Broadband third-order optical nonlinearities of layered franckeite towards mid-infrared regime

The study of nonlinear optical responses in the mid-infrared(mid-IR)regime is essential for advancing ultrafast mid-IR laser applications.However,nonlinear optical effects under mid-IR excitation are rarely reported due to the lack of suitable nonlinear optical materials.The natural van derWaals heterostructure franckeite,known for its narrow bandgap and stability in air,shows great potential for developing mid-IR nonlinear optical devices.We have experimentally demonstrated that layered franckeite exhibits a broadband wavelength-dependent nonlinear optical response in the mid-IR spectral region.Franckeite nanosheets were prepared using a liquid-phase exfoliation method,and their nonlinear optical response was characterized in the spectral range of 3000 nm to 5000 nm.The franckeite nanosheets exhibit broadband wavelengthdependent third-order nonlinearities,with nonlinear absorption and refraction coefficients estimated to be about 10^(-7)cm/W and 10^(-11)cm^(2)/W,respectively.Additionally,a passively Q-switched fluoride fiber laser operating around a wavelength of 2800 nm was achieved,delivering nanosecond pulses with a signal-to-noise ratio of 43.6 dB,based on the nonlinear response of franckeite.These findings indicate that layered franckeite possesses broadband nonlinear optical characteristics in the mid-IR region,potentially enabling new possibilities for mid-IR photonic devices.

Galerkin-Bernstein Approximations for the System of Third-Order Nonlinear Boundary Value Problems

This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in detail, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory. .

BLIND AND COMPLETE MODELING OF LINEAR SYSTEMS USING THIRD ORDER CUMULANTS

This paper presents a novel approach to structure determination of linear systems along with the choice of system orders and parameters. AutoRegressive （AR）, Moving Average （MA） or AutoRegressive-Moving Average （ARMA） model structure can be extracted blindly from the Third Order Cumulants （TOC） of the system output ts, where the unknown system is driven by an unobservable stationary independent identically distributed （i.i.d.） non-Gaussian signal. By means of the system order recursion, whether the system has an AR structure or has AR part of an ARMA structure is firstly investigated. MA features in the TOC domain is then applied as a threshold to decide if the system is an MA model or has MA part of an ARMA model. Numerical simulations illustrate the generality of the proposed blind structure identification methodology that may serve as a guideline for blind, linear system modeling.

Joint estimation of DOA, frequency, and polarization based on cumulants and UCA

A joint estimation algorithm of direction of arrival （DOA）, frequency, and polarization, based on fourth-order cumulants and uniform circular array （UCA） of trimmed vector sensors is presented for narrowband non-Gaussian signals. The proposed approach, which is suitable for applications in arbitrary Gaussian noise environments, gives a closed-form representation of the estimated parameters, without spectral peak searching. An efficient method is also provided for elimination of cyclic phase ambiguities. Simulations are presented to show the performance of the algorithm.

MULTIPLE POSITIVE SOLUTIONS OF SINGULAR THIRD-ORDER PERIODIC BOUNDARY VALUE PROBLEM

This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.

Third-order elastic, piezoelectric, and dielectric constants

The definitions of the third-order elastic,piezoelectric,and dielectric constants and the properties of the associated tensors are discussed.Based on the energy conservation and coordinate transformation,the relations among the third-order constants are obtained.Furthermore,the relations among the third-order elastic,piezoelectric,and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials.It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups,while all third-order dielectric constants are 0 for 16 kinds of point groups as well as isotropic material.The reason is that some of the point groups are centrally symmetric,and the other point groups are high symmetry.These results provide the foundation to measure these constants,to choose material,and to research nonlinear problems.Moreover,these results are helpful not only for the study of nonlinear elastic and piezoelectric problems,but also for the research on flexoelectric effects and size effects.

Accuracy of the half-power bandwidth method with a third-order correction for estimating damping in multi-DOF systems

A third-order correction was recently suggested to improve the accuracy of the half-power bandwidth method in estimating the damping of single DOF systems.This paper analyzes the accuracy of the half-power bandwidth method with the third-order correction in damping estimation for multi-DOF linear systems.Damping ratios in a two-DOF linear system are estimated using its displacement and acceleration frequency response curves,respectively.A wide range of important parameters that characterize the shape of these response curves are taken into account.Results show that the third-order correction may greatly improve the accuracy of the half-power bandwidth method in estimating damping in a two-DOF system.In spite of this,the half-power bandwidth method may significantly overestimate the damping ratios of two-DOF systems in some cases.

Conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems

This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non- conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invaxiance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.

