Page 285 - Electromagnetics

Figure 4.17: Graphical representation of the polarization of a monochromatic plane wave
e
using the Poincar´ sphere.

where the upper and lower signs correspond to LHCP and RHCP, respectively. All other
values of χ result in the general cases of left-hand or right-hand elliptical polarization.
e
The French mathematician H. Poincar´ realized that the Stokes parameters (s 1 , s 2 , s 3 )
describe a point on a sphere of radius s 0 , and that this Poincar´e sphere is useful for
visualizing the various polarization states. Each state corresponds uniquelyto one point
on the sphere, and by(4.250)–(4.252) the angles 2χ and 2ψ are the spherical angular
coordinates of the point as shown in Figure 4.17. We maytherefore map the polarization
states shown in Figure 4.16 directlyonto the sphere: left- and right-hand polarizations
appear in the upper and lower hemispheres, respectively; circular polarization appears at
the poles (2χ =±π/2); linear polarization appears on the equator (2χ = 0), with HLP
at 2ψ = 0 and VLP at 2ψ = π. The angles α and δ also have geometrical interpretations
on the Poincar´ sphere. The spherical angle of the great-circle route between the point
e
of HLP and a point on the sphere is 2α, while the angle between the great-circle path
and the equator is δ.

Uniform plane waves in a good dielectric. We maybase some useful plane-wave
c
approximations on whether the real or imaginarypart of ˜ dominates at the frequency
of operation. We assume that ˜µ(ω) = µ is independent of frequencyand use the notation
c
c
= ˜ ( ˇω), σ = ˜σ( ˇω), etc. Remember that
σ σ
c

c

c
= + j + = + j − = + j .
j ˇω ˇ ω
Bydefinition, a “good dielectric” obeys
c σ
tan δ c =− = − 1. (4.253)
c ˇ ω
© 2001 by CRC Press LLC