Three-wave Equations Research Articles

Second-order coherence across the Brillouin lasing threshold

Brillouin–Mandelstam scattering is one of the most accessible nonlinear optical phenomena and has been widely studied since its theoretical discovery one hundred years ago. The scattering mechanism is a three-wave-mixing process between two optical fields and one acoustic field and has found a broad range of applications spanning microscopy to ultra-narrow-linewidth lasers. Building on the success of utilizing this nonlinearity at a classical level, a rich avenue is now being opened to explore Brillouin scattering within the paradigm of quantum optics. Here, we take a key step in this direction by employing quantum optical techniques yet to be utilized for Brillouin scattering to characterize the second-order coherence of Stokes scattering across the Brillouin lasing threshold. We use a silica microsphere resonator and single-photon counters to observe the expected transition from bunched statistics of thermal light below the lasing threshold to Poissonian statistics of coherent light above the threshold. Notably, at powers approaching the lasing threshold, we also observe super-thermal statistics, which arise due to instability and a “flickering” in and out of lasing as the pump field is transiently depleted. The statistics observed across the transition, including the “flickering,” are a result of the full nonlinear three-wave-mixing process and cannot be captured by a linearized model. These measurements are in good agreement with numerical solutions of the three-wave Langevin equations and are well demarcated by analytical expressions for the instability and the lasing thresholds. These results demonstrate that applying second-order-coherence and photon-counting measurements to Brillouin scattering provides new methods to advance our understanding of Brillouin scattering itself and progress toward quantum-state preparation and characterization of acoustic modes.

Three-Wave Interaction Equations: Classical and Nonlocal

Three-Wave Interaction Equations: Classical and Nonlocal

Nonlinear processes and laser damage induced by a temporally modulated nanosecond UV laser beam. Part I: stimulated Brillouin scattering

Phase modulation is currently applied on pump laser beams propagating in thick optical elements to avoid optical damage induced by Brillouin backscattering. The experimental measurements show that adding a 2 GHz amplitude modulation to such a phase-modulated laser beam can compromise this optical preservation by increasing the Brillouin backscattering signal. These results are confirmed and explained with a theoretical analysis, developed in the perturbative regime, and with the numerical results obtained by solving three-wave coupling equations. Particularly, we show that when the frequency of the amplitude modulation is a harmonic of the phase modulation frequency, the Brillouin gain value is increased and can exceed the one obtained without amplitude modulation.

Benchmarking magnetised three-wave coupling for laser backscattering: analytic solutions and kinetic simulations

Understanding magnetised laser–plasma interactions is important for controlling magneto-inertial fusion experiments and developing magnetically assisted radiation and particle sources. For nanosecond pulses at non-relativistic intensities, interactions are dominated by coherent three-wave interactions, whose nonlinear coupling coefficients became known only recently when waves propagate at oblique angles with the magnetic field. In this paper, backscattering coupling coefficients predicted by warm-fluid theory are benchmarked using particle-in-cell simulations in one spatial dimension, and excellent agreements are found for a wide range of plasma temperatures, magnetic field strengths and laser propagation angles, when the interactions are mediated by electron-dominant hybrid waves. Systematic comparisons between theory and simulations are made possible by a rigorous protocol. On the theory side, the initial boundary value problem of linearised three-wave equations is solved, and the transient-time solutions allow the effects of growth and damping to be distinguished. On the simulation side, parameters are carefully chosen and calibration runs are performed to ensure that comparisons are well controlled. Fitting simulation data to analytical solutions yields numerical growth rates that match theory predictions within error bars. Although warm-fluid theory is found to be valid for a wide parameter range, genuine kinetic effects have also been observed.

Space-time exchanged symplectic integrator approach to temporal-mode selective optical three-wave mixing frequency conversion

A space-time exchanged symplectic integrator (SI) method is developed to solve the three-wave resonance interaction equations commonly used in describing a number of fundamental nonlinear optical phenomena, such as second harmonic generation, optical parametric amplification, etc. Exchanging the roles of the space and time coordinates allows us to solve these nonlinearly coupled equations with hard-wired boundary conditions using a highly accurate SI scheme instead of standard finite difference methods. The developed method is then applied to the optical three-wave mixing frequency conversion processes in which an input signal light is converted into an idler light guided by a strong pump pulse. The temporal-mode selectivity in the frequency conversion processes is examined by systematically varying system parameters and the results are summarized as contour maps of selectivity on the ‘group-slownesses’, defined by the inverse of the group velocities, of the signal pulse and of the idler pulse relative to that of the pump pulse. The areas showing the highest peaks on these undergo significant and characteristic changes in their positions as well as the heights for different values of the coupling strength γ. The observed trends are rationalized on the basis of the Schmidt modes obtained by decomposing the two-time transfer function that fully characterizes the dynamics of the signal–idler wave conversion.

