Abstract

We investigate the topological phase transition between Type-I and Type-II Weyl points (WPs) in a composite three-dimensional lattice composed of a two-dimensional brick-wall waveguide array and a synthetic frequency dimension created by dynamic modulation. By imposing different modulation amplitudes and phases in the two sublattices, we can break either parity or time-reversal symmetry and realize the phase transition between Type-I and Type-II WPs. As the array is truncated to have two edges, two Fermi-arc surface states will emerge, which propagate in opposite directions for Type-I WPs while in same directions for Type-II WPs, accompanied by bidirectional and unidirectional frequency shifts for the optical modes. Particularly at the phase transition point, we find that one of two bands becomes flat with a vanished group velocity along frequency axis in the vicinity of WPs. The study paves a way towards realizing different topological phases in the same photonic structure, which offers new opportunities to control wave transportation both in spatial and frequency domains.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]

2018 (8)

N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyl and Dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. 90(1), 015001 (2018).
[Crossref]

B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018).
[Crossref] [PubMed]

B. Dóra and R. Moessner, “Gauge field entanglement in Kitaev’s honeycomb model,” Phys. Rev. B 97(3), 035109 (2018).
[Crossref]

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum Control through Discrete Frequency Diffraction in the Presence of Photonic Gauge Potentials,” Phys. Rev. Lett. 120(13), 133901 (2018).
[Crossref] [PubMed]

C. Qin, L. Yuan, B. Wang, S. Fan, and P. Lu, “Effective electric-field force for a photon in a synthetic frequency lattice created in a waveguide modulator,” Phys. Rev. A 97(6), 063838 (2018).
[Crossref]

S. Ke, D. Zhao, Q. Liu, S. Wu, B. Wang, and P. Lu, “Optical Imaginary Directional Couplers,” J. Lightwave Technol. 36(12), 2510–2516 (2018).
[Crossref]

S. Wang, C. Qin, B. Wang, and P. Lu, “Discrete temporal Talbot effect in synthetic mesh lattices,” Opt. Express 26(15), 19235 (2018).
[Crossref]

F. Wang, S. Ke, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Topological interface modes in graphene multilayer arrays,” Opt. Laser Technol. 103, 272–278 (2018).
[Crossref]

2017 (7)

F. Li, X. Huang, J. Lu, J. Ma, and Z. Liu, “Weyl points and Fermi arcs in a chiral phononic crystal,” Nat. Phys. 14(1), 30–34 (2017).
[Crossref]

G. G. Pyrialakos, N. S. Nye, N. V. Kantartzis, and D. N. Christodoulides, “Emergence of Type-II Dirac Points in Graphynelike Photonic Lattices,” Phys. Rev. Lett. 119(11), 113901 (2017).
[Crossref] [PubMed]

Y. Zhang and Y. Zhu, “Generation of Weyl points in coupled optical microdisk-resonator arrays via external modulation,” Phys. Rev. A 96(1), 013811 (2017).
[Crossref]

Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, “Optical Interface States Protected by Synthetic Weyl Points,” Phys. Rev. X 7(3), 031032 (2017).
[Crossref]

J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, “Experimental observation of optical Weyl points and Fermi arc-like surface states,” Nat. Phys. 13(6), 611–617 (2017).
[Crossref]

B. Yang, Q. Guo, B. Tremain, L. E. Barr, W. Gao, H. Liu, B. Béri, Y. Xiang, D. Fan, A. P. Hibbins, and S. Zhang, “Direct observation of topological surface-state arcs in photonic metamaterials,” Nat. Commun. 8(1), 97 (2017).
[Crossref] [PubMed]

B. A. Bell, K. Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, “Spectral photonic lattices with complex long-range coupling,” Optica 4(11), 1433 (2017).
[Crossref]

2016 (10)

S. Tchoumakov, M. Civelli, and M. O. Goerbig, “Magnetic-Field-Induced Relativistic Properties in Type-I and Type-II Weyl Semimetals,” Phys. Rev. Lett. 117(8), 086402 (2016).
[Crossref] [PubMed]

