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

## 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.

### References

- 1.Goertz, C.K.: Dusty plasmas in the solar system. Rev. Geophys.
**27**, 271 (1989)ADSCrossRefGoogle Scholar - 2.Northrop, T.G.: Dusty plasmas. Phys. Scr.
**45**, 475 (1992)ADSCrossRefGoogle Scholar - 3.Tsytovich, V.N.: Dust plasma crystals, drops, and clouds. Usp. Fiz. Nauk
**167**, 57 (1997).**[Phys. Usp. 40, 53 (1997)]**CrossRefGoogle Scholar - 4.Bliokh, P., Sinitsin, V., Yaroshenko, V.: Dusty and self-gravitational plasmas in space. Kluwer Acad. Publ, Dordrecht (1995)CrossRefGoogle Scholar
- 5.Shukla, P.K., Mamun, A.A.: Introduction to dusty plasma physics. Institute of Physics Publishing, Bristol (2002)CrossRefGoogle Scholar
- 6.Whipple, E.C.: Potentials of surfaces in space. Rep. Prog. Phys.
**44**, 1197 (1981)ADSCrossRefGoogle Scholar - 7.Robinson, P.A., Coakley, P.: Spacecraft charging-progress in the study of dielectrics and plasmas. IEEE Trans. Electr. Insul.
**27**, 944 (1992)ADSCrossRefGoogle Scholar - 8.Tsytovich, V.N., Winter, J.: On the role of dust in fusion devices. Usp. Fiz. Nauk
**168**, 899 (1998).**[Phys. Usp. 41, 815 (1998)]**CrossRefGoogle Scholar - 9.Winter, J., Gebauer, G.: Dust in magnetic confinement fusion devices and its impact on plasma operation. J. Nucl. Mater.
**266–269**, 228 (1999)CrossRefGoogle Scholar - 10.Winter, J.: Dust: a new challenge in nuclear fusion research? Phys. Plasmas
**7**, 3862 (2000)ADSMathSciNetCrossRefGoogle Scholar - 11.Jellum, G.M., Graves, D.B.: Particulates in aluminum sputtering discharges. J. Appl. Phys.
**67**, 6490 (1990)ADSCrossRefGoogle Scholar - 12.Anderson, H.M., Jairath, R., Mock, J.L.: Particulate generation in silane/ammonia rf discharges. J. Appl. Phys.
**67**, 3999 (1990)ADSCrossRefGoogle Scholar - 13.Shivatani, M., Fukuzawa, T., Watanabe, Y.: Formation processes of particulates in helium-diluted silane RF plasmas. IEEE Trans. Plasma Sci.
**22**, 103 (1994)ADSCrossRefGoogle Scholar - 14.Cui, C., Goree, J.: Fluctuations of the charge on a dust grain in a plasma. IEEE Trans. Plasma Sci.
**22**, 151 (1994)ADSCrossRefGoogle Scholar - 15.Trajanovic, Z., Senapati, L., Sharma, R.P., Venkatesan, T.: Stoichiometry and thickness variation of YBa
_{2}Cu_{3}O_{7−x}in off-axis pulsed laser deposition. Appl. Phys. Lett.**66**, 2418 (1995)ADSCrossRefGoogle Scholar - 16.Fukushima, K., Kanka, Y., Badaye, M., Morishita, T.: Velocity distributions of ions in the ablation plume of a Y
_{1}Ba_{2}Cu_{3}O_{x}target. J. Appl. Phys.**77**, 5406 (1995)ADSCrossRefGoogle Scholar - 17.Abrahamson, J., Dinniss, J.: Ball lightning caused by oxidation of nanoparticle networks from normal lightning strikes on soil. Nature
**403**, 519 (2000)ADSCrossRefGoogle Scholar - 18.Jana, M.R., Sen, A., Kaw, P.K.: Collective effects due to charge-fluctuation dynamics in a dusty plasma. Phys. Rev. E
**48**, 3930 (1993)ADSCrossRefGoogle Scholar - 19.Sambandan, G., Tripathi, V.K., Parashar, J., Bharuthram, R.: Nonlinear interaction of a high-power electromagnetic beam in a dusty plasma: two-dimensional effects. Phys. Plasmas
**6**, 762 (1999)ADSCrossRefGoogle Scholar - 20.Sharma, S.C., Gahlot, A., Sharma, R.P.: Effect of dust on an amplitude modulated electromagnetic beam in a plasma. Phys. Plasmas
**15**, 043701 (2008)ADSCrossRefGoogle Scholar - 21.Sodha, M.S., Mishra, S.K., Misra, S.: Nonlinear dependence of complex plasma parameters on applied electric field. Phys. Plasmas
**18**, 023701 (2011)ADSCrossRefGoogle Scholar - 22.El-Taibany, W.F., Kourakis, I., Wadati, M.: Low