Quadrupole Focusing Channel Research Articles

Optimization of the arc compressor performance in the MariX free electron laser

The MariX FEL is a compact GeV-class X-ray source exploiting a two-pass two-way acceleration in a Super-Conducting linac operated in continuous wave mode. A key component of this peculiar machine layout is the Bubble Arc Compressor (BAC), a 300 m long beamline consisting of 14 “Double Bend Achromat” cells and a bidirectional quadrupole focusing channel, which allows the beam to make a U-turn while it is being compressed to greatly increase its brightness and peak current.In this paper we present the performance of the BAC of MariX and the solutions we adopted to solve the main issues that the beam dynamics encounters in a line of this kind. We show the beam dynamics in the BAC matching line which is designed to operate on beams propagating in both directions, considering the anti-symmetric quadrupole focusing behavior. We study the Coherent Synchrotron Radiation (CSR) emission in the BAC showing a scheme that preserves the low emittance granting also a linear compression in presence of strong CSR effects. Lastly, we present a strategy to correct the residual dispersion-based beam tilt that is introduced by the CSR kick and would otherwise spoil the FEL emission. Further, the projected emittance is minimized cancelling the dispersive contributions.

Mitigation of envelope instability through fast acceleration in linear accelerators

The space-charge driven envelope instability can be of great danger in high intensity accelerators. Linear accelerators were designed to avoid this instability by keeping the zero current phase advance per lattice period below 90 degrees. In this paper, we report on the acceleration effects on the instability in a periodic solenoid and radio-frequency (RF) focusing channel and a periodic quadrupole and RF focusing channel using a three-dimensional envelope model and self-consistent macroparticle simulations. The simulation results show that the envelope instability can be mitigated with a reasonable accelerating gradient in both channels. This suggests that the zero current phase advance might be above 90 degrees in linear accelerators where the accelerating gradient is sufficiently high and opens additional freedom of transverse and longitudinal focusing in the accelerator design.

Three-dimensional envelope instability in periodic focusing channels

The space-charge driven envelope instability can be of great danger in high intensity accelerators and was studied using a two-dimensional (2D) envelope model and three-dimensional (3D) macroparticle simulations before. In this paper, we propose a three-dimensional envelope instability model to study the instability for a bunched beam in a periodic solenoid and radio-frequency (RF) focusing channel and a periodic quadrupole and RF focusing channel. This study shows that when the transverse zero current phase advance is below $90$ degrees, the beam envelope can still become unstable if the longitudinal zero current phase advance is beyond $90$ degrees. For the transverse zero current phase advance beyond $90$ degrees, the instability stopband width becomes larger with the increase of the longitudinal focusing strength and even shows different structure from the 2D case when the longitudinal zero current phase advance is beyond $90$ degrees. Breaking the symmetry of two longitudinal focusing RF cavities and the symmetry between the horizontal focusing and the vertical focusing in the transverse plane in the periodic quadrupole and RF channel makes the instability stopband broader. This suggests that a more symmetric accelerator lattice design might help reduce the range of the envelope instability in parameter space.

Erratum to: Beam dynamics and stability analysis of an intense beam in a continuously twisted quadrupole focusing channel

Erratum to: Beam dynamics and stability analysis of an intense beam in a continuously twisted quadrupole focusing channel

Beam dynamics and stability analysis of an intense beam in a continuously twisted quadrupole focusing channel

This paper describes the dynamics of a space-charge-dominated beam through a continuously twisted quadrupole magnet using ten independent first-order differential equations of the beam matrix elements under the assumption of linear space-charge force. Various beam optical properties of the magnet and the evolution of the emittance that results from the coupling between the two transverse planes are studied. The perturbed equations of motion around the matched beam envelopes have been derived and utilized to analyze the stability properties of the intense beam transport by calculating the eigenvalues of the transfer map over one lattice period. Detailed analysis shows the presence of instability due to parametric resonances in a twisted quadrupole channel which generally does not appear in the FODO quadrupole channel. A 2D particle-in-cell simulation code has been developed and utilized to verify the analytical results and to examine the behavior of the intense beam with Gaussian (GA) distribution in the twisted quadrupole channel.