Oscillation Criteria for Third-order NonlinearNeutral Dynamic Equations on Time Scales

Many practical problems, such as those from electronic engineering, mechanicalengineering, ecological engineering, aerospace engineering and so on, need to bedescribed by dynamic equations on time scales, so it is important in theory andpractical significance to study these equations. In this paper, the oscillation andasymptotic behavior of third-order nonlinear neutral delay dynamic equations ontime scales are studied by using generalized Riccati transformation technique, integralaveraging methods and comparison theorems. The main purpose of this paperis to establish some new oscillation criteria for such dynamic equations. The newKamenev criteria and Philos criteria are given, and an example is considered toillustrate our main results.

Symmetry Reduction of Initial-Value Problems for a Class of Third-order Evolution Equations

Symmetry reduction of a class of third-order evolution equations that admit certain generalized conditionalsymmetries (GCSs) is implemented.The reducibility of the initial-value problem for an evolution equation to a Cauchyproblem for a system of ordinary differential equations (ODEs) is characterized via the GCS and its Lie symmetry.Complete classification theorems are obtained and some examples are taken to show the main reduction procedure.

On the Existence of Positive Solution for a Nonlinear Third-order Three-point Boundary Value Problem

An existence theorem of positive solution is established for a nonlinear third-order three-point boundary value problem. Here, we concentrated on the case that the nonlinear term is neither superlinear nor sublinear, and is not asymptotic at zero and infinity.

Existence and iteration of monotone positive solutions for a third-order two-point boundary value problem

The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.

Triple Positive Solutions for a Third-order Three-point Boundary Value Problem

In this paper,we study the existence of triple positive solutions for the nonlinear third-order three-point boundary value problem ■where η∈[0,1/2) is a constant,by using a fixed-point theorem due to Avery and Peterson,we establish results of triple positive solutions to the boundary value problem,and an example is given to illustrate the importance of result obtained.

Third-Order Stokes Wave Solutions of the Free Surface Capillary- Gravity Wave and the Interfacial Internal Wave

Based on the Stokes wave theory, the capillary-gravity wave and the interfacial internal wave in two-layer constant depth＇s fluid system are investigated. The fluids are assumed to be incompressible, inviscid and irrotational. The third-order Stokes wave solutions are given by using a perturbation method. The results indicate that the third-order solutions depend on the surface tension, the density and the depth of each layer. As expected, the first-order solutions are the linear theoretical results （the small amplitude wave theoretical results）. The second-order and the third-order solutions describe the nonlinear modification and the nonlinear interactions. The nonlinear impact appears not only in the n （n〉~2） times＇ high frequency components, but also in the low frequency components. It is also noted that the wave velocity depends on the wave number, depth, wave amplitude and surface tension.

Astronomical Cycles of the Late Permian Lopingian in South China and Their Implications for Third-Order Sea-Level Change

The late Permian(Lopingian)was a crucial climate transition period from the late Paleozoic Ice Age to the early Triassic of exceptionally high temperatures.However,the origins of the third-order sea-level changes during the Lopingian Epoch remain unclear.Here,we presented astronomically calibrated gamma-ray(GR)log and non-U GR(computed gamma ray or CGR)curves from the clastic and carbonate successions of well GFD-1 in the Pingle Depression of South China for studying the sea-level oscillations during the Lopingian.Spectral analyses of the 405 kyr-calibrated GR and CGR time data revealed periodicities close to about 405,about 100,about 44.2,about 35.1,about 21,and about 17.5 kyr,supporting the existence of Milankovitch forcing in the sedimentary records.A high-resolution astronomical time scale and high-resolution sedimentation rate curve of the Lopingian from well GFD-1 were constructed by cyclostratigraphic analysis.The eccentricity and obliquity amplitude modulation cycles suggested long periodicities of about 2.4 and about 1.2 myr,respectively.In the Wuchiapingian greenhouse of the Lopingian,the about 2.4 myr eccentricity oscillation controlled‘weak’glacio-eustasy and/or aquifer eustatic changes related to the global third-order sea-level changes and that a lowstand(W2)was initiated by an eccentricity oscillation minimum.In contrast,during the Changhsingian,which exhibited a cooling event,an about 1.2 myr obliquity cycle was probably strong,with the sea-level records highlighting the link between the‘icehouse’sea-level lowering(C2 and C1)and the obliquity nodes.Moreover,dynamic sedimentary noise model as an indicator of sea-level showed local third-order sea-level variations,the coevolution trends in the orbital power,global and local sea-level changes,and sedimentation rate had significant implications for establishing the global nature and synchronicity of these million-year-scale eustatic records and reconstructing the temporal depositional history at a regional scale.In addition,the volcanism and tectonism that continued into the early-middle Wuchiapingian probably led to a series of climate changes that drove the hydrological cycles not paced by the Milankovitch cycles.

