Betatron Oscillation Frequency 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.

Analytical and numerical analysis of longitudinally coupled transverse dynamics of Pulse Picking by Resonant Excitation in storage rings serving timing and high-flux users simultaneously

Abstract The pulse picking by resonant excitation (PPRE) method is applied at BESSY II to provide single bunch light to timing users while operating a multi-bunch filling pattern. This method can provide single bunch light to all beamlines simultaneously since the emittance of a selected bunch is increased by a quasi-resonant incoherent excitation close to the first synchrotron sideband of the betatron oscillation frequency. This gives high-flux to users by separating the synchrotron radiation from one horizontally enlarged bunch from the light of the multi-bunch beam. The properties of the excited bunch depend strongly on lattice parameters such as beta-functions and horizontal chromaticity, and the frequency and amplitude of the excitation signal. A theoretical interpretation by applying a linear transfer-matrix analysis based on the Liouville’s theorem is derived to illustrate a quantitative relation between the beam size and lattice parameters. In addition, measurements and numerical simulations show a monotonic increase of the timing bunch emittance as a function of the excitation amplitude at the first synchrotron sideband of the betatron oscillation frequency. The observed excitation-amplitude-dependent features of the transverse bunch size are confirmed by measurements with two independent diagnostics at BESSY II.

Bunch-by-bunch beam size measurement during injection at Shanghai Synchrotron Radiation Facility

Fast beam profile measurement is important in fast beam dynamic behavior investigations. A bunch-by-bunch beam size measurement system, which is presently used to measure horizontal profile, has been developed at the Shanghai Synchrotron Radiation Facility (SSRF) and is capable of measuring bunches within a separation of 2 ns. The system is based on a direct-imaging optical system and high-speed photomultiplier array detector. A high-bandwidth linear signal amplifier and acquisition module have also been designed to process bunch-by-bunch multi-channel signals from the detector. The software resampling technique and principal component analysis method were developed to obtain the synchronized data and enhance the signal-to-noise ratio. The fast injection of transients was successfully captured and analyzed. Moreover, the bunch-by-bunch positions and sizes exhibited strong oscillation after the injection at the horizontal betatron oscillation frequency of the SSRF storage ring, and this demonstrated the bunch-by-bunch measurement capability of our system.

Electromagnetic Waves Dispersion and Interaction of an Annular Beam-Ion Channel System in Plasma Waveguide

A linear theory for the electromagnetic properties and interactions of an annular beam-ion channel system in plasma waveguide is presented. The dispersion relations for two families of propagating modes, including the electrostatic and transverse magnetic modes, are derived. The dependencies of the dispersion behavior and interaction for different wave modes on the thickness of the annular beam and betatron oscillation frequency are studied in detail by numerical calculations. The results show that the inner and outer radii of the beam have different influences on propagation properties of the electrostatic and electromagnetic modes with different betatron oscillation parameters. In the weak ion channel situation, the two types of electrostatic waves, that is, space charge and betatron modes, have no interaction with the transverse magnetic modes. However, in the strong ion channel situation, the transverse magnetic modes will have two branches and a low frequency mode emerged as the new branch. In this case, compared with the solid beam case, the betatron modes not only can interact with the high frequency branch at small wavenumber but also can interact with the low frequency branch at large wavenumber.

Effects of Boundary Current on Electromagnetic Dispersion Characteristics for a Relativistic Electron Beam**supported by the National ITER Project Foundation of China (Nos. 2013GB106001 and 2013GB106003)

With betatron oscillation characteristics of the electron beam and ion channel effect taken into account, dispersion characteristics of electrostatic modes and TM modes for a relativistic electron beam guided by ion channel are studied. Dispersion relations are derived and solved numerically to investigate the dependence of the dispersion characteristics for electrostatic modes and TM modes on the betatron oscillation frequency and the ratio of the relativistic electron beam radius to the waveguide radius. The effects of the boundary current on the dispersion characteristic of the TM modes and the interaction between the betatron modes and TM modes are analyzed. When considering the boundary current, for a strong ion channel, a new low-frequency branch of the TM modes arises and the interaction frequency between the betatron modes and the TM01 modes is increased with the same parameters.

