Abstract

The onset of multi-pulsing, a ubiquitous phenomenon in laser cavities, is considered. The multi-pulsing transition is studied in a specific model that uses a waveguide array as the cavity saturable absorber. Using a low-dimensional reduction constructed by the method of proper orthogonal decomposition (principal components), a complete characterization of the multi-pulsing transition is given, including the onset of periodic solution (Hopf bifurcation) and period-doubling bifurcation and Neimark–Sacker (torus) bifurcation as routes to chaos. To the best of our knowledge, this is the first low-dimensional construction of the entire multi-pulsing transition from N to N+1 pulses per round trip. The reduced model qualitatively reproduces the dynamics observed in the multi-pulse transition of the mode-locked laser and confirms recent experimental observations of periodic and chaotic behavior preceding the multi-pulsing transition.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14, 2099–2111 (1997).
    [CrossRef]
  2. L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).
    [CrossRef]
  3. B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, “Transition dynamics for multi-pulsing in mode-locked lasers,” Opt. Express 17, 23137–23146 (2009).
    [CrossRef]
  4. J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).
    [CrossRef]
  5. T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).
    [CrossRef] [PubMed]
  6. R. A. Vicencio, M. Molina, and Y. S. Kivshar, “All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefringent waveguide arrays,” Opt. Lett. 29, 2905–2907 (2004).
    [CrossRef]
  7. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
    [CrossRef]
  8. A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
    [CrossRef] [PubMed]
  9. D. Hudson, K. Shish, T. Schibli, J. N. Kutz, D. Christodoulides, R. Morandotti, and S. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
    [CrossRef] [PubMed]
  10. M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
    [CrossRef]
  11. M. O. Williams, C. W. McGrath, and J. N. Kutz, “Light-bullet routing and control with planar waveguide arrays,” Opt. Express 18, 11671–11682 (2010).
    [CrossRef] [PubMed]
  12. J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).
    [CrossRef] [PubMed]
  13. Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
    [CrossRef] [PubMed]
  14. P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).
    [CrossRef]
  15. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
    [CrossRef]
  16. A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
    [CrossRef]
  17. H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).
    [CrossRef]
  18. U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).
    [CrossRef]
  19. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
    [CrossRef] [PubMed]
  20. A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).
    [CrossRef] [PubMed]
  21. V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).
  22. J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005).
    [CrossRef] [PubMed]
  23. J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).
    [CrossRef] [PubMed]
  24. B. G. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” in Proceedings of Photonics North 2009 (2009), Vol. 7386, paper 73862W.
  25. B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008).
    [CrossRef]
  26. E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).
    [CrossRef] [PubMed]
  27. E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).
    [CrossRef]
  28. C. R. Jones and J. N. Kutz, “Stability of mode-locked pulse solutions subject to saturable gain: computing linear stability with the Floquet–Fourier–Hill method,” J. Opt. Soc. Am. B 27, 1184–1194 (2010).
    [CrossRef]
  29. E. Shlizerman and V. Rom-Kedar, “Hierarchy of bifurcations in the truncated and forced nonlinear Schrodinger model,” Chaos 15, 013107 (2005).
    [CrossRef]
  30. E. Shlizerman and V. Rom-Kedar, “Three types of chaos in the forced nonlinear Schrödinger equation,” Phys. Rev. Lett. 96, 024104 (2006).
    [CrossRef] [PubMed]
  31. R. Scharf and A. R. Bishop, “Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations,” Phys. Rev. A 46, R2973–R2976 (1992).
    [CrossRef] [PubMed]
  32. D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
    [CrossRef]
  33. P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).
  34. M.-L. Rapún and J. M. Vega, “Reduced order models based on local pod plus Galerkin projection,” J. Comput. Phys. 229, 3046–3063 (2010).
    [CrossRef]
  35. D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).
    [CrossRef]
  36. K. Kunisch and S. Volkwein, “Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 40, 492–515 (2002).
    [CrossRef]
  37. E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).
    [CrossRef]
  38. A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808–817 (2000).
  39. E. Shlizerman and V. Rom-Kedar, “Parabolic resonance: A route to Hamiltonian spatiotemporal chaos,” Phys. Rev. Lett. 102, 033901 (2009).
    [CrossRef] [PubMed]
  40. A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).
    [CrossRef]

2010 (7)

L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).
[CrossRef]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).
[CrossRef]

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

M.-L. Rapún and J. M. Vega, “Reduced order models based on local pod plus Galerkin projection,” J. Comput. Phys. 229, 3046–3063 (2010).
[CrossRef]

