Higher-order Stokes Waves Research Articles

Nonlinear analytical solution for radiation stress of higher-order Stokes waves on a flat bottom

The radiation stress of water surface waves plays an important role in many aspects of marine hydrodynamics. The nonlinear effects of highly nonlinear waves on the radiation stress were investigated on a flat bottom under finite water depth and deep water conditions. Analytical solutions for radiation stress were derived based on different order nonlinear Stokes wave theory, respectively. The results indicate that the nonlinear radiation stresses are significantly greater than those based on the linear wave, and they increase as the nonlinearity strengthens. Through an inter-comparison of the expressions of nonlinear radiation stress of various orders of Stokes waves, it turned out that the results based on the second-order Stokes wave overestimated the radiation stress, while the expressions based on Fenton's fifth-order Stokes wave were more reasonable in reflecting the nonlinear effect of waves. In the case of extreme wave steepness, the fifth-order Stokes wave radiation stresses Sxx5 was approximately 17% larger than that of the linear wave. Finally, using the expressions of the fifth-order Stokes wave radiation stress, bound infragravity waves excited by the bichromatic wave groups formed by the superposition of two fifth-order Stokes waves were obtained.

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.

Single-frequency Brillouin distributed feedback fiber laser

We demonstrate a single-frequency Brillouin distributed feedback laser (DFB). The DFB laser cavity was a 12.4 cm long fiber Bragg grating with a π-phase shift offset from the grating center. It exhibited a threshold of 30 mW and conversion efficiency from pump to Stokes wave as high as 27%. Higher-order Stokes waves were suppressed by more than 20 dB. The Stokes output of the laser could be obtained in either the forward or backward direction, simply by changing the orientation of the offset of the discrete phase shift with respect to the pump propagation direction. The DFB laser operated over a pump frequency range of 1.2 GHz, more than 60 times larger than the SBS gain bandwidth.

On the effects of wave steepness on higher order Stokes waves

The effect of wave steepness on higher order finite amplitude Stokes waves is investigated analytically and numerically. It is shown that the phase speed increases as the wave steepness increases thereby initiating the wave instabilities. As the order increases, the phase speed also increases .However, the impact of wave steepness on phase speed is more pronounced for the wave steepness parameter &#949&#88050.09 where 0.02 &#8805 &#9490 &#8805 0.33 Beyond the upper bound, instability – wave breaking comes in. JONAMP Vol. 11 2007: pp. 369-374

Photonic crystal fibre Brillouin laser based on Bragg grating Fabry–Perot cavity

A photonic crystal fibre Brillouin laser based on fibre Bragg grating Fabry–Perot cavity is presented. A highly nonlinear photonic crystal fibre 25m in length is used as Brillouin gain medium and fibre Bragg grating Fabry–Perot cavity is chosen in order to enhance the laser conversion efficiency and suppress the higher-order Stokes waves. The laser reaches the threshold at input power of 35mW, and the experimental laser conversion efficiency achieves 18% of the input power of 140mW and does not show higher-order Stokes waves. A photonic crystal fibre Brillouin laser with shorter fibre length and lower threshold is experimentally realized.

Two-Frequency Brillouin Fiber Laser Based on Bragg Grating Fabry-Perot Cavity

A novel and simple two-frequency Brillouin fiber laser is presented. It is based on a fiber Fabry-Perot cavity with fiber Bragg gratings as reflectors. The model of stimulated Brillouin scattering in fiber grating–based Fabry-Perot resonator is investigated. The laser allows conversion efficiency of close to 100% and suppresses the higher-order Stokes waves. The theoretical prediction is presented and the experimental demonstration is realized.

Interplay between cascaded stimulated Brillouin scattering and four-wave mixing in a fiber Fabry–Perot resonator

I theoretically investigate the simultaneous generation of cascaded stimulated Brillouin scattering (CSBS) and four-wave mixing (FWM) in a fiber Fabry–Perot resonator with reflection mirrors. The coupled-wave equations for a pump, Stokes, and anti-Stokes waves are derived. To discuss the stationary interplay between the two nonlinear processes, I further deduce the coupled-power equations by introducing some approximations into them. I investigate the effect of FWM on CSBS for asymmetric and symmetric resonators by numerically solving the coupled-power equations under the boundary conditions at the two mirrors. Compared with the case without FWM, the FWM process leads to the gradual generation of higher-order Stokes waves, splitting of the intersection of two transmitted or reflected optical powers, and the generation of anti-Stokes waves. The present analysis can well explain an earlier experiment [Opt. Commun.32, 385 (1980)].

