Page 414 - Engineering Electromagnetics, 8th Edition

396 ENGINEERING ELECTROMAGNETICS

positive y direction would require a component of H in the negative x direction—thus
the minus sign. Using (91) and (92), the power density in the wave is found using (77):
1 1
∗
∗
∗ Re{E x0 H (a x × a y ) + E y0 H (a y × a x )}e −2αz
S z = Re{E s × H }= y0 x0
s
2 2
1 E x0 E ∗ E y0 E ∗
y0
= Re x0 + e −2αz a z
2 η ∗ η ∗
1 1
2
2
= Re (|E x0 | +|E y0 | )e −2αz a z W/m 2
2 η ∗
This result demonstrates the idea that our linearly polarized plane wave can be con-
sidered as two distinct plane waves having x and y polarizations, whose electric fields
are combining in phase to produce the total E. The same is true for the magnetic field
components. This is a critical point in understanding wave polarization, in that any
polarization state can be described in terms of mutually perpendicular components
of the electric field and their relative phasing.
We next consider the effect of a phase difference, φ, between E x0 and E y0 , where
φ< π/2. For simplicity, we will consider propagation in a lossless medium. The
total field in phasor form is
E s = (E x0 a x + E y0 e jφ a y )e − jβz (93)
Again, to aid in visualization, we convert this wave to real instantaneous form by
multiplying by e jωt and taking the real part:

(94)
E(z, t) = E x0 cos(ωt − βz)a x + E y0 cos(ωt − βz + φ)a y
where we have assumed that E x0 and E y0 are real. Suppose we set t = 0, in which
case (94) becomes [using cos(−x) = cos(x)]

E(z, 0) = E x0 cos(βz)a x + E y0 cos(βz − φ)a y (95)
The component magnitudes of E(z, 0) are plotted as functions of z in Figure 11.5.
Since time is fixed at zero, the wave is frozen in position. An observer can move
along the z axis, measuring the component magnitudes and thus the orientation of the
total electric field at each point. Let’s consider a crest of E x , indicated as point a in
Figure 11.5. If φ were zero, E y would have a crest at the same location. Since φ is
not zero (and positive), the crest of E y that would otherwise occur at point a is now
displaced to point b farther down z. The two points are separated by distance φ/β.
E y thus lags behind E x when we consider the spatial dimension.
Now suppose the observer stops at some location on the z axis, and time is
allowed to move forward. Both fields now move in the positive z direction, as (94)
indicates. But point b reaches the observer first, followed by point a.Sowe see that
E y leads E x when we consider the time dimension. In either case (fixed t and varying
z,or vice versa) the observer notes that the net field rotates about the z axis while
its magnitude changes. Considering a starting point in z and t,at which the field has
agiven orientation and magnitude, the wave will return to the same orientation and