Localization Of Classical Waves Research Articles

Homogeneous Dislocation-Induced Rainbow Concentrating for Elastic Waves

Defects play a crucial role in the physical properties of crystals, whether for classical or quantum systems. For example, in photonic/phononic crystals, defects can serve as precise guidance and localization of classical electromagnetic or mechanical waves. Rainbow concentrating, a recently proposed exotic wave localization [Phys. Rev. Lett. 126, 113902 (2021)] exploits defects to enable the collection and frequency routing of weak signals in real space. In this paper, using a solid-state phononic crystal (PnC) plate, we experimentally verify this phenomenon by deliberately infusing a homogeneous graded dislocation, i.e., a line defect, into the PnC. Two PnCs separated by the defect will breed deterministic interface states along with the defect, offering rainbow trapping and concentrating for elastic waves. Our PnC-based rainbow trappers and concentrators are scalable and configurable, promising for advancing applications like energy harvesting, information processing, and acoustofluidic manipulating.

Observation of a transition to a localized ultrasonic phase in soft matter

Anderson localization arises from the interference of multiple scattering paths in a disordered medium, and applies to both quantum and classical waves. Soft matter provides a unique potential platform to observe localization of non-interacting classical waves because of the order of magnitude difference in speed between fast and slow waves in conjunction with the possibility to achieve strong scattering over broad frequency bands while minimizing dissipation. Here, we provide long sought evidence of a localized phase spanning up to 246 kHz for fast (sound) waves in a soft elastic medium doped with resonant encapsulated microbubbles. We find the transition into the localized phase is accompanied by an anomalous decrease of the mean free path, which provides an experimental signature of the phase transition. At the transition, the decrease in the mean free path with changing frequency (i.e., disorder strength) follows a power law with a critical exponent near unity. Within the localized phase the mean free path is in the range 0.4–1.0 times the wavelength, the transmitted intensity at late times is well-described by the self-consistent localization theory, and the localization length decreases with increasing microbubble volume fraction. Our work sets the foundation for broadband control of localization and the associated phase transition in soft matter, and affords a comparison of theory to experiment.

Anderson localization of classical waves in weakly scattering one-dimensional Levy lattices

Anderson localization of classical waves in weakly scattering one-dimensional Levy lattices is studied analytically and numerically. The disordered medium is composed of layers with alternating refractive indices and with thickness disorder distributed according to the Pareto distribution $\ensuremath{\sim}1/{x}^{(\ensuremath{\alpha}+1)}$. In Levy lattices the variance (or both variance and mean) of a random parameter does not exist, which leads to a different functional form for the localization length. In this study an equation for the localization length is obtained, and it is found to be in excellent agreement with the numerical calculations throughout the spectrum. The explicit asymptotic equations for the localization lengths for both short and long wavelengths have been deduced. It is shown that the localization length tends to a constant at short wavelengths and it is determined by the layer interface Fresnel coefficient. At the long wavelengths the localization length is proportional to the power of the wavelength $\ensuremath{\ell}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{\alpha}}$ for $1<\ensuremath{\alpha}<2$, and it has a transcendental behavior $\ensuremath{\ell}\ensuremath{\sim}{\ensuremath{\lambda}}^{2}/ln\ensuremath{\lambda}$ for $\ensuremath{\alpha}=2$. For $\ensuremath{\alpha}>2$, where the variance of the random distribution exists, the localization length attains its classical long-wavelength asymptotic form $\ensuremath{\ell}\ensuremath{\sim}{\ensuremath{\lambda}}^{2}$.

Transmission and Anderson localization in dispersive metamaterials

Comprehensive theoretical and numerical studies of the effects of dispersion and absorption on the Anderson localization of classical waves in weakly disordered, one-dimensional stacks composed of dispersive metamaterials and normal materials are presented. An asymptotic analysis for studying the effects of dispersion and absorption is developed. It is shown that the localization of waves in random stacks composed entirely of either metamaterial or normal dielectric layers is completely suppressed at frequencies where the magnetic permeability or the dielectric permittivity is zero. In mixed stacks of alternating layers of normal and metamaterials with disorder present in either the dielectric permittivity or the magnetic permeability, localization is substantially suppressed not only at these frequencies but in essentially wider frequency ranges. When both the permittivity and the permeability are random, the localization behavior is similar to that in monotype stacks. At the transition from a double negative metamaterial to a single negative metamaterial, the transmission length drops dramatically in a manner that might be useful in optical switching. Polarization effects are also considered and it is shown that localization is suppressed at the Brewster angle, in a manner dependent on both the polarization and the nature of the disorder. Theoretical predictions are in excellent agreement with numerical calculations.

