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04/19/05 Colloquium
Stephen Watson
Florida Space Institute
Florida Space Institute
Phase-related statistics in randomly scattered waves
Abstract: The 'random walk' or 'few-scatterer' model has been used extensively to account for various scattering and propagation phenomena. Familiar examples include laser light propagation through particulate media and radar scattering from aircraft. While the approach is empirical, and by no means provides a rigorous solution to Maxwell's equations, the physics of the model is sound and it has provided much insight into both scattering and propagation problems. The field scattered from a random medium is assumed to be composed of independent real and imaginary parts whose resultant in the complex plane can be represented as a two-dimensional random walk.The problem at hand is equivalent to that posed by Pearson in 1905, namely that "A man starts from a point 0 and walks l yards [sic] in a straight line; he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times. I require the probability that after n of these stretches he is at a distance between r and r + dr from his starting point". The problem attracted immediate responses from several eminent theorists and remains of considerable interest and impact in the quantitative sciences one hundred years on. While a large body of research has focused on the resultant step length or amplitude of such a walk, as well as it's square or intensity, it would appear that relatively little attention has been paid to the phase-related, or angular directivity, properties of such walks. For example, the phase derivative of such a walk can obtained from the addition of phasors that are rotating with respect to one another, such that the random walk evolves with time. In this talk we discuss preliminary results from an investigation into phase-related statistics, including criteria to optimally extract frequency information from randomly scattered waves.
