Beam-based alignment of quadrupoles by variation of quadrupole strength is a widely used technique in accelerators today. We describe the dominant systematic limitation of such algorithms, which arises from the change in the center position of the quadrupole as the strength is varied, and derive expressions for the resulting error. In addition, we derive an expression for the statistical resolution of such techniques in a periodic transport line, given knowledge of the line's transport matrices, the resolution of the beam position monitor system, and the details of the strength variation procedure. These results are applied to the Next Linear Collider main linear accelerator, an 11 km accelerator containing 750 quadrupoles and 5 000 accelerator structures. We find that, in principle, a statistical resolution of $1\mu \mathrm{m}$ is easily achievable, but the systematic error due to variation of the magnetic centers could be several times larger.

The present paper takes one step beyond the single-resonance theory for betatron motion by summing all the members of a given resonance family and expressing the joint influence in a single driving term. As a demonstration and confirmation of this work, the family driving terms are used to derive the classic closed-orbit and betatron-modulation equations of Courant and Snyder. A more serious demonstration is made by applying the family driving terms to the compensation of linear coupling and showing how numerical matrix-based and resonance compensation schemes are related. In a final phase, the Hénon map is used to compare the efficiency of different coupling compensation schemes with respect to dynamic aperture.

We point out that, the way it is reported, the solution to the equation of motion of a relativistic electron in the field of two electromagnetic waves advanced recently by Amatuni and Pogorelsky [Phys. Rev. ST Accel. Beams 1, 034001 (1998)] does not handle the case of two copropagating waves differing in frequency. An equivalent form for that solution is rigorously developed.

Response is made to Y. I. Salamin's preceding Comment [Phys. Rev. ST Accel. Beams 3, 059001 (2000)]. We confirm the areas of applicability of the original and Salamin's general solutions and discuss new applications of the developed formalism.

In the 2.5-GeV Photon Factory storage ring at KEK, we have found that the beam lifetime can be improved by modulating the phase of an rf accelerating voltage at a frequency of 2 times the synchrotron oscillation frequency. By applying this phase modulation with a peak-to-peak amplitude of 3.2°, the beam lifetime could be improved, typically, from 22 to 36 h under a beam current of about 360 mA. At the same time, the longitudinal coupled-bunch instability could be considerably suppressed. The improvement in the beam lifetime can be explained as an improved Touschek lifetime, which was caused by a quadrupole-mode longitudinal oscillation of the stored bunches.

The transverse coherent motion of two colliding proton beams at the Large Hadron Collider is studied by multiparticle tracking. We use the beam-beam force for a Gaussian beam distribution with variable barycenters and rms beam sizes, and optionally include the effect of long-range collisions and external impedance. The simulation yields the coherent and incoherent oscillation frequencies, the emittance growth of either beam, and evidence for the existence or lack of Landau damping. For head-on collisions of beams with equal sizes, we find that the $\pi$-mode frequency lies outside of the continuum frequency spread, if the ratio of the beam-beam parameters exceeds 0.6, in accordance with predictions. For smaller ratios of the beam-beam parameters, or if, for equal beam-beam parameters, the beam sizes are widely different, the $\pi$ mode is Landau damped. When long-range collisions are also included, undamped coherent modes do still exist outside the continuum, both with and without alternating crossing planes at two interaction points. However, separating the tunes of the two beams restores the Landau damping, provided the external impedance is sufficiently small.

We recently studied the spin flipping of a 202.7 MeV vertically polarized proton beam stored in the Indiana University Cyclotron Facility cooler ring during the first polarized run with its new cooler injector synchrotron and its new cooler injector polarized ion source. We first set the vertical betatron tune to avoid the measured ${\nu }_y$ value of the $G\gamma =7-{\nu }_y$ intrinsic depolarizing resonance in the cooler ring. We then flipped the spin by ramping the frequency of an rf dipole through an rf-induced depolarizing resonance. After optimizing the rf dipole's frequency ramp parameters, we used multiple spin flips to measure a maximum spin-flip efficiency of $97.5\pm 1\%$.

