PHYSICAL REVIEW SPECIAL TOPICS—ACCELERATORS AND BEAMS 18, 110702 (2015)
Cyclotron radiation cooling of a short electron bunch kicked in an undulator with guiding magnetic field
I. V. Bandurkin,1 I. V. Osharin,1 and A. V. Savilov1,2 1Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod 603950, Russia 2Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod 603950, Russia (Received 30 June 2015; published 20 November 2015) We propose to use of an undulator with the guiding axial magnetic field as a “kicker” forming a bunch of electron gyro-oscillators with a small spread in the axial velocity. The cyclotron emission from the bunch leads to losing oscillatory velocity of electron gyrorotation, but it does not perturb the axial electron velocity. This effect can be used for transformation of minimization of the spread in electron axial velocity in the undulator section into minimization of the spread in electron energy in the cyclotron radiation section.
DOI: 10.1103/PhysRevSTAB.18.110702 PACS numbers: 41.60.Cr, 41.60.Ap
I. INTRODUCTION this dependence has the “abnormal,” increasing character, so that particles with higher input energies obtain bigger Fast development of the technique of photocathode undulator velocities. electron photoinjectors has resulted in creation of compact It is shown in [13] that the abnormal dependence of the and accessible sources of moderately relativistic (several undulator velocity on the electron energy can be used to MeV) dense (up to 1 nC in a ps pulse) electron bunches provide “axial cooling” of the electron bunch; this term [1–3]. Methods providing a decrease in the energy spread means minimization of the spread in axial electron velocity (cooling) are actual from the point of view of various inside the undulator due to transformation of the input applications of such electron bunches, including free- spread in axial velocity into the spread in undulator electron lasers (FELs) [4–6]. However, cooling methods velocity. It is mentioned also in [13] that the axial cooling are developed now basically for electron beams of very can be, in principle, transformed into “real” cooling of high energies [7–12]. As for moderately relativistic (several the electron bunch (minimization of the spread in total or tens MeV) high-dense short electron bunches, the strong electron energy) due to cyclotron emission from gyrorotat- Coulomb interaction of the particles results in a require- ing electrons, which obtain their oscillatory velocities in the ment for a short length of a cooling system. Therefore, in axial cooling undulator (so that the undulator is used as a this situation, the cooling system should possess resonant “kicker” imparting electrons their rotatory velocities). In properties, namely, a strong dependence of parameters of this paper, we study this idea in detail. In Sec. II,we the particles inside the cooling system on their input describe the process of axial cooling of a moderately energies. relativistic electron bunch in a magnetostatic undulator An example of such a resonant system is a periodic with guiding axial magnetic field, as well as discuss the use magnetistatic undulator immersed in a uniform guiding of an rf-wave undulator in the case of higher electron axial magnetic field [13]. Such undulator systems are used energies. Cyclotron emission from a short electron bunch, in mm-wavelength FELs [4,14] to decrease sensitivity of which is kicked in the axial cooling undulator, as well as these devices to the velocity spread. An important pecu- motion of electrons in the radiated field is studied in liarity of such a system is a resonant character of the Sec. III. dependence of the undulator velocity on the electron energy. This dependence has the “normal” character (namely, the decreasing one) at relatively low values of II. AXIAL COOLING UNDULATOR the guiding magnetic field, when the electron cyclotron frequency corresponding to this field is lower than the A system composed by a periodic magnetic field bounce frequency of electron oscillations in the periodic (undulator) and a uniform magnetic field can be used to undulator field. In contrast, at high axial magnetic fields provide axial cooling of an electron bunch, namely, a significant decrease in the spread in axial electron velocity by means of “transforming” this spread into the spread in transverse velocity [13]. This is possible, if the dependence Published by the American Physical Society under the terms of of the velocity of undulator electron oscillations on the the Creative Commons Attribution 3.0 License. Further distri- ð Þ bution of this work must maintain attribution to the author(s) and axial electron velocity, Vu V∥ , has the abnormal character, the published article’s title, journal citation, and DOI. ∂Vu=∂V∥ > 0.
1098-4402=15=18(11)=110702(12) 110702-1 Published by the American Physical Society I. V. BANDURKIN, I. V. OSHARIN, AND A. V. SAVILOV Phys. Rev. ST Accel. Beams 18, 110702 (2015)
(a) [Fig. 1(b)], namely, is a decreasing function at Ωu > Ωc undulator and an increasing function at Ωu < Ωc [4,14].We Ω Ω e should operate in the region u < c, where electrons with higher input energies get bigger oscillatory velocities B 0 (∂Vu=∂V∥ > 0); this is similar to the “negative-mass” effect well known in theory of cyclotron accelerators [15,16] and cyclotron resonance masers [17–24]. We consider electron motion along a helical trajectory V V u B ¼ z 0 V || in the axial uniform magnetic field, 0 B0, and in the quasiperiodic transverse field of a circularly polarized undulator, Bx þ iBy ¼ BuðzÞ expðihuzÞ. The motion equa- tion for the complex transverse momentum of a particle, (b) |Vu| pþ ¼ px þ ipy ¼ γβþ, has the following form:
