MI Gain Research Articles

Effects of residual nonlinearities on the modulational instability of three-component Bose–Einstein condensates

We study the modulation instability in a three-component Bose–Einstein condensate and the impact of higher-order interactions brought on by shape-dependent confinement. Using the modified Gross–Pitaevskii equation that accounts for the higher-order residual nonlinearity and the linear stability method, we analyze the dispersion relations and the gain spectrum for instability. In particular, we investigate the role of higher-order interactions over the MI in miscible and immiscible condensates. The observed MI spectrum shows a new regime for soliton generation in a ternary mixture of Bose–Einstein condensates. Our results show that the residual nonlinearity can effectively control the MI gain profile in both miscible and immiscible condensates. This allows one manipulate the MI in a ternary mixture of Bose–Einstein condensates to control the dynamics.

Comparative analysis between modulation instability in an erbium and non-erbium optical fiber with generalized external potentials

The effects of two-level resonant atoms and self-steepening effect on modulation instability in an erbium fiber is studied by considering the nonlinear Schrödinger Maxwell–Bloch (GINLS–MB) with generalized external potentials. By applying linear stability analysis (LSA), we examine the comparison between MI gain spectra for both with and without erbium effects. We depict certain significant results by graphically illustrating the 2D and 3D figures through suitably selecting the parameters. In addition, we provide a comparison table for adopting parameters. Obtained remarkable results are useful for the study of soliton stability in nonlinear optical communication system. Also, our investigations shows that the gain spectrum mainly influenced by erbium atoms and self-steepening term. Especially, symmetric manner of gain spectrum becomes as asymmetric due to the impact of self-steepening term. The theoretical findings carried out in this work will be beneficial for experimental observation of MI process in an erbium doped fiber.

Modulational stability brought by cubic–quartic interactions of the nearest-neighbor in FK model subjected in a parametrized on-site potential

This work extends to higher-order interactions the results of Nguetcho (2017), in which we discussed only on modulational instability in one-dimensional chain made of atoms, harmonically coupled to their nearest neighbors and subjected to an external on-site potential. Here we investigate the competition between cubic–quartic nonlinearities interactions of the nearest-neighbor and substrate’s deformability, and mainly discuss its impact on the modulational instability of the system. This makes it possible to adapt the theoretical model to a real physical system such as atomic chains or DNA lattices. The governing equation, derived from the modified Frenkel–Kontorova model, is an extended nonlinear Schrödinger equation (eNLS) containing a new higher-order nonlinear term. By employing linear stability analysis, the generic properties of the MI gain spectra of the system are demonstrated. In the presence of the new quartic nonlinearity, the combinations of the system’s parameters open a large variety of gain profiles and instability domains that cannot be explored without the quartic nonlinearity. Direct numerical simulations are performed to support our analytical results, and an excellent agreement is found.

Solitary waves and modulation instability with the influence of fractional derivative order in nonlinear left-handed transmission line

The resolution of the reduced fractional nonlinear Schrödinger equation obtained from the model describing the wave propagation in the left-handed nonlinear transmission line presented by Djidere et al recently, allowed us in this work through the Adomian decomposition method (ADM) to highlight the behavior and to study the propagation process of the dark and bright soliton solutions with the effect of the fractional derivative order as well as the Modulation Instability gain spectrum (MI) in the LHNLTL. By inserting fractional derivatives in the sense of Caputo and in order to structure the approximate soliton solutions of the fractional nonlinear Schrödinger equation reduced, ADM is used. The pipe is obtained from the bright and dark soliton by the fractional derivatives order. By the bias of MI gain spectrum the instability zones occur when the value of the fractional derivative order tends to 1. Furthermore, when the fractional derivative order takes small values, stability zones appear. These results could bring new perspectives in the study of solitary waves in left-handed metamaterials as the memory effect could have a better future for the propagation of modulated waves because in this paper the stabilization of zones of the dark and bright solitons which could be described by a fractional nonlinear Schrödinger equation with small values of fractional derivatives order has been revealed. In addition, the obtained significant results are new and could find applications in many research areas such as in the field of information and communication technologies.

Narrowband and ultra-wideband modulation instability in nonlinear metamaterial waveguides

Waveguides based on metamaterials may exhibit strongly frequency-dependent nonlinearities. In this work, we focus on the phenomenon of modulation instability in this type of waveguide, departing from a new modeling equation that ensures strict conservation of both the energy and photon number of the parametric process. In particular, we analyse the case of a waveguide with a linearly frequency-dependent nonlinear coefficient, revealing unique features such as narrowband and ultra-wideband gain spectra and the suppression of the power cutoff giving rise to an ever-growing MI gain. These markedly distinct regimes are enabled by self-steepening (SS) and manifest themselves depending upon the magnitude and sign of the SS parameter. We believe these findings to be most relevant in the context of mid-IR supercontinuum sources.

Dynamics of optical rogue wave generation in dispersion oscillating fibers.

