Probing the limits of topological protection in a. designer surface plasmon structure

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1 Probing the limits of topological protection in a designer surface plasmon structure Fei Gao 1#, Zhen Gao 1#, Xihang Shi 1, Zhaoju Yang 1, Xiao Lin 1,2, John D. Joannopoulos 3, Marin Soljačić 3, Hongsheng Chen 2,3, *, Ling Lu 3, Yidong Chong 1,4, Baile Zhang 1,4, * 1 Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore , Singapore. 2 State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou , China. 3 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 4 Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore , Singapore. #These two authors contributed equally to this work. *Authors to whom correspondence should be addressed; hschen@mit.edu (H. Chen); blzhang@ntu.edu.sg (B. Zhang) 1

2 Topological photonic states are a novel class of electromagnetic modes that are immune to scattering from imperfections This phenomenon has been demonstrated experimentally, including recently in an array of coupled on-chip ring resonators at communication wavelengths 5-7. However, the topological protection in such time-reversal-invariant photonic systems is not absolute, but applies only to certain classes of defects, and these limits have not been probed. Here, we report on the realization of similar topological states in a designer surface plasmon platform consisting of metallic sub-wavelength structures Using this tunable platform, we are able to characterize in detail the field distributions of the topological edge states, and their level of robustness against a variety of defect classes, including those that can break the topological protection. This is also the first experimental realization of anomalous Floquet topological edge states, which cannot be predicted by the usual Chern number topological invariants. Topological photonic states were firstly demonstrated in a microwave-scale photonic crystal containing magneto-optic elements biased by an external magnetic field to break the time-reversal symmetry 3,4. Two designs at optical frequencies without magnetic field followed. Firstly, Rechtsman et al. demonstrated a lattice of threedimensional (3D) coupled helical waveguides in which the paraxial propagation along the third dimension mapped formally to a periodically-driven, or Floquet, twodimensional (2D) topological insulator (TI), where the time-like-, or z -, symmetry was broken 8. Secondly, Hafezi et al. realized a time-reversal-invariant photonic TI in the form of an on-chip lattice of coupled optical ring resonators, engineered to simulate 2

3 a uniform magnetic field in the quantum Hall effect. 5-7 Many other designs have also been proposed recently However, the topological protection in such time-reversalinvariant photonic systems 5-10,17 is not absolute, whose limits require further verification, and whether a 2D Floquet TI phase can be implemented on a planar surface without 3D propagation still remains unknown. We have implemented a topological designer surface plasmon structure operating in the microwave regime. Designer surface plasmons are electromagnetic modes tunable by adjusting periodic sub-wavelength corrugations on a metal surface This highly-tunable platform allows us to not only probe the robustness of topological edge states under a wide variety of different defect conditions, including defects capable of violating topological protection, but also construct an anomalous Floquet TI phase that has not been found in reality. The designer surface plasmon system is shown in Fig. 1a. It consists of closelyspaced sub-wavelength metallic rods, placed on a flat metallic surface in an arrangement similar to the design of Hafezi et al. 5-7 (but with a significant conceptual difference, to be discussed later). Large rings, called lattice rings, are set in a square lattice, and each pair of adjacent lattice rings is connected by a smaller coupling ring. Designer surface plasmon waves can circulate clockwise (counter-clockwise) in the lattice rings. For the moment, we first consider the typical situation where the two circulations do not couple to each other. Modes of the chosen circulation can be excited via U-shaped input/output waveguides at the corners of the lattice (Fig. 1b). The field pattern is recorded by a near-field probe scanning above the metal rods, connected to a microwave network analyzer. (See Materials and Methods for details.) 3

