Quadrupole Focusing Research Articles

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 verification of resonance instability bands in quadrupole doublet focusing channels

The tabletop plasma trap experiment named “S-POD” is employed to explore the stability of intense charged-particle beams focused by a series of quadrupole doublet cells. S-POD is a compact linear Paul-trap, where we generate a single-species non-neutral ion plasma that can approximately reproduce the collective motion of an intense beam focused by periodic linear forces. Unlike conventional beam-dynamics experiments relying on large-scale transport channels and accelerators, it is straightforward in S-POD to control the functional form of quadrupole beam focusing over a wide range of variation to explore a variety of quadrupole focusing lattices. We systematically measure the loss rate of trapped particles as a function of bare betatron tune to locate resonance bands in which the plasma becomes unstable. It is confirmed that a few bands of coherent resonances appear depending on the beam intensity. When there is an imbalance between the horizontal and vertical focusing, those instability bands split. Experimental results indicate that the instability band is relatively insensitive to the phase of quadrupole focusing element placement within the doublet configuration over a significant range of parameters. Experimental observations are compared with transverse slice particle-in-cell simulations carried out using the Warp code.

Design of a Large Single-Aperture Dipole Magnet for HL-LHC Upgrade

An upgrade of the low-beta insertion system for the ATLAS and Compact Muon Solenoid experiments is proposed in the high luminosity Large Hadron Collider upgrade project. It includes final beam focusing quadrupoles, beam separation and recombination dipoles, and larger aperture matching section quadrupoles. KEK is in charge of the conceptual design of the large aperture separation dipole D1. The latest design parameters are a main field of ~ 5 T at 1.9 K with Nb-Ti superconducting technology, a coil aperture of 160 mm, and a cos-theta one-layer coil with Large Hadron Collider dipole cable. Because the new D1 is expected to be operated in a very high radiation environment, radiation resistance and a cooling scheme are being carefully considered. The collaring-yoke structure is adopted to provide the mechanical support for the single-layer Nb-Ti coil. We summarize the design study of this magnet, including i) the very large iron saturation effect on field quality due to the large aperture and limited size of the iron yoke, ii) the stray field at the outer surface of the iron cryostat, and iii) the stress management from room temperature assembly to final operation.

Symmetric achromatic low-beta collider interaction region design concept

We present a new symmetry-based concept for an achromatic low-beta collider interaction region design. A specially designed symmetric chromaticity compensation block (CCB) induces an angle spread in the passing beam such that it cancels the chromatic kick of the final focusing quadrupoles. Two such CCBs placed symmetrically around an interaction point allow simultaneous compensation of the 1st-order chromaticities and chromatic beam smear at the IP without inducing significant 2nd-order aberrations to the particle trajectory. We first develop an analytic description of this approach and explicitly formulate 2nd-order aberration compensation conditions at the interaction point. The concept is next applied to develop an interaction region design for the ion collider ring of an electron-ion collider. We numerically evaluate performance of the design in terms of momentum acceptance and dynamic aperture. The advantages of the new concept are illustrated by comparing it to the conventional distributed-sextupole chromaticity compensation scheme.

QCALT: A tile calorimeter for KLOE-2 upgrade

The upgrade of the DaΦne machine layout requires a modification of the size and position of the inner focusing quadrupoles of KLOE-2, thus asking for the realization of two new calorimeters, named QCALT, covering this area. To improve the reconstruction of KL→2π0 events with photons hitting the quadrupoles, a calorimeter with high efficiency to low energy photons (20–300MeV), time resolution of less than 1ns and space resolution of few cm, is needed. To match these requirements we are now constructing a scintillator tile calorimeter where each single tile is readout by mean of SiPM for a total granularity of 1760 channels. We show the design of the different calorimeter components and the present status of the construction.

Nonparaxial pulse broadening in a solenoid focusing element

In a standard scenario for focusing an ion beam onto a target, for example with ion beam driven inertial fusion energy, the beam is compressed longitudinally by a velocity ramp to enhance the current and then directed through a transverse focusing system to produce a small, bright spot on the target. To reach the highest levels of compression, the space-charge of the beam is neutralized, typically by the presence of a plasma with a density greater than the beam density. The system is arranged so that the peak longitudinal compression is coincident with the minimum transverse spot size. In this scenario, it has been discovered that nonparaxial effects can lead to degradation in the amount of compression. The transverse focusing causes a radially dependent variation in the axial velocity of the ions, leading to a radially dependent time delay that degrades the peak compression. This effect, nonparaxial pulse broadening, can become significant for short pulses and large focusing fields---the time delay can be comparable to the final pulse length. This pulse broadening will be present in both solenoid and quadrupole focusing systems. This paper describes this effect in solenoids, with some examples. It is expected that the size of the effect will be comparable with quadrupole focusing.

