Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion

Excitation of the Cherenkov electromagnetic radiation by a relativistic electron bunch in a dielectric waveguide is considered taking into account the frequency dispersion of the dielectric permittivity. Electric polarization in an isotropic dielectric medium and, accordingly, polarization charges a...

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Дата:	2019
Автори:	Balakirev, V.A., Onishchenko, I.N.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2019
Назва видання:	Вопросы атомной науки и техники
Теми:	Plasma electronics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/194619
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Цитувати:	Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion / V.A. Balakirev, I.N. Onishchenko // Problems of atomic science and technology. — 2019. — № 1. — С. 91-94. — Бібліогр.: 7 назв. — англ.

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spelling	irk-123456789-1946192023-11-27T21:12:26Z Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion Balakirev, V.A. Onishchenko, I.N. Plasma electronics Excitation of the Cherenkov electromagnetic radiation by a relativistic electron bunch in a dielectric waveguide is considered taking into account the frequency dispersion of the dielectric permittivity. Electric polarization in an isotropic dielectric medium and, accordingly, polarization charges and currents induced by the Coulomb electric field of a relativistic electron bunch are determined. The spatial-temporal structure of the excited wakefield in a dielectric waveguide is obtained and investigated. It is shown that the excited field consists of a potential polarization electric field and a set of eigen electromagnetic waves of the dielectric waveguide. Розглядається збудження черенківського електромагнітного випромінювання релятивістським електронним згустком у діелектричному хвилеводі з урахуванням частотної дисперсії діелектричної проникності. Визначено електричну поляризацію в ізотропному діелектричному середовищі і, відповідно, поляризаційні заряди та струми, індуковані кулонівським електричним полем релятивістського електронного пучка. Отримано та досліджено просторово-часову структуру збудженого кільватерного поля в діелектричному хвилеводі. Показано, що збуджене поле складається з потенційного поляризаційного електричного поля та набору власних електромагнітних хвиль діелектричного хвилеводу. Рассматривается возбуждение черенковского электромагнитного излучения релятивистским электронным пучком в диэлектрическом волноводе с учетом частотной дисперсии диэлектрической проницаемости. Определены электрическая поляризация в изотропной диэлектрической среде и, соответственно, поляризационные заряды и токи, индуцированные кулоновским электрическим полем релятивистского электронного пучка. Получена и исследована пространственно-временная структура возбужденного волнового поля в диэлектрическом волноводе. Показано, что возбужденное поле состоит из электрического поля потенциальной поляризации и множества собственных электромагнитных волн диэлектрического волновода. 2019 Article Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion / V.A. Balakirev, I.N. Onishchenko // Problems of atomic science and technology. — 2019. — № 1. — С. 91-94. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq http://dspace.nbuv.gov.ua/handle/123456789/194619 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	Plasma electronics Plasma electronics
spellingShingle	Plasma electronics Plasma electronics Balakirev, V.A. Onishchenko, I.N. Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion Вопросы атомной науки и техники
description	Excitation of the Cherenkov electromagnetic radiation by a relativistic electron bunch in a dielectric waveguide is considered taking into account the frequency dispersion of the dielectric permittivity. Electric polarization in an isotropic dielectric medium and, accordingly, polarization charges and currents induced by the Coulomb electric field of a relativistic electron bunch are determined. The spatial-temporal structure of the excited wakefield in a dielectric waveguide is obtained and investigated. It is shown that the excited field consists of a potential polarization electric field and a set of eigen electromagnetic waves of the dielectric waveguide.
format	Article
author	Balakirev, V.A. Onishchenko, I.N.
author_facet	Balakirev, V.A. Onishchenko, I.N.
author_sort	Balakirev, V.A.
