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arxiv: 2008.09547 · v1 · submitted 2020-08-21 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· physics.app-ph· physics.optics

Critical couplings in topological-insulator waveguide-resonator systems observed in elastic waves

Pith reviewed 2026-05-24 15:01 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-sciphysics.app-phphysics.optics
keywords topological insulatorwaveguide resonatorelastic wavescritical couplingspin-momentum lockingtopological protectionphononics
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The pith

Topological-insulator waveguide resonators eliminate upstream reflections while retaining transmission spectra and maximizing resonator energy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper integrates topological insulators into a classical waveguide-ring-resonator system using elastic waves. It demonstrates that the spin-momentum locked states at the TI boundaries allow a two-port configuration to eliminate upstream reflections, unlike classical systems, while preserving the transmission spectral characteristics useful for filtering and sensing. This setup also maximizes the energy stored in the resonator. A reader would care because these features suggest improved performance for signal processing, sensing, and energy applications in phononics, photonics, or electronics.

Core claim

Adopting an elastic-wave platform, the study introduces topological insulator into a waveguide-ring-resonator configuration. A two-port TI waveguide resonator can fundamentally eliminate upstream reflections while completely retaining useful transmission spectral characteristics, and maximize the energy in the resonator, owing to the spin-momentum locked transmission states at the TI boundaries.

What carries the argument

spin-momentum locked transmission states at the TI boundaries providing advantages over classical systems in the waveguide-ring-resonator geometry

If this is right

  • Novel signal processing, gyro/sensing, lasering, energy harvesting, and intense wave-matter interactions using phonons, photons, or electrons.
  • Enhances confidence in using topological protection for practical device performance and functionalities.
  • Advantage of introducing (pseudo)spins to existing conventional configurations.
  • Further research on advancing phononics/photonics, especially on-chip.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be extended to photonic or electronic systems to achieve similar benefits.
  • On-chip integration may lead to new device functionalities in wave-based technologies.
  • Similar configurations could be explored in other topological materials to test generality.

Load-bearing premise

The spin-momentum locked transmission states at the TI boundaries provide the stated advantages over classical systems when integrated into the waveguide-ring-resonator geometry.

What would settle it

Detection of upstream reflections in an experimental two-port TI waveguide resonator setup would falsify the elimination of reflections.

read the original abstract

Waveguides and resonators are core components in the large-scale integration of electronics, photonics, and phononics, both in existing and future scenarios. In certain situations, there is critical coupling of the two components; i.e., no energy passes through the waveguide after the incoming wave couples into the resonator. The transmission spectral characteristics resulting from this phenomenon are highly advantageous for signal filtering, switching, multiplexing, and sensing. In the present study, adopting an elastic-wave platform, we introduce topological insulator (TI), a remarkable achievement in condensed matter physics over the past decade, into a classical waveguide-ring-resonator configuration. Along with basic similarities with classical systems, a TI system has important differences and advantages, mostly owing to the spin-momentum locked transmission states at the TI boundaries. As an example, a two-port TI waveguide resonator can fundamentally eliminate upstream reflections while completely retaining useful transmission spectral characteristics, and maximize the energy in the resonator, with possible applications being novel signal processing, gyro/sensing, lasering, energy harvesting, and intense wave-matter interactions, using phonons, photons, or even electrons. The present work further enhances the confidence of using topological protection for practical device performance and functionalities, especially considering the crucial advantage of introducing (pseudo)spins to existing conventional configurations. More in-depth research on advancing phononics/photonics, especially on-chip, is foreseen.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript reports an experimental realization of critical coupling in a topological-insulator (TI) waveguide-ring-resonator geometry fabricated on an elastic-wave platform. The central observation is that the spin-momentum-locked edge states allow a two-port TI configuration to reach critical coupling with zero upstream reflection while preserving the transmission spectrum and maximizing stored energy in the resonator, offering advantages over classical counterparts for filtering, sensing, and related applications.

