Total reflection of a plane wave from a semi-infinite, one-dimensional random medium: Distribution of the phase

Abstract

It has recently been shown by Frisch and the author that, under quite general conditions, a non-absorbing, time-independent, one-dimensional, semi-infinite random medium is totally reflecting. An ergodic argument borrowed from classical statistical mechanics is used to illustrate this result. Nevertheless, ergodic theory does not give a full statistical description of the reflected wave. Assuming that the refractive index is a two-valued markovian random function of position, the distribution of the phase of the reflection coefficient is calculated both analytically and numerically. Particular attention is given to the limit where the fluctuations of the refractive index go over into a white noise. It is found that the distribution of the phase is generally not uniform and depends upon the parameters of the random medium. Finally, a conjecture is presented for the multimode problem (anisotropic media, plasmas, etc.): a semi-infinite medium is still totally reflecting but the reflected energy is not equally distributed among the modes.