3.1 WaveguidesBasic theoryOur layer of this article brings basic information about the propagation of electromagnetic waves in
waveguides.
In the layer B, we present a Dutch translation of the article to the reader in order to help him (her) to
become familiar with the terminology from the area of waveguides and microwave transmission lines.
Types of transmission lines we know from our everyday life (coaxial cables, etc.), can be used on
microwave frequencies
in limited number of applications due to unacceptable losses in the dielectrics of those lines. That is why
waveguides
are frequently used on higher frequencies.
The waveguide
usually denotes a metallic cylinder which transversal dimensions are comparable to the wavelength. Inner walls of the waveguide are processed that way
which minimizes losses in the metal (walls can be considered as a perfect electric conductor). The transversal profile of the waveguide is usually of
the rectangular or circular shape. In special cases, the profile is of the shape of letters Π or H (fig. 3.1A.1);
such waveguides are of wider broadband on one hand but they can transmit lower power on the other hand.

Fig. 3.1A.2  Longitudinally homogenous waveguide. u,v: transversal directions, z: longitudinal direction. 

Waveguides
can be met on gigahertz frequencies, because on lower frequencies, transversal dimensions of waveguides are too large.
Waveguides
are exploited in radars and in satellite communication systems. They serve here for the energy transmission from the generator to the antenna and
vice versa.
Assume a
longitudinally homogeneous
metallic waveguide of an arbitrary transversal profile (fig. 3.1A.2). Modeling electromagnetic field, which is produced by
a source inside the waveguide, is the aim of our computations.
Assume that our computations are performed is large distance from sources. Then, computations can be based on the
homogeneous wave equations
for the longitudinal component of the electric
Hertz vector
Π_{z}^{e} and the magnetic one Π_{z}^{m}. Since the field distribution in the longitudinal direction z
is not dependent on the field distribution in the transversal directions u, v, both Hertz vectors can be rewritten as the product of two functions where
the first one is dependent on transversal coordinates u, v, and the second one depends on the longitudinal coordinate z, i.e. Π_{z} =
T_{1}(u,v) × T_{2}(z). Exploiting the method of variables separation, we yield separation constants Γ
and γ, which are related as

${\gamma}^{2}+{k}^{2}={\Gamma}^{2}$.

( 3.1A.1 )

In the above relation, Γ is the separation constant that is related to the field distribution in transversal directions, γ
is related to the wave propagation in the longitudinal direction and k is wavenumber for the free space (vacuum with permitivity
ε_{0} and permeability μ_{0})

$k=\omega \sqrt{{\mu}_{0}{\epsilon}_{0}}$

( 3.1A.2 )

and ω denotes angular frequency.
The detailed mathematical description of the above approach can be found in [1].
Solving the equation for the wave propagation in the longitudinal direction in the waveguide, following relation is obtained

${T}_{2}={C}_{1}{e}^{\gamma z}+{C}_{2}{e}^{\gamma z}$

( 3.1A.3 )

C_{1} and C_{2} are integration constants. First term describes the backward wave (it propagates in the contradirection
of the axis z), the second term is related to the forward wave (it propagates in the direction of the axis z). The separation constant γ is called the
propagation constant
and it can be rewritten as

$\gamma =\beta +j\alpha $.

( 3.1A.4 )

Considering (3.1A.4), β can be understood as the
attenuation constant
and α as the
phase constant.
Since the wavenumber k is real (a lossless medium is assumed inside the waveguide) and since the separation constant G is real too (as will
be shown later), the propagation constant can be of the following values:
 γ = β for k < Γ in the longitudinal direction,
evanescent wave
propagates;
 γ = α for k > Γ in the longitudinal direction, nonattenuated wave propagates;
 γ = jk for Γ = 0 propagation properties are not influenced by the transversal profile.
Since the wavenumber linearly depends on the frequency, an interesting conclusion can be done: whereas waves of lower frequency than

${\omega}_{krit}=\Gamma /\sqrt{\mu \epsilon}$

( 3.1A.5 )

do not propagate in the waveguide, waves of higher frequency than ω_{krit} propagate without any attenuation.
Frequency (3.1A.5) is called the
critical frequency.

Fig. 3.1A.3  Dependency of the phase velocity and the group one on the frequency 

Let us examine the phenomena appearing on frequencies higher than the critical one f > f_{krit}. Substituting γ =
jα to (3.1A.1), we get

$\alpha =\sqrt{{k}^{2}{\Gamma}^{2}}$

( 3.1A.6 )

Expressing the separation constant Γ from (3.1A.5)

$\Gamma ={\omega}_{krit}\sqrt{\mu \epsilon}$


and using wavenumber k from (3.1A.2) we get the relation for the
phase constant
in the longitudinal direction

$\alpha =k\sqrt{1{\left({f}_{krit}/f\right)}^{2}}$

( 3.1A.7 )

Substituting the phase constant (3.1A.7) to the relation for the
phase velocity

${v}_{f}=\omega /\alpha $,


we get dependency of the
phase velocity
in the waveguide on the frequency

${v}_{f}=\frac{v}{\sqrt{1{\left({f}_{krit}/f\right)}^{2}}}$.

