Coding and Information Theory for Fiber-Optic Communications

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We live in a connected society where digital information is continuously exchanged across the globe. The vast majority of this information is carried by optical fibres for at least part of their journey. The development of long-haul fibre-optic communications is a fascinating story of invention in which various technological advances (the development of single-mode fibre, efficient laser transmitters and modulators, optical amplifiers, wavelength-division multiplexing schemes, coherent signal detection and digital signal processing have, over time yielded steady improvements in the information-carrying capacity of commercial systems. Many natural questions arise immediately. Can this progress continue indefinitely, or are optical fibres ultimately limited in their capacity to carry information? If so, how close are we to achieving this ultimate limit? Can existing transmission schemes be improved to approach the ultimate limit more closely, while satisfying implementation-complexity constraints? These are the research questions that motivate this work. The approach taken to answer these questions is centred upon a mathematical tool called the nonlinear Fourier transform (NFT), first devised by mathematicians and physicists in the 1970s. Pulse propagation in optical fibres is, to a very good approximation, governed by a partial differential equation called the nonlinear Schroedinger equation (NLSE). The NFT does for the NLSE what the ordinary Fourier transform does for linear time-invariant systems: it changes a complicated convolution operation in the time domain to a simple multiplication operation in the frequency domain. Our key idea is to encode information for transmission over optical fibres in the nonlinear spectrum of the propagating wave, a feature that is entirely preserved from one end of the fibre to the other. This technique, which we have termed nonlinear frequency division multiplexing, can be viewed as an analogue of orthogonal frequency-division multiplexing commonly used in linear channels. Unlike most other fibre-optic data transmission schemes, this technique deals with both dispersion and nonlinearity unconditionally without the need for additional compensation methods at the transmitter or receiver. Using the properties of the NFT, we hope to achieve an understanding of the ultimate information-carrying capacity of optical fibres. We plan to study a recently-discovered spatially and temporally discrete version of the NLSE called the Tsuchida Discretization, which is NFT-compatible and from which we hope to gain significant insights. We also plan to study optical and digital signal processing techniques in the nonlinear spectral domain. Our research results will guide us in the design of new communication schemes that are directly optimized for resilience against all fibre-optic channel impairments, both linear and nonlinear.