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IEEE Photonics Technology Letters

Modulation on Discrete Nonlinear Spectrum: Perturbation Sensitivity and Achievable Rates

H. Bülow, Vahid Aref, Laurent Schmalen

Published Jan 12, 2018• 19 citations
Publisher

Abstract

Multi-soliton pulses are studied under small perturbations. By means of mutual information, we show that the nonlinear spectrum becomes correlated when a 64-PSK modulated 2-eigenvalues soliton is perturbed by a small additive white Gaussian noise (AWGN). Transmission over two channels are considered: an AWGN channel (back-to-back scenario) and an ideal optical fiber link with distributed noise. We numerically show that the Jost coefficients b(λk) are less correlated than the discrete spectral amplitudes qd(λk) while either of them can be alternatively used for modulation. In both channels, we compare the achievable information rates (AIRs) of joint detection and separate detection if the modulation is over either qd(λk) or b(λk). We show that if the length of the link is relatively short, the separate detection of b(λk) gives almost the same total information rate as the joint detection of b(λk) or qd(λk). Moreover, the AIRs are compared with one of a first-order soliton with the same modulation.

DOI
10.1109/lpt.2018.2792785

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Using the path-integral technique we calculate the mutual information for the fiber optical channel modelled by the nonlinear Schrödinger equation with additive Gaussian noise. At large signal-to-noise ratio (SNR) we present the mutual information through the path-integral which is convenient for the perturbative expansion both in nonlinearity and dispersion. In the leading order in 1/SNR we demonstrate that the mutual information is determined through the averaged logarithm of the normalization factor Λ of the conditional probability density function P [ Y | X ]. In the limit of small noise and small nonlinearity we derive analytically the first nonzero nonlinear correction to the mutual information for the channel. For the arbitrary nonlinearity we restrict the mutual information by the low bound obtained from the Jensen's inequality and analyze the bound for the case of large dispersion.
Intrapulse quantum spectral correlation of photon numbers of a femtosecond optical pulse, which propagates a 5-m-long optical fiber, is presented in a matrix format for quantum noise study by numerical calculation. The calculation includes the higher-order optical effects such as intrapulse-stimulated Raman scattering and self-steepening effects, as well as self-phase modulation and group velocity dispersion. The quantum correlations between spectral components (spectral resolution of 0.7 nm) which evolve with the pulse energy are provided in a matrix format at soliton orders including the low orders, N=0.1N=0.1N=0.1 to 1.0 (a 0.1 step), to give the detailed properties of spectral correlation evolutions in the matrix format. It is shown that the photon-number correlations between spectral components near the pulse spectrum center form strong correlations of a square shape in the matrix, at a soliton order NNN ($\ensuremath{\le}$0.4), and then are redistributed into those between spectral components near the spectrum wings such that the cross-shaped correlation pattern in the matrix begin to be visible at N=0.5N=0.5N=0.5 before their further redistribution as $N\ensuremath{\rightarrow}1.0$. In addition, significant anticorrelations can be increasingly extended to be found between spectral components near the spectrum center as $N\ensuremath{\rightarrow}1.0$, leading to the expectation that the degree of photon-number squeezing by spectral filtering for subsolitons can be lower than that for solitons. Intrapulse-stimulated Raman scattering that starts to become visible at around N=0.6N=0.6N=0.6 produces an asymmetric correlation structure with respect to the spectrum center in the correlation matrices while the relevant pulse self-frequency shift becomes visible at $N\ensuremath{\ge}0.8$. The calculation presented indicates that the measurement of nonlinear effects which are not detectable distinctly at the classical level is possible at the quantum level.
A multistage signal estimation scheme is proposed for phase-and-amplitude modulated soliton communication systems. The estimator provides a significant mean-squared-error reduction in soliton amplitude estimation, which, in conjunction with detection of encoded phase information, results in spectral-efficiency improvements. A low-complexity version of the multistage estimation scheme is also proposed, whose performance is studied via numerical simulations under the distributed Raman amplification strategy. The achievable information rate is increased by about 1 bit/symbol for pulse bandwidth of 30 GHz, 40 GHz, and 50 GHz.