Interplay between optical nonlinearity and localization in a finite disordered Fibonacci chain

Both the average transmission coefficients and dimensionless optical resistances of a nonlinear disordered Fibonacci chain are calculated as functions of the photon flux of an incident light field with various types of chains of scatters, numbers of embedded nonlinear-optical scatterers, and energies of incident photons. If the incident optical field is very weak, the nonlinear-optical scattering in the chain becomes negligible and the chain behaves just like a transparent dielectric slab. As for the interplay between the optical nonlinearity and localization effect in the finite disordered Fibonacci chain, it is found that the localization effect introduced in the disordered Fibonacci chain exhibits a reduction in the transmission only when the incident optical field is strong. The localization effect, which increases with the number of scatterers in the chain, is found to yield an enhanced optical nonlinearity of the system. The localization effect on the incident optical wave in the chain can be accelerated by an intense light illumination in the presence of a large number of nonlinear-optical scatterers. When energetic photons fly through the chain, they tend to ignore most of the deeply embedded optical scatterers. When the number of scatterers in the system is $F_{16}=1597$, a complete localization in the chain is reached for an incident field amplitude as low as $50\mathrm{kV}∕\mathrm{cm}$.

Figure 1

Illustration of a finite nonlinear disordered Fibonacci chain embedded with optical scatterers at $z_0,z_1,\dots ,z_j,\dots ,z_{N-1},z_N$, where the positions of optical scatterers $\{z_1,z_2,\dots ,z_{N-1},z_N\}$ form a so-called disordered Fibonacci sequence (see the text) with the position of the first optical scatterer fixed at $z=z_0=0$. The linear refractive index of the chain is $n_b$, and the nonlinear refractive index for the $j\text{th}$ optical scatterer at $z=z_j$ is $n_j$. The space outside the sample ($z<z_0$ and $z>z_N$) is filled by a dielectric material with a matched refractive index $n_b$. The light is incident from the left of the sample $(z<z_0)$ and is transmitted to the right of the sample $(z>z_N)$. The field amplitudes of the incident, reflected, and transmitted optical waves in this figure are denoted as $s$, $\gamma$, and $t$, respectively.

Figure 2

Comparison of average transmission coefficients $\overline{\mathcal{T}}(s,q_c,L)$ [in (a)] and dimensionless optical resistances ${\overline{\mathcal{R}}}_{\mathit{op}}(s,q_c,L)$ [in (b)] as functions of incident photon flux ${\mid s\mid }^2$ in units of ${10}^8{\mathrm{V}}^2∕{\mathrm{cm}}^2$ for nonlinear disordered ($p=0.5$, solid curves) and pure ($p=1$, dashed curves) Fibonacci chains with $N=F_9=55$ and ${\lambda }_0=6283\mathrm{\AA }$. Here, the chain length $L=7.6\mu \mathrm{m}$. The inset in (a) shows the comparison of $\overline{\mathcal{T}}(s,q_c,L)$ for $p=0.5$ (solid curve) and $p=1$ (dashed curve) as functions of ${\mid s\mid }^2$ in units of ${10}^8{\mathrm{V}}^2∕{\mathrm{cm}}^2$ with $N=F_{16}=1597$ and $L=220.7\mu \mathrm{m}$.

Figure 3

Comparison of average transmission coefficients $\overline{\mathcal{T}}(s,q_c,L)$ [in (a)] and dimensionless optical resistances ${\overline{\mathcal{R}}}_{\mathit{op}}(s,q_c,L)$ [in (b)] as functions of incident photon flux ${\mid s\mid }^2$ in units of ${10}^8{\mathrm{V}}^2∕{\mathrm{cm}}^2$ for $N=F_9=55$ (solid curves) an $N=F_5=8$ (dashed curves) in a nonlinear disordered Fibonacci chain with $p=0.5$ and ${\lambda }_0=6283\mathrm{\AA }$. Here, the chain lengths $L$ are $7.6\mu \mathrm{m}$ for $N=55$ and $1.1\mu \mathrm{m}$ for $N=8$, respectively. The inset in (a) shows the comparison of $\overline{\mathcal{T}}(s,q_c,L)$ for $N=F_{16}=1597$ (solid curve, $L=220.7\mu \mathrm{m}$) and $N=F_{11}=144$ (dashed curve, $L=19.9\mu \mathrm{m}$) as functions of ${\mid s\mid }^2$ in units of ${10}^8{\mathrm{V}}^2∕{\mathrm{cm}}^2$ with $p=0.5$. The inset in (b) shows the comparison of ${\log }_{10}({\xi }_0∕L)$ for $N=F_{16}=1597$ (solid curve) and $N=F_{11}=144$ (dashed curve) as functions of ${\mid s\mid }^2$ in units of ${10}^8{\mathrm{V}}^2∕{\mathrm{cm}}^2$ with $p=0.5$.

Figure 4

Comparison of average transmission coefficients $\overline{\mathcal{T}}(s,q_c,L)$ [in (a)] and dimensionless optical resistances ${\overline{\mathcal{R}}}_{\mathit{op}}(s,q_c,L)$ [in (b)] as functions of incident photon flux ${\mid s\mid }^2$ in units of ${10}^8{\mathrm{V}}^2∕{\mathrm{cm}}^2$ for ${\lambda }_0=6282\mathrm{\AA }$ (solid curves) and ${\lambda }_0=4833\mathrm{\AA }$ (dashed curves) in a nonlinear disordered Fibonacci chain with $p=0.5$ and $N=F_9=55$. Here, the chain length $L=7.6\mu \mathrm{m}$.