Abstract

We investigate experimentally and theoretically the role of group-velocity dispersion and higher-order dispersion on the bandwidth of microresonator-based parametric frequency combs. We show that the comb bandwidth and the power contained in the comb can be tailored for a particular application. Additionally, our results demonstrate that fourth-order dispersion plays a critical role in determining the spectral bandwidth for comb bandwidths on the order of an octave.

© 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. W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2290 (2011).
    [CrossRef]
  3. P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 107, 063901 (2011).
    [CrossRef]
  4. Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, Opt. Lett. 36, 3398 (2011).
    [CrossRef]
  5. S. B. Papp and S. A. Diddams, Phys. Rev. A 84, 053833 (2011).
    [CrossRef]
  6. K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. Lamont, M. Lipson, and A. L. Gaeta, Opt. Express 21, 1335 (2013).
    [CrossRef]
  7. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2014).
    [CrossRef]
  8. J. S. Levy, K. Saha, Y. Okawachi, M. A. Foster, M. Lipson, and A. L. Gaeta, IEEE Photon. Technol. Lett. 24, 1375 (2012).
    [CrossRef]
  9. P.-H. Wang, F. Ferdous, H. Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 29284 (2012).
    [CrossRef]
  10. J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
    [CrossRef]
  11. L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
    [CrossRef]
  12. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
    [CrossRef]
  13. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. IlChenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
    [CrossRef]
  14. Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
    [CrossRef]
  15. S. Coen and M. Erkintalo, Opt. Lett. 38, 1790 (2013).
    [CrossRef]
  16. M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, Opt. Lett. 38, 3478 (2013).
    [CrossRef]
  17. S. Wang, H. Guo, X. Bai, and X. Zeng, Opt. Lett. 39, 2880 (2014).
  18. L. Zhang, C. Bao, V. Singh, J. Mu, C. Yang, A. M. Agarwal, L. C. Kimerling, and J. Michel, Opt. Lett. 38, 5122 (2013).
    [CrossRef]
  19. V. Torres-Company, D. Castelló-Lurbe, and E. Silvestre, Opt. Express 22, 4678 (2014).
    [CrossRef]
  20. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
    [CrossRef]
  21. T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
    [CrossRef]
  22. K. Luke, A. Dutt, C. B. Poitras, and M. Lipson, Opt. Express 21, 22829 (2013).
    [CrossRef]
  23. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, Opt. Express 16, 1300 (2008).
    [CrossRef]

2014 (4)

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2014).
[CrossRef]

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

V. Torres-Company, D. Castelló-Lurbe, and E. Silvestre, Opt. Express 22, 4678 (2014).
[CrossRef]

S. Wang, H. Guo, X. Bai, and X. Zeng, Opt. Lett. 39, 2880 (2014).

2013 (7)

2012 (2)

J. S. Levy, K. Saha, Y. Okawachi, M. A. Foster, M. Lipson, and A. L. Gaeta, IEEE Photon. Technol. Lett. 24, 1375 (2012).
[CrossRef]

P.-H. Wang, F. Ferdous, H. Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, Opt. Express 20, 29284 (2012).
[CrossRef]

2011 (6)

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]

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

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2290 (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]

2008 (1)

1992 (2)

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef]

1987 (1)

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

Agarwal, A. M.

Aksyuk, V.

Bai, X.

Bao, C.

Brasch, V.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2014).
[CrossRef]

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Castelló-Lurbe, D.

Chembo, Y. K.

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

Chen, L.

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]

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]

Dutt, A.

Erkintalo, M.

Ferdous, F.

Foster, M. A.

Freude, W.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Gaeta, A. L.

Gavartin, E.

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

Gorodetsky, M. L.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2014).
[CrossRef]

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

Guo, H.

Haelterman, M.

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef]

Hansson, T.

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

Hartinger, K.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Herr, T.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2014).
[CrossRef]

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

Hillerkuss, D.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Holzwarth, R.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[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]

IlChenko, V. S.

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, Nat. Photonics 8, 145 (2014).
[CrossRef]

Kimerling, L. C.

