Third-order Coefficients Research Articles

Analysis of the zoom system of a microscope condenser and its modification using variable focus lenses

A detailed theoretical analysis and optimization of the classical three-element zoom (pancratic) microscope condenser according to a patent from the 1930s [Reichspatentamt Nr. 713188 (29 10 1936)] is performed and formulas are derived for calculating basic parameters and the displacement of lenses during zooming. Furthermore, the modification of the classical zoom microscope condenser is investigated using a simpler optical system of two lenses with variable focal lengths and fixed positions. The relations for the calculation of the focal lengths of both variable focus lenses and the basic parameters of the zoom system have been described. The proposed two-element zoom system consisting of a system of two lenses with variable focal lengths maintains a constant distance between the object and image planes and fixed position of both lenses during zooming. The basic parameters and third-order aberration coefficients of such a system are calculated using an example.

Measurement of forward charged hadron flow harmonics in peripheral PbPb collisions at sNN=5.02TeV with the LHCb detector

Flow harmonic coefficients, vn, which are the key to studying the hydrodynamics of the quark-gluon plasma (QGP) created in heavy-ion collisions, have been measured in various collision systems and kinematic regions and using various particle species. The study of flow harmonics in a wide pseudorapidity range is particularly valuable to understand the temperature dependence of the shear viscosity to entropy density ratio of the QGP. This paper presents the first LHCb results of the second- and the third-order flow harmonic coefficients of charged hadrons as a function of transverse momentum in the forward region, corresponding to pseudorapidities between 2.0 and 4.9, using the data collected from PbPb collisions in 2018 at a center-of-mass energy of 5.02 TeV. The coefficients measured using the two-particle angular correlation analysis method are smaller than the central-pseudorapidity measurements at ALICE and ATLAS from the same collision system but share similar features. ©2024 CERN, for the LHCb Collaboration 2024 CERN

Novel polymorph of 2-amino-6-methyl pyridinium-4-methyl benzoate (2A6MP4MB) single crystals: Synthesis, crystal structure, optical, spectroscopic and self-defocusing capabilities for nonlinear optical applications

By using the slow evaporation solution growth technique, the 2-amino 6-methyl pyridinium 4-methyl benzoate (2A6MP4MB) is crystallized in a single, high-quality crystal. Single crystal X-ray diffraction (SXRD) was used to determine the crystal structure. The different functional groups of the substances were identified using Fourier Transform Infrared Spectroscopy (FTIR) analysis. The 2A6MP4MB crystal exhibits a significant optical transmittance throughout the complete visible spectrum and a lowered UV cutoff wavelength, according to studies conducted using UV-Vis spectrometers. The optical band gap energy (Eg) of a single crystal of 2A6MP4MB is 4.768[Formula: see text]eV. The compound’s photoluminescence (PL) spectrum in the title illustrates the green emission technique. We analyze the produced single crystal’s dielectric tensorial behavior at various frequencies. The produced 2A6MP4MB crystal’s third-order susceptibility, nonlinear refraction, and nonlinear absorption coefficient were evaluated using the [Formula: see text]-scan technique. A Vickers hardness tester was used to assess the created crystal’s mechanical stability.

In Situ Fatigue Damage Monitoring by Means of Nonlinear Ultrasonic Measurements

In the present work, the results of acoustic nonlinear response of ultrasonic wave propagation when monitoring the progress of damage induced by fatigue on notched C45 carbon steel specimens have been reported. Two ultrasound probes were fixed to the specimens during the tests. The input signal was sinusoidal type, while the corresponding ultrasound response signal was acquired and recorded at each stage of the test by means of a digital oscilloscope. A nonlinear frequency study was performed on the acquired data to evaluate the change in the second- and third-order nonlinearity coefficients of β1 and β2, respectively, on the tested specimens. Ultrasonic results were correlated to plastic strain at the notch tip in the initial phases of fatigue and stiffness degradation. The results showed a significant increase in second-order nonlinearity β1 in the early stages of fatigue life. Subsequently, starting from about 30–40% of the fatigue life, the nonlinearity of β1 increases. Before final failure, from 80 to 85% of fatigue life, the second-order nonlinearity further increases in the crack propagation stages. The nonlinear parameter of the third-order β2 was less sensitive to damage than the parameter β1, showing a rapid increase only starting from approximately 80 to 85% of the fatigue life. The proposed method proved to be valid for detective damage induced by fatigue and to predict the lifetime of metal materials.

