Higher-order Dispersion Effects Research Articles

A compact split step Padé scheme for higher-order nonlinear Schrödinger equation (HNLS) with power law nonlinearity and fourth order dispersion

In this paper we propose a compact split step Padé scheme (CSSPS) to solve the scalar higher-order nonlinear Schrödinger equation (HNLS) with higher-order linear and nonlinear effects such as the third and fourth order dispersion effects, Kerr dispersion, stimulated Raman scattering and power law nonlinearity. The stability of this method has been proved. It has been shown as well that the CSSPS method gives the same results as classical numerical methods like the split step Fourier method and Crank–Nicholson (CN) method but it presents many advantages over theme. It is more efficient. This proposed scheme is well suited to higher-order dispersion effects and readily generalized for nonlinear and dispersion managed fibers. We tested this scheme for the case of the quintic nonlinearity and confirmed that this effect has no significant role on the propagation of single solitons.

High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities

Near a zero-dispersion wavelength, high-order dispersion effects play a central role in a photonic crystal fiber cavity. The study of such effects reveals the existence of stable dissipative dark solitons in the anomalous dispersion regime. Such dark localized structures are not present without high-order dispersion. They consist of dips in the profile of the intensity field. The number of dips and their temporal distribution are determined by the initial conditions. A snaking bifurcation diagram associated with these solutions is constructed.

Effect of intensity dependent higher-order dispersion on femtosecond pulse propagation in quantum well waveguides

We have studied theoretically the dispersion of ultrafast coherent pulses in GaAs/AlGaAs quantum well waveguide structures as a function of optical intensity. Semiconductor Bloch equations are used to obtain the polarization induced in the medium due to an incident Gaussian electromagnetic beam. The partial differential equation describing the pulse propagation in the presence of group velocity dispersion is used to analyze the role of higher-order dispersion on femtosecond pulse propagation in the waveguide. Due consideration has been given to the intensity dependent optical susceptibility of the medium. The results of the numerical analysis manifest significant influence of higher-order dispersion on femtosecond pulse propagation over short waveguide distance.

Effects of higher-order dispersion on resonant dispersive waves emitted by solitons

Dispersive waves (DW) are generated owing to perturbation of solitons by higher-order dispersion (HOD) and nonlinearity during supercontinuum (SC) generation. The frequencies of these waves are governed by a phase-matching condition in the form of a polynomial whose coefficients depend on the numerical values of the properly normalized third- and HOD parameters. Our extensive numerical solutions show that all odd HOD terms generate a single peak on the blue or the red side of the carrier frequency, depending on the sign of the corresponding term. In contrast, positive even HOD terms create conjugate DW peaks, in both the blue and red sides. No radiation is observed for negative values of these parameters. The combination of all even and odd HOD coefficients may generate more than two DW peaks for some specific choice of parameters. The results predicted by the phase-matching condition agree well with extensive numerical simulations revealing interesting facts of SC generation.

Thermalization of incoherent nonlinear waves

The subject of incoherent nonlinear optics received a renewed interest since the first experimental demonstration of incoherent solitons in slowly responding photorefractive crystals. Several theories have been successfully developed to provide a detailed description of the dynamics of incoherent optical solitons. However, such theories leave unanswered the question regarding the long term evolution of a partially incoherent optical field propagating in a nonlinear medium. In analogy with kinetic gas theory, an incoherent optical field evolves, owing to nonlinearity, towards a thermodynamic equilibrium state. Wave turbulence theory describes the essential properties of this irreversible process of thermal wave relaxation to equilibrium. Such irreversible behavior is expressed by a H-theorem of entropy growth, whose origin is analogous to the celebrated Boltzmann’s H-theorem. In this review we analyze the thermalization of incoherent nonlinear optical fields in various circumstances. We shall begin with the simplest case where the optical field is ruled by the scalar NonLinear Schrodinger (NLS) equation. In this case wave thermalization is characterized by a condensation process of the optical field, whose thermodynamic properties are analogous to those of Bose-Einstein condensation, despite the fact that the considered optical wave is completely classical. We then study the thermalization of an ensemble of incoherent fields governed by the vector NLS equation. The influence of the relative intensity of the coupled fields reveals the existence of a process of coherence absorption: The condensation of the optical field is induced by the presence of another small-amplitude field, which absorbs almost all the noise of the system. Such a coherence absorption effect is shown to also occur in thermal quantum Bose gases. The influence of convection (group-velocity difference) on the thermalization process is also analyzed. A set of incoherent wave-packets is shown to irreversibly evolve towards an equilibrium state in which they propagate with an identical group-velocity. This velocity-locking effect has a thermodynamic origin and thus appears as a generic property of a system of incoherent nonlinear waves. The influence of a coherent coupling on wave thermalization leads to an unexpected process of spontaneous polarization of unpolarized incoherent light, without loss of energy. Finally, the influence of higher-order dispersion effects on optical thermalization is responsible for a dramatic spectral broadening phenomenon known as supercontinuum generation. This reveals that supercontinuum generation may be regarded as a consequence of the natural thermalization of an optical field to equilibrium. We finally show that the thermodynamic properties of incoherent nonlinear waves may be analyzed by means of the fundamental T dS equation of thermodynamics.

