Observation of the critical regime near Anderson localization of light

The transition from diffusive transport to localization of waves should occur for any type of classical or quantum wave in any media as long as the wavelength becomes comparable to the transport mean free path l*. The signatures of localization and those of absorption, or bound states, can, however, be similar, such that an unequivocal proof of the existence of wave localization in disordered bulk materials is still lacking. Here we present time resolved measurements of light transport through strongly scattering samples with kl* values as low as 2.5. In transmission, we observe deviations from diffusion which cannot be explained by absorption, sample geometry, or reduction in transport velocity. Furthermore, the deviations from classical diffusion increase strongly with decreasing l* as expected for a phase transition. This constitutes an experimental realization of the critical regime in the approach to Anderson localization.     

Study on localization of plane elastic waves in disordered periodic 2-2 piezoelectric composite structures

Considering the effect of mechanic-electric coupling, the propagation and localization of plane elastic waves in disordered periodic layered piezoelectric composite structures are studied. The transfer matrix between two consecutive unit cells is obtained by means of the continuity conditions and the expression of the localization factors in disordered periodic structures is presented by regarding the variables of mechanical and electrical fields as the elements of state vectors. As examples, numerical results of localization factors are presented and discussed. It can be seen from the results that ordered periodic structures possess the properties of frequency passbands and stopbands and the phenomenon of wave localization in disordered periodic structures is observed, and the larger the coefficient of variation is, the larger the localization factor or the stronger the degree of wave localization is. The characters of wave propagation and localization are very different for different sorts of piezocomposites or different structural sizes, and even for same sorts of piezocomposites and same structural sizes the characters of wave propagation and localization are also very different for different non-dimensional wavenumbers. We may design different piezocomposites or adjust the structural sizes to control the characters of wave propagation and localization.     

Wave localization in randomly disordered multi-coupled multi-span beams on elastic foundations

In this paper, the wave propagation and localization in randomly disordered periodic multi-span beams on elastic foundations are studied. For two kinds of beams, i.e. the multi-span beams on elastic foundations with periodic flexible and simple supports, the transfer matrices between two consecutive sub-spans are obtained by means of the continuity conditions. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. The localization factor characterizing the average exponential rates of growth or decay of wave amplitudes along the disordered beams is defined as the smallest positive Lyapunov exponent of the discrete dynamical system. The localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. For the two kinds of disordered periodic beams on elastic foundations, the numerical results of the localization factors are presented and analysed by comparing them with the results of the beams without elastic foundations to illustrate the effects of the elastic foundations on the wave propagation and localization. The effects of the disorder of span-length and the dimensionless torsional and linear spring stiffness on the localization factors are discussed. Moreover, the localization lengths are also calculated and discussed for certain structural parameters in disordered periodic structures. It can be observed from the results that ordered periodic multi-span beams have the characteristics of the frequency passbands and stopbands and the localization of elastic waves can occur in disordered periodic systems: the localization degree of elastic waves is strengthened with the increase of the coefficient of variation of the span-length. The influences of the elastic foundations on the wave propagation and localization are more complicated. Generally speaking, in lower-frequency regions the elastic foundations have pronounced effects on the spectral structures, but in higher-frequency regions the effects are negligible. The localization degree increases as the torsional spring stiffness increases. The linear spring has few effects on the spectral structures in higher-frequency regions, but in lower-frequency regions it has prominent effects. The larger the disorder degree, the shorter the non-dimensional localization length.     

Wave localization in randomly disordered multi-coupled multi-span beams on elastic foundations

