Lecture
Let a plane monochromatic
wave be incident on a multilayer medium (fig. 1). It is required to find the field at any point of any
layer. Since each layer is homogeneous,
i.e. within each layer n —
= const, it can be assumed that the field
in each layer is represented as
the sum of the fields of two plane waves — a direct one
and a reflected one.
It is obvious that in this case any component
of any vector of the electromagnetic
field can be represented in the form
U£/|e^'-*o'"'>,
where z — the distance, measured along the direction of propagation
of the plane wave.
But this distance, as was shown in the previous lecture, can be
written as follows:
z
pl
ng
Fig.
t x
1
where
r=xsin (p+zcos <p.
Using this notation, we obtain
U=U(z)e-j(ü''-~k'o n"s',1 f\
U(z)=\U\e-jK»zncos (1)
(2)
We wish to construct the solution of the problem from plane waves,
whose vector components have the form (1) and (2). The unknowns
in these expressions are the constant amplitude of the wave
\U\ and the angle <p of its incidence on the boundary of the layer.
It is obvious that here we must distinguish two types of polarization
— horizontal and vertical. For brevity we shall call the horizontally
polarized wave a TE-wave, and the vertically
polarized wave — a TM-wave. So, let us assume
that the sought vector field components are as follows:
TE-wave TM-wave
E=Ey=UE(z)e^i(ωt-k0xnsin(p); H=Hy=U'H(z)e^(ωt-k0xnsin(p),n(p)\
Hx=WH(z)e>{-ωt-k0xnsin(p)\; Ey= — WE(z)e^(-k0xnsin(p)\
Hz=WH(z)e^""~k0xnsin(p); Ez=—WE(z)e^(-k0xnsin(p) (3)
Ex=Ez=0, Hy=0. Hx=Hz=0, Ey=0.
Let us substitute into Maxwell's equations
rot E=—jωμ0H;
rot H=jωε0E
the expressions (3), first writing these equations in scalar
form:
TE-wave TM-wave
∂E • ∂Ex ∂E, . and
∂E . r, ∂H . r,
—=-jωμ0Hz; -^=-jωε0Ex;
∂Hx ∂Hz . ∂H . -
— ~ -jω — =jωε0Ez.
From this it follows: ~;
dUH , dVE
-^~jωμ0H\ - -%r -jk0nsin (p «jωε0sin
— +jk0nsin (p WH=jωε0nUE. WE= "m, 4 UH.
Finally we obtain the system of equations
dz n0 ' dz
dVH dVE
- y =jωε0 cos^2(p UE; -β ~jωμ0cos^2 Let us compare these two systems of equations with the telegraph equations:
dU . . ,
It is known that the solution of this system consists of the voltage
and current U and I waves along the line, where the phase coefficient equals
the characteristic impedance
* . - ) / £ • '
and the solution of the system will be the expressions
V(C)=£ I cos ß C + I, Z sin ß;
I(:)=I- ^ sin ß; +I cos ß;,
where
U0, I0 — the voltage and current at the end of the line;
C — the coordinate, measured from the end of the line:
The results of the comparison are given below:
TE-wave TM-wave
UE--U, UH-+I,
VH-+I, VE-+U,
/
b 1 Z. - l f V0 _ _
" ^ c-cos(p^-c-cos(p» * ' V cos(p=ZBcos^ZeH,
UE(C) = UE0cos ßC-VH0 . W(C) = VE0cos ßC+;U cos^ZeH cos l/H(C) = I cos (p sin ßC+VWcos ßC, c IH(C) = I- ^ sin ßC+cIH0 cos ßC (4)
(C — the coordinate, measured from the end of the layer).
k0ncos
To construct the solution of the problem from the formulas given
above, it is convenient to use matrix calculus. In this connection
let us recall the elementary rules of matrix calculus.
A square matrix of the n-th order is written as follows:
'a11 a12...a1n
| | = A.
|an1 an2...ann
The product of matrices A and B is called the matrix, the elements c,-j
of which are equal to
n
c,7 = ∑aikbkj.
A matrix may consist of a single column, for example the column
matrix
W=(u)b
The product of the square matrix A and the column matrix U is
called the column matrix P, the elements of which are equal to
n
Pi = ∑aikCk.
k=1
On close examination of expressions (4), we find
that in the case of the TE-wave these expressions represent a matrix
column
obtained as a result of multiplying the square matrix
j - ^ - sin ßC
, cos^ , nr
I -
by the column matrix
i.e.
CM0=ME(C)QE.
Similarly, as is easy to see, for the TM-wave
QH(C)=MH(C)QH,
(6)
(7)
where
jZ0cos(p sin[k
cos ߣ
c IH,
(8)
The matrices (5) and (8) are called the characteristic matrices
of the layered medium.
Note that the determinant of both matrices equals unity.
We have established a complete analogy between the problem of the propagation
of electromagnetic waves in a multilayer medium and the problem of the propagation
of voltage and current waves in a series-connected
chain of transmission-line segments. In order to calculate the voltage
and current in a given segment of this chain, it is sufficient to know the parameters
of that line segment and the load at its end. In the problem under
consideration the situation is fully analogous. In order to calculate
the field inside a given layer, one needs to know, using the terminology
of matrix theory, the characteristic matrix of that layer,
and the column matrix of the amplitudes of the field components.
The characteristic matrix of each layer can be calculated from
the values of the refractive indices in each layer, using
Snell's law for this purpose.
So, suppose it is required to determine the field inside the first layer
(fig. 1). According to formula (6) or (7) we can write
Suppose that we know the matrix Q at the boundary of the N-th layer,
then according to (9) we can write
03)
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