Optimum Constellation Size for Probabilistically Shaped Signals in the Presence of Laser Phase Noise

Higher cardinality modulation formats are generally adopted for probabilistically shaped (PS) signals to achieve the same entropy as uniformly shaped (US) signals. As a result, despite the shaping gain, PS signals are more sensitive to the laser phase noise since the phase margin is smaller than that of US signals. In this work, we investigate the optimum constellation size for PS signals by considering the effects from both shaping gain and laser phase noise. The constellation size we consider is not restricted to values given by
$2^k$
, where
$k$
is an integer. The non-
$2^k$
QAM can be generated by setting zero probability in outer constellation points of
$2^k$
QAM. Here, we compare the performances of US-64 QAM, PS-100 QAM, PS-144 QAM, PS-196 QAM and PS-256 QAM with the same information rate under the circumstance of different laser linewidths. The carrier phase recovery (CPR) is conducted by applying a modified two-stage pilot-symbols-aided carrier phase estimator. Consequently, we also study the CPR performance of the five modulation formats with different combinations of the block size in the first stage and the window size in the second stage. In our experimental demonstration, a coherent detection based optical back-to-back system is built with 100-kHz linewidth transmitter laser and local oscillator. For homodyne detection, PS-256 QAM performs best owing to the largest shaping gain. However, PS-144 QAM becomes the optimal choice for 200-kHz sum linewidth heterodyne detection benefiting from both the shaping gain and relatively large phase margin. In addition, numerical simulation has been conducted for investigating the effects of code rate and information rate on optimizing the constellation size.