Effect of Dispersion on Dispersive Wave Generation in Photonic Crystal Fibers

A theoretical investigation on the nonlinear pulse propagation and the influences of the dispersion effect on the dispersive wave generation in photonic crystal fibers is presented. By simulating the dispersive wave generation under different dispersion property conditions, it is found the origin of the dispersive wave generation is mainly due to the effect of higher-order dispersion and the pulse trapping. In addition, we also found that the intensity of the dispersive wave can be further enhanced by properly modifying the even-order dispersion coefficients to make the pulse trapping developing sufficiently.

Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers

Solitons forming inside optical fibers are perturbed by several higher-order dispersive and nonlinear effects, especially when ultrashort optical pulses are used to excite them. We study, both analytically and numerically, how the radiation emitted by solitons in the form of dispersive waves (sometimes called Cherenkov radiation) is affected by these higher-order effects. Our results show that a certain minimum amount of third-order dispersion is needed before the amplitude of the dispersive wave becomes large enough for a spectral peak to appear in the output spectrum. This minimum value depends on the soliton order $N$ and decreases with increasing $N$. The amplitude of the radiation peak increases initially with both the magnitude of the third-order dispersion and the soliton order, but then saturates to a relative power level that is typically below 10% of the launched power. Our results reveal several interesting features that should be of relevance for applications requiring an ultrabroadband optical source.

A novel dispersion-shifted single mode optical fiber design with ultra-high-bit-rate and very low loss for long-haul communications

In this paper, a novel dispersion-shifted multi-clad optical fiber with very small bending loss and ultra-high bit-rate applicable for large capacity information transmission systems is presented. To decrease dispersion and higher-order dispersion effects at λ = 1.55 μm, a weighted pulse broadening factor and genetic algorithm (GA) optimization technique is used. Compared to the works reported previously, this method can precisely set the zero-dispersion wavelength. This kind of dispersion-shifted fibers has dispersion, dispersion slope, mode field diameter (MFD), effective area and quality factor within −1.40 × 10 −4 to −8.44 × 10 −2 ps/km nm, 3.06 × 10 −2 to −4.07 × 10 −2 ps/km nm 2, 5.56−5.85 μm, 119.25−176.42 μm 2 and 3.49–5.27 at λ = 1.55 μm, respectively. Besides, by applying dispersion at λ = 1.55 μm as the cost function, dispersion of about 1.31 × 10 −8 ps/km nm is obtained. Thus, this novel optical fiber can be used in long-haul high information-transmission capacity communication systems.

The Effect of Higher-Order Corrections on the Propagation of Nonlinear Dust-Acoustic Solitary Waves in a Dusty Plasma with Nonthermal Ions Distribution

Propagation of nonlinear dust-acoustic (DA) waves in a unmagnetized collisionless mesospheric dusty plasma containing positively and negatively charged dust grains and nonthermal ion distributions are investigated. For nonlinear DA waves, a reductive perturbation method is employed to obtain a Korteweg-de Vries (KdV) equation for the first-order potential. As it is well-known, KdV equations contain the lowest-order nonlinearity and dispersion, and consequently can be adopted for only small amplitudes. As the wave amplitude increases, the width and velocity of a soliton can not be described within the framework of KdV equations. So, we extend our analysis and take higher-order nonlinear and dispersion terms into account to clarify the essential effects of higher-order corrections. Moreover, in order to study the effects of higher-order nonlinearity and dispersion on the output solution, we address an appropriate technique, namely the renormalization method.

Exact solutions of the hierarchical Korteweg–de Vries equation of microstructured granular materials

The problems under consideration are related to wave propagation in microstructured materials, characterized by higher-order nonlinear and higher-order dispersive effects; particularly, the wave propagation in dilatant granular materials. In the present paper the model equation is solved analytically by exact methods. The types of solutions are defined and discussed over a wide range of material parameters (two dispersion parameters and one microstructure parameter). The dispersion properties and the relation between group and phase velocities of the model equation are studied. The diagrams are drawn to illustrate the physical properties of the exact solutions.

Effects of higher-order dispersion on pulse splitting in the normal dispersion regime

By solving numerically the extended nonlinear Schrodinger equation we investigate the influence of higher-order dispersion effects on the propagation of optical pulses in the normal dispersion regime in a highly nonlinear fiber. Already a small amount of third-order dispersion can lead to a pulse-breakup above a certain pulse power. The splitting is followed by an expansion of the spectrum towards longer wavelengths without any impact of Raman scattering. The transfer of energy to longer wavelengths strongly depends on the dispersion profile of the fiber.

High-order dispersion in chirped-pulse oscillators

The effects of high-order dispersion on a chirped-pulse oscillator operating in the positive dispersion regime were studied both theoretically and experimentally. It was found that odd and negative even high-order dispersions impair the oscillator stability owing to resonance with the dispersion waves, but can broaden the spectrum as in the case of continuum generation in the fibers. Positive fourth-order dispersion enhances the stability and shifts the stability range into negative dispersion. The destabilization mechanism was found to be a parametrical instability which causes noisy mode locking around zero dispersion.

