  In coupled space-division multiplexing (SDM) transmission systems,
imperfections in optical amplifiers and passive devices introduce
mode-dependent loss (MDL) and gain (MDG). These effects render the channel
capacity stochastic and result in a decrease in average capacity. Several
previous studies employ multi-section simulations to model the capacity of
these systems. Additionally, relevant works derive analytically the capacity
distribution for a single-mode system with polarization-dependent gain and loss
(mode count D = 2). However, to the best of our knowledge, analytic expressions
of the capacity distribution for systems with D > 2 have not been presented. In
this paper, we provide analytic expressions for the capacity of optical systems
with arbitrary mode counts. The expressions rely on Gaussian approximations for
the per-mode capacity distributions and for the overall capacity distribution,
as well as on fitting parameters for the capacity cross-correlation among
different modes. Compared to simulations, the derived analytical expressions
exhibit a suitable level of accuracy across a wide range of practical
scenarios.
