Nonlinear Dispersion Research Articles

Nonlinear dispersion properties of metamaterial beams hosting nonlinear resonators and stop band optimization

The nonlinear dispersion properties of a metamaterial beam embedding nonlinear resonators are investigated. The metamaterial beam incorporates a distributed array of local resonators exhibiting softening or hardening cubic nonlinearity. The nonlinear dynamic behavior of the metamaterial beam is first investigated asymptotically via the method of multiple scales, which yields the closed-form nonlinear dispersion functions with the associated stop bands and the nonlinear dispersive waves as a combination of acoustic and optical bending waves. The effects of the resonators nonlinearity and the hosting beam nonlinear bending curvature on the stop bands are investigated in terms of sensitivity with respect to the strength and type of resonators nonlinearity. A full multi-variable optimization is carried out to study the nonlinear stop band size variations with respect to the resonators nonlinearity and the acoustic/optical wave amplitudes. The results show how the exploitation of the nonlinearities offers great possibilities to enlarge the size of the stop bands compared to the corresponding linear case and, hence, greatly enhance the energy absorption bandwidth.

Designing a single-mode anomalous dispersion silicon core fiber for temporal multiplet formation.

A highly nonlinear single-mode anomalous dispersion silicon core fiber (SCF) is suitably designed and optimized to generate a high repetition rate pulse train in the temporal domain from a single input pulse at a sufficiently shorter optimum length in comparison to silica-based standard fibers used for the same purpose. The large amount of Kerr-induced nonlinearity of a SCF is effectively utilized here such that input Gaussian pulses or pulse trains transform into a highly repetitive temporal multiplet. The effects of free-carrier generation-induced change in absorption and dispersion are included while studying the nonlinear pulse propagation through the SCF. To declare the generated pulse as a superior-graded triplet, a Q parameter, as a function of relative pulse parameters of the individual pulses of a triplet, is defined for the first time, to the best of our knowledge. Different pulse parameters are thoroughly optimized as well as the effect of external gain is examined from the perspective of requirement of shorter fiber length and development of quality triplets. Finally, the work is further extended for the formation of quadruplet pulses by the same type of SCF. It is to be mentioned here that such a methodical study for the generation of a temporal multiplet using a semiconductor core fiber has not been reported earlier.

Low frequency electrostatic mode generated by electromagnetic waves in the Earth’s inner magnetosphere with two distinct electrons

The generation of low frequency electrostatic mode by parametric decay of electromagnetic waves (EMWs) in the Earth’s inner magnetosphere with exponentially truncated kappa distributed hot electrons and cold electrons is studied. Nonlinear dispersion equation for the parametric process is derived from kinetic theory. The parametric instability of EMWs decay into low frequency electrostatic normal mode (ion acoustic like wave modes and electron acoustic wave modes) and electrostatic quasi–mode in the Earth’s inner magnetosphere are numerically analyzed. It is shown that parametric instability occurs only when the EMW is sufficiently strong if the collisions between ions and electrons are taken into account. The growth rate and the threshold conditions of the decay instability depend on the concentration and distribution of hot electrons. Because they change the dispersion and the damping rate of normal mode, the collisional damping of sideband EMW. In addition, the excitation of electrostatic normal mode by parametric decay of EMWs is more difficult than the excitation of electrostatic quasi–mode. The growth rate of EMWs decaying into electrostatic quasi–mode is much larger than the ones of decaying into electrostatic normal mode. But the frequency of electrostatic quasi–mode corresponding to the maximum growth rate can be as low as a few tens Hz. The mechanism may excite the electrostatic mode with frequency comparable to those of the ultra–low frequency electric fields observed in the Earth’s inner magnetosphere.

Impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)-dimensional Kundu–Mukherjee–Naskar equation

In this study, we explain the impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)-dimensional Kundu–Mukherjee–Naskar equation (KMNE). In this regard, some new optical solitons are received via the unified method to the aforesaid equation to explain such impacts of the soliton pulses. The presenting optical solitons are expressed by the dark, periodic, bell, bright, kink, and singular soliton solutions. Taking into account the two impacts help stabilize the soliton pulses during their propagation by generating new dynamics depending upon the nonlinearity and the wave dispersion parameters of the studied equation. All the characteristics of the soliton pulses are exhibited graphically. It is found from the graphical outputs that the soliton profiles are decreasing and increasing with the increase of nonlinearity and dispersion parameters, respectively. The outcomes reveal that the soliton pulses are balanced due to the influences of nonlinearity and wave dispersion parameters of the aforementioned equation. It is mentioned that the impact of wave dispersion and nonlinearity parameters on the soliton pulses has not been discussed in the past. Therefore, the applied method permits the explanation of the various wave dynamics by analyzing the attained soliton solutions in nonlinear optical fibers systems, which can be used for further studies.

Power Anisotropy, Dispersion Signature and Turbulence Diffusion Region in the 3D Wavenumber Domain of Space Plasma Turbulence

We explore the multifaceted important features of turbulence (e.g., anisotropy, dispersion, and diffusion) in the three-dimensional (3D) wavenumber domain (k ∥, k ⊥,1, k ⊥,2), by employing the k-filtering technique to high-quality measurements of fields and particles from the Magnetospheric Multiscale Mission (MMS) multi-spacecraft constellation. We compute the 3D power spectral densities (PSDs) of magnetic and electric field fluctuations (marked as PSD(δ B ( k )) and , respectively), both of which show a prominent spectral anisotropy in the sub-ion range. We give the first 3D image of the bifurcation between the power spectra of the electric and magnetic fluctuations, by calculating the ratio between and PSD(δ B ( k )), the distribution of which is related to the nonlinear dispersion relation. We also compute the ratio between electric spectra in different reference frames defined by the ion bulk velocity, , to visualize the turbulent ion diffusion region (T-IDR) in wavenumber space. The T-IDR has an anisotropy and a preferential direction of wavevectors, which is generally consistent with the plasma wave theory prediction based on the dominance of kinetic Alfvén waves. This work demonstrates the worth of the k-filtering technique in diagnosing turbulence comprehensively, especially when the electric field is involved.

Dispersive optical solitons in magneto-optic waveguides for perturbed stochastic NLSE with generalized anti-cubic law nonlinearity and spatio-temporal dispersion having multiplicative white noise

Object:The current article studies optical solitons solutions for the dimensionless form of the perturbed stochastic NLSE in magneto-optic waveguides with generalized anti-cubic (GAC) law nonlinearity and spatio-temporal dispersion (STD) having multiplicative noise in the itô sense. Methods:The generalized Jacobi-elliptic function expansion method and the addendum Kudryashov’s method are investigated. Results:The generalized Jacobi elliptic function expansion method secures the Jacobi’s elliptic functions solutions, the dark and singular soliton solutions as long as the Weierstrass elliptic functions solutions and periodic wave solutions. The addendum Kudryashov’s method deduces the combo bright-singular soliton, bright soliton and singular soliton solutions.

Elastic wave dispersion equation considering material and geometric nonlinearities

In this letter, we describe the propagation of longitudinal waves in one-dimensional nonlinear elastic thin rods with material nonlinearity and geometric nonlinearity. Mathematical analysis is used to derive the analytical dispersion relationship of longitudinal waves; subsequently, the numerical Fourier spectrum method is used to solve the nonlinear wave equation directly and the results are used to verify the correctness of the derived nonlinear dispersion relation.

Transformation and interactions among solitons in metamaterials with quadratic-cubic nonlinearity and inter-model dispersion

In this paper, we investigate multiple soliton interactions and other solitary wave solutions (SWS) for a perturbed nonlinear Schrödinger equation (NLSE) with negative index material having quadratic-cubic nonlinearity (NLSE-QCN). Due to its high order dispersion term, this model yields sub-picosecond impulses useful in mode-locked ring lasers. Hirota bilinear method (HBM) will be used to study soliton interaction. By controlling the parameters, we will obtain [Formula: see text], [Formula: see text], parabolic and anti-parabolic, butterfly, bright and dark shaped solitons. On the other hand, we will obtain some other solitary wave solutions with the help of Sine-Gordon expansion (SGE) scheme.

Complexity reduction over Bi-RNN-based nonlinearity mitigation in dual-pol fiber-optic communications via a CRNN-based approach

Bidirectional recurrent neural networks (bi-RNNs), in particular bidirectional long short term memory (bi-LSTM), bidirectional gated recurrent unit, and convolutional bi-LSTM models, have recently attracted attention for nonlinearity mitigation in fiber-optic communication. The recently adopted approaches based on these models, however, incur a high computational complexity which may impede their real-time functioning. In this paper, by addressing the sources of complexity in these methods, we propose a more efficient network architecture, where a convolutional neural network encoder and a unidirectional many-to-one vanilla RNN operate in tandem, each best capturing one set of channel impairments while compensating for the shortcomings of the other. We deploy this model in two different receiver configurations. In one, the neural network is placed after a linear equalization chain and is merely responsible for nonlinearity mitigation; in the other, the neural network is directly placed after the chromatic dispersion compensation and is responsible for joint nonlinearity and polarization mode dispersion compensation. For a 16-QAM 64 GBd dual-polarization optical transmission over 14×80km standard single-mode fiber, we demonstrate that the proposed hybrid model achieves the bit error probability of the state-of-the-art bi-RNN-based methods with greater than 50% lower complexity, in both receiver configurations.

Compacton–anticompacton collisions in the Rosenau–Hyman [formula omitted] equation by numerical simulations with hyperviscosity

Compactons are robust solitary waves with compact support, since they almost retain their shape after mutual interactions. They are solutions of evolution equations with either degenerate nonlinear dispersion or a sublinearity. The most extensively studied case is the Rosenau–Hyman K(p,p) equation, whose compactons are strong solutions for 13. This equation also has anticompactons, but compacton–anticompacton collisions has only been studied for p=2 in symmetric head-on collisions of a compacton and an anticompacton with the same absolute amplitude. Here, asymmetric compacton–anticompacton collisions for integer p≥2 are studied by using computational simulations with small hyperviscosity. For odd p=3, 5, and 7, robust compacton–anticompacton collisions are observed, even with very small hyperviscosity parameter. But for even p=2, 4, and 6, the solution blows up unless the quotient of the absolute amplitudes of the compacton and the anticompacton are below a given threshold that depends on p and the hyperviscosity parameter. Our results stress that, for physical applications, the K(p,p) equation requires the addition of a small amount of viscosity for the robustness of the compacton and anticompacton interactions.

Moiré exciton condensate: Nonlinear Dirac point, broken-symmetry Bloch waves, and unusual optical selection rules

Moir\'e exciton features tunable Dirac dispersion and spatially dependent optical selection rules. With the long lifetime due to its interlayer nature, it is promising to realize a Bose-Einstein condensation of moir\'e excitons. Here we study the properties of moir\'e exciton condensate within the mean-field theory, with special focus on exciton-exciton interaction effect on the nonlinear Bloch band and Bloch waves of exciton condensate in moir\'e potential. We find the nonlinear dispersion of the moir\'e exciton condensate exhibits complex loop structure induced by exciton-exciton interaction. A nonlinear Dirac cone emerges at $\boldsymbol{\Gamma}$ point of the Bloch band, with a three-fold degenerate nonlinear Dirac point. Each degenerate Bloch wave at nonlinear Dirac point spontaneously breaks $C3$ rotational symmetry of the moir\'e potential, and themselves differ by $C3$ rotations. Since they reside in the light cone, these nontrivial Bloch band structure and broken-symmetry Bloch waves can be experimentally detected by examining light emission from those states. Symmetry breaking of Bloch states implies unusual optical selection rules compared with single exciton case: moir\'e exciton condensate at $\boldsymbol{\Gamma}$ point emits light with all three components of polarization instead of only left or right circular polarization. We further propose that, by applying in-plane electric field on one layer to drive an initially optically dark exciton condensate towards light cone, the light polarization of final states as well as their dependence on field direction serves as the smoking gun for experimental observation.

500-nm broadband light generation in highly nonlinear dispersion shifted fiber by soliton laser

A compact and robust broadband light source is desirable for frequency combs, and broadband infrared microscopy. To address these demanding problems, we report the generation of broadband spectroscopy using highly nonlinear dispersion-shifted fiber (HNL-DSF) in an all-fiber system without any free-space optics instruments. However, due to the damage to the glass material by a large number of photons at high power, the ability of the fiber to widen will weaken over time. Therefore, the seed pulse used in this paper is the traditional soliton mode-locked pulse, which has lower energy than other types of mode-locked pulse. After two-stage amplification, the peak power, pulse width, and pulse energy of the soliton pulse are about 2 kW, 10 ps, and 18.79 nJ respectively. Amplifying the power of a low-energy pulse to pump the dispersion-shifted fiber, the ultra-wideband spectrum of more than 500 nm can be obtained without damaging the fiber, which is an efficient method to generate a wide spectrum. This method can be potentially used in astronomy, industry, medicine, etc.

Bright solitons of the model with arbitrary refractive index and unrestricted dispersion

Objective:The generalized Schrödinger equation with arbitrary refractive index and unrestricted order is considered. The objective of this paper is to demonstrate the bright solitons of this mathematical model can be found by the same formula taking into account the order of nonlinear differential equation. Method:The algorithm of constructing optical solitons of the generalized nonlinear Schrödinger equation with unrestricted dispersion using the simplest equation method is presented and applied for finding of optical solitons. Results:The optical solitons of the generalized nonlinear Schrödinger equation of the twelfth order are found at three powers on nonlinearity. The application of the method for finding solution of the generalized Korteweg–de Vries equation with arbitrary power of nonlinearity and unrestricted dispersion is discussed.

Optical soliton solutions of the Chen–Lee–Liu equation in the presence of perturbation and the effect of the inter-modal dispersion, self-steepening and nonlinear dispersion

Optical soliton solutions of the Chen–Lee–Liu equation in the presence of perturbation and the effect of the inter-modal dispersion, self-steepening and nonlinear dispersion

A design method for direct vision coaxial linear dispersion spectrometers.

A spectrometer design method based on the prism-prism-grating (PPG) dispersion module is proposed in this paper to correct the serious nonlinear dispersion that prism and grating spectrometers and other dispersive spectrometers suffer from. First, we determine the criteria for selecting the optical materials of the PPG module by analyzing the dispersion characteristics of prisms and gratings. Second, a loop traversal algorithm is used to optimize the system structure parameters after selecting optical materials. Next, the direct vision coaxial condition of the PPG module is derived according to basic optical principles and the geometrical relationship between optical elements. Then, the dispersion equation of the PPG module is used to establish the spectral linearity index of the system. Finally, combined with the design index, the structural parameters of the PPG module to meet the linear dispersion requirements are determined. A direct vision coaxial linear dispersion spectrometer is designed and realized under the condition that the working band is 400-990 nm, the deviation angle and offset of the emitted ray with a central wavelength of 695 nm with respect to the optical axis are 0, and the dispersion angle is not less than 15°. The results simulated by ZEMAX show that the actual simulation results are consistent with the theoretical calculation results, the spectral resolution of the spectrometer is less than 1.5 nm, and the spectral smile and keystone are less than 3.89% pixels. In the discussion section, the influences of the dispersion ability of optical materials and the incident angles of prisms and gratings on the spectral dispersion linearity of the PPG module are analyzed and studied. The universality of the spectrometer design method developed in this paper is discussed, and its universality is simulated and verified in the 1000-1600 nm and 1600-2200 nm bands. In addition, some advantages compared with other dispersion structures are analyzed.

Ultrafast two-dimensional imaging for surface defects measurement of mirrors based on a virtually imaged phased-array.