title	Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion
title_short	Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion
title_full	Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion
title_fullStr	Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion
title_full_unstemmed	Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion
title_sort	cherenkov radiation of the electron bunch in dielectric media with frequency dispersion
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2019
topic_facet	Plasma electronics
url	http://dspace.nbuv.gov.ua/handle/123456789/194619
citation_txt	Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion / V.A. Balakirev, I.N. Onishchenko // Problems of atomic science and technology. — 2019. — № 1. — С. 91-94. — Бібліогр.: 7 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT balakirevva cherenkovradiationoftheelectronbunchindielectricmediawithfrequencydispersion AT onishchenkoin cherenkovradiationoftheelectronbunchindielectricmediawithfrequencydispersion
first_indexed	2025-07-16T22:00:34Z
last_indexed	2025-07-16T22:00:34Z
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fulltext	PLASMA ELECTRONICS ISSN 1562-6016. ВАНТ. 2019. №1(119) PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2019, № 1. Series: Plasma Physics (25), p. 91-94. 91 CHERENKOV RADIATION OF THE ELECTRON BUNCH IN DIELECTRIC MEDIA WITH FREQUENCY DISPERSION V.A. Balakirev, I.N. Onishchenko National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine E-mail: onish@kipt.kharkov.ua Excitation of the Cherenkov electromagnetic radiation by a relativistic electron bunch in a dielectric waveguide is considered taking into account the frequency dispersion of the dielectric permittivity. Electric polarization in an isotropic dielectric medium and, accordingly, polarization charges and currents induced by the Coulomb electric field of a relativistic electron bunch are determined. The spatial-temporal structure of the excited wakefield in a dielectric waveguide is obtained and investigated. It is shown that the excited field consists of a potential polarization electric field and a set of eigen electromagnetic waves of the dielectric waveguide. PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq INTRODUCTION Electric charge (electron bunch), moving in a dielectric medium with a super light speed, radiates wake electromagnetic waves – Cherenkov radiation [1, 2]. Since the wakefields in the dielectric are slow ( ,ph phv c v – phase velocity of the wave), they can be used for acceleration charged particles. The method of acceleration of charged particles by the wakefields excited in a dielectric medium is very prospective and presently much give one's attention to it. In theoretical works devoted to this topic [3-6], as a rule, the frequency dispersion of the dielectric permittivity of the medium is not taken into account. In the framework of this approximation, the permittivity is independent of frequency and is a constant. Meanwhile, accounting of frequency dispersion leads to a number of qualitative features in the picture of excitation of wakefields in dielectric media. First of all, the permittivity can be equal to zero ( ) 0g   . The frequency g  corresponds to the longitudinal (potential) polarization oscillations of the dielectric medium. In addition, the condition for the appearance of Cherenkov radiation, its frequency spectrum and, in general, the wakefield field picture in a dielectric medium change. In this paper the process of excitation of Cerenkov radiation of electromagnetic waves by a relativistic electron bunch in a condensed medium, taking into account the frequency dispersion of the dielectric constant is investigated. For the case of a dielectric waveguide, expressions for the wakefield fields are obtained and studied. FORMULATION OF THE PROBLEM. BASIC EQUATIONS We consider a dielectric waveguide made in the form of