Changes in the Vegetation NPP of Mainland China under the Combined Actions of Climatic-Socioeconomic Factors

Under the combined impact of climatic, socioeconomic, and environmental factors, the vegetation NPP change process and its responses to drive factors in the sub-regions of Mainland China are not clear. This study analyzes the changing pattern of vegetation NPP in China from 2000 to 2022 from the perspective of zoning and clarifies its response mechanism to climate-human interaction based on the gravity center model, third-order partial correlation coefficient and geographical detector. The results showed that: (1) There was an overall decreasing trend of vegetation NPP in China from the southeast to the northwest; (2) The vegetation NPP gravity center in Northeast, Northwest, and North China migrated southwards, while that of Southwest, Central South, and East China showed northward migration.;(3) Human activities played a dominant role in zones with increasing vegetation NPP from 2000 to 2010, while climate change greatly contributed to the increase in vegetation NPP during 2011–2022; (4) Human activities, such as deforestation and overgrazing, in Northeast and North China should be reduced to prevent vegetation ecosystem degradation, and the negative impact of human activities should be reduced to maintain the growth of vegetation NPP. This study was conducted to support decision-making for the precise restoration of ecosystems.

Constructing a Regional Ionospheric TEC Model in China with Empirical Orthogonal Function and Dense GNSS Observation

Using Global Navigation Satellite Systems (GNSS) observation data for developing a high-precision ionospheric Total Electron Content (TEC) model is one of the essential subjects in ionospheric physics research and the application of satellite navigation correction. In this study, we integrate the Empirical Orthogonal Function (EOF) method with the TEC data provided by the Center for Orbit Determination in Europe (CODE), and observed by the dense GNSS receivers operated by the Crustal Movement Observation Network of China (CMONOC) to construct a regional ionospheric TEC model over China. The EOF analysis of CODE TEC in China from 1998 to 2010 shows that the first-order EOF component accounts for 90.3813% of the total variation of the ionospheric TEC in China. Meanwhile, the average value of CODE TEC is consistent with the spatial and temporal distribution characteristics of the first-order EOF base function, which mainly reflects the latitude and diurnal variations of TEC in China. The first-order coefficient after EOF decomposition shows an obvious 11-year period and semi-annual variations. The maximum amplitude of semi-annual variation mainly appears in March and October, which is closely associated with the variation in geographical longitude, the semi-annual change of the low-latitude electric field, and the ionospheric fountain effect. The second-order coefficient has an evident annual variation, the minimum amplitude mainly occurs in March, August, and September, and the amplitude values in the high solar activity years are more significant than those in the low solar activity years. The third-order coefficient mainly shows the characteristics of annual variation, and the fourth-order coefficient shows the noticeable semi-annual and annual variations. The third and fourth-order coefficients are both modulated by the solar activity index F10.7. The ionospheric TEC model in China, driven by CMONOC real-time GNSS observation data, can better reflect the latitude, local time and seasonal variation characteristics of ionospheric TEC over China. In particular, it can clearly show the spring and autumn asymmetry of ionospheric TEC in the low latitudes. The root mean square error of the absolute error between the model and the actual observation is mainly distributed around 2.45 TECU (1 TECU = 1016 electrons/m2). The values of the TEC model constructed in this study are closer to the actual observed values than those of the CODE TEC in China.

Integral precession calibration method of PIGA on linear vibration table

Linear vibration table can provide harmonic accelerations to excite the nonlinear error terms of Pendulous Integrating Gyro Accelerometer (PIGA). Integral precession calibration method is proposed to calibrate PIGA on a linear vibration table in this paper. Based on the precise expressions of PIGA’s inputs, the error calibration model of PIGA is established. Precession angular velocity errors of PIGA are suppressed by integer periodic precession and the errors caused by non-integer periods vibrating are compensated. The complete calibration process, including planning, preparation, PIGA testing, and coefficient identification, is designed to optimize the test operations and evaluate the calibration results. The effect of the main errors on calibration uncertainty is analyzed and the relative sensitivity function is proposed to further optimize the test positions. Experimental and simulation results verify that the proposed 10-position calibration method can improve calibration uncertainties after compensating for the related errors. The order of calibration uncertainties of the second- and third-order coefficients are decreased to 10−8 (rad·s−1)/g2 and 10−8 (rad·s−1)/g3, respectively. Compared with the other two classical calibration methods, the calibration uncertainties of PIGA’s nonlinear error coefficients can be effectively reduced and the proportional residual errors are decreased less than 3 × 10−6 (rad·s−1)/g by using the proposed calibration method.

