Abstract

Single solitons are a special limit of more general waveforms commonly referred to as cnoidal waves or Turing rolls. We theoretically and computationally investigate the stability and accessibility of cnoidal waves in microresonators. We show that they are robust and, in contrast to single solitons, can be easily and deterministically accessed in most cases. Their bandwidth can be comparable to single solitons, in which limit they are effectively a periodic train of solitons and correspond to a frequency comb with increased power. We comprehensively explore the three-dimensional parameter space that consists of detuning, pump amplitude, and mode circumference in order to determine where stable solutions exist. To carry out this task, we use a unique set of computational tools based on dynamical system theory that allow us to rapidly and accurately determine the stable region for each cnoidal wave periodicity and to find the instability mechanisms and their time scales. Finally, we focus on the soliton limit, and we show that the stable region for single solitons almost completely overlaps the stable region for both continuous waves and several higher-periodicity cnoidal waves that are effectively multiple soliton trains. This result explains in part why it is difficult to access single solitons deterministically.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

2018 (4)

D. C. Cole, J. R. Stone, M. Erkintalo, K. Y. Yang, X. Yi, K. J. Vahala, and S. B. Papp, “Kerr-microresonator solitons from a chirped background,” Optica 5, 1304–1310 (2018).
[Crossref]

Z. Kang, F. Li, J. Yuan, K. Nakkeeran, J. N. Kutz, Q. Wu, C. Yu, and P. K. A. Wai, “Deterministic generation of single soliton Kerr frequency comb in microresonators by a single shot pulsed trigger,” Opt. Express 26, 18563–18577 (2018).
[Crossref]

C. Joshi, A. Klenner, Y. Okawachi, M. Yu, K. Luke, X. Ji, M. Lipson, and A. L. Gaeta, “Counter-rotating cavity solitons in a silicon nitride microresonator,” Opt. Lett. 43, 547–550 (2018).
[Crossref]

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Frederick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

2017 (7)

Q. Li, T. C. Briles, D. A. Westly, T. E. Drake, J. R. Stone, B. R. Ilic, S. A. Diddams, S. B. Papp, and K. Srinivasan, “Stably accessing octave-spanning microresonator frequency combs in the soliton regime,” Optica 4, 193–203 (2017).
[Crossref]

V. Brasch, E. Lucas, J. D. Jost, M. Geiselmann, and T. J. Kippenberg, “Self-referenced photonic chip soliton Kerr frequency comb,” Light Sci. Appl. 6, e16202 (2017).
[Crossref]

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).
[Crossref]

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

C. Bao, Y. Xuan, S. Wabnitz, M. Qi, and A. M. Weiner, “Spatial mode-interaction induced single soliton generation in microresonators,” Optica 4, 1011–1015 (2017).
[Crossref]

Z. Qi, G. D’Aguanno, and C. R. Menyuk, “Nonlinear frequency combs generated by cnoidal waves in microring resonators,” J. Opt. Soc. Am. B 34, 785–794 (2017).
[Crossref]

C. Bao, Y. Xuan, J. A. Jaramillo-Villegas, D. E. Leaird, M. Qi, and A. M. Weiner, “Direct soliton generation in microresonators,” Opt. Lett. 42, 2519–2522 (2017).
[Crossref]

2016 (8)

S. Wang, B. S. Marks, and C. R. Menyuk, “Nonlinear stabilization of high-energy and ultrashort pulses in passively modelocked lasers,” J. Opt. Soc. Am. B 33, 2596–2601 (2016).
[Crossref]

Y. Shen, J. Zweck, S. Wang, and C. R. Menyuk, “Spectra of short pulse solutions of the cubic-quintic Ginzburg–Landau equation near zero dispersion,” Stud. Appl. Math. 137, 238–255 (2016).
[Crossref]

C. R. Menyuk and S. Wang, “Spectral methods for determining the stability and noise performance of passively modelocked lasers,” Nanophotonics 5, 332–350 (2016).
[Crossref]

G. D’Aguanno and C. R. Menyuk, “Nonlinear mode coupling in whispering-gallery-mode resonators,” Phys. Rev. A 93, 043820 (2016).
[Crossref]

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351, 357–360 (2016).
[Crossref]

