les Q and 0 define the field polarization. Naturally, the surface is a part of a three-sphere due to the characteristic of the fields described by Equation 9.20 to have the same power. In particular, it is that part corresponding to all the points with the same absolute phase. Due to the characteristics of this surface, it is not SO easy to use quadrature representa- tion to describe polarization modulation. In optics however, when there is the need for describing the polarization evolution, the Stokes parameters representation is frequently used. The Stokes parameters are defined as =2e1(t)es(t)+2ez(t)e4(t) (9.21) S3 2e1(t)e4(t)-2e3(t)e2(t) It is easy to verify that, for a monochromatic perfectly polarized beam it is (9.22) SO that the SOP of a field can be represented as a point in the so-called Stokes space, which is an Euclidean three dimensional space having the three Stokes parameters as coordi- nates. In this space, the equal power surface is again a sphere, as in the quadrature space, but the sphere radius in this case is equal to the square of the field power. The sphere of normalized radius (assumed to be equal to one) in the Stokes space is called the Poincare sphere. The position on the Poincare sphere of the main SOPs are shown in Figure 9.23. These positions can be derived easily by rewriting the Stokes parameters expressions using spherical coordinates in the Stokes space as follows S1 = Pcos(26)cos(20) S2 = Psin(20)cos(20) (9.23) S3 = Psin(20) Right circular 0,0,1 Linear 45° Linear vertical 0, -1, 0 -1,0,0 Linear horizontal Linear +45° 0, 1, 0 1, ,0,0 FIGURE 9.23 Left circular The Poincare sphere and the position of 0,0, -1 the main states of polarization. Telecommunication Networks The angular coordinates in the Stokes space, if defined as in Equation 9.23 and in Figure 9.23 are related to the angles determining the form of the polarization ellipse as shown in Figure 9.24 as it is possible to demonstrate starting from the definition (9.23) of the Stokes parameters. From (9.23) and the angles defi