Nonlinear Coupling Coefficient Research Articles

Role of orbital filling on nonlinear ionic Raman scattering in perovskite titanates

The linear and nonlinear phononic interactions between an optically excited infrared (IR) or hyper-Raman mode and a driven Raman mode are computed for the ${d}^{0} ({\mathrm{CaTiO}}_{3})$ and ${d}^{1} ({\mathrm{LaTiO}}_{3})$ titanates within a first-principles density functional framework. We calculate the potential energy surface expanded in terms of the ${A}_{g}$ or ${B}_{1g}$ mode amplitudes coupled to the ${A}_{u}$ or the ${B}_{3u}$ mode and determine the coupling coefficients for these multimode interactions. We find that the linear-quadratic coupling dominates the anharmonicities over the quadratic-quadratic interaction in the perovskite titanates. The IR and Raman modes both modify the electronic structure with the former being more significant but occurring on a different time scale; furthermore, the coupled-mode interactions lead to sizable perturbations to the valence bandwidth ($\ensuremath{\sim}100\phantom{\rule{0.16em}{0ex}}\text{meV}$) and band gap ($\ensuremath{\sim}50$ meV). By comparing the coupling coefficients of undoped ${\mathrm{CaTiO}}_{3}$ and ${\mathrm{LaTiO}}_{3}$ to those for electron-doped $({\mathrm{CaTiO}}_{3})$ and hole-doped $({\mathrm{LaTiO}}_{3})$ titanates, we isolate the role of orbital filling in the nonlinear coupling process. We find that with increasing occupancy of the $d$ manifold, the linear-quadratic interaction decreases by approximately 30% with minor changes induced by the cation chemistry (that mainly affect the phonon mode frequencies) or by electron correlation. We identify the importance of the Ti-O bond stiffness, which depends on the orbital filling, in governing the lattice anharmonicitiy. This microscopic understanding can be used to increase the nonlinear coupling coefficient to facilitate more facile access of nonequilibrium structures and properties through ionic Raman scattering processes.

Soliton solutions in nonlocal nonlinear coupler

We study the existence of soliton pairs in nonlocal nonlinear coupler and the effects of nonlocal nonlinearity on the stability and types of soliton pairs. Bright soliton, dipole soliton, periodic soliton, double soliton and dark soliton are found, and many kinds of antisymmetric, symmetric and asymmetric solitons can be obtained by adjusting the nonlinear parameters or coupling coefficients. In addition, the stability of the exact bright solitary wave solutions and dipole-mode solitary wave solutions with the white noise perturbation are also investigated numerically. It is investigated that the energy flow and the Hamiltonian, and the energy flow are related to the nonlocal parameter. Moreover, We also obtain some Airy-like solitons by numerical method.

Nonlinear coupling of flexural mode and extensional bulk mode in micromechanical resonators

The effect of coupling between the extensional bulk mode and the flexural mode of a cantilever beam resonator has been experimentally studied by exciting the two modes simultaneously. The modal frequency shift of linear extensional bulk mode shows a quadratic relationship with the square of flexural mode's amplitude displacement, and a frequency shift up to 1492 Hz is observed when the flexural mode is driven by a AC signal Vac of 3 V and a DC bias of 30 V. The flexural mode shows a Duffing-like behavior with a softening nonlinearity and its frequency shift is influenced not only by its own nonlinear amplitude-frequency effect but also the extensional mode's amplitude as predicted. The nonlinear coupling coefficient is found to be about two orders of magnitude larger than the softening nonlinearity.

Determination of nonlinear nanomechanical resonator-qubit coupling coefficient in a hybrid quantum system.

We have theoretically investigated a hybrid system that is composed of a traditional optomechanical component and an additional charge qubit (Cooper pair box) that induces a new nonlinear interaction. It is shown that the peak in optomechanically induced transparency has been split by the new nonlinear interaction, and the width of the splitting is proportional to the coupling coefficient of this nonlinear interaction. This may give a way to measure the nanomechanical oscillator-qubit coupling coefficient in hybrid quantum systems.

