Phase distribution in 1D localization and phase transitions in single-mode waveguides

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 (when the system length L is increased) contain three independent variables, i.e. the Landauer resistance ρ and the combined phases ψ=θ-φ and χ=θ+φ. The phase χ does not affect the evolution of ρ and was not considered in previous papers. The distribution of the phase ψ is found to exhibit an unusual phase transition at the point E_0 when changing the electron energy E, which manifests itself in the appearance of the imaginary part of ψ. The distribution of resistance P(ρ) has no singularity at the point E_0, and the transition looks unobservable in the electron disordered systems. However, the theory of 1D localization is immediately applicable to propagation of waves in single-mode optical waveguides. The optical methods are more efficient and provide possibility to measure phases ψ and χ. On the one hand, it makes observable the phase transition in the distribution P(ψ), which can be considered as a 'trace' of the mobility edge remaining in 1D systems. On the other hand, observability of the phase χ makes actual derivation of its evolution equation, which is presented below. Relaxation of the distribution P(χ) to the limiting distribution P_∞(χ) at L→∞ is described by two exponents, whose exponentials have jumps of the second derivative, when the energy E is changed.