Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor

We investigate the transition radiation of an electron crossing an interface between a dielectric and a layered superconductor. The electron’s direction of motion and the orientation of the superconductor’s layers are perpendicular to the interface. The analysis of the radiation patterns reveals the...

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Datum:	2013
Hauptverfasser:	Averkov, Yu.O., Yakovenko, V.M., Yampol’skii, V.A.
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Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2013
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Нерелятивистская электроника
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spelling	irk-123456789-1119052017-01-16T03:03:23Z Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor Averkov, Yu.O. Yakovenko, V.M. Yampol’skii, V.A. Нерелятивистская электроника We investigate the transition radiation of an electron crossing an interface between a dielectric and a layered superconductor. The electron’s direction of motion and the orientation of the superconductor’s layers are perpendicular to the interface. The analysis of the radiation patterns reveals the strong anisotropy of the radiation intensity distribution with respect to the azimuth angle in the interface plane. Досліджено перехідне випромінювання електрона, який перетинає межу поділу діелектрика і шаруватого надпровідника. Напрямок руху електрона і шари надпровідника орієнтовані перпендикулярно цієї межі. Аналіз діаграм спрямованості випромінювання показав сильну анізотропію його інтенсивності щодо азимутального кута у площині межі поділу середовищ. Исследовано переходное излучение электрона, пересекающего границу раздела диэлектрика и слоистого сверхпроводника. Направление движения электрона и слои сверхпроводника ориентированы перпендикулярно границе. Анализ диаграмм направленности излучения показал сильную анизотропию его интенсивности по азимутальному углу в плоскости границы раздела сред. 2013 Article Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor / Yu.O. Averkov, V.M. Yakovenko, V.A. Yampol’skii // Вопросы атомной науки и техники. — 2013. — № 4. — С. 15-20. — Бібліогр.: 21 назв. — англ. 1562-6016 PACS: 41.60. ±m, 74.70. ± b http://dspace.nbuv.gov.ua/handle/123456789/111905 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	Нерелятивистская электроника Нерелятивистская электроника
spellingShingle	Нерелятивистская электроника Нерелятивистская электроника Averkov, Yu.O. Yakovenko, V.M. Yampol’skii, V.A. Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor Вопросы атомной науки и техники
description	We investigate the transition radiation of an electron crossing an interface between a dielectric and a layered superconductor. The electron’s direction of motion and the orientation of the superconductor’s layers are perpendicular to the interface. The analysis of the radiation patterns reveals the strong anisotropy of the radiation intensity distribution with respect to the azimuth angle in the interface plane.
format	Article
author	Averkov, Yu.O. Yakovenko, V.M. Yampol’skii, V.A.
author_facet	Averkov, Yu.O. Yakovenko, V.M. Yampol’skii, V.A.
author_sort	Averkov, Yu.O.
title	Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor
title_short	Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor
title_full	Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor
title_fullStr	Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor
title_full_unstemmed	Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor
title_sort	transition radiation of an electron crossing an interface between a dielectric and a layered superconductor
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2013