Mode-coupling effects in anisotropic flow in heavy-ion collisions

Higher-order anisotropic flows in heavy-ion collisions are affected by nonlinear mode coupling effects. It has been suggested that the associated nonlinear hydrodynamic response coefficients probe the transport properties and are largely insensitive to the spectrum of initial density fluctuations of the medium created in these collisions. To test this suggestion, we explore nonlinear mode coupling effects in event-by-event viscous fluid dynamics, using two different models for the fluctuating initial density profiles, and compare the nonlinear coupling coefficients between the initial eccentricity vectors before hydrodynamic expansion and the final flow vectors after the expansion. For several mode coupling coefficients we find significant sensitivity to the initial fluctuation spectrum. They all exhibit strong sensitivity to the specific shear viscosity at freeze-out, but only weak dependence on the shear viscosity during hydrodynamic evolution.

Efficient second harmonic generation in internal asymmetric plasmonic slot waveguide.

We theoretically propose an internal asymmetric plasmonic slot waveguide (IAPSW), containing two different materials in the slot region. The IAPSW is used for second harmonic generation (SHG) at a wavelength of 1.55 μm. The required phase matching condition is satisfied between the 0th-order mode at the fundamental frequency and the 1st-order mode at the second harmonic frequency. By choosing appropriate slot geometry and materials, the mode field distribution is engineered to enhance the nonlinear coupling coefficient for SHG. With an 11 μm long IAPSW, a conversion efficiency of 24% (1.8 × 105 W-1cm-2 normalized conversion efficiency) is predicted. Furthermore, the SHG efficiency is more pronounced in IAPSW with thinner slot.

Landau-Ginzburg description of anomalous properties of novel room temperature multiferroics Pb(Fe1/2Ta1/2)x(Zr0.53Ti0.47)1-xO3 and Pb(Fe1/2Nb1/2)x(Zr0.53Ti0.47)1−xO3

Landau-Ginzburg thermodynamic formalism is used for the description of the anomalous ferroelectric, ferromagnetic, and magnetoelectric properties of Pb(Fe1/2Ta1/2)x(Zr0.53Ti0.47)1−xO3 and Pb(Fe1/2Nb1/2)x(Zr0.53Ti0.47)1−xO3 micro-ceramics. We calculated temperature, composition, and external field dependences of ferroelectric, ferromagnetic, and antiferromagnetic phases transition temperatures, remanent polarization, magnetization, hysteresis loops, dielectric permittivity, and magnetoelectric coupling. Special attention was paid to the comparison of developed theory with experiments. It appeared possible to describe adequately main experimental results including a reasonable agreement between the shape of calculated and measured hysteresis loops and remnant polarization. Since Landau-Ginzburg thermodynamic formalism appertains to single domain properties of a ferroic, we did not aim to describe quantitatively the coercive field under the presence of realistic poly-domain switching. Information about linear and nonlinear magnetoelectric coupling coefficients was extracted from the experimental data. From the fitting of experimental data with theoretical formula, we obtained the composition dependence of Curie-Weiss constant that is known to be inversely proportional to harmonic (linear) dielectric stiffness, as well as the strong nonlinear dependence of anharmonic parameters in free energy. Keeping in mind the essential influence of these parameters on multiferroic properties, the obtained results open the way to govern practically all the material properties with the help of suitable composition choice. A forecast of the strong enough influence of antiferrodistortive order parameter on the transition temperatures and so on the phase diagrams and properties of multiferroics are made on the basis of the developed theory.

Highly efficient second harmonic generation in hyperbolic metamaterial slot waveguides with large phase matching tolerance.

Highly efficient second harmonic generation (SHG) bridging the mid-infrared (IR) and near-IR wavelengths in a coupled hyperbolic metamaterial waveguide with a nonlinear-polymer-filled nanoscale slot is theoretically investigated. By engineering the geometrical parameters, the collinear phase matching condition is satisfied between the even hybrid modes at the fundamental frequency (3,100 nm) and the second harmonic (1,550 nm). Two modes manifest the great field overlap and the significant field enhancement in the nonlinear integration area (i.e. the slot), which leads to extreme large nonlinear coupling coefficient. For a low pumping power of 100 mW, the device length is as short as 2.19 µm and the normalized conversion efficiency comes up to more than 6.37 × 10(5) W(-1)cm(-2) which outperforms that of the plasmonic-based structures. Moreover, the efficient SHG can be achieved with great phase matching tolerance, i.e., a small theoretical fabrication-error sensitivity to filling ratio and a broad pump bandwidth in a compact device length of 2.19 µm using 100 mW pump. The proposed scheme links the mature near-IR devices to the mid-IR regime and have a great potential for integrated chip-scale all-optical signal processes.

