Light propagation in binary kagome ribbons with evolving disorder

By introducing evolving disorder in the binary kagome ribbons, we study the establishment of diffusive spreading of flat band states characterized by diffractionless propagation in regular periodic ribbons. Our numerical analysis relies on controlling strength and rate of change of disorder during light propagation while tailoring binarism of the kagome ribbons in order to isolate the flat band with the gap from the rest of the ribbon's eigenvalue spectrum and study systematically its influence on diffusion. We show that the flat band plays a dominant role in the establishment of the diffusion for a given strength and rate of change of disorder, whereas the rest of the ribbon's eigenvalue spectrum induces only quantitative differences in the light spreading regimes. Due to the universality of studied phenomena, our findings may be of interest in various disordered physical systems with flat spectral bands, ranging from photonics to ultracold matter systems and plasmonics.

Figure 1

(a) Binary KR. A unit cell that contains six sites. The FBS is a compact localized state which occupies four sites with the $\pi$ phase change between adjacent sites represented by $+$ and $-$ signs. The eigenvalue's spectra reduced to the first Brillouin zone in (b) uniform and (c) binary KRs for $V=0.25$. The bands in (c), formed due to the splitting of the corresponding bands in the uniform case (b), are depicted with the same color [41]. The gray areas around the FB illustrate its smearing in the KRs with quenched disorder.

Figure 2

Saturable values of (a) the participation number and (b) the second moment of the disordered FBSs in the uniform and binary KRs for different disorder strengths $W$. (c) The dependence of the localization length of the disordered FBSs in the uniform and binary KRs on disorder strength $W$. All figures are given on the log-log scale.

Figure 3

(a) The participation number and (b) the second moment of the FBS during propagation in the uniform and binary KRs with QD. The second moments are presented on the log-log scale.

Figure 4

The second moment and the participation number along the uniform KRs with ED for different evolution steps ${\mathrm{\Delta }}_z$ and disorder strengths (a) and (d) $W=1$, (b) and (e) $W=0.5$, and (c) and (f) $W=0.1$. The insets in (a)–(c) show values of $m_2$ as a function of $\mathrm{\Delta }z$ at different points along the light propagation direction: $z=100$ (blue circles), $z=500$ (green squares), and $z=1000$ (red diamonds). The inset in (d) illustrates the light field profile at three different points along the propagation directions $z=100$ (gray line), $z=500$ (light blue line), and $z=1000$ (magenta line) during diffusive spreading of the initially launched single FBS in the strongly disordered uniform KRs with $W=1$ for $\mathrm{\Delta }z=1.5$. The inset in (f) shows the distribution of the light field amplitudes at the output of the KRs with $W=0.1$ for $\mathrm{\Delta }z=1.5$. The second moments are presented on the log-log scale. The dashed straight lines with slopes 3,2, and 1, respectively, are guiding lines for the estimation of the growing rates of the $m_2$ curves.

Figure 5

The second moment and the participation number along binary KRs with ED for different evolution steps $\mathrm{\Delta }z$ and disorder strengths (a) and (d) $W=1$, (b) and (e) $W=0.5$, and (c) and (f) $W=0.1$. The coupling ratio of binary KRs is $V=0.25$. The insets in (a)–(c) show values of $m_2$ as a function of $\mathrm{\Delta }z$ at different points along the light propagation direction: $z=100$ (blue circles), $z=500$ (green squares), and $z=1000$ (red diamonds). The inset in (d) illustrates the light field profile at three different points along propagation directions $z=100$ (gray line), $z=500$ (light blue line), and $z=1000$ (magenta line) for the case of diffusive spreading of the initially launched single FBS for $W=1$ and $\mathrm{\Delta }z=1.5$. The inset in (f) shows the distribution of the light field amplitudes at the output of KRs with $W=0.1$ for $\mathrm{\Delta }z=1.5$. The second moments are presented on the log-log scale. The dashed straight lines with slopes 3, 2, and 1, respectively, are guiding lines for the estimation of the growing rates of the $m_2$ curves.

Figure 6

(a) The eigenstates density spectrum $\left[\rho \right(\beta )$ vs $\beta ]$ of the KRs with QD. The solid lines denote the uniform KR case ($V=1$), whereas the dashed lines denote the binary KR ($V=0.25$). The blue, red, and green lines correspond to the fixed values of the disorder strength: $W=0.1,0.5,$ and 1, respectively. The corresponding values of $\xi$ are shown in the figure. (b) The $m_2(z=1000)$ vs $\mathrm{\Delta }z$ for uniform (solid lines with solid symbols) and binary (dashed lines with open symbols) lattices with respect to $W=0.1,0.5,$ and 1.