The results of the experiments at the installation “PICO,” with thin foils heating by laser radiation pulses of nanosecond duration are reported. The Al foils with thickness in the range from 3 μ up to 40 μ were used as targets. The flux density was varied from 1013 W/cm2 to 1014 W/cm2. The sharp dependence of the portion of laser energy that passed through the target on foil thickness was observed. This phenomena was accompanied by a relatively small decrease of the passed radiation pulse duration. The anomalously high speed burning through of thin foil was observed in these experiments and the conclusion on the possible mechanism of this phenomena has been done on the base of comparison of experimental data with theoretical calculations. The observed phenomena can be interpreted with a conjecture about the local burning through of a target, in the small areas of the target surface, with many more values of flux density than the average one following laser radiation self-focusing and formation of “hot spots.”