Solutions of the Yang–Baxter equations on quadratic Lie groups: The case of oscillator groups

A Lie group is called quadratic if it carries a bi-invariant semi-Riemannian metric. Oscillator Lie groups constitute a subclass of the class of quadratic Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang–Baxter equation on a generic class of oscillator Lie algebras. Moreover, we show that any solution of the generalized classical Yang–Baxter equation (resp. classical Yang–Baxter equation) on a quadratic Lie group determines a left invariant locally symmetric (resp. flat) semi-Riemannian metric on the corresponding dual Lie groups.     

Third-Order Optical Nonlinearity in Two-Dimensional Transition Metal Dichalcogenides

We present a detailed calculation of the linear and nonlinear optical response of four types of monolayer twodimensional (2D) transition-metal dichalcogenides (TMDCs), having the formula MX₂ with M=Mo, W and X=S, Se. The calculations are based on 6-band tight-binding model of TMDCs, and then performing a semi-classical perturbation analysis of response functions. We numerically calculate the linear χ_μν⁽¹⁾ (-ω;ω) and nonlinear surface susceptibility tensors χ_μνζη⁽³⁾(-ω_Σ; ω_r; ω_s; ω_t) with ω_Σ=ω_r+ω_s+ω_t. Both non-degenerate and degenerate cases are studied for thirdharmonic generation and nonlinear refractive index, respectively. Computational results obtained with no external fitting parameters are discussed regarding two recent reported experiments on MoS₂, and thus we can confirm the extraordinarily strong optical nonlinearity of TMDCs. As a possible application, we demonstrate generation of a π/4-rotated squeezed state by means of nonlinear response of TMDCs, in a silica micro-disk resonator covered with the 2D material. Our proposed method will enable accurate calculations of nonlinear optical response, such as four-wave mixing and highharmonic generation in 2D materials and their heterostructures, thus enabling study of novel functionalities of 2D photonic integrated circuits.     

EPR and optical investigation of VO(II) in Zn(C3H3O4)2(H2O)2 single crystals: An interstitial site

EPR spectroscopic investigations on single crystals of diaquabis[malonato(1-)-κ²O,O′] zinc(II) doped with VO(II) ion have been carried out at X-band frequencies and at 300 K. The single crystal, rotated along the three mutually orthogonally axes, has yielded spin-Hamiltonian parameters g and A as: g_xx=1.980, g_yy=1.972, g_zz=1.937 and A_xx=8.4, A_yy=6.1, A_zz=18.1 mT, respectively. These spin-Hamiltonian parameters reflect a slight deviation from axial symmetry to rhombic, which is elucidated by the interstitial occupation of vanadyl ions. The isofrequency plots and powder EPR spectrum have been simulated. The percentage of metal-oxygen bond has been estimated. The optical absorption spectrum exhibits four bands at 257, 592, 720 and 764 nm suggesting a C_4v symmetry. The admixture coefficients and bonding parameters have also been calculated by collaborating EPR data with optical data.     

Crystal and magnetic structures of Y2CrS4

Chromium(II) sulfide, Y₂CrS₄, prepared by a solid-state reaction of Y₂S₃ and CrS, showed an antiferromagnetic transition at 65 K. The neutron diffraction patterns at 10 and 90 K were both well refined with the space group Pca2₁. At 90 K, cell parameters were a=12.5518(13) Å, b=7.5245(8) Å, and c=12.4918(13) Å. At 10 K, magnetic peaks were observed, which could be indexed on the same unit cell. Magnetic moments of chromium ions were parallel to the b-axis and antiferromagnetically ordered in each set of the 4a sites.     