Explicit thin-lens solution for an arbitrary four by four uncoupled beam transfer matrix

In the design of beam transport lines, one often meets the problem of constructing a quadrupole lens system that will produce desired transfer matrices in both the horizontal and vertical planes. Nowadays this problem is typically approached with the help of computer routines, but searching for the numerical solution one has to remember that it is not proven yet that an arbitrary four by four uncoupled beam transfer matrix can be represented by using a finite number of drifts and quadrupoles (representation problem) and the answer to this question is not known not only for more or less realistic quadrupole field models but also for the both most commonly used approximations of quadrupole focusing, namely thick and thin quadrupole lenses. In this paper we make a step forward in resolving the representation problem and, by giving an explicit solution, we prove that an arbitrary four by four uncoupled beam transfer matrix actually can be obtained as a product of a finite number of thin lenses and drifts.

Design and Manufacture of a Hybrid Final Focus Quadrupole Model for CLIC

A tunable hybrid quadrupole magnet design has been proposed for the final focus in the Compact Linear Collider (CLIC) that is currently under study. The proposed design is a combination of an iron dominated electromagnetic quadrupole with a bore diameter of 8.25 mm with permanent magnet blocks placed between the poles made of soft magnetic CoFe alloy “Permendur”. The possibility of using Sm2Co17 and Nd2Fe14B as material for the permanent magnet blocks has been investigated. It is shown that a very high field gradient of 530 T/m (Sm2Co17)and 590 T/m (Nd2Fe14B)can be achieved.

Design and Construction of the Final Focus Quadrupole R&D Magnet for Super-KEKB

For upgrading the electron-positron collider KEKB, the final focusing magnet system in the beam interaction region has been studied. The designed magnet system consists of five superconducting and three permanent quadrupole magnets. A superconducting quadrupole magnet has been constructed as the R&D of this system. In this paper, design and construction of this R&D magnet are reported.

Novel Hamiltonian method for collective dynamics analysis of an intense charged particle beam propagating through a periodic focusing quadrupole lattice

Identifying regimes for quiescent propagation of intense beams over long distances has been a major challenge in accelerator research. In particular, the development of systematic theoretical approaches that are able to treat self-consistently the applied oscillating force and the nonlinear self-field force of the beam particles simultaneously has been a major challenge of modern beam physics. In this paper, the recently developed Hamiltonian averaging technique [E. A. Startsev, R. C. Davidson, and M. Dorf, Phys. Rev. ST Accel. Beams 13, 064402 (2010)] which incorporates both the applied periodic focusing force and the self-field force of the beam particles, is generalized to the case of time-dependent beam distributions. The new formulation allows not only a determination of quasi-equilibrium solutions of the non-linear Vlasov-Poison system of equations but also a detailed study of their stability properties. The corrections to the well-known “smooth-focusing” approximation are derived, and the results are applied to a matched beam with thermal equilibrium distribution function. It is shown that the corrections remain small even for moderate values of the vacuum phase advance συ. Nonetheless, because the corrections to the average self-field potential are non-axisymmetric, the stability properties of the different beam quasi-equilibria can change significantly.

Rotating coil field measurement of superconducting magnet for BEPCII interaction region

Two multifunction superconducting magnets in the interaction region (IR) for Beijing electron positron collider upgrade project (BEPCII) have been measured in BESIII detector hall in 2007. Each superconducting magnet package contains multiple concentric layers with several function magnets called as a vertical focusing quadrupole (SCQ), a horizontal corrector (HDC/SCB), a vertical corrector (VDC), a skew quadrupole (SKQ) and three anti-solenoids (AS1, AS2 and AS3) to compensate the experimental detector solenoid field. All these function magnets SCQs, SCB/HDCs, VDCs and SKQs have been measured using two rotating coils. Their integral fields, their high order harmonics contents and the local fields along the beam line are obtained in detail with the rotating coil probe system. Comparing the results to the stretched wire, the differences for the integral fields are less than 0.2%. As a result, the method presented in this paper can be used as an absolute field measurement in our lab.

