Whistler wave emission by a modulated electron beam on a metal-plasma boundary

The transition radiation of a thin modulated electron beam injected from a conducting plane into a plasma along an arbitrary magnetic field normal to that plane is calculated. The radiation field is formed as a result of the interference of three waves with different wave vectors. The radiation patt...

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Datum:	2003
Hauptverfasser:	Anisimov, І.О., Kelnyk, О.І., Krafft, C., Nychyporuk, T.V.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2003
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Нелинейные процессы
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/110991
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Zitieren:	Whistler wave emission by a modulated electron beam on a metal-plasma boundary / І.О. Anisimov, О.І. Kelnyk, C. Krafft, T.V. Nychyporuk // Вопросы атомной науки и техники. — 2003. — № 4. — С. 74-77. — Бібліогр.: 7 назв. — англ.

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spelling	irk-123456789-1109912017-01-08T03:03:12Z Whistler wave emission by a modulated electron beam on a metal-plasma boundary Anisimov, І.О. Kelnyk, О.І. Krafft, C. Nychyporuk, T.V. Нелинейные процессы The transition radiation of a thin modulated electron beam injected from a conducting plane into a plasma along an arbitrary magnetic field normal to that plane is calculated. The radiation field is formed as a result of the interference of three waves with different wave vectors. The radiation pattern is mainly determined by one of those waves, depending on the parameters of the model. 2003 Article Whistler wave emission by a modulated electron beam on a metal-plasma boundary / І.О. Anisimov, О.І. Kelnyk, C. Krafft, T.V. Nychyporuk // Вопросы атомной науки и техники. — 2003. — № 4. — С. 74-77. — Бібліогр.: 7 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/110991 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Нелинейные процессы Нелинейные процессы
spellingShingle	Нелинейные процессы Нелинейные процессы Anisimov, І.О. Kelnyk, О.І. Krafft, C. Nychyporuk, T.V. Whistler wave emission by a modulated electron beam on a metal-plasma boundary Вопросы атомной науки и техники
description	The transition radiation of a thin modulated electron beam injected from a conducting plane into a plasma along an arbitrary magnetic field normal to that plane is calculated. The radiation field is formed as a result of the interference of three waves with different wave vectors. The radiation pattern is mainly determined by one of those waves, depending on the parameters of the model.
format	Article
author	Anisimov, І.О. Kelnyk, О.І. Krafft, C. Nychyporuk, T.V.
author_facet	Anisimov, І.О. Kelnyk, О.І. Krafft, C. Nychyporuk, T.V.
author_sort	Anisimov, І.О.
title	Whistler wave emission by a modulated electron beam on a metal-plasma boundary
title_short	Whistler wave emission by a modulated electron beam on a metal-plasma boundary
title_full	Whistler wave emission by a modulated electron beam on a metal-plasma boundary
title_fullStr	Whistler wave emission by a modulated electron beam on a metal-plasma boundary
title_full_unstemmed	Whistler wave emission by a modulated electron beam on a metal-plasma boundary
title_sort	whistler wave emission by a modulated electron beam on a metal-plasma boundary
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2003
topic_facet	Нелинейные процессы
url	http://dspace.nbuv.gov.ua/handle/123456789/110991
