Sergey A. Gredeskul

Propagation and scattering of nonlinear waves in disordered systems

Abstract We review recent numerical and analytical results related to the study of the coexistence of nonlinear wave propagation and disorder effects, mostly for soliton bearing systems. First of all, we briefly summarize scattering problems in the linear approximation which show many features related to the Anderson localization. Considering wavepackets instead of standard problems of the plane wave propagation, we show that the change of the asymptotic behaviour of the averaged transmission coefficient as a function of the disordered segment length may be demonstrated even at the level of the linear approach. For nonlinear problems we begin by discussing the stationary case which may be analyzed in detail. The main effect of nonlinearity, bistability, may be drastically modified by disorder in this case. However, for nonstationary problems disorder may give rise to more intensive development of nonlinear properties of physical systems via modulational instability, which is enhanced by inhomogeneities. We point out that localization effects created by disorder may vanish in the presence of strong nonlinearity, and we present examples showing that nonlinearity leads to an actual improvement of the transmission when it contributes to create soliton pulses. We analyze soliton propagation through disordered and inhomogeneous media and demonstrate that nonlinear transmission strongly depends on the soliton type: results are different for dynamical, topological, and envelope solitons.

Effect of microscopic disorder on magnetic properties of metamaterials.

We analyze the effect of microscopic disorder on the macroscopic properties of composite metamaterials and study how weak statistically independent fluctuations of the parameters of the structure elements can modify their collective magnetic response and left-handed properties. We demonstrate that even a weak microscopic disorder may lead to a substantial modification of the metamaterial magnetic properties, and a 10% deviation in the parameters of the microscopic resonant elements may lead to a substantial suppression of the wave propagation in a wide frequency range. A noticeable suppression occurs also if more than 10% of the resonant magnetic elements possess strongly different properties, and in the latter case the defects can create an additional weak resonant line. These results are of a key importance for characterizing and optimizing novel composite metamaterials with the left-handed properties at terahertz and optical frequencies.

Spectral properties and localization of an electron in a two-dimensional system with point scatterers in a magnetic field

Abstract Electron spectral properties and localization in a two-dimensional system with point potentials subject to a perpendicular magnetic field are studied. A brief review of the known results concerning electron dynamics in such systems is presented. For a set of periodic point potentials, exact dispersion laws and energy-flux diagram (Hofstadter-type butterfly) are obtained. It is shown that, in the case of one-dimensional disorder, the electron localization in a strong magnetic field is described by the random Harper equation. Energy-flux diagram for the localization length is presented and the fractal structure of the localization length is demonstrated. Near the Landau levels an exact formula for the localization length as a function of energy and disorder is obtained. The corresponding critical exponent is equal to unity which is reminiscent of one-dimensional characteristics.

Anderson localization in metamaterials and other complex media (Review Article)

This is a review of some recent (mostly ours) results on Anderson localization of light and electron waves in complex disordered systems, including: (i) left-handed metamaterials, (ii) magnetoactive optical structures, (iii) graphene superlattices, and (iv) nonlinear dielectric media. First, we demonstrate that left-handed metamaterials can significantly suppress localization of light and lead to an anomalously enhanced transmission. This suppression is essential at the long-wavelength limit in the case of normal incidence, at specific angles of oblique incidence (Brewster anomaly), and in vicinity of zero-ɛ or zero-μ frequencies for dispersive metamaterials. Remarkably, in disordered samples comprised of alternating normal and left-handed metamaterials, the reciprocal Lyapunov exponent and reciprocal transmittance increment can differ from each other. Second, we study magnetoactive multilayered structures, which exhibit nonreciprocal localization of light depending on the direction of propagation and on polarization. At resonant frequencies or realizations such nonreciprocity results in effectively unidirectional transport of light. Third, we discuss the analogy between wave propagation through multilayered samples with metamaterials and charge transport in graphene, which provides a simple physical explanation of unusual conductive properties of disordered graphene superlatices. We predict disorder-induced resonance of the transmission coefficient at oblique incidence of Dirac quasiparticles. Finally, we demonstrate that an interplay of nonlinearity and disorder in dielectric media can lead to bistability of individual localized states excited inside the medium at resonant frequencies. This results in nonreciprocity of wave transmission and unidirectional transport of light.

Effects of polarization on the transmission and localization of classical waves in weakly scattering metamaterials

We summarize the results of our comprehensive analytical and numerical studies of the effects of polarization on the Anderson localization of classical waves in one-dimensional random stacks. We consider homogeneous stacks composed entirely of normal materials or metamaterials, and also mixed stacks composed of alternating layers of a normal material and a metamaterial. We extend the theoretical study developed earlier for the case of normal incidence A. A. Asatryan et al., Phys. Rev. B 81, 075124 2010 to the case of off-axis incidence. For the general case where both the refractive indices and layer thicknesses are random, we obtain the long-wave and short-wave asymptotics of the localization length over a wide range of incidence angles including the Brewster “anomaly” angle. At the Brewster angle, we show that the long-wave localization length is proportional to the square of the wavelength, as for the case of normal incidence, but with a proportionality coefficient substantially larger than that for normal incidence. In mixed stacks with only refractive-index disorder, we demonstrate that p-polarized waves are strongly localized, while for s polarization the localization is substantially suppressed, as in the case of normal incidence. In the case of only thickness disorder, we study also the transition from localization to delocalization at the Brewster angle.

Dark solitons produced by phase steps in nonlinear optical fibers

Abstract We demonstrate analytically that dark solitons may be produced from phase steps in nonlinear optical fibers at the positive group velocity dispersion. We study the cases of a step and two steps in the phase of a cw background in detail. The parameters of general solitons are calculated. The influence of the background broadening and dissipative losses on the dark-pulse evolution is discussed too.

Nonreciprocal Anderson localization in magneto-optical random structures

We study both analytically and numerically disorder-induced localization of light in random-layered structures with magneto-optical materials. The Anderson localization in such structures demonstrates nonreciprocal features in the averaged localization length and individual transmission resonances. We employ the short-wavelength approximation where the localization effects are strong and consider both the Faraday and Voigt magneto-optical geometries. In the Faraday geometry, the transmission is strongly nonreciprocal for the circularly polarized waves, whereas in the Voigt geometry, the nonreciprocity is much weaker and it may appear only for the individual transmission resonances of the TM-polarized waves.

MULTIQUANTUM VORTICES IN CONVENTIONAL SUPERCONDUCTORS WITH COLUMNAR DEFECTS NEAR THE UPPER CRITICAL FIELD

The equilibrium vortex configuration in conventional type-II superconductors containing columnar defects is theoretically investigated. Near the upper critical field a single defect causes a strong local deformation of the vortex lattice. This deformation has

MESOSCOPIC SUPERCONDUCTING DISC WITH SHORT-RANGE COLUMNAR DEFECTS

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Electronic excitations and correlations in quantum bars

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