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2020-11-30

One-Way Topological States Along Vague Boundaries in Synthetic Frequency Dimensions Including Group Velocity Dispersion (Invited)

By Qingrou Shan, Danying Yu, Guangzhen Li, Luqi Yuan, and Xianfeng Chen
Progress In Electromagnetics Research, Vol. 169, 33-43, 2020
doi:10.2528/PIER20083101

Abstract

We recently proposed a two-dimensional synthetic space including one spatial axis and one synthetic frequency dimension in a one-dimensional ring resonator array [Opt. Lett. 41, 741 (2016)]. Nevertheless, the group velocity dispersion (GVD) of the waveguides that compose rings was ignored for simplicity. In this paper, we extend the previous work and study the topological one-way edge states in such a synthetic space involving GVD. We show that the GVD brings a natural vague boundary in the frequency dimension, so the topological edge state still propagates at several frequency modes unidirectionally along the spatial axis. Positions of such vague boundary can be controlled by changing the magnitude of the GVD. In particular, a relatively strong GVD can degrade this two-dimensional synthetic space to one-dimensional spatial lattice, but yet the one-way state is still preserved in simulations. Our work therefore exhibits the impact of the GVD on topological photonics in the synthetic space, which will be important for future practical experimental implementations.

Citation


Qingrou Shan, Danying Yu, Guangzhen Li, Luqi Yuan, and Xianfeng Chen, "One-Way Topological States Along Vague Boundaries in Synthetic Frequency Dimensions Including Group Velocity Dispersion (Invited)," Progress In Electromagnetics Research, Vol. 169, 33-43, 2020.
doi:10.2528/PIER20083101
http://jpier.org/PIER/pier.php?paper=20083101

References


    1. Lu, L., J. D. Joannopoulos, and M. Soljacic, "Topological photonics," Nature Photonics, Vol. 8, No. 11, 821-829, 2014.
    doi:10.1038/nphoton.2014.248

    2. Khanikaev, A. B. and G. Shvets, "Two-dimensional topological photonics," Nature Photonics, Vol. 11, No. 12, 763-773, 2017.
    doi:10.1038/s41566-017-0048-5

    3. Ozawa, T., H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, "Topological photonics," Reviews of Modern Physics, Vol. 91, No. 1, 015006, 2019.
    doi:10.1103/RevModPhys.91.015006

    4. Leykam, D. and L. Yuan, "Topological phases in ring resonators: Recent progress and future prospects," Nanophotonics, Vol. 9, No. 15, 4473, 2020.
    doi:10.1515/nanoph-2020-0415

    5. Kraus, Y. E., Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, "Topological states and adiabatic pumping in quasicrystals," Physical Review Letters, Vol. 109, No. 10, 106402, 2012.
    doi:10.1103/PhysRevLett.109.106402

    6. Rechtsman, M. C., J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, "Photonic Floquet topological insulators," Nature, Vol. 496, No. 7444, 196-200, 2013.
    doi:10.1038/nature12066

    7. Weimann, S., M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, "Topologically protected bound states in photonic parity-time-symmetric crystals," Nature Materials, Vol. 16, No. 4, 433-438, 2017.
    doi:10.1038/nmat4811

    8. Noh, J., S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, "Experimental observation of opticalWeyl points and Fermi arc-like surface states," Nature Physics, Vol. 13, No. 6, 611-617, 2017.
    doi:10.1038/nphys4072

    9. Stutzer, S., Y. Plotnik, Y. Lumer, P. Titum, N. H. Lindner, M. Segev, M. C. Rechtsman, and A. Szameit, "Photonic topological Anderson insulators," Nature, Vol. 560, No. 7719, 461-465, 2018.
    doi:10.1038/s41586-018-0418-2

    10. Noh, J., S. Huang, K. P. Chen, and M. C. Rechtsman, "Observation of photonic topological valley hall edge states," Physical Review Letters, Vol. 120, No. 6, 063902, 2018.
    doi:10.1103/PhysRevLett.120.063902