M. Udagawa and E. J. Bergholtz, “Field-Selective Anomaly and Chiral Mode Reversal in Type-II Weyl Materials,” Phys. Rev. Lett. 117(8), 086401 (2016).
[Crossref] [PubMed]

M. Xiao, Q. Lin, and S. Fan, “Hyperbolic Weyl Point in Reciprocal Chiral Metamaterials,” Phys. Rev. Lett. 117(5), 057401 (2016).
[Crossref] [PubMed]

H. Zheng, G. Bian, G. Chang, H. Lu, S. Y. Xu, G. Wang, T. R. Chang, S. Zhang, I. Belopolski, N. Alidoust, D. S. Sanchez, F. Song, H. T. Jeng, N. Yao, A. Bansil, S. Jia, H. Lin, and M. Z. Hasan, “Atomic-Scale Visualization of Quasiparticle Interference on a Type-II Weyl Semimetal Surface,” Phys. Rev. Lett. 117(26), 266804 (2016).
[Crossref] [PubMed]

N. Goldman, G. Jotzu, M. Messer, F. Görg, R. Desbuquois, and T. Esslinger, “Creating topological interfaces and detecting chiral edge modes in a two-dimensional optical lattice,” Phys. Rev. A 94(4), 043611 (2016).
[Crossref]

Z. Yang and B. Zhang, “Acoustic Type-II Weyl Nodes from Stacking Dimerized Chains,” Phys. Rev. Lett. 117(22), 224301 (2016).
[Crossref] [PubMed]

Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7, 13731 (2016).
[Crossref] [PubMed]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

L. Yuan, Y. Shi, and S. Fan, “Photonic gauge potential in a system with a synthetic frequency dimension,” Opt. Lett. 41(4), 741–744 (2016).
[Crossref] [PubMed]

L. Yuan and S. Fan, “Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator,” Optica 3(9), 1014–1018 (2016).
[Crossref]

2015 (11)

Y. Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan, “Prediction of Weyl semimetal in orthorhombicMoTe2,” Phys. Rev. B 92(16), 161107 (2015).
[Crossref]

J. M. Hou and W. Chen, “Hidden symmetry and protection of Dirac points on the honeycomb lattice,” Sci. Rep. 5(1), 17571 (2015).
[Crossref] [PubMed]

A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, “Type-II Weyl semimetals,” Nature 527(7579), 495–498 (2015).
[Crossref] [PubMed]

M. Trescher, B. Sbierski, P. W. Brouwer, and E. J. Bergholtz, “Quantum transport in Dirac materials: Signatures of tilted and anisotropic Dirac and Weyl cones,” Phys. Rev. B 91(11), 115135 (2015).
[Crossref]

S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
[Crossref] [PubMed]

H. M. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, “Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides,” Phys. Rev. X 5(1), 011029 (2015).
[Crossref]

B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental Discovery of Weyl Semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015).
[Crossref]

B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015).
[Crossref]

X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, “Observation of the Chiral-Anomaly-Induced Negative Magnetoresistance in 3D Weyl Semimetal TaAs,” Phys. Rev. X 5(3), 031023 (2015).
[Crossref]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

M. Xiao, W.-J. Chen, W.-Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015).
[Crossref]

2014 (2)

M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop, T. Liang, N. Haldolaarachchige, M. Hirschberger, N. P. Ong, and R. J. Cava, “Large, non-saturating magnetoresistance in WTe2,” Nature 514(7521), 205–208 (2014).
[Crossref] [PubMed]

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
[Crossref]

2013 (1)

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013).
[Crossref]

2009 (1)

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

2008 (1)

H. V. Pham, H. Murata, and Y. Okamura, “Electrooptic Modulators with Controlled Frequency Responses by Using Nonperiodically Polarization-Reversed Structure,” Adv. Optoelectron. 2008, 1–8 (2008).
[Crossref]

2007 (1)

X. Y. Feng, G. M. Zhang, and T. Xiang, “Topological characterization of quantum phase transitions in a spin-1/2 model,” Phys. Rev. Lett. 98(8), 087204 (2007).
[Crossref] [PubMed]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Ali, M. N.

