## Linear and nonlinear light propagation at the interface of two homogeneous waveguide arrays |

Optics Express, Vol. 19, Issue 2, pp. 1158-1167 (2011)

http://dx.doi.org/10.1364/OE.19.001158

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### Abstract

We investigate linear and nonlinear light propagation at the interface of two one-dimensional homogeneous waveguide arrays containing a single defect of different strength. For the linear case and in a limited region of the defect size, we find trapped staggered and unstaggered modes. In the nonlinear case, we study the dependence of power thresholds for discrete soliton formation in different channels as a function of defect strength. All experimental results are confirmed theoretically using an adequate discrete model.

© 2011 OSA

## 1. Introduction

1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**(6950), 817–823 (2003). [CrossRef] [PubMed]

4. R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. **86**(15), 3296–3299 (2001). [CrossRef] [PubMed]

5. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. **85**(9), 1863–1866 (2000). [CrossRef] [PubMed]

6. K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. **102**(12), 123905 (2009). [CrossRef] [PubMed]

7. 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**(16), 3383–3386 (1998). [CrossRef]

11. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. G. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. **92**(12), 123903 (2004). [CrossRef] [PubMed]

12. M. Stepić, C. Wirth, C. E. Rüter, and D. Kip, “Observation of modulational instability in discrete media with self-defocusing nonlinearity,” Opt. Lett. **31**(2), 247–249 (2006). [CrossRef] [PubMed]

14. R. Dong, C. E. Rüter, D. Kip, O. Manela, M. Segev, C. Yang, and J. Xu, “Spatial frequency combs and supercontinuum generation in one-dimensional photonic lattices,” Phys. Rev. Lett. **101**(18), 183903 (2008). [CrossRef] [PubMed]

15. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. **96**(6), 063901 (2006). [CrossRef] [PubMed]

18. N. Malkova, I. Hromada, X. S. Wang, G. Bryant, and Z. G. Chen, “Observation of optical Shockley-like surface states in photonic superlattices,” Opt. Lett. **34**(11), 1633–1635 (2009). [CrossRef] [PubMed]

19. H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, and A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express **11**(25), 3404–3411 (2003). [CrossRef] [PubMed]

20. F. Fedele, J. K. Yang, and Z. G. Chen, “Defect modes in one-dimensional photonic lattices,” Opt. Lett. **30**(12), 1506–1508 (2005). [CrossRef] [PubMed]

21. A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Observation of defect-free surface modes in optical waveguide arrays,” Phys. Rev. Lett. **101**(20), 203902 (2008). [CrossRef] [PubMed]

22. K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. **31**(18), 2774–2776 (2006). [CrossRef] [PubMed]

_{3}) as the nonlinear medium. First, we prove the theoretical predictions on existence and phase profile of localized linear modes. Furthermore, we systematically determine the power thresholds to form discrete solitons at and close to the interface. Depending on the width of the created defect channel, we observe a rather large increase of necessary power for light trapping, which is in good agreement with our theoretical predictions.

## 2. Model system

*w*but different separations

*d*and

_{L}*d*, are coupled by a single defect of width

_{R}*d*, which is varied in a certain range. A geometrical representation is given in Fig. 1 . According to the coupled-mode theory, the interaction of nearest neighbors can be described by a coupling constant

*C*. This parameter depends exponentially on the separation between channels and is a direct result of the individual fields’ overlap. In the case of two

*identical*semi-infinite photonic lattices separated by a defect, due to the symmetry of the system, the field overlap does not depend on which boundary channel, left or right, is excited. Hence the inter-lattice coupling constants between two boundary channels are the same (

*dissimilar*photonic lattices breaking mirror symmetry, the inter-lattice coupling constants differ, since the field overlap depends on the parameters of the system, such as channel and gap widths, refractive indices, nonlinearity of the material, etc. However, in the investigated model system (Fig. 1) differences between inter-lattice coupling constants are sufficiently small and thus can be neglected. This approximation is valid due to identical individual channel widths and relatively small differences in gap widths between the two lattices.

*z*direction, the field evolution of our system can be described by the discrete nonlinear Schrödinger equation (DNLS) [30

30. M. Stepić, D. Kip, Lj. Hadžievski, and A. Maluckov, “One-dimensional bright discrete solitons in media with saturable nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **69**(6 Pt 2), 066618 (2004). [CrossRef] [PubMed]

*n*is the channel index of a mode with amplitude

*E*,

_{n}*α*= −1 denotes the defocusing type of the nonlinearity, and

*κ*describes saturation strength. Considering that the defect is placed between channels

*n*= 0 and

*n*= 1, the following notations are used:

*C*=

_{n}*C*for

_{L}*n*< 0,

*C*=

_{n}*C*for

_{R}*n*> 1, and

*C*

_{0,1}=

*C*

_{1,0}=

*C*at the defect.

