# Negative and positive dust grain effect on the modulation instability of an intense laser propagating in a hot magnetoplasma

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## Abstract

The modulation instability of intense circularly polarized laser beam in hot magnetized dusty plasma is studied. A nonlinear equation describing the interaction of laser with dusty plasma in the quasi-neutral approximation is derived. The effect of negative and positive dust grains on the laser modulation growth rate is studied. It is shown that the existence of positive dust grains instead of ions can substantially improve the modulation growth rate.

## Keywords

Dusty plasma Laser Modulation Nonlinear ineration Growth rate Magnetoactive## Introduction

Dusty plasmas are frequently found in different places of the cosmic environment. They exist in planetary rings, comet comae and tails, and interplanetary and interstellar molecular clouds [1, 2, 3, 4, 5, 6, 7]. They can be revealed in the vicinity of aircrafts [6, 7] and in the controlled plasma fusion [8, 9, 10]. In some industrial applications of plasma such as plasma processing of materials, the formation of dusty plasma has also been observed [11, 12, 13, 14]. They can also be created during laser ablation experiments [15, 16]. In addition, complex dusty plasmas form in the flame of a humble candle, in the zodiacal light, cloud-to-ground lightings, and volcanic eruptions. Recently [17], it has been suggested that the ball lightning is the dusty plasma medium and it is created during oxidation of nanoparticle networks in the normal lightning strike on soil. Furthermore, dusty plasmas can be produced and investigated in laboratories. Dust grains not only can be intentionally added into the plasma, but can also appear because of different mechanisms in some experiments. The existence of heavy and highly ionized dust grains gives some special and extraordinary properties to the dusty plasma, providing great motivations to investigate it theoretically and experimentally. One of these interests is the study of interaction of laser with dusty plasmas and its related linear and nonlinear effects. These effects include wave dissipation [18], modulation and filamentation instabilities [19, 20, 21, 22], linear and nonlinear wave propagation [18, 23, 24, 25, 26, 27, 28, 29, 30, 31], parametric instabilities [32], self-focusing [18, 33], etc. Moreover, interaction of laser with dusty plasmas has some important industrial applications. For instance, by interaction of high power lasers with molecular or atomic clusters, during which dusty plasma is created, high-energy electrons can be produced by three processes, i.e., inner ionization, outer ionization, and Coulomb explosion [34, 35, 36]. In some experiments, lasers are used in order to study the dynamics of different phenomena in dusty plasmas which some recent experiments about investigation of different exotic phenomena can be found in [37, 38, 39, 40, 41, 42].

Here, we focus on the MI of intense lasers in magnetized dusty plasmas. The MI represents a fundamental subject in the theory of nonlinear waves. MI exists due to the interplay between the nonlinearity and dispersion/diffraction effects. The ponderomotive force created by the electromagnetic wave (EMW) stimulates low-frequency perturbations of the electrons density; then, they interact with the primary high-frequency EMW in which the amplitude of the pump wave becomes modulated and the MI of the EMW occurs. The MI of laser beams in plasmas and dielectrics has been the subject of several publications [43, 44, 45]. The MI of strong EMWs in plasmas with arbitrary large amplitude was studied by Shukla et al. in 1987 [46]. Most of the early publications about MI considered one-dimensional models in which the laser beam was represented as a plane wave [47, 48]. The MI of a laser pulse in the cold nonmagnetized plasma has been considered by several authors [46, 49, 50]. The MI of a linearly polarized laser pulse propagating in the cold magnetized plasma was studied by Jha et al. in 2005 [51]. The MI of the right-hand elliptically laser pulse in cold magnetized plasma has been investigated by Chen et al. in 2011 [52]. Recently, the MI of an intense circularly polarized laser beam in the hot magnetized electron–positron and electron–ion (e–i) plasmas as studied by Sepehri Javan [53, 54]. Our recent work [55] has extended the MI of the circularly polarized laser beam propagating along an external magnetic field in the non-Maxwellian plasma. In this article we study the MI of an intense laser beam in the magnetized hot dusty plasma. In the quasi-neutral approximation and by using a relativistic fluid model, we consider the presence of both negative and positive dust grains and investigate the effect of such grains on the MI. The organization of the paper is as follows. In Sect. 2, the basic assumptions are presented and a nonlinear wave equation is derived for the laser amplitude evolutions. An analytic expression for the growth rate of MI is obtained in Sect. 3. In Sect. 4, a numerical study of the MI of circularly polarized laser beam in the magnetized electron–ion–positive dust–negative dust (e–i–d+–d−) plasma is presented. The concluding remarks are made in Sect. 5.

## Deriving a nonlinear wave equation

*z*axis, i.e., \({\mathbf{B}}_{{\mathbf{0}}} = B_{0} {\hat{\mathbf{e}}}_{{\mathbf{z}}}\). To describe the nonlinear dynamics of the interaction of EMW with the dusty plasma, we define the electric and magnetic fields \({\mathbf{E}}\) and \({\kern 1pt} {\mathbf{B}}\) through the vector and scalar potentials \({\mathbf{A}},{\kern 1pt} \varphi\) as:

*c*is the speed of light.

*j*-type particle, and order of ionization of positive and negative dust grains, respectively.

## Derivation of nonlinear dispersion relation and MI

*a*is the nonlinear dispersion relation. In the absence of interaction between EMW and plasma, when amplitude is a real constant (\(a = a_{0}\)), we can derive the nonlinear dispersion relation for magnetoplasma with negative and positive dust grains as follows:

It is worth mentioning that in the linear approximation, there is no contribution for dust grains on the dispersion Eq. (25) because we have investigated the evolution of high-frequency EMWs where heavy ions and dust particles cannot respond to this high frequency. However, traces of dust particles can be found in the nonlinear dispersion Eq. (24) through bipolar diffusion caused by slow motion of particles under the influence of ponderomotive laser force and thermal collision force.

*ω*

_{0}and

*k*

_{0}satisfy the linear dispersion of Eqs. (23), (25) can be modified as:

*X*and

*Y*:

*K*is the modulation wave number normalized by \(\omega_{p} /c\). By substituting Eq. (32) into the set of Eq. (31), we can obtain the following nonlinear dispersion relation of MI:

## Numerical discussions

^{−1}(that corresponds to the laser wave length \(\lambda \approx 1\) µm) and \(a_{0} = 0.271\) (laser intensity \(I \approx 10^{17}\) W/cm

^{2}); also, we consider only the right-hand polarization laser in magnetized medium with \(\alpha = 0.2\) and fix the temperature \(T_{j} = 1\) keV for all the plasma components. For more clarification, we introduce two new parameters \(\eta\) and \(\xi\) as:

^{−3}.

## Conclusions

In this paper, we investigated the MI of a weakly relativistic laser propagating along an external magnetic field in the hot plasma containing positive and negative dust grains. The MI growth rate of the circularly polarized laser beam in the dusty plasma was obtained. It was found that adding the positive dust grains to plasma enhances the MI, but existence of the negative dust grains reduces it. Furthermore, the effect of the order of dust grain ionization on the MI was investigated and it was observed that its increase leads to the increase in the MI growth rate.

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