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=2\pi a/\lambda\sim20
), 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 general
\sigma_{1}^{2}\neq\sigma_{2}^{2}
and
\rho\neq 0
. However, by rotating (in phase) the quadrature axes, two simpler situations arise: (a) at one orientation,
\sigma_{1}^{2}=\sigma_{2}^{2}
but
\rho\neq0
; (b) at an orientation
45\deg
from case (a),
\sigma_{1}^{2}\neq\sigma_{2}^{2}
but
\rho\neq0
. Probability density expressions for these simpler cases exist in the literature. As the quadrature axes are rotated, the sum
\sigma_{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.