^{1,a)}, Yuri S. Kivshar

^{2}, Ara A. Asatryan

^{3}, Konstantin Y. Bliokh

^{4}, Yuri P. Bliokh

^{5}, Valentin D. Freilikher

^{6}and Ilya V. Shadrivov

^{7}

### Abstract

This is a review of some recent (mostly ours) results on Anderson localization of light and electron waves in complex disordered systems, including: (i) left-handed metamaterials, (ii) magnetoactive optical structures, (iii) graphene superlattices, and (iv) nonlinear dielectric media. First, we demonstrate that left-handed metamaterials can significantly suppress localization of light and lead to an anomalously enhanced transmission. This suppression is essential at the long-wavelength limit in the case of normal incidence, at specific angles of oblique incidence (Brewster anomaly), and in vicinity of zero-ɛ or zero-μ frequencies for dispersive metamaterials. Remarkably, in disordered samples comprised of alternating normal and left-handed metamaterials, the reciprocal Lyapunov exponent and reciprocal transmittance increment can differ from each other. Second, we study magnetoactive multilayered structures, which exhibit nonreciprocal localization of light depending on the direction of propagation and on polarization. At resonant frequencies or realizations such nonreciprocity results in effectively unidirectional transport of light. Third, we discuss the analogy between wave propagation through multilayered samples with metamaterials and charge transport in graphene, which provides a simple physical explanation of unusual conductive properties of disordered graphene superlatices. We predict disorder-induced resonance of the transmission coefficient at oblique incidence of Dirac quasiparticles. Finally, we demonstrate that an interplay of nonlinearity and disorder in dielectric media can lead to bistability of individual localized states excited inside the medium at resonant frequencies. This results in nonreciprocity of wave transmission and unidirectional transport of light.

We also thank our co-authors, especially L. C. Botten, M. A. Byrne, R. C. McPhedran, F. Nori, P. Rajan, and S. Savel’ev for the fruitful collaboration and discussions of many original results summarized in this review paper. S.G. is grateful to N. M. Makarov, P. Markos, and L. A. Pastur for useful comments and helpful discussions.

This work was partially supported by the European Commission (Marie Curie Action). V.F. acknowledges partial support from the Israeli Science Foundation (Grant No. 894/10).

I. Introduction

II. Random multilayered structures

. Transmission length and the Lyapunov exponent

. Transfer matrices and weak scattering approximation

III. Suppression of localization in metamaterials

. Model

. Mixed stack

. Homogeneous stack

. Transmission resonances

. Polarization effects

. Dispersive metamaterials

. Anomalous suppression of localization

IV. Localization in complex media

. Nonreciprocal transmission in magnetoactive optical structures

. Charge transport in disordered graphene

. Bistability of Anderson localized states in nonlinear media

V. Conclusion

### Key Topics

- Ballistics
- 59.0
- Dielectrics
- 37.0
- Refractive index
- 32.0
- Graphene
- 31.0
- Metamaterials
- 31.0

##### H01F10/08

## Figures

(Ref. 43) Two-component multilayered alternative stack.

(Ref. 43) Two-component multilayered alternative stack.

(Ref. 44) Transmission length *l _{T} * vs. λ for an M-stack (thick solid line, direct simulation and calculations based on WSA recurrence relations) and an H-stack (thick dashed line, direct simulation). Asymptotic values of the localization length

*l*: the short-wavelength asymptotic value (thin dotted line), and the long-wavelength asymptotic values (a thin solid line for the M-stack and a thin dashed line for the H-stack).

(Ref. 44) Transmission length *l _{T} * vs. λ for an M-stack (thick solid line, direct simulation and calculations based on WSA recurrence relations) and an H-stack (thick dashed line, direct simulation). Asymptotic values of the localization length

*l*: the short-wavelength asymptotic value (thin dotted line), and the long-wavelength asymptotic values (a thin solid line for the M-stack and a thin dashed line for the H-stack).

(Ref. 44) Transmission lengths *l _{T} * (solid black line), and the transmission length for a single realization

*l*(dashed blue line) vs. λ for an M-stack with

_{N}*Q*

_{ν}= 0.25,

*Q*= 0.2, and

_{d}*N*= 10

^{4}layers. Each separate point corresponds to a particular wavelength with its own realization of a random stack.

