DOE OSTI2013
All existing focusing optics for accelerators are built to be "linear", i.e. the transverse focusing force is proportional to the particle displacement from the closed orbit. Such approach is used in order to make trajectories stable in the entire area of phase space, because the potential describing linear focusing is integrable and does not produce chaotic motion. Technically, the needed field is achieved with the help of dipole and quadrupole magnets. But the linear motion is unstable to perturbations in the focusing fields and excludes any betatron tune spread over particle's energy, which is beneficial for the beam stability. Moreover, the focusing strength depends on the particle's energy deviation from the designed one. These undesired effects are usually suppressed by the addition of non- linear elements, such as sextupoles and octupoles. Unfortunately, these additions make focusing potential nonintegrabe, leading to the limitation of stable phase space area. Recent research showed, that there exist non-linear potentials that, nevertheless, are fully integrable [1, 2]. Thus, there is an opportunity to create stable non-linear focusing optics that provides high betatron tune spread and mitigate magnetic field errors. This approach is to be implemented in electron storage ring called Integrable Optics Test Accelerator (IOTA) at Fermilab [3]. The main goal is to show the possibility to implement the nonlinear integrable system in a realistic accelerator design. Development of a magnet, which creates exact non-linear potential, is a technically complicated task, that takes much time and efforts. However, it is possible to expand the non- linear potential into superposition of multipoles of even orders (i.e quadrupole, octupole, etc.). Though this series is not convergent in the entire space, a few low order terms can be used to model the true potential in some limited area. In turn, each of the multipole potentials can be generated by speci ed electrical current in PCB coiled on the surface of storage ring's vacuum tube. Such approach is much cheaper than the creation of a magnet of solid iron with high current coil and faster than the development of \exact" non-linear magnet, but it can help identify some features of non-linear integrable optics application. The goal of this research work is to find analytically the angular distribution of electrical current on the surface of the vacuum tube, which creates inside it multipole potential with given parameters. Hereafter, there is the task to develop a model of flexible printed circuit board (PCB), which could be coiled around the vacuum tube in order to provide the desired current distribution.