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 J{\Sigma}. Canonical modal families are introduced to represent the underlying beam structure, and finite-emittance beam distributions are synthesized by statistical broadening 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. The proposed framework provides a unified description of beam geometry, eigen-emittance, and canonical angular momentum, and enables arbitrary beam distributions to be interpreted in terms of dominant canonical modes. Beyond providing a canonical design representation of three-dimensional spiral injection, the proposed framework establishes a direct connection between canonical beam dynamics and experimentally realizable injection beams, thereby providing a theoretical basis for beam synthesis and high-efficiency injection design. This framework enables the systematic representation, synthesis, and evaluation of finite-emittance spiral injection beams in canonical modal space.