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NONLINEAR INTERACTION OF WAVES IN PLASMA-BEAM SUPERHETERODYNE FEL OF THE DOPPLERTRON TYPE WITH HELICAL ELECTRON BEAM V.V. Kulish1, A.V. Lysenko2, G.A. Oleksiienko2 1 National Aviation University, Kiev, Ukraine E-mail: [email protected]; 2 Sumy State University, Sumy, Ukraine E-mail: [email protected] We have constructed a cubic-nonlinear theory of a plasma-beam superheterodyne free electron laser (SFEL) of the dopplertron type with a helical electron beam. The saturation levels and mechanisms have been found for the four operating modes. We have shown that the capture of beam electrons by plasma waves causes the signal saturation. We have found that among all operating modes the highest saturation level corresponds to the interaction mode that uses an extraordinary wave as the signal. We have demonstrated the possibility of making of powerful coherent electromagnetic radiation sources in the millimeter wavelength range on the basis of such SFEL. PACS: 41.60.Cr; 52.35.Mw INTRODUCTION The devices that are capable to generate and amplify coherent electromagnetic radiation in the millimeter wavelength range constantly attract the attention of researchers. [1 - 4]. Plasma-beam SFELs belong to such devices. Superheterodyne free-electron lasers stand out among the other types of FELs due to the high amplification properties [1, 5 - 17]. An additional mechanism of electromagnetic signal amplification is a cause of these SFEL properties. The plasma-beam instability is used as the additional amplification mechanism in the plasma-beam SFEL (PBSFEL) [1, 2, 18]. Since the plasma-beam instability has extremely high growth increments, then the plasma-beam SFEL is also characterized by exceptionally high amplification properties. The idea of PBSFEL with pump in the form of slow electromagnetic wave propagating in a magnetized plasma-beam system was first proposed in [5]. Amplification properties of PBSFEL with the axial injection of electron beam along the longitudinal magnetic field were studied in the framework of the cubic nonlinear theory in [6]. It has been shown that such devices can work as powerful electromagnetic radiation sources in the millimeter wavelength range. In this paper we analyze the work PBSFEL of the Dopplertron-type with helical electron beams. A great number of articles are devoted to research of FEL properties with the helical electron beams. It is connected with the fact that FEL with the helical electron beams have a number of advantages [10 - 12, 19 - 22]. Plasmabeam SFELs of the dopplertron type with helical electron beams have been studied in the framework of the quadratic approximation in [23, 24]. It was found that PPSFEL can operate in four different modes. These devices have higher growth increments of the signal compared to PPSFEL with linear electron beams. In this paper we continue to study the properties of PBSFEL of the dopplertron type with helical electron beams in the framework of cubic-nonlinear approximation. We find out levels and mechanisms of saturation for different operation modes. The most effective operation modes have been found. We also have studied inISSN 1562-6016. ВАНТ. 2015. №6(100) fluence of helical electron beam parameters on amplifying characteristics of PBSFEL. MODEL As a model of PBSFEL we consider the plasma with the Langmuir frequency ωp and the electron beam with the Langmuir frequency ωb, which pass through it. We consider the case when ωb << ωp. The plasma-beam system is located in the longitudinal magnetic field with the induction vector B0. The magnetic field is directed along the Z axis of the system. We consider that the cyclotron frequency ωH of electron rotation in the magnetic field is much less than the plasma frequency ωH << ωp. The electrons velocity vector of the helical relativistic beam υb is directed at an angle β with respect to the Z axis. The refore tgβ = υb ⊥ / υbz , where υb ⊥ and υbz are the transversal and the longitudinal component of electron velocity. A circularly polarized intense low-frequency electromagnetic wave with frequency ω2 and wave number k2 is used as a pump. The pump wave propagates along Z axis of the system and in the opposite direction to the electron beam. The frequency of the pump wave is less