Coupled topological interface states

Topological interface states in one-dimensional electronic and photonic systems are currently intensively investigated. We demonstrate the coupling of topologically confined states: By concatenation of three substructures, where the outer embedding structures have opposite signs of reflection phases to the embedded structure, we realize a system of coupled interface states showing mode splitting. We theoretically and experimentally show that a topological transition occurs for this coupled system. The demonstration of these systems is put on a solid foundation by first realizing the substructures relevant for interface states between topologically distinct one-dimensional photonic crystals. We experimentally demonstrate band closing and band inversion for a redistribution of the optical path to the constitutive materials of the structures. The band inversion is demonstrated by the emergence of interface states at metal-dielectric interfaces and the findings are supported by ellipsometric measurements.

Figure 1

(a) The inversion symmetric unit cell is the fundamental building block of all of the structures described in this paper. (b) Five repetitions of inversion symmetric unit cells. (c) Concatenation of two structures as in (b) allows for interface states. (d) Concatenation of three structures leads to coupling of interface states.

Figure 2

Transmission of the structure depicted in Fig. 1 for a parameter sweep of $\delta \in [0,1]$ and constant refractive indices.

Figure 3

(a)–(c) Simulated and measured transmittance for different parameters $\delta$. The transmission of the unit cell is depicted by a dashed black line and shows that the band inversion occurs due to a sweep of the unit cell transmission along the band gap for varying $\delta$. The light green shaded area highlights the location of the band gap where the band inversion takes place.

Figure 4

(a) Measured transmittance for a sample with $\delta >0.5$ with and without a silver layer on top. (b) Analogous measurement for the sample with $\delta <0.5$. The light green shaded area highlights the location of the interface state resonances.

Figure 5

(a)–(c) Measurement vs theoretical prediction of ellipsometric parameters for the three structures presented in the previous section: (a) $\delta <0.5$, (b) $\delta >0.5$, (c) $\delta =0.5$.

Figure 6

(a) Calculation of reflection phases of the samples below and above band inversion. (b) Resulting ellipsometric parameters together with measured values.

Figure 7

Transmittance of the structure in Fig. 1 for a parameter sweep of ${\delta }_r$.

Figure 8

(a), (b) Measured transmittance of the fabricated samples schematically depicted in Fig. 1 compared to theory.

Figure 9

Frequency-dependent transmission characteristics and normalized reflection phases for structures with $\mathrm{Ti}{\mathrm{O}}_2$ [(a) and (b)] and with $\mathrm{Si}{\mathrm{O}}_2$ [(c) and (d)] as outer layers. The structures in (a) and (c) have a $\delta$ of 0.24 whereas the structures in (b) and (d) have a $\delta$ of 0.74. The light blue areas are bands. White areas are band gaps.

Figure 10

Transmission of the structure in Fig. 1 for a parameter sweep ${\delta }_m\in [0.0,1.0]$.

Figure 11

(a)–(c) Measured transmittance of the fabricated samples schematically depicted in Fig. 1 compared to theory.