Analysis of intense beam instability in a general quadrupole focusing channel with image charge effect

The stability properties of transverse envelopes of mismatched intense continuous charge particle beam propagating in a general quadrupole focusing channel have been investigated in the presence of image charge effect due to a cylindrical conducting pipe. Phase shifts and growth factors of the envelope oscillations in the case of instability are calculated by numerical evaluation of the eigenvalues of linearly perturbed envelope equations for small deviations from the matched beam conditions. A detailed study on the region of instability and its dependence on the system parameters like occupancy of the quadrupole focusing field, syncopation factor, zero current phase advance, beam intensity etc. have been carried out. It has been found that the strength and regions of envelope instability due to the lattice and confluent resonances in the parametric space are affected by the presence of image charge.

Analysis of continuously rotating quadrupole focusing channels using generalized Courant-Snyder theory

By extending the recently developed generalized Courant-Snyder theory for coupled transverse beam dynamics, we have constructed the Gaussian beam distribution and its projections with arbitrary mode emittance ratios. The new formulation has been applied to a continuously rotating quadrupole focusing channel because the basic properties of this channel are known theoretically and could also be investigated experimentally in a compact setup such as the linear Paul trap configuration. The new formulation retains a remarkably similar mathematical structure to the original Courant-Snyder theory, and thus, provides a powerful theoretical tool to investigate coupled transverse beam dynamics in general and more complex linear focusing channels.

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.

Experimental simulation of beam propagation over long path lengths using radio-frequency and magnetic traps

An overview is given of the novel beam-dynamics experiments based on compact non-neutral plasma traps at Hiroshima University. We have designed and constructed two different classes of trap systems, one of which uses a radio-frequency electric field (Paul trap) and the other uses an axial magnetic field (Penning trap) for transverse plasma confinement. These systems are called “S-POD” (Simulator for Particle Orbit Dynamics). The S-POD systems can approximately reproduce the collective motion of a charged-particle beam propagating through long alternating-gradient (AG) quadrupole focusing channels using the Paul trap and long continuous focusing channels using the Penning trap. This allows us to study various beam-dynamics issues in compact and inexpensive experiments without relying on large-scale accelerators. So far, the linear Paul traps have been applied for the study of resonance-related issues including coherent-resonance-induced stop bands and their dependence on AG lattice structures, resonance crossing in fixed-field AG accelerators, ultralow-emittance beam stability, etc. The Penning trap with multi-ring electrodes has been employed primarily for the study of beam halo formation driven by initial distribution perturbations. In this paper, we briefly overview the S-POD systems, and then summarize recent experimental results on resonance effects and halo formation.

Experimental study of coherent betatron resonances with a Paul trap

Linear and nonlinear resonant instabilities of charged-particle beams traveling in periodic quadrupole focusing channels are studied experimentally with a compact non-neutral plasma trap. The present experiments are based on the idea that the collective motion of a beam in an accelerator is physically similar to that of a one-component plasma in a trap. A linear Paul trap system named ``S-POD'' (simulator for particle orbit dynamics) was developed to explore a variety of space-charge-induced phenomena. To emulate lattice-dependent effects, periodic perturbations are applied to quadrupole electrodes, which gives rise to additional resonance stop bands that shift depending on the plasma density. It is confirmed that an $m$th-order resonance takes place when the corresponding tune of an $m$th-order collective mode ${\ensuremath{\Omega}}_{m}$ is close to a half integer.

Space-charge transport limits of ion beams in periodic quadrupole focusing channels

It has been empirically observed in both experiments and particle-in-cell simulations that space-charge-dominated beams suffer strong growth in statistical phase-space area (degraded quality) and particle losses in alternating gradient quadrupole transport channels when the undepressed phase advance σ 0 increases beyond about 85 ∘ per lattice period. Although this criterion has been used extensively in practical designs of strong focusing intense beam transport lattices, the origin of the limit has not been understood. We propose a mechanism for the transport limit resulting from strongly chaotic classes of halo particle resonances near the core of the beam that allow near-edge particles to rapidly increase in oscillation amplitude when the space-charge intensity and the flutter of the matched beam envelope are both sufficiently large. When coupled with a diffuse beam edge and/or perturbations internal to the beam core that can drive particles outside the edge, this mechanism can result in large and rapid halo-driven increases in the statistical phase-space area of the beam, lost particles, and degraded transport. A core–particle model is applied to parametrically analyze this process. Extensive self-consistent particle in cell simulations is employed to better quantify properties of the space-charge limits and to verify core–particle model predictions.