Glassy Properties of the Lead-Free Isovalent Relaxor BaZr0.4Ti0.6O3

Glassy dielectric properties were investigated in lead-free BaZr0.4Ti0.6O3 (BZT40) ceramic samples using dielectric spectroscopy in the frequency range of 0.003 Hz–1 MHz and at temperatures of 10 K < T < 300 K. Measurements of the quasistatic dielectric polarization in bias electric fields up to ~28 kV/cm suggested that a ferroelectric state could not be induced, in contrast to the case of canonical relaxors such as PMN. The quasistatic dielectric and freezing dynamics results for the above field amplitudes showed that BZT40 effectively behaves as a dipolar glass. The relaxation spectrum was analyzed employing a frequency–temperature plot, which showed that the longest relaxation time obeyed the Vogel–Fulcher relation τ=τ0expE0/T−T0, with a freezing temperature of 76.7 K. The shortest relaxation time, in contrast, was characterized by a freezing temperature value close to 0 K, implying an Arrhenius-type behavior. The higher value of the polarization and the nonlinear third-order dielectric coefficient ε3 indicated a shift from a pseudospin glass behavior observed for BaZr0.5Ti0.5O3 (BZT50) toward a classical relaxor ferroelectric state.

Data-driven discovery of electron continuity equations in electron swarm map for determining electron transport coefficients in argon

In this study, we develop a novel method for determining electron transport coefficients from electron swarm maps measured by a scanning drift-tube experiment. In our method, two types of electron continuity equations that describe either the spatial or the temporal evolution of an electron swarm are discovered in the electron swarm map. The electron transport coefficients can be determined from the coefficients in the discovered equations. Therefore, we can determine the Townsend ionization coefficient, ionization rate coefficient, center-of-mass drift velocity, mean arrival-time drift velocity, longitudinal diffusion coefficient, and longitudinal third-order transport coefficient. These transport coefficients in argon are determined over a wide range of reduced electric fields, E/N, from 29.7 to 1351.6 Td (1 Td = 10−21 Vm2) using our method. We establish that the consideration of high-order transport coefficients, which have been systematically ignored so far, is important for the proper determination of low-order transport coefficients, specifically the electron drift velocity and longitudinal diffusion coefficient, in the presence of ionization growth.

Influence of the blade size on the dynamic characteristic damage identification of wind turbine blades

Abstract In this article, the sensitivity coefficients of dynamic characteristic damage identification of blades with different sizes were investigated. The results show that the first third-order vibration modes of the blade before and after damage are consistent, and the wind turbine blade size has no significant effect on the vibration mode; with the increase of the blade size, the first-, third- and fourth-order natural frequency sensitivity coefficients decrease gradually, while the second-, fifth- and sixth-order natural frequency sensitivity coefficients increase gradually; as the blade size increases, the third-order displacement mode sensitivity coefficient of the blade gradually increases, which indicates that the displacement modes identification effect is better with the increase of the blade size. With the increase of the blade size, the first- and third-order strain modal sensitivity coefficients increase gradually, which indicates that when using first- and third-order strain modes for damage identification, the larger the blade size, the better the identification effect; with the increase of the blade size, the second-order strain modal sensitivity coefficient decreases gradually, which indicates that when using second-order strain mode for damage identification, the larger the blade size, the worse the identification effect. This research could provide a theoretical basis for the application of the dynamic characteristic damage identification method in the damage identification of wind turbine blades of different sizes.

