## Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays

Optics Express, Vol. 13, Issue 11, pp. 4314-4324 (2005)

http://dx.doi.org/10.1364/OPEX.13.004314

Acrobat PDF (1248 KB)

### Abstract

We investigate, experimentally and theoretically, light propagation in one-dimensional waveguide arrays exhibiting a saturable self-defocusing nonlinearity. We demonstrate low-intensity “discrete diffraction”, and the high-intensity formation of spatial gap solitons arising from the first band of the transmission spectrum. The waveguide arrays are fabricated by titanium in-diffusion in a photorefractive copper-doped lithium niobate crystal, and the optical nonlinearity arises from the bulk photovoltaic effect.

© 2005 Optical Society of America

## 1. Introduction

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4. Y. S. Kivshar, “Self-localization in arrays of defocusing waveguides,” Opt. Lett. **20**, 1147–1149 (1993). [CrossRef]

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8. S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity,” JETP **86**, 682–686 (1998). [CrossRef]

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

10. O. Manela, O. Cohen, G. Bartal, J. W. Fleischer, and M. Segev, “Two-dimensional higher-band vortex lattice solitons,” Opt. Lett. **29**, 2049–2051 (2004). [CrossRef] [PubMed]

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

^{st}Brilluoin zone [3

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

^{st}Brilluoin zone where anomalous diffraction is counteracted by self-defocusing, giving rise to spatial gap solitons [6

6. 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 (2003). [CrossRef] [PubMed]

7. 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]

23. B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M. K. Oberthaler, “Bright Bose-Einstein gap solitons of atoms with repulsive interaction,” Phys. Rev. Lett. **92**, 230401 (2004). [CrossRef] [PubMed]

24. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. **92**, 093904 (2004). [CrossRef] [PubMed]

6. 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 (2003). [CrossRef] [PubMed]

7. 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]

25. D. N. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. **93**, 083905 (2004). [CrossRef] [PubMed]

26. M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. **73**, 3211–3214 (1994). [CrossRef] [PubMed]

27. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E **66**, 046602 (2002). [CrossRef]

24. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. **92**, 093904 (2004). [CrossRef] [PubMed]

28. J. Meier, J. Hudock, D. N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. **91**, 143907 (2003). [CrossRef] [PubMed]

_{3}) crystal, and the optical nonlinearity arises from the bulk photovoltaic effect. The maximum nonlinear index change induced by the photovoltaic nonlinearity is typically very high (~0.003 [29

29. S. Orlov, A. Yariv, and M. Segev, “Nonlinear self-phase matching of optical second harmonic generation in lithium niobate,” Appl. Phys. Lett. **68**, 1610–1612 (1996). [CrossRef]

30. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. **68**, 923–926 (1992). [CrossRef] [PubMed]

31. M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. **73**, 3211–3214 (1994). [CrossRef] [PubMed]

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

33. G. Duree, J. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. **71**, 533–536 (1993). [CrossRef] [PubMed]

34. M. Taya, M. Bashaw, M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A **52**, 3095–3100 (1995). [CrossRef] [PubMed]

35. Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. **78**, 2948–2951 (1997). [CrossRef]

27. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E **66**, 046602 (2002). [CrossRef]

6. 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 (2003). [CrossRef] [PubMed]

**422**, 147–150 (2003). [CrossRef] [PubMed]

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

25. D. N. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. **93**, 083905 (2004). [CrossRef] [PubMed]

36. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” Phys. Rev. Lett. **92**, 123904 (2004). [CrossRef] [PubMed]

42. Z. Chen, A. Bezryadina, I. Makasynk, and J. Yang, “Observation of two-dimensional vector lattice solitons,” Opt. Lett. **29**, 1656–1658 (2004). [CrossRef] [PubMed]

_{3}crystals exhibiting such nonlinearity are a fairly mature technology. Namely, LiNbO

_{3}crystals doped with certain metal ions (e.g., copper or iron) exhibit strong photovoltaic effects, which lead to a nonlinear index change of a saturable self-defocusing nature. At the same time, single mode waveguide arrays in LiNbO

_{3}can be fabricated through the extensively used titanium in-diffusion method. Nonetheless, soliton lattice experiments in photovoltaic waveguide arrays have thus far not been reported. With this idea in mind, a recent theoretical paper [43

43. M. Stepic, D. Kip, Lj. Hadzievski, and A. Maluckov, “One-dimensional bright discrete solitons in media with saturable nonlinearity,” Phys. Rev. E **69**, 066618 (2004). [CrossRef]

_{3}, possessing optical power as low as several microwatts.

