Strong Nonlinear Regime Research Articles

Chaos-assisted formation of immiscible matter-wave solitons and self-stabilization in the binary discrete nonlinear Schrödinger equation

Binary discrete nonlinear Schrödinger equation is used to describe dynamics of two-species Bose–Einstein condensate loaded into an optical lattice. Linear inter-species coupling leads to Rabi transitions between the species. In the regime of strong nonlinearity, a wavepacket corresponding to condensate separates into localized and ballistic fractions. Localized fraction is predominantly formed by immiscible solitons consisted of only one species. Immiscible solitons are formed from initially non-separated states after transient chaotic regime. We calculate the finite-time Lyapunov exponent as a rate of wavepacket divergence in the Hilbert space. Appearance of immiscible solitons to spontaneous self-stabilization of the wavepacket. It is found that onset of chaos is accompanied by fast variations of interaction energy and energy of inter-site tunneling. Crossover to self-stabilization is accompanied by reduction of condensate density due to emittance of ballistically propagating waves.

Propagation Dynamics of Nonspreading Cosine-Gauss Water-Wave Pulses.

Linear gravity water waves are highly dispersive; therefore, the spreading of initially short wave trains characterizes water surface waves, and is a universal property of a dispersive medium. Only if there is sufficient nonlinearity does this envelope admit solitary solutions which do not spread and remain in fixed forms. Here, in contrast to the nonlinear localized wave packets, we present both theoretically and experimentally a new type of linearly nondispersive water wave, having a cosine-Gauss envelope, as well as its higher-order Hermite cosine-Gauss variations. We show that these waves preserve their width despite the inherent dispersion while propagating in an 18-m wave tank, accompanied by a slowly varying carrier-envelope phase. These wave packets exhibit self-healing; i.e., they are restored after bypassing an obstacle. We further demonstrate that these nondispersive waves are robust to weakly nonlinear perturbations. In the strong nonlinear regime, symmetry breaking of these waves is observed, but their cosine-Gauss shapes are still approximately preserved during propagation.

A continuation fixed-point iterative method on harmonic generations with strong nonlinear optical effects in multi-layer structures

A continuation fixed-point iterative method for harmonic generations with strong nonlinear optical effects in one-dimensional nonlinear multi-layer structures, which are governed by the nonlinear Helmholtz systems, is presented in this paper. Three kinds of harmonic generations via \(\chi ^{(2)}\) or \(\chi ^{(3)}\) nonlinear process, which are important phenomena in nonlinear optics, are modeled as the application of our proposed method. In the regime of strong nonlinearity, the theoretical and computational analysis of the model problems encounters enormous challenges due to the discontinuity across layers and the strongly nonlinear coupling of modes. To overcome the difficulties in the nonlinearity and discontinuity, a fixed-point iteration with the finite element method is used to study the nonlinear conversion efficiencies by solving the weak formulations. Furthermore, a continuation technique that depends on the weak formulations to solve three model problems, is introduced to handle the iterative procedure for cases with large nonlinearity. The theoretical framework and numerical algorithm towards a uniform process is offered to study almost all the harmonic generations in \(\chi ^{(2)}\) or \(\chi ^{(3)}\) interaction. With the continuation fixed-point iterative technique, the convergence of the iterative procedure can be ensured, even in the presence of strong nonlinearity. Three numerical experiments with high conversion efficiencies are also presented to show the accuracy and efficiency of our proposed method.

Graphene-nonlinearity unleashing at lasing threshold in graphene-assisted cavities

We investigate the nonlinear optical features of a graphene sheet embedded in an active cavity and we show that, when tuned near its lasing threshold, the cavity is able to isolate the spatially localized graphene nonlinearity thus producing a very strong nonlinear device response with multi-valued features. As opposed to standard situations where the small thickness of the graphene sheet hampers its remarkable nonlinear optical properties to be exploited, in our scheme the strong nonlinear optical regime is mainly triggered by the very intrinsic planar localization of graphene nonlinearity. The proposed strategy for exploiting graphene nonlinearity through its unleashing could open novel routes for conceiving ultra-efficient nonlinear photonic devices.

