Stokes Wave Research Articles

Compact millijoule diode-seeded two-stage fiber master oscillator power amplifier using a multipass and forward pumping scheme.

The multipass scheme for a diode-seeded fiber master oscillator power amplifier with a nanojoule-to-millijoule output energy level at a repetition rate of <100 kHz is numerically analyzed for comparison to an experimental benchmark. For a 6/125 single-mode preamplifier with a small input energy (<1 nJ), there is a significant improvement in the output energy from 0.7% to 80% and 95% of the maximum extractable energy using the double-pass and four-pass schemes, respectively. For a 30/250 large-mode-area power amplifier using the double-pass and forward pumping scheme, the required input energy is decreased from 100μJ to 18μJ for millijoule energy extraction with accompanying Stokes waves of less than 10% of the total energy. The system based on the full master oscillator power amplifier configuration with an output energy exceeding millijoule level can be optimally simplified to two stages for commercialization.

Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth

This paper describes an efficient algorithm for computing steady two-dimensional surface gravity waves in irrotational motion. The algorithm complexity is $O(N\log N)$, $N$ being the number of Fourier modes. This feature allows the arbitrary precision computation of waves in arbitrary depth, i.e. it works efficiently for Stokes, cnoidal and solitary waves, even for quite large steepnesses, up to approximately 99 % of the maximum steepness for all wavelengths. In particular, the possibility to compute very long (cnoidal) waves accurately is a feature not shared by other algorithms and asymptotic expansions. The method is based on conformal mapping, the Babenko equation rewritten in a suitable way, the pseudo-spectral method and Petviashvili iterations. The efficiency of the algorithm is illustrated via some relevant numerical examples. The code is open source, so interested readers can easily check the claims, use and modify the algorithm.

On the limiting Stokes wave of extreme height in arbitrary water depth

Both Schwartz (J. Fluid Mech., vol. 62 (3), 1974, pp. 553–578) and Cokelet (Phil. Trans. R. Soc. Lond., vol. 286 (1335), 1977, pp. 183–230) failed to gain convergent results for limiting Stokes waves in extremely shallow water by means of perturbation methods, even with the aid of extrapolation techniques such as the Padé approximant. In particular, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of$120^{\circ }$included angle first mentioned by Stokes in the 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depths, cnoidal waves in shallow water and solitary waves in extremely shallow water.

Broadband noncollinear optical parametric amplification under low‐frequency picosecond pump in homogeneous KTP crystal

AbstractThe nature of cubic and cascaded quadratic nonlinear responses in the KTiOPO4 crystal under picosecond pump is studied. The main attention is paid to determination of the origin and properties of the blue radiation accompanying the red light amplification process (the central wavelength ~685 nm, bandwidth ~20 nm) in a noncollinear optical parametric amplifier (NOPA) pumped by the second‐harmonic radiation of the picosecond Nd:YAG laser (532 nm). In NOPA, the amplified seed radiation (ωs) becomes so intense that it can participate in any nonlinear three‐wave interaction and also can play the role of the Stokes wave in the in the coherent anti‐Stokes Raman scattering‐process in which powerful pump radiation (ωp) of the main parametric process serves both as the pump wave and the probe wave. Possible causes of the blue light emanating from the KTP crystal and the mechanisms for their realization are analyzed. We interpret the ring of blue light around the axis located between the directions of propagation of the green pump beam and the red beam of the amplified seed radiation, as the emission of amplified parametric fluorescence (signal wave) under low‐frequency pump. This phenomenon can be used for effective amplification in picosecond NOPA of weak anti‐Stokes radiation in the experiments with time‐resolved broadband picosecond coherent Anti‐Stokes Raman Scattering.

On the relation between Stokes drift and the Gerstner wave

We discuss the properties of two-dimensional, nonlinear, potential, and vortex waves on the surface of an ideal liquid of infinite depth. It is shown that in the quadratic order in the amplitude, the vorticity of the Gerstner wave is equal in magnitude to and different in sign from that of the Stokes drift current in a surface layer. This allows a classic Stokes wave obtained in the framework of potential theory to be interpreted as a superposition of the Gerstner wave and Stokes drift. It is proposed that the nonlinearity coefficient in the nonlinear Schrödinger equation can be physically interpreted as the Doppler frequency shift along the vertically averaged Stokes drift current.

