3094 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 11, NOVEMBER 2013
Space Charge and Ponderomotive Force Effects inInteraction of High-Power Microwave With Plasma
Seyyed Mohammad Khorashadizadeh, Taghi Mirzaye, and Ali Reza Niknam
Abstract— The interaction between a high-power microwavebeam and a collisionless unmagnetized plasma is investigated bytaking into account the ponderomotive force and space chargeeffects. The electromagnetic wave equation coupled with Poissonand momentum transfer equations are solved for obtaining thespace charge field and plasma density profiles. It is shown thatthe increase of microwave energy flux causes steepening of theelectron density profiles, sawtooth lines in the curves of spacecharge field, and modulation of wavelength. In addition, theeffect of wave frequency, initial electron density and electrontemperature on space charge field, and electron density profilesare studied. It is indicated that the behavior of space chargefield is very much influenced by the initial electron density.Furthermore, the result of the research shows that the increaseof the electron temperature and microwave frequency cause thedecrease of steepening of the electron density profiles.
Index Terms— Microwave plasma interactions, ponderomotiveforce, space charge effect.
I. INTRODUCTION
THE interaction of a high-power microwave beam withplasma is a very important subject to research in
consideration of space and plasma physics [1]–[4]. Thisresearch includes a broad range of applications such asplasma production and heating for tokamaks, stellarators,fusion, mirror machines, charged particles acceleration,inner surface processing of small diameter tubes, controlledradical production, environmental applications, multichargedion sources, etc. [5]–[9]. Furthermore, the microwave plasmainteraction exists in the plasma waveguide [10], [11] andionosphere [2], [12]. There are also many nonlinear phenom-ena in the propagation of a strong microwave beam throughthe plasma that can be related to the ponderomotive force.The ponderomotive force changes the dielectric permittivityand the spatial distribution of electron density and modifieselectromagnetic field profiles into the plasma [13], [14]. Thisforce exists in the interaction of high intensity laser pulses andplasmas [14]–[19] and the space charge field can be generatedby this process [15]. The ponderomotive force causes theformation of cavitons [20] and solitons [19], generation ofsecond harmonic [18], magnetic field 14, shock [21], and
Manuscript received April 20, 2013; revised August 4, 2013; acceptedSeptember 17, 2013. Date of publication October 4, 2013; date of currentversion November 6, 2013.
S. M. Khorashadizadeh and T. Mirzaye are with the PhysicsDepartment, University of Birjand, Birjand 97179, Iran (e-mail:[email protected]; [email protected]).
A. R. Niknam is with the Laser and Plasma Research Institute, ShahidBeheshti University, G.C., Tehran 19839, Iran (e-mail: [email protected]).
Digital Object Identifier 10.1109/TPS.2013.2282875
X-ray [21], too. It is suitable for inertial confinement fu-sion [12], charged particles acceleration [3], [19], rarefactionand compression shock waves [14], [22].
Recently, Niknam et al. [12] have studied the modifica-tion of the electron density distribution in an unmagnetizedplasma by considering the ponderomotive force of high-powermicrowave propagating into the plasma. They have shown thatthe electric field, magnetic field, and electron density profilesdepart from a sinusoidal shape, the electron density oscillationsbecome highly steepened for the higher microwave energyflux. Also, Malik et al. [23] have investigated the effectsof ponderomotive force in microwave plasma interaction in arectangular waveguide by taking three types of electron densitydistributions. They have indicated that the plasma density ispushed away from the middle of the waveguide. Moreover, thepeaks are evolved in the density when the electron temperatureis raised and so they become significant for enormously hightemperature. Before this, they studied the wakefield excitedby microwave pulse in a plasma-filled rectangular waveguide.They showed that the wakefield produced by microwave pulseis suitable for the effective acceleration of the charged particles[24], [25].
In this paper, we investigate the space charge and pon-deromotive force effects on the modified electron densitydistribution and the electromagnetic field and space chargefield profiles. The space charge effects are very impor-tant because it is suitable for accelerating charged particles[26]–[28]. We assume a beam of high-power microwave radia-tion incident from vacuum, z < 0, on a collisionless unmagne-tized plasma in z > 0. We use the momentum transfer, Poissonand Maxwell’s equations, and the dielectric permittivity ofcold plasma. The influence of high-power microwave energyflux, initial electron density, electron temperature and mi-crowave frequency on the electromagnetic, space charge fieldsprofiles, and the electron density distribution as a function ofz will be considered. It will understood that the behavior ofspace charge field will be affected from the initial electrondensity. Thus, the wavelength and amplitude modulation willbe formed. The steepening of the electron density profiles andsawtooth lines of space charge field profiles will be observedbecause the system behavior will be strongly nonlinear.
