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 \varphi and \theta, 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 \rho and the combined phases \psi=\theta-\varphi and \chi=\theta+\varphi. The phase \chi does not affect the evolution of \rho and was not considered in previous papers. The distribution of the phase \psi 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 \psi. The distribution of resistance P(\rho) 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 \psi and \chi. On the one hand, it makes observable the phase transition in the distribution P(\psi), which can be considered as a 'trace' of the mobility edge remaining in 1D systems. On the other hand, observability of the phase \chi makes actual derivation of its evolution equation, which is presented below. Relaxation of the distribution P(\chi) to the limiting distribution P_\infty(\chi) at L\to\infty is described by two exponents, whose exponentials have jumps of the second derivative, when the energy E is changed.