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).
[CrossRef]

C. R. Jones and J. N. Kutz, “Stability of mode-locked pulse solutions subject to saturable gain: computing linear stability with the Floquet–Fourier–Hill method,” J. Opt. Soc. Am. B 27, 1184–1194 (2010).
[CrossRef]

M. O. Williams, C. W. McGrath, and J. N. Kutz, “Light-bullet routing and control with planar waveguide arrays,” Opt. Express 18, 11671–11682 (2010).
[CrossRef] [PubMed]

2009 (4)

B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, “Transition dynamics for multi-pulsing in mode-locked lasers,” Opt. Express 17, 23137–23146 (2009).
[CrossRef]

E. Shlizerman and V. Rom-Kedar, “Parabolic resonance: A route to Hamiltonian spatiotemporal chaos,” Phys. Rev. Lett. 102, 033901 (2009).
[CrossRef] [PubMed]

B. G. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” in Proceedings of Photonics North 2009 (2009), Vol. 7386, paper 73862W.

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).
[CrossRef]

2008 (3)

2007 (3)

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).
[CrossRef]

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).
[CrossRef] [PubMed]

2006 (1)

E. Shlizerman and V. Rom-Kedar, “Three types of chaos in the forced nonlinear Schrödinger equation,” Phys. Rev. Lett. 96, 024104 (2006).
[CrossRef] [PubMed]

2005 (2)

E. Shlizerman and V. Rom-Kedar, “Hierarchy of bifurcations in the truncated and forced nonlinear Schrodinger model,” Chaos 15, 013107 (2005).
[CrossRef]

J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005).
[CrossRef] [PubMed]

2004 (2)

R. A. Vicencio, M. Molina, and Y. S. Kivshar, “All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefringent waveguide arrays,” Opt. Lett. 29, 2905–2907 (2004).
[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

2003 (3)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).
[CrossRef] [PubMed]

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).
[CrossRef]

2002 (3)

2001 (1)

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[CrossRef]

2000 (1)

A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808–817 (2000).

1998 (2)

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

1997 (1)

1996 (1)

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

1995 (2)

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).
[CrossRef]

1994 (3)

A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).
[CrossRef] [PubMed]

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).
[CrossRef] [PubMed]

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

1992 (1)

R. Scharf and A. R. Bishop, “Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations,” Phys. Rev. A 46, R2973–R2976 (1992).
[CrossRef] [PubMed]

Aceves, A. B.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).
[CrossRef] [PubMed]

Aitchison, J. S.

Akhmediev, N.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).
[CrossRef]

Alonso, D.

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).
[CrossRef]

Anderson, D.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[CrossRef]

Arnold, J. M.

Bale, B. G.

Berkooz, G.

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

Berntson, A.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[CrossRef]

Bishop, A. R.

R. Scharf and A. R. Bishop, “Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations,” Phys. Rev. A 46, R2973–R2976 (1992).
[CrossRef] [PubMed]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Chatterjee, A.

A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808–817 (2000).

Christodoulides, D.

Christodoulides, D. N.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

Conti, C.

Cundiff, S.

De Angelis, C.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).
[CrossRef] [PubMed]

Dhooge, A.

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).
[CrossRef]

Ding, E.

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).
[CrossRef]

Droulias, S.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Eisenberg, H. S.

Feng, L.

L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).
[CrossRef]

Frantzeskakis, D. J.

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).
[CrossRef]

Frumker, E.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Gagnon, J.

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).
[CrossRef]

Garanovich, I. L.

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

Govaerts, W.

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).
[CrossRef]

Grapinet, M.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).
[CrossRef]

Grelu, P.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).
[CrossRef]

Haus, H. A.

Hizanidis, K.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Holmes, P.

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

Hudson, D.

Ippen, E. P.

Jones, C. R.

Kevrekidis, P. G.

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).
[CrossRef]

Kieu, K.

Kivshar, Y. S.

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

R. A. Vicencio, M. Molina, and Y. S. Kivshar, “All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefringent waveguide arrays,” Opt. Lett. 29, 2905–2907 (2004).
[CrossRef]

Kunisch, K.

K. Kunisch and S. Volkwein, “Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 40, 492–515 (2002).
[CrossRef]

Kutz, J. N.