Y. Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan, “Prediction of Weyl semimetal in orthorhombicMoTe2,” Phys. Rev. B 92(16), 161107 (2015).
[Crossref]

M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop, T. Liang, N. Haldolaarachchige, M. Hirschberger, N. P. Ong, and R. J. Cava, “Large, non-saturating magnetoresistance in WTe2,” Nature 514(7521), 205–208 (2014).
[Crossref] [PubMed]

Alidoust, N.

H. Zheng, G. Bian, G. Chang, H. Lu, S. Y. Xu, G. Wang, T. R. Chang, S. Zhang, I. Belopolski, N. Alidoust, D. S. Sanchez, F. Song, H. T. Jeng, N. Yao, A. Bansil, S. Jia, H. Lin, and M. Z. Hasan, “Atomic-Scale Visualization of Quasiparticle Interference on a Type-II Weyl Semimetal Surface,” Phys. Rev. Lett. 117(26), 266804 (2016).
[Crossref] [PubMed]

S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
[Crossref] [PubMed]

Alù, A.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum Control through Discrete Frequency Diffraction in the Presence of Photonic Gauge Potentials,” Phys. Rev. Lett. 120(13), 133901 (2018).
[Crossref] [PubMed]

Armitage, N. P.

N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyl and Dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. 90(1), 015001 (2018).
[Crossref]

Bansil, A.

H. Zheng, G. Bian, G. Chang, H. Lu, S. Y. Xu, G. Wang, T. R. Chang, S. Zhang, I. Belopolski, N. Alidoust, D. S. Sanchez, F. Song, H. T. Jeng, N. Yao, A. Bansil, S. Jia, H. Lin, and M. Z. Hasan, “Atomic-Scale Visualization of Quasiparticle Interference on a Type-II Weyl Semimetal Surface,” Phys. Rev. Lett. 117(26), 266804 (2016).
[Crossref] [PubMed]

S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
[Crossref] [PubMed]

Barr, L. E.

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Liu, R.

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F. Wang, S. Ke, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Topological interface modes in graphene multilayer arrays,” Opt. Laser Technol. 103, 272–278 (2018).
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X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, “Observation of the Chiral-Anomaly-Induced Negative Magnetoresistance in 3D Weyl Semimetal TaAs,” Phys. Rev. X 5(3), 031023 (2015).
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F. Li, X. Huang, J. Lu, J. Ma, and Z. Liu, “Weyl points and Fermi arcs in a chiral phononic crystal,” Nat. Phys. 14(1), 30–34 (2017).
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B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015).
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B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental Discovery of Weyl Semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015).
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F. Li, X. Huang, J. Lu, J. Ma, and Z. Liu, “Weyl points and Fermi arcs in a chiral phononic crystal,” Nat. Phys. 14(1), 30–34 (2017).
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B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental Discovery of Weyl Semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015).
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S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
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N. Goldman, G. Jotzu, M. Messer, F. Görg, R. Desbuquois, and T. Esslinger, “Creating topological interfaces and detecting chiral edge modes in a two-dimensional optical lattice,” Phys. Rev. A 94(4), 043611 (2016).
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Neupane, M.

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J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, “Experimental observation of optical Weyl points and Fermi arc-like surface states,” Nat. Phys. 13(6), 611–617 (2017).
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L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
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G. G. Pyrialakos, N. S. Nye, N. V. Kantartzis, and D. N. Christodoulides, “Emergence of Type-II Dirac Points in Graphynelike Photonic Lattices,” Phys. Rev. Lett. 119(11), 113901 (2017).
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M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop, T. Liang, N. Haldolaarachchige, M. Hirschberger, N. P. Ong, and R. J. Cava, “Large, non-saturating magnetoresistance in WTe2,” Nature 514(7521), 205–208 (2014).
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Peng, Y.