topic_facet	Нерелятивистская электроника
url	http://dspace.nbuv.gov.ua/handle/123456789/111905
citation_txt	Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor / Yu.O. Averkov, V.M. Yakovenko, V.A. Yampol’skii // Вопросы атомной науки и техники. — 2013. — № 4. — С. 15-20. — Бібліогр.: 21 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT averkovyuo transitionradiationofanelectroncrossinganinterfacebetweenadielectricandalayeredsuperconductor AT yakovenkovm transitionradiationofanelectroncrossinganinterfacebetweenadielectricandalayeredsuperconductor AT yampolskiiva transitionradiationofanelectroncrossinganinterfacebetweenadielectricandalayeredsuperconductor
first_indexed	2025-07-08T02:52:44Z
last_indexed	2025-07-08T02:52:44Z
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fulltext	ISSN 1562-6016. ВАНТ. 2013. №4(86) 15 NON-RELATIVISTIC ELECTRONICS TRANSITION RADIATION OF AN ELECTRON CROSSING AN INTERFACE BETWEEN A DIELECTRIC AND A LAYERED SUPERCONDUCTOR Yu.O. Averkov1, V.M. Yakovenko1, V.A. Yampol’skii1,2 1A.Ya. Usikov Institute for Radiophysics and Electronics of the National Academy of Science of Ukraine, Kharkov, Ukraine; 2V.N. Karazin Kharkov National University, Kharkov, Ukraine E-mail: yuriyaverkov@gmail.com We investigate the transition radiation of an electron crossing an interface between a dielectric and a layered su- perconductor. The electron’s direction of motion and the orientation of the superconductor’s layers are perpendicu- lar to the interface. The analysis of the radiation patterns reveals the strong anisotropy of the radiation intensity dis- tribution with respect to the azimuth angle in the interface plane. PACS: 41.60. ±m, 74.70. ± b INTRODUCTION As is known, the transition radiation effect, i.e. the effect of radiation of a uniformly moving electron due to its transition from one medium into another, was dis- covered by V.L. Ginzburg and I.M. Frank in 1945 [1]. Since then the numerous papers have been written on the subject (see, e.g., [2 - 6]). Nowadays a good deal of attention to the transition radiation effect is caused by a large number of its applications. For instance, the transi- tion radiation is used in high-energy physics to the de- tection of charged particles [2]. The electron bunch bombardment of the solids [6] and transit of the bunches through diaphragms [8, 9] allow to generate short high power pulses which are widely used for radiolocation. The transition radiation also finds application in investi- gation of two-dimensional electron systems which are the basis for many up-to-date electronic devices [10, 11]. Besides, the radiation of modulated electron beams crossing a boundary of a plasma-like medium is a very effective method for generation of surface electromag- netic waves [12 - 13]. The characteristic properties of the transition radia- tion of both the electrons and electron bunches crossing anisotropically conducting interfaces such as wire shields have been considered in [14 - 16]. Specifically, the possibility of obtaining an elliptical polarization of electromagnetic waves has been shown. V.E. Pafomov in 1959 first proved the possibility of excitation of the electromagnetic waves with negative group velocity with the aid of the Vavilov-Cherenkov radiation in the medium which possesses negative permittivity and negative permeability simultaneously (so-called left- handed medium) [17]. Namely, he first demonstrated that if an electron moves from a vacuum into the me- dium, the maximum of the intensity of the Vavilov- Cherenkov radiation is in a vacuum. A similar effect occurs when an electron crosses an interface between a vacuum and a uniaxial anisotropic conducting medium (e.g., layered superconductor) in the case where the conducting layers are parallel to the