Fourth-order Dispersion Research Articles

Internal motion within pulsating pure-quartic soliton molecules in a fiber laser

Various striking nonlinear evolution dynamics have been observed in mode-locked fiber lasers owing to the high peak power of the solitons. Herein, we numerically demonstrate the dynamical generation of dispersion-dependent pure-quartic solitons (PQS) molecules in a fiber laser. We discover the evolution from a single-pulse initial condition to different types of pulsating PQS molecules with enhanced net fourth-order dispersion, indicating that the internal motion and energy exchange can be both quasi-periodical and periodical. Furthermore, we reveal that the generation of these two types of pulsating PQS molecules is associated with a gain competition effect between the two sub pulses during mode-locking. Finally, we propose a new method that can achieve the transition between a loose PQS molecule and a tight PQS molecule. Enlightened by the numerical results, we speculate that more internal motion within the PQS molecule will be discovered, which will promote a deep insight into the physical mechanism behind.

Higher order dispersions effect on high-order soliton interactions

The object of the research is deleting the interaction of the higher order soliton interaction by introducing the third and fourth order dispersions inside an optical fiber. The results are obtained by the simulation of the nonlinear Schrödinger equation, which models the propagation of solitons in the optical fiber using the method of Fast Fourier Transform. The interaction of two higher order solitons due to the attraction of their electric field can lead to losing the solitons' properties. Hence, this can prevent the use of solitons in high-bit-rate optical fiber communication systems because it increases the bit error rate, significantly limiting the potential of the communication system. To resolve this problem, we should diminish the bit rate error by avoiding the interaction of the co-propagative solitons when they are too close. It is well known that, during higher order soliton propagation in the presence of the third order dispersion, the irregular shape of the higher order soliton disappears, and a splitting towards its fundamental constituents occurs after a considerable propagation. As for the fourth order, dispersion gives rise to two dispersive wave sidebands on the red or blue side. Our results reveal that bringing two higher order solitons together in the presence of the fourth order dispersion, a series of interactions between the components generated after their fission is obtained. In the third-order distribution, besides the fourth-order diffusion, the rare form and the supercontinuum generated by the fission of the higher-order solitons disappear, and we get two fundamental solitons propagating in parallel with a temporal shift and some inconsiderable dispersive waves. The most important aspect is that both higher-order dispersions are able to suppress the interactions of higher-order solitons thanks to the time shift induced by the third-order distribution and the intermittent compression caused by the fourth-order scattering. These results can be obtained in practice inside the dispersion-engineered photonic crystal waveguide (PhC-wg), which allows for manipulating the high order dispersion.

Dissipative pure-quartic soliton resonance in an Er-doped fiber laser

Dissipative pure-quartic soliton (DPQS) that arises from the balance of positive fourth-order dispersion (FOD), nonlinearity, gain and loss has been proposed to generate high-energy pulse. Herein, we numerically investigate the generation of dissipative pure-quartic soliton resonance (DPQSR) in an Er-doped mode-locked fiber laser with all positive FOD. Our findings indicate that the formation of DPQSR depends on the reverse saturable absorption (RSA) effect of real saturable absorber. Then, the impacts of pump power, RSA coefficient and positive FOD in generating DPQSR pulse are investigated. With the increase of pulse energy, the asymmetrical temporal profile of DPQS transforms into a rectangular one. This work can be utilized to comprehend the dissipative pure-quartic soliton resonance as well as the development of an all positive FOD Er-doped fiber laser. In addition, it lays the way for the development of highly stable, high-energy pulsed lasers.

Comprehensive analysis of pure-quartic soliton dynamics in a passively mode-locked fiber laser

The understanding of soliton dynamics promotes the development of ultrafast laser technology. High-energy pure-quartic solitons (PQSs) have gradually become a hotspot in recent years. Herein, we numerically study the influence of the gain bandwidth, saturation power, small-signal gain, and output coupler on PQS dynamics in passively mode-locked fiber lasers. The results show that the above four parameters can affect PQS dynamics. Pulsating PQSs occur as we alter the other three parameters when the gain bandwidth is 50 nm. Meanwhile, PQSs evolve from pulsating to erupting and then to splitting as the other three parameters are altered when the gain bandwidth is 10 nm, which can be attributed to the existence of the spectral filtering effect and intra-cavity fourth-order dispersion. These findings provide new insights into PQS dynamics in passively mode-locked fiber lasers.

