Structure,electronic,and nonlinear optical properties of superalkaline M_(3)O(M=Li,Na)doped cyclo[18]carbon

Cyclo[18]carbon has received considerable attention thanks to its novel geometric configuration and special electronic structure.Superalkalis have low ionization energy.Doping a superalkali in cyclo[18]carbon is an effective method to improve the optical properties of the system because considerable electron transfer occurs.In this paper,the geometry,bonding properties,electronic structure,absorption spectrum,and nonlinear optical(NLO)properties of superalkaline M_(3)O(M=Li,Na)-doped cyclo[18]carbon were studied by using density functional theory.M_(3)O and the C_(18) rings are not coplanar.The C_(18) ring still exhibits alternating long and short bonds.The charge transfer between M_(3)O and C_(18) forms stable[M_(3)O]+[C_(18)]-ionic complexes.C_(18)M_(3)O(M=Li,Na)shows striking optical nonlinearity,i.e.,their first-and second-order hyperpolarizability(βvec andγ||)increase considerably atλ=1907 nm and 1460 nm.

Electrical and nonlinear optical properties of TGZO transparent conducting films deposited by magnetron sputtering

Thin transparent oxide conducting films(TCOFs)of titanium and gallium substituted zinc oxide(TGZO)were fabricated via radio frequency(RF)magnetron sputtering technique.The effects of RF power on electrical,linear and nonlinear optical characteristics were investigated by Hall tester,Ultraviolet(UV)-visible spectrophotometer and optical characterization method.The results indicate that RF power significantly influences the electrical and optical properties of the deposited films.As RF power raises,the resistivity and Urbach energy fall initially and then rise,while the figure of merit,mean visible transmittance and optical bandgap show the reverse variation trend.At RF power of 190 W,the TGZO sample exhibits the highest electro-optical properties,with the maximum figure of merit(1.14×10^(4)Ω^(-1)∙cm^(-1)),mean visible transmittance(86.9%)and optical bandgap(3.50 eV),the minimum resistivity(6.26×10^(-4)Ω∙cm)and Urbach energy(174.23 meV).In addition,the optical constants of the deposited films were determined by the optical spectrum fitting method,and the RF power dependence of nonlinear optical properties was studied.It is observed that all the thin films exhibit normal dispersion characteristics in the visible region,and the nonlinear optical parameters are greatly affected by the RF power in the ultraviolet region.

Six New Sets of the Non-Elementary Jef-Family-Functions that Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of three functions that belong together. Differentiating these functions twice gives second-order nonlinear ODEs that have the defined set of functions as solutions. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles. Using the methods described in this paper, it is possible to define many other sets of non-elementary functions that are giving solutions to some second-order nonlinear autonomous ODEs.

Theoretical Investigation on the Second-order Nonlinear Optical Properties of Chiral Amino Acid Zinc(Ⅱ) Porphyrins

Static second-order nonlinear optical effects of amino acid zinc(II) porphyrins 1, 2, 3 and 4 were calculated by the TDHF/PM3 method based on the molecular structures optimized at the semiempirical PM3 quantum chemistry level, showing due to the cancellation of symmetric center, these amino acid zinc(II) porphyrins exhibit second order nonlinear optical response. The analysis of β components indicated that these amino acid zinc(II) porphyrins are of multipola- rizabilities, and they may be ascribed as the “mixture” of octupolar and dipoar molecules with ||βJ=3||/||βJ=1|| ≈ 5. It is found that there are no significant differences between the static β values of non-chiral and chiral amino acid zinc(II) porphyrins. However, the βxyz component, which is quite important to quadratic macroscopic χ (2) susceptibility of chiral material, is increased significantly with the increase of side chain group of amino acids.

PROPERTY (X^+) FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF GENERALIZED EULER TYPE

In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub（super） linearity are assumed. W＇e present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property （X^＋）, which is very important for the existence of periodic solutions and oscillation theory.

Theoretical Investigation on Structures and Second-order Nonlinear Optical Properties of (Thiophene) Manganese Tricarbonyl Cations

The structures of several recently reported organometallic NLO chromophores, (thio-phene) manganese tricarbonyl cations, were fully optimized at the DFT non-local (GGA) level. The calculated results show that the fragments 2-SC4H3CH=CHC6H4-R in these organometallic chromophores are not planar with dihedral angles of 42.2~59.8 between two aromatic rings, which are different from those of uncoordinated counterparts. Based on the DFT geometry opti-mization, the second-order nonlinear optical polarizabilities were calculated by using RPA method. The calculated results indicate that incorporation of +3Mn(CO) unit with thiophene leads to a sub-stantial increase in the second-order polarizability (b).

