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276 Chapter 6

(6.1) share the same propagation constant their vectors’ association is called the single wave
mode and, in general, able to carry energy independently from another propagating wave modes
in line.

The fourth assumption allows us to reduce the wave mode analysis to the solution of wave
uniform (no sources) equations (4.13) in Chapter 4. Substituting (6.1) into (4.13) and taking
2
2
2
into consideration that the second derivative ⁄ = − we obtain
2
2
� 2
∇ (, ) + � − �(, ) = 0
� (6.2)
2
� 2
2
∇ (, ) + ( − )(, ) = 0
2
2
2
2
2
Here the differential operator is reduced to ∇ = ⁄ + ⁄ and the exponential time
dependable factor is omitted as common and obvious from context. Really, we need to solve
∓ as the solution of (6.2) we
just one of the equations in (6.2). For example, having (, )
nd
can find the magnetic vector from 2 Maxwell’s equation as

= × (6.3)
0
∗
and vice versa. Then using equity (6.3), we can calculate Poynting’s vector = × =
(− ) × × proportional to the complex power carrying each mode in feed line.
∗
⁄
0
Omitting details (see (A.16) for the triple product in Appendix), we will write the final
expression as = (∓ )|| . This result is quite expected but interesting in any case:
2
⁄
0
0
any propagating mode (while is real) in any uniform and lossless feed line cannot and must
not carry any traces of reactive energy. Sometimes, it lets draw the correct conclusions about
E- and H-field structure in line avoiding Maxwell’s equations solution or controlling the
rightness of numerical analysis. We are going to follow this path investigating EM wave in
“neoclassical” mediums like plasma, metamaterial, ferrite, etc. exhibiting such unusual feature
as negative permeability or permittivity.
Comparing (6.2) and (4.13) we see that according to our last assumption all field sources are
located outside the area of solution and (, ) = 0 and (, ) = 0. We should be

punished one way or another for that because the sources transfer their energy to propagating
waves and they are the primary reason for wave existence. Meanwhile, the equation (6.2) tells
us only about the mode field distribution in lines that much more critical for practical
applications than the knowledge of real modes’ magnitude defined by source intensity. The
source presence seriously complicates the mode analysis because, in general, it causes the
simultaneous excitation in its vicinity of infinite number of different modes. Then the common
analytical approach is to find the full set of the equation (6.2) solutions combine them in some
kind of infinite sum like Fourier series or integral representation like (4.46) with unknown
coefficients or integrand. Finally, using one of diverse techniques we might (or might not) find
the analytical presentation for coefficients or integrand solving the full 3D equation (4.29).
Typically, the final expressions are so complicated that it can be evaluated just numerically. If
so, they are almost useless in engineering practice. The straight numerical approach based on
original Maxwell’s equations is superior and we will mainly resort to it in the following
discussion.
Note that the appropriate boundary and radiation conditions formulated in Section 2.3 of
Chapter 2 must be imposed to solve 2D equation (6.2). Evidently, each such solution of (6.2)