Wave propagation in one-dimensional disordered structures

Wave propagation in one-dimensional disordered structures is studied via a variable R, which represents the quantum-mechanical resistance to electronic transport, or the ratio of total-energy density to power flux for electromagnetic propagation. R is formulated by three distinct methods: (1) differential equation, (2) perturbation series, (3) state-variable representation. An explicit solution for R is obtained in the limit of small fluctuations of the potential (or dielectric constant). The configurational-average state vector is determined in the second cumulant approximation for stationary processes, and an explicit expression is given for the average 〈R〉 for any potential distribution. The electron localization length exhibits a minimum as a function of the correlation length of the random potential. The exact expression for 〈R〉 is obtained for a white-noise distribution of potentials (or dielectric constants), both from the perturbation series, and from the state-variable formulation.