Hamiltonian chaos in two coupled three-wave parametric interactions with quadratic nonlinearity

Interaction of Bessel optical vortices in quadratic nonlinear medium

We present the fascinating phenomena of resonant radiation emitted by transient rogue waves in cubic and quadratic nonlinear media, particularly those shed from Peregrine solitons, one of the main wavepackets used today to model real-world rogue waves. In cubic media, it turns out that the emission of radiation from a Peregrine soliton can be attributed to the presence of higher-order dispersion, but is affected by the intrinsic local longitudinal variation of the soliton wavenumber. In quadratic media, we reveal that a two-color Peregrine rogue wave can resonantly radiate dispersive waves even in the absence of higher-order dispersion, subjected to a phase-matching mechanism that involves the second-harmonic wave, and to a concomitant difference-frequency generation process. In both cubic and quadratic media, we provide simple analytic criteria for calculating the radiated frequencies in terms of material parameters, showing excellent agreement with numerical simulations.

Impurity solitons with quadratic nonlinearities

The paper aims at studying the influences of temperature on the suspended cables’ dynamical behaviors subjected to dual harmonic excitations in thermal environments. Significantly, the quadratic nonlinearity and the corresponding secondary resonances are considered. By introducing a tension variation factor, the nonlinear vibration equations of motion could be obtained based on the condensation model. By using Galerkin’s procedure, the continuous model of the nonlinear system is reduced to a set of infinite models with quadratic and cubic nonlinearities. By using the multiple scales method, the resultant reduced model is solved and the stability analysis is also presented in two simultaneous resonance cases. Nonlinear dynamical behaviors with thermal effects are presented using bifurcation diagrams, time‐history curves, phase portraits, frequency spectrums, and Poincaré sections. The numerical results show that thermal effects induce different scenarios. The sensitivities of linear (natural frequency) and nonlinear (quadratic and cubic) coefficients to temperature variations are different. The temperature may increase or decrease the response amplitudes depending on the excitation amplitude and the sag‐to‐span ratio. The inflection point is shifted and exhibited at a smaller or larger excitation amplitude in thermal environments. The resonant range between two Pitchfork bifurcations seems to be reduced when the temperature is decreasing. The response amplitude is very sensitive to temperature, and even an opposite spring behavior may be exhibited due to warming/cooling conditions. However, the periodic motions seem independent of temperature variations.

Self-pulsing and transients of a Fabry-Perot interferometer with quadratic nonlinear medium

A stability analysis of the nonlinear dynamics of two-wave parametric mixing in a quadratic nonlinear medium in the presence of self- and cross-phase modulation reveals a novel symmetry-breaking instability. This instability may be lead to abrupt changes, as a function of the fundamental and its second harmonic.

We investigate spatial localization in a quadratic nonlinear medium in the presence of randomness. By means of numerical simulations and theoretical analyses we show that, in the down conversion regime, the transverse random modulation of the nonlinear susceptibility generates localizations of the fundamental wave that grow exponentially in propagation. The localization length is optically controlled by the pump intensity that determines the amplification rate. The results also apply to cubic nonlinearities.

We show experimentally that a laser beam scrambled by propagation in a short segment of multimode fiber may be cleaned by the nonlinear propagation in KTP cristal with type-II second-harmonic generation.

A new type of complex periodic solutions of the nonlinear Schrödinger equation is described which can be obtained in the collinear interaction of three plane monochromatic waves (modes) in a quadratic nonlinear medium. On passing to real variables (quadrature components), the solutions of this new type describe nondegenerate two-component cnoidal waves consisting of two 'incoherent' (noninterfering) components. The amplitudes of these components perform additional (with respect to the modulus oscillations) intricate nonlinear oscillations phase-shifted by π/2, which are consistent with oscillations of the solution modulus described by an elliptic function.

The problem of generating a narrow-band biphoton field during spontaneous parametric down conversion of light in a quadratic nonlinear medium in a cavity is discussed. It is shown that, in the case of double resonance (for signal and idle fields), the form of the single-photon wave packet is biexponential and has a half-width determined by the photon life time in the cavity. An experiment is carried out on the generation of narrow-band biphotons in a crystal of lithium iodine with I-type matching in a frequency degenerate collinear regime. The second-order correlation function of the cavity-enhanced biphotons is measured using the electric nanosecond time delay line.

Boomerons are described as accelerated solitons for special integrable systems of coupled wave equations. A general formalism based on the Lax pair method is set up to introduce such systems which look of Nonlinear Schrödinger-type with linear, quadratic and cubic coupling terms. The one-soliton solution of such general systems is also briefly discussed. We display special instances of wave systems which are of potential interest for applications, including dispersion-less models of resonating waves. Among these, special attention and details are given to the celebrated equations describing the resonant interaction of three waves, in view of their application to optical pulse propagation in quadratic nonlinear media. For this particular case, we present exact solutions of the three-wave resonant interaction system, in the form of triplets moving with a common nonlinear velocity (simultons). The simultons have nontrivial phase-fronts and exist for different velocities and energy flows. We studied simulton stability upon propagation, and found that solitons with a velocity greater than a certain critical value are stable. We explore a novel consequence of the particle-like nature of three-wave simultons, namely their inelastic scattering with particular linear waves. Such phenomenon is associated with the excitation (decay) of stable (unstable) simultons by means of the absorption (emission) of the energy carried by a particular isolated pulse. Inelastic processes are exactly described in terms of boomerons. We also briefly consider collisions between different three-wave simultons.

The dynamics of two-dimensional Pythagorean lattice solitons are explored in quadratic nonlinear media. The study is focused on variation of sub-lattice depths and the strength of quadratic optical effects that specify characteristics of the considered model. The numerical existence of periodic and aperiodic lattice solitons is demonstrated, and the stability domain of solitons is determined for all parameters in the model. It is shown that, although the existence domain of periodic and aperiodic lattice solitons is identical, the stability region of periodic lattice solitons is narrower than that of aperiodic lattice solitons. It is manifested that stable solitons can exist in both periodic and aperiodic lattices, and decay of unstable solitons can be arrested by increasing the potential depth and decreasing the propagation constant.

Half-optical-cycle damped solitons in quadratic nonlinear media

By analyzing ultrashort optical pulse propagation in quadratic nonlinear media beyond the slowly varying envelope approximation, we find that the sign and magnitude of self-steepening can be controlled through the wave vector mismatch. As an example of this phenomenon's impact on ultrashort pulse propagation, we show that it may be used to cancel the propagation effects of group-velocity mismatch. We obtain quantitative agreement between theory, simulations, and experiments.

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