  The multiple-input and multiple-output (MIMO) technique is regarded as a
promising approach to boost the throughput and reliability of optical fiber
communications. However, the fundamental limits of optical fiber MIMO systems
with finite block-length (FBL) are not available in the literature. This paper
studies the fundamental limits of optical fiber multicore/multimode systems in
the FBL regime when the coding rate is a perturbation within
$\mathcal{O}(\frac{1}{\sqrt{ML}})$ of the capacity, where M and L represent the
number of transmit channels and blocklength, respectively. Considering the
Jacobi MIMO channel, which was proposed to model the nearly lossless
propagation and the crosstalks in optical fiber systems, we derive the upper
and lower bounds for the optimal error probability. For that purpose, we first
set up the central limit theorem for the information density in the asymptotic
regime where the number of transmit, receive, available channels and the
blocklength go to infinity at the same pace. The result is then utilized to
derive the upper and lower bounds for the optimal average error probability
with the concerned rate. The derived theoretical results reveal interesting
physical insights for Jacobi MIMO channels with FBL. First, the derived bounds
for Jacobi channels degenerate to those for Rayleigh channels when the number
of available channels approaches infinity. Second, the high signal-to-noise
(SNR) approximation indicates that a larger number of available channels
results in a larger error probability. Numerical results validate the accuracy
of the theoretical results and show that the derived bounds are closer to the
performance of practical LDPC codes than outage probability.
