Cherenkov radiation of a laser pulse in ion dielectrics

The process of excitation of Cherenkov electromagnetic radiation by a laser pulse in ion dielectric waveguide is investigated. Nonlinear electric polarization in isotropic ion dielectric medium and, accordingly, polarization charges and currents induced by a ponderomotive force of a laser pulse are...

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Datum:	2019
Hauptverfasser:	Balakirev, V.A., Onishchenko, I.N.
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Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2019
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Advanced methods of acceleration
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/195164
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spelling	irk-123456789-1951642023-12-03T16:36:07Z Cherenkov radiation of a laser pulse in ion dielectrics Balakirev, V.A. Onishchenko, I.N. Advanced methods of acceleration The process of excitation of Cherenkov electromagnetic radiation by a laser pulse in ion dielectric waveguide is investigated. Nonlinear electric polarization in isotropic ion dielectric medium and, accordingly, polarization charges and currents induced by a ponderomotive force of a laser pulse are determined. Frequency spectra of the excited wakefields in the infrared and microwave frequency ranges are obtained. The spatiotemporal structure of the wakefield in ion dielectric waveguide is obtained and studied. It is shown that the excited field consists of a potential polarization electric field, as well as a set of eigen electromagnetic waves of ion dielectric waveguide. Досліджено процес збудження черенковського електромагнітного випромінювання лазерним імпульсом в іонному діелектричному хвилеводі. Визначена нелiнiйна електрична поляризація в іонному діелектричному середовищі та, відповідно, поляризацiйнi заряди i струми, iндукованi пондеромоторною силою з боку лазерного імпульсу. Отримана та досліджена просторово-часова структура кільватерного поля в діелектричному хвилеводі. Показано, що збуджуване поле складається з потенціального поляризаційного електричного поля поздовжніх оптичних фононів та набору власних електромагнiтних хвиль іонного діелектричного хвилеводу. Исследован процесс возбуждения черенковского электромагнитного излучения лазерным импульсом в ионном диэлектрическом волноводе. Определена нелинейная электрическая поляризация в ионной диэлектрической среде и, соответственно, поляризационные заряды и токи, индуцированные пондеромоторной силой со стороны лазерного импульса. Получена и исследована пространственно-временная структура кильватерного поля в ионном диэлектрическом волноводе. Показано, что возбуждаемое поле состоит из потенциального поляризационного электрического поля продольных оптических фононов и набора собственных электромагнитных волн ионного диэлектрического волновода. 2019 Article Cherenkov radiation of a laser pulse in ion dielectrics / V.A. Balakirev, I.N. Onishchenko // Problems of atomic science and technology. — 2019. — № 4. — С. 39-47. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq http://dspace.nbuv.gov.ua/handle/123456789/195164 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	Advanced methods of acceleration Advanced methods of acceleration
spellingShingle	Advanced methods of acceleration Advanced methods of acceleration Balakirev, V.A. Onishchenko, I.N. Cherenkov radiation of a laser pulse in ion dielectrics Вопросы атомной науки и техники
description	The process of excitation of Cherenkov electromagnetic radiation by a laser pulse in ion dielectric waveguide is investigated. Nonlinear electric polarization in isotropic ion dielectric medium and, accordingly, polarization charges and currents induced by a ponderomotive force of a laser pulse are determined. Frequency spectra of the excited wakefields in the infrared and microwave frequency ranges are obtained. The spatiotemporal structure of the wakefield in ion dielectric waveguide is obtained and studied. It is shown that the excited field consists of a potential polarization electric field, as well as a set of eigen electromagnetic waves of ion 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 a laser pulse in ion dielectrics
title_short	Cherenkov radiation of a laser pulse in ion dielectrics
title_full	Cherenkov radiation of a laser pulse in ion dielectrics
title_fullStr	Cherenkov radiation of a laser pulse in ion dielectrics
title_full_unstemmed	Cherenkov radiation of a laser pulse in ion dielectrics
title_sort	cherenkov radiation of a laser pulse in ion dielectrics
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2019
topic_facet	Advanced methods of acceleration
url	http://dspace.nbuv.gov.ua/handle/123456789/195164
citation_txt	Cherenkov radiation of a laser pulse in ion dielectrics / V.A. Balakirev, I.N. Onishchenko // Problems of atomic science and technology. — 2019. — № 4. — С. 39-47. — Бібліогр.: 12 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT balakirevva cherenkovradiationofalaserpulseiniondielectrics AT onishchenkoin cherenkovradiationofalaserpulseiniondielectrics
first_indexed	2025-07-16T23:00:18Z
last_indexed	2025-07-16T23:00:18Z
