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J. Seeman et al.
7.6 Focusing at Interaction Point
A. Seryi · R. Tomás García
7.6.1 Final Focus Design
The main task of a Final Focus (FF) system is to focus the beams to the small sizes
required at the interaction point (IP) of a Collider. To achieve this, the FF forms
a large and almost parallel beam at the entrance to the final doublet (FD), which
contains two or more strong quadrupole lenses. However, even for a beam with a
minor energy spread of a fraction of a percent, the focused beam size will be diluted
by the chromaticity of these strong lenses. The design of a FF is therefore driven
primarily by the necessity of compensating the chromaticity of the FD.
There are two primary approaches for chromaticity compensation—the non-local
scheme, implemented particularly at FFTB [74] and B-factories [75, 76] and the
local compensation Scheme [77] at ATF2 [78, 79]. Further developments in optics
with smaller vertical beam size and larger chromaticity are also being investigated
at ATF2 [80, 81] to explore the feasibility of the local compensation scheme at
different chromaticity levels.
In the non-local FF, the chromaticity is compensated in dedicated sections
by sextupole magnets placed at maxima of dispersion and beta-functions. The
geometric aberrations generated by the sextupoles are cancelled when used in pairs
with a minus identity transformation between them.
The non-local FF is built from separated optics blocks with strictly defined
functions, and its design and analysis is relatively simple. The major drawback of
the non-local FF rests in its required length for multi-TeV colliders. This can be
partly mitigated by allowing a smaller peak dispersion function and adding extra
sextupoles to the design [82, 83].
Local compensation of chromaticity is achieved by interleaving a pair of
sextupole magnets with the quadrupoles of the final doublet, see Fig. 7.8. The
dispersion throughout the FD is created by upstream bends, and is designed to
cancel at the IP. Geometric aberrations, generated by FD sextupoles, are cancelled
by two or more sextupoles located upstream. Sextupoles placed in FD generate
second order dispersion, which, however, can be compensated simultaneously with
x and y chromaticities provided that half of the total horizontal chromaticity of the
whole FF is generated upstream. The second order aberrations are cancelled when
the x and y pairs of sextupoles are separated by transfer matrices M with blockdiagonal structure {A 0; 0 B} where A = {f 0; c − 1/f}, provided the optics is flexible
enough to adjust the coefficients and provide compensation of third and fourth order
aberrations. The FF with local compensation requires fewer bends, and allows the
design of a 3 TeV CM FF system with about half a kilometre length. The recipe for
the design of such final focus is described in [84].
J. Seeman et al.
7.6 Focusing at Interaction Point
A. Seryi · R. Tomás García
7.6.1 Final Focus Design
The main task of a Final Focus (FF) system is to focus the beams to the small sizes
required at the interaction point (IP) of a Collider. To achieve this, the FF forms
a large and almost parallel beam at the entrance to the final doublet (FD), which
contains two or more strong quadrupole lenses. However, even for a beam with a
minor energy spread of a fraction of a percent, the focused beam size will be diluted
by the chromaticity of these strong lenses. The design of a FF is therefore driven
primarily by the necessity of compensating the chromaticity of the FD.
There are two primary approaches for chromaticity compensation—the non-local
scheme, implemented particularly at FFTB [74] and B-factories [75, 76] and the
local compensation Scheme [77] at ATF2 [78, 79]. Further developments in optics
with smaller vertical beam size and larger chromaticity are also being investigated
at ATF2 [80, 81] to explore the feasibility of the local compensation scheme at
different chromaticity levels.
In the non-local FF, the chromaticity is compensated in dedicated sections
by sextupole magnets placed at maxima of dispersion and beta-functions. The
geometric aberrations generated by the sextupoles are cancelled when used in pairs
with a minus identity transformation between them.
The non-local FF is built from separated optics blocks with strictly defined
functions, and its design and analysis is relatively simple. The major drawback of
the non-local FF rests in its required length for multi-TeV colliders. This can be
partly mitigated by allowing a smaller peak dispersion function and adding extra
sextupoles to the design [82, 83].
Local compensation of chromaticity is achieved by interleaving a pair of
sextupole magnets with the quadrupoles of the final doublet, see Fig. 7.8. The
dispersion throughout the FD is created by upstream bends, and is designed to
cancel at the IP. Geometric aberrations, generated by FD sextupoles, are cancelled
by two or more sextupoles located upstream. Sextupoles placed in FD generate
second order dispersion, which, however, can be compensated simultaneously with
x and y chromaticities provided that half of the total horizontal chromaticity of the
whole FF is generated upstream. The second order aberrations are cancelled when
the x and y pairs of sextupoles are separated by transfer matrices M with blockdiagonal structure {A 0; 0 B} where A = {f 0; c − 1/f}, provided the optics is flexible
enough to adjust the coefficients and provide compensation of third and fourth order
aberrations. The FF with local compensation requires fewer bends, and allows the
design of a 3 TeV CM FF system with about half a kilometre length. The recipe for
the design of such final focus is described in [84].