Stokes Wave Research Articles

Self modulation and scattering instability of a relativistic short laser pulse in an underdense plasma

In this paper, we present an analysis framework for description of nonlinear, self-consistent laser-plasma evolutions during propagation of a short intense laser pulse in a high-density sub-critical plasma (the pulse length exceeds the plasma wavelength). In this context, the pulse evolutions are attributed to the wakefield induced self-modulation and destabilization via parametric exponentiation of the initial noise content. The self-consistent plasma evolutions are formulated in terms of used-to-be motion constants in the absence of pulse evolutions. This proves very useful both in uncovering important plasma dynamics during self-modulation and also in facilitating the instability studies in the strongly nonlinear regime through refinement of unstable plasma perturbations. General analytical solutions, at arbitrary pulse conditions, are derived for self-modulation, indicating that the envelope evolutions are driven by the induced spatial frequency-chirp. Also, these results state that the envelope attains fine modulations which produce long wavelength low-frequency modes via beating the carrier mode. The plasma wave variations are found to convect and amplify away from the pulse front. Regarding parametric instability, we assess different scattering regimes at different pulse shapes and peak intensities, manifesting anomalous behaviors ranging from wild positioning of the Stokes wave in dispersion plane to broadening in the scattered spectrum and halting the instability. Our analyses are assisted and verified by numerous fluid and particle-in-cell simulations. Based on our results, we discuss phenomena like the pulse breakup and its different regimes, and assisted particle acceleration in the presence of pulse evolutions.

Brillouin Assisted Optoelectronic Self-Narrowing of Laser Linewidth

We propose and demonstrate the possible self-narrowing of a laser line using optoelectronic and Brillouin approaches. Part of the laser is used to feed a long fiber Brillouin resonator whose Stokes wave is phase locked to the laser. The phase-locked loop which first forbids the multimode operation of the Brillouin resonator enables in a second time a dramatic reduction in the optical phase noise of the laser itself. In the case of a continuous Er,Yb:glass laser, a reduction by more than 90 dB at 100 Hz of the carrier is observed. This yields an optical linewidth reduction to 2 Hz. The method being independent of the laser wavelength, it can be implemented to almost any laser.

Direct numerical simulations of second-order Stokes wave driven smooth-walled oscillatory channel: investigation of net current formation

ABSTRACTIn wall-bounded time-periodic flows, nonlinearity, associated with higher harmonic term(s) in velocity and/or acceleration outside the boundary layer, can significantly change the wall turbulence compared with that in the linear Stokes Boundary Layer. A significant feature of a nonlinear wall-bounded turbulent time-periodic flow is the formation of a net current which has not yet been mechanistically explained. This study investigates the effects of asymmetric velocity outside the boundary layer on wall turbulence and net current formation through Direct Numerical Simulations of a smooth-walled planar channel driven by the Second-order Stokes Wave. Simulation results suggest that net current characteristics depend on whether developed turbulence is present. When turbulence is developed, asymmetric viscous length scale is found to be the primary reason of the net current whereby a vertical offset between negative and positive Reynolds shear stress profiles, associated with forward and reverse flows, respectively, is created in a cycle. After averaging over a cycle, residual Reynolds shear stress, which drives the net current, is observed to be within the offset layer.

Drifting breathers and Fermi–Pasta–Ulam paradox for water waves

One physical mechanism that is responsible for the focusing of uni-directional water waves is the modulation instability. This occurs when side-bands around the main frequency are excited either deterministically or randomly and subsequently grow exponentially. In physical space, the periodically-perturbed wave group can reach significant wave amplifications and in the case of infinite modulation period even three times the initial amplitude of the regular Stokes wave train. These periodic wave groups propagate in deep-water with a group velocity half the wave’s phase speed. In this experimental study, we investigate the dynamics of modulationally unstable wave groups that propagate with a velocity that is higher than the packet’s group velocity in deep-water. It is shown that when this additional velocity to the wave group is marginal, a very good agreement with nonlinear Schrödinger hydrodynamics is reached at all stages of propagation and the characteristic wave energy cascade remains symmetric and stationary. Otherwise, a significant deviation is observed only at the stage of large wave focusing of the isolated wave packet. In this case, the wave field experiences an almost perfect return to the initial conditions after the compression, a particular dynamics also known as the Fermi–Pasta–Ulam paradox.

