Abstract

Temporal cavity solitons (CSs) are pulses of light that can persist in coherently driven passive resonators, such as fiber ring resonators and monolithic Kerr microresonators. While these solitons can in principle occupy arbitrary positions, multisoliton configurations often appear rigidly frozen in time, seemingly insensitive to noise. Here, we elucidate this behavior by presenting theoretical and experimental evidence of a universal mechanism through which temporal CSs can form robust long-range bound states. These bound states require perturbations to the strict Lugiato–Lefever mean-field description of temporal CSs. Binding occurs when the perturbation excites a narrowband resonance in the soliton spectrum, which gives long oscillatory tails to the CSs. Those tails can then interlock for a discrete set of temporal separations between the solitons. The universality of this mechanism is demonstrated in fiber ring cavities by providing experimental observations of long-range bound states ensuing from three different perturbations: third-order dispersion (dispersive wave generation), the periodic nature of the cavity (Kelly sidebands), and the random birefringence of the resonator. Subpicosecond resolution of bound-state separations and their dynamics are obtained by using the dispersive Fourier transform technique. Good agreement with theoretical models, including a new vector mean-field model, is also reported. Our work provides a framework to better understand the many soliton bound states observed in externally driven, passive Kerr resonators, including the soliton crystals reported in microresonators.

© 2017 Optical Society of America

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References

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    [Crossref]

2017 (1)

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901(2017).

2016 (6)

2015 (4)

K. Luo, Y. Xu, M. Erkintalo, and S. G. Murdoch, “Resonant radiation in synchronously pumped passive Kerr cavities,” Opt. Lett. 40, 427–430 (2015).
[Crossref]

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]

J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, “Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons,” Nat. Commun. 6, 7370 (2015).
[Crossref]

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (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]

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).
[Crossref]

T. Herr, V. Brasch, D. Jost, J. 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).
[Crossref]

C. Milián and D. V. Skryabin, “Soliton families and resonant radiation in a micro-ring resonator near zero group-velocity dispersion,” Opt. Express 22, 3732–3739 (2014).
[Crossref]

M. Conforti, A. Mussot, A. Kudlinski, and S. Trillo, “Modulational instability in dispersion oscillating fiber ring cavities,” Opt. Lett. 39, 4200–4203 (2014).
[Crossref]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Observation of dispersive wave emission by temporal cavity solitons,” Opt. Lett. 39, 5503–5506 (2014).
[Crossref]

2013 (6)

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

F. Leo, L. Gelens, Ph. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express 21, 9180–9191 (2013).
[Crossref]

A. F. J. Runge, C. Aguergaray, N. G. R. Broderick, and M. Erkintalo, “Coherence and shot-to-shot spectral fluctuations in noise-like ultrafast fiber lasers,” Opt. Lett. 38, 4327–4330 (2013).
[Crossref]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Ultraweak long-range interactions of solitons observed over astronomical distances,” Nat. Photonics 7, 657–663 (2013).
[Crossref]

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (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]

2012 (2)

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett. 108, 263906 (2012).
[Crossref]

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).

2011 (2)

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

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

2010 (1)

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. 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]

2009 (1)

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. At. Mol. Opt. Phys. 57, 323–421 (2009).
[Crossref]

2007 (1)

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
[Crossref]

2003 (1)

2001 (1)

I. Pérez-Arjona, V. J. Sánchez-Morcillo, G. J. de Valcárcel, and E. Roldán, “Bright cavity solitons in anisotropic vectorial Kerr cavities,” J. Opt. B 3, S118–S123 (2001).
[Crossref]

2000 (2)

T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000).
[Crossref]

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref]

1999 (1)

1998 (2)

S. Coen, M. Haelterman, Ph. Emplit, L. Delage, L. M. Simohamed, and F. Reynaud, “Experimental investigation of the dynamics of a stabilized nonlinear fiber ring resonator,” J. Opt. Soc. Am. B 15, 2283–2293 (1998).
[Crossref]

