The drive for higher performance in optical fiber systems has renewed interest in coherent detection. We review detection methods, including noncoherent, differentially coherent, and coherent detection, as well as a hybrid method. We compare modulation methods encoding information in various degrees of freedom (DOF). Polarization-multiplexed quadrature-amplitude modulation maximizes spectral efficiency and power efficiency, by utilizing all four available DOF, the two field quadratures in the two polarizations. Dual-polarization homodyne or heterodyne downconversion are linear processes that can fully recover the received signal field in these four DOF. When downconverted signals are sampled at the Nyquist rate, compensation of transmission impairments can be performed using digital signal processing (DSP). Linear impairments, including chromatic dispersion and polarization-mode dispersion, can be compensated quasi-exactly using finite impulse response filters. Some nonlinear impairments, such as intra-channel four-wave mixing and nonlinear phase noise, can be compensated partially. Carrier phase recovery can be performed using feedforward methods, even when phase-locked loops may fail due to delay constraints. DSP-based compensation enables a receiver to adapt to time-varying impairments, and facilitates use of advanced forward-error-correction codes. We discuss both single- and multi-carrier system implementations. For a given modulation format, using coherent detection, they offer fundamentally the same spectral efficiency and power efficiency, but may differ in practice, because of different impairments and implementation details. With anticipated advances in analog-to-digital converters and integrated circuit technology, DSP-based coherent receivers at bit rates up to 100 Gbit/s should become practical within the next few years.

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Coherent detection in optical fiber systems

This paper reviews the fundamental concepts and basic theory of polarization mode dispersion (PMD) in optical fibers. It introduces a unified notation and methodology to link the various views and concepts in Jones space and Stokes space. The discussion includes the relation between Jones vectors and Stokes vectors, rotation matrices, the definition and representation of PMD vectors, the laws of infinitesimal rotation, and the rules for PMD vector concatenation.

https://www.pnas.org/content/pnas/97/9/4541.full.pdf

PMD fundamentals: Polarization mode dispersion in optical fibers

Digital coherent receivers have caused a revolution in the design of optical transmission systems, due to the subsystems and algorithms embedded within such a receiver. After giving a high-level overview of the subsystems, the optical front end, the analog-to-digital converter (ADC) and the digital signal processing (DSP) algorithms, which relax the tolerances on these subsystems are discussed. Attention is then turned to the compensation of transmission impairments, both static and dynamic. The discussion of dynamic-channel equalization, which forms a significant part of the paper, includes a theoretical analysis of the dual-polarization constant modulus algorithm, where the control surfaces several different equalizer algorithms are derived, including the constant modulus, decision-directed, trained, and the radially directed equalizer for both polarization division multiplexed quadriphase shift keyed (PDM-QPSK) and 16 level quadrature amplitude modulation (PDM-16-QAM). Synchronization algorithms employed to recover the timing and carrier phase information are then examined, after which the data may be recovered. The paper concludes with a discussion of the challenges for future coherent optical transmission systems.

Digital Coherent Optical Receivers: Algorithms and Subsystems

An analytical characterization of polarization dispersion measurements is presented. The authors report the solution of Poole's stochastic dynamical equation for the evolution of the polarization dispersion vector with fiber length. The authors extend this to a more complete description by considering small, second-order dispersion effects through the frequency derivative of the dispersion vector. The complete analytical solution is seen to accord with what were originally empirically derived features of the joint probability distribution of the polarization dispersion vector and its frequency derivatives. Among the analytically determined properties are the Gaussian probability densities of the three components of the dispersion vector, and the hyperbolic secant (soliton shaped) probability densities of the components of the derivative of the dispersion vector. >

Statistical theory of polarization dispersion in single mode fibers

Polarization mode dispersion and the polarization decorrelation and diffusion lengths are calculated in fibers with randomly varying birefringence. Two different physical models in which the birefringence orientation varies arbitrarily are studied and are shown to yield nearly identical results. These models are appropriate for communication fibers. We show that both the length scales for polarization mode dispersion and polarization decorrelation measured with respect to the local axes of birefringence are equal to the fiber autocorrelation length. We also show that the coupled nonlinear Schrodinger equation which describes wave evolution over long length along a communication fiber can be reduced to the Manakov equation. The appropriate averaging length for the linear polarization mode dispersion is just the fiber autocorrelation length but the appropriate averaging length for the nonlinear terms is the diffusion length in the azimuthal direction along the Poincare sphere which can be different, The implications for the nonlinear evolution are discussed.

Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence

A simple relationship is found for the evolution of the principal states of polarization (PSPs) and their differential group delay in fiber links. A simple expression is found, using the relationship, for the probability of the differential group delay (DGD), considering the evolution of the PSPs as a Brownian motion. The theory has been verified experimentally on an optical cable composed of 12 single-mode, shifted-dispersion fibers 2.2-km long. The results show that the DGD grows as the square root of the length when the length of the fiber is far larger than the correlation length of the perturbation. The measured value of DGD can vary substantially in two fibers belonging to the same ensemble, and in the same fiber, considering two frequencies differing by more than 5 nm. >

Statistical treatment of the evolution of the principal states of polarization in single-mode fibers

We report some results of measurements of the frequency dependence of the state of polarization at the end of a low-birefringence long-cabled (50-km) optical fiber. The principal states of polarization and their group-delay difference were determined by making use of the Poincare sphere representation.

Measurement of the group-delay difference between the principal states of polarization on a low-birefringence terrestrial fiber cable.

A novel multilevel coherent optical system is proposed. It is based on the exploitation of the property that the electromagnetic field propagating in a single-mode optical fiber can be represented by a four-dimensional vector whose components are the phase and quadrature terms of the two polarization components of the electrical field. This allows a wider use of the resources of the electromagnetic field for information transmission in order to obtain a spectrally efficient modulation format with a limited end. The net performance gain with respect to multilevel amplitude and phase modulation (N-APK) and N-PSK increases with an increase in the number of levels N. For instance, for N=32 the gain is 1.6 and 7.7 dB with respect to N-APK and N-PSK systems. The effect of laser phase noise on the system performance is evaluated. >

A novel multilevel coherent optical system: 4-quadrature signaling

A multilevel transmission system that allows efficient exploitation of a single-channel bandwidth in coherent optical communication systems is proposed. It is based on Stokes parameter modulation and a decision on the Poincare sphere. The block scheme of the transmitter, which is able to provide a generic output state of polarization (SOP) starting from a linearly polarized optical field, is shown and described. The propagation along the fiber is described in order to derive the expression of the optical field at the receiver input. The spectral characteristics of the transmitted and the received optical signals are analyzed. >

Multilevel coherent optical system based on Stokes parameters modulation

A multilevel coherent optical system is disclosed in which a multilevel signal with a coherent optical carrier is provided by means of a modulation of the phase and the polarization of the electromagnetic field propagating through a single-mode optical fibre, a heterodyne transmitter and receiver for said signal being also disclosed. The transmitter comprises a coherent light source (1) providing the optical carrier, a phase modulator (2) modulating the phase of the carrier, a polarization modulator (3-8), and a modulation signal generator (10) providing control signals to the phase modulator (2) and the polarization modulator. The receiver comprises a first stage (12-23) carrying out the heterodyne detection of the phase component and the phase quadrature component of the polarization of the signal received through an optical fibre (11), a second stage demodulating the received signal to provide the multilevel signal, and a processing circuit comparing the received multilevel signal with predetermined reference signals. Such a system exploits the four degrees of freedom of the electromagnetic field propagating through the optical fibre so as to approach closer to the theoretical Shannon limit than conventional systems.

Franco Curti

Papers

Statistical treatment of the evolution of the principal states of polarization in single-mode fibers

Measurement of the group-delay difference between the principal states of polarization on a low-birefringence terrestrial fiber cable.

A novel multilevel coherent optical system: 4-quadrature signaling

Multilevel coherent optical system based on Stokes parameters modulation

A multilevel coherent optical system