Rogue waves of wave resonant-interaction equations with the higher-order effects

We study a three coupled wave system which includes the vector Hirota equations and three-wave resonant interaction equations. Using the Darboux-Dressing transformation, we obtain the rational and semi-rational (mixed rational-exponential) vector solutions of such system on the finite backgrounds and mixed backgrounds for the purpose of modeling rogue waves. Besides, we show the dynamics of vector solitons to this three coupled wave system.

Novel soliton solutions for the fractional three-wave resonant interaction equations

Abstract In this article, we obtained new infinite sets of exact soliton solutions for the nonlinear evolution system of three-wave resonant interaction equations. The solved system contains the non-zero second-order dispersion coefficients, the non-zero phase velocity mismatch, and the conformable fractional time derivative of order between zero and one. The solution method is a constructed ansatz that consists of linear combinations of the tan and cotan hyperbolic functions with complex coefficients. We stated clear systematic steps toward writing an exact soliton solution for the studied system. To show the efficiency of this method, we introduced some numerical examples on each obtained set of solutions. The computations showed that similar solutions can be obtained if one replaces the tan and cotan hyperbolic functions with the tan and cotan trigonometric functions. The new obtained fractional solutions could be useful in studying the broad applications of triad resonances in plasma physics and in nonlinear optics.

Temporal-filtering dissipative soliton in an optical parametric oscillator

Abstract Dissipative solitons have been realized in mode-locked fiber lasers in the theoretical framework of the Ginzburg–Landau equation and have significantly improved the pulse energy and peak power levels of such lasers. It is interesting to explore whether dissipative solitons exist in optical parametric oscillators in the framework of three-wave coupling equations in order to substantially increase the performance of optical parametric oscillators. Here, we demonstrate a temporal-filtering dissipative soliton in a synchronously pumped optical parametric oscillator. The temporal-gain filtering of the pump pulse combined with strong cascading nonlinearity and dispersion in the optical parametric oscillator enables the generation of a broad spectrum with a nearly linear chirp; consequently, a significantly compressed pulse and high peak power can be realized after dechirping outside the cavity. Furthermore, we realized, for the first time, dissipative solitons in an optical system with a negative nonlinear phase shift and anomalous dispersion, extending the parameter region of dissipative solitons. The findings may open a new research block for dissipative solitons and provide new opportunities for mid-infrared ultrafast science.

Reducing reflectivity of stimulated Raman scattering by discretely changing phase of incident light in inertial fusion plasmas

A new method to reduce the stimulated Raman scattering (SRS) in inertial confinement fusion conditions is proposed by changing the incident light phase discretely. The proposal is first examined by three-wave coupling equations and then verified by Vlasov simulations. A remarkable decreasing in SRS reflectivity is observed when the period of phase changing is less than where is the growth rate of SRS. By contrast, some simulations with continuously changing phase of incident light are carried out to compare their influence on SRS. In addition, the proposal may suppress the stimulated Brillouin scattering.

Accurate and efficient modeling of the transverse mode instability in high energy laser amplifiers.

We study the transverse mode instability (TMI) in the limit where a single higher-order mode (HOM) is present. We demonstrate that when the beat length between the fundamental mode and the HOM is small compared to the length scales on which the pump amplitude and the optical mode amplitudes vary, TMI is a three-wave mixing process in which the two optical modes beat with the phase-matched component of the index of refraction that is induced by the thermal grating. This limit is the usual limit in applications, and in this limit TMI is identified as a stimulated thermal Rayleigh scattering (STRS) process. We demonstrate that a phase-matched model that is based on the three-wave mixing equations can have a large computational advantage over current coupled mode methods that must use longitudinal step sizes that are small compared to the beat length.

Optimization and visualization of phase modulation with filtered and amplified maximal-length sequence for SBS suppression in a short fiber system: a theoretical treatment.

