Abstract

We use numerical simulations based on an extended Lugiato–Lefever equation (LLE) to investigate the stability properties of Kerr frequency combs generated in microresonators. In particular, we show that an ensemble average calculated over sequences of output fields separated by a fixed number of resonator roundtrips allows the coherence of Kerr combs to be quantified in terms of the complex degree of first-order coherence. We identify different regimes of comb coherence, linked to the solutions of the LLE. Our approach provides a practical and unambiguous way of assessing the stability of Kerr combs that is directly connected to an accessible experimental quantity.

© 2014 Optical Society of America

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2013

2012

2011

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

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
[CrossRef]

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

Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, Opt. Lett. 36, 3398 (2011).
[CrossRef]

G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, J. Opt. Soc. Am. B 28, 2301 (2011).
[CrossRef]

2010

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

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

2008

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

2007

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

2006

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

2004

2003

2002

2000

1993

Aguergaray, C.

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Baumgartel, L.

Bellini, M.

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,” arXiv: 1211.0733 (2012).

Broderick, N. G. R.

Chembo, Y. K.

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]

Chen, L.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Coen, S.

Del’Haye, P.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Diddams, S. A.

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

S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Dudley, J. M.

Emplit, Ph.

F. Leo, L. Gelens, Ph. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, Ph. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Erkintalo, M.

Ferdous, F.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Foster, M. A.

Friberg, A. T.

Gaeta, A. L.

Gavartin, E.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Gelens, L.

Genty, G.

Gorodetsky, M. L.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[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,” arXiv: 1211.0733 (2012).

Gorza, S.-P.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Grudinin, I. S.

Gu, X.

Haelterman, M.

F. Leo, L. Gelens, Ph. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, Ph. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Hänsch, T. W.

Hartinger, K.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

Herr, T.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[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,” arXiv: 1211.0733 (2012).

Holzwarth, R.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

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

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[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]

Jang, J. K.

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[CrossRef]

Johnson, A. R.

Jost, J. D.

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,” arXiv: 1211.0733 (2012).

Kimmel, M.

Kippenberg, T. J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

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

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[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,” arXiv: 1211.0733 (2012).

Knox, W. H.

Kockaert, P.

F. Leo, A. Mussot, P. Kockaert, Ph. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Kondratiev, N. M.

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,” arXiv: 1211.0733 (2012).

Lamont, M. R. E.

Leaird, D. E.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Leo, F.

F. Leo, A. Mussot, P. Kockaert, Ph. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, L. Gelens, Ph. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

Levy, J. S.

Liang, W.

Lipson, M.

Lu, F.

Maleki, L.

Matsko, A. B.

Menyuk, C. R.

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

Miao, H.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Murdoch, S. G.

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[CrossRef]

Mussot, A.

F. Leo, A. Mussot, P. Kockaert, Ph. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Okawachi, Y.

Papp, S. B.

S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Randle, H. G.

Ratner, J.

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, Laser Photon. Rev. 7, 557565 (2013).

Rhodes, M.

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, Laser Photon. Rev. 7, 557565 (2013).

Riemensberger, J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

Runge, A. F. J.

Saha, K.

Salem, R.

Savchenkov, A. A.

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[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]

Shim, B.

Shreenath, A. P.

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]

Srinivasan, K.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Steinmeyer, G.

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, Laser Photon. Rev. 7, 557565 (2013).

Surakka, M.

Sylvestre, T.

Taki, M.

F. Leo, A. Mussot, P. Kockaert, Ph. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Trebino, R.

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, Laser Photon. Rev. 7, 557565 (2013).

X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, and R. S. Windeler, Opt. Express 11, 2697 (2003).
[CrossRef]

Turunen, J.

Varghese, L. T.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Wabnitz, S.

Wang, C. Y.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[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,” arXiv: 1211.0733 (2012).

Wang, J.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Weiner, A. M.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

Wen, Y. H.

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Windeler, R. S.

Yu, N.

J. Opt. Soc. Am. B

Laser Photon. Rev.

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, Laser Photon. Rev. 7, 557565 (2013).

Nat. Photonics

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, Nat. Photonics 5, 770 (2011).
[CrossRef]

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 6, 480 (2012).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, Ph. Emplit, and M. Haelterman, Nat. Photonics 4, 471 (2010).
[CrossRef]

J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Nat. Photonics 7, 657 (2013).
[CrossRef]

Nature

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

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]

S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Phys. Rev. Lett.

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

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, Ph. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Rev. Mod. Phys.

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Science

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

Other

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,” arXiv: 1211.0733 (2012).

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Figures (5)

Fig. 1.
Fig. 1.

Illustrative simulation results for Δ=0.5, 3, 4, and 6 [from (a) to (d)]. Left: single-roundtrip snapshot (black) and 1 million roundtrip averaged (red) spectra. Right: single-roundtrip snapshot of the corresponding temporal intensity profile.

Fig. 2.
Fig. 2.

Simulated AC traces for (a) unstable MI, (b) unstable CS, and (c) stable CS regimes. The cavity detuning values correspond to those in Figs. 1(b)1(d).

Fig. 3.
Fig. 3.

Degree of first-order coherence across simulated Kerr combs as a function of coherence delay (density maps) and for a two photon-lifetime delay (line plot at the top) for (a) unstable MI and (b) unstable CS regimes. The corresponding mean spectra are shown in Figs. 1(b) and 1(c), respectively.

Fig. 4.
Fig. 4.

Overall degree of coherence as a function of coherence delay for the unstable MI (black) and unstable CS (red) regimes.

Fig. 5.
Fig. 5.

(a) Overall degree of coherence over a wide range of cavity detunings. The corresponding mean spectra are shown in (b).

Equations (2)

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tRE(t,τ)t=[αiδ0+iLk2βkk!(iτ)k+iγL|E|2]E+θEin.
|g12(1)(λ,t1t2)|=|E˜*(t1,λ)E˜(t2,λ)|S(λ).

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