Localized modes in nonlinear binary kagome ribbons

The localized mode propagation in binary nonlinear kagome ribbons is investigated with the premise to ensure controlled light propagation through photonic lattice media. Particularity of the linear system characterized by the dispersionless flat band in the spectrum is the opening of new minigaps due to the “binarism.” Together with the presence of nonlinearity, this determines the guiding mode types and properties. Nonlinearity destabilizes the staggered rings found to be nondiffracting in the linear system, but can give rise to dynamically stable ringlike solutions of several types: unstaggered rings, low-power staggered rings, hour-glass-like solutions, and vortex rings with high power. The type of solutions, i.e., the energy and angular momentum circulation through the nonlinear lattice, can be controlled by suitable initial excitation of the ribbon. In addition, by controlling the system “binarism” various localized modes can be generated and guided through the system, owing to the opening of the minigaps in the spectrum. All these findings offer diverse technical possibilities, especially with respect to the high-speed optical communications and high-power lasers.

Figure 1

Structure of (a) binary kagome strip and (b) binary kagome ladder. Each characteristic cell comprises (a) 10 sites; (b) six sites, mutually coupled either by $V_1$ (dashed line) or by $V_2$ (thick line) coupling strength.

Figure 2

Band structures of 1D (a) uniform kagome strip, (b) uniform kagome ladder, (c) binary kagome strip, and (d) binary kagome ladder. In the case of binary configurations, the values of coupling parameters are set to $V_1=0.5$ and $V_2=1$.

Figure 3

Influence of $V_1$ coupling parameter on band structures of 1D binary kagome strip (a) and binary kagome ladder (b). Second coupling parameter $V_2$ is fixed to one. Dashed vertical lines mark the case of uniform lattice for which only semi-infinite gaps exist.

Figure 4

Stability diagrams $(P$ vs $V_1/V_2)$ of (strong) staggered rings and hourglass solutions in (a) kagome strip and (b) kagome ladder. In the greenish (dark gray) region only stable staggered rings exist, in the yellowish (light gray) region are the stable ring and unstable hourglass solutions, while in the white region the unstable ring and stable hourglass solutions are found. Amplitudes' profiles of certain modes (kagome strip, strong staggered ring $\beta =-3,P=6$; hourglass, $\beta =-3,P=2.61$; kagome ladder, strong staggered ring $\beta =-3,P=4$; hourglass, $\beta =-3,P=2.56)$ are shown on right panels and correspond to the case of uniform ribbons.

Figure 5

Uniform kagome strip configuration. (a) $P$ vs $\beta$ curves for staggered ring (red line with squares), two-peak solution (blue line with circles), and hourglass (black line with triangles) solutions; (b) amplitude profile of tp solution; (c) $\mathrm{\Delta }H$ vs $P$ curve (solid red for $H_{hg}-H_{sr}$ and dashed blue line for $H_{tp}-H_{sr})$, solid red vertical line represents the border from which the staggered rings become unstable, and black dashed line the border from which the hourglass solutions become stable; (d) the largest real eigenvalues vs $P$ for staggered ring (red line with squares), two-peak solution (blue line with circles), and hourglass (black line with triangles) solutions.

Figure 6

Stability diagrams $(P$ vs $V_1/V_2)$ of strong unstaggered ring solutions in (a) kagome strip and (b) kagome ladder. Greenish (dark gray), yellowish (light gray), and white regions correspond to the region where unstaggered rings do not exist, appear as unstable solutions, and are stable propagating modes, respectively. Amplitudes' profiles of certain modes (kagome strip, $\beta =-10,P=72.45$; kagome ladder, $\beta =-10,P=53.2)$ are shown on right panels and correspond to the case of uniform ribbons.

Figure 7

Stability diagram $(P$ vs $V_1/V_2)$ of vortex ring solutions with $S=1$ bonded between sites with $V_2$ (“strong” variant), which are found to exist in the kagome strip. The yellowish (light gray) area represents the power region that corresponds to unstable vortex modes, while the white one depicts the parameter space of stable propagation. The greenish (dark gray) colored area illustrates the region where no vortex modes exist. Phase profile of certain vortex mode $(\beta =-9.91,P=65.6,S=1)$ is shown on the right panel and corresponds to the case of the uniform kagome strip.

Figure 8

Mobility of the hg mode in the uniform kagome strip configuration. (a) Horizontal center-of-mass evolution of an unstable low-power hg solution $(P=1.02)$ in the presence of a phase gradient $\alpha =0.009$, and (b) intensity profiles taken at different positions in the $z$ direction during the modes' propagation. The corresponding positions are marked in (a) with numbers in brackets.

Figure 9

Amplitude profiles of different types of minigap solutions in the binary kagome strip (a)–(c) and binary kagome ladder (d)–(f) ribbon. Red (thick) dashed lines represent boundaries of primitive cells of binary systems. Black solid and dashed (thin) lines stand for different strengths of coupling $V_2$ and $V_1$, respectively. In all cases, parameters in simulations are set to $V_1=0.25$ and $\beta =-1.9$.