Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide

The process of wake fields excitation by a relativistic electron bunch in polar semiconductors is studied. Cylindrical semiconductor waveguide, in which relativistic electron bunch moves along axis, is considered. It is shown that the excited wake field in the terahertz and infrared frequency ranges...

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Дата:	2022
Автори:	Balakirev, V.A., Onishchenko, I.N.
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Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2022
Назва видання:	Вопросы атомной науки и техники
Теми:	Novel and non-standard acceleration technologies
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/195396
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Цитувати:	Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide / V.A. Balakirev, I.N. Onishchenko // Problems of Atomic Science and Technology. — 2022. — № 3. — С. 76-81. — Бібліогр.: 16 назв. — англ.

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spelling	irk-123456789-1953962023-12-05T11:46:39Z Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide Balakirev, V.A. Onishchenko, I.N. Novel and non-standard acceleration technologies The process of wake fields excitation by a relativistic electron bunch in polar semiconductors is studied. Cylindrical semiconductor waveguide, in which relativistic electron bunch moves along axis, is considered. It is shown that the excited wake field in the terahertz and infrared frequency ranges consists of the field of longitudinal HF and LF hybrid plasmon-phonon oscillations and the field of the HF and LF transverse polaritons, which are a set of eigen electromagnetic waves of the polar semiconductor waveguide. The spatio-temporal structure of the total excited wake field is obtained, the intensity of the excited wake waves is determined. Досліджено процес збудження кільватерних полів релятивістським електронним згустком у полярних напiвпровідниках. Розглянуто циліндричний напiвпровiдниковий хвилевод, вздовж вісі якого рухається релятивістський електронний згусток. Показано, що збуджене кільватерне поле в терагерцовому та інфрачервоному дiапазонах частот містить у собі поле повздовжніх ВЧ- i НЧ-гiбридних плазмон-фононних коливань, а також поле ВЧ- та НЧ-поперечних полярiтонiв, яке є набором власних електромагнітних хвиль напiвпровiдникового хвилеводу. Отримана просторово-часова структура результуючого кільватерного поля, визначена iнтенсивнiсть збуджених кільватерних хвиль. Исследован процесс возбуждения кильватерных полей релятивистским электронным сгустком в полярных полупроводниках. Рассмотрен цилиндрический полупроводниковый волновод, по оси которого движется релятивистский электронный сгусток. Показано, что возбуждаемое кильватерное поле в терагерцовом и инфракрасном диапазонах частот состоит из поля продольных ВЧ- и НЧ-гибридных плазмон-фононных колебаний, а также поля ВЧ- и НЧ-поперечных поляритонов, которые представляют собой набор собственных электромагнитных волн полупроводникового волновода. Получена пространственно-временная структура результирующего кильватерного поля, определена интенсивность возбуждаемых кильватерных волн. 2022 Article Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide / V.A. Balakirev, I.N. Onishchenko // Problems of Atomic Science and Technology. — 2022. — № 3. — С. 76-81. — Бібліогр.: 16 назв. — англ. 1562-6016 PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq http://dspace.nbuv.gov.ua/handle/123456789/195396 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	Novel and non-standard acceleration technologies Novel and non-standard acceleration technologies
spellingShingle	Novel and non-standard acceleration technologies Novel and non-standard acceleration technologies Balakirev, V.A. Onishchenko, I.N. Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide Вопросы атомной науки и техники
description	The process of wake fields excitation by a relativistic electron bunch in polar semiconductors is studied. Cylindrical semiconductor waveguide, in which relativistic electron bunch moves along axis, is considered. It is shown that the excited wake field in the terahertz and infrared frequency ranges consists of the field of longitudinal HF and LF hybrid plasmon-phonon oscillations and the field of the HF and LF transverse polaritons, which are a set of eigen electromagnetic waves of the polar semiconductor waveguide. The spatio-temporal structure of the total excited wake field is obtained, the intensity of the excited wake waves is determined.
