Plasma wakefield acceleration, either driven by ultra-short laser pulses or electron bunches, represents one of the most promising techniques able to overcome the limits of conventional RF technology and allows the development of compact accelerators. In the particle beam-driven scenario, ultra-short bunches with tiny spot sizes are required to enhance the accelerating gradient and preserve the emittance and energy spread of the accelerated bunch. To achieve such tight transverse beam sizes, a focusing system with short focal length is mandatory. Here we discuss the development of a compact and tunable system consisting of three small-bore permanent-magnet quadrupoles with 520 T/m field gradient. The device has been designed in view of the plasma acceleration experiments planned at the SPARC_LAB test-facility. Being the field gradient fixed, the focusing is adjusted by tuning the relative position of the three magnets with nanometer resolution. Details about its magnetic design, beam-dynamics simulations, and preliminary results are examined in the paper.
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