Photonic heterostructures with Lévy-type disorder: Statistics of coherent transmission

Abstract

We study the electromagnetic transmission T through one-dimensional (1D) photonic heterostructures whose random layer thicknesses follow a long-tailed distribution -Lévy-type distribution. Based on recent predictions made for 1D coherent transport with Lévy-type disorder, we show numerically that for a system of length L (i) the average − ln T ∝ L α for 0 < α < 1, while − ln T ∝ L for 1 ≤ α < 2, α being the exponent of the power-law decay of the layer-thickness probability distribution; and (ii) the transmission distribution P (T ) is independent of the angle of incidence and frequency of the electromagnetic wave, but it is fully determined by the values of α and ln T . Additionally we have found and numerically verified that T ∝ L −α with 0 < α < 1. Random processes characterized by density probabilities with a long tail (Lévy-type processes) have been found and studied in very different phenomena and fields such as biology, economy, and physics. One of the main features of a Lévy-type density distribution p(l) is the slow decay of its tail. More precisely, for large l,with 0 < α < 2. Note that the second moment diverges for all α and if 0 < α < 1 also the first moment diverge. This kind of distributions are also known as α-stable distributions [1]. A window on new optical materials which allow for the experimental study of Lévy flights in an outstanding controllable way was recently opened with the construction of the so-called Lévy glass [2]: titanium dioxide particles are suspended in a matrix made of glass microspheres. The distribution of the microsphere diameters is properly chosen in order that light can travel performing Lévy flights within the microspheres. The diameter distribution is characterized by the exponent α of the power-law decay of its tail; it was found [2] that when 0 < α < 1 the transport is supperdiffusive, while for α = 2 the normal diffusive transport is recovered. This experimental investigation has motivated several theoretical works on the effects of the presence of Lévy-type processes on different transport quantities in one dimension, as well as in higher dimensional systems [3][4][5][6]8].On the other hand, coherent electron transport through one-dimensional (1D) quantum wires with Lévy-type disorder was studied in Ref. [6]. It was found that for the dimensionless conductance, or transmission, T :(i) the average (over different disorder realizations) of the logarithm of the transmission behaves asand (ii) the distribution of transmission P (T ) is fully determined by the exponent α and the ensemble average ln T .We point out that although for 1 ≤ α < 2, the average ln T depends linearly on L, as in the standard Anderson localization problem, it is interesting to remark that the statistical properties of T are not those predicted by the standard scaling approach to localization, in particular by the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation [7]. That is, for 1 ≤ α < 2 the transmission fluctuations are larger than those considered in the DMPK equation. Thus, the standard statist...

Dirac wave transmission in Lévy-disordered systems

“…An anomalous localization behavior, different from the standard Anderson localization, has been obtained when the probability density of the disorder distribution has a long tail, as in the case of Lévy distributions [4][5][6][7][8][9]. In the last three decades several stochastic phenomena have been described by the statistics of Lévy distributions, such as human mobility [10], fluid dynamics [11][12][13], photons [14][15][16][17][18][19][20], random lasers [21 and 22], free-standing graphene membranes [23], and more recently, electronic transport [4][5][6][7][8][9][24][25][26][27]. These phenomena provide a venue to a deeper understanding of electronic localization.…”

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

“…The effects of Cauchy distribution on Anderson localization were studied in [26,27], while the conductance properties through quantum wires were considered in [28,29]. The effects of the long-range correlated potential on Anderson localization were considered in [30][31][32][33][34], while the effects of the Brewster anomaly were investigated numerically in [35].…”

Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder

“…There is a series of theoretical studies devoted to electron and wave transport in quantum wires with Lévy-type disorder [7][8][9][10][11][12][13]. Beenakker et al [7] calculated moments of conductance for the incoherent sequential tunneling through disordered fractal wire.…”

“…They have shown that the conductance distribution for fractal wire is also universal in the sense that it is fully determined by the Lévy index α and the average quantity ln G . Fernández-Marín et al [10] derived the statistics of electromagnetic transmission through a one-dimensional (1D) photonic heterostructure whose random layer thicknesses follow a long-tailed Lévy-type distribution. Amanatidis et al [11] investigated the conductance through a 1D disordered system where the envelope of electron wavefunction decays spatially according to the stretched exponential law.…”