A. A. Krokhin

LOCALIZATION AND THE MOBILITY EDGE IN ONE-DIMENSIONAL POTENTIALS WITH CORRELATED DISORDER

We show that a mobility edge exists in 1D random potentials provided specific long-range correlations. Our approach is based on the relation between binary correlator of a site potential and the localization length. We give the algorithm to construct numerically potentials with mobility edge at any given energy inside allowed zone. Another natural way to generate such potentials is to use chaotic trajectories of non-linear maps. Our numerical calculations for few particular potentials demonstrate the presence of mobility edges in 1D geometry.

Experimental observation of the mobility edge in a waveguide with correlated disorder

The tight-binding model with correlated disorder introduced by Izrailev and Krokhin [Phys. Rev. Lett. 82, 4062 (1999)] has been extended to the Kronig–Penney model. The results of the calculations have been compared with microwave transmission spectra through a single-mode waveguide with inserted correlated scatterers. All predicted bands and mobility edges have been found in the experiment, thus demonstrating that any wanted combination of transparent and nontransparent frequency intervals can be realized experimentally by introducing appropriate correlations between scatterers.The tight-binding model with correlated disorder introduced by Izrailev and Krokhin [PRL 82, 4062 (1999)] has been extended to the Kronig-Penney model. The results of the calculations have been compared with microwave transmission spectra through a single-mode waveguide with inserted correlated scatterers. All predicted bands and mobility edges have been found in the experiment, thus demonstrating that any wanted combination of transparent and non-transparent frequency intervals can be realized experimentally by introducing appropriate correlations between scatterers.

Photonic Crystal Optics and Homogenization of 2D Periodic Composites

We study the long-wavelength limit for an arbitrary photonic crystal (PC) of 2D periodicity. Light propagation is not restricted to the plane of periodicity. We proved that 2D PC’s are uni-axial or bi-axial and derived compact, explicit formulas for the effective (“principal”) dielectric constants; these are plotted for silicon - air composites. This could facilitate the custom design of optical components for diverse spectral regions and applications. Our method of “homogenization” is not limited to optical properties, but is also valid for electrostatics, magnetostatics, DC conductivity, thermal conductivity, etc. Thus our results are applicable to the Physics of Inhomogeneous Media where exact, compact formulas are scarce. Our numerical method yields results with very high accuracy, even for very large dielectric contrasts and filling fractions.

Enhancement of localization in one-dimensional random potentials with long-range correlations.

We experimentally study the effect of enhancement of localization in weak one-dimensional random potentials. Our experimental setup is a single-mode waveguide with 100 tunable scatterers periodically inserted into the waveguide. By measuring the amplitudes of transmitted and reflected waves in the spacing between each pair of scatterers, we observe a strong decrease of the localization length when white-noise scatterers are replaced by a correlated arrangement of scatterers.

Random 1D structures as filters for electrical and optical signals

Abstract We present experimental and theoretical studies of the transport properties of random 1D site potentials. The key result is that exponentially weak transmissivity of a disordered system may be modified by finite correlations. We show that the long-range correlations give rise to a continuum of extended states, which are separated from localized states by mobility edges. For energies (or frequencies) between the mobility edges, the disordered system is transparent, while it is not outside this interval. We propose to exploit this property for filtering of electrical and optical signals.

Effective dielectric constant of 2D photonic crystals with high dielectric contrast

Abstract We calculate the effective dielectric constant ( e eff ) of 2D photonic crystals (PCs) formed by rods of high dielectric constant arranged in a square lattice. One of the representations allows us to calculate the e eff in the limiting case of perfectly conducting cylinders in air. We compare our results with those obtained by the electrostatic method.

Screw misfit dislocations in soft substrates of epitaxial heterostructures

We derive compact analytical formulae for the elastic field induced by an anti-plane mismatch deformation in a heterostructure with different elastic moduli of the constituents. Unlike previous studies, we consider the possibility that the misfit dislocations may appear in the substrate, not in the epilayer. We show that this situation can be realized in heterostructures where the substrate is softer than the epilayer. In order to avoid cumbersome calculations, we consider screw misfit dislocations. The misfit dislocations emerge with zero density away from the interface in the body of the substrate when the epilayer reaches its critical thickness. Thus the epilayer remains free from dislocations if it is grown on a softer substrate. This property, which was recently observed experimentally, may find numerous applications in electronics, where epilayers are widely used as active elements.

Topological defects in 1D potential and superlattices

We study the energy spectrum of 1D periodic potential with a topological defect, i.e. the defect that breaks globally the translational symmetry. The topological defect appears by matching two identical periodic potentials at a point where the derivative is different from zero. At the defect the derivative (the force) suffers a discontinuous jump. The amplitude of the jump characterizes the strength of the defect. We show that topological defects may give rise to two localized states within the energy gaps. This situation is different from point defects that lead to a single discrete level in every gap.

Transport properties of 1D tight-binding disordered models : the Hamiltonian map approach

A new method is developed for the study of transport properties of 1D models with random potentials. It is based on an exact transformation that reduces discrete Schrodinger equation in the tight-binding model to a two-dimensional Hamiltonian map. This map describes the behavior of a classical linear oscillator under random parametric delta-kicks. We are interested in the statistical properties of the transmission coefficient

Optical frequencies of spectra of photonic crystals

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