C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum Control through Discrete Frequency Diffraction in the Presence of Photonic Gauge Potentials,” Phys. Rev. Lett. 120(13), 133901 (2018).
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Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
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H. V. Pham, H. Murata, and Y. Okamura, “Electrooptic Modulators with Controlled Frequency Responses by Using Nonperiodically Polarization-Reversed Structure,” Adv. Optoelectron. 2008, 1–8 (2008).
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G. G. Pyrialakos, N. S. Nye, N. V. Kantartzis, and D. N. Christodoulides, “Emergence of Type-II Dirac Points in Graphynelike Photonic Lattices,” Phys. Rev. Lett. 119(11), 113901 (2017).
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B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental Discovery of Weyl Semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015).
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C. Qin, F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alù, and P. Lu, “Spectrum Control through Discrete Frequency Diffraction in the Presence of Photonic Gauge Potentials,” Phys. Rev. Lett. 120(13), 133901 (2018).
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C. Qin, L. Yuan, B. Wang, S. Fan, and P. Lu, “Effective electric-field force for a photon in a synthetic frequency lattice created in a waveguide modulator,” Phys. Rev. A 97(6), 063838 (2018).
[Crossref]

F. Wang, S. Ke, C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Topological interface modes in graphene multilayer arrays,” Opt. Laser Technol. 103, 272–278 (2018).
[Crossref]

S. Wang, C. Qin, B. Wang, and P. Lu, “Discrete temporal Talbot effect in synthetic mesh lattices,” Opt. Express 26(15), 19235 (2018).
[Crossref]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

Ran, L.

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

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J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, “Experimental observation of optical Weyl points and Fermi arc-like surface states,” Nat. Phys. 13(6), 611–617 (2017).
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B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015).
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B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental Discovery of Weyl Semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015).
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Sanchez, D. S.

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Figures (9)

