OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 11 — May. 30, 2005
  • pp: 4314–4324
« Show journal navigation

Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays

Feng Chen, Milutin Stepić, Christian E. Rüter, Daniel Runde, Detlef Kip, Vladimir Shandarov, Ofer Manela, and Mordechai Segev  »View Author Affiliations


Optics Express, Vol. 13, Issue 11, pp. 4314-4324 (2005)
http://dx.doi.org/10.1364/OPEX.13.004314


View Full Text Article

Acrobat PDF (1248 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

The past few years have witnessed a strong interest in periodic optical systems, such as waveguide arrays, photonic lattices, and photonic crystals. Such systems exhibit many attractive features for which no counterpart exists in homogeneous media, including forbidden gaps in their transmission spectra, Bragg diffraction, and the exciting possibility of controlling diffraction (for an updated review see Ref. [1

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

]). Of particular interest are the nonlinear periodic systems, which support self-localized structures called lattice solitons, also known as “discrete solitons” [2

2. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 19, 794–796 (1988). [CrossRef]

,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]

]. As expected from such periodic systems, many of the soliton phenomena found in them cannot exist in homogeneous nonlinear media. That includes, for example, spatial gap solitons [4

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

7

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]

], dipole-like (“twisted”) solitons [8

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

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

], higher-band solitons [10

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]

,11

11. G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, “Observation of 2nd-band vortex solitons in 2D photonic lattices,” submitted to Phys. Rev. Lett.

] and breathers [12

12. 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]

], multi-band solitons [13

13. O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91, 113901 (2003). [CrossRef] [PubMed]

,14

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

], soliton trains [15

15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express 12, 2831–2837 (2004). [CrossRef] [PubMed]

], to name a few. The phenomena of lattice solitons (solitons in periodic structures) is in fact universal, manifesting itself in a variety of systems in nature, from biological molecules [16

16. A. S. Davydov, Solitons in Molecular Systems (Kluwer Academic, Dordrecht, 1991).

,17

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. 84, 5435–5438 (2000). [CrossRef] [PubMed]

], to charge density waves [18

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. 82, 3288–3291 (1999). [CrossRef]

], spin waves [19

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. 83, 223–226 (1999). [CrossRef]

], arrays of Josephson junctions [20

20. E. Trias, J. J. Mazo, and T. P. Orlando, “Discrete breathers in nonlinear lattices: experimental detection in Josephson junctions,” Phys. Rev. Lett. 84, 741–744 (2000). [CrossRef] [PubMed]

,21

21. P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Zolotaryuk, “Observation of breathers in Josephson ladders,” Phys. Rev. Lett. 84, 745–748 (2000). [CrossRef] [PubMed]

], and very recently in Bose-Einstein condensates [22

22. A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001). [CrossRef] [PubMed]

,23

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]

].

A characteristic benchmark experiment in such nonlinear periodic systems involves exciting a single channel by a narrow wavepacket (beam of light), watch the wavepacket broaden at low power, as it experiences linear lattice diffraction, and then observing it narrow down and self-trap at sufficiently high power that activates the proper nonlinearity supporting lattice solitons [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]

]. Such experiments have been carried out with wavepackets launched at the base of the 1st 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]

], where normal diffraction is balanced by self-focusing, and later on at the edge of the 1st 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

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]

]. Subsequent experiments have demonstrated gap solitons arising from the second band [23

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

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

]. Thus far, however, spatial optical gap solitons have been demonstrated in only two physical systems: photorefractives [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

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

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]

] displaying the screening nonlinearity [26

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]

] in which waveguides are optically induced [27

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]

], and fabricated GaAs waveguides possessing the optical Kerr nonlinearity [24

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

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]

].

Here, we demonstrate, experimentally and theoretically, spatial gap solitons in one-dimensional (1D) waveguide arrays exhibiting a saturable self-defocusing nonlinearity. The waveguide arrays are fabricated by titanium in-diffusion in a copper-doped lithium niobate (LiNbO3) 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]

]), and the index contrast of the fabricated waveguide array can be adjusted over a wide range. Our experiments, therefore, offer a new physical system supporting lattice (gap) solitons, offering a strong saturable nonlinearity along with the robustness of the fabricated structures.

The physical nonlinearity that may cause soliton formation in inorganic photorefractive media includes two different charge transport mechanisms: the drift mechanism in an externally applied electric field [30

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

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]

], and the bulk photovoltaic (photogalvanic) effect that leads to an internal electric field [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]

]. In bulk photorefractives, solitons supported by either one of these mechanisms have been observed since 1993 [33

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]

] and 1995 [34

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

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]

], respectively. Since then, such screening solitons and photovoltaic solitons have been demonstrated in many photorefractive materials. More recently, following the suggestion of the optical induction technique [27

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]

], the photorefractive screening nonlinearity has become a major player in lattice soliton experiments [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

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]

,9

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

,11

11. G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, “Observation of 2nd-band vortex solitons in 2D photonic lattices,” submitted to Phys. Rev. Lett.