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)

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[Crossref]

1994 (1)

D. Cai, A. R. Bishop, N. Grønbech-Jensen, and B. A. Malomed, “Bound solitons in the AC-driven, damped nonlinear Schrödinger equation,” Phys. Rev. E 49, 1677–1679 (1994).
[Crossref]

1993 (1)

B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874–2880 (1993).
[Crossref]

1992 (3)

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

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
[Crossref]

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[Crossref]

1991 (1)

B. A. Malomed, “Bound solitons in the nonlinear Schrödinger–Ginzburg–Landau equation,” Phys. Rev. A 44, 6954–6957 (1991).
[Crossref]

1987 (2)

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

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
[Crossref]

Ackemann, T.

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. At. Mol. Opt. Phys. 57, 323–421 (2009).
[Crossref]

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, “Vector solitons,” in Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003), Chap. 9, pp. 278–339.

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2013).

Aguergaray, C.

Akhmediev, N.

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
[Crossref]

J. M. Soto-Crespo, N. Akhmediev, P. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28, 1757–1759 (2003).
[Crossref]

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]

N. N. Akhmediev, “General theory of solitons,” in Soliton-Driven Photonics, A. D. Boardman and A. P. Sukhorukov, eds., vol. 31 of NATO Science Series (Springer, 2001), pp. 371–395.

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]

Andral, U.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901(2017).

Ankiewicz, A.

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]

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]

Beha, K.

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

Belhache, F.

Bishop, A. R.

D. Cai, A. R. Bishop, N. Grønbech-Jensen, and B. A. Malomed, “Bound solitons in the AC-driven, damped nonlinear Schrödinger equation,” Phys. Rev. E 49, 1677–1679 (1994).
[Crossref]

Brambilla, M.

T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000).
[Crossref]

Brasch, V.

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]

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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).
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Matsko, A. B.

Menyuk, C. R.

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]

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
[Crossref]

Merolla, J. M.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).

Milián, C.

Miller, S. A.

Mirgorodskiy, J. I.

T. Herr, V. Brasch, D. Jost, J. 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).
[Crossref]

Murdoch, S. G.

K. E. Webb, M. Erkintalo, S. Coen, and S. G. Murdoch, “Experimental observation of coherent cavity soliton frequency combs in silica microspheres,” Opt. Lett. 41, 4613–4616 (2016).
[Crossref]

J. K. Jang, M. Erkintalo, J. Schröder, B. J. Eggleton, S. G. Murdoch, and S. Coen, “All-optical buffer based on temporal cavity solitons operating at 10  Gb/s,” Opt. Lett. 41, 4526–4529 (2016).
[Crossref]

K. Luo, Y. Xu, M. Erkintalo, and S. G. Murdoch, “Resonant radiation in synchronously pumped passive Kerr cavities,” Opt. Lett. 40, 427–430 (2015).
[Crossref]

J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, “Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons,” Nat. Commun. 6, 7370 (2015).
[Crossref]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Observation of dispersive wave emission by temporal cavity solitons,” Opt. Lett. 39, 5503–5506 (2014).
[Crossref]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Ultraweak long-range interactions of solitons observed over astronomical distances,” Nat. Photonics 7, 657–663 (2013).
[Crossref]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Bound states of temporal cavity solitons,” in Nonlinear Photonics, NP’2014, Barcelona, Spain, July27–31, 2014 (Optical Society of America, 2014), paper JM5A.59.

Mussot, A.

M. Conforti, A. Mussot, A. Kudlinski, and S. Trillo, “Modulational instability in dispersion oscillating fiber ring cavities,” Opt. Lett. 39, 4200–4203 (2014).
[Crossref]

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).

Nicolis, G.

G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems: from Dissipative Structures to Order through Fluctuations (Wiley, 1977).