4 A significant departure from the design of Hafezi et al., 5-7 where the coupling rings were assigned different geometries for constructing an incommensurate magnetic vector potential, is that the coupling rings in the present lattice have identical geometries but nonetheless achieve non-trivial band topology This lattice is described by a network model which can be formally mapped onto a Floquet lattice, with the phase delay ϕ in each quarter of a lattice ring playing the role of a Floquet quasi-energy When the coupling between adjacent lattice rings is increased beyond a critical value (π/4 in Ref. 18 and Ref. 22), the lattice undergoes a topological transition from a topologically trivial quasi-energy band structure to a topologically non-trivial one with robust topological edge states. The unusual feature of the topologically non-trivial phase is that the Chern numbers are zero for all bands, within each (decoupled) circulation. Normally, in spinless condensed-matter and photonic systems, the net number of topological edge states in each band gap is equal to the sum of the Chern numbers in all bands below the gap ( bulk-edge correspondence ). 1-4,23-25 However, anomalous Floquet TI phases are an exception with vanishing Chern numbers in all bands The fundamental reason is that Floquet quasi-energies, unlike ordinary energies, are angle variables; hence, Floquet band structures are not bounded below by a unique lowest band. A recent experiment has confirmed the corresponding non-vanishing topological invariant of the anomalous Floquet TI phase. 20,30 Here, our designer surface plasmon structure explicitly constructs an anomalous Floquet topological edge state. We first calculate, using the transfer matrix method, the quasi-energy band structure of a semi-infinite strip with 50 lattices in y direction and periodic in x direction 4

5 in Fig. 1c. (See Supplementary Information for details.) By varying the spacing g between lattice and coupling rings, we can tune the effective inter-ring coupling strength. For weak couplings, the band structure is gapped; the gaps close at a critical value, and for strong couplings the gaps re-open with topologically protected edge states, while the Chern numbers of all bands are still zero, as numerically verified. Only part of the quasi-energy band structure shown in Fig. 1c is accessible, because the designer surface plasmons propagate only within a narrow frequency band Fig. 2a shows the frequency-domain band structure, computed numerically for a semi-infinite strip which has 5 lattice rings in the y direction, and is infinite in the x direction (choosing the circulation where the modes run clockwise along the lattice rings; see Supplementary Information). These results reveal a gap between 11.1 GHz and 11.7 GHz, spanned by unidirectional states localized to opposite edges of the strip. We now experimentally study the topological edge state in a finite 5 5 lattice. First, we apply a monopole source to the bulk, at a mid-gap (11.3 GHz) frequency; this produces a mode localized in the vicinity of the source (Fig. 2b), verifying that the bulk is insulating. Next, we excite the structure via one of the U-shape input/output waveguides at 11.3 GHz. This produces a mode which propagates along the edge (Fig. 2c), including around one corner of the lattice. To probe the robustness of the topological edge state, we introduce a defect by altering one of the lattice rings along the edge, decreasing the height of its rods from 5.0 mm to 3.5 mm (Fig. 2d2). The resultant edge state circumvents the defect ring, and continues propagating along the modified edge (Fig. 2d1). Finally, in order to verify that the observed robustness is a consequence of the 2D lattice, we remove all the rings except those along the bottom 5

6 and left boundaries, which form a one-dimensional (1D) chain (Fig. 2e2-3). The defect ring is left in place along this chain. The mode is now strongly reflected by the defect ring (Fig. 2e1). Fig. 2f shows the edge transmission spectra under each of the three scenarios described above. High transmission is observed within a frequency range which corresponds extremely well with the band gap predicted in Fig. 2a. With a single defect ring, transmission is somewhat reduced due to losses from a longer propagation path by circumventing the defect. For the 1D chain, the transmission is sharply lower. We further investigate the effects of three other types of defect. The first consists of fully removing a lattice ring and its surrounding coupling rings (Fig. 3a2). The edge mode circumvents the missing rings (Fig. 3a1). The second defect is a lattice ring of metallic rods with 4.3 mm height (Fig. 3b2). Since the dispersion in the modified lattice ring is very close to a regular lattice ring, this defect can be considered as a low potential barrier (substantially weaker than the similar defect in Fig. 2d). Part of the mode directly tunnels through this defect, and the remainder still circumvents the defect (Fig. 3b1). Finally, we show an absorbing defect, which is realized by gradually decreasing heights of the rods in the defect ring (Fig. 3c2). Most of the energy carried by the edge state is radiated into the surrounding medium (Fig. 3c1). We now study the effects of mixing of circulations in several ways, which should break the topological protection of edge states. First, we insert a metallic block between two metallic rods in a lattice ring (Fig. 4a2). Most of the edge state is flipped into the opposite circulation, and is back-scattered in the opposite direction along the edge (Fig. 4a1). Second, we place a metallic block near the lattice ring without touching, to 6