,25

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

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

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]

]. However, thus far the ability to employ the photovoltaic nonlinearity for lattice soliton experiments was left behind, even though LiNbO3 crystals exhibiting such nonlinearity are a fairly mature technology. Namely, LiNbO3 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 LiNbO3 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]

] has shown that discrete solitons can exist in such waveguide arrays with a saturable self-defocusing nonlinearity. Here we present the experimental observation of spatial gap solitons in photovoltaic 1D waveguide arrays in LiNbO3, possessing optical power as low as several microwatts.

2. Experimental methods

The waveguide arrays are prepared in two steps, using x-cut LiNbO3 wafers of congruently melting composition supplied by Crystal Technology Inc. In the first step, the substrates with dimensions of 1×(10-25)×7.8 mm3 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 5×1024 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 5×1024 m-3.

In the second step, a 10 nm-thin titanium layer is deposited on the top-side by sputtering, and is structured using standard lithographic techniques. In this way, we fabricate a grating of period Λ=8.4 µm, consisting of 4 µm-wide titanium stripes orientated parallel to the 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 LiNbO3 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.

Fig. 1. Refractive index profiles n(z) of a LiNbO3 waveguide array at two different depths of the in-diffused structures: at the depth of maximum field amplitude of the modes (dash-dotted line) and at a depth where the amplitude has dropped by a factor 1/e (solid line). The substrate refractive index is 2.242 (dotted line at the bottom).

Fig. 2. Experimental set-up: P, polarizer; λ/2, half-wave plate; M 1,2, mirrors; GP, thin glass plate; CL, cylindrical lens; L 1,2, microscope lenses; WA, waveguide array; CCD, CCD camera. The light source is a 514.5 nm wavelength argon ion laser.

Figure 2 shows the experimental set-up. We use the green line (wavelength 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 LiNbO3 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

Scalar paraxial wave propagation in a nonlinear 1D waveguide array is described by

idEdy+12kd2Edz2+k(n(z)+Δnn)E=0.
(1)

Here, E is the amplitude of the electric light field, k=2πn/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 Δn is the nonlinear refractive index change (|Δn|≪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 LiNbO3. The photovoltaic nonlinearity giving rise to Δn 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

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

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]

]

Δn=12n3rEpvII+Id,
(2)

with r being the electrooptic coefficient, Epv the (light-induced) photovoltaic field, I the light intensity, and Id=G/s being the dark irradiance, with a dark generation rate G and a photo-ionization cross-section s. The specific parameters for extraordinarily polarized light in LiNbO3 are n=2.242 and r=r 33=30 pm/V. Photovoltaic fields in copper-doped LiNbO3 have been found to reach values of about Epv=7 kV/mm for copper concentrations of cCu=50×1024 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]

], resulting in a maximum nonlinear refractive index change of Δn≈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 Cu2+.

We investigate first the band structure of the periodic waveguide array. To do that, we solve the linear version of Eq. (1) (with Δn=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(iKz z), where Kz is the transverse wave number and U(z)=U(z+Λ) is a periodic function. This leads to

12kd2Udz2+iKzkdUdzKz22kU+kn(z)nU=βU.
(3)

This equation can be solved numerically for the eigenvalue β as a function of the transverse Bloch wave number Kz∈[-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)+Δn) replacing n(z).

4. Results and discussion

We first carry out the “benchmark experiment” described in Section 1, by launching a low intensity light beam, of 4 µm (FWHM) horizontal width, into a single waveguide channel at normal incidence (Kz=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

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]

]. The input power in this experiment is chosen to be low and the output photograph is taken immediately after switching on the input light, so as to avoid any nonlinear effects that develop in this material rather slowly. For comparison, we also simulate the beam propagation in the waveguide array using a 4th-order FFT beam propagation method (BPM) [45

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

]. The calculated propagation results are shown at the bottom of Fig. 3, and they display a good agreement with the experimental results.

Staying within the linear case, we calculate the transmission spectrum of the 1D array and find the expected band structure shown in Fig. 4. To apply the 1D theoretical model to our 2D channel waveguide array, we use the value of the periodic index change at the proper “depth” coordinate x discussed in section 2 (see also Fig. 1).