This paper is organized as follows: In Section II, the cou-pled basic equations, i.e., the momentum transfer, wave, andPoisson equations are presented. In Section III, the numericalsimulations of these nonlinear equations are discussed and theinfluence of some physical parameters such as the microwaveenergy flux, initial electron density, electron temperature, and
KHORASHADIZADEH et al.: SPACE CHARGE AND PONDEROMOTIVE FORCE EFFECTS 3095
microwave frequency on the electromagnetic and space chargefield profiles and electron density distribution in the plasmaare analyzed. Finally, a summary and conclusions are given inSection IV.
II. BASIC EQUATIONS
Consider the propagation of a high-power microwave beamthat impinges perpendicularly on a collisionless unmagnetizedplasma. It is pointed out that by high-power microwave beam,the electron ponderomotive force is generated and it can affectthe electric permittivity of plasma. The ponderomotive forceexerted on the ions is ignored since the ion mass is muchgreater than the electron mass. The average ponderomotiveforce per unit volume acting on the electrons in plasma takesthe following form [29]:
Fpe = 1
8πne
∂ε
∂ne∇E2 (1)
where Fpe, E, ε, and ne are the ponderomotive force, electricfield strength, dielectric permittivity, and electron density,respectively.
On the other hand, in a time scale greater than τ , whichis the time required for ponderomotive nonlinearity to beestablished, there is a steady state. Therefore, using thetime-independent wave, the momentum transfer and Poissonequations, we obtain the profiles of electromagnetic fields,electron density distribution, and space charge field in plasma.These equations in the absence of external charge and currentdensities are, respectively, as
∇2E − ∇(∇ · E) +(
ω
c
)2
εE = 0 (2)
−Te∇ne − eneEs = − 1
4πne
∂ε
∂ne∇E2 (3)
∇ · Es = 4πe(ni − ne) (4)
where the electron temperature, Te, is in energy unit. Es ,ni , e, ω, and c are the space charge field, density of ion,electron charge, microwave frequency, and light velocity invacuum, respectively. The microwave electric field in plasmais considered as E(z, t) = E(z)e−iωt x̂, that x̂ is unit vector inthe x-direction and E(z) is an amplitude of the electric fieldas function of z only. Then the magnetic field, B, is obtainedfrom Faraday’s law as follows:
∂ E
∂z= iω
cB. (5)
The above equations are valid for both plasma and vacuumregions. Therefore, having the electric permittivity form andthe boundary conditions, we can solve the above equationsand obtain E, B, Es , and ne. In the present work, we considerthe cold plasma dielectric permittivity [30]
ε = 1 − ω2pe
ω2 (6)
where ω2pe = 4πnee2/me is the plasma frequency and me is
mass of electron. Considering the amplitude of the electricfield, space charge field, and the electron density in the cold
plasma being a function of z only, therefore, from (3), theelectron density distribution is as
ne(z) = A exp
(− e2 E2
mω2Te− e
∫Es(z)dz
Te
)(7)
where A is constant and it is obtained from particle conserva-tion as
Ne =∫
ne(z)dz (8)
where Ne is the number of electrons per the surface unit. The(7) shows that the electron density is dependent on both theelectric field and space charge field, so it is a complicatednonlinear equation.
