Abstract

We show a new class of complex solitary wave that exists in a nonlinear optical cavity with appropriate dispersion characteristics. The cavity soliton consists of multiple soliton-like spectro-temporal components that exhibit distinctive colors but coincide in time and share a common phase, formed together via strong inter-soliton four-wave mixing and Cherenkov radiation. The multicolor cavity soliton shows intriguing spectral locking characteristics and remarkable capability of spectrum management to tailor soliton frequencies, which would be very useful for versatile generation and manipulation of multi-octave spanning phase-locked Kerr frequency combs, with great potential for applications in frequency metrology, optical frequency synthesis, and spectroscopy.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
  4. M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Comm. 91, 401–407 (1992).
    [Crossref]
  5. 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]
  6. S. Barland, M. Giudici, G. Tissoni, J. R. Tredicce, M. Brambilla, L. Lugiato, F. Prati, S. Barbay, R. Kuszelewicz, T. Ackemann, W. J. Firth, and G.-L. Oppo, “Solitons in semiconductor microcavities,” Nature 419, 699 (2002).
    [Crossref] [PubMed]
  7. M. Stratmann, T. Pagel, and F. Mitschke, “Experimental observation of temporal soliton molecules,” Phys. Rev. Lett. 95, 143902 (2005).
    [Crossref] [PubMed]
  8. A. Hause, H. Hartwig, M. Bohm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys. Rev. A 78, 063817 (2008).
    [Crossref]
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    [Crossref]
  10. 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]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  21. 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]
  22. S. Coen and M. Erkintalo, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  29. Y. Okawachi, M. R. E. Lamont, K. Luke, D. O. Carvalho, M. Yu, M. Lipson, and A. L. Gaeta, “Bandwidth shaping of microresonator-based frequency combs via dispersion engineering,” Opt. Lett. 39, 3535–3538 (2014).
    [Crossref] [PubMed]
  30. C. Milian 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] [PubMed]
  31. S. Wang, H. Guo, X. Bai, and X. Zeng, “Broadband Kerr frequency combs and intracavity soliton dynamics influenced by high-order cavity dispersion,” Opt. Lett. 39, 2880–2883 (2014).
    [Crossref] [PubMed]
  32. P. Parra-Rivas, D. Gomila, M. A. Matias, 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]
  33. V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeier, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598v2 (2014).
  34. S.-W. Huang, H. Zhou, J. Yang, J. F. McMillan, A. Matsko, M. Yu, D.-L. Kwong, L. Maleki, and C. W. Wong, “Mode-Locked ultrashort pulse generation from on-chip normal dispersion microresonators,” Phys. Rev. Lett. 114, 053901 (2015).
    [Crossref] [PubMed]
  35. 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,” Nature Photon. 9, 594–600 (2015).
    [Crossref]
  36. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
    [Crossref] [PubMed]
  37. N. R. Newbury, “Searching for applications with a fine-tooth comb,” Nature Photon. 5, 186–188 (2011).
    [Crossref]
  38. Eq. (1) neglects other factors such as dispersion of cavity Q, Raman scattering, etc., which would have effect in practice. However, they do not alter the formation nature of soliton molecule, while they perturb the details of comb spectrum. Detailed investigations are beyond the scope of this paper and will be discussed elsewhere.
  39. More strictly speaking, a fundamental soliton located at frequency ωj exhibits a propagation constant βs(ωj) ≡ β(ωj) + γPj/2 where Pj is the peak power of the soliton [1]. Therefore, the phase matching condition for inter-soliton FWM will be given by βs(ω3) + βs(ω4) − βs(ω1) − βs(ω2) = 0. However, as the solitons exhibit similar amplitudes, the phase matching condition would be dominated by the linear phase mismatch, as shown in Eq. (6). Same argument applies to the phase matching condition for inter-soliton Cheronkov radiation given in Eq. (7).
  40. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [Crossref]
  41. D. V. Skryabin and A. V. Gorbach, “Colloquium: Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
    [Crossref]
  42. The device is assumed to be a high-Q microresonator with κ0 = κe = 0. 0027, γ = 2.5 W−1 m−1, free-spectral range FSR = 226 GHz, L = 510.6 μm. The input pump wave is launched at ω0/2π = 193.55 THz with a power of Pin = |A0|2 = 3 W. The exact values of the parameters are not essential for the underlying physics. These values are used here simply as a device example to show the phenomena.
  43. P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
    [Crossref] [PubMed]
  44. L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685 (2012).
    [Crossref] [PubMed]
  45. M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband flat dispersion tailoring of dual-slot silicon waveguides,” Opt. Express 20, 15899–15907 (2012).
    [Crossref] [PubMed]
  46. William J. Tropf and Michael E. Thomas, “Infrared refractive index and thermo-optic coefficient measurement at APL,” Johns Hopkins APL Technical Digest 19, 3 (1998).
  47. Irving H. Malitson, “Refraction and dispersion of synthetic sapphire,” J. Opt. Soc. Am. 52, 12 (1962).
    [Crossref]
  48. H. Liang, Y. He, R. Luo, and Q. Lin, “Ultra-broadband dispersion engineering of nanophotonic devices with five zero-dispersion wavelengths,” Proc. Conf. Laser and Electro-Optics, JTh2A.114, (2016).
    [Crossref]