dpþ pþ ¼ ihc þ ihuKðzÞ expðihuzÞ: ð1Þ dz p∥ V || Ω Ω Ω > Ω c < u c u Here, p ¼ γβ, β ¼ V=c, γ ¼ð1 − β2Þ−1=2 is the relativistic ¼ 2 ¼ γΩ FIG. 1. (a) Minimization of the spread in axial velocity in a electron Lorentz factor, hc eB0=mc c=c is the ð Þ¼− circular-polarized magnetostatic undulator with guiding magnetic normalized axial magnetic field, and K z eBu= 2 field. (b) Resonant dependence of the undulator velocity on the humc is the normalized undulator field (the so-called axial electron velocity. undulator factor at the zero axial field B0 ¼ 0); its depend- ence on the axial coordinate describes a smooth enter of Let us consider a scheme shown in Fig. 1(a). Electrons particles into the undulator. In the simplest model, electrons move along the axial magnetic field and enter into a have only axial velocities at the beginning of the system, β ¼ zβ β¯ − δβ β β¯ þ δβ circularly polarized undulator, where they get some rota- 0 , with some spread, 0 0 < 0 < 0 0. tory velocity. In order to avoid appearance of “parasitic” If the entry into the undulator is smooth enough, then the cyclotron oscillations, the undulator includes an input growth of the forced undulator oscillations does not induce section, where its field adiabatically slowly grows with the free cyclotron oscillations of particles. Therefore, in the ¼ the coordinate. If any electron at the input of the system regular region K const the transverse electron velocity is ¯ described as follows: possesses only axial velocity, V0 ¼ V0 þ δV0, then axial velocity in the regular undulator region is determined by the 2 2 2 βþ ¼ pþ=γ0 ¼ β expðih zÞ; ð2Þ energy conservation low: V∥ ≈ V0 − Vu, where Vu is the u u velocity of electron rotation in the undulator. Thus, where 2 ¯ 2 ¯ 2 ¯ V∥ ≈ V0 þ 2V0 × δV0 − VuðV0Þ − α × δV0; 2 βu ¼ K=γ0Δ ð3Þ α ¼ ∂Vu=∂V∥: is the velocity of electron rotation in the undulator field, The condition of the axial cooling, α ¼ 2V¯ , corresponds to and Δ ¼ 1 − Ωc=Ωu is the mismatch between the electron the minimal spread in V∥. This condition is independent on cyclotron frequency and the bounce frequency of the the initial spread, δV0. undulator oscillations. The initial axial velocity is related Evidently, in order to provide axial cooling at moderate with the axial velocity in the regular undulator by the undulator fields, one should use the close-to-resonance energy conservation law: range of parameters, where ∂Vu=∂V∥ is great enough [Fig. 1(b)]. In addition, ∂Vu=∂V∥ should be positive, so 2 2 2 β0 ¼ β∥ þ βu: ð4Þ that initial axial velocity excess, δV0, should be compen- sated by greater rotatory velocity, Vu. This is possible due to the presence of the axial magnetic field, B0 ¼ zB0.In Evidently, this case, a particle represents a free cyclotron oscillator β ≈ ðβ¯ Þþðβ − β¯ Þ 0 ð Þ (with the frequency Ωc ¼ eB0=mcγ), which is under acting ∥ f 0 0 0 f ; 5 of the periodic undulator force (with the bounce frequency Ωu ¼ huV∥, where hu is the undulator wavelength). Due to where function β∥ ¼ fðβ0Þ is determined by Eqs. (4) this fact, the dependence VuðV∥Þ has a resonant character and (2). Having differentiated Eq. (4), one obtains
110702-2 CYCLOTRON RADIATION COOLING OF A SHORT … Phys. Rev. ST Accel. Beams 18, 110702 (2015) 2 0 should be of the order of (and slightly greater than) the β ¼ β 0 − K γ3β þ hc f ð Þ 0 ∥f 3 0 0 2 : 6 undulator bounce frequency, ðγ0 − hc=β∥huÞ hu β∥ Ω ⪞ Ω : ð8Þ According to Eq. (5), the spread in β∥ is minimal, when c u 0 f ¼ ∂β∥=∂β0 ¼ 0. In the ultrarelativistic limit, γ ≫ 1, this This formula leads to the following estimation: leads to the following condition [13]: γ ⪞ ð Þ 2 ≈ −Δ3 ð Þ B0 ; 9 K : 7 λu
In this paper, the axial cooling undulator is considered as where the axial magnetic field B0 and the undulator period a specific kicking system, which imparts to electrons the λu are measured in Tesla and cm, respectively. At relatively velocity of free cyclotron rotation in the axial homogeneous low electron energies (γ ∼ 10), the undulator period λu ¼ magnetic field. According to Eq. (1), if the undulator 5 cm corresponds to a moderate (especially for pulsed section possesses a sharp exit, then the exit of electrons systems) axial magnetic field of B0 ≈ 2 T. from the undulator leads to transformation of the forced A possible way to decrease the magnetic field required at oscillations in the undulator field, βþ ¼ βu expðihuzÞ, into higher electron energies is the use a powerful rf pulse free cyclotron oscillations, propagating together with electrons instead of a magneto- Z static undulator. In the case of the rf undulator, the bounce frequency is determined as follows: β ¼ β Ω þ ⊥ exp i cdt ; Ω ¼ ω − ¼ ω ð1 − β β Þ ð Þ u u huV∥ u gr;u ∥ ; 10 with the same oscillatory velocity β⊥ ¼ β (Fig. 2). Thus, u β ¼ ¼ ω at the beginning of the radiation section the electron bunch where gr;u Vgr;u=c chu= u is the normalized group has some spread in cyclotron rotation velocity, but the velocity of the undulator wave. In this situation, the spread in axial velocity is minimal. The particles lose their Doppler conversion factor leads to a significant decrease transverse velocities due to the radiation, such that in the in the value of the required magnetic field: optimal situation electrons possess only axial velocity at the γ ⪞ ð1 − β β Þ ð Þ end. It is important that axial velocities stay constant during B0 λ gr;u ∥ ; 11 this process (see Sec. III). Thus, if the spread in axial u velocity is minimized in the beginning of the radiation where λ is the wavelength measured in cm. section [Fig. 1(a)], then the radiation process results in u minimization of the spread in the total velocity (Fig. 2). According to Eq. (7), in order to provide axial cooling at III. CYCLOTRON RADIATION SECTION moderate undulator fields, K<1, one should use the close- We consider the cyclotron rf emission from a short to-resonance range of parameters, jΔj < 1. In the case of a electron bunch moving in a circular cross-section wave- magnetostatic undulator, the axial magnetic field should be guide [Fig. 3(a)]. The electron move along helical trajec- so high that the corresponding electron cyclotron frequency tories is due to gyrorotations of particles in the guiding magnetic field B ¼ zB0 around the waveguide axis. In order to provide coherent character of the spontaneous undulator cyclotron radiation, the axial bunch length, L , should be shorter than radiation e the characteristic wavelength of the radiated wave, Le < λ. e In this situation, the bunch radiates a wave packet with an axial length, which is significantly longer that the bunch B0 length [Fig. 3(b)]. It will be shown further that the optimal (from the points of view of both the rate and the quality of the cooling process) situation is realized, when the radiated wave packet propagates together with the electron bunch, V u V⊥ V || V || such that the axial electron velocity, V∥, should be close to the group velocity of the rf wave, Vgr. Let us introduce a polar system of coordinates, þ ¼ iϕ FIG. 2. Transformation of the axial cooling (minimization of x iy re . Since every electron rotates around the the spread in axial electron velocity) in the undulator section into waveguide axis, its polar phases coincides with the gyro- the real cooling (minimization of the spread in electron energy) rotation phase, and the complex oscillatory velocity is in the radiation section. determined as follows:
110702-3 I. V. BANDURKIN, I. V. OSHARIN, AND A. V. SAVILOV Phys. Rev. ST Accel. Beams 18, 110702 (2015)
y (a) described by the assumption that the complex rf wave ϕ B0 amplitude in Eq. (12), Aðz; tÞ, is a slow function of both the e axial coordinate and the time. x e-bunch A. Equations of electron motion Let us notice that Eq. (12) describes the rf field, which is ≈ rf wave packet V gr V || similar to a plane circularly polarized wave in the center of the waveguide (in the point of injection of the electron bunch): λ (b) j ≈ i ð Þ ð ω − Þ ≈ h0 Eþ r→0 A z; t exp i 0t ih0z ;Bþ i Eþ: V ⊥ 2 k0 V || V ⊥ V || Le ð14Þ FIG. 3. (a) Motion of the electron bunch of gyrorotating electrons inside the waveguide of the radiation section, and In this case, the equation for the relativistic electron (b) a short electron bunch on the background of the radiated energy, rf-wave packet in the case of the group synchronism. 2 dγ e Zt mc ¼ − ReEþVþ; ð15Þ dz V∥ Vþ ¼ iV⊥ expðiϕÞ; ϕ ≈ ϕ0 þ Ωcdt: 0 leads to the following normalized equation:
The interaction between an axis-encircling electron 1 dγ p⊥ ¼ − Reaðz; tÞ expðiφÞ; ð16Þ beam and a mode of a circular waveguide at the funda- k0 dz 2p∥ mental cyclotron resonance is possible, if the azimuthal 2 index of the mode is equal to 1 [25,26]. We consider a where a ¼ eA=mc k0 is the dimensionless slow wave wave, which has a fixed transverse structure corresponding amplitude, and φ ¼ ω0t − h0z − ϕ is the wave phase on to the lowest possible TE1;1 transverse mode. The electric the basic frequency. and magnetic components of the field of this mode The equation for the axial electron momentum has the following form: −1 ∂A E ¼ ; B ¼ rotA c ∂t dp∥ e mc ¼ − ImBþVþ: ð17Þ dz cV∥ are described by the following vector potential: This equation together with Eqs. (14) and (15) leads A ¼ Re Aðz; tÞFðr; ϕÞ expðiω0t − ih0zÞ; ð12Þ to the following simple relations between changes in the energy and in the axial momentum [15,16,27–31]: where dp∥ γ ¼ β d ð Þ −i gr ; 18 F ¼ rot z0J1ðk⊥rÞ expð−iϕÞ dz dz k0k⊥ β ¼ ¼ where gr Vgr=c h0=k0 is the group velocity of the is the membrane function corresponding to the mode TE1;1. radiated wave normalized to the speed of light. This fact has ω ¼ ω In thesepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi formulas, 0 is a basic frequency, k0 0=c, and an evident quantum interpretation, namely, the elementary 2 2 h0 ¼ k0 − k⊥ is the axial wave number corresponding to act of radiation of the wave by a particle is emission of a the basic frequency, which is chosen so that it satisfies quant with energy ℏω and axial momentum ℏh,so δ δγ ¼ β exactly the cyclotron resonance condition, that p∥= gr. Equation (18) leads to the following equation describing ω0 ¼ h0V∥ þ Ωc0; ð13Þ the change in the axial electron velocity: where Ω 0 ¼ eB0=mcγ0 is the initial cyclotron frequency. dβ∥ d p∥ dγ β − β∥ c ¼ ¼ × gr : ð19Þ It is important that the emission from a short electron bunch dz dz γ dz γ possesses a nonmonochromatic spatiotemporal character. We suppose that the frequency spectrum of the radiated rf If the group velocity coincides with the axial electron δω ≪ ω β ¼ β signal is narrow, 0. These two statements are velocity, gr ∥, then the axial electron velocity is
110702-4 CYCLOTRON RADIATION COOLING OF A SHORT … Phys. Rev. ST Accel. Beams 18, 110702 (2015) constant, and the cyclotron radiation leads to the loss in the electron-wave interaction. If the spread is absent, then one transverse component of the electron velocity only. Let us should put Δ ¼ 0 in Eq. (22). It is seen from the formula for suppose that at the end of the radiation section the particles δ ¼ 0 (and well known from the cyclotron-resonance maser lose all their initial oscillatory velocity. If the axial cooling theory [29–31]) that the most important effect making worse (minimization of the spread in β∥) is provided at the input the cyclotron interaction of relativistic electrons with a of the radiation section, then the radiation process trans- traveling wave is the spread in axial electron velocity, β∥. forms this axial cooling into the real cooling (minimization Thus, the use of the axial cooling undulator as a kicker of the spread in the total energy) (Fig. 2). imparting to electrons their cyclotron oscillations is a way to The equation for the change in the normalized transverse solve the problem of sensitivity of the cyclotron maser to the momentum of a particle follows from the general relativ- spread in axial electron velocity. Therefore, the cyclotron 2 2 2 istic relation γ ¼ 1 þ p⊥ þ p∥ and Eq. (20): maser with the axial cooling kicker proposed in this paper can be attractive not as a way to provide a cooling cyclotron 1 γ γð1 − β β Þ radiator, but also from the more general point of view, dp⊥ ¼ d ðγ2 − 2Þ¼d gr ∥ ð Þ p∥ × : 20 namely, the realization of a relativistic electron cyclotron dz p⊥ dz dz p⊥ source of a short-wavelength radiation as itself. Having taken into account Eq. (16), one obtains One should notice that equations of electron motion used in our model do not include the rf space-charge effects. An 1 1 − β β analytical description of these effects for a moving bunch of a dp⊥ ¼ − gr ∥ ð Þ ð φÞ ×Rea z; t exp i : complicated form (a piece of helix) is difficult, but there are k0 dz 2β∥ two reasons to neglect these effects (at least in the simplest β ¼ β β ¼ model). Both of them are due to the fact that in the mechanism Since gr ∥ and ∥ const, in the ultrarelativistic approximation one obtains of electron bunching