citation_txt	Whistler wave emission by a modulated electron beam on a metal-plasma boundary / І.О. Anisimov, О.І. Kelnyk, C. Krafft, T.V. Nychyporuk // Вопросы атомной науки и техники. — 2003. — № 4. — С. 74-77. — Бібліогр.: 7 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT anisimovío whistlerwaveemissionbyamodulatedelectronbeamonametalplasmaboundary AT kelnykoí whistlerwaveemissionbyamodulatedelectronbeamonametalplasmaboundary AT krafftc whistlerwaveemissionbyamodulatedelectronbeamonametalplasmaboundary AT nychyporuktv whistlerwaveemissionbyamodulatedelectronbeamonametalplasmaboundary
first_indexed	2025-07-08T01:28:44Z
last_indexed	2025-07-08T01:28:44Z
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fulltext	UDK 533.9 WHISTLER WAVE EMISSION BY A MODULATED ELECTRON BEAM ON A METAL-PLASMA BOUNDARY І.О.Anisimov1, О.І.Kelnyk1, C.Krafft2, T.V.Nychyporuk1 1Taras Shevchenko National University of Kyiv, Radio Physics Faculty, Kyiv, Ukraine ioa@univ.kiev.ua 2Laboratoire de Physique des Gaz et des Plasmas, Université Paris Sud, Orsay Cedex, France The transition radiation of a thin modulated electron beam injected from a conducting plane into a plasma along an arbitrary magnetic field normal to that plane is calculated. The radiation field is formed as a result of the interfer- ence of three waves with different wave vectors. The radiation pattern is mainly determined by one of those waves, depending on the parameters of the model. 1. INTRODUCTION One of the possible ways for interpreting the results of active beam-plasma experiments in the ionosphere is the laboratory simulation of the observed effects. The excitation of waves by modulated electron beams inject- ed in space plasmas belongs to such effects (see, for ex- ample, [1]). In the laboratory experiment [2], whistlers excited by a modulated electron beam injected from an electron gun through a metal surface into a magneto- plasma were observed. It was shown that in some cases this excitation occurs via a transition radiation mecha- nism. The transverse length of the formation zone of the transition radiation was calculated in [3] for conditions typical of the experiment [2]. It has been shown that this length is considerably less than the dimensions of the injector. It means that the model of a radially restricted beam injected from a conductive plane is valid for the calculation of this type of radio-emission. This model was studied in the whistler approximation (ω<<ωc<<ωp) in [4]. This report presents the results of the transition radiation calculations obtained with the same model but for arbitrary parameters. 2. MODEL DESCRIPTION A sharp metal-plasma boundary is treated. The plas- ma is considered to be cold and the ambient magnetic field is directed along the z-axis. A thin modulated elec- tron beam is injected from the metal plane parallel to the magnetic field, forming the current density wave: ( ) ( )[ ]ztierjtrj zm χω −= exp)(,  , ( ) 0lim = ∞→ rjm r , ( ) ( )r r Irj m m δ π = , 0// vωχ = (1) (where ω is the modulation frequency and v0 is the elec- tron beam velocity). The current density (1) is consid- ered to be given. The permittivity tensor of the cold magnetoactive plasma has the following form: ( )           − = ⊥ ⊥ //00 0 0 ˆ ε εα αε ωε i i , 22 2 1 c p ωω ω ε − −=⊥ , 22 2 c pc ωω ω ω ω α − = , 2 2 // 1 ω ω ε p−= , (2) where ωp is the electron plasma frequency and ωc is the electron cyclotron frequency. The problem is solved in two stages. At first the transition radiation of electromagnetic waves by a radi- ally unbounded modulated electron beam forming the current density (3) ( ) ( )[ ]rtijetrj mz  χω −= exp, , { }//,,0 χχχ ⊥= (3) is examined. At the second stage, the current (1) is ex- panded into plane partial waves (3). The contributions of the separated partial waves are added to find out the transition radiation of the modulated electron beam (1). 