Simultaneous Resonances Research Articles

Grazing-angle scattering of electromagnetic waves in periodic Bragg arrays

A new powerful approximate approach for the theoretical analysis of Bragg scattering in oblique strip-like periodic arrays with the scattered wave propagating almost parallel to the array boundaries – grazing-angle scattering (GAS) – is introduced and justified. This approach is based on allowance for the diffractional divergence of the scattered wave by means of the parabolic equation of diffraction and Fourier analysis. The divergence is demonstrated to be an intrinsic physical cause of GAS. Detailed theoretical analysis of steady-state GAS is carried out for bulk and guided optical modes. It is demonstrated that the most interesting feature of GAS in arrays of width that is greater than a critical width is a unique combination of two strong simultaneous resonances with respect to frequency and angle of scattering. In such wide arrays, GAS is demonstrated to be not only unusually sensitive to angle of scattering, but also to small variations of array width and grating amplitude. Entire concentration of the resonantly strong scattered wave inside the array is shown to be possible. A relationship between GAS, conventional Bragg scattering, and extremely asymmetrical scattering (i.e. where the scattered wave propagates parallel to the array boundaries) is analysed. Applicability conditions for the used approximations and obtained results are derived and discussed.

Double-resonant extremely asymmetrical scattering of electromagnetic waves in non-uniform periodic arrays

A new type of Bragg scattering - double-resonant extremely asymmetrical scattering (DEAS) of optical waves in oblique, non-uniform, periodic Bragg arrays is analysed theoretically and numerically. Steady-state DEAS is demonstrated to occur in the extremely asymmetrical geometry where the scattered wave propagates parallel to the front array boundary. The non-uniform array is represented by two joint uniform, strip-like, periodic arrays with different phases (and amplitudes) of the grating. DEAS is characterised by a unique combination of two simultaneous resonances with respect to frequency and phase variation at the interface between the joint arrays. As a result, a strong resonant increase in the scattered wave amplitude compared with the amplitude of the incident wave is predicted and investigated theoretically. The amplitude of the incident wave inside the array is also shown to increase resonantly in the middle of the array where the step-like variation in the phase of the grating takes place. The effect of different widths of the joint arrays, and magnitudes of the grating amplitudes on DEAS is analysed. Physical explanations of this type of scattering, based on the diffractional divergence of the scattered wave from one of the joint arrays into another, are presented.

Extremely asymmetrical scattering of electromagnetic waves in gradually varying periodic arrays

This paper analyses theoretically and numerically the effect of varying grating amplitude on the extremely asymmetrical scattering (EAS) of bulk and guided optical modes in non-uniform strip-like periodic Bragg arrays with stepwise and gradual variations in the grating amplitude across the array. A recently developed new approach based on allowance for the diffractional divergence of the scattered wave is used for this analysis. It is demonstrated that gradual variations in magnitude of the grating amplitude may change the pattern of EAS noticeably but not radically. On the other hand, phase variations in the grating may result in a radically new type of Bragg scattering - double-resonant EAS (DEAS). In this case, a combination of two strong simultaneous resonances (one with respect to frequency, and another with respect to the phase variation) is predicted to take place in non-uniform arrays with a step-like phase and gradual magnitude variations of the grating amplitude. The tolerances of EAS and DEAS to small gradual variations in the grating amplitude are determined. The main features of these types of scattering in non-uniform arrays are explained by the diffractional divergence of the scattered wave inside and outside the array.

Stochastic excitation of suspended cables involving three simultaneous internal resonances using Monte Carlo simulation

In the neighborhood of three simultaneous internal resonance conditions among four normal modes, the response behavior of a suspended cable to random Gaussian excitation is studied using computational simulations. The study includes the interaction between the first two in-plane and the first two out-of-plane modes. When the first in-plane mode is externally excited its response can act as a parametric excitation to the other three modes through nonlinear coupling. The governed modal equations of motion are numerically simulated using a double-precision, initial value problem solver Adam-Moulton algorithm. The random excitation is generated numerically from a normal distribution using an inverse Cumulative Distribution Function (CDF) technique. The numerical results are processed to estimate response statistics such as mean squares power spectra, probability density functions, and autocorrelation functions. The modes, which are indirectly excited, exhibit the well-known phenomenon ‘on-off intermittency’ in the neighborhood of stochastic bifurcation. Furthermore, in the neighborhood of internal resonances the directly excited mode is found to transfer energy to other modes. Under Gaussian excitation the response coordinates of the four modes are essentially non-Gaussian narrow band random processes with a central frequency close to each mode natural frequency.

