Continuous Wave Background Research Articles

Kerr-lens mode locking of a diode-pumped Yb:KGW laser using an additional intracavity Kerr medium

Kerr-lens mode-locked operation of a Yb:KGW laser pumped by a multimode laser diode with the help of an additional Kerr medium is reported. With an enhanced Kerr lensing effect in the laser cavity operating far from the stability edge limit, the laser delivered 74 fs pulses with an average output power of 500 mW. In the high power regime of operation (>500 mW), the mode-locked laser output was accompanied by a continuous wave background radiation. Owing to the self-phase modulation effect, the laser demonstrated the generation of shorter pulses as compared to the case without an additional Kerr medium.

Nonlinear Excitation and State Transition of Multi-Peak Solitons**Supported by the National Natural Science Foundation of China under Grant Nos 11705145 and 11475135, the Guangxi Provincial Education Department Research Project of China under Grant No 2017KY0776, the Scientific Research Program Funded by Shaanxi Provincial Education Department under Grant No 17JK0767, and the Major Basic Research Program of Natural Science of Shaanxi

We study the nonlinear excitations in the integrable fifth-order nonlinear Schrödinger equation on a continuous wave background. The excited condition of each localized wave is demonstrated via concise phase diagrams. In particular, the rule of transition between asymmetric and symmetric multi-peak solitons is revealed. It is shown that the initial phase modulation can induce the transition and the transition condition is demonstrated exactly. Interestingly, our result shows that although the multi-peak solitons exhibit structural diversity, both the asymmetric and symmetric states possess an identical asymmetric spectrum structure.

Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schrödinger equation

We study the linear stability analysis and nonlinear excitations on a continuous wave background for the fifth-order nonlinear Schrödinger equation. We find that there are a modulation stability (MS) quasi-elliptic ring and a MS curve in the modulation instability (MI) regime. The concrete expressions of different types of nonlinear waves are obtained by using Darboux transformation. And we present the relations and phase diagram between MI and these waves. We discover that the antidark (AD) soliton and nonrational W-shaped soliton can only exist on the resonance line of outside of the MS quasi-elliptic ring and the right of the MS curve. We perform numerical simulation to test the stability of the AD soliton. By introducing the perturbation energy, we further show that solitons in the MI regime are caused by both the fourth- and fifth-order effects.

Phased-locked two-color single soliton microcombs in dispersion-engineered Si3N4 resonators.

We propose and theoretically investigate a dispersion-engineered Si3N4 microring resonator, based on a cross section containing a partially-etched trench, that supports phase-locked, two-color soliton microcomb states. These soliton states consist of a single circulating intracavity pulse with a modulated envelope that sits on a continuous wave background. Such temporal waveforms produce a frequency comb whose spectrum is spread over two widely-spaced spectral windows, each exhibiting a squared hyperbolic secant envelope, with the two windows phase-locked to each other via Cherenkov radiation. The first spectral window is centered near the 1550nm pump, while the second spectral window is tailored based on straightforward geometric control, and can be centered as short as 750nm and as long as 3000nm. We numerically analyze the robustness of the design to parameter variation, and consider its implications to self-referencing and visible wavelength comb generation.

Stabilization of the Peregrine soliton and Kuznetsov–Ma breathers by means of nonlinearity and dispersion management

We demonstrate a possibility to make rogue waves (RWs) in the form of the Peregrine soliton (PS) and Kuznetsov–Ma breathers (KMBs) effectively stable objects, with the help of properly defined dispersion or nonlinearity management applied to the continuous-wave (CW) background supporting the RWs. In particular, it is found that either management scheme, if applied along the longitudinal coordinate, making the underlying nonlinear Schrödinger equation (NLSE) self-defocusing in the course of disappearance of the PS, indeed stabilizes the global solution with respect to the modulational instability of the background. In the process, additional excitations are generated, namely, dispersive shock waves and, in some cases, also a pair of slowly separating dark solitons. Further, the nonlinearity-management format, which makes the NLSE defocusing outside of a finite domain in the transverse direction, enables the stabilization of the KMBs, in the form of confined oscillating states. On the other hand, a nonlinearity-management format applied periodically along the propagation direction, creates expanding patterns featuring multiplication of KMBs through their cascading fission.

Black and gray soliton interactions and cascade compression in the variable coefficient nonlinear Schrödinger equation

We present the theoretical study of dark soliton dynamics in inhomogeneous fiber system by means of a variable coefficient nonlinear Schrödinger equation (Vc-NLSE). The dark soliton interaction in the dispersion-managed system with gain/loss and cascade compression of solitons by the nonlinear tunneling phenomena are studied in detail. Numerically, we study the dark soliton propagation in the continuous wave background, and the obtained soliton solution is found to be robust against perturbations. The elastic collision behavior of the dark solitons are observed by means of asymptotic analysis. For better insight, we study the impact of the blackness factor in the phase profile and also on the interaction dynamics of the different dark soliton settings, such as black and gray soliton. Additionally, we extend the study to the investigation of tunneling of the dark soliton across the barrier/well with a particular emphasize on the role of blackness parameter in the nonlinear tunneling phenomena, and demonstrated a cascaded compression of soliton in the inhomogeneous system.

Resonant optical pulses on a continuous-wave background in two-level active media

We present exact N-soliton optical pulses riding on a continuous-wave (c.w.) beam that propagate through and interact with a two-level active optical medium. Their representation is derived via an appropriate generalization of the inverse scattering transform for the corresponding Maxwell-Bloch equations. We describe the single-soliton solutions in detail and classify them into several distinct families. In addition to the analogues of traveling-wave soliton pulses that arise in the absence of a c.w. beam, we obtain breather-like structures, periodic pulse-trains and rogue-wave–type (i.e., rational) pulses, whose existence is directly due to the presence of the c.w. beam. These soliton solutions are the analogues for Maxwell-Bloch systems of the four classical solution types of the focusing nonlinear Schrödinger equation with non-zero background, although the physical behavior of the corresponding solutions is quite different.

Extinction ratio enhancement by sinusoidal phase modulation and nonlinear spectral focusing

We propose a new approach to enhance the extinction ratio of a pulse impaired by a coherent continuous-wave background. Taking advantage of a sinusoidal phase modulation followed by a nonlinear spectral focusing and an optical bandpass filtering, we are able to significantly decrease the level of the unwanted background. The principle of the method is experimentally validated at telecommunication wavelengths.

Asymptotic expansions and solitons of the Camassa–Holm – nonlinear Schrödinger equation

We study a deformation of the defocusing nonlinear Schrödinger (NLS) equation, the defocusing Camassa–Holm NLS, hereafter referred to as CH–NLS equation. We use asymptotic multiscale expansion methods to reduce this model to a Boussinesq-like equation, which is then subsequently approximated by two Korteweg–de Vries (KdV) equations for left- and right-traveling waves. We use the soliton solution of the KdV equation to construct approximate solutions of the CH–NLS system. It is shown that these solutions may have the form of either dark or antidark solitons, namely dips or humps on top of a stable continuous-wave background. We also use numerical simulations to investigate the validity of the asymptotic solutions, study their evolution, and their head-on collisions. It is shown that small-amplitude dark and antidark solitons undergo quasi-elastic collisions.

Asymmetric W-shaped and M-shaped soliton pulse generated from a weak modulation in an exponential dispersion decreasing fiber**Project supported by the National Natural Science Foundation of China (Grant No. 11475135), the Fund from Shaanxi Province Science Association of Colleges and Universities (Grant No. 20160216), and Guangxi Provincial Education Department Research Project, China (Grant No. 2017KY0776).

We study localized waves on continuous wave background in an exponential dispersion decreasing fiber with two orthogonal polarization states. We demonstrate that asymmetric W-shaped and M-shaped soliton pulse can be generated from a weak modulation on continuous wave background. The numerical simulation results indicate that the generated asymmetric soliton pulses are robust against small noise or perturbation. In particular, the asymmetric degree of the asymmetric soliton pulse can be effectively controlled by changing the relative frequency of the two components. This character can be used to generate other nonlinear localized waves, such as dark–antidark and antidark–dark soliton pulse pair, symmetric W-shaped and M-shaped soliton pulse. Furthermore, we find that the asymmetric soliton pulse possesses an asymmetric discontinuous spectrum.

Gray and black optical solitons with quintic nonlinearity

A higher-order nonlinear Schrödinger equation with the third-order dispersion, the cubic–quintic nonlinearities, derivative non-Kerr nonlinear terms, Kerr dispersion, and stimulated inelastic scattering is considered. The model can be used to describe the subpicosecond or femtosecond optical pulse propagation in highly nonlinear optical fibers. We find the exact gray solitary wave solution (i.e., a dark localized structure with nonzero minimum in intensity) on a continuous-wave background for the equation by adopting an appropriate ansatz solution. The propagation characteristics of such solitary pulses are determined in the presence of all higher-order effects. We also find black solitary wave solutions for the higher-order evolution equation under specific conditions for its coefficients. The parametric conditions on fiber parameters for the existence of both types are given.

Modeling of matter-wave solitons in a nonlinear inductor-capacitor network through a Gross-Pitaevskii equation with time-dependent linear potential.

A lossless nonlinear LC transmission network is considered. With the use of the reductive perturbation method in the semidiscrete limit, we show that the dynamics of matter-wave solitons in the network can be modeled by a one-dimensional Gross-Pitaevskii (GP) equation with a time-dependent linear potential in the presence of a chemical potential. An explicit expression for the growth rate of a purely growing modulational instability (MI) is presented and analyzed. We find that the potential parameter of the GP equation of the system does not affect the different regions of the MI. Neglecting the chemical potential in the GP equation, we derive exact analytical solutions which describe the propagation of both bright and dark solitary waves on continuous-wave (cw) backgrounds. Using the found exact analytical solutions of the GP equation, we investigate numerically the transmission of both bright and dark solitary voltage signals in the network. Our numerical studies show that the amplitude of a bright solitary voltage signal and the depth of a dark solitary voltage signal as well as their width, their motion, and their behavior depend on (i) the propagation frequencies, (ii) the potential parameter, and (iii) the amplitude of the cw background. The GP equation derived in this paper with a time-dependent linear potential opens up different ideas that may be of considerable theoretical interest for the management of matter-wave solitons in nonlinear LC transmission networks.

Ultrashort dark solitons interactions and nonlinear tunneling in the modified nonlinear Schrödinger equation with variable coefficient

Abstract We present the study of the dark soliton dynamics in an inhomogeneous fiber by means of a variable coefficient modified nonlinear Schrodinger equation (Vc-MNLSE) with distributed dispersion, self-phase modulation, self-steepening and linear gain/loss. The ultrashort dark soliton pulse evolution and interaction is studied by using the Hirota bilinear (HB) method. In particular, we give much insight into the effect of self-steepening (SS) on the dark soliton dynamics. The study reveals a shock wave formation, as a major effect of SS. Numerically, we study the dark soliton propagation in the continuous wave background, and the stability of the soliton solution is tested in the presence of photon noise. The elastic collision behaviors of the dark solitons are discussed by the asymptotic analysis. On the other hand, considering the nonlinear tunneling of dark soliton through barrier/well, we find that the tunneling of the dark soliton depends on the height of the barrier and the amplitude of the soliton. The intensity of the tunneling soliton either forms a peak or valley and retains its shape after the tunneling. For the case of exponential background, the soliton tends to compress after tunneling through the barrier/well.

Soliton excitations on a continuous-wave background in the modulational instability regime with fourth-order effects.

We study the correspondence between modulational instability and types of fundamental nonlinear excitation in a nonlinear fiber with both third-order and fourth-order effects. Some soliton excitations are obtained in the modulational instability regime which have not been found in nonlinear fibers with second-order effects and third-order effects. Explicit analysis suggests that the existence of solitons is related to the modulation stability circle in the modulation instability regime, and they just exist in the modulational instability regime outside of the modulational stability circle. It should be emphasized that the solitons exist only with two special profiles on a continuous-wave background at a certain frequency. The evolution stability of the solitons is tested numerically by adding some noise to initial states, which indicates that they are robust against perturbations even in the modulation instability regime. Further analysis indicates that solitons in the modulational instability regime are caused by fourth-order effects.

Dark spatial solitary waves in a cubic-quintic-septimal nonlinear medium

We consider the evolution of light beams in nonlinear media exhibiting nonlinearities up to the seventh order wherein the beam propagation is governed by the cubic-quintic-septimal nonlinear Schr\odinger equation. An exact analytic solution that describes dark solitary wave propagation is obtained, based on a special ansatz. Unlike the well-known $\text{tanh}$-profile dark soliton in Kerr media, the present one has a functional form given in terms of ``${\text{sech}}^{2/3}$''. The requirements concerning the optical material parameters for the existence of this localized structure are discussed. This propagating solitary wave exists due to a balance among diffraction, cubic, quintic, and septimal nonlinearities. We also investigate its stability under initial pertubations by employing numerical simulations. The modulational stability of the continuous-wave background, which supports the dark soliton, is also studied analytically. Collisions between stable dark solitons are finally investigated.

Optical-fluid dark line and X solitary waves in Kerr media

AbstractWe consider the existence and propagation of nondiffractive and nondispersive spatiotemporal optical wavepackets in nonlinear Kerr media. We report analytically and confirm numerically the properties of spatiotemporal dark line solitary wave solutions of the (2 + 1)D nonlinear Schrödinger equation (NLSE). Dark lines represent holes of light on a continuous wave background. Moreover, we consider non-trivial web patterns generated under interactions of dark line solitary waves,which give birth to dark X solitary waves. These solitary waves are derived by exploiting the connection between the NLSE and a well-known equation of hydrodynamics, namely the (2 + 1)D type II Kadomtsev-Petviashvili (KP-II) equation. This finding opens a novel path for the excitation and control of optical solitary waves, of hydrodynamic nature.

Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations

We construct the Lax pair and Darboux transformation for the variable-coefficients coupled Hirota equations. Based on modulation instability and by taking the limit approach, we derive two types of Nth-order rogue wave solutions with different dynamic structures in compact determinant representations. The explicit first-order rogue wave solution is presented, prolific vector rogue-wave patterns such as the dark-bright, composite, three-sister, quadruple and sextuple rogue waves with multiple compression points are demonstrated. In particular, in contrast to the standard Peregrine combs, unusual vector rogue wave combs such as the dark-bright and composite rogue wave combs are revealed by choosing sufficiently large periodic modulation amplitudes. Further, some wave characteristics such as the difference between light intensity and continuous wave background, and pulse energy evolution of the dark rogue wave solution that features multiple compression points are discussed in detail.

Rational Solitons in the Parity-Time-Symmetric Nonlocal Nonlinear Schrödinger Model

In this paper, via the generalized Darboux transformation, rational soliton solutions are derived for the parity-time-symmetric nonlocal nonlinear Schr\"{o}dinger (NLS) model with the defocusing-type nonlinearity. We find that the first-order solution can exhibit the elastic interactions of rational antidark-antidark, dark-antidark, and antidark-dark soliton pairs on a continuous wave background, but there is no phase shift for the interacting solitons. Also, we discuss the degenerate case in which only one rational dark or antidark soliton survives. Moreover, we reveal that the second-order rational solution displays the interactions between two solitons with combined-peak-valley structures in the near-field regions, but each interacting soliton vanishes or evolves into a rational dark or antidark soliton as $|z|\ra \infty$. In addition, we numerically examine the stability of the first- and second-order rational soliton solutions.

Spatiotemporal optical dark X solitary waves.

We introduce spatiotemporal optical dark X solitary waves of the (2+1)D hyperbolic nonlinear Schrödinger equation (NLSE), which rules wave propagation in a self-focusing and normally dispersive medium. These analytical solutions are derived by exploiting the connection between the NLSE and a well-known equation of hydrodynamics, namely the type II Kadomtsev-Petviashvili (KP-II) equation. As a result, families of shallow water X soliton solutions of the KP-II equation are mapped into optical dark X solitary wave solutions of the NLSE. Numerical simulations show that optical dark X solitary waves may propagate for long distances (tens of nonlinear lengths) before they eventually break up, owing to the modulation instability of the continuous wave background. This finding opens a novel path for the excitation and control of X solitary waves in nonlinear optics.

Controllable behaviors of nonautonomous solitons on background of continuous wave and cnoidal wave in $$\mathcal {PT}$$ PT -symmetric dimer with inhomogeneous effect

We obtain exact \(\mathcal {PT}\)-symmetric and \(\mathcal {PT}\)-antisymmetric nonautonomous soliton solutions on background waves. These solutions indicate that dispersion and nonlinear coefficients influence form factors of nonautonomous solitons such as amplitude, width and center; however, linear coupling coefficient and gain/loss parameter only influence phase of solitons. Based on these solutions, the controllable behaviors such as postpone, sustainment and restraint on continuous wave background in an exponential decreasing dispersion system are discussed. Moreover, the propagation behaviors of solitons on the cnoidal wave background in different dispersion systems are also studied.