Conductance of photons in disordered photonic crystals

Abstract

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...

Numerical analysis of the Anderson localization

“…Special attention is deserved also for disordered systems with chiral symmetries [173]. Also, application of the concepts of the electron localization to the propagation of the electromagnetic [22,24,93,174], and acoustic [175] waves in disordered media opens a new field for studies of the localization and promise further development of the localization theory.…”

Periodic nanostructures for photonics

“…The resulting efficient modeling and simulations tools allow to optimize designs [118,119], to characterize finite-size photonic-crystal samples [120,121], and to investigate the effects of disorder on wave propagation in photonic crystals [122,123].…”

Band gap structure of disordered chiral photonic crystals

“…Despite the continual enhancement in the technology of photonic crystal fabrication, samples are inevitably disordered in a greater or lesser extent. Therefore, much research has been carried out on the disordered photonic crystals and some interesting characteristics have been reported (Kaliteevski et al 2006;Asatryan et al 2005;Vinogradov and Merzlikin 2004;Sozuer and Sevim 2005). However, all the papers on the disordered photonic crystals are based on conventional photonic crystals and there is no report about the band gap structure of disordered CPC.…”