)
R
~
(
z
)
1
+
Δ
k
~
′
(
z
)
2
k
~
(
z
)
R
~
(
z
)
.
{\displaystyle {\text{(10)}}\qquad {\tilde {R}}(z)\approx {\cfrac {{\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)}{1+{\cfrac {\Delta ~{\tilde {k}}^{'}(z)}{2~{\tilde {k}}(z)}}~{\tilde {R}}(z)}}~.}
If we assume that
Δ
k
~
′
{\displaystyle \Delta ~{\tilde {k}}^{'}}
is small such that
the denominator can be expanded in a series, we get
(11)
R
~
(
z
)
≈
[
Δ
k
~
′
(
z
)
2
k
~
(
z
)
+
R
~
(
z
)
−
Δ
R
~
′
(
z
)
+
2
Δ
i
k
z
(
z
)
R
~
(
z
)
]
[
1
−
Δ
k
~
′
(
z
)
2
k
~
(
z
)
R
~
(
z
)
+
O
(
Δ
2
)
]
.
{\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]~.}
After expanding and ingoring terms containing
Δ
2
{\displaystyle \Delta ^{2}}
, we get
(12)
R
~
(
z
)
≈
Δ
k
~
′
(
z
)
2
k
~
(
z
)
+
R
~
(
z
)
−
Δ
R
~
′
(
z
)
+
2
Δ
i
k
z
(
z
)
R
~
(
z
)
−
Δ
k
~
′
(
z
)
2
k
~
(
z
)
[
R
~
(
z
)
]
2
{\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}}
or,
(13)
R
~
′
(
z
)
=
2
i
k
z
(
z
)
R
~
(
z
)
+
k
~
′
(
z
)
2
k
~
(
z
)
{
1
−
[
R
~
(
z
)
]
2
}
.
Ricotti equation
{\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}}}
We thus get an equation that gives a continuous representation of the
generalized reflection coefficient
R
~
(
z
)
{\displaystyle {\tilde {R}}(z)}
.
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
z
1
{\displaystyle z_{1}}
is
R
~
(
z
1
)
{\displaystyle {\tilde {R}}(z_{1})}
while that at point
z
0
{\displaystyle z_{0}}
is 0. If we wish to determine the value
of
R
~
(
z
i
)
{\displaystyle {\tilde {R}}(z_{i})}
at a point inside the smoothly varying layer, then one
possibility is to assume that
μ
(
z
)
{\displaystyle \mu (z)}
and
ϵ
(
z
)
{\displaystyle \epsilon (z)}
is constant
for
z
>
z
i
{\displaystyle z>z_{i}}
and compute the value of
R
~
{\displaystyle {\tilde {R}}}
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
z
c
{\displaystyle z_{c}}
, 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
(14)
R
~
(
z
c
+
)
=
R
(
z
c
)
+
R
~
(
z
c
−
)
1
+
R
(
z
c
)
R
~
(
z
c
−
)
.
{\displaystyle {\text{(14)}}\qquad {\tilde {R}}(z_{c}^{+})={\cfrac {R(z_{c})+{\tilde {R}}(z_{c}^{-})}{1+R(z_{c})~{\tilde {R}}(z_{c}^{-})}}~.}
Also, from (4)
(15)
R
(
z
c
)
=
k
z
+
μ
+
−
k
z
−
μ
−
k
z
+
μ
+
+
k
z
−
μ
−
.
{\displaystyle {\text{(15)}}\qquad R(z_{c})={\cfrac {{\cfrac {k_{z}^{+}}{\mu ^{+}}}-{\cfrac {k_{z}^{-}}{\mu ^{-}}}}{{\cfrac {k_{z}^{+}}{\mu ^{+}}}+{\cfrac {k_{z}^{-}}{\mu ^{-}}}}}~.}
Hence we can find the generalized reflection coefficients at isolated
discontinuities within the material.
