Self-consistent Analysis of Radiation and Relativistic Electron Beam Dynamics in a Helical Wiggler Using Lienard-Wiechert Fields
M. Tecimer and L. R. Elias, Center for Research and Education in Optics and Lasers (CREOL) and Physics Department University of Central Florida, Orlando FL 32826 Lienard-Wiechert (LW) ?elds, which are exact solutions of the Many simulation codes of free electron laser ampli?ers utilize Wave Equation for a point charge in free space, are employed a single ponderomotive potential well imposing periodic to formulate a self-consistent treatment of the electron beam boundary conditions at the bucket ends, thus neglecting slipdynamics and the evolution of the generated radiation in long page effects. Based on the formalism employed here, both the undulators. In a relativistic electron beam the internal forces evolution of the multi-bucket electron phase space dynamics in leading to the interaction of the electrons with each other can the beam body as well as edges and the relative slippage of the be computed by means of retarded LW ?elds. The resulting radiation with respect to the electrons in the considered short electron beam dynamics enables us to obtain three dimensional bunch are naturally embedded into the simulation model. radiation ?elds starting from an initial incoherent spontaneous A description of the particle and ?eld dynamics underlying the emission, without introducing a seed wave at start-up. In this code is outlined in section 2 followed in section 3 the numeripaper, we present electromagnetic radiation studies, including cal results demonstrating the evolution of the radiation in the multi-bucket electron phase dynamics and angular distribution time domain and its angular distribution. Here we study the of radiation in the time and frequency domain produced by a evolution of the radiation for a monoenergetic beam with unirelativistic short electron beam bunch interacting with a circu- formly spaced electrons along the radiation wavelength as well larly polarized magnetic undulator. as a pulse with shot noise in the electron phases.
1. INTRODUCTION
2. PARTICLE AND FIELD DYNAMICS
The coherence characteristics of the radiation ?elds produced The electric and magnetic ?elds produced by a point charge q by a beam of relativistic electrons moving along a magnetic moving along a trajectory r ( t ) with relativistic energy γ mc 2 undulator depend on the degree to which electrons become can be derived from the well known LW potentials. For the organized under the in?uence of the ponderomotive force. Our electric ?eld strengths we have [1]: approach shows that making use of the complete electric and d ? × { (n ? – β) × β} magnetic ?elds produced by a point charge, the longitudinal q 1n ? –β n dt - ( 1) - ------------------------------------------- --------------------------------------------------E ( r, t ) = ------------ + -4 πε0 γ 2 ( 1 – β ? n beam dynamics of the particles is governed by the “near -” and ( 1 – β ? n )3 R ? )3 R2 c tr “far zone” ?elds. Electric ?eld components in the near zone are All quantities in (1) have to be evaluated at the retarded time tr composed of terms falling off as R-2 and R-3 whereas in the The retardation condition connects the t = t – ( R ( t ) ) ? c r r “far zone” they vary as R-1 corresponding to radiation ?elds. observation point time t to the source point time t where R repr The latter combined with the undulator ?elds gives rise to the resents the distance between the two points. The evolution of ponderomotive force. For suf?ciently high density electron the three dimensional resultant radiation ?eld can be deterbeams, depending on the pulse length and axial charge distri- mined when individual ?eld contributions of all charges in the bution within a radiation wavelength, “near zone” ?elds have beam are superimposed in any point at the observer surface. considerable effects on the longitudinal motion of the electrons Since LW ?elds are expressed in terms of particle’s retarded and the associated bunching process, altering the characteris- position and it’s time derivatives, relativistic Lorentz force tics of the produced radiation. equations for a coupled electron beam - radiation system have The purpose of this paper is to point out the way LW ?elds can to be solved in a self-consistent way to describe completly parbe exploited in obtaining spectral and temporal behavior of the ticle’s motion. The transverse motion is almost entirely deterradiated ?elds for the self ampli?ed spontaneous emission mined by the undulator magnetic ?eld, whereas the axial (SASE) process. In our formulation, rather than solving self- motion of each electron is de?ned by the combined undulator consistently the paraxial wave equation coupled to the relativ- and the resultant LW ?eld produced by other electrons in the istic single-particle equations of motion, we ?rst compute beam. The for the FEL mechanism crucial axial electron retarded LW ?elds with few assumptions to drive the electron’s dynamics can be obtained directly from the equation for the motion. Since the ?elds are evaluated in the time domain, the energy exchange between the electron and LW ?elds. The used approach allows their interaction with the electron beam equation for the energy change of the i electron :. th with no restrictions on the frequency spectrum. Knowing all dγi ? dt = ( q ? mc ) E ? β i ( 2 ) variables of the motion, such as retarded position, velocity, and acceleration of the charge, amplitude of the ?elds radiated by The electric ?eld is a resultant ?eld at the position of the ith individual electrons in the beam are determined and summed at electron obtained by summing up LW ?eld contributions from a observer surface far from the source. For simplicity, we con- the rest of the charges in the beam. The summation is implesider in our simulations a ?lamentary sub-picosecond relativis- mented by including Doppler upshifted and downshifted tic electron bunch which is substantially shorter than the parts of the radiation ?elds. The latter has, however, much smaller in?uence on the motion of the particles if the beam is slippage length.