( 3.1A.8 )

Here, v denotes phase velocity of the wave in free space (vacuum in our case)

$v=1/\sqrt{\mu \epsilon}$.

( 3.1A.9 )

Using known
phase velocity,
relation between phase velocity and wavelength

${\lambda}_{g}={v}_{f}/f$

( 3.1A.10 )

in the longitudinal direction in the waveguide can be obtained

${\lambda}_{g}=\frac{\lambda}{\sqrt{1{\left({f}_{krit}/f\right)}^{2}}}$

( 3.1A.11 )

Here, λ denotes wavelength in free space.
If we are interested in the velocity of energy propagation, the
group velocity
has to be computed. Since the product of the group velocity and the phase one has to equal to the square of the velocity of light, we can compute the
group velocity using the following relation:

${v}_{sk}=v\sqrt{1{\left({f}_{krit}/f\right)}^{2}}$,

( 3.1A.12 )

where v is
phase velocity
in free space.
The above computations were performed for the propagation of harmonic waves. If the wave consists of more harmonics then every frequency component
propagates with velocity, and therefore, the output signal differs from the input one (the signal is distorted). The described phenomenon is called
dispersion (fig.
3.1A.4).
Up to now, we dealt with analysis in the longitudinal direction. Results of this analysis do not depend on the shape of the transversal profile,
and therefore, they are valid for any
homogeneous waveguide.
In the transversal directions, the situation is totally different:
 In the transversal direction, no wave is propagating. Here, waves reflected from waveguide walls interfere, and hence, standing waves appear here.
 Since reflections from walls (and consecutively the standing waves) depend on the profile, the analysis has to be performed for the respective
shape of the profile. In our description, we are going to concentrate on the rectangular shape
The analysis is done for two types of waves, which can propagate in the waveguide, for the transversally magnetic waves (TM, components of the
magnetic intensity vector are nonzero in transversal directions only), and for the transversally electric ones (TE, components of the electric
intensity vector are nonzero in transversal directions only). Following [1],
relations for critical frequencies of waves of both types can be obtained:

${\omega}_{krit}=\frac{1}{\sqrt{\mu \epsilon}}\sqrt{{\left(\frac{m\pi}{a}\right)}^{2}+{\left(\frac{n\pi}{b}\right)}^{2}}$.

( 3.1A.13 )

Here, permitivity and permeability are related to the medium inside the waveguide, a denotes the width of the waveguide and b is its height.
Integral coefficients m and n are called
mode numbers.
Increasing mode numbers,
critical frequency
is increased (therefore, higher modes appear on higher frequencies).
Assume that a supplying generator is tuned from lower frequencies to higher ones. Reaching the
critical frequency
of the lowest
mode,
a single wave starts to propagate in the waveguide. If the
critical frequency
of the second mode is reached then two waves of two different modes propagate in the waveguide. Those waves can interfere, which can lead to many
problems, Therefore, waveguides operate in the
singlemode band
usually. The low bound of this band is given by the critical frequency of the lowest mode, the high bound equals to the critical frequency of the
second mode. The mode of the lowest critical frequency is called the
dominant mode.
Let us concentrate on the field distribution of the dominant mode TE_{10} in the moment t = t_{0}
(see animation).
Cutting the
waveguide
by a longitudinal wall that is perpendicular to the wider side, a harmonic course of the transversal component of electric intensity
E_{y} can be observed. Maximal intensity in the figure is located to coordinates z = λ_{g}/4 and
z = 3λ_{g}/4 (phase is opposite here). In z = 0 and z = λ_{g}/2, electric intensity is zero. In the points
of maximal E_{y}, longitudinal component of magnetic intensity H_{z} is zero and transversal component of magnetic
intensity H_{x} is maximal. In the transversal cut in z = λ_{g}/4, E_{y} is maximal in the center
of the waveguide and zero on walls (boundary condition met). Transversal component of magnetic intensity H_{x} is in z =
λ_{g}/4 constant.
If the cutting wall is perpendicular to the narrower side of the waveguide, magnetic intensities form ellipses (similarly as in the case of a wire
flowing by a current). In case of waveguide, a current flowing from the low wall of the waveguide to the up one through the vacuum produces magnetic
field. If the current reaches the wall, it flows in the form of the conducting current on the waveguide wall back.
Finally, we can state that the transversal electromagnetic field is distributed that way so that the
boundary conditions
on perfectly electrically conducting walls are met. This fact is illustrated by the Matlab program for the analysis of the electromagnetic field in the
rectangular waveguide by the finiteelement method. User's guide of the program can be found in the layer C.
A brief introduction to the
finiteelement method
and a brief description of the software implementation are given in the
layer D.