Stability properties of the transverse envelope equations describing intense ion beam transport

The transverse evolution of the envelope of an intense, unbunched ion beam in a linear transport channel can be modeled for the approximation of linear self-fields by the Kapchinskij-Vladimirskij (KV) envelope equations. Here we employ the KV envelope equations to analyze the linear stability properties of so-called mismatch perturbations about the matched (i.e., periodic) beam envelope in continuous focusing, periodic solenoidal, and periodic quadrupole transport lattices for a coasting beam. The formulation is analyzed and explicit self-consistent KV distributions are derived for an elliptical beam envelope in a periodic solenoidal transport channel. This derivation extends previous work to identify emittance measures and Larmor-frame transformations to allow application of standard form envelope equations to solenoidal focusing channels. Perturbed envelope equations are derived that include driving sources of mismatch excitation resulting from focusing errors, particle loss, and beam emittance growth. These equations are solved analytically for continuous focusing and demonstrate a factor of 2 increase in maximum mismatch excursions resulting from sudden driving perturbations relative to adiabatic driving perturbations. Numerical and analytical studies are carried out to explore properties of normal mode envelope oscillations without driving excitations in periodic solenoidal and quadrupole focusing lattices. Previous work on this topic by Struckmeier and Reiser [Part. Accel. 14, 227 (1984)] is extended and clarified. Regions of parametric instability are mapped, new classes of envelope instabilities are found, parametric sensitivities are explored, general limits and mode invariants are derived, and analytically accessible limits are checked. Important, and previously unexplored, launching conditions are described for pure envelope modes in periodic quadrupole focusing channels.

Analysis of a helical quadrupole focusing channel

A helical quadrupole focusing channel has continuous field symmetry. This good field symmetry allowed us to construct an explicit transfer matrix under a paraxial approximation. A paraxial analysis of the helical quadrupole focusing channel, a comparison of the beam size with a conventional quadrupole focusing channel and a comparison of the capture angle with a solenoid channel are reported.

Linear theory of transverse and longitudinal ionization cooling in a quadrupole channel

Ionization cooling is the best-known cooling mechanism for the envisioned muon colliders and neutrino factories. In this paper, using the moment-equation approach, we present a linear theory of ionization cooling dynamics in 6D phase space in a quadrupole focusing channel. A simple set of differential equations that governs the evolution of both the transverse and longitudinal emittances is derived, and closed-form solutions are given. Two new significant heating processes have been identified. This theory is analogous to the standard linear theory in electron storage rings. Multiple scattering integrals and energy straggling integrals, quantities like the synchrotron radiation integrals, are introduced to specify the cooling process and the equilibrium emittances in a periodic cooling channel.

Analysis of helical quadrupole focusing channel

A helical quadrupole focusing channel has continuous field symmetry and a stronger focusing power compared with a conventional FODO focusing channel. The good field symmetry allowed us to construct an explicit transfer matrix under the paraxial approximation. In the present paper, we report the paraxial analysis of the helical quadrupole focusing channel and compare its characteristics with those of a conventional FODO focusing channel.

Ideal matching of heavy ion beams

A novel technique is developed, and demonstrated via simulations, which allows ideal matching of an ultrahigh-brightness heavy ion beam in a linear focusing channel with general magnetic focusing field profile. Unlike the conventional rms matching used in accelerator design, this technique is based on a new class of corkscrewing elliptic beam equilibria in a linear focusing channel consisting of uniform solenoidal fields, periodic solenoidal fields, and/or alternating-gradient quadrupole magnetic focusing fields in an arbitrary arrangement including field tapering. As an illustration of the effectiveness of this technique, a perfect matching of an ultrahigh-brightness heavy ion beam from an axisymmetric particle source into alternating-gradient quadrupole focusing channel is achieved without beam hollowing and beam halo formation. Discussions are made of how to implement such a perfect beam matching to avoid beam hollowing observed in the LBNL 2-MV Heavy Ion Injector Experiment.