Coherent Raman scattering with incoherent light for a multiply resonant mixture: Theory

The theory for coherent Raman scattering (CRS) with broadband incoherent light is presented for a multiply resonant, multicomponent mixture of molecules that exhibits simultaneous multiple resonances with the frequencies of the driving fields. All possible pairwise hyperpolarizability contributions to the signal intensity are included in the theoretical treatment---(resonant-resonant, resonant-nonresonant, and nonresonant-nonresonant correlations between chromophores) and it is shown how the different types of correlations manifest themselves as differently behaved components of the signal intensity. The Raman resonances are modeled as Lorentzians in the frequency domain, as is the spectral density of the incoherent light. The analytic results for this multiply resonant mixture are presented and applied to a specific binary mixture. These analytic results will be used to recover frequencies and dephasing times in a series of experiments on multiply resonant mixtures.

Dynamics of Nonlinear Oscillators under Simultaneous Internal and External Resonances

An analysis is presented for a class of two degree of freedom weakly nonlinear oscillators, with symmetric restoring force. Conditions of one-to-three internal resonance and subharmonic external resonance of the lower vibration mode are assumed to be satisfied simultaneously. As a consequence, the second vibration mode may also be under the action of external primary resonance. Initially, a set of slow-flow equations is derived, governing the amplitudes and phases of approximate long time response of these oscillators, by applying an asymptotic analytical method. Determination of several possible types of steady-state motions is then reduced to solution of sets of algebraic equations. For all these solution types, appropriate stability analysis is also performed. In the second part of the study, this analysis is applied to an example mechanical system. First, a systematic search is performed, revealing effects of system parameters on the existence and stability properties of periodic motions. Frequency-response diagrams are presented and attention is focused on understanding the evolution and interaction of the various solution branches as the external forcing and nonlinearity parameters are varied. Finally, numerical integration of the equations of motion demonstrates that the system exhibits quasiperiodic or chaotic response for some parameter combinations.

Multiple Internal Resonance in Suspended Cables under Random In-Plane Loading

The random excitation of a suspended cable with simultaneous internal resonances is considered. The internal resonances can take place among the first in-plane and the first two out-of-plane modes. The external loading is represented by a wide-band random process. The response statistics are estimated using the Fokker-Planck-Kolmogorov (FPK) equation, together with Gaussian and non-Gaussian closures. Monte Carlo simulation is also used for numerical verification. The unimodal in-plane motion exists in regions away from the internal resonance condition. The mixed mode interaction is manifested within a limited range of internal detuning parameters, depending on the excitation power spectrum density and damping ratios. The Gaussian closure scheme failed to predict bounded solutions of mixed mode interaction. The non-Gaussian closure results are in good agreement with the Monte Carlo simulation. The on-off intermittency of the autoparametrically excited modes is observed in the Monte Carlo simulation over a small range of excitation levels. The influence of the cable parameters, such as damping ratios, sag-to-span ratio, internal detuning parameters, and excitation level on the autoparametric interaction, is studied. It is found that the internal detuning and excitation level are the two main parameters which affect the autoparametric interaction among the three modes. Due to the system's nonlinearity, the response of the three modes is strongly non-Gaussian and the coupled modes experience irregular modulation.

Random excitation of coupled oscillators with single and simultaneous internal resonances

The primary objective of this paper is to examine the random response characteristics of coupled nonlinear oscillators in the presence of single and simultaneous internal resonances. A model of two coupled beams with nonlinear inertia interaction is considered. The primary beam is directly excited by a random support motion, while the coupled beam is indirectly excited through autoparametric coupling and parametric excitation. For a single one-to-two internal resonance, we used Gaussian and non-Gaussian closures, Monte Carlo simulation, and experimental testing to predict and measure response statistics and stochastic bifurcation in the mean square. The mean square stability boundaries of the coupled beam equilibrium position are obtained by a Gaussian closure scheme. The stochastic bifurcation of the coupled beam is predicted theoretically and experimentally. The stochastic bifurcation predicted by non-Gaussian closure is found to take place at a lower excitation level than the one predicted by Gaussian closure and Monte Carlo simulation. It is also found that above a certain excitation level, the solution obtained by non-Gaussian closure reveals numerical instability at much lower excitation levels than those obtained by Gaussian and Monte Carlo approaches. The experimental observations reveal that the coupled beam does not reach a stationary state, as reflected by the time evolution of the mean square response. For the case of simultaneous internal resonances, both Gaussian and non-Gaussian closures fail to predict useful results, and attention is focused on Monte Carlo simulation and experimental testing. The effects of nonlinear coupling parameters, internal detuning ratios, and excitation spectral density level are considered in both investigations. It is found that both studies reveal common nonlinear features such as bifurcations in the mean square responses of the coupled beam and modal interaction in the neighborhood of internal resonances. Furthermore, there is an upper limit for the excitation level above which the system experiences unbounded response in the neighborhood of simultaneous internal resonances.