Fig. 3. Experimental (a) and theoretical (b,c) results showing discrete diffraction of light in a LiNbO3 waveguide array, when a single input channel is excited. The upper part in (a) shows the intensity distribution at the output of the waveguide array, as photographed with a CCD camera. The two lower parts (b) and (c) show the simulated propagation of a beam in a waveguide array under the same parameters, at the “depth” of the maximum intensity of each individual mode.
Fig. 4. Band-gap diagram of the waveguide array, relating the propagation constant β to the Bloch wave number Kz as described in section 2. The value “0” corresponds to a plane wave propagating in the substrate. The shaded regions represent the gaps where light propagation is forbidden. The black dot at the edge of the first band indicates the propagation constant of the gap soliton (shown in Fig. 7). Increasing the optical intensity creates a negative defect in the periodic structure, thereby localizing the corresponding Bloch wave by pushing its propagation constant β down into the gap, thus converting it from an extended Bloch wave into a self-localized state: a gap soliton.
Fig. 5. Probing diffraction in the periodic waveguide array by varying the angle of incidence of a four-channel input beam, from Kz=0 (solid line), to nearly diffraction-free propagation at Kz≈±π/2Λ(dotted and dashed lines).

Then, we change the angle of incidence of the input light beam, scanning the launch angle over the first Brilluoin zone, from Kz=0 to Kzπ/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

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]

]. To experimentally demonstrate almost diffraction-free propagation at Kzπ/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 Kz=0 (solid line), the beam broadens considerably into roughly 20 channels. When the incidence angle is at Kz≈±π/2Λ(dashed and dotted lines, respectively), the output beam is almost diffraction-free occupying roughly four channels.

Then, we move on to the anomalous diffraction region, launching the input beam at the Bragg angle (Kzπ/Λ). This is also where we can form first-band bright solitons in such nonlinear medium, because the photovoltaic nonlinearity in LiNbO3 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]

7

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]

]. The solitons form when the optical intensity is sufficiently high so that the propagating light beam induces (through the self-defocusing nonlinearity) a negative defect in the periodic index structure. When the beam has a structure close enough to the structure of the bound state of the induced defect, the optical beam self-traps in its own induced defect, thereby forming a spatial gap soliton.

Experimentally, to facilitate efficient excitation of a Bloch mode propagating close to the edge of the Brillouin zone (Kz=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 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

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]

]. Our excitation method facilitates a larger overlap with the gap soliton Kz=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

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.

], 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 Kz=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.

Fig. 6. Formation of a gap soliton in a 1D LiNbO3 waveguide array. The figure shows a line scan of the light intensity distribution measured by a CCD at the output facet of the array, where the dotted, dashed and solid lines represent the intensity profile at times t≈0, t=45 min, and t=160 min, respectively.
Fig. 7. Calculated wavefunction and propagation dynamic of a spatial gap soliton in our setting. Left and middle panels: amplitude and intensity of a gap soliton (solid line) plotted on the background of the waveguide structure with the light-induced (negative) defect the soliton creates (dotted lines). Right panel: simulated stable and stationary propagation of the gap soliton in the waveguide array. The intensity profile of the soliton shown propagating in the right panel corresponds to the intensity of the soliton shown on the left panel.
Fig. 8. Numerical results for the nonlinear propagation of a Gaussian beam with a π phase jump at its center (a dipole): a), b), the beam is launched normal to the waveguide array, and c), d), the same input beam is launched with a tilt of 2 mrad into the array. In both cases the input power is 16 µW and the input beam covers about half a lattice constant (FWHM of the Gaussian beam of 4.2 µm).

To simulate our experiment, we propagate a Gaussian beam (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]

], having an effect similar to that of the small tilt angle. Most probably, in order to experimentally observe the soliton pair evolving from a normal-incidence dipole-type beam, one would have to have a highly symmetric nonlinearity for which Δn(z)=Δn(-z).

Summary

We have investigated linear and nonlinear propagation of light in a 1D waveguide array possessing a saturable self-defocusing nonlinearity arising from the bulk photovoltaic effect in LiNbO3. For low input power, discrete diffraction is observed, whereas for higher input powers, the build-up of a negative nonlinear index change allows for the formation of bright gap solitons that have a propagation constant within the gap of the linear dispersion spectrum. Experimentally, such a “staggered” bright gap soliton is observed when two wave packets with a relative phase difference of 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

This work was supported by the German-Israeli DIP project, through the German Federal Ministry of Education and Research (BMBF, grant DIP-E6.1). F. Chen gratefully acknowledges the Alexander von Humboldt Foundation for financial support.

References and links

1.

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

2.

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 19, 794–796 (1988). [CrossRef]

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]

4.

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

5.

J. Feng, “Alternative scheme for studying gap solitons in infinite periodic Kerr media,” Opt. Lett. 20, 1302–1304 (1993). [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]

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]

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]

11.

