Confining continuous manipulations of accelerator beam-line optics

Altering the optics in one section of a linear accelerator beam line will in general cause an alteration of the optics in all downstream sections. In circular accelerators, changing the optical properties of any beam-line element will have an impact on the optical functions throughout the whole machine. In many cases, however, it is desirable to change the optics in a certain beam-line section without disturbing any other parts of the machine. Such a local optics manipulation can be achieved by adjusting a number of additional corrector magnets that restore the initial optics after the manipulated section. In that case, the effect of the manipulation is confined in the region between the manipulated and the correcting beam-line elements. Introducing a manipulation continuously, while the machine is operating, therefore requires continuous correction functions to be applied to the correcting quadrupole magnets. In this paper, we present an approach to calculate such continuous correction functions for six quadrupole magnets by means of a homotopy method. Besides a detailed derivation of the method, we present its application to an algebraic example, as well as its demonstration at the seeding experiment sFLASH at the free-electron laser FLASH located at DESY in Hamburg.

Figure 1

Schematic layout of the FLASH facility. The superconducting linear accelerator delivers electron bunches for the FLASH1 and FLASH2 undulator beam lines and for the seeding experiment sFLASH.

Figure 2

Schematic representation of the four-lens system in different configurations.

Figure 3

Plots of the correction functions for the initial configuration shown in Fig. 2 and $a=-2$.

Figure 4

Positions of the quadrupole magnets close to the sFLASH undulator. Note that the magnets in the SFUND section cannot be used as correctors. Magnets used in the example shown in Fig. 5 are highlighted in red. The drawing is not to scale.

Figure 5

Example of a correction function calculated using the presented method. Here, the strengths $k$ of the six selected quadrupole magnets (see legend) are used as correction parameters to compensate for the optics disturbance caused by the sFLASH undulator being closed from $K=0$ to a final value of $K=K_{\mathrm{f}}=2.63$ at a beam energy of 693 MeV.

Figure 6

Evolution of the $\beta$-functions in dependence on the longitudinal coordinate $s$. The undisturbed state (dotted) is compared to the disturbed state with (solid line) and without (dashed) application of the correction. Grey areas in the upper plot mark the position of the variable-gap undulator segments. In this example, a different set of correction magnets is used. Their positions are indicated in the lower plot. Note that at the end of the beam line the corrected optical functions agree with the open case.

Figure 7

Transverse beam sizes measured using the wire scanners between the six FLASH main undulator segments (grey boxes in the middle). The dotted curve shows the beam sizes at theory optics without any disturbance. Measurements of the disturbed case, where the seeding undulator is closed, are depicted by the dashed curve. The solid curve shows beam sizes with corrections applied. At the time of measurement, the wire scanner between the last two undulator segments was unavailable.