than the cyclotron frequency ω2 << ωH. In addition we feed a weak highfrequency circularly polarized electromagnetic signal wave with frequency ω1 and wave number k1 to the system input. In the studied device the space-charge wave (SCW) with frequency ω3 and wave number k3 is excited due to the parametric resonance of the interacting waves. The condition of the parametric resonance can be written as follows: (1) ω3 = ω1 − ω2 , k3 = k1 + k2 . In SFEL a superheterodyne amplification effect is realized. Its essence is to use an additional amplification mechanism for the one of three waves that is involved in a parametric resonance. The plasma-beam SFELs use plasma-beam instability as an additional mechanism for SCW amplification [1, 2, 18]. Parameters of the device are chosen so that the growth increment of SCW is maximal. The growth increment of the plasma-beam instability is much greater than the growth increment of the parametric instability. Therefore, the main objective of 83 the parametric resonance (1) is not the amplification of interacting waves, but the transfer of amplification from SCW to the high-frequency electromagnetic signal wave. It is known that the growth increments of the plasma-beam instability are extremely high. Therefore, the gain of the electromagnetic wave in PBSFEL is also large enough. BASIC EQUATIONS In order to describe the wave amplitudes dynamics in the studied device, we use quasi-hydrodynamic equation [1, 2, 18], the continuity equation and Maxwell's equations. The quasihydrodynamic equation we solve using the method of averaged characteristics [1, 9]. The continuity equation and the Maxwell's equations we solve using the method of slowly varying amplitudes. The velocity spread of electrons and the collision of electrons are not taken into account. In accordance with the method of averaged characteristics we carry out the transition to characteristics of the quasihydrodynamic equation. Then we express the components of the beam electron velocity υbx and υby through the transverse 2 2 and the component of the velocity υb ⊥ = υbx + υby pb 0 of the electron beam: υbx = υb ⊥ cos pb 0 , υby = υb ⊥ sin pb 0 . phase of rotation This transition is due to the fact that unperturbed electrons gyrate under the influence of the longitudinal focusing magnetic field in the transverse plane. Considering the given transition the initial motion equations have the following form: dυb ⊥ υbz e = By + cos pb 0 E x − dt me γ b c υ υ (Eυ) + sin pb 0 E y + bz Bx − b ⊥ 2 , c c dpb 0 e ω =− H + dt γ b me γ b υb ⊥ υbz Bx − cos pb 0 E y + c υ − sin pb 0 E x − bz By , c υ dυ bz e × E z + b⊥ × = c dt me γ b ( (2) ) × By cos pb 0 − Bx sin pb 0 − υ bz (Eυ) , c2 (3) d 2 E1x dE1x (6) + K1 = K3 E3 z E2 x + F1x , 2 dt dt dE1 y d 2 E1 y K2 = K3η1η2 E3 z E2 y + F1 y ,(7) + K1 2 dt dt d 2 E2 x dE2 x (8) M2 + M1 = M 3 E3*z E1x + F2 x , 2 dt dt d 2 E2 y dE2 y + M1 = M 3η1η2 E3*z E1 y + F2 y ,(9) M2 2 dt dt d 2 E3 z dE C2 + C1 3 z + D3 E3 z = 2 dt dt * (10) = C3 ( E1x E2 x + E1 y E2* y ) + F3 z . K2 It follows from the Eqs. (6) - (10) that circularly polarized electromagnetic waves are the eigenwaves for the investigated system. In the Eqs. (6) - (10) K1 = ∂D1 / ∂ (iω1 ) ; K2 = 0.5 ⋅ ∂ 2 D1 / ∂ (iω1 )2 ; M 2 = 0.5 ⋅ ∂ D2 / ∂ (iω2 ) ; 2 2 M1 = ∂D2 / ∂(iω2 ) , where D1 = D(ω1 , k1 ) , D2 = D(ω 2 , k2 ) are the dispersion functions of transverse electromagnetic waves of the signal (ω1, k1) and the pump (ω2, k2): D(ω, k) = b, p ωα2 i 2 2 2 × k c − ω + ∑ 2 cω α γ α (Ω α + ηω H / γ α ) ( υ2 × Ω α (Ω α + ηω H / γ α ) − α ⊥2 ω2 − k 2c 2 2c ) , (11) where η1 = E1 y /(iE1x ) = ±1 , η2 = E2 y /(iE2 x ) = ±1 , (4) dγ b e [E xυb ⊥ cos pb0 + Ee υb ⊥ sin pb0 + E zυbz ] . (5) = dt me γ b In Eqs. (2) - (5) E x , E y , E z are the components of electric field strength vector; Bx and By are the components of the magnetic field induction vector; ωH = eB0 /(mec) is the cyclotron frequency of electrons rotation in a magnetic field; c is the speed of light in 2 vacuum; γ b = 1 / 1 − (υbz + υb2 ⊥ ) / c 2 is the relativistic factor of the beam; e и me are the electron charge and the electron mass respectively. 