Page 282 - Electromagnetics

Polarization of a uniform plane wave. Plane-wave polarization describes the tem-
poral evolution of the vector direction of the electric field, which depends on the manner
in which the wave is generated. Completely polarized waves are produced byantennas or
other equipment; these have a deterministic polarization state which maybe described
completelybythree parameters as discussed below. Randomly polarized waves are emit-
ted bysome natural sources. Partially polarized waves, such as those produced bycosmic
radio sources, contain both completelypolarized and randomlypolarized components.
We shall concentrate on the description of completelypolarized waves.
The polarization state of a completelypolarized monochromatic plane wave propa-
gating in a homogeneous, isotropic region maybe described bysuperposing two simpler
plane waves that propagate along the same direction but with different phases and spa-
tiallyorthogonal electric fields. Without loss of generalitywe maystudypropagation
along the z-axis and choose the orthogonal field directions to be along ˆ x and ˆ y.So we
are interested in the behavior of a wave with electric field

ˇ
e
E(r) = ˆ xE x0 e jφ x − jkz + ˆ yE y0 e jφ y − jkz . (4.248)
e
The time evolution of the direction of E must be examined in the time domain where we
have
ˇ
jωt
E(r, t) = Re Ee = ˆ xE x0 cos(ωt − kz + φ x ) + ˆ yE y0 cos(ωt − kz + φ y )
and thus, bythe identity cos(x + y) ≡ cos x cos y − sin x sin y,
E x = E x0 [cos(ωt − kz) cos(φ x ) − sin(ωt − kz) sin(φ x )] ,

E y = E y0 cos(ωt − kz) cos(φ y ) − sin(ωt − kz) sin(φ y ) .
The tip of the vector E moves cyclically in the xy-plane with temporal period T = ω/2π.
Its locus maybe found byeliminating the parameter t to obtain a relationship between
E x0 and E y0 . Letting δ = φ y − φ x we note that

E x E y
sin φ y − sin φ x = cos(ωt − kz) sin δ,
E x0 E y0
E x E y
cos φ y − cos φ x = sin(ωt − kz) sin δ;
E x0 E y0
squaring these terms we find that
2 2
E x E y E x E y 2
+ − 2 cos δ = sin δ,
E x0 E y0 E x0 E y0
which is the equation for the ellipse shown in Figure 4.15. By(4.223) the magnetic field
of the plane wave is
ˇ
ˆ z × E
ˇ
H = ,
η
hence its tip also traces an ellipse in the xy-plane.
The tip of the electric field vector cycles around the polarization ellipse in the xy-
plane once every T seconds. The sense of rotation is determined bythe sign of δ, and
is described bythe terms clockwise/counterclockwise or right-hand/left-hand. There is
some disagreement about how to do this. We shall adopt the IEEE definitions (IEEE
Standard 145-1983 [189]) and associate with δ < 0 rotation in the right-hand sense: if

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