Soliton regulation in microcavities induced by fundamental–second-harmonic mode coupling
Microcomb generation with simultaneous χ(2) and χ(3) nonlinearities brings new possibilities for ultrabroadband and potentially self-referenced integrated comb sources. However, the evolution of the intracavity field involving multiple nonlinear processes shows complex dynamics that are still poorly understood. Here, we report on strong soliton regulation induced by fundamental–second-harmonic (FD-SH) mode coupling. The formation of solitons from chaos is extensively investigated based on coupled Lugiato–Lefever equations. The soliton generation shows more deterministic behaviors in the presence of FD-SH mode interaction, which is in sharp contrast with the usual cases where the soliton number and relative locations are stochastic. Deterministic single soliton transition, soliton binding, and prohibition are observed, depending on the phase-matching condition and coupling coefficient between the fundamental and second-harmonic waves. Our finding provides important new insights into the soliton dynamics in microcavities with simultaneous χ(2) and χ(3) nonlinearities and can be immediate guidance for broadband soliton comb generation with such platforms.
基金项目：National Natural Science Foundation of China (NSFC)10.13039/501100001809 (61420106003, 61690191, 61690192); Natural Science Foundation of Beijing Municipality10.13039/501100004826 (4172029).
Xiaoping Zheng：Department of Electronic Engineering, Tsinghua University, Beijing 100084, China
Bingkun Zhou：Department of Electronic Engineering, Tsinghua University, Beijing 100084, China
【1】T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332 , 555–559 (2011).
【2】A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss, S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson, S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, and R. Morandotti, “Micro-combs: a novel generation of optical sources,” Phys. Rep. 729 , 1–81 (2018).
【3】P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos, “Microresonator-based solitons for massively parallel coherent optical communications,” Nature 546 , 274–279 (2017).
【4】A. Fül?p, M. Mazur, A. Lorences-Riesgo, T. A. Eriksson, P.-H. Wang, Y. Xuan, D. E. Leaird, M. Qi, P. A. Andrekson, A. M. Weiner, and V. Torres-Company, “Long-haul coherent communications using microresonator-based frequency combs,” Opt. Express 25 , 26678–26688 (2017).
【5】J. Pfeifle, A. Coillet, R. Henriet, K. Saleh, P. Schindler, C. Weimann, W. Freude, I. V. Balakireva, L. Larger, C. Koos, and Y. K. Chembo, “Optimally coherent Kerr combs generated with crystalline whispering gallery mode resonators for ultrahigh capacity fiber communications,” Phys. Rev. Lett. 114 , 093902 (2015).
【6】A. A. Savchenkov, D. Eliyahu, W. Liang, V. S. Ilchenko, J. Byrd, A. B. Matsko, D. Seidel, and L. Maleki, “Stabilization of a Kerr frequency comb oscillator,” Opt. Lett. 38 , 2636–2639 (2013).
【7】S. B. Papp, K. Beha, P. Del’Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, “Microresonator frequency comb optical clock,” Optica 1 , 10–14 (2014).
【8】M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354 , 600–603 (2016).
【9】A. Dutt, C. Joshi, X. Ji, J. Cardenas, Y. Okawachi, K. Luke, A. L. Gaeta, and M. Lipson, “On-chip dual-comb source for spectroscopy,” Sci. Adv. 4 , e1701858 (2018).
【10】P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos, “Ultrafast optical ranging using microresonator soliton frequency combs,” Science 359 , 887–891 (2018).
【11】M.-G. Suh, and K. J. Vahala, “Soliton microcomb range measurement,” Science 359 , 884–887 (2018).
【12】S.-W. Huang, J. Yang, S.-H. Yang, M. Yu, D.-L. Kwong, T. Zelevinsky, M. Jarrahi, and C. W. Wong, “Globally stable microresonator Turing pattern formation for coherent high-power THz radiation on-chip,” Phys. Rev. X 7 , 041002 (2017).
【13】X. Xue, and A. M. Weiner, “Microwave photonics connected with microresonator frequency combs,” Front. Optoelectron. 9 , 238–248 (2016).
