EL EVANGELIO SEGÚN SAN MARCOS
Marcos 1:1–23. Cómo presentar el Evangelio de Marcos
In this chapter, the architecture, the traffic evolution, and advanced modula- tion techniques inspired by coherent detection in optical fiber networks have been reviewed. Prior to outlining the scalar nonlinear Schrödinger equa- tion (NLSE) and the Manakov equation that govern signal propagation in nonlinear dispersive fibers, linear impairments were described. Next, the nonlinearity induced channel capacity for the optical fiber channel and the compensation methods proposed to mitigate nonlinear effects in order in- crease such capacity have been presented.
One can observe that communicating through a nonlinear channel poses many practical and theoretical challenges. Basic communication methods and paradigms used in current fiber optic communications are borrowed from linear channels, thus have to be re-examined. This includes questions such as how to encode information and modulate signals, multiplex different in- formation channels, detect signals at the output of a link in the weak- and strong-nonlinear regimes, and so on. These questions necessitate an accurate modeling of nonlinear channels. The search for the solution to these ques- tions leads to the revival of eigenvalue communication systems, based on the idea of the inverse scattering transform (IST). The scope of this thesis is to discuss the implementation and experimental validation of such techniques.
Optical Transmission Systems
based on the Nonlinear Fourier
Transform
“
Everything should be made as simple as possible, but not sim- pler.”
Albert Einstein,
2.1
Introduction
I
n contrast to linear channels, optical fibers exhibit nonlinearity, and sig- nal amplification is a source of distributed noise. The interplay between dispersion, nonlinearity, and noise over the transmission length makes light propagation in fibers a complex process making it difficult to establish a channel input-output map for the simple estimation of the capacity of opti- cal fiber channels. As described earlier, the evolution of optical signals in a fiber is modeled through the scalar NLSE, Eq.(1.16), for single-polarization, and the Manakov equation, Eq.(1.25), for a dual-polarization signal. These equations do not lend themselves to an analytic solution except for some spe- cific cases, also known as soliton solutions. However, approximate analytical solutions exist for the fiber-optic channel. The NLSE, Eq.(1.17), is one of a very special class of nonlinear equations, that is an integrable nonlinear equation, which can be solved by using the NFT technique, which was first introduced by Zakharov and Shabat [60]. With the help of the NFT, a sig-2.1. Introduction
nal can be represented by its continuous (dispersive) and discrete (solitonic) nonlinear spectrum. While the evolution of signals along the fiber is remains complex in the temporal domain, all the complexity disappears in the spectral domain, where the wave evolution is linear. Based on this, Hasegawa et al. came up with the proposal to use multiple optical solitons in a communica- tion system, a technique also known as eigenvalue communication [61]. Such communication method integrates the Kerr nonlinear effect into the system design, so that the optical envelope maintains its shape along fiber propa- gation. The eigenvalue communication is preceded by the classical soliton communication. Soliton communications have been investigated intensively in the 1990s [62], and the experimental demonstration of 10 Gbit/s data transmission over one million kilometers has been provided by Nakazawa et al. [63]. However, soliton based ultra-high data rate communications suf- fer from effects such as soliton-to-soliton collisions, inter-channel cross-talk, noise, loss, high-order dispersion and, SRS, etc. [62].
With the advent of coherent detection, complex advanced digital signal pro- cessing, and the maturity in the EDFA and WDM technologies, attention was diverted from solitons to coherent long-haul transmission, in order to significantly improve the system transmission capacity. However, once again Kerr nonlinearity remained the limiting factor even with advanced high- spectral efficient modulation formats and error-correcting codes. Moreover, partial nonlinearity mitigation techniques in a weakly-nonlinear regime, such as DBP, VSTF, and others, exhibited limited performance gain due to noise- nonlinearity interactions, and inter-channel cross-talk in WDM systems. Re- cently, the idea of eigenvalue communications revived as a possibility to en- code information for nonlinear-dispersive channels, namely the optical fiber channel [64]. This proposal, simply put, combines high-order soliton solu- tions with advanced digital communication techniques to achieve high spec- tral efficiency, and its given the name of nonlinear frequency division multi- plexing (NFDM) [64] for its similarity with the orthogonal frequency-division multiplexing (OFDM) technique. The proposal to encode information in the continuous spectrum of NFDM systems was suggested by S. K. Turitsyn et.
al [93]. This is done by modulating the amplitude and the phase of the com-
plex amplitude associated with independent and parallel spectral sub-carriers that are shown to be independent upon propagation in noiseless and lossless fiber transmissions. Furthermore, the continuous spectrum is reduced to the ordinary Fourier transform as the intensity of the potential function is low. Modelling a NFDM system involves solving the so-called spectral problem associated with the integrable NLSE equation, also known as the Zakharov- Shabat spectral problem (ZSSP). For this purpose, the NFT is used to
mathematically linearize the NLSE by solving the ZSSP. The NFT oper- ation transforms the optical signal to its corresponding nonlinear spectrum, where the evolution along the fiber is linear. The inverse problem on the other hand maps the nonlinear spectral data back into the optical filed. This can be achieved by using either the Riemann-Hilbert problem [64] or the Gelfand-Levitan-Marchenko (GLM) equation [93]. Several authors have proposed numerical NFT algorithms with higher orders of accuracy, utilizing the layer-peeling [75], Ablowitz-Ladik integrable discretization [69], forward and central discretizations with first-order Euler method, the fourth-order Runge-Kutta method [76], and the piecewise-constant approximation (PCA) method [103]. In this thesis, the Ablowitz-Ladik discretization and the layer- peeling method of computing the forward NFT and inverse NFT (INFT) of the ZSSP with vanishing boundary condition, respectively, are used.
The organization of this chapter is as follows. The ZSSP is introduced in section 2.3 [85] − [88], providing an independent treatment of the computa- tion of the spectral data associated with a potential function in the ZSSP, their spatial evolution in nonlinear-dispersive channels and the spectral in- version problem. In 2.4, the numerical algorithms associated with comput- ing both forward and inverse problems are formulated. Next, the NFDM transmission architecture for a high-speed long-haul transmission system is addressed in section 2.5. Finally, the analytical formulation is extended to dual-polarization transmissions in section 2.6, considering the Manakov sys- tem in the absence of polarization induced fiber impairments (i.e., PMD or PDL).