format	Article
author	Balakirev, V.A. Onishchenko, I.N.
author_facet	Balakirev, V.A. Onishchenko, I.N.
author_sort	Balakirev, V.A.
title	Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide
title_short	Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide
title_full	Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide
title_fullStr	Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide
title_full_unstemmed	Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide
title_sort	excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2022
topic_facet	Novel and non-standard acceleration technologies
url	http://dspace.nbuv.gov.ua/handle/123456789/195396
citation_txt	Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide / V.A. Balakirev, I.N. Onishchenko // Problems of Atomic Science and Technology. — 2022. — № 3. — С. 76-81. — Бібліогр.: 16 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT balakirevva excitationofwakefieldsbyarelativisticelectronbunchinapolarsemiconductorwaveguide AT onishchenkoin excitationofwakefieldsbyarelativisticelectronbunchinapolarsemiconductorwaveguide
first_indexed	2025-07-16T23:24:15Z
last_indexed	2025-07-16T23:24:15Z
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fulltext	76 ISSN 1562-6016. ВАНТ. 2022. №3(139) NOVEL AND NON-STANDARD ACCELERATION TECHNOLOGIES https://doi.org/10.46813/2022-139-076 EXCITATION OF WAKE FIELDS BY A RELATIVISTIC ELECTRON BUNCH IN A POLAR SEMICONDUCTOR WAVEGUIDE 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 wake fields excitation by a relativistic electron bunch in polar semiconductors is studied. Cylin- drical semiconductor waveguide, in which relativistic electron bunch moves along axis, is considered. It is shown that the excited wake field in the terahertz and infrared frequency ranges consists of the field of longitudinal HF and LF hybrid plasmon-phonon oscillations and the field of the HF and LF transverse polaritons, which are a set of eigen electromagnetic waves of the polar semiconductor waveguide. The spatio-temporal structure of the total excited wake field is obtained, the intensity of the excited wake waves is determined. PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq INTRODUCTION Plasma phenomena in semiconductors and semimet- als are an inherent property of these condensed media [1 - 7]. The density of electron-hole plasma in semicon- ductors with intrinsic conductivity (for example, silicon, germanium and other) is relatively low, on the order of 10 13 310 ...10 cm , and is determined primarily by the width of the energy gap between the valence band and conduction band, and by the temperature of the material too [1, 2]. In gapless semiconductors and semimetals concentration of intrinsic carriers can be higher [8]. In doped semiconductors, the concentration of carriers (electrons in n-type semiconductors and holes in p-type semiconductors) can be significantly increased (within limits 13 18 310 ...10 cm ) and it depends mainly on the dopant concentration [9]. All crystalline compounds have a mixed covalent- ionic bond. For example, in gallium arsenide GaAs , the contribution of the ionic bond to the total bond energy is 32%. The presence of ionic bond in semiconductor compounds will influence on the polarization properties of semiconductors and, accordingly, the frequency dis- persion of the dielectric constant of semiconductors [10, 11]. The appearance of a phonon component in the die- lectric constant, in turn, will lead to a significant change spectra of longitudinal (potential) and transverse (vor- tex) electromagnetic oscillations in semiconductors. Since solid-state plasma is characterized by a high concentration of carriers and by a degree of homogenei- ty and stability, it seems very promising to use plasma of semiconductors and semimetals to realization wake methods of relativistic charged particles (primarily elec- trons and positrons) acceleration [12]. In such scheme an intense relativistic electron bunch passes through vacuum channel of a semiconductor (semimetal) and excites wake eigen oscillations (plasmons, optical elec- tromagnetic waves in the infrared range), which are in Cherenkov synchronism with a relativistic electron bunch 0( )k c   ,  is wave frequency, ( )k  is longitudinal wavenumber, 0 0 / 1v c   , c is speed of light in vacuum. Excited wake waves can be used to accelerate charged particles [13, 14]. It is possible to talk about long-lived excitations in a solid-state plasma, as in any other medium, only if their eigen frequencies significantly exceed the frequencies of collisions with phonons and impurity atoms. Further we will assume that this condition is satisfied. In the present work, the process of excitation of wake electromagnetic fields in the polar semiconductor waveguide by a relativistic electron bunch is investigat- ed. The wake field includes the longitudinal hybrid plasmon-phonon oscillations of a polar semiconductors, аnd a set of eigen bulk electromagnetic waves (bulk polaritons) of the polar semiconductor waveguide. Our aim is to obtain wake wave intensity, the frequency spectrum and a spatio-temporal structure of excited wake fields. 1. STATEMENT OF THE PROBLEM Let's consider the homogeneous polar semiconduc- tor cylinder of radius b, the side surface of which is covered with a perfectly conductive metal film. Along the axis of the polar semiconductor waveguide, an ax- isymmetric relativistic electron bunch moves uniformly and rectilinearly. Below we will only talk about iso- tropic media. These are primarily crystals with a cubic lattice. For the polar semiconductors the permittivity have the form [10, 11 ] 22 2 2 2 2 ( ) pL opt T                 , (1) opt is optical permittivity, L and T are frequencies of the longitudinal and transverse phonons, /p Len opt   is frequency of plasma oscillation, 24 e Len e n e m    is Langmuir frequency, en is free car- rier concentration, em is effective mass of electrons, e is electron charge. The first term in the expression for the permittivity (1) is due to the contribution to the total polarization of the ionic subsystem of the crystal, and the second term takes into account the polarization of free carriers (plasma). mailto:onish@kipt.kharkov.ua ISSN 1562-6016. ВАНТ. 2022. №3(139) 77 2. DISPERSION PROPERTIES OF POLAR DIELECTRIC WAVEGUIDE Let us now briefly discuss the question of the propa- gation of electromagnetic waves in an ion dielectric waveguide. Dispersion equations for potential longitu- dinal oscillations and electromagnetic waves have the forms ( ) 0   , (2) 22 2 2 2 ( ) 0n zk c b       , (3) zk is longitudinal wave number. The dielectric constant is described by the formula (1). The solution of the equation (2) is two frequencies 2 2 2 2 2 ( )2 2 2 2 2 L p L p lp T p                   . (4) As a result of the interaction of longitudinal optical phonons and plasma oscillations, high-frequency (HF) (sign +) and low-frequency (LF) (sign -) hybrid plas- mon-phonon oscillations are formed. At that, in the case L p  the frequency of HF hybrid oscillations