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fulltext	ISSN 1562-6016. ВАНТ. 2019. №4(122) 39 ADVANCED METHODS OF ACCELERATION CHERENKOV RADIATION OF A LASER PULSE IN ION DIELECTRICS 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 excitation of Cherenkov electromagnetic radiation by a laser pulse in ion dielectric waveguide is investigated. Nonlinear electric polarization in isotropic ion dielectric medium and, accordingly, polarization charg- es and currents induced by a ponderomotive force of a laser pulse are determined. Frequency spectra of the excited wakefields in the infrared and microwave frequency ranges are obtained. The spatiotemporal structure of the wake- field in ion dielectric waveguide is obtained and studied. It is shown that the excited field consists of a potential po- larization electric field, as well as a set of eigen electromagnetic waves of ion dielectric waveguide. PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq INTRODUCTION An electric charge moving in a dielectric medium with superluminal speed radiates electromagnetic waves called Cherenkov radiation [1, 2]. The electric field of a moving charge polarizes the atoms (ions) of the dielec- tric medium, which in turn coherently re-radiate elec- tromagnetic waves. A similar effect takes place when a high-power laser pulse propagates in a dielectric [3, 4]. A necessary con- dition for the appearance of a Cherenkov radiation of a laser pulse is that the group velocity of the laser pulse must exceed the phase velocity of the radiated electro- magnetic wave. The effect of Cherenkov radiation of a laser pulse in a dielectric medium is as follows. When a laser pulse propagates in a dielectric a pulsed pondero- motive force quadratic in the laser field propagating in the medium with the group velocity of the laser pulse will act on the bonded electrons of the atoms (ions) of a medium. This force, in turn, will lead to the polarization of the atoms (ions) of the dielectric. Induced polariza- tion charges and currents will coherently radiate elec- tromagnetic waves (Cherenkov radiation). The effect of the Cherenkov radiation of a laser pulse is quite similar to the Cherenkov radiation of an electron bunch moving in a dielectric medium, with the difference that the pon- deromotive force of the laser pulse plays the role of the pulse electric field of the electron bunch. The Cherenkov wakefield radiation in a dielectric medium of a high-power ultrashort laser pulse can be used to accelerate charged particles similarly to a laser- plasma wakefield acceleration method [5]. In [6, 7], the effect of the Cherenkov radiation of a laser pulse was studied using a simple model of a die- lectric medium consisting of atoms of the same type. A bright example of such a medium is diamond, whose crystal lattice consists only of carbon atoms. The carbon atoms in diamond are held by covalent forces, which are of a quantum nature and arise as a result of the bonding pairs of the valence electrons of neighboring atoms (overlapping of the wave functions of the valence elec- trons). Atoms retain their electrical neutrality. Only the electron shells of atoms contribute to the electric polari- zation of covalent dielectrics. Due to large mass the nuclei of atoms do not participate in the polarization of dielectrics. Namely, due to the electronic nature of po- larization, for covalent dielectrics, the values of dielec- tric constant in the optical frequency range and in the static limit are close. A much wider class of dielectrics is formed by ion- bonded dielectrics. No pure element of the periodic ta- ble is related to dielectrics of this class. All ion dielec- trics are chemical compounds. Ion crystals are com- posed of positive and negative ions. These ions form a crystal lattice as a result of Coulomb attraction of oppo- sitely charged ions. The traditional example of ion dielectrics are crys- tals of an alkali-halide group with the formula I VIIA B (for example, NaCl and KCl). In crystals of this group, it is energetically advantageous for an atom of alkali metal to transfer its valence electron to an adjacent hal- ide atom and fill its outer shell. As a result, an ion bond arises between the atoms of different elements. This bond is due to the interaction of oppositely charged ions. Below we restrict consideration to the simplest case of diatomic crystals. These dielectrics also include ion crystals with the formulas II VIA B and III VA B . Note also that in ion crystals a covalent bond share is always present. For example, in ion crystals of the alkali halide group, in the total binding energy, it is less than 5% [8- 10]. In determining the total electric polarization induced by a laser pulse in an ion dielectric, it is necessary to take into account both the total contribution of the po- larizations of the electron shells of all the ions which form the crystal and the total contribution of the positive and negative ions of the crystal. In this paper, a system of nonlinear equations of macroscopic electrodynamics is formulated, which de- scribes the process of excitation of Cherenkov radiation by a laser pulse in an ion dielectric medium. On the basis of these equations, the effect of the Che- renkov radiation of a laser pulse in a dielectric wave- guide (light guide) will be investigated. A complete picture of the excitation of Cherenkov radiation by a laser pulse propagating in an ion dielectric is presented. The frequency spectrum of Cherenkow radiation is de- termined. The spatiotemporal structure of the Cheren- kov electromagnetic field has been obtained and stud- ied. ISSN 1562-6016. ВАНТ. 2019. №4(122) 40 1. PROBLEM STATEMENT. BASIC EQUATIONS A laser pulse (wave packet) with electromagnetic field components propagates in a homogeneous dielec- tric medium 0 1( , ) ( , ) . . 2 Li LE r t E r t e c cψ= +     , 0 0 1( , ) ( , ) . . 2 Li LH r t rot E r t e c c ik ψ = +      , (1) L L Lk r tψ ω= −   . Lk  is wave vector, 0 /Lk cω= , Lω is carrier frequency of a laser pulse, 0 ( , )E r t   is a laser pulse envelope slowly varying in space and time. Under the action of the ponderomotive force (RF- pressure force) a polarization arises in the dielectric, slow on the carrier frequency scale, which in turn is the source of the electromagnetic field of the laser pulse (Cherenkov radiation). Maxwell's system of equations describing the electromagnetic field, which is excited by a polarization induced by a laser pulse, has the form 1 ,HrotE c t ∂ = − ∂   1 4 ,E ProtH c t c t π∂ ∂ = + ∂ ∂    4divE divPπ= −   , 0divH =  , (2) P  is vector of electric polarization. In ion dielectrics there are two mechanisms of elec- tric polarization. This is primarily an electronic polari- zation mechanism inherent in all types of dielectrics. Electron polarization is due to the