Generation and Dynamics of Cascaded Stimulated Brillouin Scattering in a High-Finesse Fiber Fabry–Perot Resonator

We theoretically investigate the dynamics of cascaded stimulated Brillouin scattering (CSBS) in a short-fiber Fabry–Perot resonator with reflection mirrors. The analysis is based on the coherent fourteen-wave model assuming the generation of up to third-order Stokes waves. Moreover, the cavity detunings for a pump wave and each Stokes wave and the optical Kerr effect are taken into account in the formulation. The temporal evolution of the pump power and each Stokes power is calculated for step and Gaussian input pulses. The threshold power for SBS and the boundary power between the stable and unstable regions are numerically calculated as a function of the amplitude reflectance of mirrors for different cavity detunings. The generation and dynamics of CSBS depend strongly on the pump power and mirror reflectance. Since higher order Stokes waves are successively generated with a finite time delay, their generation does not take place for short pulsed input where the treatment of transient phenomena is required. The experimental results obtained with a single-frequency pulsed YAG laser agree qualitatively with theoretical predictions.

Shoaling of sixth-order Stokes waves on a current

Jonsson and Arneborg (Jonsson, I.G., Arneborg, L., 1995. Energy properties and shoaling of higher-order Stokes waves on a current. Ocean Engng 22, 819-857.) combined energy flux and set-down to make shoaling predictions for fourth-order Stokes waves with and without a net volume flux. With basis in their expressions, sixth-order expressions are derived and combined to make shoaling predictions correct to sixth order with an arbitrary net volume flux. The new sixth-order results are compared with the fourth-order results and the practically exact results obtained by Sobey and Bando (Sobey R.J., Bando K., 1991. Variations on higher-order shoaling. J. Waterway, Port, Coastal Ocean Engng ASCE 117, 348-368) with a Fourier 18 model. The effects of introducing sixth-order theory rather than the fourth-order theory in shoaling calculations are in general found to be small. As expected the deviations increase with increasing wave-steepness, decreasing depth and opposing currents. Also as expected, the results obtained with the sixth-order expressions improve the results obtained with the fourth-order expressions when compared to the results of Sobey and Bando. As novelties, some considerations regarding the consistency of odd- and even-order shoaling calculations, and the magnitude of the bottom slope, are presented. Furthermore a comparison between the wave-induced current and the total current is given.

Doping of an absorbent into a Raman crystal for suppression of higher-order Stokes generation

A Raman-active crystal, lanthanum tungstate doped with praseodymium (Pr) ions, was grown to facilitate the selection of the order of Stokes waves in stimulated Raman scattering. Absorption of the second Stokes wave by the Pr ions suppressed conversion to higher-order Stokes waves, resulting in efficient energy conversion to the first Stokes wave.

Characteristic Velocities for Higher-Order Stokes Waves in Deep Water

The accuracy of asymptotic expressions for phase, wave energy, and wave action velocities for steady, plane gravity waves progressing in deep water and with no ambient currents is investigated. The calculations are to a large extent based on Fenton's fifth-order Stokes wave theory. Using the dispersion relation correct to sixth order for arbitrary water depth, the phase velocity is calculated to fourth order in wave steepness for infinitely deep water; this is in accordance with a number of previously found results. Potential and kinetic energy is calculated, which leads to the energy density correct to sixth order. Comparison with previous findings showed one discrepancy, which can however be explained. Also, the energy flux is found correct to sixth order, leading to a fourth-order expression for the wave energy velocity. From sixth-order action and action flux results, the wave action velocity is finally calculated correct to fourth order. Agreement with more accurate numerical results is excellent for all three velocities except near the wave of maximum steepness. Practical asymptotic formulas are introduced, enabling one to calculate the characteristic velocities explicitly from given wave height and period. Two new exact relations between integral properties for deep water with no current are presented.

Energy properties and shoaling of higher-order stokes waves on a current

The energy density, the energy flux, the set-down, the radiation stress, and some wave energy velocities have been derived correct to fourth order in wave steepness for waves on a vorticity-free current. The energy flux and the set-down have been used for shoaling predictions for finite amplitude waves with and without a net volume flux. The results with a zero volume flux are compared with more accurate shoaling predictions showing rather good accordance, except for large steepnesses. This also applies to the deep water wave energy transport velocity. The results with a net volume flux show that the steepness of the waves reduces the influence of this flux on the wave evolution. Some problems in connection with the orders in Stokes waves are discussed, among others concerning the dispersion relation and the orders of integral properties. Bed shear and accompanying dissipation is neglected.

Stimulated Brillouin scattering in optical fibers: the effects of optical amplification

Stimulated Brillouin scattering in optical fibers is carefully reassessed for the case of distributed optical gain (achievable in optical fibers lightly doped with rare-earth ions) instead of the usual loss. The effects of gain and amplified spontaneous emission on the threshold power are discussed, together with treatments of the effects of amplification on the line-narrowing process normally associated with stimulated Brillouin scattering in a lossy medium and the appearance of strong higher-order Stokes waves. The implications for erbium-doped fiber amplifier systems are explored.

On the theory of Stimulated Brillouin Scattering in Liquids

A calculation of Stimulated Brillouin Scattering in Liquids has been made for the steady state, analytically in the lossless case and by a machine calculation if acoustic loss is introduced. The effective interaction length in CS2 has been found to be of the order of 2×10−2 cm. Therefore no higher order Stokes waves can be produced with appreciable magnitudes, even if the first order Stokes wave has large intensity.