84 In accordance with the method of averaged characteristics [1, 9] we apply the procedure of asymptotic integration to the system of equations (2) - (5). Thus we find the velocities as the functions of the electric and magnetic fields. Then we substitute the found solutions in the continuity equation and Maxwell's equations, which are solved by the method of slowly varying amplitudes. As a result we obtain the set of differential equations for the complex amplitude of x- and y- components of the signal (E1x; E1y), x- and y- components of the pump (E2x; E2y) and SCW (E3z) in the cubic approximation: (12) are the sign functions characterizing the rotation direction of the electric field strength vector of the circularly polarized electromagnetic waves; Ω α = ω − kυαz . In Eq. (10) b, p 2 − iω3 ωα (1 − υαz 2 / c 2 ) (13) D3 = 1− 2 c α (ω3 − k3υαz ) γ α is the dispersion function of SCW, C1 = ∂D3 / ∂ (iω3 ) ; ∑ C2 = 0.5 ⋅ ∂ 2 D3 / ∂ (iω3 )2 . K3 , M 3 , C3 are the coefficients that depend on frequencies, wave numbers and parameters of the investigated system. F1x , F1 y , F2 x , F2 y , F3 z are the cubic-nonlinear components of the field amplitude of the signal, the pump and SCW reISSN 1562-6016. ВАНТ. 2015. №6(100) spectively, taking into account the cubic-nonlinear interactions. The system of equations (6) - (10) needs to be extended by the equations for constant components of the velocity and the concentration dυα / dt = Vα . dnα / dt = Nα . (14) Here Vα , Nα are the functions containing the cubic nonlinear terms. They are dependent on the wave numbers, frequencies, amplitudes of the fields, the constant components of velocities and densities (the subscript α takes the values b and p ; the index b characterizes beam parameters, the index p characterizes plasma parameters). As we mentioned above, the plasma-beam instability is realized in the studied system. This means that the dispersion equation for the space-charge wave D3 (ω3 , k3 ) = 0 has complex solutions. Therefore, if we substitute real frequencies and real wave numbers (the real components of the complex solutions) in SCW dispersion function (13), then this function will not be equal to zero D3 ≠ 0 . Left side of the equation (10) contains the term D3 which allows determining the growth increment of the plasma-beam instability. So if we consider the equation (10) in the case of absence of the parametric resonance ( C3 ( E1x E2*x + E1 y E2* y ) = 0 ) and the wave is less than the cyclotron frequency ωH. Note that previously only one operation mode of PBSFEL of the dopplertron type has been studied in the framework of a cubic nonlinear approximation [6]. The right handed circularly polarized electromagnetic wave (curve 1, point A) participates in this interaction mode. The injection angle of the electron beam is equal to zero ( β = 0° ). The modes B, C and D were studied previously only in the quadratic approximation [24]. Parameters Langmuir frequency of the ma ( ω p ), [s–1] Langmuir frequency of the beam ( ωb ), [s–1] Energy of the beam, [MeV] The focusing magnetic field induction, [G] The amplitude of the first harmonic of the pump electric field is E2 = | E2 x |2 + | E2 y |2 , [V/m] Values 1.0×1012 2.0×109 0.51 2.8×103 2.8×104 cubic-nonlinear components ( F3 z = 0 ), then the growth rate will be determined by a plasma-beam instability and will be approximately determined by the relation(here we suppose that ship (− D3 / C 2 )1 / 2 C2 d 2 E3 z / dt 2 , D3 E3 z >> C1dE3 z / dt ). ANALYSIS In this section, we analyze the wave dynamics of PBSFEL with parameters given in Table. Using the dispersion relation for the transverse D(ω1 , k1 ) = 0 , D(ω2 , k2 ) = 0 and longitudinal D3 (ω3 , k3 ) = 0 waves, we can determine the frequencies and wave numbers of waves that are involved in the three-wave parametric resonance (1). In [24] shown that the three-wave parametric interactions in the investigated plasma-beam SFEL are possible for four different cases. Fig. 1 shows the dispersion curves of the high-frequency electromagnetic signal waves (curves 1 and 2) and the SCW (curves 3) for the case when the injection angle β of the beam is equal to zero. Curve 1 corresponds to the righthanded circularly polarized electromagnetic wave (η1 = -1), curve 2 corresponds to the left-handed circularly polarized electromagnetic wave (η1 = +1), when viewed along the magnetic field direction. In Fig. 1 point O determines the frequency and the wave number of the SCW