Betatron resonance crossing during the intense beam bunching

In the process of bunching an intense beam, the betatron oscillation frequency of particles changes due to a growth in their momentum rejection (in the presence of chromaticity) and changes in the Coulomb shift. These changes lead to the intersection of nonlinear betatron resonances. A mathematical model of the dynamics of ions in the process of bunching an intense beam has been built in order to assess the influence of transverse betatron resonances on the motion of particles. This model takes into account the influence of space charge on the longitudinal and transverse oscillations of particles in the presence of one- and two-dimensional betatron resonances. An analysis of the motion of particles near the one- and two-dimensional third-order resonances is given. Results of the numerical simulation of particle losses and emittance growth in the process of bunching an intense beam in the storage ring at the Institute for Theoretical and Experimental Physics (ITEP) are presented.

Fixed field alternating gradient accelerator with small orbit shift and tune excursion

A new design principle of a nonscaling fixed field alternating gradient accelerator is proposed. It is based on optics that produce approximate scaling properties. A large field index $k$ is chosen to squeeze the orbit shift as much as possible by setting the betatron oscillation frequency in the second stability region of Hill's equation. Then, the lattice magnets and their alignment are simplified. To simplify the magnets, we expand the field profile of ${r}^{k}$ into multipoles and keep only a few lower order terms. A rectangular-shaped magnet is assumed with lines of constant field parallel to the magnet axis. The lattice employs a triplet of rectangular magnets for focusing, which are parallel to one another to simplify alignment. These simplifications along with fringe fields introduce finite chromaticity and the fixed field alternating gradient accelerator is no longer a scaling one. However, the tune excursion of the whole ring can be within half an integer and we avoid the crossing of strong resonances.

Upgrade of the Slow-Extraction System in the U-70 Proton Accelerator

The purpose of this work was to increase the efficiency of slow extraction of accelerated protons from the U-70 accelerator in order to provide the extracted beam for a physical setup requiring high intensity. On account of the limited power of the first extraction setup – an electrostatic deflector, the angle spread of the beam near the deflector barrier had to be decreased in order to decrease the losses and increase the efficiency. For this, the structure function β was increased by inserting two additional quadrupole lenses into the magnetic structure of the accelerator. As a result of the inadequate strength of the system which corrects the frequencies of the betatron oscillations at the maximum accelerator energy 70 GeV, the beam was extracted at energy 64 GeV, where the frequency of the vertical betatron oscillations could be set above the line of the resonance 4Q z = 39. As a result, the losses at the first two extraction setups increased by a factor of 3 and the efficiency of the slow-ejection system increased from 85% to 95%.

Change in the Longitudinal Shape of Bunches in the U-70 Synchrotron by Noise RF-Gymnastics

The shape of bunches in the U-70 proton accelerator needs to be changed in order to decrease the peak beam density per unit length and to decrease the Coulomb shift of the frequency of betatron oscillations on the input plateau. It is proposed that longitudinal noise pumping, based on introduction of a controllable phase noise into the accelerating rf voltage, be used for this. In the spring (2002) session of U-70, an experimental check was made of the principle of noise gymnastics on the input plateau. Measurements were performed on the beam which confirmed the workability of the method. A technical description of a model of the noise-pumping system is given. The experimental data obtained in testing the method are presented and compared with computational results.

