Beam Ellipse Research Articles

Beam dynamics framework incorporating acceleration to define the minimum aperture in two focusing schemes for proton radiotherapy linac

In this paper, a self-consistent transverse beam dynamics framework is demonstrated that incorporates acceleration into the transverse beam dynamics studies for a proton linac machine. Two focusing schemes are developed and discussed: the FODO-like scheme and the minimum aperture scheme. The FODO-like scheme is a simple scheme, requiring only one quadrupole per cavity. The scheme is analytically solved to minimize the beam size at the cavity entrance/exit and ensures a constant beam size along the lattice, with respect to adiabatic damping due to longitudinally accelerating rf cavities. The minimum aperture scheme describes the regime that matches the beam ellipse to the acceptance ellipse of a cavity, allowing for the smallest possible aperture, for a given cavity length. A simple approximation of an rf cavity map is determined to allow changes in particle energy along a lattice, and acceleration is assumed only in the longitudinal direction. Published by the American Physical Society 2024

A geometric-optical phase space formulation for paraxial light beam propagation

It is shown that the traditional classical wave picture of light beam propagation in paraxial approximation may be conveniently equivalent to phase representative points picture of ray’s-ensemble in geometric-optical phase space. In this description, the Kogelnik’s abcd law for light beam propagation in configuration space domain is replaced by the transformation equation of the beam ellipse in phase space domain, where the beam ellipse’s area plays the role of wavelength of light beam.

Generalized Courant–Snyder theory and Kapchinskij–Vladimirskij distribution for high-intensity beams in a coupled transverse focusing lattice

The Courant–Snyder (CS) theory and the Kapchinskij–Vladimirskij (KV) distribution for high-intensity beams in an uncoupled focusing lattice are generalized to the case of coupled transverse dynamics. The envelope function is generalized to an envelope matrix, and the envelope equation becomes a matrix envelope equation with matrix operations that are noncommutative. In an uncoupled lattice, the KV distribution function, first analyzed in 1959, is the only known exact solution of the nonlinear Vlasov-Maxwell equations for high-intensity beams including self-fields in a self-consistent manner. The KV solution is generalized to high-intensity beams in a coupled transverse lattice using the generalized CS invariant. This solution projects to a rotating, pulsating elliptical beam in transverse configuration space. The fully self-consistent solution reduces the nonlinear Vlasov-Maxwell equations to a nonlinear matrix ordinary differential equation for the envelope matrix, which determines the geometry of the pulsating and rotating beam ellipse. These results provide us with a new theoretical tool to investigate the dynamics of high-intensity beams in a coupled transverse lattice. A strongly coupled lattice, a so-called N-rolling lattice, is studied as an example. It is found that strong coupling does not deteriorate the beam quality. Instead, the coupling induces beam rotation and reduces beam pulsation.

Linear analysis of Hamilton perturbation theory for coupled betatron motion

The dynamics of coupled betatron motion in a circular accelerator has been studied by the Hamilton perturbation theory or the transfer matrix formalism separately. In order to incorporate the theories with each other, we develop the transfer matrix formulation on the Hamilton perturbation theory. With the help of the Hamilton perturbation theory, we can analytically solve the equation of the coupled betatron motion in the vicinity of a difference resonance. Using the solution, we can construct the transfer matrix for the coupled motion. The matrix is shown to be symplectic and then transformed to the block diagonal form which consists of two normal modes and the coupling matrix. Thus, we derive the analytical representation for the normal mode parametrization of the transfer matrix describing the coupled betatron motion. Using the formalism, we investigate the two-dimensional beam ellipse in an electron storage ring.

Cold-fluid theory of equilibrium and stability of a high-intensity periodically twisted ellipse-shaped charged-particle beam

It is shown that there exists an exact paraxial cold-fluid equilibrium of a high-intensity, space-charge-dominated charged-particle beam with a periodically twisted elliptic cross section in a nonaxisymmetric periodic magnetic field. Generalized envelope equations, which determine the beam envelopes, ellipse orientation, density, and internal flow velocity profiles, are derived. Nonrelativistic and relativistic examples of such beam equilibria are presented. The equilibrium and stability of such beams are demonstrated by self-consistent particle-in-cell (PIC) simulations.

Simulation of the ESR injection system

Abstract In order to improve the acceptance of the ESR beam injection system the influence of the fringe field of the main dipole magnet on the ion optics has to be analysed. Therefore, we perform a 3D MAFIA simulation of the dipole magnet and calculate the injection line which enters the radial fringe field in a tangential way. Starting from the given ESR beta function we trace the beam ellipse backwards in order to obtain the phase space requirements near the exit plane of the inflector magnet. Simulated phase space “monitors” defined at the start and the end of the curved reference trajectory give insight in the change of the beam emittance ellipse and indicate if aperture limitations may lead to beam losses. We obtain the second-order mapping of the phase space ellipse by calculating the first- and second-order field coefficients along the curved reference trajectory and solving the differential equations corresponding to each matrix element numerically. After an introduction into the field calculation method and the principles used to get the field index and second-order field coefficients, we will discuss our results and compare them with calculations based on measured field data (Spiller et al., GSI Scientific Report, 1996, p. 165).

Computer Control Strategies at SIN

The computers connected to the SIN control system have been programmed to be a third operator, performing primarily calculation intensive or repetitive functions. Written primarily in FORTRAN, programs exist which display the effects of trimcoils in both SIN accelerators on phase and betatron frequencies. Beamlines may be modeled by displaying beam envelopes and parameters given bending magnet and quadrupole settings and beam ellipse parameters anywhere in the beamline. Programs are used to perform a number of jobs in the normal tuning procedure. Selected subsets of device values from previous setups may be restored, using ramping functions if necessary. Horizontal and vertical waists may be formed using the quadrupole triplet and slits following extraction from the injector cyclotron. The beam may be centered at injection into the ring and in the proton channel from ring extraction to the beam dump. Proper focussing on target may be obtained and emittance measurements are possible there and in the injection channel. Finally, total operator (and program) performance is periodically measured by a Bonus number describing the quality of tune of the entire system.

Beam ellipse matching: Waist-to-waist transport

We drive the most general form of the beam transport transformation matrix that takes a given initial beam ellipse into a given final beam ellipse. The result is specialized to waist-to-waist transport and applied to the problem of drift-thin lens-drift. A large number of interpretative comments are made concerning the results of this analysis.