Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator

The process of the excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator is considered. It is shown that the potential electric field of plasma oscillations contains the field of a bulk wake plasma wave and the fields of two surface plasma oscil...

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Дата:	2023
Автори:	Balakirev, V.A., Onishchenko, I.N.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2023
Назва видання:	Problems of Atomic Science and Technology
Теми:	New methods of charged particles acceleration
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/196177
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator / V.A. Balakirev, I.N. Onishchenko // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 71-76. — Бібліогр.: 4 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1961772023-12-11T13:52:31Z Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator Balakirev, V.A. Onishchenko, I.N. New methods of charged particles acceleration The process of the excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator is considered. It is shown that the potential electric field of plasma oscillations contains the field of a bulk wake plasma wave and the fields of two surface plasma oscillations. Surface plasma oscillations are localized in the vicinity of the input and output ends of the plasma resonator. In a plasma resonator, as a result of the effect of transition radiation at the ends of the resonator, the electron bunch will also excite eigen electromagnetic oscillations of the resonator. The spatio-temporal structure of the electromagnetic field of these oscillations has been studied. Досліджено процес збудження плазмових та електромагнітних коливань релятивістським електронним згустком у плазмовому резонаторі. Показано, що потенційне електричне поле плазмових коливань містить поле об'ємної кільватерної плазмової хвилі та поля двох поверхневих плазмових коливань. Поверхневі плазмові коливання локалізовані поблизу вхідного та вихідного торців плазмового резонатора. У плазмовому резонаторі в результаті ефекту перехідного випромінювання на торцях резонатора електронний згусток також збуджує власні електромагнітні коливання резонатора. Досліджено просторово-часову структуру електромагнітного поля цих коливань. 2023 Article Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator / V.A. Balakirev, I.N. Onishchenko // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 71-76. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq DOI: https://doi.org/10.46813/2023-146-071 http://dspace.nbuv.gov.ua/handle/123456789/196177 en Problems of Atomic Science and Technology 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	New methods of charged particles acceleration New methods of charged particles acceleration
spellingShingle	New methods of charged particles acceleration New methods of charged particles acceleration Balakirev, V.A. Onishchenko, I.N. Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator Problems of Atomic Science and Technology
description	The process of the excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator is considered. It is shown that the potential electric field of plasma oscillations contains the field of a bulk wake plasma wave and the fields of two surface plasma oscillations. Surface plasma oscillations are localized in the vicinity of the input and output ends of the plasma resonator. In a plasma resonator, as a result of the effect of transition radiation at the ends of the resonator, the electron bunch will also excite eigen electromagnetic oscillations of the resonator. The spatio-temporal structure of the electromagnetic field of these oscillations has been studied.
format	Article
author	Balakirev, V.A. Onishchenko, I.N.
author_facet	Balakirev, V.A. Onishchenko, I.N.
author_sort	Balakirev, V.A.
title	Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator
title_short	Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator
title_full	Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator
title_fullStr	Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator
title_full_unstemmed	Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator
title_sort	wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2023
topic_facet	New methods of charged particles acceleration
url	http://dspace.nbuv.gov.ua/handle/123456789/196177
citation_txt	Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator / V.A. Balakirev, I.N. Onishchenko // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 71-76. — Бібліогр.: 4 назв. — англ.