a homogeneous dielectric cylinder whose lateral surface is covered by an ideally conducting film. A monoenergetic relativistic bunch of charged particles propagates uniformly and rectilinearly along the waveguide. The initial system of equations contains the Maxwell’s equations 1 H rotE c t     , ext 1 E 4 rotH j c t c      , extdivD 4  , divH 0 , (1) ,ext extj are densities of external charges and currents, in our case of an electron bunch, 4D E P  is electric induction, P is vector of electrical polarization of the dielectric medium. The Maxwell equation must be supplemented by a material equation for the electrical polarization of the P . In a condensed medium, each atom is in a local electric field locE , which can be very different from the macroscopic field E , which is described by Maxwell's equations (1). The local electric field locE includes both the external field E and the total electric field of the induced dipoles surrounding the given atom. In a crystalline medium with a cubic crystal lattice, the local electric field is described by the Lorentz formula [7] 4 3 locE E P    . (2) Taking into account the Lorentz formula (2), the material equation (3) has the form 2 2 2 2 ,d P Ze N P E mt      (3) where 2 2 2 2 2 0 4 / 3 ,d p p Z e N m        is plasma frequency, d is the frequency of the dipole oscillations of the dielectric condensed medium, which is lower than the frequency of the individual atomic dipole oscillator 0 . We shall solve the problem of exciting the wakefield by an axisymmetric electron bunch in a dielectric waveguide as follows. Let us first determine the field (Green's function) of a moving charge, which has the form of an infinitely thin ring with a charge density 0 0 0 0 0 ( )1 ( ) 2 r r z d dQ t t v r v          , (4) mailto:onish@kipt.kharkov.ua 92 ISSN 1562-6016. ВАНТ. 2019. №1(119) where 0r is ring radius, 0t is the time of flight of an elementary ring bunch into a waveguide, , 0v is bunch velocity, 0 0( , )dQ r t is the charge of particles in the ring connected with the current density of the bunch at the entrance to the dielectric waveguide ( 0)z  0 0 0( , )j t r by the relation 0 0 0 0 0 0( , )2 .dQ j t r r dr dt The current density of an elementary ring charge is related to the charges (4) by the relation 0 ,zdj v d e ze is unit vector in the longitudinal direction. We expand the values in the Fourier integrals in the Maxwell’s equations (1) and also in the material equation (3)    , , , ,i i G GE H E H e d P P e d               (5) 0 0/t z v t    , ,G GE H is electromagnetic field (Green's function) of elementary charge (4) and current. From the material equation (3) we find the expression for the Fourier component of the polarization ( ) 1 , 4 P E       (6) where 2 2 2 ( ) 1 p d         . (7) The system of Maxwell's equations (1), taking into account the relations (7), is conveniently reduced to equations for the longitudinal Fourier component of the electric field 2 2 0 0 ( )1 ( ) z z dE r rkd i r k E dQ r dr dr r            , (8) 2 2 2 0 0 0( ) , / , /k k k k v k c        . On the ideally conducting side surface of the dielectric waveguide r b , the longitudinal component of the electric field is equal zero ( ) 0zE r b   . (9) On the surface 0r r along which the ring electron bunch propagates, the longitudinal Fourier component of the electric field satisfies the boundary conditions 0 0( 0) ( 0),z zE r r E r r      2 0 0 0 ( ) . ( ) z z r r r r dE dE ki dQ dr dr r             (10) The solution of equation (8) taking into account the boundary conditions (9)-(10) has the form 2 0 0 0 ( ) ( , ) ( , ) , 2 ( ) ( ) i Gz k G k r k ri E r dQ d e J k b                (11) where 0 0( , )G k r k r  = 0 0 0 0 0 0 0 0 ( ) ( , ), , ( ) ( , ), , J k r k r k b r r J k r k r k b r r            0 0 0 0 0( , ) ( ) ( ) ( ) ( )k r k b J k r N k b J k b N k r        . The integrand in (11) has simple poles 0g i     2 2 g p d    , located in the lower half-plane of the complex variable  and being zeros of the dielectric constant ( ) 0   . As noted above, these poles correspond to the excitation of longitudinal polarization oscillations of the dielectric medium. Calculating the residues at these poles, we find the expression for the polarization wakefield. excited by a ring electron bunch 2 ( ) 0 02 0 ( , ) 3 ( )cos ( , ), ppol Gz g p pE r dQ k r k r v        0 0 0 0 0 0 0 0 0 00 ( ) ( , ), ,1 ( , ) ( ) ( , ), ,( ) p p p p p p p pp I k r F k r k b r r k r k r I k r F k r k b b r rI k b       0 0 0 0 0( , ) ( ) ( ) ( ) ( )p p p p p pF k r k b K k r I k b K k b I k r  , where 0/ ,p gk v ( )  is unit function. The integrand in (11) has also simple poles, which are determined from the equations 22 2 2 2 2 0 ( ) ( ) 0.n nD c v b         (12) These roots determine the frequency spectrum of the Cerenkov electromagnetic waves excited by an electron ring bunch in a dielectric waveguide. In the spectral equation (12), n are the roots of the Bessel function 0 ( )J x . The spectral equation (12), taking into account the explicit expression for the dielectric constant (7), can be given a form more convenient for analysis 2 2 2 2 2 0 2 2 0 0 1 1 ( ) ( )( ) 0,n chn stn d D v              where 2 2 1 0 0 0 0(1 ) , / , , ,chn d chn stn d stnv c x x           2 2 2 2 2 0 0 1 1 ( ) ( ) , 2 4 chn n n nx b y b y y      (13) 2 2 2 2 2 0 0 1 1 ( ) ( ) , 2 4 stn n n nx b y b y y     (14) 2 2 0 0( 1)g gb     , ,n n g g d c y b      2 2 0 1 /p d    is static permittivity of the dielectric medium 0 ( 0)    . The poles 0chn i    are also located in the lower half-plane of the complex variable  nearly the real axis. The frequencies chn  are the frequencies of the electromagnetic waves of the dielectric waveguide, which are in the Cerenkov synchronism 2 0 ( ) 1chn    with the electron bunch. Since these frequencies are always real, the Cherenkov radiation of electromagnetic waves excited by an electron bunch in a dielectric waveguide takes place for all values of the electron bunch velocity and parameters of the dielectric waveguide (in our case, the values of the static dielectric constant 0 and the radius of the waveguide b ). In addition to the real Cherenkov poles, there are also a pair of complex conjugate poles stni   in the integrand (11) located on the imaginary axis. These poles correspond to a quasi-static electromagnetic field localized in the region of the electron bunch. Calculating the residues at all these poles of the integrand (11), we find the expression for the Cherenkov electromagnetic field excited by a thin ring electron bunch ISSN 1562-6016. ВАНТ. 2019. №1(119) 93 ( ) 02 1 2 ( , ) ( , ) stnem Gz n n stn n dQ E r r r sign e b             2 ( )cosstn chn     , (15) where 2 2 2 2 n d n d stn y       , 2 2 2 ,d chn chn chn chn        2 2 2 ,d chn stn chn stn        2 2 0 2 2 ,nch d chn nch d          2 2 0 2 2 ,stn d chn stn d          0 0 0 2 1 ( / ) ( / ) ( , ) . ( ) n n n n J r b J r b r r J      The Cerenkov electromagnetic field includes a bipolar field pulse which propagates in the waveguide with the bunch velocity and localizes in the vicinity of the ring electron bunch, as well as a set of eigen monochromatic waves of the dielectric waveguide propagating behind the bunch. Let us investigate the expression for the electromagnetic field (15) in the quasistatic approximation 2 2 d  . In this approximation, the permittivity is independent of frequency 0ch  . Consider the most interesting case 0 0b  or 0 0/v c  , when the Cerenkov radiation condition is satisfied in the static approximation. In this case, from expressions (13), (14) we find 2 0 0 2 0 0 , 1 1 . n gd chn g stn d d g g st v b b b                   Accordingly, for the expression of the Cerenkov electric field, we obtain ( ) 02 10 4 ( , ) ( , ) ( )cosem Cz n chn n dQ E r r r b             2 2 2 2 0 0 0 2 2 2 1 ( 1) 2 chn chn d n b sign e c                . The amplitude and width of the quasi-static electromagnetic pulse in the considered limiting case are small. In the case 0 0b  or 0 0/v c  , when the Cherenkov radiation condition is not satisfied in the quasistatic approximation we have 2 0 01 ,chn d d g gb         2 0 0 . 