displacement of a shell of bound electrons relative to their nuclei under the action of an electric field. The second polarization mechanism is ionic; it is caused by the relative dis- placement of oppositely charged ions. It should be noted that such a separation of the polarization mechanisms is not quite rigorous. A more adequate is the polarization model, in which the ions are not only displaced, but also deformed (the model of deformable ions [9]). Under the action of an electric field, the electron shell of each ion will be deformed and displaced relative to the nucleus, so that an internal dipole moment is forms in the ion, which will weaken the applied electric field. According- ly, the force causing the displacement of the ions will decrease and, as a result, the ion polarization will de- crease. Qualitatively, this weakening effect can be taken into account by renormalizing the ion charge or by in- troducing the effective Scigetty charge [9]. For most ion dielectrics, the Scigetti charge is 0.7…0.9 of the ion charge. However, to simplify the analysis of the Che- renkov effect of a laser pulse, we restrict ourselves to the model of hard (non-deformable) ions. First of all, we formulate equations describing the electron polarization of diatomic ionic crystals induced by a laser pulse. Induced electron polarization can be described in the framework of the following model [10]. An atom is represented as a point nucleus surrounded by an electron cloud. When the electron cloud is displaced as a whole relative to the nucleus, a dipole moment of the atom p Zer= −   arises, where r is the radius-vector of the electron cloud center, Ze is charge nucleus. Ac- cordingly, a dipole returning force will act on the cloud [11] ( )2 3 0 de Ze F r R = −   , which leads to harmonic dipole oscillations of an atom with its eigen frequency 2 3 0 de Ze mR ω = , (3) 0R is the radius of the atom. In a condensed medium, each atom is in a local (act- ing) electric field locE  , which can differ substantially from the macroscopic field E  included in Maxwell's equations (2). The local electric field locE  includes both the external field and the total electric field of the in- duced dipoles surrounding a given atom (ion). In a crys- tal medium with a cubic crystal lattice, the local electric field is described by the Lorentz formula [8 - 12] loc 4E E P 3 π = +    , (4) Taking into account the local field effect, the expres- sion for the ponderomotive force acting on the electrons of the crystal ion shell from the side of the laser pulse has the form [6, 7] 2 ( ) L pon 2 2 de( ) L 2e 1F 4m 3 ± ± e + = Π ω −ω   , (5) ( ) ( )2 L 0 0 0 0 0E E E 1 E E 3 e − ∇ ∇ Π = ∇ + +       . (6) The indexes ( ± ) correspond to positive and negative ions, ( )deω ± are frequencies (3) of the dipole oscillations of the electron shells of ions, Le is the dielectric con- stant of the medium at the frequency of the laser pulse. The first term in (6) describes the gradient force of HF- pressure. The second term appears only in the case of a crystal medium and is caused by the difference between the local electric field in a crystal and the electric field of a laser pulse in vacuum. In dielectric media where the active field coincides with the external field, for exam- ple, in the gas dielectric or plasma this term is absent. Under the action of ponderomotive force in dielec- tric electron polarization appears ( ) ( ) e e eP P P+ −= +    , where ( ) eP ±  are the partial electron polarizations of posi- tive and negative ions. Partial electron polarizations are described by the following equations [6, 7] 2 ( ) 2 ( ) 2 2 ( ) ( ) ( )2 ( )0 1 1 3 4 2 , 4 3 e de e pe pe L L P P P E t eN m ω ω ω π e α ± ± ± ± ± ± ∂ + − = − ∂ + − Π      (7) where ( ) ( ) 2 2 ( ) 1 L de L eq m α ω ω ± ± ± = − are electron polarizabilities of individual positive and negative ions at laser pulse frequencies, ( ) 2 0 ( ) 4 pe eq N m π ω ± ± = is square of the effective plasma frequency, ( )q ± is full charge of the electron shell of the ISSN 1562-6016. ВАНТ. 2019. №4(122) 41 corresponding ion, 0N is concentration of ions of each type. The left-hand sides of equations (7) for electron po- larizations include complete polarization of the ion die- lectric. ( ) ( ) e e iP P P P+ −= + +     , (8) which also includes ion polarization iP  . Ion polarization occurs as a result of the relative displacement of posi- tive and negative ions under the action of an electric field. If the ions are not deformed, then the dipole mo- ment of the unit cell of a crystal containing two ions of opposite sign is ( ) ( ), ,i i i i i ip q R R R R+ −= = −      where ( ) iR ±  are the displacements of positive and nega- tive ions from the equilibrium position, iq is ion charge. If the crystal deformation is smooth over the microscop- ic scale of the crystal (unit cell size), then the displace- ments of positive and negative ions obey to the equa- tions [6] ( ) 2 ( ) ( ) ( ) ( ) 2 4 , 3 i i i i d R M K R R q E P dt π+ + + −  + − = +         ( ) 2 ( ) ( ) ( ) ( ) 2 4 , 3 i i i i d R M K R R q E P dt π− − − +  + − = − +         which are reduced to one equation for the relative dis- placement of ions 2 2 2 4 3 i i di i d R q R E P Mdt πω  + = +        , (9) ( )M ± are ion masses, K is force parameter, is reduced mass, /di K Mω = is the eigen frequency of ion di- pole oscillations. Note that since the ponderomotive force acting on ions is inversely proportional to the mass of the ion, then it is small and we neglected it in equation (9). The equation for motion (9) implies the following equation for ion polarization 2 2 2 2 2 1 1 , 3 4 i di i pi pi P P P E t ω ω ω π ∂ + − = ∂     where 2 2 04 i pi q N M π ω = is the square of the ion plasma frequency. Thus, partial polarizations are described by a sys- tem of coupled linear oscillators. ( ) 2 ( ) 2 ( ) 2 ( ) ( ) 2 2 ( )0 1 3 21 , 4 4 3 e de e pe e e i L pe L P P P P P t eN E m ω ω e ω α π + + + − + + + + ∂ + − + + = ∂ + = − Π        ( ) 2 ( ) 2 ( ) 2 ( ) ( ) 2 2 ( )0 1 3 21 , 4 4 3 e de e pe e e i L pe L P P P P P t eN E m ω ω e ω α π − − + − − − − − ∂ + − + + = ∂ + = − Π        (10) ( ) 2 2 2 ( ) ( ) 2 2 1 1 . 