(Ref. 44) Transmission lengths *l _{T} * (solid black line), and the transmission length for a single realization

*l*(dashed blue line) vs. λ for an M-stack with

_{N}*Q*

_{ν}= 0.25,

*Q*= 0.2, and

_{d}*N*= 10

^{4}layers. Each separate point corresponds to a particular wavelength with its own realization of a random stack.

(Ref. 44) Transmission length *l _{T} * vs. λ for H-stacks of

*N*= 10

^{3}(solid line) and 10

^{4}(dotted line) layers (numerical simulation and WSA). Long-wave asymptotic values for the ballistic length in the near and far ballistic regions are plotted in thin solid lines.

(Ref. 44) Transmission length *l _{T} * vs. λ for H-stacks of

*N*= 10

^{3}(solid line) and 10

^{4}(dotted line) layers (numerical simulation and WSA). Long-wave asymptotic values for the ballistic length in the near and far ballistic regions are plotted in thin solid lines.

(Ref. 44) Transmission lengths *l _{T} * (solid black line) and the transmission length for a single realization

*l*(dashed blue line) vs. λ for an H-stack with

_{N}*Q*

_{ν}= 0.25,

*Q*= 0.2, and

_{d}*N*= 10

^{4}layers. Each separate point corresponds to a particular wavelength with its own realization of a random stack.

(Ref. 44) Transmission lengths *l _{T} * (solid black line) and the transmission length for a single realization

*l*(dashed blue line) vs. λ for an H-stack with

_{N}*Q*

_{ν}= 0.25,

*Q*= 0.2, and

_{d}*N*= 10

^{4}layers. Each separate point corresponds to a particular wavelength with its own realization of a random stack.

(Ref. 43) Transmittance |*T*|^{2} vs. λ for a single realization (*Q* = 0.25, *N* = 10^{3}). Solid: normal H-stack, dotted: M-stack.

(Ref. 43) Transmittance |*T*|^{2} vs. λ for a single realization (*Q* = 0.25, *N* = 10^{3}). Solid: normal H-stack, dotted: M-stack.

(Ref. 44) Single realization transmittance |*T*|^{2} vs. wavelength λ for RID M-stacks with *Q* _{ν} = 0.25 and *Q _{d} * = 0 for

*N*= 10

^{5}layers (solid line) and

*N*= 10

^{3}layers (dotted line).

(Ref. 44) Single realization transmittance |*T*|^{2} vs. wavelength λ for RID M-stacks with *Q* _{ν} = 0.25 and *Q _{d} * = 0 for

*N*= 10

^{5}layers (solid line) and

*N*= 10

^{3}layers (dotted line).

(Ref. 44) Single realization transmittance |*T*|^{2} vs. λ for an M-stack of *N* = 10^{3} layers with *Q* _{ν} = 0.25. Solid line corresponds to an M-stack with *Q _{d} * = 0.2, and the dashed line – to an M-stack with no thickness disorder, i.e.,

*Q*= 0.0.

_{d}(Ref. 44) Single realization transmittance |*T*|^{2} vs. λ for an M-stack of *N* = 10^{3} layers with *Q* _{ν} = 0.25. Solid line corresponds to an M-stack with *Q _{d} * = 0.2, and the dashed line – to an M-stack with no thickness disorder, i.e.,

*Q*= 0.0.

_{d}(Ref. 44) Ratio *s* vs. wavelength λ for *Q* _{ν} = 0.25 and stack length *N* = 10^{3}. Solid and dashed curves are for the RID H-stack and H-stack with *Q _{d} * = 0.2, respectively. The middle dashed-dotted curve is for an M-stack with

*Q*= 0.25, and the bottom dotted line is for a RID M-stack.

_{d}(Ref. 44) Ratio *s* vs. wavelength λ for *Q* _{ν} = 0.25 and stack length *N* = 10^{3}. Solid and dashed curves are for the RID H-stack and H-stack with *Q _{d} * = 0.2, respectively. The middle dashed-dotted curve is for an M-stack with

*Q*= 0.25, and the bottom dotted line is for a RID M-stack.

_{d}(Ref. 45) Transmission length *l _{T} * vs. λ for an M-stack in

*p*-polarized light with

*Q*

_{ν}= 0.1,

*Q*= 0.2, and

_{d}*N*= 10

^{6}, at the Brewster angle θ = 45° (red solid line). The blue dashed line shows results for

*s*-polarization and an H-stack, replotted for comparison.