Numerical Investigations into the Effects of Multiple Parameters on Nonlinear Piezoelectric Vibration Energy Harvesters

It is very promising to improve the performance of piezoelectric vibration energy harvesters (PVEHs) using nonlinearities in energy harvesting applications. However it is almost impossible to analytically obtain their nonlinear dynamic behaviors. The goal of this paper is to investigate the relations among power outputs, vibration characteristics, and systematic nonlinearities by numerical analysis. A mathematical model of nonlinear PVEH is built by combining nonlinear damping, stiffness, and piezoelectric coupling coefficient. An iterative algorithm is proposed to solve the nonlinear model. Then the effects of these nonlinear parameters on the performance of the PVEH are studied deeply. The results demonstrate that the power output of a nonlinear PVEH will be overestimated or subestimated using a linear model. And nonlinearities can be used for broadband and low frequency vibration energy harvesting.

Dynamics of correlation of quadrature amplitudes fluctuations and of correlation of photon-number fluctuations of field modes in intracavity third-harmonic generation process

The dynamics of correlation function of quantum fluctuations of quadrature amplitudes, as well as the dynamics of correlation function of quantum fluctuations of photon-numbers of fundamental and the third-harmonics mode are investigated for the process of intracavity generation of the third-harmonic. In the case of the evolution of system from initial vacuum state of both modes into the region of great times of interaction, both correlation functions tend to the stationary value. It is shown that these correlations depend strongly on the non-linear coupling coefficient between these modes. In the case of small coupling coefficients the correlation of fluctuations of the quadrature amplitudes as well as the correlation of fluctuations of photon-number are weak. As the coupling coefficient is rising both correlations are also rising (the state of the sub-systems becomes more entangled by the corresponding variables). By further rise of the coupling coefficient the correlation of quadrature amplitudes of fluctuations vanishes (the entanglement of subsystem states over the quadrature amplitudes is also vanishing) but the correlation of fluctuations of the photon-numbers is not vanishing (the subsystems remain entangled over the photon-number variables).

Identification of mode couplings in nonlinear vibrations of the steelpan

The vibrations and sounds produced by two notes of a double second steelpan are investigated, the main objective being to quantify the nonlinear energy exchanges occurring between vibration modes that are responsible of the peculiar sound of the instrument. A modal analysis first reveals the particular tuning of the modes and the systematic occurence of degenerate modes, from the second one, this feature being a consequence of the tuning and the mode localization. Forced vibrations experiments are then performed to follow precisely the energy exchange between harmonics of the vibration and thus quantify properly the mode couplings. In particular, it is found that energy exchanges are numerous, resulting in complicated frequency response curves even for very small levels of vibration amplitude. Simple models displaying 1:2:2 and 1:2:4 internal resonance are then fitted to the measurements, allowing to identify the values of the nonlinear quadratic coupling coefficients resulting from the geometric nonlinearity. The identified 1:2:4 model is finally used to recover the time domain variations of an impacted note in normal playing condition, resulting in an excellent agreement for the temporal behaviour of the first four harmonics.

Quantum dynamics of interacting modes in the process of intracavity subharmonic generation

The dynamics of correlation between fluctuations of the number of photons of interacting modes is investigated for the process of intracavity subharmonic generation. It is demonstrated that this correlation strongly depends on the nonlinear mode coupling coefficient. For small values of the coupling coefficient, the correlation between fluctuations of the number of photons is small. With an increase in the coupling coefficient, the correlation increases (the state of subsystems becomes entangled) and, starting from a particular value of the coupling coefficient, starts to decrease with further increase in the coupling coefficient, gradually approaching zero (entanglement of subsystem states decreases). The quantum dynamics of the number of photons, quantum entropy, and Wigner function of the stationary state of the fundamental and subharmonic modes is investigated. It is demonstrated that the dynamics of these quantities also strongly depends on the coupling coefficient of the interacting modes. We show that, for large values of the mode coupling coefficient and long interaction times, the subharmonic mode becomes localized in a two-component state with equal probabilities of finding it in each component. Quantum entropy of this state is smaller than maximum entropy of the two-component state equal to ln2, which suggests that quantum-mechanical interference takes place between the subharmonic mode components.

