  In this paper, we propose a practical adaptive coding modulation scheme to
approach the capacity of free-space optical (FSO) channels with intensity
modulation/direct detection based on probabilistic shaping. The encoder
efficiently adapts the transmission rate to the signal-to-noise ratio,
accounting for the fading induced by the atmospheric turbulence. The
transponder can support an arbitrarily large number of transmission modes using
a low complexity channel encoder with a small set of supported rates. Hence, it
can provide a solution for FSO backhauling in terrestrial and satellite
communication systems to achieve higher spectral efficiency. We propose two
algorithms to determine the capacity and capacity-achieving distribution of the
scheme with unipolar M-ary pulse amplitude modulation (M-PAM) signaling. Then,
the signal constellation is probabilistically shaped according to the optimal
distribution, and the shaped signal is channel encoded by an efficient binary
forward error correction scheme. Extensive numerical results and simulations
are provided to evaluate the performance. The proposed scheme yields a rate
close to the tightest lower bound on the capacity of FSO channels. For
instance, the coded modulator operates within 0.2 dB from the M-PAM capacity,
and it outperforms uniform signaling with more than 1.7 dB, at a transmission
rate of 3 bits per channel use.
