Localization-length calculations in alternating metamaterial-birefringent disordered layered stacks

O. del Barco, V. Gasparian, and Zh. Gevorkian

Phys. Rev. A 91, 063822 – Published 18 June 2015

Abstract

A detailed theoretical and numerical analysis of the localization length in alternating metamaterial-birefringent random layered stacks, under uncorrelated thickness disorder, has been performed. Similar structures have recently been reported to suppress the Brewster delocalization for $p$ -polarized light when “standard” isotropic layers (with a positive index of refraction) are considered instead of metamaterial layers, providing a generic means to produce polarization-insensitive, broadband reflections. However, this enhancement of localization is valid for short wavelengths $λ$ compared to the mean layer thickness $a_{0}$ . At higher wavelengths, we recover the Brewster anomalies for $p$ -polarized states impeding a remarkable localization of light. To achieve a better localization for a wider range of wavelengths, we replaced the conventional isotropic layers by negative-index metamaterials presenting low losses and constant index of refraction over the near-infrared range. As a result, our numerical calculations exhibit a linear dependence of the localization length on $λ$ (in the region $5 < λ / a_{0} < 60$ ), reducing the Brewster anomalies by more than two orders of magnitude with respect to the standard isotropic scheme at oblique incidence. This enhancement of localization is practically independent of the thickness disorder kind and is also held under weak refractive-index disorder.

Received 8 August 2014

DOI:https://doi.org/10.1103/PhysRevA.91.063822

Authors & Affiliations

O. del Barco^1,*, V. Gasparian², and Zh. Gevorkian^3,4

¹Departamento de Física, Universidad de Murcia, 30100, Spain
²Department of Physics, California State University, Bakersfield, California 93311-1099, USA
³Yerevan Physics Institute, Alikhanian Brothers Street 2, 0036 Yerevan, Armenia
⁴Institute of Radiophysics and Electronics, Ashtarak-2, 0203, Armenia

^*Corresponding author: obn@um.es

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Issue

Vol. 91, Iss. 6 — June 2015

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Images

Figure 1
Our disordered layered stack of alternating uniaxial positive ( $n_{+}$ ) and uniaxial negative ( $n_{-}$ ) layers. Dark gray regions correspond to left-handed isotropic layers with negative index of refraction $n_{0}$ . The principal axes of the birefringent layers are aligned to the $x$ and $y$ directions, while light enters from the left at angle $θ$ and propagates through $z$ axis.
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Figure 2
Inverse of the localization length $ξ_{p}^{- 1}$ [calculated via Eq. (16)] vs the parameter $f$ and the angle of incidence $θ$ . Two different values of the positive-layer birefringence $Δ_{+} = n_{2} - n_{1}$ have been chosen.
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Figure 3
Localization length $ξ_{s, p}$ vs the angle of incidence $θ$ in the short-wavelength regime ( $λ / a_{0} = 0.032$ ) for a Poisson-thickness distribution for (top) $s$ -polarized states and (bottom) $p$ -polarized light. Solid lines symbolize our numerical calculations, whereas dashed lines show our theoretical results [Eq. (16)].
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Figure 4
Localization length $ξ_{s, p}$ vs the incident wavelength $λ$ for two different angles of incidence, $θ = 0^{\circ}$ and $θ = 28^{\circ}$ . (a) and (b) Standard isotropic-birefringent stacks (with positive indices of refraction) and (c) and (d) LH isotropic-birefringent disordered structures. The birefringence of the positive layers has been chosen to be $Δ_{+} = 2.33$ , and the ratio $f = 0.75$ in all cases. Solid lines represent our numerical calculations [Poisson-thickness distribution, Eq. (17)], whereas the dotted lines symbolize the localization-length calculations performed via Eqs. (18) and (19) (top panels) and Eqs. (21) and (22) (bottom panels).
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Figure 5
Localization length $ξ_{s, p}$ vs the angle of incidence $θ$ in the short-wavelength regime ( $λ / a_{0} = 0.032$ ) for a uniform-thickness distribution for (left) $s$ polarization (right) $p$ -polarized light. Solid lines symbolize our numerical calculations, whereas dashed lines show our theoretical results [Eq. (16)].
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Figure 6
Enhancement of localization under a mixed scheme of uniform thickness and weak refractive-index disorder for LH isotropic-birefringent stacks. Numerical calculations for $ξ_{s, p}$ vs the wavelength $λ$ and the angle of incidence $θ$ are shown. The birefringence of positive layers was chosen to be $Δ_{+} = 2.33$ , while the ratio $f = 0.75$ in all cases. (left) Our numerical results for isotropic-birefringent stacks with $| n_{0} | = 1.33$ and (right) our left-handed isotropic stacks with $n_{0} = - 1.33$ . The simulations were averaged over 800 random configurations of the layered stacks.
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