series	Problems of Atomic Science and Technology
work_keys_str_mv	AT balakirevva wakeexcitationofplasmaandelectromagneticoscillationsbyarelativisticelectronbunchinaplasmaresonator AT onishchenkoin wakeexcitationofplasmaandelectromagneticoscillationsbyarelativisticelectronbunchinaplasmaresonator
first_indexed	2025-07-17T00:40:43Z
last_indexed	2025-07-17T00:40:43Z
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fulltext	ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 71 https://doi.org/10.46813/2023-146-071 WAKE EXCITATION OF PLASMA AND ELECTROMAGNETIC OSCILLATIONS BY A RELATIVISTIC ELECTRON BUNCH IN A PLASMA RESONATOR V.A. Balakirev, I.N. Onishchenko National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine E-mail: onish@kipt.kharkov.ua The process of the excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator is considered. It is shown that the potential electric field of plasma oscillations contains the field of a bulk wake plasma wave and the fields of two surface plasma oscillations. Surface plasma oscillations are localized in the vicinity of the input and output ends of the plasma resonator. In a plasma resonator, as a result of the effect of transition radiation at the ends of the resonator, the electron bunch will also excite eigen electromagnetic oscillations of the resonator. The spatio-temporal structure of the electromagnetic field of these oscillations has been studied. PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq INTRODUCTION The influence of the transverse boundedness of the plasma on the process of wakefield excitation by a rela- tivistic electron bunches has been studied for the case of plasma waveguides [1, 2]. Meanwhile, the plasma is always limited not only in the transverse direction, but also in the longitudinal one. In this work, using the ex- ample of the plasma placed in a perfectly conducting metal resonator, we study the influence of the longitudi- nal plasma boundaries on the pattern of the field of ex- cited wake plasma oscillations by a relativistic electron bunch. In case of a plasma resonator, the process of wake plasma oscillations excitation by a relativistic electron bunch will always be accompanied by the excitation of eigen electromagnetic oscillations of the resonator. The process of excitation of these oscillations is based on the effect of coherent transition radiation of a relativistic electron bunch [3], which crosses the ideally conducting input and output ends of the resonator. Attention is also paid to the transient excitation of eigen electromagnetic oscillations in the plasma resonator. 1. STATEMENT OF THE PROBLEM. BASIC EQUATIONS The plasma resonator is made in the form of a per- fectly conducting metal resonator, the volume of which is completely filled with a homogeneous isotropic cold plasma. Along the axis, an axisymmetric relativistic electron bunch moves uniformly and rectilinearly. For simplicity, we will consider a relativistic electron bunch in the form of an infinitely thin charged ring. The cur- rent density of such a bunch has the form  0 0 0 ( ) / 2 b z r r j e Q t z v r        , (1) where ze is unit vector in the direction of the longitudi- nal axis z , Q is total bunch charge. r is radial coordi- nate, 0r is ring bunch radius, z is longitudinal coordi- nate, t is time, 0v is electron bunch velocity. It is convenient to reduce the original inhomogene- ous system of Maxwell's equations to the following equations for determining the excited wake electric field 2 2 2 2 2 1 4 ( ),bjD E E tc t c         (2) 4 bdivD  , D E ,  is integral operator of plasma permittivity. The current and charge densities of the electron bunch are in the following relation 0b z bj e v  . We expand the quantities in the Maxwell equations into the Fourier integrals over frequencies    , , ,i tE H E H e d         where 0/t t z v  ,     1 , ( ), ( ) 2 i tE H E t H t e dt        . The system of equations (2) for the longitudinal Fou- rier component of the electric field can be transformed to the equation 2 2 2 0 02 0 0 ( )1 lik zz z z E E r ri Q r k E e r r r k c rz                 , (3) where 2 2 2 0 0 