Noninstantaneous Nonlinearity Research Articles

Intermodal synchronization effects in multimode fibers with noninstantaneous nonlinearity

The synchronization of spatial modes in the formation of multimode solitons can be considered as the transverse analog of mode locking. In this process, the inherent Kerr nonlinearity in the fiber core binds different spatial fiber modes together. However, compared to their temporal counterparts, the binding mechanism in multimode solitons is rather weak and susceptible to perturbation. In order to mitigate this effect, the propagation in multimode fibers with noninstantaneous Kerr media (NKM) is theoretically investigated. Our study focuses on the dynamics of intermodal energy transfer for the two cases of weak and strong walk-off. Both the moment method and numerical simulations of a multimode expansion of the nonlinear Schr\"odinger equation are employed. Scenarios of pure, hybrid, and varying NKM are investigated. For pure NKM, it is found that for weak walk-off, higher order spatial modes are accelerated in the time domain when their energy is transferred to lower order modes, that is, an effect that has previously been discussed as self-cleaning. In the hybrid case, when an additional instantaneous Kerr effect is present, this results in an enhancement of intermodal nonlinear coupling, leading to prominent oscillating evolutions of the derivatives of energy and pulse center. For varying NKM, the nonlinear refractive index and the proportion of noninstantaneous Kerr nonlinearity may both vary with pulse width. In this case, energy transfer and temporal shift are essentially determined by the magnitude of nonlinear response time of NKM. Significant temporal self-splitting at the trailing edge is observed for the lowest order mode provided only that the response time is large enough and irrespective of strong or weak walk-off.

Thermo-optical dynamics of a nonlinear GaInP photonic crystal nanocavity depend on the optical mode profile

We measure the dynamics of the thermo-optical nonlinearity of both a mode-gap nanocavity and a delocalized mode in a Ga0.51In0.49P photonic crystal membrane. We model these results in terms of heat transport and thermo-optical response in the material. By step-modulating the optical input power we push the nonlinear resonance to jump between stable branches of its response curve, causing bistable switching. An overshoot of the intensity followed by a relaxation tail is observed upon bistable switching. In this way, the thermal relaxation of both the localized resonance and the delocalized resonance is measured. Significant difference in decay time is observed and related to the optical mode profile of the resonance. We reproduce the observed transient behavior with our thermo-optical model, implementing a non-instantaneous nonlinearity, and taking into account the optical mode profile of the resonance, as experimentally measured.

Resonance in modulation instability from non-instantaneous nonlinearities

To explore resonance phenomena in the nonlinear region, we show by experimental measurements and theoretical analyses that resonance happens in modulation instability from non-instantaneous nonlinearities in photorefractive crystals. With a temporally periodic modulation in the external bias voltage, corresponding to a modulation in the nonlinear strength, an enhancement in the visibility of MI at resonant frequency is reported through spontaneous optical pattern formations. Theoretical curves obtained from a nonlinear non-instantaneous Schrödinger equation give good agreement to experimental data.

Measurement of nonlinear refractive indices of air, oxygen, and nitrogen in capillary by changing the temporal width of short laser pulses

Nonlinear refractive indices of air and its constituents were measured as a function of the pulse width in a capillary used to confine the gas and to increase the interaction length. This study was carried out with nonlinear ellipse rotation measurements using an amplified Ti:sapphire laser system providing pulses at 800 nm, continuously tunable from ∼60 fs up to ∼3 ps. Due to the presence of noninstantaneous orientational response, nonlinearities in molecular gases strongly depend on the pulse width. The pulse width dependence and discrimination between instantaneous and noninstantaneous nonlinearities of air gases were obtained with a simple exponential growth model.

Airy-type solitary wave in highly noninstantaneous Kerr media.

We investigate the dynamics of a decelerating Airy pulse in the highly noninstantaneous Kerr media. It is found that the deceleration of the Airy pulse can be counteracted by the highly noninstantaneous nonlinearity. When the power of the pulse is specifically chosen, the deceleration of the Airy pulse can be totally restrained, and an Airy-type solitary wave is observed within several dispersion lengths.

Non-instantaneous optical nonlinearity of an a-Si:H nanowire waveguide.

We use pump-probe spectroscopy and continuous wave cross-phase and cross-amplitude modulation measurements to study the optical nonlinearity of a hydrogenated amorphous silicon (a-Si:H) nanowire waveguide, and we compare the results to those of a crystalline silicon waveguide of similar dimensions. The a-Si:H nanowire shows essentially zero instantaneous two-photon absorption, but it displays a strong, long-lived non-instantaneous nonlinearity that is both absorptive and refractive. Power scaling measurements show that this non-instantaneous nonlinearity in a-Si:H scales as a third-order nonlinearity, and the refractive component possesses the opposite sign to that expected for free-carrier dispersion.

Spectral dynamics of incoherent waves with a noninstantaneous nonlinear response

We study the influence of a constant background noise on the dynamics of spectral incoherent solitons, which are incoherent structures sustained by a noninstantaneous (Raman-like) nonlinearity. As the level of the noise background increases, the incoherent wave enters a novel nonlinear regime characterized by oscillatory dynamics of the incoherent spectrum, which develop within a spectral cone during the propagation. In contrast to the conventional Raman-like spectral red shift, such incoherent spectral dynamics can be characterized by a significant spectral blue shift. On the basis of the kinetic wave theory, we derive explicit analytical expressions of these incoherent oscillatory spectral dynamics.