with maximal growth increment (ω3, k3). The points A, B, C and D determine the frequencies and wave numbers of high-frequency electromagnetic signal waves ((ω1, k1) that can participate in the resonant interactions (1). The parametric coupling of the signal waves (ω1, k1) and the SCW ((ω3, k3) is caused by the pump wave (ω2, k2). We assume that the frequency of this ISSN 1562-6016. ВАНТ. 2015. №6(100) Fig. 1 Dispersion curves of high-frequency electromagnetic signal wave (curves 1 and 2) and the SCW (curve 3) Using standard numerical methods, we analyze dynamics of waves in the studied system in the framework of the cubic-nonlinear approximation (6) - (10), (14) for the different operation modes of PBSFEL (the modes A , B , C and D ) and for the different injection angles of the beam β . Fig. 2 shows the dependences of the first harmonic amplitudes of the electric field signal wave E1 = | E1x | 2 + | E1 y, | 2 on the normalized time τ = t ⋅ δω , (15) for four possible operation modes. The injection angle of the electron beam is β = 30° . In Eq. (15) δω is the growth rate of the plasma-beam instability at an injection angle of the beam β = 0° [1, 2, 18]. It should be noted that we consider the case when the beam energy and the beam velocity modulus are constant. Therefore, when the beam injection angle β is changing, then its 85 longitudinal velocity υ bz and the growth rate of the plasma-beam instability are also changing. We take constant multiplier as the normalization factor: the growth rate of the plasma-beam instability at the beam injection angle β = 0° . We do it in order to correctly compare the wave dynamics at the different injection angles β and the different operation modes in the studied system. In Fig. 2 the curve A corresponds to the interaction mode A (Fig. 1), curve B corresponds to the interaction mode B , curve C corresponds to the interaction mode C , curve D corresponds to the interaction mode D . harmonic amplitude of the plasma concentration n p3 / n p0 . Fig. 3. First harmonic amplitudes of the electric field SCW E3 z as a function of normalized time τ = t ⋅ δω Fig. 2. First harmonic amplitudes of the electric field signal wave E1 as a function of normalized time τ = t ⋅ δω . Curve A corresponds to the interaction mode A , curve B corresponds to the mode B , curve C corresponds to the mode C , curve D corresponds to the mode D (Fig. 1) We can see in Fig. 2 that the maximal saturation level of the signal wave electric field strength is achieved in the mode D . Although the amplification rate of the signal amplitude in the mode D at the initial interaction stage is less than in the mode C . It should be noted that the growth increment of the plasma-beam instability has the same values for all modes shown in Fig. 2. The parametric growth increment is different in the modes A, B, C and D. Therefore, the parametric interaction defines the different signal wave dynamics for the different modes. We also note that the signal electric field strength reaches the value of ~ 2.5 MV/m in the saturation region of the mode D . The signal wavelength for this mode is λ1 = 2πc / ω1 ≈ 1.8 mm. Thus, the PBSFEL can work as a powerful source of electromagnetic millimeter wavelength range radiation in the interaction mode D . Let's find out the signal saturation mechanism in the PBSFEL. Fig. 3 shows the dependences of the first harmonic amplitudes of the electric field SCW E3 z on normalized time τ = t ⋅ δω for the different operation modes. Fig. 4 shows the dependence of the first harmonic amplitude of the plasma concentration n p 3 on normalized time τ = t ⋅ δω for the modes A , B , C and D . The plasma concentration is normalized to constant component n p 0 . Comparing these figures, we see that the dependences of the SCW electric field strength E3 z are similar to the dependences of the normalized first 86 Fig. 4. First harmonic amplitude of the plasma concentration n p 3 is normalized to constant component n p 0 as a function of normalized time τ = t ⋅ δω They are correlated for all modes of operation. This means that the plasma electrons determine the dynamics and the saturation level of the SCW electric field strength. The beam electrons give the negligible contribution to the SCW electric field strength E3 z dynamics. Also, Figs. 