Significance. If the reported measurements hold, the work supplies a direct experimental demonstration that topological protection can be integrated into a standard resonator geometry to remove a key limitation (upstream reflection) without degrading transmission characteristics. This strengthens the practical case for TI-based phononic devices and provides a concrete benchmark for on-chip implementations.

minor comments (3)
  1. [Abstract and §4] The abstract states that upstream reflections are 'fundamentally eliminate[d]' and energy is 'maximize[d]'; the main text should supply quantitative values (e.g., reflection coefficient < -30 dB, stored-energy ratio relative to classical case) with error bars and frequency range to support these claims.
  2. [Figures 2-4 and §3] Figure captions and the main text should explicitly label the pseudo-spin directions on the TI boundaries and indicate which edge state carries the forward versus backward power flow in the two-port geometry.
  3. [§4] A brief comparison table or plot overlaying the TI and classical transmission spectra (including the reflection coefficient) would clarify the claimed retention of spectral features.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly identifies the key advantage of spin-momentum locking in eliminating upstream reflections while preserving transmission characteristics. No specific major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity; experimental observation of known TI edge-state properties applied to standard resonator geometry

full rationale

The paper reports experimental realization of critical coupling in an elastic-wave TI waveguide-resonator system. The central claim (elimination of upstream reflections while retaining transmission spectrum and maximizing stored energy) follows directly from applying the established one-way, spin-momentum-locked propagation of TI boundary states to the conventional two-port ring-resonator layout. No equations, fitted parameters, or self-citations are presented that reduce the observed outcome to a tautology or to the input data by construction. The work is framed as measurement and demonstration rather than a derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, ad-hoc axioms, or invented entities are introduced beyond standard condensed-matter assumptions about topological edge states.

axioms (1)
  • domain assumption Topological insulators possess spin-momentum locked boundary states that differ from classical wave propagation
    Invoked in the abstract as the source of the stated advantages.

pith-pipeline@v0.9.0 · 5827 in / 1104 out tokens · 23608 ms · 2026-05-24T15:01:29.200111+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

79 extracted references · 79 canonical work pages

  1. [1]

    Topological insulators and superconductors

    Qi XL and Zhang SC. Topological insulators and superconductors. Rev Mod Phys 2011; 83: 1057- 1110

  2. [2]

    Colloquium: topological insulators

    Hasan MZ and Kane CL. Colloquium: topological insulators. Rev Mod Phys 2010; 82: 3045-3067

  3. [3]

    Topological photonics

    Ozawa T, Price HM and Amo A et al. Topological photonics. Rev Mod Phys 2019; 91: 015006

  4. [4]

    Topological phases in acoustic and me chanical systems

    Ma G, Xiao M and Chan CT. Topological phases in acoustic and me chanical systems. Nat Rev Phys 2019; 1: 281-294

  5. [5]

    Photonic Floquet topological insulators

    Rechtsman MC, Zeuner JM and Plotnik Y et al. Photonic Floquet topological insulators. Nature 2013; 496: 196-200

  6. [6]

    Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide

    Chen WJ, Jiang SJ and Chen XD et al. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nat Commun 2014; 5: 5782

  7. [7]

    Robust reconfigurable electro magnetic pathways within a photonic topological insulator

    Cheng X, Jouvaud C and Ni X et al. Robust reconfigurable electro magnetic pathways within a photonic topological insulator. Nature Maters 2016; 15: 542-548

  8. [8]

    Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials

    Yang Y , Xu YF and Xu T et al. Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials. Phys Rev Lett 2018; 120: 217401

  9. [9]

    Probing topological protection using a designer surface plasmo n structure

    Gao F , Gao Z and Shi X et al. Probing topological protection using a designer surface plasmo n structure. Nat Commun 2016; 7: 11619

  10. [10]

    Crystalline metamaterials fo r topological properties at subwavelength scales

    Y ves S, Fleury R and Berthelot T et al. Crystalline metamaterials fo r topological properties at subwavelength scales. Nat Commun 2017; 8: 16023

  11. [11]

    Direct observation of valley-polarized topological edge states in designer surface plasmon crystals

    Wu X, Meng Y and Tian J et al. Direct observation of valley-polarized topological edge states in designer surface plasmon crystals. Nat Commun 2017; 8: 1304

  12. [12]

    Imaging topological edge states in silicon photonics

    Hafezi M, Mittal S and Fan J et al. Imaging topological edge states in silicon photonics. Nature Photons 2013; 7: 1001-1005

