## Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica

Optics Express, Vol. 14, Issue 13, pp. 6055-6062 (2006)

http://dx.doi.org/10.1364/OE.14.006055

Acrobat PDF (345 KB)

### Abstract

The observation of a two-dimensional discrete soliton in a cubic 5×5 fs laser written waveguide array in fused silica is reported for the first time. In addition to the localization the sharp edges of the array allow to study the influence of the array’s boundaries. The results are in excellent agreement with theoretical predictions and provide the basis for a variety of future applications for nonlinear two-dimensional integrated optical devices.

© 2006 Optical Society of America

## 1. Introduction

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

2. H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. **81**, 3383–3386 (1998). [CrossRef]

3. J. Meier, G. Stegeman, D. Christodoulides, R. Morandotti, M. Sorel, H. Yang, G. Salamo, J. Aitchison, and Y. Silberberg, “Nonlinear beam interactions in 1D discrete Kerr systems,” Opt. Exp. **13**, 1797–1807 (2005). [CrossRef]

4. R. Iwanow, R. Schieck, G. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. **93**, 1139021–4 (2004). [CrossRef]

5. A. Fratalocchi, G. Assanto, K. Brzdakiewicz, and M. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. **29**, 1530–1532 (2004). [CrossRef] [PubMed]

6. N. Efremidis, S. Sears, D. Christodoulides, J. Fleischer, and M. Segev, “Discrete Solitons in photorefractive optically induced nonlinear photonic lattices,” Phys. Rev. E **66**, 04660211–5 (2002). [CrossRef]

7. D. Christodoulides and E. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. **87**, 2339011–4 (2001). [CrossRef]

8. E. Eugenieva, N. Efremidis, and D. Christodoulides, “Design of switching junctions for two-dimensional discrete soliton networks,” Opt. Lett. **26**, 1978–1980 (2001). [CrossRef]

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

10. J. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Exp. **13**, 1780–1796 (2005). [CrossRef]

12. S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A. **77**, 109–111 (2003). [CrossRef]

13. W. Watanabe, T. Asano, K. Yamada, and K. Itoh, “Wavelength division with three-dimensional couplers fabricated by filamentation of femtosecond laser pulses,” Opt. Lett. **28**, 2491–2493 (2003). [CrossRef] [PubMed]

## 2. Fabrication of the waveguides

14. K. Davies, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a fs-laser,” Opt. Lett. **21**, 1729–1731 (1996). [CrossRef]

15. D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica,” Opt. Exp. **14**, 2151–2157 (2006). [CrossRef]

16. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. **29**, 468–470 (2004). [CrossRef] [PubMed]

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

18. A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, “Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica,” Opt. Exp. **13**, 10552–10557 (2005). [CrossRef]

19. I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured near-field intensity,” J. Lightwave Technol. **14**, 423–428 (1996). [CrossRef]

*A*(

*x,y*) is the modal field and

*n*

_{eff}is the effective refractive index of the propagating mode. The maximum index change obtained was Δ

*n*≈1×10

^{-3}with a size of 3 m×14 µm

^{2}(Fig. 2b). The transmission losses of a single waveguide, measured by a cut-back method, were <0.4 dB/cm and the waveguides showed no polarization dependency.

## 3. Experimental results

*n=m*=3) but the waveguide aside (

*m*=4,

*n*=3). Therefore non-symmetrical effects at the left and the right boundaries are obtained. Furthermore, due to the elliptical shape of the waveguides, there is a strong influence of asymmetrical coupling between horizontal and vertical waveguide neighbours. The resulting output pattern obtained in the experiment, with strong boundary effects and asymmetrical coupling, is shown in Fig. 4(a).