3 4 i di i pi e e i pi P P P P P E t ω ω ω π + −∂ + − + + = ∂       The external force exciting these oscillators is the ponderomotive force from the side of the laser pulse. The Maxwell equations (2), together with the equa- tions for partial polarizations (10) and the relation (8) for the full polarization, are closed and describe the Cherenkov excitation of electromagnetic radiation of a laser pulse in an ion dielectric. We will solve this system of equations by the meth- od of Fourier transform ( , ) ( ) , ( , ) ( )i t i tE r t E r e d P r t P r e dω ω ω ωω ω ∞ ∞ − − −∞ −∞ = =∫ ∫       , where ( )E rω  , ( )P rω   are Fourier-components of the cor- responding quantities. For example 1( ) ( , ) . 2 i tE r E r t e dtω ω π ∞ −∞ = ∫    From the system of coupled equations for partial po- larizations (10) we find the expression for the Fourier components of the full polarization vector ( ) 1 , 4 P Eω ω ω e ω µ π − = − Π    (11) where 21 ( ) 3( ) , 11 ( ) 3 ω e ω ω + Λ = − Λ (12) 2 2 2 2 2 2 2 2 2( ) ,pi pe pe di de de ω ω ω ω ω ω ω ω ω ω − + − + Λ = + + − − − ( ) ( ) 2 2 2 2( ) .L L de de α α ω ω ω ω ω + − + − Γ = + − − ωΠ  is Fourier-component of the quadratic depend- ence of the ponderomotive force (6) on the intensity of the electric field of a laser pulse. The value ( )e ω is the dielectric constant of a diatomic dielectric with an ion bond. Note that the expression for the dielectric constant (12) follows the Loretz-Lorentz relation [12] ( )( ) ( ) 0 ( ) 1 4 , ( ) 2 3 e e iNe ω π α α α e ω − +− = + + + (13) where ( ) ( ) 2 2 0 1( )e eq m α ω ω ω ± ± = − are electron polarizabilities of ions, 2 2 2 1( ) i i di q M α ω ω ω = − is ion polarizability of a pair of oppositely charged ions in the unit cell. The relation (13) establishes a relation- ship between the dielectric constant and the sum of the polarizations of all particles forming the crystal. Maxwell's system of equations for Fourier- component of the electromagnetic field, taking into ac- count the relation for the full polarization (11) can be represented as 0 0 4( ) ,polrotH ik E j rotE ik H cω ω ω ω ω πe ω= − + =      , ( ) 4 , 0poldivE divHω ω ωe ω πρ= =   , (14) 0 /k cω= . The Fourier-components of the polarization currents and charges induced in the dielectric by the ISSN 1562-6016. ВАНТ. 2019. №4(122) 42 ponderomotive force of a laser pulse are described by the expressions ,pol polj i divω ω ω ωωµ ρ µ= Π = Π    . (15) The resulting working system of equations makes it possible to investigate Cherenkov radiation in a wide variety of physical situations: the model of an infinite dielectric medium, dielectric waveguides and cavities. 2. CHERENKOV RADIATION OF A LASER PULSE IN A DIELECTRIC WAVEGUIDE We consider the dielectric waveguide, made in the form of a homogeneous dielectric cylinder, the lateral surface of which is covered with a perfectly conductive metal film. A circularly polarized laser pulse with elec- tric field components propagates along the axis of the waveguide ( )0 0x I E r, 2 = ψ τ , 0y 0xE iE= , (16) ( ) ( ) 1 2 R r Tψ = τ   . The function ( )R r describes the radial profile of the laser pulse intensity 2 0 0I E=  , ( )R 0 1,= ( )R r b 0= = , b is the waveguide radius, the function ( )T τ describes the longitudinal profile, / ,gt z vτ = − gv is the group velocity, ( ) 1,maxT τ = 0I is the maximum intensity. From the system of Maxwell equations (14) the wave equation for the longitudinal Fourier component of the Cherenkov electric field follows pol2 0 z 0 z zpol k1E k ( )E 4 i j ( ) z c ω ω ω ω ∂ρ  ∆ + e ω = π − e ω ∂  . (17) Fourier-components of polarization charges and cur- rents polωρ , zpolj ω are defined by expressions (15). For a circularly polarized laser pulse (16), these expressions take the form ( ) ( ) gik z2 L pol g 1 ( k )I r I r e , 6ω ⊥ ω ⊥ ω e − ρ = µ ∆ − + ∆   (18) ( ) gik z zpol gj k I r eω ω= −ω µ , (19) where /g gk vω= , ⊥∆ is the transverse part of Laplaci- an, I ( )rω is Fourier component of the intensity of the laser pulse field. We introduce a function ( )z z gD ( )E 4 ik I r .ω ω= e ω − π µ (20) For this function, instead of equation (17), taking in- to account relations (18), (19), we obtain the equation ( )gik z2 L z 0 z g 1 D k ( )D 4 ik e I r . 6ω ω ⊥ ω e − ∆ + e ω = π µ ∆ (21) The function zDω has a simple physical meaning and is a longitudinal Fourier-component of the longitu- dinal electric induction z z zD E 4 P= + π , taking into account the polarization (11) caused by the action of the ponderomotive force of the laser pulse. The longitudinal component of electrical induction should be sought as a series of Bessel functions. ( )gik z z n 0 n n 0 rD e C J b ∞ ω =  = ω λ    ∑ , (22) where nλ are the roots of the Bessel function ( )0J x . Using the orthogonality of the Bessel functions ( )0 nJ r / bλ , from the equation (21) we find the expan- sion coefficients 0 1 ( ) 4 ( ) . 6 ( ) nL n g n C ik I T δe ω π µ ω ω − == − ∆ (23) Here ( ) 2 2 2 1 02 0 , ( ), , 2 b n n n n n n n n b rN J R r J rdr N bb λ ρ δ λ ρ λ = = =    ∫ 1( ) ( ) 2 iT T e dωτω τ τ π ∞ −∞ = ∫ , 2 2 2 n n 0 g 2( ) k ( ) k b λ ∆ ω = e ω − − . (24) Taking into account relations (22) - (24), we obtain the following expression of Fourier component of the longitudinal electric field 0( ) ( ) ( , ) .gik z zE r A T G r eω ω ω= (25) Here 2 0 02 1 ( ) 2( , ) ( ) ( ) 3 ( ) ( 1)[ ( ) 2] ( ) , 18 ( ) L n n n n n G r i r ki rJ bk ω e ωω ω ω e ω ω e e ω ω δ λ ω ∞ = + = Γ Φ − − +  Γ  ∆   ∑ (26) 02 1 1 ( ) ( ) 6 nL n n n rr R r J bkω δe λ ∞ = −  Φ = +     ∑ , 2 2 2 2/n g nk k bλ= + . Accordingly, the longitudinal component of the ex- cited electric field can be represented as a convolution 0 0 0 0 1( , ) ( ) ( , ) 2zE r A T G r dτ τ τ τ τ π ∞ −∞ = −∫ , (27) where 0( ) 0 1( , ) ( , ) 2 iG r G r e dω τ ττ τ ω ω π ∞ − − −∞ − = ∫ (28) is Green function. For further analysis, we will present the Green function in the