M. O. Williams, C. W. McGrath, and J. N. Kutz, “Light-bullet routing and control with planar waveguide arrays,” Opt. Express 18, 11671–11682 (2010).
[CrossRef] [PubMed]

L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).
[CrossRef]

C. R. Jones and J. N. Kutz, “Stability of mode-locked pulse solutions subject to saturable gain: computing linear stability with the Floquet–Fourier–Hill method,” J. Opt. Soc. Am. B 27, 1184–1194 (2010).
[CrossRef]

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).
[CrossRef]

B. G. Bale, K. Kieu, J. N. Kutz, and F. Wise, “Transition dynamics for multi-pulsing in mode-locked lasers,” Opt. Express 17, 23137–23146 (2009).
[CrossRef]

B. G. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” in Proceedings of Photonics North 2009 (2009), Vol. 7386, paper 73862W.

B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008).
[CrossRef]

D. Hudson, K. Shish, T. Schibli, J. N. Kutz, D. Christodoulides, R. Morandotti, and S. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
[CrossRef] [PubMed]

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).
[CrossRef] [PubMed]

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005).
[CrossRef] [PubMed]

Kuznetsov, Y. A.

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).
[CrossRef]

Laedke, E. W.

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).
[CrossRef]

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).
[CrossRef] [PubMed]

Lahini, Y.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Lisak, M.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[CrossRef]

Lumley, J. L.

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

Luther, G. G.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

Malomed, B. A.

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).
[CrossRef]

Matuszewski, M.

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

McGrath, C. W.

Mezentsev, V. K.

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).
[CrossRef]

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

Molina, M.

Molina, M. I.

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

Morandotti, R.

Muschall, R.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Musher, S. L.

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

Namiki, S.

Pennelli, G.

Pertsch, T.

Peschel, T.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Peschel, U.

Proctor, J.

Rapún, M. -L.

M.-L. Rapún and J. M. Vega, “Reduced order models based on local pod plus Galerkin projection,” J. Comput. Phys. 229, 3046–3063 (2010).
[CrossRef]

Rebenchik, A. M.

Rom-Kedar, V.

E. Shlizerman and V. Rom-Kedar, “Parabolic resonance: A route to Hamiltonian spatiotemporal chaos,” Phys. Rev. Lett. 102, 033901 (2009).
[CrossRef] [PubMed]

E. Shlizerman and V. Rom-Kedar, “Three types of chaos in the forced nonlinear Schrödinger equation,” Phys. Rev. Lett. 96, 024104 (2006).
[CrossRef] [PubMed]

E. Shlizerman and V. Rom-Kedar, “Hierarchy of bifurcations in the truncated and forced nonlinear Schrodinger model,” Chaos 15, 013107 (2005).
[CrossRef]

Rubenchik, A. M.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

Ryzhenkova, I. V.

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

Sandstede, B.

B. G. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” in Proceedings of Photonics North 2009 (2009), Vol. 7386, paper 73862W.

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).
[CrossRef] [PubMed]

Scharf, R.

R. Scharf and A. R. Bishop, “Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations,” Phys. Rev. A 46, R2973–R2976 (1992).
[CrossRef] [PubMed]

Schibli, T.

Shish, K.

Shlizerman, E.

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).
[CrossRef]

E. Shlizerman and V. Rom-Kedar, “Parabolic resonance: A route to Hamiltonian spatiotemporal chaos,” Phys. Rev. Lett. 102, 033901 (2009).
[CrossRef] [PubMed]

E. Shlizerman and V. Rom-Kedar, “Three types of chaos in the forced nonlinear Schrödinger equation,” Phys. Rev. Lett. 96, 024104 (2006).
[CrossRef] [PubMed]

E. Shlizerman and V. Rom-Kedar, “Hierarchy of bifurcations in the truncated and forced nonlinear Schrodinger model,” Chaos 15, 013107 (2005).
[CrossRef]

Silberberg, Y.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).
[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Soto-Crespo, J. M.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).
[CrossRef]

Spatchek, K. H.

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).
[CrossRef]

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).
[CrossRef] [PubMed]

Sukhorukov, A. A.

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

Trillo, S.

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Turitsyn, S. K.

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).
[CrossRef]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, A. M. Rebenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).
[CrossRef] [PubMed]

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

Vega, J.

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).
[CrossRef]

Vega, J. M.

M.-L. Rapún and J. M. Vega, “Reduced order models based on local pod plus Galerkin projection,” J. Comput. Phys. 229, 3046–3063 (2010).
[CrossRef]

Velazquez, A.

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).
[CrossRef]

Vicencio, R. A.

