Nonlinear Periodic Structures Research Articles

Gap solitons in spatiotemporal photonic crystals

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.

The development of slugging in two-layer hydraulic flows

In this paper, we develop a hydraulic theory to describe the occurrence and structure of slugging in a confined two-layer gas-liquid flow generated by prescribed, constant, upstream volumetric flow rates in each layer. A linearized theory for the uniform flow is established, after which we use bifurcation theory to study fully non-linear periodic travelling wave structures. We find that a two-parameter family of such travelling wave solutions exists. Under given conditions, the volumetric flow rate constraint provides a relation between these two parameters. To select a unique periodic travelling wave solution, we require a further relation. We first investigate the conjecture that the periodic travelling wave solution selected in the initial value problem has the same wavelength as the linearly most temporally unstable mode. To do this, we solve the initial value problem numerically on a periodic domain. We find that the separation of the liquid slugs that form is much longer than the wavelength of the most unstable temporal mode. We then develop a different conjecture based on the convective instability of the long ‘tails’ of the available periodic travelling wave solutions, which leads to a better understanding of the wavelength selection process.

Localization and trapping of light in one- and two-dimensional nonlinear periodic structures

Novel photonic crystals have been proposed and built where the phenomenon of light localization occurs. This localization takes many forms, including discrete and continuous gap solitons, trapping via defect modes, localization due to randomness, just to name some. In this paper we discuss pulse dynamics in nonlinear one-dimensional fiber Bragg gratings and two-dimensional periodic waveguides. Here, gap solitons propagate and are trapped into defect via resonant interactions. A third case we consider is that of light propagation in nonlinear fiber arrays where we add randomness that model imperfections in the separation between fibers. The model which describes the dynamics is the discrete nonlinear Schroedinger equation with random coupling coefficients. The question we address is whether light localization as a balance of discrete diffraction and nonlinearity is robust when random effects are present.

Bright and dark solitons in quadratic nonlinear periodic structures and application to an all-optical logic gate

By studying one-dimensional periodic structures with modulation of both linear and quadratic nonlinear susceptibilities, we obtain the stable propagation of a fundamental soliton pair with both bright and dark shapes and the stable propagation of a fundamental soliton series with bright shapes. Furthermore, we find that optical logic gate functions can be constructed by using these kinds of periodic structures and by setting different parameters. As examples, OR and XOR gates are presented.

FDTD analysis of nonlinear Bragg grating based optical devices

In this paper, several nonlinear media with different instantaneous third-order nonlinearity responses are analysed. A Finite-Difference-Time-Domain method (FDTD), developed for nonlinear structures, is used to analyse the nonlinear waveguides and periodic nonlinear structures that exhibit attractive properties that make them suitable for novel devices with wavelength tuneable characteristics.

Generation of Bessel beams by parametric frequency doubling in annular nonlinear periodic structures

We analyze the second-harmonic generation in two-dimensional photonic structures with radially periodic domains created by poling of a nonlinear quadratic crystal. We demonstrate that the parametric conversion of the Gaussian fundamental beam propagating along the axis of the annular structure leads to the axial emission of the second-harmonic field in the form of the radially polarized first-order Bessel beam.

Molding Light in Two-Dimensional Photonic Lattices

F abricated dielectric periodic structures such as photonic crystals are attracting increasing attention, opening new possibilities for engineering transmission and refl ection of light and developing novel applications of photonic-crystal devices for all-optical signal processing and switching.1 Conventional approaches to guiding light in such periodic structures are based on wave transport through permanent waveguides, combined with resonant Bragg refl ection and total internal refl ection. However, these approaches may only allow limited fl exibility in the light molding and routing that is inherently restricted by the fabricated geometry. Recently, we demonstrated, both theoretically and experimentally, that increased fl exibility can be achieved when the light itself induces its own waveguide through the nonlinear response of the material.2 Th e internal structure and symmetry of the generated nonlinear selftrapped state selects itself the propagation direction in defect-free periodic structures. Th e symmetry of such localized optical waves is intrinsically defi ned by the physical mechanisms responsible for light localization—i.e., by total internal refl ection and Bragg scattering. In particular, we demonstrated that, in two-dimensional periodic optically induced photonic lattices created in biased photorefractive crystals, it is possible to use both localization mechanisms to obtain self-trapped states with diff erent mobility properties along the two principal directions in a square lattice. We described theoretically the families of such highly anisotropic gap solitons and studied them experimentally. An example of such a localized mode generated experimentally is shown in the fi gure (left). It originates from the x-symmetry point of the lattice bandgap spectrum, and possesses a reduced symmetry with highly anisotropic diff raction properties. Because of this anisotropy, such modes exhibit high mobility along the direction of their spatial modulation, and they are trapped by the lattice in the other transverse direction, enabling directional wave transport with possible applications for optical routing and switching in nonlinear periodic photonic structures. We predicted and verifi ed experimentally the unique mobility of these nonlinear modes and a novel mechanism for directional nonlinear wave transport in symmetric lattices. Th e fi gure summarizes our results. We imposed an initial tilt of the input beam along the diff erent directions and studied the beam displacement at the output. In simulations, an initial tilt of 20 mrad (15 percent of the Bragg angle) along x moves the output by two lattice sites (b), whereas the same tilt in y leads to no motion of the output state (c). Even with both tilts superimposed, the soliton moves the same two lattice sites along the x-axis only (d). Two examples of the corresponding experimental results O P T IC A L B E A M S

