Localized gap modes in nonlinear dimerized Lieb lattices

Compact localized modes of ring type exist in many two-dimensional lattices with a flat linear band, such as the Lieb lattice. The uniform Lieb lattice is gapless, but gaps surrounding the flat band can be induced by various types of bond alternations (dimerizations) without destroying the compact linear eigenmodes. Here, we investigate the conditions under which such diffractionless modes can be formed and propagated also in the presence of a cubic on-site (Kerr) nonlinearity. For the simplest type of dimerization with a three-site unit cell, nonlinearity destroys the exact compactness, but strongly localized modes with frequencies inside the gap are still found to propagate stably for certain regimes of system parameters. By contrast, introducing a dimerization with a 12-site unit cell, compact (diffractionless) gap modes are found to exist as exact nonlinear solutions in continuation of flat band linear eigenmodes. These modes appear to be generally weakly unstable, but dynamical simulations show parameter regimes where localization would persist for propagation lengths much larger than the size of typical experimental waveguide array configurations. Our findings represent an attempt to realize conditions for full control of light propagation in photonic environments.

Figure 1

Schematic representations of two classes of dimerized Lieb lattices: (a) three-site unit cell with general coupling coefficients; (b) special case of (a) with a diagonal symmetry axis through $b$ sites (“type I”); (c) 12-site unit cell (“type II”) with diagonal symmetry axis through sites $f$ and $j$, and antidiagonal symmetry axis through sites $d$ and $l$. The primitive cells are denoted by dashed squares. Amplitudes within the characteristic sites are denoted by $a,b,c,d,e,f,g,h,i,j,k$, and $l$, respectively. Here, $p$ ($q$) denotes cell index in the $x$ ($y$) direction, where $p,q\in [1,N]$. The Bloch vector corresponding to translations between two neighboring fundamental blocks is described by $k_x$ in the horizontal and $k_y$ in the vertical direction, respectively. Different colors of lines in (a) represent different strengths of coupling between sites, while solid and dashed lines stand for the two coupling constants in (b) and (c).

Figure 2

Real parts of amplitude for: (a) minimal 4-peak ring mode; (b) 6-peak ring mode—superposition of two “out-of-phase” 4-peak modes that cancel each other at the central $c$ site; (c) 7-peak ring mode—superposition of two “in-phase” 4-peak modes. The amplitude changes sign from site to site, according to the relation (5). Couplings $V_{x1}=V_{y1}=0.5$ are marked by orange (gray) lines, while $V_{x2}=V_{y2}=1$ are marked by black ones. The indices $m$ and $n$ represent the site numbers of the lattice.

Figure 3

Dispersion relations (energy spectra) for the cases of (a) uniform Lieb lattice, and dimerized Lieb lattices of (b) type I, and (c) type II, when $V_1=1$ and $V_2=2$. Plot (d) shows influence of strength of dimerization $V_1/V_2$ on gap width of both dimerized Lieb lattices of I and II type. Second coupling parameter $V_2$ is fixed to one. Dashed vertical line marks the case of uniform lattice for which only semi-infinite gaps exist.

Figure 4

The amplitude distribution of a 4-ring mode in the dimerized Lieb lattice of type II. Orange (gray) and black solid lines depict coupling strength $V_1$ and $V_2$, respectively. The dashed square frames a block with the particular couplings between sites that provides existence of ring modes.

Figure 5

Nonlinear localized modes in the minigap of type I dimerized Lieb lattice: (a) Power vs propagation constant for two families ($A$ and $B$) and corresponding amplitude profiles of a typical small-amplitude mode from (b) family $A$ ($\beta =0.101$) and (c) family $B$ ($\beta =0.3$) when $V_1/V_2=0.25$. Orange (gray) solid lines stand for $V_1=0.25$ coupling strength, while black ones represent $V_2$ coupling. The $m$ and $n$ represent the site numbers of the lattice. Power curves in (a) show the existence range of solutions until the upper gap edge, located at $\beta =0.75\sqrt{2}\approx 1.06$ (solid curves, $V_1=0.25$) and $\beta =0.5\sqrt{2}\approx 0.71$ (dashed curves, $V_1=0.5$), respectively.