1 n gd n stn st g vy b b           Then for the longitudinal component of the electric field we obtain the following expression ( ) 02 10 4 1 ( , ) ( , ) 2 stnem Cz n n dQ E r r r sign e b               2 2 2 2 0 0 0 2 2 2 2 ( 1) ( )coschn stn ch d n d b c                  . The electromagnetic field contains a set of bipolar pulses with a characteristic width 1/ stn and a monochromatic high-frequency wakefield with a frequency ch and a small amplitude. Consider an electron bunch with a current density    0 0 0 0 0 0( , ) / /b bj r t j R r r T t t , (16) where the function  0 / bR r r describes the dependence of the bunch density on the radius (transverse profile), br is the characteristic transverse dimension, and the function  0 / bT t t describes the longitudinal profile of bunch density, bt is the characteristic duration of the bunch. The value 0j is related to the total charge Q by the relation 0 * / ( )j Q s t , where ,s t are the effective area of the bunch transverse section and bunch duration. The resulting electromagnetic field of the electron bunch of the axisymmetric form (16) is found by summing fields of the elementary ring charges (4) ( ) ( )( , ) ( , ) ( , )pol em z z zE r E r E r    , ( ) 2 20 0 0 * 1 1 ( , ) 2 ( ) ( ),pol z p pol polE r E k b P r Z s t       0 0 0 0 02 0 4 , ( ) ( , ) ( ) . b pol p p Q E P r G k r k r R r r dr b     0 0 0( ) / cos ( ) ,pol b gZ T t t t dt        ( ) 0 0 2 1* * 1 ( / )1 ( , ) 2 ( ) ( ) em n z n n chn chn n n J r b E r E P Z s t J               0 0 ( ) , / / ,   b stn stn n b n bZ P R r r J r r rdr   (17)  0 0 0( ) / cos ( ) ,chn b chnZ T t t t dt          0 0 0 0( ) / ( ) .stn t stn bZ T t t sign t e dt           The electric field ( )pol zE describes the potential polarization field excited in the dielectric waveguide by an electron bunch, ( )em zE takes into account the contribution to full field of the electromagnetic (vortex) component of field, which is the sum of the eigen radial harmonics of the dielectric waveguide. Let us investigate the nature of the distribution of the electric field in the dielectric waveguide, depending on the shape of the bunch. First of all we consider a symmetric bunch in the longitudinal direction, whose density has a maximum at the entry time 0 0t  and decreases to zero at 0t  , 0 0( ) ( )T t T t  . In the wave zone bt  , the wakefield has the form of a set of monochromatic waves 2 20 0 * * 0 11 ( , ) 2 ( ) ( )cosz p pol g gE r E k b P r T s t            0 2 1 1 ( / ) ( )cos , ( ) n n chn n chn chn n n J r b PT J               where  0 0 0 0 ( ) 2 / cosbT T t t t dt      94 ISSN 1562-6016. ВАНТ. 2019. №1(119) are amplitudes of the Fourier component of the function 0( / )bT t t on eigen frequencies ,g chn   . To determine the field in the volume of the waveguide, including the region of the bunch, it is necessary to specify specific longitudinal and transverse density profiles. As an example, for a bunch with the model profile    0 0/ exp /b bT t t t t  we have 2 2 ( ) 2 ( )cos exp( / ) , 1 b pol g b g b t Z sign t t              2 2 ( ) 2 ( )cos exp( / ) , 1 b chn chn b chn b t Z sign t t               * 2 bt t . As follows from these expressions the electric field of potential polarization oscillations, as well as the fields of all radial harmonics, have the same structure. Behind a bipolar solitary impulse, a monochromatic wake eigen wave of a dielectric waveguide propagates Amplitudes of the wakefield waves essentially depend on the duration of the bunch. For a fixed number of particles in the bunch, the amplitudes of wake waves are the larger, the shorter the bunch. If