We present an in-depth investigation of optical rogue waves (ORWs) during picosecond supercontinuum generations in photonic crystal fibers with periodic landscapes of group velocity dispersion and nonlinearity, namely dispersion oscillating fibers (DOFs). Specifically, it is shown via ensembles of numerical simulations that during supercontinuum generations, the rogue nature of extreme and rare events formed in uniform fibers can be effectively manipulated in DOFs. This is also verified by comparing single evolution dynamics in different dispersion longitudinal profiles. For investigating the influence of slow dynamics of ORW generation in DOFs, we increase the propagating distance and find out MI gain is still the major factor that influences the generation of ORWs. In addition, analytical results associated with simulations indicate the rogue manipulations in DOFs are attributed to the adjustable modulation-instability-gain due to periodic dispersion variation along fiber length. Finally, unlike MI in uniform fiber, MI gain side lobes result from quasi-phase-matching (QPM) relation in DOFs provide additional degree of freedom to control generations of ORWs. We believe our results will provide not only a novel insight of understanding ORW dynamics in presence of dispersion modulations, but also a new way of harnessing rogue waves in oceanology.

Chirped soliton solutions of Fokas–Lenells equation with perturbation terms and the effect of spatio-temporal dispersion in the modulational instability analysis

In the present work, exact chirped solitons are derived from the Fokas–Lenells equation with the perturbation terms which describes the ultrashort pulse in the optical fiber. The obtained chirps could help for either pulse compression or amplification in optical fiber. We also analyzed the effect of spatio-temporal dispersion in the modulational instability. As result, the modulational instability analysis shows that the gain is increasing when the spatio-temporal dispersion coefficient decreases; besides, the region of MI gain apparition is well determined.

Spatio-temporal instability in negative index materials

In this paper, we investigate spatiotemporal instability in negative index materials (NIMs) made of split ring resonators and arrays of wires embedded in a Kerr medium. The pulse propagation in NIM is modeled by nonlinear Schrödinger type equation. The linear stability analysis is carried out to obtain the instability gain. We study the influence of electric and magnetic self steepening effects and coupling between electric and magnetic fields on MI gain. We report additional ways to generate solitons or ultrashort pulses in negative index artificial materials.

Optical solitons and modulation instability analysis with (3 + 1)-dimensional nonlinear Shrödinger equation

This paper addresses the (3 + 1)-dimensional nonlinear Shrödinger equation (NLSE) that serves as the model to study the propagation of optical solitons through nonlinear optical fibers. Two integration schemes are employed to study the equation. These are the complex envelope function ansatz and the solitary wave ansatz with Jaccobi elliptic function methods, we present the exact dark, bright and dark-bright or combined optical solitons to the model. The intensity as well as the nonlinear phase shift of the solitons are reported. The modulation instability aspects are discussed using the concept of linear stability analysis. The MI gain is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

Theoretical investigation of incoherently coupled solitons in centrosymmetric photorefractive crystals

A theory on incoherently coupled soliton pairs for photorefractive solitons is developed that gives rise to incoherently coupled soliton pairs and multi-component spatial solitons in biased centrosymmetric photorefractive crystals in all three realizations, bright-bright, dark-dark and grey-grey. The properties and characteristics of these soliton pairs are studied under different conditions like the bias field strength, external and internal resistance, and the intensity of the beams. This theory is extended to incorporate incoherently coupled multicomponent solitons. The stability of the soliton pairs is investigated by the modulation instability theory. A quantitative assessment is made out for the MI gain in all three realizations of the incoherently coupled soliton pairs. These can be established provided the incident beams have the same polarization, wavelength and are mutually incoherent. Relevant examples are also provided to illustrate our results.

Modulation instabilities in randomly birefringent two-mode optical fibers**Project supported by the Natural Science Foundation of Jiangsu Provincial Universities (Grant No. 14KJB140009), the National Natural Science Foundation of China (Grant No. 11447113), and the Startup Foundation for Introducing Talent of NUIST (Grant No. 2241131301064).

Modulation instabilities in the randomly birefringent two-mode optical fibers (RB-TMFs) are analyzed in detail by accounting the effects of the differential mode group delay (DMGD) and group velocity dispersion (GVD) ratio between the two modes, both of which are absent in the randomly birefringent single-mode optical fibers (RB-SMFs). New MI characteristics are found in both normal and anomalous dispersion regimes. For the normal dispersion, without DMGD, no MI exists. With DMGD, a completely new MI band is generated as long as the total power is smaller than a critical total power value, named by Pcr, which increases significantly with the increment of DMGD, and reduces dramatically as GVD ratio and power ratio between the two modes increases. For the anomalous dispersion, there is one MI band without DMGD. In the presence of DMGD, the MI gain is reduced generally. On the other hand, there also exists a critical total power (Pcr), which increases (decreases) distinctly with the increment of DMGD (GVD ratio of the two modes) but varies complicatedly with the power ratio between the two modes. Two MI bands are present for total power smaller than Pcr, and the dominant band can be switched between the low and high frequency bands by adjusting the power ratio between the two modes. The MI analysis in this paper is verified by numerical simulation.

