High gradient quadrupoles are necessary for different applications such as laser plasma acceleration, colliders, and diffraction limited light sources. Permanent magnet quadrupoles provide a higher field strength and compactness than conventional electro-magnets. An original design of permanent magnet based quadrupole (so-called QUAPEVA), composed of a Halbach ring placed in the center with a bore radius of 6 mm and surrounded by four permanent magnet cylinders capable of providing a gradient of 210 T/m, is presented. The design of the QUAPEVAs, including magnetic simulation modeling, and mechanical issues are reported. Magnetic measurements of seven systems of different lengths are presented and confirmed the theoretical expectations. The variation of the magnetic center while changing the gradient strength is +/- 10 micrometer. A triplet of three QUAPEVA magnets are used to focus a beam with large energy spread and high divergence that is generated by Laser Plasma Acceleration source for a free electron laser demonstration.
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