Revisiting the optical PTPT-symmetric dimer

Optics has proved a fertile ground for the experimental simulation of quantum mechanics. Most recently, optical realizations of $\mathcal{PT}$-symmetric quantum mechanics have been shown, both theoretically and experimentally, opening the door to international efforts aiming at the design of practical optical devices exploiting this symmetry. Here, we focus on the optical $\mathcal{PT}$-symmetric dimer, a two-waveguide coupler were the materials show symmetric effective gain and loss, and provide a review of the linear and nonlinear optical realizations from a symmetry based point of view. We go beyond a simple review of the literature and show that the dimer is just the smallest of a class of planar $N$-waveguide couplers that are the optical realization of Lorentz group in 2+1 dimensions. Furthermore, we provide a formulation to describe light propagation through waveguide couplers described by non-Hermitian mode coupling matrices based on a non-Hermitian generalization of Ehrenfest theorem.Comment: 25 pages, 12 figure

Observation of surface states with algebraic localization

We introduce and experimentally demonstrate a class of surface bound states with algebraic decay in a one-dimensional tight-binding lattice. Such states have an energy embedded in the spectrum of scattered states and are structurally stable against perturbations of lattice parameters. Experimental demonstration of surface states with algebraic localization is presented in an array of evanescently-coupled optical waveguides with tailored coupling rates.Comment: revised version with Supplemental Material, to appear in Phys. Rev. Let

The effect of disorder on the wave propagation in one-dimensional periodic optical systems

The influence of disorder on the transmission through periodic waveguides is studied. Using a canonical form of the transfer matrix we investigate dependence of the Lyapunov exponent $\gamma$ on the frequency $\nu$ and magnitude of the disorder $\sigma$. It is shown that in the bulk of the bands $\gamma \sim \sigma^2$, while near the band edges it has the order $\gamma \sim \sigma^{2/3}$. This dependence is illustrated by numerical simulations.Comment: 15 pages, 4 figure

Dynamic versus Anderson wavepacket localization

We address the interplay between two fundamentally different wavepacket localization mechanisms, namely resonant dynamic localization due to collapse of quasi-energy bands in periodic media and disorder-induced Anderson localization. Specifically, we consider light propagation in periodically curved waveguide arrays on-resonance and off-resonance, and show that inclusion of disorder leads to a gradual transition from dynamic localization to Anderson localization, which eventually is found to strongly dominate. While in the absence of disorder, the degree of localization depends critically on the bending amplitude of the waveguide array, when the Anderson regime takes over the impact of resonant effects becomes negligible.Comment: 13 pages, 5 figures, to appear in Physical Review