Page 491 - Engineering Electromagnetics, 8th Edition

CHAPTER 13 Guided Waves 473

The field configuration for a given mode can be found through the superposition
of the fields of all the reflected waves. We can do this for the TE waves, for example,
by writing the electric field phasor in the guide in terms of incident and reflected fields
through

E ys = E 0 e − jk u ·r − E 0 e − jk d ·r (45)
where the wavevectors, k u and k d , are indicated in Figure 13.12. The minus sign in
front of the second term arises from the π phase shift on reflection. From the geometry
depicted in Figure 13.14, we write

k u = κ m a x + β m a z (46)
and

k d =−κ m a x + β m a z (47)
Then, using

r = xa x + za z
Eq. (45) becomes

E ys = E 0 (e − jκ m x − e jk m x )e − jβ m z = 2 jE 0 sin(κ m x)e − jβ m z = E sin(κ m x)e − jβ m z

0
(48)
where the plane wave amplitude, E 0 , and the overall phase are absorbed into E .In

0
real instantaneous form, (48) becomes

E y (z, t) = Re E ys e jωt = E sin(κ m x) cos(ωt − β m z) (TE mode above cutoff)

0
(49)
We interpret this as a wave that propagates in the positive z direction (down the guide)
5
while having a field profile that varies with x. The TE mode field is the interference
pattern resulting from the superposition of the upward and downward plane waves.
Note that if ω< ω cm , then (42) yields an imaginary value for β m , which we may
write as − j|β m |=− jα m . Eqs. (48) and (49) then become
E ys = E sin(κ m x)e −α m z (50)

0
E(z, t) = E sin(κ m x)e −α m z cos(ωt) (TE mode below cutoff) (51)

0
This mode does not propagate, but simply oscillates at frequency ω, while exhibiting
a field pattern that decreases in strength with increasing z. The attenuation coefficient,
α m ,is found from (42) with ω< ω cm :

2 2
nω cm ω 2πn λ cm
α m = 1 − = 1 − (52)
c ω cm λ cm λ

5 We can also interpret this field as that of a standing wave in x while it is a traveling wave in z.