Betatron Frequency Research Articles

Development of Digital Signal Processing Techniques with BPM

In order to ensure the continuity of operation of the VEPP-2000 collider, accurate measurement of the betatron frequency is necessary. To do this, this work proposes to use methods that refine the Fourier transform, such as parabola interpolation (Gassior method), NAFF and window functions. The refined frequency is subsequently used in construction of phase portraits of the beam to control the interference of high-order magnetic fields. In addition, the work discusses methods for extracting a signal from a mixture for subsequent analysis – PCA and ICA. Finally, to improve the accuracy of frequency determination, the paper considers the simplest implementation of a Kalman filter to improve the accuracy of subsequent harmonic analysis. In addition to all of the above, the paper briefly discusses a method for monitoring the operation of the beam position monitors themselves.

Alignment & stability challenges for FCC-ee

In order to achieve its ultra-low vertical emittance (1 pm) and high luminosity (of up to 230 times 10^{34}text{ cm}^{-2}text{ s}^{-1} per collision point), the e+e− Future Circular Collider (FCC-ee) requires a well-informed alignment strategy, powerful correction methods, and good understanding of the impact of vibrations. The large ring size, high natural chromaticity, small beta ^{*}, and the low coupling ratio make the FCC-ee design susceptible to misalignment and field errors, which if not properly addressed, threaten to increase the horizontal and vertical emittances and adversely affect the luminosity. Tight alignment tolerances around the 100 km ring would be a major cost driver and therefore alignment and stability need to be carefully studied. In this paper we present a status update, in which we apply analytical estimate methods, verified with simulation data, to determine the influence of the alignment of specific magnet types with the result informing the relative alignment tolerances. This is followed by simulations of a correction strategy that includes a large set of magnet misalignments and field errors. Finally, we also consider the tolerances on vibrations of quadrupoles through evaluating three cases: coherent vibration due to external seismic motion, vibrations resonant with the betatron frequency, and non-resonant, incoherent vibration.

Modeling of three-dimensional betatron oscillation and radiation reaction in plasma accelerators

Betatron oscillation is a commonly known phenomenon in laser or beam driven plasma wakefield accelerators. In the conventional model, the plasma wake provides a linear focusing force to a relativistic electron, and the electron oscillates in one transverse direction with the betatron frequency proportional to $1/\sqrt{\gamma}$, where $\gamma$ is the Lorentz factor of the electron. In this work, we extend this model to three-dimensional by considering the oscillation in two transverse and one longitudinal directions. The long-term equations, with motion in the betatron time scale averaged out, are obtained and compared with the original equations by numerical methods. In addition to the longitudinal and transverse damping due to radiation reaction which has been found before, we show phenomena including the longitudinal phase drift, betatron phase shift and betatron polarization change based on our long-term equations. This work can be highly valuable for future plasma based high-energy accelerators and colliders.

Multi-layer structure formation of relativistic electron beams in plasmas

A two-dimensional electromagnetic particle-in-cell simulation model is proposed to study the density evolution and collective stopping of electron beams in background plasmas. We show here the formation of the multi-layer structure of the relativistic electron beam in the plasma due to the different betatron frequency from the beam front to the beam tail. Meanwhile, the nonuniformity of the longitudinal wakefield is the essential reason for the multi-layer structure formation in beam phase space. The influences of beam parameters (beam radius and transverse density profile) on the formation of the multi-layer structure and collective stopping in background plasmas are also considered.

On the possibility of footprint compression with one lens in nonlinear accelerator lattice

Electromagnetic interaction of colliding beams along with other nonlinear fields often limits the beams' lifetimes and luminosities. Nonlinearities result in the spread of betatron frequencies (footprint) and, thus, may enhance dynamic diffusion of particles due to high order resonances. One of the possible ways to eliminate nonlinearities and overcome the corresponding difficulties is compensation of nonlinear forces, but, in practice, it is hardly possible to obtain exact linearity of the system. The compensation with a single nonlinear lens cannot cope with distributed nonlinearities, nonlinearities due to parasitic crossings, etc. In the article, we present a method to compute parameters of nonlinear element (lens) that eliminates both the footprint and resonance strength without achieving full compensation.