## 2. Experimental methods

*x*-cut LiNbO

_{3}wafers of congruently melting composition supplied by Crystal Technology Inc. In the first step, the substrates with dimensions of 1×(10-25)×7.8 mm

^{3}along the crystallographic

*x, y, z*axes are doped with copper ions to increase the photorefractive effect. The ferroelectric

*c*-axis points along the

*z*-direction. A thin copper layer of 20 nm thickness is vacuum-deposited on top of the substrate and in-diffused for 2 hours at 1000 °C in a wet argon atmosphere. For a total diffusion time of 4 hours (this time includes the additional titanium in-diffusion of another 2 hours, see description below), a nearly constant copper concentration of 5×10

^{24}m

^{-3}is obtained at the surface region. Alternatively, to avoid the increase in surface roughness caused by copper in-diffusion, copper may be also in-diffused from the back side of the crystal. For that, a 70 nm-thick layer of copper is in-diffused for 24 hours. In this case the concentration beneath the surface is again 5×10

^{24}m

^{-3}.

*y*-axis of the substrate and separated by 4.4 µm. Subsequently the stripes are in-diffused for 2 hours at 1000 °C in air. The titanium increases the refractive index of the LiNbO

_{3}substrate and gives rise to waveguiding in its vicinity. Each separate channel forms a single-mode waveguide for TE polarized light of wavelength 514.5 nm. Each such channel is evanescently coupled to its nearest neighbours. The corresponding coupling constant is given by the overlap integral of the modes with the index profile and has been calculated to be ≈1 mm

^{-1}.

*x*=0. Figure 1 shows the calculated periodic index potential

*n*(

*z*)=

*n*(

*z*+Λ) at the depth below the surface where the maximum amplitude of the modes occurs and at the depth where the modal amplitude has dropped by a factor 1/

*e*. For the theoretical modelling in the next section, the index profiles are fairly well approximated by a cos

^{2}function, which yields the profile

*n*(

*z*)=2.242+0.00052cos

^{2}(

*πz*/Λ) for a depth close to the one where the intensity of the waveguide mode has its “center of gravity”.

*λ*=514.5 nm) of an argon ion laser as our light source. A combination of half-wave plate

*λ*/2 and polarizer

*P*allows for precise adjustment of power and polarization of the light. We first install a thin tilted glass plate (denoted as

*GP*) into half of the optical beam, so as to retard the phase of one half of the beam by

*π*with respect to the other half, thus generating a dipole-like structure. Such a dipole-like beam proves to be better suited for exciting spatial gap solitons arising from the edge of the Brillouin zone of the first band, because such solitons possess an inherent staggered structure. Then, we adjust the distance between the cylindrical lens

*CL*and the input microscope lens

*L*

_{1}(40×magnification), so that the input beam into the waveguide array

*WA*attains an elliptic shape of proper dimensions, thus facilitating the excitation of a well defined number of input channels. We choose the polarization of the input light to be extraordinary with respect to the crystalline axes of the LiNbO

_{3}crystal, thus using its largest electrooptic coefficient

*r*

_{33}. In all our experiments, we maintain a constant value for the beam diameter in the non-periodic direction of ~2.5 µm (FWHM), fitting nicely to the diffusion depth of each channel waveguide. On the other hand, in the direction of the grating vector, we use various beam diameters in the range from 4 to 100 µm. We vary the propagation angle of the input beam by moving the input lens

*L*

_{1}perpendicular to the beam. Alternatively, in some cases we use a rotatable thick glass plate located in front of the cylindrical lens. Finally, at the end face of the array we use another microscope lens

*L*

_{2}(20×magnification) to image the light distribution onto a CCD camera.

