Anomalous beam diffusion near beam-beam synchro-betatron resonances

The diffusion process near low order synchro-betatron resonances driven by beam-beam interactions at a crossing angle is investigated. Macroscopic observables such as beam emittance, lifetime and beam profiles are calculated. These are followed with detailed studies of microscopic quantities such as the evolution of the variance at several initial transverse amplitudes and single particle probability distribution functions. We present evidence to show that the observed diffusion is anomalous and the dynamics follows a non-Markovian continuous time random walk process. We derive a modified master equation to replace the Chapman-Kolmogorov equation in action-angle space and a fractional diffusion equation to describe the density evolution for this class of processes.Comment: 23 pages, 12 figure

A two-step method for retrieving the longitudinal profile of an electron bunch from its coherent radiation

The coherent radiation emitted by an electron bunch provides a diagnostic signal that can be used to estimate its longitudinal distribution. Commonly only the amplitude of the intensity spectrum can be measured and the associated phase must be calculated to obtain the bunch profile. Very recently an iterative method was proposed to retrieve this phase. However ambiguities associated with non-uniqueness of the solution are always present in the phase retrieval procedure. Here we present a method to overcome the ambiguity problem by first performing multiple independent runs of the phase retrieval procedure and then second, sorting the good solutions by mean of cross-correlation analysis. Results obtained with simulated bunches of various shapes and experimental measured spectra are presented, discussed and compared with the established Kramers-Kronig method. It is shown that even when the effect of the ambiguities is strong, as is the case for a double peak in the profile, the cross-correlation post-processing is able to filter out unwanted solutions. We show that, unlike the Kramers-Kronig method, the combined approach presented is able to faithfully reconstruct complicated bunch profiles.Comment: 22 pages, 5 figure

Diffusion measurement from observed transverse beam echoes

We study the measurement of transverse diffusion through beam echoes. We revisit earlier observations of echoes in RHIC and apply an updated theoretical model to these measurements. We consider three possible models for the diffusion coefficient and show that only one is consistent with measured echo amplitudes and pulse widths. This model allows us to parameterize the diffusion coefficients as functions of bunch charge. We demonstrate that echoes can be used to measure diffusion much quicker than present methods and could be useful to a variety of hadron synchrotrons.Comment: 25 pages, 10 figure

Nonlinear theory of transverse beam echoes

Transverse beam echoes can be excited with a single dipole kick followed by a single quadrupole kick. They have been used to measure diffusion in hadron beams and have other diagnostic capabilities. Here we develop theories of the transverse echo nonlinear in both the dipole and quadrupole kick strengths. The theories predict the maximum echo amplitudes and the optimum strength parameters. We find that the echo amplitude increases with smaller beam emittance and the asymptotic echo amplitude can exceed half the initial dipole kick amplitude. We show that multiple echoes can be observed provided the dipole kick is large enough. The spectrum of the echo pulse can be used to determine the nonlinear detuning parameter with small amplitude dipole kicks. Simulations are performed to check the theoretical predictions. In the useful ranges of dipole and quadrupole strengths, they are shown to be in reasonable agreement.Comment: 32 pages, 11 figure

Observations and Open Questions in Beam-Beam Interaction

A review article on the physics of beam-beam interactions in circular colliders.Comment: 20 pages, 8 figure

Density distributions of tune shifts from space charge or beam-beam interactions in Gaussian bunches

The amplitude dependent tune shifts from either space charge or beam-beam interactions are calculated analytically with the inclusion of synchrotron oscillations and multiple interactions around the ring. Simpler formulae are derived under limits of bunches longer than the transverse sizes, equal and unequal transverse sizes etc. This is used to derive semi-analytical forms for the density distribution of the tune shifts. The tune spread and the density distribution are needed to understand beam decoherence or Landau damping with either interaction. The tune footprints due to space charge in IOTA are simulated using pyorbit and found to be in good agreement with the theoretical predictions.Comment: 21 pages, 13 figure

Estimates of HE-LHC beam parameters at different injection energies

A future upgrade to the LHC envisions increasing the top energy to 16.5 TeV and upgrading the injectors. There are two proposals to replace the SPS as the injector to the LHC. One calls for a superconducting ring in the SPS tunnel while the other calls for an injector (LER) in the LHC tunnel. In both scenarios, the injection energy to the LHC will increase. In this note we look at some of the consequences of increased injection energy to the beam dynamics in the LHC

Optical diffraction radiation from a beam off a circular target

The use of optical diffraction radiation (ODR) as a diagnostic tool has increased in recent years. The potential of this technique has been demonstrated in several experiments at KEK [1], APS [2], FLASH [3] and possibly other facilities. These experiments were performed in extraction beam lines of lepton machines. However this technique can also be applied to high energy hadron beams. In this report we consider the ODR produced by such beams with the target as a round hole and apply the results to the Tevatron. This radiation is produced when a beam passes in the vicinity of a conducting target. The electro-magnetic fields due to the beam induce currents on the target and as the beam propagates, the currents change in time producing radiation both in the direction of beam propagation and along the direction of specular reflection from the target. This latter radiation, also termed backward diffraction radiation (BDR), is more useful for diagnostics since it can be directed out at the same longitudinal location as the target. This radiation is different from optical transition radiation (OTR) in which the beam passes through a metal target. Transition radiation is not suitable for continuous monitoring of a beam in a collider due to the beam energy loss and emittance growth and the fact that the target may be damaged. However the techniques for analyzing ODR are similar in many respects to those for OTR. Measurements of the radiation intensity either in the near field or far-field have been used to determine beam positions and sizes. For example, the beam size and beam position of a 1.28 GeV electron beam were measured in an extraction beam line at KEK [1] using the far-field angular distribution of the radiation. The near-field image was used to monitor the relative beam size of a 7 GeV electron beam in the extraction line at APS [2]. In principle, measurements of the beam divergence are also possible using the interference of ODR between two targets, as has been done with OTR. This paper is motivated by the desire to use this technique in colliders, especially for the LHC and possibly for future colliders envisaged such as the muon collider. A brief report on these prospects was presented earlier [4]. If the technique yields beam measurements with sufficient accuracy and reliability then the non-invasive nature would allow continuous monitoring during the length of a luminosity run. This would be valuable if the beam can be imaged close to the interaction points. Synchrotron radiation is already used as a non-invasive diagnostic tool in the Tevatron and will also be used in the LHC. The principal advantage of ODR is that it can be generated in a straight section and therefore used for imaging in an experimental insertion. The disadvantage is that the ODR flux is less copious than synchrotron radiation (OSR) and imaging will take longer than with OSR. In Section 2 we briefly discuss the parameters of different hadron colliders. In Section 3 we derive the basic results for the angular differential spectrum of ODR from a round hole due to a bunch. We apply these results in Section 4 to find the sensitivity of the spectrum to beam size and offset changes. In Section 5, we calculate the expected photon yield from a bunch per turn as a function of frequency and we use this to find the frequency range where a sufficiently strong ODR signal can be obtained. In Section 6 we do a brief comparison of the ODR spectrum with the OTR spectrum. We briefly list in Section 7 the experimental issues associated with measuring ODR when two beams are present. We end with our conclusions in Section 8. We will use CGS units throughout