SELF-STIMULATED UNDULATOR KLYSTRON E.G Bessonov, A.L. Osipov, Lebedev Phys. Inst. RAS, Moscow, Russia A.A.Mikhailichenko, Cornell University, CLASSE, Ithaca, NY 14853, U.S.A.

Abstract The Self Stimulated Undulator Klystron (SSUK) and its possible applications in the Particle Accelerator Physics, incoherent Self-Stimulated Undulator Radiation Sources (SSUR) and Free-Electron Lasers (FEL) are discussed.

INTRODUCTION The system of two undulators located one by one in a Figure 1. Scheme of the installation and its sequence at some distance along the straight line is called equivalent optical analog. an Undulator Klystron (UK). It was invented by We would like to underline here, that in OK there is no R.M.Phillips in 1960 for generation of spontaneous co- requirements that the particle should interact with its own herent UR [1]. In the first undulator (modulator) the elec- wavelet, so here is the principal difference between OK tron beam was modulated by energy in the field of and SSUK. The SSUK scheme is similar to ones consi- copropagated wave, then in a straight section it was dered in [11], [12]. It has two undulators also but does not grouped in the bunches and in the second undulator (ra- include an X-ray monochromator or an amplifier located diator) the bunched beam emitted a coherent undulator in between. The first undulator stage in [11] operates as radiation (UR) on the lowest or higher harmonics. Inverse the FEL amplifier in the SASE linear regime. After exit- FEL-accelerator scheme and tapered undulators were ing out from the first undulator the electron bunch is suggested there as well. Note that the both spontaneous guided through a non-isochronous bypass and the X-ray incoherent and coherent UR sources were suggested by beam enters the monochromator. The main function of V.L.Ginsburg in 1947 [2]. Later the spontaneous coherent this bypass is to suppress the modulation of the electron UR sources were named by parametric (superradiant, pre- beam induced in the first undulator. At the entrance in the bunched) FELs [3], [4]. Below we will use more suitable second undulator the radiation power from the monoch- term “pre-bunched” suggested by A.Gover [5]. The UR romator dominates significantly over the shot noise and emitted in the UK consisted of Nu undulators located at the power determined by the residual electron bunching. some distances along a straight line was investigated in As a result, the second stage of the FEL amplifier oper- [6], [7]. The UK with a dispersion element located in its ates in a steady-state regime. Phase correlations of the straight section for enhancement the bunching process for URWs emitted in the first and second undulators by the the ultrarelativistic particles was called an Optical Kly- same particle was not considered in [11]. The URW emit- stron (OK); it was suggested in 1977 [8]. ted in the first undulator in [12] was amplified and used in SSUK is further modification of UK with controlled the second undulator for production a coherent kick to the delay of the Undulator Radiation Wavelets (URWs) mov- particle for cooling of the beam. URW emitted in the ing between the undulators [9], [10]. The optical delay second undulator was not used. line is arranged with the mirrors and lenses. It serves for Using the SSUK in our case means the generation of proper phasing of the URWs with particles for their fur- coherent URWs by every particle of the beam in the sys- ther interaction in the following undulator. The special tem of Nu undulators of the SSUK under conditions of magnet system installed between the undulators (kicker) incoherent emission of coherent URWs by particles of the serves for separation of the URWs from the particle beam (incoherent superposition of coherent URWs emit- beam, see Fig.1. The URWs and particle beams are fo- cused back to the location at the entrance of the down- ted by N particles of the beam in Nu undulators of the stream undulator. The optical and particle’s delays are SSUK). These elements are playing a key role in our pro- chosen so that the particle enters the following undulator posal of SSUK. Usage of SSUK in FEL where the cohe- in a decelerating phase at the front edge of its own URW, rence among the particles of the beam exists already is emitted in a preceding undulator. Under such conditions useful as well. the superposition of the URW emitted in the first undula- tor and the URW emitted in the following one occurs, MAGNETIC LATTICE PROPERTIES OF which yields the