  This paper presents design methods for highly efficient optimisation of
geometrically shaped constellations to maximise data throughput in optical
communications. It describes methods to analytically calculate the
information-theoretical loss and the gradient of this loss as a function of the
input constellation shape. The gradients of the \ac{MI} and \ac{GMI} are
critical to the optimisation of geometrically-shaped constellations. It
presents the analytical derivative of the achievable information rate metrics
with respect to the input constellation. The proposed method allows for
improved design of higher cardinality and higher-dimensional constellations for
optimising both linear and nonlinear fibre transmission throughput.
Near-capacity achieving constellations with up to 8192 points for both 2 and 4
dimensions, with generalised mutual information (GMI) within 0.06 bit/2Dsymbol
of additive white Gaussian noise channel (AWGN) capacity, are presented.
Additionally, a design algorithm reducing the design computation time from days
to minutes is introduced, allowing the presentation of optimised constellations
for both linear AWGN and nonlinear fibre channels for a wide range of
signal-to-noise ratios.
