Foundation of Three-Dimensional Spiral Beam Injection Using Canonical Angular Momentum and Symplectic Eigen-Modes
Aiming for high injection efficiency in three-dimensional spiral injection, the underlying physical principles governing beam formation and matching should be systematically organized within a unified canonical framework. However, a general theoretical framework explaining why particular beam distributions become naturally matched has not yet been established. In this work, a canonical description of three-dimensional spiral injection is developed based on the eigensystem of the symplectic covariance matrix $JΣ$. Canonical modal families are introduced to represent the underlying beam structure, and statistically broadened beam distributions are synthesized around the corresponding modal skeletons while preserving their canonical topology. Unlike conventional beam-matching methods based on Twiss parameters or eigen-emittance analysis, the proposed framework employs canonical symplectic modes as design variables for beam-family synthesis. It provides a unified description of beam geometry and canonical angular momentum in terms of canonical symplectic modes, and enables beam distributions to be interpreted in terms of their dominant modal structures. Beyond providing a canonical design representation of three-dimensional spiral injection, this approach establishes a direct connection between canonical beam dynamics and experimentally realizable injection beams, thereby providing a theoretical basis for systematic beam synthesis and injection-beam design. The framework further enables the systematic representation, synthesis, and evaluation of statistical distributions of spiral-injection beams in canonical modal space.