always exceeds the frequency of longitudinal optical phonons ( ) lp L   , and the frequency of LF hybrid oscillations is always lower than the plasma frequency ( ) lp p   . At a low concentration of free carriers 2 2 L p  , one of them ( ( ) lp  ) is close to the frequency of longitudinal optical phonons ( )2 2 2 st opt lp L p st           , and the other ( ( ) lp  ) to the frequency ( ) opt lp p st       , where st is static permittivity ( st opt  ). If p L  , then the frequencies L and p change places. The frequency ( ) lp  is higher than the plasma frequency, and the frequency ( ) lp  is lower than the frequency of longitudinal optical phonons. At high plasma density 2 2 p L  , the frequency ( ) lp  is close to the plasma frequency ( )2 2 2 lp p L st          , st opt     , and the frequency ( ) lp  is approximately equal to the frequency ( ) opt lp L st       . Let us now consider the dispersion properties of transverse (vortex) electromagnetic waves in polar sem- iconductors (transverse polaritons), which are described by equation (3). Taking into account the expression for the dielectric constant (1), this equation can be reduced to the following 2 2 2 2 2 2 2 2 L np s T k v           , (5) where /s optv c  , 2 2 2 2 2 n np p opt c b       . Roots of the biquadratic equation (5) 2 2 2 2 ( )2 2 L np s ntp k v        2 2 2 2 2 2 2 2 2( ) 2 L np s T np s k v k v                (6) describe the dispersion of HF (sign +) and LF (sign -) of transverse polaritons. The dispersion curve of HF polaritons starts at the cutoff frequency 2 2 2 2 2 ( )2 2 2 2 2 L np L np nc T np                   . Further, with an increase in the longitudinal wave number k , the frequency increases and at k  , the dispersion curve asymptotically approaches from above to the straight line skv  . The dispersion curve of low-frequency polaritons ( ) ( )ntp k  also begins with cut- off frequency 2 2 2 2 2 ( )2 2 2 2 2 L np L np nc T np                   , then enters on the section of a straight line / stkc  and, at k  asymptotically approaches from below to the frequency of transverse optical pho- nons. The presence of free carriers in a polar semiconduc- tor raises the cutoff frequencies, and in general, the qualitative picture of the dispersion of transverse polaritons in polar semiconductor waveguides is the same as in ionic dielectric waveguides [15]. 2.1. DETERMINATION OF THE GREEN FUNCTION We will solve the problem of wake field excitation by an axisymmetric relativistic electron bunch in the polar semiconductor waveguide with permittivity (1) as follows [12, 16]. We will find wake field GE (Green function) of elementary charge, having the form of a thin ring with charge dQ . Elementary charge density of an infinitely thin ring has the form 0 0 0 0 0 0 ( , ) ( ) ( ) 2 b o dQ r t r r z d t t v r v          , (7) 0 0 0 0 0 0( , )2dQ j t r r dr dt ,    0 0 0 0 0( , ) / /b b eff eff Q j r t R r r T t t s t  , (8) where Q is full charge of bunch, 0t is time of entry of elementary charge, 0r is radius ring, 0v is bunch veloci- ty, ,b bt r are characteristic duration and transverse bunch size,  0 / bR r r is function described transversal profile 78 ISSN 1562-6016. ВАНТ. 2022. №3(139) of bunch density, effs is characteristic square of bunch transverse section, / 2 0 0 0 0 ˆ ˆ, 2 ( ) . bb r eff bs r R d        The function  0 / bT t t describes longitudinal pro- file, efft is effective bunch duration, 0 0 0 ˆ ˆ, 2 ( ) .eff bt t T d        Let us represent the electromagnetic field excited by an