Manley-Rowe Relations Research Articles

Triad mode resonant interactions in suspended cables

A triad mode resonance, or three-wave resonance, is typical of dynamical systems with quadratic nonlinearities. Suspended cables are found to be rich in triad mode resonant dynamics. In this paper, modulation equations for cable’s triad resonance are formulated by the multiple scale method. Dynamic conservative quantities, i.e., mode energy and Manley-Rowe relations, are then constructed. Equilibrium/dynamic solutions of the modulation equations are obtained, and full investigations into their stability and bifurcation characteristics are presented. Various bifurcation behaviors are detected in cable’s triad resonant responses, such as saddle-node, Hopf, pitchfork and period-doubling bifurcations. Nonlinear behaviors, like jump and saturation phenomena, are also found in cable’s responses. Based upon the bifurcation analysis, two interesting properties associated with activation of cable’s triad resonance are also proposed, i.e., energy barrier and directional dependence. The first gives the critical amplitude of high-frequency mode to activate cable’s triad resonance, and the second characterizes the degree of difficulty for activating cable’s triad resonance in two opposite directions, i.e., with positive or negative internal detuning parameter.

An extra invariant for the system of four wave packets in one spatial dimension

The system of four wave packets (like light pulses in optical fibers) is shown to have an extra adiabatic-like invariant, in addition to the energy, momentum, and Manley–Rowe relations. A theoretical argument is suggested showing that the 4-wave interaction in one-dimensional systems can restrict spectral broadening.

Three-wave interaction and Manley–Rowe relations in quantum hydrodynamics

The theory for nonlinear three-wave interaction in magnetized plasmas is reconsidered using quantum hydrodynamics. The general coupling coefficients are calculated for the generalized Bohm de Broglie term. It is found that the Manley–Rowe relations are fulfilled only if the form of the particle dispersive term coincides with the standard expression. The implications of our results are discussed.

Geometric interpretation of four-wave mixing

The nonlinear phenomenon of four-wave mixing (FWM) is investigated using a method, where, without the need of calculus, both phase and amplitudes of the mixing fields are visualized simultaneously, giving a complete overview of the FWM dynamics. This is done by introducing a set of Stokes-like coordinates of the electric fields, which reduce the FWM dynamics to a closed two-dimensional surface, similar to the Bloch sphere of quantum electrodynamics or the Pointcar\'e sphere in polarization dynamics. The coordinates are chosen so as to use the gauge invariance symmetries of the FWM equations which also give the conservation of action flux known as the Manley-Rowe relations. This reduces the dynamics of FWM to the one-dimensional intersection between the closed two-dimensional surface and the phase-plane given by the conserved Hamiltonian. The analysis is advantageous for visualizing phase-dependent FWM phenomena which are found in a large variety of nonlinear systems and even in various optical communication schemes.

A Phase-Coherent Upconverting Parametric Amplifier

A phase-coherent negative-resistance upconverting parametric amplifier is presented that upconverts to three times the source input frequency and exhibits over 600 MHz, or 33%, fractional usable bandwidth at the upconverted output. An analytical model for the transducer gain is presented, enabling an initial selection of the termination resistors and pump signal. A prototype board was constructed and measured data matches well with both simulated results and those predicted by the analytical model. Simulations, measurements, and the analytic derivation demonstrate that when the source and pump frequencies are phase coherent, it is possible to achieve a transducer gain greater than the theoretical limit imposed by the Manley-Rowe relations for phase-incoherent upconverters.

Nonlinear wave interaction and spin models in the magnetohydrodynamic regime

Here we consider the influence on the electron spin in the magnetohydrodynamic (MHD) regime. Recently developed models that include spin–velocity correlations are taken as the starting point. A theoretical argument is presented, suggesting that in the MHD regime a single-fluid electron model with spin correlations is equivalent to a model with spin-up and spin-down electrons constituting different fluids, but where the spin–velocity correlations are omitted. Three-wave interaction of two shear Alfvén waves and a compressional Alfvén wave is then taken as a model problem to evaluate the asserted equivalence. The theoretical argument turns out to be supported, because the predictions of the two models agree completely. Furthermore, the three-wave coupling coefficients obey the Manley–Rowe relations, which further support the soundness of the models and the validity of the assumptions made in the derivation. Finally, we point out that the proposed two-fluid model can be incorporated in standard particle-in-cell schemes with only minor modifications.