Determining the coefficients Aj
edit
Recall equation (1):
E
~
y
j
(
z
)
=
A
j
[
exp
(
−
i
k
z
j
z
)
+
R
~
j
,
j
+
1
exp
(
i
k
z
j
z
+
2
i
k
z
j
d
j
)
]
.
{\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]~.}
So far we have determine the value of
R
~
{\displaystyle {\tilde {R}}}
is this equation. But
how do we determine the coefficients
A
j
{\displaystyle A_{j}}
in multilayered media?
Let us start with the coefficients for a single layer that we determined
in the previous lecture. We had
(16)
A
2
=
T
12
A
1
exp
[
i
(
k
z
1
−
k
z
2
)
d
1
]
1
−
R
21
R
23
exp
[
2
i
k
z
2
(
d
2
−
d
1
)
]
{\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})]}}}
where
T
12
=
2
μ
2
k
z
1
μ
2
k
z
1
+
μ
1
k
z
2
.
{\displaystyle T_{12}={\cfrac {2~\mu _{2}~k_{z1}}{\mu _{2}~k_{z1}+\mu _{1}~k_{z2}}}~.}
We can rewrite (16) as
(17)
A
2
exp
(
i
k
z
2
d
1
)
=
T
12
A
1
exp
(
i
k
z
1
d
1
)
1
−
R
21
R
23
exp
[
2
i
k
z
2
(
d
2
−
d
1
)
]
{\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})]}}}
Using the same arguments as before, we can generalize (17) to
a medium with
N
{\displaystyle N}
layers. Thus, for the
j
{\displaystyle j}
-th layer, we have
(18)
A
j
exp
(
i
k
z
,
j
d
j
−
1
)
=
T
j
−
1
,
j
A
j
−
1
exp
(
i
k
z
,
j
−
1
d
j
−
1
)
1
−
R
j
,
j
−
1
R
~
j
,
j
+
1
exp
[
2
i
k
z
,
j
(
d
j
−
d
j
−
1
)
]
.
{\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})]}}~.}
Define
S
j
−
1
,
j
:=
T
j
−
1
,
j
1
−
R
j
,
j
−
1
R
~
j
,
j
+
1
exp
[
2
i
k
z
,
j
(
d
j
−
d
j
−
1
)
]
.
{\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})]}}~.}
Then we can write (18) as
(19)
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
)
]
.
{\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}}}
The second of equations (19) gives us a recurrence relation
that can be used to compute the other
A
j
{\displaystyle A_{j}}
s. Thus, we can write
(20)
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
)
]
=
…
=
A
1
exp
(
i
k
z
1
d
1
)
∏
m
=
1
j
−
1
S
m
,
m
+
1
exp
[
i
k
z
,
m
(
d
m
−
d
m
−
1
)
]
.
{\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}}}
We can introduce a generalized transmission coefficient
T
~
1
N
:=
∏
m
=
1
N
−
1
S
m
,
m
+
1
exp
[
i
k
z
,
m
(
d
m
−
d
m
−
1
)
]
.
{\displaystyle {\tilde {T}}_{1N}:=\prod _{m=1}^{N-1}S_{m,m+1}~\exp[i~k_{z,m}~(d_{m}-d_{m-1})]~.}
Then,
A
N
exp
(
i
k
z
,
N
d
N
−
1
)
=
T
~
1
N
A
1
exp
(
i
k
z
1
d
1
)
.
{\displaystyle {A_{N}~\exp(i~k_{z,N}~d_{N-1})={\tilde {T}}_{1N}~A_{1}~\exp(i~k_{z1}~d_{1})~.}}
So the downgoing wave amplitude in region
N
{\displaystyle N}
at
z
=
−
d
n
+
1
{\displaystyle z=-d_{n+1}}
is
T
~
1
N
{\displaystyle {\tilde {T}}_{1N}}
times the downgoing amplitude in region 1 (
z
=
−
d
1
{\displaystyle z=-d_{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:
(21)
[
ϵ
(
z
)
d
d
z
(
ϵ
(
z
)
−
1
d
d
z
)
+
ω
2
ϵ
(
z
)
μ
(
z
)
−
k
x
2
]
H
~
y
=
0
{\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}
where
H
y
(
x
,
z
)
=
H
~
y
(
z
)
e
±
i
k
x
x
.