Single-shot measurement of surface defects of mirrors is vital for monitoring the operating states of high power lasers systems. While conventional methods suffer from low speed and small dynamic range. Here, we demonstrate a method for high speed two-dimensional (2D) surface amplitude-type defects measurement based on ultrafast single-pixel imaging assisted by a virtually imaged phased-array. Together with an optical grating, 2D wavelength to space mapping is achieved based on Fraunhofer far field diffraction, and the uniform broad spectrum of a home-made dissipative soliton is uniformly dispersed into the targeted mirror with one-to-one wavelength-to-space mapping. The surface amplitude-type defects are modulated into the intensity variation of the reflected spectrum. Then, we build a dispersive Fourier transform module for wavelength to time mapping, through which modulated spectral information is time stretched into the temporal domain, and recorded by a high speed photodetector together with a real time oscilloscope. Finally, to diminish the distortions induced by nonlinear dispersion during the wavelength-time mapping, we utilize the interpolation, and reconstruct the 2D surface with a frame rate of 7.6 MHz. A two-dimensional image with widths of 1.5 × 2 mm can be obtained within 10 ns, with a y direction spatial resolution of 180 µm and a x direction spatial resolution of 140 µm. This ultrafast 2D surface defects measurement scheme is promising for real-time monitoring of surface defects mirrors with large aperture, which are widely utilized in various high power laser systems.

Effects of frequency-dependent Kerr nonlinearity on higher-order soliton evolution in a photonic crystal fiber with one zero-dispersion wavelength

We show that the higher-order soliton evolution can be considerably affected by the relative positions of the zero-nonlinearity wavelength (ZNW) and zero-dispersion wavelength (ZDW) with respect to the input wavelength in photonic crystal fibers (PCFs) with frequency-dependent Kerr nonlinearity. By regulating the magnitude of nonlinear dispersion, the position of the ZNW and the strength of the doping Kerr nonlinearity can vary to a large extent, which significantly changes the general process of higher-order soliton evolution in the PCFs. Particularly, some interesting phenomena observed numerically in the PCFs with one ZDW and one ZNW are as follows: the formation of a blueshifted fundamental soliton that satisfies the phase-matching and group-velocity-matching conditions with a redshifted fundamental soliton, the tunneling effect of a fundamental soliton that transfers from one solitonic region to another one through a spectrally limited nonsolitonic region, and the generation of the high-intensity blueshifted dispersive wave and redshifted dispersive wave.

Quiescent Optical Solitons with Kudryashov’s Generalized Quintuple-Power and Nonlocal Nonlinearity Having Nonlinear Chromatic Dispersion

The paper derives stationary optical solitons with nonlinear chromatic dispersion. A nonlocal form of nonlinearity and quintuple power–law of nonlinearity are considered. The Kudryashov’s integration scheme enables to retrieve such solitons. A plethora of solitons come with this algorithm.

High-speed PAM4 transmission using directly modulated laser and artificial neural network nonlinear equaliser

Due to its simplicity, low-cost and small footprint, intensity modulation and direct detection (IM/DD) using directly modulated lasers (DMLs) remains the most widely-adopted optical transmission scheme for short-reach applications. However, when the data rate increases to 100 Gb/s and beyond, the DML nonlinearity and fibre chromatic dispersion become major factors that limit system performance. In this paper, we show that artificial neural networks (ANNs) are an effective nonlinear equaliser for enhancing the transmission performance of high-speed IM/DD systems using DML. More specifically, for 56 Gbaud PAM-4 transmission system, a low-complexity ANN equaliser with only 1 hidden layer and 10 neurons can increase the receiver sensitivity at KP-4 FEC limit in comparison with FFE-15 by approximately 1 dB at dispersion of −60 ps/nm and ＋20 ps/nm. In addition, we also show that ANN equaliser outperforms the commonly used Volterra equaliser for 56 Gbaud PAM-4 transmissions.

Enhancing the Spectral Efficiency of Nonlinear Frequency Division Multiplexing Systems via Hermite-Gaussian Subcarriers

The nonlinear frequency division multiplexing (NFDM) is a promising concept in the field of optical fiber communications. Based on a well established mathematical technique of the nonlinear Fourier transform (NFT), a data modulation scheme has been suggested, which is intrinsically immune to Kerr nonlinearity and chromatic dispersion. However, most NFDM systems still suffer from low values of spectral efficiency due to the large burst sizes and the nonlinear interaction of the signal and the inline amplifier noise. Here, we show that this problem can be partially circumvented by using the Hermite-Gaussian spectral carriers. Using this approach, we are able to report a significantly higher spectral efficiency for NFDM schemes approaching 4.5 bits/s/Hz for a single polarization while keeping the bit error rate of uncoded input below the 7% hard decision forward error correction threshold.