2015 (2)

S.-W. Huang, H. Zhou, J. Yang, J. F. McMillan, A. Matsko, M. Yu, D.-L. Kwong, L. Maleki, and C. W. Wong, “Mode-Locked ultrashort pulse generation from on-chip normal dispersion microresonators,” Phys. Rev. Lett. 114, 053901 (2015).
[Crossref] [PubMed]

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,” Nature Photon. 9, 594–600 (2015).
[Crossref]

2014 (8)

S. Coen and M. Erkintalo, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014).
[Crossref] [PubMed]

W. Loh, P. Del’Haye, S. B. Papp, and S. A. Diddams, “Phase and coherence of optical microresonator frequency combs,” Phys. Rev. A 89, 053810 (2014).
[Crossref]

P. Parra-Rivas, D. Gomila, F. Leo, S. Coen, and L. Gelens, “Third-order chromatic dispersion stabilizes Kerr frequency combs,” Opt. Lett. 39, 2971–2974 (2014).
[Crossref] [PubMed]

Y. Okawachi, M. R. E. Lamont, K. Luke, D. O. Carvalho, M. Yu, M. Lipson, and A. L. Gaeta, “Bandwidth shaping of microresonator-based frequency combs via dispersion engineering,” Opt. Lett. 39, 3535–3538 (2014).
[Crossref] [PubMed]

C. Milian 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] [PubMed]

S. Wang, H. Guo, X. Bai, and X. Zeng, “Broadband Kerr frequency combs and intracavity soliton dynamics influenced by high-order cavity dispersion,” Opt. Lett. 39, 2880–2883 (2014).
[Crossref] [PubMed]

P. Parra-Rivas, D. Gomila, M. A. Matias, 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,” Nature Photon. 8, 145–152 (2014).
[Crossref]

2013 (7)

2012 (7)

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-pump-power, low-phase-noise, and microwave to millimeter-wave repetition rate operation in microcombs,” Phys. Rev. Lett. 109, 233901 (2012).
[Crossref]

M. Erkintalo, Y. Q. Xu, S. G. Murdoch, J. M. Dudley, and G. Genty, “Cascaded phase matching and nonlinear symmetry breaking in fiber frequency combs,” Phys. Rev. Lett. 109, 223904 (2012).
[Crossref]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A 85, 023830 (2012).
[Crossref]

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nature Photon. 6, 84–92 (2012).
[Crossref]

P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
[Crossref] [PubMed]

L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685 (2012).
[Crossref] [PubMed]

M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband flat dispersion tailoring of dual-slot silicon waveguides,” Opt. Express 20, 15899–15907 (2012).
[Crossref] [PubMed]

2011 (5)

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

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).
[Crossref] [PubMed]

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]

Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, “Octave-spanning frequency comb generation in a silicon nitride chip,” Opt. Lett. 36, 3398–3400 (2011).
[Crossref] [PubMed]

N. R. Newbury, “Searching for applications with a fine-tooth comb,” Nature Photon. 5, 186–188 (2011).
[Crossref]

2010 (2)

D. V. Skryabin and A. V. Gorbach, “Colloquium: Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[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,” Nature Photon. 4, 471–476 (2010).
[Crossref]

2009 (1)

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative soliton molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80, 043829 (2009).
[Crossref]

2008 (1)

A. Hause, H. Hartwig, M. Bohm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys. Rev. A 78, 063817 (2008).
[Crossref]

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

2005 (1)

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental observation of temporal soliton molecules,” Phys. Rev. Lett. 95, 143902 (2005).
[Crossref] [PubMed]

2002 (2)

S. Barland, M. Giudici, G. Tissoni, J. R. Tredicce, M. Brambilla, L. Lugiato, F. Prati, S. Barbay, R. Kuszelewicz, T. Ackemann, W. J. Firth, and G.-L. Oppo, “Solitons in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[Crossref] [PubMed]

1998 (1)

William J. Tropf and Michael E. Thomas, “Infrared refractive index and thermo-optic coefficient measurement at APL,” Johns Hopkins APL Technical Digest 19, 3 (1998).