{\displaystyle H_{y}(x,z)={\tilde {H}}_{y}(z)~e^{\pm i~k_{x}~x}~.}
Define
φ
:=
H
~
y
{\displaystyle \varphi :={\tilde {H}}_{y}}
. Also,
k
z
2
=
ω
2
ϵ
(
z
)
μ
(
z
)
−
k
x
2
.
{\displaystyle k_{z}^{2}=\omega ^{2}~\epsilon (z)~\mu (z)-k_{x}^{2}~.}
Therefore, (21) can be written as
(22)
[
ϵ
(
z
)
d
d
z
(
ϵ
(
z
)
−
1
d
d
z
)
+
k
z
2
]
φ
=
0
.
{\displaystyle {\text{(22)}}\qquad \left[\epsilon (z)~{\cfrac {d}{dz}}\left(\epsilon (z)^{-1}~{\cfrac {d}{dz}}\right)+k_{z}^{2}\right]~\varphi =0~.}
To reduce (22) to a first order differential equation,
introduce the quantity
(23)
ψ
:=
1
i
ω
ϵ
d
φ
d
z
⟹
d
φ
d
z
=
i
ω
ϵ
ψ
.
{\displaystyle {\text{(23)}}\qquad \psi :={\cfrac {1}{i\omega \epsilon }}~{\cfrac {d\varphi }{dz}}\qquad \implies \qquad {\cfrac {d\varphi }{dz}}=i\omega \epsilon ~\psi ~.}
Clearly,
ψ
{\displaystyle \psi }
has to be continuous across the interface for the
differential equation (22) to be satisfied.
Plugging (23) into (22) gives
(24)
d
ψ
d
z
=
i
k
z
2
ϵ
ω
φ
.
{\displaystyle {\text{(24)}}\qquad {\cfrac {d\psi }{dz}}={\cfrac {i~k_{z}^{2}}{\epsilon ~\omega }}~\varphi ~.}
Therefore, (23)
2
{\displaystyle _{2}}
and (24) for a system
of differential equations which can be written as
(25)
d
d
z
[
φ
ψ
]
=
[
0
i
ω
ϵ
i
k
z
2
ϵ
ω
0
]
[
φ
ψ
]
.
{\displaystyle {\text{(25)}}\qquad {\cfrac {d}{dz}}{\begin{bmatrix}\varphi \\\psi \end{bmatrix}}={\begin{bmatrix}0&i\omega \epsilon \\{\cfrac {i~k_{z}^{2}}{\epsilon ~\omega }}&0\end{bmatrix}}~{\begin{bmatrix}\varphi \\\psi \end{bmatrix}}~.}
Define
v
:=
[
φ
ψ
]
;
H
¯
:=
[
0
i
ω
ϵ
i
k
z
2
ϵ
ω
0
]
.
{\displaystyle \mathbf {v} :={\begin{bmatrix}\varphi \\\psi \end{bmatrix}}~;~~{\overline {\mathbf {H} }}:={\begin{bmatrix}0&i\omega \epsilon \\{\cfrac {i~k_{z}^{2}}{\epsilon ~\omega }}&0\end{bmatrix}}~.}
Then, equations (25) can be written as
(26)
d
v
d
z
=
H
¯
v
.
{\displaystyle {\text{(26)}}\qquad {\cfrac {d\mathbf {v} }{dz}}={\overline {\mathbf {H} }}~\mathbf {v} ~.}
If
H
¯
{\displaystyle {\overline {\mathbf {H} }}}
is constant, particular solutions to (26) can be
sought of the form
(27)
v
=
v
0
e
λ
z
.