Quantitative characterization of high molecular weight polymer solutions in microfluidic hyperbolic contraction flow

Nonlinear flows of polyacrylamide (PAAm) ( $$M_{\text{w}} = 5.7 \times 10^{6}\,{\text{g}}/{\text{mol}}$$ ) aqueous solutions through a micro-fabricated, hyperbolic contraction geometry with high Hencky strain ( $$\varepsilon_{{\text{H}}} = 3.7$$ ) have been characterized by micro-particle image velocimetry ( $$\mu $$ -PIV). Various flow dynamics regimes in a range of Weissenberg number (Wi) and Reynolds number (Re) are presented in a Wi–Re diagram. The symmetric corner vortices are only observed in the flow of low concentration PAAm solution ( $$c/c^{*}=3.3$$ ). In a higher concentration ( $$c/c^{*}=8.3$$ ), PAAm solution exhibits chaotic-like flow patterns in the strong nonlinear flow regime ( $$Wi>350$$ ). Extensional deformation in nonlinear flows of Wi up to 860 has been analyzed. Furthermore, the local stretch experienced by the polymer chain in complex flow is systematically quantified and linked to the corresponding velocity vector fields, which are valuable for understanding the highly nonlinear flow phenomena.

Oscillation dependence of two-wave mixing gain for unidirectional ring resonator in photorefractive materials

Dependence of the coupling strength of two-wave mixing gain in photorefractive materials for the single unidirectional ring resonator on oscillation conditions has been analyzed in the strong nonlinear regime. In this regime, difference between the frequency of the pump beam and oscillating beam is proportional to the cavity-length detuning, which can be explained in terms of the photorefractive phase-shift. This phase-shift results due to slightly non-degenerate two-wave mixing that compensates for cavity detuning and satisfies the round-trip phase condition for the steady-state oscillation. The presence of such a phase-shift allows the possibility of the nonreciprocal steady-state energy transfer between the pump and oscillating beams. If the gain due to the beam coupling is large enough to overcome the cavity losses then the signal beam is amplified in the presence of material absorption. Such amplification is responsible for the oscillations. For the single unidirectional ring resonator, the effects of cavity-length detuning, energy coupling coefficient, crystal thickness of the material, reflectivity of the cavity mirrors and material's absorption coefficient on the frequency and intensity of oscillations have also been studied in detail. It has been found that for the smaller value of absorption coefficient ( α) of the photorefractive crystal, the unidirectional ring resonator can oscillate at almost any cavity-length detuning (Δ Γ) whereas for the larger value of α oscillation occurs only when the cavity-length detuning is limited to small region (around Δ Γ=0). But reverse of the case is found for energy coupling coefficient ( γ 0).

Chaotic dynamics of a Bose-Einstein condensate coupled to a qubit

We study numerically the coupling between a qubit and a Bose-Einstein condensate (BEC) moving in a kicked optical lattice using Gross-Pitaevskii equation. In the regime where the BEC size is smaller than the lattice period, the chaotic dynamics of the BEC is effectively controlled by the qubit state. The feedback effects of the nonlinear chaotic BEC dynamics preserve the coherence and purity of the qubit in the regime of strong BEC nonlinearity. This gives an example of an exponentially sensitive control over a macroscopic state by internal qubit states. At weak nonlinearity quantum chaos leads to rapid dynamical decoherence of the qubit. The realization of such coupled systems is within reach of current experimental techniques.

Strong Bragg backscattering in reflectometry

The reflection of the probing microwave occurring in the vicinity of the backscattering Bragg resonance point (far from the cut-off) at a high enough density fluctuation level and leading to a large jump of the reflected wave phase and a corresponding time delay is described analytically using a 1D model. Explicit expressions for the reflection and transmission coefficients are derived and compared against results of numerical modelling. The criteria for transition to the nonlinear regime of strong Bragg backscattering (BBS) is obtained for both O-mode and X-mode reflectometry. It is shown that a strong nonlinear regime of BBS may occur in ITER at the 0.5–2% relative density perturbation level both for the ordinary and extraordinary mode probing. The possibility of probing wave trapping leading to strong enhancement of the electric field and associated high phase variation of the reflected wave due to BBS is demonstrated.

Dependence of gain and phase-shift on crystal parameters and pump intensity in unidirectional photorefractive ring resonators

The steady-state amplification of light beam during two-wave mixing in photorefractive materials has been analysed in the strong nonlinear regime. The oscillation conditions for unidirectional ring resonator have been studied. The signal beam can be amplified in the presence of material absorption, provided the gain due to the beam coupling is large enough to overcome the cavity losses. Such amplification is responsible for the oscillations. The gain bandwidth is only a few Hz. In spite of such an extremely narrow bandwidth, unidirectional oscillation can be observed easily at any cavity length in ring resonators by using photorefractive crystals as the medium and this can be explained in terms of the photorefractive phase-shift. The presence of such a phase-shift allows the possibility of the non-reciprocal steady-state transfer of energy between the two light beams. Dependence of gain bandwidth on coupling constant, absorption coefficient of the material’s cavity length (crystal length) and modulation ratio have also been studied.