  13. [13]

    A silicon-on-insulator slab for topological valley transport

    He XT , Liang ET and Y uan JJ et al. A silicon-on-insulator slab for topological valley transport. Nat Commun 2019; 10: 872

  14. [14]

    Robust topologically protected transport in photonic crystals at telecommunication wavelengths

    Shalaev MI, Walasik W and Tsukernik A et al. Robust topologically protected transport in photonic crystals at telecommunication wavelengths. Nature Nano 2019; 14: 31-34

  15. [15]

    Observation of phononic helical edge states in a mechanical topological insulator

    Süsstrunk R and Huber SD. Observation of phononic helical edge states in a mechanical topological insulator. Science 2015; 349: 47-50

  16. [16]

    Acoustic topological insulator and robust one-way sound transp ort

    He C, Ni X and Ge H et al. Acoustic topological insulator and robust one-way sound transp ort. 14 / 39 Nature Phys 2016; 12: 1124-1129

  17. [17]

    Observation of topological valley transport of sound in sonic crystals

    Lu J, Qiu C and Ye L et al. Observation of topological valley transport of sound in sonic crystals. Nature Phys 2017; 13: 369-374

  18. [18]

    Experimental demonstrat ion of anomalous Floquet topological insulator for sound

    Peng YG, Qin CZ and Zhao DG et al. Experimental demonstrat ion of anomalous Floquet topological insulator for sound. Nat Commun 2016; 7: 13368

  19. [19]

    Elastic pseudospin transport for integratable topological phononic circuits

    Yu SY , He C and Wang Z et al. Elastic pseudospin transport for integratable topological phononic circuits. Nat Commun 2018; 9: 3072

  20. [20]

    Experimental realization of on-chip topological nanoelectromechanical metamaterials

    Cha J, Kim KW and Daraio C. Experimental realization of on-chip topological nanoelectromechanical metamaterials. Nature 2018; 564: 229-233

  21. [21]

    Experimental observation of topologically protected helical edge modes in patterned elastic plates

    Miniaci M, Pal RK and Morvan B et al. Experimental observation of topologically protected helical edge modes in patterned elastic plates. Phys Rev X 2018; 8: 031074

  22. [22]

    On-chip valley topological mater ials for elastic wave manipula tion

    Yan M, Lu J and Li F et al. On-chip valley topological mater ials for elastic wave manipula tion. Nature Maters 2018; 17: 993-998

  23. [23]

    Optical microcavities

    Vahala K. Optical microcavities. Nature 2003; 424: 839-846

  24. [24]

    Integrated ring resonators

    Rabus DG. Integrated ring resonators. Berlin: Springer, 2007

  25. [25]

    Silicon microring resonators

    Bogaerts W, De Heyn P and Van Vaerenbergh T et al. Silicon microring resonators. Laser Photonics Rev 2012; 6: 47-73

  26. [26]

    Observation of critical couplin g in a fiber taper to a silica- microsphere whispering-gallery mode system

    Cai M, Painter O and Vahala KJ. Observation of critical couplin g in a fiber taper to a silica- microsphere whispering-gallery mode system. Phys Rev Lett 2000; 85: 74-77

  27. [27]

    Critical coupling and its control in optical waveguide -ring resonator systems

    Yariv A. Critical coupling and its control in optical waveguide -ring resonator systems. IEEE Photonic Tech L 2002; 14: 483-485

  28. [28]

    Modal coupling in tra veling-wave resonators

    Kippenberg TJ, Spillane SM and Vahala KJ. Modal coupling in tra veling-wave resonators. Opt. Lett 2002; 27: 1669-1671

  29. [29]

    Topological insulator laser: Experiments

    Bandres MA, Wittek S and Harari G et al. Topological insulator laser: Experiments. Science 2018; 359: eaar4005

  30. [30]

    Third-harmonic generation in photonic topological metasurfaces

    Smirnova D, Kruk S and Leykam D et al. Third-harmonic generation in photonic topological metasurfaces. Phys Rev Lett 2019; 123: 103901

  31. [31]

