Beam Envelope Equation Research Articles

Predicting the transverse emittance of space charge dominated beams using the phase advance scan technique and a fully connected neural network

The transverse emittance of a charged particle beam is an important figure of merit for many accelerator applications, such as ultra-fast electron diffraction, free electron lasers and the operation of new compact accelerator concepts in general. One of the easiest to implement methods to determine the transverse emittance is the phase advance scan method using a focusing element and a screen. This method has been shown to work well in the thermal regime. In the space charge dominated laminar flow regime, however, the scheme becomes difficult to apply, because of the lack of a closed description of the beam envelope including space charge effects. Furthermore, certain mathematical, as well as beamline design criteria must be met in order to ensure accurate results. In this work we show that it is possible to analyze phase advance scan data using a fully connected neural network (FCNN), even in setups, which do not meet these criteria. In a simulation study, we evaluate the perfomance of the FCNN by comparing it to a traditional fit routine, based on the beam envelope equation. Subsequently, we use a pre-trained FCNN to evaluate measured phase advance scan data, which ultimately yields much better agreement with numerical simulations. To tackle the confirmation bias problem, we employ additional mask-based emittance measurement techniques.

Theoretical analysis and experimental verification of electron beam transmission with low guiding magnetic field in V-band coaxial transit-time oscillator

The transmission of an electron beam with a low magnetic field is theoretically and experimentally investigated. In a high frequency band, due to a narrow tunnel of the electron beam, the transmission of the electron beam highly depends on beam parameters. The transmission of the electron beam is analyzed through the beam envelope equation, and accordingly, the relationship between beam parameters and transversal diffusion of the electron beam is derived. By particle-in-cell simulation, the influence of the current density, loading position, and guiding magnetic field on the electron beam is discussed. An electron beam with a beam voltage of 400 kV and a beam current of 5.0 kA in the guiding magnetic field of 1.0 T is applied in the experiment. The experiment consists of the electron beam bombardment on the nylon target and measurements of the beam current. The experimental results show that the electron beam can be stably and uniformly transmitted at a low loss under the given conditions.

Six-dimensional matching of intense beam with linear accelerating structure

Beam matching is a common technique that is routinely employed in accelerator design with the aim of minimizing beam losses and preservation of beam brightness. Despite being widely used, a full theoretical understanding of beam matching in 6D remains elusive. Here, we present an analytical treatment of 6D beam matching of a high-intensity beam onto an RF structure. We begin our analysis within the framework of a linear model, and apply the averaging method to a set of 3D beam envelope equations. Accordingly, we obtain a matched solution that is comprised of smoothed envelopes and periodic terms, describing envelope oscillations with the period of the focusing structure. We then consider the nonlinear regime, where the beam size is comparable with the separatrix size. Stating with a Hamiltonian analysis in 6D phase space, we attain a self-consistent beam profile and show that it is significantly different from the commonly used ellipsoidal shape. Subsequently, we analyze the special case of an equilibrium with equal space charge depression between all degrees of freedom. Comparison of beam dynamics for equipartitioned, equal space charge depression, and equal emittances beams is given. Finally, we present experimental results on beam matching in the LANSCE linac.

Evolution of bunch envelope and emittance in photo-injectors with application to the AWAKE experiment

We investigate on time evolution of envelopes and emittance of bunched beams in photo-injectors. We carry out a detailed beam dynamic study with focus on calculations of space charge forces. We take the average of the forces over the whole ensemble of particles. This results in space charge coefficients that are largely independent of the bunch distribution. By implementing these coefficients in the beam envelope equations we find a set of modified equations as a model for describing the evolution of the bunch envelope. Our equations crucially decrease the consumed time in particle-in-cell codes for designing today’s photo-injectors. The feasibility and accuracy of the model have been benchmarked against the simulation results from existing particle-in-cell codes with very good agreement. We apply our model for the design of the ultra-short bunch photo-injector of the AWAKE experiment. Moreover, by using assumption of linear external fields we develop the governing equations of the bunch emittances in photo-injectors. These equations allow us to partly quantitatively and partly qualitatively study the effects of external and space charge fields on emittance growth phenomena.