An asymptotic analysis of the 1:3:4 Hamiltonian normal form

The normal form of the Hamiltonian 1:3:4 resonance, which exhibits two simultaneous resonances of differing orders, is studied asymptotically. Since the two resonances have different strengths, the exact solution of the primary single resonance system may be used to construct an action-angle transformation. The resulting standard form system is solved asymptotically by canonical near-identity averaging transformations. In addition to the Hamiltonian itself and its unperturbed part, which are two exact constants of the motion, a third independent adiabatic invariant of the original Hamiltonian system is constructed. The results apply directly to the problem of a free electron laser with weak self-fields. A specific model problem is studied numerically to verify the asymptotic validity of the results over long times.

Third harmonic generation spectrum of a degenerate ground state conjugated polymer. Direct evidence of simultaneous two- and three-photon resonance

We report the third harmonic generation (THG) spectrum of poly(dioctyldipropargylmalonate) (PDDPM), covering pump energies from 0.6 to 1.4 eV. Like trans-polyacetylene, PDDPM has a bond-alternated structure and a twofold degenerate ground state. The phase of the nonlinear optical susceptibility, χ (3)(3 ω), undergoes a 2π shift, demonstrating that two nearly simultaneous resonances (at 0.7 and 0.9 eV) are traversed, as expected for degenerate ground state polymers where the two-photon mid-gap state formed by neutral soliton-antisoliton pairs plays a dominant role in χ (3) processes.

Asymptotic analysis of a class of three-degree-of-freedom Hamiltonian systems near stable equilibria

Abstract A conservative near-integrable Hamiltonian dynamical system is examined, which to leading order consists of three uncoupled harmonic oscillators with constant frequencies in the ratio 1:2:α for certain rational α. Formally, the problem considered can arise by perturbing any three-degree-of-freedom Hamiltonian near a stable equilibrium point, so that the Hamiltonian consists of a power series expansion in a small parameter, where successive terms are homogeneous polynomials of increasing degree in the coordinates and the momenta. The special case of two exact simultaneous resonances, one in the first perturbation term and one in the second, is examined and explicit asymptotic solutions are obtained. The solution procedure involves reducing the original Hamiltonian to two degrees of freedom using one integral of the motion; then transforming to standard form to find two additional adiabatic invariants by near-identity averaging canonical transformations. A specific example is studied numerically to verify the asymptotic validity of the results over long times.

Experimental Investigation of Isolated and Simultaneous Internal Resonances in Suspended Cables

The near resonant response of suspended elastic cables driven by harmonic, planar excitation is investigated experimentally. Measurements of large amplitude cable motions confirm previous theoretical predictions of fundamental classes of internally-resonant responses. For particular magnitudes of equilibrium curvature, strong modal interactions arise through isolated (two-mode) or simultaneous (three-mode) internal resonances. Four qualitatively different periodic responses are observed: (1) pure planar response, (2) 2:1 internally resonant nonplanar response, (3) 1:1 internally resonant nonplanar response, and (4) simultaneous, 2:2:1 internally resonant nonplanar response. Quasiperiodic responses are also observed.

Three-dimensional oscillations of suspended cables involving simultaneous internal resonances

The near resonant response of suspended, elastic cables driven by planar excitation is investigated using a three degree-of-freedom model. The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1:1 and 2:1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions. The perturbation solutions are compared with results obtained by numerically integrating the equations of motion. Furthermore, numerical simulations demonstrate the existence of quasiperiodic responses.

Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions

Abstract A four-degree-of-freedom model able to capture the main phenomena which are likely to occur in the non-planar finite dynamics of an elastic suspended cable subjected to external forcings and support motions is developed from the continuum equations. It contains two in-plane and two out-of-plane components, one symmetric and one antisymmetric. An order two multiple time scales solution is obtained for the case of primary external resonance and three simultaneous internal resonances (cable at crossover frequency). Possible classes of steady state (planar/non-planar, unimodal/multimodal) regular motions of the system are identified, and their linearized stability analysis is performed. Some results are presented to highlight the role played by key perturbations on the onset of various classes, the rich behaviour of the system and the influence of some control parameters on the response.

