## Discrete light propagation and self-trapping in liquid crystals

Optics Express, Vol. 13, Issue 6, pp. 1808-1815 (2005)

http://dx.doi.org/10.1364/OPEX.13.001808

Acrobat PDF (1019 KB)

### Abstract

We investigate light propagation and self-localization in a voltage-controlled array of channel waveguides realized in undoped nematic liquid crystals. We report on discrete diffraction and solitons, as well as all-optical angular steering and the formation of multiband vector breathers. The results and are in excellent agreement with both coupled mode theory and full numerical simulations.

© 2005 Optical Society of America

## 1. Introduction

*et al*., who experimentally demonstrated light tunneling between neighboring channels [1

1. 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]

2. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. **QE-18**, 1580–1583 (1982). [CrossRef]

5. S. Trillo and S. Wabnitz, “Coupling instability and power-induced switching with two-core dual-polarizations fiber nonlinear couplers,” J. Opt. Soc. Am. B **5**, 483–491 (1985). [CrossRef]

*et al*. in the late 1980’s [6

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

7. A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. **39**, 31–50 (2003). [CrossRef]

28. 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 (2004). [CrossRef] [PubMed]

11. A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E **53**, 1172–1189 (1996). [CrossRef]

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

11. A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E **53**, 1172–1189 (1996). [CrossRef]

12. M. Matsumoto, “Optical switching in nonlinear waveguide arrays with a longitudinally decreasing coupling coefficient,” Opt. Lett. **20**, 1758–1760 (1995). [CrossRef] [PubMed]

14. T. Peschel, R. Muschall, and F. Lederer, “Power-controlled beam steering in non equidistant arrays of nonlinear waveguides,” Opt. Comm. **136**, 16–21 (1997). [CrossRef]

20. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E **70**, 026602 (2004). [CrossRef]

17. O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. **21**, 1105–1107 (1996). [CrossRef] [PubMed]

19. W. Krolikowsky and Y. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B **13**, 876–880 (1996). [CrossRef]

27. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. **90**, 053902 (2003). [CrossRef] [PubMed]

13. A. A. Sukhorukov and Y. S. Kivshar, “Generation and stability of discrete gap solitons,” Opt. Lett. **28**, 2345–2347 (2003). [CrossRef] [PubMed]

22. A. A. Sukhorukov and Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. **91**, 113902 (2003). [CrossRef] [PubMed]

24. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. **90**, 253902 (2003). [CrossRef] [PubMed]

27. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. **90**, 053902 (2003). [CrossRef] [PubMed]

30. M. Peccianti, G. Assanto, A. de Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. **77**, 7–9 (2000). [CrossRef]

33. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. **92**, 113902 (2004). [CrossRef] [PubMed]

*director*, with a significant degree of orientational order. An external electric field (either static, or radio-frequency or optical) induces dipoles and tends to realign the NLC molecules with major axes parallel to its direction of oscillation (or polarization) in order to minimize the system energy. Although molecular reorientation is slow compared to electronic nonlinearities, the power level required to produce self-focusing is usually low (typically mW or less) and, therefore, quite amenable to all-optical switching in network reconfiguration applications (where bandwidth is not the crucial issue) [34

34. M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. **81**, 3335–3337 (2002). [CrossRef]

35. M. Peccianti and G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. **26**, 1690–1692 (2002). [CrossRef]

## 2. Sample and model equations

*d*-is sandwiched between two glass plates, which provide planar anchoring in the direction of light propagation. The plates are coated with an equally-spaced array of parallel Indium Tin Oxide (ITO) electrodes which, through the electro-optic response of the medium, allow to defining a set of identical channel waveguides supporting quasi-TM guided modes. [36

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

*V*, in fact, corresponds to an electric field distribution which reorients the director in the (

*x,z*) plane, yielding a refractive index modulation of period

*Λ*across the sample. A typical index distribution is displayed in Fig. 1 (right panel).