Suppression of Stimulated Brillouin Scattering in Single-Frequency Fiber Raman Amplifier Through Pump Modulation

We propose and implement the use of an intensity-modulated pump, counter-directional to a forward-propagating signal for suppressing unwanted stimulated Brillouin scattering (SBS) of spectrally narrow signals in optical fiber Raman amplifiers. The modulated Raman pump cross-phase modulates and thus spectrally broadens the parasitic Brillouin Stokes wave traveling in the same (backward) direction. The parasitic SBS then decreases when the broadening exceeds the Brillouin gain bandwidth. By contrast, the modulated pump does not induce any significant cross-phase modulation on the signal, which therefore can remain spectrally narrow. We study the effect of the pump modulation frequency and modulation format and experimentally obtain nearly 5 dB of SBS threshold enhancement and over 50 $\%$ pump-to-signal conversion efficiency.

Simultaneous generation of guided-acoustic-wave Brillouin scattering and stimulated-Brillouin-scattering in hybrid As2Se3-PMMA microtapers.

We demonstrate simultaneous generation of Stimulated Brillouin Scattering (SBS) and guided-acoustic-wave Brillouin scattering (GAWBS) from electrostriction of optical waves in a 60cm As2Se3-PMMA microtaper waveguide. The GAWBS in the microtaper couples with SBS through a complex energy transfer between weak Stokes and Anti-Stokes (AS) continuous waves in the presence of a high power pulsed pump wave. This results in an amplification of Stokes wave at 7.4 GHz due to modulation of the optical fiber by GAWBS at 211 MHz generated by the pump in addition to a strong Stokes peak at 7.62 GHz and a secondary peak at 7.8 GHz that are contributed by SBS for a 2μm As2Se3 core radius. Such strong coupling of forward and backward Brillouin scattering due to large acoustic impedance between the core and cladding in such compact, highly nonlinear fibers plays a vital role in simultaneously sensing longitudinal and trasnverse strain within the core as well as its surroundings.

Modeling the optical nonlinear effects on DFB-RF laser based on the transfer matrix method

• The operation of a DFB-RF laser above threshold condition is analyzed numerically. • Three coupled nonlinear wave equations are solved using transfer matrix method. • The nonlinear optical effects, XPM and SPM are considered on the behavior of laser. • The XPM and SPM effects cause to output power saturation and a red shift in the laser operation wavelength. • The optimum length is presented for DFB-RF laser with considering nonlinear optical effects. In this paper, the operation of π-phase shifted distributed feedback Raman fiber (DFB-RF) laser above threshold condition is analyzed theoretically. The nonlinear optical phenomena such as self phase modulation (SPM) and cross phase modulation (XPM) have significant effect on the performance of DFB-RF laser. The numerical results show that the nonlinear effects cause to the saturation of output power and the value of saturated power is dependent on the fiber length. It is found that, the operation wavelength of stokes modes of DFB-RF laser varies in above threshold condition as a result of nonlinear optical properties of the fiber. Simulation is performed by using transfer matrix method to solve three coupled nonlinear wave equations which describe the propagation of pump, forward and backward Stokes waves. The nonlinear SPM and XPM effects are considered in the presented theoretical model.

Numerical simulation of an oscillating water column device installed over a submerged breakwater

In this paper, an example of a symbiotic combination of marine structures is studied. The base hydrodynamic performance of a traditional offshore oscillating water column (OWC) device is compared against the case in which it is constructed over a submerged breakwater. Numerical simulations are performed with the open source package OpenFOAM and the toolbox waves2Foam. The classical free surface capturing method volume of fluid (VOF) is employed to model first-order Stokes waves. The effects of the dimensions (height and length) of the submerged breakwater on the hydrodynamic characteristics, such as wave energy conversion efficiency (ξ), reflection coefficient (Cr), transmission coefficient (Ct), and energy dissipation ratio (ED) are explored thoroughly. Moreover, the effect of wave nonlinearity induced by increasing incident wave heights on the OWC device is examined. The results show that a proper configuration of the submerged breakwater is significantly helpful to optimize the energy conversion curve. Varying breakwater length will lead to a periodic variation in energy absorption efficiency. In addition, a higher incident wave height will lead to a lower energy absorption efficiency but a stronger dissipation ratio.