Determining band offsets with triple quantum-well structures

Coupled triple quantum-well GaAs/AlGaAs heterostructures have been designed to exhibit simultaneous electron and hole tunneling from the central well to opposite side wells at a fixed applied bias. Band offset is the critical parameter for the design of structures with simultaneous resonances. Photocurrent measurements reveal which triple quantum-well structures exhibit simultaneous resonances. A band offset ratio near 62:38 is required to correctly engineer structures with simultaneous resonances.

On the Occurrence of Simultaneous Resonances in Parametrically-Excited Rectangular Plates

The present work deals primarily with the problem of the occurrence of simultaneous resonances in parametrically-excited rectangular plates. The analysis is based on the dynamic analog of the von Karman’s large-deflection theory and the governing equations are satisfied using the orthogonality properties of the assumed functions. The temporal response of the damped system is determined by the generalized asymptotic method and various types of simultaneous resonances are investigated. An experimental investigation is performed to verify the analytical predictions and to possibly discover new phenomena not predicted by the theory. Experiments are conducted for four different sets of boundary conditions. Analytically defined simultaneous resonances are experimentally observed and isolated.

Nonlinear line narrowing of excited state resonances in perylene doped poly(methylmethacrylate) using multiply enhanced nonparametric spectroscopy

Line narrowing of perylene vibronic transitions in poly(methylmethacrylate) (PMMA) samples has been demonstrated using multiply enhanced nonparametric spectroscopy. Three lasers are used to generate a four wave mixing signal that is resonantly enhanced by three simultaneous perylene resonances involving a vibrational state, a vibrationless excited electronic state, and a vibronic state of the excited electronic state. There is a correlation between the position of the electronic and vibronic states within the inhomogeneously broadened absorption spectrum. The correlation can be used to achieve line narrowing by fixing one of the resonances to a subset of perylene sites within the inhomogeneous profile so the other resonance of those sites will be selectively enhanced. The position of the line narrowed vibronic peaks depends upon the electronic detuning from the center of the inhomogeneously broadened absorption band as different sites within the band are probed. Excited state coherent processes involving resonance with higher singlet states are shown to be unimportant from the detuning dependence on a simultaneous vibrational enhancement.

Novel method for accurate g measurements in electron-spin resonance

In high-accuracy work, electron-spin-resonance (ESR) g values are generally determined by calibrating against the accurately known proton nuclear magnetic resonance (NMR). For that method—based on leakage of microwave energy out of the ESR cavity—a convenient technique is presented to obtain accurate g values without needing conscientious precalibration procedures or cumbersome constructions. As main advantages, the method allows the easy monitoring of the positioning of the ESR and NMR samples while they are mounted as close as physically realizable at all time during their simultaneous resonances. Relative accuracies on g of ≊2×10−6 are easily achieved for ESR signals of peak-to-peak width ΔBpp≲0.3 G. The method has been applied to calibrate the g value of conduction electrons of small Li particles embedded in LiF—a frequently used g marker—resulting in gLiF:Li=2.002 293±0.000 002.

Effect of boundary conditions on the dynamic instability and non-linear response of rectangular plates, Part II: Experiment

Experiments are performed to investigate the dynamic stability and responses of four rectangular plates subjected to periodic in-plane loads and under four different sets of boundary conditions. Special attention is paid to satisfy the boundary conditions assumed in the analytical models so as to draw conclusion with sufficient degree of confidence. In general, the experimental data exhibit good agreement with the theoretical predictions. A loosely clamped plate is the most susceptible to various resonances and all steady state response curves associated with the lower mode shapes for this type of boundary condition exhibit hardest spring effects. On the other hand, a simply supported plate is the most prone to the existence of large amplitudes of vibration and this kind of boundary condition is the only one easily reproducible in laboratory. The results also show that a parametric combination resonance is not important as compared with a principal parametric resonance for the problem under investigation. Some physical aspects of the problem, such as the drop-out beyond the instability zone, the effects of initial imperfections and the occurrence of simultaneous resonances, are also discussed.

The effect of nonlinearities on the response of a single-machine-quasi-infinite busbar system

A single-machine quasi-infinite busbar power system is formulated taking into consideration quadratic and cubic nonlinearities. The model equation contains parametric (time-varying coefficients) and external (inhomogeneous terms) excitations. The method of multiple scales is used to approximate the response of the system to simultaneous principal parametric resonances and subharmonic resonances of order one-half. In contrast with the linear analysis, the nonlinear analysis shows that the response can exhibit: (1) limit cycles instead of infinite motions; (2) multivaluedness that can lead to jumps; (3) subcritical instabilities; and (4) constructive and destructive interference of resonances.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>