Title Simulation of electromagnetic waves in the magnetized coldplasma by a DGFETD method

Author(s) LI, P; Jiang, L

Citation IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS,2013, v. 12, p. 1244-1247

Issued Date 2013

URL http://hdl.handle.net/10722/202836

Rights IEEE Antennas and Wireless Propagation Letters. Copyright ©Institute of Electrical and Electronics Engineers

1244 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 12, 2013

Simulation of Electromagnetic Waves in theMagnetized Cold Plasma by a DGFETD Method

Ping Li, Student Member, IEEE, and Li Jun Jiang, Senior Member, IEEE

Abstract—The properties of electromagnetic waves propagatingin magnetized plasma slabs and scattering from plasma coatedspheres are investigated using the discontinuous Galerkin finite-el-ement time-domain (DGFETD) method. The magnetized plasmais a kind of gyroelectric anisotropic dispersive medium. Thisanisotropy must be described by an electric permittivity tensorwith both nonzero diagonal and off-diagonal elements. Instead ofdoing the convolution directly in the time domain, an auxiliarydifferential equations (ADE) method is employed to representthe constitutive relation. The algorithm is validated by studyingthe transmission characteristics via electron-composed plasmaslabs and radar cross section (RCS) of plasma-coated spheres. Allnumerical results demonstrate very good agreements with exactsolutions and comply with the plasma physics.

Index Terms—Anisotropic magnetized plasma, auxiliary differ-ential equation (ADE), discontinuousGalerkin finite-element time-domain (DGFETD) method, gyroelectric dispersive media.

I. INTRODUCTION

O VER the past several decades, studies of electromagneticwave propagation in cold plasma have continued in an

increasing scale because of its importance in many practicalareas [1]–[3] such as the propagation of radio waves in Earth’sionosphere and magnetosphere [1], [12].Many finite-difference time-domain (FDTD) models are

extensively exploited to model the radio wave propagationthrough magnetized plasma slabs [2], [3]. Various treatmentsusing FDTD have been developed [3], [4]. However, the spatialand time staggering of the electric/magnetic vector fields andplasma related parameters in FDTD models make the FDTDmore cumbersome. An alternative to FDTD is finite-elementtime-domain (FETD) method that is also capable of handlinganisotropic dispersive media [5]. The discontinuous Galerkinfinite-element time-domain (DGFETD) method was firstdeveloped to solve the neutron transport problem in 1970s.Recently, it attracted extensive attention due to its variousmerits. DGFETD is a highly efficient method for solving thetwo first-order time-dependent Maxwell’s equations [6], [7]. Itsupports various types and shapes of elements and unstructuredmeshes with high-order accuracy. Furthermore, all the opera-

Manuscript received August 21, 2013; revised September 04, 2013; acceptedSeptember 18, 2013. Date of publication September 20, 2013; date of currentversion October 09, 2013. This work was supported by the GRF under Grants713011 and 712612 and the NSFC under Grant 61271158, and in part by theAoE/P-04/08.The authors are with the Department of Electrical and Electronic Engineering,

The University of Hong Kong, Hong Kong (e-mail: liping@eee.hku.hk;jianglj@hku.hk).Color versions of one or more of the figures in this letter are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LAWP.2013.2282955

tions are local since solutions are allowed to be discontinuousat the neighboring element interfaces. In this way, the resultantmass matrix is block-diagonal, which enables a very efficientexplicit time marching scheme, at the cost of added degreesof freedom. The aim of this letter is to study the propagationof electromagnetic waves in a magnetized plasma slab viaDGFETD method. Recently, DGFETD has been employed tosimulate the propagation of electromagnetic waves in isotropicdispersive materials [8], [9] such as unmagnetized plasma [14].In [10], DGFETD is further extended to handle anisotropicmedia with only nonzero diagonal elements by reformulatingthe numerical flux. In our work, however, auxiliary differentialequation (ADE) technique is applied while the formulation ofnumerical flux is the same as the isotropic case.In this letter, a DGFETD algorithm is developed to explore

the propagation of the electromagnetic wave in magnetizedplasma. The method is based on the ADE formulation forthe constitutive relationships. Via ADE method, an auxiliarypolarization current is introduced to equivalently representthe effects of the gyroelectric medium. As far as the authors’knowledge, it is the first time that DGFETD is employed tomodel such kind of gyroelectric medium.