Study of orbit properties and intense beam dynamics in a high current separated sector proton cyclotron

This paper outlines a simple method based on analytical formulations to find out the physics design of a separated sector cyclotron. Formulations are developed to determine the magnet sector shape and equilibrium orbit properties for a chosen orbital frequency and a peak magnetic field in the magnet sector. Stability analysis of single particle motion without space charge effect has been performed and allowed values of sector angles are determined. The dynamics of space charge dominated beam in the separated sector proton cyclotron are explored by using beam envelope equations that includes radial and longitudinal coupling. Evolution of the rms beam envelopes and emittances has been studied as a function of the beam current for various sector geometry. The envelope equations and perturbed equations of motion around the matched beam envelopes have been utilized to obtain the matched beam conditions at different values of the beam current.

The 3-D numerical simulation research of vacuum injector for linear induction accelerator

Simulation method for voltage in-feed and electron injection of vacuum injector is given, and verification of the simulated voltage and current is carried out. The numerical simulation for the magnetic field of solenoid is implemented, and a comparative analysis is conducted between the simulation results and experimental results. A semi-implicit difference algorithm is adopted to suppress the numerical noise, and a parallel acceleration algorithm is used for increasing the computation speed. The RMS emittance calculation method of the beam envelope equations is analyzed. In addition, the simulated results of RMS emittance are compared with the experimental data. Finally, influences of the ferromagnetic rings on the radial and axial magnetic fields of solenoid as well as the emittance of beam are studied.

New beam-tracking simulation code using bulk-to-point calculation technique for space charge fields

A new two-dimensional beam-tracking simulation code for electron injectors using a bulk-to-point calculation technique for space charge fields was developed. The calculated space charge fields are produced not by a point charge but by a hollow cylinder that has a volume. Each tracked electron is a point charge. This bulk-to-point calculation technique for space charge fields is based on that used in the multiple beam envelope equations, which were developed by the author. The multiple beam envelope equations are a set of differential equations for investigating the beam dynamics of electron injectors and can be used to calculate bunched beam dynamics with high accuracy. However, there is one limitation. The bunched beam is assumed to be an ensemble of several segmentation pieces in both the transverse and longitudinal directions. In this bunch model, each longitudinal segmentation slice in a bunch must not warp; consequently, the accuracy of the calculated emittance is reduced in the case of a highly charged beam for calculations of a typical rf gun injector system. This limitation is related to the calculation model of longitudinal space charge fields. In the newly developed beam-tracking simulation code, the space charge field calculation scheme is upgraded and the limitation has been overcome. Therefore, the applicable range is extended while maintaining the high accuracy of emittance calculations. Simultaneously, the calculation time is markedly shortened because the emittance dependence on the segmentation number is extremely weak. In this paper, several examples of beam dynamics that cannot be calculated accurately using the multiple beam envelope equations are demonstrated using the new beam-tracking simulation code. The accuracy of the calculated emittance is also discussed.

Studies on space charge neutralization and emittance measurement of beam from microwave ion source.

A 2.45 GHz microwave ion source together with a beam transport system has been developed at VECC to study the problems related with the injection of high current beam into a compact cyclotron. This paper presents the results of beam profile measurement of high current proton beam at different degrees of space charge neutralisation with the introduction of neon gas in the beam line using a fine leak valve. The beam profiles have been measured at different pressures in the beam line by capturing the residual gas fluorescence using a CCD camera. It has been found that with space charge compensation at the present current level (∼5 mA at 75 keV), it is possible to reduce the beam spot size by ∼34%. We have measured the variation of beam profile as a function of the current in the solenoid magnet under the neutralised condition and used these data to estimate the rms emittance of the beam. Simulations performed using equivalent Kapchinsky-Vladimirsky beam envelope equations with space charge neutralization factor are also presented to interpret the experimental results.

Transverse dynamics of space charge dominated beam during bunching

Abstract In this paper we present a technique to optimize the transport parameters for transverse matching of the space charge dominated beams in the presence of bunching. The increase in the current within the specified bunch width during the transport is obtained from the disc model and is included in the optimization method. A 2D particle in cell (PIC) code is then developed to study the details transverse dynamics of the bunch taking into account the effect of longitudinal compression by reweighting the charge and mass of macroparticles. The evolution of rms beam size obtained using PIC simulation agrees well with the results obtained by solving beam envelope equation. The PIC simulation shows emittance growth during bunching induced by the space charge effect for nonuniform distribution. The growth in the emittance increases with the nonlinearity of distribution and peaks near the time focus.

