We study the effect of localized modes in lattices of size N with parity-time (PT) symmetry. Such modes are arranged in pairs of quasidegenerate levels with splitting delta approximately exp(-N/xi) where xi is their localization length. The level "evolution" with respect to the PT breaking parameter gamma shows a cascade of bifurcations during which a pair of real levels becomes complex. The spontaneous PT symmetry breaking occurs at gammaPT approximately min{delta}, thus resulting in an exponentially narrow exact PT phase. As N/xi decreases, it becomes more robust with gammaPT approximately 1/N2 and the distribution P(gammaPT) changes from log-normal to semi-Gaussian. Our theory can be tested in the frame of optical lattices.

We propose a new class of optical synthetic materials that are described by non-Hermitian Hamiltonians. The building blocks of such systems are coupled PT-symmetric elements (dimers), with coupling t. Despite the lack of global PT-symmetry, these systems have a robust parameter region of real spectra (exact phase) even in cases where the complex refractive index n = β + iγ of each PT dimer is random. The validity of our proposition is confirmed for representative cases where we calculate the borders of the exact phase in terms of β, γ and t.

We consider waveguides formed by single or multiple two-dimensional chaotic cavities connected to leads. The cavities are chaotic in the sense that the ray (or equivalently, classical particle) dynamics within them is chaotic. Geometrical parameters are chosen to produce a mixed phase space (chaotic regions surrounding islands of stability where motion is regular). Incoming rays (or particles) cannot penetrate into these islands but incoming plane waves dynamically tunnel into them at a certain discrete set of frequencies (energies). The support of the corresponding quasi-bound states is along the trajectories of periodic orbits trapped within the cavity. We take advantage of this difference in the ray/wave behavior to demonstrate how chaotic waveguides can be used to design beam splitters and microlasers. We also present some preliminary experimental results in a microwave realization of such chaotic waveguide.

We propose the construction of electromagnetic (or electronic) switches and beam splitters by use of chaotic two-dimensional multiport waveguides. A prototype two-port waveguide is locally deformed to produce a ternary incomplete horseshoe characteristic of mixed phase space (chaotic regions surrounding islands of stability where motion is regular). Owing to tunneling to the phase-space stability islands, quasi-bound states (QBS) appear. Then we attach transversal ports to the waveguide in the deformation region in positions where the phase-space structure is only slightly perturbed. We show how QBS can be guided out of the waveguide through the attached transversal ports,giving rise to frequency-selective switches and beam splitters.