Figure 6

Compactness vs $V_1/V_2$ with $V_1<1$ and $V_2=1$ for (a) family $A$ and (b) nonlinear ring solitons, characterized with $\beta =0.001$. The upper (red) lines show compactness on all four ring sites, while the lower (blue) lines show compactness on the two ring sites belonging to a strongly bonded trimer. Panels on the right show amplitude distributions of modes for different coupling strengths: trimer solution ($V_1/V_2=0$) and gap soliton of family $A$ for $V_1/V_2=0.25$—first row, and gap solitons of family $A$ for $V_1/V_2=0.9$ and $V_1/V_2=1$—second row. Amplitude distributions of nonlinear ring solitons for $V_1/V_2=0.96$ and uniform lattice are depicted in the third row.

Figure 7

Stability diagram for the solution of $A$ family of the type I dimerized Lieb lattice for $V_1<1$ and $V_2=1$.

Figure 8

(a) Maximum values of real parts of stability EVs vs $\beta$ for the solution of family $A$ in the type I dimerization with $V_1=0.5$ and $V_2=1$, and different system sizes with the number of unit cells denoted by $N\times N$, where $N$ varies between 11 and 29 (periodic boundary conditions). (b) Magnification of (a) for smaller $\beta$. The dashed lines depict growth trend of EVs.

Figure 9

Dynamics of gap solitons (family $A$) for $V_1=0.5$ and $V_2=1$. (a) The plot of the upper part of full spectrum of stability EVs [positive Im(EV)] vs $\beta$. Examples of oscillating dynamics of gap solitons for (b) $\beta =0.04$ (no apparent decay of oscillations) and (c) $\beta =0.42$ (unstable regime). Red (gray) lines in plots (b) and (c) depict profiles of amplitude on $a$ site $\left[\right(m,n)=(12,13)$ in Fig. 5] with added perturbation, while the black one stands for the case when the mode evolves without presence of perturbations (only affected by numerical round-off errors). Green (left) and blue (right) vertical dashed lines in (a) correspond to the modes depicted in (b) and (c), respectively. Plot (d) depicts spectrum of amplitude presented in (b) with corresponding positive Im(EVs).

Figure 10

Upper part of full spectrum of stability EVs [positive Im(EVs)] vs $\beta$ for coupling ratios (a) $V_1/V_2=0.1$, (b) $V_1/V_2=0.5$, and (c) $V_1/V_2=0.9$, and corresponding positive Re(EVs) vs $\beta$ in (d), (e), and (f). Size of the system is $7\times 7$.

Figure 11

Moduli of amplitudes on ring sites with the corresponding power of a perturbed nonlinear ring mode, characterized by $\beta =0.1$, in a dimerized Lieb lattice of type II with (a), (b) $V_1/V_2=0.1$, (c), (d) $V_1/V_2=0.5$, (e), (f) $V_1/V_2=0.9$, and (g), (h) $V_1/V_2=1$. In all cases $V_2$ is set to 1.

Figure 12

Moduli of amplitude distributions of perturbed ring modes at $z=1045$ in a dimerized Lieb lattice of type II with (a) $V_1/V_2=0.1$ and (b) $V_1/V_2=0.5$.

Figure 13

Parameter diagram depicting regimes of preservation [light blue (gray) color] and breaking [deep blue (dark gray) color] of four-site nonlinear ring formation when $0<V_1<1$ and $V_2=1$, of the type II dimerized Lieb lattice. Regimes are defined upon two criteria: regular exchange of energy along the propagation length $z=500$ and power preservation of at least $2/3$ of total modes' power on four main sites. Line between deep blue (dark gray) and white region represents upper border of minigap.