Temporal and statistical properties of the ytterbium doped fiber laser

Recently, temporal and statistical properties of quasi-CW fiber lasers have attracted a great attention. In particular, properties of Raman fiber laser (RFLs) have been studied both numerically and experimentally [1,2]. Experimental investigation is more challengeable, as the full generation optical bandwidth (typically hundreds of GHz for RFLs) is much bigger than real-time bandwidth of oscilloscopes (up to 60GHz for the newest models). So experimentally measured time dynamics is highly bandwidth averaged and do not provide precise information about overall statistical properties. To overpass this, one can use the spectral filtering technique to study temporal and statistical properties within optical bandwidth comparable with measurement bandwidth [3] or indirect measurements [4]. Ytterbium-doped fiber lasers (YDFL) are more suitable for experimental investigation, as their generation spectrum usually 10 times narrower. Moreover, recently ultra-narrow-band generation has been demonstrated in YDFL [5] which provides in principle possibility to measure time dynamics and statistics in real time using conventional oscilloscopes. © 2013 IEEE.

Nonlinear mixing and mode correlations in a short Raman fiber laser

In the present paper we experimentally demonstrate a generation in a short Raman fiber laser having 10 000 different longitudinal modes only. We design the laser using 12 meters of commercially available fiber. Contrary to the recently demonstrated single longitudinal mode DFB Raman laser and short DBR Raman laser, in the laser under study the number of modes is high enough for efficient nonlinear interactions. Experimentally measured time dynamics reveals the presence of mode correlations in the radiation: the measured extreme events lasts for more than 10 round-trips.

On-off and multistate intermittencies in cascaded random distributed feedback fibre laser

A range of physical and engineering systems exhibit an irregular complex dynamics featuring alternation of quiet and burst time intervals called the intermittency. The intermittent dynamics most popular in laser science is the on-off intermittency [1]. The on-off intermittency can be understood as a conversion of the noise in a system close to an instability threshold into effective time-dependent fluctuations which result in the alternation of stable and unstable periods. The on-off intermittency has been recently demonstrated in semiconductor, Erbium doped and Raman lasers [2-5]. Recently demonstrated random distributed feedback (random DFB) fiber laser has an irregular dynamics near the generation threshold [6,7]. Here we show the intermittency in the cascaded random DFB fiber laser. We study intensity fluctuations in a random DFB fiber laser based on nitrogen doped fiber. The laser generates first and second Stokes components 1120 nm and 1180 nm respectively under an appropriate pumping. We study the intermittency in the radiation of the second Stokes wave. The typical time trace near the generation threshold of the second Stokes wave (Pth) is shown at Fig. 1a. From the number of long enough time-traces we calculate statistical distribution between major spikes in time dynamics, Fig. 1b. To eliminate contribution of high frequency components of spikes we use a low pass filter along with the reference value of the output power. Experimental data is fitted by power law, ~(P-Pth)y, where is a mean time between pikes. There are two different intermittency regimes. Just above Pth, the mean time is approximated by the -3/2 power law. The -3/2 power law is typical to the on-off intermittency with hopping between two states (first and second Stokes waves in our case) [7]. At higher power, the mean time is approximated by -4 power law, that indicates a change in intermittency type to multistate. Multistable dynamics is observed in erbium-doped fiber lasers [8]. The origin of multiples states in our system could be probably connected with polarization hopping or other reasons and should be further investigated. We have presented a first experimental statistical characterisation of the on-off and multistate intermittencies that occur in the generation of the second Stokes wave in nitrogen doped random DFB fiber laser. References [1] H. Fujisaka and T. Yamada, “A New Intermittency in Coupled Dynamical Systems,” Prog. Theor. Phys. 74, 918 (1985). [2] S. Osborne, A. Amann, D. Bitauld, and S. O’Brien, “On-off intermittency in an optically injected semiconductor laser,” Phys. Rev. E 85, 056204 (2012). [3] S. Sergeyev, K. O'Mahoney, S. Popov, and A. T. Friberg, “Coherence and anticoherence resonance in high-concentration erbium-doped fiber laser,” Opt. Lett. 35, 3736 (2010). [4] A.E. El-Taher, S.V. Sergeyev, E.G. Turitsyna, P. Harper, and S. K. Turitsyn, “Intermittent Self-Pulsing in a Fiber Raman Laser”, In proc. Conf. Nonlin. Photon., paper ID 1367139, Colorado Springs, USA, 2012 [5] S.K. Turitsyn, S.A. Babin, A.E. El-Taher, P. Harper, D.V. Churkin, S.I. Kablukov, J.D. Ania-Castañón, V. Karalekas, and E.V. Podivilov, “Random distributed feedback fibre laser”, Nat. Photon..4, 231 (2010). [6] I. D. Vatnik, D. V. Churkin, S. A. Babin, and S. K. Turitsyn, "Cascaded random distributed feedback Raman fiber laser operating at 1.2 μm," Opt. Express 19, 18486 (2011). [7] W. Feller, An introduction to probability theory and its applications, Vol. 1, 3rd ed. (Wiley, New-York, 1968). [8] G. Huerta-Cuellar, A.N. Pisarchik, and Y.O. Barmenkov, “Experimental characterization of hopping dynamics in a multistable fiber laser,” Phys. Rev. E 78, 035202(R) (2008).