Kippenberg, T. J.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2014).
[CrossRef]

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[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. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef]

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2014).
[CrossRef]

Koos, C.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Lamont, M. R.

Lamont, M. R. E.

Lauermann, M.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Leaird, D. E.

Lefever, R.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

Leuthold, J.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Levy, J. S.

Li, J.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Liang, W.

Lipson, M.

Lugiato, L. A.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

Luke, K.

Maleki, L.

Matsko, A. B.

Menyuk, C. R.

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

Miao, H.

Michel, J.

Modotto, D.

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

Mu, J.

Okawachi, Y.

Papp, S. B.

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

Pfeifle, J.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Poitras, C. B.

Saha, K.

Salem, R.

Savchenkov, A. A.

Schindler, P.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Schmogrow, R.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Seidel, D.

Shim, B.

Silvestre, E.

Singh, V.

Srinivasan, K.

Torres-Company, V.

Trillo, S.

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef]

Turner, A. C.

Wabnitz, S.

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

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (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, Nat. Photonics 8, 145 (2014).
[CrossRef]

Wang, J.

Wang, P.-H.

Wang, S.

Wegner, D.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Weimann, C.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Weiner, A. M.

Wen, Y. H.

Yang, C.

Yu, Y.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

Zeng, X.

Zhang, L.

IEEE Photon. Technol. Lett. (1)

J. S. Levy, K. Saha, Y. Okawachi, M. A. Foster, M. Lipson, and A. L. Gaeta, IEEE Photon. Technol. Lett. 24, 1375 (2012).
[CrossRef]

Nat. Photonics (2)

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, Nat. Photonics 8, 375 (2014).
[CrossRef]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2014).
[CrossRef]

Opt. Commun. (1)

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

Opt. Express (5)

Opt. Lett. (8)

Phys. Rev. A (3)

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

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

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

Phys. Rev. Lett. (2)

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

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

Science (1)

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

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Figures (4)

Fig. 1.
Fig. 1.

(a) Simulated GVD for silicon–nitride waveguides with cross sections 690×1900nm (blue) and 910×1800nm (green). (b) Simulated comb spectra and (c) temporal plot for 260 GHz free spectral range (FSR) Si3N4 microring resonators with cross sections 690×1900nm (top) and 910×1800nm (bottom).

Fig. 2.
Fig. 2.

(a) Simulated GVD and FOD for 690×1900nm cross section Si3N4 waveguide. (b) Simulated comb spectra for pump wavelengths of 1550, 1400, and 1200 nm.

Fig. 3.
Fig. 3.

Measured comb spectra in Si3N4 microrings with cross sections (a) 690×1900nm and (b) 910×1800nm.

Fig. 4.
Fig. 4.

(a) Simulated GVD for 950×1400nm cross section Si3N4 microrings for a 100 and 46 μm radius. (b) Simulated comb spectra in 100 and 46 μm radius microrings, corresponding to 260 and 530 GHz FSRs, respectively.

Equations (7)

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

tRE(t,τ)t=[αiδ0+iLn2βnn!(iτ)n+iγL(1+iω0τ)|E(t,τ)|2]E(t,τ)+θEin,
tRA0t=(α+iδ0)A0+iγL|A0|2A0+2iγLA0*A+A+θEin,
tRA+t=[α+iΔκ+iLn=3,5,βnn!Ωn+2iLΩω0γ|A0|2]A++iγL(1Ωω0)A02A*,
tRAt=[α+iΔκiLn=3,5,βnn!Ωn2iLΩω0γ|A0|2]A+iγL(1+Ωω0)A02A+*,
Δκ=δ0Ln=2,4,βnn!Ωn2γL|A0|2.
A+(t)=A+(0)exp[Λ(Ω)ttR]×exp[i(Ln=3,5,βnn!Ωn+2γLΩω0|A0|2)ttR],
A(t)=A(0)exp[Λ(Ω)ttR]×exp[i(Ln=3,5,βnn!Ωn+2γLΩω0|A0|2)ttR],

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