Nonlinear Landau-Zener tunneling under higher-order dispersion

We studied nonlinear Landau-Zener tunneling under the effect of higher-order dispersion, transformed the Gross-Pitaevskii (GP) equation with higher-order dispersion terms into a nonlinear two-energy level form using a two-mode approximation, and comprehensively analyzed the loop structure of the lowest-energy band and the nonlinear Landau-Zener tunneling. The results show that, when third-order dispersion coefficient is not zero, there is a spatial inversion symmetry breaking of the energy band structure and unbalanced Landau-Zener tunneling. By analyzing the fixed point's nature of the classical Hamiltonian, we obtain a law for the variation of the loop structure and adiabatic tunneling probability with the dispersion coefficient, consistent with our numerical analysis results. In addition, we analyzed the real-time evolution of solitons with transverse bias and observed the tunneling phenomenon. Consistent with our analysis, the adjustment of the dispersion term can effectively control optical tunneling, which provides an alternative idea for optical switching.

Ethyl acetoacetate and acetylacetone appended hexabromo porphyrins: synthesis, spectral, electrochemical, and femtosecond third-order nonlinear optical studies.

Asymmetrically substituted porphyrins possessing ethyl acetoacetate or acetylacetone (EAA or acac) with six bromine atoms at β-positions were synthesized and then characterized by various spectroscopic techniques, such as UV-Vis, fluorescence and NMR, and also by CV, DFT, MALDI-TOF-MS and elemental analysis. The mechanistic pathway followed the nucleophilic substitution reaction (nucleophile: EAA and acac) with MTPP(NO2)Br6 (M = 2H, Cu(II), and Ni(II)), and the resultant β-heptasubstituted porphyrins exhibited keto-enol tautomerism, as supported by 1H NMR spectroscopy. The six bulky bromo and EAA/acac groups made the macrocyclic ring highly electron deficient and nonplanar such that the quantum yield and fluorescence intensity for H2TPP[EAA]Br6 and H2TPP[acac]Br6 were severely reduced in contrast to those values for H2TPP. The poor electron density and nonplanarity over the porphyrin ring shifted the first oxidation potential from 11 to 521 mV anodically for MTPP[X]Br6 [M = 2H, Cu(II), and Ni(II); X = EAA or acac] as compared to corresponding MTPPs. Notably, density functional theory proved the nonplanarity of the synthesized porphyrins as Δ24 spans from ±0.546 to ± 0.559 Å while ΔCβ stretches from ±0.973 to ±1.162 Å. The third-order nonlinear optical measurements were performed using the femtosecond pulsed laser Z-scan technique at 800 nm and 1 kHz repetition rate to acquire insights into nonlinear absorption and nonlinear refraction of the porphyrins. The three-photon absorption coefficients (γ) are in the range of 2.2 × 10-23-2.8 × 10-23 cm3 W-2 and the nonlinear refractive index values were in the range of 3.7 × 10-16-5.1 × 10-16 cm2 W-1. The higher-order nonlinear absorption exhibited by porphyrins helps improve resolution at depth for various photonic and optoelectronic applications.

Investigation of in-medium effects of charmonia using azimuthal anisotropy and jet fragmentation function in PbPb collisions at √SNN = 5.02 TeV with the CMS experiment

Quarkonia have been long proposed as the golden probes to study quark-gluon plasma (QGP) in heavy-ion collisions. In this presentation, we present the second- and third-order Fourier coefficients, v2 and v3, for prompt and nonprompt J/ψ mesons in PbPb collisions as functions transverse momentum (pT) and PbPb collision centrality. Also, we report the first measurements of v2 and v3 for the prompt ψ(2S) mesons in PbPb collisions. The results provide v2 and v3 values over the wide studied kinematics regions of transverse momentum and collision centrality.

Mechanism of optical limiting in metalloporphyrins under visible continuous radiation.

Here, we present an experimental investigation on the nonlinear optical (NLO) and optical limiting properties of metalloporphyrin compounds (Cu-1-OH, Zn-1-OH, Cu-1-E and Zn-1-E) using spatial self-phase modulation (SSPM) method in the visible range. It is found that all of the samples show a large self-defocusing effect at 532 nm, which is attributed to the thermal nonlinear optical effects with negative nonlinear refractive index coefficient n2 due to the relatively high absorption at 532 nm. In contrast, at 780 nm where absorption is weak for both Zn- and Cu-porphyrins, Zn-porphyrins still exhibit visible self-defocusing while Cu-porphyrins do not show any nonlinear diffraction pattern. Such a phenomenon can be explained by the Kerr effect of Cu-porphyrins at 780 nm. As the thermal nonlinear optical effects (of negative n2) at 780 nm are reduced due to the low absorption, the Kerr effect with positive n2 becomes comparable and the overall nonlinearity is reduced. The Kerr effect of Cu-porphyrins is stronger than that of Zn-porphyrins because of the enhanced π-electron delocalization effect as Cu(II) has a variable number of valence electrons and incompletely filled d atomic orbitals. Finally, the optical limiting performance of Zn-porphyrins is demonstrated as a representative and its dependence on sample position is examined. This work not only enriches the understanding of the physical mechanism of optical limiting in porphyrin materials, but also provides a significant reference to improve the third-order NLO coefficient by adjusting the structure of compounds.