## 3. Fundamentals

*E*is the amplitude of the electric light field,

*k*=2

*πn/λ*and n are the wave number and refractive index of the light in the substrate,

*n(z)*is the periodically-modulated refractive index defining the waveguide array, and Δ

*n*is the nonlinear refractive index change (|Δ

*n*|≪

*n*). In this equation,

*y*is the propagation direction and

*z*is the transverse coordinate (

*x*is the “depth” coordinate and does not play any role here; see also sketch in Fig. 2). We use this notation to conform the definition of the crystalline axes of LiNbO

_{3}. The photovoltaic nonlinearity giving rise to Δ

*n*is of saturable form [32

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

34. M. Taya, M. Bashaw, M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A **52**, 3095–3100 (1995). [CrossRef] [PubMed]

35. Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. **78**, 2948–2951 (1997). [CrossRef]

*r*being the electrooptic coefficient,

*E*the (light-induced) photovoltaic field,

_{pv}*I*the light intensity, and

*I*being the dark irradiance, with a dark generation rate

_{d}=G/s*G*and a photo-ionization cross-section

*s*. The specific parameters for extraordinarily polarized light in LiNbO

_{3}are

*n*=2.242 and

*r*=

*r*

_{33}=30 pm/V. Photovoltaic fields in copper-doped LiNbO

_{3}have been found to reach values of about

*E*=7 kV/mm for copper concentrations of

_{pv}*c*=50×10

_{Cu}^{24}m

^{-3}[44

44. K. Peithmann, J. Hukriede, K. Buse, and E. Krätzig: “Photorefractive properties of lithium niobate volume crystals doped by copper diffusion,” Phys. Rev. B **61**, 4615–4620 (2000). [CrossRef]

*n*≈10

^{-3}. This value may be considerably lower for smaller total copper concentration and samples that have been reduced by annealing treatment, i.e. that have a smaller concentration of Cu

^{2+}.

*n*=0, corresponding to a vanishing low intensity), seeking solutions of the form

*E*(

*y, z*)=

*A*(

*z*) exp (

*iβ y*) with mode amplitude

*A*and propagation constant

*β*. Then, following the translation symmetry

*n*(

*z*)=

*n*(

*z*+Λ), we apply the Floquet-Bloch theorem and seek solutions of the form

*A*(

*z*)=

*U*(

*z*) exp(

*iK*), where

_{z}z*K*is the transverse wave number and

_{z}*U*(

*z*)=

*U*(

*z*+Λ) is a periodic function. This leads to

*β*as a function of the transverse Bloch wave number

*K*∈[-

_{z}*π*/Λ,

*π*/Λ] z K giving the diffraction/dispersion relation (band structure) of the periodic medium, where allowed values of the propagation constant

*β*are separated by band-gaps. After solving for the linear Bloch modes of the system and obtaining its transmission spectrum, we investigate nonlinear propagation and the formation of 1D lattice solitons by solving Eq. (3) with (

*n*(

*z*)+Δ

*n*) replacing

*n*(

*z*).

## 4. Results and discussion

*K*=0). A photograph of the output intensity is shown at the top of Fig. 3. Clearly, the output intensity distribution is symmetric about the excited channel, covering 35 channels, and possesses the characteristic twin lobes on such an experiment [3

_{z}**81**, 3383–3386 (1998). [CrossRef]

^{th}-order FFT beam propagation method (BPM) [45

45. H. Yoshida, “Construction of higher order sympletic integrators,” Phys. Lett. A **150**, 262–269 (1990). [CrossRef]

*x*discussed in section 2 (see also Fig. 1).

*K*=0 to

_{z}*K*≈

_{z}*π*/Λ. Subsequently, diffraction varies from being normal, to almost no diffraction, and then to being anomalous [46

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

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

*K*≈

_{z}*π*/2Λ, we launch an input beam with a width (FWHM) of 30 µm, exciting roughly four channels of the array. As shown by the intensity profiles in Fig. 5, which are taken from the output face of the array, for a normal-incidence input beam with

*K*=0 (solid line), the beam broadens considerably into roughly 20 channels. When the incidence angle is at

_{z}*K*≈±

_{z}*π*/2Λ(dashed and dotted lines, respectively), the output beam is almost diffraction-free occupying roughly four channels.

