Statistics of phase quadrature components of microwave fields transmitted through a random medium

The phase quadrature components of the total microwave field transmitted through a random volume distribution of Styrofoam spheres have been measured in the laboratory. The radii (a) of the spheres were large compared to the wavelengths (ka=2pi a/lambdasim20), and their index of refraction was close to unity (eta = 1.019). The statistical results lead in general to the bivariate normal distribution with correlation (rho) to describe the scattered incoherent field, rather than to the simpler Rayleigh distribution. The quadrature components of the incoherent field are Gaussian, but in generalsigma_{1}^{2}neqsigma_{2}^{2}andrhoneq 0. However, by rotating (in phase) the quadrature axes, two simpler situations arise: (a) at one orientation,sigma_{1}^{2}=sigma_{2}^{2}butrhoneq0; (b) at an orientation45degfrom case (a),sigma_{1}^{2}neqsigma_{2}^{2}butrhoneq0. Probability density expressions for these simpler cases exist in the literature. As the quadrature axes are rotated, the sumsigma_{1}^{2}+sigma_{2}^{2}remains equal to a constant (the incoherent power), as it should. These departures of the incoherent field from the Rayleigh distribution are a function of the sum of the reciprocal transmitter and receiver distances. This behavior suggests that the departures are related to the sphericity of the transmitting and receiving beams.