Breather Turbulence: Exact Spectral and Stochastic Solutions of the Nonlinear Schrödinger Equation

I address the problem of breather turbulence in ocean waves from the point of view of the exact spectral solutions of the nonlinear Schrödinger (NLS) equation using two tools of mathematical physics: (1) the inverse scattering transform (IST) for periodic/quasiperiodic boundary conditions (also referred to as finite gap theory (FGT) in the Russian literature) and (2) quasiperiodic Fourier series, both of which enhance the physical and mathematical understanding of complicated nonlinear phenomena in water waves. The basic approach I refer to is nonlinear Fourier analysis (NLFA). The formulation describes wave motion with spectral components consisting of sine waves, Stokes waves and breather packets that nonlinearly interact pair-wise with one another. This contrasts to the simpler picture of standard Fourier analysis in which one linearly superposes sine waves. Breather trains are coherent wave packets that “breath” up and down during their lifetime “cycle” as they propagate, a phenomenon related to Fermi-Pasta-Ulam (FPU) recurrence. The central wave of a breather, when the packet is at its maximum height of the FPU cycle, is often treated as a kind of rogue wave. Breather turbulence occurs when the number of breathers in a measured time series is large, typically several hundred per hour. Because of the prevalence of rogue waves in breather turbulence, I call this exceptional type of sea state a breather sea or rogue sea. Here I provide theoretical tools for a physical and dynamical understanding of the recent results of Osborne et al. (Ocean Dynamics, 2019, 69, pp. 187–219) in which dense breather turbulence was found in experimental surface wave data in Currituck Sound, North Carolina. Quasiperiodic Fourier series are important in the study of ocean waves because they provide a simpler theoretical interpretation and faster numerical implementation of the NLFA, with respect to the IST, particularly with regard to determination of the breather spectrum and their associated phases that are here treated in the so-called nonlinear random phase approximation. The actual material developed here focuses on results necessary for the analysis and interpretation of shipboard/offshore platform radar scans and for airborne lidar and synthetic aperture radar (SAR) measurements.

A Microwave Signal Generation Technique Based on Brillouin-Erbium Fiber Laser

An all-optical microwave signal generator based on multiwavelength Brillouin-Erbium fiber laser (MBEFL) is proposed. The MBEFL unit is designed at fixed wavelength spacing of 0.084 nm, which corresponds to ~10 GHz carrier signal. The underlying mechanism MBEFL unit is by recycling the backscattered Stokes waves into the cavity to generate higher-order Stokes waves. Heterodyning process is then applied to the Brillouin pump (BP) consisted of first-order Brillouin Stokes (BS1) (0.084 nm spacing) and second-order Brillouin Stokes (BS2) (0.168 nm spacing) signals by using a photodetector (PD). The heterodyned outputs are microwave signals at the frequencies of 10.56 GHz and 21.2 GHz, relative to first order and second order Stokes waves, respectively. These microwave signals are experimentally achieved by controlling the EDF pump power where the EDF pump power can be as low as 20 mW.

The Investigation on the Beam Spatial Intensity Distributions in the Injection-Seeded Terahertz Parametric Generator

In terahertz (THz) wave parametric sources, the large angle between the pump and terahertz waves (about 65° in LiNbO3 crystal) will affect the spatial intensity distributions. To investigate how the noncollinear phase-matching affects the spatial intensity properties of the pump, Stokes, and THz waves, we take the injection-seeded THz-wave parametric generator (is-TPG) as an example to study the beam spatial intensity properties via the numerical solutions of the coupled-wave equations. The simulated results demonstrate that the pump and Stokes intensities have significant changes along the THz-wave propagation direction. It is the first time that the spatial intensity distributions of the three waves in the THz-wave parametric sources have been studied in theory. In addition, the output intensity distributions of the pump, Stokes, and THz waves are measured in the designed is-TPG. The measured spatial intensity distributions are in coincidence with the simulated results. Investigations on the beam spatial intensity properties provide a potential approach to improve the THz-wave outputs and the pump conversion efficiency for the is-TPG and other THz-wave parametric sources.

