Bandwidth programmable optical Nyquist pulse generation in passively mode-locked fiber laser

We propose and numerically demonstrate a novel simple method to produce optical Nyquist pulses based on pulse shaping in a passively mode-locked fiber laser with an in-cavity flat-top spectral filter. The proposed scheme takes advantage of the nonlinear in-cavity dynamics of the laser and offers the possibility to generate high-quality sinc-shaped pulses with widely tunable bandwidth directly from the laser oscillator. We also show that the use of a filter with a corrective convex profile relaxes the need for large nonlinear phase accumulation in the cavity by offsetting the concavity of the nonlinearly broadened pulse spectrum.

Nonlinear dynamics of cilia and flagella

Cilia and flagella are hairlike extensions of eukaryotic cells which generate oscillatory beat patterns that can propel micro-organisms and create fluid flows near cellular surfaces. The evolutionary highly conserved core of cilia and flagella consists of a cylindrical arrangement of nine microtubule doublets, called the axoneme. The axoneme is an actively bending structure whose motility results from the action of dynein motor proteins cross-linking microtubule doublets and generating stresses that induce bending deformations. The periodic beat patterns are the result of a mechanical feedback that leads to self-organized bending waves along the axoneme. Using a theoretical framework to describe planar beating motion, we derive a nonlinear wave equation that describes the fundamental Fourier mode of the axonemal beat. We study the role of nonlinearities and investigate how the amplitude of oscillations increases in the vicinity of an oscillatory instability. We furthermore present numerical solutions of the nonlinear wave equation for different boundary conditions. We find that the nonlinear waves are well approximated by the linearly unstable modes for amplitudes of beat patterns similar to those observed experimentally.

Nonlinear dynamics of a periodically driven Duffing resonator coupled to a Van der Pol oscillator

We explore the dynamics of a periodically driven Duffing resonator coupled elastically to a van der Pol oscillator in the case of 1?:?1 internal resonance in the cases of weak and strong coupling. Whilst strong coupling leads to dominating synchronization, the weak coupling case leads to a multitude of complex behaviours. A two-time scales method is used to obtain the frequency-amplitude modulation. The internal resonance leads to an antiresonance response of the Duffing resonator and a stagnant response (a small shoulder in the curve) of the van der Pol oscillator. The stability of the dynamic motions is also analyzed. The coupled system shows a hysteretic response pattern and symmetry-breaking facets. Chaotic behaviour of the coupled system is also observed and the dependence of the system dynamics on the parameters are also studied using bifurcation analysis.

Synchronization processes in nonlinear systems and their relation to cortical oscillatory dynamics

This thesis was focused on theoretical models of synchronization to cortical dynamics as measured by magnetoencephalography (MEG). Dynamical systems theory was used in both identifying relevant variables for brain coordination and also in devising methods for their quantification. We presented a method for studying interactions of linear and chaotic neuronal sources using MEG beamforming techniques. We showed that such sources can be accurately reconstructed in terms of their location, temporal dynamics and possible interactions. Synchronization in low-dimensional nonlinear systems was studied to explore specific correlates of functional integration and segregation. In the case of interacting dissimilar systems, relevant coordination phenomena involved generalized and phase synchronization, which were often intermittent. Spatially-extended systems were then studied. For locally-coupled dissimilar systems, as in the case of cortical columns, clustering behaviour occurred. Synchronized clusters emerged at different frequencies and their boundaries were marked through oscillation death. The macroscopic mean field revealed sharp spectral peaks at the frequencies of the clusters and broader spectral drops at their boundaries. These results question existing models of Event Related Synchronization and Desynchronization. We re-examined the concept of the steady-state evoked response following an AM stimulus. We showed that very little variability in the AM following response could be accounted by system noise. We presented a methodology for detecting local and global nonlinear interactions from MEG data in order to account for residual variability. We found crosshemispheric nonlinear interactions of ongoing cortical rhythms concurrent with the stimulus and interactions of these rhythms with the following AM responses. Finally, we hypothesized that holistic spatial stimuli would be accompanied by the emergence of clusters in primary visual cortex resulting in frequency-specific MEG oscillations. Indeed, we found different frequency distributions in induced gamma oscillations for different spatial stimuli, which was suggestive of temporal coding of these spatial stimuli. Further, we addressed the bursting character of these oscillations, which was suggestive of intermittent nonlinear dynamics. However, we did not observe the characteristic-3/2 power-law scaling in the distribution of interburst intervals. Further, this distribution was only seldom significantly different to the one obtained in surrogate data, where nonlinear structure was destroyed. In conclusion, the work presented in this thesis suggests that advances in dynamical systems theory in conjunction with developments in magnetoencephalography may facilitate a mapping between levels of description int he brain. this may potentially represent a major advancement in neuroscience.