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 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.

SELF-CONSISTENT THEORY OF ANDERSON LOCALIZATION: GENERAL FORMALISM AND APPLICATIONS

The self-consistent theory of Anderson localization of quantum particles or classical waves in disordered media is reviewed. After presenting the basic concepts of the theory of Anderson localization in the case of electrons in disordered solids, the regimes of weak and strong localization are discussed. Then the scaling theory of the Anderson localization transition is reviewed. The renormalization group theory is introduced and results and consequences are presented. It is shown how scale-dependent terms in the renormalized perturbation theory of the inverse diffusion coefficient lead in a natural way to a self-consistent equation for the diffusion coefficient. The latter accounts quantitatively for the static and dynamic transport properties except for a region near the critical point. Several recent applications and extensions of the self-consistent theory, in particular for classical waves, are discussed.

Anderson localization of classical waves in weakly scattering metamaterials

We study the propagation and localization of classical waves in one-dimensional disordered structures composed of alternating layers of left- and right-handed materials (mixed stacks) and compare them with structures composed of different layers of the same material (homogeneous stacks). For weakly scattering layers, we have developed an effective analytical approach and have calculated the transmission length within a wide range of the input parameters. This enables us to describe, in a unified way, the localized and ballistic regimes as well as the crossover between them. When both refractive index and layer thickness of a mixed stack are random, the transmission length in the long-wave range of the localized regime exhibits a quadratic power wavelength dependence with different coefficients of proportionality for mixed and homogeneous stacks. Moreover, the transmission length of a mixed stack differs from the reciprocal of the Lyapunov exponent of the corresponding infinite stack. In both the ballistic regime of a mixed stack and in the near long-wave region of a homogeneous stack, the transmission length of a realization is a strongly fluctuating quantity. In the far long-wave part of the ballistic region, the homogeneous stack becomes effectively uniform and the transmission length fluctuations are weaker. The crossover region from the localization to the ballistic regime is relatively narrow for both mixed and homogeneous stacks. In mixed stacks with only refractive-index disorder, Anderson localization at long wavelengths is substantially suppressed, with the localization length growing with wavelength much faster than for homogeneous stacks. The crossover region becomes essentially wider and transmission resonances appear only in much longer stacks. The effects of absorption on one-dimensional transport and localization have also been studied, both analytically and numerically. Specifically, it is shown that the crossover region is particularly sensitive to losses, so that even small absorption noticeably suppresses frequency dependent oscillations in the transmission length. All theoretical predictions are in an excellent agreement with the results of numerical simulations.

Anderson localization of polar eigenmodes in random planar composites

Anderson localization of classical waves in disordered media is a fundamental physicalphenomenon that has attracted attention in the past three decades. More recently,localization of polar excitations in nanostructured metal–dielectric films (also known asrandom planar composites) has been subject of intense studies. Potential applications ofplanar composites include local near-field microscopy and spectroscopy. A numberof previous studies have relied on the quasistatic approximation and a directanalogy with localization of electrons in disordered solids. Here I consider thelocalization problem without the quasistatic approximation. I show that localization ofpolar excitations is characterized by algebraic rather than by exponential spatialconfinement. This result is also valid in two and three dimensions. I also showthat the previously used localization criterion based on the gyration radius ofeigenmodes is inconsistent with both exponential and algebraic localization. Analternative criterion based on the dipole participation number is proposed. Numericaldemonstration of a localization–delocalization transition is given. Finally, it is shown that,contrary to the previous belief, localized modes can be effectively coupled to runningwaves.

Localized modes in random arrays of cylinders

Anderson localization of classical waves in random arrays of dielectric cylinders is numerically investigated as a function of the distribution of their diameters. We show that using polydispersed resonant scatterers increases the localization length, while using identical resonant scatterers fosters Anderson localization. We discuss this collective process and link it to the effect of proximity resonances that has been studied in the case of a small number of resonant scatterers.

