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The first investigation of the transverse emittance of a hot-cavity laser ion source based on all-solid-state Ti:sapphire lasers is presented. The emittances of (63)Cu ion beams generated by three-photon resonant ionization are measured and compared with that of the (69)Ga and (39)K ion beams resulting from surface ionization in the same ion source. A self-consistent unbiased elliptical exclusion method is adapted for noise reduction and emittance analysis. Typical values of the rms and 90% fractional emittances of the Cu ion beams at 20 keV energy are found to be about 2 and 8 pi mm mrad, respectively, for the ion currents of 2-40 nA investigated. The emittances of the laser-produced Cu ion beams are smaller than those of the surface-ionized Ga and K ion beams.