## X - waves in nonlinear normally dispersive waveguide arrays

Optics Express, Vol. 13, Issue 6, pp. 1827-1832 (2005)

http://dx.doi.org/10.1364/OPEX.13.001827

Acrobat PDF (734 KB)

### Abstract

We theoretically demonstrate that optical discrete X-waves are possible in normally dispersive nonlinear waveguide arrays. We show that such X-waves can be effectively excited for a wide range of initial conditions and in certain occasions can be generated in cascade. The possibility of observing this family of waves in AlGaAs array systems is investigated in terms of pertinent examples.

© 2005 Optical Society of America

## 1. Introduction

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]

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]

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. **85**, 1863–1866 (2000). [CrossRef] [PubMed]

3. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**, 794–796 (1988). [CrossRef] [PubMed]

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. **85**, 1863–1866 (2000). [CrossRef] [PubMed]

3. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**, 794–796 (1988). [CrossRef] [PubMed]

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. **85**, 1863–1866 (2000). [CrossRef] [PubMed]

4. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. **83**, 2726–2729 (1999). [CrossRef]

5. J. Meier, J. Hudock, D. N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. **91**, 143907 (2003). [CrossRef] [PubMed]

6. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147–150 (2003). [CrossRef] [PubMed]

7. 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 **66**, 046602 (2002). [CrossRef]

8. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902 (2004). [CrossRef] [PubMed]

*χ*

^{(2)}) arrays [

*9*]. In addition, other processes like discrete modulational instability [10

10. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. **92**, 163902 (2004). [CrossRef] [PubMed]

11. 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**, 023902 (2003). [CrossRef] [PubMed]

12. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. **92**, 093904 (2004). [CrossRef] [PubMed]

13. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. **92**, 123904 (2004). [CrossRef] [PubMed]

14. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. **92**, 123903 (2004). [CrossRef] [PubMed]

15. J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control **39**, 19 (1992). [CrossRef] [PubMed]

16. D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three- dimensional bidispersive optical systems,” Opt. Lett. **29**, 1446–1448 (2004). [CrossRef] [PubMed]

17. 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**, 093904 (2003). [CrossRef] [PubMed]

18. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett. **90**, 093904 (2003). [CrossRef]

17. 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**, 093904 (2003). [CrossRef] [PubMed]

## 2. System analysis

*U*in the n-th waveguide evolves according to [3

_{n}3. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**, 794–796 (1988). [CrossRef] [PubMed]

19. 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**, 11, 2637–2644 (2002). [CrossRef]

*β*

_{2}is the group velocity dispersion,

*C*the linear coupling strength,

*n*

_{2}=(

*n*̂

_{2}

*n*)/(2

*η*

_{0}),

*n*̂

_{2}=2.47×10

^{-13}

*cm*

^{2}/

*W*is the AlGaAs self-focusing Kerr coefficient and

*k*

_{0}=2

*π/λ*

_{0}. n is the linear refractive index of AlGaAs and

*η*

_{0}=(

*µ*

_{0}/

*ε*

_{0})

^{½}. We consider a

*1cm*long array consisting of

*41*waveguide elements with an effective cross-sectional area of

*5µm*. The distance between waveguides is taken to be

^{2}*10µm*. This array is used at

*λ0=1.55 µm*where the AlGaAs normal dispersion is

*β*. The coupling constant is assumed to be

_{2}=+1.3 ps^{2}/m*C*=

*728 m*which is close to that encountered in previous experiments [2

^{-1}**85**, 1863–1866 (2000). [CrossRef] [PubMed]

4. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. **83**, 2726–2729 (1999). [CrossRef]

5. J. Meier, J. Hudock, D. N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. **91**, 143907 (2003). [CrossRef] [PubMed]

*P*and the Hamiltonian

*H*of the system, i.e.,

## 3. Discrete X-wave formation

*0.5kW*to

*1.5kW*, where the 2- and 3-photon absorption is negligible compared to the Kerr-effect in AlGaAs waveguides. The initial pulse duration was taken to be either

*1ps*or

*200fs*. In all the figures, |

*U*|

_{n}^{2}=1 represents the peak-power initially used.

### 3.1 Single waveguide excitation

*1ps*and by varying only the input peak-power. For a low input peak-power (

*0.5kW*) the coupling among waveguides dominates the process and the initial pulse spreads in the transverse direction, as shown in Fig. 1. It is evident that the process of X-wave formation has already begun even at this power level. Clearly the nonlinearity is not strong enough to prevent the initial pulse from spreading in space and as a result it discretely diffracts.

*0.7kW*), discrete X-waves are more effectively excited. The interplay between diffraction and nonlinearity causes this pattern to arise in the following way. During propagation the initial pulse reaches a stage of maximum time-compression and edge-steepening occurs, leading to an X-like (or H-like) pattern. This pattern owes its shape to the confinement of the highly nonlinear part of the pulse in the central waveguide and to the linear diffraction of the low-power pulse edges. After this compression, both the linear edges (that diffract far from the central waveguide) as well as the main part of the pulse continue to broaden due to normal dispersion. The broadening of the main body of the pulse reduces the level of nonlinearity and it is therefore more amenable to diffraction. As a result, the central part of the remaining pulse diffracts, giving birth to two ultrashort spikes that keep moving apart. The X-like pattern that is generated by this procedure is depicted in Fig. 2.

