Dissertations, Theses, and Capstone Projects

Date of Degree


Document Type


Degree Name





Azriel Z. Genack

Subject Categories

Physics | Statistics and Probability


conductance, focusing, quasi mode, random media, speckles, transmission matrix


This thesis describes the measurement and analysis of the transmission matrix (TM) for microwave radiation propagating through multichannel random waveguides in the crossover to Anderson localization. Eigenvalues of the transmission matrix and the associated eigenchannels are obtained via a singular value decomposition of the TM. The sum of the transmission eigenvalues yields the transmittance T, which is the classical analog of the dimensionless conductance g. The dimensionless conductance g is the electronic conductance in units of the quantum conductance, G/(e^2/h).

For diffusive waves g>1, approximately g transmission eigenchannels contribute appreciably to the transmittance T. In contrast, for localized waves with g<1, T is dominated by the highest transmission eigenvalue, &tau&1. For localized waves, the inverse of the localization lengths of different eigenchannels are found to be equally spaced.

Measurement of the TM allows us to explore the statistics of the transmittance T. A one-sided log-normal distribution of T is found for a random ensemble with your g=0.37 and explained using an intuitive Coulomb gas model for the transmission eigenvalues. Single parameter scaling (SPS) predicted for one dimension random system is approached in multichannel systems once T is dominated by a single transmission eigenchannel.

In addition to the statistics of the TM for ensembles of random samples, we investigated the statistics of a single TM. The statistics within a large single TM are found to depend upon a single parameter, the eigenchannel participation number, M. The variance of the total transmission normalized by its averaging in the TM is equal to M-1. We found universal fluctuation of M, reminiscent of the well known universal conductance fluctuations for diffusive waves.

We demonstrate focusing of steady state and pulse transmission through a random medium via phase conjugation of the TM. The contrast between the focus and the background is determined by M and the size of the transmission matrix N. The spatio-temporal profile of focused radiation in the diffusive limit is shown to be the square of the field-field correlation function in space and time.

We determine the density of states (DOS) of a disordered medium from the dynamics of transmission eigenchannels and from the quasi-normal modes of the medium for localized samples. The intensity profile of each eigenchannel within the random media is closely linked to the dynamics of transmission eigenchannels and an analytical expression for intensity profile of each of the eigenchannel based on numerical simulation was provided.