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. .

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.

Structure and Nonlinear Refractive Index of Bismuth Borate Glass

The method of conventional glass melting is used to study the glass formation region of Bi2O3-B2O3-TiO2-La2O3 system. The instrument of Differential Scanning Calorimeter （DSC） is used to research the glass stability. Raman spectra and IR spectra are used to speculate on the structure of glasses. The refractive index of glass is measured by prism coupler. With increase of Bi2O3, the glass stability, the amount of [BiO3] group and boron-oxygen loops decrease, while the content of B-O- bond, refractive index and nonlinear refractive index increase.

Synthesis of Poly (3-acetylpyrrolyl methine) with Azobenzene Side Groups and Study on Its Third-order Nonlinear Optical Property

A novel soluble π-conjugated polymer, poly [（3-acetylpyrrole-2, 5-diyl） p-（N, N-dimethylamino） azobenzylidene] （PAPDMAABE）, was synthesized by condensation of 3-acetylpyrrole with 4-aldehyde-4＇-dimethylaminoazobenzene （ADMAA）. The chemical structure of PAPDMAABE was characterized by Fourier transform infrared spectroscopy （FTIR）, ^1H-NMR, and UV-Vis-NIR spectra. Transmission electron microscope （TEM） analysis for PAPDMAABE indicates that part of PAPDMAABE is in crystal state, due to the short-range order of the polymer. Thermogravimetric analysis （TGA） curve shows that the polymer has good thermal stability and its decomposition temperature is 248℃. The optical band gap of PAPDMAABE obtained from the optical absorption spectrum is about 1.73 eV. The resonant third-order nonlinear optical property of PAPDMAABE at 532 nm was studied using degenerate four-wave mixing （DFWM） technique. The resonant third-order nonlinear optical susceptibility of the polymer is about 7.48×10^-8 esu.

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.

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.

UV-Vis Spectrum and the Third-order Nonlinear Optical Properties of the Chiral Camphorderived β-diketonate Platinum Complexes

UV-Vis spectrum and the third-order nonlinear optical properties of the chiral camphor-derived β-diketonate have been studied at the B3LYP/6-31G＊ level. The results showed that the introduction of electron-drawing group -CF3 and -C3F7 on β-diketonate made the strongest absorption peak red-shift and the lowest energy absorption blue-shied. Introduction of -OC2H5 on the benzene or pyridine ring made the lowest energy absorption blue-shift. When the -C2H3 was introduced on the benzene or pyridine ring, the lowest energy absorption was red-shifted. Introduction of electron-donating group on β-diketonate can enlarge their nonlinear optical properties. On the contrary, the introduction of electron-drawing group dropped it down.

Electronic Absorption Spectra and Third-Order Nonlinear Optical Property of Dinaphtho[2,3-b:2’,3’-d]Thiophene-5,7,12,13- Tetraone (DNTTRA) and Its Phenyldiazenyl Derivatives: DFT Calculations

Third-order nonlinear optical (NLO) materials have broad application prospects in high-density data storage, optical computer, modern laser technology, and other high-tech industries. The structures and frequencies of Dinaphtho[2,3-b:2’,3’-d]thiophene-5,7,12,13-tetraone (DNTTRA) and its 36 derivatives containing azobenzene were calculated by using density functional theory B3LYP and M06-2X methods at 6-311++g(d, p) level, respectively. Besides, the atomic charges of natural bond orbitals (NBO) were analyzed. The frontier orbitals and electron absorption spectra of A-G5 molecule were calculated by TD-DFT (TD-B3LYP/6-311++g(d, p) and TD-M06-2X/6-311++g(d, p)). The NLO properties were calculated by effective finite field FF method and self-compiled program. The results show that 36 molecules of these six series are D-π-A-π-D structures. The third-order NLO coefficients γ (second-order hyperpolarizability) of the D series molecules are the largest among the six series, reaching 107 atomic units (10-33 esu) of order of magnitude, showing good third-order NLO properties. Last, the third-order NLO properties of the azobenzene ring can be improved by introducing strong electron donor groups (e.g. -N(CH3)2 or -NHCH3) in the azobenzene ring, so that the third-order NLO materials with good performance can be obtained.

