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 . time TL= / v, where Ldss =+Δuu /cos[θθ ( )] is the euu uu∫0 in 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 .