Landau Damping with Electron Lenses in Space-Charge Dominated Beams

Further progress of fundamental physics requires accelerated beams of high intensity. The intensities, however, are limited by many factors, including coherent beam instabilities. Usual methods to control the instabilities, such as octupole magnets, beam feedback dampers and employment of chromatic effects, may be ineffective or insufficient. Electron lenses were proposed as a means to provide stabilizing spread in the beam betatron frequencies. It was shown that electron lenses are uniquely effective for Landau damping of transverse beam instabilities in high energy particle accelerators, and that their employment does not compromise incoherent (single particle) stability, dynamic aperture and the beam lifetime. Here we consider effectiveness of the Landau damping with electron lenses when the space charge tune shift cannot be neglected. We demonstrate that the desired stability can be assured with proper choice of the electron beam parameters and current distributions.

Betatron frequency and the Poincaré rotation number

Symplectic maps are routinely used to describe single-particle dynamics in circular accelerators. In the case of a linear accelerator map, the rotation number (the betatron frequency) can be easily calculated from the map itself. In the case of a nonlinear map, the rotation number is normally obtained numerically, by iterating the map for given initial conditions, or through a normal form analysis, a type of a perturbation theory for maps. Integrable maps, a subclass of symplectic maps, allow for an analytic evaluation of their rotation numbers. In this paper we propose an analytic expression to determine the rotation number for integrable symplectic maps of the plane and present several examples, relevant to accelerators. These new results can be used to analyze the topology of the accelerator Hamiltonians as well as to serve as the starting point for a perturbation theory for maps.

Luminosity of an Ion Collider

A formula is obtained for the luminosity of a collider at the collision of two beams that differ, generally speaking, by their parameters (asymmetric colliders). The formula is valid for counterpropagating and merging beams with coincident longitudinal axes. Three special cases of the formula are considered: collision of two identical axially symmetric bunched beams, collision of a bunch with a coasting beam, and collision of two coasting beams. Collision of intersecting beams is briefly considered, and the method for the luminosity calculation is formulated. The synchronization problem is considered for collisions of asymmetric beams. A method is presented for optimizing parameters of a cyclic collider by minimizing betatron frequency shifts caused by the effect of the space charge of the beams. Numerical examples of luminosity calculations for several types of asymmetric colliders are given.

Dynamic aperture limitation in e+e− colliders due to synchrotron radiation in quadrupoles

In a lepton storage ring of very high energy (e.g. in the $e^+e^-$ Higgs factory) synchrotron radiation from quadrupoles constrains transverse dynamic aperture even in the absence of any magnetic nonlinearities. This was observed in tracking for LEP and the Future Circular $e^+e^-$ Collider (FCC-ee). Here we describe a new mechanism of instability created by modulation of the particle energy at the double betatron frequency by synchrotron radiation in the quadrupoles. Energy modulation varies transverse focusing strength at the same frequency and creates a parametric resonance of the betatron oscillations with unusual properties. It occurs at arbitrary betatron frequency (the resonant detuning is always zero) and the magnitude of the parameter modulation of the betatron oscillation (strength of the resonance driving term) depends on the oscillation amplitude. Equilibrium between the radiation damping and the resonant excitation gives the boundary of the stable motion. Starting from 6d equations of motion we derive and solve the relevant differential equation describing the resonance, and show good agreement between analytical results and numerical simulation.

Stable Particle Acceleration in Coaxial Plasma Channels

The attainable transformer ratio in plasma accelerators is limited by instabilities. Using three-dimensional particle-in-cell simulations, we demonstrate that these can be controlled using a hollow plasma channel with a coaxial plasma filament. The driver scatters electrons from the filament, and the slow pinch of the ions leads to a strong chirp of the effective betatron frequency, preventing beam breakup. We demonstrate the monoenergetic acceleration of an electron bunch to 20GeV over 4.4m, achieving a transformer ratio of 10, an energy efficiency of 40%, and an emittance of 1.8 μm.

Optical Structure and Dynamic Aperture of the NICA Сollider

The optimized optical structure of the NICA collider is considered. Studies and calculation methods of the stability region of the motion of charged particles in the collider—the dynamic aperture—are presented. The influence of the magnetic-field nonlinearities, in particular, the fringe fields of the magnetic elements on the dynamic aperture, is considered. The dependences of the dynamic aperture on optical properties of the collider, the choice of the working point of betatron frequencies, and tuning parameters of the beam-interaction region are shown. The methods for compensating the effect of the fringe fields on the stability of beam dynamics are discussed.

