Localization length of nearly periodic layered metamaterials

We have analyzed numerically the localization length of light $\xi$ for nearly periodic arrangements of homogeneous stacks (formed exclusively by right-handed materials) and mixed stacks (with alternating right- and left-handed metamaterials). Layers with index of refraction $n_1$ and thickness $L_1$ alternate with layers of index of refraction $n_2$ and thickness $L_2$. Positional disorder has been considered by shifting randomly the positions of the layer boundaries with respect to periodic values. For homogeneous stacks, we have shown that the localization length is modulated by the corresponding bands and that $\xi$ is enhanced at the center of each allowed band. In the limit of long wavelengths $\lambda$, the parabolic behavior previously found in purely disordered systems is recovered, whereas for $\lambda \ll L_1+L_2$ a saturation is reached. In the case of nearly periodic mixed stacks with the condition $|n_1{L}_1|=|n_2{L}_2|$, instead of bands there is a periodic arrangement of Lorenztian resonances, which again is reflected in the behavior of the localization length. For wavelengths of several orders of magnitude greater than $L_1+L_2$, the localization length $\xi$ depends linearly on $\lambda$ with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers. When the condition $|n_1{L}_1|=|n_2{L}_2|$ is no longer satisfied, the transmission spectrum is very irregular and this considerably affects the localization length.

Figure 1

A periodic arrangement of layers with index of refraction $n_1$ and thickness $L_1$ alternating with layers of index of refraction $n_2$ and thickness $L_2$. The grating period is $\Lambda =L_1+L_2$.

Figure 2

(a) The transmission coefficient $T$ and (b) the parameter $\cos \left(\beta \Lambda \right)$ versus the frequency $\omega$ for the homogeneous periodic system described in the text (99 layers).

Figure 3

Localization length $\xi$ vs the wavelength $\lambda$ for different values of the disorder parameter $\delta$. The H stack corresponds to the arrangement represented in Fig. 2 but now 50 000 layers have been considered. The dashed line stands for the “total disorder” case. All lengths are expressed in units of the grating period $\Lambda$.

Figure 4

Localization length $\xi$ vs the wavelength $\lambda$ for (a) the first and (b) the second gaps depicted in Fig. 3.

Figure 5

Three-dimensional graphs of the localization length $\xi$ vs the wavelength $\lambda$ and the disorder $\delta$ for (a) the first and (b) the third allowed bands. The H stack is the same as in Fig. 2. All lengths are expressed in units of the grating period $\Lambda$.

Figure 6

(a) The transmission coefficient $T$ and (b) the parameter $\cos \left(\beta \Lambda \right)$ vs the frequency $\omega$ for the asymmetric periodic H stack described in the main text (99 layers) and (c) the corresponding localization length $\xi$ vs the wavelength $\lambda$ for different disorder parameters $\delta$ (50 000 layers).

Figure 7

(a) The transmission coefficient $T$ and (b) the parameter $\cos \left(\beta \Lambda \right)$ vs the frequency $\omega$ for the mixed periodic system described in the text (99 layers).

Figure 8

Localization length $\xi$ vs the wavelength $\lambda$ for different disorder parameters $\delta$. The M stack corresponds to the one represented in Fig. 7 but here 50 000 layers have been considered. The dashed line stands for the “total disorder” case.

Figure 9

Numerical calculations of the slope $a$ vs $\left|r\right|$ for several values of ${\Lambda }_{\mathrm{op}}$ (expressed in units of the grating period $\Lambda$). The solid lines correspond to the results obtained via Eq. (18).

Figure 10

(a) The transmission coefficient $T$ and (b) the parameter $\cos \left(\beta \Lambda \right)$ vs the frequency $\omega$ for the asymmetric periodic M stack described in the main text (99 layers) and (c) the corresponding localization length $\xi$ vs the wavelength $\lambda$ for different disorder parameters $\delta$ (50 000 layers).