G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, “Observation of 2nd-band vortex solitons in 2D photonic lattices,” submitted to Phys. Rev. Lett.

12.

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.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91, 113901 (2003). [CrossRef] [PubMed]

14.

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

15.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express 12, 2831–2837 (2004). [CrossRef] [PubMed]

16.

A. S. Davydov, Solitons in Molecular Systems (Kluwer Academic, Dordrecht, 1991).

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. 84, 5435–5438 (2000). [CrossRef] [PubMed]

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. 82, 3288–3291 (1999). [CrossRef]

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. 83, 223–226 (1999). [CrossRef]

20.

E. Trias, J. J. Mazo, and T. P. Orlando, “Discrete breathers in nonlinear lattices: experimental detection in Josephson junctions,” Phys. Rev. Lett. 84, 741–744 (2000). [CrossRef] [PubMed]

21.

P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Zolotaryuk, “Observation of breathers in Josephson ladders,” Phys. Rev. Lett. 84, 745–748 (2000). [CrossRef] [PubMed]

22.

A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001). [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]

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]

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]

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]

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]

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. 92, 123903 (2004). [CrossRef] [PubMed]

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 433, 500–503 (2005). [CrossRef] [PubMed]

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. 92, 123902 (2004). [CrossRef] [PubMed]

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. 92, 143902 (2004). [CrossRef] [PubMed]

41.

J. Yang, I. Makasynk, A. Bezryadina, and Z. Chen, “Dipole solitons in optically-induced two-dimensional photonic lattices,” Opt. Lett. 29, 1662–1664 (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]

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]

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]

45.

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

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]

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 14, 1772–1781 (1997). [CrossRef]

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


Sort:  Journal  |  Reset  

References

  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, �??Discretizing light behavior in linear and nonlinear waveguide lattices,�?? Nature 424, 817-823 (2003). [CrossRef] [PubMed]
  2. D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 19, 794-796 (1988). [CrossRef]
  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]
  4. Y. S. Kivshar, �??Self-localization in arrays of defocusing waveguides,�?? Opt. Lett. 20, 1147-1149 (1993). [CrossRef]
  5. J. Feng, �??Alternative scheme for studying gap solitons in infinite periodic Kerr media,�?? Opt. Lett. 20, 1302-1304 (1993). [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]
  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]
  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]
  11. G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, �??Observation of 2nd-band vortex solitons in 2D photonic lattices,�?? submitted to Phys. Rev. Lett.
  12. 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. O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, �??Multiband vector lattice solitons,�?? Phys. Rev. Lett. 91, 113901 (2003). [CrossRef] [PubMed]
  14. A. A. Sukhorukov and Y.S. Kivshar, �??Multigap discrete vector solitons,�?? Phys. Rev. Lett. 91, 113902 (2003). [CrossRef] [PubMed]
  15. Y. V. Kartashov, V. A. Vysloukh, L. Torner, �??Soliton trains in photonic lattices,�?? Opt. Express 12, 2831-2837 (2004). [CrossRef] [PubMed]
  16. A. S. Davydov, Solitons in Molecular Systems (Kluwer Academic, Dordrecht, 1991).
  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. 84, 5435-5438 (2000). [CrossRef] [PubMed]
  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. 82, 3288-3291 (1999). [CrossRef]
  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. 83, 223-226 (1999). [CrossRef]
  20. E. Trias, J. J. Mazo, and T. P. Orlando, �??Discrete breathers in nonlinear lattices: experimental detection in Josephson junctions,�?? Phys. Rev. Lett. 84, 741-744 (2000). [CrossRef] [PubMed]
  21. P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Zolotaryuk, �??Observation of breathers in Josephson ladders,�?? Phys. Rev. Lett. 84, 745-748 (2000). [CrossRef] [PubMed]
  22. A. Trombettoni and A. Smerzi, �??Discrete solitons and breathers with dilute Bose-Einstein condensates,�?? Phys. Rev. Lett. 86, 2353-2356 (2001). [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]
  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]
  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]
  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]
  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]
  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. 92, 123903 (2004). [CrossRef] [PubMed]
  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 433, 500-503 (2005). [CrossRef] [PubMed]
  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. 92, 123902 (2004). [CrossRef] [PubMed]
  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. 92, 143902 (2004). [CrossRef] [PubMed]
  41. J. Yang, I. Makasynk, A. Bezryadina, and Z. Chen, �??Dipole solitons in optically-induced two-dimensional photonic lattices,�?? Opt. Lett. 29, 1662-1664 (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]
  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]
  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]
  45. H. Yoshida, �??Construction of higher order sympletic integrators,�?? Phys. Lett. A 150, 262-269 (1990). [CrossRef]
  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]
  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 14, 1772-1781 (1997). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited