Electron Motion in the Three-Dimensional Field of Undulator

In this paper we consider the motion of relativistic electrons in an ideal three-dimensional magnetic field of an undulator. The ideality of the magnetic field means that, on the undulator axis, the field is directed strictly vertically upward and has a strictly sinusoidal shape. In the overwhelming majority of cases, only this leading component of the field is taken into account in calculating the electron trajectory. In this paper, in the equations of motion of an electron in the magnetic field of an undulator, all three components of the field are taken into account, so that the undulator field under consideration satisfies the stationary Maxwell equations. In this case, the differential equations of motion of the electron are solved analytically with the help of perturbation theory, and not by the method of averaging over fast oscillations of the electron, as was done in a number of previous papers. These analytic expressions for trajectories describe the behavior of particles in the focusing magnetic field of an undulator much more completely. An analysis of these expressions shows that the behavior of electrons in such a three-dimensional field of the undulator is much more complicated than what follows from the equations obtained by the averaging method. In particular, there is a cross effect when changes in the initial vertical parameters of the electron trajectory cause changes in the horizontal component of its trajectory and vice versa. A comparison of the solutions obtained analytically with the results of numerical calculations of electron trajectories using the Runge–Kutta method demonstrates their high accuracy.     

Theoretical and numerical analysis of electron motion in a three-dimensional undulator field

The results of numerical calculations of electron trajectories in the three-dimensional ideal magnetic field of an undulator are given. These trajectories can be regarded as exact results in our case. It is shown that, in a series of cases, results of the calculation of trajectories using approximate analytical formulas significantly differ from those obtained using numerical methods. The obtained numerical results show that the influence of the undulator on the dynamics of the electron beam is very complicated and cannot be reduced only to the focusing properties of the undulator magnetic field or wiggler.     

Basic features of a charged particle dynamics in a laser beam with static axial magnetic field

In this paper, the trajectory and kinetic energy of a charged particle, subjected to interaction from a laser beam containing an additionally applied external static axial magnetic field, have been analyzed. We give the rigorous analytical solutions of the dynamic equations. The obtained analytical solutions have been verified by performing calculations using the derived solutions and the well known Runge-Kutta procedure for solving original dynamic equations. Both methods gave the same results. The simulation results have been obtained and presented in graphical form using the derived solutions. Apart from the laser beam, we show the results for a maser beam. The obtained analytical solutions enabled us to perform a quantitative illustration, in a graphical form of the impact of many parameters on the shape, dimensions and the motion direction along a trajectory. The kinetic energy of electrons has also been studied and the energy oscillations in time with a period equal to the one of a particle rotation have been found. We show the appearance of, so-called, stationary trajectories (hypocycloid or epicycloid) which are the projections of the real trajectory onto the (x, y) plane. Increase in laser or maser beam intensity results in the increase in particle’s trajectory dimension which was found to be proportional to the amplitude of the electric field of the electromagnetic wave. However, external magnetic field increases the results in shrinking of the trajectories. Performed studies show that not only amplitude of the electric field but also the static axial magnetic field plays a crucial role in the acceleration process of a charged particle. At the authors of this paper best knowledge, the precise analytical solutions and theoretical analysis of the trajectories and energy gains by the charged particles accelerated in the laser beam and magnetic field are lacking in up to date publications. The authors have an intention to clarify partly some important aspects connected with this process. The presented theoretical studies apply for arbitrary charged particle and the attached figures-for electrons only.     

Optimization of the lattice function in the planar undulator applied for terahertz FEL oscillators

Since the beta function of the electron beam within the undulator has a great influence on the power gain of the free electron laser (FEL), optimization of the undulator lattice becomes important. In this paper, the transfer matrix of the planar undulator is obtained from differential equations of the electron motion. Based on this, the lattice function of the planar undulator in a terahertz FEL oscillator proposed by Huazhong University of Science and Technology (HUST-FEL) is optimized and the expressions of the average beta function are derived. The accuracy of the optimization result was confirmed well by the numerical method. The application range of this analytical method is given as well. At last, the emittance growth in the horizontal direction due to the attenuation of the magnetic field is discussed.     

Momentum spread in a relativistic electron beam in an undulator

The motion of the relativistic electron beam in the spatially periodic magnetic field of an undulator has been considered taking into account the effect of the incoherent field of the spontaneous undulator radiation on the motion of the electrons. An expression for the rms momentum of the electrons has been obtained. It has been shown that the momentum spread in the ultrarelativistic electron beam increases in the spontaneous incoherent emission mode. Conditions for the self-amplification of the spontaneous undulator radiation in ultrashort-wavelength free-electron lasers have been discussed.     

