Betatron Motion Research Articles

Electron energy gain due to a laser frequency modulation experienced by electron during betatron motion

Direct laser acceleration of electrons is an important energy deposition mechanism for laser-irradiated plasmas that is particularly effective at relativistic laser intensities in the presence of quasi-static laser-driven plasma electric and magnetic fields. These radial electric and azimuthal magnetic fields provide transverse electron confinement by inducing betatron oscillations of forward-moving electrons undergoing laser acceleration. Electrons are said to experience a betatron resonance when the frequency of betatron oscillations matches the average frequency of the laser field oscillations at the electron position. In this paper, we show that the modulation of the laser frequency as seen by an electron performing betatron oscillations can be another important mechanism for net energy gain that is qualitatively different from the betatron resonance. Specifically, we show that the frequency modulation experienced by the electron can lead to net energy gain in the regime where the laser field performs three oscillations per betatron oscillation. There is no net energy gain in this regime without the modulation because the energy gain is fully compensated by the energy loss. The modulation slows down the laser oscillation near transverse stopping points, increasing the time interval during which the electron gains energy and making it possible to achieve net energy gain.

Operational tests of CRYRING@ESR without electron cooler solenoid compensation

We have tested operation of FAIR’s low-energy ion storage ring CRYRING@ESR with uncompensated electron cooler solenoid. With its standard working point on the lowest-order difference resonance, a second solenoid is normally used to cancel betatron coupling introduced by the cooler’s magnetic field. In operation with a D+ test beam, we found that omission of the compensation solenoid did not lead to a notable deterioration of beam intensity, quality, or cooling time, though the expected coupling of betatron motion is then clearly observed.

Parametrizations of coupled betatron motion for strongly coupled lattices

The coupling of transverse motion is a natural occurrence in particle accelerators, either in the form of a residual coupling arising from imperfections or originating by design from strong systematic coupling fields. While the first can be treated perturbatively, the latter requires a robust approach adapted to strongly coupled optics, and a parametrization of the linear optics must be performed to explore beam dynamics in such peculiar lattices. This work highlights the key physical interpretations of the main parametrization formalisms to describe linear coupled optics, along with explicit links and comparisons of these parametrizations. Concepts rarely illustrated in other works, such as forced mode flips and local coupling, are explored in detail, clarifying some anomalies that can arise in lattice functions. The analytical methods have been implemented in a reference Python package and connected with ray-tracing and integration codes to explore examples of strongly coupled lattices, which are discussed in detail to highlight the key physical interpretations of the parametrizations and characteristics of the lattices.

3D beam-envelope approach for an asymmetric beam in a high intensity linac with a solenoid channel

Presented here is a theory model based on the beam-envelope approach to determine the global behavior of a charged beam propagating through a high intensity linac with a solenoid channel, namely, the orbital evolution of the root-mean-square beam size in the transverse and longitudinal direction. The model is applied to the Gaussian particle distribution, where the motion of an individual particle is governed by linear focusing/defocusing forces in the solenoid channel and RF fields, except for space-charge forces. In this model, highly nonlinear effects that originated from space-charge forces disappear by averaging over the harmonic motions such as the betatron motion and phase motion in the moving direction and the Gaussian distribution. All particles making up a 3D Gaussian bunch obey the same linear equations of motion where all information of nonlinearity in the space-charge forces and the particle distribution are carried in. The theory is validated using Gaussian macro-particle simulation and one of the widely used photonic integrated circuit codes, TraceWin. As an example, the model is applied to the case of an international fusion materials irradiation facility superconducting linac with solenoid guiding and half wave RF acceleration. The results are compared with results obtained by this model and other two approaches.

Enhanced betatron x-ray emission in a laser wakefield accelerator and wiggler due to collective oscillations of electrons

We report x-ray emission caused by betatron motion of electrons in a laser wakefield accelerator driven by high intensity ($\ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}{10}^{19}\text{ }\text{ }\mathrm{W}/{\mathrm{cm}}^{2}$) laser pulse interaction with a 4-mm long helium gas-jet and study the role of collective oscillation of electrons as well as plasma density. The collective oscillations of electrons, inferred from the observed wavy pattern in the electron beam spectrum (up to 300 MeV) led to enhance peak x-ray flux which was found to be further enhanced at higher plasma densities. A similar correlation between the enhancement in x-ray emission and collective oscillation was also observed in nitrogen gas target. Possible physical pictures for the above observations are also suggested and discussed. For the helium gas jet target, the typically observed x-ray source brightness is $\ensuremath{\ge}1\ifmmode\times\else\texttimes\fi{}{10}^{20}\text{ }\text{ }\mathrm{photons}/\mathrm{s}\text{ }{\mathrm{mm}}^{2}\text{ }{\mathrm{mrad}}^{2}$ in 0.1% BW, with a critical energy of $35\ifmmode\pm\else\textpm\fi{}5\text{ }\text{ }\mathrm{keV}$. We have also demonstrated the application of the betatron x-ray source in high-resolution imaging.