*K*≈

_{z}*π*/Λ). This is also where we can form first-band bright solitons in such nonlinear medium, because the photovoltaic nonlinearity in LiNbO

_{3}is of the self-defocusing type. Such solitons, arising from the Bloch wave at the edge of the first Brilluoin zone, have a staggered phase structure, and their propagation constant lies within the first gap (between the first and second transmission bands). Thus, such solitons are often called “spatial gap solitons” or “staggered solitons” [4

4. Y. S. Kivshar, “Self-localization in arrays of defocusing waveguides,” Opt. Lett. **20**, 1147–1149 (1993). [CrossRef]

**422**, 147–150 (2003). [CrossRef] [PubMed]

*K*=π/Λ), we divide the input light beam into two parts using the tilted thin glass plate (see Fig. 2) that covers half of the beam, thus creating a dipole with a relative phase of

_{z}*π*. This beam is then carefully adjusted to be launched, at normal incidence, into two neighboring waveguides, with a total input power of ≈16 µW. The idea behind this method is to generate an input beam that has a larger overlap with the amplitude profile of a gap soliton arising from the edge of the Brillouin zone of the first band. Such solitons have the “phase signature” of the Bloch mode associated with the same transverse momentum [6

**90**, 023902 (2003). [CrossRef] [PubMed]

*K*=

_{z}*π*/Λ wavefunction. At the same time, our method avoids difficulties associated with setting two-beam interference to generate nn appropriate input beam, which then has to be matched carefully to the lattice. The response time of the photovoltaic nonlinearity in our sample is rather long (≈1 hour) [48], thus we monitor the intensity distribution of the beam exiting the array as it evolves in time, eventually forming a spatial gap soliton. Consequently, we monitor the linear propagation of the input beam through the waveguide by monitoring the intensity distribution at the sample’s output face immediately after switching on the light. This output intensity distribution is shown for linear propagation at

*K*=

_{z}*π*/Λ in Fig. 6 (dotted line). After several minutes of illumination, the pattern starts to narrow (dashed line) by the action of the self-defocusing photovoltaic nonlinearity. Finally, the output intensity distribution reaches steady state (solid line) which does not change its intensity profile even after several hours. This structure is a spatial gap soliton. We compare these experimental results with the theory, and find in Fig. 7 that the calculated amplitude and intensity (left and middle diagrams, solid lines) and simulated propagation (right diagram) of such a gap soliton in a sample with 50 channel waveguides, under the same parameters as used in the experiment, occupies roughly 5 channels, and exhibits stable stationary propagation in the waveguide array. The experimental results are in good agreement with the simulation.

*x*exp(-

*x*

^{2}) with a

*π*phase jump at its center (a dipole). The beam’s FWHM is half a lattice constant (4.2 µm), it is centered between two waveguides, and its input power is 16 µW. When the beam is launched exactly normal to the waveguide array, two staggered solitons are created propagating in opposite directions. As the two solitons propagate, their transverse velocities decrease (see Figs. 8(a), (b)). However, when the beam is launched with some minor tilt (in our simulation: 2 mrad which is 1/6 of the angle to the edge of the first Brillouin zone), the power is divided unevenly between the two lobes of the beam. As a consequence, only one staggered soliton is created from the intense lobe, while the second lobe of the beam, being underpowered, radiates slowly its energy and disappears (see Figs. 8(c), (d)). In this process, approximately a quarter of the beam’s initial power was radiated and the surviving soliton’s power is approximately 12 µW. In our experiments, we generally observe only a single gap soliton (similar to Fig. 8(d)), and not the soliton pair. Apart from a very small asymmetry in the incoupling geometry (small tilt angle, slightly unequal intensity in the two lobes), which hardly can be avoided, the reason is, most likely, higher order terms in the photovoltaic nonlinearity that break the symmetry [49

49. M. Segev, G. C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B **14**, 1772–1781 (1997). [CrossRef]

*n*(

*z*)=Δ

*n*(-

*z*).