form 0 0 0( , ) ( , ) ( , ),l trG r G r G rτ τ τ τ τ τ− = − + − (29) 0( ) 0 ( ) 2 ( )( , ) ( ) , 3 ( ) i lG r i r e dω τ τ ω e ω ωτ τ ω ω e ω ∞ − − −∞ + Γ − = Φ∫ 0 0 0 1 1 ( , ) ( ) 6 L tr n n n n rG r i J S b e τ τ δ λ τ τ ∞ = −  − = − −    ∑ , 0 2 ( )0 0 2 ( )( ) 2( ) 3 ( ) i n n n k S i e d k ω τ τωe ωτ τ ω ω ω ∞ − − −∞ Γ+ − = ∆∫ . The Green function actually describes the structure of the wakefield in a dielectric medium excited by a laser pulse with a δ − shaped longitudinal intensity pro- file. Moreover, the term 0( , )lG r τ τ− takes into account the excitation of potential longitudinal oscillations of the ionic dielectric, and the term 0( , )trG r τ τ− describes the excitation of transverse electromagnetic waves. ISSN 1562-6016. ВАНТ. 2019. №4(122) 43 2.1. FREQUENCY DISPERSION OF DIELECTRIC PERMEABILITY The Green's function (28) and, accordingly, the wakefield (27) are largely determined by the value and frequency dispersion of the dielectric constant ( )e ω determined by the formula (12). For the qualitative analysis of this dependence, the expression for the die- lectric constant can be conveniently represented as 2 2 2 2 2 2 2 2 2 2 2 2 ( )( )( ) ( ) , ( )( )( ) Li Le Le Ti Te Le ω ω ω ω ω ω e ω ω ω ω ω ω ω − − + − − − = − − − (30) where ( ),Li Leω ω ± are the roots of the cubic equation with respect to the square of the frequency 3( ) 2 ωΛ = − . (31) It is easy to show that all three roots are positive, i.e. the frequencies are real. At these frequencies, the dielec- tric permeability is zero. The frequency Liω is the fre- quency of longitudinal optical phonons and belongs to the infrared frequency range. Frequencies ( )Leω ± are the frequencies of longitudinal polarization electron oscilla- tions and are in the optical or even ultraviolet frequency ranges. The specified frequencies are in the intervals , ,de Li di de Le de de Leω ω ω ω ω ω ω ω− + − − − +> > > > < . For definiteness, we assumed that .de deω ω+ −> Since the frequencies of the longitudinal ion and electron os- cillations are very different, the roots of the cubic equa- tion (30) can be found approximately 2 2 222 , 9 opt Li di pi opt e ω ω ω e + = + (32) ( )22 2 2 2 2 2 2 ( ) 1 16 2 9Le ge ge ge ge pe peω ω ω ω ω ω ω± + − + − + −   = + ± + +    . Here 2 2 2 ( ) ( ) ( ) 2 , 3ge de peω ω ω± ± ±= + 2 2 2 2 21 3 , , 11 3 opt pe pe opt opt de de opt ω ω e ω ω + − + + + Λ = Λ = + − Λ opte is dielectric permeability of an ion crystal in the optical frequency range 2 2 2 2max( , )de pi diω ω ω ω+ >> >> . The poles of the dielectric constant (30) are the roots of the cubic equation with respect to the square of the frequency ( ) 3ωΛ = . (33) This cubic equation has three positive roots too. These roots correspond to the frequencies ( ),Ti Teω ω ± . These frequencies are the absorption lines of the elec- tromagnetic waves of an ion crystal. In the vicinity of these frequencies, the imaginary part of the dielectric constant and, accordingly, the energy losses of electro- magnetic waves increase greatly. The frequency of ab- sorption by the ion subsystem is the frequency of trans- verse optical phonons. Note that the optical longitudinal and transverse ion oscillation branches are characterized by the fact that in the unit cell of the crystal oppositely charged ions are displaced towards each other. At the same time, the center of gravity of the unit cell remains motionless. As in the case of longitudinal optical pho- nons, the frequencies of transverse optical phonons lie in the infrared range. Electron resonance absorption frequencies are in the optical ranges. For the indicated frequencies from the cubic equation (33) we find the following approximate expressions 2 2 2 23 , 3 2 optst Ti di di opt st e ω ω ω e +−Λ = ≡ −Λ + (34) 2 2 ,pi st opt di ω ω Λ = + Λ 21 3 11 3 st st st e + Λ = − Λ , ( )22 2 2 2 2 2 2 ( ) 1 4 2 9Te he he he he pe peω ω ω ω ω ω ω± + − + − + −   = + ± + +    , where 2 2 2 ( ) ( ) ( ) 1 . 3he de peω ω ω± ± ±= − From the obvious requirement 1ste > from equality (35) it follows that for ion crystal dielectrics the condi- tion on the parameter value 3 1st> Λ > is always satis- fied. Note also that the expression for the frequency of transverse optical phonons (34) implies that when it tends to zero 3stΛ → , and the static dielectric constant increases indefinitely ste → ∞ (the phenomenon of "polarization catastrophe" [12]). In the frequency range Liω ω<< (35) the dielectric permeability of the ion crystal frequency independent and has constant value ste e= , where 2 2 2 2 2 2 Li Le Le st Ti Te T ω ω ω e ω ω ω − + − = (36) is the static dielectric constant. On the other hand in the optical frequency range 2 2 2 Te Liω ω ω− >> >> (37) dielectric permeability is also constant opte e= 2 2 2 2 Le Le opt Te T ω ω e ω ω − + − = . (38) And for all ion dielectrics always st opte e> . We note that from the expressions (36) and (38) imply the well- known Liddane-Sachs-Teller relation [12] 2 2 Li st optTi ω e eω = , relating the ratio of the frequency of longitudinal and transverse optical phonons with the values of the static and optical dielectric constants. From inequality st opte e> important conclusion follows. Since Cheren- kov radiation appears for a laser pulse when the condi- tion 2 2 1g st v c e > is satisfied and the group velocity of the laser pulse in the optical range is equal 1/g optv e= , then in the ion ISSN 1562-6016. ВАНТ. 2019. №4(122) 44 crystal the condition for the appearance of Cherenkov radiation in the microwave and terahertz ranges is al- ways fulfilled. The expression for the dielectric constant of the ion dielectric (12) can be given the usual and comfortable look 2 2 2 2 2 2 2 2 2( ) 1 pi pe pe Ti Te Te e ω ω ω ω ω ω ω + − − + Ω Ω Ω − − − − − − − . Here, the plasma frequencies are defined as follows 2 2( 2) 9 opt pi pi e ω + Ω = , 2 2 2 2 2 2 2 ( 1)Te pe Te opt Le Te Te Te ω ω e ω ω ω ω + + − + −  Ω = − − + − , 2 2 2 2 2 2 2 ( 1)Te pe Te opt Le Te Te Te ω ω e ω ω ω ω − − + + −  Ω = − − + − , 2 2 2 2 2 2,Le Le Le Te