) and horizontally (
α
=0)
polarized waves. Note that one can speak of “plane polarized plane waves,” meaning linearly polarized waves with an
exp
(
j
(
ωt
−
kz
))
dependence.
43
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2.4.2
Circular polarization
Another important special case is given by
E
1
=
E
2
,
ψ
=
±
π/
2
so that
E
=
(ˆ
x
±
j
ˆ
y
)
E
1
e
j
(
ωt
−
kz
)
(2.12)
In instantaneous notation, this is
E
(
z,t
)
=
ℜ
bracketleftBig
(ˆ
x
±
j
ˆ
y
)
E
1
e
j
(
ωt
−
kz
)
bracketrightBig
=
E
1
[ˆ
x
cos(
ωt
−
kz
)
∓
ˆ
y
sin(
ωt
−
kz
)]
The electric field vector therefore traces out a circle with a radius
E
1
such that
E
2
x
+
E
2
y
=
E
2
1
and an angle with
respect to
ˆ
x
given by
α
= tan
−
1
E
y
/E
x
= tan
−
1
∓
sin(
ωt
−
kz
)
/
cos(
ωt
−
kz
)
=
∓
(
ωt
−
kz
)
.
Imagine holding the spatial coordinate
z
fixed. At a single location, increasing time decreases (increases)
α
given a
plus (minus) sign in (2.12). In the example in Figure 2.10, which illustrates the plus sign, the angle
α
decreases as time
advances with
z
held constant, exhibiting rotation with a lefthand sense (left thumb in the direction of propagation,
fingers curling in the direction of rotation). The plus sign is therefore associated with left circular polarization in most
engineering and RF applications, where the polarization of a wave refers to how the electric field direction varies in
time. This text follows this convention.
Consider though what happens when time
t
is held fixed. Then the angle
α
increases (decreases) with increasing
z
given the plus (minus) sign in the original field representation (2.12). Referring again to Figure 2.10, the angle
α
increases as
z
advances with
t
held constant, exhibiting rotation with a righthand sense (right thumb in the direction
of propagation, fingers curling in the direction of rotation). In many physics and optics texts, polarization refers to
how the electric field direction varies in space, and the plus sign is associated with right circular polarization.
This is
not the usual convention in engineering, however.
2.4.3
Elliptical polarization
This is the general case and holds for
ψ
negationslash
=
±
π/
2
or
E
1
negationslash
=
E
2
. We are left with
E
(
z,t
)
=
ℜ
bracketleftBig
(
E
1
ˆ
x
+
E
2
e
±
jψ
ˆ
y
)
e
j
(
ωt
−
kz
)
bracketrightBig
=
ˆ
xE
1
cos(
ωt
−
kz
) + ˆ
yE
2
cos(
ωt
−
kz
±
ψ
)
At fixed
z
= 0
, for example, this is
E
(
z
= 0
,t
)
=
ˆ
xE
1
cos(
ωt
) + ˆ
yE
2
cos(
ωt
− ±
ψ
)
which is a parametric equation for an ellipse. With a little effort, one can solve for the major and minor axes and tilt
angle of the ellipse in terms of
E
1
,
2
and
ψ
. One still has left and right polarizations, depending on the sign of
ψ
.
The following principles are important for understanding freespace wave polarization:
•
H
is in every case normal to
E
, with
E
×
H
in the direction of propagation and

E

/

H

=
Z
◦
.
•
A wave with any arbitrary polarization can be decomposed into two waves having orthogonal polarizations.
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 Spring '13
 HYSELL
 The Land, power density, Solid angle

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