*E*

_{m,n}evolve. Since the dispersion length exceeds the length of the waveguide array by a factor of 10, one can also neglect temporal effects. This leads to a system of coupled ordinary differential equations [1

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

*E*

_{m,n}is the amplitude in the waveguide with coordinates

*m*in horizontal and

*n*in vertical direction,

*β*is the propagation constant and

*c*

^{h}and

*c*

^{v}are the coupling constants in horizontal and vertical direction, respectively. The value

*κ*is a measure for the waveguide’s effective third-order nonlinearity which is determined by the nonlinear refractive index

*n*

_{2}. It can be calculated as

*A*

_{eff}the mode’s effective area and

*v*the vacuum speed of light. Making an ansatz for a scaled plane wave

*c*

^{h}≠

*c*

^{v}). A strong influence of the array’s boundaries is obvious. For comparison an infinite cubic waveguide array with the same coupling properties between the single waveguides is shown in Fig. 4(c) which exhibits a considerably different discrete diffraction pattern. Furthermore there is a strong influence of asymmetrical coupling between horizontal and vertical waveguide neighbours, so that in Eq. (2) it is

*c*

^{h}≠

*c*

^{v}. In comparison in Fig. 4(d) an output pattern is shown that would be obtained if the coupling was isotropic. The non-isotropic coupling results from the shape of the waveguides which is nearly elliptical. Therefore the mode profile is also non-isotropic which yields a different mode overlap between the waveguides in horizontal and vertical directions. Since the profile of the focus of the writing objective can be chosen very precisely [20

20. M. Ams, G. Marshall, D. Spence, and M. Withford, “Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses,” Opt. Express **13**, 5676–5681 (2005). [CrossRef] [PubMed]

*c*

^{h}and

*c*

^{v}the nonlinear term compensates the linear coupling resulting in a localization of the propagating light. For sufficient high peak powers almost all of the light will be “trapped” in the excited waveguide due to the suppression of the linear coupling.

*P*

_{peak}=40kW and b) corresponding theoretical result

*P*

_{peak}=700kW and d) corresponding theoretical result

*P*

_{peak}=1000kW and f) corresponding theoretical result.

*P*

_{peak}≈40kW. At

*P*

_{peak}≈700kW (Fig. 5c) the linear coupling is reduced and at

*P*

_{peak}≈1000kW (Fig. 5e) almost all of the guided energy is “trapped” in the excited waveguide. To simulate this result one has to solve Eq. (2) numerically. However, for the calculation of

*κ*in Eq. (3) one has to consider the influence of the fs writing process on the effective nonlinearity in the waveguides. In recent experiments it turned out, that the nonlinear refractive index

*n*

_{2}in the waveguides is a function of the writing velocity [14

14. K. Davies, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a fs-laser,” Opt. Lett. **21**, 1729–1731 (1996). [CrossRef]

^{-20}

*m*

^{2}/

*W*, which is only 0.5×

*κ*(Eq. 3). The result agrees well with our experimental data, showing the reduction of the evanescent coupling (Fig. 5d) at

*P*

_{peak}≈700kW and the formation of a discrete soliton at

*P*

_{peak}≈1000kW (Fig. 5f).

## 4. Conclusion

## References and links

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

2. | H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. |

3. | J. Meier, G. Stegeman, D. Christodoulides, R. Morandotti, M. Sorel, H. Yang, G. Salamo, J. Aitchison, and Y. Silberberg, “Nonlinear beam interactions in 1D discrete Kerr systems,” Opt. Exp. |

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

5. | A. Fratalocchi, G. Assanto, K. Brzdakiewicz, and M. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. |

6. | N. Efremidis, S. Sears, D. Christodoulides, J. Fleischer, and M. Segev, “Discrete Solitons in photorefractive optically induced nonlinear photonic lattices,” Phys. Rev. E |

7. | D. Christodoulides and E. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. |

8. | E. Eugenieva, N. Efremidis, and D. Christodoulides, “Design of switching junctions for two-dimensional discrete soliton networks,” Opt. Lett. |

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

10. | J. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Exp. |

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

12. | S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A. |

13. | W. Watanabe, T. Asano, K. Yamada, and K. Itoh, “Wavelength division with three-dimensional couplers fabricated by filamentation of femtosecond laser pulses,” Opt. Lett. |