field growth ~ N and the energy density u THE SSUK 2 N growth in emitted radiation becomes ~ u . So the Self- We considered the case when the optical delays are tuned Stimulated UR (SSUR) is emitted by each particle in the so that the wavelets emitted by the particles are congruent SSUK in the self-fields of its own wavelets emitted at and all particles stay at decelerating phase. For this pur- earlier times in the upstream undulators. poses the beam delay system in the SSUK must be quasi- are the energy and angular spreads of the particle beam. It isochronous. To be optimally effective, the optical part of follows from here that the requirements to the beam pa- a system must use appropriate focusing elements such as rameters should be lenses and/or focusing mirrors. The mirrors and lenses 22 Δ+γλγλbm uu(1K ) form a crossover in the middle of the undulators with the  22= , γ 2(1)4(1)LKuu++ kick mLK uu kick ZM≅ λ /2 λ Rayleigh length Ru, where u is the undulator 2 / λλmuu1 (1+ K ) period, M is the number of undulator periods. σ b  = , (3) The URWs emitted by each electron on the harmonic LmLuuγ 2 uu with the number m are overlapped effectively at the exit where Ku is the undulator deflection parameter. We took of the SSUK by the superposition one by another if theirs into account the equation for the emitted UR wavelength longitudinal shifts satisfy the following condition 22 λmu=+λγ(1K u )/2m . In this case the local slip factor Δ=lcT||/2, ⋅ΔeURW, − nλ m <<λ m (1) η = (/γδγTT )(∂∂= )/ (1 + K 22)/γ . where λ = λ / m is the wavelength of the UR emitted by cloc, e e kick m 1 The requirements to the beam parameters for SSUK (3) the electron on the m-th harmonic in the direction of its are much easier than the ones for SSUR source based on average velocity, Δ=−TTT is the difference eURW, e URW the storage ring [10]. Note that a small slip factor system between the entrance time of the URW and the electron to of two undulators separated by the bending magnetic sys- tem was used to study the radiation coherency conditions the next undulator, TURW=+Δ=()/ L uu l c const , Te = in optical region [13], [14]. It means that technical reali- TKekickin(,ε ,θ ), Luu is a distance between the undulators, zation of tuning of the URWs is possible in the optical and even harder wavelength regions. L + Δl is the length of the light way in the optical delay uu For a kicker with small bending magnet lengths system, ε is the particle energy, K is the deflection kick ( lllbbb132==/2 << L uu), the value Kkick= ϑ b , where parameter of the kicker (the kicker can be considered as a 2 −4 θ = eBl/ mc γ  610⋅ Bgs [ ]⋅ l [ cm ]/γ is the bending half - period undulator), θ is the initial angle between bb1 b1 in angle of the first and third kicker magnets (we neglected the particle velocity and the UK axis, the influence of magnetic fields of the quadrupole lenses). nnmM=±±0, 1, 2,... | | ≤ is the synchronicity condition In this case the orbit will be deviated from the SSUK axis number. at its center by the value aL= uuθ b /2. It must be 5-10 The electron pass on the way between undulators for the σ L times higher than the rms particle beam size b . TL=  / v Ldss =+Δuu /cos[θθ ( )] time euu, where uu∫0 in is the Example. Let the SSUK has 2 identical helical undula- length of particle’s trajectory between the undulators, v is tors with the period λu =3 cm, number of periods M=30, s the particle velocity, Δ=θβε()s ∫0 [eB ()/ s ] ds is the bending the distance between the undulators Luu = 1 m, bending β =vc/ angle along the kicker’s axis at the position s, , 3 Bs() is the kicker’s transverse bending magnet field magnet lengths lllbbb132= ==/2 3 cm, γ = 10 , the 2 strength. In the relativistic case βγ=−vc/11/2 , deflection parameters Ku = 2 , the transverse beam di- 2 |()|1θθin +Δ s  , cos[θθin +Δ (s )] 1 − (())/2θθin +Δ s the val- σ  1 mension b mm. ues In this example the bending angle of the first and third 2 2 Ls  θin e uu 2 −2 LLuu=++ uu (1 ) [ Bsdsds ( ) ] , kicker magnets θ =⋅210 , the deviation of orbit in the 24 2 2∫∫00 b 22mce βγ Ls center of the SSUK aL= θ /2= 1 cm, K = 20 , the 22uu buu kick 2 eBsdsds[()] Luuθ in 1 ∫∫00 4 TLv==+++ /(1) , bending magnet field strength B  1.1⋅ 10 gs, the local euu cmc22γβγ22532 2 e −4 slip factor ηcloc, =⋅410 . The electron beam for every n Luu 22 δTTTeee=−,0 =2 (1 + K kickin +ϑ ) , (2) −4 2cγ and m=1 must have the energy spread Δ<γγb /10, the 2 /4− where γε=>>/1mc is the particle’s relativistic fac- σ <⋅2.7 10 e angular spread b .