Nonlinear Refractive Index and Absorption Coefficient of Single-Shell Semiconductor Carbon Nanotube

The nonlinear refractive index and absorption coefficient of single\|shell semiconductor carbon nanotubes(CN s ) are calculated based on the two\|band approximation and Genkin\|Mednis approach. The results of nonlinear refractive index and absorption coefficient reach the order of 10 -8 and 10 -4 cm 2\5W -1 separately, which indicates that CN s have wonderful nonlinear optical properties. Taking into account the temperature effect and overlapping of σ and π orbits, the effect of relaxation term and chiral angle is discussed. The results show that the smaller the relaxation term, the larger the nonlinear absorption coefficient and refractive index. At the same time, CN s with different chiral angles have different results due to their different energy gap.

Ultrafast third-order optical nonlinearity of several sandwich-type phthalocyaninato and porphyrinato europium complexes

The third-order optical nonlinearity of two sandwich-type phthalocyaninato and porphyrinato europium com- plexes, including double- and triple-deckers （Eu[Pc（OC5H11）8]2, Eu2（Pc）（TPP）2, Pc=phthalocyanine, TPP=5, 10, 15, 20- tetraphenylporphyrinate）, was investigated by using the femtosecond time-resolved optical Kerr gate method at 830 nm wavelength. Their second-order hyperpolarizability is estimated to be 0.74× 10^-30esu and 3.0× 10^-3esu respectively. This exhibits an evident enhancement in comparison with 0.47×10^-30esu for one-decker Eu（Pc）（acac） （acac=acetylacetonate）, which is also measured under the same conditions. The enhancement is attributed to the introduction of lanthanide metal to the large π-conjugated system, intermacrocycle interaction and two-photon resonance etc.

Ultrashort pulse breaking in optical fiber with third-order dispersion and quintic nonlinearity

The optical wave breaking （OWB） characteristics in terms of the pulse shape, spectrum, and frequency chirp, in the normal dispersion regime of an optical fiber with both the third-order dispersion （TOD） and quintic nonlinearity （QN） are numerically calculated. The results show that the TOD causes the asymmetry of the temporal- and spectral-domain, and the chirp characteristics. The OWB generally appears near the pulse center and at the trailing edge of the pulse, instead of at the two edges of the pulse symmetrically in the case of no TOD. With the increase of distance, the relation of OWB to the TOD near the pulse center increases quickly, leading to the generation of ultra-short pulse trains, while the OWB resulting from the case of no TOD at the trailing edge of the pulse disappears gradually. In addition, the positive （negative） QN enhances （weakens） the chirp amount and the fine structures, thereby inducing the OWB phenomena to appear earlier （later）. Thus, the TOD and the positive （negative） QN are beneficial （detrimental） to the OWB and the generation of ultra-short pulse trains.

Crystal Structure and Third-order Nonlinear Optical Property Studies on Copper(Ⅱ) Complex Containing NS Donor Ligand

A novel Schiff base ligand (HL) derived from S-methyldithiocabazate and pmethoxylbenzaldehyde was prepared and characterized. The Schiff base ligand acts as a single negatively charged bidentate ligand fondng D-M-D type comPlex (D=donor, M=metal). Single crystal X-ray diffraction analysis of the copper(Ⅱ) complex established that the geometry around Cu (Ⅱ) is square-planar with two equivalent M-N and M-S bonds. The two phenyl rings and the coordinated plane are almost in one plane fotheng an electronic delocalization system. Their thirdorder response was also studied.

A New Newton-Type Method with Third-Order for Solving Systems of Nonlinear Equations

In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.

Nonlinear Change in Refractive Index and Transmission Coefficient of ZnSe:Fe²⁺at Long-Pulse 2.94-μm Excitation

An experimental study of the nonlinear changes in refractive index and transmission coefficient of single-crystal ZnSe:Fe2+, fabricated through the Fe-diffusion method, at long-pulse (~300 ns), sub-mJ, 2.94-mm Z-scan probing is reported. As well, a theoretical model based on the generalized Avizonis-Grotbeck equations is developed and applied for straightforward fitting of the open- and closed-aperture Z-scans, obtained for ZnSe:Fe2+ with different Fe2+ centers concentrations. The modeling results reveal that the contributions in the absorption and refractive index nonlinearities of ZnSe:Fe2+ are “common” resonant-absorption saturation (the minor part) and pulse-induced heating of the samples (the major part), which are strongly dependent on Fe2+ concentrations. Large values of the index change (>~10-3) and partial resonant-absorption bleaching (limited by ~50%), both produced via the thermal effect mainly, are the features of the ZnSe:Fe2+ samples inherent to this type of excitation.