3. TRANSITION RADIATION OF THE PLANE CURRENT WAVE It is convenient to use the vector-potential instead of the field components of the emitted electromagnetic wave by imposing the calibration condition ϕ=0. From the Maxwell's equations one can obtain the wave equa- tion for the vector-potential corresponding to the current density wave (3) taking into account the permittivity tensor (2). Using the vector-potential in the form ( ) ( )ritiAtrA m  χω −′= exp, one obtains the relation for the wave as : ( ) mmmm j c AkAA  πεχχχ 42 0 2 −=′+′−′ . (4) Hence the normalized components of the vector –poten- tial amplitude Am (Ax,Ay,Az) can be presented as: ( )( ) ∆ αεε 22 // 2 // −−−−=≡ ⊥⊥ nnAA z , ( )( ) ∆ +− −= ⊥ αε yx x innnn A 2 // , ( )( ) ∆ +−− = ⊥ αε xy y inninn A 2 // , m m m j Ack A π4 2 0 ′ =   , n k   = 0 χ , znn =// , ( )( )[ ] ( )[ ]22222 // 2 // αεεαεεε∆ +−+−−−= ⊥⊥⊥⊥⊥ nnnn , (5) where ∆=0 is the dispersion relation for eigenmodes of the cold magnetized plasma. On the metal-plasma boundary the tangential com- ponent of the electric field vanishes 0=τE  , and then 0=τA  , Hence it results in 0=Σ ± , (6) where ( ) ( )( ) 22 // 2 //// 22 αεε ε −−− − =Σ ⊥⊥ ⊥−+ + nn Annnn , ( )( ) 22 // 2 ////2 αεε α −−− =Σ ⊥⊥ −+ − nn Annn , 2 yx inn n ± =± . Ordinary and extraordinary electromagnetic waves propagating away from the metal plane should also be taken into account besides the current wave. Conse- quently the boundary conditions on the metal-plasma border can be written as:    =Σ+Σ+Σ =Σ+Σ+Σ −−− +++ 0 ,0 21 21 B B (7) where the index 1 refers to the ordinary wave, the index 2 to the extraordinary wave, and the index B to the cur- rent wave. The amplitudes of the electromagnetic waves excit- ed by the radially unbounded modulated electron beam have the form: ( )( )[ ] ( )     ∆− −−−    − −= ⊥⊥ β β αεε β 1 1 2,1// 2 1,2// 2 2,1// 22 2,1// 2 2,12 2 1,2// 2,1 nnn nnn Az , β 1 // =Bn . (8) The transformation coefficient of the current wave into the electromagnetic waves, determined as the ratio of the denominate longitudinal component of the vector- potential and the amplitude of the current density wave, is specified by the formula: ( ) ( )( ) ( )     − −−−     −−= ⊥⊥ ⊥ β ∆β αεε β πΚ 1 14 2,1// 2 1,2// 2 2,1// 22 2,1// 2 2,1 2 2 1,2//2 0 2,1 nnn nnn ck n (9) 4. TRANSITION RADIATION OF THE THIN MODULATED ELECTRON BEAM After expanding the current (1) into plane partial waves (3) and taking into account (9), one can obtain the expressions for the radiation field components (in cylindrical coordinates): ( ) ( ) ( ) ( )( )ziti rJjdA mz ⊥ ⊥⊥⊥⊥ ∞ ⊥ −× ×Κ= ∫ χχω χχχχχ // 02,1 0 2,1 exp ( ) ( ) ( ) ×−= ⊥⊥⊥⊥ ∞ ⊥∫ rJjdiA mr χχχΚχχ 12,1 0 2,1 ( ) ( )( ) ( )( ) 4 0 22 0 2 // 2 0 2 // 2 0 2 // exp kkk zitik αεχεχ χχωεχχχ −−− −−× ⊥⊥ ⊥⊥⊥ ( ) ( ) ( ) ( )( ) ( )( ) 4 0 22 0 2 // 2 0 2 // 2 0// 12,1 0 2,1 exp kkk zitik rJjdiA m αεχεχ χχωαχχ χχχχχϕ −−− − × ×Κ= ⊥⊥ ⊥⊥ ⊥⊥⊥⊥ ∞ ⊥∫ , (10) where J0 and J1 are the Bessel functions of the zeroth and the first order, respectively. Then the components of that field in spherical coordinates can be written as: θθ sincos 2,12,12,1 rzR AAA += , θθθ cossin 2,12,12,1 rz AAA +−= , 2,12,1 ϕϕ AA = , (11) where θ is the azimuthal angle, the angle between the magnetic field and the direction of observation. For the calculation of the integrals (10) the method of the stationary phase is applied. As a result: ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )       Θ      Θ++Θ− +       + Θ      Θ−−Θ− × × ΘΘ ΘΘΚ× ×     Θ+Θ Θ = =+= − −− + ++ Θ= ∑ ∑ jij jijjij jij jijjij jji jimjii i i i j ti zzz S SiRSik S SiRSik k j d d Rk e AAA j // // 0 // // 0 0 2 1 0 21 sgn1 4 exp sgn1 4 exp sin sin sinsin cossin π π θ χ χχ χ χω ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )            −−− −       −      ++− × × −−− −× ×× ×     += =+= − −− + ++ ⊥⊥ ⊥ = ∑ ∑ jij jijjij jij jijjij ii ijji jji jimjii i i i j ti rrr S SiRSik S SiRSik kkk k kj d d Rk e AAA j Θ ΘπΘ Θ ΘπΘ αεχεχ εχΘΘχ θ ΘΘχΘχΘχΚ ΘχΘ Θ χ Θ ω // // 0 // // 0 4 0 22 0 2 // 2 0 2 2 0 22 0 2 1 0 21 sgn3 4 exp sgn3 4 exp cossin sin sinsinsin cossin ( ) ( ) ( ) ×× ×     += =+= ∑ ∑ = θ ΘΘχΘχΘχΚ ΘχΘ Θ χ Θ ω ϕϕϕ sin sinsinsin cossin 0 2 1 0 21 kj d d Rk e AAA jji jimjii i i i j ti j ( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )( ) ( )    −−− −     − ++− × × −−− × − −− + ++ ⊥⊥ jij jijjij jij jijjij ii jji S SiRSik S SiRSik kkk k Θ ΘπΘ Θ ΘπΘ αεχεχ αΘΘχ // // 0 // // 0 4 0 22 0 2 // 2 0 2 2 0 2 sgn34exp sgn34exp cossin (12) where ( ) ( )θ−ΘΘ=± jjiij nS cos , (13) and where the values Θj correspond to the stationary phase points. In fact Θ is the propagation angle of the electromagnetic wave. From (12) one can see that the radiation field is formed as an interference of several waves with differ- ent wave vectors. 5. STATIONARY PHASE POINTS To find out the stationary phase points it is necessary to solve the equation: 0= Θ ± d dS ij (14) that can be presented in the form: Θ+ Θ− ±= tgf tgf tg 2,1 2,1 1 θ ; Θ = d dn n f 2 2,1 2 2,1 2,1 2 1 , (15) a b Fig. 1: a - S+ versus the propagation angle of the electromagnetic wave for H=60G, np=1.4 1011cm-3, fm=50MHz, b – azimuthal angle versus the propa- gation angle of the electromagnetic wave for the same parameters. Stationary phase points are indi- cated where n1,2 are the roots of the dispersion relation (5). The equation (15) cannot be solved analytically and therefore numerical methods are used. The dependence S(Θ) is shown on the Fig.1,a. The extrem p1, p2 and p3 correspond to the stationary phase points. The point p1 has an analogue for electromagnet- ic waves in vacuum, the points p2 and p3 are specific for magnetoactive plasma. They appear due to the sharp increase of the wave number near the angle Θ corre- sponding to the resonance cone. The values of the stationary phase points versus the azimuthal angle for the observation point have been cal- culated numerically. They are shown on the Fig.1,b that plots the angle of observation as a function of the propa- gation angle. Figs.2,a-b illustrate the influence of the magnetic field H and the modulation frequency on the stationary phase points’ values. a b Fig. 2. Number of stationary points versus the mag- netic field H (in Gauss) for np = 1.2 1011cm-3, fm=50 МHz (a) and versus the modulation frequency ω=2π fm (in rad/s) for Н=60G, np = 1.2 1011cm-3 (b) A subsequent calculation shows that the point p3 corresponding to the largest curvature gives the main contribution to the radio-emission. But this point does not exist for all possible values of the model parameters (see Fig.2). In particular it disappears when the usual conditions for whistler approximation pc ωωω < << < . (16) are satisfied. 