Lsuu 22 tor, Kkick=e∫∫[()]// BsdsdsL uu mc e , ϑ = θγ , T = 00 e,0 POSSIBLE APPLICATIONS TBsein[0,()0]θ ==. The shifts of URWs, according to (1), / Spontaneous incoherent UR sources (2) are Δ=lc(/)(()/) ∂δγγδ T ∂ Δ + c ∂ T ∂ ϑσδ =− 2() cT eebeinb based on SSUK /2 2 / (/)()/(1)Δ+γγcT σ + K << λ /2 Δγ σ be b kick m , where b and b All properties of spontaneous incoherent radiation emitted by the particle beam in an undulator and in the SSUK SSUK FEL regime can be used here. In this case the based on such undulators under main synchronicity condi- power of the emitted coherent radiation is tion n = 0 are identical, except intensity, which becomes coh incoh 22 incoh PPNN≈ 1 u , where P is the power of incohe- 2 higher by Nu times. If the centers of URWs at the exit of rent radiation of the unbunched beam emitted in one un- the last SSUK undulator are displaced in the transverse dulator. direction inside some area with a dimension d >λmγ then The system of a modulator undulator (in a combination the additional degree of directionality will appear in the with the driving laser beam) and the radiator SSUK in- stalled in a storage ring (as well as in ordinary or energy 2 UR beam: Δθ~/d λm < 1K+ u /γ . At the same time recovering linacs and recirculators) tuned on the main or an increase in the power will be lesser than N 2 (phased higher harmonics of the microbunched beam can be used u by analogy with the scheme considered in [17]. antenna array analogy in prebunched FEL). It follows 2. Ordinary FELs. Using SSUK at the condition (1) in from the general theory of such FELs [15]-[16]. Two ordinary FELs will permit to decrease the threshold cur- bending magnets with opposite polarity located between rent of FELs and to increase their power. the undulators can be used for the transverse displace- ments of URWs. If URWs is emitted by an electron at the Cooling of particle beams −<≤mM n mM collateral synchronicity conditions are 1. Optical cooling. Usage of SSUK klystrons as pickups shifted in the longitudinal direction at the exit of the last for optical stochastic cooling [18], [12] and enhanced ±±λ , 2λλ ... ±mN undulator by the distances mm umthen the optical cooling of particle beams [19] will permit to in- 2 2 additional directionality Δθγ 1K+ / crease the number of photons in the sample Nu times u min{ M ,Nu } and the cooling rate N times. Using SSUK both as appears in the UR emitted by every electron (director- u type antenna analogy) [15, [16]. Moreover the intensity pickup and kicker undulator will increase the rate of cool- 22 ing N 2 times. will be increased by min{NMu , } times for nmM . u The intensity will be dropped and the monochromaticity 2. Cooling based on incoherent SSUR. If the revolution will be increased N times if the number period of a particle in a storage ring is multiple to the u round trip period of the URWs emitted by the particle in nmMnmM→≤(). The angular spread of the beam in the undulator installed in the straight section of the ring / this case must be small σ < Δθ . (see Fig. 2) and circulating in the optical resonator then b such URWs will be effectively stored and overlapped in In both cases we deal with the self-stimulated UR for the resonator under resonance conditions in some energy every particle of the beam and spontaneous incoherent interval [10]. Frictional losses of energy and cooling ap- UR between particles of the beam. There is no require- pear in this interval. Using an optical amplifier in the ment for the coherence in radiation among the different electrons in the bunch like it is required for the pre- bunched FELs. The stimulated process of radiation for each electron is