7 simulate a semi-transparent scatterer 5 (Fig. 4b2). The field pattern reveals partial mixing between the two circulations, with a portion of the mode continuing to propagate to the left and another portion back-reflected to the right (Fig. 4b1). Thirdly, we insert a dielectric block into the gap between two metallic rods in a lattice ring (Fig. 4c2); the resulting field pattern (Fig. 4c1) is similar to Fig. 4b1. Finally, we demonstrate a topologically trivial phase, when the coupling between lattice rings is sufficiently weak. To accomplish this, we increase the inter-ring separation g from 5.0 mm to 7.5 mm. An extended field pattern (Fig. 5a) is observed at 11.3 GHz (in a bulk band). (See Supplementary Information.) When the excitation is tuned to GHz (in the band gap), the field pattern shows a mode that is localized in the vicinity of excitation and does not propagate (Fig. 5b). The above results demonstrate the tolerance of topological protection in a timereversal-invariant topological photonic system, supported by surface plasmon-like waves. Anomalous Floquet topological edge states are demonstrated. In view of the tunability of designer surface plasmon structure, intriguing avenues to explore include the possibility of topologically protected mode amplification 18, which can be achieved by integrating microwave amplifiers, and topological many-body physics by incorporating non-linear components. 7

8 Materials and Methods The 5 5 lattice of lattice rings contains a total of 3,320 metal rods, each having diameter 2.5 mm and height 5.0 mm, standing on a flat aluminum plate which is 1 m 1 m in size and 5.0 mm thick. Each lattice ring consists of 56 rods arranged in a circle of radius of R1=44.56 mm. Adjacent lattice rings are coupled through coupling rings, each consisting of 48 rods arranged in a circle of radius R2=38.20 mm. The excitation source is a single-mode cable-to-waveguide adaptor with a rectangular port. Half of the port is covered with aluminum foil to increase the cutoff frequency of the waveguide port. Exciting the U-shape waveguide at different legs (input 1 or 2) can excite different circulations of surface EM wave in lattice rings. The field pattern of Ez component is recorded with a 3 mm-in-length monopole probe which scans in the xy plane, 1 mm above the top of the metal rods. Acknowledgements This work was sponsored by Nanyang Technological University for Start-up Grants, Singapore Ministry of Education under Grant No. Tier 1 RG27/12 and Grant No. MOE2011-T , and the Singapore National Research Foundation under Grant No. NRFF The work at MIT was supported by the U. S. Army Research Laboratory and the U. S. Army Research Office through the Institute for Soldier Nanotechnologies, under contract number W911NF-13-D-0001, in part by the MRSEC Program of the NSF under Award No. DMR , and in part by the MIT S3TEC EFRC of DOE under Grant No. DE-SC The work at ZJU was supported by the NNSFC (Grants Nos and ), the National Top-Notch Young Professionals Program, the NCET , and the FRFCU-2014XZZX We are grateful to 8

9 M. Rechtsman for helpful comments. Author Contributions All authors contributed extensively to this work. F. G. and Z. G. designed structures and performed measurements. F. G. implemented transfer matrix method. X. S., Z. Y., and X. L. involved in scanner setup and assembling sample. F. G., Z. G., J. D. J., M. S, H. C., L. L., Y. C. and B. Z. analyzed data, discussed and interpreted detailed results, and prepared the manuscript with input from all authors. H. C., L. L., Y. C. and B. Z. supervised the project. Competing Financial Interests statement The authors declare no competing financial interests. 9