3 and 4 show that the SCW saturation levels have the same values for all operation modes. The SCW electric field E3 z reaches the sufficiently high values of ~ 5 MV/m at saturation. Therefore, the electron beam with kinetic energy of 0.5 MeV is captured by plasma wave at the saturation. The beam electrons translational motion energy is the same for all four operation modes of PBSFEL. So the electric field strength level has the same value of saturation and capture of the electron beam. Figs. 3 and 4 also show that the SCW amplification rate of mode D is the least compared to other modes A , B and C . This happens due to the features of the parametric interaction and the different coefficient values C3 in the equation (10) for the different operating modes. Since the capture of the beam electrons in the mode D occurs later than in other modes, then the SCW electric field strength level is also achieved later. The saturation time τ D in the mode D is longer than in the modes A , B and C ( τ A , τ B and τC respectively, see. Figs. 3 and 4). Therefore, the increase of the signal wave is terminated in the modes A , B and C (Fig. 2) ISSN 1562-6016. ВАНТ. 2015. №6(100) earlier than in the mode D . The signal saturation level is the highest in this mode (Fig. 2) because the electromagnetic signal rise time (the saturation time) of the mode D is the largest. It should also be noted that this saturation mechanism (when the beam electrons are captured by the plasma waves) often occurs in different plasma electronics devices [2]. Рис. 5. First harmonic amplitude of the electric field signal wave as a function of normalized time τ = t ⋅ δω at different beam injection angles β with respect to the magnetic field in the interaction mode D . Curve 1 corresponds to injection angle β = 0° , curve 2 corresponds to injection angle β = 10° , curve 3 corresponds to injection angle β = 20° , curve 4 corresponds to injection angle β = 30° Let us find out the changes of the electromagnetic signal wave saturation levels depending on the electron beam injection angle β . Fig. 5 shows the dependence of the first harmonic amplitude of the electric field signal wave on the normalized time τ = t ⋅ δω at the different injection angles β of the beam with respect to the magnetic field in the interaction mode D. Curve 1 corresponds to the injection angle β =0°, curve 2 corresponds to the injection angle β =10°, curve 3 corresponds to the injection angle β =20°, curve 4 corresponds to the injection angle β =30°. From figure 5 follows that the gain of the signal wave increases with the increase of injection angle β at the initial interaction stage. The gain increase is primarily associated with the change of the plasmabeam instability growth increment. The growth increment of the plasma-beam instability can be found from the equation D3 = 0 (where D3 is given by Eq. (13)). From the Eq. (13) follows that D3 depends on the longitudinal velocity of the beam υbz . The longitudinal velocity of the beam decreases in the case of the off-axis beam injection (the case of the constant beam energy). It leads to the growth rate increase of the plasma-beam instability. As a result the gain of the electromagnetic signal wave increases. On the other hand, it leads to the increase of the SCW electric field amplification rate because the beam electrons are captured by the plasma waves. As we see in Fig. 5, the signal wave saturation level rises insignificantly with the increase of the beam injection angle. In addition, the signal wave saturation at the injection angle β = 30° happens earlier then at the injection angle β = 0°. Thus, the use of the helical relativistic electron beams in PBSFEL is possible. It allows ISSN 1562-6016. ВАНТ. 2015. №6(100) strengthening of the powerful electromagnetic signals in the millimeter wavelength range. CONCLUSIONS We have constructed the cubic nonlinear theory of the wave interaction in a PBSFEL of the dopplertron type with a helical electron beam. The saturation levels are found for the four the operation modes. We found that interaction mode D has the highest saturation levels of the high-frequency electromagnetic signal (the parametric resonance interaction with extraordinary circularly polarized electromagnetic signal wave) among the other possible SFEL