Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium

Kogan, Eugene; Kaveh, M.

doi:10.1103/physrevb.52.r3813

Cited by 98 publications

(129 citation statements)

References 11 publications

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“…32 for a few values of the mean conductance. Distribution p(s a ) possesses universal variance, Universality of the distribution (131) was confirmed experimentally in experiment with microwave electromagnetic waves [24,95]. Numerically, it was studied in Ref.…”

Section: Other Universal Relationsmentioning

confidence: 93%

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Numerical analysis of the Anderson localization

Markoš¹

2006

Acta Physica Slovaca. Reviews and Tutorials

135

242

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The aim of this paper is to demonstrate, by simple numerical simulations, the main transport properties of disordered electron systems. These systems undergo the metal insulator transition when either Fermi energy crosses the mobility edge or the strength of the disorder increases over critical value. We study how disorder affects the energy spectrum and spatial distribution of electronic eigenstates in the diffusive and insulating regime, as well as in the critical region of the metal-insulator transition. Then, we introduce the transfer matrix and conductance, and we discuss how the quantum character of the electron propagation influences the transport properties of disordered samples. In the weakly disordered systems, the weak localization and anti-localization as well as the universal conductance fluctuation are numerically simulated and discussed. The localization in the one dimensional system is described and interpreted as a purely quantum effect. Statistical properties of the conductance in the critical and localized regimes are demonstrated. Special attention is given to the numerical study of the transport properties of the critical regime and to the numerical verification of the single parameter scaling theory of localization. Numerical data for the critical exponent in the orthogonal models in dimension 2 < d ≤ 5 are compared with theoretical predictions. We argue that the discrepancy between the theory and numerical data is due to the absence of the self-averaging of transmission quantities. This complicates the analytical analysis of the disordered systems. Finally, theoretical methods of description of weakly disordered systems are explained and their possible generalization to the localized regime is discussed. Since we concentrate on the one-electron propagation at zero temperature, no effects of electronelectron interaction and incoherent scattering are discussed in the paper.

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Section: Other Universal Relationsmentioning

confidence: 93%

“…[94,95]. The distribution p(s a ) is determined only by the mean conductance, and is given by the following analytical formula,…”

Section: Other Universal Relationsmentioning

confidence: 99%

Numerical analysis of the Anderson localization

Markoš¹

2006

Acta Physica Slovaca. Reviews and Tutorials

135

242

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“…The variance of relative fluctuations of conductance or of transmission of classical waves increases as hTi falls (7,(10)(11)(12)(13), where h. . .i indicates averaging over an ensemble of samples.…”

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confidence: 99%

Microwave conductance in random waveguides in the cross-over to Anderson localization and single-parameter scaling

Shi

Wang

Genack

2014

Proc. Natl. Acad. Sci. U.S.A.

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The nature of transport of electrons and classical waves in disordered systems depends upon the proximity to the Anderson localization transition between freely diffusing and localized waves. The suppression of average transport and the enhancement of relative fluctuations in conductance in one-dimensional samples with lengths greatly exceeding the localization length, L > > ξ, are related in the single-parameter scaling (SPS) theory of localization. However, the difficulty of producing an ensemble of statistically equivalent samples in which the electron wave function is temporally coherent has so-far precluded the experimental demonstration of SPS. Here we demonstrate SPS in random multichannel systems for the transmittance T of microwave radiation, which is the analog of the dimensionless conductance. We show that for L ∼ 4 ξ, a single eigenvalue of the transmission matrix (TM) dominates transmission, and the distribution of the ln T is Gaussian with a variance equal to the average of −ln T , as conjectured by SPS. For samples in the cross-over to localization, L ∼ ξ, we find a one-sided distribution for ln T . This anomalous distribution is explained in terms of a charge model for the eigenvalues of the TM τ in which the Coulomb interaction between charges mimics the repulsion between the eigenvalues of TM. We show in the localization limit that the joint distribution of T and the effective number of transmission eigenvalues determines the probability distributions of intensity and total transmission for a single-incident channel.T he suppression of transport of coherent quantum or classical waves in disordered media known as Anderson localization (1-3) has been observed for electrons in solids (4), atoms in laser speckle patterns (5), and electromagnetic and acoustic waves in random dielectric and metallic structures (6-9). The variance of relative fluctuations of conductance or of transmission of classical waves increases as hTi falls (7, 10-13), where h. . .i indicates averaging over an ensemble of samples. Large variations in conductance relative to its average value occur for localized waves because the wave can be on-or off-resonance with modes of the medium with centers of localization that are at different positions within the sample (14-16). In addition, modes of the medium may occasionally overlap to enhance coupling through the sample (17). Such fluctuations make it impossible to predict transport in an individual disordered sample. However, fluctuations of the field within disordered samples may be exploited to sharpen focusing in random systems (18,19) and lower the threshold for lasing (20,21). A full account of transport in a random system would begin with the measurement of the probability distributions of conductance in ensembles of random samples with different physical dimensions in the cross-over to Anderson localization.The importance of fluctuations in electronics was first recognized in calculations of conductance mediated by localized states (3). The first observations of the impact o...