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.

Nonlinear pulse shaping and polarization dynamics in mode-locked fibre lasers

We review our recent progress on the study of new nonlinear mechanisms of pulse shaping in passively mode-locked fibre lasers. These include a mode-locking regime featuring pulses with a triangular distribution of the intensity, and spectral compression arising from nonlinear pulse propagation. We also report on our recent experimental studies unveiling new families of vector solitons with precessing states of polarization for multipulsing and bound-state soliton operations in a carbon nanotube mode-locked fibre laser with anomalous dispersion cavity. © 2013 IEEE.

Nonlinear pulse shaping and polarization dynamics in mode-locked fibre lasers

We review our recent progress on the study of new nonlinear mechanisms of pulse shaping in passively mode-locked fibre lasers. These include a mode-locking regime featuring pulses with a triangular distribution of the intensity, and spectral compression arising from nonlinear pulse propagation. We also report on our recent experimental studies unveiling new families of vector solitons with precessing states of polarization for multipulsing and bound-state soliton operations in a carbon nanotube mode-locked fibre laser with anomalous dispersion cavity. © 2013 IEEE.

Nonlinear pulse shaping and polarization dynamics in mode-locked fiber lasers

We review our recent progress on the study of new nonlinear mechanisms of pulse shaping in passively mode-locked fiber lasers. These include a mode-locking regime featuring pulses with a triangular distribution of the intensity, and spectral compression arising from nonlinear pulse propagation. We also report on our recent experimental studies unveiling new types of vector solitons with processing states of polarization for multi-pulse and tightly bound-state soliton (soliton molecule) operations in a carbon nanotube (CNT) mode-locked fiber laser with anomalous dispersion cavity. © 2014 World Scientific Publishing Company.

Modeling of CW Yb-doped fiber lasers with highly nonlinear cavity dynamics

We develop a theoretical framework for modeling of continuous wave Yb-doped fiber lasers with highly nonlinear cavity dynamics. The developed approach has shown good agreement between theoretical predictions and experimental results for particular scheme of Yb-doped laser with large spectral broadening during single round trip. The model is capable to accurately describe main features of the experimentally measured laser outputs such as power efficiency slope, power leakage through fibre Bragg gratings, spectral broadening and spectral shape of generated radiation. © 2011 Optical Society of America.

Mode-locking intracavity dynamics:modeling nonlinear power and energy fluctuations

Modern high-power, pulsed lasers are driven by strong intracavity fluctuations. Critical in driving the intracavity dynamics is the nontrivial phase profiles generated and their periodic modification from either nonlinear mode-coupling, spectral filtering or dispersion management. Understanding the theoretical origins of the intracavity fluctuations helps guide the design, optimization and construction of efficient, high-power and high-energy pulsed laser cavities. Three specific mode-locking component are presented for enhancing laser energy: waveguide arrays, spectral filtering and dispersion management. Each component drives a strong intracavity dynamics that is captured through various modeling and analytic techniques.

Interaction dynamics in small networks of nonlinear elements

We study a small circuit of coupled nonlinear elements to investigate general features of signal transmission through networks. The small circuit itself is perceived as building block for larger networks. Individual dynamics and coupling are motivated by neuronal systems: We consider two types of dynamical modes for an individual element, regular spiking and chattering and each individual element can receive excitatory and/or inhibitory inputs and is subjected to different feedback types (excitatory and inhibitory; forward and recurrent). Both, deterministic and stochastic simulations are carried out to study the input-output relationships of these networks. Major results for regular spiking elements include frequency locking, spike rate amplification for strong synaptic coupling, and inhibition-induced spike rate control which can be interpreted as a output frequency rectification. For chattering elements, spike rate amplification for low frequencies and silencing for large frequencies is characteristic

Amplitude response of a unilaterally constrained nonlinear micromechanical resonator

Dynamical systems that involve impacts frequently arise in engineering. This Letter reports a study of such a system at microscale that consists of a nonlinear resonator operating with an unilateral impact. The microresonators were fabricated on silicon-on-insulator wafers by using a one-mask process and then characterised by using the capacitively driving and sensing method. Numerical results concerning the dynamics of this vibro-impact system were verified by the experiments. Bifurcation analysis was used to provide a qualitative scenario of the system steady-state solutions as a function of both the amplitude and the frequency of the external driving sinusoidal voltage. The results show that the amplitude of resonant peak is levelled off owing to the impact effect and that the bandwidth of impacting is dependent upon the nonlinearity and the operating conditions.