Modulation instability in noninstantaneous Kerr media with walk-off and cross-phase modulation for mixed group-velocity-dispersion regimes

Taking into account relaxing Kerr nonlinearity and walk-off effects, the conditions and gain spectra of cross-phase modulation-induced modulational instability (XPM-MI) of two incoherently copropagating optical waves of different frequencies and same polarization are investigated. We devote particular attention to the mixed case in which one pulse propagates under the normal group-velocity dispersion (GVD) regime, while the second one is under an anomalous GVD regime. We unveil that in the limit of an instantaneuous nonlinear response, the typical frequency with maximum gain converges to a finite value in the mixed GDV regime, while it continuously grows with the group-velocity mismatch in the normal GVD regime. As a result, the maximum gain typically decreases with the group-velocity mismatch in the mixed regime, contrasting with the opposite trend in the normal GVD regime. Further, we show that besides the mode having maximum gain at a frequency decaying with $1/{\ensuremath{\tau}}^{1/3}$ in the slow response limit, there is a second mode having maximum gain with a distinct scaling behavior ${\mathrm{\ensuremath{\Omega}}}_{\text{max}}\ensuremath{\propto}1/\ensuremath{\tau}$ in the absence of group-velocity mismatch. The associated maximum gains scale, respectively, as $1/{\ensuremath{\tau}}^{2/3}$ and $1/\ensuremath{\tau}$, thus signaling the corresponding quadratic and linear dispersion relation of these modes in the low-frequency limit. A detailed analysis of the influence of the nonlinear response time and group-velocity dispersion on the MI gain spectrum is also provided.

Modified Kerr-type saturable nonlinearity effect on the modulational instability of nonlinear coupler with a negative-index metamaterial channel

We consider a nonlinear oppositely direct coupler with a negative-index metamaterial channel and a modified Kerr-type saturable nonlinearity (MSN). We presented an analytical expression for MI gain. Special attention is paid to investigation of the influence of system parameters such as the forward to backward-propagating wave's, the power, nonlinear parameters and nonlinear saturable parameter on the MI. The MSN behaves in a perceptible manner such that the gain and the unstable region inherently decrease. In the normal and anomalous dispersion region, the gain of the instability spectrum increases (decreases) monotonously with increases (decreases) of saturable nonlinearity. In other term, when the saturable parameter or the input power increases, the MI peak gain and the gain band-width may increase and then decrease.

Modulational instability and energy localization of twisted DNA with solvent interaction

We investigate the generation of soliton-like pulses along a DNA chain which takes into account both torsional and solvent interaction effects. Interactions between neighboring base pairs are described by a twist angle. Twisting is essential in the model to capture the importance of nonlinear effects for the thermodynamical properties. The nonlinear dynamics of the DNA is then modeled in the Hamiltonian approach by the generalized Dauxois–Peyrard–Bishop model (combination of several models). We introduce the generalized discrete nonlinear Schrödinger equation describing the dynamics of modulated wave through the twisted DNA with solvent interaction. The modulational instability is studied and we present an analytical expression for the MI gain to show the effects of twist angle on MI gain spectra as well as on stability diagram. With the increase of the twist angle the MI gain decreases then increases. Some interesting MI phenomena appear with an additional new MI region as the twist angle increases. The instability and stability diagrams are also affected. Numerical simulations are carried out to show the validity of the analytical approach. The result is that the initial wave breaks into a train of ultrashort pulses with repetition rates, which are trapped in some sites. The impact of the twist angle is investigated and we obtain that the twist angle affects the dynamics of stable patterns generated through the molecule. Thereafter, we study energy localization in the framework of twisted DNA with solvent interaction. While the twist angle leads to a stronger localization of energy, the solvent interaction delocalizes energy along the molecule.

Effects of higher order nonlinearities on modulational instability in nonlinear oppositely directed coupler

We are motivated by recent studies in medium formed by two tunnel-coupled waveguides. One of the waveguides is manufactured from an ordinary dielectric, while the second has negative refraction. We present an investigation of the gain spectrum permitting modulation instability in the nonlinear optical coupler with a negative-index metamaterial channel whose non-linear response includes third- and fifth-order terms. The principal motivation for our analysis stems from the impact of the inevitable presence of the effective cubic–quintic nonlinearity. We emphasize the influence of higher order nonlinear terms, over the MI phenomena, and the outcome of its development achieved by using linear stability analysis. Gain spectrum investigation has been carried out for both anomalous and normal dispersion regime in the focusing and defocusing cases of nonlinearity and near-zero dispersion regime where higher order linear dispersive effects emerge. Our results show that the MI gain spectra consist of multiple spectral region which are symmetric to the zero point. Moreover, some spectra have a high cut-off frequency but a narrow spectral width, which is obviously beneficial to the generation of high-repetition-rate pulse trains.