  We analyze the amplification processes occurring in a nonlinear fiber, either
driven with one or two pumps. After determining the solution for the signal and
idler fields resulting from these amplification processes, we analyze the
physical transformations that these fields undergo. To this aim, we use a
Bloch-Messiah decomposition for the symplectic transformation governing the
fields evolution. Although conceptually equivalent to other works in this area
[McKinstrie and Karlsson, Opt. Expr. 21, 1374 (2013)], this analysis is
intended to be particularly simple, gathering results spread in the literature,
which is useful for guiding practical implementations. Furthermore, we present
a study of the correlations of the signal-idler fields at the amplifier output.
We show that these fields are correlated, study their correlations as a
function of the pump power, and stress the impact of these correlations on the
amplifier noise figure. Finally, we address the effect of losses. We determine
whether it is advantageous to consider a link consisting in an amplifying
non-linear fiber, followed by a standard fiber based lossy transmission line,
or whether the two elements should be reversed, by comparing the respective
noise figures.