A pulsed single-frequency Nd:GGG/BaWO4 Raman laser

A single-frequency pulsed laser at 1178.3 nm was demonstrated in a crystalline Raman laser. A crystal combination of Nd:GGG and BaWO4 was selected to realize Raman conversion from a 1062.5 nm fundamental wave to a 1178.3 nm Stokes wave. An entangled cavity was specially designed to form an intracavity Raman configuration. Single-longitudinal-mode operation was realized by introducing two Fabry–Perot etalons into the Raman laser cavity. This laser operated at a pulse repetition rate of 50 Hz with 2 ms long envelopes containing micro pulses at a 30 kHz repetition rate. The highest output power was 41 mW with the micro pulse duration of 15 ns. The linewidth was measured to be less than 130 MHz.

Multi-Wavelength Brillouin Random Fiber Laser via Distributed Feedback From a Random Fiber Grating

Multi-wavelength (MW) laser sources with high optical signal-to-noise ratios (OSNR) are of great interest for optical communications with ultrahigh data capacity as well as microwave and terahertz photonics. In this paper, we report the first demonstration of MW Brillouin random fiber laser in the telecom spectral window around 1.5 μm based on a unique random fiber grating (RFG). Random feedback by enhanced distributed Rayleigh scattering from the RFG basically enables a high-efficiency random lasing resonance of the Stokes wave via stimulated Brillouin scattering in optical fibers. A subfiber loop eventually delivers the cascading process for the high-efficiency generation of up to a 14-order Stokes’ comb with an ultrahigh OSNR of 41 dB and an optimized peak power discrepancy of 3.4 dB. Thanks to Rayleigh scattered random feedback from the RFG, single longitudinal mode operation of the Stokes laser emission is achieved.

Nonlinear phase-resolved reconstruction of irregular water waves

We develop and validate a high-order reconstruction (HOR) method for the phase-resolved reconstruction of a nonlinear wave field given a set of wave measurements. HOR optimizes the amplitude and phase of $L$ free wave components of the wave field, accounting for nonlinear wave interactions up to order $M$ in the evolution, to obtain a wave field that minimizes the reconstruction error between the reconstructed wave field and the given measurements. For a given reconstruction tolerance, $L$ and $M$ are provided in the HOR scheme itself. To demonstrate the validity and efficacy of HOR, we perform extensive tests of general two- and three-dimensional wave fields specified by theoretical Stokes waves, nonlinear simulations and physical wave fields in tank experiments which we conduct. The necessary $L$, for general broad-banded wave fields, is shown to be substantially less than the free and locked modes needed for the nonlinear evolution. We find that, even for relatively small wave steepness, the inclusion of high-order effects in HOR is important for prediction of wave kinematics not in the measurements. For all the cases we consider, HOR converges to the underlying wave field within a nonlinear spatial-temporal predictable zone ${\mathcal{P}}_{NL}$ which depends on the measurements and wave nonlinearity. For infinitesimal waves, ${\mathcal{P}}_{NL}$ matches the linear predictable zone ${\mathcal{P}}_{L}$, verifying the analytic solution presented in Qi et al. (Wave Motion, vol. 77, 2018, pp. 195–213). With increasing wave nonlinearity, we find that ${\mathcal{P}}_{NL}$ contains and is generally greater than ${\mathcal{P}}_{L}$. Thus ${\mathcal{P}}_{L}$ provides a (conservative) estimate of ${\mathcal{P}}_{NL}$ when the underlying wave field is not known.

Oil Droplet Transport under Non-Breaking Waves: An Eulerian RANS Approach Combined with a Lagrangian Particle Dispersion Model

Oil droplet transport under a non-breaking deep water wave field is investigated herein using Computational Fluid dynamics (CFD). The Reynolds-averaged Navier–Stokes (RANS) equations were solved to simulate regular waves in the absence of wind stress, and the resulting water velocities agreed with Stokes theory for waves. The RANS velocity field was then used to predict the transport of buoyant particles representing oil droplets under the effect of non-locally generated turbulence. The RANS eddy viscosity exhibited an increase with depth until reaching a maximum at approximately a wave height below the mean water level. This was followed by a gradual decrease with depth. The impact of the turbulence was modeled using the local value of eddy diffusivity in a random walk framework with the added effects of the gradient of eddy diffusivity. The vertical gradient of eddy viscosity increased the residence time of droplets in the water column region of high diffusivity; neglecting the gradient of eddy diffusivity resulted in a deviation of the oil plume centroid by more than a half a wave height after 10 wave periods.

Kadomtsev–Petviashvili type equation for entire range of relative water depths

ABSTRACTA Kadomtsev–Petviashvili type equation valid for the entire range of relative water depths; namely, shallow, intermediate, and deep, is derived. The new equation is capable of simulating the shallow water cnoidal waves and the deep water second-order Stokes waves equally well besides accounting for wave shoaling due to varying water depths. Linear shoaling properties of the equation for unidirectional sinusoidal waves are in complete agreement with the energy flux concept. A finite-difference scheme is adopted for numerical solution of the equation to demonstrate its performance against test cases and its wide range of possible applications.

