Effects of quadrupole fringe fields in final focus systems for linear colliders

Quadrupole fringe fields in the final focus system can be a source of aberrations in the interaction point transverse beam sizes. This paper investigates the fringe field impact on the transverse beam size in the ATF2, ILC, and CLIC lattices in the linear and non-linear regimes. The linear effects are studied by replacing the hard-edge quadrupolar field by the more realistic gradient fall-off. To address the nonlinear effects, the fringe fields are represented as high order kicks added to both sides of the hard-edge magnets. It will be shown that the linear fringe fields effects can be easily cured by tuning the quadrupole strengths. On the other hand, mitigation of the nonlinear fringe fields effects is more difficult and requires use of octupole magnets or, alternatively, increasing the value of interaction point horizontal beta function ${\beta }_x^{*}$.

Figure 1

Hard-edge and fringe fields models. Note the different longitudinal scale for each case, which depends on the aperture.

Figure 2

Linear fringe field impact on the horizontal (left) and vertical (right) beam size in the CLIC BDS lattice. The horizontal axis stands for the polynomial order of the transfer map.

Figure 3

Linear fringe field impact on the horizontal (left) and vertical (right) beam size in the ILC BDS lattice.

Figure 4

Linear fringe field impact on the horizontal (left) and vertical (right) beam size in the ATF2 Nominal lattice, together with a correction of the fringe field effect using the FD quadrupoles as reported in Table 3.

Figure 5

Linear fringe field impact on the horizontal (left) and vertical (right) beam size in the ATF2 Ultra Low ${\beta }^{*}$ lattice, together with a correction of the fringe field effect using the FD quadrupoles as reported in Table 3.

Figure 6

$\beta$ functions and dispersion along the FF line with the two octupoles locations under consideration.

Figure 7

Layout of ATF2 final focus line, with the two octupoles locations under consideration.

Figure 8

The vertical phase space distributions in IP for the case of the ultra low ${\beta }^{*}$ lattice. The upper left plot corresponds to the case without fringe fields and without octupoles. The upper right plot corresponds to the case with fringe fields but without octupoles. Both bottom plots correspond to the case with fringe fields and octupoles included.

Figure 9

Nonlinear fringe field impact on the horizontal (left) and vertical (right) beam size in the CLIC BDS lattice.

Figure 10

Nonlinear fringe field impact on the horizontal (left) and vertical (right) beam size in the ILC BDS lattice.

Figure 11

Nonlinear fringe field impact on the horizontal (left) and vertical (right) beam size in the ATF2 nominal lattice.

Figure 12

Nonlinear fringe field impact on the horizontal (left) and vertical (right) beam size in the ATF2 ultra low ${\beta }^{*}$ lattice.

Figure 13

Nonlinear fringe field impact on the horizontal (left) and vertical (right) beam size in the ATF2 $10{\beta }_{\mathrm{x}}$ lattice.