Localized nonlinear edge states in honeycomb lattices

Two-dimensional localized edge modes in optical honeycomb lattices are found and analyzed analytically and computationally. Weak nonlinearity and transverse modulation are found to introduce self-phase modulation in the phase and create internal nonlinear interactions as the electromagnetic field propagates through the lattice. Even with relatively strong nonlinearity localization and persistence of modes along the edge are found.

Figure 1

Indexing scheme for HC lattice with row ($m$)/column ($n$) format. The vertical bars indicate separate columns of lattice sites.

Figure 2

In (a) we see the bearded edge, while in (b) the zigzag edge is shown. In each figure $\mathbf{A}$ and $\mathbf{B}$ are zero to the left of the edge. ${\mathbf{A}}_0$ refers to the 0th column of $\mathbf{A}$ sites and ${\mathbf{B}}_0$ the 0th column of $\mathbf{B}$ sites.

Figure 3

Typical stationary bearded edge modes. In (a) we have plotted (7) for $\nu =0.2$, and in (b) we have plotted (7) for $\nu =0.5$. In both panels, $\rho =1$ and ${\theta }_1={\theta }_2=\pi /4$.

Figure 4

Comparison of envelopes along a bearded edge ($n=0$). Smaller values of $\nu$ correspond to wider modes. In this figure, $\rho =1$ and ${\theta }_1={\theta }_2=\pi /4$.

Figure 5

The shaded areas represent (17); the horizontal lines represent zero-energy modes. In (a), for $\rho =1>1/2$, the zero-energy bearded edge modes exist at frequencies in ${\mathcal{I}}_{\tilde{\theta }}$; they touch the Dirac points at $\pm \tilde{\theta }$, $-\pi +\tilde{\theta }$, and $\pi -\tilde{\theta }$. In (b) we see that, for $\rho =0.4<1/2$, the zero-energy bearded edge states no longer exist and the bands have completely separated. In both figures we have chosen ${\phi }_{-}=0$.

Figure 6

The shaded areas represent (17); the horizontal lines represent zero-energy modes. In (a), for $\rho =1>1/2$, the zero-energy zigzag edge modes exist at frequencies in ${\mathcal{I}}_{\tilde{\theta }}^c$. In (b) we see that, for $\rho =0.4<1/2$, the zigzag edge zero-energy modes exist for all possible frequencies. In both figures we have chosen ${\phi }_{-}=0$.

Figure 7

Edge modes at $z=5$ for $\rho =1,{\theta }_1={\theta }_2=\pi /4$ on the $\mathbf{A}$ sites in (a) and the $\mathbf{B}$ sites in (b). We take weak nonlinearity ($\epsilon =0.2$) and a wide beam ($\nu =0.2$). Localization along the edge is maintained up to $z=5=1/\epsilon$. The size of the induced $B$ component is small.

Figure 8

Graph of $\rho =1,{\theta }_1={\theta }_2=\pi /4$. Comparing (a), the error between the asymptotics and numerics for $\epsilon =\nu =0.2$, and (b), which gives the error for $\epsilon =\nu =0.1$, by halving $\epsilon$ and $\nu$, the error is reduced by half even though we propagate twice as far in (b).

Figure 9

Graph of $\rho =1,{\theta }_1={\theta }_2=\pi /4$. In (a) we have plotted the analytically and numerically computed phases for $\epsilon =\nu =0.2$; in (b) we plot the case $\epsilon =\nu =0.1$. By comparing (a) and (b), by halving $\epsilon$ and $\nu$, the numerically computed phase is reduced by half even though we propagate twice as far in (b).

Figure 10

Edge modes at $z=25$ for $\rho =1,{\theta }_1={\theta }_2=\pi /4$, and weak nonlinearity ($\epsilon =0.2$) and wide beam ($\nu =0.2$) on the $\mathbf{A}$ sites in (a) and the $\mathbf{B}$ sites in (b). Localization along the edge is maintained up to $z=25=1/\left(\nu \epsilon \right)$. The size of the induced $B$ component is small.

Figure 11

Edge modes at $z=5$ for $\rho =0.55,{\theta }_1={\theta }_2=\pi /4$, and weak nonlinearity ($\epsilon =0.2$) and wide beam ($\nu =0.2$) on the $\mathbf{A}$ sites in (a) and the $\mathbf{B}$ sites in (b). The nonlinearity does not introduce any significant further delocalization of the edge mode.

Figure 12

Graph of ${\theta }_1={\theta }_2=\pi /4$. Maximum error between asymptotics and numerics for $\epsilon =\nu =0.2$ and $\rho =0.55$.

Figure 13

Edge modes at $z=10$ for $\rho =1,{\theta }_1={\theta }_2=\pi /4$, and strong nonlinearity ($\epsilon =1$) and narrow beam ($\nu =0.5$) on the $\mathbf{A}$ sites in (a) and the $\mathbf{B}$ sites in (b). Localization along the edge is maintained up to $z=10$.

Figure 14

Graph of $\rho =1,{\theta }_1={\theta }_2=\pi /4$. For large nonlinearity ($\epsilon =1$) and narrow beam ($\nu =0.5$), in (a) we plot $[{\sum }_m|{\mathbf{A}}_{m0}\left(z\right){{|}^2]}^{1/2}$, or the energy on the $\mathbf{A}$ edge. In (b) we plot the total energy, $[{\sum }_{m,n}|{\mathbf{B}}_{mn}\left(z\right){{|}^2]}^{1/2}$, in all $\mathbf{B}$ sites. The majority ($\sim$$82\%$) of the energy is localized along $\mathbf{A}$ edge sites as shown in (a). Some energy is leaked into the bulk along the $\mathbf{B}$ sites as seen in (b).

Figure 15

Edge modes at $z=10$ for $\rho =0.55,{\theta }_1={\theta }_2=\pi /4$, and strong nonlinearity ($\epsilon =1$) and narrow beam ($\nu =0.5$) on the $\mathbf{A}$ sites in (a) and the $\mathbf{B}$ sites in (b). For $\rho \sim 1/2$, more energy is scattered into the bulk along both $\mathbf{A}$ and $\mathbf{B}$ sites.

Figure 16

Graph of $\rho =0.55,{\theta }_1={\theta }_2=\pi /4$. For large nonlinearity ($\epsilon =1$) and narrow beam ($\nu =0.5$), in (a) we plot $[{\sum }_m|{\mathbf{A}}_{m0}\left(z\right){{|}^2]}^{1/2}$, or the energy on the $\mathbf{A}$ edge. In (b) we plot the total energy, $[{\sum }_{m,n}|{\mathbf{B}}_{mn}\left(z\right){{|}^2]}^{1/2}$, in all $\mathbf{B}$ sites. From (a), we see less than $1\%$ of the energy on the edge is scattered into the bulk. As seen in (b), energy scatters into $\mathbf{B}$ sites in a steadier way, but the amount of energy scattered into the $\mathbf{B}$ sites is still small.