elementary ring charge (7) as 0 0 0 0( , , , ) ( , , , )GE r r z t t dQ r r z t t    . Then the full electromagnetic field, excited by an electron bunch of finite dimensions, is found by sum- ming (integrating) the fields of elementary ring bunches 0 0 0 0 0 0 0 0 ( , , ) 2 ( , ) ( , , , ) b t E r z t r dr dt j r t r r z t t      . Taking into account relation (8), this expression can be written as follows 0 0 0 0 2 ( , , ) b eff eff b rQ E r z t R r dr s t r          0 0 0 0( , , , ) t b t T r r z t t dt t           . The Green's function for the considered dielectric waveguide with a permittivity ( )  was obtained in [12, 16] in the form of a series in Bessel functions 0 0 0 2 2 1 1 2 ( , ) ( ) ( ) n n Gz n n n r r J J i b b E r t dQ S t b J                   , (9) where 2 ( ) ( ) ( ) ( ) i t n n kd S t e D             , (10) 2 2 2 0 2 ( ) ( ) n n lD k k b       , (11) 0 0/ , /lk v k c   , n are the roots of the Bessel function 0 ( )J x , 0 0/t t t z v   . The zeros of the dielectric constant ( ) 0   are the poles of the integrand (10). Calculating the residues at the poles ( ) 0,lp i     ( ) 0lp i     , we find the potential part of the Green's function ( ) ( )2 ( ) ( ) ( ) ( ) 0( , ) 2 ( ) ( , )cosl Gz lp lp lp lp lp opt dQ E r t t k L G k r k r t         ( )2 ( ) ( ) ( ) ( ) 0( , )coslp lp lp lp lpk L G k r k r t       , (12) ( )2 2 ( ) ( )2 ( )2 lp T lp lp lp L            , 2 ( )2 ( ) ( )2 ( )2 T lp lp lp lp L            , 1, 0, ( ) 0, 0, t t t      ( ) ( ) 0/lp lpk v  , 0 0 0 0 0 0 0 0 0 0 0 ( ) ( , ), , ( ) ( , ) ( ) ( , ), , ( ) I kr kr kb r r I kb G kr kr I kr kr kb r r I kb           (13) 0 0 0 0 0( , ) ( ) ( ) ( ) ( )kr kb I kb K kr I kr K kb   . In the limiting case 1kb  , the expression for the function 0( , )G kr kr is simplified 0 0 0 0 0 0 0 0 0 ( ) ( ), , ( , ) ( ) ( ), . I kr K kr r r G kr kr I kr K kr r r     The potential wake field, excited by an ring bunch, contains two waves: HF and LF plasmon-phonon waves, the frequencies of which are determined by ex- pression (4). The integrand in (10) also has poles that are the roots of the equation 2 2 2 0 2 ( ) ( ) 0n n lD k k b        . (14) Equation (14) determines the frequency spectrum of the radial harmonic with the number n of electromag- netic waves excited by the relativistic electron bunch in ion dielectric waveguide. With respect to the square of the frequency 2 , the spectrum equation (14) reduces to determining the roots of the quadratic equation. The frequencies ( ) ntp  , corresponding to the roots of this equation, lie in the microwave and infrared ranges. The spectrum equation (19) can be written as follows 2 22 2 2 2 2 2 2 2 2 0 1 ,nL opt n Len T c b                   where 0 0 /v c  . The roots of this equation are of the form ( ) 2 21 2 ntp T st n opt d d         2 2 2 2 24T st n n T std d          . (15) Here 2 2 0 0,opt opt st std d        . For the frequency ( ) ntp  it is always ( ) ntp L   , and for the frequency ( ) ntp  we have ( ) ntp T   . In the limit- ing case 2 2 T st nd  expression for frequencies (15) are simplified 2 2 2 2 2 ( ) 20 2 2 0 1 n n ntp Len st st c d b                  , (16) 2( ) 2 2st ntp T n opt opt st d d d d        . (17) In the limiting case 2 2 2 st n L T opt        expressions for the frequencies of the eigen waves of the semiconductor waveguide follow from (15) and have the form ISSN 1562-6016. ВАНТ. 2022. №3(139) 79 2( ) 2 ,ntp T   2 2 2 2 ( ) 0 2 0 1 n n ntp opt optd           . Frequency ( ) ntp  (16) is known in the theory of wake fields excitation by