Fig. 1
Fig. 1 (a) Schematic diagram of a dynamically modulated brick-wall waveguide array consisting of evanescently coupled waveguide A (red) and B (blue) arranged in a square lattice. One of the four coupling bonds is blocked (denoted by the green dashed arrow). Both A and B waveguides are subject to the travelling-wave index modulation n1(2)(z, t) with phase difference Δϕ = ϕ2ϕ1. (b) Equivalent lattice model which consists of the in-plane brick-wall lattice and perpendicular frequency lattice, with lattice constants a and Ω. The in-plane coupling coefficients Jx and Jy have real values and frequency-domain coupling coefficients are complex values with amplitudes Jω1 (Jω2) and phases ± ϕ1 ( ± ϕ2) in waveguides A and B. (c) The Brillouin zone (mark in blue) for the equivalent lattice of (b), with kx, ky ∈ (− π/a, π/a) and kω ∈ (− π/Ω, π/Ω). (d) Projected band structure ε(kx, ky) for the brick-wall lattice in the absence of dynamic modulation. D1, D2 denotes two Dirac points by choosing a = 1 and Jx = Jy.
Fig. 2
Fig. 2 Phase diagram for Type-I and Type-II Weyl points versus the modulation phase difference Δϕ = ϕ2ϕ1 and coupling strength ratio Jω2/Jω1. In the simulation, we fix Jω2/Jω1 = 0.5 as denoted by the black dashed line. The red and blue circles represent the situations of in-phase (Δϕ = 0) and out-of-phase modulations (Δϕ = π), respectively. Black circle denotes Δϕ = 0.383π, which denotes the transition point between Type-I and Type-II WPs.
Fig. 3
Fig. 3 Projected band structures and isoenergy contours for Type-I WPs under Δϕ = π. (a) Projected band structure ε(kx, kω) at ky0 = 0 with W1, W2, W3, and W4 locating at ( ± 2π/3, ± π/2). The parameters are a = Ω = 1, Jx = Jy = Jω1 = 1 and Jω2 = 0.5. (b) (c) Isoenergy contours for the upper and lower bands of the projected band structure of (a). The signs “+” and “−” denote the sign of the Chern number for each WP. (b) Projected band structure ε(ky, kω) at kx0 = 2π/3 with W1(2), W4(3) locating at (0, ± π/2). (e) (f) Isoenergy contours for the upper and lower bands.
Fig. 4
Fig. 4 Projected band structures and isoenergy contours for Type-II WPs under Δϕ = 0. All other parameters are kept the same with those in Fig. 3. (a) Projected band structure ε(kx, kω) at ky0 = 0 with four WPs W1, W2, W3, and W4 locating at ( ± 2π/3, ± π/2). (b) (c) Isoenergy contours for the upper and lower bands, with the signs “+” and “−” at each WP denotes its chirality. (b) Projected band structure ε(ky, kω) at kx0 = 2π/3 with W1(2), W3(4) locating at (0, ± π/2). (e) (f) Isoenergy contours for the upper and lower bands of the projected band structure in (d).
Fig. 5
Fig. 5 Projected band structures and isoenergy contours at the phase transition point of Δϕ = 0.383π. (a) Projected band structure ε(kx, kω) at ky0 = 0 with the four WPs locating at kω0 = 0.335π and − 0.665π. (b) (c) Isoenergy contours where the signs “+” and “−” at each WP denotes its chirality. (b) Projected band structure ε(ky, kω) at kx0 = 2π/3 with W1(2), W3(4) locating at kω0 = 0.335π and − 0.665π at ε0 = ± 0.989, respectively. (e) (f) Isoenergy contours for the upper and lower bands of the projected band structure in (d).
Fig. 6
Fig. 6 Schematic diagrams of the A-A (left) and A-B (right) truncated lattice structures. The lattice is truncated along ( x y ) direction and kept periodic along ( x + y ) direction and the frequency dimension. The orange dashed rectangles denote a unit cell for the truncated lattice, which contains odd and even number of waveguides for A-A and A-B lattices, respectively. The black dashed squares denote the primitive cell for the in-plane brick-wall lattice.
Fig. 7
Fig. 7 (a) Projected band structure of Type-I WPs for the A-B truncated array under Δϕ = π. The two surface states are denoted by the green surfaces and the red (blue) surfaces are the bulk bands. The red circles represent the four WPs connected by the Fermi arcs denoted by the red lines. (b) A sliced projected band structure at kp = 5π/6 with the black arrows denoting the group velocities of the two surface states. (c) Projected band structure of Type-II WPs under Δϕ = 0. (d) Sliced projected band structure at kp = 5π/6. (e) Projected band structure of Type-II WPs with Δϕ = 0.383 π. (f) Sliced projected band structure at kp = 5π/6. (g) (h) Eigen-mode amplitudes for the two surface states as kp = 3π/4 and 5π/6 under Δϕ = π and kω = π/2.
Fig. 8
Fig. 8 (a) Projected band structure of Type-I WPs for A-A type array under Δϕ = π. The green surface denotes the only surface state and the red (blue) surfaces denote the bulk bands. (b) The sliced projected band structure at kp = 5π/6. (c) (d) 2D Projected band structure for Type-II WPs with Δϕ = 0 and sliced projected band structure at kp = 5π/6. (e) (f) 2D projected band structure at Δϕ = 0.383π and sliced projected band structure at kp = 5π/6. (g) (h) Eigen-mode amplitudes for the only surface state as kp = 3π/4 and 5π/6 under Δϕ = π and kω = π/2.
Fig. 9
Fig. 9 (a) Schematic diagram of a unit cell for the brick-wall waveguide array. The refractive indices of the waveguide and the cladding medium are n0 = 1.45 and n1 = 1.2, respectively. The metallic slab is composed of Au, which is used to block the coupling between waveguide A and the upper waveguide B. (b) Simulated field distribution at different sections at z = 0, 1 and 4 μm under the single mode input from the central A waveguide.

