  Momentum correlation functions at small relative momenta are calculated for
light particles $\left(n, p, d, t\right)$ emitted from $^{197}$Au + $^{197}$Au
collisions at different impact parameters and beam energies within the
framework of the isospin-dependent quantum molecular dynamics model
complemented by the $Lednick\acute{y}$ and $Lyuboshitz$ analytical method. We
first make sure our model is able to reproduce the FOPI data of proton-proton
momentum correlation in a wide energy range from 0.4$A$ GeV to 1.5$A$ GeV. Then
we explore more physics insights through the emission times and momentum
correlations among different light particles. The specific emphasize is the
effects of total pair momentum among different light particles, impact
parameters and in-medium nucleon-nucleon cross section. Both two-deuteron and
two-triton correlation functions are anti-correlation due to the final state
interaction, and they are affected by in-medium nucleon-nucleon cross section
for the higher total momentum of the particle pairs, but not for the lower
ones. In addition, impact parameter and in-medium nucleon-nucleon cross section
dependences of the emission source radii are extracted by fitting the momentum
correlation functions. The results indicate that momentum correlation functions
gating with total pair momentum is stronger for the smaller in-medium
nucleon-nucleon cross section factor $\left(\eta\right)$ or impact parameter
$\left(b\right)$. Non-identical particle correlations ($np, pd, pt, $ and $dt$)
are also investigated by the velocity-gated correlation functions which can
give information of the particles' emission sequence, and the result indicates
that heavier ones $\left(deuteron/triton\right)$ are, one the average, emitted
earlier than protons, in the small relative momentum region.
