Principles of Modern Radar - Volume 2 1891121537

13.3 Scattering Matrix 601left-hand elliptical polarization state at the spherical angles (π/4,π/4). This ellipse isinclined 22.5 ◦ relative to the horizontal, as depicted in Figure 13-5 [32].As an application of the Poincaré sphere representation, it may be shown that voltageat the terminals of a receiving antenna due to an incident wave of arbitrary polarization(radiated by a transmitting antenna) is given by [27]V = C cos MM a2∣= C∣ĥ · ĥ ∗ a∣ (13.23)where M and M a are points on the Poincaré sphere corresponding to polarization states ĥand ĥ a , for the incident wave and the antenna, respectively, MM a is the angle subtendedby great-circle line from the polarization state M to M a , and C is a constant that dependson the field strength of the wave and the effective aperture of the antenna. In particular,if the angle MM a = 0, the polarization states of the transmit and receive antennas arematched, and the response in (13.23) is maximized. This happens, for example, at the polesof the Poincaré sphere, when two circularly polarized states of the same sense overlap,or on the equator, between two co-polarized linear states. If the angle MM a = 180 ◦ , thepolarization states are orthogonal, and no signal is received. Thus, for maximum signalat the terminals of the receiving antenna, the transmit and receive antennas must be copolarized.Conversely, no signal is received if the two antennas are cross-polarized.It is convenient to calculate the polarization efficiency of a communication link from(13.23) asp = cos 2 α 2 ,α = ̸ MM a (13.24)Assuming completely polarized states, the angle α may be calculated in terms of Stokesparameters of the transmit and receive antennas ascos α = gt 1 gr 1 + gt 2 gr 2 + gt 3 gr 3g0 t (13.25)gr 0with the superscripts t and r denoting the transmitter and receiver, respectively. Using thepolarization efficiency, receive antenna aperture, and the incident power density, we cancalculate the net power received [3].13.3 SCATTERING MATRIXIn microwave circuit theory, the term ‘scattering matrix’ is used to described voltage measurementsin a multi-port network. However, in radar polarimetry, the scattering matrixcharacterizes the target-scattered response in terms of either complex voltage measurementsor relative power measurements. For voltage measurements, two coordinate systemsare used to specify the incident wave at the transmitter, the scattered wave at the target, andthe received wave. The choice of the coordinate system defines the polarization of thesewave components, and influences the scattering matrix. In the forward-scatter alignment(FSA) convention, two local right-handed coordinate systems are defined at the transmitterand the target relative to the propagating wave, with the z-axis pointed along thedirection of propagation [11]. A third co-ordinate system is defined at the receiver withthe z-axis pointed along the line of sight from the target into the receiver. FSA systemis employed in bistatic measurements of the scattering matrix, when the transmitter andthe receiver antennas are not collocated. In the back-scatter alignment (BSA) coordinate