Nonlinear parametric resonance of relativistic electrons with a linearly polarized laser pulse in a plasma channel

The direct laser-acceleration mechanism, nonlinear parametric resonance, of relativistic electrons in a linearly polarized laser-produced plasma channel is examined by a self-consistent model including the relativistic laser dispersion in plasmas. Nonlinear parametric resonance can be excited, and the oscillation amplitude of electrons grows exponentially when the betatron frequency of electron motion varies roughly twice the natural frequency of the oscillator. It is shown analytically that the region of parametric resonance is defined by the self-similar parameter nenca0. The width of this region decreases with nenca0, but the energy gain and oscillation amplitude increases. In this regime, the electron transverse momentum grows faster than that in the linear classical resonance regime.

Single-particle dynamics in a nonlinear accelerator lattice: attaining a large tune spread with octupoles in IOTA

Fermilab is constructing the Integrable Optics Test Accelerator (IOTA) as the centerpiece of the Accelerator R&D Program towards high-intensity circular machines. One of the factors limiting the beam intensity in present circular accelerators is collective instabilities, which can be suppressed by a spread of betatron frequencies (tunes) through the Landau damping mechanism or by an external damper, if the instability is slow enough. The spread is usually created by octupole magnets, which introduce the tune dependence on the amplitude and, in some cases, by a chromatic spread (tune dependence on particle's momentum). The introduction of octupoles usually has both beneficial (improved Landau damping) and harmful properties, such as a resonant behavior and a reduction of the dynamic aperture. One of the research goals at the IOTA ring is to achieve a large betatron tune spread, while retaining a large dynamic aperture, using conventional octupole magnets in a special but realistic accelerator configuration. The configuration, although not integrable by design, approximates an autonomous 2D Hamiltonian system. In this paper, we present results of computer simulations of an electron beam in the IOTA by particle tracking and the Frequency Map Analysis. The results show that the ring's octupole magnets can be configured to provide a betatron tune shift of 0.08 (for particles at large amplitudes) with the dynamical aperture of over 20 beam sigma for a 150-MeV electron beam. The influence of the synchrotron motion, lattice errors, and magnet imperfections is insignificant for the parameters and levels of tolerances set by the design of the ring. The described octupole insert could be beneficial for enhancing Landau damping in high intensity machines.

Electromagnetic dispersion characteristics of a high energy electron beam guided with an ion channel

Taking self-fields into consideration, dispersion properties of two types of electromagnetic modes for a high energy electron beam guided with an ion channel are investigated by using the linear perturbation theory. The dependences of the dispersion frequencies of electromagnetic waves on the electron beam radius, betatron frequency and boundary current are revealed. It is found that the electron beam radius and betatron frequency have different influences on the electromagnetic waves dispersion behavior by compared with the previous works. As the boundary current is taken into account, the TM modes will have two branches and a low-frequency branch emerged as the new branch in strong ion channel case. This new branch has similar dispersion behavior to the betatron modes. For TE modes, there are two branches and they have different dispersion behaviors in strong ion channel case. However, in weak ion channel case, the dispersion behaviors for both of the low frequency and high frequency branches are similar.

REDESIGN OF NESTOR STORAGE RING LATTICE

NESTOR is the hard X-ray source that is under commissioning at NSC KIPT. NESTOR based on the Compton scattering of laser photons on relativistic electrons. The structure of the facility can be represented as the following components: a linear accelerator, a transport channel, a storage ring, and a laser-optical system. Electrons are stored in the storage ring for energy of 40-225 MeV, laser photons are accumulated in the optical resonator. The frequencies of laser and electron bunches are synchronized, the interaction point located in the optical resonator. The measurements showed that parameters of the manufactured bending magnets different from the design ones. The bending magnets have different field index, besides the vertical focusing forces are significantly lower than designed ones. The beam dynamics simulation showed that due to decreasing of the vertical focusing the vertical betatron frequency becomes close to the frequency of the integer resonance Q = 2, as a result the vertical motion in the ring is unstable [1]. The second reason for optimization is the need to increase the length of the drift space for optical resonator. The results of optimization are presented in this paper.