in the cyclotron masers significantly differ from this mechanism taking place in the electron maser 1 −1 based with the axial electron bunching, namely, free-electron dp⊥ ≈ ð Þ ð φÞ ð Þ 2 ×Rea z; t exp i : 21 masers and Cherenkov (Smith-Purcell) devices. In cyclotron k0 dz 2γ 0 masers, the electron bunching the 2D character, namely, this In the right-hand part of the following equation for the is a combination of the axial bunching (caused by the electron phase with respect to the radiated wave, dependence of the axial electron velocity on the energy) and the azimuthal bunching (caused by the dependence of the 1 φ 1 1 Ω γ electron cyclotron velocity on the energy). It is important that d ¼ d ðω − − ϕÞ¼ − β − c0 0 0t h0z gr × ; axial and azimuthal bunching mechanisms have “opposite k0 dz k0 dz β∥ β∥ω0 γ signs,” so that there is a partial compensation of these two only the last term varies in the process of the rf-wave kinds of bunching; this compensation is described by the ð1 − β β Þ ≈ γ−2 radiation. Having taken into account both cyclotron reso- difference gr ∥ 0 in Eq. (22). nance condition (13) and group synchronism condition Thus, the first reason to neglect the rf space-charge effects β ¼ β is their weakness. Actually, due to the partial compensation gr ∥, one obtains of the two different kinds of bunching, a “compensation” −2 1 φ Ω ∂ γ γ − γ factor ð1 − β β∥Þ ≈ γ appears in the space-charge param- d ¼ − c0 0 ðγ − γ ÞþΔ ≈ 0 þ Δ gr 0 × × 0 3 : eter in the theory of the cyclotron resonance maser [24]. k0 dz β∥ω0 ∂γ γ γ0 Moreover, in the case of the cyclotron excitation of a wave ð Þ 22 with the group velocity comparable with the speed of light, there is one more effect weakening the rf space-charge Let us assume that at the beginning of the radiation forces. This is a complicated (helical) spatial structure of the ¼ 0 section, z , electrons have the same phases of their electron bunches [18–24] caused by rotating character of ϕ cyclotron rotation, . Then the initial phase of a particle, the excited wave. According to the theory [24], this effect is φðz ¼ 0Þ¼φ0 ¼ ω0t0, is determined by the time of its −2 described also by a factor of ≈γ0 . Finally, as compared to entry into the radiation section. Thus, the phase size of the the cases of both free-electron and Cherenkov masers, the electron bunch is determined by the ratio between its length space-charge parameter in the case of the cyclotron maser is and the characteristic wavelength of the radiated wave: −4 smaller by a factor of γ0 . The second reason to neglect the “ ” 0 ≤ φ ≤ δφ δφ ¼ 2π λ ð Þ rf space-charge effects is the negative-mass character of 0 0; 0 Le= : 23 these effects [18–24]. Since the azimuthal bunching is stronger than the axial one, the quasi-Coulomb interaction In Eq. (22), the mismatch of the cyclotron resonance, 1 Ω of electrons leads not to their mutual repulsion (as it Δ ¼ − β − c0 , describes the possible effect of β∥ gr β∥ω0 happens free-electron and Cherenkov masers) but to their the spread in electron velocity of the process of the effective phase attraction.
110702-5 I. V. BANDURKIN, I. V. OSHARIN, AND A. V. SAVILOV Phys. Rev. ST Accel. Beams 18, 110702 (2015)
B. rf wave emission fðζÞ ≈ k0LeδðζÞ ≈ δφ0δðζÞ; In order to describe the spatiotemporal evolution of the slow rf wave amplitude aðz; tÞ, we consider the wave and wave equation (26) is reduced as follows [32,33]: equation, ∂a ∂a ∂2a 2i þ 2iε − ¼ iG × jˆðτÞ × δðζÞ; ð28Þ 1 ∂2A ∂2A 4π ∂τ ∂ζ ∂ζ2 − − Δ A ¼ j 2 2 2 ⊥ ; c ∂t ∂z c 2I with G ¼ δφ0. Let us notice that in this delta-function NIA with the electron current density, approximation, the electron current harmonic is determined by the same formula, ¼ þ ¼ ρh i ð − Þ δð Þ jþ jx ijy Vþ × f z V∥t × r : ˆ jðτÞ¼hβ⊥ expð−iφÞi; ð29Þ
Here, ρ is the electron density, hVþi is the averaged h i complex transverse velocity, h i denotes averaging over but denotes averaging over all electrons of the bunch. Thus, in order to provide the start of the spontaneous rf- all electrons having the same z-coordinate, and fðz − V∥tÞ describes the position of the electron bunch: f ¼ 1 at wave emission, the initial phase size of the electron bunch should be small enough: δφ0 < 2π and, therefore, Le < λ. 0 ≤ z − V∥t ≤ L , otherwise f ¼ 0. The wave equation 2 e ∂ a leads to the following equation for the slow rf wave In the left-hand part of Eq. (28), ∂τ2 has been omitted, as ∂a ∂2a amplitude: this is small as compared to ∂τ. As for the term ∂ζ2 cannot be omitted if the slippage factor ε is small. Otherwise, in the 2 2 2 1 ∂ a 2iω0 ∂a ∂a ∂ a ∂a ∂ a approximation of a large slippage factor (ε ≫ 2 ), one þ þ 2ih0 − ∂ζ ∂ζ 2 ∂ 2 2 ∂ ∂ ∂ 2 c t c t z z easily obtains the following solution of Eq. (28): 2 ¼ 2 I ð1 − β2 Þ ˆ ð − Þ ð Þ ik0 gr × j × f z V∥t : 24 G NIA ≈ ˆðτ − ζ εÞ ð Þ a 2ε × j = ; 30 Here, N ≈ 0.404 is the norm of the TE1;1 transverse ˆ which describes quasistationary spontaneous emission mode, and jðz; tÞ¼hβ⊥ expð−iφÞi is the electron-current harmonic. from the electron bunch. Similar to works [32,33] devoted to the cyclotron In the general case, the solution of Eq. (28) cam be superradiation of a short electron bunch, we introduce obtained after the following substitution: the following normalized variables: 2 aðζ; τÞ¼a1ðζ; τÞ expð−iε τ=2 − iεζÞ: 2 ζ ¼ k0 ðz − V∥tÞ; τ ¼ ω0 ðt − V∥z=c Þ: ð Þ × × 25 This transforms Eq. (27) into the Schrödinger equation,
Then, Eq. (24) is transformed as follows: ∂ ∂2 2 a1 − a1 ¼ ðζ τÞ ð Þ i ∂τ ∂ζ2 S ; ; 31 ∂2 ∂2 ∂ ∂ 2 a − a þ 2 a − 2 ε a ¼ I ˆ ðζÞ ð Þ 2 2 is i ig × j × f : 26 ∂τ ∂ζ ∂τ ∂ζ NIA with the following source:
¼ð1−β2 Þ ð1−β2Þ ¼ð1 − β β Þ ð1 − β2Þ ˆ 2 Here, g gr = ∥ , s gr ∥ = ∥ , Sðζ; τÞ¼iG × jðτÞ × expðiε τ=2 þ iεζÞ × δðζÞ: 2 and ε ¼ðβ∥ − β Þ=ð1 − β∥Þ are factors determined by gr ðτ ¼ 0 ζÞ¼0 the axial electron velocity and the group velocity of the At the zero initial condition, a1 ; , the radiated wave. solution of Eq. (31) can be found analytically [34], Further, we consider the situation, when the group Zt þZ∞ synchronism condition is fulfilled at least approximately, 1 1 0 0 β ≈ β ≈ ≈ 1 a1ðζ; τÞ¼ pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Sðζ ; τ Þ gr ∥. In this case, g s , whereas the factor describ- 2 iπ τ − τ0 −∞ ing the slippage of the wave with respect to electrons, 0 pffiffiffiffi 0 2 0 0 0 ε ≈ γ2ðβ − β Þ ð Þ × exp½− 2iðζ − ζÞ =4ðτ − τ Þ dζ dτ : 0 ∥ gr ; 27 can be small. In addition, we consider emission from a very This leads to the following formula for the slow rf short electron bunch, Le ≪ λ. In this case, function fðζÞ wave amplitude in the point of the electron beam, can be approximated as follows [see also Eq. (23)]: abðτÞ¼aðτ; ζ ¼ 0Þ:
110702-6 CYCLOTRON RADIATION COOLING OF A SHORT … Phys. Rev. ST Accel. Beams 18, 110702 (2015)
pffiffi Zτ 0 cooling” in the process of the cyclotron emission) and to iG jˆðτ Þ a ðτÞ¼expð−iατÞ × pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi expðiατ0Þdτ0; ð32Þ increase of the rate of this process. b 2 π τ − τ0 0 C. Estimations and numerical simulations where α ¼ ε2=2 describes a small slippage of the radiated rf In equations of the electron motion (21) and (22), the wave with respect to the electron bunch. In the simplest transition to normalized variables ðζ; τÞ [see Eq. (25)] can approximation of the exact group synchronism (α ¼ 0) and ˆ ˆ be provided in the following way: the stationary state of the electron bunch (j ¼ const ¼ j0), Eq. (32) reduces to the following formula: 1 − β2 1 d 1 ∂ 1 ∂ ∥ ∂ −2 ∂ pffiffi ≈ þ ¼ ≈ γ0 : pffiffiffi k0 dz k0 ∂z V∥ ∂t β∥ ∂τ ∂τ ðτÞ¼ piffiffiffiG ˆ τ ð Þ ab π j0 × : 33 Thus, Eqs. (21) and (22) are transformed as follows: Let us compare solutions (30) and (33). If the difference ∂ −1 ∂φ γ − γ between the wave group velocity and the electron axial p⊥ 0 ≈ ×ReabðτÞ expðiφÞ; ≈ ; ð34Þ velocity is significant, then the radiated wave leaves the ∂τ 2 ∂τ γ0 electron bunch region fast, so that different “quanta” emitted from the bunch in different moments of time are with the initial conditions radiated independently [Fig. 4(a)]. In this case, the rf wave ðτ ¼ 0Þ¼ ¼ γ β φðτ ¼ 0Þ¼φ ð Þ emission has a quasistationary character [abðtÞ ≈ const] p⊥ p⊥0 0 ⊥0; 0; 35 described by Eq. (30). In contrast, if the group synchronism β ¼ β takes place ( gr ∥), then quanta emitted in different where the initial phases are distributed uniformly over moments of time do not leave the bunch region, so that the interval described by Eq. (23). The electron energy is rf field is accumulated close to the bunch [Fig. 4(b)]; related with the transverse momentum by the general γ2 ¼ 1 þ 2 þ 2 according to Eq. (33), this is describedpffiffi by the growth of the relativistic relation p⊥ p∥; in the case of rf wave amplitude, abðtÞ ∝ t. This is analogous to the β∥ ¼ const, this leads to the following formula: superradiation effects, studied both theoretically and exper- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi imentally for various types of electron masers [32,33]. 1 þ p2 Evidently, the use of this superradiant regime should lead to γ ¼ ⊥: ð Þ 1 − β2 36 a faster rate of the loss of the transverse electron velocity in ∥ the radiation process. Thus, the group synchronism regime is required to provide both conservation of the axial Let us use the “superradiant” approximation (33) for the ðτÞ electron velocity (this is required for the “proper” trans- rf wave amplitude, ab , as well as the approximation of a formation of the “axial velocity cooling” into the “energy small phase size of the electron beam, δφ0 → 0. Moreover, we assume that φ ¼ const ¼ 0, so that the radiation process a(z) does not result in a significant change in the electron energy (and, therefore, in the electron phase); this is true in the case t1 < t2 t2 < t3 t3 rf of small enough initial transverse momentum. Then one (a) V gr < V || t=t3 z obtains e V || β⊥0G ðτÞ ≈ γ β − pffiffiffiffiffiffi τ3=2 ð Þ a(z) p⊥ 0 ⊥0 : 37 (b) a(z) 3 2π a(z) t=t3 2 t=t2 The normalized time τ ¼ ω0ðt − V∥z=c Þ, which corre- t=t1 z sponds to a particle with the axial coordinate z ¼ V∥t,is t ≈ 1 V gr V || t1 t related to the length of the path of this particle, L,as t rf 2 1 t2 t3 follows:
e V || τ ≈ γ−2 × 2πL=λ: FIG. 4. (a) Portions of the rf wave radiated by the electron bunch in different moments of the time t1 110702-7 I. V. BANDURKIN, I. V. OSHARIN, AND A. V. SAVILOV Phys. Rev. ST Accel. Beams 18, 110702 (2015) 3=2 2 1 L L I 8π As an example, we consider an electron bunch with the 0.4 total charge 1 nC and the duration 1 ps; this corresponds ¼ 1 ¼ 0 3 ¼ λ 3 0.2 to I kA and Le . mm. We assume also Le = , (a) λ ≈ 1 so that the wavelength of the radiated wave is mm. 0 For these parameters, estimation (38) is reduced as 0 200 400 600 800 follows: |ab | 8=3 L=λ ≈ γ0 : ð39Þ (b) In the case of a 5 MeV electron bunch (γ0 ≈ 10), the ∼ 500λ ¼ 50 radiation section length amounts to L cm. An 0 200 400 600 800 increase in the electron energy up to γ0 ≈ 50 results in L ∼ 35000 λ ¼ 35 m 0.05 , . γ γ γ Let us notice that parameters mentioned above corre- < 0- >/ 0 spond to a waveguide diameter of 6 mm at γ0 ≈ 10 and to a waveguide diameter of 3 cm at γ0 ≈ 50. This is true in the case of the lowest TE1;1 operating transverse mode; naturally, the system could be more oversized, if the (c) “ ” β ¼ β operating group synchronism condition gr ∥ is 0 organized for a higher transverse mode. In this situation, 0 200 400 600 800 λ it is natural (at least, in the simplest model) to neglect the Axial coordinate, z/ transverse-position-dependent effects (wakefields, inhomo- FIG. 5. Results of simulations for the electron energy γ0 ¼ 10: geneity of the transverse structure of the radiated wave), (a) averaged transverse electron energy hw⊥i, (b) rf wave which could be the source of an additional spread in amplitude in the point of the electron bunch, and (c) the averaged electron energy. change in the total electron energy versus the axial coordinate of Estimations (38) and (39) are followed from formula the bunch. Solid curves illustrate the case of the exact group (33), which is obtained in the approximation of the constant synchronism (α ¼ 0), and dashed curves correspond to the case α ¼ 0 072 transverse electron velocity, β⊥ ≈ β⊥0. More important, of the optimal mismatch of the group synchronism ( . ). estimation (38) is obtained in the approximation of a small phase size of the electron bunch, δφ0 → 0. This approxi- δγ 1 ∂γ 2 ≈ δðβ2 Þ¼p⊥0 δ mation