(Ref. 45) Transmission length *l _{T} * vs. λ for an M-stack in

*p*-polarized light with

*Q*

_{ν}= 0.1,

*Q*= 0.2, and

_{d}*N*= 10

^{6}, at the Brewster angle θ = 45° (red solid line). The blue dashed line shows results for

*s*-polarization and an H-stack, replotted for comparison.

(Ref. 45) Transmission length *l _{T} * vs. λ for an M-stack in

*s*-polarized light with

*Q*

_{ν}= 0.1,

*Q*= 0.2, and

_{d}*N*= 10

^{4}, and for the supercritical incidence angle θ = 75°. Red solid curve: numerical simulations; blue dash curve: analytical form.

(Ref. 45) Transmission length *l _{T} * vs. λ for an M-stack in

*s*-polarized light with

*Q*

_{ν}= 0.1,

*Q*= 0.2, and

_{d}*N*= 10

^{4}, and for the supercritical incidence angle θ = 75°. Red solid curve: numerical simulations; blue dash curve: analytical form.

(Ref. 45) Transmission length *l _{T} * vs. incidence angle θ for a mixed stack with

*Q*

_{ν}= 0.1,

*Q*= 0.2, for λ = 0.1 (a), and λ = 1 (

_{d}*b*). The top and bottom curves are, respectively, for

*p*- and

*s*-polarizations.

(Ref. 45) Transmission length *l _{T} * vs. incidence angle θ for a mixed stack with

*Q*

_{ν}= 0.1,

*Q*= 0.2, for λ = 0.1 (a), and λ = 1 (

_{d}*b*). The top and bottom curves are, respectively, for

*p*- and

*s*-polarizations.

(Color online) (Ref. 46) Transmission length *l _{T} * vs. frequency

*f*at normal incidence (θ

_{0}= 0) for a metamaterial stack without absorption (top curve) and in the presence of absorption (bottom curves). Red solid curves display numerical simulations, while blue dashed curves show analytical predictions. Inset: the real (red solid line) and imaginary (green dashed line) part of the metamaterial layer refractive index.

(Color online) (Ref. 46) Transmission length *l _{T} * vs. frequency

*f*at normal incidence (θ

_{0}= 0) for a metamaterial stack without absorption (top curve) and in the presence of absorption (bottom curves). Red solid curves display numerical simulations, while blue dashed curves show analytical predictions. Inset: the real (red solid line) and imaginary (green dashed line) part of the metamaterial layer refractive index.

(Ref. 46) Transmission length *l _{T} * vs. frequency

*f*for θ

_{0}= 30° for a metamaterial stack: without absorption,

*p*-polarization (top curves),

*s*-polarization (middle curves); in the presence of absorption (bottom curves).

(Ref. 46) Transmission length *l _{T} * vs. frequency

*f*for θ

_{0}= 30° for a metamaterial stack: without absorption,

*p*-polarization (top curves),

*s*-polarization (middle curves); in the presence of absorption (bottom curves).

(Ref. 46) Transmission length *l _{T} * vs. angle of incidence for a homogenous metamaterial stack at

*f*= 10.7 GHz with permittivity disorder: in the absence of absorption (upper curve) and for

*p*-polarization; middle curve is for

*s*-polarization; and in the presence of absorption and for both polarizations (lower curves).

(Ref. 46) Transmission length *l _{T} * vs. angle of incidence for a homogenous metamaterial stack at

*f*= 10.7 GHz with permittivity disorder: in the absence of absorption (upper curve) and for

*p*-polarization; middle curve is for

*s*-polarization; and in the presence of absorption and for both polarizations (lower curves).

(Ref. 43) Characteristic length *l* _{ξ} vs. wavelength λ for *Q* = 0.25 and *N* = 10^{9} layers; the solid line is for the M-stack, while the dashed line is for the corresponding (normal) H-stack.

(Ref. 43) Characteristic length *l* _{ξ} vs. wavelength λ for *Q* = 0.25 and *N* = 10^{9} layers; the solid line is for the M-stack, while the dashed line is for the corresponding (normal) H-stack.

(Ref. 45) Transmission length *l _{T} * vs. λ for an M-stack with

*Q*

_{ν}= 0.25,

*Q*= 0, and θ = 30° for

_{d}*p*-polarized light (cyan dashed dotted curve,

*N*= 10

^{6}) and

*s*-polarized light (red solid curve,

*N*= 10

^{5}; green dashed curve,

*N*= 10

^{7}; blue dotted curve,

*N*= 8·10

^{8}).