Effect of Si δ-Doping on the Linear and Nonlinear Optical Absorptions and Refractive Index Changes in InAlN/GaN Single Quantum Wells

In the framework of effective mass approximation, we theoretically investigate the electronic structure of the Si δ-doped InAIN/GaN single quantum well by solving numerically the coupled equations Schrodinger-Poisson self-consistently. The linear, nonlinear optical absorption coefficients and relative refractive index changes are calculated as functions of the doping concentration and its thickness. The obtained results show that the position and the amplitude of the linear and total optical absorption coefficients and the refractive index changes can be modified by varying the doping concentration and its thickness. In addition, it is found that the maximum of the optical absorption can be red-shifted or blue-shifted by varying the doping concentration. The obtained results are important for the design of various electronic components such as high-power FETs and infrared photonic devices.

Investigation on Third-Order Optical Nonlinearities of Two Organometallic Dmit2- Complexes Using Z-Scan Technique

The third-order nonlinear optical properties of two dmit organometallic complexes, [(CH3)4N] [Au(C3S5)2] (MeAu) and [(CH3)4N][Ni(C3S5)2] (Me Ni) in acetone solutions, were characterized us- ing a short pulse Z-scan technique at 1064 nm wavelength. Self-defocusing effects were found in both samples and stronger saturable absorp-tion was observed in MeNi solution comparing with that of MeAu. The origins were analyzed for the differences between the results. Two figures of merit W and T were also calculated to evalu-ate the suitability of two materials for all-optical integrated devices. The results of W=22.84 and T≈0 of MeAu make it an excellent candidate for the all-optical applications.

Single Beam Z-Scan Measurement of Nonlinear Refractive Index of Crude Oils

Nonlinear properties of two different crude oils from west-south Iran reservoirs have been investigated using the single beam Z-scan technique. The nonlinear refractive indices of the samples were measured by use of diode laser in the CW regime and at the wavelength 660 nm. The experiment results showed that the samples have large and negative refractive index nonlinear refraction coefficient. These results showed that the crude oil has significant nonlinear properties and it could be a candidate for photonic and nonlinear optical devices and also this information can be used in petroleum science and oceanology and etc.

Nonlinear Optical Susceptibility and Refractive Index of Two Polyaniline Families

By use of the Keldysh non-equilibrium Green’s-function methods, the third harmonic susceptibilities of two polyaniline families, PANI-HCl and PANI-H 3PO 4, are calculated [ x (3) ( ω )≈10 -12 esu]. It was found that the third harmonic susceptibility of polyaniline strongly depends on the delocalization of the electrons. The refractive indices n ( λ =589 nm) of PANI-HCl and PANI-H 3PO 4 are calculated by use of three common methods (the Lorentz-Lorentz theoretical model, the Gladstone-Dale group contribution and the Vogel group correlation) based on group contributions to molar refraction. The calculated n values are varied from 1.31 to 1.42 for PANI-HCl and 1.36 to 1.45 for PANI-H 3PO 4.

Synthesis of Poly[(3-octanoylpyrrole-2,5-diyl)-p-nitrobenzylidene] and Its Third-order Nonlinear Optical Property

A novel soluble π-conjugated polymer, poly[(3-octanoylpyrrole-2,5-diyl)-p-nitrobenzylidene] (POPNBE), was synthesized by a five-step reaction. As a result of introducing a long acyl to pyrrole at 3-position, the polymer can be easily solved in polar solvents and be processed into films. The chemical structures of the intermediates and POPNBE were characterized by FTIR, 1H NMR, and UV-Vis-NIR spectrometries. The absorption band in the range of 400-600 nm in the UV-Vis-NIR spectrum is due to the π-π * band gap transition. The optical band gap of POPNBE obtained from the optical absorption is about 1.72 eV. The resonant third-order nonlinear optical property of POPNBE at 532 nm was studied by using the degenerate four-wave mixing(DFWM) technique. The resonant third-order nonlinear optical susceptibility of POPNBE is about 3.42×10 -8 esu.