## Interaction dynamics of X-waves in waveguide arrays |

Optics Express, Vol. 18, Issue 20, pp. 21252-21260 (2010)

http://dx.doi.org/10.1364/OE.18.021252

Acrobat PDF (606 KB)

### Abstract

The interaction dynamics of X-waves in an AlGaAs waveguide array is theoretically considered. The nonlinear discrete diffraction dynamics of a waveguide array mediates the generation of spatio-temporal X-waves from pulsed initial conditions. The interactions between co-propagating and counter-propagating X-waves are studied. For the co-propagating case, the initial phase relation between the X-waves determine the attractive or repulsive behavior of the X-wave interaction. For the counter-propagating case, the collisions between X-waves generate a nonlinear phase-shift. These dynamics show that X-waves interact in a manner similar to solitons.

© 2010 Optical Society of America

## 1. Introduction

1. J. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar waveequation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferrelec. Freq. contr. **39**, 19–31 (1992); [CrossRef]

3. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X-waves,” Phys. Rev. Lett. **90**, 170406 (2003); [CrossRef] [PubMed]

*normal*group-velocity dispersion (GVD) in bulk. Indeed, envelope X-waves were first observed in second-harmonic generation [4

4. 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]

5. M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X-waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. **92**253901 (2004). [CrossRef] [PubMed]

6. D. Faccio, M. 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**, 193901 (2006). [CrossRef] [PubMed]

7. C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice,” Phys. Rev. Lett. **92**, 120404 (2004). [CrossRef] [PubMed]

8. S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B **70**, 235123 (2004). [CrossRef]

9. Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, “Discrete X-Wave Formation in Nonlinear Waveguide Arrays,” Phys. Rev. Lett. **98**, 023901 (2007); [CrossRef] [PubMed]

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

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

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

12. K. Staliunas and M. Tlidi, “Hyperbolic Transverse Patterns in Nonlinear Optical Resonators,” Phys. Rev. Lett. **94**, 133902 (2005); [CrossRef] [PubMed]

14. 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]

*linear*collision dynamics aside from nonlinear phase changes that are intensity dependent. In the context of co-propagating X-waves, the initial phase difference determines an effective attraction or repulsion dynamics. Thus, the spatial-temporal X-waves behave in much the same manner as solitons.

## 2. Nonlinear X-wave Interactions

14. 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]

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

*c*= 0.82 mm

^{−1}and the nonlinear self-phase modulation parameter is taken to be γ = 3.6 m

^{−1}W

^{−1}. In waveguides, chromatic dispersion also is present. However in keeping with previous works, dispersion has been neglected in this system [11

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

*D*= 0 in all that follows. Additionally, simulations with the inclusion of realistic dispersion values as measured experimentally [10

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

**33**, 1440–1442 (2008). [CrossRef] [PubMed]

14. 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]

9. Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, “Discrete X-Wave Formation in Nonlinear Waveguide Arrays,” Phys. Rev. Lett. **98**, 023901 (2007); [CrossRef] [PubMed]

**33**, 1440–1442 (2008). [CrossRef] [PubMed]

**81**, 3383–3386 (1998); [CrossRef]

### 2.1. Co-propagating X-waves

**81**, 3383–3386 (1998); [CrossRef]

*A*represents the electric field envelope in the

_{n}*n*−th of the

*2N*+ 1 waveguides

*(n=*−

*N*, ···,−1,0,1,···,

*N)*where

*N =*20 for 41 waveguides. This set of governing equations has been shown to accurately reproduce experimental findings for pulses with kilowatt peak powers and pulsewidths of hundreds of femtoseconds [10

**33**, 1440–1442 (2008). [CrossRef] [PubMed]

*A*(0,

_{n}*T*) = 0 for

*n*≠ 0,1 where

*η*

_{0},

*η*

_{1}= 2.0 and Δ

*T*= 1. At this value, fully nonlinear X-waves are formed during the propagation in the WGA. The dynamics of the X-waves depend on the relative phase difference between the injected pulses. Figure 1 demonstrates a time-history of propagation of (1) with an initial phase difference of Δ

*θ*= 0. From the pulse-initial condition, the characteristic X-wave structure forms. Although the majority of the energy in the X-wave remains confined in the initial waveguide, the low amplitude sections interact and perturb the X-waves. Ultimately, a pair of X-waves results with a slightly larger separation than the original pair of pulses,

*i.e. they repel.*In contrast, consider the initial conditions in (2) with identical amplitudes but with Δ

*θ*=

*π*. A time-history of (1) with these initial conditions is shown in Fig. 2. The WGA still generates an X-wave from each of the initial pulses as shown in the second panel in Fig. 2. However, the separation in

*T*between the resulting X-waves is negligible,

*i.e. they attract.*Therefore, for X-waves with identical amplitudes, the resulting dynamics depends on the initial phase-difference between the pulses.

*T*= 0.5, 1.0, and 1.5. In all cases, for Δ

*θ*near zero the X-waves repel, and for Δ

*θ*near

*π*the X-waves attract. The strength of interaction does depend upon the initial separation. As shown in Figure 3, X-waves that have a small initial separation will both repel more strongly and also attract at smaller values of Δ

*θ*. Likewise, X-waves with larger initial separations will repel and attract less strongly. Indeed, this agrees with the limiting case of X-waves that are so far separated that they are non-interacting and will neither attract nor repel.

