## Light bullets in waveguide arrays: spacetime-coupling, spectral symmetry breaking and superluminal decay [Invited] |

Optics Express, Vol. 19, Issue 23, pp. 23171-23187 (2011)

http://dx.doi.org/10.1364/OE.19.023171

Acrobat PDF (4366 KB)

### Abstract

We investigate the effects of the space-time coupling (STC) on the nonlinear formation and propagation of Light Bullets, spatiotemporal solitons in which dispersion and diffraction along all dimensions are balanced by nonlinearity, through periodic media with a weak transverse modulation of the refractive index, i.e. waveguide arrays. The STC arises from wavelength dependence of the strength of inter-waveguide coupling and can be tuned by variation of the array geometry. We show experimentally and numerically that the STC breaks the spectral symmetry of Light Bullets to a considerable degree and modifies their group velocity, leading to superluminal propagation when the Light Bullets decay.

© 2011 OSA

## 1. Introduction

1. A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A **63**(5), 053806 (2001). [CrossRef]

5. M. D. Stenner, D. J. Gauthier, and M. A. Neifeld, “The speed of information in a ‘fast-light’ optical medium,” Nature **425**(6959), 695–698 (2003). [CrossRef] [PubMed]

6. G. Nimtz, “Evanescent modes are not necessarily Einstein causal,” Eur. Phys. J. B **7**(4), 523–525 (1999), doi:. [CrossRef]

5. M. D. Stenner, D. J. Gauthier, and M. A. Neifeld, “The speed of information in a ‘fast-light’ optical medium,” Nature **425**(6959), 695–698 (2003). [CrossRef] [PubMed]

7. M. Nakazawa, T. Yamamoto, and K. Tamura, “1.28 tbit/s–70 km OTDM transmission using third- and fourth-order simultaneous dispersion compensation with a phase modulator,” Electron. Lett. **36**(24), 2027–2029 (2000). [CrossRef]

9. T. D. Vo, H. Hu, M. Galili, E. Palushani, J. Xu, L. K. Oxenløwe, S. J. Madden, D.-Y. Choi, D. A. P. Bulla, M. D. Pelusi, J. Schröder, B. Luther-Davies, and B. J. Eggleton, “Photonic chip based transmitter optimization and receiver demultiplexing of a 1.28 Tbit/s OTDM signal,” Opt. Express **18**(16), 17252–17261 (2010). [CrossRef] [PubMed]

10. A. Stepanov, J. Kuhl, I. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Opt. Express **13**(15), 5762–5768 (2005). [CrossRef] [PubMed]

12. D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. D. Trapani, and A. Couairon, “Stimulated Raman X waves in ultrashort optical pulse filamentation,” Opt. Lett. **32**(2), 184–186 (2007). [CrossRef] [PubMed]

13. O. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. **25**(12), 2464–2468 (1989). [CrossRef]

14. P. Saari and K. Reivelt, “Evidence of *x*-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. **79**(21), 4135–4138 (1997). [CrossRef]

15. A. V. Gorbach and D. V. Skryabin, “Cascaded generation of multiply charged optical vortices and spatiotemporal helical beams in a Raman medium,” Phys. Rev. Lett. **98**(24), 243601 (2007). [CrossRef] [PubMed]

16. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. **91**(9), 093904 (2003). [CrossRef] [PubMed]

17. D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. **96**(19), 193901 (2006). [CrossRef] [PubMed]

19. I. Blonskyi, V. Kadan, O. Shpotyuk, and I. Dmitruk, “Manifestations of sub- and superluminality in filamented femtosecond laser pulse in fused silica,” Opt. Commun. **282**(9), 1913–1917 (2009). [CrossRef]

20. C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express **17**(7), 5879–5884 (2009). [CrossRef] [PubMed]

22. A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A **82**(4), 041802 (2010). [CrossRef]

23. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. **15**(22), 1282–1284 (1990). [CrossRef] [PubMed]

24. S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-dimensional light bullets in arrays of waveguides,” Phys. Rev. Lett. **105**(26), 263901 (2010). [CrossRef] [PubMed]

25. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. **83**(1), 247–306 (2011). [CrossRef]

33. J. E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. **17**(8), 583–585 (1992). [CrossRef] [PubMed]

34. A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “Pt-symmetry in honeycomb photonic lattices,” Phys. Rev. A **84**(2), 021806 (2011). [CrossRef]

35. F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. Di Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. **33**(24), 2952–2954 (2008). [CrossRef] [PubMed]

