A State-Variable Approach to Submarine Links Capacity Optimization

Abstract

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 <italic>et al.</italic> <xref ref-type= bibr rid= ref1 >[1]</xref> 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 <xref ref-type= bibr rid= ref1 >[1]</xref>, we concentrate on a single spatial mode of a spatial division multiplexed (SDM) link at low EDFA pump power <inline-formula><tex-math notation= LaTeX >$P_{p}$</tex-math></inline-formula>, 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 <inline-formula><tex-math notation= LaTeX >$x_{1}$</tex-math></inline-formula> of the first EDFA (the state-variable of the link) and analytically finding capacity <inline-formula><tex-math notation= LaTeX >$C(x_{1})$</tex-math></inline-formula> by searching over the <inline-formula><tex-math notation= LaTeX >$x_{1}$</tex-math></inline-formula>-feasible input wavelength division multiplexed (WDM) PSD distributions. Then the optimum inversion <inline-formula><tex-math notation= LaTeX >$x_{1}$</tex-math></inline-formula> that maximizes <inline-formula><tex-math notation= LaTeX >$C(x_{1})$</tex-math></inline-formula> 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 <inline-formula><tex-math notation= LaTeX >$x_{1}$</tex-math></inline-formula>, CIP and CSNR achieve essentially the same capacity as the optimal allocation. At sufficiently large pump <inline-formula><tex-math notation= LaTeX >$P_{p}$</tex-math></inline-formula> (<inline-formula><tex-math notation= LaTeX >$\\gtrsim 30$</tex-math></inline-formula> mW) the optimal inversion <inline-formula><tex-math notation= LaTeX >$x_{1}$</tex-math></inline-formula> is such that the EDFA gain at 1538 nm equals the span attenuation, for EDFA emission and absorption as in <xref ref-type= bibr rid= ref1 >[1]</xref>. When span attenuations increase to 20 dB, then we start seeing an advantage of the optimal allocation.