frequency localized wavepackets in dusty plasmas with opposite charge polarity dust components. Plasma Phys. Control. Fusion
**50**, 074003 (2008)ADSCrossRefGoogle Scholar - 23.Varma, R.K., Shukla, P.K., Krishan, V.: Electrostatic oscillations in the presence of grain-charge perturbations in dusty plasmas. Phys. Rev. E
**47**, 3612 (1993)ADSCrossRefGoogle Scholar - 24.Tripathi, K.D., Sharma, S.K.: Self-consistent charge dynamics in magnetized dusty plasmas: low-frequency electrostatic modes. Phys. Rev. E
**53**, 1035 (1996)ADSCrossRefGoogle Scholar - 25.Das, C., Janaki, M.S., Dasgupta, B.: Dust cyclotron instability in presence of dust charge fluctuations. Phys. Scr. T
**75**, 216 (1998)ADSCrossRefGoogle Scholar - 26.Dwivedi, C.B., Pandey, B.P.: Electrostatic shock wave in dusty plasmas. Phys. Plasmas
**2**, 4134 (1995)ADSMathSciNetCrossRefGoogle Scholar - 27.Popel, S.I., Yu, M.Y., Tsytovich, V.N.: Shock waves in plasmas containing variable-charge impurities. Phys. Plasmas
**3**, 4313 (1996)ADSCrossRefGoogle Scholar - 28.Paul, S.N., Mondal, K.K., Roychowdhury, A.: Effects of streaming and attachment coefficients of ions and electrons on the formation of soliton in a dusty plasma. Phys. Lett. A
**257**, 165 (1999)ADSCrossRefGoogle Scholar - 29.Janaki, M.S., Dasgupta, B.: Surface waves in a dusty plasma. Phys. Scr.
**58**, 493 (1998)ADSCrossRefGoogle Scholar - 30.Alam, M.K., Roy Chowdhury, A.: Surface wave propagation in a magnetized dusty plasma with charge fluctuation. Phys. Plasmas
**6**, 3765 (1999)ADSCrossRefGoogle Scholar - 31.Bhat, J.R., Pandey, B.P.: Self-consistent charge dynamics and collective modes in a dusty plasma. Phys. Rev. E
**50**, 3980 (1994)ADSCrossRefGoogle Scholar - 32.Burman, S., Paul, S.N., Roy Chowdhury, A.: Stimulated brillouin scattering in a magnetized dusty plasma with charge fluctuation. Phys. Plasmas
**9**, 3752 (2002)ADSCrossRefGoogle Scholar - 33.Mishra, S.K., Mishra, S., Sodha, M.S.: Self-focusing of a Gaussian electromagnetic beam in a complex plasma. Phys. Plasmas
**18**, 043702 (2011)ADSCrossRefGoogle Scholar - 34.Nantel, M., Ma, G., Gu, S., Cote, C.Y., Itatani, J., Umstadter, D.: Pressure ionization and line merging in strongly coupled plasmas produced by 100-fs laser pulses. Phys. Rev. Lett.
**80**, 4442 (1998)ADSCrossRefGoogle Scholar - 35.Springate, E., Hay, N., Tisch, J.W.G., Mason, M.B., Ditmire, T., Hutchinson, M.H.R., Marangos, J.P.: Explosion of atomic clusters irradiated by high-intensity laser pulses: scaling of ion energies with cluster and laser parameters. Phys. Rev. A
**61**, 063201 (2000)ADSCrossRefGoogle Scholar - 36.Last, I., Jortnera, J.: Electron and nuclear dynamics of molecular clusters in ultraintense laser fields. I. Extreme multielectron ionization. J. Chem. Phys.
**120**, 1336 (2004)ADSCrossRefGoogle Scholar - 37.Stoffels, E., Stoffels, W.W., Vender, D., Kroesen, G.M.W., de Hoog, F.J.: Laser-particulate interactions in a dusty RF plasma. IEEE Trans. Plasma Sci.
**22**, 116 (1994)ADSCrossRefGoogle Scholar - 38.Melzer, A.: Laser-experiments on particle interactions in strongly coupled dusty plasma crystals. Phys. Scr.
**T89**, 33 (2001)ADSCrossRefGoogle Scholar - 39.Nefedov, A.P., Petrov, O.F., Molotkov, V.I., Fortovs, V.E.: Formation of liquidlike and crystalline structures in dusty plasmas. JETP Lett.
**72**, 218 (2000)ADSCrossRefGoogle Scholar - 40.van de Wetering, F.M.J.H., Oosterbeek, W., Beckers, J., Nijdam, S., Gibert, T., Mikikian, M., Rabat, H., Kovačević, E., Berndt, J.: Interaction of nanosecond ultraviolet laser pulses with reactive dusty plasma. Appl. Phys. Lett.
