Abstract

We theoretically investigate the phase locking phenomena between the spectral components of Kerr optical frequency combs in the dynamical regime of Turing patterns. We show that these Turing patterns display a particularly strong and robust phase locking, originating from a cascade of phase locked triplets which asymptotically lead to a global phase locking between the modes. The local and global phase locking relationships defining the shape of the comb are analytically determined. Our analysis also shows that solitons display a much weaker phase locking that can be destroyed more easily than in the Turing pattern regime. Our results indicate that Turing patterns are generally the most suitable for applications requiring the highest stability. Experimental generation of such combs is also discussed in detail, and is in excellent agreement with the numerical simulations.

© 2014 Optical Society of America

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  1. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
    [CrossRef]
  2. P. Del’Haye, S. B. Papp, and S. A. Diddams, Phys. Rev. Lett. 109, 263901 (2012).
    [CrossRef]
  3. J. Li, H. Lee, T. Chen, and K. Vahala, Phys. Rev. Lett. 109, 233901 (2012).
    [CrossRef]
  4. A. B. Matsko and L. Maleki, Opt. Express 21, 28862 (2013).
    [CrossRef]
  5. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
    [CrossRef]
  6. Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
    [CrossRef]
  7. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
    [CrossRef]
  8. I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).
  9. C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: case of normal dispersion,” arXiv:1308.2539 (2013).
  10. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Phys. Rev. Lett. 93, 083904 (2004).
    [CrossRef]
  11. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
    [CrossRef]
  12. Y. K. Chembo, D. V. Strekalov, and N. Yu, Phys. Rev. Lett. 104, 103902 (2010).
    [CrossRef]
  13. Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
    [CrossRef]
  14. A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
    [CrossRef]
  15. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
    [CrossRef]
  16. T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
    [CrossRef]
  17. M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, Opt. Lett. 38, 3478 (2013).
    [CrossRef]

2013 (6)

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
[CrossRef]

M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, Opt. Lett. 38, 3478 (2013).
[CrossRef]

A. B. Matsko and L. Maleki, Opt. Express 21, 28862 (2013).
[CrossRef]

2012 (2)

P. Del’Haye, S. B. Papp, and S. A. Diddams, Phys. Rev. Lett. 109, 263901 (2012).
[CrossRef]

J. Li, H. Lee, T. Chen, and K. Vahala, Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

2011 (2)

2010 (2)

Y. K. Chembo, D. V. Strekalov, and N. Yu, Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef]

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

2008 (1)

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

2005 (1)

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

2004 (1)

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef]

Balakireva, I.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: case of normal dispersion,” arXiv:1308.2539 (2013).

Chembo, Y. K.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Y. K. Chembo, D. V. Strekalov, and N. Yu, Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef]

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: case of normal dispersion,” arXiv:1308.2539 (2013).

Chen, T.

J. Li, H. Lee, T. Chen, and K. Vahala, Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

Coen, S.

Coillet, A.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: case of normal dispersion,” arXiv:1308.2539 (2013).

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).

Del’Haye, P.

P. Del’Haye, S. B. Papp, and S. A. Diddams, Phys. Rev. Lett. 109, 263901 (2012).
[CrossRef]

Diddams, S. A.

P. Del’Haye, S. B. Papp, and S. A. Diddams, Phys. Rev. Lett. 109, 263901 (2012).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

Dudley, J.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

Erkintalo, M.

Gaeta, A. L.

Godey, C.

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: case of normal dispersion,” arXiv:1308.2539 (2013).

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).

Hansson, T.

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Henriet, R.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Kippenberg, T. J.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef]

Lamont, M. R. E.

Larger, L.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

Lee, H.

J. Li, H. Lee, T. Chen, and K. Vahala, Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

Li, J.

J. Li, H. Lee, T. Chen, and K. Vahala, Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

Liang, W.

Maleki, L.

A. B. Matsko and L. Maleki, Opt. Express 21, 28862 (2013).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Matsko, A. B.

A. B. Matsko and L. Maleki, Opt. Express 21, 28862 (2013).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Menyuk, C.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

Menyuk, C. R.

Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

Modotto, D.

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Okawachi, Y.

Papp, S. B.

