  In this work, we present a new and general method for measuring the
astrophysical S-factor of nuclear reactions in laser-induced plasmas and we
apply it to d(d,n)$^{3}$He. The experiment was performed with the Texas
Petawatt laser, which delivered 150-270 fs pulses of energy ranging from 90 to
180 J to D$_{2}$ or CD$_{4}$ molecular clusters. After removing the background
noise, we used the measured time-of-flight data of energetic deuterium ions to
obtain their energy distribution. We derive the S-factor using the measured
energy distribution of the ions, the measured volume of the fusion plasma and
the measured fusion yields. This method is model-independent in the sense that
no assumption on the state of the system is required, but it requires an
accurate measurement of the ion energy distribution especially at high energies
and of the relevant fusion yields. In the d(d,n)$^{3}$He and
$^{3}$He(d,p)$^{4}$He cases discussed here, it is very important to apply the
background subtraction for the energetic ions and to measure the fusion yields
with high precision. While the available data on both ion distribution and
fusion yields allow us to determine with good precision the S-factor in the d+d
case (lower Gamow energies), for the d+$^3$He case the data are not precise
enough to obtain the S-factor using this method. Our results agree with other
experiments within the experimental error, even though smaller values of the
S-factor were obtained. This might be due to the plasma environment differing
from the beam target conditions in a conventional accelerator experiment.
