Philippe Mounaix

Space and time behavior of parametric instabilities for a finite pump wave duration in a bounded plasma

The space and time behavior of the decay waves is computed analytically in the regime of standard parametric decay. The plasma is assumed to be homogeneous and bounded. The pump wave has a finite pulse duration. The propagation of the pump wave is taken into account, its depletion is ignored. The parametric growth is solved in terms of fluctuating initial and boundary conditions corresponding to thermal noise at equilibrium. Fluctuating source terms, representing noise emission, are accordingly retained in the coupled mode equations. The initial stage of parametric growth is investigated in detail; the time from which the asymptotic concept of absolute or convective instability applies is computed. The connection between the Manley–Rowe and flux conservation relations is discussed.

Numerical simulation of filamentation and its interplay with SBS in underdense plasmas

We present results of a new code which models the interaction of intense electromagnetic beams with the low-frequency dynamics of the plasma fluid in two or three spatial dimensions. The light propagation is treated without the restriction of the paraxial optics approximation. The numerical scheme is based on spatial discretisation in the direction along the axis of the incident laser light whereas the transversal directions are treated spectrally. The specific cases discussed here refer to conditions found in recent experiments where filamentation was diagnosed [1]. In this study we first focus on effects originating from filamentation and/or forward stimulated Brillouin scattering (SBS), which can hardly be distinguished in the nonlinear case. We discuss then the impact of side- and backward SBS on these effects, observing the onset of an absolute filamentation instability predicted by Luther et al. [2].

Space and time behavior of backscattering instabilities in the modified decay regime

The space and time behavior of parametric backscattering instabilities is computed analytically in the so‐called modified decay regime. The plasma is assumed to be homogeneous and of finite length. The propagation of the pump wave and its finite pulse duration both are taken into account, its depletion is ignored. The parametric growth is solved in terms of fluctuating initial and boundary conditions corresponding to thermal noise at equilibrium. Fluctuating source terms, representing spontaneous emission of waves, are accordingly retained in the coupled mode equations. The initial stage of the instability is investigated in detail; the time from which the time asymptotic concept of absolute or convective instability applies is computed. Approximate expressions for the fluctuations of the waves, that are uniformly valid for any gain factor and any time, are derived.

Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks

We investigate the statistics of the gap G(n) between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration L(n) which separates the occurrence of these two extremal positions. The distribution of the jumps η(i)s of the RW, f(η), is symmetric and its Fourier transform has the small k behavior 1-f[over ^](k)~|k|(μ), with 0<μ≤2. For μ=2, the RW converges, for large n, to Brownian motion, while for 0<μ<2 it corresponds to a Lévy flight of index μ. We compute the joint probability density function (PDF) P(n)(g,l) of G(n) and L(n) and show that, when n→∞, it approaches a limiting PDF p(g,l). The corresponding marginal PDFs of the gap, p(gap)(g), and of L(n), p(time)(l), are found to behave like p(gap)(g)~g(-1-μ) for g>>1 and 0<μ<2, and p(time)(l)~l(-γ(μ)) for l>>1 with γ(1<μ≤2)=1+1/μ and γ(0<μ<1)=2. For l, g>>1 with fixed lg(-μ), p(g,l) takes the scaling form p(g,l)~g(-1-2μ)p[over ˜](μ)(lg(-μ)), where p[over ˜](μ)(y) is a (μ-dependent) scaling function. We also present numerical simulations which verify our analytic results.

Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrödinger Equation Driven by the Square of a Gaussian Field

Solutions to the equation are investigated, where S(x, t) is a complex Gaussian field with zero mean and specified covariance, and m≠0 is a complex mass with Im(m) ≥ 0. For real m this equation describes the backscattering of a smoothed laser beam by an optically active medium. Assuming that S(x, t) is the sum of a finite number of independent complex Gaussian random variables, we obtain an expression for the value of λ at which the qth moment of w.r.t. the Gaussian field S diverges. This value is found to be less or equal for all m ≠ 0, Im(m) ≥ 0 and |m|<+∞ than for |m| = +∞, i.e. when the term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem.