Synthesis, Structure, and the Second-order Nonlinear Optical Properties of a Borate-carboxylic Acid Compound: KB[L-（-）-C4H4O5]2·H2O

A new organic-inorganic hybrid noncentrosymmetric potassium bis(malic acid)-borates KB(L-(-)-C4H4O5)2·H2O was synthesized by solution method. It was characterized by elemental analysis, FT-IR, TGA and single-crystal X-ray crystallography. It crystallizes in the monoclinic space group P21/c with a = 5.546(6), b = 11.985(13), c = 9.952(11) ?, β = 97.522(17)°, V = 655.8(13) ?3, Z = 2, Dc = 1.682 g/cm3, μ(MoKa) = 0.46 mm–1 and F(000) = 340. 5009 reflections were measured and 2893 independent reflections(Rint = 0.051) were used for further refinement. Single-crystal X-ray diffraction reveals that the complex exhibits a threedimensional pseudo tunnel structure consisting of fundamental building block [B(L-(-)-C4H4O5)2]– anions. The small cavities are occupied by the H2O molecules, which stabilize the whole structure by O–H×××O hydrogen bonds. The complex exhibits nonlinear optical effect as high as 1.5 times that of KDP standard.

Distributed Containment Control of Networked Nonlinear Second-order Systems With Unknown Parameters

In this paper, we study the containment control problem for nonlinear second-order systems with unknown parameters and multiple stationary/dynamic leaders. The topologies that characterize the interaction among the leaders and the followers are directed graphs. Necessary and sufficient criteria which guarantee the control objectives are established for both stationary leaders(regulation case) and dynamic leaders(dynamic tracking case) based protocols. The final states of all the followers are exclusively determined by the initial values of the leaders and the topology structures. In the regulation case, all the followers converge into the convex hull spanned by the leaders,while in the dynamic tracking case, not only the positions of the followers converge into the convex hull but also the velocities of the followers converge into the velocity convex hull of the leaders.Finally, all the theoretical results are illustrated by numerical simulations.

Nonlinear analysis on electrical properties in a bended composite piezoelectric semiconductor beam

In this paper,the interactions between the transverse loads and the electrical field quantities are investigated based on the nonlinear constitutive relation.By considering a composite beam consisting of a piezoelectric semiconductor and elastic layers,the nonlinear model is established based on the phenomenological theory and Euler’s beam theory.Furthermore,an iteration procedure based on the differential quadrature method(DQM)is developed to solve the nonlinear governing equations.Before analysis,the convergence and correctness are surveyed.It is found that the convergence of the proposed iteration is fast.Then,the transverse pressure induced electrical field quantities are investigated in detail.From the calculated results,it can be found that the consideration of nonlinear constitutive relation is necessary for a beam undergoing a large load.Compared with the linear results,the consideration of the nonlinear constitutive relation breaks the symmetry for the electric potential,the electric field,and the perturbation carrier density,and has little influence on the electric displacement.Furthermore,the non-uniform pressures are considered.The results show that the distributions of the electric field quantities are sensitively altered.It indicates that the electrical properties can be manipulated with the design of different transverse loads.The conclusions in this paper could be the guidance on designing and manufacturing electronic devices accurately.

An 8-Connected Chiral Lanthanide Metal-organic Framework Constructed from Naturally Camphoric Acid:Crystal Structure,Vibrational Circular Dichroism Spectroscopy and Second-order Nonlinear Optical Effect

A chiral lanthanide metal-organic framework based on enantiopure camphoric acid （D-H2cam）, [Nd3（D-cam）8（H2O）4Cl]n （1）, has been synthesized and characterized by single-crystal X-ray structural analysis, elemental analysis, IR, thermal gravimetric, and X-ray powder diffraction. Crystal data for the title compound are as follows： orthorhombic system, space group P212121 with a = 13.8287（7）, b = 14.0715（7）, c = 25.7403（12） A^°, V = 5008.8（4） A^°3, Mr = 1333.08, Z = 4, F（000） = 2644, Dc = 1.768 g/cm^3, μ（MoKα） = 3.189 mm^-1, the final R = 0.0351 and wR = 0.0814 （I 〉 2σ（I））. Compound 1 displays an 8-connected bcu topology 3D framework and hydrogen-bonding interactions stabilize the solid-state structure. The vibrational circular dichroism （VCD） spectrum and second-order nonlinear optical effect of compound 1 have been studied in the solid state.

QUALITATIVE ANALYSIS FOR A CLASS OF SECOND-ORDER NONLINEAR SYSTEM WITH DELAY

The second-order nonlinear system with delay x ' (t) + f(x(t),x ' (t)) + g(x(t),x ' (t))psi (x(t-tau)) = p(t) being considered. Four theorems on the stability of zero solution, the boundedness of the solutions, the existence of the periodic solutions, the existence and uniqueness of the stationary oscillation are obtained by means of the Liapunov's second method, The conclusion in the literatures are generalized.

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation

In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.

An Attempt to Make Non-Elementary Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

In this paper, we define an exponential function whose exponent is the product of a real number and the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of functions. Differentiating these functions twice give second-order nonlinear ODEs that have the defined set of functions as solutions.