【14】W. Liang, D. Eliyahu, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6 , 7957 (2015).
【15】S. Diallo, and Y. K. Chembo, “Optimization of primary Kerr optical frequency combs for tunable microwave generation,” Opt. Lett. 42 , 3522–3525 (2017).
【16】X. Xue, Y. Xuan, H.-J. Kim, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Programmable single-bandpass photonic RF filter based on Kerr comb from a microring,” J. Lightwave Technol. 32 , 3557–3565 (2014).
【17】X. Xue, Y. Xuan, C. Bao, S. Li, X. Zheng, B. Zhou, M. Qi, and A. M. Weiner, “Microcomb-based true-time-delay network for microwave beamforming with arbitrary beam pattern control,” J. Lightwave Technol. 36 , 2312–2321 (2018).
【18】T. G. Nguyen, M. Shoeiby, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Integrated frequency comb source based Hilbert transformer for wideband microwave photonic phase analysis,” Opt. Express 23 , 22087–22097 (2015).
【19】X. Xu, J. Wu, M. Shoeiby, T. G. Nguyen, S. T. Chu, B. E. Little, R. Morandotti, A. Mitchell, and D. J. Moss, “Reconfigurable broadband microwave photonic intensity differentiator based on an integrated optical frequency comb source,” APL Photon. 2 , 096104 (2017).
【20】T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8 , 145–152 (2014).
【21】X. Xue, Y. Xuan, Y. Liu, P.-H. Wang, S. Chen, J. Wang, D. E. Leaird, M. Qi, and A. M. Weiner, “Mode-locked dark pulse Kerr combs in normal-dispersion microresonators,” Nat. Photonics 9 , 594–600 (2015).
【22】J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4 , 37–40 (2010).
【23】H. Jung, C. Xiong, K. Y. Fong, X. Zhang, and H. X. Tang, “Optical frequency comb generation from aluminum nitride microring resonator,” Opt. Lett. 38 , 2810–2813 (2013).
【24】M. Pu, L. Ottaviano, E. Semenova, and K. Yvind, “Efficient frequency comb generation in AlGaAs-on-insulator,” Optica 3 , 823–826 (2016).
【25】B. J. M. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, and M. Loňcar, “Diamond nonlinear photonics,” Nat. Photonics 8 , 369–374 (2014).
【26】S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giorgis, “Second-harmonic generation in amorphous silicon nitride microcavities,” Appl. Phys. Lett. 81 , 4706–4708 (2002).
【27】T. Ning, H. Pietarinen, O. Hyv?rinen, J. Simonen, G. Genty, and M. Kauranen, “Strong second-harmonic generation in silicon nitride films,” Appl. Phys. Lett. 100 , 161902 (2012).
【28】A. Kitao, K. Imakita, I. Kawamura, and M. Fujii, “An investigation into second harmonic generation by Si-rich SiNx thin films deposited by RF sputtering over a wide range of Si concentrations,” J. Phys. D 47 , 215101 (2014).
【29】S. Miller, K. Luke, Y. Okawachi, J. Cardenas, A. L. Gaeta, and M. Lipson, “On-chip frequency comb generation at visible wavelengths via simultaneous second- and third-order optical nonlinearities,” Opt. Express 22 , 26517–26525 (2014).
【30】H. Jung, R. Stoll, X. Guo, D. Fischer, and H. X. Tang, “Green, red, and IR frequency comb line generation from single IR pump in AlN microring resonator,” Optica 1 , 396–399 (2014).
【31】X. Xue, F. Leo, Y. Xuan, J. A. Jaramillo-Villegas, P.-H. Wang, D. E. Leaird, M. Erkintalo, M. Qi, and A. M. Weiner, “Second-harmonic assisted four-wave mixing in chip-based microresonator frequency comb generation,” Light Sci. Appl. 6 , e16253 (2017).
【32】X. Guo, C.-L. Zou, H. Jung, Z. Gong, A. Bruch, L. Jiang, and H. X. Tang, “Efficient visible frequency comb generation via Cherenkov radiation from a Kerr microcomb,” arXiv:1704.04264 (2017).
