Tight-binding methods for general longitudinally driven photonic lattices: Edge states and solitons

A systematic approach for deriving tight-binding approximations in general longitudinally driven lattices is presented. As prototypes, honeycomb and staggered square lattices are considered. Time-reversal symmetry is broken by varying and/or rotating the waveguides, longitudinally, along the direction of propagation. Different sublattice rotation and structure are allowed. Linear Floquet bands are constructed for intricate sublattice rotation patterns such as counter-rotation, phase offset rotation, as well as different lattice sizes and frequencies. An asymptotic analysis of the edge modes, valid in a rapid-spiraling regime, reveals linear and nonlinear envelopes which are governed by linear and nonlinear Schrödinger equations, respectively. Nonlinear unidirectional edge modes, referred to as topologically protected edge solitons, are identified. Direct numerical simulations for both the linear and nonlinear edge states agree with asymptotic theory. Topologically protected modes are found; they possess unidirectionality and do not scatter at lattice defect boundaries.

Figure 1

The honeycomb lattice consists of two triangular sublattices $V_1\left(\mathbf{r}\right)$ and $V_2\left(\mathbf{r}\right)$ with minima (zeros) located at the black circles and white circles, respectively. The defining lattice vectors are given in Eq. (15).

Figure 2

The staggered square lattice consists of two interpenetrating square sublattices $V_1\left(\mathbf{r}\right)$ and $V_2\left(\mathbf{r}\right)$ with minima (zeros) located at the black circles and white circles, respectively. The defining lattice vectors are given in Eq. (16).

Figure 3

The honeycomb lattice linear band structure (27) of Eqs. (20) and (21) for different sublattice rotation patterns. The value of $\eta$ is (a–d) $2/3$, (e) $1.7/15$, (f) $2/15$, (g) 1/10, and (h) $2/15$. The lattice parameters are $V_0^2=45,\epsilon =0.75/\pi ,\mathrm{and}{\sigma }_x={\sigma }_y=0.3$, except (b) where ${\sigma }_x=0.5,{\sigma }_y=0.25$. Red curves correspond to the asymptotic solution given in Eq. (C10).

Figure 4

Honeycomb lattice edge modes corresponding to the (a) left and (b) right zig-zag edges of Fig. 1. The lattice parameters are the same as those used to generate Fig. 3 at $\omega =5\pi /8$ and correspond to the curves with (a) negative and (b) positive slope.

Figure 5

Floquet bands (27) used to create the $\pi$-phase offset band structure shown in Fig. 3. The fundamental ($\tau =0$) and its periodic extensions ($\tau =\pm 1$) are combined at the Floquet edge $\alpha =\pm \pi /T\approx \pm 2.09$.

Figure 6

Left column: Evolution of linear edge mode magnitude $\left|b_{m,0}\left(z\right)\right|$ in the honeycomb lattice. Right column: Corresponding evolution of ${max}_{m,n}\left|a_{mn}\left(z\right)\right|$ (blue curve) and ${max}_{m,n}\left|b_{mn}\left(z\right)\right|$ (black curve). The corresponding Floquet bands are (a,b) Fig. 3, (c,d) Fig. 3, and (e,f) Fig. 3.

Figure 7

The staggered square lattice linear band structure (27) of Eqs. (25) and (26) for different rotation patterns. The value of $\eta$ is (a) $2/3$, (b) $2/15$, (c) $2/3$, (d) $2/3$, (e) $1.1/15$, (f) $1.7/15$, and (g) $2/15$. The lattice parameters are $V_0^2=45,\epsilon =0.75/\pi ,$ and ${\sigma }_x={\sigma }_y=0.3.$

Figure 8

Left column: Evolution of linear edge mode magnitude $\left|b_{m,0}\left(z\right)\right|$ in the staggered square lattice. Right column: Corresponding evolution of ${max}_{m,n}\left|a_{mn}\left(z\right)\right|$ (blue curve) and ${max}_{m,n}\left|b_{mn}\left(z\right)\right|$ (black curve). The corresponding Floquet bands are (a,b) Fig. 7, (c,d) Fig. 7, and (e,f) Fig. 7.

Figure 9

Evolution of nonlinear edge magnitude, $|b_{m,0}\left(z\right)|$, found by solving (17) and (18) using initial condition (36). The parameters in panels (a) and (b) are the same as those in Figs. 3 and 3, respectively, except $\sigma =\epsilon ,\nu =0.2,$ and (a) ${\omega }_0=\pi /2,{\alpha }_0^{\prime }=-0.190,{\alpha }_0^{\prime \prime }=0,{\alpha }_0^{\prime \prime \prime }=0.748,{\alpha }_{\mathrm{nl}}=0.238$ and (b) ${\omega }_0=3\pi /4,{\alpha }_0^{\prime }=0,{\alpha }_0^{\prime \prime }=0.362,{\alpha }_0^{\prime \prime \prime }=0,{\alpha }_{\mathrm{nl}}=0.172.$

Figure 10

Profile comparison between the discrete solution (blue stems), $b_{m,0}\left(z\right)$, and envelope (red curve), $C(y,z)$. The parameters in panels (a) and (b) are the same as those in Fig. 9, and similarly for panels (c) and (d) and Fig. 9.

Figure 11

Floquet bands for a $\pi$-offset honeycomb lattice rotation. The value of $\epsilon$ is (a) $0.75/\pi \approx 0.239$, (b) 0.2, and (c) 0.18. The other parameters are the same as those in Fig. 3.

Figure 12

Intensity snapshots, $|b_{mn}{\left(z\right)|}^2$, for a (top row) topologically protected mode and (bottom row) nontopologically protected linear modes. The defect barrier is located in the region $[-46,-40]\times [0,4].$

Figure 13

Maximum intensity evolution of $|a_{mn}{\left(z\right)|}^2$ (blue curve) and $|b_{mn}{\left(z\right)|}^2$ (black curve) for the (a) topologically and (c) nontopologically protected modes shown in Fig. 12. The corresponding participation ratios (42) are displayed in panels (b) and (d), respectively. Vertical red lines correspond to the snapshots in Fig. 12.

Figure 14

Intensity snapshots, $|b_{mn}{\left(z\right)|}^2$, for a topologically protected edge soliton ($\sigma =\epsilon$). The defect barrier is located in the region $[-46,-40]\times [0,4].$

Figure 15

(a) Maximum intensity evolution of $|a_{mn}{\left(z\right)|}^2$ (blue curve) and $|b_{mn}{\left(z\right)|}^2$ (black curve) for the topologically protected edge soliton shown in Fig. 14. (b) The corresponding participation ratio (42). Vertical red lines correspond to the snapshots in Fig. 14.