A Unified Approach for Coupled Beam Optics in Accelerators

Abstract

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.

Figures (11)

• Figure 1: Schematic representation of a coupled linear symplectic transfer map $\mathcal{M}\in Sp(4)$ acting on the transverse phase space $\mathbb{R}^4=(x,p_x,y,p_y)$. For a stable map, the phase space admits a decomposition into two $\mathcal{M}$-invariant two-dimensional symplectic subspaces (normal-mode or eigenmode planes) $\mathcal{P}_1$ and $\mathcal{P}_2$, such that $\mathbb{R}^4=\mathcal{P}_1\oplus\mathcal{P}_2$ and $\mathcal{M}\mathcal{P}_k=\mathcal{P}_k$. Motion corresponds to an elliptic linear map (conjugate to a rotation) on each plane with phase advances $\mu_1$ and $\mu_2$.
• Figure 2: Derbenev's Adapter optics computation for creation of circular mode and comparison with different optics codes. Top left plot shows the coupled $\beta$ functions computed from MADX PTC, OptiMX, and our formalism. Top right plot shows the coupling phases $\nu_{1,2}$ computed from our formalism and OptiMX. Bottom left plot shows the signed symplectic area measures $A_{x,y}(W_{1,2})$ and Lebedev-Bogacz coupling strength parameter $u$. Bottom right plot shows the gauge invariant coupling measures for both eigenmode planes, $u_{1,2,inv}$.
• Figure 3: Ellipse Projections from the invariant planes $\mathcal{P}_1$ and $\mathcal{P}_2$ onto phase spaces for Derbenev's Adapter. Top plot is at initial location $s=0.0$ and bottom plot is at final location.
• Figure 4: Skew quadrupole triplet eigenmode optics computation and comparison with Lebedev-Bogacz formalism. Top left plot shows the coupled $\beta$ functions, top right plot shows the coupling phases, bottom left shows coupling measures $A_{x,y}(W_{1,2})$ and $u,1-u$ parameters, bottom right plot shows the invariant coupling fractions $u_{1,2,\mathrm{inv}}$.
• Figure 5: Solenoid and skew doublet cell. Top left plot shows the coupled $\beta$ functions, top right plot shows the coupling phases $\nu_{1,2}$, bottom left shows coupling emasures $A_{x,y}(W_{1,2})$ and $u$ from Lebedev--Bogacz parametrization, bottom right plot shows the gauge invariant coupling fractions $u_{1,2,\mathrm{inv}}$.