Nithyanandan, K.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901(2017).

Obrzud, E.

E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” arXiv:1612.08993 (2016).

Okawachi, Y.

Oppo, G.-L.

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. At. Mol. Opt. Phys. 57, 323–421 (2009).
[Crossref]

Papp, S. B.

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” arXiv:1610.00080 (2016).

Parra-Rivas, P.

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]

P. Parra-Rivas, D. Gomila, P. Colet, and L. Gelens, “Interaction of solitons and the formation of bound states in the generalized Lugiato-Lefever equation,” arXiv:1705.02619 (2017).

Pérez-Arjona, I.

I. Pérez-Arjona, V. J. Sánchez-Morcillo, G. J. de Valcárcel, and E. Roldán, “Bright cavity solitons in anisotropic vectorial Kerr cavities,” J. Opt. B 3, S118–S123 (2001).
[Crossref]

Pfeiffer, M. H. P.

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]

Prigogine, I.

G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems: from Dissipative Structures to Order through Fluctuations (Wiley, 1977).

Randle, H. G.

Reynaud, F.

Roldán, E.

I. Pérez-Arjona, V. J. Sánchez-Morcillo, G. J. de Valcárcel, and E. Roldán, “Bright cavity solitons in anisotropic vectorial Kerr cavities,” J. Opt. B 3, S118–S123 (2001).
[Crossref]

Romagnoli, M.

Ropers, C.

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90  MHz frame rate,” Nat. Photonics 10, 321–326 (2016).
[Crossref]

Runge, A. F. J.

Sánchez-Morcillo, V. J.

I. Pérez-Arjona, V. J. Sánchez-Morcillo, G. J. de Valcárcel, and E. Roldán, “Bright cavity solitons in anisotropic vectorial Kerr cavities,” J. Opt. B 3, S118–S123 (2001).
[Crossref]

Savchenkov, A. A.

Schäpers, B.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref]

Schröder, J.

Scroggie, A. J.

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[Crossref]

Seidel, D.

Simohamed, L. M.

Skryabin, D. V.

Smirnov, Y. S.

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]

Socci, L.

Solli, D. R.

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90  MHz frame rate,” Nat. Photonics 10, 321–326 (2016).
[Crossref]

Soto-Crespo, J. M.

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
[Crossref]

J. M. Soto-Crespo, N. Akhmediev, P. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28, 1757–1759 (2003).
[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]

Stefani, A.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).

Suh, M.-G.

Sylvestre, T.

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

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).

Tchofo-Dinda, P.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901(2017).

Trillo, S.

M. Conforti, A. Mussot, A. Kudlinski, and S. Trillo, “Modulational instability in dispersion oscillating fiber ring cavities,” Opt. Lett. 39, 4200–4203 (2014).
[Crossref]

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

Turaev, D.

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett. 108, 263906 (2012).
[Crossref]

Vahala, K.

Vladimirov, A. G.

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett. 108, 263906 (2012).
[Crossref]

Wabnitz, S.

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

Wang, C. Y.

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).
[Crossref]

Webb, K. E.

Wetzel, B.

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).

Xu, Y.

Yang, K. Y.

Yang, Q.-F.

Yi, X.

Zelik, S.

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett. 108, 263906 (2012).
[Crossref]

Adv. At. Mol. Opt. Phys. (1)

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. At. Mol. Opt. Phys. 57, 323–421 (2009).
[Crossref]

Electron. Lett. (1)

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[Crossref]

J. Opt. B (1)

I. Pérez-Arjona, V. J. Sánchez-Morcillo, G. J. de Valcárcel, and E. Roldán, “Bright cavity solitons in anisotropic vectorial Kerr cavities,” J. Opt. B 3, S118–S123 (2001).
[Crossref]

J. Opt. Soc. Am. B (3)

Nat. Commun. (2)

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, “Temporal tweezing of light through the trapping and manipulation of temporal cavity solitons,” Nat. Commun. 6, 7370 (2015).
[Crossref]

Nat. Photonics (5)

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).
[Crossref]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. 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]

G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90  MHz frame rate,” Nat. Photonics 10, 321–326 (2016).
[Crossref]

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (2013).
[Crossref]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Ultraweak long-range interactions of solitons observed over astronomical distances,” Nat. Photonics 7, 657–663 (2013).
[Crossref]

Opt. Commun. (1)

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

Opt. Express (2)

Opt. Lett. (12)

A. F. J. Runge, C. Aguergaray, N. G. R. Broderick, and M. Erkintalo, “Coherence and shot-to-shot spectral fluctuations in noise-like ultrafast fiber lasers,” Opt. Lett. 38, 4327–4330 (2013).
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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]

C. Joshi, J. K. Jang, K. 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).
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J. K. Jang, M. Erkintalo, J. Schröder, B. J. Eggleton, S. G. Murdoch, and S. Coen, “All-optical buffer based on temporal cavity solitons operating at 10  Gb/s,” Opt. Lett. 41, 4526–4529 (2016).
[Crossref]

K. E. Webb, M. Erkintalo, S. Coen, and S. G. Murdoch, “Experimental observation of coherent cavity soliton frequency combs in silica microspheres,” Opt. Lett. 41, 4613–4616 (2016).
[Crossref]

M. Conforti, A. Mussot, A. Kudlinski, and S. Trillo, “Modulational instability in dispersion oscillating fiber ring cavities,” Opt. Lett. 39, 4200–4203 (2014).
[Crossref]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Observation of dispersive wave emission by temporal cavity solitons,” Opt. Lett. 39, 5503–5506 (2014).
[Crossref]

K. Luo, Y. Xu, M. Erkintalo, and S. G. Murdoch, “Resonant radiation in synchronously pumped passive Kerr cavities,” Opt. Lett. 40, 427–430 (2015).
[Crossref]

J. M. Soto-Crespo, N. Akhmediev, P. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28, 1757–1759 (2003).
[Crossref]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011).
[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 model,” Opt. Lett. 38, 37–39 (2013).
[Crossref]

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
[Crossref]

Optica (1)

Phys. Rev. A (3)

B. A. Malomed, “Bound solitons in the nonlinear Schrödinger–Ginzburg–Landau equation,” Phys. Rev. A 44, 6954–6957 (1991).
[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]

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]

Phys. Rev. E (5)

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
[Crossref]

B. A. Malomed, “Bound states of envelope solitons,” Phys. Rev. E 47, 2874–2880 (1993).
[Crossref]

T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000).
[Crossref]

D. Cai, A. R. Bishop, N. Grønbech-Jensen, and B. A. Malomed, “Bound solitons in the AC-driven, damped nonlinear Schrödinger equation,” Phys. Rev. E 49, 1677–1679 (1994).
[Crossref]

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]

Phys. Rev. Lett. (7)

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[Crossref]

T. Herr, V. Brasch, D. Jost, J. 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).
[Crossref]

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901(2017).

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]

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

D. Turaev, A. G. Vladimirov, and S. Zelik, “Long-range interaction and synchronization of oscillating dissipative solitons,” Phys. Rev. Lett. 108, 263906 (2012).
[Crossref]

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[Crossref]

Sci. Rep. (1)

B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).

Science (2)

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. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

Other (9)

P. Parra-Rivas, D. Gomila, P. Colet, and L. Gelens, “Interaction of solitons and the formation of bound states in the generalized Lugiato-Lefever equation,” arXiv:1705.02619 (2017).

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, “Bound states of temporal cavity solitons,” in Nonlinear Photonics, NP’2014, Barcelona, Spain, July27–31, 2014 (Optical Society of America, 2014), paper JM5A.59.