Application of Three Order Cumulants in Fault Diagnosis

The three-order cumulants’ complex forms of different definitions include different coupling information of signals, and the information can be used to diagnose fault. In the experiment of pressure reducing valve’s fault diagnosis, through these different coupling information, the features of fault signals and normal signals were extracted by wavelet in different directions, then these features were inputted to diagnose the fault. The experiment shows that this method can achieve a satisfactory result.

Signal Classification Method Based on Support Vector Machine and High-Order Cumulants

In this paper, a classification method based on Support Vector Machine (SVM) is given in the digital modulation signal classification. The second, fourth and sixth order cumulants of the received signals are used as classification vectors firstly, then the kernel thought is used to map the feature vector to the high dimensional feature space and the optimum separating hyperplane is constructed in space to realize signal recognition. In order to build an effective and robust SVM classifier, the radial basis kernel function is selected, one against one or one against rest of multi-class classifier is designed, and method of parameter selection using cross- validation grid is adopted. Through the experiments it can be concluded that the classifier based on SVM has high performance and is more robust.

Giant and light modifiable third-order optical nonlinearity in a free-standing h-BN film

Recently,hexagonal boron nitride(h-BN)has become a promising nanophotonic platform for on-chip information devices due to the practicability in generating optically stable,ultra-bright quantum emitters.For an integrated information-processing chip,high optical nonlinearity is indispensable for various fundamental functionalities,such as all-optical modulation,high order harmonic generation,optical switching and so on.Here we study the third-order optical nonlinearity of free-standing h-BN thin films,which is an ideal platform for on-chip integration and device formation without the need of transfer.The films were synthesized by a solution-based method with abundant functional groups enabling high third-order optical nonlinearity.Unlike the highly inert pristine h-BN films synthesized by conventional methods,the free-standing h-BN films could be locally oxidized upon tailored femtosecond laser irradiation,which further enhances the third-order nonlinearity,especially the nonlinear refraction index,by more than 20 times.The combination of the free-standing h-BN films with laser activation and patterning capability establishes a new promising platform for high performance on-chip photonic devices with modifiable optical performance.

The Third-Order Nonlinear Optical Properties in Cobalt-Doped ZnO Films

We quantitatively investigate the third-order optical nonlinear response of Co-doped ZnO thin films prepared by magnetron sputtering using the Z-scan method. The two-photon absorption and optical Kerr effect are revealed to contribute to the third-order nonlinear response of the Co-doped ZnO films. The nonlinear absorption coefficient β is determined to be approximately 8.8 × 10-5 cm/W and the third-order nonlinear susceptibility X（3） is 2.93 × 10-6 esu. The defect-associated energy levels within the band gap are suggested to be responsible for the enhanced nonlinear response observed in Co-doped ZnO films.

Recognition Analysis and Simulation Implementation Based on High-Order Cumulants of Wireless Digital Modulation Mode

This paper mainly studies the data characteristics of high order cumulants using digitally modulated signals, and constructs the identification feature parameters that can distinguish the signal modulation type by the high-order cumulants data of the digital modulation signal. Set the identification signal modulation type determination threshold based on the value of the identification feature parameter. The identification feature parameter value of the signal modulation type is compared with the set determination threshold, to realize the recognition of digital modulation signal. This identification method is implemented based on MATLAB design, with a 2ASK (2-ary Amplitude Shift Keying) signal, 4ASK (4-ary Amplitude Shift Keying) signal, 2PSK (2-ary Phase Shift Keying) signal, 4PSK (4-ary Phase Shift Keying) signal, 2FSK (2-ary Frequency Shift Keying) signal, 4FSK (4-ary Frequency Shift Keying) signal. The second, fourth and sixth order cumulants of the six signals were analyzed. Calculate the selected identification feature parameter value and the determination threshold to identify the six signals. The six signals have made MATLAB identification simulation. Simulation results show that this method is feasible and has high recognition rate. Simulation results verify that such recognition methods maintain a high recognition rate under conditions with low signal-to-noise ratio. This identification method can be extended to more MASK (M-ary Amplitude Shift Keying), MPSK (M-ary Phase Shift Keying), MFSK (M-ary Frequency Shift Keying), MQAM (M-ary Quadrature Amplitude Modulation) signal identification.