Analytic theory of nonlinearly coupled electrokinetics in nanochannels

The nonlinear electrokinetic response of ionic solutions is important in nanofluidics. However, quantitatively understanding the mechanisms is still a challenging problem because of a lack of analytic approaches. Here, a general framework for calculating the nonlinear electrokinetic coefficients of strongly confined electrolytes is constructed using a perturbation scheme of the pressure and voltage differences across a nanochannel. The theory is applied to an electrically neutral nanochannel filled with electrolytes, and analytic expressions for the first- and third-order electrokinetic coefficients are obtained. We demonstrate that the combination of high hydrodynamic permeability and ion–wall friction plays an essential role in nonlinear electrokinetics. Furthermore, we analytically demonstrate that the external flow induces uniform excess charge inside the nanochannel.

High polarizability enhanced EPR, optical linear & nonlinear and electric conductivity: Role of NiFe2O4 nanocrystals in transparent tellurite glass-ceramics

Glass containing magnetic nanocrystals is attractive to obtain high nonlinear and high polarizability. In this study, for the first time, 10 nm cubic NiFe2O4 (NFO) nanocrystals doped tellurite glass was studied in terms of the influence on glass structure, chemical bonds, EPR spectra, optical linear and nonlinearity, complex impedance, and dielectric constant. X-ray diffraction, FT-IR, and X-ray photoelectron spectroscopy revealed the single phase and chemical energy of NFO NCs and NFO- doped glasses. The introduction of NFO modified the glass structure through conversion of TeO4→TeO3, and PO4→PO3 units and producing non-bridging oxygen, which in turn enhanced the polarizability according. EPR analysis revealed the super-exchange interaction between multi-valence states of Ni and Fe ions, contributing to the deformation of NFO lattice and NBO defects in the glass network. The energy band gap of glass was red-shifted by NFO due to the multi-valence states of 3d ions. In addition, the linear and nonlinear refractive index, third-order absorption coefficient, and susceptibility were significantly enhanced because of the high polarizability. 3%NFO glass demonstrated huge third-order nonlinear susceptibility of 9.37 × 10−7 esu and giant DC electric conductivity of 8.84 × 10−4 S/cm at room temperature. The values obtained in the present study were much superior to values from the literature, and the obtained glasses are promising for high ultra-fast laser, supercontinuum generation, and holography applications.

Stereochemically Active Lone Pairs and Nonlinear Optical Properties of Two-Dimensional Multilayered Tin and Germanium Iodide Perovskites.

Two-dimensional (2D) metal halide perovskites are promising tunable semiconductors. Previous studies have focused on Pb-based structures, whereas the multilayered Sn- and Ge-based analogues are largely unexplored, even though they potentially exhibit more diverse structural chemistry and properties associated with the more polarizable ns2 lone-pair electrons. Herein, we report the synthesis and structures of 2D tin iodide perovskites (BA)2(A)Sn2I7, where BA = n-butylammonium and A = methylammonium, formamidinium, dimethylammonium, guanidinium, or acetamidinium, and those of 2D germanium iodide perovskites (BA)2(A)Ge2I7, where A = methylammonium or formamidinium. By comparing these structures along with their Pb counterparts, we establish correlations between the effect of group IV-cation's lone-pair stereochemical activity on the perovskite crystal structures and the resulting semiconducting properties such as bandgaps and carrier-phonon interactions and nonlinear optical properties. We find that the strength of carrier-phonon interaction increases with increasing lone-pair activity, leading to a more prominent photoluminescence tail on the low-energy side. Moreover, (BA)2(A)Ge2I7 exhibit strong second harmonic generation with second-order nonlinear coefficients of ∼10 pm V-1 that are at least 10 times those of Sn counterparts and 100 times those of Pb counterparts. We also report the third-order two-photon absorption coefficients of (BA)2(A)Sn2I7 to be ∼10 cm MW-1, which are one order of magnitude larger than those of the Pb counterparts and traditional inorganic semiconductors. These results not only highlight the role of lone-pair activity in linking the compositions and physical properties of 2D halide perovskites but also demonstrate 2D tin and germanium iodide perovskites as promising lead-free alternatives for nonlinear optoelectronic devices.