0( ) , / , / ,l lk k k v k c         2 2 1 p     is plasma permittivity. On the perfectly conducting side surface of the plasma resonator r b , the longitudinal component of the electric field vanishes ( ) 0zE r b   . (4) The solution of the partial differential equation (3), which satisfies the boundary condition (4), is convenient to find in the form of a series in Bessel functions 0 1 ( ) ( / )z z n n n E E z J r b      , (5) where b is radius of the resonator, n are roots of the Bessel function 0 ( )J x . Then from the partial differen- tial equation (3) we obtain the following ordinary dif- ferential equation 2 2 2 0 0 2 0 ( / ) lik zz n n n z n n d E J r bi k E Q e k c Ndz       , (6) mailto:onish@kipt.kharkov.ua 72 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 2 2 2 2 0 /n nk k b   is the square of the longitudinal wave number of the electromagnetic wave of the plasma waveguide forming the plasma resonator, 2 2 1 ( ) 2 n n b N J  is eigen electromagnetic wave norm. The total solution of the linear inhomogeneous equation (6) has the form ( ) ( )( ) n nik z ik z z n n nE z E e E e     2 0 0 2 2 0 ( / )1 lik zn nn l J r bi Q e k c Nk k     . (7) Here ( ) nE  are constants which should be deter- mined. Accordingly, for the expansion coefficients of the radial component of the electric field in terms of Bessel functions 1 1 ( ) ( / )r r n n n E E z J r b      (8) we have the following expression ( ) ( )( ) ( )n n nik z ik z ik zn r n n n n n k b E z i E e E e e       , (9) 0 0 2 2 0 ( / )1n n n nn l J r b Q bv Nk k        . On the perfectly conducting ends of the plasma res- onator, which have longitudinal coordinates of the input end 0z  and output end z L , L is the length of the resonator, the radial component of the electric field van- ishes. Accordingly, all coefficients (9) of expansion (8) also vanish ( 0) ( ) 0r n r nE z E z L     . From these conditions we find constants ( ) nE  and, accordingly, expressions for the Fourier expansion coef- ficients ( ), ( )z n r nE z E z  . Ultimately, for the expansion coefficients of the longitudinal and radial components of the electric field, we obtain the following integral Fourier representations 2 2 2 0 2 ( , ) ( , ) ( )( ) i t zn n z n l Q e d E z t z b v k k              , (10) 2 2 2 0 2 ( , ) ( , ), ( )( ) i t rn n r n l Q e d E z t z b v k k              (11) where   0 0 2 1 n n n r J b J           , 2 2 1 ( , ) cos ( ) cos sin nik Ln z n n nn z k z L e k L k Lk b           2 2 0 lik z l l i k k e k   , 1 ( , ) sin ( ) sin sin nik Ln r n n n z k z L e k L b k L             lik z e . The integrands in expressions (10), (11) have three groups of simple poles. First of all, these are the poles 0p i    , (12) which are the roots of the plasma permittivity ( ) 0   and correspond to the wake field of plasma oscillations, excited by a relativistic electron bunch moving in the volume of the plasma resonator. The poles (12) are lo- cated in the lower half-plane of the complex variable  near the real axis. The poles 0nm i    , (13) determined from the equation nk L m , 1,2,3,...m  , correspond to the frequencies of eigen electromagnetic oscillations of the plasma resonator 2 2 2 2 2 2 2 n nm p m c b L             . These poles are also located in the lower half-plane near the real axis. And finally, the roots of the equation 2 2 2 0 2 0n lk k b      determine the complex conjugated poles located on the imaginary axis ,nsti   where 0 0nst nc    , 2 2 2 2/nc p n c b    is the cut- off frequency of the plasma waveguide forming the res- onator, 0 0 0/ ,v c  is the relativistic factor. These poles are responsible for the own quasi-static electro- magnetic field of the relativistic electron bunch in the plasma resonator. 