Title of Dissertation : STATISTICS OF IMPEDANCE AND SCATTERING MATRICES IN CHAOTIC MICROWAVE CAVITIES : THE RANDOM COUPLING MODEL

Title of Dissertation: STATISTICS OF IMPEDANCE AND SCATTERING MATRICES IN CHAOTIC MICROWAVE CAVITIES: THE RANDOM COUPLING MODEL Xing Zheng, Doctor of Philosophy, 2005 Dissertation directed by: Professor Edward Ott Professor Thomas M. Antonsen, Jr. Department of Physics A model is proposed for the study of the statistical properties of the impedance (Z) and scattering (S) matrices of open electromagnetic cavities with several transmission lines or waveguides connected to the cavity. The model is based on assumed properties of the eigenfunctions for the closed cavity. Analysis of the model successfully reproduces features of the random matrix model believed to be universal, while at the same time incorporating features which are specific to individual systems. Universal statistical properties of the cavity impedance Z are obtained in terms of the radiation impedance. These universal properties are independent of system-specific details and shared by the members of the general class of systems whose corresponding ray trajectories are chaotic. In the single channel case, I obtained the normalized impedance and scattering coefficients whose probability density functions (PDF) are predicted to be universal. In the multiple-channel case, I focused on correlations in the phases of the eigenvalues of the S-matrix, and derived a formula for the averaged reflection coefficients in terms of the port radiation impedance. Effects of time-reversal symmetry and wall absorption are discussed. Furthermore, I study the characterization of statistical fluctuations of the scattering matrix S and the impedance matrix Z, through their variance ratios. The variance ratio for the impedance matrix is shown to be a universal function of distributed losses within the scatterer, which contrasts with variance ratio of the scattering matrix for which universality applies only in the large loss limit. Theoretical predictions are tested by direct comparison with numerical solutions for a specific system, and also agree with experimental results obtained from scattering measurements on a chaotic microwave cavity. STATISTICS OF IMPEDANCE AND SCATTERING MATRICES IN CHAOTIC MICROWAVE CAVITIES: THE RANDOM COUPLING MODEL