Fig. 2: Measured Poincaré surface of section.

a–d, Four projections of a Poincaré surface of section. The blue markers are the scaled beam coordinates obtained experimentally from 3,000 passages through the selected longitudinal observation point. a, Projection of horizontal position and momentum expressed in scaled Courant–Snyder coordinates. The circular shape shows that the amplitude \({\bar{a}}_{x}\) is constant. b, Projection in the (\(\bar{y},{\bar{p}}_{y}\)) plane. The orbit is circular showing that \({\bar{a}}_{y}\) is constant. c,d, In the mixed coordinates, (\(\bar{x},\bar{y}\)) (c) and (\({\bar{p}}_{x},{\bar{p}}_{y}\)) (d), the motion of the beam follows a Lissajous curve. The spread of the markers yields direct information on the random error created by the four BPMs, with standard deviations σ_BPM,x ≈ 0.66 mm and σ_BPM,y ≈ 0.5 mm. The red line is the best fit of equation (1) to the experimental data, which confirms that the topology is consistent with that of a fixed line.