*T*= 1, but the same qualitative results occur even if other separations are used.

*η*

_{0}= 2.0 and

*η*

_{1}= 1.0 and Δ

*T*= 1 and Δ

*θ*=

*π*. Unlike the equally sized pulses, the larger pulse dominates the dynamics and incorporates the smaller pulse into a single X-wave structure regardless of the phase difference. Therefore, it is the amplitude difference and not the phase-difference that determines the resulting dynamics.

### 2.2. Counter-propagating X-waves

*A*is the forward-propagating field of the

_{n}*n*th waveguide and

*B*is the backward-propagating field of the

_{n}*nth*waveguide. In these equations, the nearest-neighbor coupling and self-phase modulation terms are retained from the co-propagating case, but an additional cross-phase modulation term appears along with a group-velocity term determining the forward (+ σ) and backward (−σ) directions of propagation.

*Z*measures the initial spatial separation between the right moving (forward-propagting) and left moving (backward propagating) X-waves. It must be stressed that unlike the co-propagating interaction the governing equations of this system

*occur in a stationary lab frame*and the initial condition is defined for all

*Z*at

*T*= 0 and not in the usual optical coordinate system. In this case, the waves collide at

*Z*= 0. Figure 6 demonstrates the basic collision dynamics. The two X-waves pass through each other without visible deformation. Thus the collision appears to be

*linear*. However, there is an induced nonlinear phase shift due to the collision. The resulting phase shift depends upon the initial launch intensity of the counter-propagating pulses.

*Z*. Figure 8 shows the phase-shift due to X-wave interaction for five different initial values of

*η*

_{+},

*η*

_{−}. In all cases, there is no measurable delay due to the interaction, but the amount of phase-shift is dependent upon both the initial separation and the initial pulse height. The results of Figure 8 may be broken down into two separate regimes, the low-amplitude initial conditions and the high-amplitude initial conditions.

*η*

_{+},

*η*

_{−}= 1.00, 0.75, 0.50, and are characteristic of lower amplitude solutions as well. In these case, the amount of phase-shift decreases as the distance between the initial pulses increase. At relatively short initial separations, the low-amplitude X-wave has little time to distribute energy to neighboring waveguides through discrete diffraction. Because the phase shift is generated solely by nonlinear effects, the short distances allow the X-waves to be as nonlinear as possible given their initial conditions. Indeed, as the separation is increased the phase-shift decreases because the X-wave distributes energy to the outer waveguides and acts in a linear fashion. With an initial separation of 1.5, there is essentially no noticeable phase-shift.

## 3. Conclusions

**33**, 1440–1442 (2008). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar waveequation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferrelec. Freq. contr. |

2. | E. Recami, M. Zamboni-Rached, and H. E. Hernandez-Figueroa, |

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

4. | 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. |

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

6. | D. Faccio, M. 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. |

7. | C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice,” Phys. Rev. Lett. |

8. | S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B |

9. | Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, “Discrete X-Wave Formation in Nonlinear Waveguide Arrays,” Phys. Rev. Lett. |

10. | D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. |

11. | J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express |

12. | K. Staliunas and M. Tlidi, “Hyperbolic Transverse Patterns in Nonlinear Optical Resonators,” Phys. Rev. Lett. |

13. | L. Mollenauer and J. Gordon, |

14. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(320.7090) Ultrafast optics : Ultrafast lasers

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: August 2, 2010

Revised Manuscript: September 10, 2010

Manuscript Accepted: September 10, 2010

Published: September 22, 2010

**Citation**

Matthew O. Williams, Colin W. McGrath, and J. Nathan Kutz, "Interaction dynamics of X-waves in waveguide arrays," Opt. Express **18**, 21252-21260 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21252

Sort: Year | Journal | Reset

### References

- J. Lu and J. F. Greenleaf, "Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferrelec. Freq. contr. 39, 19-31 (1992). [CrossRef]
- E. Recami, M. Zamboni-Rached, and H. E. Hernandez-Figueroa, Localized waves (Wiley, 2007).
- C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003). [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). [CrossRef] [PubMed]
- M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92, 253901 (2004). [CrossRef] [PubMed]
- D. Faccio, M. 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, 193901 (2006). [CrossRef] [PubMed]
- C. Conti, and S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice," Phys. Rev. Lett. 92, 120404 (2004). [CrossRef] [PubMed]
- S. Longhi, and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235123 (2004). [CrossRef]
- Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007). [CrossRef] [PubMed]
- D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, "Nonlinear femtosecond pulse reshaping in waveguide arrays," Opt. Lett. 33, 1440-1442 (2008). [CrossRef] [PubMed]
- J. N. Kutz, C. Conti, and S. Trillo, "Mode-locked X-wave lasers," Opt. Express 15, 16022-16028 (2007). [CrossRef] [PubMed]
- K. Staliunas, and M. Tlidi, "Hyperbolic Transverse Patterns in Nonlinear Optical Resonators," Phys. Rev. Lett. 94, 133902 (2005). [CrossRef] [PubMed]
- L. Mollenauer, and J. Gordon, Solitons in Optical Fibers: Fundamentals and Applications, (Springer, 2006).
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Supplementary Material

» Media 1: MOV (2284 KB)

» Media 2: MOV (2388 KB)

» Media 3: MOV (2223 KB)

» Media 4: MOV (2484 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.