## Summary

*π*are used to excite, at normal incidence, two adjacent waveguides at the input face. The diffraction properties of our sample are modelled using Floquet-Bloch waves and calculating the corresponding band-gap structure. These numerical calculations confirm our experimental results and show a stable propagation of gap solitons in the array. This is to our knowledge the first observation of lattice solitons supported by the photovoltaic nonlinearity.

## Acknowledgments

## References and links

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13. | O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. |

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

15. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express |

16. | A. S. Davydov, |

17. | A. H. Xie, L. van der Meer, V. Hoff, and R. H. Austin, “Long-lived amide I vibrational modes in myoglobin,” Phys. Rev. Lett. |

18. | B. I. Swanson, J. A. Brozik, S. P. Love, G. F. Strouse, A. P. Shreve, A. R. Bishop, W.-Z. Wang, and M. I. Salkola, “Observation of intrinsically localized modes in a discrete low-dimensional material,” Phys. Rev. Lett. |

19. | U. T. Schwartz, L. Q. English, and A. J. Sievers, “Experimental generation and observation of intrinsic localized spin wave modes in an antiferromagnets,” Phys. Rev. Lett. |

20. | E. Trias, J. J. Mazo, and T. P. Orlando, “Discrete breathers in nonlinear lattices: experimental detection in Josephson junctions,” Phys. Rev. Lett. |

21. | P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Zolotaryuk, “Observation of breathers in Josephson ladders,” Phys. Rev. Lett. |

22. | A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein condensates,” Phys. Rev. Lett. |

23. | B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M. K. Oberthaler, “Bright Bose-Einstein gap solitons of atoms with repulsive interaction,” Phys. Rev. Lett. |

24. | D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. |

25. | D. N. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. |

26. | M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. |

27. | N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E |

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

29. | S. Orlov, A. Yariv, and M. Segev, “Nonlinear self-phase matching of optical second harmonic generation in lithium niobate,” Appl. Phys. Lett. |

30. | M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. |

31. | M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. |

32. | G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. Fejer, and M. Bashaw, “Bright and dark photovoltaic spatial solitons,” Phys. Rev. A |

33. | G. Duree, J. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. |

34. | M. Taya, M. Bashaw, M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A |

35. | Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. |

36. | J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” Phys. Rev. Lett. |

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

38. | O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature |

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

40. | Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,” Phys. Rev. Lett. |

41. | J. Yang, I. Makasynk, A. Bezryadina, and Z. Chen, “Dipole solitons in optically-induced two-dimensional photonic lattices,” Opt. Lett. |

42. | Z. Chen, A. Bezryadina, I. Makasynk, and J. Yang, “Observation of two-dimensional vector lattice solitons,” Opt. Lett. |

43. | M. Stepic, D. Kip, Lj. Hadzievski, and A. Maluckov, “One-dimensional bright discrete solitons in media with saturable nonlinearity,” Phys. Rev. E |

44. | K. Peithmann, J. Hukriede, K. Buse, and E. Krätzig: “Photorefractive properties of lithium niobate volume crystals doped by copper diffusion,” Phys. Rev. B |

45. | H. Yoshida, “Construction of higher order sympletic integrators,” Phys. Lett. A |

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

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

48. | The long response time is a result of the Cu doping and the low photoconductivity of our sample. Very recently we have fabricated samples with Fe doping where the response time can be shortened to about 100 s. |

49. | M. Segev, G. C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B |

**OCIS Codes**

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

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(230.7370) Optical devices : Waveguides

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 22, 2005

Revised Manuscript: May 18, 2005

Published: May 30, 2005

**Citation**

Feng Chen, Milutin Stepi�?, Christian Rüter, Daniel Runde, Detlef Kip, Vladimir Shandarov, Ofer Manela, and Mordechai Segev, "Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays," Opt. Express **13**, 4314-4324 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4314

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### References

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- T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, �??Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,�?? Opt. Lett. 29, 468-470 (2004). [CrossRef] [PubMed]
- The long response time is a result of the Cu doping and the low photoconductivity of our sample. Very recently we have fabricated samples with Fe doping where the response time can be shortened to about 100 s.
- M. Segev, G. C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, �??Photovoltaic spatial solitons,�?? J. Opt. Soc. Am. B 14, 1772-1781 (1997). [CrossRef]

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