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Spatially and temporally resolved measurements of atomic hydrogen concentration above the dielectric of coplanar barrier discharge are presented for atmospheric pressure in 2.2% H2/Ar. The measurements were carried out in the afterglow phase by means of two-photon absorption laser-induced fluorescence (TALIF). The difficulties of employing the TALIF technique in close proximity to the dielectric surface wall were successfully addressed by taking measurements on a suitable convexly curved dielectric barrier, and by proper mathematical treatment of parasitic signals from laser–surface interactions. It was found that the maximum atomic hydrogen concentration is situated closest to the dielectric wall from which it gradually decays. The maximum absolute concentration was more than 10^22 m-3. In the afterglow phase, the concentration of atomic hydrogen above the dielectric surface stays constant for a considerable time (10 us - 1 ms), with longer times for areas situated farther from the dielectric surface. The existence of such a temporal plateau was explained by the presented 1D model: the recombination losses of atomic hydrogen farther from the dielectric surface are compensated by the diffusion of atomic hydrogen from regions close to the dielectric surface. The fact that a temporal plateau exists even closest to the dielectric surface suggests that the dielectric surface acts as a source of atomic hydrogen in the afterglow phase.