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 τn of the transmission matrix tt† whose sum equals g. In diffusive samples, the highest eigenvalue, τ1, is close to unity corresponding to a transmission of nearly 100%, while for localized waves, the average of τ1, is nearly equal to g. We find that the spacing between average values of ln⁡τn is constant and demonstrate that when surface interactions are taken into account it is equal to the inverse of the bare conductance.