**108**, 213103 (2016)ADSCrossRefGoogle Scholar - 41.Nosenko, V., Avinash, K., Goree, J., Liu, B.: Laser-excited mach cones in a dusty plasma crystal. Phys. Rev. E
**62**, 4162 (2000)CrossRefGoogle Scholar - 42.van de Wetering, F.M.J.H., Oosterbeek, W., Beckers, J., Nijdam, S., Kovačević, E., Berndt, J.: Laser-induced incandescence applied to dusty plasmas. J. Phys. D Appl. Phys.
**49**, 295206 (2016)CrossRefGoogle Scholar - 43.Esarey, E., Sprangle, P., Krall, J., Ting, A.: Overview of plasma-based accelerator concepts. IEEE Trans. Plasma Sci.
**24**, 252 (1996)ADSCrossRefGoogle Scholar - 44.Esarey, E., Sprangle, P., Krall, J., Ting, A.: Self-focusing and guiding of short laser pulses in ionizing gases and plasmas. IEEE J. Quantum Electron.
**33**, 1879 (1997)ADSCrossRefGoogle Scholar - 45.Shukla, P.K., Marklund, M., Eliasson, B.: Nonlinear dynamics of intense laser pulses in a pair plasma. Phys. Lett. A
**324**, 193 (2004)ADSCrossRefMATHGoogle Scholar - 46.Shukla, P.K., Bharuthram, R.: Modulational instability of strong electromagnetic waves in plasmas. Phys. Rev. A
**35**, 4889 (1987)ADSCrossRefGoogle Scholar - 47.McKinstrie, C.J., Bingham, R.: Stimulated Raman forward scattering and the relativistic modulational instability of light waves in rarefied plasma. Phys. Fluids B
**4**, 2626 (1992)ADSCrossRefGoogle Scholar - 48.Sprangle, P., Esarey, E., Hafizi, B.: Intense laser pulse propagation and stability in partially stripped plasmas. Phys. Rev. Lett.
**79**, 1046 (1997)ADSCrossRefGoogle Scholar - 49.Shukla, P.K., Rao, N.N., Yu, M.Y., Tsintsadze, N.L.: Relativistic nonlinear effects in plasmas. Phys. Rep.
**138**, 1 (1986)ADSCrossRefGoogle Scholar - 50.Esarey, E., Kralland, J., Sprangle, P.: Envelope analysis of intense laser pulse self-modulation in plasmas. Phys. Rev. Lett.
**72**, 2887 (1994)ADSCrossRefGoogle Scholar - 51.Jha, P., Kumar, P., Raj, G., Upadhyaya, A.K.: Modulation instability of laser pulse in magnetized plasma. Phys. Plasmas
**12**, 123104 (2005)ADSCrossRefGoogle Scholar - 52.Chen, H.Y., Liu, S.Q., Li, X.Q.: Self-modulation instability of an intense laser beam in a magnetized pair plasma. Phys. Scr.
**83**, 035502 (2011)ADSCrossRefMATHGoogle Scholar - 53.Sepehri Javan, N.: Modulation instability of an intense laser beam in the hot magnetized electron-positron plasma in the quasi-neutral limit. Phys. Plasmas
**19**, 122107 (2012)ADSCrossRefGoogle Scholar - 54.Sepehri Javan, N.: Competition of circularly polarized laser modes in the modulation instability of hot magnetoplasma. Phys. Plasmas
**20**, 012120 (2013)ADSCrossRefGoogle Scholar - 55.Etemadpour, R., Sepehri, N.: Javan, Effect of super-thermal ions and electrons on the modulation instability of a circularly polarized laser pulse in magnetized plasma. Laser Part. Beams
**33**, 265 (2015)ADSCrossRefGoogle Scholar - 56.Sepehri Javan, N., Nasirzadeh, Zh: Self-focusing of circularly polarized laser pulse in the hot magnetized plasma in the quasi-neutral limit. Phys. Plasmas
**19**, 112304 (2012)ADSCrossRefGoogle Scholar - 57.Sepehri Javan, N., Rouhi Erdi, F.: Effect of dynamical non-neutrality on the modulational instability of laser propagating through hot magnetoplasma. Phys. Plasmas
**22**, 062116 (2015)ADSCrossRefGoogle Scholar - 58.Sepehri Javan, N., Hosseinpour Azad, M.: Thermal behavior change in the self-focusing of an intense laser beam in magnetized electron-ion-positron plasma. Beams
**32**, 321 (2014)Google Scholar - 59.Malomed, B.: Nonlinear Schrödinger equations. In: Scott, A. (ed.) Encyclopedia of nonlinear science, pp. 639–643. Routledge, New York (2005)Google Scholar
- 60.Pitaevskii, L., Stringari, S.: Bose-Einstein condensation. Clarendon, Oxford (2003)MATHGoogle Scholar
- 61.Gurevich, A.V.: Nonlinear phenomena in the ionosphere. Springer, Berlin (1978)CrossRefGoogle Scholar
- 62.Balakrishnan, R.: Soliton propagation in nonuniform media. Phys. Rev. A.
**32**, 1144 (1985)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.