2.5 Wave propagation in layered mediaAdvanced theoryIn the layer A, we introduced two approaches,
which enable to solve electromagnetic wave propagation in a layered medium. The first
approach was based on the distribution of field intensities and on the basic
boundary condition.
The second approach was based on the transformation of reflection coefficient and was
identical with the method, which is used for solving similar tasks on transmission lines,
in a fact. In this layer, we are going to introduce another approach, which makes the
solution more formal, and therefore, it is suitable for more complicated situations.
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Fig. 2.5B.1 | Wave propagation in layered medium |
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The explanation is done for
three media only (two boundaries). The situation is depicted in fig. 2.5B.1. We can simply imagine multiple
reflections
appearing in the structure. Therefore, there are an infinite number of waves propagating
in each layer in both directions. All the waves are coherent, and therefore, all the waves
propagating in a given direction interfere into a single harmonic wave. Hence, only
two waves in each medium are considered as depicted in fig. 2.5B.1. The media are
denoted as 1, 2, and 3. Next, A is the surface of the boundary 1-2 in the medium 1;
B is the surface of the boundary 1-2 in the medium 2, etc. Using indexes
A, B, C, D, we denote the surface, where the intensity is
computed, and using indexes in brackets (p), (z), we denote the direction
of propagation (forward wave, backward wave). E.g., EB(z) denotes the intensity of the
backward wave on the surface B, i.e. on the boundary 1-2 in the medium 2.
The layered medium consists of boundaries and layers; we examine separately the wave
transition through the boundary and through the layer.
We consider the boundary 1-2 and express wave intensities,
which travel outwards this boundary. The reflected wave of the intensity
EA(z) is created by the reflection of the incident wave EA(p),
and by the transition of the wave EB(z) from the medium 2 to the medium 1. Similarly, the wave of the intensity
EB(p) is created by the transition of the wave EA(p)
from the medium 1 to the medium 2, and by the transition of the wave EB(z). This is expressed by the equation:
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( 2.5B.1 )
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where ρ12 is
reflection coefficient
of the wave propagation in the medium 1 and reflecting from the boundary with the medium 2, and ρ21 is reflection coefficient
of the wave propagating in the medium 2 and reflecting from the boundary with the medium 1. Similarly,
coefficients of transmission
are indexed as τ12 and τ21. Eqn. (2.5B.1) can be rewritten to the matrix form:
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( 2.5B.2 )
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The right-hand side matrix, which is composed of coefficients of reflection and transmission, is called the scattering matrix of the boundary.
The scattering matrix can be simply built, and moreover, their elements are of the obvious physical meaning
(coefficients of reflection and transmission).
Unfortunately, eqn. (2.5B.2) is not suitable for the solution of the problem because intensities of input and output waves are mixed
in input and output column matrices. Eqn. (2.5B.2) is therefore solved for
EA(p) and EA(z). Rearranging the relation, we consider
ρ21 = -ρ12 and τ12 τ21 - r12 ρ21 = 1.
The result is expressed in the matrix form:
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( 2.5B.3 )
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The right-hand matrix is called the cascade boundary matrix. The left-hand column matrix consists
of input-wave intensities, and the right-hand column matrix consists of output-wave
intensities. The cascade matrix is suitable for cascade connection of elements
(two-ports), which are present even in our task. If an arbitrary number of elements
are connected into the cascade, then the cascade matrices are simply mutually multiplied.
Now, the cascade matrix of the dielectric layer has to be
derived. This aim can be met without any need for computing the scattering matrix.
In the dielectric layer, there are two traveling waves propagating in opposite directions. Obviously,
EB(z) = EC(z)exp(-jk2d),
EB(p) = EC(p)exp(+jk2d), and therefore
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( 2.5B.4 )
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where k2 is
wave number
in the second medium.
In the system of plan-parallel layers, single boundaries and layers are lined in parallel way. The final equation can be therefore obtained
by multiplying respective matrices. For the situation from fig. 2.5B.1, we get:
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( 2.5B.5 )
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The final result supposes a traveling wave in the third medium. In the opposite case, the intensity ED(z)
appears in the last column matrix. The way of extending the principle to an arbitrary number of layers is obvious.