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]

1992 (1)

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

1987 (1)

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

1962 (1)

Irving H. Malitson, “Refraction and dispersion of synthetic sapphire,” J. Opt. Soc. Am. 52, 12 (1962).
[Crossref]

Ackemann, T.

S. Barland, M. Giudici, G. Tissoni, J. R. Tredicce, M. Brambilla, L. Lugiato, F. Prati, S. Barbay, R. Kuszelewicz, T. Ackemann, W. J. Firth, and G.-L. Oppo, “Solitons in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Agarwal, A. M.

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Akhmediev, N.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nature Photon. 6, 84–92 (2012).
[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]

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]

Bai, X.

Bao, C.

Barbay, S.

S. Barland, M. Giudici, G. Tissoni, J. R. Tredicce, M. Brambilla, L. Lugiato, F. Prati, S. Barbay, R. Kuszelewicz, T. Ackemann, W. J. Firth, and G.-L. Oppo, “Solitons in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Barland, S.

S. Barland, M. Giudici, G. Tissoni, J. R. Tredicce, M. Brambilla, L. Lugiato, F. Prati, S. Barbay, R. Kuszelewicz, T. Ackemann, W. J. Firth, and G.-L. Oppo, “Solitons in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Beausoleil, R. G.

Bohm, M.

A. Hause, H. Hartwig, M. Bohm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys. Rev. A 78, 063817 (2008).
[Crossref]

Brambilla, M.

S. Barland, M. Giudici, G. Tissoni, J. R. Tredicce, M. Brambilla, L. Lugiato, F. Prati, S. Barbay, R. Kuszelewicz, T. Ackemann, W. J. Firth, and G.-L. Oppo, “Solitons in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Brasch, V.

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,” Nature Photon. 8, 145–152 (2014).
[Crossref]

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeier, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598v2 (2014).

Carvalho, D. O.

Chembo, Y. K.

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]

Chen, S.

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,” Nature Photon. 9, 594–600 (2015).
[Crossref]

Chen, T.

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-pump-power, low-phase-noise, and microwave to millimeter-wave repetition rate operation in microcombs,” Phys. Rev. Lett. 109, 233901 (2012).
[Crossref]

Coen, S.

S. Coen and M. Erkintalo, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014).
[Crossref] [PubMed]

P. Parra-Rivas, D. Gomila, F. Leo, S. Coen, and L. Gelens, “Third-order chromatic dispersion stabilizes Kerr frequency combs,” Opt. Lett. 39, 2971–2974 (2014).
[Crossref] [PubMed]

P. Parra-Rivas, D. Gomila, M. A. Matias, 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 model,” Opt. Lett. 38, 37–39 (2013).
[Crossref] [PubMed]

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]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Colman, P.

P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
[Crossref] [PubMed]

Combrié, S.

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Nature (2)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
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Opt. Comm. (1)

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Opt. Lett. (10)

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Other (6)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Eq. (1) neglects other factors such as dispersion of cavity Q, Raman scattering, etc., which would have effect in practice. However, they do not alter the formation nature of soliton molecule, while they perturb the details of comb spectrum. Detailed investigations are beyond the scope of this paper and will be discussed elsewhere.

More strictly speaking, a fundamental soliton located at frequency ωj exhibits a propagation constant βs(ωj) ≡ β(ωj) + γPj/2 where Pj is the peak power of the soliton [1]. Therefore, the phase matching condition for inter-soliton FWM will be given by βs(ω3) + βs(ω4) − βs(ω1) − βs(ω2) = 0. However, as the solitons exhibit similar amplitudes, the phase matching condition would be dominated by the linear phase mismatch, as shown in Eq. (6). Same argument applies to the phase matching condition for inter-soliton Cheronkov radiation given in Eq. (7).

The device is assumed to be a high-Q microresonator with κ0 = κe = 0. 0027, γ = 2.5 W−1 m−1, free-spectral range FSR = 226 GHz, L = 510.6 μm. The input pump wave is launched at ω0/2π = 193.55 THz with a power of Pin = |A0|2 = 3 W. The exact values of the parameters are not essential for the underlying physics. These values are used here simply as a device example to show the phenomena.

H. Liang, Y. He, R. Luo, and Q. Lin, “Ultra-broadband dispersion engineering of nanophotonic devices with five zero-dispersion wavelengths,” Proc. Conf. Laser and Electro-Optics, JTh2A.114, (2016).
[Crossref]

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeier, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598v2 (2014).