highly relativistic. To compute the ?eld contribution of individual electrons to the ? , β , and dβ ? dt have summation, the retarded values of R, n to be determined from the present position of the particles in the undulator and be inserted into (1). The retarded distance R is given by R = ( x – x' ) 2 + ( y – y' ) 2 + ( z – z' ) 2 where x, y, z and x', y', z' represent the coordinates of the electron’s present and retarded positions. At this point we introduce new dimensionless longitudinal position variables θ = ku z , θ' = ku z' , retarded distance ρ = ku R , retarded axial distance χR = ku ( zi – z'j ) , and present axial distance χp = ku ( zi – zj ) where prime denotes evaluation at retarded time and i, j refer to the particle’s index in the beam. With the assumption of having a uniform longitudinal motion ( βz constant) during a numerical integration step for the energy exchange and employing the retardation condition, χ R can be obtained by solving the equation: au ?2 2 ( 1 – β 2 ) – 2 χ χ + χ 2 – 2 ? -----χR z p ? γβ - ? ( 1 – cos χR ) = 0 R p z The solution establishes the relation between the position of the source electrons at the time of emission and the position of the observer electron at the time of reception where the Lorentz force is exerted on that particular electron by the resultant ?eld. Arranged in inverse powers of ρ , the rate of the energy exchange takes the form:
dγi = r 0 ku dτ j≠i 1 ∑ ------------------------------? )3 ( 1 – βj ? n 1 ----- {( au 2 ? γ 4 βzj ) ( 1 + au 2 ) sin χR ρ3
describes the Coulomb interaction between the particles. It competes with the ponderomotive force reducing the depth of the ponderomotive potential well thus preventing particles from being trapped in the potential buckets. The rest of the “near ?eld” terms are combined with the undulator ?elds and contributes to the energy modulation of the beam provided that the distance between the electrons are much smaller than the radiation wavelength. These ?elds would vanish in the abscence of an undulator. The numerical integration of (3) 2 ) ? γ 2) 1 / 2 together with the relation βz = ( 1 – ( 1 + au allows us ?nally to obtain the longitudinal motion of the electrons. Having the knowledge of the position of each electron and it’s time derivatives we can elaborate on the radiation evolution at a spherical surface of observation making use of the radiative part of (1). Superposition of ?elds emitted by each electron yields the resultant ?eld:
q Er ( r, t ) = --------------4 πε0 c
∑
i
? ? r ( tr ) Introducing the far-?eld assumption R ( tr ) ? D – n where D is the distance from the observer to the entrance of the ? specifying the observation undulator and the unit vector n ? = { sin θ cos φ, sin θ sin φ, cos θ } , 1 – β ? n ? and direction n R(tr) may be approximated as: R ( tr ) = ( D – z ? cos θ – ( au ? γ k u β z ) θ sin ku z ) and ? = 1 – ( au ? γ ) θ cos ku z – βz cos θ 1–β?n respectively. We use here the fact that the radiation pattern emitted by a charge in a helical undulator is azimuthally symmetric around z-axis, and con?ned to angles of the order of θ ≈ ( 1 ? γ ) ? 1 . Noticing that k u l b ? 2 π for the considered short bunch , where l b is the bunch length, we can evaluate time structure and frequency composition of the radiated ?eld components Ex and Ey at any point over the observer surface. The time evolution of the angular distribution is deter2 mined by d W ? d? dt = ε0 c RE 2 The total radiated power can be obtained performing the integral over the solid angle: 2π π 2 dW E R2θdθdφ ( 4) ≈ ε0 c dt 00 where d ? ≈ θ d θ d φ . The time integration of (4) over the radiation pulse yields the total radiated energy. Evaluating Jackson’s formula for the energy spectrum, we can also determine energy radiated per frequency ω , per solid angle ? during the time the beam travels through the undulator: ? ? ri ( tr) ? 2 n 2 -? iω? ? t – ---------------------r0 mc ω 2 T d W c ? ×n ?× = -------------------n βie dt d?dω 0 4π2 i
d ? d ? ?n ? – β )i – ( 1 – β ? n )i ? ? d t β ?i ? ? ? dt β? i ( n ---------------------------------------------------------------------------------------------------3 R ( 1 – β ? n )i i tr