more realistic conditions 1, 1g b chn bt t   are fulfilled, then the polarization field of the electron bunch excites noncoherence and its level is small. On the other hand, for an electromagnetic wakefield, the electron bunch is a completely coherent source. The wakefield field (17) consists of a set of electromagnetic eigen modes of the dielectric waveguide. For Gaussian transverse profile model of the laser pulse intensity  2 2( / ) exp / b bR r r r r we obtained  2 2 2 2( / 4)exp / 4 n b n bP r r b . So, a finite number of radial harmonics are efficiently excited, if / 2 1n br b  , and 1chn bt  . CONCLUSIONS The process of excitation of the wake Cherenkov radiation by an electron bunch in a dielectric waveguide is investigated. The polarization of the dielectric medium induced by the electric field of the REB is determined. It is shown that the excited electric field consists of a potential field of polarization oscillations and a set of eigen electromagnetic waves of a dielectric waveguide. REFERENCES 1. V.P. Zrelov. Vavilov-Cherenkov radiation and its application in physics of high energy. M.: “Atomizdat”, 1968. 2. D.V. Dgelli. Cherenkov radiation // UFN. 1956, v. 58, № 2, p. 231-283. 3. I.N. Onishchenko, V.A. Kiselev, A.F. Linnik, G.V. Sotnikov. Concept of dielectric wakefield accelerator driven by a long sequence of electron bunches // Proc. IPAC2013, Shanghai, China TUPEA056. 2013, p. 12569-1261. 4. V. Kiselev, A. Linnik, V. Mirny, N. Zemliansky, R. Kochergov, I. Onishchenko, G. Sotnikov, Ya. Fainberg. Dielectric wake-field generator // Proc. of BEAMS’98, Haifa, Israel, June 7-12, 1998, v. II, p. 756- 759. 5. I.N. Onishchenko, G.V. Sotnikov. Synchronization of wakefield modes in the dielectric resonator // Technical Physics. 2008, v. 53, № 10, p. 1344-1349. 6. V.A. Balakirev, I.N. Onishchenko, D.Y. Sidorenko, G.V. Sotnikov. Charged particles accelerated by wake fields in a dielectric resonator with exciting electron bunch channel // Technical Physics Letters. 2003, v.29, № 7, p. 589-591. 7. C. Kittel. Introduction to solid statу physics. М.: “Science”, 1978, 791 p. Article received 28.09.2018 ЧЕРЕНКОВСКОЕ ИЗЛУЧЕНИЕ ЭЛЕКТРОННОГО СГУСТКА В ДИЭЛЕКТРИЧЕСКОЙ СРЕДЕ С ЧАСТОТНОЙ ДИСПЕРСИЕЙ В.А. Балакирев, И.Н. Онищенко Рассматривается возбуждение черенковского электромагнитного излучения релятивистским электронным пучком в диэлектрическом волноводе с учетом частотной дисперсии диэлектрической проницаемости. Определены электрическая поляризация в изотропной диэлектрической среде и, соответственно, поляризационные заряды и токи, индуцированные кулоновским электрическим полем релятивистского электронного пучка. Получена и исследована пространственно-временная структура возбужденного волнового поля в диэлектрическом волноводе. Показано, что возбужденное поле состоит из электрического поля потенциальной поляризации и множества собственных электромагнитных волн диэлектрического волновода. ЧЕРЕНКІВСЬКЕ ВИПРОМIНЮВАННЯ ЕЛЕКТРОННОГО ЗГУСТКА В ДIЕЛЕКТРИЧНОМУ СЕРЕДОВИЩI З ЧАСТОТНОЮ ДИСПЕРСIЄЮ В.А. Балакiрев, I.М. Онiщенко Розглядається збудження черенківського електромагнітного випромінювання релятивістським електронним згустком у діелектричному хвилеводі з урахуванням частотної дисперсії діелектричної проникності. Визначено електричну поляризацію в ізотропному діелектричному середовищі і, відповідно, поляризаційні заряди та струми, індуковані кулонівським електричним полем релятивістського електронного пучка. Отримано та досліджено просторово-часову структуру збудженого кільватерного поля в діелектричному хвилеводі. Показано, що збуджене поле складається з потенційного поляризаційного електричного поля та набору власних електромагнітних хвиль діелектричного хвилеводу. javascript:void(0) javascript:void(0) javascript:void(0)

Cherenkov radiation of the electron bunch in dielectric media with frequency dispersion

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