operation modes. Moreover, the saturation level of the signal in the mode D exceeds the saturation levels in the operation modes A , B and C more than twice. We have found that the saturation is associated with the capture of beam electrons by plasma waves in the investigated SFEL. We have shown that the gain of the electromagnetic signal rises with increase of the electron beam injection angle. Such increase of the signal amplification is determined by the increase of the growth rate of the plasma-beam instability. It is associated with the decrease of the longitudinal electron energy. At the same time, the saturation level of the electromagnetic signal increases slightly with the increase of the injection angle of the electron beam. Also, the saturation time decreases with the increase of electron beam injection angle. It means that SFELs with the high-current helical electron beams may use beams with shorter duration of the current pulse compared to SFELs with coaxial beam. Thus, the plasma beam SFEL of the dopplertron type with the helical electron beams can be used as a powerful source of coherent electromagnetic radiation in the millimeter wavelength range. REFERENCES 1. V.V. Kulish. Hierarchic Electrodynamics and Free Electron Lasers. Boca Raton, London, New York: CRC Press, 2011. 2. М.V. Kuzelev, A.A. Ruhadze, P.S. Strelkov. The relativistic plasma SHF electronic. Moscow: Bauman МSТU, 2002. 3. S.E. Tsimring. Electron beams and microwave vacuum electronics. Hoboken, New Jersey: Wiley, 2007. 4. J.H. Booske, R.J. Dobbs, C.D. Joye, C.L. Kory, G.R. Neil, Gun-Sik Park; Park Jaehun, R.J. Temkin. Vacuum electronic high power terahertz sources // IEEE Transactions on Terahertz Science and Technology. 2011, v. 1, № 1, p. 54-75. 5. N.Ya. Kotsarenko and V.V. Kulish // Radiotekh. Elektron. 1980, v. 25, p. 2470-2471. 6. V.V. Kulish, A.V. Lysenko, V.V. Koval. 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Nonlinear theory of a free electron laser with a helical wiggler and an axial guide magnetic field // Phys. Rev. ST Accel. Beams. 2013, v. 16, № 9, p. 090701. 23. V.V. Kulish, A.V. Lysenko, G.A. Oleksiienko, V.V. Koval, M.Yu. Rombovsky. Plasma-beam superheterodyne FELs with helical electron beams // Applied Physics. 2014, № 5, p. 24-28. 24. V.V. Kulish, A.V. Lysenko, G.A. Oleksiienko, V.V. Koval, M.Yu. Rombovsky. Nonlinear theory of plasma-beam superheterodyne free electron laser of dopplertron type with non-axial injection of electron beam // Acta Physica Polonica A. 2014, v. 126, № 6, p. 1263-1268. Article received 02.09.2015 . ISSN 1562-6016. ВАНТ. 2015. №6(100) НЕЛИНЕЙНЫЕ ВЗАИМОДЕЙСТВИЯ ВОЛН В СУПЕРГЕТЕРОДИННОМ ПЛАЗМЕННОПУЧКОВОМ ЛСЭ ДОПЛЕРТРОННОГО ТИПА С ВИНТОВЫМ ЭЛЕКТРОННЫМ ПУЧКОМ В.В. Кулиш, А.В. Лысенко, Г.А. Алексеенко Построена кубически-нелинейная теория плазменно-пучкового супергетеродинного лазера на свободных электронах (СЛСЭ) доплертронного типа с винтовым электронным пучком. Определены уровни и механизм насыщения для четырех режимов работы. Показано, что насыщение сигнала связано с захватом электронов пучка плазменными волнами. Выяснено, что среди всех режимов работы СЛСЭ наиболее высоким уровнем насыщения сигнала обладает режим взаимодействия, в котором в качестве сигнала используется необыкновенная электромагнитная волна. Продемонстрирована возможность создания на базе такого СЛСЭ источников мощного когерентного электромагнитного излучения в миллиметровом диапазоне длин волн. НЕЛІНІЙНІ ВЗАЄМОДІЇ ХВИЛЬ У СУПЕРГЕТЕРОДИННОМУ ПЛАЗМОВО-ПУЧКОВОМУ ЛВЕ ДОПЛЕРТРОННОГО ТИПУ З ГВИНТОВИМ ЕЛЕКТРОННИМ ПУЧКОМ В.В. Куліш, О.В. Лисенко, Г.А. Олексієнко Побудована кубічно-нелінійна теорія плазмово-пучкового супергетеродинного лазера на вільних електронах (СЛВЕ) доплертронного типу з гвинтовим електронним пучком. Визначені рівні та механізм насичення для чотирьох режимів роботи. З'ясовано, що насичення сигналу пов'язане із захопленням електронів пучка плазмовими хвилями. Показано, що серед усіх режимів роботи СЛВЕ найбільш високий рівень насичення сигналу має режим взаємодії, у якому як сигнал використовується незвичайна електромагнітна хвиля. Продемонстрована можливість створення на базі такого СЛВЕ джерел потужного когерентного електромагнітного випромінювання в міліметровому діапазоні довжин хвиль. ISSN 1562-6016. ВАНТ. 2015. №6(100) 89