Hierarchical mapping distribution matching for probabilistic shaping in coherent optical communication systems.

This Letter proposes a hierarchical mapping distribution matching (HMDM) scheme for probabilistic shaping (PS) systems. Using a lowest-energy first scheduling (LEFS) approach, the proposed scheme achieves superior energy efficiency over enumerative sphere shaping (ESS) without increasing complexity. A trellis construction approach is also proposed to support LEFS. Experimental and simulation results demonstrate that, under short blocklengths, HMDM approximates the Maxwell-Boltzmann distribution more closely than ESS, achieving superior performance over both ESS and uniform QAM. Specifically, HMDM offers SNR gains of up to 0.19 dB and 0.53 dB, respectively, along with an achievable information rate (AIR) gain of 0.2 bits/sym over uniform QAM in nonlinear channels.