Self-trapping of polychromatic light in nonlinear periodic photonic structures

We study dynamical reshaping of polychromatic beams due to collective nonlinear self-action of multiple-frequency components in periodic and semi-infinite photonic lattices and predict the formation of polychromatic gap solitons and polychromatic surface solitons due to light localization in spectral gaps. We show that the self-trapping efficiency and structure of emerging polychromatic solitons depend on the input spectrum due to the lattice-enhanced dispersion, including the effect of crossover from localization to diffraction in media with defocusing nonlinearity.

Finite-dimensional model for defect-trapped light in planar periodic nonlinear structures

We study the dynamics of 2D gap solitons (GSs) in Bragg resonant nonlinear (photonic) gratings in the presence of localized defects. Previous work [Stud. Appl. Math.115, 209 (2005)] explains the mechanism of trapping the GS-carried energy at a defect via a resonant energy transfer from the GS into defect modes. We derive a finite-dimensional model that describes the evolution of the defect-trapped state as an interaction of linear defect modes and show that this model approximates the full dynamics very well in the regime when moderate amounts of GS energy are trapped.

Spatial Four Wave Mixing in Nonlinear Periodic Structures

We present the first experimental study of spatial four-wave mixing in photonic lattices, demonstrating universal aspects of nonlinear processes in periodic media, such as engineered phase matching, Bloch-wave folding, and continuous control over the band at which the interaction products emerge.

Nonlinear responses of the periodic structure composed of single negative materials

We investigate the nonlinear responses of the one-dimensional periodic structure containing alternate single negative materials. The transmission property of this nonlinear periodic structure is studied and the bistable behavior is found. Similar as in the linear periodic structure, the transmission property of this nonlinear structure in the zero phase (zero–ϕ) gap region is found to be relatively insensitive to the incident angle from 0° to 30°, in comparison with that in the Bragg gap. Especially, the transmissions associated with the zero–ϕ gap reach the peaks at almost the same input intensity for different incident angles. Another merit of the transmission property in the zero–ϕ gap is that a lower threshold is needed to achieve the bistability than in the Bragg gap. The electric field distribution in the system is further studied. We observe the zero–ϕ gap soliton and compare it with the usual Bragg gap soliton. The differences between these two kinds of solitons are analyzed, and the zero–ϕ gap soliton is shown to be insensitive to the incident angle. In the end, we find that the transmission and the soliton in the zero–ϕ gap are also weakly dependent on the loss and the scaling in contrast to that in the Bragg gap.

Optical limiting in defective quadratic nonlinear photonic crystals

By using a nonlinear transfer matrix method and an iteration method, we investigated an optical limiting in a defective quadratic nonlinear photonic crystal under strong pump-depletion conditions. Due to a giant enhancement of the second harmonic (SH) generation induced by the strong localization of light near the defective band, the progress of the energy exchange from the forward fundamental frequency (FF) channel to the SH channels (both forward and backward) and the backward FF channel is speeded up. The structure can therefore be used as a high-efficient compact optical power limiter. We have also studied the effects of two important parameters of the defective nonlinear periodic structure, the number of the periods and the nonlinear efficiency of the composite material, on the optical limiting.

Analysis of stable all-optical limiting in nonlinear periodic structures

We have studied the propagation of coherent light through a passive optical power limiter (OPL) consisting of two alternating layers with different linear and nonlinear refractive indices. We have obtained general formulas for the transmittance and the optical limiting output power.