Equations (31)

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n 1 ( 2 ) ( z , t ) = n 0 + Δ n 1 ( 2 ) cos [ Ω t q z + ϕ 1 ( 2 ) ] ,
H = i , n ( J x a r i , n b r i + e 1 , n + J x a r i , n b r i + e 2 , n + J y a r i , n b r i + e 3 , n + h . c . ) + i , n ( J ω 1 e i ϕ 1 a r i , n a r i , n + 1 + h . c . ) + j , n ( J ω 2 e i ϕ 2 b r j , n b r j , n + 1 + h . c . ) ,
H ( k ) = [ 2 J x cos ( k x a ) + J y cos ( k y a ) ] σ x + J y sin ( k y a ) σ y + [ J ω 1 cos ( k ω Ω ) J ω 2 cos ( k ω Ω Δ ϕ ) ] σ z + [ J ω 1 cos ( k ω Ω ) + J ω 2 cos ( k ω Ω Δ ϕ ) ] σ 0 ,
H 2 D ( k ) = [ 2 J x cos ( k x a ) + J y cos ( k y a ) ] σ x + J y sin ( k y a ) σ y ,
H 2 D ( q ) = [ 2 J x cos ( k x 0 a ) + J y ] σ x 2 J x sin ( k x 0 a ) q x a σ x + J y q y a σ y ,
H 2 D ( q ) = ± v x q x σ x + v y q y σ y ,
{ v x = 4 J x 2 J y 2 a v y = J y a
H ( k ) = H 2 D ( k ) + { cos ( k ω Ω ) ( J ω 1 J ω 2 ) ( σ z + τ σ 0 ) , ( Δ ϕ = 0 ) cos ( k ω Ω ) ( J ω 1 + J ω 2 ) ( σ z + 1 τ σ 0 ) , ( Δ ϕ = π )
H W ( q ) = ± v x q x σ x + v y q y σ y ± { v ω , 0 q ω ( σ z + τ σ 0 ) , ( Δ ϕ = 0 ) v ω , π q ω ( σ z + 1 τ σ 0 ) , ( Δ ϕ = π )
H ( k ) = [ 2 J x cos ( k x a ) + J y cos ( k y a ) ] σ x + J y sin ( k y a ) σ y J ω 1 2 2 J ω 1 J ω 2 cos ( Δ ϕ ) + J ω 2 2 sin ( k ω Ω α ) σ z + J ω 1 2 + 2 J ω 1 J ω 2 cos ( Δ ϕ ) + J ω 2 2 sin ( k ω Ω + β ) σ 0 ,
tan ( α ) = J ω 1 J ω 2 cos ( Δ ϕ ) J ω 2 sin ( Δ ϕ ) , tan ( β ) = J ω 1 + J ω 2 cos ( Δ ϕ ) J ω 2 sin ( Δ ϕ ) ,
H W 1 ( q ) = v x q x σ x + v y q y σ y + v ω q ω σ z + ε 0 σ 0 + v 0 q ω σ 0 + ε 0 σ 0 [ 1 2 q ω 2 Ω 2 + o ( q ω 2 Ω 2 ) ] ,
{ v ω = J ω 1 2 2 J ω 1 J ω 2 cos ( Δ ϕ ) + J ω 2 2 Ω v 0 = cos ( α + β ) J ω 1 2 + 2 J ω 1 J ω 2 cos ( Δ ϕ ) + J ω 2 2 Ω ε 0 = sin ( α + β ) J ω 1 2 + 2 J ω 1 J ω 2 cos ( Δ ϕ ) + J ω 2 2 ,
ε ± ( q ) = ε 0 + v 0 q ω + ε 0 [ 1 2 q ω 2 Ω 2 + o ( q ω 2 Ω 2 ) ] ± v x 2 q x 2 + v y 2 q y 2 + v ω 2 q ω 2 ,
{ H U ( q ) = v x q x σ x + v y q y σ y + v ω q ω σ z H T ( q ) = v 0 q ω σ 0 + ε 0 σ 0 [ 1 2 q ω 2 Ω 2 + o ( q ω 2 Ω 2 ) ]
{ U ( q ) = v x 2 q x 2 + v y 2 q y 2 + v ω 2 q ω 2 T ( q ) = ε ± ( q ) ε 0 U ( q )
v g , ± ω ( q ) = ε ± ( q ) q ω = v 0 + ε 0 [ q ω Ω 2 + o ( q ω Ω 2 ) ] ± q ω v ω 2 v x 2 q x 2 + v y 2 q y 2 + v ω 2 q ω 2 ,
