Gino Biondini

Importance sampling for polarization-mode dispersion

We describe the application of importance sampling to Monte-Carlo simulations of polarization-mode dispersion (PMD) in optical fibers. The method allows rare differential group delay (DGD) events to be simulated much more efficiently than with standard Monte-Carlo methods and, thus, it can be used to assess PMD-induced system outage probabilities at realistic bit-error rates. We demonstrate the technique by accurately calculating the tails of the DGD probability distribution with a relatively small number of Monte-Carlo trials.

On a family of solutions of the Kadomtsev?Petviashvili equation which also satisfy the Toda lattice hierarchy

We describe the interaction pattern in the x–y plane for a family of soliton solutions of the Kadomtsev–Petviashvili (KP) equation, The solutions considered also satisfy the finite Toda lattice hierarchy. We determine completely their asymptotic patterns for y → ±∞, and we show that all the solutions (except the 1-soliton solution) are of resonant type, consisting of arbitrary numbers of line solitons in both asymptotics; that is, arbitrary N− incoming solitons for y → −∞ interact to form arbitrary N+ outgoing solitons for y → ∞. We also discuss the interaction process of those solitons, and show that the resonant interaction creates a web-like structure having (N− − 1)(N+ − 1) holes.

Soliton solutions of the Kadomtsev-Petviashvili II equation

We study a general class of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form of its tau-function. We show that, in addition to the previously known line soliton solutions of KPII, this class also contains a large variety of multisoliton solutions, many of which exhibit nontrivial spatial interaction patterns. We also show that, in general, such solutions consist of unequal numbers of incoming and outgoing line solitons. From the asymptotic analysis of the tau function, we explicitly characterize the incoming and outgoing line solitons of this class of solutions. We illustrate these results by discussing several examples.

Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions

The inverse scattering transform for the vector defocusing nonlinear Schrodinger NLS equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros poles of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert RH problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system.

Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions

The inverse scattering transform for the focusing nonlinear Schrodinger equation with non-zero boundary conditions at infinity is presented, including the determination of the analyticity of the scattering eigenfunctions, the introduction of the appropriate Riemann surface and uniformization variable, the symmetries, discrete spectrum, asymptotics, trace formulae and the so-called theta condition, and the formulation of the inverse problem in terms of a Riemann-Hilbert problem. In addition, the general behavior of the soliton solutions is discussed, as well as the reductions to all special cases previously discussed in the literature.

Importance sampling for polarization-mode dispersion: techniques and applications

The basic theory of importance sampling (IS) as relevant to polarization-mode dispersion (PMD) in optical fibers is discussed, and its application to Monte Carlo (MC) simulations of PMD-induced transmission impairments is demonstrated. The use of IS allows rare PMD events to be simulated much more efficiently than with standard MC methods. As a consequence, methods employing IS provide natural and effective tools to assess PMD-induced impairments and outages in optical transmission systems at realistic probability levels.

Multiple importance sampling for first- and second-order polarization-mode dispersion

A simulation method that targets all possible combinations of first- and second-order polarization-mode dispersion (PMD) is described. Use of this method in importance-sampled Monte Carlo simulations yields a more comprehensive determination of PMD-induced system penalties than first-order biasing alone and significantly speeds up the calculation of outage probabilities, particularly when PMD compensation is employed. The technique is demonstrated by using it to calculate the probability distribution function (pdf) of second-order PMD and the joint pdf of the magnitude of first- and second-order PMD.

Four-wave mixing in wavelength-division-multiplexed soliton systems: damping and amplification

Four-wave mixing in wavelength-division-multiplexed soliton systems with damping and amplification is studied. An analytical model is introduced that explains the dramatic growth of the four-wave terms. The model yields a resonance condition relating the soliton frequency and the amplifier distance. It correctly predicts all essential features regarding the resonant growth of the four-wave contributions.

Quasi-linear optical pulses in strongly dispersion-managed transmission systems

A unified analytical description of the evolution of quasi-linear optical pulses and solitons in strongly dispersion-managed transmission systems is developed. Asymptotic analysis of the nonlocal equation that describes the averaged dynamics of a dispersion-managed system shows that the nonlinearity decreases for large map strength s , as O(log s/s) . The spectral intensity is found to be an invariant of the propagation, which allows the phase shift to be computed. These findings provide a clear description of pulse propagation in the quasi-linear regime, which is characterized by much lower energies than those required for stable dispersion-managed soliton transmission with the same dispersion map.

Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions

The inverse scattering transform for an integrable discretization of the defocusing nonlinear Schrodinger equation with nonvanishing boundary values at infinity is constructed. This problem had been previously studied, and many key results had been established. Here, a suitable transformation of the scattering problem is introduced in order to address the open issue of analyticity of eigenfunctions and scattering data. Moreover, the inverse problem is formulated as a Riemann–Hilbert problem on the unit circle, and a modification of the standard procedure is required in order to deal with the dependence of asymptotics of the eigenfunctions on the potentials. The discrete analog of Gel’fand–Levitan–Marchenko equations is also derived. Finally, soliton solutions and solutions in the small-amplitude limit are obtained and the continuum limit is discussed. (Some figures in this article are in colour only in the electronic version)