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

Abstract

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.