Four-wave Interaction Research Articles

Quantum optical effects in semiconductor microcavities

Investigations of quantum effects in semiconductor quantum-well microcavities interacting with laser light in the strong-coupling regime are presented. Modifications of quantum fluctuations of the outgoing light are expected due to the non-linearity originating from coherent exciton–exciton scattering. In the strong-coupling regime, this scattering translates into a four-wave mixing interaction between the mixed exciton–photon states, the polaritons. Squeezing and giant amplification of the polariton field and of the outgoing light field fluctuations are predicted. However, polariton–phonon scattering is shown to yield excess noise in the output field, which may destroy the non-classical effects. Experiments demonstrate evidence for giant amplification due to coherent four-wave mixing of polaritons. Noise reduction below the thermal noise level was also observed. To cite this article: E. Giacobino et al., C. R. Physique 3 (2002) 41–52

The nonlinear dynamics of the modulational instability of drift waves and the associated zonal flows

The linear and nonlinear dynamics of zonal flows and their interactions with drift wave turbulence is considered in the simple but illuminating generalized Charney–Hasegawa–Mima model due to Smolyakov et al. [Phys. Plasmas 7, 1349 (2000)]. Two positive definite, exact, integral invariants associated with the full generalized Charney–Hasegawa–Mima system are derived. For an initial monochromatic drift wave pump with small but finite amplitude, a modulational instability can occur, characterized by growing zonal flow and sideband perturbations (i.e., a four-wave interaction). The pump threshold for instability is readily satisfied, depending on the zonal flow wave number. The fully nonlinear Charney–Hasegawa–Mima equations are solved with a numerical scheme which is validated by demonstrating the conservation of the two exact invariants. The simulations show that the validity of the four-wave model is limited to approximately three instability growth times. The radial structure of the zonal flow can be “jet-like” or highly oscillatory depending upon the ratio of the system size to the density scale length and initial conditions. It is found that zonal flows can be dramatically reduced if the most unstable zonal flow wave number does not fit into the system.

Multiple four-wave mixing–induced modulational instability in highly birefringent fibers

Theoretical and experimental results are presented that illustrate efficient generation of new optical frequencies by means of induced modulational instability in birefringent fibers for an initially highly phase-mismatched process. Modulational instability is assisted by multiple four-wave mixing interactions. This technique relaxes the strict spectral limitations imposed by the phase-matching conditions on the signal used for frequency conversion by means of modulational instability.

Parametric excitations of fast plasma waves by counterpropagating laser beams.

Short- and long-wavelength plasma waves can become strongly coupled in the presence of two counterpropagating laser pump pulses detuned by twice the cold-plasma frequency omega(p). What makes this four-wave interaction important is that the growth rate of the plasma waves occurs much faster than in the more obvious copropagating geometry.

Nonlinear coherent four-wave interaction in space plasma

According to a widespread point of view, intensive electrostatic structures in the E-region of the auroral ionosphere can be a consequence of the excitation of the modified two-stream or Farley-Buneman (FB) plasma turbulence. But in spite of the successes of the theoretical and experimental research of the auroral radar scattering, it is impossible to explain the existence of auroral echoes with large aspect angles (> 2 deg.), the wave propagation perpendicular to the electron drift velocity and wave scales less than 1 m. In this paper the coherent nonlinear interactions of three and four electrostatic FB-waves are considered analytically and numerically. The evolution of the nonlinear waves is described by a system of magnetohydrodynamic equations. 1) It is shown that the interaction of three and four coherent waves is the main physical mechanism which leads to the saturation of the FB-instability. 2) If no dissipative and dispersive effects occur, an explosive instability may be excited. 3) The main result of the interaction of coherent waves is the generation of nonlinear waves and nonlinear structures when the waves are damped linearly and propagate perpendicular to the electron drift velocity. This region corresponds to large aspect angles of the small-scale waves. 4) Further, the wave interaction causes a nonlinear stabilization of the growth of the high-frequency waves and a formation of local density structures of the charged particles. The results of the numerical models allow to analyse the possibility of scenarios of the two-stream plasma instability in the collisional auroral E-region.