C. Joshi, J. K. Jang, L. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi, M. Lipson, and A. L. Gaeta, “Thermally controlled comb generation and soliton modelocking in microresonators,” Opt. Lett. 41, 2565–2568 (2016).
[Crossref]

X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, “Active capture and stabilization of temporal solitons in microresonators,” Opt. Lett. 41, 2037–2040 (2016).
[Crossref]

P.-H. Wang, J. A. Jaramillo-Villegas, X. Yuan, X. Xue, C. Bao, D. E. Leaird, M. Qi, and A. M. Weiner, “Intracavity characterization of micro-comb generation in the single-soliton regime,” Opt. Express 24, 10890–10897 (2016).
[Crossref]

2015 (6)

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
[Crossref]

M. Pang, X. Jiang, W. He, G. K. L. Wong, G. Onishchukov, N. Y. Joly, G. Ahmed, C. R. Menyuk, and P. St.J. Russell, “Stable subpicosecond soliton fiber laser passively mode-locked by gigahertz acoustic resonance in photonic crystal fiber core,” Optica 2, 339–342 (2015).
[Crossref]

J. Pfeifle, A. Coillet, R. Henriet, K. Saleh, P. Schindler, C. Weimann, W. Freude, I. V. Balakireva, L. Larger, C. Koos, and Y. Chembo, “Optimally coherent Kerr combs generated with crystalline whispering gallery mode resonators for ultrahigh capacity fiber communications,” Phys. Rev. Lett. 114, 093902 (2015).
[Crossref]

H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
[Crossref]

W. Liang, D. Eliyahu, V. S. Ilchenko, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

J. A. Jaramillo-Villegas, X. Xue, P.-H. Wang, D. E. Leaird, and A. M. Weiner, “Deterministic single soliton generation and compression in microring resonators avoiding the chaotic region,” Opt. Express 23, 9618–9626 (2015).
[Crossref]

2014 (6)

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, “Dynamics of localized and patterned structures in the Lugiato–Lefever equation determine the stability and shape of optical frequency combs,” Phys. Rev. A 89, 043813 (2014).
[Crossref]

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugtiato–Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

S. Wang, A. Docherty, B. S. Marks, and C. R. Menyuk, “Boundary tracking algorithms for determining stability of mode-locked pulses,” J. Opt. Soc. Am. B 31, 2914–2930 (2014).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

T. Kaladze and S. Mahmood, “Ion-acoustic cnoidal waves in plasmas with warm ions and kappa distributed electrons and positrons,” Phys. Plasmas 21, 032306 (2014).
[Crossref]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Rogue waves and solitons on a cnoidal background,” Eur. Phys. J. Spec. Top. 223, 43–62 (2014).
[Crossref]

2013 (6)

2011 (2)

2010 (1)

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics 4, 471–476 (2010).
[Crossref]

2007 (1)

2006 (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[Crossref]

2003 (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Cnoidal wave patterns in quadratic nonlinear media,” Phys. Rev. E 67, 066612 (2003).
[Crossref]

2000 (2)

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Cnoidal wave compression by means of multisoliton effect,” Opt. Commun. 185, 305–314 (2000).
[Crossref]

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[Crossref]

1998 (1)

I. V. Barashenkov, Y. S. Smirnov, and N. V. Alexeeva, “Bifurcation to multisoliton complexes in the ac-driven, damped nonlinear Schrödinger equation,” Phys. Rev. E 57, 2350–2364 (1998).
[Crossref]

1997 (1)

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg–Landau equation,” Phys. Rev. Lett. 79, 4047–4051 (1997).
[Crossref]

1996 (1)

I. V. Barashenkov and Y. S. Smirnov, “Existence and stability for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E 54, 5707–5725 (1996).
[Crossref]

1992 (1)

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun. 91, 401–407 (1992).
[Crossref]

1990 (1)

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[Crossref]

1987 (1)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[Crossref]

1983 (1)

1976 (1)

H. A. Haus, “Parameter ranges for CW passive modelocking,” IEEE J. Quantum Electron. 12, 169–176 (1976).
[Crossref]

1975 (1)

H. Schamel, “Analytical BGK modes and their modulational instability,” J. Plasma Phys. 13, 139–145 (1975).
[Crossref]

1952 (1)

A. M. Turing, “The chemical basis of morphogenesis,” Philos. Trans. R. Soc. B 237, 37–72 (1952).
[Crossref]

1895 (1)

D. J. Korteweg and G. de Vries, “On the change of form of long waves in a rectangular canal, and on a new type of long stationary waves,” Philos. Mag. 39(240), 422–443 (1895).
[Crossref]

Adibi, A.

