時(shí)空非均勻等離子體鞘套中太赫茲波的傳播特性?

陳偉 郭立新 李江挺 淡荔

(西安電子科技大學(xué)物理與光電工程學(xué)院,西安710071)

時(shí)空非均勻等離子體鞘套中太赫茲波的傳播特性?

陳偉 郭立新?李江挺 淡荔

(西安電子科技大學(xué)物理與光電工程學(xué)院,西安710071)

(2016年12月12日收到;2017年1月13日收到修改稿)

高超聲速飛行器再入地面的過(guò)程中,其周圍等離子體的電子密度是非均勻且隨時(shí)間變化的.對(duì)于不同的再入高度,飛行器周圍的溫度和壓強(qiáng)也會(huì)發(fā)生改變.因此,研究電磁波在時(shí)空非均勻等離子體鞘套中的傳播特性意義重大.首先建立了時(shí)變非均勻的等離子體鞘套模型,然后通過(guò)經(jīng)驗(yàn)公式得到溫度、壓強(qiáng)與碰撞頻率三者的關(guān)系.采用時(shí)域有限差分方法計(jì)算了太赫茲波段中不同電子密度弛豫時(shí)間、溫度、壓強(qiáng)時(shí)的反射系數(shù)、透射系數(shù)和吸收率.研究結(jié)果表明:在太赫茲波段中,電子密度的弛豫時(shí)間越長(zhǎng),溫度越高,壓強(qiáng)越大,電磁波越容易穿透等離子體;弛豫時(shí)間越短,溫度越低,壓強(qiáng)越小,等離子體對(duì)電磁波吸收率的變化越明顯.這些結(jié)果為解決“黑障”問(wèn)題提供了理論依據(jù).

時(shí)空非均勻等離子體,等離子體鞘套,太赫茲波,時(shí)域有限差分方法

1 引言

臨近空間高超聲速飛行器再入地面的過(guò)程中會(huì)與大氣發(fā)生摩擦,使飛行器周圍的溫度迅速升高,從而導(dǎo)致稠密的空氣發(fā)生電離.隨著飛行器周圍的電子濃度逐漸上升,最終會(huì)形成一個(gè)含有大量自由電子的等離子體包覆流場(chǎng),即等離子體鞘套[1?4].等離子體鞘套會(huì)改變飛行器天線的阻抗特性,影響天線的正常工作,導(dǎo)致飛行器與地面測(cè)控站之間通信中斷,使飛行器處于盲監(jiān)控狀態(tài),造成通信“黑障”現(xiàn)象[5?8],嚴(yán)重時(shí)甚至危及飛行員的生命安全.因此,突破“黑障”對(duì)飛行器測(cè)控的影響是目前的一項(xiàng)國(guó)際難題.

20世紀(jì)70年代,美國(guó)國(guó)家航空航天局(NASA)針對(duì)20—80 km的臨近空間做了很多再入實(shí)驗(yàn)[9?11].國(guó)內(nèi)外許多專家和學(xué)者也開(kāi)展了大量的研究并且提出了很多減輕“黑障”效應(yīng)的方法[12?14].近幾年來(lái),由于太赫茲(THz)技術(shù)的興起,THz波的一些特殊性質(zhì)越來(lái)越被人們所采納.因此,利用THz波來(lái)穿透等離子體鞘套也成為了解決“黑障”問(wèn)題的一種有效方法.例如,袁承勛等[15]研究了太赫茲波在磁化、均勻等離子體中的功率吸收特性.中國(guó)空間技術(shù)研究院李拴濤等[16]分析了電子密度在空間呈雙指數(shù)分布時(shí)太赫茲波的傳輸特性.南華大學(xué)陳文波等[17]分析了等離子體在時(shí)變情況下THz波的反射系數(shù)、透射系數(shù)及吸收率.電子科技大學(xué)鄭靈等[18]通過(guò)理論和實(shí)驗(yàn)對(duì)THz波在等離子體中的反射、透射以及衰減特性做了大量研究.飛行器在大氣層中高速飛行時(shí),其周圍等離子體流場(chǎng)必然是空間非均勻且隨時(shí)間快速變化的[19],因此研究電磁波在時(shí)空非均勻等離子體鞘套中的傳輸特性很有意義.

