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Basic Physical Processes and Principles of Free Electron Lasers(FELs)Dagnachew W. WorkieDepartment of PhysicsUniversity of CincinnatiCincinnati, Ohio 45221November 26, 01AbstractA tunable coherent and high peak power sources have been very usefull in many ap-plications. In this paper the physical process of new type of coherent, flexible and verybright radiation source called Free-Electron Lasers (F ELs) is descreibed. The motion ofelectron beam in the presence of the optical and undulator fields leads to energy exchange.This energy exhange leads to formation of bunches of electrons which will emit very brightradiation. The resonance wavelength can, in principle,be tuned to any radiation regions.This is possible by changing the undulatorparameter and/or electron beam energy. Theline-shape of the emitted spectrum has the characteristic form of the short time emissionprocess. Different regions of the emitted radiation have found very wide applications.11 IntroductionThe Free electron lasers (F ELs) was invented by by J.M.J. Madey in 1971[5]. Free ElectronLasers are very flixible sources of coherent radiation. Due to their wide range tunabilty andhigh brghtness they have a growing applications. The F ELssystem consists of an electronaccelerator, an undulator in which the electrons emit the syncrotron radiation, and an opticalresonator.2 Principles of operation of FELIn an FEL a beam of relativistic electrons produced by an accelerator passess through atransverse periodic field produced by a magnet called undulator and exchanges energy withan electromagnetic field (Fig. 1). As a result of energy exchange, the electrons that gainFigure 1: Basic components of FEL [3].energy begin to move ahead of the average electron, while the electrons that lose energy beginto fall behind the average . The beam then becomes bunched on the scale of the radiationwavelength and this collective motion of the bunches radiatespowerful coherent synchrotronradiation. Giga watts peak power have been demonstarted. The wavelength of the emittedradiation at the resonance depends on the electron energy and the magnitude and periodicityof the undulator field according to the relation [1, 3]λo=λu2γ2(1 + κ2) (1)where λuis the undulator period, γ is the relativistic factor and κ is the so called undulatorparameter which is proportional tothe magnetic field inside the undulator.Typical values for electron beam enery ranges from few MeV (γ ∼ 1) toa few GeV (γ ∼1000); and the peak current varies from several ampersto many hundred of ampers. The peakundulator field strength Bois usually several KG (Bo∼ 2 − 7KG). The undulator period λuis in the range of a few centimeters (λu∼ 2 − 10cm), whereas the undulator length Lu= Nλuextends for a few meters (Lu∼ 2 − 20m), so that the number of periods N can vary from 20to several hundreds[1].23 Basic Mechanism of FEL InteractionsThe motion of the electron in the presence of the optical and the undulator field is governedby the Lorentz force equations [1, 3]:ddt(γ~β) = −emc(~E +~β × (~Bu+~β)) (2)ddtγ = −emc~β ·~E (3)Where~E and~B are the electric and magnetic componenets of the e.m fields. Writting eqns.(2)and (3) interms of transverse and longitudinal components we obtain:ddt(γ~β⊥) =emc(~βz×~Bu) −emc(1 − βz)~E (4)ddt[γ(1 − βz)] = −emc(~z ×~Bu) ·~β⊥(5)Where,~βz=1cdZdtˆz. The first term on the right of eqn. (4) accounts for the interaction ofthe electrons with the undulator magnetic field ; whereas the the second term summerizesthe effect of the transverse e.m fields,which combines almost to cancel out each other in therelativistic limit (βz→ 1). Eqns. (4) and (5) desplay the complementary role of the undulatorand optical fields in the FEL interaction.Considering the ideal plane -polarized undulator magnetic field,~Bu= ˆyBosin kuz, ku=2πλuand linearly ploarized emitted fields with a complex amplitude E(~r, t) =| E(~r, t) | eiφ(~r,t),propagating along the z-direction,~E(~r, t) = ReE(~r, t)ei(kz−ωt)ˆx,~B(~r, t) = ˆz ×~E (6)eqns. (4)and (5) becomes, respectively:ddt(γβ⊥) = −ˆx(κddtcos kuz − KR(~r, t)[ddtsin Ψ(~r, t) −ddtcosΨ(~r, t)]) (7)ddtγ =ωγKR[κ cos kuz cos Ψ(~r, t) − KRsin Ψ(~r, t) cos Ψ(~r, t) (8)Assuming that the electron to experience a nearly constant optical field (over λu) eqns ( 7)and (8) can easily be integrated to get the longitudinal velocity (βz= [1 − β⊥2−1γ2]1/2)βz= [1 −1γ2(1 + κ2cos2kuz + K2Rsin2Ψ(~r, t) − 2κKRcos kuz sin Ψ(~r, t))]1/2(9)where,κ =eBomcωu; and KR(~r, t) =e | E(~r, t) |mcω,are the undulator parameter and the dimenssionless optical vector potential, respectively; andΨ(~r, t) = (kz − ωt) + φ(~r, t)3is the phase angle combining of the fast and slow varing componnents of the e.m fields.Since usually κ > KRby the order of magnitude, the average value¯βz(t) is given by:¯βz(t) = 1 −1 + κ2/22γ2(t)(10)On the other hand, requiring~β⊥to be in phase with the optical field during the motionassures that the electron transverse velocity retains its orentation relative to the optical electricfield, thus simplifying that the energy exchange persists in the same direction. The resonantcondition demands therefore for vanishing the difference between the optical phase and theelectron transverse velocity phase at the instantaneous electron position,thus providing theexpression[1]ω − kc¯βz= ωu¯βz(11)For propagation in vacuum (ω = kc ) and using eqn. (10) in the above relation one finds theresonant frequency ωoand wavelength λ0given byω0=2γ20ωu1 + κ2/2, and λ0=λu2γ20(1 +κ22) (12)In general the electron phase in the combined optical and undulator fields is defined byξ(t) = kZ − (ω − ωu)t = (k + ku)¯Z − ωt (13)Introducing a dimensionless time τ =ctLu, eqns. (10) and (12) gives us the parametermeasuring the degree of resonanace (phase velocity):υ(τ) =ddτξ(τ) = 2πN[1 −1 + κ2/22γ2(t)]ω0(τ) − ωω0(τ)(14)where, ω0(τ) = 2γ2(τ)ωu1+k2/2is the instantaneous resonance frequency.4 Spectral charactersitics of the Emmitted RadaitionsFor relativistic electron moving on arbitrary curved trajectory, the electric field is given bythe standard expression[2]:~E(t) = e[ˆn − βR2γ2(1 − ˆn · β)3]ret+ec[ˆn × (ˆn − β) ×˙βR(1 − ˆn · β)3]ret(15)Where the subscript ret means that the quantieties in the square braket are evaluated at theretarded time t0= t − R(t0) and R(t0) =~X − ~r(t0) is