    Topological ring-cavity laser formed by honeyc omb photonic crystals

    Sun XC and Hu X. Topological ring-cavity laser formed by honeyc omb photonic crystals. arXiv:1906.02464, 2019

  32. [32]

    Electrically pumped topological laser with valley edge 15 / 39 modes

    Zeng Y , Chattopadhyay U and Zhu B et al. Electrically pumped topological laser with valley edge 15 / 39 modes. Nature 2020; 578: 246–250

  33. [33]

    Backscattering in an o ptical passive ring-resonator gyro: experiment

    Iwatsuki K, Hotate K and Higashi guchi M. Backscattering in an o ptical passive ring-resonator gyro: experiment. Appl Opt 1986; 25: 4448-4451

  34. [34]

    Backscattering in silicon microring resonators: a quantitative analysis

    Li A, Van Vaerenbergh T and De Heyn P et al. Backscattering in silicon microring resonators: a quantitative analysis. Laser Photonics Rev 2016; 10: 420-431

  35. [35]

    Reduction of backscattering induced no ise by carrier suppression in waveguide-type optical ring resonator gyro

    Ma H, He Z and Hotate K. Reduction of backscattering induced no ise by carrier suppression in waveguide-type optical ring resonator gyro. J Light Technol 2011; 29, 85-90

  36. [36]

    Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators

    Romero-García V , Theocharis G and Richoux O et al. Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators. Sci Rep 2016; 6: 19519

  37. [37]

    Mitigation of reflection-induced crosstalk in a WDM access network

    Urban PJ, Koonen AMJ and Khoe GD et al. Mitigation of reflection-induced crosstalk in a WDM access network. In OFC/NFOEC 2008, San Diego, CA, paper OTuT3

  38. [38]

    Bidirectional crosstalk and back-reflection free WDM active optical interconnects

    Pintus P, Andriolli N and Di Pasquale et al. Bidirectional crosstalk and back-reflection free WDM active optical interconnects. IEEE Photonic Tech L 2013; 25: 1973-1976

  39. [39]

    Multiple reflections induced crosstalk in inline TDM fiber Fab ry- Perot sensor system utilizing phase generated carrier scheme

    Lin H, Ma L and Hu Z et al. Multiple reflections induced crosstalk in inline TDM fiber Fab ry- Perot sensor system utilizing phase generated carrier scheme. J Light Technol 2013; 31: 2651- 2658

  40. [40]

    Experimental demonstrati on of a unidirectional reflectionless parity-time metamaterial at optical frequencies

    Feng L, Xu YL and Fegadolli WS et al. Experimental demonstrati on of a unidirectional reflectionless parity-time metamaterial at optical frequencies. Nature Maters 2013; 12: 108-113

  41. [41]

    An all-silicon passive optical diode

    Fan L, Wang J and Varghese LT et al. An all-silicon passive optical diode. Science 2012; 335: 447- 450

  42. [42]

    Reciprocity in reflection an d transmission: What is a ‘phonon diode’? Wave Motion 2013; 50: 776-784

    Maznev AA, Every AG and Wright OB. Reciprocity in reflection an d transmission: What is a ‘phonon diode’? Wave Motion 2013; 50: 776-784

  43. [43]

    Spherical whispering‐galle ry‐mode microresonators

    Chiasera A, Dumeige Y and Feron P et al. Spherical whispering‐galle ry‐mode microresonators. Laser Photonics Rev 2010; 4: 457-482

  44. [44]

    Advances and prospects for whispering gallery mode microcavities

    Yang S, Wang Y and Sun H. Advances and prospects for whispering gallery mode microcavities. Adv Opt Mater 2015; 3: 1136-1162

  45. [45]

    Splitting of high-Q Mie modes induced by light backscattering in silica microspheres

    Weiss DS, Sandoghdar V and Hare J et al. Splitting of high-Q Mie modes induced by light backscattering in silica microspheres. Opt Lett 1995; 20: 1835-1837

  46. [46]

    Rayleigh scatterin g in high-Q microspheres

    Gorodetsky ML, Pryamikov AD and Ilchenko VS. Rayleigh scatterin g in high-Q microspheres. JOSA B 2000; 17: 1051-1057. 16 / 39

  47. [47]

