Linear and nonlinear traveling edge waves in optical honeycomb lattices

Traveling unidirectional localized edge states in optical honeycomb lattices are analytically constructed. They are found in honeycomb arrays of helical waveguides designed to induce a periodic pseudomagnetic field varying in the direction of propagation. Conditions on whether a given pseudofield supports a traveling edge mode are discussed; a special case of the pseudofields studied agrees with recent experiments. Interesting classes of dispersion relations are obtained. Envelopes of nonlinear edge modes are described by the classical one-dimensional nonlinear Schrödinger equation along the edge. Nonlinear states termed edge solitons are predicted analytically and are found numerically.

Figure 1

The semi-infinite honeycomb lattice with zigzag boundary conditions. The $A$ and $B$ sites are indexed following Ref. [22]. The primitive lattice vectors ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$ and the intersite distance $\mathbf{d}$ are also labeled.

Figure 2

The $(\kappa ,\lambda )$ plane partitioned according to the combinations of cases (I)–(III) and $N=0\text{–}2$, represented as $(\mathrm{case},N)$ pairs with parameters $(\rho ,D)$: (i) $(1,2)$, (ii) $(0.4,2)$, (iii) $(0.6,2)$, (iv) $(2,2)$, and (v) $(1,1)$. The full dispersion relations at the labeled points (a)–(f) are shown in Fig. 3.

Figure 3

Dispersion relations for points in Fig. 2 with parameters $(\rho ,D,\kappa ,\lambda )$: (a) $(1,2,1.4,0)$, (b) $(1,2,6,2.4)$, (c) $(0.4,2,1.4,0)$, (d) $(0.4,2,6,2.4)$. (e) $(0.6,2,3.6,2)$ using 80 lattice sites and $\epsilon =0.3/\left(2\pi \right)$, and (f) $(1,1,2,3)$. The black curve shows Eq. (16).

Figure 4

Plots of the pseudofield $\mathbf{A}\left(\zeta \right)=(A_1\left(\zeta \right),A_2\left(\zeta \right))$ corresponding to the parameters used in (a) Figs. 3 and 3, (b) Figs. 3 and 3, (c) Fig. 3, and (d) Fig. 3.

Figure 5

Plot of $|b_{m0}\left(z\right)|$ (at the edge) for parameters $(\rho ,\nu ,{\omega }_0)$: (a) and (b) $(1,0.2,\pi /2)$, (c) and (d) $(0.4,0.2,\pi /4)$, and (e) and (f) $(0.4,0.2,3\pi /4)$. The pseudofield parameters are fixed at $(D,\kappa ,\lambda )=(2,1.4,0)$, and the edge modes are linear $\left(\sigma =0\right)$. Periodic boundary conditions in $m$ are used. The left panels are calculated from the 2D discrete system Eqs. (8) and (9); the right panels are found from the 1D LS equation (22).

Figure 6

Plot of $|b_{m0}\left(z\right)|$ (at the edge) for the same parameters as in Fig. 5, except that (a) and (b) $\sigma =0.005$; (c)–(f) $\sigma =0.02$. The left panels are calculated from the 2D discrete system Eqs. (8) and (9); the right panels are found from the full 1D NLS equation (22) with $\tilde{\sigma }\ne 0$.

Figure 7

Plot of $|b_{m0}\left(z\right)|$ (at the edge) for the same parameters as in Fig. 6, except that ${\omega }_0=5\pi /8$ and (a) $\rho =0.6$; (b) $\rho =0.4$.