本文根據(jù)NASA再入實(shí)驗(yàn)的相關(guān)數(shù)據(jù)建立了一維時(shí)變非均勻等離子體鞘套模型,通過(guò)經(jīng)驗(yàn)公式得到等離子體鞘套中溫度、壓強(qiáng)和碰撞頻率三者的關(guān)系.利用時(shí)域有限差分(finite-difference timedomain,FDTD)方法計(jì)算了不同電子密度的弛豫時(shí)間,以及不同溫度和壓強(qiáng)時(shí)的反射系數(shù)、透射系數(shù)及吸收率.結(jié)合數(shù)值結(jié)果分析了THz波段中時(shí)空非均勻等離子體鞘套對(duì)電磁波傳播特性的影響,為解決“黑障”問(wèn)題提供了理論依據(jù).

2 時(shí)空非均勻等離子體的FDTD迭代公式

各向異性色散介質(zhì)碰撞磁化等離子體中,Maxwell方程組和相關(guān)的本構(gòu)方程為[20?22]

式中E為電場(chǎng),其強(qiáng)度單位為V/m;H為磁場(chǎng),其強(qiáng)度單位為A/m;J為電流密度,單位為A/m2;ε0為真空中的介電常數(shù),ε0=8.85×10?12F/m;μ0為真空中的磁導(dǎo)率,μ0=4π×10?7H/m;為等離子體頻率,ne(r,t)為電子密度,這里表示空間和時(shí)間的量;ν為等離子體的碰撞頻率;ωb=e B0/Me為電子旋轉(zhuǎn)頻率,B0為外部靜態(tài)磁場(chǎng);e和Me分別為電子電量和電子質(zhì)量.

對(duì)于一維情況下的橫磁(TEM)波,設(shè)外磁場(chǎng)的方向?yàn)?Z方向,即ωb=ωb,在笛卡兒坐標(biāo)下的各矢量可表示為E=Ex+Ey,H=Hx+Hy,J=Jx+Jy,ωb=ωbz.

在上述坐標(biāo)下,則(1)式和(2)式可寫(xiě)為

將(3)式改寫(xiě)為矩陣可得

式中

對(duì)(1)式和(2)式進(jìn)行差分離散,得到的FDTD方程為

根據(jù)文獻(xiàn)[23],在時(shí)變等離子體中J(t)可以利用拉普拉斯及逆拉普拉斯變換得到:

J(t)的FDTD離散形式可以寫(xiě)為

式中

結(jié)合(11)式、(12)式和(14)式即可得到電場(chǎng)E的迭代公式,磁場(chǎng)H的迭代公式不變.

3 數(shù)值驗(yàn)證

對(duì)于時(shí)空非均勻等離子體的算例驗(yàn)證,我們將多層模型退化到一層時(shí)變等離子體模型.計(jì)算時(shí),入射波為微分高斯脈沖,網(wǎng)格長(zhǎng)度δ=75μm,時(shí)間步?t=δ/(2c),c是真空中的光速,兩邊采用MUR吸收邊界,計(jì)算時(shí)間步為15000步.等離子體厚度為200個(gè)網(wǎng)格.電子密度的弛豫時(shí)間Tr=1500.利用本文的計(jì)算方法(LTJEC-FDTD)和SO-FDTD兩種方法進(jìn)行了計(jì)算對(duì)比.計(jì)算模型如圖1所示,結(jié)果如圖2所示.

圖1物理模型Fig.1.PhysicalModel.

圖2 所示為L(zhǎng)TJEC-FDTD方法和SO-FDTD方法的反射系數(shù)和透射系數(shù)對(duì)比結(jié)果,從圖中可知兩種方法得到的結(jié)果基本一致,這也證明了本文方法的可行性.

4 數(shù)值模擬結(jié)果

以NASA再入實(shí)驗(yàn)[7]中47 km高度的電子密度分布規(guī)律為例進(jìn)行數(shù)值模擬.對(duì)于空間非均勻問(wèn)題,采用將電子密度進(jìn)行分層近似處理的方法,如圖3所示,等離子體厚度為6.7 cm,共分為20層,每一層的電子密度看作均勻的.對(duì)于時(shí)變問(wèn)題,將每層電子密度乘以弛豫時(shí)間?t/Tr,則電子密度的時(shí)變表達(dá)式為[17,24]

圖2 計(jì)算結(jié)果對(duì)比(a)反射系數(shù);(b)透射系數(shù)Fig.2.CoMparison of calcu lation resu lts:(a)Refl ection coeffi cient;(b)transMission coeffi cient.