静电六极装置电场分布的数值计算方法及在N2O分子转动态选择、取向中的应用

阐述求解极性分子转动态选择及取向静电六极装置中势能分布、电场分布的数值计算方法.为了获得电场分布公式,需通过数值迭代求解势能满足的Laplace方程,获取数值分布点,通过数值分布点,由待定系数的多级展开势能解析表达式进行最小二乘拟合获得势能分布公式,由势能对空间向量的微分获得电场分布.分子在六极电场中的运行轨迹采用经典Newton方程描述,并通过四阶龙格-库塔方法(Four Order Runge-Kutta Method)实现数值求解,其中能量处理采用量子力学方法.应用此方法给出静电六极装置的电场分布公式,运用获得的电场分布公式计算和讨论电场对极性分子N₂O的静电六极转动态选择、取向所带来的影响.     

大回旋半径电子枪渐变线圈磁场的设计及优化

对大回旋半径电子枪的渐变线圈磁场进行了设计,采用3个线圈实现所需要的渐变磁场分布,增加了线圈磁场系统的调节能力,理论和计算机仿真的磁场分布结果符合得很好。将实现的渐变磁场分布同给定的静电场分布相结合,通过求解带电粒子的运动方程得到了粒子轨迹,在此基础上建立大回旋半径电子枪的3维粒子仿真模型,在给定静电场分布条件下分析了3个线圈安匝数对电子束参数的影响,完成了工作电压为40 kV、工作电流为1 A的大回旋半径电子枪的参数优化,得到了横纵速度比为1.4～2.5,纵向速度离散小于8%(横纵速度比为1.9时)的大回旋半径电子束。     

Spherical aberration from trajectories in real and hard-edge solenoid fields

For analytical, real and hard-edge solenoidal axial magnetic fields, the low-energy electron trajectories are obtained using the third-order paraxial ray equation. Using the particle trajectories, it is shown that the spherical aberration in the hard-edge model is high and it increases monotonously with hard edginess, although the focal length converges, in agreement with a recent field and spherical aberration model. The model paved the way for a hard-edge approximation that gives correct focal length and spherical aberration, which is verified here by the trajectory method. In essence, we show that exact hard-edge fields give infinite spherical aberrations.     

Iterative solutions of the Lotka-Volterra equations

Summary Lotka-Volterra equations for two-species predator-prey system are solved by an iteration method. The first iteration step reproduces elliptical trajectories known from the linearization treatment. The second iteration step leads to analytical expressions describing large-amplitude motions and lying beyong the elliptical approximation. The range of validity of the method is discussed. It is shown that deviations from the exact trajectories increase if the initial state is far from the centre of motion. For illustration some trajectories are calculated showing a good agreement with trajectories obtained by numerical methods.     

Acceleration and focusing of intense ion beams in rf undulator structures

Bunching, acceleration, and transverse focusing of intense ion beams in an undulator linac are considered. Such an accelerator features the absence of an rf field harmonic synchronous with the beam. A 3D equation of motion in the Hamiltonian form is derived in the smooth approximation, and the general conditions for ion beam acceleration and transverse focusing in the undulator linac are formulated. Basic analytical results are compared with the results of numerical simulation of the beam dynamics in the polyharmonic field of an accelerating cavity.     

The parallel permittivity of magnetized toroidal plasmas with elliptic magnetic surfaces

The asymptotic solution of the Vlasov equation under the drift approximation, for an axially symmetric toroidal plasma configuration with an elliptic cross section of magnetic surfaces, is presented. The analytical expressions for the parallel component of the dielectric permittivity tensor are obtained. These expressions are used for theoretical analyses of the trapped and untrapped electron influence on the collisionless wave dissipation. The evaluated dielectric tensor components can be used for computer calculations of the radio frequency field structure and the collisionless dissipated power related to trapped and untrapped electrons in tokamak plasmas.This work was supported by the State University of Rio de Janeiro (UERJ), Rio de Janeiro Research Foundation (FAPERJ) and Brazilian National Council of Research (CNPq).     

Effect of energy spread on micro-bunching from shot noise in SASE FELs

The motion of Ne electrons moving along a helical undulator and interacting with each other through Lienard-Wiechert fields is considered. The numerical solution for initially mono-energetic electrons shows the emergence and growth of micro-bunching. For the case of initially non-mono-energetic electrons, calculations show that the presence of energy spread results in partial suppression of the micro-bunching. The calculations also show that the initial energy spread would shift the micro-bunching maximum to deeper positions within the undulator. Analytical treatment of equations of motion demonstrates that the micro-bunching fall and shift is mainly due to the random motion of the interacting electrons.     