relativistic electron bunch in semi- conductor waveguides [12] and is in the microwave (terahertz) range. The frequency ( ) ntp  (17) lies in the infrared range and the process of wake fields excitation at this frequency, as it seems to us, has not been previ- ously studied. For further analysis, the Fourier integral (10) is con- veniently represented as      2 2 2 22 2 2 ( ) 2 ( ) 0 ( )1 ( ) ( ) i t T n opt ntp ntp ek S t d d k                     . By calculating the residues in the poles ( ) 0,ntp i     ( ) 0,ntp i     we find the electro- magnetic part of the Green's function ( ) ( ) ( ) 0 0 0( , , ) ( , , ) ( , , )t Gz tz tzE r r t dE r r t dE r r t   , (18) ( ) ( ) ( ) 0 0 1 ( , , ) ( , ) ( )cosw tz n n ntp nopt dE dE r r t r r t t d           , (19) 2 2 2 2 ( ) 22 ( ) 2 ( ) 2 ( ) ( ) ntp Tn n n ntp ntp ntpk b                , 2 2 2 2 2 ( )2 ( ) 2 ( ) 2 ( ) ( ) T ntpn n n ntp ntp ntpk b                , ( ) ( ) 0/ntp ntpk v , 0 0 0 0 2 1 ( / ) ( / ) ( , ) ( ) n n n n J r b J r b r r J      , 2 4 w dQ dE b  . Expressions (18), (19) describe bulk wake electro- magnetic field excited by the infinitely thin electron ring bunch in polar semiconductor waveguide. Thus, we obtained the Green function, which con- tains the longitudinal (potential) and electromagnetic (vortex) parts. The potential part is a field of longitudi- nal bulk plasmon-phonon oscillations. As for the elec- tromagnetic part of the Green function, it contains a set of radial electromagnetic waves of polar semiconductor waveguide. 2.2. EXCITATION OF WAKEFIELDS BY AN ELECTRON BUNCH OF FINITE SIZE The resulting electromagnetic field ( , )E r  of the electron bunch can be determined by summing the fields GE of elementary electron ring charges. We first con- sider the excitation of wake plasmon-phonon oscilla- tions by the relativistic electron bunch of finite dimen- sions. For the wake field of plasmon-phonon oscillations we obtain the following expression ( ) ( )2 ( ) ( ) ( )2 ( , ) ( ) ( )l z lp lp lp lp opt Q E r k L k r Z          ( )2 ( ) ( ) ( )( ) ( )lp lp lp lpk L k r Z         , (20) where 0/t z v   ,  0 0 0 1 ( ) / cos ( )b eff Z T t d t           , (21)  0 0 0 0 0 2 ( ) / ( , ) b b eff kr R r r G kr kr r dr s     . (22) The function ( )Z  describes the distribution of the wake field at a frequency  in the longitudinal di- rection at each moment of time and function ( )kr de- scribes the distribution of the wake field by cross sec- tion of the waveguide. We will consider an electron bunch with a symmetric longitudinal profile 0 0( ) ( )T T   . Behind a bunch   , the wake field (20) of plas- ma oscillations has the form of a monochromatic wave ( ) ( )2 ( ) ( ) ( ) ( )2 ˆ( , ) ( ) ( ) ( )cosl z lp lp lp lp lp opt Q E r k L T k r              ( )2 ( ) ( ) ( ) ( )ˆ( ) ( )coslp lp lp lp lpk L T k r          , (23) where 0 2ˆ( ) cos( ) eff b t T T t dt t t            (24) is the Fourier component of a function  / bT t t on fre- quency  . We present the expressions for the Fourier component ( )T  for Gaussian longitudinal profile of the electron bunch 2 2 2 0 / ( ) /4 0 ˆ( / ) , ( )b bt t bT t e T e       . Wake HF and LF plasmon-phonon waves is most ef- ficiently radiated when the coherence condition is ful- filled ( ) 2( ) 1lp bt   . If the inequality ( ) 2( ) 1lp bt   holds, then the electron bunch radiates incoherently and the amplitude of the wake plasma wave is exponentially small. Let’s consider an electron bunch with a Gaussian transverse profile 2 2/ ( ) br r R r e   . (25) When the condition 1Lk b  is