## 3. Sample fabrication and experimental methods

_{3}crystals. This material possesses a defocusing nonlinearity due to the photorefractive effect at moderate light intensities. Sample dimensions are 1 × 20 × 7.8 mm

^{3}with the ferroelectric

*c*-axis pointing along the 7.8 mm-long direction. Arrays of parallel-aligned channel waveguides, each being 5 μm wide, are fabricated by patterning a 10 nm-thick Ti layer formed by sputtering on the sample surface, using standard photolithographic techniques. In-diffusion of the Ti stripes takes place for 2 hours at a temperature of 1000 °C in wet Ar atmosphere. Finally, input and output facets are polished to optical quality to allow for direct coupling of light into the 20 mm-long channels. In order to ensure equal (nonlinear) waveguide properties, on a single substrate 11 different waveguide arrays are formed at the same time. Each array consists of two homogeneous parts with separations (gaps) of 4 μm (left array, grating period Λ

*= 9 μm) and 3 μm (right array, grating period Λ*

_{L}*= 8 μm), respectively, separated by a defect. The defect width*

_{R}*d*is varied between 2 µm and 4.5µm in steps of 0.25 µm.

31. E. Smirnov, C. E. Rüter, M. Stepić, V. Shandarov, and D. Kip, “Dark and bright blocker soliton interaction in defocusing waveguide arrays,” Opt. Express **14**(23), 11248–11255 (2006). [CrossRef] [PubMed]

_{4}laser with wavelength 532 nm as the light source. By using a chrome-glass mask with sets of adjacent pinholes, either single or multiple neighbored channels of the waveguide array can be excited. A small optional tilt angle of the input light distribution allows for either unstaggered or staggered input conditions. The output intensity distribution is imaged onto a CCD camera with a microscope objective. With the help of a Mach-Zehnder interferometer, which interferes the output amplitude with a plane reference wave, we are able to monitor the phase distribution of the out-coupled light.

## 4. Linear propagation

*C*

_{0,1}=

*C*

_{1,0}=

*C*for light coupling from channel 0 to channel 1 and vice versa. This assumption is confirmed in the example given in Fig. 2 when comparing experimentally obtained linear discrete diffraction patterns in the fabricated lattices with corresponding numerical results.

*C*on the defect width

*d*is calculated from Eq. (1) for the linear case (

*α*= 0), using the experimentally obtained diffraction patterns, and the results are depicted in Fig. 3 . As can be seen, the results are well described by an exponential function of the form

*C*~exp(−

*d*/

*d*

_{0}) with

*d*

_{0}being a constant.

*β*being the nonlinear propagation constant. In the linear low-power regime (

*α*= 0), we find localized solutions of the form

*d*in the range from 2 μm to 3 μm. As can be seen, linear localized states exist only for

*d*≤ 2.5 μm.

*d*a higher average refractive index occurs, resulting in a stronger localization of the corresponding modes. By slightly tilting the incident angle we find both, staggered and unstaggered light distributions, for each guided mode. As an example, Fig. 6c and 6d monitor the out-of-phase and in-phase output patterns obtained by interference for the case

*d*= 2 μm given in Fig. 6b.

## 5. Nonlinear propagation

*n*= 0) have higher power threshold compared to the homogeneous arrays, which represent waveguides placed far from the defect. Also, lower power is necessary to achieve light localization in the homogeneous array with 4 μm gaps than in an array with 3 μm gaps. In these regions no localized linear modes can occur. The waveguide

*n*= 1 exhibits a different behavior, since there is no power threshold for achieving surface modes, i.e. there is continuous transition from linear to nonlinear localized states. The three insets show the formation of highly localized states by increasing the input power. The bottom one represents the linear mode, the middle one monitors a solution where 60% of total soliton power is localized in the central waveguide

*n*= 1, while the top inset shows a solution with 95% of total power concentrated in the excited waveguide. On the other hand, Fig. 7b depicts power versus nonlinear propagation constant for a defect too wide to achieve localization in the linear regime. Now, a rather high power threshold appears for (nonlinear) modes localized at

*n*= 1. It is also evident that the increase of the defect width reduces the repulsive potential for channel

*n*= 0 (and all other neighboured ones).