In this paper, the wave propagation and localization in randomly disordered periodic multi-span beams on elastic foundations are studied. For two kinds of beams, i.e. the multi-span beams on elastic foundations with periodic flexible and simple supports, the transfer matrices between two consecutive sub-spans are obtained by means of the continuity conditions. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. The localization factor characterizing the average exponential rates of growth or decay of wave amplitudes along the disordered beams is defined as the smallest positive Lyapunov exponent of the discrete dynamical system. The localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. For the two kinds of disordered periodic beams on elastic foundations, the numerical results of the localization factors are presented and analysed by comparing them with the results of the beams without elastic foundations to illustrate the effects of the elastic foundations on the wave propagation and localization. The effects of the disorder of span-length and the dimensionless torsional and linear spring stiffness on the localization factors are discussed. Moreover, the localization lengths are also calculated and discussed for certain structural parameters in disordered periodic structures. It can be observed from the results that ordered periodic multi-span beams have the characteristics of the frequency passbands and stopbands and the localization of elastic waves can occur in disordered periodic systems: the localization degree of elastic waves is strengthened with the increase of the coefficient of variation of the span-length. The influences of the elastic foundations on the wave propagation and localization are more complicated. Generally speaking, in lower-frequency regions the elastic foundations have pronounced effects on the spectral structures, but in higher-frequency regions the effects are negligible. The localization degree increases as the torsional spring stiffness increases. The linear spring has few effects on the spectral structures in higher-frequency regions, but in lower-frequency regions it has prominent effects. The larger the disorder degree, the shorter the non-dimensional localization length.     

Localization in disordered,nonlinear dynamical systems

We study localization and wave trapping in disordered, nonlinear dynamical systems. For some models of classical, disordered anharmonic crystal lattices, we prove that, with large probability, there are quasiperiodic lattice vibrations of finite total energy which lie on some infinite-dimensional, compact invariant tori in phase space. Such vibrations remain localized, for all times, and there is no transport of energy through the lattice. Our general concepts and techniques extend to other systems, such as disordered, nonlinear Schrödinger equations, or randomly coupled rotors.     

Localization length of wave functions in one-dimensional disordered systems with periodic modulation

The localization length of wave functions in one-dimensional disordered systems with periodic modulation is studied. The role of spatial inhomogeneity in the problem is considered. We calculate the localization length varying with energy for the compositionally modulated systems with either the disorder of Anderson distributions or the randomness of periods. The results show that the non-uniformity of disordered systems leads to some different properties of the localized wave functions.     

Localization of elastic waves in randomly disordered multi-coupled multi-span beams

The wave localization in randomly disordered periodic multi-span continuous beams is studied. The transfer matrix method is used to deduce transfer matrices of two kinds of multi-span beams. To calculate the Lyapunov exponents in discrete dynamical systems, the algorithm for determining all the Lyapunov exponents in continuous dynamical systems presented by Wolf et al is employed. The smallest positive Lyapunov exponent of the corresponding discrete dynamical system is called the localization factor, which characterizes the average exponential rates of growth or decay of wave amplitudes along the randomly mistuned multi-span beams. For two kinds of disordered periodic multi-span beams, numerical results of localization factors are given. The effects of the disorder of span-length, the non-dimensional torsional spring stiffness and the non-dimensional linear spring stiffness on the wave localization are analysed and discussed. It can be observed that the localization factors increase with the increase of the coefficient of variation of random span-length and the degree of localization for wave amplitudes increases as the torsional spring stiffness and the linear spring stiffness increase.     

Localization of elastic waves in randomly disordered multi-coupled multi-span beams

Abstract

The wave localization in randomly disordered periodic multi-span continuous beams is studied. The transfer matrix method is used to deduce transfer matrices of two kinds of multi-span beams. To calculate the Lyapunov exponents in discrete dynamical systems, the algorithm for determining all the Lyapunov exponents in continuous dynamical systems presented by Wolf et al is employed. The smallest positive Lyapunov exponent of the corresponding discrete dynamical system is called the localization factor, which characterizes the average exponential rates of growth or decay of wave amplitudes along the randomly mistuned multi-span beams. For two kinds of disordered periodic multi-span beams, numerical results of localization factors are given. The effects of the disorder of span-length, the non-dimensional torsional spring stiffness and the non-dimensional linear spring stiffness on the wave localization are analysed and discussed. It can be observed that the localization factors increase with the increase of the coefficient of variation of random span-length and the degree of localization for wave amplitudes increases as the torsional spring stiffness and the linear spring stiffness increase.     

Statistical properties of wave transport through surface-disordered waveguides

We review some general statistical properties of wave transport through surface disordered waveguides. These systems are shown to present both striking similarities and differences with respect to quasi-one-dimensional waveguides with volume disorder. The statistical properties are analysed using extensive numerical calculations and random matrix theory results. The transport properties are characterized by the statistical behaviour of different transport coefficients that can be defined for both classical (light, microwaves, sound, etc.) and quantum (electrons) waves. In analogy with bulk-disordered systems, the behaviour of the waveguide conductance/resistance (defined for both classical and quantum waves) as a function of the system length defines three different transport regimes: ballistic, diffusive and localization. However, the coupling between waveguide modes presents significant differences with respect to the coupling induced by volume defects. For any incoming mode, there is a strong preference for the forward propagation through the lowest mode. For narrow waveguides, the statistics of reflection coefficients (reflected speckle pattern) present strong finite-size effects which can be surprisingly well described by random matrix theory. Special attention is paid to the fundamental problem of the transition between different regimes. The long-standing problems of the phase randomization process between ballistic and diffusive regimes and the evolution of the conductance statistical distribution in the transition from diffusion (Gaussian statistics) to localization (log normal statistics) are also discussed.     

Anderson localization or nonlinear waves: a matter of probability

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems (localization versus propagation) is under intense theoretical debate and experimental study. We resolve this dispute showing that, unlike in the common hypotheses, the answer is probabilistic rather than exclusive. At any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results generalize to higher dimensions as well.     

Wave localization in two-dimensional periodic systems with randomly disordered size

The expression of the localization factor in the two-dimensional periodic systems is derived based on the plane-wave expansion, transfer matrix and matrix eigenvalue methods. A comprehensive study is performed for the wave localization in the phononic crystal which is composed of steel cylinders embedded in epoxy matrix with the randomly disordered rod size. From the results, it can be observed that with the increase of the disorder degree, the localization phenomenon is strengthened. Furthermore, the filling fraction has significant effects on the wave localization characteristics.     

Statistical properties of wave transport through surface-disordered waveguides

We review some general statistical properties of wave transport through surface disordered waveguides. These systems are shown to present both striking similarities and differences with respect to quasi-one-dimensional waveguides with volume disorder. The statistical properties are analysed using extensive numerical calculations and random matrix theory results. The transport properties are characterized by the statistical behaviour of different transport coefficients that can be defined for both classical (light, microwaves, sound, etc.) and quantum (electrons) waves. In analogy with bulk-disordered systems, the behaviour of the waveguide conductance/resistance (defined for both classical and quantum waves) as a function of the system length defines three different transport regimes: ballistic, diffusive and localization. However, the coupling between waveguide modes presents significant differences with respect to the coupling induced by volume defects. For any incoming mode, there is a strong preference for the forward propagation through the lowest mode. For narrow waveguides, the statistics of reflection coefficients (reflected speckle pattern) present strong finite-size effects which can be surprisingly well described by random matrix theory. Special attention is paid to the fundamental problem of the transition between different regimes. The long-standing problems of the phase randomization process between ballistic and diffusive regimes and the evolution of the conductance statistical distribution in the transition from diffusion (Gaussian statistics) to localization (log normal statistics) are also discussed.     

Localization and Relaxation in Anisotropically Disordered Systems

We discuss localization and the scattering of excitations in bifractals, a model of anisotropically disordered systems. The localization behavior is anisotropic. With the increase of energy, the excitation crosses over from an extended wave to a wave extended in one subspace while localized in another, then to a wholly-localized wave. The loffe-Regel frequency is shown to be in the wholly-localized regime. Relaxation processes are calculated for the emission and absorption of localized vibrational excitations by a localized electronic state.The anisotropy makes effects on the results.     

Frequency-dependent localization length of SH-wave in randomly disordered piezoelectric phononic crystals

In this paper, the localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. Numerical results of the localization lengths of SH-wave are presented and discussed in ordered and disordered piezoelectric phononic crystals to identify the different effect degrees for the decay of electrical potential in the polymers and the randomness on the localization level. For the disordered case, disorder in the thickness of the polymers and disorder in the elastic constant of the piezoelectric ceramics are all considered. The results show that some parameters such as the incident angle of elastic wave, the randomness degree and the piezoelectricity of piezoelectric ceramics and so on have pronounced effects on the frequency-dependent localization length.     