Te Teω ω ω ω ω ω+ − + −= + = + . Fig. 1 shows the qualitative dependence of the die- lectric constant on frequency, described by formula (30). Fig. 1. Dependence of the dielectric constant on frequency 2.2. DISPERSION PROPERTIES OF ION 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 form ( ) 0e ω = , (39) 22 2 2 2( ) 0n zk c b λω e ω − − = , (40) zk is longitudinal wave number. The dielectric constant is described by the formula (30). Fig. 2 shows the qualitative dependences of the fre- quency on the longitudinal wave number zk . In total, there are three branches of longitudinal oscillations ( ),Li Leω ω ω ±= and four branches (1) - (4) of electromagnetic waves. The low-frequency branch corresponds to the longitudi- nal optical phonons, and the other two branches are po- larization electron oscillations. As for the electromag- netic branches, the lowest frequency (ion) branch1 is in the infrared and microwave ranges. 1( ) ,Ti z cikω ω ω −> > /ci n stc bω λ e− = is low frequency ion cutoff frequen- cy. In the frequency range Ti ciω ω ω −>> >> , the disper- sion curve has a linear plot /z stk cω e= . Fig. 2. Dispersion curves of ion dielectric for longitudinal oscillations and electromagnetic waves on the plane ( , zkω ) The frequencies of the electromagnetic branch 2 are within 2 ( )Te z сi Likω ω ω ω− +> > ≈ , сiω + is high frequen- cy ion cutoff frequency The low-frequency section of this branch corresponds to the infrared frequency range and the high-frequency region corresponds to the optical one. This branch also has a linear dispersion region /z орtk cω e= . The tilt angle of this line exceeds the tilt angle of the straight section of branch 1. And finally, branches 3 and 4 are purely electron branches and locate in the optical and ultraviolet frequency ranges. The phase velocity of electromagnetic waves belonging to the fourth branch exceeds the speed of light and in the limiting case approaches it. 2.3. CALCULATION OF GREEN'S FUNCTION The Green function (29) contains two terms that de- scribe the excitation of longitudinal potential oscilla- tions and electromagnetic waves. The potential Green's function 0( , )lG r τ τ− has only simple poles, which are the zeros of the dielectric constant ( ) 0e ω = . The fre- quency spectrum of longitudinal oscillations contains the frequency of longitudinal optical phonons Liω and the frequencies ( )Leω  of electron polarization oscilla- tions. Below we restrict ourselves to the study of wake fields in the infrared and lower frequency ranges. This is due to the fact that for effective wake field excitation by a laser pulse necessary to achieve coherency of excita- tion. For this, it is necessary that the longitudinal and transverse dimensions of the laser pulse be smaller (sub- stantially less) than the length of the radiated wave. For the optical and especially the ultraviolet frequency ranges, this requirement is very problematic. And if this requirement is not satisfied, the amplitude of the wake waves will be negligible. ISSN 1562-6016. ВАНТ. 2019. №4(122) 45 Calculating the residues in the integral 0( , )lG r τ τ− at the poles 0Li iω ω= ± − , we find the following ex- pression for the potential Green function 2 0 0 0 4 ( ) ( ) cos ( ), 3 st opt l Li i Li st opt G r e eπ ω ϑ τ τ ω τ τ e e − = Γ Φ − − (41) where 0( )ϑ τ τ− is the Heaviside function, 2 0 0 0 0 0 1 ( ) ( ) ( , ) ( ) , 6 b L i i rr R r k G r r R r r dre − Φ = + ∫ 0 0 0 0 0 0 0 0 00 ( ) ( ), ,1( , ) ( ) ( ), ,( ) i i r i ii I k r k r r r G r r I k r k r r rI k b ∆ < =  ∆ > 0 0 0 0 0( ) ( ) ( ) ( ) ( )i i i i ik r I k r K k b I k b K k r∆ = − , / ,i Li gk vω= ( ) ( ) 0 2 2 .L L de de α α ω ω + − + − Γ = + Term in the total Green's function (29) 0( , )trG r τ τ− describes the Cherenkov excitation of the eigen elec- tromagnetic waves of the dielectric waveguide. Inte- grands of Fourier integrals 0( )nS τ τ− contain only sim- ple poles, which are the roots of the equation ( ) 0n ω∆ = . (42) As we are interested in the infrared (microwave) frequency range equation (42) three pairs of roots. Two of them are located on the real axis 2 10, 1 n g in in g st v i b λ ω ω ω b e = ± − = − , (43) 1/4 2 2 20, n en en Te Te opt Te Le i ω ω ω ω ω ω e ω ω+ −   = ± − =   −  , (44) n n c b λ ω = , and one pair eniω ω= ± on the imaginary axis. Calculat- ing the residues in these poles we find the expression for the Green function [ 0 2 0 0 0 1 2 0 0 0 1 0 ( ) cos ( ) 9 ( ) cos ( ) 18 1 ( ) , (45) 2 en tr st st in n n in n opt st en n n en n rG e J b re J b sign e ω τ τ π ϑ τ τ ω s λ ω τ τ π ω s λ ϑ τ τ ω τ τ τ τ ∞ = ∞ = − −  = − Γ − − −     − Γ − − −    − −  ∑ ∑ where ,n n nN ρ s = ( 1)( 2)L st st st e e e e − + = , ( 1)( 2)L opt opt opt e e e e − + = . The first term in the expression for the electromag- netic Green's function (45) describes the electric field in the microwave (terahertz) frequency range 2 2 Li inω ω>> (ion branch 1) and is a set of eigen electromagnetic waves with frequencies inω . The second term in expres- sion (45) describes a purely electron electromagnetic field and belongs to branch 2 in the infrared frequency range 2 2 2 Te en Liω ω ω− >> >> . The longitudinal structure of this field is more complicated. Each radial harmonic contains a wake monochromatic wave, as well as a bi- polar antisymmetric solitary pulse. Moreover, the height of this pulse is exactly two times smaller than the ampli- tude of the wake wave. The characteristic width of the polarization pulse is equal to the reverse frequency of the wake wave 1/ enτ ω∆ = . We also note an important point. Since the amplitudes of the waves entering the Green function are proportional to the square of their frequencies, the electron electromagnetic waves will have a larger amplitude compared to the ion waves. 