John Zweck

Optimization of the split-step Fourier method in modeling optical-fiber communications systems

We studied the efficiency of different implementations of the split-step Fourier method for solving the nonlinear Schro/spl uml/dinger equation that employ different step-size selection criteria. We compared the performance of the different implementations for a variety of pulse formats and systems, including higher order solitons, collisions of soliton pulses, a single-channel periodically stationary dispersion-managed soliton system, and chirped return to zero systems with single and multiple channels. We introduce a globally third-order accurate split-step scheme, in which a bound on the local error is used to select the step size. In many cases, this method is the most efficient when compared with commonly used step-size selection criteria, and it is robust for a wide range of systems providing a system-independent rule for choosing the step sizes. We find that a step-size selection method based on limiting the nonlinear phase rotation of each step is not efficient for many optical-fiber transmission systems, although it works well for solitons. We also tested a method that uses a logarithmic step-size distribution to bound the amount of spurious four-wave mixing. This method is as efficient as other second-order schemes in the single-channel dispersion-managed soliton system, while it is not efficient in other cases including multichannel simulations. We find that in most cases, the simple approach in which the step size is held constant is the least efficient of all the methods. Finally, we implemented a method in which the step size is inversely proportional to the largest group velocity difference between channels. This scheme performs best in multichannel optical communications systems for the values of accuracy typically required in most transmission simulations.

Evaluation of the very low BER of FEC codes using dual adaptive importance sampling

We evaluate the error-correcting performance of a low-density parity-check (LDPC) code in an AWGN channel using a novel dual adaptive importance sampling (DAIS) technique based on multicanonical Monte Carlo (MMC) simulations, that allows us to calculate bit error rates as low as 10/sup -19/ for a (96,50) LDPC code without a priori knowledge of how to bias. Our results agree very well with standard MC simulations, as well as the union bound for the code.

Computation of bit error ratios for a dense WDM system using the noise covariance matrix and multicanonical Monte Carlo methods

We extend the noise covariance matrix method to dense wavelength-division-multiplexed (DWDM) systems in order to efficiently and accurately compute the probability density function of the received voltage in the central channel of a DWDM 10-Gb/s chirped return-to-zero transmission system with a channel spacing of 50 GHz and a transmission distance of 6120 km. The results agree with those that we obtain using a multicanonical Monte Carlo method, which mutually validates both methods.

Analysis of four-wave mixing between pulses in high-data-rate quasi-linear subchannel-multiplexed systems

We study four-wave mixing between pulses in two subchannels of a quasi-linear 40-Gbit/s subchannel-multiplexed system. For a pseudorandom bit string there are resonances in the mean of the ghost pulse energy and in the jitter of the energy in the marks as functions of the subchannel frequency spacing. However, away from these resonances the effect of four-wave mixing decreases as the subchannel spacing increases, permitting propagation over longer distances.

Statistics of the system performance in a scrambled recirculating loop with PDL and PDG

We have demonstrated that with the loop-synchronous scrambling technique, the Q distribution of a recirculating loop closely resembles that of a straight-line system. By carefully choosing the scrambling rate, we show that slow scrambling at the transmitter improves the system performance and reduces the performance variation. We investigate the system performance for different polarization-dependent loss (PDL) levels and obtain excellent agreement between the experimental and simulation results. Our results show, for the first time, that the repolarization of the noise due to significant PDL causes an asymmetric Q-factor distribution.

Euclidean Group Invariant Computation of Stochastic Completion Fields Using Shiftable-Twistable Functions

We describe a method for computing the likelihood that a completion joining two contour fragments passes through any given position and orientation in the image plane, that is, a method for completing the boundaries of partially occluded objects. Like computations in primary visual cortex (and unlike all previous models of contour completion in the human visual system), our computation is Euclidean invariant. This invariance is achieved in a biologically plausible manner by representing the input, output, and intermediate states of the computation in a basis of shiftable-twistable functions. The spatial components of these functions resemble the receptive fields of simple cells in primary visual cortex. Shiftable-twistable functions on the space of positions and directions are a generalization of shiftable-steerable functions on the plane.

System performance variations due to partially polarized noise in a receiver

We systematically investigate effects of partially polarized noise in a receiver. We introduce a relationship between the Q factor and the signal-to-noise ratio (SNR) that accounts for effects of partially polarized noise. We derive an expression for the distribution of the Q factor for a fixed SNR, and we validate our results by comparison to back-to-back experiments. We show that the system performance varies depending on the angle between the Stokes vectors of the signal and the noise as well as the degree-of-polarization of the noise. Highly polarized noise will cause a larger variation of the system performance.

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

In this paper, dependence of both the polarization decorrelation length and the diffusion length on the amount of ellipticity present in an optical fiber was described. A small elliptical birefringence in optical fibers yields the same differential group delay as in linearly birefringent fiber. However, ellipticity may affect the interaction of nonlinearity and polarization mode dispersion during propagation.

Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling

We propose a technique that uses Monte Carlo simulations with importance sampling and a reduced Stokes model to compute the probability density function of the Q factor and the outage probability for a channel in a long-haul wavelength-division-multiplexed optical-fiber transmission system due to the combination of polarization mode dispersion, polarization dependent loss, and polarization dependent gain. This technique allows us to compute outage probabilities as small as 10/sup -6/ at a fraction of the computational cost required by standard Monte Carlo simulations.

Stiefel—Whitney currents

A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v1,…,vm, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called “atomicity.” Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ2-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle.More is true if q is odd or q = n. In this case a linear dependency current which is twisted by the orientation of the bundle can be associated to the collection v. The mod 2 reduction of this current is the mod 2 linear dependency current. The cohomology class of the linear dependency current is 2-torsion and is the qth twisted integral Stiefel-Whitney class of the bundle.In addition, higher dependency and general degeneracy currents of bundle maps are studied, together with applications to singularities of projections and maps.These results rely on a theorem of Federer which states that the complex of integrally flat currents mod p computes cohomology mod p. An alternate approach to Federer’s theorem is offered in an appendix. This approach is simpler and is via sheaf theory.