High-Power Microwave Generation by a Periodic Focusing Quadrupole Transport System

High-power microwave generation via a relativistic electron beam propagating through a periodic focusing quadrupole transport (PFQT) system is studied. Transverse confinement of e-beam in a PFQT system is provided by segmented cylindrical electrodes with applied oscillatory voltages <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">osc</sub> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> ) over 90 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> segments. With regard to transverse confinement, there is no applied axial guide magnetic field. An analytic theory of laser gain in the low-gain-per-pass limit is developed. Numerical computations of electron trajectories and gain are also presented. It is shown that the presence of transverse oscillation frequency due to PFQT leads to significant gain enhancement under appropriate conditions.

Approximate kinetic quasiequilibrium distributions for intense beam propagation through a periodic focusing quadrupole lattice

The transverse dynamics of an intense charged particle beam propagating through a periodic quadrupole focusing lattice is described by the nonlinear Vlasov-Maxwell system of equations, where the propagation distances play the role of time. To determine matched-beam quasiequilibrium distribution functions, one needs to determine a dynamical invariant for the beam particles moving in the combined applied and self-generated fields. In this paper, a perturbative Hamiltonian transformation method is developed which is an expansion in the particle’s vacuum phase advance � � � � v=2� , treated as a small parameter, which is used to transform away the fast particle orbit oscillations and obtain the average Hamiltonian accurate to order � � 3 . The average Hamiltonian is an approximate invariant of the original system, and can be used to determine self-consistent beam quasiequilibrium solutions that are matched to the focusing channel. The equation determining the average self-field potential is derived for general boundary conditions by taking into account the average contribution of the charges induced on the boundary. It is shown for a cylindrical conducting boundary that the average self-field potential acquires an octupole component, which results in the average motion of some beam particles being nonintegrable and their trajectories chaotic. This chaotic behavior of the beam particles may significantly change the nature of the Landau damping (or growth) of collective excitations supported by an intense charged particle beam.

Progress on the Design and Fabrication of the MICE Focusing Magnets

The Muon Ionization Cooling Experiment (MICE) focusing solenoid magnets focus the muon beam within the MICE cooling channel on a liquid or solid absorber that is within the warm bore of solenoid. The focusing magnet has a warm bore of 470 mm. This magnet consists of two coils 210-mm long that is separated by an aluminum mandrel that is 200 mm long. Each of the coils has its own leads. The coils may be operated in either the non-flip mode (solenoid mode with both coils at the same polarity) or the flip mode (quadrupole focusing mode where both coils are at opposite polarity). This report describes the focusing solenoid magnet design that will be built by the vendor. The progress on the construction of the first of the focusing magnets will also be discussed in this report. Ultimately three of these magnets will be built. These magnets will be cooled using a pair 1.5 W (at 4.2 K) pulse tube coolers.

Determination of electrodynamic parameters of the period of an accelerating structure with spatially periodic radio-frequency quadrupole focusing

The abandonment of a constant radio-frequency voltage across acceleration periods of the structure with a spatially periodic radio-frequency quadru pole focusing substantially expands the applicability range but complicates the tuning process of dynamic parameters. The possibility and necessity of applying the calculation is shown, when the accelerating structure is tuned. A procedure for calculating electrodynamic parameters of the acceleration period is proposed. The theoretical and experimental results are compared.

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.

Generalized phase-space tomography for intense beams

Tomographic phase-space mapping in an intense particle beam is reviewed. The diagnostic is extended to beams with space-charge by assuming linear forces and is implemented using either solenoidal or quadrupole focusing lattices. The technique is benchmarked against self-consistent simulation and against a direct experimental sampling of phase-space using a pinhole scan. It is demonstrated that tomography can work for time-resolved phase-space mapping and slice emittance measurement. The technique is applied to a series of proof-of-principle tests conducted at the University of Maryland.

Adiabatic formation of a matched-beam distribution for an alternating-gradient quadrupole lattice

The formation of a quasiequilibrium beam distribution matched to an alternating-gradient quadrupole focusing lattice by means of the adiabatic turn-on of the oscillating focusing field is studied numerically using particle-in-cell simulations. Quiescent beam propagation over several hundred lattice periods is demonstrated for a broad range of beam intensities and vacuum phase advances describing the strength of the oscillating focusing field. Properties of the matched-beam distribution are investigated. In particular, self-similar evolution of the beam density profile is observed over a wide range of system parameters. The numerical simulations are performed using the WARP particle-in-cell code.