【33】T. Hansson, F. Leo, M. Erkintalo, J. Anthony, S. Coen, I. Ricciardi, M. D. Rosa, and S. Wabnitz, “Single envelope equation modeling of multi-octave comb arrays in microresonators with quadratic and cubic nonlinearities,” J. Opt. Soc. Am. B 33 , 1207–1215 (2016).
【34】X. Xue, X. Zheng, and A. M. Weiner, “Soliton trapping and comb self-referencing in a single microresonator with χ(2) and χ(3) nonlinearities,” Opt. Lett. 42 , 4147–4150 (2017).
【35】M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. P. Pfeiffer, M. Zervas, M. Geiselmann, and T. J. Kippenberg, “Raman self-frequency shift of dissipative Kerr solitons in an optical microresonator,” Phys. Rev. Lett. 116 , 103902 (2016).
【36】T. Herr, V. Brasch, J. D. Jost, I. Mirgorodskiy, G. Lihachev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode spectrum and temporal soliton formation in optical microresonators,” Phys. Rev. Lett. 113 , 123901 (2014).
【37】X. Yi, Q.-F. Yang, X. Zhang, K. Y. Yang, X. Li, and K. Vahala, “Single-mode dispersive waves and soliton microcomb dynamics,” Nat. Commun. 8 , 14869 (2017).
【38】C. Bao, Y. Xuan, D. E. Leaird, S. Wabnitz, M. Qi, and A. M. Weiner, “Spatial mode-interaction induced single soliton generation in microresonators,” Optica 4 , 1011–1015 (2017).
【39】S. Fujii, A. Hori, T. Kato, R. Suzuki, Y. Okabe, W. Yoshiki, A. C. Jinnai, and T. Tanabe, “Effect on Kerr comb generation in a clockwise and counter-clockwise mode coupled microcavity,” Opt. Express 25 , 28969–28982 (2017).
【40】E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” Nat. Photonics 11 , 600–607 (2017).
【41】V. Ulvila, C. R. Phillips, L. Halonen, and M. Vainio, “Frequency comb generation by a continuous-wavepumped optical parametric oscillator based on cascading quadratic nonlinearities,” Opt. Lett. 38 , 4281–4284 (2013).
【42】I. Ricciardi, S. Mosca, M. Parisi, P. Maddaloni, L. Santamaria, P. D. Natale, and M. D. Rosa, “Frequency comb generation in quadratic nonlinear media,” Phys. Rev. A 91 , 063839 (2015).
【43】X. Guo, C.-L. Zou, H. Jung, and H. X. Tang, “On-chip strong coupling and efficient frequency conversion between telecom and visible optical modes,” Phys. Rev. Lett. 117 , 123902 (2016).
【44】C. Bao, and C. Yang, “Mode-pulling and phase-matching in broadband Kerr frequency comb generation,” J. Opt. Soc. Am. B 31 , 3074–3080 (2014).
【45】D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” Nat. Photonics 11 , 671–676 (2017).
【46】Y. Wang, F. Leo, J. Fatome, M. Erkintalo, S. G. Murdoch, and S. Coen, “Universal mechanism for the binding of temporal cavity solitons,” Optica 4 , 855–863 (2017).
【47】H. Taheri, A. B. Matsko, and L. Maleki, “Optical lattice trap for Kerr solitons,” Eur. Phys. J. D 71 , 153 (2017).
【48】D. J. Wilson, S. H?nl, K. Schneider, M. Anderson, T. J. Kippenberg, and P. Seidler, “Gallium phosphide microresonator frequency combs,” in Conference on Lasers and Electro-Optics , OSA Technical Digest (online) (Optical Society of America, 2018), paper?SW3A.1.
Xiaoxiao Xue, Xiaoping Zheng, and Bingkun Zhou, "Soliton regulation in microcavities induced by fundamental–second-harmonic mode coupling," Photonics Research 6(10), 948-953 (2018)