Nonlinear coherent four-wave interaction in space plasma

Nonlinear coherent four-wave interaction in space plasma

Autowave instability in refractive crystals

Within the framework of four-wave interactions, an analysis was made of the autowave instability in crystals with cubic nonlinearity. The analysis takes into account the transverse nonuniformity of the interacting fields. The conditions of the existence of this phenomenon were found. The influence of the photovoltaic effect and the charge carrier diffusion on the frequency spectrum of autowaves were studied.

Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: Saturation and polarization effects

The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.

Amplification of a probe field without population inversion in a two-level atom driven by a strong bichromatic pump field

The polarization spectrum of a two-level atom driven by a symmetrical bichromatic pump field is examined. It is shown that the heterodyne parametric four-wave interactions result in the amplification of a weak probe field without population inversion at the line center in a wide range of pump and probe field intensities.

REDUCTION PROBLEM FOR FOUR WAVE INTERACTION AND THE CLASSICAL r MATRIX

Integrable nonlinear partial differential equations, describing the process of four-wave interaction are derived by means of the reduction procedure. Explicit multisoliton solutions are constructed by formulating a degenerate Reimann–Hilbert problem; while the analytic structure of the Lax operators in the complex plane is used to deduce general Backlund transformations. Conservation laws are derived and the Hamiltonian structure is used to deduce a new form of the classical r matrix, which in turn is used to prove the integrability of our model in case of nontrivial boundary conditions. Lastly the corresponding Quantum R-matrix is derived in the lattice approximation and is seen to be related to an eight-vertex model.

Hard soliton excitation regime: Stability investigation

The problem of the stability of one-dimensional solitons in the hard regime of soliton excitation, where the matrix element of the four-wave interaction has an additional smallness, is studied. It is that shown for optical solitons striction can weaken the Kerr nonlinearity. It is shown that solitons with a finite amplitude discontinuity at the critical soliton velocity, equal to the minimum phase velocity of linear waves, are unstable while solitons with a soft transition remain stable with respect to one-dimensional perurbations. Two-and three-dimensional solitons near threshold are unstable with respect to modulation perturbations.

Four-wave interaction in gas and vacuum: definition of a third-order nonlinear effective susceptibility in vacuum: χvacuum(3)

Semiclassical methods are used to study the nonlinear interaction of light in vacuum in the context of four wave mixing. This study is motivated by a desire to investigate the possibility of using recently developed powerful ultrashort (femtosecond) laser pulses to demonstrate the existence of nonlinear effects in vacuum, predicted by quantum electrodynamics (QED). An approach, similar to classical nonlinear optics in a medium, is developed in this article. A third-order nonlinear effective susceptibility of vacuum is then introduced.

Study of nonlinear four-wave interactions

The nonlinear interaction between four electrostatic waves in a plasma media is investigated analytically in the presence of linear damping or growth and also frequency mismatch, using a nonlinear perturbation method. Depending on the various initial conditions of the wave amplitudes, solutions of different types are discussed.

Modelling shallow water wave generation and transformation in an intertidal estuary

An experiment is described in which wave growth was measured in Manukau Harbour, a New Zealand estuary with relatively large fetches and extensive intertidal flats. Wave spectra were obtained from pressure sensors and current meters placed at six sites across the estuary. The SWAN third-generation spectral model was then used to simulate wave transformation during a part of the study period during which consistent south-westerly winds blew along the instrument transect. The simulations incorporated refraction by currents using output from a circulation model of the estuary. Measured wave variance spectra were compared with the model results, and the contributions of the various processes represented by source terms within the model were compared. It was found that, along with whitecapping, bed friction and exponential growth from wind input, four-wave nonlinear interactions played a dominant role. Some limitations were noted in the discrete interaction approximation which the SWAN model uses to compute the four-wave nonlinear interaction term.

