## Crossover from self-defocusing to discrete trapping in nonlinear waveguide arrays

Optics Express, Vol. 14, Issue 1, pp. 254-259 (2006)

http://dx.doi.org/10.1364/OPEX.14.000254

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### Abstract

We predict a sharp crossover from nonlinear self-defocusing to discrete self-trapping of a narrow Gaussian beam with the increase of the refractive index contrast in a periodic photonic lattice. We demonstrate experimentally nonlinear discrete localization of light with defocusing nonlinearity by single site excitation in LiNbO_{3} waveguide arrays.

© 2006 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]

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

3. R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. **86**, 3296–3299 (2001). [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]

*n*) is increased. We also observe experimentally the formation of a self-trapped state from a single-site excitation in the defocusing regime.

## 2. Discrete self-trapping in photonic lattices

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]

3. R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. **86**, 3296–3299 (2001). [CrossRef] [PubMed]

4. 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]

5. 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**, 113902-4 (2004). [CrossRef] [PubMed]

6. F. Chen, M. Stepic, C. E. Ruter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO_{3} waveguide arrays,” Opt. Express **13**, 4314–4324 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4314. [CrossRef] [PubMed]

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

8. 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**, 023902-4 (2003). [CrossRef] [PubMed]

9. 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]

10. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. **28**, 710–712 (2003). [CrossRef] [PubMed]

11. H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902-4 (2004). [CrossRef] [PubMed]

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

*z*is the propagation coordinate,

*a*(

_{n}*z*) is the mode amplitude in the

*n*-th waveguide [Fig. 1(a)], and

*β*is the propagation constant. Coefficient

*C*stands for the nearest-neighbor coupling between the waveguides, and the last term in Eq. (1) accounts for mode detuning through the intensity-dependent change of the refractive index.

*unstaggered discrete solitons*[1

**424**, 817–823 (2003). [CrossRef] [PubMed]

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

*staggered solitons*when the amplitude of the modes in neighboring waveguides is out of phase [14

14. Yu. S. Kivshar, “Self-localization in arrays of defocusing wave-guides,” Opt. Lett. **18**, 1147–1149 (1993). [CrossRef] [PubMed]

*a*→ (-1)

_{n}

^{n}*a*

_{n}^{*}and

*γ*→ -

*γ*. It follows that the beam dynamics in the framework of Eq. (1) is fully equivalent for positive (

*γ*> 0) and negative (

*γ*<0) nonlinearities, with the only difference being in the phase structure, provided only a single site is excited at the input, i.e.

*a*(

_{n}*z*= 0) = 0 for

*n*≠ 0. In particular, when the input intensity is high enough, a discrete soliton should form for either type of nonlinear response.

## 3. Nonlinear localization in continuous periodic photonic structures

15. G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrodinger equation with a periodic potential,” Phys. Rev. E **66**, 46608-6 (2002). [CrossRef]

*E*(

*x*,

*z*),

*D*=

*z*/(4

_{s}λ*πn*

_{0}

*x*

_{s}^{2}) is the diffraction coefficient,

*ρ*= 2

*πz*/

_{s}*λ*,

*x*is normalized to

*x*,

_{s}*z*is normalized to

*z*. We choose the parameters to match the conditions of our experiments described below. The linear refractive index change Δ

_{s}*n*is taken as Δ

*n*(

*x*) =

*ξ*∑

*exp[-(*

_{n}*x*-

*n d*)

^{2}/

*w*

^{2}], where

*ξ*defines the modulation depth. We take the values of waveguide width

*w*= 12

*μ*m and the waveguide spacing

*d*= 19

*μ*m, and then the corresponding refractive index contrast is Δ

*n*= 0.442

_{max}*ξ*. We note that this index profile [Fig. 1(b)] is defined by the experimental realization, however we have verified that our conclusions are valid for different lattice profiles. Other parameters are:

*λ*= 0.532

*μ*m,

*n*

_{0}= 2.234,

*x*= 1

_{s}*μ*m,

*z*= 1mm, and ℱ(

_{s}*I*) = 1.5(1+

*I*)

^{-1}for photovoltaic defocusing nonlinearity [17

17. G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A **50**, R4457–R4460 (1994). [CrossRef] [PubMed]

18. A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. **92**, 093901-4 (2004). [CrossRef] [PubMed]

*n*. In the case of defocusing nonlinearity, solitons can form at the bottom edge of the first band, and their propagation constant is shifted deeper into the gap for larger intensities. This scenario is predicted correctly by the tight-binding model (1), however this model does not account for the existence of the second band, that defines the gap extent. On the other hand, the gap size limits the minimum width of self-trapped beams, see examples in Figs. 2(c,d). The width of the gap is smaller for weaker refractive index contrast and increases for larger Δ

_{max}*n*. The calculated minimal width of the gap soliton,

_{max}*W*= 3 ∫ |

*x*||

*ψ*|

^{2}

*dx*/ ∫ |

*ψ*|

^{2}

*dx*, is plotted in Fig. 2(e) vs. the refractive index contrast. In the case of small refractive index contrast and a narrow band gap, the narrowest soliton spans over several waveguides. The situation changes when the contrast of index modulation increases, and the narrowest soliton is localized at a single waveguide.