– ( au 4 ? γ 4 βzj ) sin χR cos χR } 1 - { ( β zi ? γ 2 ) ( nz – β zj ) ( 1 + au 2 ( 1 – cos χR ) ) + ----ρ2 + ( au ? γ ) 4 ( 1 + sin2 χR ) – ( au2 ? γ 4 ) ( 1 + au2 ) cos ( χR ) } 1 - { ( 1 – nz β zj ) ( au ? γ ) 2 β zj sin χR } ( 3 ) – -ρ
where we introduced the dimensionless time variable τ = cku t . ρ and the axial unit vector component nz are given by: χ R ρ = χR 1 + ( au ? γβz ) 2 sin2 ( χR ? 2 ) ? ( χ ? 2 )2 , nz = ----ρ R ? , can be respectively. The factor in the denominator, 1 – β ? n ? = 1 – nz β z – ( au 2 ? γ 2 βz ) sin χR ? ρ expressed by 1 – β ? n Accounting for all forces involved in the electron-wave interaction, except for self radiation reaction, (3) gives a complete description of the change of electron’s energy along the undulator. The term falling off as ρ – 1 (~ R – 1 ) contains the combined undulator-radiation ?elds causing modulation of the beam energy with subsequent bunching of the electron beam. The terms decreasing as ρ– 2 and ρ– 3 dominate at a distance close to the source. Thus we refer them as “near zone” ?elds. ? )3 The factor ( 1 ? ρ 2 ) ( β zi ? γ 2 ) ( n z – β zj ) ? ( 1 – β j ? n
∫∫
∫
∑
where ri and βi are position and velocity of the ith electron obtained from the describtion of the self-consistent motion of electrons. In order to give a correct account of the conservation of energy, energy loss of each electron due to its self ?eld is included
into (3). Using Lienard formula for the radiated power from a single electron we have:
q2 6? ? d ? 2 ? d 2? -γ β – ? β × β? P = --------------6 πε0 c ? ? d t ? dt ? ?
3. NUMERICAL RESULTS
The theoretical derivations presented in the previous section have been implemented in a simulation code to obtain angular and temporal characteristics of the three dimensional radiation ?elds emitted by a 35 MeV, 14 A, 70 ? m subpico- second ?lamentary electron bunch propagating trough a 4.5 m long, 1.5 cm period helical undulator. The undulator parameter au is chosen to be 1.12. To the parameter set corresponding radiation
wavelength is 3.5 ? m . The code computes the time evolution of the three dimensional ?eld amplitude with no initial bunching and seed wave at start-up. For a typical run ,a ?at-top pro?le electron beam is used. The simulation particles are uniformly distributed along the electron pulse. The bunch , twenty radiation wavelength long , is much shorter than the slippage distance. To study effects of the shot noise in the electron phase , simulations with randomly spaced electrons are carried out.
4. REFERENCES
[1]J. D. Jackson, Classical Electrodynamics,p.657 (J. Wiley & Sons , 1975)
Fig. 1 a-c. show phase versus exchanged energy of initially uniformly and randomly distributed monoenergetic electron beam after traveling trough a 4.5 m long helical undulator. Plot 1.c is obtained from the numerical integration of (19) by considering the 1/R dependent (ponderomotive force) term alone.In plot 1.a-b the in?uence of the “near-zone” ?elds on the longitudinal beam dynamics is included. In Plot 1.b the periodicity of the potential wells is perturbed due to the initial random distribution of the charge along the bunch length.
Figures 2. a-c illustrate the angular distribution of the radiated energy by the initially randomly phased electron bunch. The angular distribution observed at a surface far from the source is divided into three different time domains; leading - and trailing edge of the radiation pulse, ?gures 2a. ,2 c respectively, and the steady state regime ?g. 2b, where θ is the observation angle.Due to the (partial) bunching of the electron beam at the end of the undulator, the coherent superposition of radiation pulses from all electrons in the beam produce, as a result of wave interference effects a narrower cone of radiation than the radiation generated by the electrons at the entrance of the undulator.Dependent on the distribution of the electron phases, by summing the amplitude of the ?elds radiated by individual electrons one can obtain interferece patterns at the observer surface where constructive (destructive) interference takes place for certain observation angles.Fig.2d. shows the angular distribution of the radiated energy in the absence of the “near-zone” ?elds . Due to the increased coherence in the resultant ?eld, the angular radiation cone becomes narrower than the ones shown in ?gures 2.a-c.