P. Del’Haye, S. B. Papp, and S. A. Diddams, Phys. Rev. Lett. 109, 263901 (2012).
[CrossRef]

Randle, H. G.

Saleh, K.

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

Savchenkov, A. A.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Seidel, D.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Solomatine, I.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Spillane, S. M.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef]

Strekalov, D.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Strekalov, D. V.

Y. K. Chembo, D. V. Strekalov, and N. Yu, Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef]

Sylvestre, T.

Vahala, K.

J. Li, H. Lee, T. Chen, and K. Vahala, Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

Vahala, K. J.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef]

Wabnitz, S.

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Yu, N.

Y. K. Chembo, D. V. Strekalov, and N. Yu, Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef]

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

IEEE Photon. J. (1)

A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. Dudley, C. Menyuk, and Y. K. Chembo, IEEE Photon. J. 5, 6100409 (2013).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (4)

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Phys. Rev. Lett. (5)

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Y. K. Chembo, D. V. Strekalov, and N. Yu, Phys. Rev. Lett. 104, 103902 (2010).
[CrossRef]

P. Del’Haye, S. B. Papp, and S. A. Diddams, Phys. Rev. Lett. 109, 263901 (2012).
[CrossRef]

J. Li, H. Lee, T. Chen, and K. Vahala, Phys. Rev. Lett. 109, 233901 (2012).
[CrossRef]

Science (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

Other (2)

I. Balakireva, A. Coillet, C. Godey, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part II: case of anomalous dispersion,” arXiv:1308.2542 (2013).

C. Godey, I. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs. Part I: case of normal dispersion,” arXiv:1308.2539 (2013).

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Figures (6)

Fig. 1.
Fig. 1.

Experimental setup.

Fig. 2.
Fig. 2.

Comparison between experimental [(a), (c), and (e)] and theoretical [(b), (d), and (f)] multiple-FSR spectra of Kerr combs corresponding to Turing patterns. Various line spacings are observed ranging from 7 to 88 FSR (41–516 GHz).

Fig. 3.
Fig. 3.

Simulated spectra of (a) a Turing pattern with L=12 and (b) a cavity soliton.

Fig. 4.
Fig. 4.

Time evolution of the relative phases of the 4 first modes of the Turing pattern of Fig. 3(a). After a delay increasing with the mode number, the relative phases reach a constant value, the Kerr comb becoming phase locked.

Fig. 5.
Fig. 5.

Phase locking behavior for different noisy initial conditions (represented by different colors). (a) Phase difference, ϕ12=φ12φ0. Evidence of triplet phase locking [(b), ξ2 is constant], and global phase locking [(c), Ξ5 is constant].

Fig. 6.
Fig. 6.

Variation of the relative phases, Δϕl, of different spectral modes of the Kerr combs while the pump signal undergoes an abrupt phase transition. (a) and (c) Case of the Turing pattern of Fig. 3(a). (b) and (d) Case of the bright soliton comb of Fig. 3(b). At t=0, the phase of the pump is shifted by π/5 in (a) and (b) and by π/2 for (c) and (d). In the π/5 phase shift case, oscillations in the relative phases of the comb’s lines are observed, but the phase locking returns to its previous state and value. The amplitude and duration of these oscillations are much larger in the case of the bright soliton compared to the Turing patterns case. In the π/2 phase shift case, while the phase locking of the Turing pattern is maintained, the soliton is destroyed, and thus the phase locking.

Equations (6)

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

ψτ=(1+iα)ψ+i|ψ|2ψiβ22ψθ2+F,
ψ(θ,τ)=lΨl(τ)eilθ,
dΨldτ=[(1+iα)+iβ2l2]Ψl+δ(l)F+im,n,pδ(mn+pl)ΨmΨn*Ψp,
κkLΨkL=δ(k)F+|ΨkL|2ΨkL+2{|Ψ(k1)L|2ΨkL+|Ψ(k+1)L|2ΨkL+Ψ(k1)LΨkL*Ψ(k+1)L},
ξk=φ(k1)L2φkL+φ(k+1)L=constant.
ΞN=12k=(N1)k=N1[1(1)N+k]ξk=φNL+2k=(N1)k=N1(1)N+kφkL+φNL=Constant.

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