Mechanisms and control of beam halo formation in intense microwave sources and accelerators

Halo formation and control in space-charge-dominated electron and ion beams are investigated in parameter regimes relevant to the development of high-power microwave (HPM) sources and high-intensity electron and ion linear accelerators. In particular, a mechanism for electron beam halo formation is identified in high-power periodic permanent magnet (PPM) focusing klystron amplifiers. It is found in self-consistent simulations that large-amplitude current oscillations induce mismatched beam envelope oscillations and electron beam halo formation. Qualitative agreement is found between simulations and the 50 MW 11.4 GHz PPM focusing klystron experiment at Stanford Linear Accelerator Center (SLAC) (D. Sprehn, G. Caryotakis, E. Jongewaard, and R. M. Phillips, “Periodic permanent magnetic development for linear collider X-band klystrons,” Proceedings of the XIXth International Linac Conference, Argonne National Laboratory Report ANL-98/28, 1998, p. 689). Moreover, a new class of cold-fluid corkscrewing elliptic beam equilibria is discovered for ultrahigh-brightness, space-charge dominated electron or ion beam propagation through a linear focusing channel consisting of uniform solenoidal magnetic focusing fields, periodic solenoidal magnetic focusing fields, and/or alternating-gradient quadrupole magnetic focusing fields in an arbitrary arrangement including field tapering. As an important application of such new cold-fluid corkscrewing elliptic beam equilibria, a technique is developed and demonstrated for controlling of halo formation and beam hollowing in a rms-matched ultrahigh-brightness ion beam as it is injected from an axisymmetric Pierce diode into an alternating-gradient magnetic quadrupole focusing channel.

Studies on stability of charge-compensated ion beams

The propagation stability of the charge-compensated two-component ion beam in the periodical quadrupole focusing channel are investigated in a linear approximation. The frequencies of transversal noncoherent oscillations are determined. Also, the frequencies of coherent dipole and quadrupole oscillations have been found. Compared with the noncoherent case these frequencies rapidly increase with current. It is shown that the most dangerous is the coherent quadrupole asymmetric mode, which determines the upper limit of the ion current.

Antiproton Production and Collection for the CERN Antiproton Accumulator

Antiprotons are produced for the CERN Antiproton Accumulator (AA) by focusing 26 GeV/c protons onto a 3 mm diameter, 11 cm long copper wire. Negatively charged particles with momenta about 3.5 GeV/c are focused by a short focal length coaxial horn and transported to the AA by a normal quadrupole focusing channel. The yield of antiprotons was found to be considerably less than anticipated (factor about 2) and the reason is presumed to be the assumption of too large a production cross-section in the original machine design proposal. Studies involving new horn design, introduction of an axial current e 150 kA) along the target and use of lithium lenses as an alternative to the magnetic horn are under way. Some preliminary measurements involving some of these techniques have been made, both to confirm the validity of calculations and to test the feasibility of building targets and focusing systems to withstand the mechanical forces and heat load due to the proton beam and the high pulsed currents.

Wide Band Neutrino Beams with Quadrupole Focusing

For high energy neutrino beams the requirements of large solid angle acceptance and wide bandpass can be achieved with a focusing channel composed of quadrupole cells where each cell is sequentially scaled in aperture, length and field gradient. A formalism for the design of these quadrupole focused neutrino beams with wide bandpass has been developed. The theory will be presented and the results of calculations will be given for some quadrupole focused neutrino beams. Compared to more exotic focusing techniques quadrupoles offer inherent reliability and easy maintenance. The system, which can be used in either long or short spill operation, can also be used as the front end of a muon beam for a facility that would be shared between neutrino and muon experiments. The quadrupole focusing channel does not, however, distinguish between parent particle charge and, therefore, cannot produce a pure neutrino or antineutrino beam.