## 2. Space-time focusing, Light Bullets, and fibre arrays

*z*-axis and periodic in

*x*and

*y*one can explicitly write the dispersion relation in the form

33. J. E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. **17**(8), 583–585 (1992). [CrossRef] [PubMed]

36. A. Zozulya and S. Diddams, “Dynamics of self-focused femtosecond laser pulses in the near and far fields,” Opt. Express **4**(9), 336–343 (1999). [CrossRef] [PubMed]

38. S. Malaguti, G. Bellanca, and S. Trillo, “Two-dimensional envelope localized waves in the anomalous dispersion regime,” Opt. Lett. **33**(10), 1117–1119 (2008). [CrossRef] [PubMed]

33. J. E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. **17**(8), 583–585 (1992). [CrossRef] [PubMed]

38. S. Malaguti, G. Bellanca, and S. Trillo, “Two-dimensional envelope localized waves in the anomalous dispersion regime,” Opt. Lett. **33**(10), 1117–1119 (2008). [CrossRef] [PubMed]

39. A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tünnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A **75**(5), 053814 (2007). [CrossRef]

40. A. Szameit, D. Blömer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tünnermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B **82**(4), 507–512 (2006). [CrossRef]

21. C. J. Benton, A. V. Gorbach, and D. V. Skryabin, “Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires,” Phys. Rev. A **78**(3), 033818 (2008). [CrossRef]

22. A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A **82**(4), 041802 (2010). [CrossRef]

41. F. Eilenberger, S. Minardi, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Evolution dynamics of discrete-continuous light bullets,” Phys. Rev. A **84**(1), 013836 (2011). [CrossRef]

44. N. Vakhitov and A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. **16**(7), 783–789 (1973). [CrossRef]

45. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **66**(4), 046602 (2002). [CrossRef] [PubMed]

47. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. **90**(2), 023902 (2003). [CrossRef] [PubMed]

40. A. Szameit, D. Blömer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tünnermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B **82**(4), 507–512 (2006). [CrossRef]

48. U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, “Fiber waveguide arrays as model system for discrete optics,” Appl. Phys. B **104**(3), 481–486 (2011), doi:. [CrossRef]

49. U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, “Two-dimensional high-precision fiber waveguide arrays for coherent light propagation,” Opt. Express **15**(11), 6894–6899 (2007). [CrossRef] [PubMed]

50. J. C. Knight, “Photonic crystal fibres,” Nature **424**(6950), 847–851 (2003). [CrossRef] [PubMed]

51. P. Russell, “Photonic crystal fibers,” Science **299**(5605), 358–362 (2003). [CrossRef] [PubMed]

48. U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, “Fiber waveguide arrays as model system for discrete optics,” Appl. Phys. B **104**(3), 481–486 (2011), doi:. [CrossRef]

49. U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, “Two-dimensional high-precision fiber waveguide arrays for coherent light propagation,” Opt. Express **15**(11), 6894–6899 (2007). [CrossRef] [PubMed]

52. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. **93**(5), 053901 (2004). [CrossRef] [PubMed]

53. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature **446**(7131), 52–55 (2007). [CrossRef] [PubMed]

54. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(3), 036604 (2004). [CrossRef] [PubMed]

56. P. Kinsler, “Unidirectional optical pulse propagation equation for materials with both electric and magnetic responses,” Phys. Rev. A **81**(2), 023808 (2010). [CrossRef]

41. F. Eilenberger, S. Minardi, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Evolution dynamics of discrete-continuous light bullets,” Phys. Rev. A **84**(1), 013836 (2011). [CrossRef]

## 3. Spectral Symmetry Breaking of LBs

### 3.1 Experimental Observation

35. F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. Di Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. **33**(24), 2952–2954 (2008). [CrossRef] [PubMed]

63. M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. D. Trapani, “Three dimensional imaging of short pulses,” Opt. Commun. **229**(1-6), 381–390 (2004). [CrossRef]

64. S. Minardi, J. Trull, and M. A. C. Potenza, “Holographic properties of parametric image conversion for spatiotemporal imaging of ultrashort laser pulses,” J. Hologr. Speckle **5**(1), 85–93 (2009). [CrossRef]

24. S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-dimensional light bullets in arrays of waveguides,” Phys. Rev. Lett. **105**(26), 263901 (2010). [CrossRef] [PubMed]

41. F. Eilenberger, S. Minardi, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Evolution dynamics of discrete-continuous light bullets,” Phys. Rev. A **84**(1), 013836 (2011). [CrossRef]