III. DISCUSSION
It is well known that the spatial distribution of the electrondensity and the dielectric permittivity of the plasma can changeunder the action of the ponderomotive force of a strongmicrowave beam. Therefore, in this section, considering theponderomotive force and solving (2)–(4), we describe the mi-crowave propagation into a collisionless unmagnetized plasma.These equations are strongly nonlinear and must be solvednumerically. We use fourth-order Runge–Kutta technique forsolving these equations. By using this technique, we obtain theelectric, space charge fields Profiles, and electron density dis-tribution in plasma. Then, we find the magnetic field profiles inthe plasma by (5). In Fig. 1(a)–(d), the effect of high-powermicrowave energy flux on the profiles of the electric field,magnetic field, electron density, and space charge field havebeen plotted as a function of z. In these figures, the initial elec-tron density is n0 = 4 × 109 cm−3, the electron temperatureis Te = 4 eV and the microwave frequency is 8 GHz. Theseprofiles are also plotted for the different energy fluxes I = 1× 104 W/cm2 (solid line), I = 5 × 104 W/cm2 (dashed line),and I = 10 × 104 W/cm2 (dotted line). These figures showthat, by increasing microwave energy flux, the amplitude of theelectric and magnetic fields oscillations increases because themicrowave energy flux is proportional to square of amplitudes.Furthermore, by increasing the microwave energy flux, theponderomotive force and consequently the nonlinear behaviorof system can be enhanced. From Fig. 1(c) and (d) one cansee the increasing microwave energy flux, the steepening ofthe electron density distribution, and sawtooth lines in spacecharge field profiles are amplified, which can be suitable foracceleration of charged particles. The sawtooth lines in spacecharge field profiles have been obtained by Jha et al. [26] inthe interaction of the laser pulse with uniformly magnetizedplasma, too. We can also see that by increasing the microwaveenergy flux, the wavelength and amplitude modulation occur,and hence the wavelength of oscillations and the peaks ofspace charge field become longer and higher, respectively.Although the modulation in wavelength and amplitude hasbeen observed by Niknam and Shokri [12], it is shown thatthe space charge effect caused a stronger modulation thanbefore especially for the electron density distribution and spacecharge field profiles.
3096 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 11, NOVEMBER 2013
Fig. 1. (a) Electric field E, (b) magnetic field B, (c) electron density distribution, and (d) space charge field as a function of z for different energy fluxes,I = 1 × 104 W/cm2 (solid line), I = 5 × 104 W/cm2 (dashed line), and I = 10 × 104 W/cm2 (dotted line) with n0 = 4 × 109 cm−3, Te = 4 eV, andf = 8 GHz.
Fig. 2. (a) Electron density distribution and (b) space charge field as afunction of z for different initial electron densities, n0 = 1 × 109 cm−3
(dotted line) with I = 5 × 104 W/cm2, Te = 4 eV and f = 8 GHz.
The influence of initial electron density on the electrondensity distribution and the space charge field profiles is shownin Fig. 2(a) and (b). The parameters chosen in Fig. 2 arethe same as those chosen in Fig. 1 except for the microwaveenergy flux that is I = 5 × 104 W/cm2 and for the different
initial electron density that is n0 = 1 × 109 cm−3 (solid line),n0 = 2 × 109 cm−3 (dashed line), and n0 = 4 × 109 cm−3
(dotted line). Fig. 2(a) and (b) show that by increasing theinitial electron density, the amplitude of space charge fieldis approximately constant and the oscillations wavelength ofelectron density distribution and space charge field becomeless. According to Poisson equation and considering that theamplitudes of microwave and space charge fields are constant,the increase of the initial electron density causes that thecharacteristic length decreases and consequently, the numberof electron bunches increases. Therefore, the behavior of spacecharge field is very much influenced by the initial electrondensity.
In Fig. 3(a)–(d), the effect of microwave frequency onthe profiles of electric field, magnetic field, electron den-sity. and space charge field has been shown, respectively.In these figures, the high-power microwave energy flux isI = 5 × 104 W/cm2 and three different microwave frequenciesare 8 GHz (solid line), 10 GHz (dashed line), and 12 GHz(dotted line) and other parameters are similar to Fig. 1.Fig. 3(a) and (b) show that by increasing the microwavefrequency, the amplitude of electric field and magnetic fieldin the plasma decreases and increases, respectively. Of course,the decrease of the amplitude of electric field is not veryobservable. These figures indicate that the energy conservationis valid. According to Fig. 3(c) and (d), one can see that byincreasing the microwave frequency, a decrease in the wave-length of oscillations and a decrease in the amplitude of elec-tron density distribution and space charge field was formed.Thus, steepening in the electron density profiles was de-creased. Because by increasing the microwave frequency,
KHORASHADIZADEH et al.: SPACE CHARGE AND PONDEROMOTIVE FORCE EFFECTS 3097
Fig. 3. (a) Electric field E, (b) magnetic field B, (c) electron density distribution, and (d) space charge field as a function of z for the different microwavefrequencies, 8 GHz ( solid line), 10 GHz (dashed line), and 12 GHz (dotted line) with n0 = 4 × 109 cm−3, Te = 4 eV, and I = 5 × 104 W/cm2.