Developments of a Pulse Kicker System for the Three-Dimensional Spiral Beam Injection of the J-PARC Muon g-2/EDM Experiment

The development of a vertical kicker system is an important part of a three-dimensional spiral-injection scheme. The beam injected at a pitch angle of a few tens of degrees was controlled by applying a pulsed magnetic field using a vertical kicker system inside the storage magnet. The stored muon beam then followed a vertical betatron motion around the midplane of the storage magnet owing to the weak focusing magnetic field. Accurate control of the pulsed kick is crucial for minimizing the amplitude of the vertical betatron oscillation. The specifications of the pulsed magnetic field and detailed design of the kicker system at the test beam bench are discussed herein.

Time-domain study of the synchrotron radiation emitted from electron beams in plasma focusing channels.

This paper sheds light on the time structure of betatron radiation, emitted by electrons that undergo betatron oscillations as they accelerate under the action of plasma wakefields. It is a common practice to assume that the betatron pulses are as short as the electron bunch length, however we show that this is not a general rule. Indeed, the betatron pulse length is affected by the betatron motion, which stretches and modulates the radiation pulses already at the source level. Propagation in a vacuum, therefore, can greatly lengthen the betatron pulses by orders of magnitude. In the wake of the above, the coherent emission of betatron radiation is studied. Coherent betatron radiation has been found to propagate in an underdense region created by ponderomotive forces, thus not suppressed by the overdense plasma absorption. This could be observed experimentally, revealing information on the acceleration process and on key beam parameters.

Beam dynamics corrections to the Run-1 measurement of the muon anomalous magnetic moment at Fermilab

This paper presents the beam dynamics systematic corrections and their uncertainties for the Run-1 data set of the Fermilab Muon g-2 Experiment. Two corrections to the measured muon precession frequency $\omega_a^m$ are associated with well-known effects owing to the use of electrostatic quadrupole (ESQ) vertical focusing in the storage ring. An average vertically oriented motional magnetic field is felt by relativistic muons passing transversely through the radial electric field components created by the ESQ system. The correction depends on the stored momentum distribution and the tunes of the ring, which has relatively weak vertical focusing. Vertical betatron motions imply that the muons do not orbit the ring in a plane exactly orthogonal to the vertical magnetic field direction. A correction is necessary to account for an average pitch angle associated with their trajectories. A third small correction is necessary because muons that escape the ring during the storage time are slightly biased in initial spin phase compared to the parent distribution. Finally, because two high-voltage resistors in the ESQ network had longer than designed RC time constants, the vertical and horizontal centroids and envelopes of the stored muon beam drifted slightly, but coherently, during each storage ring fill. This led to the discovery of an important phase-acceptance relationship that requires a correction. The sum of the corrections to $\omega_a^m$ is 0.50 $\pm$ 0.09 ppm; the uncertainty is small compared to the 0.43 ppm statistical precision of $\omega_a^m$.

Using betatron coupling to suppress horizontal intrinsic spin resonances driven by partial snakes

A method for using betatron coupling to suppress partial snake driven resonances in a synchrotron is described. While the partial snakes permit avoidance of the strong resonances associated with the vertical betatron motion of the particles, they also excite numerous weak resonances associated with the horizontal betatron motion. These resonances occur whenever the fractional spin tune equals the fractional horizontal betatron tune. Since these are the same frequencies at which depolarizing resonances from betatron coupling occur, coupling resonances can be intentionally driven to exactly cancel the resonance driving terms of the partial snakes, thereby avoiding polarization loss due to these resonance crossings. Presented here is an explicit derivation of the partial snake driven resonances as betatron sidebands of strong imperfection resonances and a description of a practical scheme for their correction using the Brookhaven AGS as an example. Tracking results that verify the suppression via coupling are presented.

Scaling laws for direct laser acceleration in a radiation-reaction dominated regime

We study electron acceleration within a sub-critical plasma channel irradiated by an ultra-intense laser pulse (a0 > 100 or I > 1022 W cm−2). In this regime, radiation reaction significantly alters the electron dynamics. This has an effect not only on the maximum attainable electron energy but also on the phase-matching process between betatron motion and electron oscillations in the laser field. Our study encompasses analytical description, test-particle calculations and two-dimensional particle-in-cell simulations. We show single-stage electron acceleration to multi-GeV energies within a 0.5 mm-long channel and provide guidelines how to obtain energies beyond 10 GeV using optimal initial configurations. We present the required conditions in a form of explicit analytical scaling laws that can be applied to plan the future electron acceleration experiments.