Kinetic equilibrium of space charge dominated beams in a misaligned quadrupole focusing channel

The dynamics of intense beam propagation through the misaligned quadrupole focusing channel has been studied in a self-consistent manner using nonlinear Vlasov-Maxwell equations. The equations of motion of the beam centroid have been developed and found to be independent of any specific beam distribution. A Vlasov equilibrium distribution and beam envelope equations have been obtained, which provide us a theoretical tool to investigate the dynamics of intense beam propagating in a misaligned quadrupole focusing channel. It is shown that the displaced quadrupoles only cause the centroid of the beam to wander off axis. The beam envelope around the centroid obeys the familiar Kapchinskij-Vladimirskij envelope equation that is independent of the centroid motion. However, the rotation of the quadrupole about its optical axis affects the beam envelope and causes an increase in the projected emittances in the two transverse planes due to the inter-plane coupling.

Optimization of beam line parameters for space charge dominated multi-species beam using random search method

In this Letter we present a method based on random search optimization technique to transport and match a space charge dominated multi-species beam using the recently developed beam envelope equations. This technique optimizes the transport of primary species and maximizes the loss of other unwanted species. To illustrate the utility of the procedure, we have studied and optimized the transport of proton beam in the presence of H2+ and H3+ in a solenoid based transport line. Numerical results of the beam selection and optimized transport parameters have been presented for various values of the total beam current and different fractions of p, H2+, and H3+ species.

Multiple beam envelope equations for electron injectors using a bunch segmentation model

A new semianalytical method of investigating the beam dynamics for electron injectors was developed. In this method, a short bunched electron beam is assumed to be an ensemble of several segmentation pieces in both the longitudinal and the transverse directions. The trajectory of each electron in the segmentation pieces is solved by the beam envelope equations while taking into account the space charge fields produced by all the pieces, the electromagnetic fields of an rf cavity, and the image charge fields at a cathode surface. The shape of the entire bunch is consequently calculated, and thus the emittances can be obtained from weighted mean values of the solutions for the obtained electron trajectories. The advantage of this method is its unique assumption for the beam parameters. We assume that each segmentation slice is not warped in the calculations. Although if the beam energy is low and the charge density is large, this condition is not satisfied, in practice, this condition is usually satisfied. We have performed beam dynamics calculations to obtain traces in free space and in the BNL-type rf gun cavity by comparing the analytical solutions with those obtained by simulation. In most cases, the emittances obtained by the simulation become closer to those obtained analytically with increasing the number of particles used in the simulation. Therefore, the analytically obtained emittances are expected to coincide with converged values obtained by the simulation. The applicable range of the analytical method for the BNL-type rf gun cavity is under 0.5 nC per bunch. This range is often used in recently built x-ray free electron laser facilities.

Investigation on beam envelope oscillations and amplitude growth in a high-current compact cyclotron

In this paper we have studied the behavior of space charge dominated transverse beam envelopes in a compact cyclotron along the accelerated orbits using the coupled beam envelope equations and assuming the beam as a uniform ellipsoidal bunch. A detailed investigation on the amplitude growth and oscillations in the beam envelopes has been carried out by changing the input beam sizes and phase ellipse orientations at the injection in both transverse planes. We have developed a method to obtain the parameters of a matched beam with space charge including the effect of acceleration in a cyclotron in linear approximation for a given beam current and emittance. We have also estimated the maximum beam current that can be transported through a given focusing channel of a compact cyclotron.

Effect of nonuniform radial density distribution on the space charge dominated beam bunching

Beam dynamics of a space charge dominated beam during the bunch compression is studied self consistently for the case of fixed shape non-uniform bell shape and hollow shape density distributions in the transverse direction. We have used thick slices at different parts of the beam to account for variation in the beam radius in the study of the transverse dynamics. The longitudinal dynamics has been studied using the disc model. The axial variation of the radius of the slices and emittance growth arising from the phase dependence of the transverse rf forces are also included in the simulation. We have modified the beam envelope equation to take into account the longitudinal space charge effect on the transverse motion which arises due to the finite bunch size. To demonstrate the application of the theoretical formulations developed, we have studied a sinusoidal beam bunching system and presented detailed numerical results.