Theory of the Alternating-Gradient Synchrotron

The equations of motion of the particles in a synchrotron in which the field gradient indexn=−(r/B)∂B/∂rvaries along the equilibrium orbit are examined on the basis of the linear approximation. It is shown that if n alternates rapidly between large positive and large negative values, the stability of both radial and vertical oscillations can be greatly increased compared to conventional accelerators in which n is azimuthally constant and must lie between 0 and 1. Thus aperture requirements are reduced. For practical designs, the improvement is limited by the effects of constructional errors; these lead to resonance excitation of oscillations and consequent instability if 2νx or 2νz or νx+νz is integral, where νx and νz are the frequencies of horizontal and vertical betatron oscillations, measured in units of the frequency of revolution. The mechanism of phase stability is essentially the same as in a conventional synchrotron, but the radial amplitude of synchrotron oscillations is reduced substantially. Furthermore, at a “transition energy” E1≈νxMc2 the stable and unstable equilibrium phases exchange roles, necessitating a jump in the phase of the radiofrequency accelerating voltage. Calculations indicate that the manner in which this jump is performed is not very critical.

Emittance tailoring of electron beams for propagation in dense gas

An intense relativistic electron beam injected into dense gas characteristically propagates in a self-pinched mode but is susceptible to the resistive hose instability. This convective instability typically leads to large amplitude beam motion and the disruption of propagation. Theory and computation suggest that, although resistive hose cannot be completely suppressed, its convective growth can be reduced by varying the average betatron oscillation frequency from head to tail in the beam pulse. We report here on experiments designed to implement this variation by tailoring the beam emittance using an ion-focused regime ‘‘conditioning’’ cell. Conditioning effectiveness is assessed by using measured beam quantities to evaluate a detuning parameter η(t). This information is correlated with beam propagation measurements to determine the optimum conditioning for resistive hose suppression.

Three-dimensional free electron laser dispersion relation including betatron oscillations

Abstract We have developed a 3-D FEL theory based upon the Maxwell-Vlasov equations including the effects of the energy spread and emittance of the electron beam, and of betatron oscillations. The radiation field is expressed in terms of Green's function of the inhomogeneous wave equation and the distribution function of the electron beam. The distribution function is expanded in terms of a set of orthogonal functions determined by the unperturbed particle distribution. The coupled Maxwell-Vlasov equations are then reduced to a matrix equation, from which a dispersion relation for the eigenvalues is derived. In the limit of small betatron oscillation frequency, the present dispersion relation reduces to the well-known cubic equation of the one-dimensional theory in the limit of large beam size, and it gives the correct gain in the limit of small beam size. Comparisons of our numerical results with other approaches show good agreement. We present a handy empirical formula for the FEL gain of a 3-D Gaussian beam, as a function of the scaled parameters, that can be used for a quick estimate of the gain.

Multifrequency effects in an inverse FEL accelerator

The electron dynamics in the inverse FEL accelerator is investigated by a 1D FEL simulation code. The electron bunch length in acceleration buckets and the trapping fraction are found to vary when the gradient of wiggler field tapering is changed. When an infrared laser of 60 GW is focused to a 200 μm diameter laser spot, the maximum acceleration rate reaches 100 MeV/m and the bunch length is about 115 of the laser wavelength. As a realistic case study, FInite bandwidth of the laser field has to be taken into account. The critical bandwidth is found to be approximately a betatron oscillation frequency.

A Quadrupole Beam Transport Experiment for Heavy Ions under Extreme Space Charge Conditions

A Cs ion-beam transport experiment is in progress to study beam behavior under extreme space charge conditions. A five-lens section matches the beam into a periodic electrostatic quadrupole FODO channel and its behavior is found to agree with predictions. With the available parameters (?200 keV, ?20 mA, ??n ? 10-7 ? rad-m, up to 41 periods) the transverse (betatron) oscillation frequency (?) can be depressed down to one-tenth of its zero current value (?o), where ?2 = ?2o - ?2p/2, and ?p is the beam plasma frequency. The current can be controlled by adjustment of the gun and the emittance can be controlled independently by means of a set of charged grids.