Numerical modeling of breaking wave induced seawall scour

Scour is recognised as one of the major causes of seawall failure. This paper presents numerical modelling of seawall scour due to wave impact on a vertical seawall. The modelling of waves hydrodynamics is based on the solution of the Reynolds-Averaged Navier-Stokes equations with the k-ω model. Fifth-order Stokes waves are generated in a numerical wave tank (NWT) using the relaxation method. The free surface under the breaking wave is captured with the level set method. The wave field is coupled with a sediment transport algorithm to simulate seawall scour. The model is validated for an accurate wave field and the seawall scour due to the wave impact. The accuracy of the simulations is assured by comparing the numerical results with the theory and the experimental observations. The numerical results give insight into the seawall scour process through different stages. The results show that the seawall scour is governed by the breaking wave impact on the seawall. The development of the standing wave due to the reflected wave energy from the seawall leads to further sediment transport seawards. Finally, the model is used to simulate seawall scour for different scenarios of seawall locations, incident wave height and seabed slope. The study examines the wave impact on the seawall toe and resulting scour. The maximum seawall scour is observed when the seawall is located at the intersection point of the still water depth and the bed slope. A displacement of the seawall from the intersection point leads to a decrease in the wave impact. The surf similarity parameter ξ0 is recognised as an important factor affecting the seawall scour.

Constructive Study of Modulational Instability in Higher Order Korteweg-de Vries Equations

Our present study is devoted to the constructive study of the modulational instability for the Korteweg-de Vries (KdV)-family of equations u t + s u p u x + u x x x (here s = ± 1 and p > 0 is an arbitrary integer). For deducing the conditions of the instability, we first computed the nonlinear corrections to the frequency of the Stokes wave and then explored the coefficients of the corresponding modified nonlinear Schrödinger equations, thus deducing explicit expressions for the instability growth rate, maximum of the increment and the boundaries of the instability interval. A brief discussion of the results, open questions and further research directions completes the paper.

Stokes waves with constant vorticity: I. Numerical computation

AbstractPeriodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.

Improving performance of a semi-analytical model for nonlinear water waves

An efficient semi-analytical solution to the problem of nonlinear water waves propagating at a constant depth is derived using parallel computing techniques. The approach is based on the assumptions of an irrotational flow of an inviscid and incompressible fluid. Velocity potential and free-surface displacement functions are represented in a form based on Fourier series satisfying a formulated initial boundary-value problem by a proper selection of time-dependent coefficients. The eigenfunction expansions are resolved by means of a Fast Fourier Transform technique and a higher-order time-stepping procedure to provide an efficient spectral collocation method of computation of wave-induced free-surface flows. The application of the Fourier transform qualifies the semi-analytical algorithm for efficient parallelization. Hence, the numerical model performance is improved using the computing capabilities of a GPU. The derived semi-analytical approach is easy to implement and may be straightforwardly extended to cover the generation of waves in a flume or a basin. The ability of the model to reproduce nonlinear phenomena is demonstrated in simulations of two- and three-dimensional instabilities of a modulated Stokes wave. The model can be used to solve standard as well as complex wave problems in large computational domains considerably faster than its CPU-based counterpart.

Nonstationary Stimulated Raman Scattering by Polaritons in Cubic Crystals

The purpose of this article is to consider two aspects of the nonstationary stimulated Raman scattering by polaritons in cubic crystals. The first feature is related to the pump field, which, by deforming the permittivity of the medium, changes its symmetry. As a result, for example, the cubic crystal becomes anisotropic. The second one results from the possibility of exciting anomalous longitudinal waves at the frequency of the mechanical phonons which is the fundamental difference between scattering by dipole-active (polar) phonons and that of by dipole-inactive (nonpolar) ones. When the phonon frequency is approached, the amplitude of the transverse polariton wave decreases due to increased absorption and the wave mismatch. The polariton wave becomes practically longitudinal. Such a wave is maintained by the pump field and exists only in a pumped medium. The system of four shortened nonstationary equations (two for the Stokes waves with perpendicular polarizations and two for both transverse and longitudinal polariton waves) is obtained. The analysis is carried out for a given stationary pump field which is assumed to be a linearly polarized plane electromagnetic wave. Principal attention was paid to the calculation and analysis of the gain factor which defines the intensities of both stimulated (SRS) and spontaneous Raman scattering. The expressions for two proper gain factors gμ are obtained for Stokes waves in nonstationary case. It was shown that the pumped cubic crystal becomes anisotropic. It is also shown that the values of intensities calculated by using the expression for gμ are consistent with the experimental results for spectra of ZnS.