interface [18]. In the present paper we theoretically investigate the transition radiation of an electron crossing a layered superconductor in the case where its layers are perpen- dicular to the interface. Specifically, it turned out that unlike the results of work [18], in our system the excita- tion of the electromagnetic waves with negative group velocity in the layered superconductor is principally impossible, i.e. the energy flux density vector in the superconductor does not form an obtuse angle with the direction of the electron motion. In addition, we have found that energy flux distribution of transition radia- tion exhibits a strong anisotropy on azimuth angle in the interface plane. 1. STATEMENT OF THE PROBLEM AND BASIC EQUATIONS Let the structure under study be comprised of the di- electric (the region with index 1) with permittivity 1ε and the layered superconductor (the region with index 2) with dielectric permittivity given by the diagonal permittivity tensor with components ,abyyxx εεε == czz εε = (Fig. 1), where ( )ωνγωγεε abab i 2221= +− , (1) ( )ωνωεε cc i+− 211= . (2) Fig. 1. Geometry of the problem Here, we introduce the dimensionless parameters ,Jωωω = )(4= 2γεωπσν Jabab and )(4= Jcc εωπσν , 1>>= abc λλγ is the current-anisotropy parameter, cλ and abλ are the magnetic-field penetration depths along and across the layers, respectively. The relaxation fre- quencies abν and cν are proportional to the averaged quasi-particle conductivities abσ (along the layers) and cσ (across the layers), respectively, ( )1/28= επω hcJ eDJ ISSN 1562-6016. ВАНТ. 2013. №4(86) 16 is the Josephson plasma frequency, e is the electron charge, D is the period of the layered structure, cJ is the critical Josephson current density, ε is the inter- layer dielectric constant. All media are supposed to be nonmagnetic. We define our coordinate system so that the dielectric occupies the half-space ,0<y the layered superconductor occupies the half-space ,0>y and the z -axis coincides with the crystallographic c r -axis of superconductor. An electron moves uniformly in the dielectric along the y -axis at a velocity cv << (where c is the speed of light in a vacuum) and crosses the interface. The electron-charge density ( )trn , r is deter- mined by the formula ( ) ( ) ( ) ( ),, zytvxetrn δδδ −= r (3) where ( )xδ is the Dirac delta function. The electromag- netic fields of the electron are expressed in terms of Fourier integrals ( ) ( ) ( )[ ] ,exp,, ωωω dkdtrkikEtrE ee rrrrrrr ll −= ∫ (4) where 2 ,1=l is the number of the medium, ( ), ,x y zk k k k= r . The Fourier components for the elec- tric and magnetic fields of the electron in the dielectric are ( ) ( ) ( ), 2 , 1 22 2 1 1 21 vk ck kcviekE y e − − − = ωδ εω ωε επ ω rrrr (5) ( ) ( )1 1 1, ,e eH k v E k c ε ω ω= × r rr rr . (6) The corresponding field components in the super- conductor are ( ) ( ) ( ) ( ), ΔΔ2 Λ , eo2 1 2 vk kievk kE y yxe x −−= ωδ ωπ ω r (7) ( ) ( ) ( ) ( ), ΔΔ2 Λ, eo2 2 2 vkievkE y e y −−= ωδ ωπ ω r (8) ( ) ( ) ( ), Δ2 , e22 vk kievk kE y yze z −−= ωδ ωπ ω r (9) ( ) ( ),,1, 22 ωω kDv c kH ee rrrrr ×= (10) where ( ) ( ) ( ),,, 22 ωωεω kEkD e kjk e j rr = ,Λ 2 2 2 1 cc k εω −= (11) ( ),Λ 2 2 2 2 2 2 2 2 2 2 abczcaby c k c k c k εεωεωεω −−⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −= (12) ( ) ,Δ 2 2 2 ab o c k εω −= (13) ( ) ( ) .2 2 222Δ cababyxcz e c kkk εεωεε −++= (14) The radiation fields we express in terms of the fol- lowing Fourier integrals ( ) ( ) ( )[ ] ,exp,, ωκωρκωκ ddtykiEtrE y rrrrrrr lll −+= ∫ (15) where ( ),x zk kκ = r is the wave vector in the xz plane, ( )yx,=ρ r is the radius vector in the xz plane. We assume that the radiation field in the dielectric is the superposition of the ordinary and extraordinary electro- magnetic waves with