Phase Statistics in Random Media

Abstract

Because the ensemble average of the field transmitted through a random medium tends to zero as the sample length increases, studies of wave propagation in random media have focused upon the field amplitude or its square, the intensity, whose statistical properties have been extensively investigated. In this paper, we show that, beyond the random phase approximation, the phase remains a rich statistical quantity which is central to the understanding of wave transport in random systems. We define the cumulative phase φ and present measurements of the frequency dependence of its average, its probability distribution and the correlation function of the phase derivative for microwave radiation transmitted through random collections of polystyrene spheres. Static field measurements are made using a network analyzer. The phase and amplitude of the wave as a function of frequency is obtained for an ensemble of samples by rotating the sample after each spectrum is taken. The probability distribution is calculated using the phase measurement at each frequency for every configuration and is found to reduce to a single gaussian in terms of the variable (φ-)/σ, where σ = (var(φ))½. We show that this is a direct result of the rapid decay of the frequency correlation function of phase derivative found at any frequency. We compare the dwell time Δτ(ν), which is the time for a gaussian pulse with carrier frequency ν to traverse the sample, with phase derivative dφ/dω(ν), where ω = 2πν. We demonstrate theoretically and experimentally that these two quantities are identical if the frequency width of the gaussian pulse is narrow enough (Fig. 1). Finally we outline some of the ways in which φ reflects wave dynamics and relate the phase derivative with the density of states.