Order and chaos in nonlinear wave interactions in astrophysical and space plasmas

We discuss the observation and theory of nonlinear wave interactions in astrophysical and space plasmas. Observational evidence of nonlinear mode-mode coupling in solar type III radio bursts, auroral whistler-mode radiation and active experiments in space is reviewed. The transition from order to chaos in nonlinear plasma wave interactions is analysed. For Hamiltonian four-wave interactions, the nonlinear dynamical process may undergo a chaotic transition via the quasi-periodicity route. For dissipative three-wave interactions, the nonlinear system may evolve from order to chaos via two different routes: period doubling and intermittency. It is shown that chaos in dissipative nonlinear three-wave coupling can be controlled by applying a small sinusoidal wave to the system.

Wave Front Conjugation in Bleachable Dye Doped Liquid Crystals

Wave-front conjugation (WFC) in four-wave interaction (FWI) of monopulse radiation of ruby laser in dye-doped liquid crystals (DDLC) was obtained. The amplification of conjugation wave was observed. The amplitude gratings and the phase gratings on thermal nonlinearity give the contribution in wave front conjugation.

Four-wave mixing of a chirped signal with bandwidth-limited pump waves in a resonant medium

An investigation is reported of the characteristic features of four-wave interaction of a chirped signal with bandwidth-limited pump waves in a resonant medium modelled by a two-level scheme. Analytic estimates are obtained and a numerical analysis is made of the combined influence of various mechanisms (spatial phase matching, a finite nonlinear response time, and phase cross-modulation) on the spectral composition and on the temporal behaviour of the fourth pulse. Conditions are found for achieving, with practically undetectable distortions, phase conjugation of a chirped signal accompanied by shortening of the pulse envelope and narrowing of the spectrum of the phase-conjugate wave.

Parametric frequency conversion with amplification of a weak spin wave in a ferrite film

We report on experiments of parametric frequency conversion of a weak signal spin wave by a copropagating strong pump spin wave. An effective spin-wave frequency conversion with amplification occurred when the carrier frequencies of the co-propagating waves were in the spectral region of high negative dispersion in the dipole-exchange spin-wave spectrum of the film. A theoretical explanation of the observed phenomenon is given in terms of a four-wave parametric interaction process.

Hexagonal optical structures in photorefractive crystals with a feedback mirror

A nonlinear theory is presented for the formation of hexagonal optical structures in a photorefractive medium equipped with a feedback mirror. Oppositely directed beams in photorefractive crystals are unstable against the excitation of sideband waves. It is shown here that as this instability evolves to its nonlinear stage, the three-wave interaction between weak sideband beams does not stabilize it, but rather leads to explosive growth of the amplitudes of beams whose transverse wave vectors form angles that are multiples of π/3. As a result, sideband beams at these angles are found to be correlated. A range of parameters is found in which four-wave interactions saturate the explosive instability, which explains the appearance of stable hexagons in the experiment. Outside this region, nonlinearities of higher order saturate the explosive instability, and the process of hexagon generation must be studied numerically. Matrix elements are obtained for the three-and four-wave interactions as functions of the distance to the feedback mirror, and an equation for the time evolution of the sideband wave amplitudes is derived that describes the hexagon generation. A comparison is made with experimental results for the photorefractive crystals KNbO3 and BaTiO3.

Classical r-matrices and the four-wave interaction equation

A classical r-matrix in conjunction with a zero-curvature equation is used to generate a new nonlinear integrable system known as a four-wave interaction equation. On reduction it yields a new integrable Klein-Gordon system. These equations turn out to be bi-Hamiltonian. Finally, it is indicated how the modified equations can be obtained by the zero-curvature equation. Our system is very similar to those studied by Tsuchida and Wadati.