## Ultrafast transverse undulation of self-trapped laser beams

Optics Express, Vol. 16, Issue 21, pp. 16935-16940 (2008)

http://dx.doi.org/10.1364/OE.16.016935

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### Abstract

The propagation of a self-trapped laser beam in a planar waveguide that exhibits a Kerr nonlinearity and a normal chromatic dispersion is considered. We demonstrate experimentally for the first time to our knowledge that such a beam undergoes an undulation responsible for ultrafast transverse oscillations of its axis. This phenomenon, called “snake instability”, was predicted theoretically in 1973 by Zakharov and Rubenchik on the basis of a study of the soliton solutions of the hyperbolic nonlinear Schrödinger equation. The signature of this instability is observed in the spatially resolved temporal spectrum.

© 2008 Optical Society of America

2. P. M. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. **15**, 1005–1007 (1965). [CrossRef]

5. A. De Rossi, S. Trillo, A. V. Buryak, and Y. S. Kivshar, “Snake instability of one-dimensional parametric spatial solitons,” Opt. Lett. **22**, 868–870 (1997). [CrossRef] [PubMed]

6. A. V. Mamaev, M. Saffman, and A.A. Zozulia, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. **35**, 25–30 (1996). [CrossRef]

12. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. **70**, 1633–1713 (2007). [CrossRef]

*A*of an optical wave obeys the (2+1)D NLS equation:

*A*exp[

*i*(

*kz*-

*w*

_{0}

*t*)],

*w*

_{0}being the central frequency of the laser beam and

*k*its wavenumber. The notations here and below are as follows:

*z*is the coordinate in the propagation direction of the laser beam,

*x*is the in-plane transverse coordinate,

*t*is the time,

*β*

_{2}is the group velocity dispersion coefficient and

*γ*>0 is the nonlinearity coefficient. The soliton solution of Eq.1

*A*=

*A*

_{0}sech(

*x*/

*X*

_{0})exp(

*i*ϕ

_{NL}), where

*ϕ*

_{NL}=

*γA*

^{2}

_{0}

*z*/2 and

*A*

^{2}

_{0}=1/(

*γkX*

^{2}

_{0}), represents a shape-invariant monochromatic laser beam. Yet, in dispersive media (

*β*

_{2}≠0), this solution is unstable, which is revealed by the exponential growth of perturbations of the forme

*ε*(

*x, z, t*)=

*ε*

_{0}(

*u*

_{1}+

*iu*

_{2})exp(

*iϕ*) where

_{NL}*u*

_{1,2}=[

*U*

_{1,2}(

*x*)exp(

*iΩt*+Γ

*z*)+

*c.c*.] [3, 16]. When the group velocity dispersion is normal (

*β*

_{2}>0), i.e. when the NLS equation is hyperbolic, the eigenmodes

*U*

_{1,2}(

*x*) are antisymmetric. The net effect of these unstable eigenmodes is therefore to shift the lateral position of the soliton, the direction of the lateral shift being reversed every half-period Δ

*t*=π/

*Ω*(see Fig. 1).

*µ*m width (FWHM) by means of a cylindrical lens (f=200mm) and a ×60 microscope objective. The semiconducor planar waveguide is 12mm-long and is made up of a 1.6

*µ*m-thick guiding layer of Al

_{0.18}Ga

_{0.82}As on the top of a 4

*µ*m-thick cladding of Al

_{0.24}Ga

_{0.76}As. In order to avoid any back-reflection into the amplifier, a free space isolator is used. This isolator also enables us to set the polarization horizontally to excite the TE

_{0}mode of the waveguide and to control the input power by means of a half-wave plate. At the output of the waveguide, the beam is collected by a ×60 microscope objective. Part of the light is used to image the output laser beam on an infrared vidicon camera. The remaining part of the beam is imaged by a lens (f=140mm) on the entrance of a home made two-dimensional spectrometer made up of a collimating lens (f=200mm), a grating (1200 groves/mm) and an imaging lens (f=80mm). The resolution of this spectrometer is 1.1 nm. The two-dimensional spectrum was recorded by the same infrared vidicon camera. The response time of this camera is in the millisecond range and determines the integration time of the experimental time-averaged spectra. The calibration of the spectrometer was performed by using a tunics source and an optical spectrum analyser.

