The focusing of particle beams for collider experiments is crucial for maximizing the luminosity and thus the discovery potential of these machines. In recent years, plasma wakefield acceleration has emerged as a leading candidate for achieving higher energy collisions with smaller facility footprints due to the large accelerating gradients in the plasma. This higher beam energy poses significant challenges for the final focusing system of the collider. Here, we discuss the various challenges of final focusing for TeV-scale plasma accelerators and propose possible solutions. Finally, we present the first design of a final focusing system for a 10 TeV linear wakefield collider, evaluate its performance, and discuss its shortcomings as well as improvements for future designs.
The role of the final focusing system (FFS) in a collider is to demagnify the beams to the desired spot sizes at the interaction point (IP) in order to maximize the luminosity L:
where σ x and σ y are the root-mean-square beam sizes in the x and y planes, respectively. For flat beam collisions with σ x ≫ σ y , this focusing can be achieved by arranging pairs of quadrupole magnets into a telescope, which provides point-to-point and parallel-to-parallel imaging [1].
Complications arise when considering beams with a finite energy spread δ, as particles with different energies are focused to different longitudinal positions near the IP. These chromatic aberrations, if left untreated, result in an increase of the beam size at the IP and degrade the luminosity of the collider. Furthermore, the strong focusing fields of the quadrupoles closest to the IP, referred to as the final doublet (FD), induce the emission of incoherent synchrotron radiation (SR), which further increases the energy spread of the beam, known as the Oide effect. Schemes for handling the chromatic effects have been demonstrated at the SLAC Linear Collider (SLC) [2] and Final Focus Test Beam (FFTB) [3] using dedicated chromatic correction sections. A more recent compact approach [4] relies on local chromaticity correction by utilizing sextupoles interleaved with the FD, decreasing the overall length of the FFS while providing larger FD bandwidth than the schemes with dedicated chromatic correction sections. First experimental demonstrations of this approach are being carried out at ATF2 [5][6][7][8]. In general, the FFS designs for future colliders need further experimental demonstrations in terms of operational reproducibility, and achieved bunch charge and IP parameters [9]. A dedicated comparison of FFS designs with the traditional and the compact approaches is presented in reference [10]. Plasma wakefield acceleration (PWFA) [11] has emerged as a leading candidate for achieving compact, multi-TeV scale colliders due to the high acceleration gradients with respect to state-of-the-art metallic superconducting radio-frequency (SRF) technology [12,13]. The drastic length reduction of the plasma linac may lead to a situation where the beam delivery system (BDS), which includes the FFS, becomes the largest contribution to the facility footprint. The scaling of the BDS system for multi-TeV colliders was examined in ref. [14]. In the FFS design with the local compact chromatic correction scheme, the length of the FFS scales approximately as [4]:
where E is the energy of the particles. Another scheme, arXiv:2602.15777v1 [physics.acc-ph] 17 Feb 2026
derived for scaling the LHC interaction region to FCC-hh beam energies under the assumption of a constant beam stay clear, utilizes a different scaling [15]:
A similar procedure was also utilized for scaling the CLIC 3 TeV FFS to the 7 TeV design [16,17]. The optimal scaling for the FFS length with energy is expected to fall somewhere between those shown in Eq. 2 and Eq. 3 by allowing some luminosity loss due to synchrotron radiation. As a result, higher energy colliders will require a longer (and thus more expensive) FFS. Compact, high-energy linear colliders must balance luminosity and length considerations for the FFS.
The desire for high intensity electron beams at future PWFA colliders also poses a problem for the achievable luminosity. Particles in colliding beams interact strongly with the field produced by the opposing beam, resulting in the emission of SR. This process, known as beamstrahlung, causes each beam to radiate away its energy and reduce the center-of-mass energy of the colliding particles, broadening the luminosity spectrum [18].
In the following sections we discuss each of these challenges in more detail and propose possible solutions. Finally, we present the first design of a FFS as part of the 10 TeV Wakefield Collider Design Study [19] and assess its performance in the context of a future PWFA collider design.
In this section, we review the physical processes that limit our ability to focus beams to arbitrarily small spot sizes. These processes include limitations of beam transport systems as well as dynamic radiative effects.
In practice, all particle beams have some finite energy spread, δ. The bending of an individual particle in a magnetic field depends on the energy of the particle itself, and, due to this, particles of different energies are focused differently in a quadrupole field. This is illustrated in Figure 1 for particles with an energy offset ∆p with respect to the nominal particle energy p 0 ; the particles are focused to different points along the longitudinal coordinate, which increases the spot size at the focal point for the reference particle energy (the IP).
Typically, the largest focusing gradients in the FFS are present in the final doublet (FD), which demagnify the beams at the IP to maximize the luminosity. As a result, the largest contribution to
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