H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Ahmed, G.

Akhmediev, N.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Rogue waves and solitons on a cnoidal background,” Eur. Phys. J. Spec. Top. 223, 43–62 (2014).
[Crossref]

N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg–Landau and Swift–Hohenberg equations,” in Dissipative Solitons, N. Akhmediev and A. Ankiewicz, eds. (Springer, 2005).

Akhmediev, N. N.

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg–Landau equation,” Phys. Rev. Lett. 79, 4047–4051 (1997).
[Crossref]

Aleshkevich, V.

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Cnoidal wave compression by means of multisoliton effect,” Opt. Commun. 185, 305–314 (2000).
[Crossref]

Alexeeva, N. V.

I. V. Barashenkov, Y. S. Smirnov, and N. V. Alexeeva, “Bifurcation to multisoliton complexes in the ac-driven, damped nonlinear Schrödinger equation,” Phys. Rev. E 57, 2350–2364 (1998).
[Crossref]

Ankiewicz, A.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Rogue waves and solitons on a cnoidal background,” Eur. Phys. J. Spec. Top. 223, 43–62 (2014).
[Crossref]

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg–Landau equation,” Phys. Rev. Lett. 79, 4047–4051 (1997).
[Crossref]

N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg–Landau and Swift–Hohenberg equations,” in Dissipative Solitons, N. Akhmediev and A. Ankiewicz, eds. (Springer, 2005).

Balakireva, I.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photon. J. 5, 6100409 (2013).
[Crossref]

Balakireva, I. V.

J. Pfeifle, A. Coillet, R. Henriet, K. Saleh, P. Schindler, C. Weimann, W. Freude, I. V. Balakireva, L. Larger, C. Koos, and Y. Chembo, “Optimally coherent Kerr combs generated with crystalline whispering gallery mode resonators for ultrahigh capacity fiber communications,” Phys. Rev. Lett. 114, 093902 (2015).
[Crossref]

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugtiato–Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

Bao, C.

Barashenkov, I. V.

I. V. Barashenkov, Y. S. Smirnov, and N. V. Alexeeva, “Bifurcation to multisoliton complexes in the ac-driven, damped nonlinear Schrödinger equation,” Phys. Rev. E 57, 2350–2364 (1998).
[Crossref]

I. V. Barashenkov and Y. S. Smirnov, “Existence and stability for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E 54, 5707–5725 (1996).
[Crossref]

Bluestone, A.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Frederick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Bowers, J. E.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Frederick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Brasch, V.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

V. Brasch, E. Lucas, J. D. Jost, M. Geiselmann, and T. J. Kippenberg, “Self-referenced photonic chip soliton Kerr frequency comb,” Light Sci. Appl. 6, e16202 (2017).
[Crossref]

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351, 357–360 (2016).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Briles, T. C.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Frederick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Q. Li, T. C. Briles, D. A. Westly, T. E. Drake, J. R. Stone, B. R. Ilic, S. A. Diddams, S. B. Papp, and K. Srinivasan, “Stably accessing octave-spanning microresonator frequency combs in the soliton regime,” Optica 4, 193–203 (2017).
[Crossref]

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, 1971).

Chang, L.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Frederick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Chembo, Y.

J. Pfeifle, A. Coillet, R. Henriet, K. Saleh, P. Schindler, C. Weimann, W. Freude, I. V. Balakireva, L. Larger, C. Koos, and Y. Chembo, “Optimally coherent Kerr combs generated with crystalline whispering gallery mode resonators for ultrahigh capacity fiber communications,” Phys. Rev. Lett. 114, 093902 (2015).
[Crossref]

Chembo, Y. K.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugtiato–Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photon. J. 5, 6100409 (2013).
[Crossref]

Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato–Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013).
[Crossref]

Coen, S.

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, “Dynamics of localized and patterned structures in the Lugiato–Lefever equation determine the stability and shape of optical frequency combs,” Phys. Rev. A 89, 043813 (2014).
[Crossref]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato–Lefever equation,” Opt. Lett. 38, 37–39 (2013).
[Crossref]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics 4, 471–476 (2010).
[Crossref]

Coillet, A.