Eye-Safe KGd(WO4)2 Raman Lasers: Comparative Study of Pumping of Nd:KGd(WO4)2 Lasers With 4F3/2 → 4I11/2 and 4F3/2 → 4I13/2 Working Transitions**

This is a comparative study of eye-safe extracavity KGd(WO4)2 (KGW) Raman lasers that convert multimode emission from pulsed Nd:KGW lasers operating on the 4F3/2 → 4I11/2 and 4F3/2 → 4I13/2 transitions into the third Stokes component at λ = 1500 nm and the first Stokes component at λ = 1507 or 1538 nm. For equal sizes of the Raman and laser sections of a Raman laser emitting the third Stokes component, the pulse energy is higher (~14.2 mJ) and the divergence of the Stokes beam is lower (~9.4 mrad) for a lower energy of the Nd:KGW laser (~6.7 J). For pump pulse energies of 29–34 mJ, all the Raman lasers have essentially the same optical lasing efficiency of up to 36%. In terms of the electrical energy delivered to the flashlamp, however, the overall efficiency of the Raman laser for the third Stokes component is a factor of ~1.7 higher. The duration of the Raman pulses is comparable to that of the pump pulses and is ~20 ns for the first Stokes components. The pulse in the third Stokes component is shorter by a factor of 2.5–3. Given the different pump levels and the differences in the thermallyinduced optical distortions of the Nd:KGW laser components, the energy density distribution for beams of the first Stokes components is elliptical, while that of the third Stokes components is almost circular. The possibility of simultaneous lasing at wavelengths of 1507 and 1538 nm with parallel and mutually orthogonal polarizations of the Stokes waves is demonstrated.

Dynamics of a Wave Packet on the Surface of an Inhomogeneously Vortical Fluid (Lagrangian Description)

A nonlinear Schrodinger equation (NSE) describing packets of weakly nonlinear waves in an inhomogeneously vortical infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is shown that the modulational instability criteria for the weakly vortical waves and potential Stokes waves on deep water coincide. The effect of vorticity manifests itself in a shift of the wavenumber of high-frequency filling. A special case of Gerstner waves with a zero coefficient at the nonlinear term in the NSE is noted.

Numerical analysis on random wave-induced porous seabed response

ABSTRACTResearch on the response of random wave on offshore structures has received great deal of attention of many researchers and engineers in the design of marine structures. Most previous investigations have been limited to the regular waves. In this paper, based on Longuet–Higgins random wave theory and finite element method, a numerical model for random wave-induced seabed response is established. The seabed is treated as poroelastic medium and characterized by Biot’s partly dynamic equations (u–p model). The JONSWAP spectrum is adopted in Longuet–Higgins model, which is based on the cumulative superposition of linear diffraction solution. Based on the numerical results, the effects of random wave on seabed response are investigated by comparing with the corresponding Stokes wave and cnoidal wave. Then, a parametric study is conducted to examine the effect of wave and soil characteristic on the seabed.

High sensitivity distributed temperature fiber sensor using stimulated Brillouin scattering

Optical fiber technology has become a very powerful tool for distributed temperature (strain and refractive index) sensing, and can be used to monitor critical infrastructures such as bridges, aircrafts, pipelines, etc. Stimulated Brillouin scattering (SBS) in optical fibers used for distributed sensing utilizing the first Stokes order, is limited to a fixed material property, 1.1 MHz/°C for SMF-28. We demonstrate a distributed higher order Stokes SBS temperature fiber-sensor increasing the achievable sensitivity by several folds to over 4 MHz/°C. The proposed system uses time-gating for distributed sensing. This allows the increase in sensitivity by the order of the Stokes waves generated while maintaining a fairly normal spatial resolution over a few kilometers of sensing length. Increased sensitivity on these types of sensors may allow an earlier detection which could prevent failure of the monitored structure.

Dynamic-pressure distributions under Stokes waves with and without a current.

To investigate changes in the instability of Stokes waves prior to wave breaking in shallow water, pressure data were recorded vertically over the entire water depth, except in the near-surface layer (from 0 cm to -3 cm), in a recirculating channel. In addition, we checked the pressure asymmetry under several conditions. The phase-averaged dynamic-pressure values for the wave-current motion appear to increase compared with those for the wave-alone motion; however, they scatter in the experimental range. The measured vertical distributions of the dynamic pressure were plotted over one wave cycle and compared to the corresponding predictions on the basis of third-order Stokes wave theory. The dynamic-pressure pattern was not the same during the acceleration and deceleration periods. Spatially, the dynamic pressure varies according to the faces of the wave, i.e. the pressure on the front face is lower than that on the rear face. The direction of wave propagation with respect to the current directly influences the essential features of the resulting dynamic pressure. The results demonstrate that interactions between travelling waves and a current lead more quickly to asymmetry.This article is part of the theme issue 'Nonlinear water waves'.