2.4. THE EXCITATION OF WAKE FIELD BY LASER PULSE The wakefield excited by a laser pulse is described by convolution (27), in which the Green function is the key element. We first consider the excitation of longitu- dinal optical phonons. Using the potential polarization part of the Green function (41), we obtain the following expression for the wake field of longitudinal optical phonons ( , ) ( ) ( )iz Li i LiE r E r Zτ ω τ= Φ , (46) where 0 0 0 1( ) cos ( ) L L Z T d t t τ τ ωτ ω τ τ τ −∞   = −    ∫ , 2 2 02 ( 2)( )2 , 9 opt st opt g lLi Li L st opt clg v te E a rv e e e ωπ κ e e + − = Lt is characteristic duration of a laser pulse. 2 0L st LNκ ω= Γ , 2 2/ ,clr e mc= 2 2 0 0 L ea I mcω   =     . The function ( )Z ωτ describes the distribution of the wakefield on frequencyω in the longitudinal direction at each moment of time. We will consider a laser pulse with a symmetric longitudinal profile 0 0( ) ( )T Tτ τ= − . The wake function ( )Z ωτ is conveniently represented as ( ) ( ) ( ) cos ( ),Z T Xωτ ϑ τ ωτ τ= Ω −  (47) where , /L Lt tω τ τΩ = = 0 ( ) 2 ( ) cos( ) .T T s s ds ∞ Ω = Ω∫  The first term in (47) describes the wake wave prop- agating behind the laser pulse. The amplitude of the wake wave is equal to the Fourier amplitude function ( )0 / LT tτ , which describes the longitudinal profile of the laser pulse. The second term in (47) describes a bi- polar antisymmetric pulse of a polarization field local- ized in the region of a laser pulse. The field of this pulse decreases and tends to zero with increasing distance from the laser pulse. Behind a laser pulse, the wakefield (46) of longitudi- nal optical phonons has the form of a monochromatic wave ( , ) ( ) ( ) cos ,iz Li i Li LiE r E r Tτ ω τ= Φ Ω  Li Li LtωΩ = . Let us give expressions for the Fourier amplitude ( )LiT Ω  for two model longitudinal profiles of a laser pulse: a Gaussian and a power ones 2 2 2 0 / / 4 0 ˆ( / ) , ( ) ,Lt LT t e T eττ π− −Ω= Ω = (48) ISSN 1562-6016. ВАНТ. 2019. №4(122) 46 0 2 2 0 1 ˆ( / ) , ( ) 1 /L L T t T e t τ π τ −Ω= Ω = + . Longitudinal optical phonons are most efficiently ra- diated when the coherence condition 1Li Ltω ≤ is satis- fied. If the condition 1Li Ltω >> is satisfied, then the lon- gitudinal optical phonons are radiated incoherently and the amplitude of the wake wave is exponentially small. Let us now consider the excitation of electromagnet- ic waves by a laser pulse. Taking advantage of the elec- tromagnetic Green's function, we obtain the wake elec- tromagnetic field as a superposition of radial harmonics 2 02 1 0 2 02 1 0 ( , ) ( ) 1( ) ( ) , 2 in tz ti n n in n en te n n in in n rE r E J Z b rE J Z Y b ω τ s λ ω τ ω ω s λ ω τ ω τ ω ∞ = ∞ =  = − −       − −       ∑ ∑ where ( ) 0 0 0 0( ) / ( ) en en LY T t sign e dω τ τω τ τ τ τ τ ∞ − − −∞ = −∫ , 2 20 02 2 18 3 opt g L ti st L cl g v t e E e a r v e ωπ κ + = , 0 /c bω = , 2 20 02 2 36 3 opt g L ti opt L cl g v t e E e a r v e ωπ κ + = . Behind the laser pulse / 1,Ltτ >> , 1ni eω τ >> , the pulse fields are negligible and only the set of eigen waves of the dielectric waveguide remains 2 02 1 0 2 02 1 0 ˆ( , ) ( ) cos( ) ˆ( ) cos( ). in tz ti n n in in n en te n n en en n rE r E J T b rE J T b ω τ s λ ω τ ω ω s λ ω τ ω ∞ = ∞ =  = − Ω     − Ω    ∑ ∑ (49) Let us consider, for example, a laser pulse that has a Gaussian profile both in the longitudinal direction (48) and in the transverse one 2 2 0 0( / ) exp( / )L LR r r r r= − moreover, the radius of the laser pulse is small com- pared with the radius of the dielectric waveguide Lr b<< . In this case, for the expansion coefficients in the series (49) we have 2 2 2 2 4 2 2 1 1 ( ) n Lr L b n n r e b J λ s λ − = . Accordingly, for the wake electromagnetic field in- stead of (49) we obtain 2 2 2 2 2 4 0 2 1 0 2 4 2 0 ( , ) cos( ) cos( ) . (50) in L en L t in tz n n ti in n t en te en rE r J E e b E e ω ω ω τ π s λ ω τ ω ω ω τ ω ∞ − = −  = − +      +   ∑ Amplitudes of wake electromagnetic waves are pro- portional to the square of their frequencies. Therefore, a short laser pulse will predominantly excite electron electromagnetic waves, since their frequencies greatly exceed the frequencies of ion electromagnetic waves en inω ω>> . But the number of these waves is limited by inequality 1en Ltω ≤ . If the laser pulse is long at the scale of the minimum period of electron electromagnet- ic waves 1 1e Ltω > , but short compared with the periods of ion electromagnetic waves 1in Ltω << , then low- frequency ion electromagnetic waves will be most ef- fectively excited. Under these conditions, only low- frequency waves are emitted coherently by a laser pulse. CONCLUSIONS In this work, the process of excitation of wake Ce- renkov radiation by a laser pulse in an ion dielectric waveguide is investigated. For definiteness, a diatomic ion crystal medium is considered. The nonlinear electric polarization of the ion dielectric medium, induced by the ponderomotive force with the side of the laser pulse, is determined. The total electric polarization in the ion dielectric includes the electron polarization of the elec- tron shells of ions of opposite charges, as well as the ion polarization proper, due to the displacement of ions in the electric field. A system of three strongly coupled linear oscillator equations is obtained, which describes the excitation of partial electric polarizations of an ion dielectric by a ponderomotive force from the side of a laser pulse. The solution of these equations is obtained and the complete polarization in a diatomic ion dielec- tric medium is determined. Accordingly, expressions are obtained for polarization charges and currents, which, in turn, are the source of Cerenkov wake waves. The fre- quency spectrum and the space-time structure of the Cherenkov wake field, excited by a laser pulse in an ion dielectric waveguide, is determined. It is shown that in the infrared (microwave) frequency range, the excited wake electric field consists of a potential field of longi- tudinal optical phonons and a set of eigen wake elec- tromagnetic waves of a dielectric waveguide. The die- lectric constant in the infrared (microwave) frequency range in ion dielectrics always exceeds the dielectric constant in the optical range. Therefore, the condition of the Cherenkov radiation of a laser pulse in ion dielec- trics is always satisfied. REFERENCES 1. V.P. Zrelov. Vavilov-Cherenkov radiation and its application in high-energy physics. M.: “Atomiz- dat”, 1968, 302 p. 2. J.V. Jelly. Cherenkov radiation // UFN. 1956, v. 58, № 2, p. 231-283. 