Neural network applied to reconstruction of complex objects based on fringe projection

The neural network has been introduced into the reconstruction of the complex object based on fringe projection. In this method, the neural network with powerful property of approximation is used to get the continuous approximate function of a discrete fringe pattern captured by an image frame grabber. The depth-related phase of the measured object modulated into the fringe pattern can be demodulated by dealing with the approximate function. Compared with the Fourier transform profilometry (FTP), in the network method, one deformed fringe pattern is needed to reconstruct the tested object, and a high spatial resolution is maintained for no filtering process. Therefore, this method performs better than FTP in the measurement of the complex object. Moreover, the network method is capable of demodulating more depth-related phase even in the case that the local shadow exists in the fringe pattern. Computer simulations and experiments validate the feasibility of this method.     

Spectroscopic and photoluminescence characteristics of Dy ions in lead containing sodium fluoroborate glasses for laser materials

Lead containing calcium zinc sodium fluoroborate (LCZSFB) glasses doped with different concentrations of trivalent dysprosium ions were prepared and investigated by the XRD, FTIR, optical absorption, photoluminescence and decay curve analysis. The experimentally determined oscillator strengths have been determined by measuring the areas under the absorption peaks and the Judd–Ofelt (J–O) intensity parameters were calculated using the least squares fit method. From the evaluated J–O parameters the radiative transition probability rates, radiative lifetimes and branching ratios were calculated for ⁴F_9/2 excited level. Room temperature photoluminescence spectra for different concentrations of Dy³⁺-doped LCZSFB glasses were obtained by exciting the glass samples at 386 nm. The intensity of Dy³⁺ emission spectra increases with increasing concentration of 0.1, 0.25, 0.5 and 1.0 mol% and beyond 1.0 mol% the concentration quenching is observed. The measuring branching ratios are reasonably high for transitions ⁴F_9/2→⁶H_15/2 and ⁶H_13/2, suggesting that the emission at 484 and 576 nm, respectively, can give rise to lasing action in the visible region. From the visible emission spectra, yellow–blue (Y/B) intensity ratios and chromaticity color coordinates were also estimated. The lifetimes of ⁴F_9/2 metastable state for the samples with different concentrations were also measured and discussed.     

Photoluminescence of CdGeP2 and (Cd,Mn)GeP2

Photoluminescence of CdGeP₂ (112) single crystal and CdGeP₂ epitaxial film grown on GaAs (001) substrate have been studied and their spectral similarity found. Spectral bands associated with donor/acceptor transitions peak at close energies for both substances and all are lower than the energy gap of the chalcopyrite crystal.On the other hand, the growth of (Cd,Mn)GeP₂ ferromagnetic layer on CdGeP₂ (112) single crystal was performed to make it possible observation of PL from both the ferromagnetic layer and substrate. The green laser excitation (514, 532 nm) produces a proper photoluminescence similar to that in the undoped CdGeP₂ crystal and film. An extra emission from the ferromagnetic-nonmagnetic heterojunction occurs to extend up to photon energies exceeding E_g of the host semiconductor. The short wavelength photoluminescence is to be due to (Cd,Mn)GeP₂ dilute magnetic semiconductor (DMS). This fact states that Mn-doped II-IV-V₂ chalcopyrites are closer to II-VI DMS than to another group III-V DMS, where the heavy Mn-doping suppresses photoluminescence at all. Features of the observed short wavelength emission are discussing based on the temperature and spectral analyses.     

Isogroups of differential ideals of vector-valued differential forms: Application to partial differential equations

An older geometric technique for the study of invariance groups of partial differential equations, originally proposed by one of the authors and F. B. Estabrook, is generalized and extended to problems involving exterior equations for vector-valued or Lie algebra-valued exterior differential forms. Use of the method is demonstrated in the study of the symmetry groups of the two-dimensional Dirac equation and the full Yang-Mills free-field equations in Minkowski spacetime.     

On longitudinal mode spacing of laser cavities with pairs of gratings

It has recently been suggested that a laser cavity formed by a pair of gratings would show anomalously large longitudinal mode spacing which should favor single-longitudinal-mode emission. An experimental verification of that prediction with a TEA-CO₂ laser is presented. The repeatedly negative results obtained under various conditions are given a tentative explanation, based on the well-known analysis of a closely related problem: the grating-pair pulse compression scheme.