  Wavefront shaping techniques allow the control of the transport of light
through many types of scattering or complex media, among them multimode fibers.
The case of a multimode fiber which is also a gain medium presents further
intriguing prospects for control, due to the possibility of wavefront shaping
not only the signal light amplified by the medium, but also the pump light
absorbed within it. Provided that the lightwave used for pumping is coherent,
its shaping prior to injection will affect the complex, speckle-like spatial
patterns of excitation within the amplifier volume, which in turn will act upon
the signal as heterogeneous gain. We introduce a new theoretical model which
captures the essential features of a multimode amplifying fiber pumped by a
coherent beam with configurable modal content, allowing the calculation of the
gain-dependent transmission matrix. We numerically implement our model to
explore the extent to which one may control the amplified light in the
spatial-domain, by shaping the pump; we demonstrate a significant ability of
manipulation, as well as several interesting physical mechanisms limiting it.
In particular, we show how taking into account the cross-contributions between
all guided modes within the fiber is essential for understanding the full range
of the signal sensitivity to the choice of pump shaping.