The dynamic pressure in deep-water extreme Stokes waves.

In this paper, we consider the dynamic pressure in a deep-water extreme Stokes wave. While the presence of stagnation points introduces a number of mathematical complications, maximum principles are applied to analyse the dynamic pressure in the fluid body by means of an excision process. It is shown that the dynamic pressure attains its maximum value beneath the wave crest and its minimum beneath the wave trough, while it decreases in moving away from the crest line along any streamline.This article is part of the theme issue 'Nonlinear water waves'.

Third-order stokes wave solutions of the free surface capillary-gravity wave and the interfacial internal wave

Based on the Stokes wave theory, the capillary-gravity wave and the interfacial internal wave in two-layer constant depth’s fluid system are investigated. The fluids are assumed to be incompressible, inviscid and irrotational. The third-order Stokes wave solutions are given by using a perturbation method. The results indicate that the third-order solutions depend on the surface tension, the density and the depth of each layer. As expected, the first-order solutions are the linear theoretical results (the small amplitude wave theoretical results). The second-order and the third-order solutions describe the nonlinear modification and the nonlinear interactions. The nonlinear impact appears not only in the n (n≥2) times’ high frequency components, but also in the low frequency components. It is also noted that the wave velocity depends on the wave number, depth, wave amplitude and surface tension.

Theoretical investigation on mid-infrared cascaded Raman fiber laser based on tellurite fiber.

A numerical simulation of third-order cascaded Raman fiber laser based on tellurite fiber at the 2-5 μm waveband is presented. The Raman fiber laser can be optimized with the most suitable tellurite fiber length of 0.5-1.0 m and the most reasonable reflectivity of the third-order Stokes output FBG32 of 10%-20%. We demonstrate numerically that the third-order Stokes wave can reach the maximum average power of 45.2 W and the maximum optical conversion efficiency of 45.2%, corresponding to the FBG32 reflectivity of 10% and the tellurite fiber length of 0.3 m with the attenuation of 0.85 dB/m, when pumped by 2 μm light with the average power of 100 W. Our simulated results provide valuable theoretical guidance for the design and experiment of tellurite Raman fiber laser at a mid-infrared waveband.

16.7 W 885 nm diode-side-pumped actively Q-switched Nd:YAG/YVO4 intracavity Raman laser at 1176 nm

We proposed and experimentally demonstrated the generation of high-power 1176 nm Stokes wave by frequency shifting of a 885 nm diode-side-pumped Nd:YAG laser using a YVO4 crystal in a Z-shaped cavity configuration. Employing the 885 nm diode-side-pumped scheme and the Z-shaped cavity, for the first time to our knowledge, we realized the thermal management effectively, achieving excellent 1176 nm Stokes wave consequently. With an incident pump power of ~190.0 W, a maximum average output power of 16.7 W was obtained at the pulse repetition frequency of 10 kHz. The pulse duration and spectrum linewidth of the Stokes wave at the maximum output power were 20.3 ns and ~0.08 nm, respectively.

Demonstration of a 150-kW-peak-power, 2-GHz-linewidth, 1.9-μm fiber gas Raman source.

We report here for the first time, to the best of our knowledge, a 100-kW-peak-power, GHz-linewidth, sub-nanosecond, 1.9-μm laser source by gas stimulated Raman scattering (SRS) in hollow-core fiber. A H2-filled, anti-resonance, hollow-core fiber is pumped with a sub-nanosecond, high-peak-power, 1064-nm microchip laser, generating a 1908-nm Stokes wave by vibrational SRS of H2 molecules. A maximum peak power of about 150kW (average power 55mW, pulse energy 55μJ) is achieved with a 1.4-m fiber length and only 3bar H2 pressure. The maximum quantum efficiency is about 54%, and the corresponding slope efficiency is about 37%. The linewidth of the Stokes wave is about 2GHz, which decreases about 1-2 orders compared with the rare-earth-doped fiber lasers of the same peak-power level. Operation close to atmospheric pressure makes it more convenient in future applications. If a tunable pump laser is used, a high-peak-power, narrow-linewidth, broadly tunable, 2-μm fiber laser source can be easily achieved.