{\displaystyle {\text{(27)}}\qquad \mathbf {v} =\mathbf {v} _{0}~e^{\lambda ~z}~.}
Plugging (27) into (25) leads to the eigenvalue
problem
(
H
¯
−
λ
1
)
v
0
=
0
.
{\displaystyle ({\overline {\mathbf {H} }}-\lambda ~\mathbf {1} )~\mathbf {v} _{0}=\mathbf {0} ~.}
Solutions exist only if
det
(
H
¯
−
λ
1
)
=
0
⟹
λ
2
=
−
k
z
2
⟹
λ
=
±
i
k
z
.
{\displaystyle \det({\overline {\mathbf {H} }}-\lambda ~\mathbf {1} )=0\qquad \implies \qquad \lambda ^{2}=-k_{z}^{2}\qquad \implies \qquad \lambda =\pm i~k_{z}~.}
Therefore, the general solution of (26) is
(28)
v
(
z
)
=
A
+
e
i
k
z
z
n
+
+
A
−
e
−
i
k
z
z
n
−
{\displaystyle {\text{(28)}}\qquad \mathbf {v} (z)=A_{+}~e^{i~k_{z}~z}~\mathbf {n} _{+}+A_{-}~e^{-i~k_{z}~z}~\mathbf {n} _{-}}
where
n
+
{\displaystyle \mathbf {n} _{+}}
and
n
−
{\displaystyle \mathbf {n} _{-}}
are the eigenvectors corresponding to the
eigenvalues
i
k
z
{\displaystyle i~k_{z}}
and
−
i
k
z
{\displaystyle -i~k_{z}}
, respectively.
Equation (28) can be written more compactly in the form
(29)
v
(
z
)
=
[
n
+
n
−
]
[
e
i
k
z
z
0
0
e
−
i
k
z
z
]
[
A
+
A
−
]
{\displaystyle {\text{(29)}}\qquad \mathbf {v} (z)={\begin{bmatrix}\mathbf {n} _{+}&\mathbf {n} _{-}\end{bmatrix}}{\begin{bmatrix}e^{i~k_{z}~z}&0\\0&e^{-i~k_{z}~z}\end{bmatrix}}{\begin{bmatrix}A_{+}\\A_{-}\end{bmatrix}}}
or,
(30)
v
(
z
)
=
n
~
K
¯
(
z
)
a
~
{\displaystyle {\text{(30)}}\qquad \mathbf {v} (z)={\tilde {\mathbf {n} }}~{\overline {\mathbf {K} }}(z)~{\tilde {\mathbf {a} }}}
where
n
~
:=
[
n
+
n
−
]
;
K
¯
(
z
)
:=
[
e
i
k
z
z
0
0
e
−
i
k
z
z
]
;
a
~
:=
[
A
+
A
−
]
.
{\displaystyle {\tilde {\mathbf {n} }}:={\begin{bmatrix}\mathbf {n} _{+}&\mathbf {n} _{-}\end{bmatrix}}~;~~{\overline {\mathbf {K} }}(z):={\begin{bmatrix}e^{i~k_{z}~z}&0\\0&e^{-i~k_{z}~z}\end{bmatrix}}~;~~{\tilde {\mathbf {a} }}:={\begin{bmatrix}A_{+}\\A_{-}\end{bmatrix}}~.}
Note that, for a point
z
′
{\displaystyle z'}
that is different from
z
{\displaystyle z}
,
K
¯
(
z
−
z
′
)
=
[
e
i
k
z
(
z
−
z
′
)
0
0
e
−
i
k
z
(
z
−
z
′
)
]
=
[
e
i
k
z
z
0
0
e
−
i
k
z
z
]
[
e
−
i
k
z
z
′
0
0
e
i
k
z
z
′
]
=
K
¯
(
z
)
K
¯
(
−
z
′
)
.
