ARTÍCULO ORIGINAL

Anomalous Localization of Light in One-Dimensional Disordered Photonic Superlattices

Localización anómala de la luz en superredes fotónicas unidimensionales desordenadas

D. Aristizábal-Giraldo¹ and E. Reyes-Gómez²

1 Physics, deisyaristizabal@fisica.udea.edu.co, Universidad de Antioquia, Medellín, Colombia.

2 PhD. in Physical Sciences, ereyes@fisica.udea.edu.co, Universidad de Antioquia, Medellín, Colombia.

Received: 20-09-2012, Accepted: 26-08-2013 Available online: 05-11-2013

PACS:78.67.Pt, 42.25.-p, 46.65.+g, 72.15.Rn

Abstract

The Anderson localization of light in one-dimensional disordered photonic superlattices is theoretically studied. The system is considered to be made of alternating dispersive and nondispersive layers of different randomthickness. Dispersive slabs of the heterostructure are characterized by Drude-like frequency-dependent electric permittivities and magnetic permeabilities. Numerical results for the localization length are obtained via an analytical model, only valid in the case of weak disorder, and also through its general definition involving the transmissivity of the multilayered system. Anomalous λ⁴- and λ^-4-dependencies of the localization length in positive-negative disordered photonic superlattices are obtained, in certain cases, in the long and short wavelength limits, respectively.

Key words: anderson localization; brewster anomaly; photonic superlattices.

Resumen

La localización de Anderson de la luz en superredes fotónicas desordenadas unidimensionales es estudiada teóricamente. El sistema se considera compuesto de capas alternadas dispersivas y no dispersivas de diferentes espesores aleatorios. Las capas dispersivas de la heteroestructura están caracterizadas por permitividades eléctricas y permeabilidades magnéticas tipo Drude dependientes de la frecuencia. Los resultados numéricos para la longitud de la localización son obtenidos mediante un modelo analítico, solo válido en caso de desorden débil, y también a través de la definición general que involucra la transmisividad del sistema multicapas. Las dependencias anómalas λ⁴ y λ^-4 de la longitud de localización en superredes fotónicas desordenadas son obtenidas, en ciertos casos, en los límites de longitudes de onda larga y corta, respectivamente.

Palabras clave: localización de anderson; anomalías de brewster; superredes fotónicas.

1 Introduction

The Anderson localization of light in one-dimensional (1D) photonic heterostructures has been the subject of a considerable amount of work in the last few years [1],[2],[3],[4],[5],[6],[7]. Disorder affects a wide variety of physical properties of the heterostructure, causes multiple light scattering, originates the extinction of coherent waves propagating through the photonic superlattice, and leads to a dramatic change of the localization properties of the electromagnetic modes. In this sense, the Anderson localization of light in disordered photonic crystals has been widely studied both from the experimental and theoretical points of view [8, 9],[10],[11],[12],[13],[14]. The advent of metamaterial, also known as left-handed material (LHM) in allusion to the vectorial product between the electric and magnetic fields intensities [15], has opened up the possibility of investigate new phenomena with no counterpart in usual right-handed materials (RHM). For instance, a study on one-dimensional (1D) heterostructures composed of alternate layers of air and a non-dispersive LHM has evidenced strong suppression of Anderson localization due to the lack of phase accumulation during wave propagation [11],[16].

The aim of the present work is to theoretically investigate the asymptotic behavior, in both the long and short wavelength limits, of the Anderson localization length in 1D heterostructures obtained by the stacking of non-dispersive RHM (A) and Drude-like dispersive LHM (B) layers. Both the slabs A and B are characterized by electric permittivities and magnetic permeabilities ϵ_A and μ_A, and ϵ_B and μ_B, respectively. Here we assume that the system is sandwiched between two semi-infinite layers of material A. The absorption effects are not taken into account. The width a_j (b_j) of the layer A (B) at the j−th site of the 1D system is defined as a_j = a+ (b_j = b+), where and are random variables uniformly distributed in the interval [−Δ/2,Δ/2]. We also suppose that there is no correlation between the disorder of the heterostructure slabs [13]. One may note that a = [a_j] and b = [b_j], where the symbol [...] represents the configurational average of a given geometrical or physical variable.