J. Pfeifle, A. Coillet, R. Henriet, K. Saleh, P. Schindler, C. Weimann, W. Freude, I. V. Balakireva, L. Larger, C. Koos, and Y. Chembo, “Optimally coherent Kerr combs generated with crystalline whispering gallery mode resonators for ultrahigh capacity fiber communications,” Phys. Rev. Lett. 114, 093902 (2015).
[Crossref]

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugtiato–Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photon. J. 5, 6100409 (2013).
[Crossref]

Cole, D. C.

D. C. Cole, J. R. Stone, M. Erkintalo, K. Y. Yang, X. Yi, K. J. Vahala, and S. B. Papp, “Kerr-microresonator solitons from a chirped background,” Optica 5, 1304–1310 (2018).
[Crossref]

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).
[Crossref]

Copie, F.

Cundiff, S. T.

S. T. Cundiff, “Soliton dynamics in mode-locked lasers,” in Dissipative Solitons, N. Akhmediev and A. Ankiewicz, eds. (Springer, 2005).

D’Aguanno, G.

Z. Qi, G. D’Aguanno, and C. R. Menyuk, “Nonlinear frequency combs generated by cnoidal waves in microring resonators,” J. Opt. Soc. Am. B 34, 785–794 (2017).
[Crossref]

G. D’Aguanno and C. R. Menyuk, “Nonlinear mode coupling in whispering-gallery-mode resonators,” Phys. Rev. A 93, 043820 (2016).
[Crossref]

Z. Qi, S. Wang, J. A. Jaramillo-Villegas, M. Qi, A. M. Weiner, G. D’Aguanno, and C. R. Menyuk, “Stability of cnoidal wave frequency combs in microresonators,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2018), paper SF2A.6.

de Vries, G.

D. J. Korteweg and G. de Vries, “On the change of form of long waves in a rectangular canal, and on a new type of long stationary waves,” Philos. Mag. 39(240), 422–443 (1895).
[Crossref]

Del Bino, L.

Del’Haye, P.

Devaney, R. L.

M. W. Hirsch, S. Smale, and R. L. Devaney, Dynamical Systems and an Introduction to Chaos (Academic, 2013).

Diddams, S. A.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Frederick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

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).
[Crossref]

Q. Li, T. C. Briles, D. A. Westly, T. E. Drake, J. R. Stone, B. R. Ilic, S. A. Diddams, S. B. Papp, and K. Srinivasan, “Stably accessing octave-spanning microresonator frequency combs in the soliton regime,” Optica 4, 193–203 (2017).
[Crossref]

Docherty, A.

Drake, T.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Frederick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Drake, T. E.

Dudley, J. M.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photon. J. 5, 6100409 (2013).
[Crossref]

Eftekhar, A. A.

H. Taheri, A. A. Eftekhar, K. Wiesenfeld, and A. Adibi, “Soliton formation in whispering-gallery-mode resonators via input phase modulation,” IEEE Photon. J. 7, 2200309 (2015).
[Crossref]

Eliyahu, D.

W. Liang, D. Eliyahu, V. S. Ilchenko, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

Emplit, P.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics 4, 471–476 (2010).
[Crossref]

Erkintalo, M.

Frederick, C.

D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C. Frederick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T. Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M.-G. Suh, K. Y. Yang, M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan, K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A. Diddams, and S. Papp, “An optical-frequency synthesizer using integrated photonics,” Nature 557, 81–85 (2018).
[Crossref]

Freude, W.

J. Pfeifle, A. Coillet, R. Henriet, K. Saleh, P. Schindler, C. Weimann, W. Freude, I. V. Balakireva, L. Larger, C. Koos, and Y. Chembo, “Optimally coherent Kerr combs generated with crystalline whispering gallery mode resonators for ultrahigh capacity fiber communications,” Phys. Rev. Lett. 114, 093902 (2015).
[Crossref]

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, 1971).

Gaeta, A. L.

Geiselmann, M.

V. Brasch, E. Lucas, J. D. Jost, M. Geiselmann, and T. J. Kippenberg, “Self-referenced photonic chip soliton Kerr frequency comb,” Light Sci. Appl. 6, e16202 (2017).
[Crossref]

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351, 357–360 (2016).
[Crossref]

Gelens, L.