10 References 1. Lu, L., Joannopoulos, J. D. and Soljacic, M. Topological photonics. Nature Photon. 8, (2014). 2. Haldane, F. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, (2008). 3. Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljacic, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, (2009). 4. Poo, Y., Wu, R. X., Lin, Z. F., Yang, Y. & Chan, C. T. Experimental Realization of Self-Guiding Unidirectional Electromagnetic Edge States. Phys. Rev. Lett. 106, (2011). 5. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nature Phys. 7, (2011). 6. Hafezi, M., Mittal, S., Fan, J., Migdall, A. and Taylor, J. M. Imaging topological edge states in silicon photonics. Nature Photon. 7, (2013). 7. Mittal, S., Fan, J., Faez, S., Migdall, A., Taylor, J. M., and Hafezi, M. Topologically Robust Transport of Photons in a Synthetic Gauge Field. Phys. Rev. Lett. 113, (2014). 8. Rechtsman, M. C. et al. Photonic floquet topological insulators. Nature 496, (2013). 9. Khanikaev, A. B. et al. Photonic topological insulators. Nature Mater. 12, (2012). 10

11 10. Chen, W., Jiang, S., Chen, X., Zhu, B., Zhou, L., Dong, J., and Chan, C. T. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nature Commun. 5, 5782 (2014). 11. Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature Photon. 6, (2012). 12. Harvey, A. F., Periodic and Guiding Structures at Microwave Frequencies. IRE Trans. Microwave. Theory Tech. 8, (1960) 13. Pendry, J. B., Martin-Cano, D. and Garcia-Vidal, F. J. Mimicking Surface Plamons with Structured Surfaces. Science 305, (2004). 14. Hibbins, A. P., Evans, B. R. and Sambles, J. R. Experimental Verification of Designer Surface Plasmons. Science 308, (2005). 15. Martin-Cano, D. et al. Domino plasmons for subwavelength terahertz circuitry. Opt. Express 18, (2010). 16. Yu, N. et al. Designer spoof surface plasmon structures collimate terahertz laser beams. Nature Mater. 9, (2010). 17. Umucalılar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, (2011) 18. Liang, G. Q. and Chong, Y. D. Optical Resonator Analog of a Two-Dimensional Topological Insulator. Phys. Rev. Lett. 110, (2013). 19. Liang, G. Q., and Chong, Y. D. Optical Resonator Analog of a Photonic Topological Insulator: A Finite-Difference Time-Domain Study. Int. J. Mod. Phys. B 28, (2014). 20. Pasek, M. and Chong, Y. D. Network models of photonic Floquet topological insulator. 11

12 Phys. Rev. B 89, (2014). 21. Chalker, J. T. and Coddington, P. D. Percolation, quantum tunneling and the integer Hall effect. J. Phys. C: Solid State Phys. 21, (1988). 22. Ho, C. M. and Chalker, J. T. Models for the integer quantum Hall effect: The network model, the Dirac equation, and a tight-binding Hamiltonian. Phys. Rev. B 54, 8708 (1996). 23. Hasan, M. Z. and Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, (2010). 24. Qi, X. L. and Zhang, S. C. Topological insulators and superconductors. Rev. Mod. Phys. 83, (2011). 25. Moore, J. E. The birth of topological insulators. Nature 464, (2010). 26. Kitagawa, T., Berg, E., Rudner, M., and Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, (2010). 27. Kitagawa, T., Rudner, M., Berg, E. and Demler, E. Exploring topological phases with quantum walks. Phys. Rev. A 82, (2010) 28. Rudner, M. S., Lindner, N. H., Berg, E., and Levin, M. Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems. Phys. Rev. X 3, (2013). 29. Lindner, N. H., Refael, G. and Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nature Phys. 7, (2011). 30. Hu, W., Pillay, J. C., Wu, K., Pasek, M., Shum, P. P., and Chong, Y. D. Measurement of a Topological Edge Invariant in a Microwave Network. Phys. Rev. X 5, (2015). 12

Figure 1 Construction of a topological designer surface plasmon structure and its topological transition. a, Photo of metallic rods with diameter 2.5 mm and height 5.

