Page 284 - Electromagnetics

Figure 4.16: Polarization states as a function of tilt angle ψ and ellipse aspect ratio angle
χ. Left-hand polarization for χ> 0, right-hand for χ< 0.

Each of these parameter sets is somewhat inconvenient since in each case the units
differ among the parameters. In 1852 G. Stokes introduced a system of three independent
quantities with identical dimension that can be used to describe plane-wave polarization.
Various normalizations of these Stokes parameters are employed; when the parameters
are chosen to have the dimension of power densitywe maywrite them as

1 2 2
s 0 = E x0 + E y0 , (4.249)
2η
1 2 2
s 1 = E x0 − E y0 = s 0 cos(2χ) cos(2ψ), (4.250)
2η
1
s 2 = E x0 E y0 cos δ = s 0 cos(2χ) sin(2ψ), (4.251)
η
1
s 3 = E x0 E y0 sin δ = s 0 sin(2χ). (4.252)
η

2
2
2
2
Onlythree of these four parameters are independent since s = s + s + s . Often the
0 1 2 3
Stokes parameters are designated (I, Q, U, V ) rather than (s 0 , s 1 , s 2 , s 3 ).
Figure 4.16 summarizes various polarization states as a function of the angles ψ and
χ. Two interesting special cases occur when χ = 0 and χ =±π/4. The case χ = 0
corresponds to b = 0 and thus δ = 0. In this case the electric vector traces out a straight
line and we call the polarization linear. Here

E = ˆ xE x0 + ˆ yE y0 cos(ωt − kz + φ x ).
When ψ = 0 we have E y0 = 0 and refer to this as horizontal linear polarization (HLP);
when ψ = π/2 we have E x0 = 0 and vertical linear polarization (VLP).
The case χ =±π/4 corresponds to b = a and δ =±π/2.Thus E x0 = E y0 , and E
traces out a circle regardless of the value of ψ.If χ =−π/4 we have right-hand rotation
of E and thus refer to this case as right-hand circular polarization (RHCP). If χ = π/4
we have left-hand circular polarization (LHCP). For these cases

E = E x0 [ˆ x cos(ωt − kz) ∓ ˆ y sin(ωt − kz)] ,

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