式中ne_layer(n)表示每一層電子密度的最大值,?t為時(shí)間步,Tr為一個(gè)定值,控制電子密度隨時(shí)間變化的速率.

等離子體鞘套中碰撞頻率的經(jīng)驗(yàn)公式為[25?27]

式中T和P分別為等離子體鞘套的溫度和壓強(qiáng),T的單位為K,P的單位為atm.

在采用FDTD方法分析電磁波傳播特性時(shí),入射波的頻率范圍取0—0.2 THz,入射波為微分高斯脈沖,網(wǎng)格長(zhǎng)度δ=75μm,時(shí)間步?t=cδ/2,兩邊采用MUR吸收邊界,計(jì)算時(shí)間步為15000步.在時(shí)變非均勻等離子體鞘套模型中分別計(jì)算了不同弛豫時(shí)間、溫度和壓強(qiáng)時(shí)的反射系數(shù)、透射系數(shù)和吸收率,計(jì)算參數(shù)如表1所示.計(jì)算結(jié)果如圖4—圖6所示.

圖3 等離子體鞘套電子密度空間分布(a)不同再入高度的電子密度分布;(b)47 km的電子密度分布Fig.3.Spatial d istribution of electron density for p lasMa sheath:(a)Electron density distribution at d iff erent reentry heights;(b)electron density d istribution at 47 km.

表1 不同結(jié)果圖對(duì)應(yīng)的等離子體鞘套計(jì)算參數(shù)Tab le 1.Calcu lation paraMeters of p lasMa sheath for d iff erent figu res.

圖4所示為弛豫時(shí)間對(duì)電磁波傳播特性的影響.從圖4(a)和圖4(b)可以看出,Tr越大,反射系數(shù)越小,透射系數(shù)越大.根據(jù)(15)式可知,隨著Tr的變大,電子密度隨時(shí)間上升的速度變慢,同一時(shí)間段內(nèi)電子密度會(huì)減小,因而電磁波更容易穿透等離子體,導(dǎo)致反射系數(shù)減小.在0—0.1 THz波段,反射系數(shù)的振蕩周期較大,而到了THz波段(0.1—0.2 THz)振蕩周期變得更加緊密和劇烈.在THz波段,電磁波的衰減在10 dB以內(nèi),因此在這種情況下能夠穿透等離子體鞘套.另外,從圖4(c)可以看出,隨著Tr的增大,吸收峰在變小且有逐漸向低頻方向移動(dòng)的趨勢(shì).隨著頻率的增大,等離子體鞘套對(duì)電磁波的吸收作用也越來(lái)越小.這是因?yàn)楫?dāng)電子密度的弛豫時(shí)間尺度小于或等于THz波的周期時(shí),快速產(chǎn)生的等離子體能夠從電磁波中吸收更多的能量.反之,當(dāng)電子密度弛豫時(shí)間大于THz波的周期時(shí),等離子體對(duì)電磁波能量的吸收會(huì)減弱,導(dǎo)致吸收率下降,進(jìn)而加強(qiáng)了電磁波穿透等離子體層的能力.

圖4 (網(wǎng)刊彩色)弛豫時(shí)間對(duì)電磁波傳播特性的影響(a)反射系數(shù);(b)透射系數(shù);(c)吸收率Fig.4.(color on line)E ff ect of relaxation tiMe on electroMagnetic wave p ropagation p roperties:(a)Refl ection coeffi cient;(b)transMission coeffi cient;(c)absorp tivity.