Extraction of new optical solitons in presence of fourth-order dispersion and cubic-quintic nonlinearity

Extraction of new optical solitons in presence of fourth-order dispersion and cubic-quintic nonlinearity

Bright and dark optical modulated soliton solutions for the fourth-order (2+1)-dimensional Schrödinger equation with higher-order odd and even terms

This work addresses a (2+1)-dimensional nonlinear Schrödinger (NLS) equation (NLSE) with fourth-order nonlinearity and dispersion. Some different solutions in the form of bright and dark optical modulated solitons (OMSs) to the current problem are derived using different approaches. Moreover, based on some certain conditions for the related parameters, some, bright, dark, periodic, exponential, and singular modulated solutions are obtained. The derived solutions can be helpful in understanding the bright and dark OMSs features of identical models in optical fields, fluid mechanics, and plasmas.

Pure–quartic optical solitons and modulational instability analysis with cubic–quintic nonlinearity

We analyze the propagation dynamics of intense light pulses in an optical medium exhibiting cubic–quintic nonlinearity and pure fourth-order dispersion. We show that this physical system may support a rich variety of periodic wave solutions in the presence of all physical processes. Interestingly, in the long-wave limit, two different types of quartic dark solitons with equal amplitudes and wave numbers but different widths are identified for the first time. The obtained dark soliton solutions are shown to be independent of any free parameters and their characteristics are uniquely determined by system parameters. The numerical examples are presented for illustrating the soliton evolution dynamics in the waveguiding system. The formation of these quartic dark solitons occurs in normal pure fourth-order dispersion when the quintic and cubic nonlinearities are competitive. The modulational instability of a continuous wave is also analyzed under the combined effects of quartic dispersion and cubic–quintic nonlinearities. Additionally, the role of quintic nonlinearity and fourth-order dispersion effects on the modulational instability has been discussed.

Radial and non-radial multiple solutions to a general mixed dispersion NLS equation

We study the following nonlinear Schrödinger equation with a fourth-order dispersion term Δ2u−βΔu=g(u)in RN in the positive and zero mass regimes: in the former, and , where m > 0 depends on g; in the latter, and β > 0. In either regimes, we find an infinite sequence of solutions under rather generic assumptions about g; if N = 2 in the positive mass case, or N = 4 in the zero mass case, we need to strengthen such assumptions. Our approach is variational.

Creeping and erupting dynamics in a pure-quartic soliton fiber laser.

Pure-quartic solitons (PQSs) are gradually becoming a hotspot in recent years due to their potential advantage to achieve high energy. Meanwhile, the fundamental research of PQSs is still in the fancy stage, and exploring soliton dynamics can promote the development of PQSs. Herein, we comprehensively and numerically investigate the impact of saturation power, small-signal gain, and output coupler on PQS dynamics in passively mode-locked fiber lasers. The result indicates that altering the above parameters makes PQSs exhibit pulsating or creeping dynamics similar to traditional solitons. Moreover, introducing an intra-cavity filter combined with intra-cavity large fourth-order dispersion makes PQSs go through stationary, pulsating to erupting. That is, the intra-cavity filter changes PQS dynamics. These findings provide new insights into PQS dynamics in fiber lasers.

Research progress of pure quartic soliton fiber laser

The pure-quartic soliton fiber laser is an innovative ultra-short pulse laser that can maintain a stable pulse shape through a balance between fourth-order dispersion effect and self-phase modulation effect. Comparing with traditional soliton laser that is dominated by second-order dispersion, the mode-locked pulse energy of pure-quartic soliton laser can be 1–2 orders of magnitude higher. This provides researchers with new ideas for developing high-energy and high-peak-power fiber lasers. Here, the generation and transmission characteristics of pure-quartic solitons in nonlinear optical systems such as fiber lasers in recent years are systematically reviewed. It also explores some special transient dynamic phenomena. Furthermore, in this article, the latest achievements of our research group in this area are also presented. Finally, the application prospect and development trend of pure-quartic soliton fiber lasers are prospected. These results will contribute to a more comprehensive understanding of the basic physical properties of pure-quartic soliton fiber lasers.