R. A. Vicencio, M. Molina, and Y. S. Kivshar, “All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefringent waveguide arrays,” Opt. Lett. 29, 2905–2907 (2004).
[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

Volkwein, S.

K. Kunisch and S. Volkwein, “Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 40, 492–515 (2002).
[CrossRef]

Wabnitz, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Wai, P. K. A.

L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).
[CrossRef]

Williams, M. O.

Wise, F.

Yu, C. X.

ACM Trans. Math. Softw. (1)

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Matcont: A MATLAB package for numerical bifurcation analysis of odes,” ACM Trans. Math. Softw. 29, 141–164 (2003).
[CrossRef]

Chaos (1)

E. Shlizerman and V. Rom-Kedar, “Hierarchy of bifurcations in the truncated and forced nonlinear Schrodinger model,” Chaos 15, 013107 (2005).
[CrossRef]

Comput. Methods Appl. Mech. Eng. (1)

D. Alonso, A. Velazquez, and J. Vega, “A method to generate computationally efficient reduced order models,” Comput. Methods Appl. Mech. Eng. 198, 2683–2691 (2009).
[CrossRef]

Curr. Sci. (1)

A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808–817 (2000).

J. Comput. Phys. (1)

M.-L. Rapún and J. M. Vega, “Reduced order models based on local pod plus Galerkin projection,” J. Comput. Phys. 229, 3046–3063 (2010).
[CrossRef]

J. Op. Soc. Am. B (1)

L. Feng, P. K. A. Wai, and J. N. Kutz, “A geometrical description of the onset of multi-pulsing in mode-locked laser cavities,” J. Op. Soc. Am. B 27, 2068–2077 (2010).
[CrossRef]

J. Opt. Soc. Am. B (5)

JETP Lett. (1)

V. K. Mezentsev, S. L. Musher, I. V. Ryzhenkova, and S. K. Turitsyn, “Two-dimensional solitons in discrete systems,” JETP Lett. 60, 829–835 (1994).

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. A (3)

E. Ding, E. Shlizerman, and J. N. Kutz, “Modeling multi-pulsing transition in ring cavity lasers with proper orthogonal decomposition,” Phys. Rev. A 82, 023823 (2010).
[CrossRef]

R. Scharf and A. R. Bishop, “Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations,” Phys. Rev. A 46, R2973–R2976 (1992).
[CrossRef] [PubMed]

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).
[CrossRef]

Phys. Rev. E (5)

P. G. Kevrekidis, J. Gagnon, D. J. Frantzeskakis, and B. A. Malomed, “x, y, and z waves: Extended structures in nonlinear lattices,” Phys. Rev. E 75, 016607 (2007).
[CrossRef]

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2010).
[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

E. W. Laedke, K. H. Spatchek, S. K. Turitsyn, and V. K. Mezentsev, “Analytic criterion for soliton instability in a nonlinear fiber array,” Phys. Rev. E 52, 5549–5554 (1995).
[CrossRef]

Phys. Rev. Lett. (6)

E. W. Laedke, K. H. Spatchek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994).
[CrossRef] [PubMed]

E. Shlizerman and V. Rom-Kedar, “Parabolic resonance: A route to Hamiltonian spatiotemporal chaos,” Phys. Rev. Lett. 102, 033901 (2009).
[CrossRef] [PubMed]

E. Shlizerman and V. Rom-Kedar, “Three types of chaos in the forced nonlinear Schrödinger equation,” Phys. Rev. Lett. 96, 024104 (2006).
[CrossRef] [PubMed]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, “Discrete x-wave formation in nonlinear waveguide arrays,” Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Pramana (1)

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

K. Kunisch and S. Volkwein, “Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 40, 492–515 (2002).
[CrossRef]

Other (2)

P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1998).

B. G. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” in Proceedings of Photonics North 2009 (2009), Vol. 7386, paper 73862W.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Surface and pseudo-color plots of possible behaviors in the WGAML system. From left to right, single-pulse, breather, chaotic, and two-pulse solutions are shown. The particular behaviors are determined by the values of g 0 which are 2.3, 2.5, 2.6, and 2.7 for the single-pulse, breather, chaotic, and two-pulse solutions, respectively.

Fig. 2
Fig. 2

Plots of the POD mode profile and the error as defined in Eq. (10) using the first N POD modes in a low-dimensional model. Left: the POD modes of the single-pulse solutions for g 0 = 2.3 . Right: the POD modes of the breather solutions for g 0 = 2.5 . For the breather solutions, the error was measured at the point of maximum amplitude for both the PDE and ODE solutions due to differences in the period of the limit cycles.

