  We investigate the target normal sheath acceleration of protons in thin
aluminum targets irradiated at relativistic intensity by two time-separated
ultrashort (35 fs) laser pulses. For identical laser pulses and target
thicknesses of 3 and 6 $\mu$m, we observe experimentally that the second pulse
boosts the maximum energy and charge of the proton beam produced by the first
pulse for time delays below $\sim0.6-1$ ps. By using two-dimensional
particle-in-cell simulations we examine the variation of the proton energy
spectra with respect to the time-delay between the two pulses. We demonstrate
that the expansion of the target front surface caused by the first pulse
significantly enhances the hot-electron generation by the second pulse arriving
after a few hundreds of fs time delay. This enhancement, however, does not
suffice to further accelerate the fastest protons driven by the first pulse
once three-dimensional quenching effects have set in. This implies a limit to
the maximum time delay that leads to proton energy enhancement, which we
theoretically determine.
