Abstract:
We study spatial optical solitons in a one-dimensional nonlinear photonic crystal created by an array of thin-film nonlinear waveguides, the so-called Dirac-comb nonlinear lattice. We analyze modulational instability of the extended Bloch-wave modes and also investigate the existence and stability of bright, dark, and ``twisted'' spatially localized modes in such periodic structures. Additionally, we discuss both similarities and differences of our general results with the simplified models of nonlinear periodic media described by the discrete nonlinear Schrodinger equation, derived in the tight-binding approximation, and the coupled-mode theory, valid for shallow periodic modulations of the optical refractive index.

Abstract:
We analyze the existence, stability, and mobility of gap solitons in a periodic photonic structure with nonlocal nonlinearity. Within the Bragg region of band gaps, gap solitons exhibit better stability and higher mobility due to the combinations of non-locality effect and the oscillation nature of Bloch waves. Using linear stability analysis and calculating the Peierls-Nabarro potentials, we demonstrate that gap solitons can revive a non-trivial elastic-like collision even in the periodic systems with the help of nonlocal nonlinearity. Such interesting behaviors of gap solitons in nonlocal nonlinear photonic crystals are believed to be useful in optical switching devices.

Abstract:
We generalize the concept of nonlinear periodic structures to systems that show arbitrary spacetime variations of the refractive index. Nonlinear pulse propagation through these spatiotemporal photonic crystals can be described, for shallow nonstationary gratings, by coupled mode equations which are a generalization of the traditional equations used for stationary photonic crystals. Novel gap soliton solutions are found by solving a modified massive Thirring model. They represent the missing link between the gap solitons in static photonic crystals and resonance solitons found in dynamic gratings.

Abstract:
The dynamics of the coupled solitons in one\|dimensional photonic crystals with quadratic and cubic nonlinearities is studied. Starting from the Maxwell equation, the coupled\|mode equations for the envelopes of two fundamental frequency mode and one low\|frequency mode components due to the optical rectification are derived by multiple scales method. A set of coupled soliton solutions of the coupled\|mode equations is provided. The results show that there exists a modulation of the fundamental frequency modes by the optical rectification field resulting from the quadratic nonlinearity, which makes the fundamental frequency mode components appear as soliton pairs of bright\|bright, brigh\|dark and dark\|dark types. The optical rectification field will disappear when the frequencies of the fundamental frequency fields approach to the frequency of the photonic band boundary.

Abstract:
We study two-color surface solitons in two-dimensional photonic lattices with quadratic nonlinear response. We demonstrate that such parametrically coupled optical localized modes can exist in the corners or at the edges of a square photonic lattice, and we analyze the impact of the phase mismatch on their properties, stability, and the threshold power for their generation.

Abstract:
We report results of a systematic analysis of spatial solitons in the model of 1D photonic crystals, built as a periodic lattice of waveguiding channels, of width D, separated by empty channels of width L-D. The system is characterized by its structural "duty cycle", DC = D/L. In the case of the self-defocusing (SDF) intrinsic nonlinearity in the channels, one can predict new effects caused by competition between the linear trapping potential and the effective nonlinear repulsive one. Several species of solitons are found in the first two finite bandgaps of the SDF model, as well as a family of fundamental solitons in the semi-infinite gap of the system with the self-focusing nonlinearity. At moderate values of DC (such as 0.50), both fundamental and higher-order solitons populating the second bandgap of the SDF model suffer destabilization with the increase of the total power. Passing the destabilization point, the solitons assume a flat-top shape, while the shape of unstable solitons gets inverted, with local maxima appearing in empty layers. In the model with narrow channels (around DC =0.25), fundamental and higher-order solitons exist only in the first finite bandgap, where they are stable, despite the fact that they also feature the inverted shape.

Abstract:
We analyze two-color spatially localized modes formed by parametrically coupled fundamental and second-harmonic fields excited at quadratic (or chi-2) nonlinear interfaces embedded into a linear layered structure --- a quasi-one-dimensional quadratic nonlinear photonic crystal. For a periodic lattice of nonlinear interfaces, we derive an effective discrete model for the amplitudes of the fundamental and second-harmonic waves at the interfaces (the so-called discrete chi-2 equations), and find, numerically and analytically, the spatially localized solutions --- discrete gap solitons. For a single nonlinear interface in a linear superlattice, we study the properties of two-color localized modes, and describe both similarities and differences with quadratic solitons in homogeneous media.

Abstract:
We study the properties of two-color nonlinear localized modes which may exist at the interfaces separating two different periodic photonic lattices in quadratic media, focussing on the impact of phase mismatch of the photonic lattices on the properties, stability, and threshold power requirements for the generation of interface localized modes. We employ both an effective discrete model and continuum model with periodic potential and find good qualitative agreement between both models. Dynamics excitation of interface modes shows that, a two-color interface twisted mode splits into two beams with different escaping angles and carrying different energies when entering a uniform medium from the quadratic photonic lattice. The output position and energy contents of each two-color interface solitons can be controlled by judicious tuning of

Abstract:
Starting from the Maxwell's equations and without resort to the paraxial approximation, we derive equations describing stationary (1+1)-dimensional beams propagating at an arbitrary direction in an optical crystal with cubic symmetry and purely quadratic nonlinearity. The equations are derived separately for beams with the TE and TM polarizations. In both cases, they contain and cubic nonlinear terms, the latter ones generated via the cascading mechanism. The final TE equations and soliton solutions to them are quite similar to those in previously known models with mixed quadratic-cubic nonlinearities. On the contrary to this, the TM model is very different from previously known ones. It consists of four first-order equations for transverse and longitudinal components of the electric field at the fundamental and second harmonics. Fundamental-soliton solutions of the TM model are also drastically different from the usual "quadratic" solitons, in terms of the parity of their components. In particular, the transverse and longitudinal components of the electric field at the fundamental harmonic in the fundamental TM solitons are described, respectively, by odd and single-humped even functions of the transverse coordinate. Amplitudes of the longitudinal and transverse fields become comparable for very narrow solitons, whose width is commensurate to the carrier wavelength.

Abstract:
We discuss a novel method for generating octave-spanning supercontinua and few-cycle pulses in the important mid-IR wavelength range. The technique relies on strongly phase-mismatched cascaded second-harmonic generation (SHG) in mid-IR nonlinear frequency conversion crystals. Importantly we here investigate the so-called noncritical SHG case, where no phase matching can be achieved but as a compensation the largest quadratic nonlinearities are exploited. A self-defocusing temporal soliton can be excited if the cascading nonlinearity is larger than the competing material self-focusing nonlinearity, and we define a suitable figure of merit to screen a wide range of mid-IR dielectric and semiconductor materials with large effective second-order nonlinearities $d_{\rm eff}$. The best candidates have simultaneously a large bandgap and a large $d_{\rm eff}$. We show selected realistic numerical examples using one of the promising crystals: in one case soliton pulse compression from 50 fs to 15 fs (1.5 cycles) at $3.0\mic$ is achieved, and at the same time a 3-cycle dispersive wave at $5.0\mic$ is formed that can be isolated using a long-pass filter. In another example we show that extremely broadband supercontinua can form spanning the near-IR to the end of the mid-IR (nearly 4 octaves).