Dimension-Reduced Model for Deep-Water Waves

Starting from the 2D Euler equations for an incompressible potential flow, a dimension-reduced model describing deep-water surface waves is derived. Similar to the Shallow-Water case, the z-dependence of the dependent variables is found explicitly from the Laplace equation and a set of two one- dimensional equations in x for the surface velocity and the surface elevation remains. The model is nonlocal and can be formulated in conservative form, describing waves over an infinitely deep layer. Finally, numerical solutions are presented for several initial conditions. The side-band instability of Stokes waves and stable envelope solitons are obtained in agreement with other work. The conservation of the total energy is checked.

Highly nonlinear wind waves in Currituck Sound: dense breather turbulence in random ocean waves

We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of nonlinear Fourier analysis (NLFA) methods (Osborne 2010) to analyze the data set: The approach spectrally decomposes a nonlinear wave field into sine waves, Stokes waves, and phase-locked Stokes waves otherwise known as breather trains. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over cycle times of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrodinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the nonlinear spectral representation. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its nonlinear coherent structures (Stokes waves and breathers) rather than just sine waves. This is done by the implementation of multidimensional, quasi-periodic Fourier series, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height Hs, the peak period Tp, and the length of the time series T. The time series analyzed here have 8192 points and T =1677.72 s = 27.96 min. Near the peak of the storm, we find Hs ≈ 0.55 m, Tp ≈ 2.4 s so that for the wave steepness of a near Gaussian process, \({S} = \left (\pi ^{5/2}/g\right )H_{s}/{T}_{p}^{2}\), we find S ≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process, \({I_{BF}} = \left (\pi ^{5/2}/g \right ) H_{s} T/{T}_{p}^{3}\), and we find IBF ≈ 119. Since the BF parameter describes the nonlinear behavior of the modulational instability, leading to the formation of breather packets in a measured wave train, we find the IBF for these storm waves to be a surprisingly high number. This is because IBF, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a nonlinear spectral point of view, the presence of large numbers of breather trains and we determine many of their properties, including the rise time for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of breather turbulence, highly nonlinear integrable turbulence theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.

Relative intensity noise in a multi-Stokes Brillouin laser.

We investigate the relative intensity noise (RIN) properties of a multi-Stokes Brillouin fiber ring laser. We experimentally analyse the intensity noise of each Stokes wave and study the noise dynamics of the cascaded Brillouin scattering process. We observe up to 20 dB/Hz intensity noise reduction compared to that of the RIN input pump laser. We examine the impact of the fiber ring quality factor on the laser RIN features such as amplitude reduction and relaxation frequency. We also investigate the influence of the Brillouin gain detuning on the RIN reduction. A numerical model based on a set of coupled-mode equations replicates the experimental observations. Our study enables us to determine the optimal parameter values to operate the multi-Stokes laserin the low noise regime.

Stimulated Raman Amplification in GaAs / AlAs Intermixed Superlattices

We show the effectiveness of a χ(3)-quasi-phase-matched structure made of a GaAs/AlAs inter-mixed superlattice in stimulated Raman amplification for an enhancement in the output efficiency of anti-Stokes waves, where the efficiency is originally extremely small due to a phase mismatch for the anti-Stokes waves (although the output efficiency of Stokes waves is sufficiently large because of no such phase mismatch). We demonstrate that our quasi-phase matched structure enhances the anti-Stokes output efficiency up to the order of 103 when compared with that without the quasi-phase matched structure, particularly at a small pump intensity. Here, the effects of four-wave mixing, self- and cross-phase modulations are included in addition to Raman amplification with linear optical losses. In this efficiency enhancement, we see that the output efficiency of the anti-Stokes waves is boosted via simultaneous four-wave mixing. Moreover, we show how much the effects of the self- and cross-phase-modulations degrade this efficiency enhancement. With all those effects taken into account, we show that there is an optimal superlattice length that provides the highest efficiency. In addition, we analyze the efficiency degradation due to two-photon absorption, and point out that in order to keep the highest efficiency at the optimal length, the Stokes, anti-Stokes, and pump wavelengths should be set larger than the half-bandgap wavelength of the intermixed superlattice.