We describe experimental, theoretical, and simulation activities testing Derbenev’s 1998 proposal for using flat-to-round-to-flat (FTRTF) transformations to enable electron synchrotrons for ion beam cooling. FTRTF systems have also been proposed for storage-ring and single-pass light sources (FELs), beam sources, and microwave tubes. The experiment—based on a low-energy (5–10 keV) linear electron transport system—includes an electron source, beam-shaping aperture plate, quadrupole matching section, Derbenev skew-quadrupole vortex sections, and a long solenoid. Our theoretical efforts explore the optical conditions required to optimize the canceling of angular momenta at the core of the Derbenev system. The complexity of the beam dynamics requires the use of simulation codes—here WARP and OPAL—to model the system for comparison with experiment. To reduce the computational effort required for optimization, we introduce the use of the adjoint technique, well-known in plasma physics but not beam physics. Using 5–10 keV beams allows us to study beam dynamics over a broad range of space charge in an environment readily accessible to students.

The design of accelerator lattices involves evaluating and optimizing Figures of Merit (FoMs) that characterize a beam's properties. These properties (hence the FoMs) depend on the many parameters that describe a lattice, including the strengths, locations, and possible misalignments of focusing elements. We have developed efficient algorithms to determine the multi-parameter dependence of an FoM, taking advantage of recent developments in adjoint techniques that facilitate the efficient computation of FoM derivatives with respect to the many parameters that describe a lattice. One algorithm applies to lattices and beams for which the paraxial approximation holds and particle motion is described as 4D in transverse phase space with distance along the beam path as the independent variable. Another algorithm—appropriate for implementation in a code such as OPAL—applies to beams in which particle trajectories are calculated in 6D phase space with time as the independent variable. We describe both the underlying adjoint theory and the numerical implementation of these algorithms.