The design of a detuned and damped accelerating structure implies variations in the geometry which induce, in turn, a variation of the group velocity $v_g$ and the impedance per unit length $R^{\prime }$, divided by the quality factor $Q$. The resulting differential equation for the longitudinal electric field (fundamental mode) contains coefficients that depend on the distance $z$ along the structure. This report describes a possible method to solve this nonlinear, first-order differential equation analytically and how to obtain approximate closed algebraic forms, by using the sequence of Gauss integration methods. Analytical expressions of the longitudinal field profile in a loaded or unloaded accelerating section are deduced for both linear and arbitrary variations of $v_g$ and $R^{\prime }/Q$. Simple relations between the average field $〈E〉$ and the field at the entrance of the structure $E\left(0\right)$ make it possible to provide the dependence of the field function $E\left(z\right)$ on the design value for $〈E〉$ and on the structure parameters. The results are in good agreement with the direct numerical integration. Applications are presented for particular structure designs.

Magnetic dispersion followed by off-phase rf acceleration and an energy spectrometer were used to measure the longitudinal density profile of bunched relativistic electron beams with a resolution approaching 100 fs. Various electron bunch shapes were measured by this method and compared to bunch length measurements derived from the spectra of coherent transition radiation produced as the bunches traversed a thin foil. A reasonable agreement was found between the two bunch length methods. The off-phase rf acceleration method was then used to observe the modulated profile of a microbunched electron beam at the exit of a far-infrared free electron laser oscillator. In a preliminary study of the relationship between optical cavity power and the microbunch distribution, the density modulation and the energy spread of the electron beam were seen to increase as the optical field strength increased.

We use the linearized Vlasov-Poisson equations to study the response of a Kapchinskij-Vladimirskij beam to magnetic multipole errors in a circular lattice. This work extends the calculation of Gluckstern [Proceedings of the Linac Conference, 1970 (Fermilab, Batavia, IL, 1970), p. 811] to the case of nonideal periodic lattices. The smooth approximation is assumed. We determine the resonance conditions as well as the amplitude of the excited collective modes as a function of the error size outside the stopbands. We find that the frequencies associated with lattice resonances are a subset of the beam natural eigenfrequencies. The result is used to study the motion of test particles crossing the boundary of the beam core. Close to resonance the model predicts the emergence of a halo if sufficiently large gradient errors are present. Application is made to the University of Maryland Electron Ring.

The Vlasov equation embodies the smooth field approximation of the self-consistent equation of motion for charged particle beams. This framework is fundamentally altered if we include the fluctuating forces that originate from the actual charge granularity. We thereby perform the transition from a reversible description to a statistical mechanics description covering also the irreversible aspects of beam dynamics. Taking into account contributions from fluctuating forces is mandatory if we want to describe effects such as intrabeam scattering or temperature balancing within beams. Furthermore, the appearance of “discreteness errors” in computer simulations of beams can be modeled as “exact” beam dynamics that are being modified by fluctuating “error forces.” It will be shown that the related emittance increase depends on two distinct quantities: the magnitude of the fluctuating forces embodied in a friction coefficient, $\gamma$, and the correlation time dependent average temperature anisotropy. These analytical results are verified by various computer simulations.

An advanced facility for the production of nuclei far from stability could be based on a high-power driver accelerator providing ion beams over the full mass range from protons to uranium. A beam power of several hundred kilowatts is highly desirable for this application. At present, however, the beam power available for the heavier ions would be limited by ion source capabilities. A simple and cost-effective method to enhance the available beam current would be to accelerate multiple charge states through a superconducting ion linac. This paper presents results of numerical simulation of multiple charge state beams through a 1.3 GeV ion linac, the design of which is based on current state-of-the-art superconducting elements. The dynamics of multiple charge state beams are detailed, including the effects of possible errors in rf field parameters and misalignments of transverse focusing elements. The results indicate that operation with multiple charge state beams is not only feasible but straightforward and can increase the beam current by a factor of 3 or more.