(Ref. 45) Transmission length *l _{T} * vs. λ for an M-stack with

*Q*

_{ν}= 0.25,

*Q*= 0, and θ = 30° for

_{d}*p*-polarized light (cyan dashed dotted curve,

*N*= 10

^{6}) and

*s*-polarized light (red solid curve,

*N*= 10

^{5}; green dashed curve,

*N*= 10

^{7}; blue dotted curve,

*N*= 8·10

^{8}).

(Ref. 46) (a) Transmission length *l _{T} * vs. frequency

*f*for a mixed stack with

*N*= 10

^{7}layers (top dotted blue curve), and only dielectric permittivity disorder. The bottom curves on all the panels (a, b, c) are for a stack with

*N*= 10

^{7}layers with both permittivity and permeability disorder (the cyan, solid curve displays simulation results, while the dashed, black curve is for the analytical prediction); (b) is the same as (a), but plotted as a function of free space wavelength λ

_{0}, while on panel (c) we plot transmission length as a function of the averaged wavelength inside the stack normalized to the thickness of the layer, for

*N*= 10

^{7}layers (blue dotted top curve),

*N*= 10

^{6}layers (dashed green curve) and for

*N*= 10

^{5}layers (red solid curve), respectively.

(Ref. 46) (a) Transmission length *l _{T} * vs. frequency

*f*for a mixed stack with

*N*= 10

^{7}layers (top dotted blue curve), and only dielectric permittivity disorder. The bottom curves on all the panels (a, b, c) are for a stack with

*N*= 10

^{7}layers with both permittivity and permeability disorder (the cyan, solid curve displays simulation results, while the dashed, black curve is for the analytical prediction); (b) is the same as (a), but plotted as a function of free space wavelength λ

_{0}, while on panel (c) we plot transmission length as a function of the averaged wavelength inside the stack normalized to the thickness of the layer, for

*N*= 10

^{7}layers (blue dotted top curve),

*N*= 10

^{6}layers (dashed green curve) and for

*N*= 10

^{5}layers (red solid curve), respectively.

(Ref. 52) (a) The phase space trajectory generated using Eq. ((3.45)) for an H-array with *N* = 10^{4}, ϕ = π/15, for zero disorder (solid circle), and for σ^{2} = 0.003 (scattered points). (b) One trajectory for an M-array with *N* = 10^{6}, ϕ = 2π/5, σ^{2} = 0.003. (c) ρ(θ) from Eq. ((3.45)) for an H-array (histogram), and Eq. ((3.56)) (horizontal line); (d) ρ(θ) from Eq. ((3.45)) for an M-array (histogram), and Eq. ((3.57)) (solid curve).

(Ref. 52) (a) The phase space trajectory generated using Eq. ((3.45)) for an H-array with *N* = 10^{4}, ϕ = π/15, for zero disorder (solid circle), and for σ^{2} = 0.003 (scattered points). (b) One trajectory for an M-array with *N* = 10^{6}, ϕ = 2π/5, σ^{2} = 0.003. (c) ρ(θ) from Eq. ((3.45)) for an H-array (histogram), and Eq. ((3.56)) (horizontal line); (d) ρ(θ) from Eq. ((3.45)) for an M-array (histogram), and Eq. ((3.57)) (solid curve).

(Ref. 52) (a) Phase space trajectory in new variables ; (b) distribution ρ(Θ) generated by the transformed map with Eqs. ((3.58) and (3.62)), for γ = 0, ϕ = 2π/5, σ^{2} = 0.02, and *N* = 10^{7}.

(Ref. 52) (a) Phase space trajectory in new variables ; (b) distribution ρ(Θ) generated by the transformed map with Eqs. ((3.58) and (3.62)), for γ = 0, ϕ = 2π/5, σ^{2} = 0.02, and *N* = 10^{7}.

(Ref. 34) A schematic picture of wave transmission and reflection from a random-layered structure consisting of two types of alternating layers “α” (here—a magnetoactive material) and “β” (here—air) with random widths. Magnetization of the medium, wave polarizations, and directions of propagation are shown for the Faraday and Voigt geometries.

