Transmission characteristics of mesoscopic dichotomic Markov filters

The transmission characteristics of a statistical ensemble of multilayer filters for electrons (or electromagnetic waves) is studied for potentials (dielectric constants) that may be represented by dichotomic Markov processes, both continuous and discrete. For the ratio R of the wave reflectivity to transmissivity, three approximate methods, direct perturbation, state-variable-cumulant expansion, and state-variable-marginal averaging, are critically compared with each other and with exact numerical configurational averaging. The probability distributions P(lnR) and P(η) (η is the input-wave impedance), obtained numerically, exhibit salient oscillatory structure, and large deviations from the normal distribution. The connection between the spatial correlation function of the potential and the probability distributions of the transmission characteristics is investigated. It is proved that an often-referenced expression for the average 〈R〉 is valid only in the limit ${\mathit{l}}_0$≫λ/2π, independently of l, where ${\mathit{l}}_0$ and λ are the localization length and wavelength of an electron, and l is the filter length.