Transmission Eigenvalues and the Bare Conductance in the Crossover to Anderson Localization

We measure the field transmission matrix t for microwave radiation propagating through random waveguides in the crossover to Anderson localization. From these measurements, we determine the dimensionless conductance g and the individual eigenvalues tau(n) of the transmission matrix tt(dagger) whose sum equals g. In diffusive samples, the highest eigenvalue, tau(1), is close to unity corresponding to a transmission of nearly 100%, while for localized waves, the average of tau(1), is nearly equal to g. We find that the spacing between average values of ln tau(n) is constant and demonstrate that when surface interactions are taken into account it is equal to the inverse of the bare conductance.