14. | K. Davies, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a fs-laser,” Opt. Lett. |

15. | D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica,” Opt. Exp. |

16. | T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. |

17. | A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, “Hexagonal waveguide arrays written with fs-laser pulses,” Appl. Phys. B. |

18. | A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, “Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica,” Opt. Exp. |

19. | I. Mansour and F. Caccavale, “An improved procedure to calculate the refractive index profile from the measured near-field intensity,” J. Lightwave Technol. |

20. | M. Ams, G. Marshall, D. Spence, and M. Withford, “Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses,” Opt. Express |

**OCIS Codes**

(130.4310) Integrated optics : Nonlinear

(140.7090) Lasers and laser optics : Ultrafast lasers

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(230.7370) Optical devices : Waveguides

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: May 12, 2006

Revised Manuscript: June 7, 2006

Manuscript Accepted: June 7, 2006

Published: June 26, 2006

**Citation**

Alexander Szameit, Jonas Burghoff, Thomas Pertsch, Stefan Nolte, Andreas Tünnermann, and Falk Lederer, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Express **14**, 6055-6062 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-6055

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### References

- D. Christodoulides and R. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988). [CrossRef] [PubMed]
- H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998). [CrossRef]
- J. Meier, G. Stegeman, D. Christodoulides, R. Morandotti, M. Sorel, H. Yang, G. Salamo, J. Aitchison, and Y. Silberberg, "Nonlinear beam interactions in 1D discrete Kerr systems," Opt. Exp. 13, 1797-1807 (2005). [CrossRef]
- R. Iwanow, R. Schieck, G. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of discrete quadratic solitons," Phys. Rev. Lett. 93, 1139021-4 (2004). [CrossRef]
- A. Fratalocchi, G. Assanto, K. Brzdakiewicz, and M. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530-1532 (2004). [CrossRef] [PubMed]
- N. Efremidis, S. Sears, D. Christodoulides, J. Fleischer, and M. Segev, "Discrete Solitons in photorefractive optically induced nonlinear photonic lattices," Phys. Rev. E 66, 04660211-5 (2002). [CrossRef]
- D. Christodoulides and E. Eugenieva, "Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays," Phys. Rev. Lett. 87, 2339011-4 (2001). [CrossRef]
- E. Eugenieva, N. Efremidis, and D. Christodoulides, "Design of switching junctions for two-dimensional discrete soliton networks," Opt. Lett. 26, 1978-1980 (2001). [CrossRef]
- J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003). [CrossRef] [PubMed]
- J. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Exp. 13, 1780-1796 (2005). [CrossRef]
- T. Pertsch, U. Peschel, S. Nolte, A. Tuennermann, F. Lederer, J. Kobelke, K. Schuster, and H. Bartelt, "Nonlinearity and disorder in two-dimensional fiber arrays," Phys. Rev. Lett. 39, 468-470 (2004).
- S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to threedimensional integrated optics," Appl. Phys. A. 77, 109-111 (2003). [CrossRef]
- W. Watanabe, T. Asano, K. Yamada, and K. Itoh, "Wavelength division with three-dimensional couplers fabricated by filamentation of femtosecond laser pulses," Opt. Lett. 28, 2491-2493 (2003). [CrossRef] [PubMed]
- K. Davies, K. Miura, N. Sugimoto, and K. Hirao, "Writing waveguides in glass with a fs-laser," Opt. Lett. 21, 1729-1731 (1996). [CrossRef]
- D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006). [CrossRef]
- T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004). [CrossRef] [PubMed]
- A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B. 82, 507-512 (2006). [CrossRef]
- A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10552-10557 (2005). [CrossRef]
- I. Mansour and F. Caccavale, "An improved procedure to calculate the refractive index profile from the measured near-field intensity," J. Lightwave Technol. 14, 423-428 (1996). [CrossRef]
- M. Ams, G. Marshall, D. Spence, and M. Withford, "Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses," Opt. Express 13, 5676-5681 (2005). [CrossRef] [PubMed]

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