Principles of Modern Radar - Volume 2 1891121537

13.2 Polarization 599zFIGURE 13-4Poincaré sphere.LCPg 1PVHg 2g 02t2yg 3yxRCP√where ∗ denotes the complex conjugate, g 0 = g1 2 + g2 2 + g2 3 denotes the total power fora completely polarized wave, and therefore, the parameters g 1 ,g 2 ,g 3 are independent. Ingeneral, √the wave is partially polarized, and the total average power satisfies the conditiong 0 > g1 2 + g2 2 + g2 3 , and g 1 = g 2 = g 3 = 0 for an unpolarized wave [34]. Stokesparameters for partial polarization will be reviewed in Section 3.3. The derivation of theStokes vector in terms of tilt and ellipticity angles of the polarization ellipse (the lastcolumn in (13.21)) is left as an exercise. Equation (13.21) suggests a simple geometricalinterpretation of all polarization states by recognizing that g 1 ,g 2 ,g 3 are the rectangularcomponents of a point on a sphere with radius g 0 (see Figure 13-4), known as the Poincarésphere. The elevation and azimuth angles in the spherical coordinate system are given byθ = 2τ and φ = 2ψ, respectively. Alternatively, 2ψ and 2τ are the longitude and latitude,respectively, of the point on the sphere. Since positive τ occurs for left-hand polarizationand negative τ for right-hand polarization (see (13.11)), the former polarization states mapon to the upper hemisphere and the latter to the lower. These locations on the sphere will bereversed if exp(−iωt) time convention is employed. In fact, all corresponding orthogonalpolarizations are antipodal on the Poincaré sphere. Some examples will be discussed next.Case 13.1: Linear PolarizationIf τ = 0, the wave is linearly polarized along a ray inclined at the tilt angle ψ. It followsfrom (13.21) that the Stokes vector is given by⎡ ⎤1g = g 0⎢ cos 2ψ⎥⎣ sin 2ψ ⎦ (13.22)0Thus, any point on the equator of the Poincaré sphere represents a state of linear polarization.For (τ,ψ) = (0,0), the wave is horizontally polarized along the x-direction, and thepolarization state is located at (g 1 ,g 2 ,g 3 ) = (g 0 ,0,0). Likewise, (τ,ψ) = (0,π/4) resultsin a polarization state on the y-axis with an orientation at 45 ◦ relative to the horizontal,