## Soliton shedding from Airy pulses in Kerr media |

Optics Express, Vol. 19, Issue 18, pp. 17298-17307 (2011)

http://dx.doi.org/10.1364/OE.19.017298

Acrobat PDF (1494 KB)

### Abstract

We simulate and analyze the propagation of truncated temporal Airy pulses in a single mode fiber in the presence of self-phase modulation and anomalous dispersion as a function of the launched Airy power and truncation coefficient. Soliton pulse shedding is observed, where the emergent soliton parameters depend on the launched Airy pulse characteristics. The Soliton temporal position shifts to earlier times with higher launched powers due to an earlier shedding event and with greater energy in the Airy tail due to collisions with the accelerating lobes. In spite of the Airy energy loss to the shed Soliton, the Airy pulse continues to exhibit the unique property of acceleration in time and the main lobe recovers from the energy loss (healing property of Airy waveforms).

© 2011 OSA

## 1. Introduction

1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

3. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics **2**(11), 675–678 (2008). [CrossRef]

4. I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. **35**(10), 1581–1583 (2010). [CrossRef] [PubMed]

5. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science **324**(5924), 229–232 (2009). [CrossRef] [PubMed]

6. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. **103**(12), 123902 (2009). [CrossRef] [PubMed]

7. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics **4**(2), 103–106 (2010). [CrossRef]

8. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. **105**(25), 253901 (2010). [CrossRef] [PubMed]

10. R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A **82**(4), 043832–043835 (2010). [CrossRef]

*β*is the dispersion coefficient,

_{2}*γ*is the nonlinear coefficient and

*A*is the wave amplitude that depends on local time-

*T*, and distance-

*z*. Due to the addition of the nonlinear potential (or SPM term) in the NLSE, the Airy function is no longer a valid solution and we cannot predict analytically the Airy pulse evolution. The Soliton, on the other hand, is a well-known solution of the NLSE. For the canonical first order case, its profile is

*P*is peak power and

_{0}*T*is duration and it is obtained only when there is equilibrium between the dispersion and the nonlinear effect, leading to the condition

_{0}11. N. Belanger and P. A. Belanger, “Bright solitons on a cw background,” Opt. Commun. **124**(3-4), 301–308 (1996). [CrossRef]

15. H. A. Haus, F. I. Khatri, W. S. Wong, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. **32**(6), 917–924 (1996). [CrossRef]

### 1.1. Normalization terms

*|β*=

_{2}|*γ*=

*T*= 1, and the launched Airy pulse profile is defined as:

_{0}*K*(

_{p}*a*) is a truncation-dependent factor that sets the pulse peak intensity to 1 for any

*a*value . This factor was numerically calculated and found to be in parabolic dependence with the truncation coefficient.

*T*is the time variable in a frame of reference that moves with the wave group velocity, i.e.

*R*is a dimensionless parameter we vary for scaling the Airy power. At

*R*=1 the Airy main lobe intensity profile looks quite similar to the fundamental soliton, as shown in Fig. 1(b).

*L*units, defined as

_{d}## 2. Effects of launched Airy power

*R*in the range 0.1-2 and for every

*R*value we propagated the pulse using the SSFM algorithm. Figure 2 shows pulse evolution examples for select

*R*values. At low launched power, the Airy pulse performs the acceleration in time and subsequently it succumbs to dispersion. However, when

*R*is sufficiently large (above 0.9) a stationary soliton pulse is formed out of the centered energy about the Airy main lobe. The soliton exhibits periodic oscillations in the soliton amplitude and width as a function of propagation distance. In addition, we witness the resilience of the temporal Airy waveform to shedding of a fraction of the energy as a soliton; the wavefront continues to propagate along a parabolic trajectory. Similar resilience has been shown in main lobe masking for spatial Airy beams [17

17. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express **16**(17), 12880–12891 (2008). [CrossRef] [PubMed]

### 2.1. The emergent soliton

^{2}product oscillates about the equilibrium condition (= 1) defined in Eq. (2). These oscillations about the stable soliton are known to arise as a result of interference between dispersive background radiation and the formed soliton [11

11. N. Belanger and P. A. Belanger, “Bright solitons on a cw background,” Opt. Commun. **124**(3-4), 301–308 (1996). [CrossRef]

12. M. W. Chbat, P. R. Prucnal, M. N. Islam, C. E. Soccolich, and J. P. Gordon, “Long-range interference effects of soliton reshaping in optical fibers,” J. Opt. Soc. Am. B **10**(8), 1386–1395 (1993). [CrossRef]

*R*values. The pulse width narrows and the oscillations period decreases with higher launch power. The decreasing oscillation period with increasing launch power is depicted in Fig. 3(b). Similar behavior was reported in [12

12. M. W. Chbat, P. R. Prucnal, M. N. Islam, C. E. Soccolich, and J. P. Gordon, “Long-range interference effects of soliton reshaping in optical fibers,” J. Opt. Soc. Am. B **10**(8), 1386–1395 (1993). [CrossRef]

13. A. Hasegawa and Y. Kodama, “Amplification and reshaping of optical solitons in a glass fiber-I,” Opt. Lett. **7**(6), 285–287 (1982). [CrossRef] [PubMed]

14. E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D **87**(1-4), 201–215 (1995). [CrossRef]

*R*a relatively long time is required for accumulation of enough energy by SPM for the soliton formation and shedding, and during this time the Airy pulse is accelerating and ‘carries’ the accumulating energy with it to later times. For larger

*R*values there is enough energy in the Airy main lobe for soliton formation and shedding at an early point.

