Analytical approach to chromatic correction in the final focus system of circular colliders

A conventional final focus system in particle accelerators is systematically analyzed. We find simple relations between the parameters of two focus modules in the final telescope. Using the relations, we derive the chromatic Courant-Snyder parameters for the telescope. The parameters are scaled approximately according to $(L^{*}/{\beta }_y^{*})\delta$, where $L^{*}$ is the distance from the interaction point to the first quadrupole, ${\beta }_y^{*}$ the vertical beta function at the interaction point, and $\delta$ the relative momentum deviation. Most importantly, we show how to compensate its chromaticity order by order in $\delta$ by a traditional correction module flanked by an asymmetric pair of harmonic multipoles. The method enables a circular Higgs collider with 2% momentum aperture and illuminates a path forward to 4% in the future.

Figure 1

Focusing points and geometrical parameters in a conventional optical system.

Figure 2

A schematic layout of a doublet.

Figure 3

A schematic layout of a telescope system that consists of two focusing modules.

Figure 4

An element in the beam line represented by its transfer map of the canonical coordinates $z=(x,p_x,y,p_y,\delta ,\ell )$ from position $s_1$ to $s_2$.

Figure 5

The Courant-Synder parameters in a final telescope with demagnification factors $M$: 10 and 25 in the horizontal and vertical planes respectively. The parameters are $L^{*}=2\mathrm{m}$, $L=40\mathrm{m}$, $L_2=1\mathrm{m}$, ${\beta }_x^{*}=0.1\mathrm{m}$, and ${\beta }_y^{*}=1\mathrm{mm}$.

Figure 6

(Left) The vertical phase advance in unit of $2\pi$, namely ${\nu }_y={\psi }_y/2\pi$ and (right) the Sands’ mismatch parameter $\mathrm{\Upsilon }$ as a function of $\delta$.

Figure 7

A schematic layout of a chromatic correction module in the vertical plane. The focusing length $f=L_c/2\sqrt{2}$ for 90° cells.

Figure 8

Lattice functions in a final focusing system with local chromatic compensation section. The parameters in the chromatic correction module are $L_c=30\mathrm{m}$ and $\varphi =\pi /500$.

Figure 9

W function and the second-order dispersion in the final focusing system with the local chromatic compensation.

Figure 10

Chromatic bandwidth of the final focus system with different order of corrections shown on (left) the vertical phase advance in unit of $2\pi$ and (right) the Sands’ mismatch parameter $\mathrm{\Upsilon }$.

Figure 11

Improved chromatic bandwidth of the final focus system with higher order of corrections shown on (left) the vertical phase advance in unit of $2\pi$ and (right) the Sands’ mismatch parameter $\mathrm{\Upsilon }$.

Figure 12

Lattice functions in a final focusing system with local chromatic compensation section.

Figure 13

W function and the second-order dispersion in the final focusing system with the local chromatic compensation.

Figure 14

The second derivative of the Sands’ mismatch parameter $\mathrm{\Upsilon }$ as a function of $s$ in the horizontal (left) and vertical (right) planes.

Figure 15

Dynamic aperture of the collider with various momentum deviations.