*n*in which the localized state is excited is shown in Fig. 8 . For defect widths

*d*< 3 μm (Fig. 8a, where

*d*is smaller than the gaps of the array with Λ

*= 8 μm) strong coupling between lattices causes high thresholds for nonlinear localized states for all waveguides closer to the defect, reaching maximum values for boundary channels (*

_{R}*n*= 0 and

*n*= 1, solid lines). Here, for waveguide

*n*= 1, we adopted the criterion of 90% of total soliton power in the central waveguide element for nonlinear localization. This was done because of the existence of linear modes in this channel, which means that its real power threshold equals zero (dashed lines). These linear localized solutions can be obtained for structures with defect widths less or equal to 2.5 μm. Also, the increase of the defect width causes a decrease of the threshold for light localization in every channel. While the highest threshold for

*d*= 3 μm is obtained for

*n*= 0, in the range of 3 μm <

*d*< 4 μm this maximum transits to waveguide

*n*= 1 (Fig. 8b). Further increase of the defect width does not affect the power threshold, since the lattices act as two nearly-independent semi-infinite arrays (Fig. 8c). The shape of the power threshold curves for

*d*> 4 μm is dictated by the interplay between a repulsive edge potential and Bragg reflection inside the arrays causing Tamm-like oscillations [32

32. M. Stepi, E. Smirnov, C. E. Rüter, D. Kip, A. Maluckov, and L. O. Hadžievski, “Tamm oscillations in semi-infinite nonlinear waveguide arrays,” Opt. Lett. **32**(7), 823–825 (2007). [CrossRef] [PubMed]

*n*. Coupling light into only one element provides both, simple and stable input conditions, and has proven to be an effective method to excite discrete (surface) solitons in waveguide arrays fabricated in LiNbO

_{3}[16

16. E. Smirnov, M. Stepić, C. E. Rüter, D. Kip, and V. Shandarov, “Observation of staggered surface solitary waves in one-dimensional waveguide arrays,” Opt. Lett. **31**(15), 2338–2340 (2006). [CrossRef] [PubMed]

17. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. **97**(8), 083901 (2006). [CrossRef] [PubMed]

*n*, and (ii) this situation is achieved by a rather small increase of necessary input power. In fact, using these criteria, at and above threshold more than 80% of the total output power is located in the excited element.

*n*= − 6 (left array, Λ

*= 9 μm) and*

_{L}*n*= 8 (right array, Λ

*= 8 μm). Around the border between channels*

_{L}*n*= 0 and

*n*= 1, the repulsive character of the defect causes increased power thresholds for all investigated defect widths, as predicted by our numerical modeling. With decreasing width

*d*, the measured power thresholds increase. For small defect sizes

*d*< 2.75 μm, we are not able to trap light in the excited channel anymore, which is due to limited input power available with our setup (low transmission of the pinhole mask).

*d*= 2.75 μm is given in Fig. 9 , showing the two cases of excitation of channel

*n*= 0 (Fig. 9a) and

*n*= 1 (Fig. 9b). After the input light is switched on, the growth of the (defocusing) nonlinearity Δ

*n*follows a well-known exponential law, Δ

*n*(

*t*) = Δ

*n*(1−exp(−

_{sat}*t*/

*τ*)), where

*τ*is the Maxwell time constant and Δ

*n*is the nonlinear index change in saturation, i.e. for

_{sat}*t*→ ∞.

## 6. Summary

## Acknowledgments

## References and links

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

2. | P. St, J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in |

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

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

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

6. | K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. |

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

8. | 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 |

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

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

11. | D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. G. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. |

12. | M. Stepić, C. Wirth, C. E. Rüter, and D. Kip, “Observation of modulational instability in discrete media with self-defocusing nonlinearity,” Opt. Lett. |

13. | 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. |

14. | R. Dong, C. E. Rüter, D. Kip, O. Manela, M. Segev, C. Yang, and J. Xu, “Spatial frequency combs and supercontinuum generation in one-dimensional photonic lattices,” Phys. Rev. Lett. |

15. | S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. |

16. | E. Smirnov, M. Stepić, C. E. Rüter, D. Kip, and V. Shandarov, “Observation of staggered surface solitary waves in one-dimensional waveguide arrays,” Opt. Lett. |

17. | C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. |

18. | N. Malkova, I. Hromada, X. S. Wang, G. Bryant, and Z. G. Chen, “Observation of optical Shockley-like surface states in photonic superlattices,” Opt. Lett. |

19. | H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, and A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express |

20. | F. Fedele, J. K. Yang, and Z. G. Chen, “Defect modes in one-dimensional photonic lattices,” Opt. Lett. |

21. | A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Observation of defect-free surface modes in optical waveguide arrays,” Phys. Rev. Lett. |

22. | K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. |

23. | M. I. Molina and Y. S. Kivshar, “Nonlinear localized modes at phase-slip defects in waveguide arrays,” Opt. Lett. |

24. | M. I. Molina and Yu. S. Kivshar, “Interface localized modes and hybrid lattice solitons in waveguide arrays,” Phys. Lett. A |

25. | Y. Hu, R. Egger, P. Zhang, X. S. Wang, and Z. G. Chen, “Interface solitons excited between a simple lattice and a superlattice,” Opt. Express |