6. RADIATION PATTERN FOR DIFFERENT PARAMETERS OF THE MODEL The radiation pattern (i.e. the angular dependence of the radial component of the Pointing vector) is shown on the Figs.3,a-b. Fig.3,a shows this dependence for the case when the conditions (16) are satisfied and the point p3 disappears. The shape of the radiation pattern for this case conforms to the results obtained in [4]. Fig.3,b is plotted for parameters corresponding to the experimental conditions [2]. For this case the point p3 gives the main contribution to radioemission. a b Fig.3 Angular dependence of the radial component of the Pointing vector (in arbitrary units): a – H=300G, np =3.5 1012cm-3, fm=50MHz; b – H=60G, np =1.2 1011cm-3, fm=100MHz Fig. 4. Maximum energy flow density (in arbitrary units) versus the plasma density (in cm-3) for H=60G, fm=100MHz The dependence of the maximum energy flow density versus the plasma density for that case is shown on the Fig.4. One can see that the increase of the plasma densi- ty results in the decrease of the transition radiation in- tensity. The Fig.5 [2], shows the variation of the intensity of the transition radiation harmonics as a function of time, that is, as a function of the decreasing plasma density (the appearance of upper harmonics is caused by the an- harmonicity of the modulation beam current). One can see that the intensity of the harmonics radiation (particu- larly for the second and the third harmonics) increases (to some degree). These results qualitatively conform to our calculations (Fig.4). 7. CONCLUSION The transition radiation of a thin modulated electron beam injected from a conducting plane into a plasma along an arbitrary magnetic field normal to that plane is calculated. The radiation field is formed as a result of the interference of three waves with different wave vec- tors. The radiation pattern is mainly determined by one of those waves (it depends on the parameters of the model). Fig. 5. a –Variation of the plasma density np (in cm-3) as a function of the time; amplitudes of the whistler waves (in arbitrary units) as function of time for H=60G, fm=50MHz (b), 2 fm =100MHz (c), 3 fm=150MHz (d) For the case of the whistler approximation, the ob- tained results coincide with our previous calculations [4]. The calculated dependence of the radiation intensity versus the plasma density qualitatively agrees with the experimental data [2]. In order to perform a precise comparison of the cal- culation results and the experimental data it is necessary to take into account the finite radius of the electron beam, to calculate the radiation in the near-field region and to examine the case of the beam injection at some angle to the magnetic field (this case corresponds to the conditions of the experiment [5]). References 1. Artificial particle beams in space plasma studies. Ed. B. Grandal. NY, London: Plenum Press, 1984. 2. M. Starodubtsev, C. Krafft, P. Thevenet, A. Kostrov. Whistler wave emission by a modulat- ed electron beam through transition radiation // Physics of Plasmas. 1999, v. 6, №5, p. 1427-1434. 3. I.O. Anisimov, O.I. Kelnyk. On the transversal length of the transition radiation formation zone in the magnetoactive plasma // Kyiv University Bul- letin. Radio Physics and Electronics. 2001, Issue 3, p. 5-9. (In Ukrainian). 4. 5. I.O. Anisimov, O.I. Kelnyk. On the possibility of observation of whistler transition radiation in active beam-plasma experiments in the ionosphere// Kyiv University Bulletin. 1998, Issue 4, p. 238-242. (In Ukrainian). 6. M. Starodubtsev, C. Krafft. Whistler emission through transition radiation by a modulated electron beam spiraling in a magnetoplasma // J. Plasma Physics. 2000, v. 63, part 3, p. 285-295. 7. І.О.Anisimov1, О.І.Kelnyk1, C.Krafft2, T.V.Nychyporuk1

Whistler wave emission by a modulated electron beam on a metal-plasma boundary

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