*1.5kW*) nonlinearity dominates over coupling, thus confining the major part of the initial pulse within the central waveguide, as shown in Fig. 3. The side lobes again diffract in a manner similar to that described previously, but they are now weaker. In a way, the high nonlinearity in the central waveguide decouples it from the rest of the array and the evolution of the initial pulse resembles that of pulse broadening in normal-dispersive nonlinear fibers. In the array however, this pulse profile is considerably steeper at the edges, due to the discrete diffraction of the low power lobes in the transverse direction. This process shows promise in terms of generating square-like pulses.

*1.5kW*and by using a shorter

*200fs*pulse at the input, the effect of group velocity dispersion becomes even more pronounced. As depicted in Fig. 4, appreciable pulse broadening occurs and this fact leads to a totally different scenario. In particular, this leads to an X-formation and after that high-intensity spikes are generated sequentially. These spikes in turn produce other new X-waves in cascade.

### 3.2 Multiple waveguide excitation

*Δφ*among adjacent waveguides [2

**85**, 1863–1866 (2000). [CrossRef] [PubMed]

*Δφ=0*), the X-pattern is again observed as shown in Fig. 5 when the peak power in the central waveguide is

*0.5kW*and the initial temporal width of the pulse is set to be

*1ps*.

*Δφ=π*, where the diffraction in the system is anomalous, the beam discretely diffracts (Fig. 6(a)), being unable to support any kind of X-like formation. For

*Δφ=π/2*, where the diffraction in the system is zero, a more complicated V-like pattern is observed (Fig. 6(b)) due to higher-order diffraction effects. Again, the non-bidispersive nature of this configuration inhibits any X-like pattern.

## 4. Conclusions

## Acknowledgments

## References and links

1. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

2. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. |

3. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

4. | R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. |

5. | J. Meier, J. Hudock, D. N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. |

6. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

7. | 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 |

8. | H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. |

9. | R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. |

10. | J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. |

11. | 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. |

12. | D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. |

13. | J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. |

14. | D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. |

15. | J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control |

16. | D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three- dimensional bidispersive optical systems,” Opt. Lett. |

17. | 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. |

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

19. | 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 |

**OCIS Codes**

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Focus Issue: Discrete solitons in nonlinear optics

**History**

Original Manuscript: January 13, 2005

Revised Manuscript: January 19, 2005

Published: March 21, 2005

**Citation**

Sotiris Droulias, Kyriakos Hizanidis, Joachim Meier, and Demetrios Christodoulides, "X �?? waves in nonlinear normally dispersive waveguide arrays," Opt. Express **13**, 1827-1832 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-1827

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### References

- D. N. Christodoulides, F. Lederer, and Y. Silberberg, �??Discretizing light behaviour in linear and nonlinear waveguide lattices,�?? Nature 424, 817-823 (2003). [CrossRef] [PubMed]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Diffraction management,�?? Phys. Rev. Lett. 85, 1863-1866 (2000). [CrossRef] [PubMed]
- D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 13, 794�??796 (1988). [CrossRef] [PubMed]
- R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, �??Dynamics of discrete solitons in optical waveguide arrays,�?? Phys. Rev. Lett. 83, 2726�??2729 (1999). [CrossRef]
- J. Meier, J. Hudock, D. N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Discrete vector solitons in Kerr nonlinear waveguide arrays,�?? Phys. Rev. Lett. 91, 143907 (2003).J [CrossRef] [PubMed]
- J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, �??Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,�?? Nature 422, 147�??150 (2003). [CrossRef] [PubMed]
- 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 66, 046602 (2002). [CrossRef]
- H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, �??Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,�?? Phys. Rev. Lett. 92, 123902 (2004). [CrossRef] [PubMed]
- R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, �??Observation of discrete quadratic solitons,�?? Phys. Rev. Lett. 93, 113902 (2004). [CrossRef] [PubMed]
- J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, �??Experimental observation of discrete modulational instability,�?? Phys. Rev. Lett. 92, 163902 (2004). [CrossRef] [PubMed]
- 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, 023902 (2003). [CrossRef] [PubMed]
- D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, �??Gap solitons in waveguide arrays,�?? Phys. Rev. Lett. 92, 093904 (2004). [CrossRef] [PubMed]
- J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, �??Observation of vortex-ring �??discrete�?? solitons in 2D photonic lattices,�?? Phys. Rev. Lett. 92, 123904 (2004). [CrossRef] [PubMed]
- D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, �??Observation of discrete vortex solitons in optically induced photonic lattices,�?? Phys. Rev. Lett. 92, 123903 (2004). [CrossRef] [PubMed]
- J. Y. Lu and J. F. Greenleaf, �??Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,�?? IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992). [CrossRef] [PubMed]
- D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, �??Bessel X waves in two- and three-dimensional bidispersive optical systems,�?? Opt. Lett. 29, 1446�??1448 (2004). [CrossRef] [PubMed]
- 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, 093904 (2003).,�?? Phys. Rev. Lett. 91, 093904 (2003). [CrossRef] [PubMed]
- C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, �??Nonlinear Electromagnetic X Waves,�?? Phys. Rev. Lett. 90, 093904 (2003). [CrossRef]
- U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, T. Pertsch and F. Lederer, �??Optical discrete solitons in waveguide arrays. 2. Dynamic properties,�?? J. Opt. Soc. Am. B 19, 11, 2637-2644 (2002). [CrossRef]

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