Nonlinear Coupled-mode Equations Research Articles

Vortex formation in a non-uniform self-gravitating dusty magnetoplasma

AbstractThe dynamics of low-frequency, short-wavelength electrostatic (SWES) drift waves in a self-gravitating, non-uniform, collisional dusty magnetoplasma with equilibrium dust-velocity gradients is studied in the present work. By employing the dust continuity and momentum equations to describe the dust dynamics and Boltzmann distribution for the electrons and ions, we have derived a new set of nonlinear mode coupling equations. In the linear limit, it is found that SWES drift waves are subjected to dissipative instability in the presence of an equilibrium dust sheared flow and self-gravitation effect. On the other hand, in the nonlinear case, it is shown that possible stationary solutions of the nonlinear equations are dipolar vortices.

Dynamic characteristics of nonlinear Bragg gratings in photonic crystal fibres

We study numerically the temporal and spatial dynamics of light in Bragg gratings in highly nonlinear photonic crystal fibre for a CW input signal. Our numerical model is based on the plane wave mode solver and a set of nonlinear coupled-mode equations which we solve using a variation of implicit fourth order Runge-Kutta method. We observe not only bistability of the intensity versus transmitted and reflected light but also complex dynamics. We demonstrate that for values of input intensity above the bistable region the steady state may undergo a supercritical Hopf bifurcation. For some ranges of the input intensity we also observe a coexistence of two periodic attractors. The dynamics found, in particular the features in the bifurcation diagram, strongly depend on the parameters of the fibre. Consequently, we suggest that by proper design of the photonic crystal in the cladding we can adjust such nonlinear features of the Bragg gratings as the width of the bistable region, the intensity at which the bifurcation occurs and also the characteristics of the dynamics at high values of input intensity.

Dispersive destabilization of nonlinear light propagation in fiber Bragg gratings: A numerical verification

This paper presents some numerical simulations of the full one-dimensional Maxwell-Lorentz equations that describe light propagation in fiber Bragg gratings in order to confirm that the standard nonlinear coupled mode equations fail to predict the weakly nonlinear dynamics of the system when dispersive instabilities come into play, and that, in this case, the correct slow envelope description of the system requires us to consider higher order dispersion effects.

Double optical bistability and its application in nonlinear chalcogenide-fiber Bragg gratings

We study the continuous wave (CW) nonlinear transmission characteristics of chalcogenide-fiber Bragg gratings (c-FBGs). The set of Cubic quintic septic nonlinear coupled mode equations (CQSNLCMEs) is derived analytically to govern nonlinear wave propagation in c-FBGs by counting until seventh-order nonlinearity in chalcogenide glass. Double-optical bistability is observed in the CW nonlinear transmission characteristics of π phase-shifted c-FBG. Based on unstable state principle in the hysteresis, all-optical transistor operation is shown as a prominent application of optical bistability in such device.

Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations

Extending the general approach for first-order hyperbolic systems developed in [D. Appelö, T. Hagstrom, G. Kreiss, Perfectly matched layers for hyperbolic systems: general formulation, well-posedness and stability, SIAM J. Appl. Math., 2006, to appear], we construct PML equations for the mixed-type system governing propagation of optical wave packets in both 1D and 2D Bragg resonant photonic waveguides with a cubic nonlinearity, i.e. the coupled mode equations. We prove that in the linear case the layer equations are absorbing and perfectly matched. We also prove they are stable for constant parameters. A number of numerical experiments are performed to assess the layer’s performance in both the linear and nonlinear regimes.

Nonlinear dynamics of electrostatic ion-temperature-gradient modes in a dust-contaminated plasma with variable charge and sheared ion flows

By employing Braginskii transport equations for ions and Boltzmann distribution for electrons in a dust-contaminated plasma with equilibrium density, temperature, and magnetic field gradients, the nonlinear set of equations are derived. New ion-temperature-gradient driven modes are obtained and various limiting cases are discussed. It is shown that the ion-temperature-gradient driven mode of drift-waves are attenuated in the presence of dust-charge fluctuations. It has been found that dust charging is always dissipative and the growth rate of various modes are damped. Furthermore, the possible stationary solution of the nonlinear mode coupling equations can be represented in the form of dipolar and vortex chains type solutions. The results of the present investigation should be helpful in understanding the fluctuations and transport phenomena in magnetically confined dustcontaminated tokamak plasma.

