  We optimize modulation formats for the additive white Gaussian noise channel
with nonnegative input, also known as the intensity-modulated direct-detection
channel, with and without confining them to a lattice structure. Our
optimization criteria are the average electrical, average optical, and peak
power. The nonnegative constraint on the input to the channel is translated
into a conical constraint in signal space, and modulation formats are designed
by sphere packing inside this cone. Some dense packings are found, which yield
more power-efficient modulation formats than previously known. For example, at
a spectral efficiency of 1.5 bit/s/Hz, the modulation format optimized for
average electrical power has a 2.55 dB average electrical power gain over the
best known format to achieve a symbol error rate of 10^-6. The corresponding
gains for formats optimized for average and peak optical power are 1.35 and
1.72 dB, respectively. Using modulation formats optimized for peak power in
average-power limited systems results in a smaller power penalty than when
using formats optimized for average power in peak-power limited systems. We
also evaluate the modulation formats in terms of their mutual information to
predict their performance in the presence of capacity-achieving error-
correcting codes, and finally show numerically and analytically that the
optimal modulation formats for reliable transmission in the wideband regime
have only one nonzero point.