Pulse dynamics in inhomogeneous optically coupled amplifying optical fibers with high-order dispersion effects taken into account

The evolution of the envelope of a two-mode wave packet in an inhomogeneous-over-length amplifying optical fiber is analyzed with the third-order dispersion effects taken into account, the presence of which leads to modulations of one or both edges of the wave packet. High degrees of compression of the pulse and the complete suppression of oscillations of its time envelope can be attained by an appropriate choice of the type of inhomogeneity of the dispersion parameters.

Induced modulational instability in EDFAs in the presence of higher-order nonlinear and dispersive effects

We derive the dispersion equation for erbium doped fiber amplifiers (EDFAs), assuming the supplementary effects of time-derivative nonlinearities including intrapulse Raman scattering and self-steepening incorporated with third order dispersion and product of gain and square of dipole relaxation time parameter, in the extended form of nonlinear Schrodinger equation and investigate their impacts on gain and cut-off frequency of modulational instability (MI) in EDFAs. Also, the influences of those effects on instability of steady-state amplitude, in the form of generation of terahertz pulse train of femtosecond solitons from a weak-modulated signal by induced MI, are numerically simulated and discussed within EDFAs in normalized form.

Critical ratio between the amplitudes of two overtaking solitary water waves

In this paper, we study the critical ratio between the amplitudes of two overtaking solitary waves on a layer of water with uniform depth. At the center of encounter, the wave profile is fore-and-aft symmetric, but it could have a single peak or double peaks. The critical ratio separates these two regimes. At the critical point, the wave peak is flat with zero slope and curvature. We solve the full water wave problem numerically by using a fully nonlinear and highly dispersive Boussinesq model. The model is numerically justified to be a good approximation of the Euler equations for solitary waves with very large amplitude. For small amplitude water waves, our calculated critical ratio reduces to the well-known result of 3 predicted by the Korteweg-de Vries equation, a weakly nonlinear and weakly dispersive model. For large amplitude water waves, the nonlinear effect is significant; we find that the critical ratio deviates significantly from 3. For water waves with very high amplitude, e.g., 0.6 relative wave height, the critical ratio could be as large as 4. Our result suggests that higher-order nonlinear and dispersive effects are important when modeling the strong interaction between large amplitude water waves.

STUDY OF PULSE EVOLUTION AND OPTICAL BISTABILITY UNDER THE INFLUENCE OF CUBIC-QUINTIC NONLINEARITY AND THIRD ORDER DISPERSION

Using the variational method, we examine the evolution of different parameters to characterize an optical pulse propagating in a non-Kerr type medium having cubic-quintic nonlinearity in the vicinity of zero dispersion wavelength. The validity of the derived results is tested by applying different ansatz functions over a critical propagation distance. Furthermore, the results are compared with numerically simulated data generated through the Split Step Fourier beam propagation method, solving the dynamic nonlinear cubic-quintic Schrödinger equation (NLCQSE). Subsequently, the effect of higher order dispersion on optical bistability under cubic-quintic nonlinearity is studied in the framework of our proposed model.

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.

Effect of third-order dispersion on subpicosecond pulse propagation in photonic-crystal waveguides

We have experimentally investigated the effects of dispersion on subpicosecond pulse propagation over two-dimensional (2D) photonic-crystal-slab line-defect waveguides by time domain measurements. Using a heterodyne detection technique, pulse shapes after propagating along the waveguide were recorded as cross-correlated traces. When the pulse central frequency approached the band edge of the guided mode, the pulses were temporally delayed and their shapes were found to be broadened asymmetrically; this effect originated from the group velocity dispersion (GVD), encompassing higher-order dispersion effects. Theoretical pulse shapes, including third-order dispersion (TOD), agreed well with the experimental results. When the group velocity (vg) reached 0.08c, c being the speed of light, we found a GVD of ∼5.0ps2∕mm and a TOD of ∼0.3ps3∕mm. Experimentally obtained vg, GVD, and TOD characteristics agreed well with those obtained from 2D plane-wave-expansion band calculations.

Interplay of nonlinearity and gain shaping in femtosecond fiber amplifiers

A study of chirped-pulse amplification in the presence of large nonlinear phase shifts (as large as approximately 12pi) and finite gain bandwidth is presented. Numerical simulations that include the effect of nonlinearity, group-velocity dispersion, higher-order dispersion, and finite gain bandwidth predict the spectral signature of the interplay of nonlinearity with gain shaping. Experimental results obtained for up to approximately 0.4 microJ pulse energies from a Yb fiber amplifier agree with the numerical calculations.

Analytical studies of the steady solution for optical soliton equation with higher-order dispersion and nonlinear effects

The exact analytical solution of the optical soliton equation with higher-order dispersion and nonlinear effects has been obtained by the method of separating variables. The new type of optical solitary wave solution, which is quite different from the bright and dark soliton solutions, has been found under two special cases. The stability of the solitary wave solutions for the optical soliton equation is discussed. Some new conclusion of the stability are obtained, for the solitary wave solutions of the nonlinear wave equations, by using the Liapunov direct method.