## Nonlinear diffusion and beam self-trapping in diffraction-managed waveguide arrays

Optics Express, Vol. 15, Issue 15, pp. 9547-9552 (2007)

http://dx.doi.org/10.1364/OE.15.009547

Acrobat PDF (546 KB)

### Abstract

We study nonlinear propagation of light in diffraction-managed photonic lattices created by periodically-curved arrays of optical waveguides. We identify different regimes of the nonlinear propagation of light in such structures depending on the input power. We start from the regime of self-collimation at low powers and demonstrate that, as the beam power increases, nonlinearity destroys the beam self-imaging and leads to *nonlinear diffusion*. At higher powers, we observe a sharp transition to the self-trapping and the formation of discrete diffraction-managed solitons.

© 2007 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]

*negative*allowing for focusing of diverging beams [3

3. T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. **88**, 093901–4 (2002). [CrossRef] [PubMed]

4. P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nature Materials **5**, 93–96 (2006). [CrossRef] [PubMed]

5. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, and P. Laporta, “Observation of Dynamic Localization in Periodically Curved Waveguide Arrays,” Phys. Rev. Lett. **96**, 243901–4 (2006). [CrossRef] [PubMed]

7. R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact dynamic localization in curved AlGaAs optical waveguide arrays,” Opt. Express **15**, 3212–3223 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3212. [CrossRef] [PubMed]

8. I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, “Broadband diffraction management and self-collimation of white light in photonic lattices,” Phys. Rev. E **74**, 066609-4 (2006). [CrossRef]

9. 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]

12. M. J. Ablowitz and Z. H. Musslimani, “Discrete diffraction managed spatial solitons,” Phys. Rev. Lett. **87**, 254102–4 (2001). [CrossRef] [PubMed]

13. K. Staliunas, R. Herrero, and G. J. Valcarcel, de, “Arresting soliton collapse in two-dimensional nonlinear Schrodinger systems via spatiotemporal modulation of the external potential,” Phys. Rev. A **75**, 011604-4 (2007). [CrossRef]

*all*orders. In the latter case, the modulational instability is suppressed [6

6. S. Longhi, “Self-imaging and modulational instability in an array of periodically curved waveguides,” Opt. Lett. **30**, 2137–2139 (2005). [CrossRef] [PubMed]

*nonlinear diffusion*in other types of periodically-curved photonic lattices; this regime has

*no analogies*with the nonlinear beam self-focusing or self-defocusing in a bulk medium [11], or discrete self-trapping of light in arrays of straight waveguides [1

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. Propagation of light beams in arrays of curved waveguides

*d*in the

*x*direction, where the waveguide axes are periodically curved in the propagation direction

*z*with the period

*L*≫

*d*[see Fig. 1(a)]. When the tilt of beams and waveguides at the input facet is less than the Bragg angle, the beam propagation is primarily characterized by coupling between the fundamental modes of the individual waveguides, and it can be described by the tight-binding equations taking into account the periodic waveguide bending [5

5. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, and P. Laporta, “Observation of Dynamic Localization in Periodically Curved Waveguide Arrays,” Phys. Rev. Lett. **96**, 243901–4 (2006). [CrossRef] [PubMed]

6. S. Longhi, “Self-imaging and modulational instability in an array of periodically curved waveguides,” Opt. Lett. **30**, 2137–2139 (2005). [CrossRef] [PubMed]

*id*Ψ

*/*

_{n}*dz*+

*C*

_{0}(Ψ

_{n+1}+Ψ

_{n-1})=

*ωẍ*

_{0}(

*z*)

*n*Ψ

_{n}-

*γ*|Ψ

*|*

_{n}^{2}Ψ

*, where Ψ*

_{n}*(*

_{n}*z*) is the amplitude of the

*n*-th waveguide,

*ω*=2

*π*n

_{0}

*d*/

*λ*is the dimensionless frequency,

*λ*is the vacuum wavelength, n

_{0}is the average refractive index of the medium,

*γ*is an effective nonlinear coefficient which accounts for the Kerr-type nonlinear response of the waveguide material, and the dots stand for the derivatives. Transverse shift

*x*

_{0}(

*z*)≡

*x*

_{0}(

*z*+

*L*) defines the periodic longitudinal bending profile of the waveguide axis. Coefficient

*C*

_{0}defines the coupling strength between the neighboring waveguides, and it characterizes diffraction in a straight waveguide array with

*x*

_{0}≡0 [14

14. S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. **22**, 46–47 (1973). [CrossRef]

*E*(

*x*,

*z*) is represented as a superposition of the modes

*E*

_{0}(

*x*) of the individual waveguides,

*E*(

*x*,

*z*)=Σ

*Ψ*

_{n}

_{n}*E*

_{0}[

*x*-

*nd*-

*x*

_{0}(

*z*)].

5. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, and P. Laporta, “Observation of Dynamic Localization in Periodically Curved Waveguide Arrays,” Phys. Rev. Lett. **96**, 243901–4 (2006). [CrossRef] [PubMed]

6. S. Longhi, “Self-imaging and modulational instability in an array of periodically curved waveguides,” Opt. Lett. **30**, 2137–2139 (2005). [CrossRef] [PubMed]

*γ*≥0. In our simulations presented below, we use the following values which are typical for the experiments with optical waveguide arrays:

*d*=9 µm, n

_{0}=2.35, λ=532 nm,

*C*

_{0}=0.13 mm

^{-1},

*γ*=1.9. Normalization chosen is such that

*x*is measured in

*µ*m and

*z*is measured in mm. We use the discrete model for the calculations presented in this paper, however we have confirmed the validity of our results by simulating the full parabolic equations for the continuous electric field envelopes. We note that in case of the strong coupling between the waveguides, the long range coupling between non-nearest neighbours becomes important. In this case, exact dynamical localization can also be ralized in arrays of curved waveguides [7

7. R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact dynamic localization in curved AlGaAs optical waveguide arrays,” Opt. Express **15**, 3212–3223 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3212. [CrossRef] [PubMed]

## 3. Discrete diffraction and self-collimation

*x*

_{0}(

*z*-

*z*)=

_{a}*x*

_{0}(

*z*-

_{a}*z*) for a given coordinate shift

*z*, since asymmetry may introduce other effects due to the modification of refraction [15

_{a}15. Y. V. Kartashov, L. Torner, and D. N. Christodoulides, “Soliton dragging by dynamic optical lattices,” Opt. Lett. **30**, 1378–1380 (2005). [CrossRef] [PubMed]

19. I. Tsopelas, Y. Kominis, and K. Hizanidis, “Soliton dynamics and interactions in dynamically photoinduced lattices,” Phys. Rev. E **74**, 0366139-8 (2006). [CrossRef]

*z*→

*z*+

*L*) the beam diffraction remains the same as in a straight waveguide array with the effective coupling coefficient [5

**96**, 243901–4 (2006). [CrossRef] [PubMed]

**30**, 2137–2139 (2005). [CrossRef] [PubMed]

*C*

_{eff}=

*C*

_{0}

*L*

^{-1}∫

^{L}_{0}cos[

*ωẋ*

_{0}(ζ)]

*d*ζ. As a specific example, we consider a curved waveguide array with the sinusoidal bending of the form

*x*

_{0}(

*z*)=

*A*{cos[2

*πz*/

*L*]-1}, for which the effective coupling coefficient is

*C*

_{eff}=

*C*

_{0}J

_{0}(2

*πωA*/

*L*), where J

_{0}is the Bessel function of the first kind of zero order. The effective diffraction can be made either normal, zero, or anomalous depending on the value of the bending amplitude [see Fig. 1(b)].

*A*=

*A*

_{sc}=ξ

*L*[2

*πω*]

^{-1}, where ξ⋍2.405 is the first root of the function J

_{0}. For example, for the bending period

*L*=15 mm self-collimation occurs when

*A*

_{sc}=23.0 µm [see Fig. 2(a), where we use the modal representation described above in order to reconstruct the optical field distribution]. The beam width is determined as the width of the transverse cross-section function centered at the current center of mass of the beam where 75% of the beam power is concentrated. When the bending amplitude differs from the self-collimation value, the beam experiences discrete diffraction at low powers, similar to the effect observed in straight waveguide arrays [1

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]

## 4. Nonlinear beam propagation and control

*nonlinearity destroys the self-collimation condition*of light by changing the refractive index of the waveguide material. Initially, we observe that the beam shape experiences irregular distortion, such that the periodicity of the self-collimation is lost [see Fig. 3(a)]. However, the beam does not broaden significantly, and it still experiences approximate self-restoration at some points.

*nonlinear diffusion*takes place, where the beam experiences significant broadening and self-defocusing, as shown in Fig. 3(b). This self-defocusing is intrinsically limited due to the diffraction cancellation in the waveguide array. After propagation over some distance the beam broadens and its intensity is reduced accordingly. Therefore, the further beam spreading stops when the average beam width achieves a certain value. Such a peculiar nonlinear beam dynamics has no analogies in bulk media [11] or discrete systems [1

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

*a sharp transition from the nonlinear diffusion to the discrete self-trapping*over the whole length of the array, and the discrete lattice soliton is formed [see Fig. 3(d)].