{\displaystyle {\overline {\mathbf {K} }}(z-z')={\begin{bmatrix}e^{i~k_{z}~(z-z')}&0\\0&e^{-i~k_{z}~(z-z')}\end{bmatrix}}={\begin{bmatrix}e^{i~k_{z}~z}&0\\0&e^{-i~k_{z}~z}\end{bmatrix}}{\begin{bmatrix}e^{-i~k_{z}~z'}&0\\0&e^{i~k_{z}~z'}\end{bmatrix}}={\overline {\mathbf {K} }}(z)~{\overline {\mathbf {K} }}(-z')~.}
Also,
K
¯
(
z
−
z
′
)
K
¯
(
z
′
)
=
[
e
i
k
z
(
z
−
z
′
)
0
0
e
−
i
k
z
(
z
−
z
′
)
]
[
e
−
i
k
z
z
′
0
0
e
i
k
z
z
′
]
=
K
¯
(
z
)
.
{\displaystyle {\overline {\mathbf {K} }}(z-z')~{\overline {\mathbf {K} }}(z')={\begin{bmatrix}e^{i~k_{z}~(z-z')}&0\\0&e^{-i~k_{z}~(z-z')}\end{bmatrix}}{\begin{bmatrix}e^{-i~k_{z}~z'}&0\\0&e^{i~k_{z}~z'}\end{bmatrix}}={\overline {\mathbf {K} }}(z)~.}
Therefore we can write (30) in the form
v
(
z
)
=
n
~
K
¯
(
z
−
z
′
)
K
¯
(
z
′
)
a
~
=
n
~
K
¯
(
z
−
z
′
)
n
~
−
1
n
~
K
¯
(
z
′
)
a
~
{\displaystyle \mathbf {v} (z)={\tilde {\mathbf {n} }}~{\overline {\mathbf {K} }}(z-z')~{\overline {\mathbf {K} }}(z')~{\tilde {\mathbf {a} }}={\tilde {\mathbf {n} }}~{\overline {\mathbf {K} }}(z-z')~{\tilde {\mathbf {n} }}^{-1}~~{\tilde {\mathbf {n} }}~{\overline {\mathbf {K} }}(z')~{\tilde {\mathbf {a} }}}
or,
(31)
v
(
z
)
=
P
¯
(
z
,
z
′
)
v
(
z
′
)
{\displaystyle {\text{(31)}}\qquad {\mathbf {v} (z)={\overline {\mathbf {P} }}(z,z')~\mathbf {v} (z')}}
where
P
¯
(
z
,
z
′
)
:=
n
~
K
¯
(
z
−
z
′
)
n
~
−
1
.
{\displaystyle {{\overline {\mathbf {P} }}(z,z'):={\tilde {\mathbf {n} }}~{\overline {\mathbf {K} }}(z-z')~{\tilde {\mathbf {n} }}^{-1}~.}}
The matrix
P
¯
{\displaystyle {\overline {\mathbf {P} }}}
is called the propagator matrix or the
transition matrix that related the fields at
z
{\displaystyle z}
and
z
′
{\displaystyle z'}
.
In a multilayered system (see Figure~3), since the
vector
v
{\displaystyle \mathbf {v} }
is discontinuous, we can show that
P
¯
(
−
d
2
,
0
)
=
P
¯
(
−
d
2
,
−
d
1
)
P
¯
(
−
d
1
,
0
)
{\displaystyle {\overline {\mathbf {P} }}(-d_{2},0)={\overline {\mathbf {P} }}(-d_{2},-d_{1})~{\overline {\mathbf {P} }}(-d_{1},0)}
where
P
¯
(
−
d
2
,
−
d
1
)
{\displaystyle {\overline {\mathbf {P} }}(-d_{2},-d_{1})}
depends on
ϵ
3
,
μ
3
{\displaystyle \epsilon _{3},\mu _{3}}
and
P
¯
(
−
d
1
,
0
)
{\displaystyle {\overline {\mathbf {P} }}(-d_{1},0)}
depends on
ϵ
2
,
μ
2
{\displaystyle \epsilon _{2},\mu _{2}}
.
Figure 3. Multilayered medium.
↑ 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.