    11. Hafezi, M., S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, "Imaging topological edge states in silicon photonics," Nature Photonics, Vol. 7, No. 12, 1001-1005, 2013.
    doi:10.1038/nphoton.2013.274

    12. Bandres, M. A., S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, "Topological insulator laser: Experiments," Science, Vol. 359, No. 6381, eaar4005, 2018.
    doi:10.1126/science.aar4005

    13. Leykam, D., S. Mittal, M. Hafezi, and Y. D. Chong, "Recon¯gurable topological phases in next-nearest-neighbor coupled resonator lattices," Physical Review Letters, Vol. 121, No. 2, 023901, 2018.
    doi:10.1103/PhysRevLett.121.023901

    14. Mittal, S., V. V. Orre, G. Zhu, M. A. Gorlach, A. Poddubny, and M. Hafezi, "Photonic quadrupole topological phases," Nature Photonics, Vol. 13, No. 10, 692-696, 2019.
    doi:10.1038/s41566-019-0452-0

    15. Mittal, S., V. V. Orre, D. Leykam, Y. D. Chong, and M. Hafezi, "Photonic anomalous quantum Hall effect," Physical Review Letters, Vol. 123, No. 4, 043201, 2019.
    doi:10.1103/PhysRevLett.123.043201

    16. Wang, Z., Y. Chong, J. D. Joannopoulos, and M. Soljacic, "Observation of unidirectional backscattering-immune topological electromagnetic states," Nature, Vol. 461, No. 7265, 772-775, 2009.
    doi:10.1038/nature08293

    17. Lu, L., Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljacic, "Experimental observation of Weyl points," Science, Vol. 349, No. 6248, 622, 2015.
    doi:10.1126/science.aaa9273

    18. Gao, F., H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, "Topologically protected refraction of robust kink states in valley photonic crystals," Nature Physics, Vol. 14, No. 2, 140-144, 2018.
    doi:10.1038/nphys4304

    19. Yang, B., 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, Vol. 359, No. 6379, 1013, 2018.
    doi:10.1126/science.aaq1221

    20. Yang, Y., Z. Gao, H. Xue, L. Zhang, M. He, Z. Yang, R. Singh, Y. Chong, B. Zhang, and H. Chen, "Realization of a three-dimensional photonic topological insulator," Nature, Vol. 565, No. 7741, 622-626, 2019.
    doi:10.1038/s41586-018-0829-0

    21. Khanikaev, A. B., S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, "Photonic topological insulators," Nature Materials, Vol. 12, No. 3, 233-239, 2013.
    doi:10.1038/nmat3520

    22. Fu, J.-X., R.-J. Liu, and Z.-Y. Li, "Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces," Applied Physics Letters, Vol. 97, No. 4, 041112, 2010.
    doi:10.1063/1.3470873

    23. Poo, Y., R.-X. Wu, Z. Lin, Y. Yang, and C. T. Chan, "Experimental realization of self-guiding unidirectional electromagnetic edge states," Physical Review Letters, Vol. 106, No. 9, 093903, 2011.
    doi:10.1103/PhysRevLett.106.093903

    24. Skirlo, S. A., L. Lu, Y. Igarashi, Q. Yan, J. Joannopoulos, and M. Soljacic, "Experimental observation of large chern numbers in photonic crystals," Physical Review Letters, Vol. 115, No. 25, 253901, 2015.
    doi:10.1103/PhysRevLett.115.253901

    25. Blanco-Redondo, A., I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, "Topological optical waveguiding in silicon and the transition between topological and trivial defect states," Physical Review Letters, Vol. 116, No. 16, 163901, 2016.
    doi:10.1103/PhysRevLett.116.163901

    26. Lu, L., H. Gao, and Z. Wang, "Topological one-way fiber of second Chern number," Nature Communications, Vol. 9, No. 1, 5384, 2018.
    doi:10.1038/s41467-018-07817-3

    27. Pilozzi, L. and C. Conti, "Topological lasing in resonant photonic structures," Physical Review B, Vol. 93, No. 19, 195317, 2016.
    doi:10.1103/PhysRevB.93.195317

    28. Zhang, W., X. Xie, H.-M. Hao, J. Dang, S. Xiao, S. Shi, H.-Q. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, "Low-threshold topological nanolasers based on the second-order corner state," Light, Science & Applications, Vol. 9, 2020.

