A Formalism for the Transport and Matching of Coupled Beams in Accelerators

TL;DR

This work develops an explicit, complete formalism for transporting and matching coupled beam optics, building on the Mais-Ripken and Lebedev-Bogacz parametrizations to treat both coupled and uncoupled lattices. It introduces generating vectors and a transfer-matrix framework that yields explicit expressions for coupled optics functions, eigenmode emittances, and phase advances, and it clarifies the relationship to Edwards-Teng parametrization while fixing gauge ambiguities. The authors demonstrate practical applications, including Derbenev's Adapter, mode flipping with skew quadrupoles, skew-FODO cells, solenoid-based cells, and fully coupled periodic rings, plus methods for diagnosing coupling and extracting eigenmode emittances from measurements. The formalism enables periodic matching, integration with existing codes (e.g., MAD-X, OptiMX), and potential future use in AI/ML-driven lattice design, with implications for advanced beam control and circular-mode beam concepts. Overall, the paper provides a rigorous, transferable toolkit for designing and diagnosing strongly coupled accelerator lattices and offers a pathway to novel operating regimes and beam configurations.

Abstract

Understanding transverse coupling dynamics is crucial for beam physics, accelerator design, and operations. Currently, most accelerators are designed for uncoupled beams, and coupling is treated as an error or perturbation. Many transverse ($x$-$y$) coupling parametrizations exist: Edward-Teng, Mais-Ripken, Levedev-Bogacz, and others. Here, we present an explicit and complete formalism for transporting coupled beam optics functions based on Mais-Ripken and Lebedev-Bogacz parametrizations. The formalism allows for matching generally coupled beam optics functions but applies to uncoupled optics as well. A complete transformation method for coupled optics provides easy matching routines that can be added to known beam optics codes that lack this feature. For fully coupled lattices, we present methods for extracting eigenmode emittances and other beam parameters from observables that can be measured, which is essential to diagnose and characterize the beam in a real machine. We express the linear difference resonance in terms of coupled optics functions and relate it to the coupling strength parameter with explicit emittance exchange formalism realized from the generating functions discussed here.

Figures (16)

• Figure 1: Creation of coupled optics functions for uncoupled periodic cells.
• Figure 2: Coupled beta functions in normal quadrupole triplets with: $u=0.1$ (top-left), $u=0.25$ (top-right), $u=0.4$ (bottom-left), $u=0.5$ (bottom-right).
• Figure 3: Derbenev Adapter Matching: Initial uncoupled beta functions, $\beta_{x,y}=5.0\mathrm{m}$ and $\alpha_{x,y}=0.0$ converts to coupled optics functions $\beta_{1x},\beta_{2x},\beta_{1y},\beta_{2y}=2.5\mathrm{m}$ and all $\alpha$ functions zero. Yellow elements represent skew quadrupoles. Coupled beta functions plot (left). Coupling strength parameter, $u$ (right).
• Figure 4: Mode flipping process: creation and flipping of modal beta functions (top-left). Coupling strength parameter $u$ (top-right). Initial flat beam with $\epsilon_{y}\ll\epsilon_{x}$ (bottom-left) flipped into flat beam with $\epsilon_{x}\ll\epsilon_{y}$ (bottom-right) from particle tracking simulation using the TRACK code.
• Figure 5: Optics of a FODO cell with a skew quadrupole component. Normal quadrupoles are in red, and the skew quadrupole is in yellow. Focusing quadrupole strength $k_{f}=3.043$, defocusing quadrupole $k_{d}=-3.043$, and skew quadrupole $k_{s}=0.01$.