### 2.2. The accelerating wavefront

*R*parameter. Note that the linear Airy pulse evolution is identical for every intensity value.

10. R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A **82**(4), 043832–043835 (2010). [CrossRef]

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

*R*values, the energy evolution of the linear propagations coincides to one curve that asymptotically approaches the value of half launched pulse energy, according to its linear nature. For the nonlinear propagations we clearly see that as

*R*grows the fractional energy amount that is delayed is decreasing, where the oscillatory behavior is due to the soliton oscillations which take place in the boundary of the right half propagation plane. Those curves and those of Fig. 6(b), which chart the energy evolution of the formed soliton for different

*R*values, show the fact that the formed soliton not only has more intensity when

*R*is growing, but also carries a larger energy fraction from the whole pulse. This can also be seen in Fig. 6(c), where the mean soliton relative energy was calculated for every

*R*value. From Figs. 6(b-c) we also see the energy preservation—the normalized delayed energy is missing energy that is about half of the shed soliton energy, where the other half originates from the faster propagating energy components. When

*R*= 2, for example, the soliton energy fraction is about 0.39 and the missing fractional energy amount from the delayed energy is about 0.19, half of 0.39.

## 3. Truncation coefficient effect

*R*to 1.5 while varying the truncation coefficient in the range 0.01-0.1, as shown in Fig. 7(a) , and propagate the apodized Airy for every truncation value. Figures 7(b-c) show two examples of the Airy pulse evolution in time-distance space. We see that when the truncation is small the Airy original features as self-similarity and acceleration in time are more noticeable. The influence of the truncation degree on emergent soliton properties and on the accelerating wavefront was examined in the same manner as in the previous section.

### 3.1. The emergent soliton

**·**) fit process was started from a different propagation distance for every truncation value.

15. H. A. Haus, F. I. Khatri, W. S. Wong, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. **32**(6), 917–924 (1996). [CrossRef]

### 3.2. The accelerating wavefront

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

## 4. Soliton time position for power and truncation

## 5. Summary

8. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. **105**(25), 253901 (2010). [CrossRef] [PubMed]

*R*= 4). We see that for such intense excitation three solitons are shed, the main soliton in a consistent manner to that described here, and two additional weaker soliton s at both higher and lower center frequencies. This result was still obtained with the standard nonlinear Schrodinger equation (Eq. (4)). However, for proper simulation of intense Airy pulse excitation, one should also add additional terms to account for higher-order nonlinear effects such as Raman scattering and self-steepening.

## References

1. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

2. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. |

3. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics |

4. | I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. |

5. | P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science |

6. | P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. |

7. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

8. | D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. |

9. | C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with self-healing Airy pulses,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC9. |

10. | R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A |

11. | N. Belanger and P. A. Belanger, “Bright solitons on a cw background,” Opt. Commun. |

12. | M. W. Chbat, P. R. Prucnal, M. N. Islam, C. E. Soccolich, and J. P. Gordon, “Long-range interference effects of soliton reshaping in optical fibers,” J. Opt. Soc. Am. B |

13. | A. Hasegawa and Y. Kodama, “Amplification and reshaping of optical solitons in a glass fiber-I,” Opt. Lett. |

14. | E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D |

15. | H. A. Haus, F. I. Khatri, W. S. Wong, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. |

16. | G. P. Agrawal, |

17. | J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express |

**OCIS Codes**

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

(190.0190) Nonlinear optics : Nonlinear optics

(190.3270) Nonlinear optics : Kerr effect

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 2, 2011

Revised Manuscript: August 5, 2011

Manuscript Accepted: August 12, 2011

Published: August 18, 2011

**Citation**

Yiska Fattal, Amitay Rudnick, and Dan M. Marom, "Soliton shedding from Airy pulses in Kerr media," Opt. Express **19**, 17298-17307 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17298

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### References

- M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef] [PubMed]
- J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]
- I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). [CrossRef] [PubMed]
- P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef] [PubMed]
- P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009). [CrossRef] [PubMed]
- A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]
- D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010). [CrossRef] [PubMed]
- C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with self-healing Airy pulses,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC9.
- R.-P. Chen, C.-F. Yin, X.-X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A 82(4), 043832–043835 (2010). [CrossRef]
- N. Belanger and P. A. Belanger, “Bright solitons on a cw background,” Opt. Commun. 124(3-4), 301–308 (1996). [CrossRef]
- M. W. Chbat, P. R. Prucnal, M. N. Islam, C. E. Soccolich, and J. P. Gordon, “Long-range interference effects of soliton reshaping in optical fibers,” J. Opt. Soc. Am. B 10(8), 1386–1395 (1993). [CrossRef]
- A. Hasegawa and Y. Kodama, “Amplification and reshaping of optical solitons in a glass fiber-I,” Opt. Lett. 7(6), 285–287 (1982). [CrossRef] [PubMed]
- E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, “Nonlinear interaction of solitons and radiation,” Physica D 87(1-4), 201–215 (1995). [CrossRef]
- H. A. Haus, F. I. Khatri, W. S. Wong, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. 32(6), 917–924 (1996). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).
- J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]

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