Vol. 169

Latest Volume
All Volumes
All Issues

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


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.


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.


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

    2. Khanikaev, A. B. and G. Shvets, "Two-dimensional topological photonics," Nature Photonics, Vol. 11, No. 12, 763-773, 2017.

    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.

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

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    26. Lu, L., H. Gao, and Z. Wang, "Topological one-way fiber of second Chern number," Nature Communications, Vol. 9, No. 1, 5384, 2018.

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

    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.

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

    31. Ozawa, T. and H. M. Price, "Topological quantum matter in synthetic dimensions," Nature Reviews Physics, Vol. 1, No. 5, 349-357, 2019.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    57. Saleh, B. E. A. and M. C. Teich, Fundamentals of Photonics, Vol. 32, Wiley, 1991.

    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.

    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.