Intrinsic energy spread and bunch length growth in plasma-based accelerators due to betatron motion

Plasma-based accelerators (PBAs), having demonstrated the production of GeV electron beams in only centimetre scales, offer a path towards a new generation of highly compact and cost-effective particle accelerators. However, achieving the required beam quality, particularly on the energy spread for applications such as free-electron lasers, remains a challenge. Here we investigate fundamental sources of energy spread and bunch length in PBAs which arise from the betatron motion of beam electrons. We present an analytical theory, validated against particle-in-cell simulations, which accurately describes these phenomena. Significant impact on the beam quality is predicted for certain configurations, explaining previously observed limitations on the achievable bunch length and energy spread. Guidelines for mitigating these contributions towards high-quality beams are deduced.

Calculating the polarization lifetime from the Thomas-Bargmann-Michel-Telegdi equation

It was observed at the Relativistic Heavy Ion Collider that the polarization of the stored beam decays by 0.17 to 3% per hour. This effect could not be simulated up to now since the simulation time for an exact calculation exceeded the computer capacity. To solve this problem the action of the whole arc was considered as one element acting on the spin and not, as before, each optical element in the arc individually (called lattice independent integration). In addition to former simulations the coupling of synchrotron and betatron motions in the beam interaction regions and their action on the polarization were taken into account (called addition of longitudinal dynamics). It is shown that with these two changes the action on the polarization life time can be explained.

Transverse dynamics considerations for microbunched electron cooling

The technique of microbunched electron cooling (MBEC) is an attractive coherent cooling scheme with potential applications in future high-energy circular colliders. In our previous work, we analyzed the cooling of the energy spread using a one-dimensional (1D) technique that tracks the dynamics of microscopic fluctuations in the hadron and electron beams. In this paper, we extend this approach so that it covers the transverse emittance cooling as well. In order to do so, it is necessary to consider the betatron motion of the hadron beam and take into account effects of the momentum dispersion in the modulator and kicker regions. We derive relatively simple analytical expressions for the emittance and energy spread cooling times in terms of the various beam and lattice parameters, allowing us to perform fast optimization studies for an MBEC configuration. Verified through comparison with simulation, our theory can also incorporate features such as plasma amplification stages, which are crucial components of a realistic cooling system.

X-ray sources using a picosecond laser driven plasma accelerator

Laser-plasma-based accelerators are now able to provide the scientific community with novel high-energy light sources that are essential to study high-energy density matter, inertial confinement fusion, astrophysical systems, and fundamental plasma physics. Due to the transient and high-density properties of these systems, it is essential to develop light sources that are in the hard x-ray energy range (0.01–1 MeV) and directional and have high yield, low divergence, and short duration (ps and sub-ps). In this work, we show that by using a Laser plasma accelerator, it is possible to generate a broadband (0.01–1 MeV) hard x-ray source that satisfies the previous requirements. A series of experiments were conducted on the Titan laser at the Lawrence Livermore National Laboratory where a 10 nC electron beam in the 10–380 MeV energy range was generated through a laser plasma accelerator. The electrons generate x-rays via their betatron motion (few-30 keV) and hard x-rays through inverse Compton scattering (10–250 keV) and/or Bremsstrahlung (up to 1 MeV). Due to its unique characteristics, this source can be an important tool for many applications in large-scale international laser facilities.

High-resolution phase-contrast imaging of biological specimens using a stable betatron X-ray source in the multiple-exposure mode

Phase-contrast imaging using X-ray sources with high spatial coherence is an emerging tool in biology and material science. Much of this research is being done using large synchrotron facilities or relatively low-flux microfocus X-ray tubes. An alternative high-flux, ultra-short and high-spatial-coherence table-top X-ray source based on betatron motions of electrons in laser wakefield accelerators has the promise to produce high quality images. In previous phase-contrast imaging studies with betatron sources, single-exposure images with a spatial resolution of 6–70 μm were reported by using large-scale laser systems (60–200 TW). Furthermore, images obtained with multiple exposures tended to have a reduced contrast and resolution due to the shot-to-shot fluctuations. In this article, we demonstrate that a highly stable multiple-exposure betatron source, with an effective average source size of 5 μm, photon number and pointing jitters of <5% and spectral fluctuation of <10%, can be obtained by utilizing ionization injection in pure nitrogen plasma using a 30–40 TW laser. Using this source, high quality phase-contrast images of biological specimens with a 5-μm resolution are obtained for the first time. This work shows a way for the application of high resolution phase-contrast imaging with stable betatron sources using modest power, high repetition-rate lasers.