2. EXCITATION OF WAKE PLASMA OSCILLATIONS The components of the total wakefield excited in the plasma resonator by a relativistic electron bunch are described by Fourier frequency integrals (10), (11). The calculation of the residues in the poles (12) gives analyt- ical expressions for the expansion coefficients of the components (10), (11) of the wakefield and, ultimately, the final expressions for the components of the electric field of plasma oscillations ( ) 0 1 0 0 0 1 ( , ) ( / )sin ( / ) 2 l r p pE E k bG r r t z v t z v     0 1( , ) ( / )sin pE W r L z t z с t     1 0 0 ( , ) sin p L z L L W r z t t v c v                   , (14) ( ) 0 0 0 0 0 1 ( , ) ( / )cos ( / ) 2 l z p pE E k bG r r t z v t z v     0 0 ( , ) ( / )sin pE W r L z t z с t     0 0 0 ( , ) sin p L z L L W r z t t v c v                   . (15) Here 0 4 /pE Qk b , 0/ ,p pk v ( )x is unit function 1, 0, ( ) 0, 0, x x x      ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 73 0 0 1 ( , ) , n n n n n y ch r b W r y S J Lb sh b                         1 1 1 ( , ) , n n n n n y sh r b W r y S J Lb sh b                         0L y  ,  0 0 2 2 2 2 1 / ( ) n n n n n p J r b S J k b       , 0 0 0 0 0 0 0 0 0 0 ( ) ( , ), , ( , ) ( ) ( , ), , p p p p p p I k r k b k r r r G r r I k r k b k r r r       0 0 0 0 0( , ) ( ) ( ) ( ) ( )p p p p p pk b k r I k b K k r I k r K k b   , 0 0 1 0 1 0 1 0 0 0 ( ) ( , ), , ( , ) ( ) ( , ), , p p p p p p I k r k b k r r r G r r I k r k b k r r r        1 0 1 1 0( , ) ( ) ( ) ( ) ( )p p p p p pk b k r I k b K k r I k r K k b   . For large values of the argument y b , the func- tions 0,1( , )W r y have asymptotic representations 1 ( , ) n L y b n n n r W r y S J e b                . It is easy to verify that the radial component of the electric field of the wake plasma oscillations at each moment of time satisfies the boundary conditions at the perfectly conducting ends of the plasma resonator. The unit functions included in the expressions for the com- ponents of the wakefield (14), (15) describe the motion of the wave fronts of the wakefield. Note that the total wakefield electric field is potential. Expressions (13), (14) for the components of the wakefield of plasma oscillations excited by a relativistic electron bunch in a cylindrical plasma resonator with perfectly conducting walls contain three terms. The first term describes the volume wakefield of plasma oscilla- tions. This field has the same spatio-temporal structure and amplitude as in a longitudinally infinite plasma waveguide. A monochromatic plasma wave propagates behind the electron bunch with a phase velocity which is equal to the bunch velocity. When the bunch leaves the resonator, the bulk wake wave exists along the entire length of the resonator. The electric field of only a trav- eling plasma wave cannot satisfy the boundary condi- tions at the perfectly conducting ends of the cavity at any time. That is why, there are two more terms in the expressions the wakefield components (14), (15) in the plasma resonator. Each of them has a simple physical meaning. The first term in each of the expressions de- scribes the transition electromagnetic radiation that oc- curs when the electron bunch crosses the input end of the plasma cavity. The second term describes surface plasma oscillations located near the output end. The amplitude of these oscillations decreases with distance from the input end into the plasma volume. On the sur- face of the input end, the amplitude of the radial com- ponent of the electric field of surface plasma oscillations is equal to the amplitude of the bulk plasma wake wave at this end, but is always opposite in phase. Therefore, the total radial electric field on the surface of the input end at each moment of time is identically equal to zero. The leading front of the surface wave propagates in the volume of the resonator at the speed of light and slightly overtakes the relativistic electron bunch. At the moment of time, the leading front of the surface oscillations ap- proaches to the output end and a stationary configura- tion of the input surface oscillations is established. At the moment of time when the electron bunch crosses the output end, the second surface plasma wave is excited, which is localized near to the output end of the plasma resonator. This surface field of plasma oscillations en- sures the fulfillment of the boundary condition at the output end of the plasma resonator. The leading front of this surface wave also propagates at the speed of light from the output end