Pure-quartic solitons in presence of weak nonlocality

In this paper, we study the characteristics of pure-quartic optical solitons in a nonlinear medium having a weakly nonlocal response. We propose an nonlinear Schrödinger equation incorporating pure fourth-order diffraction, Kerr nonlinearity and weak nonlocality to describe the transmission of solitons in the system. We obtain analytic pure-quartic soliton solutions to this model in a variety of shapes and forms, unlike the case of the model without taking into account the weak nonlocal contribution. Besides, we find that the obtained localized structures have both bright and dark type waveforms in the transmission system. Interestingly, the functional forms of these solitons are expressed in terms of sech and tanh functions which do not possess oscillating tails, markedly different from the conventional pure-quartic solitons in the absence of weak nonlocality. These structures exist for both positive and negative quartic diffraction coefficient whereas pure-quartic solitons in the absence of nonlocality occur only for negative fourth-order dispersion. The numerical examples are presented for illustrating the soliton evolution dynamics in the optical waveguide.

Cherenkov radiation emitted by Kuznetsov–Ma solitons

Cherenkov radiation emitted by Kuznetsov–Ma soliton (KMS) with an arbitrary propagation constant in the presence of higher-order dispersions is studied analytically and numerically. We show that the third-order dispersion (TOD) yields asymmetric radiated bands, while the fourth-order dispersion (FOD) gives rise to symmetric radiated bands only when the value of FOD is positive. In contrast to radiations emitted by other localized waves, such a radiation emerges periodically in propagation, and can exhibit multi-frequency bands which depends strongly on the propagation constant of the KMS. We presented radiation conditions to calculate different frequency bands, which shows great agreement with numerical simulations. Important radiation features such as radiation frequencies, velocities, and distances are shown in phase diagrams. Our results could be helpful for controllable radiations in nonlinear fiber and other nonlinear systems.

Periodic and localized waves in parabolic-law media with third- and fourth-order dispersions.

Considering the higher-order nonlinearity is essential in a broad range of real physical media as it significantly influences the wave dynamics in these systems. We study the propagation of femtosecond light pulses inside an optical fiber medium exhibiting higher-order dispersion and cubic-quintic nonlinearities. Pulse evolution in such a system is governed by a higher-order nonlinear Schrödinger equationincorporating second-, third-, and fourth-order dispersions as well as cubic and quintic nonlinearities. The periodic and solitary wave solutions are identified using the equationmethod. Results presented indicated the potentially rich set of periodic waves in the system under the combined influence of higher-order dispersive effects and cubic-quintic nonlinearity. The velocity of these structures is uniquely dependent on all orders of dispersion. Conditions on the optical fiber parameters for the existence of these exact stable solutions are found by analytical stability analysis.

Switching dynamics of femtosecond solitons in parity-time-symmetric coupled optical waveguides

We report a detailed study on soliton steering dynamics in a parity-time-symmetric directional coupler in the femtosecond domain, which requires incorporation of higher-order perturbative effects such as third-order and fourth-order dispersions, self-steepening, and intrapulse Raman scattering. With a high gain/loss, the combination of all these effects is found to stabilize the soliton pulse evolution in the coupler from the chaotic behavior of unperturbed evolution. This work demonstrates that efficient soliton steering can be achieved at very low critical power and a relatively higher gain/loss even in the femtosecond regime.

Obtaining optical soliton solutions of the cubic–quartic Fokas–Lenells equation via three different analytical methods

In this study, we have focused on finding soliton solutions of the cubic–quartic Fokas–Lenells equation, which models the nonlinear pulse transmission through optical fiber, is a pretty new and updated model. The main motivation of this study is to produce new solutions with previously unused methods for data transmission models in fiber optic cables together with the developing technology, which has been used very frequently today. We have used three different efficient analytical methods, namely, the Sinh–Gordon expansion method, enhanced modified extended tanh expansion method and unified Riccati equation expansion method. By modifying our proposed method and using it effectively, we have obtained much more optical solutions. We have supported the results with the 3D surface, 2D and contour plots of the soliton solutions, such as singular, periodic, periodic-singular, dark and M-shaped solitons. The originality of the method we propose is that it has never been used before. With the innovation of the method, we have obtained M-shaped solitons, which are very rarely obtainable in soliton waves. On the other hand, enhanced modified extended tanh expansion method have provided us more extra solutions. As the last and most important part, in order to examine physical behaviors, we have investigated the effect of the coefficient of the fourth order dispersion parameter on wave propagation by presenting the graphical representation.

