Principles of Modern Radar - Volume 2 1891121537

598 CHAPTER 13 Introduction to Radar PolarimetryAlternatively, the ellipticity angle can be written succinctly in terms of the auxiliary angleγ (see (13.9)) assin 2τ = sin 2γ sin δ (13.17)In summary, the polarization ellipse is completely specified by (a) the tilt angle given by(13.8), (b) the ellipticity angle in (13.16), and (c) the sense of rotation specified by thesign on the ellipticity angle as determined from (13.11).The elliptically polarized field components in the rotated (ξ,η) coordinate system (seeFigure 13-3) may be written in complex vector form as[ ] [ ]Eξ cos τE(ξ,η) = = m eE η j sin τjα 0(13.18)√where m = e1 2 + e2 2 is the field amplitude, the angle τ is negative for right-handedEP and positive for left-handed EP (see (13.11)), and α 0 is an arbitrary reference phaseangle common to both E ξ and E η . Converting the phasor fields in (13.18) into real timedependentform using E(t) = Re[(E ξ ˆξ + E η ˆη)e jωt ], it can be readily verified that thetime-dependent fields satisfy the equation to the ellipse with semi-major axis, e 1 , andsemi-minor axis, e 2 , given byE 2 ξ (t)e 2 1+ E 2 η (t)e 2 2= 1 (13.19)The complex field components in the (x,y) coordinate system follow from (13.5) and(13.18) as[ ] [ ][ ]ExE(x,y) = = meE jα 0 cos ψ − sin ψ cos τ= mey sin ψ cos ψ j sin τjα 0ĥ(x,y) (13.20)where the complex unit vector ĥ represents the polarization state, and is known as theJones vector [3]. Equation (13.20) defines the field components in the (x,y) coordinatesystem entirely in terms of the tilt angle and the ellipticity angle and forms the basis forscattering matrix representation of polarization.13.2.2 Stokes Parameters and Poincaré SphereIt is convenient to represent polarization of a monochromatic wave in terms of real-valuedStokes parameters that may be derived from the polarization ellipse specified by thetilt angle and the ellipticity angle. Stokes parameters permit an interpretation of variouspolarizations states as points on the Poincaré sphere. The polarization states can be definedin terms of a Stokes vector, comprising the Stokes parameters, as [14, 32],⎡⎡ ⎤g|E x | 2 + ∣∣ ∣E y 2 ⎤⎡⎤0g = ⎢ g 1⎥⎣ g 2⎦ = |E x | 2 − ∣∣ ∣E y 21⎢⎣ 2Re(E x Ey ∗g ) ⎥ = g ⎢ cos 2τ cos 2ψ⎥0⎣⎦ cos 2τ sin 2ψ ⎦ (13.21)32Im(E x Ey ∗) sin 2τ