*x*, experiences a refractive index

*n*

_{e}=

*n*

_{//}

*n*

_{⊥}(

^{2}

*θ*+

^{2}

*θ*)

^{-1/2}, which depends on the mean angular molecular orientation

*θ*between the (propagation) axis z and the major molecular axis or director (subscripts//and ⊥ refer to field polarizations parallel or normal to the director, respectively). The steady-state director distribution can be calculated with the Frank free-energy formulation, [37–38] evaluating the NLC energy density and its minimum through the Euler-Lagrange equation:

*E*

_{x}the

*x*-component of the static or low-frequency field,

*K*the elastic constant (single constant approximation [37]) and Δ

*ε*

_{RF}the low-frequency birefringence. Maxwell equations are used to obtain the potential distribution

*V*(

*x,y*):

*k*

_{y}the Bloch wavenumber,

*k*

_{z}the propagation constant and

*∏*(

*x,y*) the beam envelope,

*Λ*-periodic across y. By substituting Eq. (3) into Maxwell equations, we obtain:

*k*

_{z}as a function of

*k*

_{y}) and the corresponding FB modes

*∏*

_{k}. The presence of the refractive index

*n*(

*x,y*) suggests that, in an electro-optic material such as NLC, the eigenvalue spectrum of the array can be adjusted through the external voltage. To demonstrate this peculiar feature, we numerically solved Eq. (4) with (1)-(2) in the bias range 0.7<

*V*<2.0V for a cell with

*Λ=d*=6µm, and show the results in Fig. 2. The band-gap spectrum (Fig. 2, left) consists of permitted bands (color lines), separated by gaps in which propagating modes are forbidden. As the bias increases, the width of each band-gap changes, as well. More specifically, as the index modulation grows for

*V*<1.3V (Fig. 2, right), the gaps widen in the spectrum (Fig. 2, center) due to an increasing refractive contrast. Conversely, for

*V*>1.3

*V*non locality intervenes by reducing the index difference in-between waveguides (Fig. 2, right), thereby lowering the index modulation and shrinking the gaps once again. Discrete solitons (next section) appear above the first linear band (band 0), while spatial gap-solitons exist in each of the lower gaps. [13

13. A. A. Sukhorukov and Y. S. Kivshar, “Generation and stability of discrete gap solitons,” Opt. Lett. **28**, 2345–2347 (2003). [CrossRef] [PubMed]

27. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. **90**, 053902 (2003). [CrossRef] [PubMed]

## 3. Discrete light localization in NLC

### 3.1 Discrete solitons

*λ*=1.064µm) Nd:YAG laser, collecting the light scattered out of the plane (

*y,z*) by a high resolution CCD. Results from a sample with

*Λ*=8µm and thickness

*d*=6µm are displayed in Fig. 3. At low launch-power (

*P*=1mW), when a single channel is excited, light couples from waveguide to waveguide and gives rise to the characteristic pattern of discrete diffraction (Fig. 3(a)). By varying the bias in the range 0.65<

*V*<1V while keeping constant the input power

*P*=1mW, we monitored the beam evolution in the two cross sections at

*z*

_{0}=1.4mm and

*z*

_{1}=1.5mm (Fig. 3(b)). As the index modulation grows versus voltage, light experiences continuous (

*V*<0.7V) or discrete diffraction (

*V*>0.7V). In the latter regime the coupling distance (i.e., the coherence length of each two-channel directional coupler) increases with the bias, because of the improved confinement afforded by each channels. Conversely, if the input power grows for a given bias

*V*=0.74V (Fig. 3(c)), the all-optical increase in refractive index detunes the excited waveguide. Figure 3(c) shows light evolution in versus power in 1.40<z<1.55mm. Eventually, when the excitation is large enough (P=10mW), light gets completely trapped in one waveguide and a discrete soliton is generated in the array, as visible in Fig. 3(d).

