  We investigate the role of weak-field homodyne (WF) measurement for quantum
communications over a lossy bosonic channel with coherent state encoding. This
kind of receiver employs photon-number resolving (PNR) detectors with finite
resolution and low-intensity local oscillator. As a figure of merit, we
consider the mutual information for a Gaussian input modulation. We prove an
enhancement over Shannon capacity in the photon starved regime, obtained by
exploiting information on the mean signal energy to suitably optimize the local
oscillator intensity. Thereafter, we investigate the performance of
non-Gaussian modulation, by considering a Gamma distribution of the energy of
the encoded pulses, and achieve an increase in the information rate with
respect to the Gaussian modulation case in the intermediate energy regime,
being more accentuated for low values of the PNR resolution.
