A State-Variable Approach to Submarine Links Capacity Optimization

We consider the capacity optimization of submarine links when including a realistic model of the constant-pump erbium doped fiber amplifiers (EDFA) with gain-shaping filters (GSF). While Perin et al. [1] numerically attacked this optimization for Constant-Signal (CS) amplified links, we extend the analysis also to constant power-spectral-density (CPSD) links, which mimic the way modern submarine links are gain-designed at cable assembly. Given the practical tolerances in GSF fabrication, the CS and CPSD approaches will be shown to essentially model the same link at large-enough pumps, but the CPSD approach yields a much simpler analysis. As in [1], we concentrate on a single spatial mode of a spatial division multiplexed (SDM) link at low EDFA pump power $P_{p}$, and thus consider only the impairments of amplified spontaneous emission noise. Here we adopt a novel semi-analytical approach which consists of fixing the inversion $x_{1}$ of the first EDFA (the state-variable of the link) and analytically finding capacity $C(x_{1})$ by searching over the $x_{1}$-feasible input wavelength division multiplexed (WDM) PSD distributions. Then the optimum inversion $x_{1}$ that maximizes $C(x_{1})$ is numerically obtained. This approach enables us to get both approximate (for CS links) and exact (for CPSD links) capacity-maximizing WDM input distributions, which vary inversely with the EDFA gain profile. For CS links the optimal WDM allocation is called the gain-shaped water-filling. Other practical allocations are analyzed, such as the signal to noise ratio equalizing allocation (CSNR), and the constant input power (CIP) allocation which uses a flat WDM distribution. We find that, for typical submarine span attenuations around 10 dB and when the link works at the optimal inversion $x_{1}$, CIP and CSNR achieve essentially the same capacity as the optimal allocation. At sufficiently large pump $P_{p}$ ($\gtrsim 30$ mW) the optimal inversion $x_{1}$ is such that the EDFA gain at 1538 nm equals the span attenuation, for EDFA emission and absorption as in [1]. When span attenuations increase to 20 dB, then we start seeing an advantage of the optimal allocation.