## Soliton families and resonant radiation in a micro-ring resonator near zero group-velocity dispersion |

Optics Express, Vol. 22, Issue 3, pp. 3732-3739 (2014)

http://dx.doi.org/10.1364/OE.22.003732

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### Abstract

We report theoretical and numerical study of the dynamical and spectral properties of the conservative and dissipative solitons in micro-ring resonators pumped in a proximity of the zero of the group velocity dispersion. We discuss frequency and velocity locking of the conservative solitons, when dissipation is accounted for. We present theory of the dispersive radiation emitted by such solitons, report their Hopf instability and radiation enhancement by multiple solitons.

© 2014 Optical Society of America

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

2. P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. **107**, 063901 (2011). [CrossRef]

3. Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A **82**, 033801 (2010). [CrossRef]

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

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

8. M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. **38**, 3478–3481 (2013). [CrossRef] [PubMed]

9. L. Zhang, C. Bao, V. Singh, J. Mu, C. Yang, A.M. Agarwal, L.C. Kimerling, and J. Michel, “Generation of two-cycle pulses and octave-spanning frequency combs in a dispersion-flattened micro-resonator,” Opt. Lett. **38**, 5122–5125 (2013). [CrossRef] [PubMed]

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

8. M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. **38**, 3478–3481 (2013). [CrossRef] [PubMed]

10. D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. **82**, 1287–1299 (2010). [CrossRef]

14. M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. **35**, 306–309 (2010). [CrossRef] [PubMed]

15. M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A **88**, 035802 (2013). [CrossRef]

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

15. M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A **88**, 035802 (2013). [CrossRef]

**38**, 37–39 (2013). [CrossRef] [PubMed]

8. M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. **38**, 3478–3481 (2013). [CrossRef] [PubMed]

10. D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. **82**, 1287–1299 (2010). [CrossRef]

16. 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,” Nature Photon. **4**, 471–476 (2010). [CrossRef]

*t*=

*Tτ*,

*τ*=

*c*/[2

*πRn*] (so that the group velocity coefficient is unity), and distance

_{g}*z*=

*ZL*,

*L*= 2

*πR*, we get a more handy form of the dimensionless gLLE: where

*γ*= Γ

*τ*,

*δ*= (

*ω*

_{0}−

*ω*)

_{p}*τ*,

*T*,

*Z*) =

*ψ*(

*Z*− (

*v*+ 1)

*T*), where

*ψ*obeys The amplitude of the single mode state at

*Q*= 0, found from (

*iγ*−

*δ*+ 2|Ψ

_{0}|

^{2})Ψ

_{0}+

*h*= 0, is multivalued (bistable) in the soliton existence range and the solitons are nested on the background given by the root with the smallest value of |Ψ

_{0}|

^{2}, see Fig. 1(a).

*γ*= 0 corresponds to the Hamiltonian (conservative) case and is a convenient starting point to understand physical properties of solitons and their radiation. The field momentum

*v*of the group velocity, while all other parameters are kept fixed [18

18. I. V. Barashenkov and E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrdinger equation,” J. Phys. A: Math. Theor. **44**, 1–23 (2011). [CrossRef]

*v*-family is known in the analytical form for

*h*=

*B*

_{3}= 0:

*β*

_{2}< 0 and

*B*

_{2}> 0. Numerically calculated plots of

*M*vs

*v*for several pump values are shown in Fig. 1(b). All branches are plotted only up to their turning points,

*∂*→ ∞, where the multi-hump solitons emerge, since our interest is focused here on the single hump solitons only (see Ref. [18

_{v}M18. I. V. Barashenkov and E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrdinger equation,” J. Phys. A: Math. Theor. **44**, 1–23 (2011). [CrossRef]

*h*= 5 × 10

^{−4}are shown in Fig. 2(a–d). Spectra of these solutions provide a useful physical insight. As

*v*increases, the soliton core spectrally separates from the pump, so that the pump ability to supply energy into the soliton diminishes. Thus, when losses are accounted for, the soliton adiabatically decays in time without any change in its momentum, see Fig. 2(g,h). If, however, the spectral offset is moderate to small, then the soliton is fed by the pump efficiently, so its velocity shift

*v*evolves towards zero and its momentum converges to the pump momentum, see Fig. 2(e,f). For

*γ*≠ 0, the momentum is not conserved and there exists only one soliton state with

*v*= 0 [19

19. I. V. Barashenkov and Yu. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrdinger solitons,” Phys. Rev. E **54**, 5707–5725 (1996). [CrossRef]

*B*

_{3}= 0. Before we proceed further, we describe a physical system, which is used to present numerical data in this work including the ones in the already discussed Fig. 2. We consider a silicon nitride ring resonator of radius

*R*= 0.9 mm with the cross section 500 × 730 nm

^{2}, see Fig. 1(c). The zero GVD wavelength (

*β*

_{2}= 0) is at ≈ 1586 nm.