components ( ) ( )( ),0,, 11 o y o x EE ( ) ( ) ( )( )o z o y o x HHH 111 ,, and ( ) ( ) ( )( ),,, 111 e z e y e x EEE ( ) ( )( )0,, 11 e y e x HH , respectively. From the Maxwell equations we attain the following relationships between the Fourier components of electric and magnetic fields of o and e types: ( ) ( ),,=, )( 1 1 )( 1 ωκωκ rr o x y xo y E k k E − (16) ( ) ( ),,=, )( 1 1 )( 1 ωκ ω ωκ rr o x y zxo x E k kck H (17) ( ) ( ),,=, )( 1 )( 1 ωκ ω ωκ rr o x zo y EckH (18) ( ) ( ),,=, )( 1 1 2 1 2 )( 1 ωκ ω ωκ rr o x y yxo z E k kkcH + − (19) ( ) ( ),,=, )( 1 1)( 1 ωκωκ rr e x x ye y E k k E (20) ( ) ( ),,=, )( 1 2 1 2 )( 1 ωκωκ rr e x zx yxe z E kk kk E + − (21) ( ) ( ),,=, )( 11 1)( 1 ωκεωωκ rr e x zx ye x E kk k c H − (22) ( ) ( ),,=, )( 11 )( 1 ωκεωωκ rr e x z e y E ck H (23) where .= 22 1 2 1 κεω −− ck y In the layered superconductor the radiation field we represent as the superposition of electromagnetic waves of o and e types with following Fourier components: ( ) ( ),,=, )( 2)( 2 )( 2 ωκωκ rr o xo y xo y E k k E − (24) ( ) ( ),,=, )( 2)( 2 )( 2 ωκ ω ωκ rr o xo y zxo x E k kck H (25) ( ) ( ),,=, )( 2 )( 2 ωκ ω ωκ rr o x zo y EckH (26) ( ) ( ),,=, )( 2)( 2 2)( 2 2 )( 2 ωκ ω ωκ rr o xo y o yxo z E k kkcH + − (27) ( ) ( ),,=, )( 2 )( 2)( 2 ωκωκ rr e x x e ye y E k k E (28) ( ) ( ),,=, )( 2 2)( 2 2 )( 2 ωκ ε ε ωκ rr e x zx e yx c abe z E kk kk E + − (29) ( ) ( ),,=, )( 2 )( 2)( 2 ωκεωωκ rr e xab zx e ye x E kk k c H − (30) ( ) ( ),,=, )( 2 )( 2 ωκεωωκ rr e xab z e y E ck H (31) where ,= 222)( 2 κεω −ck ab o y (32) .= 2222)( 2 abzcxc e y kkck εεεω −− (33) ISSN 1562-6016. ВАНТ. 2013. №4(86) 17 From the continuity conditions for the tangential components of the electric and magnetic fields at the interface we derive the following expressions for the total radiation fields in the dielectric: ( ) , Δ , 1321 1 OSW x QQ c iE ααωωκ + −= r (34) ( ) ( ) ( ) 1 1 3 2 1 1 4 2 , Δ , y y OSW x z x z iE ck k k Q k k Q ωκ ω α α α α = × × + + −⎡ ⎤⎣ ⎦ r (35) ( ) , Δ , 1224 1 OSW z QQ c iE ααωωκ − = r (36) ( ) ( ) ( ) ( ) 1 1 1 1 2 2 12 2 1 4 1 1 1 1 , , , x x y z y x z z y y cH E k i k k k k k k Q k κ ω κ ω α ω α α α = × + ⎡ ⎤× − + −⎣ ⎦ r r (37) ( ) ( ) ,,, 1 1 1 1 4 1 Q ik EkkcH x xxzy α ωκ α α ω ωκ +⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += rr (38) ( ) ( ) ( ) 1 1 1 1 2 2 1 1 4 1 1 1 , , , z x y x z x y x z y cH E k ik kk k k k Q k κ ω κ ω α ω α α α = − × ⎡ ⎤× + − +⎣ ⎦ r r (39) where ( ) ( ) , 21 12 1 o yy y o y zx kk kk kk − −=α ( ),12 εεα −−= abzxkk (40) ( ) ( )( ),2111 2 3 e yyyababz kkkk −+−= εεεα (41) ( ) ,2 12 2 1 1 2 2 21 24 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −= −− zyzab o y k c kk c k εωεωα (42) ,Δ 4321 αααα +=OSW (43) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )} 1 o2 1 2 1 2 2 o 2 2 1 1 2 2 2 o 2 2 1 12 2 o2 1 1 2 Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ , x o e y e c ab z o e y ab c e z eckQ v k k k c c v k k c c k π ωε ω ωε ε ε ω ωε ε ε ε = − × ⎧⎡ ⎤⎛ ⎞ ⎛ ⎞⎪× − − − −⎨⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎪⎣ ⎦⎩ ⎡ ⎤⎛ ⎞ − − − +⎢ ⎥⎜ ⎟ ⎝ ⎠⎣ ⎦ + − (44) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) 1 2 o2 2 1 2 o 2 1 2 2 o2 2 1 2 o 2 1 1 1 2 Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ , z y e e c ab z e x ab c e eab y eck k Q v v k c vk k c k π ω ωε ε ωε ε ωε ε ε = × ⎧ ⎛ ⎞⎪× − − −⎨ ⎜ ⎟ ⎪ ⎝ ⎠⎩ ⎡ ⎤⎛ ⎞ − − − −⎢ ⎥⎜ ⎟ ⎝ ⎠⎣ ⎦ ⎫ − − ⎬ ⎭ (45) ,Δ 12 2 2 1 εω c k −= .2 2 222 v kkk zx ω ++= (46) In order to derive the spatial and time dependencies of the total electromagnetic fields in the explicit forms, we need to integrate expressions (34) - (39) with respect to xk and zk by means of the stationary phase method for two-dimensional integrals [19]. According to this method, we obtain following stationary points: ,sinsin10 ϕϑεω c kx = (47) ,cossin10 ϕϑεω c kz = (48) where ϑ is the angle relative to the y -axis, ϕ is the azimuth angle in the interface plane (see Fig. 1). To calculate the energy losses by the electron to ra- diation in the dielectric we find the energy flux of the total electromagnetic wave in the dielectric across the remote hemisphere using the following time-integrated Poynting vector: ( ) ( )[ ].,,,Re 4 ∫ ∞ ∞− = trHtrEdtcS rrrrr π . (49) Eventually, we will get the following expression for the spectral density ( )ϕϑω ,,Π of the radiation energy (in units of ce2 0Π = ) per unit solid angle ϕϑϑ ddsinΩ =d averaged over all transit time of the electron: ( ) ( ) ( ){ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) } 2 2 2 1 1 1 * 1 1 * 1 1 * 1 1 * 1 1 * 1 1 Π , , 2 cos Re , , , , , , , , sin sin , , , , , , , , sin cos , , , , , , , , cos , y z z y x y y x z x x z E H E H E H E H E H E H ω ϑ φ π ε ω ϑ ω ϑ φ ω ϑ φ ω ϑ φ ω ϑ φ ϑ φ ω ϑ φ ω ϑ φ ω ϑ φ ω ϑ φ ϑ φ ω ϑ φ ω ϑ φ ω ϑ φ ω ϑ φ ϑ = × ⎡× −⎣ ⎤− +⎦ ⎡+ −⎣ ⎤− −⎦ ⎡− −⎣ ⎤− ⎦ (50) where ( ),,,1 ϕϑωjE ( )ϕϑω ,,1 jH are dimensionless Fourier components for the radiation fields (in units of Je ω ) given by equations (34) - (39) and expressed in terms of dimensionless frequency ω with due account of equations (47), (48). 2. ANALYSIS OF THE SPECTRUM Let us analyze the dependences of the spectral den- sity ( )ϕϑω ,,Π on the tilt angle ϑ and the azimuth an- gle .ϕ Hereafter, we will make use of the following parameters of the adjacent media: ,11 =ε ,16=ε ,200=γ ,0=abν .10 5−=cν (51) Fig. 2 shows the ϑ dependence of Π for a number of ϕ values at 0.7,=ω 3.0== cvβ (the electron enters into the superconductor). In Fig. 2 curve 1 corre- sponds to ,90o=ϕ curve 2 is for ,60o=ϕ curve 3 is for ,30o=ϕ and curve 4 is for .0o=ϕ As seen from ISSN 1562-6016. ВАНТ. 2013. №4(86) 18 Fig. 2, at o90=ϕ the maximum of the spectral energy density is located at .90o≈ϑ Physically this implies that radiation energy flux at o90≈ϑ is directed at a grazing angle to the interface. 0 15 30 45 60 75 90 0 1 2 3 4 ϑ Π , 1 0-3 (i n un its o f e 2 /c ) (degree) 1 2 3 4 Fig. 2. The dependences of ( )ϑΠ for a number of ϕ values at 0.7,=ω 3.0=β It easily seen from Fig. 2 that the maximum of the spectral energy density decreases and shifts towards lower values of the angle ϑ with decreasing the angle .ϕ Note that at angles o0≈ϕ and o90≈ϕ the radiation field is TM polarized. At o0≈ϕ the radiation field has components ( )zy EE 11 ,,0 , ( )0,0,1xH , while at o90≈ϕ it has components ( )0,, 11 yx EE , ( )zH1,0,0 . At angles oo 900 <<ϕ the radiation field, as mentioned earlier, is the superposition of o and e polarized waves and pos- sesses all components of the electric and magnetic fields. The dependences of ( )ϑΠ for a number of frequen- cies ω at ,0o=ϕ 3.0=β are shown in Fig. 3. 0 15 30 45 60 75 90 0 1 2 3 4 Π , 1 0-3 (i n un its o f e 2 /c ) ϑ (degree) 1 2 3 4 5 6 7 Fig. 3. The dependences of ( )ϑΠ for a number of ω values at ,0o=ϕ 3.0=β In Fig. 3 curve 1 corresponds to ,0→ω curve 2 is for ,1.0=ω curve 3 is for ,5.0=ω curve 4 is for ,8.0=ω curve 5 is for ,1=ω curve 6 is for ,2=ω and curve 7 is for .10=ω From Fig. 3 it follows that at o0=ϕ and the low frequencies ( 0→ω ) the maximum of the spectral energy density is located at .90o≈ϑ As the frequency further increases, the maximum decreases and tends to a certain minimum value at 1=ω (curve 5). At 1>ω the value of the maximum grows with the in- crease of the frequency and tends to the limit at 1>>ω (curve 7). Fig. 4 presents the analogous dependences at ,90o=ϕ .3.0=β In Fig. 4 curve 1 corresponds to ,0→ω curve 2 is for ,2=ω and curve 3 is for .10=ω 0 15 30 45 60 75 90 0 1 2 3 4 5 80 85 90 1 2 3 4.05 4.50 Π , 1 0-3 (i n un its o f e 2 /c ) ϑ (degree) 1 2 3 Fig. 4. The dependences of ( )ϑΠ for a number of ω values at ,09 o=ϕ 3.0=β The dependences demonstrate that for all frequen- cies, which have a physical meaning, the maximums of the spectral energy density are directed at very large angles close to 90° with respect to the normal to the interface. It is worthwhile to consider the Vavilov-Cherenkov radiation generated by the electron in the layered super- conductor. As suggested earlier, the electron enters into the superconductor. In this case, the excited bulk elec- tromagnetic wave in the superconductor has extraordi- nary polarization and we can write the following exact formula for : 2)( 2 e yk ,ImRe 2)( 2 2)( 2 2)( 2 e y e y e y kikk += (52) where ,Re 2 22 2 2 2 22)( 2 ab сab zx ab ab zc e y kkk c k ε εε ε εωε ′′′′ −−⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ −′= (53) ,Im 2 2 2 2 2 22)( 2 ab cab z ab ab zc e y kk c k ε εε ε εωε ′′′ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ −′′= (54) cabcab ,, Reεε =′ , сabсab ,, Imεε =′′ . (55) From equations (52) - (54) it becomes evident that the excited extraordinary electromagnetic wave is a ho- mogeneous one over the frequency range where ,0<′abε 0>′cε and the condition ,0Re 2)( 2 >e yk in principle, can be satisfied. It is seen that over the same frequency range the condition 0Im 2)( 2 >e yk is satisfied as well. Using the equations (52) - (54), we can easily demonstrate that over the frequency range where ,0<′abε 0>′cε the real and imaginary parts of )( 2 e yk are simultaneously positive. Physically this implies that the Vavilov-Cherenkov radiation generated by the elec- tron entering the layered superconductor cannot be re- versed, i.e. the energy flux density vector cannot form an obtuse angle with the direction of the electron mo- tion. In other words, the superconductor does not be- have like a left-handed medium. On the other hand, in case where the layers of the superconductor are parallel to the interface the reversed Vavilov-Cherenkov radia- ISSN 1562-6016. ВАНТ. 2013. №4(86) 19 tion is possible [18]. Hence, the orientation of the su- perconductor layers with respect to the interface plane plays a critical role for the formation of the reversed Vavilov-Cherenkov radiation. Let us consider the case where the electron escapes from the layered superconductor. In this case, the above-described characteristics of the transition radia- tion hold true. At the same time, the Vavilov-Cherenkov radiation can escape from the superconductor and its dependences of ( )ϑΠ are shown in Fig. 5. 0 15 30 45 60 75 90 0.0 0.2 0.4 0.6 Π , 1 0-3 (i n un its o f e 2 /c ) ϑ (degree) 1 2 3 4 Fig.5. The dependences of ( )ϑΠ for a number of ϕ values at 1.81,≈ω 3.0−=β In Fig. 5 curve 1 is for ,0o=ϕ curve 2 is for ,1.0 o=ϕ curve 3 is for ,2.0 o=ϕ and curve 4 is for .3.0 o=ϕ It is seen that the Vavilov-Cherenkov radia- tion escapes form the semiconductor at very small an- gles .ϕ So, we can say that the almost all Vavilov- Cherenkov radiation is mainly concentrated in the yz - plane. In order for the Vavilov-Cherenkov radiation to arise, the condition vk e y ω=)( 2 must be satisfied. This condition allows one to derive the following representa- tive values of the azimuth angles ϕΔ , the radiation fre- quency VCHω , and the electron velocity :β ,1tan2 <<< ab c ε ε ϕ (56) ( ) ,1 2112 VCH −− ⎥⎦ ⎤ ⎢⎣ ⎡ −≈ εβω (57) where .12 εβ > The numerical estimates of the condi- tion vk e y ω=)( 2 for the abovementioned parameters of the system show that the Vavilov-Cherenkov radiation exists over the narrow range of frequencies -510Δ ∝ω in the vicinity of 81.1VCH ≈ω at .3.0−=β These cir- cumstances are regarded to be new as compared to the known results obtained from the analysis of the transi- tion radiation in the case where an electron crosses an interface