Observation of a Gradient Catastrophe Generating Solitons

We investigate the propagation of a dark beam in a defocusing medium in the strong nonlinear regime. We observe for the first time a shock fan filled with noninteracting one-dimensional gray solitons that emanates from a gradient catastrophe developing around a null of the optical intensity. This scenario turns out to be very robust, persisting also when the material nonlocal response averages the nonlinearity over dimensions much larger than the emerging soliton filaments.

Influence of the non-uniformity of the grating along sample thickness and of grating period variation, on the diffraction efficiency optimization in a sillenite crystal (Bi 12SiO 20) in a strong non-linear regime under an external dc field

We solved numerically the set of partial, non-linear, material rate differential equations for different values of grating period in a photorefractive, sillenite crystal (Bi 12SiO 20). These solutions were used to include the non-uniformity of the grating along sample thickness to calculate the energy exchange in two wave mixing. We employed a vectorial approach to calculate the beam energy exchange. This calculation was performed for each value of grating period. We included optical activity, light absorption, birefringence and polarization angle of the interacting beams. Two crystal orientations were considered: when the grating vector is parallel to the face [0 0 1] and when it is perpendicular to the same face. We found a large influence of the non-uniformity of the grating on the diffraction efficiency and on the optimal sample thickness.

Scaling properties of a simplified bouncer model and of Chirikov's standard map

Scaling properties of Chirikov's standard map are investigated by studying the average value of I2, where I is the action variable, for initial conditions in (a) the stability island and (b) the chaotic component. Scaling behavior appears in three regimes, defined by the value of the control parameter K: (i) the integrable to non-integrable transition (K ≈ 0) and K < Kc (Kc ≈ 0.9716); (ii) the transition from limited to unlimited growth of I2, K ≳ Kc; (iii) the regime of strong nonlinearity, K ≫ Kc. Our scaling results are also applicable to Pustylnikov's bouncer model, since it is globally equivalent to the standard map. We also describe the scaling properties of a stochastic version of the standard map, which exhibits unlimited growth of I2 even for small values of K.

Selective Vibrational Detachment of Microspheres Using Optically Excited In-Plane Motion of Nanomechanical Beams

Optical excitation plays an important role in the actuation of higher flexural and torsional modes of nanoelectromechanical oscillators. We show that optical fields are efficient for excitation, direct control, and measurement of in-plane motion of cantilever-type nanomechanical oscillators. As a model system, 200- and 250-nm-thick single-crystal silicon cantilevers with dissimilar lengths and widths ranging from 6 to 12 microm and 500 nm to 1 microm, respectively, were fabricated using surface micromachining and dynamically analyzed using optical excitation and interferrometric detection. Three-dimensional finite element analysis incorporating shear, rotational inertia, cross-sectional deplanation, and nonideal boundary conditions due to the structural undercut describe the dynamics of the nanomechanical structures adequately. The quality factor of a particular in-plane harmonic was consistently higher than the transverse mode. The increased dissipation of the out-of-plane mode was attributed to material and acoustic loss mechanisms. The in-plane mode was used to demonstrate vibrational detachment of submicrometer polystyrene spheres on the oscillator surface. In contrast, the out-of-plane motion, even in the strong nonlinear impact regime, was insufficient for the removal of bound polystyrene spheres. Our results suggest that optical excitation of in-plane mechanical modes provide a unique mechanism for controlled removal of particles bound on the surface of nanomechanical oscillators.

Beam energy exchange in sillenite crystals (Bi12SiO20 and Bi12TiO20), considering the variation of light modulation along sample thickness in a strong non-linear regime

The energy exchange during two-wave mixing can be increased in photorefractive sillenite crystals considering thick samples (i.e., large interaction lengths). Nevertheless, in thick samples under a non-linear regime, the light modulation does not remain constant along wave propagation, and there is a spatial non-uniform grating along sample thickness. Here, we study the optimization of the effective gain for non-moving gratings in thick samples, taking into account the spatial non-uniformity of the grating on the beam energy exchange. Optical activity, absorption of light, birefringence, polarization angles of the incident beams, different values of a static applied d.c. field were also considered. We did this by first solving numerically the material rate equations to obtain the time evolution of the space charge field up to the steady state, and then by numerically solving the coupled wave equations in a self-consistent way in order to take into account the variation of the magnitude and phase of the light modulation along sample thickness. We obtained the variation of the complex amplitudes of the interacting beams along sample thickness for both, when the grating vector is parallel to the face [001] and when it is perpendicular to the same face. We predict that an effective gain of 45 can be obtained for samples >2cm. We also found that the influence of the initial polarization angles on the value of the gain is very different when considering the spatial non-uniformity of the grating.