圖5所示為不同溫度對(duì)電磁波傳播特性的影響.從圖中可知,電磁波在時(shí)空非均勻等離子體中的傳播特性與溫度密切相關(guān).溫度越高,反射系數(shù)越小,到了THz波段,溫度的變化對(duì)反射系數(shù)幾乎沒(méi)有影響.從圖5(b)來(lái)看,透射系數(shù)隨頻率的變化曲線在0.05 THz處發(fā)生了明顯的改變.這與等離子體的截止頻率有關(guān),從圖3(b)可知等離子體的截止頻率約為0.05 THz,對(duì)比圖5(c),當(dāng)入射波的頻率小于等離子體截止頻率時(shí),等離子體對(duì)電磁波的吸收作用最強(qiáng),因此電磁波很難穿透等離子體,當(dāng)入射波頻率大于等離子體截止頻率時(shí),等離子體對(duì)電磁波的吸收作用開(kāi)始減弱,電磁波很容易穿透等離子體.當(dāng)入射波頻率大于等離子體截止頻率時(shí),溫度越高,透射系數(shù)越小,根據(jù)(16)式可知,溫度與等離子體中碰撞頻率成正比,溫度越高,碰撞頻率越大,此時(shí)等離子體的碰撞吸收作用對(duì)電磁波產(chǎn)生了很大的影響.在電磁波穿透等離子體時(shí),溫度越高,等離子體對(duì)電磁波的碰撞吸收作用越明顯,因此透射系數(shù)會(huì)相應(yīng)減小.但是當(dāng)入射波頻率遠(yuǎn)大于等離子體截止頻率時(shí),電磁波的周期減小使等離子體的弛豫時(shí)間遠(yuǎn)大于電磁波的周期,導(dǎo)致吸收率下降,到了THz波段,電磁波可以幾乎可以無(wú)損耗穿透等離子體.這說(shuō)明提高入射波的頻率能改善溫度對(duì)電磁波的影響.

圖5 (網(wǎng)刊彩色)溫度對(duì)電磁波傳播特性的影響(a)反射系數(shù);(b)透射系數(shù);(c)吸收率Fig.5.(color on line)E ff ect of teMperatu re on electroMagnetic wave p ropagation p roperties:(a)Refl ection coeffi cient;(b)transMission coeffi cient;(c)absorp tivity.

圖6 (網(wǎng)刊彩色)壓強(qiáng)對(duì)電磁波傳播特性的影響(a)反射系數(shù);(b)透射系數(shù);(c)吸收率Fig.6.(color on line)E ff ect of p ressu re on electroMagnetic wave p ropagation p roperties:(a)Refl ection coefficient;(b)transMission coeffi cient;(c)absorp tivity.

圖6所示為不同壓強(qiáng)對(duì)電磁波傳播特性的影響,當(dāng)入射波頻率為0—0.05 THz時(shí),透射系數(shù)與壓強(qiáng)成正比.這是因?yàn)楫?dāng)入射波頻率小于等離子體的截止頻率時(shí),等離子體對(duì)電磁波的反射作用大于碰撞吸收作用,此時(shí)入射波大部分能量會(huì)被等離子體反射.當(dāng)入射波頻率為0.05—0.2 THz時(shí),由于入射波頻率大于等離子體的截止頻率,電磁波能夠穿透等離子體.此時(shí)等離子體的吸收作用會(huì)對(duì)電磁波產(chǎn)生很大的影響.根據(jù)(16)式可知,溫度、壓強(qiáng)和等離子體的碰撞頻率三者存在線性關(guān)系.因此,圖6中反射系數(shù)、透射系數(shù)和吸收率的變化趨勢(shì)與圖5具有一致性.另外,從圖6(b)和圖6(c)可以看出,壓強(qiáng)的變化對(duì)透射系數(shù)和吸收率的影響比溫度的影響大.當(dāng)入射波為0.1—0.2 THz時(shí),壓強(qiáng)的變化對(duì)電磁波吸收率的影響明顯減小.所以,在THz波段同樣可以改善壓強(qiáng)對(duì)電磁波造成的影響.

5 結(jié)論

本文首先通過(guò)NASA相關(guān)數(shù)據(jù)建立了一維時(shí)空非均勻等離子體鞘套模型,基于此模型并結(jié)合等離子體鞘套中碰撞頻率的經(jīng)驗(yàn)公式,利用FDTD方法,計(jì)算了不同電子密度弛豫時(shí)間、溫度和壓強(qiáng)下的反射系數(shù)、透射系數(shù)和吸收率.分析了時(shí)空非均勻等離子體鞘套對(duì)太赫茲波傳播特性的影響.結(jié)果表明:電子密度的弛豫時(shí)間、溫度和壓強(qiáng)會(huì)對(duì)電磁波的反射、透射和吸收產(chǎn)生很大的影響,當(dāng)入射波頻率在太赫茲波段時(shí)能有效地減輕這種影響,提高電磁波穿透等離子體鞘套的能力.這些結(jié)果為解決“黑障”問(wèn)題提供了理論依據(jù).

