Effects of polarization on the transmission and localization of classical waves in weakly scattering metamaterials

We summarize the results of our comprehensive analytical and numerical studies of the effects of polarization on the Anderson localization of classical waves in one-dimensional random stacks. We consider homogeneous stacks composed entirely of normal materials or metamaterials, and also mixed stacks composed of alternating layers of a normal material and a metamaterial. We extend the theoretical study developed earlier for the case of normal incidence [A. A. Asatryan et al., Phys. Rev. B 81, 075124 (2010)] to the case of off-axis incidence. For the general case where both the refractive indices and layer thicknesses are random, we obtain the long-wave and short-wave asymptotics of the localization length over a wide range of incidence angles (including the Brewster “anomaly” angle). At the Brewster angle, we show that the long-wave localization length is proportional to the square of the wavelength, as for the case of normal incidence, but with a proportionality coefficient substantially larger than that for normal incidence. In mixed stacks with only refractive-index disorder, we demonstrate that $p$-polarized waves are strongly localized, while for $s$ polarization the localization is substantially suppressed, as in the case of normal incidence. In the case of only thickness disorder, we study also the transition from localization to delocalization at the Brewster angle.

Figure 1

(Color online) The geometry of the structure.

Figure 2

(Color online) Transmission length $l_N$ versus wavelength $\lambda$ for $Q_{\nu }=0.1$, $Q_d=0.2$, and $\theta =45°$ for $s$-polarized waves; panels (a) localized part of the spectrum and (b) ballistic part of the spectrum. Red solid curve: numerical simulation; blue dashed curve analytic form in Eq. (21).

Figure 3

(Color online) Transmission length $l_N$ versus $\lambda$ for $Q_{\nu }=0.1$, $Q_d=0.2$ for $p$-polarized waves at the Brewster anomaly angle $\theta =45°$. Panel (a) localized part of the spectrum; panel (b) the transition from localization to ballistic propagation. Red solid curve: numerical simulation; blue dashed curve: analytic form in Eq. (21).

Figure 4

(Color online) Transmission length $l_N$ of the homogeneous stack with $N={10}^4$, $Q_{\nu }=0.1$, and $Q_d=0.2$ versus wavelength $\lambda$ for $s$-polarized waves at the supercritical incidence angle $\theta =75°$. Red solid curve: numerical simulation; blue dashed curve analytic form in Eq. (21).

Figure 5

(Color online) Transmission length versus $\lambda$ for a M-stack in $s$-polarized light with $Q_{\nu }=0.1$, $Q_d=0.2$, and $N={10}^4$ for $\theta =45°$. Red solid curve: numerical simulations; blue dashed curve: analytic form in Eq. (15).

Figure 6

(Color online) Transmission length versus $\lambda$ for a M-stack in $p$-polarized light with $Q_{\nu }=0.1$, $Q_d=0.2$, and $N={10}^6$, at the Brewster angle $\theta =45°$ red solid line. The blue dashed line shows results for $s$ polarization and a H-stack, replotted for comparison.

Figure 7

(Color online) Transmission length versus $\lambda$ for a M-stack in $s$-polarized light with $Q_{\nu }=0.1$, $Q_d=0.2$, and $N={10}^4$, and for the supercritical incidence angle $\theta =75°$. Red solid curve: numerical simulations; blue dashed curve: analytic form in Eq. (15).

Figure 8

(Color online) Transmission length $l_N$ versus $\lambda$ for a M-stack with $Q_{\nu }=0.25$, $Q_d=0$, and $\theta =30°$ for $p$-polarized light (cyan dashed dotted curve, $N={10}^6$) and $s$-polarized light (red solid curve, $N={10}^5$; green dashed curve, $N={10}^7$; and blue dotted curve, $N=8\times {10}^8$).

Figure 9

(Color online) Transmission length $l_N$ versus incidence angle $\theta$ for a homogeneous stack with $Q_{\nu }=0.1$, $Q_d=0.2$ for (a) $\lambda =0.1$ (upper panel), (b) $\lambda =10$ (lower panel). Red solid curve: numerical simulations; blue dashed curve analytic form in Eq. (22). The top set of curves in each of the panels are for $p$ polarization while the bottom set of curves are for $s$ polarization.

Figure 10

(Color online) Transmission length $l_N$ versus incidence angle $\theta$ for a mixed stack with $Q_{\nu }=0.1$, $Q_d=0.2$, for (a) $\lambda =0.1$ (upper panel) and (b) $\lambda =1$ (lower panel). The top and bottom curves are, respectively, for $p$ and $s$ polarizations.

Figure 11

(Color online) Delocalization at the Brewster angle ${\theta }_B$: (a) at a short wavelength $\lambda =0.1$ and (b) an intermediate wavelength $\lambda =0.5$. Top and bottom curves are, respectively, for $p$ and the $s$ polarizations.