Spectral imaging of topological edge states in plasmonic waveguide arrays

We report on the observation of a topologically protected edge state at the interface between two topologically distinct domains of the Su-Schrieffer-Heeger model, which we implement in arrays of evanescently coupled dielectric-loaded surface-plasmon polariton waveguides. Direct evidence of the topological character of the edge state is obtained through several independent experiments: Its spatial localization at the interface as well as the restriction to one sublattice is confirmed by real-space leakage radiation microscopy. The corresponding momentum-resolved spectrum obtained by Fourier imaging reveals the midgap position of the edge state as predicted by theory.

Figure 1

(a) The two distinct dimerizations of the SSH model. The intra- and interunit-cell hopping amplitudes are characterized by $J$ and $J^{\prime }$, respectively. (b) Band structure of the SSH model for $J^{\prime }=J/2$. (c) The $x\text{−}y$ components of the eigenstates $\mathbf{d}\left(k\right)$ for both dimerizations. The same color scheme is used in (b) and (c) to indicate quasimomentum $k$ in the Brillouin zone.

Figure 2

(a) Waveguide array fabricated out of PMMA on top of a Cr- and Au-coated glass substrate. Alternating center-to-center separations, 600 and 1000 nm, implement the bulk SSH model. (b) Leakage radiation image of the spatial evolution of SPPs propagating along the structure depicted in (a). The fading intensity along the $z$ axis is induced by radiation losses and absorption (propagation length $\approx 16\mu \mathrm{m}$). The color scale representing the SPP intensity is the same as in (e). (c) rms width of the intensity distribution in (b), demonstrating ballistic expansion. Ballistic expansion for $z<0$ corresponds to free SPPs propagating at the gold-air interface. (d) Plasmonic waveguide array incorporating a topological defect where the long separation is repeated twice. Three different excitation sites, I, II, and III, are highlighted. (e) Leakage radiation image of SPPs propagating along the structure depicted in (d) with excitation at site I. A pronounced central feature is visible around the topological defect at $x=0$. The ballistically expanding background results from SSH bulk modes that are also excited. The inset shows the intensity distribution integrated along the $z$ axis in the region $50\mu \mathrm{m}<z<130\mu \mathrm{m}$.

Figure 3

Leakage radiation image of propagating SPPs, with SPPs excited at the excitation site II (a) and excitation site III (b). The color scale is the same as in Fig. 1. (c) Probability amplitude of the topologically protected edge state as predicted by the SSH theory for the waveguide geometry in Fig. 2.

Figure 4

(a) Measurement scheme for recording momentum-resolved spectra of SPPs via Fourier imaging of leakage radiation. Leakage radiation preserves the SPP momentum component parallel to the glass substrate of index $n_{\text{glass}}$. Principal plane $H_1$ and sine condition are also shown. (b) Momentum-resolved spectrum of SPPs propagating in the bulk [see Fig. 2], acquired by Fourier imaging of the leakage radiation (similar color scale as in Fig. 1). As explained in the text, we interpret the momentum distribution along the $y$ axis as the energy spectrum of the SSH model; the vertical units can be gauged taking into account that the dashed circle denotes the maximum detected momentum ($k_0\text{NA}$). Thin dashed lines are the expected energy bands for the SSH model extended to include next-nearest-neighbor coupling $J^{\prime \prime }$ ($J^{\prime }\approx 0.5J$ and $J^{\prime \prime }\approx 0.1J$). (c) The dotted curve is the energy band of a simple lattice ($J=J^{\prime }$, lattice constant $a/2$), and the solid and dashed curves are the energy band in the presence of a weak imbalance ($J\ne J^{\prime }$) (see text). We use the same color scheme as in Fig. 1. (d) Like (b) but for SPPs excited at the topological defect (site I) [see Fig. 2].

Figure 5

Calculated SPP electric field distributions for solutions of the SSH model with quasimomentum $k=0$ for (a) the lower and (b) the upper band, with effective refractive indices $n_{\text{eff}}=1.085+0.004i$ and $n_{\text{eff}}=0.936+0.005i$, respectively. (c) Electric field distribution of the topological edge mode with $n_{\text{eff}}=1.044+0.004i$. The solid black lines depict the geometry of the waveguide arrays used in the experiments without [(a), (b)] and with (c) the topological defect. The color scale shows the norm in arbitrary units.

Figure 6

Calculated SPP real-space intensity distributions and momentum-resolved spectra. From top to bottom, the calculations correspond to the experiments presented in Figs. 2, 4 2, and 4, respectively. Units and color scales are the same as in the corresponding experimental graphs. The calculations are based on a finite element method in frequency space (see text).