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Figures (7)

Fig. 1
Fig. 1 Schematic of a multicolor cavity soliton, composed of soliton-like spectro-temporal components with distinctive colors. The figure also shows the the corresponding sinusoidally oscillatory GVD (red), group index (green), phase mismatch for FWM (orange), and phase mismatch for Cherenkov radiation (blue).
Fig. 2
Fig. 2 Formation dynamics of a three-color cavity soliton. (a) Laser-cavity detuning Δ0 increases linearly from 0 to 0.4 within 9.29 ns and stays constant afterwards. (b) Phase mismatch ΔβFWM for FWM process. (c) Spectral growth of the multicolor cavity soliton. The scaling of the time axis changes at t = 5 ns (the vertical dashed line) to show the formation details of soliton and its stability, which is also responsible for the kink in (a). (d) Schematic of the mode locking mechanism. The device parameters are given in [42].
Fig. 3
Fig. 3 Temporal and spectral structure of a three-color cavity soliton. (a) Pulse waveform. (b) Spectrogram. (c) Spectrum (blue) and phase (magenta). The arrow indicates the discrete phase of the pump mode. The dashed curves are hyperbolic-secant fittings of the three distinct spectral components. (d) GVD (blue) and corresponding ΔβCR(ω0,ω) (red), where ω0 is the pump frequency.
Fig. 4
Fig. 4 Spectra of three-color cavity solitons when the pump frequency varies from 189.93 to 198.97 THz, with a step of 0.904 THz (corresponding to 4 FSRs). Each spectrum is relatively shifted by 20 dB for better comparison. The stars denote the spectral peaks of the solitons. Right panel: zoom-in spectra in the range of 187 203 THz.
Fig. 5
Fig. 5 (a) Spectrum (blue) and phase (magenta) of a four-color cavity soliton. The arrow indicates the pump phase, and dashed hyperbolic-secant curves fit the four spectral components. (b) GVD (blue) and ΔβCR(ω0,ω) (red). Device parameters are given in [42].
Fig. 6
Fig. 6 Formation dynamics of a four-color cavity soliton. The figure structure is the same as Fig. 2. In (a), Δ0 increases linearly from 0 to 0.4 within 13.27 ns and stays constant afterwards. Note again the change of time scaling at t = 5 ns.
Fig. 7
Fig. 7 (a) Schematic of a suspended microring coupled with a phase-matched bus waveguide. (b) Cross section of the suspended microring made of SiC and Al2O3, and the field profile of its fundamental quasi-TM mode (at pump wavelength). (c) GVD (blue) and ΔβCR (red). The GVD curve is simulated by the finite element method. The dimensions of the waveguide structure are: h1 = 200 nm, h2 = 100 nm, h3 = 254 nm, h4 = 200 nm, and w = 500 nm. The material dispersions are from Refs. [46] and [47]. (d) Spectrum of the two-color cavity soliton produced inside the microring, simulated via Eq. (1). The dashed curves are hyperbolic-secant fittings of two solitons. The pump is launched at 1600 nm with Pin = |A0|2 = 1.1 W. Other parameters are Δ0 = 0.086, κ0 = κe = 0.0029, γ = 2.5 W−1m−1, FSR = 200 GHz, and L = 522.2 μm. (e) Schematic of the mode locking mechanism. The figure is shown as a function of wavelength instead of frequency, simply in a direct correspondence to (d).

Equations (7)

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t R E t = ( κ t 2 i Δ 0 ) E + m = 2 i m + 1 β m m ! m E τ m + i γ L ( 1 + i ω 0 τ ) ( | E | 2 E ) + κ e A 0 ,
E ( t , τ ) E 0 + n = 1 N E n sech ( τ / T n ) e i ( ω n ω 0 ) τ e i ϕ n ,
n g ( ω ) = n g ( ω 0 ) + B c Ω sin ( ω ω 0 Ω ) ,
β ( ω ) = β ( ω 0 ) + n g ( ω 0 ) c ( ω ω 0 ) + 2 B Ω 2 sin 2 ( ω ω 0 2 Ω ) .
Δ β FWM ( ω s ) β ( ω s ) + β ( ω i ) 2 β ( ω 0 ) = 4 B Ω 2 sin 2 ( ω s ω 0 2 Ω ) ,
β ( ω 3 ) + β ( ω 4 ) β ( ω 1 ) β ( ω 2 ) 0 ,
Δ β CR ( ω i , ω j ) β ( ω i ) β ( ω j ) n g ( ω i ) c ( ω i ω j ) 0 .

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