is true, when the wavelength significantly exceeds 2 ⊥ w⊥: γ0 γ0 ∂ðβ⊥Þ 2 the bunch length, λ ≫ Le. However, according to Eq. (38), an increase in the wavelength leads to the increase in the radiation section length. On the other hand, if the bunch Figures 5 and 6 show results of numerical simulations for γ ¼ 10 ¼ 1 ¼ λ 3 phase size is not small, then electrons with different phases an electron bunch with 0 , I kA and Le = , have a different rate of the radiation loss of their transverse when the initial phase size of the bunch is quite large δφ 2π ¼ 1 3 velocities. At the same time, in order to provide minimi- 0= = , and the initial transverse momentum γ β ¼ 0 3 zation of the spread in energy, one should provide the imparted by the kicking undulator is 0 ⊥0 . . Solid situation, when all electrons lose their oscillatory velocity curves in Fig. 5 illustrate averaged (over the whole electron h i almost in the same point. ensemble) transverse kinetic energy w⊥ , rf wave ampli- In order to study this problem, we solve numerically tude in the point of the electron bunch and the averaged – change in gamma factor, hγ − γ0i=γ0, versus the axial motion equations (34) (36) together with formula (32) 2 for the rf wave amplitude. We introduce the normal- coordinate of the bunch, z=λ ¼ γ τ=2π, in the case of β ¼ β ized “transverse kinetic energy” of a particle, w⊥ ¼ the exact group synchronism, gr ∥, when the rf wave 2 α ¼ 0 ðβ⊥=β⊥0Þ . Since β∥ ¼ const in the radiation process, amplitude is determined by formula (32) at . The and the spread in initial axial velocity has been minimized behavior of transverse kinetic energies w⊥ of electrons with different initial phases 0 ≤ φ0 ≤ δφ0 is illustrated in at the beginning of thisq process,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the spread in relativistic Fig. 6(a). γ ¼ 1 1 − β2 − β2 electron energy, = ∥ ⊥, is determined by In the beginning of the radiation process, the rf wave the spread in w⊥: amplitude increases as predicted by formula (33), 110702-8 CYCLOTRON RADIATION COOLING OF A SHORT … Phys. Rev. ST Accel. Beams 18, 110702 (2015) this value, the averaged transverse energy of all electrons of the bunch, hw⊥i, becomes very close to zero at some point of the radiation section [Fig. 5(a)]. After this point, the radiation process should be stopped (for instance, by changing the group velocity of the radiated waveguide mode). In the case of γ0 ¼ 10 and Le ¼ λ=3, the optimal value of the slippage factor amounts to α ¼ 0.072. In this case, at the length of the radiation section z=λ ¼ 600–800, trans- verse energies of all electrons of the bunch becomes close to zero [Fig. 6(b)]. This length is slightly longer than the optimal length in the case of the exact group synchronism (α ¼ 0); according to Fig. 5(b), this is due to a smaller amplitude of the rf wave in the case, when there is a slippage between the electron bunch and the radiated rf signal. As for the loss in the total electron energy caused by the cooling radiation hγ − γ0i=γ0, this is as small as several percent [Fig. 5(c)]. Figures 7 and 8 are similar to Figs. 5 and 6, respectively, but they illustrate the cooling process for an electron bunch with a higher energy, γ0 ¼ 50. As compared to the case of 1 FIG. 6. Results of simulations for the electron energy γ0 ¼ 10. 0.2 pffiffiffi (a) abðzÞ ∝ z. However, abðzÞ goes to saturation, when the 0 loss in the averaged transverse energy of electrons, hw⊥i, 0 50000 100000 becomes significant. Estimations (38) and (39) predict quite exactly the radiation section length corresponding |ab | to the maximal loss in the averaged transverse energy, hw⊥i [Fig. 5(a)]. However, this loss is not total; the particles of the bunch lose approximately 70% of their initial transverse energy. The reason has been mentioned above, namely, this (b) is a different rate of the emission of electrons with different initial phases, φ0. According to Fig. 6, every particle loses 0 50000 100000 ¼ 0 0.05 its initial transverse energy, w⊥ , at some point of the γ γ γ axial coordinate. After this point, the cyclotron emission of < 0- >/ 0 this particle is changed to the absorption of the energy of the radiated wave propagating together with the particle, such that the transverse energy of this electron increases. β ¼ β Since in the case of the exact group synchronism ( gr ∥, (c) α ¼ 0 ) the points of the total loss of the transverse energy, 0 w⊥ ¼ 0, are very different for particles with different initial 0 50000 100000 phases [Fig. 6(a)], the averaged transverse energy of all Axial coordinate, z/λ electrons of the bunch, hw⊥i, cannot reach zero. γ ¼ 50 In order to solve this problem, one should provide the FIG. 7. Results of simulations for the electron energy 0 : (a) averaged transverse electron energy hw⊥i, (b) rf wave situation, when the group synchronism condition is fulfilled β ≈ β amplitude in the point of the electron bunch, and (c) the averaged only approximately, gr ∥. In this case, the slippage change in the total electron energy versus the axial coordinate of factor α in formula (32) for the rf wave amplitude is not the bunch. Solid curves illustrate the case of the exact group equal to zero. If the bunch phase size is not too large, δφ0, synchronism (α ¼ 0), and dashed curves correspond to the case then there is an optimal value of the slippage factor, α;at of the optimal mismatch of the group synchronism (α ¼ 0.014). 