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A method for enhancing the brightness of an intense slow positron beam produced by an electron linear accelerator (LINAC) in order to obtain an intense positron microbeam was developed. The developed brightness enhancement system is simple and consists only of a few beam optics and a transmission remoderator. The slow positron beam produced by the LINAC is magnetically guided from the positron source to an experimental room. The beam is extracted from the magnetic field and is focused by a lens on the remoderator to enhance its brightness. The brightness-enhanced beam is extracted from the remoderator and focused on a sample by a lens. The beam size at the sample was 90 μm, which was two orders of magnitude smaller than that in the magnetic transport system that was about 10 mm. The efficiency of the transmission remoderator was about 5%. Adiabatic rules in the magnetic transport and the paraxial-ray equation were used to estimate the beam size that could be produced using this method. [ABSTRACT FROM AUTHOR]