Correlation of fluctuations of the quadrature amplitudes of interacting modes in the instability region of intracavity third-harmonic generation

Correlation of fluctuations of quadrature amplitudes of interacting modes is studied for the process of intracavity third-harmonic generation in the region of instability. It is shown that this correlation strongly depends on the value of the nonlinear mode coupling coefficient. For small values of the coupling coefficient, correlation of fluctuations of quadrature amplitudes is small. With increasing coupling coefficient, the correlation increases (states of the subsystems become entangled). Then, starting from a certain value of the coupling coefficient, it starts decreasing gradually, tending to zero (entanglement of states of the subsystems decreases). Correlation of fluctuations of quadrature amplitudes decreases also when the amplitude of the fundamental mode perturbing field is decreased.

Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models

We propose a direct method for computing modal coupling coefficients—due to geometrically nonlinear effects—for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. These coupling coefficients arise when considering a discrete expansion of the unknown displacement onto the eigenmodes of the linear operator. The evolution problem is thus projected onto the eigenmodes basis and expressed as an assembly of oscillators with quadratic and cubic nonlinearities. The nonlinear coupling coefficients are directly derived from the FE formulation, with specificities pertaining to the shell elements considered, namely, here elements of the “Mixed Interpolation of Tensorial Components” family. Therefore, the computation of coupling coefficients, combined with an adequate selection of the significant eigenmodes, allows the derivation of effective reduced-order models for computing—with a continuation procedure —the stable and unstable vibratory states of any vibrating shell, up to large amplitudes. The procedure is illustrated on a hyperbolic paraboloid panel. Bifurcation diagrams in free and forced vibrations are obtained. Comparisons with direct time simulations of the full FE model are given. Finally, the computed coefficients are used for a maximal reduction based on asymptotic nonlinear normal modes, and we find that the most important part of the dynamics can be predicted with a single oscillator equation.

Thermalized polarization dynamics of a discrete optical-waveguide system with four-wave mixing

Statistical mechanics of two coupled vector fields is studied in the tight-binding model that describes propagation of polarized light in discrete waveguides in the presence of the four-wave mixing. The energy and power conservation laws enable the formulation of the equilibrium properties of the polarization state in terms of the Gibbs measure with positive temperature. The transition line $T=\infty$ is established beyond which the discrete vector solitons are created. Also in the limit of the large nonlinearity an analytical expression for the distribution of Stokes parameters is obtained which is found to be dependent only on the statistical properties of the initial polarization state and not on the strength of nonlinearity. The evolution of the system to the final equilibrium state is shown to pass through the intermediate stage when the energy exchange between the waveveguides is still negligible. The distribution of the Stokes parameters in this regime has a complex multimodal structure strongly dependent on the nonlinear coupling coefficients and the initial conditions.

Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations

Nonlinear vibrations of thin rectangular plates are considered, using the von Karman equations in order to take into account the effect of geometric nonlinearities. Solutions are derived through an expansion over the linear eigenmodes of the system for both the transverse displacement and the Airy stress function, resulting in a series of coupled oscillators with cubic nonlinearities, where the coupling coefficients are functions of the linear eigenmodes. A general strategy for the calculation of these coefficients is outlined, and the particular case of a simply supported plate with movable edges is thoroughly investigated. To this extent, a numerical method based on a new series expansion is derived to compute the Airy stress function modes, for which an analytical solution is not available. It is shown that this strategy allows the calculation of the nonlinear coupling coefficients with arbitrary precision, and several numerical examples are provided. Symmetry properties are derived to speed up the calculation process and to allow a substantial reduction in memory requirements. Finally, analysis by continuation allows an investigation of the nonlinear dynamics of the first two modes both in the free and forced regimes. Modal interactions through internal resonances are highlighted, and their activation in the forced case is discussed, allowing to compare the nonlinear normal modes (NNMs) of the undamped system with the observable periodic orbits of the forced and damped structure.

Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics

We theoretically investigate the possible increase of the second harmonic generation (SHG) efficiency in silicon compatible waveguides by considering an asymmetrical plasmonic slot waveguide geometry and a χ((2)) nonlinear polymer infiltrating the slot. The needed phase matching condition is satisfied between the fundamental waveguide mode at the fundamental frequency (FF) and second-order waveguide mode at the second harmonic frequency (SHF) by an appropriate design of the waveguide opto-geometrical parameters. The SHG signal generated in our starting waveguide is three orders of magnitude higher than those previously reported for a FF corresponding to λ = 1550 nm. Then, the SHG performance was further improved by increasing the asymmetry of the structure where nonlinear coupling coefficients as large as 292 psm(-1)W(-1/2) are predicted. The device length is shorter than 20 µm and the normalized SHG conversion efficiency comes up to more than 1 × 10(5) W(-1)cm(-2).

Interplay between relaxation of nonlinear response and coupling coefficient dispersion in the instability spectra of dual core optical fiber

The modulational instability of a dual core optical fiber under the combined effect of coupling coefficient dispersion (CCD) and relaxation of nonlinear response is presented. The effect of wavelength dependence of coupling coefficient is effectively included in the pair of generalized linearly coupled nonlinear Schrödinger equation (CNLSE). In order to account for the finite response time, a time dependent nonlinear response is incorporated in the system of CNLSE using the Debye relaxation model. The results from the linear stability analysis show that in anomalous dispersion regime for any finite relaxation time, the nonlinearity becomes complex and there exist two unstable modes. In the normal dispersion regime, a single unstable band at higher detuning frequency is observed which is identified as the Raman band. In symmetric case the CCD is found to be immaterial, whereas in asymmetric case the CCD is found to be crucial and brings new spectral bands. A critical value of CCD is predicted where the dynamics evolve dramatically in different manners. Thus in this paper, the MI dynamics of a dual core optical fiber is fully explored and the interplay between CCD and relaxation is clearly highlighted.

Modulational instability in a twin-core fiber with the effect of saturable nonlinear response and coupling coefficient dispersion

Modulational instability (MI) in a twin-core optical fiber with the combined effect of saturation of nonlinearity (SNL) and coupling coefficient dispersion (CCD) is presented. The CCD does not dramatically modify the spectrum of the symmetric or antisymmetric continuous wave (CW) case, but the SNL on the other hand behaves in a perceptible manner such that the gain and the unstable region inherently decrease. In the anomalous dispersion region, the gain of the instability spectrum increases (decreases) monotonously with power (coupling coefficient). The effective nonlinearity and the power threshold for the sustained CW solution becomes a function of power for the SNL case. The so-called nonlinear factor behaves in a unique way such that there exist two powers for the same value of nonlinear factor. The interplay between CCD and SNL is emphasized, where any nonzero value of CCD leads to new instability bands and the saturation on the other hand suppresses the gain of the instability band. We identified a pair of power corresponding to the constant value of the nonlinear factor, where the system becomes invariant, such that the number of instability bands, the gain, and the range of the unstable region are all preserved. In the case of the normal dispersion case, the MI is achieved purely by means of the coupling coefficient. In both cases, a critical CCD is predicted, where the system evolves dramatically in a different manner.

Phase regeneration of phase-shift keying signals in highly nonlinear hybrid plasmonic waveguides

Phase regeneration of phase-shift keying signals is theoretically proposed, we believe for the first time, based on the efficient optical parametric amplification (OPA) process in a highly nonlinear symmetric hybrid plasmonic waveguide. This optimized stacked waveguide with nonlinear organic materials has a relatively low loss of about 0.005 dB/μm and an effective nonlinear OPA coupling coefficient up to 60 ps/m/W(1/2). The phase-recovery process was achieved in this waveguide within a length as short as 150 μm.