110702-9 I. V. BANDURKIN, I. V. OSHARIN, AND A. V. SAVILOV Phys. Rev. ST Accel. Beams 18, 110702 (2015) estimated from Eq. (22). The spread in initial electron energy, δγ, induces the following spread in mismatch: (a) 2 Ω 0 δγ 1 δγ δγ 1.5 δΔ ≈ c ≈ − β ≈ : w⊥ gr 3 β∥ω0 γ0 β∥ γ0 γ0 1 0.5 The radiation process is sensitive to this spread, if the spread induces a significant spread in electron phases 0 during the motion of the electron bunch through the entire 0 50000 100000 radiation section, k0L × δΔ ≈ 2π. This leads to the follow- 1 ing simple estimation for the permissible spread in electron energy: w⊥ δγ 2 λ 0.5 ≈ γ0 : ð40Þ γ0 L γ ¼ 10 (b) In the case of 0 , the radiation section length 0 amounts to L=λ ¼ 600–800 [Fig. 5(a)]; this leads to a 0 50000 100000 reasonable value of the permissible energy spread Axial coordinate, z/λ (δγ=γ0 ≈ 13–17%). As for the case of γ0 ¼ 50, the λ ≈ 105 FIG. 8. Results of simulations for the electron energy γ0 ¼ 50. radiation section length L= [Fig. 7(a)] corresponds Transverse kinetic energies w⊥ of electrons with different initial to a permissible energy spread of δγ=γ0 ≈ 2.5%. According phases versus the axial coordinate of the bunch in the cases of to estimations (39) and (40), the further increase in the (a) the exact group synchronism (α ¼ 0) and (b) the optimal energy should lead to a decrease in the relative permissible mismatch of the group synchronism (α ¼ 0.014). −2=3 energy spread, δγ=γ0 ∝ γ0 . An important peculiarity of the electron-wave inter- action illustrated by Figs. 5 and 7 is the slowness of this γ0 ¼ 10, the interaction of more heavy electrons with the process at the end region, where transverse momenta of radiated wave is weaker; correspondingly, the required all electrons are small and, therefore, their interaction length of the radiation region is longer. In this situation, the with the rf wave (either the further radiation or the ¼ λ 3 same phase size of the electron bunch (Le = ) leads to a absorption of the wave energy) is weak. Due to this fact, significantly bigger spread in the electron transverse the length of the radiation section is not strongly fixed. energy, w⊥, arisen in the radiation process. According to Inthecaseofγ0 ¼ 10 the length can be varied within a Fig. 8(a), in the case of the exact group synchronism range of L=λ ¼ 600–800 [Fig. 5(a)], such that the (α ¼ 0)atz=λ > 50000 the bunch contains both of the possible relative variation is as big as δL=L ≈ 30%. emitting electrons, which have lost almost the whole of This fact makes the cooling process weakly sensitive to their initial transverse energies, and absorbing particles nonstability (jitter) of parameters of the electron bunch. with significantly increased transverse energies. In this Actually, according to Eq. (38), the permissible varia- situation, the radiation provides the loss of only about half tions of the electron current in the bunch, δI,andofthe of the initial transverse energy of the whole bunch, hw⊥i bunch length, δLe, are related with a variation of the [Fig. 7(a)]. However, even in this case the proper choice of radiation section length as follows: the slippage factor (α ¼ 0.014) provides the situation, when all electrons of the bunch lose their transverse δI δL 3 δL energies almost simultaneously [Fig. 8(b)], and the aver- ¼ e ≈ ≈ 45 2 %: aged transverse energy of the bunch becomes close to zero I Le L [Fig. 7(a)]. Similar to the case of γ0 ¼ 10, the loss in the total electron energy caused by the cooling radiation In the case of γ0 ¼ 50 the radiation section length can be amounts to several percent [Fig. 7(c)]. varied within a range of L=λ ¼ð1.0–1.2Þ × 105 [Fig. 7(a)]; Let us discuss these results from the point of view of thus corresponds to δL=L ≈ 18% and to the following sensitivity of the proposed system to electron bunch permissible variations of the electron bunch parameters: parameters. Although the spread in axial electron velocity δI=I ¼ δLe=Le ≈ 27%. Evidently, the further increase in is minimized by the kicker (axial cooling undulator), the the electron energy should lead to an increase in the spread in initial electron energy (in electron cyclotron sensitivity of the cooling process to a jitter of bunch frequency) exists. The effect of this spread can be easily parameters. 110702-10 CYCLOTRON RADIATION COOLING OF A SHORT … Phys. Rev. ST Accel. Beams 18, 110702 (2015) IV. CONCLUSION of all, the energy determines the length of the radiation region required to provide the entire loss of the transverse In this paper, we propose to use of an undulator with the energy of all particles. In the example considered in the guiding axial magnetic field as a kicker, which transforms a paper, this length amounts to several hundreds of wave- bunch of rectilinear electrons into a bunch of particles, lengths in the case of a 5 MeV electron bunch, whereas an which move in a uniform axial magnetic field and perform increase in the electron energy up to 50 MeV results in free cyclotron oscillations. At definite conditions, such a 105 kicker can form a bunch with very special properties, length as long as wavelengths. Naturally, longer namely, the initial (at the input of the kicker) spread in lengths of the radiation section cause stronger sensitivity electron energy can be transformed into the spread in of the cooling process to the spread in electron energy and oscillatory (transverse) velocity, whereas the spread in the to a jitter of the electron bunch parameters. Evidently, this axial velocity is minimized at the output of the kicking is the main factor limiting the usage of the proposed system. This effect (axial cooling) can be provided, if the approach for high-energy electron bunches. cyclotron electron frequency corresponding to the axial In conclusion, we could state that the system described in magnetic field exceeds the bounce frequency of electron this paper can be interesting from the point of view of not oscillations in the periodic undulator field. In this situation, only electron cooling, but also as a way for creation of a electrons with higher input energies obtain bigger undu- submillimeter-wavelength source of coherent cyclotron lator oscillatory velocities. In the case of a magnetostatic radiation from a relativistic electrons bunch. Actually, undulator, the value of the axial magnetic field required to the proposed way to impart cyclotron rotation to electrons provide this “negative-mass” effect is proportional to the in the axial cooling undulator is attractive, as it provides electron gamma factor. The problem of high magnetic minimization of the spread in axial electron velocity; the “ fields required to provide axial cooling at high electron strong sensitivity to this kind of spread is the Achilles ” energies can be solved by the use of an rf-wave undulator heel of relativistic electron cyclotron masers with a high – copropagating together with the electron bunch. Doppler up-conversion of the electron cyclotron fre- The cyclotron emission