Numerical modelling of spectral, temporal and statistical properties of Raman fiber lasers

Efficient numerical modelling of the power, spectral and statistical properties of partially coherent quasi-CW Raman fiber laser radiation is presented. XPM between pump wave and generated Stokes wave is not important in the generation spectrum broadening and XPM term can be omitted in propagation equation what sufficiently speeds-up simulations. The time dynamics of Raman fiber laser (RFL) is stochastic exhibiting events several times more intense that the mean value on the ps timescale. However, the RFL has different statistical properties on different time scales. The probability density function of spectral power density is exponential for the generation modes located either in the spectrum centre or spectral wings while the phases are distributed uniformly. The pump wave preserves the initial Gaussian statistics during propagation in the laser cavity. Intense pulses in the pump wave are evolved under the SPM influence and are not disturbed by the dispersion. Contrarily, in the generated wave the dispersion plays a significant role that results in stochastic behavior. © 2012 Elsevier B.V. All rights reserved.

Complex temporal patterns in molecular dynamics:a direct measure of the phase-space exploration by the trajectory at macroscopic time scales

Computer simulated trajectories of bulk water molecules form complex spatiotemporal structures at the picosecond time scale. This intrinsic complexity, which underlies the formation of molecular structures at longer time scales, has been quantified using a measure of statistical complexity. The method estimates the information contained in the molecular trajectory by detecting and quantifying temporal patterns present in the simulated data (velocity time series). Two types of temporal patterns are found. The first, defined by the short-time correlations corresponding to the velocity autocorrelation decay times (â‰0.1â€ps), remains asymptotically stable for time intervals longer than several tens of nanoseconds. The second is caused by previously unknown longer-time correlations (found at longer than the nanoseconds time scales) leading to a value of statistical complexity that slowly increases with time. A direct measure based on the notion of statistical complexity that describes how the trajectory explores the phase space and independent from the particular molecular signal used as the observed time series is introduced. © 2008 The American Physical Society.

Computational mechanics reveals nanosecond time correlations in molecular dynamics of liquid systems

Statistical complexity, a measure introduced in computational mechanics has been applied to MD simulated liquid water and other molecular systems. It has been found that statistical complexity does not converge in these systems but grows logarithmically without a limit. The coefficient of the growth has been introduced as a new molecular parameter which is invariant for a given liquid system. Using this new parameter extremely long time correlations in the system undetectable by traditional methods are elucidated. The existence of hundreds of picosecond and even nanosecond long correlations in bulk water has been demonstrated. © 2008 Elsevier B.V. All rights reserved.

Modelling nonlinear stochastic dynamics in financial time series

For analysing financial time series two main opposing viewpoints exist, either capital markets are completely stochastic and therefore prices follow a random walk, or they are deterministic and consequently predictable. For each of these views a great variety of tools exist with which it can be tried to confirm the hypotheses. Unfortunately, these methods are not well suited for dealing with data characterised in part by both paradigms. This thesis investigates these two approaches in order to model the behaviour of financial time series. In the deterministic framework methods are used to characterise the dimensionality of embedded financial data. The stochastic approach includes here an estimation of the unconditioned and conditional return distributions using parametric, non- and semi-parametric density estimation techniques. Finally, it will be shown how elements from these two approaches could be combined to achieve a more realistic model for financial time series.

Real-time heterodyned measurements of spatio-frequency dynamics in fibre lasers

We report high-resolution real-time measurements of spectrum evolution in a fibre. The proposed method combines optical heterodyning with a technique of spatio-temporal intensity measurements revealing fast spectral dynamics of cavity-based systems.

Real-time heterodyne-based measurements of fiber laser spectral dynamics

We present a method to measure in real-time the instantaneous generation spectrum of fiber lasers over consecutive cavity round-trips. © 2015 OSA.

