Analytic formulas for the rapid evaluation of the orbit response matrix and chromatic functions from lattice parameters in circular accelerators

TL;DR

This paper introduces analytic formulas that significantly accelerate the evaluation of orbit response matrices and chromatic functions in circular accelerators, enhancing efficiency in design and correction processes.

Contribution

It extends the Lie algebra formalism to off-momentum regimes and derives explicit formulas for chromatic functions based on lattice parameters, improving speed and robustness.

Findings

Formulas greatly speed up orbit response matrix evaluation

Analytic expressions for chromatic functions are derived

Formulas are robust across different lattice configurations

Abstract

Measurements and analysis of orbit response matrix have been providing for decades a formidable tool in the detection of linear lattice imperfections and their correction. Basically all storage-ring-based synchrotron light sources across the world make routinely use of this technique in their daily operation, reaching in some cases a correction of linear optics down to 1% beta beating and 0.1% coupling. During the design phase of a new storage ring it is also applied in simulations for the evaluation of magnetic and mechanical tolerances. However, this technique is known for its intrinsic slowness compared to other methods based on turn-by-turn beam position data, both in the measurement and in the data analysis. In this paper analytic formulas are derived and discussed that shall greatly speed up this second part. The mathematical formalism based on the Lie algebra and the resonance driving terms is extended to the off-momentum regime and explicit analytic formulas for the evaluation of chromatic functions from lattice parameters are also derived. The robustness of these formulas, which are linear in the magnet strengths, is tested with different lattice configurations.