v g , ± ω ( 0 , 0 , q ω ) = v 0 + ε 0 [ q ω Ω 2 + o ( q ω Ω 2 ) ] ± v ω ,
{ v g , + ω ( 0 , 0 , q ω 0 ) = v 0 + v ω v g , ω ( 0 , 0 , q ω 0 ) = 0 ,
H AB = p , n ( J x a 1 , p , n b 1 , p + 1 , n + J y a 1 , p , n b 1 , p , n + h . c . ) + m = 2 , p , n M ( J x a m , p , n b m , p + 1 , n + J x a m , p , n b m 1 , p , n + J y a m , p , n b m , p , n + h . c . ) + m = 1 , p , n M ( J ω 1 e i ϕ 1 a m , p , n a m , p , n + 1 + J ω 2 e i ϕ 2 b m , p , n b m , p , n + 1 + h . c . ) ,
H AB = k ( J x e i k p 2 a + J y ) a 1 , k b 1 , k + k ( J x e i k p 2 a + J y ) b 1 , k a 1 , k + m = 2 , k M [ ( J x e i k p 2 a + J y ) a m , k b m , k + ( J x e i k p 2 a + J y ) b m , k a m , k + J x a m , k b m 1 , k + J x b m 1 , k a m , k ] + m = 1 , k M [ ( J ω 1 e i ϕ 1 e i k ω Ω + J ω 1 e i ϕ 1 e i k ω Ω ) a m , k a m , k + ( J ω 2 e i ϕ 2 e i k ω Ω + J ω 2 e i ϕ 2 e i k ω Ω ) b m , k b m , k ] ,
H AB = ( a 1 , k b 1 , k a 2 , k b 2 , k ... a m , k b m , k ) H AB ( k ) ( a 1 , k b 1 , k a 2 , k b 2 , k ... a m , k b m , k ) T ,
H AB ( k ) = ( 2 J ω 1 cos ( k ω Ω ϕ 1 ) J x e i k p 2 a + J y 0 0 0 J x e i k p 2 a + J y 2 J ω 2 cos ( k ω Ω ϕ 2 ) J x 0 0 0 J x 2 J ω 1 cos ( k ω Ω ϕ 1 ) 0 0 0 0 0 0 0 2 J ω 1 cos ( k ω Ω ϕ 1 ) J x e i k p 2 a + J y 0 0 0 J x e i k p 2 a + J y 2 J ω 2 cos ( k ω Ω ϕ 2 ) )
2 E ( x , y , z , t ) ε d c 2 2 t 2 E ( x , y , z , t ) = Δ ε c 2 2 t 2 [ cos ( Ω t q z + ϕ ) E ( x , y , z , t ) ] ,
n 2 i β n a n ( z ) z ψ n ( x , y ) e i ( ω n t β n z ) ,
2 t 2 ( 1 2 ( e i [ Ω t q z + ϕ ] + e i [ Ω t q z + ϕ ] ) n a n ( z ) ψ n ( x , y ) e i ( ω n t β n z ) ) = 1 2 n a n ( z ) ψ n ( x , y ) ( ω n + 1 2 e i ( ω n + 1 t β n + 1 z ) e i ϕ + ω n 1 2 e i ( ω n 1 t β n 1 z ) e i ϕ ) ,
n a n ( z ) ψ n ( x , y ) ω n ± 1 2 e i ( ω n ± 1 t β n ± 1 z ) = n a n 1 ( z ) ψ n 1 ( x , y ) ω n 2 e i ( ω n t β n z ) ,
i a n ( z ) z = J ω ( e i ϕ a n 1 ( z ) + e i ϕ a n + 1 ( z ) ) ,
J ω = Δ ε ω n 2 4 β n c 2 = Δ n β 0 2 n 0 ,
H = n J ω ( e i ϕ a ^ n a ^ n + 1 + h . c . ) ,
H = n ( J ω 1 e i ϕ 1 a ^ n a ^ n + 1 + J ω 2 e i ϕ 2 b ^ n b ^ n + 1 + h . c . ) .

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