Drain voltage induced barrier increasing of quantum-wire transistors

In-plane quantum-wire transistors based on a modulation-doped GaAs/AlGaAs heterostructure are studied in the threshold regime. It is found that the threshold voltage VT of a quantum wire transistor depends on the applied bias voltage in a parabolic way. In particular, VT increases in the strong nonlinear transport regime with increasing bias voltage, which is in contrast to the drain voltage induced barrier lowering observed for submicron transistors. The increase of VT by a barrier increase along the channel with increasing bias voltage due to a self-depletion is explained.

Four-wave acousto-electromagnetic interactions in crystals with a nonlinear electrostriction

The four-wave parametric interaction of a microwave sound with a powerful electromagnetic radiation in a strong nonlinear regime is studied numerically. The interaction occurs due to account of the fourth order terms in expansion of the free energy of the anisotropic crystal. It is shown that both instability and parametric amplification of the wave amplitude in a system take place. This leads to generation of the periodical structures of short nonlinear pulses of sound. The account of small noise in a bounded crystal leads to the quasi-coherent dynamics and results in arising of self-organizing of acousto-electromagnetic waves.

Nonlinear atomic Fabry-Perot interferometer: From the mean-field theory to theatom blockadeeffect

We have investigated nonlinear atom optical effects which arise from atom-atom collisional interactions in a single-mode atomic Fabry-Perot cavity driven by a coherent cw atom laser beam. When the nonlinear interaction energy per single atom is small, the exact numerical solution of the master equation is well reproduced by a mean-field treatment in which quantum fluctuations are included linearizing the stochastic equations of the Positive-P representation. On the other hand, when the damping of the cavity mode is very weak and its wave-function is tightly confined in space, a regime of strong nonlinearity can be achieved. For the specific case of an incident atom laser frequency at resonance with the empty cavity, the numerical calculations predict a sort of atom blockade effect, which is a sort of atom optical analog of the well-known Coulomb blockade effect of electronic transport through microscopic structures: only one atom can occupy the cavity mode at a time and the statistical properties of the transmitted beam, being very similar to the resonance fluorescence from a single two-level system, show definitely nonclassical behaviors such as antibunching. Only at very large incident intensities, more than one atom can be simultaneously forced inside the cavity mode: in this regime, the results of the numerical calculations can be successfully interpreted using a dressed cavity model. From the formal analogy between atomic matter waves and optical light waves in nonlinear media, it follows that the same results hold for photonic systems.

Impedance of a thin metal film in the regime of strong magnetodynamic nonlinearity

The nonlinear response of a metallic film to the self-magnetic field of an ac transport current is investigated theoretically. The nonlinearity is due to a magnetodynamic mechanism which derives from the influence of the magnetic field of the current on the dynamics of the conduction electrons and is most typical for pure metals at low temperatures. The nonlinearity leads to a dependence of the surface impedance on the amplitude of the ac current. It is shown that the real part of the surface impedance, which is responsible for the Joule losses, has a smooth maximum as a function of the amplitude. The imaginary part of the surface impedance is also a nonmonotonic function of the amplitude and has a minimum.

Buckling of thin plates in the weakly and strongly nonlinear regimes

Abstract The buckling of thin plates is interesting from the point of view of materials science, because it provides a rather unique opportunity of exploring the predictions of elasticity theory in non-trivial conditions and allows in a specific example to disentangle new physical effects from effects that can be explained in the framework of the fundamental equations. I review these basic Föppl-von Karman equations for the elasticity of thin plates and then study the two limits of weak and strong nonlinear regimes for buckling.

Buckling of thin plates in the weakly and strongly nonlinear regimes

Abstract The buckling of thin plates is interesting from the point of view of materials science, because it provides a rather unique opportunity of exploring the predictions of elasticity theory in non-trivial conditions and allows in a specific example to disentangle new physical effects from effects that can be explained in the framework of the fundamental equations. I review these basic Föppl-von Karman equations for the elasticity of thin plates and then study the two limits of weak and strong nonlinear regimes for buckling.