Fig. 3
Fig. 3

Plots of POD mode profiles for reconstructing the chaotic and double-pulse solutions of the WGAML. Left: the first six of the 22 modes of the chaotic transition needed to capture 99.9% of the energy at g 0 = 2.6 ; right: the first six modes of the double-pulse solution at g 0 = 2.7 . Only two of these modes are needed to capture 99.9% of the energy.

Fig. 4
Fig. 4

Comparison of the number of POD modes needed to capture 99.9% of the energy for the even WGAML model and the unrestricted-WGAML model using data taken for 0 Z Z max . The gains used for these results are g 0 = 2.65 and g 0 = 3.1 , respectively.

Fig. 5
Fig. 5

POD modes taken from the combined data set for the zeroth waveguide that are capable of qualitatively reproducing the dynamics observed in the full WGAML model.

Fig. 6
Fig. 6

Top: Surface and pseudo-color plots of the single-pulse, breather, chaotic, and double-pulse solutions computed for the finite-dimensional model at g 0 = 1.5 , 2.5, 3.495, and 3.5, respectively. Bottom: The same plots for the even WGAML model taken at g 0 = 2.3 , 2.5, 3.1, and 3.2, respectively. The reduced model accurately reproduces the four behaviors observed. These solutions should be compared to their equivalent behavior in the full WGAML model shown in the same order in Fig. 1.

Fig. 7
Fig. 7

Top: The bifurcation diagram of energy ( L 2 norm) versus g 0 for the multi-pulse transition in the POD model. Bottom: The bifurcation diagram of the multi-pulse transition of the WGAML model. The different plots show the same diagram with emphasis on different regions of the transition. For the stationary solutions, emphasized on the left, linearly unstable regions are dashed (red) curves, while linearly stable regions are solid (blue) curves. Stationary solutions that were not computed explicitly are denoted with dots. For the periodic solutions, the x’s are the local extrema of the energy ( L 2 norm).

Tables (1)

Tables Icon

Table 1 The Values of g 0 Associated with Each Bifurcation in the ODE and PDE. The Labels Correspond to the Bifurcations of the ODE in Fig. 7. Values in Parentheses are Estimated from the Same Figure.

Equations (18)

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

i A 0 Z + D 2 2 A 0 T 2 + β | A 0 | 2 A 0 i g ( Z ) ( 1 + τ 2 T 2 ) A 0 + i γ 0 A 0 + C A 1 = 0 ,
i A 1 Z + C ( A 0 + A 2 ) + i γ 1 A 1 = 0 ,
i A 2 Z + C A 1 + i γ 2 A 2 = 0 ,
g ( Z ) = 2 g 0 1 + A 0 2 / e 0 ,
( D , β , C , γ 0 , γ 1 , γ 2 , τ , e 0 ) = ( 1 , 8 , 5 , 0 , 0 , 10 , 0.1 , 1 ) .
X = U Σ V ,
X = [ u 1 u i u m ] [ σ 1 σ j σ n 0 0 ] [ ϕ 1 ϕ j ϕ n ] .
x k = j = 1 n σ j u k j ϕ j ,
E j = σ j 2 i = 1 n σ i 2 ,
A 0 ( T , Z ) = j = 1 N a j ( Z ) ϕ 0 , j ( T ) ,
A 1 ( T , Z ) = j = 1 N b j ( Z ) ϕ 1 , j ( T ) ,
A 2 ( T , Z ) = j = 1 N c j ( Z ) ϕ 2 , j ( T ) ,
a n Z = ( i D 2 + g τ ) j = 1 N ϕ 0 , n , T 2 ϕ 0 , j a j + ( g γ 0 ) a n + i C j = 1 N ϕ 0 , n , ϕ 0 , j b j + i β j = 1 N k = 1 N m = 1 N ϕ 0 , n , ϕ 0 , k ϕ 0 , j ϕ 0 , m a j a k a m ,
b n Z = i C j = 1 N ( ϕ 1 , n , ϕ 0 , j a j + ϕ 1 , n , ϕ 2 , j c j ) γ 1 b n ,
c n Z = i C j = 1 N ϕ 2 , n , ϕ 1 , j b j γ 2 c n ,
g = 2 g 0 1 + j = 1 N | a j | 2 / e 0 .
Error ( N ) = A 0 , WGAML A 0 , POD ( N ) 2 .
A 2 = A 0 2 + A 1 2 + A 2 2 ,

Metrics