from the electron bunch leads to quency. The use of the coherent spontaneous character losing oscillatory velocity of electron gyrorotation in a of the cyclotron radiation (instead of the stimulated magnetic field. However, if the group velocity of the emission) is also attractive, as it provides a high efficiency radiated wave (waveguide mode) coincides with the axial and a high rate of the extraction of the kinetic electron electron velocity (the group synchronism condition), then energy. In addition, the phase of the radiated rf signal is the cyclotron emission does not lead to a change in the axial fixed by the short electron bunch; this property can be electron velocity. This effect can be used for transformation useful for some applications. of the axial cooling of the electron bunch (minimization of the spread in electron axial velocity) in the undulator ACKNOWLEDGMENTS section into the “real cooling” (minimization of the spread in electron energy) in the cyclotron radiation section. The This work is supported by the Russian Foundation for group synchronism condition is needed also to provide the Fundamental Research, Projects No. 14-08-00803, No. 14- maximal rate of the radiation and, therefore, of the cooling 02-00691, and No. 15-42-00260. process. Another condition of the fast cooling process is the coherent character of the cyclotron radiation; this occurs if the length of the electron bunch is smaller than the [1] J. G. Power, in Proceedings of 14th Advanced Accelerator wavelength of the radiated wave. Concepts Workshop (AAC 2010), edited by S. H. Gold and It is important that the cooling of the electron bunch is G. S. Nusinovich (AIP, Melville, 2010), p. 163; AIP Conf. provided, if all electrons lose their transverse velocities Proc. 1299, 20 (2010). almost in the same point of the cyclotron radiation section. [2] B. Dunham, J. Barley, A. Bartnik, I. Bazarov, L. Cultrera, This is provided automatically, if the phase size of the J. Dobbins, G. Hoffstaetter, B. Johnson, R. Kaplan, S. electron bunch is small (the bunch length is negligibly short Karkare et al., Appl. Phys. Lett. 102, 034105 (2013). as compared to the wavelength of the radiated wave). [3] F. Stephan, C. H. Boulware, M. Krasilnikov, J. Bahr, G. However, shorter wavelengths are more attractive from the Asova, A. Donat, U. Gensch, H. J. Grabosch, M. Hanel, L. point of view of the length of the radiation section. It is Hakobyan et al., Phys. Rev. ST Accel. Beams 13, 020704 shown in this paper that even in the case when the bunch (2010). [4] H. P. Freund and T. M. Antonsen, Principles of Free- length is comparable with the wavelength, the simultaneous Electron Lasers (Chapman & Hall, London, 1996). loss of the transverse velocities of all electrons of the bunch [5] Z. Huang and K.-J. Kim, Phys. Rev. ST Accel. Beams 10, can be provided by an optimization of parameters of the 034801 (2007). microwave system of the radiation section. [6] C. Bostedt, J. D. Bozek, P. H. Bucksbaum, R. N. Coffee, The process of cyclotron radiation cooling studied in this J. B. Hastings, Z. Huang, R. W. Lee, S. Schorb, J. N. paper is strongly dependent on the energy of electrons. First Corlett, P. Denes et al., J. Phys. B 46, 164003 (2013). 110702-11 I. V. BANDURKIN, I. V. OSHARIN, AND A. V. SAVILOV Phys. Rev. ST Accel. Beams 18, 110702 (2015) [7] A. M. Sessler, Report No. LBL-38278 UC-427, 1996; [24] V. L. Bratman, A. E. Fedotov, and A. V. Savilov, Int. J. Proceedings of the 31st Workshop: Crystalline beams Infrared Millim. Waves 19, 939 (1998). and related issues, Erice, Sicily, Italy, 1995 (World [25] H. Jory, R&D Technology Report No. ECOM- Scientific, Singapore, 1996). 01873-F (Varian Associates, Paolo Alto, CA, 1968). See [8] J. L. Hirshfield and G. S. Park, Phys. Rev. Lett. 66, 2312 National Technical Information Service Document (1991). No. AD675509 (H. Jory, Investigation of Electronic [9] T. Ohgaki and I. Endo, Phys. Rev. ST Accel. Beams 4, Interaction with Optical Resonators for Microwave Gen- 111001 (2001). eration and Amplification). Copies may be ordered from [10] D. Mohl and A. M. Sessler, Nucl. Instrum. Methods Phys. the National Technical Information Service, Springfield, Res., Sect. A 532, 1 (2004). VA 22161. [11] E. G. Bessonov, M. V. Gorbunkov, and A. A. [26] V. L. Bratman, A. E. Fedotov, Y. K. Kalynov, V. N. Mikhailichenko, Phys. Rev. ST Accel. Beams 11, Manuilov, M. M. Ofitserov, S. V. Samsonov, and A. V. 011302 (2008). Savilov, IEEE Trans. Plasma Sci. 27, 456 (1999). [12] H. Deng and C. Feng, Phys. Rev. Lett. 111, 084801 (2013). [27] M. I. Petelin, Radiophys. Quantum Electron. 17, 686 [13] I. V. Bandurkin, S. V. Kuzikov, and A. V. Savilov, Appl. (1974). Phys. Lett. 105, 073503 (2014). [28] V. L. Bratman, N. S. Ginzburg, and M. I. Petelin, Opt. [14] N. S. Ginzburg and N. Yu. Peskov, Phys. Rev. ST Accel. Commun. 30, 409 (1979). Beams 16, 090701 (2013). [29] V. L. Bratman, N. S. Ginzburg, G. S. Nusinovich, [15] C. Nielsen and A. Sessler, Rev. Sci. Instrum. 30, 80 (1959). M. I. Petelin, and P. S. Strelkov, Int. J. Electron. 51, 541 [16] A. A. Kolomensky and A. N. Lebedev, Atom Energy 7, 549 (1981). (1959). [30] V. L. Bratman, G. G. Denisov, N. S. Ginzburg, and M. I. [17] A. V. Gaponov, M. I. Petelin, and V. K. Yulpatov, Radio- Petelin, IEEE J. Quantum Electron. 19, 282 (1983). phys. Quantum Electron. 10, 686 (1967). [31] A. V. Savilov, Phys. Rev. E 64, 066501 (2001). [18] V. L. Bratman, Tech. Phys. 46, 2030 (1976). [32] N. S. Ginzburg, I. V. Zotova, and A. S. Sergeev, JETP Lett. [19] A. Bondeson and T. M. Antonsen, Jr., Int. J. Electron. 61, 60, 513 (1994). 855 (1986). [33] N. S. Ginzburg, A. S. Sergeev, I. V. Konoplev, I. V. Zotova, [20] H. Li and T. M. Antonsen, Jr.,, Phys. Plasmas 1, 714 (1994). A. D. R. Phelps, A. W. Cross, S. Cook, P. Aitken, V. G. [21] V.L.Bratman and A. V.Savilov, Phys. Plasmas2, 557 (1995). Shpak, M. I. Yalandin, and S. A Shunailov, Phys. Rev. Lett. [22] A. V. Savilov, Phys. Plasmas 4, 2276 (1997). 78, 2365 (1997). [23] V. L. Bratman, O. Dumbrajs, P. Nikkola, and A. V. Savilov, [34] R. D. Richtmyer, Principles of Advanced Mathematical IEEE Trans. Plasma Sci. 28, 633 (2000). Physics (Springer-Verlag, Berlin, 1978), Vol. 1. 110702-12