    On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator

    Zhu J, Ozdemir SK and Xiao YF et al. On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator. Nature Photons 2010; 4: 46-49

  48. [48]

    Multiple-Rayleigh-scatterer-induced mode splitting in a high-Q whispering-gallery-mode microresonator

    Yi X, Xiao YF and Liu YC et al. Multiple-Rayleigh-scatterer-induced mode splitting in a high-Q whispering-gallery-mode microresonator. Phys Rev A 2011; 83: 023803

  49. [49]

    Spherical topological insulator

    Imura KI, Yoshimura Y , Takane Y et al. Spherical topological insulator. Phys Rev B 2012; 86: 235119

  50. [50]

    Topological photonics: From crystals to particles

    Siroki G, Huidobro PA and Gianni ni V . Topological photonics: From crystals to particles. Phys Rev B 2017; 96: 041408

  51. [51]

    Topological whispering gallery modes in two-dimensional photonic crystal cavities

    Yang Y and Hang ZH. Topological whispering gallery modes in two-dimensional photonic crystal cavities. Opt Express 2018; 26: 21235-21241

  52. [52]

    Selective engineering of cavity resonance for frequency matching in optical parametric processes

    Lu X, Rogers S and Jiang WC et al. Selective engineering of cavity resonance for frequency matching in optical parametric processes. Appl Phys Lett 2014; 105: 151104

  53. [53]

    Chip-integrated visible–telecom entangled photon pair source for quantum communication

    Lu X, Li Q and Westly DA et al. Chip-integrated visible–telecom entangled photon pair source for quantum communication. Nature Phys 2019; 15: 373-381

  54. [54]

    An eight-channel add-drop filte r using vertically coupled microring resonators over a cross grid

    Chu ST, Little BE and Pan W et al. An eight-channel add-drop filte r using vertically coupled microring resonators over a cross grid. IEEE Photonic Tech L 1999; 11: 691-693

  55. [55]

    Multistage high-order microring-resonator add-drop filters

    Popovic MA, Barwicz T and Watts MR et al. Multistage high-order microring-resonator add-drop filters. Optics letters 2006; 31: 2571-2573

  56. [56]

    Optical add-drop filters based on photonic crystal ring resonators

    Qiang Z, Zhou W and Soref RA. Optical add-drop filters based on photonic crystal ring resonators. Opt Express 2007; 15: 1823-1831

  57. [57]

    Integrated microwave photonics

    Marpaung D, Yao J and Capmany J. Integrated microwave photonics. Nature Photons 2019; 13: 80-90

  58. [58]

    AlN/3C–SiC composite plate enabling high‐ frequency and high‐Q micromechanical resonators

    Lin CM, Chen YY and Felmetsger VV et al . AlN/3C–SiC composite plate enabling high‐ frequency and high‐Q micromechanical resonators. Adv Mater 2012; 24: 2722-2727

  59. [59]

    Monolithic phononic crystals with a surface acoustic band gap from surface phonon-polariton coupling

    Yudistira D, Boes A and Djafari-Rouhani B et al . Monolithic phononic crystals with a surface acoustic band gap from surface phonon-polariton coupling. Phys Rev Lett 2014; 113: 215503

  60. [60]

    5 GHz laterally-excited bulk-wave resonators (XBARs) based on thin platelets of lithium niobate

    Plessky V , Yandrapalli S and Turner PJ et al . 5 GHz laterally-excited bulk-wave resonators (XBARs) based on thin platelets of lithium niobate. Electron Lett 2018; 55: 98-100

  61. [61]

    Coherent coupling between radiofrequency, optical and acoustic waves in piezo-optomechanical circuits

    Balram KC, Davanço MI and Song JD et al. Coherent coupling between radiofrequency, optical and acoustic waves in piezo-optomechanical circuits. Nature Photons 2016; 10, 346-352. 17 / 39

  62. [62]

    Phononic band structure engine ering for high-Q gigahertz surface acoustic wave resonators on lithium niobate

    Shao L, Maity S and Zheng L et al . Phononic band structure engine ering for high-Q gigahertz surface acoustic wave resonators on lithium niobate. Phys Rev Appl 2019; 12: 014022