Rewriting (2.5B.5), we get:
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( 2.5B.6 )
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The ratio EA(z)/ EA(p) gives the real value of the
reflection coefficient
on the boundary 1-2. The ratio ED(p)/ EA(p) equals to the total
transmission coefficient.
Both the quantities can be computed using the last equations.
Let us note that in the case of the normal incidence of a
perpendicularly polarized wave
to a single boundary (no reflected wave can exists behind that), following relations are valid for the reflection coefficient and the transmission one:
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( 2.5B.7 )
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The described method is suitable for more complicated structures. Common tasks can be numerically solved using Smith chart or directly considering
boundary conditions
on the boundary (see layer A.)
As already mentioned in the introduction, provision of the reflection-free transition of the electromagnetic wave from one medium to another one is
an example of the application of layered media. Such requirement can appear when exciting waves in the ground or in the other media, when exciting
waves in human body, at dielectric radomes of antennas, etc.; the wave is required to enter the medium and to exit it with the same intensity and
the same equiphase surface. Covering the boundary between the media by one or several dielectric layers of exactly given permittivity and thickness
can fulfill the requirement. The layers play the role of the matching circuitry (transformer) in analogy to the transmission line. In both the cases,
electromagnetic wave propagates in the structure. On the transmission line, voltages and currents represent the wave. In the dielectric layer, the wave
is represented by electric-field intensity and magnetic-field intensity. Voltage U corresponds to electric intensity E, and current I
corresponds to magnetic intensity H.
Characteristic impedance
of the transmission line corresponds to wave impedance of the medium (layer). The difference between the above-described cases is hidden in the structure
of circuits. On the transmission line, matching circuitry could consist not only of cascade-connected segments of the transmission line but too of parallel
or serial shunts. Matching layers can be connected in cascade only. The reason for that is fundamental. In the part of the transmission line, where the
parallel reactance is connected, a current jump appears (a part of current flows to the reactance). In the layered equivalent, a jump of magnetic-field
intensity should appear in the respective part of the boundary. This is impossible from the physical point of view (it contradicts the basic boundary
condition). In general, jump of magnetic-field intensity H can appear on the boundary if electric current is admitted. Then, the
boundary condition
is of the form n × H1 = K + n × H2. Indexes distinguish between the first medium and the second one, vector products of the normal
n result in tangential components and K [A m-1] is surface density of electric current on the boundary. In practical life, this solution can be used. However, the elected technology has to
admit the flow of electric currents on the surface of the boundary.
Routinely, we can design matching circuits consisting of a single one-quarter-wavelength thick dielectric layer or of a first dielectric layer of
a given thickness plus of a second one, which is one-quarter-wavelength thick. No matter whether segments of the transmission line or dielectric
layers are used. If efficient, Smith chart can be used.
The above-described matching circuits can be modeled by the computer program, which is described in the
layer C. The program cannot be used for the design of those circuits, it analyses a circuitry of given parameters
(permittivity, thickness of layers) and computes the distribution of E and H in layers,
reflection coefficient and
standing wave ratio
in various points of the structure. Observing numerical values, the circuit operation can be investigated. The matching circuits can consists of up to
four dielectric layers. The operation of the circuitry can be observed in a wider frequency band.
Frequency filters are another application area of layered media. As a basic element, the half-wavelength-thick dielectric layer is used. This layer
(in analogy with the half-wavelength-long transmission line) transforms intensities and impedances in the ratio 1:1. Other words, the wave reflected
at the beginning of the layer cancels out with the wave reflected from the end of layer. If the incident wave is of different frequency (the layer is
not half-wavelength thick), the total cancellation does not appear. The layer reflects the part of energy, and therefore, the transmitted energy is lower.
In front of the layer, the standing wave appears.
In the practical life, filters are composed of several half-wavelength-thick layers, which are mutually separated by layers of another permittivity
and thickness (l/4, e.g.).
Properties of layered structures as filters can be demonstrated by another program in the
layer C. In this layer, even the more detailed description of the program is given.
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