Wave-packet spreading dynamics under a noninstantaneous nonlinearity: Self-trapping, defocusing, and focusing

Special localized wave modes show up in several physical scenarios including BEC in optical lattices, nonlinear photonic crystals, and systems with strong electron-phonon interaction. These result from an underlying nonlinear contribution to the wave equation that is usually assumed to be instantaneous. Here we demonstrate that the relaxation process of the nonlinearity has a profound impact in the wave-packet dynamics and in the formation of localized modes. We illustrate this phenomenology by considering the one-electron wave packet spreading in a C60 buckball structure whose dynamics is governed by a discrete nonlinear Schrödinger equation with a Debye relaxation of the nonlinearity. We report the full phase diagram related to the spacial extension of the asymptotic wave packet and unveil a complex wave-packet dynamical behavior.

Kinetic Description of Random Optical Waves and Anomalous Thermalization of a Nearly Integrable Wave System

This article is composed of two parts. The first part is aimed at providing an overview on the kinetic description of random nonlinear waves considering the one-dimensional nonlinear Schr¨odinger (NLS) equation as a representative model of optical wave propagation. We expose, in particular, the key problem of achieving a closure of the infinite hierarchy of moment equations for the random field. The hierarchy is closed at the first order when the statistics of the random wave is non-stationary or when the response time of the nonlinearity is non-instantaneous, which, respectively, leads to the Vlasov kinetic equation and the weak-Langmuir turbulence equation. When the amount of nonstationary statistics is comparable to the amount of non-instantaneous nonlinearity, we derive a generalized Vlasov-Langmuir equation that provides a unified formulation of the Vlasov and Langmuir approaches. On the other hand, when the statistics of the random wave is stationary and the nonlinear response instantaneous, the closure of the hierarchy of moment equations requires a second-order perturbation expansion procedure, which leads to the Hasselmann (or wave turbulence) kinetic equation. Contrarily to the Vlasov and Langmuir equations, the Hasselmann equation is irreversible, a feature which is expressed by a H-theorem of entropy growth that describes wave thermalization toward the thermodynamic equilibrium distribution, i.e. the Rayleigh-Jeans (RJ) spectrum. In the second part of the paper, we discuss a process of anomalous thermalization by considering the example of the scalar NLS equation whose integrability is broken by the presence of third-order dispersion. The anomalous thermalization is characterized by an irreversible evolution of the wave toward an equilibrium state of a fundamental different nature than the conventional RJ equilibrium state. The wave turbulence kinetic equation reveals that the anomalous thermalization is due to the existence of a local invariant in frequency space Jω, which originates in degenerate resonances of the system. In contrast to integral invariants that lead to a generalized RJ distribution, here, it is the local nature of the invariant Jω that makes the new equilibrium states fundamentally different than the usual RJ equilibrium states. We study in detail the anomalous thermalization by means of numerical simulations of the NLS equation and of the wave turbulence equation by using an improved criterion of applicability of the kinetic theory. The spectrum of the field is shown to exhibit an intriguing asymmetric deformation, which is characterized by the unexpected emergence of a constant spectral pedestal in the long-term evolution of the field. It turns out that the local invariant Jω explains all the essential properties of the anomalous thermalization of the wave.

Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response

This article presents a unified kinetic formulation of partially coherent nonlinear optical waves propagating in a noninstantaneous response Kerr medium. We derive a kinetic equation that combines the weak Langmuir turbulence kinetic equation and a Vlasov-like equation within a general framework: It describes the evolution of the spectrum of a random field that exhibits a quasistationary statistics in the presence of a noninstantaneous nonlinear response. The kinetic equation sheds new light on the dynamics of partially coherent nonlinear waves and allows for a qualitative interpretation of the interplay between the noninstantaneous nonlinearity and the nonstationary statistics of the incoherent field. It is shown that the incoherent modulational instability of a random nonlinear wave can be suppressed by the noninstantaneous nonlinear response. Moreover, incoherent modulational instability can prevent the generation of spectral incoherent solitons.

Elliptic incoherent solitons in strongly nonlocal media with anisotropic Kerr nonlinearity

The propagation of elliptic incoherent beam in strongly nonlocal media with noninstantaneous anisotropic Kerr nonlinearity is systematically investigated. Using the mutual coherence function approach, we obtain the existence curve of such solitons and an interesting outcome: correlation characteristics of the elliptic incoherent solitons can be anisotropic as well as isotropic. When the existence conditions of solitons are not satisfied, the elliptic incoherent beam will undergo periodic oscillation. Nonstationary evolution behaviors of the elliptic beam are shown in detail by numerical calculation. It is also obtained that the oscillation periods of the beam in x and y direction are different and the elliptic beam will become circular at some propagation distance under special conditions.