26. | Z. Y. Xu, M. I. Molina, and Yu. S. Kivshar, “Interface solitons in quadratic nonlinear photonic lattices,” Phys. Rev. A |

27. | S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, R. Morandotti, M. Volatier, V. Aimez, R. Arès, C. E. Rüter, and D. Kip, “Optical modes at the interface between two dissimilar discrete meta-materials,” Opt. Express |

28. | A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. |

29. | S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, R. Morandotti, M. Volatier, V. Aimez, R. Arès, E. H. Yang, and G. Salamo, “Optical spatial solitons at the interface between two dissimilar periodic media: theory and experiment,” Opt. Express |

30. | M. Stepić, D. Kip, Lj. Hadžievski, and A. Maluckov, “One-dimensional bright discrete solitons in media with saturable nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

31. | E. Smirnov, C. E. Rüter, M. Stepić, V. Shandarov, and D. Kip, “Dark and bright blocker soliton interaction in defocusing waveguide arrays,” Opt. Express |

32. | M. Stepi, E. Smirnov, C. E. Rüter, D. Kip, A. Maluckov, and L. O. Hadžievski, “Tamm oscillations in semi-infinite nonlinear waveguide arrays,” Opt. Lett. |

**OCIS Codes**

(130.3730) Integrated optics : Lithium niobate

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

(230.7370) Optical devices : Waveguides

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 4, 2010

Revised Manuscript: September 13, 2010

Manuscript Accepted: September 13, 2010

Published: January 11, 2011

**Citation**

A. Kanshu, C. E. Rüter, D. Kip, P. P. Beličev, I. Ilić, M. Stepić, and V. M. Shandarov, "Linear and nonlinear light propagation at the interface of two homogeneous waveguide arrays," Opt. Express **19**, 1158-1167 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1158

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### References

- D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003). [CrossRef] [PubMed]
- P. St, J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, Eds., (Plenum Press, 1995), pp. 585–633.
- Yu. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).
- R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001). [CrossRef] [PubMed]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000). [CrossRef] [PubMed]
- K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102(12), 123905 (2009). [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(16), 3383–3386 (1998). [CrossRef]
- 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(6928), 147–150 (2003). [CrossRef] [PubMed]
- J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005). [CrossRef] [PubMed]
- F. Chen, M. Stepić, C. E. Rüter, 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(11), 4314–4324 (2005). [CrossRef] [PubMed]
- D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. G. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004). [CrossRef] [PubMed]
- M. Stepić, C. Wirth, C. E. Rüter, and D. Kip, “Observation of modulational instability in discrete media with self-defocusing nonlinearity,” Opt. Lett. 31(2), 247–249 (2006). [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(2), 023902 (2003). [CrossRef] [PubMed]
- R. Dong, C. E. Rüter, D. Kip, O. Manela, M. Segev, C. Yang, and J. Xu, “Spatial frequency combs and supercontinuum generation in one-dimensional photonic lattices,” Phys. Rev. Lett. 101(18), 183903 (2008). [CrossRef] [PubMed]
- S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. 96(6), 063901 (2006). [CrossRef] [PubMed]
- E. Smirnov, M. Stepić, C. E. Rüter, D. Kip, and V. Shandarov, “Observation of staggered surface solitary waves in one-dimensional waveguide arrays,” Opt. Lett. 31(15), 2338–2340 (2006). [CrossRef] [PubMed]
- C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. 97(8), 083901 (2006). [CrossRef] [PubMed]
- N. Malkova, I. Hromada, X. S. Wang, G. Bryant, and Z. G. Chen, “Observation of optical Shockley-like surface states in photonic superlattices,” Opt. Lett. 34(11), 1633–1635 (2009). [CrossRef] [PubMed]
- H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, and A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express 11(25), 3404–3411 (2003). [CrossRef] [PubMed]
- F. Fedele, J. K. Yang, and Z. G. Chen, “Defect modes in one-dimensional photonic lattices,” Opt. Lett. 30(12), 1506–1508 (2005). [CrossRef] [PubMed]
- A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Observation of defect-free surface modes in optical waveguide arrays,” Phys. Rev. Lett. 101(20), 203902 (2008). [CrossRef] [PubMed]
- K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. 31(18), 2774–2776 (2006). [CrossRef] [PubMed]
- M. I. Molina and Y. S. Kivshar, “Nonlinear localized modes at phase-slip defects in waveguide arrays,” Opt. Lett. 33(9), 917–919 (2008). [CrossRef] [PubMed]
- M. I. Molina and Yu. S. Kivshar, “Interface localized modes and hybrid lattice solitons in waveguide arrays,” Phys. Lett. A 362(4), 280–282 (2007). [CrossRef]
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