going in the undulator with their own URW fields only. The considered phenomena of the power, directionality and monochromaticity increase in the SSUK installed in a storage ring or in the linear accelerators and recirculators takes place both for broad band and narrow band mirrors used in the SSUK in the optical up to to X-ray regions. Figure 2. Schematic diagram of SSUR installation The accuracy of the SSUK lattice tuning is built around a storage ring. λ (1+ K 2 ) Δ l  uu. (4) mN(1+ K 2 ) optical resonator system, switching it on for a short time ukick (dozens of particle revolutions) and switching it off for a It follows from the necessity to maintain by the optical small number of revolutions can lead to the high rate of delay line the distance between next URW relative to the / the particle energy loss, its strong dependence versus par- previous one with the accuracy of Δ lN λmu/ . ticle energy and, following the analogy with the Robinson theorem [21], to the high rate of particle beam cooling in Free-electron lasers the longitudinal phase space. In this case the losses of the 1. Prebunched FELs. In spontaneous incoherent SSUR energy in the undulator occur with equal probability and sources the URWs are emitted by each particle indepen- independently on the sign of transverse deviation of the dently from the other particles of the beam. Single micro particle from its instantaneous orbit. The jumps of the bunch with the number of particles N1 and the length particle amplitudes of betatron oscillations have different l << λ is equivalent to one particle with the charge signs and hence in the first approximation the cooling in mkb m the transverse plane is absent. Using SSUK instead of an eN1 . The trains of such micro bunches in the prebunched undulator may increase the cooling rate of particles in the [8] N.A.Vinokurov, A.N.Skrinsky. Preprint INP No77-59, storage ring. We suppose that the current of the particle Novosibirsk (1977). beam is less than a threshold current for such FEL-like [9] E.G.Bessonov, M.V.Gorbunkov, A.A.Mikhailichenko, scheme. Cooling is possible in such a way at the main and A.L.Osipov, “Self-Stimulated Emission of Undulator collateral synchronicity conditions as well [10]. Radiation”, Journal of Instrumentation The energy interval of the main and collateral synchro- JINST_012P_0510, 2010, p.1-4; arXive: nicity conditions depends on the slip factor of the ring. http://arxiv.org/abs/1003.3747. Quasi-isochronous storage rings are desirable in this case. [10] E.G.Bessonov, M.V.Gorbunkov, A. Mikhailichenko, A.L.Osipov, A.V.Vinogradov, “Self-Stimulated Un- CONCLUSION dulator Radiation and its Possible Applications”, We hope that the SSUK can be used as a high efficiency http://arxiv.org/ftp/arxiv/papers/1009/1009.3724.pdf. pickup for cooling of the proton, muon and ion beams in [11] J.Feldhaus, E.L.Saldin, J.R.Schneider, E.A. Schneid- the storage rings, as highly effective SSUR source based miller, M.V. 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Rev E, V. 50, linacs, International linear collider and FEL sources will No 4, 1994, p. 3087-3091. permit to enhance novel X-ray quantum optics experi- [13] G.N.Kulipanov, V.N.Litvinenko, A.S.Sokolov, ments in the femtosecond regime and generate γ rays N.A.Vinokurov, “On Mutual Coherency of Sponta- carring the Orbital Angular Momentum for nuclear and neous Radiation from two Undulators Separated by particle physics [22]. Achromatic Bend”, IEEE Journal of Quantum This work was supported in part by RFBR under Electronics, VOL. 27, No 12, 1991, p. 2566-2568. Grants No 09-02-00638a, 09-02-01190a. 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