Variational formulation for weakly nonlinear perturbations of ideal magnetohydrodynamics

AbstractA new equation of motion that governs weakly nonlinear phenomena in ideal magnetohydrodynamics (MHDs) is derived as a natural extension of the well-known linearized equation of motion for the displacement field. This derivation is made possible by expanding the MHD Lagrangian explicitly up to third order with respect to the displacement of plasma, which necessitates an efficient use of the Lie series expansion. The resultant equation of motion (i.e. the Euler–Lagrange equation) includes a new quadratic force term which is responsible for various mode–mode coupling due to the MHD nonlinearity. The third-order potential energy serves to quantify the coupling coefficient among resonant three modes and its cubic symmetry proves the Manley–Rowe relations. In contrast to earlier works, the coupling coefficient is expressed only by the displacement vector field, which is already familiar in the linear MHD theory, and both the fixed and free boundary cases are treated systematically.

Fragments of the theory of nanotransistors: self-switching of a plane transverse hypersonic wave in nonlinearly elastic nanocomposite materials

We present fragments of the theory of nanotransistors concerning general information about transistors, models of nanocomposite materials, effects of nonlinear interaction of waves, and a theoretical analysis of the interaction of cubically nonlinear elastic plane harmonic waves in materials whose nonlinear properties are described by the Murnaghan potential. Using the method of slowly varying amplitudes, we investigate the interaction of two harmonic vertical transverse plane waves. Shortened and evolution equations, and Manley-Rowe relations, are obtained. We analyze analytically and numerically the mechanism of repumping of the energy of a strong pumping wave that propagates at a frequency of ω to a weak signal wave propagating at a frequency of 3ω . The described mechanism of switching of hypersonic waves in a nonlinearly elastic nanomaterial is similar to the mechanism of switching observed in optical and other transistors.

NONDEGENERATE PARAMETRIC AMPLIFICATION IN SUPERLATTICES AND THE LIMITS OF STRONG AND WEAK DISSIPATION

We develop a semiclassical theory of the nondegenerate parametric amplification in a single miniband of superlattice. We present the formulas describing absorption and gain of signal and idler fields in superlattice and analyze the limiting cases of strong and weak dissipation. We show how the well-known Manley-Rowe relations arise in the tight-binding lattice in the weak dissipation limit. Our results can be applied to an amplification of THz signals in semiconductor superlattices and a control of nonlinear transport of cold atoms in optical lattices.

Manley–Rowe relations for an arbitrary discrete system

Manley–Rowe relations are formulated for a discrete Hamiltonian system with an arbitrary number of resonances. Assuming that the resonances are defined as R ˆ | ω 〉 = 0 , where R ˆ is an n × n integer matrix of rank r < n , and | ω 〉 ≡ ( ω 1 , … , ω n ) T is the frequency vector, the projection of the action vector | J 〉 on ker R ˆ is an adiabatic invariant. Hence n − r independent integrals, from where the conventional Manley–Rowe relations for a single resonance follow as a particular case.

Diffusion paths in resonantly driven Hamiltonian systems

A diffusive Hamiltonian flow triggered by a resonant drive is confined to a phase subspace determined by the resonance structure. The diffusion path is found for an arbitrary, possibly nonstationary discrete system by applying generalized Manley–Rowe relations to an extended Hamiltonian.

Spatially autoresonant stimulated Raman scattering in nonuniform plasmas

New solutions to the coupled three-wave equations in a nonuniform plasma medium are presented that include both space and time dependence of the waves. By including the dominant nonlinear frequency shift of the material wave, it is shown that if the driving waves are sufficiently strong (in relation to the medium gradient), a nonlinearly phase-locked solution develops that is characteristic of autoresonance. In this case, the material (electrostatic) wave develops into a front starting at the linear resonance point and moving with the wave group velocity in a manner such that the intensity increases linearly with the propagation distance. The forms of the other two (electromagnetic) waves follow naturally from the Manley–Rowe relations.

