..............................................................Observation of two-dimensionaldiscrete solitons in opticallyinduced nonlinear photonic latticesJason W. Fleischer*†, Mordechai Segev*†, Nikolaos K. Efremidis‡& Demetrios N. Christodoulides‡* Physics Department, Technion—Israel Institute of Technology, Haifa 32000,Israel† Electrical Engineering Department, Princeton University, New Jersey 08544,USA‡ School of Optics/CREOL, University of Central Florida, Florida 32816-2700,USA.............................................................................................................................................................................Nonlinear periodic lattices occur in a large variety of systems,such as biological molecules1, nonlinear optical waveguides2,solid-state systems3and Bose–Einstein condensates4. The under-lying dynamics in these systems is dominated by the interplaybetween tunnelling between adjacent potential wells and non-linearity1–15. A balance between these two effects can result ina self-localized state: a lattice or ‘discrete’ soliton1,2.Directobservation of lattice solitons has so far been limited to one-dimensional systems, namely in arrays of nonlinear opticalwaveguides2,9–17. However, many fundamental features areexpected to occur in higher dimensions, such as vortex latticesolitons18, bright lattice solitons that carry angular momentum,and three-dimensional collisions between lattice solitons. Here,we report the experimental observation of two-dimensional (2D)lattice solitons. We use optical induction, the interference of twoor more plane waves in a photosensitive material, to create a 2Dphotonic lattice in which the solitons form11,12. Our results pavethe way for the realization of a variety of nonlinear localizationphenomena in photonic lattices and crystals19–23. Finally, ourobservation directly relates to the proposed lattice solitons inBose–Einstein condensates4, which can be observed in opticallyinduced periodic potentials24,25.In general, wave propagation in periodic lattices (such as anarray of optical waveguides) is fundamentally different from thatoccurring in a homogeneous medium. For example, when light isfocused into one waveguide, linear propagation along the wave-guides results in tunnelling to adjacent sites, exhibiting a charac-teristic diffraction pattern with the intensity mainly concentrated inthe outer lobes. For a sufficiently high nonlinearity, self-focusingcan balance this effect, leading to a lattice (discrete) soliton2,9,10. Forlight propagating at an angle v ¼ kx=k < kx=kzwith respect to thearray, the periodicity of the lattice becomes important, as thecorresponding ‘Bloch momentum’ kxcan satisfy Bragg reflectionconditions within the Brillouin zone (defined in the range jkxDj #p; where D is the lattice spacing). Near the edge of this zone,diffraction becomes anomalous (‘negative’), leading to such effectsas diffraction management13,14and staggered (pout-of-phase)solitons11,15,17. These are some of the fundamental aspects of solitonsin nonlinear periodic structures, which were originally discoveredthrough the theoretical paradigm of the discrete nonlinear Schro¨-dinger equation1–4,16, hence the term ‘discrete soliton’1,2. Exper-imentally, self-localized ‘breathers’ have been observed in variousphysical settings5–8, but lattice (discrete) solitons have thus far beenreported only in nonlinear optical systems and only in one-dimen-sional (1D) configurations9–11,13,17. In what follows we demonstratebright 2D lattice solitons in their simplest realization: in-phasesolitons at the base of the first Brillouin zone. In addition, wedemonstrate bright self-trapped wave packets at the edge of the firstBrillouin zone.The formation of the 2D nonlinear photonic lattice relies on anoptical induction technique11,12in which a 2D array of waveguides isinduced in a nonlinear medium. We proposed this method theo-retically12and recently demonstrated experimentally111D latticesolitons in a 1D waveguide array. The waveguide array is induced, inreal time, in a photosensitive material by interfering two or moreplane waves. A separate ‘probe’ beam is launched into the periodicwaveguide array, where it exhibits discrete diffraction and, at asufficiently high nonlinearity, forms a lattice soliton. For this systemto work, it is essential that the waveguides are as uniform as possible,implying that the interference pattern (inducing the lattice) mustnot change in the propagation direction. For this to happen in thenonlinear medium, the interfering waves themselves should not beaffected by the nonlinearity. At the same time, the probe (soliton-forming) beam must experience the highest possible nonlinearity. Aphotorefractive material with a strong electro-optic anisotropyallows this scenario; the interfering beams are polarized in a non-electro-optic direction and the probe is polarized along the crystal-line c axis. In this arrangement, the interfering beams will propagatemostly linearly, while the signal beam will experience both aperiodic potential and a significant (screening) nonlinearity26,27.In this way we have demonstrated both on-axis and staggered 1Dlattice solitons11. However, 1D systems, although quite instructive,cannot host a variety of fascinating nonlinear phenomena thatrequire a higher lattice dimensionality. Our method of optical latticeinduction allows for dynamic, reconfigurable arrays of almost anygeometry.This method of optical induction is quite general, allowing theFigure 1 Experimental scheme and a typical photonic lattice. a, Diagram of ourexperimental set-up. We use a photosensitive (photorefractive) crystal with electro-opticanisotropy: two interfering pairs of ordinarily polarized plane waves induce the photonicarray, while the extraordinarily polarized probe (soliton-forming) beam is focused into asingle waveguide. b, Typical observation of a waveguide array at the exit face of thecrystal. Each waveguide is approximately 7mm in diameter, with an 11mm spacingbetween nearest neighbours.Figure 2 Numerical simulation results depicting the induced photonic lattice and thestructure of an on-axis lattice soliton. a, Calculated 2D structure of the induced indexchange (photonic lattice) for the self-focusing nonlinearity. b, Simulated intensity structureof the on-axis lattice soliton.letters to natureNATURE | VOL 422 | 13 MARCH 2003 | www.nature.com/nature 147©