*β*

_{3}≠ 0, in optical fibers leads to the emission of the resonant radiation by solitons [10

10. D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. **82**, 1287–1299 (2010). [CrossRef]

9. L. Zhang, C. Bao, V. Singh, J. Mu, C. Yang, A.M. Agarwal, L.C. Kimerling, and J. Michel, “Generation of two-cycle pulses and octave-spanning frequency combs in a dispersion-flattened micro-resonator,” Opt. Lett. **38**, 5122–5125 (2013). [CrossRef] [PubMed]

12. V. V. Afanasjev, Y. S. Kivshar, and C. R. Menyuk, “Effect of third-order dispersion on dark solitons,” Opt. Lett. **21**, 1975–1977 (1996). [CrossRef] [PubMed]

*ψ*(

*x*) +

*g*(

*t*,

*x*), where

*g*we find Then, the dispersion law for the radiation is

*Q*) = 0. The corresponding physical frequencies are found as

_{r}*ω*

_{0}+

*Q*[

_{r}*v*+ 1]

*c*/[2

*πRn*]. Graphical solutions of Ω(

_{g}*Q*) = 0 are shown in Fig. 3(a). One can see that there exist two resonances symmetrically located with respect to the pump. The existence of two real nonzero roots is common to all radiating nonlinear waves which are embedded in a non-vanishing background and in particular this has been reported in the case of fiber dark solitons [11

_{r}11. V. I. Karpman, “Stationary and radiating dark solitons of the third order nonlinear Schrodinger equation,” Phys. Lett. A **181**, 211–215 (1993). [CrossRef]

13. C. Milián, D. V. Skryabin, and A. Ferrando, “Continuum generation by dark solitons,” Opt. Lett. **34**, 2096–2098 (2009). [CrossRef] [PubMed]

*x*(equivalently, in

*z*) makes the exact resonance possible only for

*Q*=

_{r}*q*/(2

*π*) (since dimensionless cavity length is 1),

*q*= 0, ±1,..., implying that in practice we are dealing with the quasi-resonant radiation in this system.

*W*=

*Ŵ*(

*Q*=

*Q*,

_{r}*∂*= 0). Note, that for the wavelength interval considered here

_{x}*β*

_{3}< 0 and the short wavelength resonance is driven through the nonlinear mixing of the soliton background with the primary resonance, and therefore its amplitude is relatively small and tends to zero together with

*h*. The long wavelength resonance is much stronger and it retains the non-zero amplitude in the limit

*h*= 0, corresponding to the case of free (no background) propagation [10

**82**, 1287–1299 (2010). [CrossRef]

*β*

_{3}> 0 the short wavelength resonance becomes domineering.

*δ*are shown in Fig. 3(b). While the pump frequency varies roughly within the cavity free spectral range (FSR), the resonance frequency swipes across many FSRs. The question to consider now is will the periodic boundary conditions imposed by the cavity have an appreciable impact on the radiation amplitude, when its frequency resonates with the cavity mode. In quantitative terms, this problem is best addressed numerically, since the soliton core is expected to be altered when the radiation feeds back on the soliton through the periodic boundaries and the radiation frequency is going to be influenced by the soliton being on its way. Therefore we solve Eq. (3), for

*γ*= 0, numerically and find the soliton solutions nesting on top of the radiation wave, see Figs. 4(a,b). Conservative solitons shown here are numerical continuation along

*B*

_{3}and

*δ*of those at

*h*= 5 × 10

^{−4}and

*v*= −1.25 in Fig. 1(b) (

*v*is also a free parameter for

*B*

_{3}≠ 0). The plot of the radiation amplitude,

*α*, vs

*δ*is shown in Fig. 3(d), one can see the series of peaks that appear for the values of

*δ*producing the resonant radiation nearly matching the cavity resonances, see Fig. 3(b). Matching is not exact, primarily due to the effect of the soliton core on the effective length of the cavity. The maxima of the radiation amplitude corresponds to the amplitude of the soliton background, which drops with

*δ*.

*γ*≠ 0 for

*β*

_{3}≠ 0 has the two fold impact on solitons. First, it selects the value of the group velocity shift

*v*to be non-zero, unlike

*v*= 0 selection for

*β*

_{3}= 0, see Fig. 5 (note

*γ*= 0 for

*t*≤ 500 and

*γ*> 0 for

*t*> 500). Second, it damps the radiation in space, so that solitons propagate together with the extended, but localized, radiation tail attached to them, see Fig. 6.

*v*≠ 0 implies that the soliton carrier frequency is detuned from the pump. This can be interpreted as the spectral recoil effect from the radiation on the soliton [10

**82**, 1287–1299 (2010). [CrossRef]

*β*

_{2}= 0 wavelength. It is also important to notice that the soliton state persists, when the pump frequency shifts into the normal GVD range, while the soliton spectral maximum remains in the range of anomalous GVD, see Fig. 6(c), 7(a). The corresponding changes of the soliton velocity with the wavelength of the reference cavity mode approaching

*β*

_{3}= 0 are shown in Fig. 7(b), where the pump frequency is assumed to change simultaneously, so that

*δ*is kept constant.