Behavior of space charge dominated beam during longitudinal compression in a bunching system

Longitudinal and transverse beam dynamics during the bunch compression in a beam bunching system have been studied using disc model and envelope equation. The major contribution here is the inclusion of the effect of axial variation of the longitudinal electric field in the space charge term of beam envelope equation. Also included is a random Gaussian energy spread in the beam before the compression begins. In addition, the emittance growth resulting from the time dependant rf field in the transverse direction has been taken into account and the effective value of beam emittance has been used in the beam envelope equation. Finally, we have studied the beam bunching system of 100 keV proton to be used in the injection line of the 10 MeV high current compact cyclotron. Various numerical results of the study are presented for different geometrical and beam parameters.

Universal envelope equation and emittance evolution of high-brightness beam in linac

We report a universal beam envelope equation that governs the transverse linear dynamics of high-intensity and high-brightness relativistic beams under constant acceleration in axisymmetric linear accelerators. This dimensionless and almost parameter-free nonlinear equation is useful for understanding scaling properties and for investigating nonlinear behaviors that are beyond analytical analysis. Particularly, we explore emittance compensation in high-brightness beams evolving from the space-charge regime to the thermal-emittance regime, a transition that commonly occurs during acceleration but is not well studied. A new formula is given for correctly computing the rms bunch emittance from slice envelopes, which is different from the commonly used quadratic sum of the thermal emittance and the rms emittance in the envelope phase space.

Adiabatic warm-fluid equilibrium theory of thermal charged-particle beams in alternating-gradient focusing fields

An adiabatic warm-fluid equilibrium theory for a thermal charged-particle beam in an alternating-gradient focusing field is presented. Warm-fluid equilibrium equations are solved in the paraxial approximation. The theory predicts that the four-dimensional rms thermal emittance of the beam is conserved, but the two-dimensional rms thermal emittances are not constant. The rms beam envelope equations and the self-consistent Poisson equation, governing the beam density and potential distributions, are derived. Although the presented rms beam envelope equations have the same form as the previously known rms beam envelope equations, the evolution of the rms emittances in the present theory is given by analytical expressions. The density does not have the simplest elliptical symmetry, but the constant-density contours are ellipses, and the aspect ratio of the elliptical constant-density contours decreases as the density decreases along the transverse displacement from the beam axis. For high-intensity beams, the beam density profile is flat in the center of the beam and falls off rapidly within a few Debye lengths, and the rate at which the density falls is approximately isotropic in the transverse directions.

Control of Beam Halo-Chaos for an Intense Charged-Particle Beam Propagating Through Double Periodic Focusing Field by Soliton

We study an intense beam propagating through the double periodic focusing channel by the particle-core model, and obtain the beam envelope equation. According to the Poincare–Lyapunov theorem, we analyze the stability of beam envelope equation and find the beam halo. The soliton control method for controlling the beam halo-chaos is put forward based on mechanism of halo formation and strategy of controlling beam halo-chaos, and we also prove the validity of the control method, and furthermore, the feasible experimental project is given. We perform multiparticle simulation to control the halo by using the soliton controller. It is shown that our control method is effective. We also find the radial ion density changes when the ion beam is in the channel, not only the halo-chaos and its regeneration can be eliminated by using the nonlinear control method, but also the density uniformity can be found at beam's centre as long as an appropriate control method is chosen.

Comment on the invariant-envelope solution in rf photoinjectors

The beam envelope equation has been used to address the beam dynamics in rf photoinjectors. A special solution of the envelope equation, known as the invariant envelope, plays a critical role in the theory of emittance compensation. In this comment, I will present a different view of the invariant envelope solution that better delineates its properties and simplifies the picture of beam dynamics.

Coherent error study in a retarding field energy analyzer

A novel cylindrical retarding electrostatic field energy analyzer for low-energy beams has been designed, simulated, and tested with electron beams of several keV, in which space charge effects play an important role. A cylindrical focusing electrode is used to overcome the beam expansion inside the device due to space-charge forces, beam emittance, etc. In this paper, we present the coherent error analysis for this energy analyzer with beam envelope equation including space charge and emittance effects. The study shows that this energy analyzer can achieve very high resolution (with relative error of around 10 −5) if taking away the coherent errors by using proper focusing voltages. The theoretical analysis is compared with experimental results.