# Topological Photonics

Since the discovery of topological insulator [1], exploring various topological phases of matter has become a major goal of research in condensed matter physics. The exotic phase of topological insulator allows us to transport an electron on the interface without any backscattering when it encounters surface impurities. In the recent work, we demonstrated that metamaterials represent a promising platform for photonic analog of topological insulator [2]. Following up, we propose a new design of photonic topological insulator based on a bi-anisotropic waveguide [4] to take this idea steps further to experimental realization. Like the well-known advantage of topological insulator in electronic system, we present the outperformance of photonic topological insulator (PTI) in guiding wave in photonic device compared to traditional method (i.e. Guided mode present in photonic crystal defects [3]). As shown in Fig. 2, the topological edge mode is excited at the top horizontal domain wall seamlessly changes its propagation direction by 120 degree and continues to follow the domain wall without back-reflection [4]. The traditional photonic defect mode on the other hand can only achieve 100% transmission with narrow-band resonance tunneling.

The specific platform used in the recent work [5] is shown in Fig. 3a. The QSH PTI is comprised of the parallel-plate metal waveguide sandwiching a periodically arranged hexagonal array of metallic cylinders attached to one of the two metal plates and separated by a finite gap from the opposite plate. The simulated photonic band structure (PBS) of the PTI shown in Fig. 3b (see the caption and Methods for the physical dimensions, and the details of numerical simulations and measurements) reveals a complete topological band gap (gray-shaded area) of the bulk PTI. The bandgap was demonstrated by measuring a 30 dB transmission drop in the $|f-f_G^{(top)} | \textless \Delta f_{BG}/2$ frequency range (black line in Fig. 3c: $f_G^{(top)} \approx 6.1GHz$ and $\Delta f_{BG} \approx 0.07f_G$) when all rods are attached to the top plate. The topological spin-Chern index of the electromagnetic modes propagating below the bandgap changes sign when the rods are re-attached from the top to the bottom plate. Therefore, the Chern number is reversed across the wall between two QSH PTIs domains (“claddings”) with the rods attached to the opposite plates as shown in Fig. 3a. Such topological waveguide is expected3 to support four TPSWs plotted in Fig. 3b as red lines: two spin-up states propagating in the forward, and two spin-down states propagating in the backward directions. In the absence of spin-ﬂipping perturbations, backscattering is prohibited for the TPSWs. Teir existence across the entire bandgap is experimentally demonstrated (Fig. 3c) by measuring the ~30 dB transmission enhancement (red line) over that through the bulk PTI (black line). Therefore, the electromagnetic waves excited by the launching antenna inside the bandgap do not evanescently tunnel through the PTI’s bulk. Instead, they couple to the surface mode and propagate unimpeded towards the probe antenna. The spatial localization of the surface mode to a small fraction of the wavelength on either side of the interface is established by mapping the field profile in the y-direction (Fig. 4a). To our knowledge, this is the first experimental evidence of the wavelength-scale confinement of a surface wave propagating at the interface between two PTIs.

Finally, we demonstrate the topological protection of the surface wave by experimentally observing its most important physical property: that reﬂection-free energy ﬂow can occur despite encountering a broad class of possible lattice defects along its propagation path that maintain spin-degeneracy and preserve the spin DOF. Within this class falls the detour defect shown in the inset of Fig. 4b, where the rods are re-attached to the opposite plate so as to bend the interface between PTIs. The defect contains four 120° bends, each of which is capable of reﬂecting most of the incident surface wave in the absence of topological protection. As we show below, the addition of the defect creates a reﬂection-free single-channel delay line. The topological protection is apparent from Fig. 4b, where the transmission spectra along the uninterrupted interface (red line) and the same interface interrupted by a detour-type defect (green line) are plotted as a function of frequency. Outside of the bandgap (e.g., at the frequencies marked by black arrows) the transmission is reduced by almost an order of magnitude because the defect blocks the propagating bulk modes from the receiving antenna. Inside the bandgap, however, the forward-propagating spin-up TPSW ﬂows around the defect (Fig. 4c) because the defect does not ﬂip the spin, and no back-reﬂection is allowed. The almost negligible ~1 dB decrease in transmission (red arrows) serves as a clear experimental signature of topologically robust transport.