## Acknowledgments

## References and links

1. | T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator based optical frequency combs,” Science |

2. | P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. |

3. | Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A |

4. | Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A |

5. | A. B. Matsko, A. A. Savachenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. |

6. | 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 cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. |

7. | S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field LugiatoLefever model,” Opt. Lett. |

8. | M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. |

9. | L. Zhang, C. Bao, V. Singh, J. Mu, C. Yang, A.M. Agarwal, L.C. Kimerling, and J. Michel, “Generation of two-cycle pulses and octave-spanning frequency combs in a dispersion-flattened micro-resonator,” Opt. Lett. |

10. | D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. |

11. | V. I. Karpman, “Stationary and radiating dark solitons of the third order nonlinear Schrodinger equation,” Phys. Lett. A |

12. | V. V. Afanasjev, Y. S. Kivshar, and C. R. Menyuk, “Effect of third-order dispersion on dark solitons,” Opt. Lett. |

13. | C. Milián, D. V. Skryabin, and A. Ferrando, “Continuum generation by dark solitons,” Opt. Lett. |

14. | M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. |

15. | M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A |

16. | 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,” Nature Photon. |

17. | F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. |

18. | I. V. Barashenkov and E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrdinger equation,” J. Phys. A: Math. Theor. |

19. | I. V. Barashenkov and Yu. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrdinger solitons,” Phys. Rev. E |

20. | A. B. Matsko, A. A. Savchenkov, and L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett. |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(320.6629) Ultrafast optics : Supercontinuum generation

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 18, 2013

Revised Manuscript: January 27, 2014

Manuscript Accepted: January 29, 2014

Published: February 7, 2014

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3732

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### References

- T. J. Kippenberg, R. Holzwarth, S. A. Diddams, “Microresonator based optical frequency combs,” Science 332, 555–559 (2011). [CrossRef] [PubMed]
- P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011). [CrossRef]
- Y. K. Chembo, N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010). [CrossRef]
- Y. K. Chembo, C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Phys. Rev. A 87, 053852 (2013). [CrossRef]
- A. B. Matsko, A. A. Savachenkov, W. Liang, V. S. Ilchenko, D. Seidel, L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011). [CrossRef] [PubMed]
- A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, Y. K. Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with whispering-gallery-mode resonators,” IEEE Photonics J. 5, 6100409 (2013). [CrossRef]
- S. Coen, H. G. Randle, T. Sylvestre, M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field LugiatoLefever model,” Opt. Lett. 38, 37–39 (2013). [CrossRef] [PubMed]
- M. R. E. Lamont, Y. Okawachi, A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. 38, 3478–3481 (2013). [CrossRef] [PubMed]
- L. Zhang, C. Bao, V. Singh, J. Mu, C. Yang, A.M. Agarwal, L.C. Kimerling, J. Michel, “Generation of two-cycle pulses and octave-spanning frequency combs in a dispersion-flattened micro-resonator,” Opt. Lett. 38, 5122–5125 (2013). [CrossRef] [PubMed]
- D. V. Skryabin, A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010). [CrossRef]
- V. I. Karpman, “Stationary and radiating dark solitons of the third order nonlinear Schrodinger equation,” Phys. Lett. A 181, 211–215 (1993). [CrossRef]
- V. V. Afanasjev, Y. S. Kivshar, C. R. Menyuk, “Effect of third-order dispersion on dark solitons,” Opt. Lett. 21, 1975–1977 (1996). [CrossRef] [PubMed]
- C. Milián, D. V. Skryabin, A. Ferrando, “Continuum generation by dark solitons,” Opt. Lett. 34, 2096–2098 (2009). [CrossRef] [PubMed]
- M. Tlidi, L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. 35, 306–309 (2010). [CrossRef] [PubMed]
- M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Phys. Rev. A 88, 035802 (2013). [CrossRef]
- F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nature Photon. 4, 471–476 (2010). [CrossRef]
- F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, M. Taki, “Nonlinear symmetry breaking induced by third order dispersion in optical fiber cavities,” Phys. Rev. Lett. 110, 104103 (2013). [CrossRef]
- I. V. Barashenkov, E. V. Zemlyanaya, “Travelling solitons in the externally driven nonlinear Schrdinger equation,” J. Phys. A: Math. Theor. 44, 1–23 (2011). [CrossRef]
- I. V. Barashenkov, Yu. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrdinger solitons,” Phys. Rev. E 54, 5707–5725 (1996). [CrossRef]
- A. B. Matsko, A. A. Savchenkov, L. Maleki, “On excitation of breather solitons in an optical microresonator,” Opt. Lett. 37, 4856–4858 (2012). [CrossRef] [PubMed]

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