On Undulator Radiation in Infrared and Millimeter Wave Ranges: Part 1—Classical Description

In this paper the results of theoretical and experimental research of the motion and radiation of low-voltage electron beams in Motz undulator are given. Constant magnetic field of undulator ordinary magnet under some conditions can be substituted for field of magnet dipole. Such substitution allows us to accurately integrate the electron trajectory. We can with the high accuracy consider this field as chaotic and this fact allow us to use well-known Ginsburg theory for estimation of power of braking radiation. There are longitudinal and transversal components of magnetic fields of described structure. The longitudinal component can be gained by covering of external longitudinal constant magnetic field. In this connection certain resonance effects are observed.     

太赫兹自由电子激光波荡器的设计、测量与优化

波荡器电子轨迹中心偏移和磁场误差对CTFEL装置性能影响很大,通过前期设计和后期测量与优化将其限制在指标要求范围内。在前期设计中尽量避免引入全局性的系统误差:磁结构具有平面反对称结构,保证电子轨迹中心和波荡器磁轴重合;磁结构端部的特殊设计减弱了间隙对出口磁场二次积分的影响;机械系统的大梁和立柱具有良好的刚性,闭环控制系统保证了高的波荡器间隙控制精度,这些措施降低了间隙不一致引入的磁场误差。在后期测量与优化中削弱了磁场的残存全局系统误差和局部随机误差:利用磁场点测台测量了波荡器磁场的纵向和横向分布,通过调节标准单元组件位置对磁场进行了垫补和优化,优化后电子轨迹中心偏移、峰峰值误差、相位误差、好场区及其误差均满足指标要求。     

Design and optimization of an electromagnet undulator

In this paper we discuss the design and optimization of an electromagnet undulator. A short period electromagnet undulator has been reported and used for free electron laser experiment under the approximation of infinite permeability of the lamination core. Here we remove this approximation and present analytical expressions for an alternate design concept on the electromagnet undulator.     

Nonlocal electron kinetics in collisional gas discharge plasmas

Nonlocal phenomena in electron kinetics of collisional gas discharge plasmas, their kinetic treatment by a nonlocal approach, and relevant experimental results are reviewed in this paper. Using the traditional two-term approximation for the electron distribution function, a general method to analyze electron kinetics in nonuniform plasmas in DC and RF fields for atomic gases is presented for the nonlocal case, when the electron energy relaxation length exceeds the characteristic spatial scale of bounded plasmas. The nonlocal method, which is based on the great difference between the electron mean free path for the momentum transfer and the electron energy relaxation length, considerably simplifies the solution of the kinetic equation and, in a number of cases, allows one to obtain analytical and semi-analytical solutions. The main simplification is achieved for trapped electrons by averaging the Boltzmann equation over space and fast electron motion. Numerous examples of spatial nonlocality are considered in the positive column and near the electrodes of DC glow discharges, in spatial relaxation of the electron distribution and in striations, and in capacitively and inductively coupled low-pressure RF discharges. The modeling of fast beam-like electrons is based on a continuous-energy-loss approximation with the assumption of forward scattering. Simple analytic expressions for the fast electron spectrum are obtained in cathode regions of DC discharges with planar and hollow cathodes     

摆动器场误差对自由电子激光器自发辐射谱的影响

在考虑平面型摆动器（Ｗｉｇｇｌｅｒ）场误差△Ｂ（ｚ）的情况下，求解电子在磁场中运动的洛仑兹（Ｌｏｔｅｎｔｚ）方程，得到场误差对电子横向速度的改变，然后作傅里叶变换即为电子自发辐射谱的改变，并讨论了各种场误差对自由电子激光器自发辐射谱的影响，选择北京自由电子激光器（ＢＦＥＬ）参数，进行模拟，最后确定出各种场误差的可接受条件。     

托卡马克电场电子注入的漂移电子束研究

 研究了漂移电子束中电子的运动规律, 得到了电子在外加电场、纵磁场和电子束自场中的运动轨迹和漂移速度表达式, 并就它们的成束条件、成束形状和电压要求以及束的稳定性条件进行了比较和讨论, 认为阴极垂直于纵磁场放置更好一些。     

Influence of a constant magnetic field on the radiation of a planar undulator

The influence of a constant magnetic field on the radiation of a planar undulator is studied analytically with the use of generalized Bessel functions. The behavior of on-axis radiation intensity in the undulator and the influence of a magnetic field on off-axis radiation are investigated. The critical magnitude of the constant component of the magnetic field, which has a significant effect on the motion of electrons in the undulator, is calculated analytically. The examples of several planar undulators are discussed, and the influence of the Earth’s magnetic field on their spectrum is evaluated.