Study On The Discrete Spatial Solitons In Optically Induced Photonic Lattices In Photorefractive Crystals

Light propagation in periodic optical systems such as waveguide arrays,optically-induced photonic lattices, and photonic crystals has attracted a lot ofattention. Compared to homogeneous media, several fundamental features exist insuch periodic systems, including allowed bands and forbidden gaps. Of particularinterest are the nonlinear periodic systems, in which the interplay between linearcoupling effects among adjacent potential wells and nonlinearity plays a dominantrole. The balance between these two processes can result in a self-localized structureknown as a discrete or lattice soliton.In this paper, we theoretically study lattice solitons, defet modes, and defectsolitons in the two-dimensional optical lattice based on photovoltaic-photorefractivenonlinearity.1. We theoretically analyze two families of two-dimensional bright lattice solitonsin photovoltaic-photorefractive crystals. We find that self-focusing and-defocusinglattice solitons are possible only when their power level exceeds a critical threshold.For the focusing nonlinearity, no complete band gap exists when the potential depth isless than a certain threshold value, at which the first band gap starts to occur. Whenthe potential depth is increased, more band gaps occur. Self-focusing lattice solitonsexist not only in the semi-infinite band gap, but also in the first band gap. Comparedwith lattice solitons in the semi-infinite band gap, the tails of the lattice soliton in thefirst band gap occupy many lattice sites. For the defocusing nonlinearity, no completeband gap exists when the potential depth exceeds a certain threshold value. When thecertain threshold value is decreased, only one gap is generated. When aself-defocusing lattice soliton is close to the second band, the self-defocusing latticesoliton is more confined, and its tails occupy a few lattice sites.2. We analyze defect modes in optically induced two-dimensional lattices inphotovoltaic-photorefractive crystals. We show that both the positive and negativedefects can support various types of two-dimensional defect modes such as fundamental, dipole, quadrupole, and sixpole modes. These modes reside in variousbandgaps of the photonic lattice. For positive defects, we find that defect modebranches exist not only in the semi-infinite gap, but also in the first, second, and thirdbandgaps. These defect mode branches stay inside their respective bandgaps. At eachpoint on branches, there is one defect mode in the semi-infinite bandgap and are twodefect modes in the first, second, and third bandgaps. For negative defects, we findalso that defect mode branches exist only in the first, second, and third bandgaps.These branches march to edges of the other Bloch bands and then reappear in higherband gaps. When these branches are born on edges of Bloch bands at small negativevalues of the defect strength parameter, there is one defect mode at each point on itsbranch in the first bandgap and are two defect modes at each point on their branchesin the second and third bandgaps.3. We study on the existence and stability of solitons in a defect embedded in asquare optical lattice based on a photovoltaic-photorefractive crystal. For differentintensity of defects, these solitons exist in different bandgaps. For a positive defect,the solitons only exist in the semi-infinite gap and can be stable in the low powerregion but not in the high power region. For a negative defect with moderate intensity,the solitons can exit not only in the semi-infinite gap, but also in the first gap. Withincreasing the defect depth, these solitons are found in more bandgaps, and theirstability become more complicated.