Localization of classical waves in weakly scattering two-dimensional media with anisotropic disorder

We study the localization of classical waves in weakly scattering two-dimensional systems with anisotropic disorder. The analysis is based on a perturbative path-integral technique combined with a spectral filtering that accounts for the first-order Bragg scattering only. It is shown that in the long-wavelength limit the radiation is always localized, and the localization length is independent of the direction of propagation, the latter in contrast to the predictions based on an anisotropic tight-binding model. For shorter wavelengths that are comparable to the correlation scales of the disorder, the transport properties of disordered media are essentially different in the directions along and across the correlation ellipse. There exists a frequency-dependent critical value of the anisotropy parameter, below which waves are localized at all angles of propagation. Above this critical value, the radiation is localized only within some angular sectors centered at the short axis of the correlation ellipse and is extended in other directions.

A General Framework for Localization of Classical Waves: II. Random Media

We study localization of classical waves in random media in the general framework introduced in Part I of this work. This framework allows for two random coefficients, encompasses acoustic waves with random position dependent compressibility and mass density, elastic waves with random position dependent Lame moduli and mass density, electromagnetic waves with random position dependent magnetic permeability and dielectric constant, and allows for anisotropy. We show exponential localization (Anderson localization) and strong Hilbert–Schmidt dynamical localization for random perturbations of periodic media with a spectral gap.

Explicit finite volume criteria for localization in continuous random media and applications

We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application to Anderson Hamiltonians on the continuum yields localization at the bottom of the spectrum in an interval of size Cλ for large λ, where λ stands for the disorder parameter. A more sophisticated application proves localization for two-dimensional random Schrodinger operators in a constant magnetic field (random Landau Hamiltonians) up to a distance \( C \frac{\log B}{B} \) from the Landau levels for large B, where B is the strength of the magnetic field.

About wave localization in two-dimensional random media

In this Letter, we wish to discussion some basic questions pertinent to the phenomenon of Anderson localization of classical waves in two-dimensional random media. Although a definite answer to the two-dimensional localization is not yet found, a common consensus has been reached based upon the scaling analysis Abraham et al. [Phys. Rev. Lett. 43 (1979) 679]. That is, all waves are localized in two dimensions for any given amount of disorders. This view has been prevalent for more than two decades. Here, we explain some recent results and considerations that tend to be contrary to this view or the consequences of it.

Localization of classical waves in two-dimensional random media: a comparison between the analytic theory and exact numerical simulation.

The localization length of classical waves in two-dimensional random media is calculated exactly, and is compared with the theoretical prediction from the current analytic theory. Significant discrepancies are observed. It is also shown that as the frequency varies, critical changes in the localization behavior can occur. However, by a rescaling of parameters the two results tend to match each other for weak scattering. Possible reasons for the discrepancies are discussed.

A General Framework for Localization of Classical Waves: I. Inhomogeneous Media and Defect Eigenmodes

We introduce a general framework for studying the localization of classical waves in inhomogeneous media, which encompasses acoustic waves with position dependent compressibility and mass density, elastic waves with position dependent Lame moduli and mass density, and electromagnetic waves with position dependent magnetic permeability and dielectric constant. We also allow for anisotropy. We develop mathematical methods to study wave localization in inhomogeneous media. We show localization for local perturbations (defects) of media with a spectral gap, and study midgap eigenmodes.

Resonances and localization of classical waves in random systems with correlated disorder.

An original approach to the description of classical wave localization in weakly scattering random media is developed. The approach accounts explicitly for the correlation properties of the disorder, and is based on the idea of spectral filtering. According to this idea, the Fourier space (power spectrum) of the scattering potential is divided into two different domains. The first one is related to the global (Bragg) resonances and consists of spectral components lying within a limiting sphere of the Ewald construction. These resonances, arising in the momentum space as a result of a self-averaging, determine the dynamic behavior of the wave in a typical realization. The second domain, consisting of the components lying outside the limiting sphere, is responsible for the effect of local (stochastic) resonances observed in the configuration space. Combining a perturbative path-integral technique with the idea of spectral filtering allows one to eliminate the contribution of local resonances, and to distinguish between possible stochastic and dynamical localization of waves in a given system with arbitrary correlated disorder. In the one-dimensional (1D) case, the result, obtained for the localization length by using such an indirect procedure, coincides exactly with that predicted by a rigorous theory. In higher dimensions, the results, being in agreement with general conclusions of the scaling theory of localization, add important details to the common picture. In particular, the effect of the high-frequency localization length saturation is predicted for 2D systems. Some possible links with the problem of wave transport in periodic or near-periodic systems (photonic crystals) are also discussed.