Lattice approach to the dynamics of phase-coded soliton trains

We address the collective dynamics of a soliton train propagating in a medium described by the nonlinear Schrödinger equation. Our approach uses the reduction of train dynamics to the discrete complex Toda chain (CTC) model for the evolution of parameters for each train constituent: such a simplification allows one to carry out an approximate analysis of the dynamics of positions and phases of individual interacting pulses. Here, we employ the CTC model to the problem which has relevance to the field of fibre optics communications where each binary digit of transmitted information is encoded via the phase difference between the two adjacent solitons. Our goal is to elucidate different scenarios of the train distortions and the subsequent information garbling caused solely by the intersoliton interactions. First, we examine how the structure of a given phase pattern affects the initial stage of the train dynamics and explain the general mechanisms for the appearance of unstable collective soliton modes. Then we further discuss the nonlinear regime concentrating on the dependence of the Lax scattering matrix on the input phase distribution; this allows one to classify typical features of the train evolution and determine the distance where the soliton escapes from its slot. In both cases, we demonstrate deep mathematical analogies with the classical theory of crystal lattice dynamics.

Identification of nonlinear systems by correlation using pseudorandom signals

This thesis is concerned with the measurement of the characteristics of nonlinear systems by crosscorrelation, using pseudorandom input signals based on m sequences. The systems are characterised by Volterra series, and analytical expressions relating the rth order Volterra kernel to r-dimensional crosscorrelation measurements are derived. It is shown that the two-dimensional crosscorrelation measurements are related to the corresponding second order kernel values by a set of equations which may be structured into a number of independent subsets. The m sequence properties determine how the maximum order of the subsets for off-diagonal values is related to the upper bound of the arguments for nonzero kernel values. The upper bound of the arguments is used as a performance index, and the performance of antisymmetric pseudorandom binary, ternary and quinary signals is investigated. The performance indices obtained above are small in relation to the periods of the corresponding signals. To achieve higher performance with ternary signals, a method is proposed for combining the estimates of the second order kernel values so that the effects of some of the undesirable nonzero values in the fourth order autocorrelation function of the input signal are removed. The identification of the dynamics of two-input, single-output systems with multiplicative nonlinearity is investigated. It is shown that the characteristics of such a system may be determined by crosscorrelation experiments using phase-shifted versions of a common signal as inputs. The effects of nonlinearities on the estimates of system weighting functions obtained by crosscorrelation are also investigated. Results obtained by correlation testing of an industrial process are presented, and the differences between theoretical and experimental results discussed for this case;

Assessing interactions of linear and nonlinear neuronal sources using MEG beamformers:a proof of concept

Objective: This study aimed to explore methods of assessing interactions between neuronal sources using MEG beamformers. However, beamformer methodology is based on the assumption of no linear long-term source interdependencies [VanVeen BD, vanDrongelen W, Yuchtman M, Suzuki A. Localization of brain electrical activity via linearly constrained minimum variance spatial filtering. IEEE Trans Biomed Eng 1997;44:867-80; Robinson SE, Vrba J. Functional neuroimaging by synthetic aperture magnetometry (SAM). In: Recent advances in Biomagnetism. Sendai: Tohoku University Press; 1999. p. 302-5]. Although such long-term correlations are not efficient and should not be anticipated in a healthy brain [Friston KJ. The labile brain. I. Neuronal transients and nonlinear coupling. Philos Trans R Soc Lond B Biol Sci 2000;355:215-36], transient correlations seem to underlie functional cortical coordination [Singer W. Neuronal synchrony: a versatile code for the definition of relations? Neuron 1999;49-65; Rodriguez E, George N, Lachaux J, Martinerie J, Renault B, Varela F. Perception's shadow: long-distance synchronization of human brain activity. Nature 1999;397:430-3; Bressler SL, Kelso J. Cortical coordination dynamics and cognition. Trends Cogn Sci 2001;5:26-36]. Methods: Two periodic sources were simulated and the effects of transient source correlation on the spatial and temporal performance of the MEG beamformer were examined. Subsequently, the interdependencies of the reconstructed sources were investigated using coherence and phase synchronization analysis based on Mutual Information. Finally, two interacting nonlinear systems served as neuronal sources and their phase interdependencies were studied under realistic measurement conditions. Results: Both the spatial and the temporal beamformer source reconstructions were accurate as long as the transient source correlation did not exceed 30-40 percent of the duration of beamformer analysis. In addition, the interdependencies of periodic sources were preserved by the beamformer and phase synchronization of interacting nonlinear sources could be detected. Conclusions: MEG beamformer methods in conjunction with analysis of source interdependencies could provide accurate spatial and temporal descriptions of interactions between linear and nonlinear neuronal sources. Significance: The proposed methods can be used for the study of interactions between neuronal sources. © 2005 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.