Covariance-Aware Phase-Space Matching and AC-Septum Trajectory Control for Hybrid Nonlinear-Kicker Injection

This work presents a covariance-aware extension of the hybrid nonlinear-kicker (NLK) injection framework. While previous baseline studies established centroid action reduction and hardware-aware NLK placement for idealized beams, the present study addresses finite-beam optimization using a realistic, position-dependent eight-wire kick profile. A realistic NLK field not only corrects the injected-beam centroid but also induces an amplitude-dependent shear across the internal phase-space ellipse. Depending on the injected covariance and ellipse orientation, this shear can either reduce or enhance beam filamentation. To exploit this mechanism, the AC-septum angle is treated as a trajectory-control knob for the injected centroid, while transfer-line matching is used to control the injected covariance matrix. We use ID2-downstream as the primary optics-favored case, as its high βN simultaneously increases nonlinear-field sampling and reduces the required correction kick. Furthermore, at this location, the injected beam is close to a matched, near-upright covariance condition (αbeam≈0). We also examine ID1 middle and ID1 downstream as trajectory-controllable alternatives, and formulate a combined optimization over septum angle, transfer-line optics, and NLK current. Operational measurements of AC-septum stability, first-turn trajectory reproducibility, septum response, and injection-bump response validate the available trajectory-control authority required for realistic NLK optimization. The analysis identifies a residual oscillation floor proportional to the square of the remaining centroid offset and inversely proportional to the local beta function, demonstrating that the physically relevant objective is minimization of the final mean action after the realistic NLK kick, rather than percentage action reduction alone. This framework establishes a practical optimization strategy under realistic aperture, stability, and hardware constraints. In a representative covariance-matched ID2D case, the realistic eight-wire NLK reduces both mean horizontal action (from 3.372 μm to 0.382 μm) and finite-beam spread contribution (from 0.475 μm to 0.039 μm). Operationally, first-turn rms trajectory variations remain under 0.4 mm, and the measured trajectory response of 19.4 mm/mrad closely matches the 19.8 mm/mrad model.