Relativistic and ponderomotive effects on stimulated Raman scattering of intense laser radiation in plasma

The effects of relativistic nonlinearities on stimulated Raman scattering (SRS) of mildly relativistic laser radiation propagating through preionized underdense plasma is studied. The nonlinear mode coupling equations and linearized growth rates of Raman instabilities in presence of combined effects of relativistic and ponderomotive nonlinearities are derived. The temporal evolution of SRS process is then studied and analyzed using numerical techniques in presence of the two nonlinearities. It is observed that relativistic effects now taken into account significantly affect the growth rates and the nonlinear evolution of SRS process as compared to the case where they are neglected. The study shows that inclusion of relativistic nonlinearities along with the ponderomotive nonlinearities destabilizes SRS process

Chaotic behavior of nonlinearly coupled electrostatic and electromagnetic modes in electron-positron-ion magnetoplasma with equilibrium flows

A new set of nonlinear equations has been derived to study the temporal behavior of low frequency electrostatic and electromagnetic ion-temperature-gradient driven modes in an electron-positron-ion (e-p-i) magnetoplasma. The temporal behavior of the nonlinear mode coupling equations, under certain conditions, are governed by the coupled equations, which are the generalization of Lorenz and Stenflo type equations, admitting chaotic behavior. The linear stability of the generalized Lorenz-Stenflo system of equations is also presented for electrostatic and electromagnetic cases. The results of present investigation should be useful to understand the nonlinear properties of electromagnetic/electrostatic waves in an e-p-i magnetoplasma.

Soliton compression and pulse-train generation by use of microchip Q-switched pulses in Bragg gratings

Pulse compression and pulse-train generation are demonstrated by use of kilowatt 580 ps pulses generated by a compact (15 cm x 3 cm x 3 cm) microchip Q-switched laser followed by a fiber Bragg grating. A 12-fold pulse compression to 45 ps with five times peak power enhancement is achieved at 1.4 kW through soliton effect compression in the fiber grating. At 2.5 kW, modulational instability leads to a train of high-contrast sub-100 ps pulses. These demonstrations take advantage of the ultrastrong dispersion at frequencies close to the edge of the photonic bandgap. Experimental results are discussed in the context of the nonlinear Schrödinger equation and are compared with simulations of the nonlinear coupled-mode equations.

Optical Soliton Bullets in (2+1)D Nonlinear Bragg Resonant Periodic Geometries

We consider light propagation in a Kerr‐nonlinear 2D waveguide with a Bragg grating in the propagation direction and homogeneous in the transverse direction. Using Newton's iteration method we construct both stationary and travelling solitary wave solutions of the corresponding mathematical model, the 2D nonlinear coupled mode equations (2D CME). We call these solutions 2D gap solitons due to their similarity with the gap solitons of 1D CME (fiber grating). Long‐time stable evolution preserving the solitary fashion is demonstrated numerically despite the fact that, as we show, for the 2D CME no local constrained minima of the Hamiltonian functional exist. Building on the 1D study of [1], we demonstrate trapping of slow enough 2D gap solitons at localized defects. We explain the mechanism of trapping as resonant transfer of energy from the soliton to one or more nonlinear defect modes. For a special class of defects, we construct a family of nonlinear defect modes by numerically following a bifurcation curve starting at analytically or numerically known linear defect modes. Compared to 1D the dynamics of trapping are harder to fully analyze and the existence of many defect modes for a given defect potential causes that slow solitons store a part of their energy for virtually all of the studied attractive defects.

Analysis of femtosecond optical pulse propagation in one-dimensional nonlinear photonic crystals using finite-difference time-domain method

In this paper, we have used the finite-difference time-domain (FDTD) method to analyze the optical pulse propagation in a nonlinear, one-dimensional photonic crystal (1DPC). Hyperbolic secant pulses with various carrier wavelengths are utilized in this study. In a nonlinear regime, a 1DPC introduces a photonic band-gap whose central wavelength and width depend on the input pulse intensity. In the present work, three different cases are considered. These correspond to the carrier wavelengths of the incident pulses being out of, near to, and partially in the band-gap. For each case, the effect of nonlinearity on pulse propagation is investigated. Also, we have analyzed the two-frequency regime, in which each of the two pulses has a different carrier frequency (wavelength). This kind of study can be done directly with FDTD without any further computational burden but it is somewhat complicated using nonlinear coupled-mode equations (NLCME) and nonlinear Schrödinger equation (NLSE), which require separate treatments for each carrier wavelength.

Dispersive destabilization of nonlinear light propagation in fiber Bragg gratings

The effect of retaining the material dispersion terms in the nonlinear coupled mode equations (NLCME) that describe light propagation in fiber Bragg gratings is analyzed. It is found that dispersion is responsible for new instabilities of the uniform states and gives rise to new complex spatio-temporal dynamics that is not captured by the standard NLCME formulation. A detailed analysis of the effect of dispersion on the linear stability characteristics of the uniform solutions is presented and some numerical integrations of the NLCME with dispersion are also performed in order to corroborate the theoretical results.

Bright and dark Bragg solitons in a fiber Bragg grating

We investigate the nonlinear pulse propagation through the fiber Bragg grating structure where the pulse dynamics are governed by the nonlinear-coupled mode (NLCM) equations. Using multiple scale analysis, we reduce the NLCM equations into the perturbed nonlinear Schrodinger (PNLS) type equation. To construct the bright and dark Bragg solitons in the upper and lower branches of the dispersion curve, we solve the PNLS equation using the coupled amplitude-phase method.