*P*normalized to the power

*P*

_{0}required for the formation of one-site discrete lattice soliton in the straight array, where

*P*

_{0}⋍5

*C*

_{0}/

*γ*[20

20. A. C. Scott and L. Macneil, “Binding energy versus nonlinearity for a “small” stationary soliton,” Phys. Lett. A **98**, 87–88 (1983). [CrossRef]

## 5. Conclusions

## 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. | T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. |

4. | P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nature Materials |

5. | S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, and P. Laporta, “Observation of Dynamic Localization in Periodically Curved Waveguide Arrays,” Phys. Rev. Lett. |

6. | S. Longhi, “Self-imaging and modulational instability in an array of periodically curved waveguides,” Opt. Lett. |

7. | R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact dynamic localization in curved AlGaAs optical waveguide arrays,” Opt. Express |

8. | I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, “Broadband diffraction management and self-collimation of white light in photonic lattices,” Phys. Rev. E |

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

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

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

12. | M. J. Ablowitz and Z. H. Musslimani, “Discrete diffraction managed spatial solitons,” Phys. Rev. Lett. |

13. | K. Staliunas, R. Herrero, and G. J. Valcarcel, de, “Arresting soliton collapse in two-dimensional nonlinear Schrodinger systems via spatiotemporal modulation of the external potential,” Phys. Rev. A |

14. | S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. |

15. | Y. V. Kartashov, L. Torner, and D. N. Christodoulides, “Soliton dragging by dynamic optical lattices,” Opt. Lett. |

16. | I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, “Soliton control in modulated optically-induced photonic lattices,” Opt. Express |

17. | C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Demonstration of all-optical beam steering in modulated photonic lattices,” Opt. Lett. |

18. | Y. Kominis and K. Hizanidis, “Continuous-wave-controlled steering of spatial solitons,” J.Opt. Soc. Am. B |

19. | I. Tsopelas, Y. Kominis, and K. Hizanidis, “Soliton dynamics and interactions in dynamically photoinduced lattices,” Phys. Rev. E |

20. | A. C. Scott and L. Macneil, “Binding energy versus nonlinearity for a “small” stationary soliton,” Phys. Lett. A |

**OCIS Codes**

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

(190.5940) Nonlinear optics : Self-action effects

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 1, 2007

Revised Manuscript: July 12, 2007

Manuscript Accepted: July 15, 2007

Published: July 17, 2007

**Citation**

Ivan L. Garanovich, Andrey A. Sukhorukov, and Yuri S. Kivshar, "Nonlinear diffusion and beam self-trapping in diffraction-managed
waveguide arrays," Opt. Express **15**, 9547-9552 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9547

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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]
- T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, "Anomalous refraction and diffraction in discrete optical systems," Phys. Rev. Lett. 88, 093901-4 (2002). [CrossRef] [PubMed]
- P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nature Mater. 5, 93-96 (2006). [CrossRef] [PubMed]
- S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, and P. Laporta, "Observation of Dynamic Localization in Periodically Curved Waveguide Arrays," Phys. Rev. Lett. 96, 243901-4 (2006). [CrossRef] [PubMed]
- S. Longhi, "Self-imaging and modulational instability in an array of periodically curved waveguides," Opt. Lett. 30, 2137-2139 (2005). [CrossRef] [PubMed]
- R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, C. M. de Sterke, "Exact dynamic localization in curved AlGaAs optical waveguide arrays," Opt. Express 15, 3212-3223 (2007). [CrossRef] [PubMed]
- I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, "Broadband diffraction management and selfcollimation of white light in photonic lattices," Phys. Rev. E 74, 066609 (2006). [CrossRef]
- 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]
- 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]
- Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
- M. J. Ablowitz and Z. H. Musslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102-4 (2001). [CrossRef] [PubMed]
- K. Staliunas, R. Herrero, and G. J. Valcarcel, de, "Arresting soliton collapse in two-dimensional nonlinear Schrodinger systems via spatiotemporal modulation of the external potential," Phys. Rev. A 75, 011604 (2007). [CrossRef]
- S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-47 (1973). [CrossRef]
- Y. V. Kartashov, L. Torner, and D. N. Christodoulides, "Soliton dragging by dynamic optical lattices," Opt. Lett. 30, 1378-1380 (2005). [CrossRef] [PubMed]
- I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, "Soliton control in modulated optically-induced photonic lattices," Opt. Express 13, 5704-5710 (2005). [CrossRef] [PubMed]
- C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, "Demonstration of all-optical beam steering in modulated photonic lattices," Opt. Lett. 31, 1498-1500 (2006). [CrossRef] [PubMed]
- Y. Kominis and K. Hizanidis, "Continuous-wave-controlled steering of spatial solitons," J. Opt. Soc. Am. B 21, 562-567 (2004). [CrossRef]
- I. Tsopelas, Y. Kominis, and K. Hizanidis, "Soliton dynamics and interactions in dynamically photoinduced lattices," Phys. Rev. E 74, 036613 (2006). [CrossRef]
- A. C. Scott and L. Macneil, "Binding energy versus nonlinearity for a "small" stationary soliton," Phys. Lett. A 98, 87-88 (1983). [CrossRef]

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