Topological and flat-band states induced by hybridized linear interactions in one-dimensional photonic lattices

We report on a study of a one-dimensional linear photonic lattice hosting, simultaneously, fundamental and dipolar modes at every site. We show how, thanks to the coupling between different orbital modes, this minimal model exhibits rich transport and topological properties. By varying the detuning coefficient we find a regime where bands become flatter (with reduced transport) and a second regime, where both bands connect at a gap-closing transition (with enhanced transport). We detect an asymmetric transport due to the asymmetric intermode coupling and a linear energy exchange mechanism between modes. Further analysis shows that the bands have a topological transition with a nontrivial Zak phase which leads to the appearance of edge states in a finite system. Finally, for zero detuning, we found a symmetric condition for coupling constants, where the linear spectrum becomes completely flat, with states fully localized in space occupying only two lattice sites.

Figure 1

(a) A 1D lattice of horizontally oriented elliptical waveguides. (b) On-row coupling interactions. (c) Effective model, with a unitary cell shown in gray.

Figure 2

(a) ${\lambda }_{\pm }$ vs $k_{x}a$ and $\mathrm{\Delta }\beta$, where guidelines are located at $\mathrm{\Delta }\beta =0,2,4,6,8,10,12,14$. (b, c) Output intensity profiles at $z_{\text{max}}=20$ vs $\mathrm{\Delta }\beta$ for $s$ and $p$ modes, respectively. (d) Output center of mass for profiles shown in (b) and (c), using red and black lines, respectively. $\{C_s,C_p,C_{sp}\}=\{1,-2,1.5\}$. (e, f) Output intensity profiles at $z_{\text{max}}=20$ vs $\mathrm{\Delta }\beta$ for $s$ and $p$ modes, respectively, for $\{C_s,C_p,C_{sp}\}=\{1,-2,0\}$.

Figure 3

(a) $V_x/a$ vs $k_x$ for $\mathrm{\Delta }\beta =0$ (black), 2 (gray), 6 (red), and 10 (blue), for the upper (full lines) and lower (dashed lines) bands. Output intensity profiles at $z_{\text{max}}=20$ vs $\mathrm{\Delta }\beta$, for $(s,p)$ modes: (b1, b2) for $k_x=0$, (c1, c2) for $k_x=\pi /2$, and (d1, d2) for $k_x=\pi$, respectively. (b3, c3, d3) Output center of mass for profiles shown in (b), (c), and (d), using red and black lines for $s$ and $p$ modes, respectively. $\{C_s,C_p,C_{sp}\}=\{1,-2,1.5\},\alpha =0.002$ and $n_i=101$.

Figure 4

(a) $\lambda$ vs $\mathrm{\Delta }\beta$ for $N=30$. The participation ratio is plotted using different colors for each eigenvalue. (b1) [(b2)], (c1) [(c2)], and (d1) [(d2)] show the edge state profiles at the left [right] border for $\mathrm{\Delta }\beta =0$, 2, and 4, respectively. Red and black lines correspond to the $s$ and $p$ mode amplitudes. $\{C_s,C_p,C_{sp}\}=\{1,-2,1.5\}$.

Figure 5

(a) $\lambda$ vs $k_{x}a$ for $\mathrm{\Delta }\beta =0$. (b1) and (b2) show the FB modes for $\lambda =2C_s$ and $-2C_s$, respectively. (c1) and (c2) show the propagation of $s$ and $p$ modes, respectively, for an input condition $u_n\left(0\right)=A{\delta }_{n,no}$, $v_n\left(0\right)=0$, for $\mathrm{\Delta }\beta =0$. (d1) and (d2) Same as (c) but for $\mathrm{\Delta }\beta =1$. $C_s=-C_p=C_{sp}=1$.