Measurement and Compensation of Coherent Laslett Tune Shifts in the Fermilab Main Ring

The coherent Laslett tune shifts are studied by measuring the dependence on circulating intensity of the coherent betatron oscillation frequency at 8 GeV. The power supply control was modified to adjust the quadrupole currents so as to cancel beam induced tune changes. The article reports on running experience with this modification as well as on the data, its acquisition and reconciliation with the Laslett formulas. 3 refs.

Experimental Research of the Closed Orbit Expansion Effect in the Strong Focusing Cyclotron

The theoretical and experimental study of beam extraction method using the closed orbit expansion effect is described. The steep slope of the magnetic field variation magnitude is used to change the momentum compaction factor for the limit radial area. The orbit separation is found by computing the dynamical equations, since the behaviour of the betatron oscillation frequencies and the beam phase are investigated in this radial region. The experimental study of the effect is carried out with the ring cyclotron electron model, which is a strong focusing eight sector isochronous cyclotron. Calculated orbit separation in the extraction area is about 2+4 cm. The magnet system which is to obtain the proper gradient of the magnetic field variation and the magnet measurements results are described. The measured value of the orbit separation equals 4 cm. At the same time the space between two orbits without current (free of particles) is about 2 cm and the full current on the separated orbit is equal to that before separation. The theoretically predicted beam phase shift is found to be equal to 30+40°. The obtained results confirm the possibility of the 100% beam extraction from the accelerator with space magnetic field variation.

Extension of the CERN ISR Energy by Acceleration of Stacked Beams

The proton beams in the CERN Intersecting Storage Rings (ISR) have been accelerated to achieve colliding beams at 31.4 GeV/c, equivalent to a momentum of above 2000 GeV/c if a conventional accelerator with a stationary target were used. High density stacks were made at 26.5 GeV/c from which a part was rebunched with the maximum available RF voltage. The rebunched beam was kept near central orbit while the magnetic field was gradually increased to its maximum value. The major problem was to keep the betatron oscillation frequencies unchanged while changing the magnetic field thus avoiding any blow-up of the beam due to resonances. The corrections of the highly non-linear saturation effects of the main magnet were achieved by a computer controlled tuning of the field by excitation of the poleface windings and sextupole lenses. Closed orbit corrections were applied throughout the acceleration. Colliding beams with currents above 0.5 A were obtained.

Stimulation and Measurement of Radial Betatron Oscillations in the Bevatron Using the RF Accelerating System

Recent activity in resonant extraction of the circulating proton beam at the Bevatron spurred interest in accurately measuring the radial betatron oscillation frequency. Calculations indicated that it would be practical to use the existing RF system tuned to the betatron-oscillation frequency for excitation of the desired oscillations. Measurement of the proton rotational frequency and the betatron oscillation frequency has been carried out with the aid of a small digital data processor. The system we have used allows dynamic on-line presentation of beam radius, betatron frequency, and rotational frequency to precision of approximately one part in 10 000.

Theoretical and experimental beam studies for the Michigan state university cyclotron

Numerical orbit studies using Michigan State University cyclotron fields have been performed to determine the validity of use of the separated longitudinal equations of motion under conditions obtained in actual cyclotron operation. These results have been applied in a systematic study of phase measurement procedures using frequency and trim coil detuning. Direct experimental measurements were performed to determine the phase and phase width of the beam, as well as the betatron oscillation frequencies v r and v z , as a function of radius. From the phase history data the radial dependence of the rf dee voltage has been obtained, and the nominal initial phase ϕ 0 has been determined to be +20° ± 1° for the MSU central region geometry. Use of necessary computer calculations with this result gives an rf starting time τ 0 of −28° ± 1° for maximum beam intensity. Results of experimental measurements are compared with pre-computed orbit behavior where possible.

A method for measuring the frequency of radial betatron oscillations in a synchrotron

A method for measuring the frequency of radial betatron oscillations in a synchrotron