II. ADE AND DGFETD FORMULATION

The well-known permittivity expression for the unmagne-tized plasma contributed by electrons is [11]:

, where denotes the plasma frequency andis the electron collision frequency. Notice that below , the

permittivity is negative, meaning that the plasma is opaque forwaves at this frequency band. When a plasma is subjected toa static magnetic field, such as in the Earth’s ionosphere, theelectrons will rotate around the magnetic field vector due tothe Lorentz force. The electric permittivity of the magnetizedplasma (gyroelectric medium) will become a full tensor .Supposing a plasma is subjected to an arbitrary-oriented static

magnetic field, direct derivation of the ADEs is difficult. Totackle this problem, we split the biasing magnetic field into- -, and -components: . Natu-rally, the total effects contributed by this biasing magnetic fieldare the linear superposition of , and -directed biasing mag-netic fields. Via this decomposition, the permittivity tensor canbe rewritten as , where corresponds to the-directed biasing magnetic field

(1)

(2)

1536-1225 © 2013 IEEE

LI AND JIANG: SIMULATION OF EMWAVES IN MAGNETIZED COLD PLASMA BY DGFETD 1245

(3)

corresponds to the -directed biasing magnetic field

(4)

(5)

(6)

corresponds to the -directed biasing magnetic field

(7)

(8)

(9)

In (1)–(9), denotes the cyclotron frequencydue to the static magnetic biasing. Based on the Fourier trans-form, the time changing rate of the electric flux density can beexpressed in the frequency domain as:

, where the auxiliary variablecan be regarded as a polarization current and is expressed as

, where, and . To

derive the auxiliary equations, we take the component as anexample.From (1)–(3), we can get three equations corresponding to the- -, and -components of . Namely

(10)

(11)

(12)

By combining (11) and (12), the following two equations canbe obtained:

(13)

(14)

Next, by performing the inverse Fourier transform, (10), (13),and (14) can be reformulated as differential equations in timedomain

(15)

(16)

(17)

Via similar operation, other two group auxiliary equations canbe obtained for and . By linear combination of these threegroup equations, a general expression for an arbitrary-orientedmagnetic field can be formulated as

(18)

(19)

(20)

Through some mathematical operations, we can finally get

(21)

With (21) and two first-order curl Maxwell’s equations

(22)

(23)

From (22) and (23), the anisotropic and dispersive behavioris fully embedded into the auxiliary field . In this case, theoriginal frequency-dependent wave impedance becomesconstant, which significantly simplifies the problem sincefrequency-independent wave impedance means no convolutionis involved in computing the numerical flux.To solve (21)–(23) via DGFETD, we first suppose that the

interest of computational domain of bounded by .We consider a partition into a number of nonoverlappingpolyhedral elements with boundary . Inside each el-

ement, the unknown values

and are approximated bya linear combination of independent edge vector basis functions

and and denote the local numberof degrees of freedom for and , respectively.The approximated fields are allowed to be completely discon-tinuous across element boundaries. By applying discontinuousGalerkin testing to (21)–(23), a linear of semi-discrete systemequations can be obtained

(24)

(25)

(26)

The semi-discrete equations (24)–(26) are explicitly solvedby fourth-order Runge–Kutta (RK)marching scheme. To ensurestability, the time-step size is determined in terms of

1246 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 12, 2013

Fig. 1. LCP wave (a) reflection and (b) transmission coefficients for plasmawith a biasing magnetic field parallel to the propagation direction.

[8], where is the free-space lightspeed, is the minimum edge length, and is the order ofthe basis function.