to the input end. When the leading front of this wave reaches the input end at the moment of time 0/ /t L v L c  , in the plasma resonator the stationary structure of the wake field of plasma oscilla- tions is established, which is described by the following expressions for the components of the electric field ( ) 0 1 0 0 1 ( , )sin ( / ) 2 l r p pE E k bG r r t z v   (16) 0 1 1 0 ( , )sin ( , )sinp p L E W r L z t W r z t v                , ( ) 0 0 0 0 1 ( , )cos ( / ) 2 l z p pE E k bG r r t z v   (17) 0 0 0 0 ( , )sin ( , )sinp p L E W r L z t W r z t v                . Let us consider a special case when the time of flight of an electron bunch through a plasma resonator is a multiple of an even number of half-periods of plasma oscillations 0 2 p L l v    , 0,1,2...l  . This condition is also equivalent to the requirement that an even number of spatial half-periods of the bulk wake plasma wave fit within the resonator length. In this case, from (16), (17) the following expressions fol- low for the components of the electric field of surface plasma oscillations ( ) ( ) 0 1 ( , )sin ,l sm rs pE E U r x t  ( ) ( ) 0 0 ( , )sin ,l as zs pE E U r x t  where / 2x z L  , ( ) 1 1 1( , ) ( , / 2 ) ( , / 2 )smU r x W r L x W r L x     1 1 2 n n n n n x ch r b S J Lb ch b                         , (18) ( ) 0 0 0( , ) ( , / 2 ) ( , / 2 )asU r x W r L x W r L x     0 1 . 2 n n n n n x sh r b S J Lb ch b                         (19) 74 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) The distribution of the radial component of the elec- tric field of surface plasma oscillations along the axis of the plasma resonator has symmetrical form ( ) ( ) 1 1( , ) ( , )sm smU r x U r x  with respect to the central plane / 2,( 0)z L x  . The dependence of the ampli- tude of each radial harmonic (18) of the surface plasma wave is described by the function     ( ) / ( ) / 2 nsm n n ch x b C x ch L b    . For the values / 2x L corresponding to the input and output ends of the resonator, we have ( ) ( / 2) 1sm nC x L  , and for the central plane of the resonator 0x  we obtain  ( ) ( 0) 1/ / 2sm n nC x ch L b  . It follows from this formula that the most pro- nounced surface character of the field is manifested in the case of a long plasma resonator / 2 1L b  . In turn, it follows from expressions (18), (20) that the distribu- tion of the longitudinal component of the surface field of plasma oscillations is antisymmetric ( ) ( ) 0 0( , ) ( , )as asU r x U r x   . Let us now consider a special case when the time of flight of an electron bunch through a plasma resonator is a multiple of an odd number of half-periods of plasma oscillations 0 (2 1) p L l v     , 0,1,2...l  . In this case, the expressions for the components of the electric field of surface plasma oscillations have the following form ( ) ( ) 0 1 ( , )sin ,l as rs pE E U r x t  ( ) ( ) 0 0 ( , )sin ,l sm zs pE E U r x t  ( ) 1 1 1( , ) ( , / 2 ) ( , / 2 )asU r x W r L x W r L x     1 1 2 n n n n n x sh r b S J Lb ch b                         , (20) ( ) 0 0 0( , ) ( , / 2 ) ( , / 2 )smU r x W r L x W r L x     0 1 . 2 n n n n n x ch r b S J Lb ch b                         In the case under consideration, the dependence of the radial component of the surface plasma field ampli- tude in the resonator is antisymmetric with respect to the central plane / 2,( 0)z L x  , ( ) ( ) 1 1( , ) ( , )as asU r x U r x   and a similar dependence for the longitudinal component of the surface field is sym- metric ( ) ( ) 0 0( , ) ( , )sm smU r x U r x  . Thus, a relativistic electron bunch always excites in a plasma resonator a conglomerate of a bulk plasma wake wave and two surface plasma oscillations local- ized near the input and output ends of the plasma reso- nator. Only in combination these oscillations provide the fulfillment of the boundary conditions for the total wake field at the ideally conducting ends of the plasma resonator. 