Self-switching of a transverse plane wave propagating through a two-component elastic composite

The paper discusses the results of theoretical and numerical analysis of the interaction of nonlinear elastic plane harmonic waves in a composite material whose nonlinear properties are described by modeling it with a two-phase mixture. The interaction of two transverse vertically polarized harmonic waves is studied using the method of slowly varying amplitudes. The truncated and evolutionary equations as well as the Manley-Rowe relations are derived. The mechanism of energy pumping from a strong pumping wave with frequency ω to a weak signal wave with frequency 3ω is analyzed. The switching mechanism for hypersonic waves in a nonlinear elastic composite is similar to the switching mechanism observed in transistors

About quantum and classical schemes of LWI and the violation of Manley-Rowe relations

The analogy between quantum and classical systems of lasing without inversion (LWI) is developed. The realization of different regimes of simultaneous “nonresonant” amplification (or absorption) of two coherent waves coupled parametrically are demonstrated for a quantum five-level scheme. This effect conflicting with Manlew-Rowe relations is confirmed with the attraction of a quantum description of radiation and medium.

Vlasov models for the study of stimulated Raman scattering and beatwave acceleration scenario

Relativistic Vlasov–Maxwell model was developed to address the particle trapping or particle acceleration in phase space, including situation relevant to Stimulated Raman Scattering and plasma beatwave acceleration. In particular the case of realistically high ratio of driver frequency to plasma wave frequency ( ω 0 / ω p ∼ 30 ) was also considered using a hybrid version called Vlasov–Hilbert model. Some of the more striking features that have emerged from the Vlasov simulations are discussed in this paper, with particular emphasis on time-dependent matching relations in frequency and wavevector allowing the maintenance of the parametric resonance over a long time, particle dynamics in phase space and action transfer results obtained from the derivation of the integrated Manley–Rowe relations derived for a finite causal system.

Parametric cascade downconverter for intense ultrafast mid-infrared generation beyond the Manley-Rowe limit

We propose a scheme to generate intense, ultrafast mid-infrared pulses with conversion efficiencies exceeding the upper bound for single-stage difference-frequency mixing as predicted by the Manley-Rowe relations. Finite-element fast Fourier transform simulations of the mixing process show that the parametric cascade downconverter generates 1.7 times more photons (at 10 microm) than in the initial pump pulse (center wavelength of 1.48 microm, duration of 130 fs, and pulse energy of 50 microJ), with negligible pulse spatial and temporal distortion.

On the interaction of cubically nonlinear transverse plane waves in an elastic material

The paper presents theoretical results on the interaction of cubically nonlinear harmonic elastic plane waves in a nonlinear material described by the Murnaghan potential. The interaction of two harmonic transverse waves is studied using the method of slowly varying amplitude. Reduced and evolution equations and the Manley-Rowe relations are derived. An analysis is made of the mechanism of energy transfer from the strong pumping wave, which has frequency ω, to the weak signal wave, which has frequency 3ω because of this interaction. A switching mechanism for hypersonic waves in a nonlinear elastic material is described, which is similar to the switching mechanism observed in transistors

Negative-index metamaterials: second-harmonic generation, Manley–Rowe relations and parametric amplification

Second harmonic generation and optical parametric amplification in negative-index metamaterials (NIMs) are studied. The opposite directions of the wave vector and the Poynting vector in NIMs results in a "backward" phase-matching condition, causing significant changes in the Manley-Rowe relations and spatial distributions of the coupled field intensities. It is shown that absorption in NIMs can be compensated by backward optical parametric amplification. The possibility of distributed-feedback parametric oscillation with no cavity has been demonstrated. The feasibility of the generation of entangled pairs of left- and right-handed counter-propagating photons is discussed.

Nonlinear interaction between three kinetic Alfvén waves

Using Hall-MHD theory, we focus on the resonant interaction between three kinetic Alfvén waves. We thus derive three coupled equations which govern this process. It turns out that these equations contain the same coupling coefficient, directly showing that they satisfy the Manley–Rowe relations. The coupling coefficient can be written in a comparatively very simple form, that has not been deduced before. The decay rate, when a pump kinetic Alfvén wave decays into two similar Alfvén waves, is therefore easily estimated for plasmas of astrophysical interest.

Evolution Equations for Plane Cubically Nonlinear Elastic Waves

Consideration is given to the nonlinear theory of elastic waves with cubic nonlinearity. This nonlinearity is separated out, and the interaction of four harmonic waves is studied. The method of slowly varying amplitudes is used. The shortened and evolution equations, the first integrals of these equations (Manley–Rowe relations), and energy balance law for a set of four interacting waves (quadruplet) are derived. The interaction of waves is described using the wavefront reversal scheme