Enhanced betatron radiation in strongly magnetized plasma

Betatron radiation in strongly magnetized plasma is investigated by two dimensional (2D) particle-in-cell (PIC) simulations. The results show that the betatron radiation in magnetized plasmas is strongly enhanced and is more collimated compared to that in unmagnetized plasma. Single particle model analysis shows that the frequency and the amplitude of the electrons's betatron oscillation are strongly influenced by the axial external magnetic field and the axial self-generated magnetic field. And the 2D PIC simulation shows that the axial magnetic field is actually induced by the external magnetic field and tends to increase the betatron frequency. By disturbing the perturbation of the plasma density in the laser-produced channel, the hosing instability is also suppressed, which results in a better angular distribution and a better symmetry of the betatron radiation.

Effect of imperfections of the radial component of a magnetic field on beam dynamics in medical cyclotron C235-V3

This paper presents numerical simulations and experimental results related to the effect of imperfections of the radial component of a magnetic field on the beam dynamics in the medical cyclotron C235-V3 of the Dimitrovgrad Proton Therapy Center. These imperfections in the region of the minimal axial betatron frequency lead to a transformation of coherent motion of the center of gravity of the beam to the incoherent motion of separate particles. The radial component increases the axial size of the beam by a factor of 2 at a radius of 20 cm, which produces additional losses of protons. To reduce undesirable actions of the radial component on the axial motion, the magnetic system in the central region has been optimized using two procedures: the positioning of shim correctors on sectors and selecting a special asymmetric arrangement of the upper and lower central plugs. This led to a twofold reduction in the axial size of the beam and a decrease in proton losses. Eventually, the beam transmission in C235-V3 has been increased to 72% without a limiting aperture diaphragm, which is commonly used in cyclotrons of this type. This makes it possible to reduce the irradiation dose of machine elements and increase the beam current at a deflector entrance of the cyclotron C235-V3 by a factor of 1.5 when compared to a serial C235 cyclotron.

Enhancement of laser-driven electron acceleration in an ion channel

A laser beam with duration longer than the period of plasma oscillations propagating through an underdense plasma produces a steady-state positively charged channel in the electron density. We consider a test electron in the two-dimensional plane channel under the combined action of the laser field and the transverse static electric field of the channel. At ultrarelativistic laser wave amplitude (a≫1), the electron is pushed primarily forward. As the electron gradually dephases from the wave, the field it samples and its relativistic γ-factor strongly oscillate. The natural frequency of electron oscillations across the channel (betatron frequency) depends on γ, which couples the betatron oscillations to the longitudinal motion induced by the wave. We show that the modulation of the natural frequency makes the oscillations unstable. The resulting amplification of the oscillations across the channel reduces the axial dephasing between the electron and the wave, leading to a considerable electron energy enhancement well above the ponderomotive energy. We find that there is a well-pronounced laser amplitude threshold a*, above which the enhancement takes place, that scales as a*∝1/n0, where n0 is the ion density. The presented mechanism of energy enhancement is robust with respect to a longitudinal variation of the density, because it relies on a threshold phenomenon rather than on a narrow linear resonance.

Radio frequency quadrupole for Landau damping in accelerators

We propose using a radio frequency quadrupole (RFQ) to introduce both the longitudinal spread of betatron frequency and the transverse spread of synchrotron frequency for Landau damping of transverse beam instabilities in accelerators. The existing theory of stability diagrams for Landau damping is applied to the case of a RFQ. As an example, the required quadrupolar strength is calculated for stabilizing the Large Hadron Collider beams at 7 TeV. It is shown that this strength can be provided by a superconducting rf device only a few meters long.

Generating overcritical dense relativistic electron beams via self-matching resonance acceleration.

We show a novel self-matching resonance acceleration regime for generating dense relativistic electron beams by using ultraintense circularly polarized laser pulses in near-critical density plasmas. When the self-generated quasistatic axial magnetic field is strong enough to pinch and trap thermal relativistic electrons, an overdense electron bunch is formed in the center of the laser channel. In the trapping process, the electron betatron frequencies and phases can be adjusted automatically to match the resonance condition. The matched electrons are accelerated continuously and a collimated electron beam with overcritical density, helical structure, and plateau profile energy spectrum is hence generated.