Ronald Holzlöhner

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.

Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization

We present a novel linearization method to calculate accurate eye diagrams and bit error rates (BERs) for arbitrary optical transmission systems and apply it to a dispersion-managed soliton (DMS) system. In this approach, we calculate the full nonlinear evolution using Monte Carlo methods. However, we analyze the data at the receiver assuming that the nonlinear interaction of the noise with itself in an appropriate basis set is negligible during transmission. Noise-noise beating due to the quadratic nonlinearity in the receiver is kept. We apply this approach to a highly nonlinear DMS system, which is a stringent test of our approach. In this case, we cannot simply use a Fourier basis to linearize, but we must first separate the phase and timing jitters. Once that is done, the remaining Fourier amplitudes of the noise obey a multivariate Gaussian distribution, the timing jitter is Gaussian distributed, and the phase jitter obeys a Jacobi-/spl Theta/ distribution, which is the periodic analogue of a Gaussian distribution. We have carefully validated the linearization assumption through extensive Monte Carlo simulations. Once the effect of timing jitter is restored at the receiver, we calculate complete eye diagrams and the probability density functions for the marks and spaces. This new method is far more accurate than the currently accepted approach of simply fitting Gaussian curves to the distributions of the marks and spaces. In addition, we present a deterministic solution alternative to the Monte Carlo method.

Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems

We apply the multicanonical Monte Carlo (MMC) method to compute the probability distribution of the received voltage in a chirped return-to-zero system. When computing the probabilities of very rare events, the MMC technique greatly enhances the efficiency of Monte Carlo simulations by biasing the noise realizations. Our results agree with the covariance matrix method over 20 orders of magnitude. The MMC method can be regarded as iterative importance sampling that automatically converges toward the optimal bias so that it requires less a priori knowledge of the simulated system than importance sampling requires. A second advantage is that the merging of different regions of a probability distribution function to obtain the entire function is not necessary in many cases.

Optimization of cw sodium laser guide star efficiency

Context. Sodium laser guide stars (LGS) are about to enter a new range of laser powers. Previous theoretical and numerical methods are inadequate for accurate computations of the return flux, hence for the design of the next-generation LGS systems. Aims. We numerically optimize the cw (continuous wave) laser format, in particular, the light polarization and spectrum. Methods. Using Bloch equations, we simulate the mesospheric sodium atoms, including Doppler broadening, saturation, collisional relaxation, Larmor precession, and recoil, taking all 24 sodium hyperfine states into account and 100–300 velocity groups. Results. LGS return flux is limited by “three evils”: Larmor precession due to the geomagnetic field, atomic recoil due to radiation pressure, and transition saturation. We study their impact and show that the return flux can be boosted by repumping (simultaneous excitation of the sodium D2 aa nd D 2b lines with 10−20% of the laser power in the latter). Conclusions. We strongly recommend the use of circularly polarized lasers and repumping. As a rule of thumb, the bandwidth of laser radiation in MHz (at each line) should approximately equal the launched laser power in Watts divided by six, assuming a diffraction-limited spot size.

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.

Modeling of pulsed-laser guide stars for the Thirty Meter Telescope project

The Thirty Meter Telescope (TMT) has been designed to include an adaptive optics system and associated laser guide star (LGS) facility to correct for the image distortion due to Earth’s atmospheric turbulence and achieve diffraction-limited imaging. We have calculated the response of mesospheric sodium atoms to a pulsed laser that has been proposed for use in the LGS facility, including modeling of the atomic physics, the light–atom interactions, and the effect of the geomagnetic field and atomic collisions. This particular pulsed-laser format is shown to provide comparable photon return to a continuous-wave (cw) laser of the same average power; both the cw and pulsed lasers have the potential to satisfy the TMT design requirements for photon return flux.

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.

Efficient and accurate computation of eye diagrams and bit-error rates in a single-channel CRZ system

We use linearization to compute the noise evolution in a 10-Gb/s single-channel chirped return-to-zero (CRZ) system propagating over 6100 km, transmitting 32 bits. Linearization allows us to efficiently and accurately compute eye diagrams and bit error rates (BERs) without the use of Monte Carlo simulations. Phase jitter prevents the successful application of linearization unless it is removed, and we describe a method to separately remove it from each pulse in the signal. We show that the BER in our test system is dominated by the bit pattern that leads to the smallest eye opening in a noise-free simulation.

Dependence of sodium laser guide star photon return on the geomagnetic field

Aims. The efficiency of optical pumping that increases the backscatter emission of mesospheric sodium atoms in continuous wave (cw) laser guide stars (LGSs) can be significantly reduced and, in the worst case, eliminated by the action of the geomagnetic field. Our goal is to present an estimation of this effect for several telescope sites. Methods. Sodium atoms precess around magnetic field lines that cycle the magnetic quantum number, reducing the effectiveness of optical pumping. Our method is based on calculating the sodium magnetic sublevel populations in the presence of the geomagnetic field and on experimental measurements of radiance return from sodium LGS conducted at the Starfire optical range (SOR). Results. We propose a relatively simple semi-empirical formula for estimating the effect of the geomagnetic field on enhancing the LGSs photon return due to optical pumping with a circularly polarized cw single-frequency laser beam. Starting from the good agreement between our calculations and the experimental measurements for the geomagnetic field effect, and in order to more realistically estimate the sodium LGSs photon return, we introduce the effect of the distance to the mesospheric sodium layer and the atmospheric attenuation. The combined effect of these three factors is calculated for several telescope sites. Conclusions. In calculating the return flux of LGSs, only the best return conditions are often assumed, relying on strong optical pumping with circularly polarized lasers. However, one can only obtain this optimal return along one specific laser orientation on the sky, where the geomagnetic field lines are parallel to the laser beam. For most of the telescopes, the optimum can be obtained at telescope orientations beyond the observation limit. For the telescopes located close to the geomagnetic pole, the benefit of the optical pumping is much more important than for telescopes located close to the geomagnetic equator.

Intersymbol interference and timing jitter measurements in a 40-Gb/s long-haul dispersion-managed soliton system

In this work, we have measured timing jitter in a 40-Gb/s dispersion-managed soliton system. By accurately separating out the Gordon-Haus timing jitter from the overall timing jitter, we have demonstrated that intersymbol interference limits the error-free propagation distance in our system. Finally, we show that amplitude noise can enhance the measurement of timing jitter, which can lead to inaccuracy in determining the true system limitations.