Fig. 4. (a) Electron density distribution and (b) space charge field as afunction of z for the different electron temperatures Te = 4 eV (solid line),Te = 7 eV (dashed line), Te = 10 eV (dotted line) with n0 = 4 × 109 cm−3,f = 8 GHz, and I = 5 × 104 W/cm2.
the ponderomotive force decreases. The decrease of amplitudein space charge profiles by increasing the microwave frequencyis similar in [24] for wakefield generation in a plasma-filledrectangular waveguide.
The influence of electron temperature on the electron den-sity distribution and space charge field profiles is shown inFig. 4(a) and (b), respectively. In these figures, the microwavefrequency is f = 8 GHz and the different electron tem-peratures are Te = 4 eV (solid line), Te = 7 eV (dashedline), Te = 10 eV (dotted line), and other parameters aresimilar to Fig. 3. These figures indicate that by increasingthe electron temperature, the width of peaks and distancebetween peaks in the electron density distribution and thespace charge field profiles are increased. This result is similarto those obtained by Malik and Aria [23] in a plasma-filledrectangular waveguide. Then, the increase of the electrontemperature was caused the decrease of steepening of theelectron density profiles. Because by increasing the electrontemperature, the effective dielectric permittivity of plasmadecreases and, consequently, the wavelength of oscillationsincreases.
IV. CONCLUSION
In this paper, we investigated the nonlinear interaction ofa high-frequency electromagnetic wave with a collisionlessunmagnetized plasma. Taking into account the space chargeand ponderomotive force effects and by solving a systemof equations (the equations of wave propagation, momentumtransfer, and Poisson), we showed that, by changing themicrowave energy flux, wavelength and amplitude modulationhave taken place. Also, it was shown that the amplitudes ofelectron density variations and space charge field profiles wereincreased by increasing the high-power microwave intensity. Inthe other words, the steepening of the electron density profilesand sawtooth lines in the curves of space charge field were
3098 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 11, NOVEMBER 2013
observed by increasing microwave energy flux. Moreover, itwas found that the behavior of space charge field was verymuch influenced by the initial electron density and a decreasein the oscillations wavelength of electron density distributionand space charge field was caused by increasing the initialelectron density. Also, by increasing the microwave frequency,the wavelength of oscillations and the amplitude of electrondensity distribution and space charge field decrease. Thus,the steepening of the electron density profiles was decreased.Finally, it was indicated that width of peaks and distancebetween peaks in the electron density distribution profiles wereenhanced by increasing the electron temperature.
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Seyyed Mohammad Khorashadizadeh was born inBirjand, Iran, on September 20, 1959. He receivedthe B.Sc. degree in physics from the University ofBirjand, Birjand, Iran, in 1988, the M.Sc. degree inphysics from the Ferdowsi University of Mashhad,Mashhad, Iran, in 1993, and the Ph.D. degree inplasma physics from Shahid Beheshti University,Tehran, Iran, in 2005.
He is currently an Associate Professor with theDepartment of Physics, University of Birjand. Hiscurrent research interests include beam–plasma in-
teractions, dusty plasmas, and quantum plasmas.
Taghi Mirzaye was born in Torbat Heidariyeh, Iran,on January 28, 1984. He received the B.Sc. degreein atomic physics from Qom University, Qom, Iran,in 2006 and the M.Sc. degree in plasma physicsfrom the Sahand University of technology, Tabriz,Iran, 2009. He is currently pursuing the Ph.D. degreein plasma physics from Birjand University, Birjand,Iran.
His current research interests include microwave-plasma interactions.
Ali Reza Niknam was born in Shahrood, Iran,in 1965. He received the B.Sc., M.Sc., and Ph.D.degrees in physics from Shahid Beheshti University,Tehran, Iran.
He has been an Associate Professor with the Laserand Plasma Research Institute, Shahid Beheshti Uni-versity, since 2005. His current research interestsinclude laser–plasma interactions and plasma insta-bilities.