satisfied on the axis 0r  the function ( )kr takes on the value   2 2 1 (0) , 2 4 b b b b k r e Ei        , (26)   z te Ei z dt t    is integral exponential function. For thin 1b  and wide 1b  bunches the asymptotic representations for function (26) are 1 1 ln , 1, 2 (0) 1 , 1. b b b b                    Thus, with the full coherence of the Cherenkov exci- tation of wake plasma wave ( ) 1lp bt   , ( ) 1lp bk r  the wake field on the axis of the waveguide takes the max- imum value 80 ISSN 1562-6016. ВАНТ. 2022. №3(139) ( ) ( )2 ( ) ( ) ( ) 2 2 ( , ) ( ) ln cosl z lp lp lp opt lp b Q E r k L k r             ( )2 ( ) ( ) ( ) 2 ln coslp lp lp lp b k L k r          . (27) Let us now consider the excitation of wake electro- magnetic waves by an electron bunch in the dielectric waveguide. Using the electromagnetic Green function (23), we obtain the wake electromagnetic field as a su- perposition of radial modes 0 ( ) ( ) ( ) 2 2 1 1 4 ( , ) ( ) ( ) n t z n n ntp n n r J Q b E r Z b J                    ( ) ( )( )n ntpZ      , (28) 0 0 0 0 0 0 2 b n n eff b r r R J r dr s r b                 . For a symmetric electron bunch in the “wave zone” ( ) 1ntp   , the wake field (28) is a superposition of radial monochromatic modes of the semiconductor waveguide 0 ( ) ( ) ( ) ( ) 2 2 1 1 4 ˆ( , ) cos ( ) n t z n n n ntp n n r J Q b E r T b J                      ( ) ( ) ( )ˆ cosn n ntpT       , (29) where ( ) ( )ˆ ˆ( )n ntpT T   is Fourier component (22). We also present an expression for the power of the wake electromagnetic radiation, which we define as the component of the total Poiting vector along the dielec- tric waveguide axis ( ) ( ) 0 2 4 b t t r c P E H rdr     . Angle brackets mean averaging over high-frequency wake field oscillations. As a result, for the radiated power we obtain the following expression ( ) ( )2 2 ( ) ( )2 ( )2 2 2 1 1 ˆ2 ( ) ( ) ntp ntbn ntp n n n n n k P Q c T cJ                ( ) ( ) ( ) ( )2 ( )2ˆ( ) ntp ntb ntp n n k T c              . (30) For an electron bunch with a Gaussian longitudinal and transverse profiles the coefficients n and ˆ nT , which are determined by the specific form of the trans- verse and longitudinal density profiles of the bunch, have the form ( )2 2 ( )ˆ exp 4 ntp b n t T             , (31) 2 0 1/ 0 0 0 0 0 2 ( ) , b b n n b b r J e d b           . When the condition 1b  is satisfied, the expres- sion for the coefficient n is simplified 2 21 exp 4 n n b          . (32) Accordingly, expression (29) for a Gaussian bunch, taking into account relations (31), (32), takes the form ( ) 22 2 0 ( ) ( ) ( )4 4 2 2 1 1 4 ( , ) cos ( ) ntp bn b tn t z n ntp n n r J Q b E r e e b J                           ( ) 2 ( ) ( )4 cos ntp bt n ntpe            . (33) For the radiated power (30), we have 2 2 ( )2 2 ( ) ( ) 2 2 ( ) ( )2 2 2 2 1 1 2 ( ) ( ) n b ntp bt ntp ntb ntp n n n n ke P Q c e cJ                   ( )2 2 ( ) ( ) ( ) ( )2 2( ) ntp bt ntp ntb ntp n k e c               . From expression (33) it follows that an electron bunch excites finite number of radial modes of electro- magnetic radiation, for which the coherence condition ( )2 2 2 2 21, / 1ntp b n bt r b    for excitation by an electron bunch is satisfied. CONCLUSIONS Polar semiconductors are the plasma-like media, in which wake electromagnetic fields can be excited by the relativistic electron bunches. In the semiconductors crystals of this type, there are two groups of oscillation branches of in the infrared and terahertz frequency ranges. These are, first of all, longitudinal HF and LF