Unusual phase transition in 1D localization and its observability in optics

Localization of electrons in 1D disordered systems is usually described in the random phase approximation, when distributions of phases φ and θ, entering the transfer matrix, are considered as uniform. In the general case, the random phase approximation is violated, and the evolution equations are written in terms of the Landauer resistance ρ and the combined phases ψ = θ − φ and χ = θ + φ. The distribution of the phase ψ is found to exhibit an unusual phase transition at the point ε ₀ when changing the electron energy ε , which manifests itself in the appearance of the imaginary part of ψ. The distribution of resistance P(ρ) has no singularity at the point ε ₀ , and the transition seems unobservable in the framework of condensed matter physics. However, the theory of 1D localization is immediately applicable to the scattering of waves propagating in a single-mode optical waveguide. Modern optical methods open a way to measure phases ψ and χ. As a result, the indicated phase transition becomes observable.