Experimental observation of edge states in SSH-Stub photonic lattices

We reveal unconventional edge states in a one-dimensional Stub lattice of coupled waveguides with staggered hoppings. The edge states appear for the same values of hoppings as topological edge states in the Su-Schrieffer-Heeger model. They have different energies depending on the lattice termination and present a remarkable robustness against certain types of disorder. We evidence experimentally the phase transition at which these edge states appear, opening the door to the engineering of one-dimensional lattices with localized edge modes whose energy and location can be controlled at will.

Figure 1

(a) Scheme of a finite SSH-Stub lattice. The unit cell is marked with a rectangle. (b) Dispersion relation of an infinite lattice for $\delta =0.5$ (blue), $\delta =1.0$ (black), and $\delta =1.5$ (red). The FB at ${\beta }_z=0$ exists for any value of $\delta$. (c) Spectrum for different values of $\delta$. (d1)–(d3) Amplitude profiles of left (${\beta }_z=\pm 1$) and right (${\beta }_z=0$) edge states. In (b) and (c) $t_1=t_3=1$.

Figure 2

(a) Femtosecond laser writing technique. (b) Bright-field microscopy of dimerized Stub photonic lattices for $d_1=16\mu \mathrm{m}$, $d_3=18\mu \mathrm{m}$, and $d_2=22$ (top), 16 (center), and 12 (bottom) $\mu \mathrm{m}$, respectively. (c) HeNe characterization setup. (d)–(f) Intensity output profiles for edge-$B$, edge-$C$, and edge-$A$ excitations (see yellow circle), respectively. $d_2$ and $\delta$ are indicated directly in the figure. A dashed horizontal line indicates the transition region. Insets in (d) and (e): $P_r$ vs distance $d_2$.

Figure 3

(a) Averaged spectrum of a disordered SSH-Stub lattice vs the disorder strength $t_0/t_1$, for $\delta =0.5$. Blue, green, and red dots indicate the energies of the edge states, while gray dots correspond to the energies of bulk states. (b) Scheme for a SSH-Stub lattice including a coupling defect $t_d$ at the first unit cell. (c) Microscope image of a SSH-Stub photonic lattice for $\{d_1,d_2,d_3,d_d\}=\{16,14,18,12\}\mu \mathrm{m}$. (d) Intensity output profiles for an $A$-site excitation (see yellow circle). (d1)–(d6) $d_d=19,17,16,15,14,12\mu \mathrm{m}$, respectively.

Figure 4

The frequency spectrum vs parameter $\delta$ for (a) left-$B$, (b) left-$C$, and (c) right-$A$ input conditions, as indicated directly in the insets. The dashed line marks $\delta =1$ as the limit where edge modes exist. Simulations were performed considering 31 unit cells and a propagation length of $z_{\text{max}}=1000$

Figure 5

Dimerized Stub lattice including disorder in coupling constants.

Figure 6

(a) Inverse participation ratio (IPR) vs $t_d/t_1$ for both Tamm states, for $\delta =1.4$ and $t_3=1$. (b1)–(b3) Intensity profiles for Tamm states for $t_d/t_1=1.85$, 2.1, and 3.0, for the same parameters used in (a) and using an avocado color scale. (c)–(e) Frequencies excited vs $\delta$, for $A$-, $B$-, or $C$-edge input conditions, as sketched in the insets. Simulations were performed considering 31 unit cells and a propagation length of $z_{\text{max}}=1000$.

Figure 7

Spectrum of a finite SSH-Stub lattice of 31 unit cells as a function of $t_3$ coupling when $\delta =0.5$. Gray lines correspond to the energies of bulk states. Blue, red, and green lines correspond to the energies of the edge states.