D. C. Cole, E. S. Lamb, P. Del’Haye, S. A. Diddams, and S. B. Papp, “Soliton crystals in Kerr resonators,” arXiv:1610.00080 (2016).

E. Obrzud, S. Lecomte, and T. Herr, “Temporal solitons in microresonators driven by optical pulses,” arXiv:1612.08993 (2016).

G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems: from Dissipative Structures to Order through Fluctuations (Wiley, 1977).

N. N. Akhmediev, “General theory of solitons,” in Soliton-Driven Photonics, A. D. Boardman and A. P. Sukhorukov, eds., vol. 31 of NATO Science Series (Springer, 2001), pp. 371–395.

W. J. Firth, “Theory of cavity solitons,” in Soliton-Driven Photonics, A. D. Boardman and A. P. Sukhorukov, eds., vol. 31 of NATO Science Series (Springer, 2001), pp. 459–485.

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2013).

Y. S. Kivshar and G. P. Agrawal, “Vector solitons,” in Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003), Chap. 9, pp. 278–339.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic illustration of the universal temporal CS binding mechanism. A perturbation leads to a sideband in the soliton spectrum (blue arrow). That sideband is associated with an extended oscillatory tail in the temporal intensity profile. An adjacent CS gets trapped by the tail oscillations.

Fig. 2.
Fig. 2.

Experimental setup. PC, polarization controller; PD, photodiode; other acronyms defined in the text. DSF: β2=1.95  ps2/km, β3=0.2  ps3/km, and γ=2/(Wkm); SMF: β2=21.4  ps2/km, β3=0.1  ps3/km, and γ=1.2/(Wkm).

Fig. 3.
Fig. 3.

(a) Dynamical evolution of the separation between pairs of temporal CSs. Overlapping data are shown in different colors for clarity (black, green, and blue). Red curves are measured in the presence of additional ASE noise from EDFA2. (b) Measured (blue) and simulated (purple) spectra of an isolated CS. (c) Observation of a transition between two bound states when a short burst of ASE noise is injected (at time t7  s).

Fig. 4.
Fig. 4.

(a) Simulated temporal intensity profiles of an isolated temporal CS (blue) as well as a stable bound state with 7.31 ps separation (red; shifted vertically for clarity). (b) Vertical enlargement of the tail of the isolated CS. Blue dots denote maxima, while red (green) lines indicate the separations of simulated (experimentally observed) bound states.

Fig. 5.
Fig. 5.

Spectra of a single temporal CS (top) and dynamical evolution of the separation between a pair of such CSs (bottom) (a), (b) in a 100 m long all-SMF cavity, where CS binding is primarily induced by the Kelly sidebands (K), and (c), (d) in a 315 m long all-SMF cavity, where binding arises through excitation of orthogonally polarized resonances (P, P’). Notice the different time scales in (b) and (d) (ms versus s).

Fig. 6.
Fig. 6.

(a) Simulated temporal intensity profile of an isolated temporal CS (blue curves) and that of a particular bound state (red curve) obtained for the parameters of the experimental results shown in Figs 5(a)5(b) for the Kelly-sideband perturbed cavity. (b) Corresponding simulated spectrum of the isolated soliton.

Fig. 7.
Fig. 7.

Simulated temporal intensity profiles of the (a) driven (+) and (b) cross-polarized (−) modes of an isolated temporal CS (blue curves) and that of a particular bound state (red curve) in a cavity perturbed by fiber birefringence. The green curve in (a) is the ellipticity angle Ψ of the field with respect to the driving field. (b) Simulated (black lines) and measured (green lines) allowed bound-state separations. (c) Corresponding spectra.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

E+t=[1+i(|E+|2+B|E|2Δ)iη2τ2]E++iD[q1/2*E+2E*+2q1/2|E+|2E]+S,
Et=[1+i(|E|2+B|E+|2Δ)vτiη2τ2]E+iCq*E+2E*+iDq1/2*|E+|2E+,

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