Statistics of the Lyapunov Exponent in 1D Random Periodic-on-Average Systems

By means of Monte Carlo simulations we show that there are two qualitatively different modes of localization of classical waves in 1D random periodic-on-average systems. States from pass bands and band edges of the underlying band structure demonstrate single parameter scaling with universal behavior. States from the interior of the band gaps do not have universal behavior and require two parameters to describe their scaling properties. The transition between these two types of behavior occurs in an extremely narrow region of frequencies. When the degree of disorder exceeds a certain critical value the single parameter scaling is restored for an entire band gap.

Asymptotic analysis of classical wave localization in multiple-scattering random media

In this work we consider the localization of classical waves propagating in random continuum. We apply the method of proper time for the transfer from the elliptic-type wave equation to the generalized parabolic one. Presenting the solution of the latter equation in the form of the Feynman path integral allows us to estimate the so-called wave correction terms. These corrections are related to coherent backscattering and recurrent multiple-scattering events, i.e., to phenomena that cannot be described within the framework of the conventional theories of radiative transfer or small-angle scattering. We evaluate the wave correction to the mean intensity of a point source located in a statistically homogeneous Gaussian random medium. Our results confirm that there is an essential difference between two- and three-dimensional systems. We consider both isotropic and anisotropic media and show in particular that in the latter case there is a critical value of the anisotropy parameter, below which the system behaves basically as a three-dimensional isotropic medium, i.e., the wave correction is positive for all observation angles. Above this critical value the properties of the medium are similar to those of a layered structure.

Path-integral analysis of scalar wave propagation in multiple-scattering random media.

In this work we consider the general problem of scalar wave propagation in a continuously inhomogeneous random medium, applying the approach originally proposed by Fock for the integration of quantum-mechanical equations. The principal idea of the method is based on the introduction of an additional pseudotime variable and the transfer to a higher-dimensional space, in which the propagation process is described by the generalized parabolic equation similar to the nonstationary Schr\"odinger equation in quantum mechanics. We present its solution in a form of Feynman path integral, the asymptotic evaluation of which in the far field allows us to estimate the so-called wave correction terms. These corrections are related to coherent backscattering and repeated multiple-scattering events on the same inhomogeneities, i.e., to the phenomena that are not described in the framework of the conventional theories of radiative transfer or small-angle scattering. As an example of the approach we consider the first statistical moments of the field for a point source located in a statistically homogeneous Gaussian random medium. The correction term obtained for the mean field coincides exactly with the classical result of the Bourret approximation for the Dyson equation, but with a much weaker restriction on the value of wave number k, which allows us to analyze the correction as a function of k. The main feature of the result obtained for the second moment of the field is that the normalized mean intensity is not equal to unity. We relate such behavior to the localization phenomenon. The dependence of the correction term does not differ significantly from that obtained in works concerning the localization of classical waves in discrete random media. \textcopyright{} 1996 The American Physical Society.

Localization of classical waves I: Acoustic waves

We consider classical acoustic waves in a medium described by a position dependent mass density ϱ(x). We assume that ϱ(x) is a reandom perturbation of a periodic function ϱ0(x) and that the periodic acoustic operator\(A_0 = - \nabla \cdot \tfrac{1}{{\varrho _0 (x)}}\nabla \) has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the self-adjoint operators\(A = - \nabla \cdot \tfrac{1}{{\varrho _0 (x)}}\nabla \) onL2(ℝd). We prove that, in the random medium described by ϱ(x), the random operatorA exhibits Anderson localization inside the gap in the spectrum ofA0. This is shown even in situations when the gap is totally filled by the spectrum of the random opertor; we can prescribe random environments that ensure localization in almost the whole gap.