## 4. Crossover from self-defocusing to discrete self-trapping

*n*, self-trapping with almost 100% efficiency becomes possible, as predicted by the discrete model (1). This feature represents a transition from self-defocusing to discrete self-trapping as the index contrast exceeds a certain threshold.

_{max}## 5. Experimental observation of single-site self-trapping with defocusing nonlinearity

5. 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**, 113902-4 (2004). [CrossRef] [PubMed]

6. F. Chen, M. Stepic, C. E. Ruter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO_{3} waveguide arrays,” Opt. Express **13**, 4314–4324 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4314. [CrossRef] [PubMed]

8. 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**, 023902-4 (2003). [CrossRef] [PubMed]

9. 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]

5. 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**, 113902-4 (2004). [CrossRef] [PubMed]

8. 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**, 023902-4 (2003). [CrossRef] [PubMed]

9. 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]

6. F. Chen, M. Stepic, C. E. Ruter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO_{3} waveguide arrays,” Opt. Express **13**, 4314–4324 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4314. [CrossRef] [PubMed]

*μ*m and refractive index contrast Δ

*n*= 2.8 × 10

_{max}^{-4}, which was chosen above the threshold for a crossover to discrete self-trapping predicted in numerical simulations [Fig. 2(f)]. In the fabrication process, 100 Å of Ti was deposited on the X-cut LiNbO

_{3}using electron beam evaporation. The Ti layer was then photolithographically patterned and etched in a buffered hydrofluoric acid solution. The diffusion was conducted at 1050 °C for 3 hours in a wet oxygen environment. The waveguides were verified as single mode using a prism coupling technique. The array was then diced to a total length of 5cm and both facets were mechanically polished.

_{3}sample exhibits a strong photovoltaic effect which leads to the negative (self-defocusing) nonlinear response to laser beams at a photosensitive wavelength. In our experiments, we tightly focused an extraordinary polarized laser beam from a cw Nd:YVO

_{4}laser into a single guide of the array by a microscope objective (×20). The input and output facets were monitored by two CCD cameras. The array was externally illuminated with white light in order to reduce the nonlinear response time of the photovoltaic material to less than a minute. At low laser power (~10 nW) the propagating beam experienced typical discrete diffraction [1

**424**, 817–823 (2003). [CrossRef] [PubMed]

**424**, 817–823 (2003). [CrossRef] [PubMed]

*π*shift of the interference fringes at the zero intensity lines is clearly observed.

## 6. Conclusions

## Acknowledgements

## 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. | R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. |

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

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

6. | F. Chen, M. Stepic, C. E. Ruter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO |

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

8. | 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. |

9. | 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 |

10. | D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. |

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

12. | N. W. Ashcroft and N. D. Mermin, |

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

14. | Yu. S. Kivshar, “Self-localization in arrays of defocusing wave-guides,” Opt. Lett. |

15. | G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrodinger equation with a periodic potential,” Phys. Rev. E |

16. | Yu. S. Kivshar and G. P. Agrawal, |

17. | G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A |

18. | A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. |

**OCIS Codes**

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

(190.5940) Nonlinear optics : Self-action effects

**ToC Category:**

Nonlinear Optics

**Citation**

Michal Matuszewski, Christian R. Rosberg, Dragomir N. Neshev, Andrey A. Sukhorukov, Arnan Mitchell, Marek Trippenbach, Michael W. Austin, Wieslaw Krolikowski, and Yuri S. Kivshar, "Crossover from self-defocusing to discrete trapping in nonlinear waveguide arrays," Opt. Express **14**, 254-259 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-254

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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]
- R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, "Self-focusing and defocusing in waveguide arrays," Phys. Rev. Lett. 86, 3296-3299 (2001). [CrossRef] [PubMed]
- 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]
- 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, 113902-4 (2004). [CrossRef] [PubMed]
- F. Chen, M. Stepic, C. E. Ruter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, "Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays," Opt. Express 13, 4314-4324 (2005). [CrossRef] [PubMed]
- A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530-1532 (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, 023902-4 (2003). [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]
- D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003). [CrossRef] [PubMed]
- H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92, 123902-4 (2004). [CrossRef] [PubMed]
- N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart And Winston, New York, 1976).
- D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled wave- guides," Opt. Lett. 13, 794-796 (1988). [CrossRef] [PubMed]
- Yu. S. Kivshar, "Self-localization in arrays of defocusing wave-guides," Opt. Lett. 18, 1147-1149 (1993). [CrossRef] [PubMed]
- G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, "Wannier functions analysis of the nonlinear Schrodinger equation with a periodic potential," Phys. Rev. E 66, 46608-6 (2002). [CrossRef]
- Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
- G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, "Dark and bright photovoltaic spatial solitons," Phys. Rev. A 50, R4457-R4460 (1994). [CrossRef] [PubMed]
- A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Yu. S. Kivshar, "Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices," Phys. Rev. Lett. 92, 093901-4 (2004). [CrossRef] [PubMed]

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