When a gap in the electron bunch train prevents the trapping of ions, a transverse electron-ion instability may result from the ions created and lost during a single passage of the bunch train. A spread-frequency model is used to study this instability when the ions have a broad distribution of natural oscillation frequencies about the center of the electron beam. A growing disturbance saturates from Balakin-Novokhatsky-Smirnov damping at approximately the same time, and with the same total growth, as in the case without an ion frequency spread. At the tail of the bunch train, an unstable disturbance is amplified by a factor $\sim \exp \left({\omega }_i{\tau }_b\right)$ before saturation occurs, where ${\omega }_i$ is a typical ion oscillation frequency and ${\tau }_b$ is the duration of the bunch train. Initially, the instability displays exponential growth in time, unlike the case where the ion-frequency spread is neglected. For a broad distribution of ion frequencies, instability may be prevented by a betatron damping rate that exceeds the incoherent betatron frequency shift induced by ions at the tail of the bunch train.

Studies of multipactor resonances in coaxial and hollow waveguide via standard particle trajectory and secondary emission numerical simulation techniques are presented. Agreement between simulations and experimentally observed multipactor barriers gives confidence in results of simulation studies of a wedge-shaped hollow waveguide geometry that shows promise of being far less susceptible to multipactor resonances than standard coax and hollow rectangular waveguides.

Transverse beam profiles are observed to broaden with increasing intensity in the Proton Storage Ring at the Los Alamos Neutron Scattering Center. Measured profiles are simulated with an ${\mathrm{H}}^{-}$ injection model that includes a 2D particle-in-cell space charge calculation. Inclusion of space charge effects in the simulation improves the agreement between the experimentally observed profiles and the calculated profiles. The comparisons are made for a range of injected intensities.

Measurements of the beam's bunch length in the Cornell Electron-Positron Storage Ring (CESR) have been made using a streak camera. The streak camera uses visible synchrotron radiation produced by the beam to measure its longitudinal distribution. A description of CESR, the experimental setup, the streak camera used, and systematic errors and analysis techniques of the streak camera are described in this paper. The dependence of the bunch distribution on the current and accelerating rf voltage for a single bunch CESR was measured and compared with a theoretical model of CESR. The CESR vacuum chamber impedance is determined from the measured bunch distributions and is presented in this paper.

The nature of off-axis undulator radiation is discussed. Of particular interest is coherent off-axis radiation, where the wavelengths of emission are longer than the electron bunch length. We show how this off-axis radiation may be used to measure relative electron bunch lengths. The theory is presented, and calculated spectra are presented in a number of cases of interest.

Three dimensional test particle simulations are applied to optimization of the plasma-channeled laser wakefield accelerator (LWFA) operating in a weakly nonlinear regime. Electron beam energy spread, emittance, and luminosity depend upon the proportion of the electron bunch size to the plasma wavelength. This proportion tends to improve with the laser wavelength increase. We simulate a prospective two-stage $\sim 1\mathrm{GeV}$ LWFA with controlled energy spread and emittance. The input parameters correspond to realistic capabilities of the BNL Accelerator Test Facility that features a picosecond-terawatt ${\mathrm{CO}}_2$ laser and a high-brightness electron gun.

We have modified the beam-beam simulation code CBI to handle asymmetric beams and used it to look at energy transparency and symmetries in the beam-beam interaction. We find that even a small violation of energy transparency, or of the symmetry between the two beams, changes the character of the collective (coherent) motion; in particular, period-$n$ oscillations are no longer seen. We speculate that the one-time observation of these oscillations at LEP, and the more ubiquitous observation of the flip-flop instability in colliders around the world, may be a consequence of breaking the symmetry between the electron and positron beams. We also apply this code to the asymmetric collider PEP-II, and find that for the nominal parameters of PEP-II, in particular, the nominal tune-shift parameter of ${\xi }_0=0.03\mathrm{}$, there are no collective beam-beam issues. Collective quadrupole motion sets in only at ${\xi }_0=0.06\mathrm{}$ and above, consistent with earlier observations for symmetric beams.