(Ref. 34) A schematic picture of wave transmission and reflection from a random-layered structure consisting of two types of alternating layers “α” (here—a magnetoactive material) and “β” (here—air) with random widths. Magnetization of the medium, wave polarizations, and directions of propagation are shown for the Faraday and Voigt geometries.

(Ref. 34) Localization decrement κ vs. magnetooptical parameter *Q* for opposite modes propagating through a two-component random structure in the Faraday geometry (see details in the text). The modes with ς = ±1 correspond to either opposite circular polarizations or propagation directions. Numerical simulations of exact equations (symbols) and the theoretical formula (4.8) (lines).

(Ref. 34) Localization decrement κ vs. magnetooptical parameter *Q* for opposite modes propagating through a two-component random structure in the Faraday geometry (see details in the text). The modes with ς = ±1 correspond to either opposite circular polarizations or propagation directions. Numerical simulations of exact equations (symbols) and the theoretical formula (4.8) (lines).

(Ref. 34) Transmission spectra of a random magnetooptical sample in the Faraday geometry (see details in the text) for waves with ς = ±1. While the averaged localization decrements are only slightly different (Fig. 22), all individual resonances are shifted significantly as compared with their widths, Eq. (4.10).

(Ref. 34) Transmission spectra of a random magnetooptical sample in the Faraday geometry (see details in the text) for waves with ς = ±1. While the averaged localization decrements are only slightly different (Fig. 22), all individual resonances are shifted significantly as compared with their widths, Eq. (4.10).

(Ref. 34) Differential transmittance, ^{+} − ^{−}, for two resonances from Fig. 22 as dependent on the value of magnetooptical parameter *Q*, cf. Eq. (4.10).

(Ref. 34) Differential transmittance, ^{+} − ^{−}, for two resonances from Fig. 22 as dependent on the value of magnetooptical parameter *Q*, cf. Eq. (4.10).

(Ref. 34) Differential transmittance, ^{+} − ^{−}, in the vicinity of a single resonance in the Voigt geometry (see Sec. ??? for details) as dependent on the magnetooptical parameter *Q*.

(Ref. 34) Differential transmittance, ^{+} − ^{−}, in the vicinity of a single resonance in the Voigt geometry (see Sec. ??? for details) as dependent on the magnetooptical parameter *Q*.

(Ref. 47) Transmission coefficient *T*(θ) for periodic (thin black line) and disordered (bold blue line) graphene.

(Ref. 47) Transmission coefficient *T*(θ) for periodic (thin black line) and disordered (bold blue line) graphene.

(Ref. 47) Spatial distribution of wave function localized inside the sample for θ, marked by red arrow in Fig. 26.

(Ref. 47) Spatial distribution of wave function localized inside the sample for θ, marked by red arrow in Fig. 26.

(Ref. 31) Nonlinear deformations of transmission spectra of two random resonances at different intensities of the incident wave. Numerical simulations of the Eq. (4.17) (curves) and theoretical Eq. (4.18) (symbols) are shown for the case of defocusing nonlinearity, χ > 0. Light-grey stripes indicate three-valued regions for the high-intensity curves, where only two of them (corresponding to the lower and upper branches) are stable.

(Ref. 31) Nonlinear deformations of transmission spectra of two random resonances at different intensities of the incident wave. Numerical simulations of the Eq. (4.17) (curves) and theoretical Eq. (4.18) (symbols) are shown for the case of defocusing nonlinearity, χ > 0. Light-grey stripes indicate three-valued regions for the high-intensity curves, where only two of them (corresponding to the lower and upper branches) are stable.

(Ref. 31) Stationary and FDTD simulations showing hysteresis loops in the output vs input power dependence for three different resonances. Panel (d) shows deformation of the transmitted Gaussian pulse corresponding to the hysteresis switching on resonance 2.

(Ref. 31) Stationary and FDTD simulations showing hysteresis loops in the output vs input power dependence for three different resonances. Panel (d) shows deformation of the transmitted Gaussian pulse corresponding to the hysteresis switching on resonance 2.

(Ref. 31) (a) Nonreciprocal transmission through the nonlinear disordered structure, showing different output powers for identical waves incident from different directions. (b) Corresponding shape of the incident pulse, and pulses transmitted in different directions.

(Ref. 31) (a) Nonreciprocal transmission through the nonlinear disordered structure, showing different output powers for identical waves incident from different directions. (b) Corresponding shape of the incident pulse, and pulses transmitted in different directions.

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