3. TRANSITION EXCITATION OF ELECTROMAGNETIC OSCILLATIONS When a relativistic electron bunch crosses perfectly conducting ends of a plasma resonator, transition elec- tromagnetic radiation arises. In the frequency Fourier integral describing the total electromagnetic field excit- ed by a relativistic electron bunch in a plasma resonator, poles (13) are responsible for the excitation of eigen electromagnetic oscillations in the plasma resonator. The calculation of residues in these poles gives the fol- lowing expressions for the components of the field of electromagnetic oscillations ( ) 02 1 00 8 ( , )em nm z n n n nm n m nm LQ L r E R J Z z t b bb                  cos sin ,nm z m t L          (21) ( ) 12 1 10 8 ( , )em nm r n n nm n m nm mLQ r E R J Z z t bb                  sin sin ,nm z m t L          (22) ( ) 12 1 00 8 ( , )em nm n n nm nm n m LQ r H R J L Z z t b cb                   cos cos ,nm z m t L          (23) where 0 ( , ) ( 1) nmik Lm nm z L z L Z t z t e t c v c                  , (24) 0 1 , ( ), ,nm nm nm nm nm nm nm c L k b d v         2 2 2 2 nm nmd k L m  , 0 0 2 1 ( / ) ( ) n n n n J r b R J     . The function ( , )nmZ t z contains two terms and de- scribes the propagation of the fronts of two pulses of transition electromagnetic radiation. The structure of the transient electromagnetic field and the dynamics of its excitation are described by ex- pressions (21)-(24). As follows from these expressions, at the moment when the electron bunch enters the plas- ma resonator 0t  , the first pulse of transition electro- magnetic radiation is formed. The leading front of this radiation propagates at the speed of light from the input end of the plasma resonator to the output one. The lead- ing front of the transition radiation pulse is formed by the high-frequency component of the electromagnetic pulse. These frequencies are much higher than the cut- off frequencies of the eigen electromagnetic waves of the plasma waveguide forming the resonator. At the moment when the electron bunch crosses the output end, the second pulse of transition electromagnetic radi- ation is excited, which propagates from the output end of the plasma resonator to the input one. The phase fac- tor in (24) 0/nmi L v e  relates the phases of the transition ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 75 electromagnetic oscillations to the moment when the electron bunch leaves the plasma resonator. When the leading front of the input end of the plasma resonator is reached, a stationary picture of the electromagnetic os- cillations excitation is established at the moment of time 0/ /t L v L c  . In this state, the field of transition electromagnetic radiation is a set of eigen electromag- netic oscillations with constant amplitudes. The factor (24) takes the form 1 ( 1) nmik Lm nmZ e   . This factor takes into account the effect of interfer- ence of electromagnetic oscillations excited by an elec- tron bunch at the input and output ends of the plasma resonator. Oscillations with even indices m (for which the condition 2nmk L l , where l is an integer, is not fulfilled) are not excited. For odd indices m , the total oscillations are completely extinguished at (2 1)nmk L l  . On the other hand, the maximum am- plification of electromagnetic oscillations takes place for even indices m at (2 1)nmk L l  , and for odd m , when 2nmk L l . As is known [4], the fundamental E type oscillation of the cylindrical resonator is the oscillation with indi- ces 0, 1m n  . The eigen frequency of this oscillation is equal to the cutoff frequency of the cylindrical plasma waveguide. 2 2 2 2 10 1 /p c b    . The longitudinal component of the electric field and the azimuthal component of the magnetic field are ho- mogenious along the longitudinal axis of the plasma. There is no radial component of the electric field. For the fundamental oscillation, formulas (21)-(23) give expres- sions for the components of the electromagnetic field. 