    29. Leykam, D. and Y. D. Chong, "Edge solitons in nonlinear-photonic topological insulators," Physical Review Letters, Vol. 117, No. 14, 143901, 2016.
    doi:10.1103/PhysRevLett.117.143901

    30. Yuan, L., Q. Lin, M. Xiao, and S. Fan, "Synthetic dimension in photonics," Optica, Vol. 5, No. 11, 1396-1405, 2018.
    doi:10.1364/OPTICA.5.001396

    31. Ozawa, T. and H. M. Price, "Topological quantum matter in synthetic dimensions," Nature Reviews Physics, Vol. 1, No. 5, 349-357, 2019.
    doi:10.1038/s42254-019-0045-3

    32. Yuan, L., Y. Shi, and S. Fan, "Photonic gauge potential in a system with a synthetic frequency dimension," Opt. Lett., Vol. 41, No. 4, 741-744, 2016.
    doi:10.1364/OL.41.000741

    33. Ozawa, T., H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, "Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics," Physical Review A, Vol. 93, No. 4, 043827, 2016.
    doi:10.1103/PhysRevA.93.043827

    34. Bell, B. A., K.Wang, A. S. Solntsev, D. N. Neshev, A. A. Sukhorukov, and B. J. Eggleton, "Spectral photonic lattices with complex long-range coupling," Optica, Vol. 4, 1433-1436, 2017.
    doi:10.1364/OPTICA.4.001433

    35. Qin, C., F. Zhou, Y. Peng, D. Sounas, X. Zhu, B. Wang, J. Dong, X. Zhang, A. Alu, and P. Lu, "Spectrum control through discrete frequency di®raction in the presence of photonic Gauge potentials," Physical Review Letters, Vol. 120, No. 13, 133901, 2018.
    doi:10.1103/PhysRevLett.120.133901

    36. Yuan, L., M. Xiao, Q. Lin, and S. Fan, "Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation," Physical Review B, Vol. 97, No. 10, 104105, 2018.
    doi:10.1103/PhysRevB.97.104105

    37. Yuan, L., Q. Lin, A. Zhang, M. Xiao, X. Chen, and S. Fan, "Photonic Gauge potential in one cavity with synthetic frequency and orbital angular momentum dimensions," Physical Review Letters, Vol. 122, No. 8, 083903, 2019.
    doi:10.1103/PhysRevLett.122.083903

    38. Luo, X.-W., X. Zhou, C.-F. Li, J.-S. Xu, G.-C. Guo, and Z.-W. Zhou, "Quantum simulation of 2D topological physics in a 1D array of optical cavities," Nature Communications, Vol. 6, No. 1, 7704, 2015.
    doi:10.1038/ncomms8704

    39. Zhou, X.-F., X.-W. Luo, S. Wang, G.-C. Guo, X. Zhou, H. Pu, and Z.-W. Zhou, "Dynamically manipulating topological physics and edge modes in a single degenerate optical cavity," Physical Review Letters, Vol. 118, No. 8, 083603, 2017.
    doi:10.1103/PhysRevLett.118.083603

    40. Luo, X.-W., X. Zhou, J.-S. Xu, C.-F. Li, G.-C. Guo, C. Zhang, and Z.-W. Zhou, "Synthetic-lattice enabled all-optical devices based on orbital angular momentum of light," Nature Communications, Vol. 8, No. 1, 16097, 2017.
    doi:10.1038/ncomms16097

    41. Regensburger, A., C. Bersch, B. Hinrichs, G. Onishchukov, A. Schreiber, C. Silberhorn, and U. Peschel, "Photon propagation in a discrete fiber network: An interplay of coherence and losses," Physical Review Letters, Vol. 107, No. 23, 233902, 2011.
    doi:10.1103/PhysRevLett.107.233902