17. M. Sheik-Bahae, D. C. Hutching, D. J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solid,” IEEE J. Quantum Electron. **27**, 1296–1309 (1991). [CrossRef]

*µ*m broadens up to 190

*µ*m FWHM at the output of the waveguide, which corresponds to approximately 10 diffraction lengths. Figure 3 clearly shows that, as the power increases, the width of the output profile decreases, with the narrowest profile of 40

*µ*m FWHM occurring at the peak power of 780W. Note that the waveguide losses decrease the beam power during propagation, which explains why the optimal output width remains larger than the input width. However, losses do not prevent the snake instability from developing because the snake instability gain, Γ=22 dB/cm, is much larger than the measured loss coefficient

*α*=3.8 dB/cm.

21. J. R. Myra and C. S. Liu, “Self-modulation of ion Berstein waves,” Phys. Fluids **23**, 2258–2264 (1980). [CrossRef]

22. N. R. Pereira, A. Sen, and A. Bers, “Nonlinear development of lower hybrid cones,” Phys. Fluids **21**, 117–120 (1978). [CrossRef]

## References and links

1. | S. Trillo and W. Torruellas, |

2. | P. M. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. |

3. | V. E. Zakharov and A. M. Rubenchik, “Instablity of waveguides and solitons in nonlinear media,” Sov. Phys. JETP |

4. | K. Germaschewski, R. Grauer, L. Bergé, V. K. Mezentsev, and J. J. Rasmussen, “Splittings, coalescence, bunch and snake patterns in the 3D nonlinear Schrödinger equation with anisotropic dispersion,” Physica |

5. | A. De Rossi, S. Trillo, A. V. Buryak, and Y. S. Kivshar, “Snake instability of one-dimensional parametric spatial solitons,” Opt. Lett. |

6. | A. V. Mamaev, M. Saffman, and A.A. Zozulia, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. |

7. | V. Tikhonenko, J. Christou, B. Luther-Davies, and Y. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. |

8. | S.-P. Gorza, N. Roig, Ph. Emplit, and M. Haelterman, “Snake Instability of a Spatiotemporal Bright Soliton Stripe,” Phys. Rev. Lett. |

9. | S.-P. Gorza, Ph. Emplit, and M. Haelterman, “Observation of the snake instability of a spatially extended temporal bright soliton,” Opt. Lett. |

10. | X. Liu, K. Beckwitt, and F. Wise, “Transverse Instability of Optical Spatiotemporal Solitons in QuadraticMedia,” Phys. Rev. Lett. |

11. | S. D. Jenkins, D. Salerno, S. Minardi, S. Tamošauskas, T. A. B. Kennedy, and P. Di Trapani, “Quantum-noise initiated symmetry breaking of spatial solitons,” Phys. Rev. Lett. |

12. | L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. |

13. | B. P. Anderson et al. “Watching dark soliton decay into vortex rings in a Bose-Einstein condensate,” Phys. Rev. Lett. |

14. | Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, “Observation of quatum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condenstate,” Science |

15. | E. D. Brown, S. B. Buchsbaum, R. E. Hall, J. P. Penhune, K. F. Schmitt, K. M. Watson, and D. C. Wyatt, “Observation of a nonlinear solitary wave packet in the Kelvin wake of a ship,” J. Fluid Mech. |

16. | B. Deconink, D. E. Pelinovsky, and J. D. Carter, “Transverse instabilities of deep-water solitary waves,” Proc. Roy. Soc. A |

17. | M. Sheik-Bahae, D. C. Hutching, D. J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solid,” IEEE J. Quantum Electron. |

18. | S. Adachi, “Optical properties of AlxGa1-xAs alloys,” Phys. Rev. B |

19. | G. P. Agrawal, |

20. | H. C. Yuen and B. M. Lake, |

21. | J. R. Myra and C. S. Liu, “Self-modulation of ion Berstein waves,” Phys. Fluids |

22. | N. R. Pereira, A. Sen, and A. Bers, “Nonlinear development of lower hybrid cones,” Phys. Fluids |

**OCIS Codes**

(190.3100) Nonlinear optics : Instabilities and chaos

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 27, 2008

Revised Manuscript: September 30, 2008

Manuscript Accepted: October 3, 2008

Published: October 8, 2008

**Citation**

S.-P. Gorza and M. Haelterman, "Ultrafast transverse undulation of self-trapped laser beams," Opt. Express **16**, 16935-16940 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16935