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(Received 12 DeceMber 2016;revised Manuscrip t received 13 January 2017)

PACS:41.20.Jb,52.25.Os,52.40.Db,52.40.KhDOI:10.7498/aps.66.084102

*Project supported by the National Natural Science Foundation of China(G rant Nos.61431010,61301065)and the Foundation for Innovative Research G roups of the National Natu ral Science Foundation of China(G rant No.61621005).

?Corresponding author.E-Mail:lxguo@xidian.edu.cn

P ropagation characteristics of terahertz w aves in teMporally and spatially inhoMogeneous p lasMa sheath?

Chen Wei Guo Li-Xin?Li Jiang-Ting Dan Li

(School of Physics and Optoelectronic Engineering,Xidian University,Xi’an 710071,China)

The p lasMa sheath is produced by high-teMperature heating during the reentry of a hypersonic vehicle to the Earth atMosphere.TeMperature around the vehicle rises rapid ly because of severe friction w ith air.The vehicle teMperature behind friction ishigh enough to excite various realgaseff ects including cheMical reactionsof air,which containsab lation particles of vehicle,free electrons,and ions.The p lasMa sheath greatly aff ects the transMission of electroMagnetic waves and has very strong interference on the communication signals,which results in interrup t between the target and the ground station,namely,b lackout.The electron density of p lasma sheath surrounding the aircraft is inhomogeneous and varies w ith tiMe.TeMperature and pressure w ill also change at diff erent altitudes.Therefore,it isMeaningful to investigate the propagation characteristics of electromagnetic waves in teMporally and spatially inhomogeneous p lasma sheath.The teMporally and spatially inhoMogeneous p lasMa sheath Model is introduced and the electron density data of the National Aeronautics and Space AdMinistration(NASA)reentry vehicle is eMp loyed.The relationships aMong teMperature,p ressure,and collision frequency are obtained w ith the eMpirical formula of collision frequency.Then,the refl ection coeffi cient and transMission coeffi cient of tiMe-varying single layer p lasMa are calcu lated w ith the shift operator finite-diff erence tiMe-doMain(SO-FDTD)Method.These resu lts are coMpared to verify the correctness of the p roposed method.Finally,the LTJEC-FDTD method is used to calcu late the refl ection coeffi cient,transMission coeffi cient and absorp tivity at diff erent relaxation tiMe,teMperature,and pressure in the terahertz(THz)band.The resu lts show that the higher teMperature and pressure w ill enable the electroMagnetic wave to penetrate the p lasMa sheath at high relaxation time of electron density.If the incident wave frequency is lower than the cut-off frequency of p lasMa,the refl ection of electroMagnetic wavew ill beMore obvious.However,when the incident wave frequency is in the THz band,the eff ects of teMperature and pressure on the propagation of electromagnetic wave are obviously weakened.The absorption of electroMagnetic wave by p lasMa w ill be More obvious when the relaxation tiMe,teMperature,and pressure decrease.If the relaxation tiMe of electron density is shorter than or equal to the period of THz wave,more energy of electromagnetic wave w ill be absorbed by the p lasma sheath.Contrarily,if the relaxation time of electron density ismuch longer than the period of THz wave,the absorption of electroMagnetic energy w ill decrease.This study gives soMe insight into the teMporally and spatially inhoMogeneous p lasMa sheath,and p rovides a theoretical basis for solving the b lackout problem.

teMporally and spatially inhomogeneous p lasMa,p lasma sheath,terahertz wave,finitediff erence time-domain method

10.7498/aps.66.084102

?國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào):61431010,61301065)和國(guó)家自然科學(xué)基金創(chuàng)新研究群體科學(xué)基金(批準(zhǔn)號(hào):61621005)資助的課題.

?通信作者.E-Mail:lxguo@xid ian.edu.cn

?2017中國(guó)物理學(xué)會(huì)C h inese P hysica l Society

http://w u lixb.iphy.ac.cn