Nonlinear wave interaction in photonic band gap materials

We present detailed analytical and numerical studies of nonlinear wave interaction processes in one-dimensional (1D) photonic band gap (PBG) materials with a Kerr nonlinearity. We demonstrate that some of these processes provide efficient mechanisms for dynamically controlling so-called gap-solitons. We derive analytical expressions that accurately determine the phase shifts experienced by nonlinear waves for a large class of non-resonant interaction processes. We also present comprehensive numerical studies of inelastic interactions, and show that rather distinct regimes of interaction exist. The predicted effects should be experimentally observable, and can be utilized for probing the existence and parameters of gap solitons. Our results are directly applicable to other nonlinear periodic structures such as Bose–Einstein condensates in optical lattices.

Reduced-Symmetry Two-Dimensional Solitons in Photonic Lattices

We demonstrate theoretically and experimentally a novel type of localized beam supported by the combined effects of total internal and Bragg reflection in nonlinear two-dimensional square periodic structures. Such localized states exhibit strong anisotropy in their mobility properties, being highly mobile in one direction and trapped in the other, making them promising candidates for optical routing in nonlinear lattices.

Generation of Bragg solitons through modulation instability in a Bragg grating structure

In this article, we consider the continuous wave (cw) propagation through the nonlinear periodic structure that consists of alternating layers of both positive and negative Kerr coefficients along the propagation direction. We investigate the modulational instability (MI) conditions required for the generation of ultrashort pulses for the nonlinearity management system. We study the occurrence of MI at the top and bottom edges of the photonic band gap (PBG) where the forward and backward propagating waves are strongly coupled because of the presence of the grating structure. We also study the MI when cw is detuned from the edges of the PBG into the anomalous and normal dispersion regimes. In addition, we discuss the existence of gap solitons for the nonlinearity management system in the upper and lower branches of the dispersion curve through the MI gain spectra. We observe the generation of higher order solitons in the nonlinear periodic structure when the input power is increased beyond a certain critical level. Finally, we discuss the generation of higher order Bragg grating solitons through the intensity evolution of the forward and backward propagating fields.

Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales

We study the attenuation, caused by weak damping, of harmonic waves through a discrete, periodic structure with frequency nominally within the Propagation Zone (i.e., propagation occurs in the absence of the damping). The period of the structure consists of a linear stiffness and a weak linear/nonlinear damping. Adapting the transfer matrix method and using harmonic balance for the nonlinear terms, a four-dimensional linear/nonlinear map governing the dynamics is obtained. We analyze this map by applying the method of multiple scales upto first order. The resulting slow evolution equations give the amplitude decay rate in the structure. The approximations are validated by comparing with other analytical solutions for the linear case and full numerics for the nonlinear case. Good agreement is obtained. The method of analysis presented here can be extended to more complex structures.

Numerical solution of nonlinear diffraction problems

An adaptive finite element method is developed for solving Maxwell's equations in a nonlinear periodic structure. The medium or computational domain is truncated by a perfect matched layer (PML) technique. Error estimates are established. Numerical examples are provided, which illustrate the efficiency of the method.

Optical bistability of nonlinear multilayered structure containing left-handed materials

We investigate the optical bistability in a multilayered structure containing alternating left-handed materials (LHM) and conventional (right-handed) materials. Numerical calculations show that optical bistability can appear in both Bragg and zero- 〈 n 〉 gaps inside the structure created by alternating right-handed materials (RHM) and LHM with an embedded nonlinear LHM (or nonlinear RHM) defect layer. We find that even if the properties of nonlinear defect layer change from double-negative to single-negative materials with increasing the incident field intensity, the bistability can still be obtained. Moreover, we also analyzed optical bistability in nonlinear periodic structure created by alternating RHM and nonlinear LHM. The bistability threshold values in three microstructures are also studied and compared.

Optical bistability in periodic nonlinear structures containing left handed materials

We study the transmission properties of a nonlinear peri- odic structure containing alternating slabs of a nonlinear right-handed material and a linear left-handed material. We find that the transmission associated with the zero-averaged refractive-index gap exhibits a bista- ble characteristic that is relatively insensitive to incident angles. This is in contrast to the nonlinear behavior of the usual Bragg gap. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 528 -530, 2005; Published online in Wiley InterScience (www.interscience.wiley. com). DOI 10.1002/mop.21037