We use a model to investigate both the temporal and spectral characteristics of a signal lightwave which has been spectrally broadened through phase modulation with a maximal-length sequence (MLS), which is a common type of pseudo-random bit sequence. The enhancement of the stimulated Brillouin scattering (SBS) threshold of the modulated lightwave in a fiber system is evaluated by numerically simulating the coupled three-wave SBS interaction equations. We find that SBS can build up on a nanosecond-level time scale in a short fiber, which can reduce the SBS suppressing capability of MLS modulation waveforms with GHz-level clock rate, if the sub-sequence ("run") lengths with the same symbol (zero or one) of the MLS extend over several nanoseconds. To ensure the SBS buildup is perturbed and thus suppressed also during these long sub-sequences, we introduce a low-pass filter to average the signal over several bits so that the modulation waveform changes gradually even during long runs and amplify the RF modulation waveforms to the level required for sufficient spectral broadening and carrier suppression of the optical signal. We find that the SBS suppression depends non-monotonically on the parameters of the filtered and amplified MLS waveform such as pattern length, modulation depth, and the ratio of low-pass filter cutoff frequency to clock rate for maximum SBS mitigation. We optimize the SBS suppression through numerical simulations and discuss it in terms of the temporal and spectral characteristics of the lightwave and modulation waveform using derived analytical expressions and numerical simulations. The simulations indicate that the normalized SBS threshold reaches a maximum for a RMS modulation depth of 0.56π and a ratio of filter cutoff frequency to clock rate of 0.54 and that MLS9 is superior to other investigated patterns.

Influences of sinusoidal density modulation on stimulated Raman scattering in inhomogeneous plasmas

The influence of sinusoidal density modulation on stimulated Raman scattering (SRS) reflectivity in inhomogeneous plasmas is studied using three-wave coupling equations, fully kinetic Vlasov simulations, and particle-in-cell (PIC) simulations. Through a numerical solution of the three-wave coupling equations, we find that the sinusoidal density modulation is capable of inducing absolute SRS even though the Rosenbluth gain is smaller than π, and we describe a region of modulational wavelength and amplitude in which absolute SRS can be induced, which agrees with earlier studies. The average reflectivity obtained by the Vlasov simulations has the same trend as the growth rate of absolute SRS obtained from the three-wave equations. Instead of causing absolute instability, a modulation wavelength shorter than the basic gain length can suppress the inflation of SRS through harmonic waves. In addition, the PIC simulations qualitatively agree with our Vlasov simulations. Our results offer an alternative explanation for high reflectivity in experimental underdense plasmas, which is due to long-wavelength modulation, and a potential method to suppress SRS using short-wavelength modulation.

General rogue waves in the three-wave resonant interaction systems

Abstract General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction, respectively. It is shown that while the first family of solutions associated with a simple root exists for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, the ones generated by a $2\times 2$ block determinant in the double-root case, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained.

Temperature Sensing Utilizing Stimulate Brillouin Scattering Fast Light in Liquid-Filled Photonic Crystal Fibers

A novel temperature sensor designed on stimulate Brillouin scattering fast light in liquid-filled photonic crystal fibers is proposed. The time advancement and the Brillouin frequency shift of fast light are simulated according to the three-wave coupling equations of stimulate Brillouin scattering, and the temperature sensing characteristics of the fast light in liquid-filled hexagonal photonic crystal fibers with three different air filling factors are simulated from 20 °C to 70 °C by using the full-vector finite element method. The alcohol-filled photonic crystal fibers exhibit rather sensitive responses to temperature. With temperature varying from 20 °C to 70 °C, the variation of the effective mode area is 2.75 µm at the air filling factor of 0.6, the Brillouin frequency shift is about 11 GHz and its average modification is 1.15 MHz. The time advancement increases with the rise of temperature, its increment can reach up to 4.53 ns at the air filling factor of 0.6 and the pump power of 60 mW, the temperature sensitivity of the time advancement is 0.272 ns/°C.

Spatiotemporal chaos and quasipatterns in coupled reaction–diffusion systems

In coupled reaction–diffusion systems, modes with two different length scales can interact to produce a wide variety of spatiotemporal patterns. Three-wave interactions between these modes can explain the occurrence of spatially complex steady patterns and time-varying states including spatiotemporal chaos. The interactions can take the form of two short waves with different orientations interacting with one long wave, or vice verse. We investigate the role of such three-wave interactions in a coupled Brusselator system. As well as finding simple steady patterns when the waves reinforce each other, we can also find spatially complex but steady patterns, including quasipatterns. When the waves compete with each other, time varying states such as spatiotemporal chaos are also possible. The signs of the quadratic coefficients in three-wave interaction equations distinguish between these two cases. By manipulating parameters of the chemical model, the formation of these various states can be encouraged, as we confirm through extensive numerical simulation. Our arguments allow us to predict when spatiotemporal chaos might be found: standard nonlinear methods fail in this case. The arguments are quite general and apply to a wide class of pattern-forming systems, including the Faraday wave experiment.