Stigmatic optical system with corrected third-order spherical aberration for an arbitrary position of the object

Our paper is focused on a problem of analysis and design of a stigmatic optical system that has corrected third-order spherical aberration for an arbitrary position of the object. The relations for Seidel aberration coefficients of the system for an object at infinity that must be satisfied to ensure that the third-order spherical aberration does not depend on the position of the object are given. The method for obtaining design parameters of the initial optical system that can serve as a good starting point for further refinement using numerical optimization methods is proposed. Based on the use of modified formulas for third-order aberration coefficients, this method enables one to decide if the individual members of the optical system can be simple lenses or if these should be more complex elements (cemented doublets, triplets, etc.). As a final result, one obtains the design parameters of the above-mentioned optical system (radii of curvature, optical materials, axial separations between individual elements). The analysis is performed for a thin-lens representation of the system. The transition to the thick-lens optical system then can be done by mathematical methods of numerical optimization using commercially available optical design software. The proposed method is shown on a practical example of calculation of parameters of such an optical system.

Femtosecond Third-Order Non-Linear Optical Properties of Unconstrained Green Fluorescence Protein Chromophores

We report herein results on the third-order non-linear optical (NLO) properties of four structurally unconstrained green fluorescence protein (GFP) chromophores, namely, 1, 2, 3, and 4. Using experimental techniques and theoretical calculations such as UV–visible spectroscopy, density functional theory (DFT), time-dependent density functional theory (TDDFT), and Z-scan techniques, we have investigated the linear absorption, ultrafast non-resonant third-order optical non-linearities, and the onset of optical-limiting thresholds of these benzylidenedimethylimidazolinone (BDI) dyes. The Z-scan measurements were performed at a wavelength of 800 nm with ∼70 femtosecond (fs) pulses. We have witnessed a strong reverse saturable absorption (fitted to three-photon absorption) for all of the molecules with fs pulse excitation. The valley–peak curves obtained from the closed-aperture Z-scan technique revealed the positive non-linear refractive index (self-focusing) nature of these molecules. We have evaluated the various third-order NLO coefficients (second hyperpolarizability, γ ∼10–33 esu), which were found to be larger than those of similar molecules reported in the recent literature.

The effect of Ag on the structural, dielectric, linear and third-order nonlinear optical properties of graphitic carbon nitride nanosheets

In this work, g-C3N4 and Ag/g-C3N4 nanomaterials were prepared and their structural, dielectric and optical properties are compared. The UV–Vis spectrophotometry, XRD pattern, FESEM image, FTIR spectroscopy, KK method and Z-scan measurements were applied to characterize the obtained nanomaterials. The Z-scan method was used to analyze third-order nonlinearity studies such as the nonlinear refractive (NLR) index, n2, and nonlinear absorption (NLA) coefficient, β by using Nd: YAG laser with 50 mW power at 532 nm under different concentration of Ag/g-C3N4 nanostructure. The impact of concentration on the nonlinear optical (NLO) properties have been investigated in detail. NLO properties of the Ag/g-C3N4 nanostructure are enhanced, reaching values of 2.99 × 10−8 cm2/W and 6.53 × 10−4 cm/W for the third-order NLR and NLA coefficients, respectively. In addition, this material possesses large optical characteristics in the IR region. The complex refractive index and dielectric coefficient of Ag/g-C3N4 nanostructure is investigated. Moreover, the Ag/g-C3N4 nanostructure's IR light responsiveness has significantly improved. The red shifts of longitudinal and transversal frequencies for Ag/g-C3N4 is more visible than g-C3N4. This research might lead to fresh ideas and a new way to make high-stability supported Ag optical devices.