A Unified Approach for Coupled Beam Optics in Accelerators

Coupled beam optics can be geometrically described in terms of invariant eigenmode planes of a stable symplectic ``one-turn'' map $\mathcal M\in Sp(4)$. We show that the non-uniqueness of symplectically normalized bases within each eigenmode plane constitutes an in-plane gauge freedom $Sp(2)\times Sp(2)$, and that many coupled-optics parametrizations differ primarily by gauge choice. Building on this fact, we identify basis-independent descriptors of lattice and beam optics and introduce bounded, gauge-invariant coupling parameters or fractions $u_{k,\mathrm{inv}}$ computed from orthogonal projectors onto the eigenmode planes. To obtain smooth $s$-dependent optics functions and consistent mode labeling, we present a unifying and practical approach based on an $SO(2)$ continuity gauge (Procrustes alignment), together with diagnostics for stability and invariance. We further relate Edwards--Teng, Mais--Ripken, Lebedev--Bogacz, Wolski, and Sagan--Rubin parametrizations as gauge-equivalent representations within the respective $Sp(2)\times Sp(2)$ gauge freedom. Numerical examples of coupled lattices and beam optics illustrate the proposed invariants and show how representation-dependent scalar coupling parameters (e.g.\ in the Lebedev--Bogacz gauge) can leave their nominal bounds while $u_{k,\mathrm{inv}}$, defined here, remain bounded and physically interpretable.