Optical solitary waves in a photonic band gap material

A detailed analysis of finite-energy solitary waves in two- and three-dimensional nonlinear photonic band gap (PBG) structures is presented. Solitary waves in a photonic crystal exhibiting a nonresonant Kerr response with a two-dimensional (2d) square and triangular symmetry group as well as a 3d fcc symmetry group, are described in terms of an effective nonlinear Dirac equation derived using the slowly varying envelope approximation for the electromagnetic field. Unlike the case of one dimension, the multiple symmetry points of the 2d and 3d Brillouin Zones give rise to two distinct classes of solitary wave solutions. Solutions associated with a higher-order symmetry point of the crystal exist for both positive and negative Kerr coefficient, whereas solutions associated with a two-fold symmetry point occur only for positive Kerr nonlinearity. We obtain approximate solutions using a variational method. The nonlinear wave equations are then solved numerically using the Ritz-Galerkin method. An analytical stability criterion is obtained for a spinor field satisfying a nonlinear Dirac type of equation. Our study suggests that, for an ideal Kerr medium, 2d solitary waves in a band gap are stable whereas 3d ones are stable only in a certain region of the band gap.; We derive the properties of self-induced transparency (SIT) solitary waves in a one-dimensional periodic structure doped uniformly with two-level atoms. In our model, the electromagnetic field is treated classically and the dopant atoms are described quantum mechanically. Solitary wave formation involves the combined effects of group-velocity dispersion (GVD), nonresonant Kerr nonlinearity, and resonant interaction with dopant atoms. We find three distinct types of propagating solitary wave pulses. Far from Bragg resonance, we recapture the usual McCall-Hahn soliton with hyperbolic secant profile when the Kerr coefficient is set to zero. However, when the host Kerr coefficient is nonzero, the optical envelope function deviates from the hyperbolic secant profile and pulse propagation requires nontrivial phase modulation. When the laser frequency and atomic transition frequencies are near the photonic band edge, the additional effect of the GVD facilitates the propagation of a SIT-Gap soliton. The soliton structure changes dramatically as the laser frequency is tuned through the atomic resonance. A distinct type of near-band-edge solitary wave can propagate when the Kerr coefficient is zero. This third type of solution arises from the balance between GVD and the resonance interaction with the dopant atoms.