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, “Dynamics of localized and patterned structures in the Lugiato–Lefever equation determine the stability and shape of optical frequency combs,” Phys. Rev. A 89, 043813 (2014).
[Crossref]

Ghalanos, G. N.

Godey, C.

C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal Lugtiato–Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes,” Phys. Rev. A 89, 063814 (2014).
[Crossref]

Gomila, D.

P. Parra-Rivas, D. Gomila, M. A. Matías, S. Coen, and L. Gelens, “Dynamics of localized and patterned structures in the Lugiato–Lefever equation determine the stability and shape of optical frequency combs,” Phys. Rev. A 89, 043813 (2014).
[Crossref]

Gordon, J. P.

Gorodetsky, M. L.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351, 357–360 (2016).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Gorza, S.-P.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics 4, 471–476 (2010).
[Crossref]

Guo, H.

H. Guo, M. Karpov, E. Lucas, A. Kordts, M. H. P. Pfeiffer, V. Brasch, G. Lihachev, V. E. Lobanov, M. L. Gorodetsky, and T. J. Kippenberg, “Universal dynamics and deterministic switching of dissipative Kerr solitons in optical microresonators,” Nat. Phys. 13, 94–102 (2017).
[Crossref]

Haelterman, M.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics 4, 471–476 (2010).
[Crossref]

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun. 91, 401–407 (1992).
[Crossref]

Han, K.

J. A. Jaramillo-Villegas, C. Wang, P.-H. Wang, C. Bao, Y. Xuan, K. Han, D. E. Leaird, M. Qi, and A. M. Weiner, “Experimental characterization of pump power and detuning in microresonator frequency combs,” in Latin America Optics & Photonics Conference (2016), paper LTh3B.2.

Haus, H. A.

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[Crossref]

H. A. Haus, “Parameter ranges for CW passive modelocking,” IEEE J. Quantum Electron. 12, 169–176 (1976).
[Crossref]

He, W.

Helmberg, G.

G. Helmberg, Introduction to Spectral Theory in Hilbert Space (Dover, 1997).

Henriet, R.

J. Pfeifle, A. Coillet, R. Henriet, K. Saleh, P. Schindler, C. Weimann, W. Freude, I. V. Balakireva, L. Larger, C. Koos, and Y. Chembo, “Optimally coherent Kerr combs generated with crystalline whispering gallery mode resonators for ultrahigh capacity fiber communications,” Phys. Rev. Lett. 114, 093902 (2015).
[Crossref]

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K. Chembo, “Azimuthal Turing patterns, bright and dark solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photon. J. 5, 6100409 (2013).
[Crossref]

Herr, T.

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip-based optical frequency comb using soliton Cherenkov radiation,” Science 351, 357–360 (2016).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Hirsch, M. W.

M. W. Hirsch, S. Smale, and R. L. Devaney, Dynamical Systems and an Introduction to Chaos (Academic, 2013).

Ilchenko, V. S.

W. Liang, D. Eliyahu, V. S. Ilchenko, A. B. Matsko, D. Seidel, and L. Maleki, “High spectral purity Kerr frequency comb radio frequency photonic oscillator,” Nat. Commun. 6, 7957 (2015).
[Crossref]

A. B. Matsko, A. A. Savchenko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011).
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Supplementary Material (1)

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Figures (13)

Fig. 1.
Fig. 1. Schematic illustration of the boundary-tracking algorithm. The blue-dashed line indicates the boundary of the stable region. The red circle is the starting point. We obtain the complete stability boundary numerically by moving along the boundary while tracking its location, as shown with the blue arrows.
Fig. 2.
Fig. 2. Schematic illustration of a basin of attraction in the phase space that consists of all possible waveforms. If the initial condition, shown as a red dot, starts inside the basin of attraction of a stationary waveform, shown in gray, then the solution will converge to the stationary solution. If the initial condition starts outside the basin, shown as a blue dot, then the solution evolves to another stationary solution or never becomes stationary.
Fig. 3.
Fig. 3. Stable regions of the cnoidal waves for (a) L=10, (b) L=25, and (c) L=50. We show a selection of the stable regions, labeled with Nper. The red-dashed curves show the limit below which continuous waves are stable.
Fig. 4.
Fig. 4. L/Nper versus L for four values of (α,F): α=2, F=3.5 (red); α=1, F=2.6 (green); α=0, F=1.7 (blue); α=1, F=1.2 (black). Dashed lines show the asymptotic values in the limit L. We obtained the stationary solution by using an evolutionary approach with a small random initial seed.
Fig. 5.
Fig. 5. (a) Waveform, (b) frequency spectrum, and (c) dynamical spectrum with Nper=8, L=50, α=4.4, and F=2.3. (d) Expanded view of the dashed rectangular region near λ=(0,0) in (c). (e) Waveform and (f) frequency spectrum with α=1 and F=1.15. Figures 5(b) and 5(c) are modified from Fig. 2 of [69].
Fig. 6.
Fig. 6. (a) Stable regions for the Nper=8 (red curve) and Nper=9 (black curve) cnoidal waves. The red-dashed curve plots the limit below which continuous waves are stable. A, B, C, and D indicate the four trajectories through the parameter space that we consider. Dynamical spectra and pulse intensities for (b) and (c) trajectory A, (d) and (e) trajectory B, (f) and (g) trajectory C, and (h) and (i) trajectory D.
Fig. 7.
Fig. 7. Evolution of pulse intensity corresponding to the final points in trajectories (a) B, (b) C, and (c) D in Fig. 6.
Fig. 8.
Fig. 8. Comparison of the stability map for L=50 with evolutionary simulations from which the stable regions can be inferred. Figure 8(a) is modified from Fig. 2(a) in Ref. [70]. The color bar shows the number of intensity maxima in the plot. The solid curves in (a) and the red-dashed curve in (b) show trajectories through the parameter space that never pass through the chaotic region. This figure is modified from Fig. 1 in [69].
Fig. 9.
Fig. 9. Contour plots of (a) the exponential decrease in the frequency spectrum Pn/Pn+1 (dB), n>3, and (b) the ratio of the total power in the comb lines to the input pump power ρ for Nper=8. Figure 9(a) is modified from Fig. 2(a) in Ref. [69].
Fig. 10.
Fig. 10. Stable regions for the Nper=1 (black), Nper=3 (green), and Nper=5 (cyan) cnoidal waves. The Nper=1 cnoidal wave is a single bright soliton. Continuous waves are stable below the red-dashed curve. The curves for Nper=1, 3, and 5 almost completely overlap except in the black dashed rectangular region, of which an expanded view is shown in (b).
Fig. 11.
Fig. 11. Schematic illustration of three trajectories through the parameter space to obtain single solitons. The gray region indicates where single solitons are stable, and continuous waves are stable below the red-dashed line.
Fig. 12.
Fig. 12. Waveforms (left), frequency spectra (middle), and the corresponding dynamical spectra (right) of single solitons for α=5.5 and (a) F=2.11, (b) F=2.45, and (c) F=2.9 at L=50.
Fig. 13.
Fig. 13. Evolution of the waveform when the system parameters move along the red-dashed curve in Fig. 8(a). (a) The lower white line corresponds to t=100, and the upper white line corresponds to t=250, inside of which the detuning shifts from α=0 to α=6. At the end of the evolution, a single soliton appears. (b) The lower white line corresponds to t=100, and the upper white line corresponds to t=1.201×105, inside of which the detuning again shifts from α=0 to α=6. (c) The lower white line corresponds to t=1.201×105. At the end of the time, the solution has collapsed to a continuous wave.

Tables (1)

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Table 1. Physical Device Parameters for Different Experimental Devicesa

Equations (10)

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TRAτ=iβ222Aθ2+iγ|A|2A+(iσl2)A+iPin,
ψt=iβ22ψθ2+i|ψ|2ψ(iα+1)ψ+F.
ψt=i2ψx2+i|ψ|2ψ(iα+1)ψ+F.
ψt=i2ψx2+2i|ψ|2ψ(i+δ)ψ+ih=0.
0=i2ψx2+i|ψ|2ψ(iα+1)ψ+F.
ψ(x,t)=ψ0(x)+Δψ(x,t),ψ¯(x,t)=ψ0*(x)+Δψ¯(x,t),
ΔΨt=LΔΨ,
ΔΨ=[ΔψΔψ¯],
L=[i2/x2+2i|ψ0|2iα1iψ02iψ0*2i2/x22i|ψ0|2+iα1].
F=1+(α1)2.

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