High-energy photon emission and radiation reaction effects in the ultra-high intensity laser bubble regime

We investigate high-energy photon emission and related radiation back-reaction effects in the bubble regime of ultra-short, extreme intensity (1023–1024 ) laser interactions with under-dense plasma. Both the spontaneous emission by the accelerated electron beam in the focusing fields of the cavitated bubble (betatron motion) and the stimulated emission by oscillating electrons in the fields of the guided laser pulse are discussed. Our analyses are in good agreement with our particle-in-cell simulations performed using the contemporary particle code EPOCH (Arber et al 2015 Plasma Phys. Control. Fusion 57 113001). Intense bursts of hard photons () at density with a dominant fraction of gamma ray () content are observed at a typical intensity and plasma density 0.01 critical density. The intensity and energy of photons become highly enhanced by using higher plasma densities. Moreover, spontaneous emissions by the accelerated electron beam become partially collimated along the pulse propagation direction. At high plasma densities, the self-consistent plasma fields and plasma profile are noticeably modified by radiation backreactions, and strong effects are observed on the accelerated electron beam.

Gamma-ray emission from wakefield-accelerated electrons wiggling in a laser field

Ultra-fast synchrotron radiation emission can arise from the transverse betatron motion of an electron in a laser plasma wakefield, and the radiation spectral peak is limited to tens of keV. Here, we present a new method for achieving high-energy radiation via accelerated electrons wiggling in an additional laser field whose intensity is one order of magnitude higher than that for the self-generated transverse field of the bubble, resulting in an equivalent wiggler strength parameter K increase of approximately twenty times. By calculating synchrotron radiation, we acquired a peak brightness for the case of the laser wiggler field of 1.2 × 1023 ph/s/mrad2/mm2/0.1%BW at 1 MeV. Such a high brilliance and ultra-fast gamma-ray source could be applied to time-resolved probing of dense materials and the production of medical radioisotopes.

Multiparticle time-domain analysis of coherent undulator radiation

The physics involving the emission of coherent undulator radiation from a bunch of correlated electrons wiggling inside an undulator has been very well established in the frequency domain. With the aim of describing the process of emission of coherent undulator radiation in time domain while incorporating the effect of hyperbolic field profile of the undulator and various electron beam parameters, we present a time-domain analysis which can serve as a complimentary formulation to the frequency-domain analysis of the coherent undulator radiation. The formulation requires revisiting the equation of motion of an electron moving inside a planar undulator. An electron beam with substantial spread along the direction of the magnetic field will undergo betatron oscillations due to the hyperbolic profile of the undulator fields. We show that the betatron motion can be conveniently expressed in terms of Mathieu functions. The new set of equations of motion have been used to study the effect of betatron motion on the emitted undulator radiation. The single-particle time-domain analysis is then easily extended to multiparticle systems by correctly accounting for the phase-difference between the individual electromagnetic waves emitted by different electrons occupying different phase-space in the bunch.

Toward low energy spread in plasma accelerators in quasilinear regime

In this paper, we address the energy spread and slice energy spread of an externally injected electron beam in plasma wakefield accelerators operating in the linear or quasilinear regime. The energy spread is first derived taking into account the phase dependence of the wakefield along the finite-length bunch together with the dephasing during acceleration and found to be strongly dependent on the bunch length. This could be compensated by the beam loading effect, the energy spread from which is then derived and found to be nearly independent of the bunch length. However, the transverse dependence of the beam loading effect also makes the particles at the same longitudinal position experience different accelerating fields, introducing a significant slice energy spread. To estimate the slice energy spread, a theoretical analysis was conducted by taking the transverse betatron motion into account. As a study case, 3D simulations for the 5 GeV laser-plasma acceleration stage of the European Plasma Research Accelerator with eXcellence in Applications project have been performed. Careful optimization of the parameters allows one to obtain an energy spread of $\ensuremath{\le}1%$ and a slice energy spread of $\ensuremath{\le}0.1%$, with good agreement between theories and simulations.

Far-field constant-gradient laser accelerator of electrons in an ion channel

We predict that electrons in an ion channel can gain ultra-relativistic energies by simultaneously interacting with a laser pulse and, counter-intuitively, with a decelerating electric field. The crucial role of the decelerating field is to maintain high-amplitude betatron oscillations, thereby enabling constant rate energy flow to the electrons via the inverse ion channel laser mechanism. Multiple harmonics of the betatron motion can be employed. Injecting electrons into a decelerating phase of a laser wakefield accelerator is one practical implementation of the scheme.