Incoherent solitons in strongly nonlocal media: The coherent density theory

Abstract We study the properties of one dimension incoherent accessible solitons in strongly nonlocal media with noninstantaneous Kerr nonlinearity. Following the coherent density theory, we obtain an exact solution of such incoherent solitons. The spatial width of the incoherent solitons is related to the incoherent angular power spectrum θ 0 as well as the incident power. The evolution properties of the intensity profile and the coherence characteristics are also discussed in detail when the solitons undergo periodic harmonic oscillation.

Elliptic incoherent accessible solitons in strongly nonlocal media

We study the propagation of elliptic incoherent accessible solitons in strongly nonlocal media with noninstantaneous Kerr nonlinearity. For this soliton to exist, the coherence properties of the incoherent beam should be anisotropic. The total power of the incident beam should also equal to a critical value which depend on the beam width as well as the coherence properties. When initial parameters of the beam do not satisfy the existence conditions, the elliptic incoherent accessible solitons will undergo linear harmonic oscillation in different states. Corresponding properties are studied in detail.

Incoherent accessible white-light solitons in strongly nonlocal Kerr media

We present a theoretical investigation of incoherent accessible white-light solitons in strongly nonlocal medium with noninstantaneous Kerr nonlinearity. This soliton has elliptic Gaussian intensity profile and anisotropic spatiotemporal coherence properties. For this soliton to exist, the spatial coherence distance should be larger for lower frequencies and shorter for higher frequencies. When the incident power differs from the critical value, we demonstrate the periodic harmonic oscillations of the accessible white-light solitons.

Cochlear gain control

The nonlinear auditory phenomena of compression, suppression, and distortion are known to have a cochlear-mechanical origin. An instantaneous nonlinear transfer function is often assumed to underlie these phenomena, but there are experimental indications that auditory nonlinearity is sluggish rather than instantaneous. This study analyzes the consequences of such sluggishness, using automatic gain control (AGC) as a model noninstantaneous nonlinearity. The distinctive characteristic of AGC, its delayed action, is shown to produce a number of observable and measurable effects that distinguish AGC from instantaneous nonlinearities. A major class of such AGC-specific effects concerns the phase of aural distortion products. For example, the phase of the cancellation tone in the classical psychoacoustic cancellation paradigm is linearly related to the frequency spacing of the primary tones in an AGC, as opposed to the square-law relationship produced by an instantaneous nonlinearity. These and other predictions are confronted with experimental data from the literature. The impact of putative AGC-related delays on the interpretation of distortion product otoacoustic emissions (DPOAEs) is discussed. Detailed suggestions are made for experiments specifically aimed at determining whether cochlear nonlinearity is instantaneous or delayed.

Dissipative solitons and their critical slowing down near a supercritical bifurcation

The existence of a novel kind of soliton in dissipative systems with competing, noninstantaneous nonlinearities has been predicted and experimentally verified. These subcritically bifurcating solitons exist near a supercritical continuous wave bifurcation. We show that their peculiarities originate from different saturation behavior of nonlinear loss and gain with regard to power and energy. Branches of stable and unstable solitary waves have been identified. For the first time to our best knowledge, we have experimentally proved critical slowing down of solitons.

Braiding of two spiraling laser beams due to plasma wave wakes.

We study how two Gaussian laser beams interact through plasma wave wakes produced when they co-propagate in a plasma. Using a variational principle, we derive equations of motion for the centroid of each beam, and find braided centroid solutions. These results can be generalized to other nonlinear optical media with non-instantaneous nonlinearity.

Dynamic Soliton-Like Modes

Incoherent optical spatial solitons require noninstantaneous nonlinearity, i.e., the local intensity fluctuation of the solitons must be faster than the medium can respond. Observing partially incoherent bicomponent solitons, we find that there exists a threshold speed. When the fluctuation of the soliton intensity, resulting from the time-varying interference of its constituent modes, is below the threshold, the soliton beam and its induced waveguide oscillate violently. Just above the threshold, the soliton-induced waveguide is observed to be dragged by the soliton beam.

Investigating nonlinear femtosecond pulse propagation with frequency-resolved optical gating

Frequency-resolved optical gating (FROG) is used to investigate nonlinear pulse propagation in normally dispersive media. We present high-dynamic-range measurements of broad-bandwidth femtosecond pulses that result from nonlinear propagation in fused silica and compare these measurements with a (3+1)-dimensional modified nonlinear Schrodinger equation. We also demonstrate the ability of FROG to provide information about a noninstantaneous nonlinearity in methanol. In this case, the instantaneous nonlinear index and the time response of the noninstantaneous nonlinearity are used as fit parameters in a (1+1)-dimensional model.

Ultrashort-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating

Ultrashort-pulse-characterization techniques generally require instantaneously responding media. We show that this is not the case for frequency-resolved optical gating (FROG). We include, as an example, the noninstantaneous Raman response of fused silica, which can cause errors in the retrieved pulse width of as much as 8% for a 25-fs pulse in polarization-gate FROG. We present a modified pulse-retrieval algorithm that deconvolves such slow effects and use it to retrieve pulses of any width. In experiments with 45-fs pulses this algorithm achieved better convergence and yielded a shorter pulse than previous FROG algorithms.