A hybrid molecular dynamics/fluctuating hydrodynamics method for modelling liquids at multiple scales in space and time

A new 3D implementation of a hybrid model based on the analogy with two-phase hydrodynamics has been developed for the simulation of liquids at microscale. The idea of the method is to smoothly combine the atomistic description in the molecular dynamics zone with the Landau-Lifshitz fluctuating hydrodynamics representation in the rest of the system in the framework of macroscopic conservation laws through the use of a single "zoom-in" user-defined function s that has the meaning of a partial concentration in the two-phase analogy model. In comparison with our previous works, the implementation has been extended to full 3D simulations for a range of atomistic models in GROMACS from argon to water in equilibrium conditions with a constant or a spatially variable function s. Preliminary results of simulating the diffusion of a small peptide in water are also reported.

Real-time intensity domain characterization of fibre lasers using spatio-temporal dynamics

Fibre lasers are light sources that are synonymous with stability. They can give rise to highly coherent continuous-wave radiation, or a stable train of mode locked pulses with well-defined characteristics. However, they can also exhibit an exceedingly diverse range of nonlinear operational regimes spanning a multi-dimensional parameter space. The complex nature of the dynamics poses significant challenges in the theoretical and experimental studies of such systems. Here, we demonstrate how the real-time experimental methodology of spatio-temporal dynamics can be used to unambiguously identify and discern between such highly complex lasing regimes. This two-dimensional representation of laser intensity allows the identification and tracking of individual features embedded in the radiation as they make round-trip circulations inside the cavity. The salient features of this methodology are highlighted by its application to the case of Raman fibre lasers and a partially mode locked ring fibre laser operating in the normal dispersion regime.

Real-time high-resolution heterodyne-based measurements of spectral dynamics in fibre lasers

Conventional tools for measurement of laser spectra (e.g. optical spectrum analysers) capture data averaged over a considerable time period. However, the generation spectrum of many laser types may involve spectral dynamics whose relatively fast time scale is determined by their cavity round trip period, calling for instrumentation featuring both high temporal and spectral resolution. Such real-time spectral characterisation becomes particularly challenging if the laser pulses are long, or they have continuous or quasi-continuous wave radiation components. Here we combine optical heterodyning with a technique of spatiooral intensity measurements that allows the characterisation of such complex sources. Fast, round-trip-resolved spectral dynamics of cavity-based systems in real-time are obtained, with temporal resolution of one cavity round trip and frequency resolution defined by its inverse (85 ns and 24 MHz respectively are demonstrated). We also show how under certain conditions for quasi-continuous wave sources, the spectral resolution could be further increased by a factor of 100 by direct extraction of phase information from the heterodyned dynamics or by using double time scales within the spectrogram approach.

Description and training of neural network dynamics

Attractor properties of a popular discrete-time neural network model are illustrated through numerical simulations. The most complex dynamics is found to occur within particular ranges of parameters controlling the symmetry and magnitude of the weight matrix. A small network model is observed to produce fixed points, limit cycles, mode-locking, the Ruelle-Takens route to chaos, and the period-doubling route to chaos. Training algorithms for tuning this dynamical behaviour are discussed. Training can be an easy or difficult task, depending whether the problem requires the use of temporal information distributed over long time intervals. Such problems require training algorithms which can handle hidden nodes. The most prominent of these algorithms, back propagation through time, solves the temporal credit assignment problem in a way which can work only if the relevant information is distributed locally in time. The Moving Targets algorithm works for the more general case, but is computationally intensive, and prone to local minima.

Noisy fitness evaluation in genetic algorithms and the dynamics of learning

A theoretical model is presented which describes selection in a genetic algorithm (GA) under a stochastic fitness measure and correctly accounts for finite population effects. Although this model describes a number of selection schemes, we only consider Boltzmann selection in detail here as results for this form of selection are particularly transparent when fitness is corrupted by additive Gaussian noise. Finite population effects are shown to be of fundamental importance in this case, as the noise has no effect in the infinite population limit. In the limit of weak selection we show how the effects of any Gaussian noise can be removed by increasing the population size appropriately. The theory is tested on two closely related problems: the one-max problem corrupted by Gaussian noise and generalization in a perceptron with binary weights. The averaged dynamics can be accurately modelled for both problems using a formalism which describes the dynamics of the GA using methods from statistical mechanics. The second problem is a simple example of a learning problem and by considering this problem we show how the accurate characterization of noise in the fitness evaluation may be relevant in machine learning. The training error (negative fitness) is the number of misclassified training examples in a batch and can be considered as a noisy version of the generalization error if an independent batch is used for each evaluation. The noise is due to the finite batch size and in the limit of large problem size and weak selection we show how the effect of this noise can be removed by increasing the population size. This allows the optimal batch size to be determined, which minimizes computation time as well as the total number of training examples required.