A coupling ring formed by 48 metallic rods is with radius R 2 =38.2 mm. The ring-ring distance is g = 5.0 mm. ϕ denotes the phase delay of electromagnetic waves along a quarter of a lattice ring.

13 Figure 1 Construction of a topological designer surface plasmon structure and its topological transition. a, Photo of metallic rods with diameter 2.5 mm and height 5.0 mm distributed with centercenter distance 5.0 mm on a flat metallic surface. A lattice ring formed by 56 metallic rods is with radius R 1 = mm. A coupling ring formed by 48 metallic rods is with radius R 2 =38.2 mm. The ring-ring distance is g = 5.0 mm. ϕ denotes the phase delay of electromagnetic waves along a quarter of a lattice ring. b, Schematic of a 5 5 lattice in experiment. A network analyzer records the field pattern by scanning a near-field probe above the metallic rods. The red meandering curve above the structure represents edge states. c, Topological transition as the inter-ring coupling strength that links to the ringring distance g is tuned from weak to strong in a finite strip of the lattice. Before and after the transition, all bands have zero Chern number C=0. Red and blue lines denote edge states confined to the upper and lower edges of the strip respectively. 13

c, Observed edge state at frequency 11.3 GHz.

14 Figure 2 Demonstration of topological edge state. a, Simulated band diagram of the narrow-band designer surface plasmon structure. b, Observed field pattern when the excitation is inside the bulk at frequency 11.3 GHz. c, Observed edge state at frequency 11.3 GHz. d1, The edge state circumvents a defect lattice (shown in d2) consisting of metallic rods with height s = 3.5 mm. All other rods have the height h = 5.0 mm. e1, The defect (shown in e3) causes strong reflection for a 1D lattice (e2). f, Transmission spectra of above configurations in c-e. 14

All other rods maintain their height h = 5 mm.

15 Figure 3 Demonstration of circulation-conserved defects. a1, Observed field pattern when removing a lattice ring and its surrounding coupling rings (shown in a2). All other rods maintain their height h = 5 mm. b1, Observed field pattern when the height of metallic rods of a lattice ring is decreased to s = 4.3 mm (shown in b2). c1, Observed field pattern when heights of metallic rods of a lattice ring gradually decrease to zero (shown in c2; from s 1 to s 7, s 1 = 4.7 mm, s 2 = 4.1 mm, s 3 = 3.5 mm, s 4 = 2.9 mm, s 5 = 2.3 mm, s 6 = 1.7 mm, and s 1 = 1.0 mm). 15

ring. b1, Observed field pattern when the metallic block (shown in b2) is located near a

16 Figure 4 Demonstration of circulation mixing defects. a1, Observed field pattern when a metallic block (shown in a2) is inserted in a lattice ring. b1, Observed field pattern when the metallic block (shown in b2) is located near a lattice ring without touching. c1, Observed field pattern when a dielectric block (shown in c2) is inserted in a lattice ring. 16

17 Figure 5 Demonstration of a trivial insulator phase. a, Observed field pattern at 11.3 GHz in the bulk band. b, Observed field pattern at GHz in the topologically trivial band gap. 17

Transport properties of photonic topological insulators based on microring resonator array

Transport properties of photonic topological insulators based on microring resonator array Xiaohui Jiang, Yujie Chen*, Chenxuan Yin, Yanfeng Zhang, Hui Chen, and Siyuan Yu State Key Laboratory of Optoelectronic