**84**(1), 013836 (2011). [CrossRef]

### 3.2 Numerical Verification

*z*and the input energy

24. S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-dimensional light bullets in arrays of waveguides,” Phys. Rev. Lett. **105**(26), 263901 (2010). [CrossRef] [PubMed]

**84**(1), 013836 (2011). [CrossRef]

## 4. Superluminal Decay of LBs

22. A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A **82**(4), 041802 (2010). [CrossRef]

### 4.1 Experimental Observation

**84**(1), 013836 (2011). [CrossRef]

### 4.2 Numerical Verification

**105**(26), 263901 (2010). [CrossRef] [PubMed]

## 5. Conclusions

## References and links

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13. | O. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. |

14. | P. Saari and K. Reivelt, “Evidence of |

15. | A. V. Gorbach and D. V. Skryabin, “Cascaded generation of multiply charged optical vortices and spatiotemporal helical beams in a Raman medium,” Phys. Rev. Lett. |

16. | P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. |

17. | D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. |

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34. | A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “Pt-symmetry in honeycomb photonic lattices,” Phys. Rev. A |

35. | F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. Di Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. |

36. | A. Zozulya and S. Diddams, “Dynamics of self-focused femtosecond laser pulses in the near and far fields,” Opt. Express |

37. | M. A. Porras, I. Gonzalo, and A. Mondello, “Pulsed light beams in vacuum with superluminal and negative group velocities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

38. | S. Malaguti, G. Bellanca, and S. Trillo, “Two-dimensional envelope localized waves in the anomalous dispersion regime,” Opt. Lett. |

39. | A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tünnermann, U. Peschel, and F. Lederer, “Light evolution in arbitrary two-dimensional waveguide arrays,” Phys. Rev. A |

40. | A. Szameit, D. Blömer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tünnermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B |

41. | F. Eilenberger, S. Minardi, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Evolution dynamics of discrete-continuous light bullets,” Phys. Rev. A |

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47. | J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. |

48. | U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, “Fiber waveguide arrays as model system for discrete optics,” Appl. Phys. B |

49. | U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, “Two-dimensional high-precision fiber waveguide arrays for coherent light propagation,” Opt. Express |

50. | J. C. Knight, “Photonic crystal fibres,” Nature |

51. | P. Russell, “Photonic crystal fibers,” Science |

52. | T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. |

53. | T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature |

54. | M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

55. | I. Babushkin, A. Husakou, J. Herrmann, and Y. S. Kivshar, “Frequency-selective self-trapping and supercontinuum generation in arrays of coupled nonlinear waveguides,” Opt. Express |

56. | P. Kinsler, “Unidirectional optical pulse propagation equation for materials with both electric and magnetic responses,” Phys. Rev. A |

57. | R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 å via four-photon coupling in glass,” Phys. Rev. Lett. |

58. | J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. |

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60. | C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. |

61. | M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. |

62. | M. Heinrich, A. Szameit, F. Dreisow, R. Keil, S. Minardi, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Observation of three-dimensional discrete-continuous |

63. | M. A. C. Potenza, S. Minardi, J. Trull, G. Blasi, D. Salerno, A. Varanavicius, A. Piskarskas, and P. D. Trapani, “Three dimensional imaging of short pulses,” Opt. Commun. |

64. | S. Minardi, J. Trull, and M. A. C. Potenza, “Holographic properties of parametric image conversion for spatiotemporal imaging of ultrashort laser pulses,” J. Hologr. Speckle |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(190.6135) Nonlinear optics : Spatial solitons

(320.6629) Ultrafast optics : Supercontinuum generation

**ToC Category:**

Waveguide Arrays

**History**

Original Manuscript: September 1, 2011

Revised Manuscript: October 18, 2011

Manuscript Accepted: October 21, 2011

Published: November 1, 2011

**Virtual Issues**

Nonlinear Optics (2011) *Optical Materials Express*

(2011) *Advances in Optics and Photonics*

**Citation**

Falk Eilenberger, Stefano Minardi, Alexander Szameit, Ulrich Röpke, Jens Kobelke, Kay Schuster, Hartmut Bartelt, Stefan Nolte, Andreas Tünnermann, and Thomas Pertsch, "Light bullets in waveguide arrays: spacetime-coupling, spectral symmetry breaking and superluminal decay [Invited]," Opt. Express **19**, 23171-23187 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23171

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