III. NUMERICAL RESULTS

The developed algorithm is employed to analyze the propa-gation of electromagnetic (EM) waves in a 1-D plasma slab andscattering from a 3-D plasma-coated perfectly electrical con-ducting (PEC) sphere.To study the wideband propagation properties of EM waves

through a plasma slab, a -polarized Gaussian-derivativepulsed plane wave propagating along -axis with the band-width more than 100 GHz is applied as the excitation. Weassume that the biasing static magnetic field is also alongthe -axis. The parameters of this magnetized plasma slabare rad/s, rad/s, and

rad/s [11]. To reduce the computationalcost, Silver–Müller absorbing boundary condition (SM-ABC)is used to terminate the end of the plasma slab. The linear-po-larized plane wave will be decomposed into the superpositionof left-handed circularly polarized (LCP) and right-handedcircularly polarized (RCP) waves along the propagation di-rection. The associated propagation properties of LCP andRCP waves are quite different based on the dispersion prop-erty in [1] and [13]. Figs. 1 and 2 show the transmissionand reflection coefficients for LCP and RCP waves, respec-tively. For comparison, the analytical reference based on

Fig. 2. RCP wave (a) reflection and (b) transmission coefficients for plasmawith a biasing magnetic field parallel to the propagation direction.

Fig. 3. Relative error of LCP reflection coefficient for different basis orders.

scattering matrix method [12] is provided. The coefficientsfor LCP and RCP are: and

, where and arecalculted from the recorded time domain data using fast Fouriertransform (FFT) method. For LCP wave, it is nonpropagatingbelow . For RCP wave, the nonpropagating band is fromto . The wave in the frequency band below is called

the whistler mode and is of extreme importance in the studyof ionospheric phenomena [1], [13]. The L-2 norm error of thereflection coefficient for basis functions with different orders ispresented in Fig. 3. The error peaks above 1 GHz correspond to

LI AND JIANG: SIMULATION OF EMWAVES IN MAGNETIZED COLD PLASMA BY DGFETD 1247

Fig. 4. Faraday rotation effect when a biasing magnetic field is applied alongthe propagating direction.

Fig. 5. RCS (dBsm) of a plasma-coated sphere. Red and black lines denotethe RCS of uncoated sphere by DGFETD and Mie theory, respectively. Pinkline represents RCS of the unmagnetized plasma coated sphere with

rad/s and rad/s. Blue line is the RCS of magnetizedplasma coated sphere with rad/s.

deeps in the reflection coefficients in Fig. 1(a), while the largeerror below 1 GHz is mainly attributable to the poor absorptionproperty of SM-ABC for low-frequency waves.To demonstrate the Faraday rotation effect, a single DGFETD

simulation is performed with a -polarized sinusoidal planewave. The transmitted fields are recorded at different distanceaway from the incident wave. The traces in Fig. 4 are obtainedby plotting the recorded time-domain electric field. The rotationof this linear-polarized wave along the propagating path canbe understood in terms of the difference in the phase velocityof the RCP and LCP waves [1], [13]. It can be observed thatthe amount of rotation angle is proportional to the plasmathickness.To verify the 3-D capability of the proposed method, the

scattering from a PEC sphere with 10-cm radius is investigated.This PEC sphere is coated with a 5-cm plasma layer. A si-nusoidal modulated Gaussian pulse propagating along the

-direction is applied as the incident wave. To have a better ab-sorption of the scattered wave, perfectly matched layer (PML)is employed to truncate the computational domain. The calcu-lated radar cross section (RCS) around 1.0 GHz in the -planeis presented in Fig. 5. It is observed that the RCS with coatedplasma is profoundly reduced since electromagnetic waves areabsorbed by the plasma. This interesting phenomenon is calledplasma stealth, which can be potentially exploited for stealthyaircrafts. Also, it is noted that the RCS is nonsymmetrical whenthe plasma is magnetized, which is due to the existence ofLorentz force.

IV. CONCLUSION

A DGFETD algorithm is developed to model the interac-tion of EM wave in magnetized plasma. ADEs are employedto tackle the constitutive relationship between the electric per-mittivity tensor and the electric field . The DGFETD methodis applied to study the EM wave propagation through a plasmaslab with electrons only and scattering from a plasma-coatedsphere.

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