Figure 1. (a) The domain wall between two BMWs with reversed bianisotropy at the center. The 3D view of the domain wall has part of top metal plate removed to see the structure below. χ indicates the sign of bi-anisotropy in each domain (b) The corresponding PBS of the 1D supercell formed by two BMWs. In (b) black dots: bulk modes; blue lines: edge modes; arrows: effective spin of the edge modes [4].

Figure 2. (a) The left inset shows the energy density of resonant-tunneling photonic defect mode through a zigzag channel between two trivial-gapped photonic crystal. The frequency of the mode indicated by red arrow at the peak in (b). The right inset shows the case of off-resonance. (b) Transmission spectrum of the defect mode with the frequency range in the band gap. (c) Reflection-free routing of the PTI edge mode. (d) Transmission spectrum of the PTI edge mode [4].

Figure 3. The platform for a quantum spin-Hall photonic topological insulator (QSH PTI): a bianisotropic metawaveguide. (a) Schematic of a QSH PTI. Part of the top metal plate is removed to reveal the cylinders attached to the top plate (purple) and to the bottom plate (green), leaving a gap of thickness g to the other plate. The 2-port VNA is connected to the feeding source (yellow double arrow) and to the receiving probe (red double arrow) for transmission measurement. Top inset: geometric parameters of the PTI. Bottom inset: picture of the assembled structure showing an interface between the two the PTIs which serve as two topological claddings. (b) Calculated 1D projected PBS of the PTI with topologically non-trivial interface. Cyan area: bulk bands, gray area: a complete band gap around the doubly degenerate Dirac cones, red curve: TPSWs supported by the topologically non-trivial interface. (c) Measured transmission spectra through the bulk PTI (all rods attached to the top plate: black curve) and along the interface between two PTIs shown in Fig. 1a (red curve). The transmission is enhanced by nearly 30dB in the 5.87< f< 6.29 GHz frequency range by the presence of the interface, indicating that the surface propagation dominates over the bulk PTI propagation. QSH PTI parameters defined in the inset: h= a= 36.8 mm, d= 0.345a= 12.7 mm, g= 0.15a= 5.5 mm.

Figure 4. Spatial localization and topological protection of the surface waves. (a) Frequency-dependent spatial profiles of the electric field intensity measured at the end of the photonic structure. The surface mode is transversely confined to a small fraction of its wavelength. The zero of the scanning axis (y= 0) is at the interface between the two PTIs. (b) Comparison of measured transmission spectra between the straight (red curve) and the delay line (green curve) interfaces. The receiving probe is at y= 0. Inset: schematic of the interface inside the dashed white box in Fig. 1a for the straight interface (red box) and the delay line interface (green box) containing a large detour-type defect. (c) Simulated energy density at f= 6.08 GHz showing the TPSW ﬂowing around the defect without scattering.

Relevant publications:

[1] C. L. Kane and E. J. Mele, “Z2 Topological Order and the Quantum Spin Hall Effect,” Phys. Rev. Lett. 95, 146802:1-4 (2005)

[2] A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, G. Shvets, “Photonic Topological Insulators”, Nature Materials 12, 233–239 (2013). DOI: 10.1038/NMAT3520 (published online).

[3] J. D. Joannopoulos et al, Photonic Crystals 2nd ed, chap. 3, 4 (Princeton University Press, 2008)

[4] Ma, T., Khanikaev, A. B., Mousavi, S. H., Shvets, G., Topologically protected photonic transport in bi-anisotropic meta-waveguide, eprint arXiv:1401.1276 (2014)

[5] Lai, K., Ma, T., Bo, X., Anlage, S., Shvets, G., “Experimental Realization of a Reflections-Free Compact Delay Line Based on a Photonic Topological Insulator”, Sci. Rep. 6, 28453 (2016)