2 Theoretical framework

The localization length ξ may be evaluated through the expression [4],[17]

where N is the number of double layers (AB) in the photonic system and L = (a_j + b_j) is its corresponding length. The light-transmission coefficient T of the photonic heterostructure in Eq. (1) may be computed via the transfer-matrix formalism [14],[18],[19]. For weakly disordered systems it is possible to derive an analytical expression for ξ in terms of parameters corresponding to a 1D finite photonic superlattice without disorder, with slabs A and B of widths a and b, respectively. It has been shown that [13],[14]

where d = a +b, k is the 1D Bloch wave vector in the perfect superlattice,

with the functions f_x (x = A, B), for transversal-electric (TE) and transversalmagnetic (TM) polarizations, given by

respectively. In the above expressions one has u_x = , Q_x = (ω/c)u_x, and θ is the incidence angle relative to the semi-infinite RHM (normal) A material.

We are interested in obtaining the frequency values at which the localization length diverges. We denote such frequency values as ωc critical frequencies. It is then useful to rewrite Eq. (3) as

where

and g is defined as

for X = TE or X = TM modes, with R_TE(ω) = μ_B(ω)/μ_A(ω) and R_TM(ω) = ϵ_B(ω)/ϵ_A(ω). For a given value of the incidence angle, the critical frequencies satisfy

It is possible to show that the frequency values corresponding to the zeroes of h(ω, θ) are not critical frequencies. In this case, the critical frequencies are the positive real values of ω satisfying the equation g^X(ω, θ) = 0, with X = TE or X = TM. Consequently, the TE (or TM) modes may be delocalized in such case, and delocalization may be interpreted as a Brewster anomaly. The incidence angle may then be identified as the Brewster angle θ_B at the frequency ω_c. Delocalization of light in similar photonic heterostructures was previously studied in recent papers [14],[18] for oblique incidence.

For normal incidence (θ = 0) the critical frequencies come from the zeroes of g = g(ω, 0). One may see that the condition is fulfilled in this case, where

is the optical impedance of medium x = A or x = B. In other words, delocalization of light for normal incidence would be due to the matching of the square of the optical impedance throughout the heterostructure [20].

Now we consider RHM-LHM multilayered systems in which layers A are non-dispersive RHM materials and layers B consist of dispersive LHM metamaterials with both electric permittivity and magnetic permeability given by the Drude model, i.e.,

and

where ω_e and ω_m are the electric and magnetic plasmon frequencies, respectively, and ϵ_∞ and μ_∞ are the positive electric permittivity and magnetic permeability, respectively, of material B in the limit ω → ∞.

As a consequence of Eq. (2), it may be shown that, for normal incidence, the critical frequency is given by

under any of the following four pair of conditions: (i) ω_m < ω_eZ_A/Z_∞ and Z_∞ < Z_A; (ii) ω_m > ω_eZ_A/Z_∞ and Z_∞ > Z_A; (iii) ω_m = ω_eZ_A/Z_∞ and Z_∞ ≠ Z_A; and (iv) ω_m ≠ ω_eZ_A/Z_∞ and Z_∞ = Z_A. In the above equations we have defined Z_∞ = as the optical impedance of layers B at ω → ∞.

The first two cases lead to finite and nonzero values of the critical frequency, whereas the third and fourth cases lead to critical frequencies ω_c = 0 and ω_c → ∞, respectively. The asymptotic behavior of the localization length as a function of the frequency (or wavelength λ) may be obtained by taking the corresponding limits λ → ∞ or λ → 0 in Eq. (2). In this sense, if conditions (iii) are accomplished, Eq. (2) leads to

where

and

One may note from Eq. (14) that the oscillatory part of the asymptotic localization length is modulated by λ⁴.

If conditions (iv) are fulfilled, then one may obtain from Eq. (2) that

where

and

Again, the localization length may be expressed as a bounded and highly oscillatory function of λ modulated by a power-of-λ function.