Dipole and quadrupole nonparaxial solitary waves.

The cubic nonlinear Helmholtz equation with third and fourth order dispersion and non-Kerr nonlinearity, such as the self steepening and the self frequency shift, is considered. This model describes nonparaxial ultrashort pulse propagation in an optical medium in the presence of spatial dispersion originating from the failure of slowly varying envelope approximation. We show that this system admits periodic (elliptic) solitary waves with a dipole structure within a period and also a transition from a dipole to quadrupole structure within a period depending on the value of the modulus parameter of a Jacobi elliptic function. The parametric conditions to be satisfied for the existence of these solutions are given. The effect of the nonparaxial parameter on physical quantities, such as amplitude, pulse width, and speed of the solitary waves, is examined. It is found that by adjusting the nonparaxial parameter, the speed of solitary waves can be decelerated. The stability and robustness of the solitary waves are discussed numerically.

Existence of Blow-up Solutions to Nonlinear Schrödinger Equations with Anisotropic Fourth-Order Dispersion

We consider the Cauchy problem for nonlinear Schrödinger equations with fourth-order anisotropic dispersion. In this paper, we show the existence of blow-up solutions in the mass-supercritical case and in the mass-critical case.

Analytical soliton solutions of the higher order cubic-quintic nonlinear Schrödinger equation and the influence of the model’s parameters

In this paper, we present the higher-order nonlinear Schrödinger equation (NLSE) with third order dispersion (3OD), fourth-order dispersion (4OD), and cubic-quintic nonlinearity (CQNL) terms that define the propagation of ultrashort pulses. Two analytical methods, which are the new Kudryashov’s method and the unified Riccati equation expansion method, are implemented to extract the analytical soliton solutions of the presented equation for the first time. Thus, bright, dark, and singular soliton solutions are acquired. To illustrate the physical behavior of some of the obtained solutions, 3D, 2D, and contour graphs are depicted. In particular, to understand the effects of the group velocity dispersion, 3OD, 4OD, CQNLs, self-steepening coefficient terms, and group velocity term of the traveling wave transformation on the soliton dynamics of the proposed equation, 2D plots for different values of coefficients are represented. The obtained results provide us with the knowledge that the presented model can be examined from a physical perspective. It can be concluded that the used methods are effective approaches to derive the analytical solutions for the NLSE.

Stabilisation of higher-order solitons by the compensation of quintic nonlinearity, fourth-order dispersion and intrapulse Raman scattering in optical fibres

Stabilisation of higher-order solitons by the compensation of quintic nonlinearity, fourth-order dispersion and intrapulse Raman scattering in optical fibres

The Bilinear Formula in Soliton Theory of Optical Fibers

Solitons are wave phenomena or pulses that can maintain their shape stability when propagating in a medium. In optical fibers, they become general solutions of the Non-Linear Schrödinger Equation (NLSE). Despite its mathematical complexity, NLSE has been an interesting issue. Soliton analysis and mathematical techniques to solve problems of the equation keep doing. Yan & Chen (2022) introduced them based on bilinear formula for the case of the generalized NLSE extended models into third and fourth-order dispersions and cubic-quintic nonlinearity. In this paper, we review the form of the bilinear formula for the case. We re-observed a one-soliton solution based on the formula and verified the work of the last researcher. Here, the mathematical parameters of position α(0) and phase η are verified to become features of change in horizontal position and phase of one soliton in the (z, t) plane during propagation. In addition, we notice the soliton has established stability. Finally, for the condition Kerr effect focusing or the group velocity dispersion β2 more dominates, we present like the soliton trains in optical fibers under modulation instability of plane wave.