*ε*

_{0}(

*θ*/4 Eq. (1) with

### 3.2 Discrete Beam steering

11. A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E **53**, 1172–1189 (1996). [CrossRef]

17. O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. **21**, 1105–1107 (1996). [CrossRef] [PubMed]

19. W. Krolikowsky and Y. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B **13**, 876–880 (1996). [CrossRef]

*w*

_{y}=10µm) impinges on the waveguide array (

*Λ*=8µm,

*V*=0.77V) with an input tilt

*γ=λ*/4

*Λ*=1.90° along

*y*as to excite the array at maximum transverse velocity and minimum diffraction. [8

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

*P=P*

_{L}=0.1mW), light couples between adjacent waveguides and “walks” across the array (Fig. 5(a)), as also expected based on CMT. As the power increases, the nonlinear detuning of the central waveguide (i.e., the one excited by the peak beam intensity) at sufficiently high power (

*P=P*

_{H}=2mW) causes light trapping and the excitation of a discrete soliton, which is forced to propagate straight along the input channel, as in Fig. 5(b). After propagating over z=2mm, the input beam in the two cases displays a significant lateral shift as the power changes from

*P*

_{L}to

*P*

_{H}, as graphed in Fig. 5(c).

*w*

_{y}=10µm and an array with

*Λ*=8µm and

*V*=0.77V. Light discretely diffracts for a low power

*P=P*

_{L}=1mW (Fig. 6(a)), and localizes in a single channel when the power reaches

*P=P*

_{H}=7mW (Fig. 6(b)). The intensity cross-sections in z=2mm clearly show nonlinear beam steering (Fig. 6(c)).

*V*<1V) and device compactness (

*L*<2mm). Array optimization (beyond the scope of this paper) would rely on the reduction of period

*Λ*(decreasing device length because of a higher overlap between evanescent fields) and the minimization of propagation losses (mostly due to scattering with a 1/

*λ*

^{2}dependence in NLC) by operating at longer wavelengths. [37]

## 4. Multiband vector breathers

24. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. **90**, 253902 (2003). [CrossRef] [PubMed]

*w*

_{y}=8µm across

*y*) launched on-axis and equidistant from two neighboring channels. In this case, light propagation for

*P*=0.2mW (Fig. 7(b), top) exhibits a pattern characteristic of discrete diffraction. Modes of band 1, conversely, are characterized by maxima between channels (Fig. 7(a), bottom) and can be excited by a narrow Gaussian input centered between two adjacent waveguides (Fig. 7(b), bottom). When co-launched with a total power of

*P*=0.4mW, the previous FB modes originate a symmetric breather, oscillating in a periodic fashion as it propagates along z (Fig. 7(c)).

*Λ=d*=6µm. Fig. 8 displays our results. To ensure an adequate spatial overlap with modes of the first two bands, we used a single Gaussian beam of waist

*w*

_{y}=5µm centered between neighboring waveguides. In this configuration, the spatial overlap is larger between the input and modes in band 1, because the intensity peaks between channels. At low power (

*P*=0.2mW), in fact, we observed light spreading across the array (Fig. 8(a)) as in the numerical experiment of Fig. 7(b), bottom panel. At high power (

*P*=7mW), a symmetric breather is formed via cross phase modulation and propagates in the array (Fig. 8(b)). The beating and its period depend on the slight difference between the propagation constants of the sourcing FB modes. Since their position in the dispersion diagram can be electro-optically adjusted, we expected to be able to tune such period by varying the applied bias, i.e., by altering the width of gap 1. The calculated gap-width and the measured breathing period are graphed in Fig. 8(c) versus applied voltage. Clearly, as the width of gap 1 has a maximum in

*V*=1.3V (red line), correspondingly the period exhibits a minimum around the same value (blue dots). As

*V*>1.3V, non locality of NLC mediates reorientation between neighboring channels, shrinking the gap and lengthening the period once again.

## 5. Conclusions

## References and Links

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

2. | S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. |

3. | N. Finlayson and G. I. Stegeman, “Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler,” Appl. Phys. Lett. |

4. | C. Schmidt-Hattenberger, U. Trutschel, and F. Lederer, “Nonlinear switching in multiple-core couplers,” Opt. Lett. |

5. | S. Trillo and S. Wabnitz, “Coupling instability and power-induced switching with two-core dual-polarizations fiber nonlinear couplers,” J. Opt. Soc. Am. B |

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

7. | A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. |

8. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. |

9. | R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. |

10. | D. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

11. | A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E |

12. | M. Matsumoto, “Optical switching in nonlinear waveguide arrays with a longitudinally decreasing coupling coefficient,” Opt. Lett. |