    42. Regensburger, A., C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, "Parity-time synthetic photonic lattices," Nature, Vol. 488, No. 7410, 167-171, 2012.
    doi:10.1038/nature11298

    43. Wimmer, M., A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, "Observation of optical solitons in PT-symmetric lattices," Nature Communications, Vol. 6, No. 1, 7782, 2015.
    doi:10.1038/ncomms8782

    44. Wimmer, M., H. M. Price, I. Carusotto, and U. Peschel, "Experimental measurement of the Berry curvature from anomalous transport," Nature Physics, Vol. 13, No. 6, 545-550, 2017.
    doi:10.1038/nphys4050

    45. Chen, C., X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, "Observation of topologically protected edge states in a photonic two-dimensional quantum walk," Physical Review Letters, Vol. 121, No. 10, 100502, 2018.
    doi:10.1103/PhysRevLett.121.100502

    46. Dutt, A., M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, "Experimental band structure spectroscopy along a synthetic dimension," Nature Communications, Vol. 10, No. 1, 3122, 2019.
    doi:10.1038/s41467-019-11117-9

    47. Lustig, E., S.Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, "Photonic topological insulator in synthetic dimensions," Nature, Vol. 567, No. 7748, 356-360, 2019.
    doi:10.1038/s41586-019-0943-7

    48. Dutt, A., Q. Lin, L. Yuan, M. Minkov, M. Xiao, and S. Fan, "A single photonic cavity with two independent physical synthetic dimensions," Science, Vol. 367, No. 6473, 59, 2020.
    doi:10.1126/science.aaz3071

    49. Yu, D., L. Yuan, and X. Chen, "Isolated photonic flatband with the effective magnetic flux in a synthetic space including the frequency dimension," Laser & Photonics Reviews, Vol. 14, No. 11, 2000041, 2020.
    doi:10.1002/lpor.202000041

    50. Malitson, L. H., "Interspecimen comparison of the refractive index of fused silica," Journal of the Optical Society of America (1917--1983), Vol. 55, 1205, 1965.
    doi:10.1364/JOSA.55.001205

    51. Yuan, L. and S. Fan, "Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator," Optica, Vol. 3, No. 9, 1014-1018, 2016.
    doi:10.1364/OPTICA.3.001014

    52. Haus, H. A., Waves and Fields in Optoelectronics, Prentice-Hall, 1984.

    53. Little, B. E., S. T. Chu, H. A. Haus, J. Foresi, and J. Laine, "Microring resonator channel dropping filters," Journal of Lightwave Technology, Vol. 15, No. 6, 998-1005, 1997.
    doi:10.1109/50.588673

    54. Minkov, M., Y. Shi, and S. Fan, "Exact solution to the steady-state dynamics of a periodically modulated resonator," APL Photonics, Vol. 2, No. 7, 076101, 2017.
    doi:10.1063/1.4985381

    55. Gardiner, C. W. and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Physical Review A, Vol. 31, No. 6, 3761-3774, 1985.
    doi:10.1103/PhysRevA.31.3761

    56. Fan, S., S. E. Kocabas, and J.-T. Shen, "Input-output formalism for few-photon transport in one-dimensional nanophotonic waveguides coupled to a qubit," Physical Review A, Vol. 82, No. 6, 063821, 2010.
    doi:10.1103/PhysRevA.82.063821

    57. Saleh, B. E. A. and M. C. Teich, Fundamentals of Photonics, Vol. 32, Wiley, 1991.
    doi:10.1002/0471213748

    58. Zhang, M., B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. M. Kahn, and M. Loncar, "Broadband electro-optic frequency comb generation in a lithium niobate microring resonator," Nature, Vol. 568, No. 7752, 373-377, 2019.
    doi:10.1038/s41586-019-1008-7

    59. Hu, Y., C. Reimer, A. Shams-Ansari, M. Zhang, and M. Loncar, "Realization of high-dimensional frequency crystals in electro-optic microcombs," Optica, Vol. 7, No. 9, 1189-1194, 2020.
    doi:10.1364/OPTICA.395114