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“…The distribution of the photon conductance was also investigated experimentally [17]. In the diffusive regime p(g) is approximately Gaussian, while in the strong scattering regime (g 1), theory [14,15] and experiment [18] are consistent. To date, however, none of the models for the conductance distribution for dimensions d ≥ 2 with bulk defects have fully incorporated multiple scattering.…”

mentioning

confidence: 93%

“…Calculations of p(g) have been reported in the diffusive approximation [14], using random matrix theory [15] and for surface corrugated waveguides [16]. The distribution of the photon conductance was also investigated experimentally [17].…”

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confidence: 99%

Conductance of photons in disordered photonic crystals

et al. 2005

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The conductance of photons in two-dimensional disordered photonic crystals has been calculated using an exact multipole/plane wave method that includes all multiple scattering processes. The importance of evanescent coupling between adjacent layers is demonstrated and reveals that the widely used Pichard theory of electron conductance and the probability density distribution for the conductance of electrons based on the Dorokhov-Mello-Pereyra-Kumar equation do not apply to photons in a strongly scattering medium. The variance of the conductance is shown numerically to be independent of sample size for weak disorder, in accordance with the phenomenon of Universal Conductance Fluctuations established earlier for electrons. The conductance variance is also a strong function of disorder with the region of the Universal Conductance Fluctuation being very narrow. We show also that the relevant transfer matrix belongs to the class of complex symplectic matrices Sp(2N, C). Since the scaling theory of Anderson localization [5] is based on the scaling properties of the averaged conductance g , the discovery of UCF initiated a discussion about the validity of scaling theory itself, and it has been suggested that it should be reformulated in terms of conductance distributions p(g) [6]. However, no such theory exists at this time. In quasi-1D systems, the evolution of p(g) as a function of conductor length L is governed by the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation [7]. This elegant theory was recently extended to higher dimensions [8] for broken time reversal symmetry. The electronic conductance distribution was calculated numerically [9] for both insulating and metallic regimes and at the mobility edge [10,11]. Subsequently, nonanalytic behavior of p(g) was reported at the crossover point on the mobility edge (g ≈ 1) [12]. Despite substantial research, complete characterization of the conductance distribution remains open.Originally developed to describe the transport properties of electrons in disordered wires, the concept of conductance can also be applied to photons [13]. Calculations of p(g) have been reported in the diffusive approximation [14], using random matrix theory [15] and for surface corrugated waveguides [16]. The distribution of the photon conductance was also investigated experimentally [17]. In the diffusive regime p(g) is approximately Gaussian, while in the strong scattering regime (g 1), theory [14,15] and experiment [18] are consistent. To date, however, none of the models for the conductance distribution for dimensions d ≥ 2 with bulk defects have fully incorporated multiple scattering. Here we undertake such a calculation. While the modelling of non-interacting photons differs from that electrons, the majority of electronic models do not take into account electron-electron interactions. Accordingly, the results obtained in the photon case could be also relevant to electrons.Our aim here is to investigate the conductance fluctuations of photons and its distribution for two-dimensional disordere...

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Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium

Abstract: Statistical properties of coherent radiation propagating in a quasi-one-dimensional random medium are studied in the framework of random-matrix theory. Distribution functions for the total transmission coefficient and the angular transmission coefficient are obtained.

Cited by 98 publications

References 11 publications

Numerical analysis of the Anderson localization

Numerical analysis of the Anderson localization

Microwave conductance in random waveguides in the cross-over to Anderson localization and single-parameter scaling

Conductance of photons in disordered photonic crystals

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