plasma-phonon oscillations of a polar semiconductor. And also in the infrared range there are the two branch- es corresponding to HF and LF transverse bulk polaritons of the media, which is a set of electromagnet- ic eigen waves of a polar semiconductor waveguide. For all of these branches, analytical expressions for the wake fields excited by a relativistic electron bunch are obtained and investigated. REFERENCES 1. V.L. Bonch-Bruevich, S.G. Kalashnikov. Semicon- ductor physics. M.: “Nauka”, 1977, 672 p. 2. I.A. Anselm. Introduction to the theory of semicon- ductors. M.: “Nauka”, 1968, 616 p. 3. F. Platzman, P. Wolf. Waves and interactions in solid plasma. M.: “Mir”, 1975, 427 p. 4. N.B. Brandt, V.A. Kulbchinsky. Quasiparticles in condensed matter physics. M.: “Fizmatlit”, 2005, 632 p. 5. Yu.K. Pozhela. Plasma and current instability in semiconductors. M.: “Fizmatlit”, 1977, 378 p. 6. N.N. Beletsky, A.A. Bulgakov, S.I. Khankina, V.M. Yakovenko. Plasma instabilities and nonline- ar phenomena in semiconductors. 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Kyiv: “Naukova Dumka”, 1993, 207 p. 14. V.A. Balakirev, I.N. Onishchenko, D.Yu. Sidorenko, G.V. Sotnikov. Chardge particle accelerated by wakefields in a dielectric resonator with exciting electron bunch channel // Technical Physics Letters. 2003, v. 29, № 7, p. 589-591. 15. V.A. Balakirev, I.N. Onishchenko. Wakefield exci- tation by a short laser pulse in ion dielectrics. arXiv:2002.09335 [physics.acc-ph]. 16. V.A. Balakirev, I.N. Onishchenko. Wake excitation of volume and surface waves by a relativistic elec- tron bunch in ion dielectric // Problems of Atomic Science and Technology. Series “Plasma Electron- ics and New Methods of Acceleration”. 2020, № 1, p. 270-275. Article received 19.10.2021 ЗБУДЖЕННЯ КІЛЬВАТЕРНИХ ПОЛІВ РЕЛЯТИВІСТСЬКИМ ЕЛЕКТРОННИМ ЗГУСТКОМ У ПОЛЯРНОМУ НАПIВПРОВIДНИКОВОМУ ХВИЛЕВОДІ В.A. Балакiрєв, I.М. Онiщенко Досліджено процес збудження кільватерних полів релятивістським електронним згустком у полярних напiвпровідниках. Розглянуто циліндричний напiвпровiдниковий хвилевод, вздовж вісі якого рухається релятивістський електронний згусток. Показано, що збуджене кільватерне поле в терагерцовому та інфрачервоному дiапазонах частот містить у собі поле повздовжніх ВЧ- i НЧ-гiбридних плазмон-фононних коливань, а також поле ВЧ- та НЧ-поперечних полярiтонiв, яке є набором власних електромагнітних хвиль напiвпровiдникового хвилеводу. Отримана просторово-часова структура результуючого кільватерного поля, визначена iнтенсивнiсть збуджених кільватерних хвиль. ВОЗБУЖДЕНИЕ КИЛЬВАТЕРНЫХ ПОЛЕЙ РЕЛЯТИВИСТСКИМ ЭЛЕКТРОННЫМ СГУСТКОМ В ПОЛЯРНОМ ПОЛУПРОВОДНИКОВОМ ВОЛНОВОДЕ В.A. Балакирев, И.Н. Онищенко Исследован процесс возбуждения кильватерных полей релятивистским электронным сгустком в поляр- ных полупроводниках. Рассмотрен цилиндрический полупроводниковый волновод, по оси которого движет- ся релятивистский электронный сгусток. Показано, что возбуждаемое кильватерное поле в терагерцовом и инфракрасном диапазонах частот состоит из поля продольных ВЧ- и НЧ-гибридных плазмон-фононных ко- лебаний, а также поля ВЧ- и НЧ-поперечных поляритонов, которые представляют собой набор собственных электромагнитных волн полупроводникового волновода. Получена пространственно-временная структура результирующего кильватерного поля, определена интенсивность возбуждаемых кильватерных волн.

Excitation of wake fields by a relativistic electron bunch in a polar semiconductor waveguide

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