Plasma optics in the context of high intensity lasers

The use of plasmas provides a way to overcome the low damage threshold of classical solid-state based optical materials, which is the main limitation encountered in producing and manipulating intense and energetic laser pulses. Plasmas can directly amplify or alter the characteristics of ultra-short laser pulses via the three-wave coupling equations for parametric processes. The strong-coupling regime of Brillouin scattering (sc-SBS) is of particular interest: recent progress in this domain is presented here. This includes the role of the global phase in the spatio-temporal evolution of the three-wave coupled equations for backscattering that allows a description of the coupling dynamics and the various stages of amplification from the initial growth to the so-called self-similar regime. The understanding of the phase evolution allows control of the directionality of the energy transfer via the phase relation between the pulses. A scheme that exploits this coupling in order to use the plasma as a wave plate is also suggested.

Auto-resonant stimulated Brillouin backscattering in supersonic flowing plasmas by fully kinetic Vlasov simulations

The auto-resonant behavior of ion acoustic waves (IAWs) driven by stimulated Brillouin backscattering in supersonic flowing plasmas is self-consistently investigated by Vlasov simulations. Vlasov simulations show that the peak amplitude of the auto-resonant growing IAWs in the positive flow gradient is much larger than that in the negative flow gradient, which is contrary to the previous analysis (Wang et al 2018 Plasma Phys. Control. Fusion 60 025016). It is found that the IAW phase velocity remains unchanged in supersonic flowing plasmas, which significantly influences wave-particle interaction by changing the Landau damping and the coefficient of the kinetic nonlinear frequency shift in different position. As a result, the Landau damping of IAWs in the peak region in the negative case is stronger than that in the positive case, which has never been considered before. After considering its effects on particle trapping and, particularly, Landau damping of IAWs, theoretical predictions from three-wave mode equations are consistent with results by Vlasov simulations.

Cascade random-quasi-phase-matched harmonic generation in polycrystalline ZnSe

We experimentally and numerically investigated the cascade random-quasi-phase-matched harmonic generation in polycrystalline zinc selenide with tightly focused mid-infrared nanosecond laser pulses. We observed a sharp transition between two distinct intensity scaling regimes as a function of pump energy from sub IN to super IN, where I is the pump intensity and N is the harmonic order. To gain an understanding of the observations of this complex nonlinear process, we carried out simulations of the coupled three-wave mixing equations with random grain sizes and orientations. We found a grain size dependence of the effective phase-matching condition, suggesting that manipulation of grain size can be an effective pathway to phase-matching optimization. This approach may provide a framework for effectively enhancing high harmonic generation and other wave mixing effects in polycrystalline materials.

Large-Scale Compressibility in Rotating Flows of Astrophysical Plasma in the Shallow Water Approximation

Two systems of magnetohydrodynamic equations in the shallow water approximation are proposed as a basis for studies in the field of plasma astrophysics: the system of equations with a full allowance for the Coriolis force and the system of equations on the β-plane in which the changes of the Coriolis parameter are linear in coordinate. Both systems of equations take into account such fundamentally important phenomena in plasma astrophysics as the compressibility and external magnetic field effects, increasing significantly the potential for applying these equations to study astrophysical objects. Compressibility in plasma astrophysics is shown to change significantly the dispersion laws for magneto-Poincare, magnetostrophic, and magneto-Rossby waves. The same nonlinear interactions as those in the absence of compressibility have been found to be realized in the case of a compressible rotating plasma. Three-wave equations in the weak nonlinearity approximation, in which the interaction coefficients depend on plasma large-scale compressibility and thermodynamic characteristics, have been derived by the method of multiscale expansions. Expressions for the growth rates of the parametric instabilities of three-wave interactions with large-scale compressibility have been derived.

The interplay between the kinetic nonlinear frequency shift and the flowing gradient in stimulated Brillouin scattering

The effect of the kinetic nonlinear frequency shift (KNFS) on backward stimulated Brillouin scattering (SBS) in homogeneous plasmas and inhomogeneous flowing plasmas is investigated by three-wave coupled-mode equations. When the positive contribution to the KNFS from electrons as well as the negative contribution from ions is included, the net KNFS can become positive at a large electron-ion temperature ratio . In homogeneous plasmas, KNFS can greatly reduce the SBS reflectivity at low or large but has a weak effect on SBS at where the positive frequency shifts from electrons almost cancels out the negative shifts from ions. In inhomogeneous plasmas, the net negative frequency shift can enhance the backward SBS reflectivity for the negative gradient of the plasma flowing, and can suppress the reflectivity for the positive case. On the contrary, the net positive frequency can suppress the reflectivity for the negative case of the flowing gradient and enhance the reflectivity for the positive case. This indicates that the SBS in inhomogeneous flowing plasmas can be controlled by changing the sign of the nonlinear frequency shift.