The Constant Power Spectral Density Model for Capacity Optimization of Submarine Links

Abstract

We review the fundamentals of the recently proposed Constant Power Spectral Density (CPSD) model for very long-haul space-division multiplexed submarine links, and highlight its use for Erbium-doped fiber amplifier (EDFA) optimization to maximize the achievable information rate (AIR) of the link. The CPSD line is an abstraction of modern submarine lines, where all EDFAs have identical physical parameters, among which the doped-fiber length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> , and the same pump and Erbium inversion, with identical gain-shaping filters that reproduce the line input power spectral density at the output of each span. The key idea in the CPSD line analysis is to use the hidden state-variable of the EDFA, namely the Erbium population inversion <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> , as a free variable. When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> is known, so is the EDFA gain and its noise figure frequency profiles. Thus in the CPSD line we derive a simple expression of the received signal to noise ratio and thus of the AIR in the assumption of Gaussian noises. Among the set of input wavelength-division multiplexed signals that achieve inversion x at EDFA length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> we can find analytically the one maximizing AIR(x, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> ). We finally look numerically for the best (x, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> ) values that maximize AIR, i.e., we optimize the line EDFAs for maximum AIR. The major novelty of this invited paper is the extension of the analysis to include nonlinear effects into the AIR optimization.