Nonlinear delocalization on disordered Stark ladder

We study effects of weak nonlinearity on localization of waves in disordered Stark ladder corresponding to propagation in presence of disorder and a static field. Our numerical results show that nonlinearity leads to delocalization with subdiffusive spreading along the ladder. The exponent of spreading remains close to its value in absence of the static field. The delocalization implies the existence of statistical entanglement between far away parts of the spreading wave packet indicating importance of long-range effects.     

Density,phase and coherence properties of a low dimensional Bose-Einstein systems moving in a disordered potential

We present a detailed numerical study of the dynamics of a disordered one-dimensional Bose-Einstein condensates in position and momentum space. We particularly focus on the region where non-linearity and disorder simultaneously effect the time propagation of the condensate as well as the possible interference between various parts of the matter wave. We report oscillation between spatially extended and localized behavior for the propagating condensate which dies down with increasing non-linearity. We also report intriguing behavior of the phase fluctuation and the coherence properties of the matter wave. We also briefly compare these behavior with that of a two-dimensional condensate. We mention the relevance of our results to the related experiments on Anderson localization and indicate the possibility of future experiments     

Analytical study of the propagation of acoustic waves in a 1D weakly disordered lattice

This paper presents an analytical approach of the propagation of an acoustic wave through a normally distributed disordered lattice made up of Helmholtz resonators connected to a cylindrical duct. This approach allows to determine analytically the exact transmission coefficient of a weakly disordered lattice. Analytical results are compared to a well-known numerical method based on a matrix product. Furthermore, this approach gives an analytical expression of the localization length apart from the Bragg stopband which depends only on the standard deviation of the normal distribution disorder. This expression permits to study on one hand the localization length as a function of both disorder strength and frequency, and on the other hand, the propagation characteristics on the edges of two sorts of stopbands (Bragg and Helmholtz stopbands). Lastly, the value of the localization length inside the Helmholtz stopband is compared to the localization length in the Bragg stopband.     

Statistical description of eigenfunctions in chaotic and weakly disordered systems beyond universality

We present a semiclassical approach to eigenfunction statistics in chaotic and weakly disordered quantum systems which goes beyond random matrix theory, supersymmetry techniques, and existing semiclassical methods. The approach is based on a generalization of Berry's random wave model, combined with a consistent semiclassical representation of spatial two-point correlations. We derive closed expressions for arbitrary wave-function averages in terms of universal coefficients and sums over classical paths, which contain, besides the supersymmetry results, novel oscillatory contributions. Their physical relevance is demonstrated in the context of Coulomb blockade physics.     

On the stability of the polaron in one dimensional disordered systems

A one-dimensional disordered system of electrons described by a tight binding model interacting with vibrational degrees of freedom (in harmonic approximation) is considered. A stable configuration is determined by a numerical minimization of the total energy which is based on the adiabatic approximation. The behaviour of the electron density (charge density wave) and the density of states is analysed. The localization properties are investigated as well. In contrast to the corresponding disordered system with vanishing electron-phonon coupling the present model has an energy gap. The formation of the gap and the polaron band is shown to be quite different for both onsite and intersite types of coupling terms. For large disorder, the lattice distortion and the gap disappear if only the vibrational contribution to the intersite coupling is important. They increase, however, if only the vibrational contribution to the site energies is taken into account. In both cases the localization length decreases upon increasing the electron-phonon coupling energy. The results are discussed with respect to low dimensional organic materials and amorphous semiconductors.     

Role of optical nonlinearity on transverse localization of light in a disordered one-dimensional optical waveguide lattice

We report a detailed numerical investigation on transverse localization of light in a 1D disordered lattice consisting of a large array of coupled waveguides in the presence of nonlinearity in the medium. Our study reveals that the presence of a focusing type of nonlinearity favors faster localization of light while a defocusing type of nonlinearity degrades the quality of localization. It is shown that presence of either of these could over-shadow localization of light unless the strength of disorder is sufficiently strong. Influence of the input beam width on propagation of light in such a disordered nonlinear medium has also been discussed. The present study should be useful in potential applications, in which one could exploit dominance of focusing nonlinearity on transverse localization of light in a disordered medium.