3. S.A. Akhmanov, V.A. Fold. Optics of femtosecond laser pulses. M.: “Nauka”, 1988, 388 p. 4. V.L. Ginzburg, V.I. Tsytovich. Transition radiation and transition scattering. M.: “Science”. 1964, 360 p. 5. T. Tajima, J.M. Dawson. Laser electron acceleration // Phys. Rev. Letter, 1979, v. 43, № 4, p. 267-270. 6. V.A. Balakirev, I.N. Onishchenko. Cherenkov radia- tion of a laser pulse in a dielectric waveguide // Problems of Atomic Science and Technology. Series “Plasma Physics”. 2018, № 6, p. 147-151. 7. V.A. Balakirev, I.N. Onishchenko. Wakefield exci- tation by a laser pulse in a dielectric medium // Problems of Atomic Science and Technology. Series “Plasma Electronics and New Methods of Accelera- tion”. 2018, № 4, p. 76-82. ISSN 1562-6016. ВАНТ. 2019. №4(122) 47 8. M. Born, H. Kun. Dynamic theory of cristal lattice. M.: “Foreign literature”, 1958, 488 p. 9. J. Sleter. Dielectrics, semiconductors, metals. M.: “Mir”, 1969, 647 p. (in Russian). 10. N. Ashkfort, N. Merlin. Fizika tverdogo tela. M.: “Mir”, 1978, v. 2, 392 p. (in Russian). 11. C. Worth, R. Thomson. Solid State Physics. M.: “Mir”, 1969, 558 p. 12. I. Kittel. Introduction to Solid State Physics. M.: “Science”, 1978, 791 p. Article received 29.06.2019 ЧЕРЕНКОВСКОЕ ИЗЛУЧЕНИЕ ЛАЗЕРНОГО ИМПУЛЬСА В ИОННОМ ДИЭЛЕКТРИКЕ В.А. Балакирев, И.Н. Онищенко Исследован процесс возбуждения черенковского электромагнитного излучения лазерным импульсом в ионном диэлектрическом волноводе. Определена нелинейная электрическая поляризация в ионной диэлек- трической среде и, соответственно, поляризационные заряды и токи, индуцированные пондеромоторной силой со стороны лазерного импульса. Получена и исследована пространственно-временная структура киль- ватерного поля в ионном диэлектрическом волноводе. Показано, что возбуждаемое поле состоит из потен- циального поляризационного электрического поля продольных оптических фононов и набора собственных электромагнитных волн ионного диэлектрического волновода. ЧЕРЕНКОВСЬКЕ ВИПРОМІНЮВАННЯ ЛАЗЕРНОГО ІМПУЛЬСУ В ІОННОМУ ДІЕЛЕКТРИКУ В.А. Балакiрєв, I.М. Онiщенко Досліджено процес збудження черенковського електромагнітного випромінювання лазерним імпульсом в іонному діелектричному хвилеводі. Визначена нелiнiйна електрична поляризація в іонному діелектрично- му середовищі та, відповідно, поляризацiйнi заряди i струми, iндукованi пондеромоторною силою з боку лазерного імпульсу. Отримана та досліджена просторово-часова структура кільватерного поля в діелектрич- ному хвилеводі. Показано, що збуджуване поле складається з потенціального поляризаційного електричного поля поздовжніх оптичних фононів та набору власних електромагнiтних хвиль іонного діелектричного хви- леводу. Advanced methods of acceleration CHERENKOV RADIATION OF A LASER PULSE IN ION DIELECTRICS V.A. Balakirev, I.N. Onishchenko E-mail: onish@kipt.kharkov.ua The wakefield excited by a laser pulse is described by convolution (27), in which the Green function is the key element. We first consider the excitation of longitudinal optical phonons. Using the potential polarization part of the Green function (41)... , (46) where , is characteristic duration of a laser pulse. , . The function describes the distribution of the wakefield on frequency in the longitudinal direction at each moment of time. We will consider a laser pulse with a symmetric longitudinal profile . The wake function is conveniently represented as (47) where The first term in (47) describes the wake wave propagating behind the laser pulse. The amplitude of the wake wave is equal to the Fourier amplitude function , which describes the longitudinal profile of the laser pulse. The second term in (47) describ... Behind a laser pulse, the wakefield (46) of longitudinal optical phonons has the form of a monochromatic wave . Let us give expressions for the Fourier amplitude for two model longitudinal profiles of a laser pulse: a Gaussian and a power ones (48) . Longitudinal optical phonons are most efficiently radiated when the coherence condition is satisfied. If the condition is satisfied, then the longitudinal optical phonons are radiated incoherently and the amplitude of the wake wave is exponentially ... Let us now consider the excitation of electromagnetic waves by a laser pulse. Taking advantage of the electromagnetic Green's function, we obtain the wake electromagnetic field as a superposition of radial harmonics where , , , . Behind the laser pulse , the pulse fields are negligible and only the set of eigen waves of the dielectric waveguide remains (49) Let us consider, for example, a laser pulse that has a Gaussian profile both in the longitudinal direction (48) and in the transverse one moreover, the radius of the laser pulse is small compared with the radius of the dielectric waveguide . In this case, for the expansion coefficients in the series (49) we have . Accordingly, for the wake electromagnetic field instead of (49) we obtain Amplitudes of wake electromagnetic waves are proportional to the square of their frequencies. Therefore, a short laser pulse will predominantly excite electron electromagnetic waves, since their frequencies greatly exceed the frequencies of ion electr... In this work, the process of excitation of wake Cerenkov radiation by a laser pulse in an ion dielectric waveguide is investigated. For definiteness, a diatomic ion crystal medium is considered. The nonlinear electric polarization of the ion dielectri... ЧЕРЕНКОВСКОЕ ИЗЛУЧЕНИЕ ЛАЗЕРНОГО ИМПУЛЬСА В ИОННОМ ДИЭЛЕКТРИКЕ В.А. Балакирев, И.Н. Онищенко ЧЕРЕНКОВСЬКЕ ВИПРОМІНЮВАННЯ ЛАЗЕРНОГО ІМПУЛЬСУ В ІОННОМУ ДІЕЛЕКТРИКУ В.А. Балакiрєв, I.М. Онiщенко

Cherenkov radiation of a laser pulse in ion dielectrics

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