10 10 0 1 10sinz r E E J t b          , 10 0rE  , 1 10 10 1 1 10 10 cos r H E J t k b b            , where     0 1 00 10 102 2 10 1 1 /8 J r bvQ E Z Lb J     . Note that the fundamental electromagnetic oscilla- tion of a cylindrical plasma resonator can be used to accelerate relativistic charged particles. Really, if the time of flight of the accelerated particle is equal to the half-period of the fundamental electromagnetic oscilla- tion 10 0/L v  , then the particle will be constantly in the accelerating phase during its presence in the plasma resonator. From the equations of motion of a accelerated relativistic particle 10 10sin dp eE t dt  we find that the increasing of the relativistic factor of an accelerated particle during its flight through the plasma resonator is equal 0 102a   , where 01 10 2 10 10 16eE eQ a mc b mc     is parameter of the electromagnetic oscillation strength. Obviously, when this parameter is 10 1a  , then there will be an effective acceleration of relativistic particles by the field of the fundamental electromagnetic oscilla- tion in the plasma resonator. CONCLUSIONS The process of excitation of plasma and electromag- netic oscillations by a relativistic electron bunch in a plasma resonator is considered. First of all, the excited field includes the potential field of Langmuir oscillations. In turn, Langmuir oscillations include a bulk wake plas- ma wave, which has the same spatio-temporal structure as it is in a plasma waveguide that is infinite in the longi- tudinal direction and used for resonator formation. In addition to the bulk wake wave, two surface plasma oscil- lations are excited in the plasma resonator. One of them is localized in the vicinity of the input end of the resonator, and the second is localized in the vicinity of the output end. The amplitudes of surface waves decrease with dis- tance from the ends deep into the plasma. Only a con- glomerate from the field of a bulk wake wave and the fields of two surface plasma oscillations satisfies the boundary conditions at the ends of the plasma resonator. In addition to potential plasma oscillations, the elec- tron bunch will also excite a vortex electromagnetic field in the plasma resonator. The analytical expression for the electromagnetic field contains two terms. Each of them has a simple physical meaning. The first term describes the transition electromagnetic radiation that occurs when the electron bunch crosses the input end of the plasma resonator. The second term in the expression for the total electromagnetic field describes the transi- tion electromagnetic radiation excited by the relativistic electron bunch when it crosses the output end of the plasma resonator with perfectly conducting walls. Final- ly, after the transient process, the field of electromagnet- ic radiation is established as a superposition of eigen electromagnetic oscillations of the plasma resonator with constant amplitudes. REFERENCES 1. V.A. Balakirev, N.I. Karbushev, A.J. Ostrovskij, Yu.V. Tkach. Theory of amplifiers and generators on relativistic beams. Kiev: “Naukova dumka”, 1993, 208 p. 2. V.A. Balakirev, V.I. Karas, A.P. Tolstoluzskii. Exci- tation of wake field in a plasma with radial density variation // Plasma Physics Reports. 1997, v. 23, № 4, p. 290-298. 3. V.I. Ginzburg, V.I. Tsytovich. Transition radiation and transition scattering. M.: “Science”, 1964, 360 p. 4. L.A. Vainstein. Electromagnetic waves. M.: “Radio and svyaz”, 1988, 440 p. Article received 15.05.2023 76 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) КІЛЬВАТЕРНЕ ЗБУДЖЕННЯ ПЛАЗМОВИХ ТА ЕЛЕКТРОМАГНІТНИХ КОЛИВАНЬ РЕЛЯТИВІСТСЬКИМ ЕЛЕКТРОННИМ ЗГУСТКОМ У ПЛАЗМОВОМУ РЕЗОНАТОРІ В.А. Балакiрeв, I.М. Онiщенко Досліджено процес збудження плазмових та електромагнітних коливань релятивістським електронним згустком у плазмовому резонаторі. Показано, що потенційне електричне поле плазмових коливань містить поле об'ємної кільватерної плазмової хвилі та поля двох поверхневих плазмових коливань. Поверхневі плаз- мові коливання локалізовані поблизу вхідного та вихідного торців плазмового резонатора. У плазмовому резонаторі в результаті ефекту перехідного випромінювання на торцях резонатора електронний згусток та- кож збуджує власні електромагнітні коливання резонатора. Досліджено просторово-часову структуру елект- ромагнітного поля цих коливань.

Wake excitation of plasma and electromagnetic oscillations by a relativistic electron bunch in a plasma resonator

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