Temporal behavior of backscattering instabilities in a three-dimensional cylindrical hot spot: I. Standard decay regime

The space and time behavior of backscattering instabilities in a three-dimensional cylindrical hot spot of finite lifetime is computed analytically in the so-called standard decay (or weak coupling) regime. A uniform expression for the instantaneous backscattered power, valid for any time and any hot spot intensity in the standard decay regime, is proposed. It is shown that the finite transverse size of the hot spot leads to important diffraction effects that can significantly reduce the hot spot backscattered energy. The plasma is assumed to be homogeneous and the hot spot depletion is ignored. The coupled mode equations are solved in terms of fluctuating source terms corresponding to the noise from which the instability grows. Diffraction effects on the growth of the absolute instability inside a long lifetime hot spot is investigated.

Harmonic generation and departure from quasineutrality in ion sound and Langmuir wave coupling

The validity conditions of the Zakharov equations are reconsidered by investigating the linear stability analysis of a dipolar Langmuir wave. It is shown that the corresponding dispersion relation has a wider range of applicability, W/NT<(kλD)−2, than the usual domain of validity of the Zakharov equations, W/NT<1 (W/NT is the ratio of the Langmuir wave energy to the particle thermal energy, k is the characteristic wave number of the low‐frequency perturbation). This result follows from an exact cancellation between the corrections as a result of the departure from quasineutrality and the contributions from the harmonics of the Langmuir wave. Such a cancellation is interpreted in terms of absence of nonlinear frequency shift caused by a dipolar pump wave in the dispersion relation of the Langmuir waves. The next‐order corrections to the linearized Zakharov equations are computed; the proper renormalization of the low‐frequency part of the dispersion relation is shown to result from the slow time variation ...

Diffraction-controlled backscattering threshold and application to Raman gap

In most classic analytical models of linear stimulated scatter, light diffraction is omitted, a priori. However, modern laser optic typically includes a variant of the random phase plate [Y. Kato et al., Phys. Rev. Lett. 53, 1057 (1984)], resulting in diffraction limited laser intensity fluctuations—or localized speckles—which may result in explosive reflectivity growth as the average laser intensity approaches a critical value [H. A. Rose and D. F. DuBois, Phys. Rev. Lett. 72, 2883 (1994)]. Among the differences between stimulated Raman scatter (SRS) and stimulated Brillouin scatter is that the SRS scattered light diffracts more strongly than the laser light with increase of electron density. This weakens the tendency of the SRS light to closely follow the most amplified paths, diminishing gain. Let G0 be the one-dimensional power gain exponent of the stimulated scatter. In this paper we show that differential diffraction gives rise to an increase of G0 at the SRS physical threshold with increase of elec...

Nonequilibrium stationary state of a truncated stochastic nonlinear Schrödinger equation: formulation and mean-field approximation.

We investigate the stationary state of a model system evolving according to a modified focusing truncated nonlinear Schrödinger equation used to describe the envelope of Langmuir waves in a plasma. We restrict the system to have a finite number of normal modes each of which is in contact with a Langevin heat bath at temperature T . Arbitrarily large realizations of the field are prevented by restricting each mode to a maximum amplitude. We consider a simple modeling of wave breaking in which each mode is set equal to zero when it reaches its maximum amplitude. Without wave breaking the stationary state is given by a Gibbs measure. With wave breaking the system attains a nonequilibrium stationary state which is the unique invariant measure of the time evolution. A mean-field analysis shows that the system exhibits a transition from a regime of low-field values at small |lambda| , to a regime of higher-field values at large |lambda| , where lambda<0 specifies the strength of the nonlinearity in the focusing case. Field values at large |lambda| are significantly smaller with wave breaking than without wave breaking.

Note on a diffraction-amplification problem

We investigate the solution of the equation ∂ t e(x,t) - iD∂ 2 x e(x,t) = λ|S(x, t)| 2 e(x, t), for x in a circle and S(x, t) a Gaussian stochastic field with a covariance of a particular form. It is shown that the coupling λ c at which diverges for t ≥ 1 (in suitable units), is always less or equal for D > 0 than D = 0.