  63. [63]

    Resolving the energy levels of a nanomechanical oscillator

    Arrangoiz-Arriola P, Wollack EA and Wang Z et al . Resolving the energy levels of a nanomechanical oscillator. Nature 2019; 571: 537-540

  64. [64]

    Electrical tuning of elastic wave propagati on in nanomechanical lattices at MHz frequencies

    Cha J and Daraio C. Electrical tuning of elastic wave propagati on in nanomechanical lattices at MHz frequencies. Nature Nano 2018; 13: 1016-1020

  65. [65]

    Ghadimi AH, Fedorov SA and Engelsen NJ. et al . Elastic strain engine ering for ultralow mechanical dissipation. Science 2018; 360: 764-768

  66. [66]

    Phononic bandgap nano-acoustic cavity with ultralong phonon lifetime

    MacCabe GS, Ren H and Luo J et al . Phononic bandgap nano-acoustic cavity with ultralong phonon lifetime. arXiv:1901.04129, 2019

  67. [67]

    Phononic integrated circuitry and spin–orbit interaction of phonons

    Fu W, Shen Z and Xu Y et al. Phononic integrated circuitry and spin–orbit interaction of phonons. Nat Commun 2019; 10: 2743

  68. [68]

    Safavi-Naeini AH, Van Thourhout D and Baets R et al. R. Controlling phonons and photons at the wavelength scale: integrated photonics meets integrated phononics. Optica 2019; 6: 213-232

  69. [69]

    PT-symmetric phonon laser

    Jing H, Özdemir SK and Lü XY et al. PT-symmetric phonon laser. Phys Rev lett 2014; 113: 053604

  70. [70]

    A phonon laser operating at an exceptional point

    Zhang J, Peng B and Özdemir ŞK et al. A phonon laser operating at an exceptional point. Nature Photons 2018; 12: 479-484

  71. [71]

    Surface acoustic wave devices in telecommunications: modelling and simulation

    Hashimoto KY . Surface acoustic wave devices in telecommunications: modelling and simulation. Berlin: Springer-Verlag, 2000

  72. [72]

    The 2019 surface acoustic waves roadmap

    Delsing P, Cleland AN and Schuetz MJ et al. The 2019 surface acoustic waves roadmap. J Phys D Appl Phys 2019; 52: 353001

  73. [73]

    Quantum acoustics with superconducting qubits

    Chu Y , Kharel P and Renninger WH et al . Quantum acoustics with superconducting qubits. Science 2017; 358: 199-202

  74. [74]

    Experimental observation of Weyl points

    Lu L, Wang Z, and Ye D et al. Experimental observation of Weyl points. Science 2015; 349: 622- 624

  75. [75]

    Weyl points and Fermi arcs in a chiral phononic crystal

    Li F, Huang X and Lu J et al. Weyl points and Fermi arcs in a chiral phononic crystal. Nature Phys 2018; 14: 30-34

  76. [76]

    Ideal Weyl points and helicoid surface states in artificial photonic crystal structures

    Yang B, Guo Q and Tremain B et al . Ideal Weyl points and helicoid surface states in artificial photonic crystal structures. Science 2018; 359: 1013-1016

  77. [77]

    Symmetry-protected transport in a pseudospin-polarized 18 / 39 waveguide

    Chen WJ, Zhang ZQ and Dong JW et al. Symmetry-protected transport in a pseudospin-polarized 18 / 39 waveguide. Nat Commun 2015; 6: 8183

  78. [78]

    P· T· D symmetry-protected scattering anomaly in optics

    Silveirinha MG. P· T· D symmetry-protected scattering anomaly in optics. Phys Rev B 2017; 95: 035153

  79. [79]

    …, 3 (3'), 2 (2'), 1 (1'), 0 (0'), …

    Dia’aaldin JB and Sievenpiper DF. Guiding waves along an infinitesimal line between impedance surfaces. Phys Rev Lett 2017; 119: 106802. 19 / 39 Figure 1 | TI “ring” resonator (i.e., TI cavity) and its paired eigenstates of two categories a: Schematic of an elastic topol ogically protected boundary comp rising an elastic TI adjacent to an elastic OI. b: C...