Symmetry-breaking instability in gap soliton

We derive the existence of bright gap soliton solution for nonlinear-coupled mode equations, which governs the pulse propagation in fiber Bragg grating. We find the occurrence of symmetry-breaking instability in gap soliton for arbitrary values of detuning parameter. We confirm this instability by linear stability analysis, potential well plots and phase-plane diagrams.

Numerical investigation of a self-induced transparency soliton in a nonlinear photonic bandgap structure doped uniformly with two-level atoms

We numerically study the characteristics of a nonlinear pulse that propagates through a one-dimensional photonic bandgap (PBG) structure uniformly doped with two-level atoms. The numerical model adopted is the coupled system of nonlinear coupled-mode equations and atomic Bloch equations. The simulation results show that a self-induced transparency (SIT) soliton and a gap soliton can coexist in a nonlinear PBG structure uniformly doped with resonant atoms. Although this mixed state, known as a SIT-gap soliton, near the PBG edge has been theoretically predicted, we numerically show that such a solitary wave still exists even if its central frequency is located deep inside the PBG. The propagating characteristics of the SIT-gap soliton are also discussed.

Simple analytical expression for the effect of initial interaction in - cerenkov second-harmonic generation

We analyze the nonlinear coupled-mode equations describing the Cerenkov second-harmonic generation in a nonlinear dielectric waveguide. An approximate analytic solution in a closed form is newly obtained for the pump depletion and nonlinear phase-shift of the fundamental field. Our solution, valid over the whole length, shows that the conversion efficiency is reduced by the feature of the initial process of the nonlinear interaction. It gives a simple expression for the efficiency with better accuracy for short device length. We have compared our solution with the previous analytic solutions derived by others. The validity of our solution is also checked by the nonlinear beam propagation method. Furthermore, the solution can be expressed in terms of a leaky mode phase-mismatch, and it shows explicitly that a resonant pump depletion and nonlinear phase-shift of the fundamental field occurs when the phase-matching to the lowest order leaky mode is accomplished. Extending our analysis method, we have also performed higher order numerical analysis for the nonlinear coupled-mode equations.

Nonlinear switching of optical pulses in fiber bragg gratings

We study numerically the nonlinear switching characteristics of optical pulses transmitted though fiber Bragg gratings. We consider both the uniform and phase-shifted gratings and compare their performance as a nonlinear switch. The nonlinear coupled-mode equations were solved numerically to obtain the pulse-switching characteristics. The steady-state behavior known to occur for continuous-wave optical beams is realized only for pulses wider than 10 ns with long tails. For pulsewidths in the range 0.1-1 ns, the use of phase-shifted gratings reduces the switching threshold, but the on-off contrast is generally better for uniform gratings. We also quantify the effects of rise and fall times associated with an optical pulse on nonlinear switching by considering the Gaussian pulses with smooth tails and nearly rectangular pulses with sharp leading and trailing edges.

Dynamics of nonlinear pulse propagation through a fiber Bragg grating with linear coupling

We consider nonlinear pulse propagation through a fiber Bragg grating with Kerr nonlinearity and linear coupling effects. Using the Stokes parameters, we obtain a set of four equations from the nonlinear coupled-mode equations. From these equations we find the relation for energy density of a gap soliton and construct a quartic anharmonic oscillator equation after finding the appropriate conserved quantities. With the help of the anharmonic oscillator equation we investigate the bright gap soliton solution and as a special case we also obtain another bright gap soliton solution that resembles the solutions of two coupled nonlinear equations for the envelopes of the fundamental and the second-harmonic field components. Moreover, we analyze the motion of the photon in two different potential wells. Finally, we carry out a linear stability analysis to study the stability of the system.

Hamiltonian formulation of the nonlinear coupled mode equations.

We derive a canonical Hamiltonian formulation of the nonlinear coupled mode equations (CME) that govern the dynamics of pulse propagation in a one-dimensional, periodic Kerr medium when the frequency content of the pulse is in the vicinity of a photonic band gap, and sufficiently narrow relative to a carrier frequency. The Hamiltonian is equal to the energy in the electromagnetic field. We show that even for large photonic band gaps (25% of the Bragg frequency), the CME give an excellent approximation to the dispersion relation of the linear, periodic medium. This suggests that two- and three-dimensional photonic band-gap materials, which necessarily have large index contrasts, might be effectively described by a set of generalized CME.

Rigorous derivation of coupled mode equations for short, high-intensity grating-coupled, co-propagating pulses

We present a rigorous derivation of coupled mode equations for the grating-assisted coupling of short, co-propagating pulses. Since this is a copropagating geometry, the required grating period is much longer than the wavelength of the light and is typically 0.1–10 mm. This can exceed the spatial extent of the pulses substantially, and the validity of the usual assumptions that lead to coupled mode theory is therefore not guaranteed. Here we show rigorously that the standard nonlinear coupled mode equations can be used to describe these experiments and we give expressions for the parameters.