of two isotropic media [2]. It is worthwhile to emphasis that the electron excites not only the above-considered bulk electromagnetic waves, but also the so-called oblique surface electro- magnetic waves (OSWs) in the interface plane. The properties of the OSWs have recently been investigated in [20, 21]. Indeed, the value OSWΔ in the denomina- tors of equations (34) - (36) corresponds to the disper- sion relation of the OSWs 0ΔOSW = . In order to derive the spatial and time dependences of the OSW fields in the explicit form, we need to integrate the expressions (34) - (39) with respect to ,xk ,zk and ω taking into account the poles of the integrands in equations (34) - (39). We will consider the case of the OSWs excitation by means of the transition radiation effect in a subse- quent paper. CONCLUSIONS The problem of the transition radiation of an elec- tron moving along the normal to the interface between an isotropic dielectric and a layered superconductor has been theoretically examined. It has been assumed that the layers of the superconductor are perpendicular to the interface. At azimuth angles oo 900 << ϕ the radiation field of excited bulk electromagnetic waves is shown to be a superposition of the ordinary and extraordinary electromagnetic waves. At angles o0≈ϕ and o90≈ϕ the radiation field becomes TM polarized. At o0=ϕ the position of the maximum of the spectral energy den- sity varies with frequency over a wide range of the tilt angle .ϑ At o90=ϕ the position of the maximum of the spectral energy density is close to o90≈ϑ for all frequencies which have a physical meaning. It has been established that for the case under study the Vavilov- Cherenkov radiation generated by the electron entering the layered superconductor cannot be reversed. In the case where the electron escapes from the layered super- conductor the Vavilov-Cherenkov radiation is mainly concentrated in the plane that comprises the normal to the interface and the crystallographic c r -axis. Besides, the so-called oblique surface electromagnetic waves can also be excited along with the bulk waves. REFERENCES 1. V.L. Ginzburg, I.M. Frank. 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Conversion of terahertz wave po- larization at the boundary of a layered superconduc- tor due to the resonance excitation of oblique surface waves // Physical review letters. 2012, v. 109, № 2, p. 027005(5). 21. Yu.O. Averkov, V.M. Yakovenko, V.A. Yam- pol’skii, F. Nori. Oblique surface Josephson plasma waves in layered superconductors // Physical Review B. 2013, v. 87, № 5, p. 054505(8). Article received 06.03.2013. ПЕРЕХОДНОЕ ИЗЛУЧЕНИЕ ЭЛЕКТРОНА, ПЕРЕСЕКАЮЩЕГО ГРАНИЦУ РАЗДЕЛА ДИЭЛЕКТРИКА И СЛОИСТОГО СВЕРХПРОВОДНИКА Ю.О. Аверков, В.М. Яковенко, В.А. Ямпольский Исследовано переходное излучение электрона, пересекающего границу раздела диэлектрика и слоистого сверхпроводника. Направление движения электрона и слои сверхпроводника ориентированы перпендику- лярно границе. Анализ диаграмм направленности излучения показал сильную анизотропию его интенсивно- сти по азимутальному углу в плоскости границы раздела сред. ПЕРЕХІДНЕ ВИПРОМІНЮВАННЯ ЕЛЕКТРОНА, ЯКИЙ ПЕРЕТИНАЄ МЕЖУ ПОДІЛУ ДІЕЛЕКТРИКА І ШАРУВАТОГО НАДПРОВІДНИКА Ю.О. Аверков, В.М. Яковенко, В.О. Ямпольський Досліджено перехідне випромінювання електрона, який перетинає межу поділу діелектрика і шаруватого надпровідника. Напрямок руху електрона і шари надпровідника орієнтовані перпендикулярно цієї межі. Аналіз діаграм спрямованості випромінювання показав сильну анізотропію його інтенсивності щодо азиму- тального кута у площині межі поділу середовищ.

Transition radiation of an electron crossing an interface between a dielectric and a layered superconductor

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