2.1 Results and discussion

Now we compare the asymptotic behavior of the localization length, obtained from the above described analytical model [cf. Eq. (2), Eq. (14), and Eq. (17)], with the numerical results computed from the general Eq. (1). In Figure 1 we depict the normal-incidence localization length, as a function of the wavelength, corresponding to photonic superlattices of N = 106 double layers with thickness a = b = 12 mm. Solid and dashed lines correspond to numerical results obtained from Eq. (1) for Δ = 1 mm and Δ = 12 mm, respectively. The configurational average in Eq. (1) was taken for 100 realizations of disorder. Calculations shown in Figure 1(a) were computed for μ_A = ϵ_A = 1, ϵ_∞ = 1.21, and μ_∞ = 1, a case which corresponds to the third pair of conditions above discussed. Results displayed in Figure 1(b) were computed for the same set of geometrical parameters, but for ϵA = ϵ1 = 1.21 and μ_A = μ_∞ = 1. Such physical situation agrees with the fourth pair of conditions already mentioned. Dark and white areas in both Figure 1(a) and 1(b) represent the regions of localized and delocalized states, respectively. From a full statistical analysis of the numerical results of Figure 1(a), it is apparent that the localization length, in the long wavelength limit, follows the approximated expression ξ/[L] = Ξ_∞ (λ/d)^α∞. For Δ = 1 mm one may find α_∞ = 4.06 ± 0.03, and a similar result may be obtained for Δ = 12 mm. The values of α_∞ are found in good agreement with the λ⁴ behavior predicted by Eq. (14). In the same way, results for the localization length shown in Figure 1(b) may be approximated, in the short wavelength limit, by the expression ξ/[L] = Ξ₀ (λ/d)^α0 . In this case one has α₀ = −3.94 ± 0.01 for Δ = 1 mm, and a very similar value of α0 in the case of Δ = 12 mm is also obtained. The asymptotic behavior of the localization length, numerically computed from Eq. (1) in the limit λ → 0, is quantitatively different from the λ^-2 behavior predicted by Eq. (17). Discrepancies are related with the fact that the condition of weak disorder σ_xQ_x << 1 (with x = A and x = B) [13], used to derive Eq. (2), is violated in this case. In other words, both σ_AQ_A and σ_BQ_B diverge in the limit ω → ∞ (or λ → 0). The above results suggest that, even though Eq. (2) correctly predicts the values of critical frequencies, its description of the asymptotic behavior of the localization length in the limits of short and long wavelength is only qualitative. Numerical results obtained from Eq. (1) indicate the existence of ξ ∼ λ⁴ and ξ ∼ λ^-4 asymptotic behaviors of the localization length in the long and short wavelength limits, respectively. In the long wavelength limit the obtained behaviors of ξ is far from the classical dependence ξ ∼ λ² [17]. In addition, the obtained divergence of the localization length in the short wavelength region has not been previously observed. We would like to stress that anomalous behaviors of localization length reported here are direct consequences of the Drude-like electric and magnetic responses of the dispersive LHM slabs B.

3 Conclusions

Summing up, we have investigated the asymptotic behavior of the Anderson localization length of electromagnetic waves in 1D disordered photonic superlattices in which the electric permittivity and magnetic permeability of the dispersive slabs B composing the heterostructure may depend on the wave frequency according to the Drude model. We have carried out a theoretical study of the localization length by using an analytical model valid for weakly-disordered photonic heterostructures [13],[14] which predicts, under certain conditions discussed above, a λ⁴- and λ^-2-dependence of ξ in the limits λ → ∞ and λ → 0, respectively. Moreover, we performed numerical calculations of the localization length by using its general definition [cf. Eq. (1)] involving the transmissivity of the heterostructure, and a ξ ∼ λ⁴ (ξ ∼ λ^-4) asymptotic behavior was obtained in the long (short) wavelength limit. Present theoretical results indicate that, in spite of the analytical model correctly predicts the critical-frequency values at which the localization length diverges, its description of the asymptotic behavior of the localization length is only qualitative. Our results suggest that the asymptotic behavior of the localization length is essentially determined by both the electric and magnetic responses characterizing the LHM slabs of the heterostructure.

Acknowledgements

We would like to thank the Colombian Agencies CODI, Vicerrectoría de Investigación, Sistema de Bienestar Universitario, and Instituto de Física - University of Antioquia, for partial financial support.

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