Second-order Nonlinearities of CdS Nanoparticles Studied by Hyper-Rayleigh Scattering Technique

A series of CdS nanoparticles with different surfaces were prepared by colloidal chemical method and reverse micelle method. Their second-order nonlinear optical (NLO) properties were experimentally studied in solution by newly developed hyper-Rayleigh scattering (HRS) technique. The results show that 'per particle' first-order hyperpolarizability beta values are sensitive To the synthetic method and the surface chemical modification.

Synthesis of Imidazole Derivatives for Their Second-order Nonlinear Optics

The design and the synthesis of two conjugated donor acceptor imidazole derivatives(1, 2) were carried out for second order nonlinear optics. The thermal properties, the transparency and second order nonlinear optical properties of the molecules were investigated. The experimental results indicate that a good nonlinearity transparency thermal stability trade off is achieved for them.

Synthesis,Structure and Second-order Nonlinear Optical Activities of a New Tungsten(0) Compound with Pyridine-2-thiolato Ligand,[Et_4N][W(pyS)(CO)_4](pys=C_5H_4NS^-)

The reaction of W（CO）6 with pyOSNa （C5H4NOSNa） and Et4NCl in MeCN affords a new tungsten（0）complex [Et4N][W（pyS）（CO）4] 1 （Mr = 536.29）. The crystal and molecular structures have been determined by X-ray single-crystal diffraction. Complex 1 crystallizes in the orthorhombic system, space group P212121 with α= 8.2429（5）, b = 9.1045（4）, c = 26.8851（14） A, β= 90.00°, V=2017.66（18） A3^, Z = 4, Dc = 1.765 g/cm^3, μ= 58.51 cm^-1, F（000） = 1048, the final R = 0.0204 and wR = 0.0400 for 4432 observed reflections with I 〉 2σ（I）. X-ray structure analysis revealed that the molecule is acentric and has large first=order hyperpolarizability （7.2 × 10^-30 esu）, so it could be an IR second=order nonlinear optical candidate material.

Fluorine-Mediated Benzothiadiazole Derivatives for Second-Order Nonlinear Optics

With increasing research interests in the field of light-matter interactions,various methods have been developed for regulating nonlinear optical(NLO)materials.However,the design and synthesis of organic molecular materials for second-order nonlinear optics remain a great challenge because of the strict requirement of the materials to possess a noncentrosymmetric structure.In this work,two benzothiadiazole(BTD)derivatives referred to as BTD-H and BTD-F were synthesized,and their NLO properties in the crystalline states were studied.It was found that introducing fluorine into the BTD backbone effectively tuned the crystal packing styles of BTD derivatives to a noncentrosymmetric system for effective second-order NLO responses.Such a strategy to induce the noncentrosymmetric structure by introducing the fluorine atoms and halogen interactions may provide guidance for future engineering of organic NLO molecular materials.

Unconventional photon blockade in a three-mode system with double second-order nonlinear coupling

The unconventional photon blockade(UPB)for low-frequency mode is investigated in a three-mode system with double second-order nonlinearity.By analyzing the Hamiltonian of the system,the optimal analytic condition of UPB in low-frequency mode is obtained.The numerical results are calculated by solving the master equation in a truncated Fock space,which agrees well with the analytic conditions.Through the numerical analysis of the system,it is found that the weak driving strength is favorable for the system to realize the UPB effect,and the system is insensitive to the changes of attenuation rate and environmental temperature.The comparison with the two-mode system and another similar threemode system shows that,under similar system parameters,the UPB effect of this double two-order nonlinear system is more obvious.

Study of second-order nonlinear hyperpolarisability of all-trans-β-carotene in solutions by linear spectroscopic technique

This paper demonstrates the second-order nonlinear hyperpolarisability γ of all-trans-β-carotene in different solvents by linear spectroscopic technique that is based on resonance Raman scattering and UV-VIS （Ultraviolet-visible） absorption spectroscopy. Owing to the two-level model well describing the link that exists between the resonance Raman scattering and stimulated Raman scattering, the stimulated Raman polarisability αR can be calculated through the two-photon resonance system. The value of γ of all-trans-β-carotene in carbon bisulfide solution is 6.435×10^-33 esu （1 esu of resistance = 8.98755×10^11Ω） that is close to the true value, because the solution of all-trans-β-carotene in carbon bisulfide satisfies the rigid resonance Raman scattering condition. This method is expected to be worthy of applications to measure the second-order nonlinear hyperpolaxisability of a conjugate organic molecule.

Hyers-Ulam Stability of a Generalized Second-Order Nonlinear Differential Equation

In this paper we have established the stability of a generalized nonlinear second-order differential equation in the sense of Hyers and Ulam. We also have proved the Hyers-Ulam stability of Emden-Fowler type equation with initial conditions.