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### References

- S. Trillo and W. Torruellas, Spatial solitons (Springer, Berlin, 2001).
- P. M. Kelley, "Self-focusing of optical beams," Phys. Rev. Lett. 15, 1005-1007 (1965). [CrossRef]
- V. E. Zakharov and A. M. Rubenchik, "Instablity of waveguides and solitons in nonlinear media," Sov. Phys. JETP 38, 494-500 (1974).
- K. Germaschewski, R. Grauer, L. Bergé, V. K. Mezentsev, and J. J. Rasmussen, "Splittings, coalescence, bunch and snake patterns in the 3D nonlinear Schrödinger equation with anisotropic dispersion," Physica D 151, 175-198 (2001).
- A. De Rossi, S. Trillo, A. V. Buryak, and Y. S. Kivshar, "Snake instability of one-dimensional parametric spatial solitons," Opt. Lett. 22, 868-870 (1997). [CrossRef] [PubMed]
- A. V. Mamaev, M. Saffman, and A.A. Zozulia, "Break-up of two-dimensional bright spatial solitons due to transverse modulation instability," Europhys. Lett. 35, 25-30 (1996). [CrossRef]
- V. Tikhonenko, J. Christou, B. Luther-Davies, and Y. S. Kivshar, "Observation of vortex solitons created by the instability of dark soliton stripes," Opt. Lett. 21, 1129-1131 (1996). [CrossRef] [PubMed]
- S.-P. Gorza, N. Roig, Ph. Emplit, and M. Haelterman, "Snake Instability of a Spatiotemporal Bright Soliton Stripe," Phys. Rev. Lett. 92, 084101 (2004). [CrossRef] [PubMed]
- S.-P. Gorza, Ph. Emplit, and M. Haelterman, "Observation of the snake instability of a spatially extended temporal bright soliton," Opt. Lett. 31, 1280-1282 (2006). [CrossRef] [PubMed]
- X. Liu, K. Beckwitt, and F. Wise, "Transverse Instability of Optical Spatiotemporal Solitons in QuadraticMedia," Phys. Rev. Lett. 85, 1871-1874 (2000). [CrossRef] [PubMed]
- S. D. Jenkins, D. Salerno, S. Minardi, S. Tamo�?sauskas, T. A. B. Kennedy, and P. Di Trapani, "Quantum-noise initiated symmetry breaking of spatial solitons," Phys. Rev. Lett. 95, 203902 (2005). [CrossRef] [PubMed]
- L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70, 1633-1713 (2007). [CrossRef]
- B. P. Anderson et al. "Watching dark soliton decay into vortex rings in a Bose-Einstein condensate," Phys. Rev. Lett. 86, 2926-2929 (2001). [CrossRef] [PubMed]
- Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, "Observation of quatum shock waves created with ultracompressed slow light pulses in a Bose-Einstein condenstate," Science 293, 663-668 (2001). [CrossRef] [PubMed]
- E. D. Brown, S. B. Buchsbaum, R. E. Hall, J. P. Penhune, K. F. Schmitt, K. M. Watson, and D. C. Wyatt, "Observation of a nonlinear solitary wave packet in the Kelvin wake of a ship," J. Fluid Mech. 204, 263-293 (1989). [CrossRef]
- B. Deconink, D. E. Pelinovsky, and J. D. Carter, "Transverse instabilities of deep-water solitary waves," Proc. Roy. Soc. A 462, 2039-2061 (2005).
- M. Sheik-Bahae, D. C. Hutching, D. J. Hagan, and E. W. Van Stryland, "Dispersion of bound electronic nonlinear refraction in solid," IEEE J. Quantum Electron. 27, 1296-1309 (1991). [CrossRef]
- S. Adachi, "Optical properties of AlxGa1�??xAs alloys," Phys. Rev. B 38, 12345-12352 (1988). [CrossRef]
- G. P. Agrawal, Nonlinear fiber optics (Academic Press, San-Diego, 2001).
- H. C. Yuen and B. M. Lake, Non-linear dynamics of deep-water gravity waves (Accademic Press, New York, 1982).
- J. R. Myra and C. S. Liu, "Self-modulation of ion Berstein waves," Phys. Fluids 23, 2258-2264 (1980). [CrossRef]
- N. R. Pereira, A. Sen, and A. Bers, "Nonlinear development of lower hybrid cones," Phys. Fluids 21, 117-120 (1978). [CrossRef]

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