Theories of shaping for lattice codes have been developed for systems (optical or non-optical) using coherent detection with additive white Gaussian noise (AWGN) and for direct-detection optical systems with AWGN. This paper considers shaping for amplified direct-detection optical systems in which signal-spontaneous beat noise, a form of signal-dependent noise, is dominant. An N-dimensional (N-D) signal is formed by modulating the intensities (squares of field magnitudes) of a sequence of N time-disjoint pulses. In field magnitude coordinates, signal energy is represented by a L2 norm, and the optimal constellation bounding region is the nonnegative orthant bounded by an N-sphere. Under a continuous approximation, as N ,
Wei Mao, Joseph M. Kahn