13. | A. A. Sukhorukov and Y. S. Kivshar, “Generation and stability of discrete gap solitons,” Opt. Lett. |

14. | T. Peschel, R. Muschall, and F. Lederer, “Power-controlled beam steering in non equidistant arrays of nonlinear waveguides,” Opt. Comm. |

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

16. | T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. |

17. | O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. |

18. | T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. |

19. | W. Krolikowsky and Y. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B |

20. | R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E |

21. | N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. |

22. | A. A. Sukhorukov and Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. |

23. | O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband Vector Lattice Solitons,” Phys. Rev. Lett. |

24. | D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. |

25. | J. Hudock, P. G. Kevrekidis, B. A. Malomed, and D. N. Christodoulides, “Discrete vector solitons in two-dimensional nonlinear waveguide arrays: solutions, stability, and dynamics,” Phys. Rev. E |

26. | J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. |

27. | D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. |

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

29. | K. Sakoda, |

30. | M. Peccianti, G. Assanto, A. de Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. |

31. | M. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E |

32. | G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. |

33. | C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. |

34. | M. Peccianti, C. Conti, G. Assanto, A. de Luca, and C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. |

35. | M. Peccianti and G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. |

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

37. | I. C. Khoo, |

38. | D. A. Dumm, A. Fukuda, and G. R. Luckhurst, |

**OCIS Codes**

(160.3710) Materials : Liquid crystals

(190.0190) Nonlinear optics : Nonlinear optics

**ToC Category:**

Focus Issue: Discrete solitons in nonlinear optics

**History**

Original Manuscript: November 16, 2004

Revised Manuscript: January 25, 2005

Manuscript Accepted: January 25, 2005

Published: March 21, 2005

**Citation**

Andrea Fratalocchi, Gaetano Assanto, Kasia A. Brzda̢kiewicz, and Mirek A. Karpierz, "Discrete light propagation and self-trapping in liquid crystals," Opt. Express **13**, 1808-1815 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-1808

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### References

- S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973). [CrossRef]
- S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982). [CrossRef]
- N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990). [CrossRef]
- C. Schmidt-Hattenberger, U. Trutschel, F. Lederer, “Nonlinear switching in multiple-core couplers,” Opt. Lett. 16, 294–296 (1991). [CrossRef] [PubMed]
- S. Trillo, S. Wabnitz, “Coupling instability and power-induced switching with two-core dual-polarizations fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1985). [CrossRef]
- D. N. Christodoulides, R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]
- A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003). [CrossRef]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000). [CrossRef] [PubMed]
- R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999). [CrossRef]
- D. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] [PubMed]
- A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996). [CrossRef]
- M. Matsumoto, “Optical switching in nonlinear waveguide arrays with a longitudinally decreasing coupling coefficient,” Opt. Lett. 20, 1758–1760 (1995). [CrossRef] [PubMed]
- A. A. Sukhorukov, Y. S. Kivshar, “Generation and stability of discrete gap solitons,” Opt. Lett. 28, 2345–2347 (2003). [CrossRef] [PubMed]
- T. Peschel, R. Muschall, F. Lederer, “Power-controlled beam steering in non equidistant arrays of nonlinear waveguides,” Opt. Comm. 136, 16–21 (1997). [CrossRef]
- D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001). [CrossRef] [PubMed]
- T. Pertsch, U. Peschel, F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003). [CrossRef] [PubMed]
- O. Bang, P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. 21, 1105–1107 (1996). [CrossRef] [PubMed]
- T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002). [CrossRef]
- W. Krolikowsky, Y. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–880 (1996). [CrossRef]
- R. A. Vicencio, M. I. Molina, Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004). [CrossRef]
- N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91, 213906 (2003). [CrossRef] [PubMed]
- A. A. Sukhorukov, Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003). [CrossRef] [PubMed]
- O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, D. N. Christodoulides, “Multiband Vector Lattice Solitons,” Phys. Rev. Lett. 91, 113901 (2003). [CrossRef] [PubMed]
- D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 253902 (2003). [CrossRef] [PubMed]
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