Page 394 - Engineering Electromagnetics, 8th Edition

376 ENGINEERING ELECTROMAGNETICS

are invariant with field orientation). The Helmholtz equation is

2 2
∇ E s =−k E s (36)
wherethewavenumberisafunctionofthematerialproperties,asdescribedbyµand :

√ √
k = ω µ = k 0 µ r r (37)
For E xs we have

2
d E xs 2
=−k E xs (38)
dz 2
An important feature of wave propagation in a dielectric is that k can be complex-
valued, and as such it is referred to as the complex propagation constant.A general
solution of (38), in fact, allows the possibility of a complex k, and it is customary to
write it in terms of its real and imaginary parts in the following way:

jk = α + jβ (39)
A solution to (38) will be:

e
E xs = E x0 e − jkz = E x0 e −αz − jβz (40)
Multiplying (40) by e jωt and taking the real part yields a form of the field that can
be more easily visualized:

E x = E x0 e −αz cos(ωt − βz) (41)

We recognize this as a uniform plane wave that propagates in the forward z direction
with phase constant β,but which (for positive α) loses amplitude with increasing z
according to the factor e −αz . Thus the general effect of a complex-valued k is to yield a
traveling wave that changes its amplitude with distance. If α is positive, it is called the
attenuationcoefficient.Ifα isnegative,thewavegrowsinamplitudewithdistance,and
α is called the gain coefficient. The latter effect would occur, for example, in laser am-
plifiers.Inthepresentandfuturediscussionsinthisbook,wewillconsideronlypassive
media, in which one or more loss mechanisms are present, thus producing a positive α.
The attenuation coefficient is measured in nepers per meter (Np/m) so that the
exponent of e can be measured in the dimensionless units of nepers. Thus, if α =
0.01 Np/m, the crest amplitude of the wave at z = 50 m will be e −0.5 /e −0 = 0.607
of its value at z = 0. In traveling a distance 1/α in the +z direction, the amplitude of
−1
the wave is reduced by the familiar factor of e ,or 0.368.
The ways in which physical processes in a material can affect the wave electric
field are described through a complex permittivity of the form

= − j = 0 ( − j ) (42)

r
r