Nonlinear Optical Pulses in Media with Asymmetric Gain.

A generic novel model governing optical pulse propagation in a nonlinear dispersive amplifying medium with asymmetric (linear spectral slope) gain is introduced. We examine the properties of asymmetric optical pulses formed in such gain-skewed media, both theoretically and numerically. We derive a dissipative optical modification of the classical shallow water equations that highlights an analogy between this phenomenon and hydrodynamic wave breaking. These findings provide insight into the nature of asymmetric optical pulses capable of accumulating large nonlinear phase without wave breaking, a crucial aspect in the design of nonlinear fiber amplifiers.

Nonlinear beam tapering and two-dimensional ring solitons.

We examine a possibility to exploit the nonlinear lens effect-the initial stage of self-focusing to localize initially broad field distribution into the small central area where wave collapse is arrested-the nonlinear beam tapering. We describe two-dimensional localized solitary waves (ring solitons) in a physical system that presents a linear medium in the central core, surrounded by the cladding with the focusing Kerr nonlinearity. The standard variational analysis demonstrates that such solitons correspond to the minimum of the Hamiltonian.

Nonlinear spectral blueshift in semiconductor optical amplifiers.

We demonstrate that spectral peak power of negatively chirped optical pulses can acquire a blueshift after amplification by a semiconductor optical amplifier. The central wavelength of a transform limited optical pulse translates over 20 nm towards a shorter wavelength after propagation in a single-mode fiber and semiconductor optical amplifier. A chirped Gaussian pulse with full width at half maximum 1 ps and dimensionless chirp parameter C=-20 can be blueshifted by 5 THz.

Nonlinear Fourier transform for analysis of optical spectral combs.

The nonlinear Fourier transform (NFT) is used to characterize the optical combs in the Lugiato-Lefever equation with both anomalous and normal dispersion. We demonstrate that the NFT signal processing technique can simplify analysis of the formation of dissipative dark solitons and regimes exploiting modulation instability for a generation of coherent structures, by approximating the comb with several discrete eigenvalues, providing a platform for the analytical description of dissipative coherent structures.

Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems.

Using the cubic Ginzburg-Landau equation as an example, we demonstrate how the inverse scattering transform can be applied to characterize coherent structures in dissipative nonlinear systems. Using this approach one can reduce the number of the effective degrees of freedom in the system when the dynamic is dominated by the coherent structures, even if they are embedded in the dispersive waves and demonstrate unstable behavior.

Polarization-multiplexed nonlinear inverse synthesis with standard and reduced-complexity NFT processing.

In this work, we study the performance of polarization division multiplexing nonlinear inverse synthesis transmission schemes for fiber-optic communications, expected to have reduced nonlinearity impact. Our technique exploits the integrability of the Manakov equation-the master model for dual-polarization signal propagation in a single mode fiber-and employs nonlinear Fourier transform (NFT) based signal processing. First, we generalize some algorithms for the NFT computation to the two- and multicomponent case. Then, we demonstrate that modulating information on both polarizations doubles the channel information rate with a negligible performance degradation. Moreover, we introduce a novel dual-polarization transmission scheme with reduced complexity which separately processes each polarization component and can also provide a performance improvement in some practical scenarios.

Publisher Correction: Wave kinetics of random fibre lasers.

This corrects the article DOI: 10.1038/ncomms7214.

All-optical multilevel amplitude regeneration in a single nonlinear optical loop mirror.

We experimentally demonstrate all-optical amplitude regeneration of 4-level pulse amplitude modulated signals (PAM4) based on a single nonlinear optical loop mirror (NOLM). Four power-plateau regions are achieved using return-to-zero (RZ) pulses of narrow pulse-width, enabling large nonlinear phase shifts within the highly nonlinear fiber (HNLF). We quantify noise suppression characteristics at each amplitude level and obtain an overall EVM improvement of 0.92dB by optimizing input power and distortion strength. A theoretical analysis has been also carried out matching the experimental results and revealing the design characteristics of the regenerator's nonlinear transfer function.

Log-log growth of channel capacity for nondispersive nonlinear optical fiber channel in intermediate power range.

We consider a model nondispersive nonlinear optical fiber channel with an additive Gaussian noise. Using Feynman path-integral technique, we find the optimal input signal distribution maximizing the channel's per-sample mutual information at large signal-to-noise ratio in the intermediate power range. The optimal input signal distribution allows us to improve previously known estimates for the channel capacity. We calculate the output signal entropy, conditional entropy, and per-sample mutual information for Gaussian, half-Gaussian, and modified Gaussian input signal distributions. We demonstrate that in the intermediate power range the capacity (the per-sample mutual information for the optimal input signal distribution) is greater than the per-sample mutual information for half-Gaussian input signal distribution considered previously as the optimal one. We also show that the capacity grows as loglogP in the intermediate power range, where P is the signal power.

New Approaches to Coding Information using Inverse Scattering Transform.

Remarkable mathematical properties of the integrable nonlinear Schrödinger equation (NLSE) can offer advanced solutions for the mitigation of nonlinear signal distortions in optical fiber links. Fundamental optical soliton, continuous, and discrete eigenvalues of the nonlinear spectrum have already been considered for the transmission of information in fiber-optic channels. Here, we propose to apply signal modulation to the kernel of the Gelfand-Levitan-Marchenko equations that offers the advantage of a relatively simple decoder design. First, we describe an approach based on exploiting the general N-soliton solution of the NLSE for simultaneous coding of N symbols involving 4×N coding parameters. As a specific elegant subclass of the general schemes, we introduce a soliton orthogonal frequency division multiplexing (SOFDM) method. This method is based on the choice of identical imaginary parts of the N-soliton solution eigenvalues, corresponding to equidistant soliton frequencies, making it similar to the conventional OFDM scheme, thus, allowing for the use of the efficient fast Fourier transform algorithm to recover the data. Then, we demonstrate how to use this new approach to control signal parameters in the case of the continuous spectrum.

Simple geometric interpretation of signal evolution in phase-sensitive fibre optic parametric amplifier.

Visualisation of complex nonlinear equation solutions is a useful analysis tool for various scientific and engineering applications. We have re-examined the geometrical interpretation of the classical nonlinear four-wave mixing equations for the specific scheme of a phase sensitive one-pump fiber optical parametric amplification, which has recently attracted revived interest in the optical communications due to potential low noise properties of such amplifiers. Analysis of the phase portraits of the corresponding dynamical systems provide valuable additional insight into field dynamics and properties of the amplifiers. Simple geometric approach has been proposed to describe evolution of the waves, involved in phase-sensitive fiber optical parametric amplification (PS-FOPA) process, using a Hamiltonian structure of the governing equations. We have demonstrated how the proposed approach can be applied to the optimization problems arising in the design of the specific PS-FOPA scheme. The method considered here is rather general and can be used in various applications.

Calculation of mutual information for nonlinear communication channel at large signal-to-noise ratio.

Using the path-integral technique we examine the mutual information for the communication channel modeled by the nonlinear Schrödinger equation with additive Gaussian noise. The nonlinear Schrödinger equation is one of the fundamental models in nonlinear physics, and it has a broad range of applications, including fiber optical communications-the backbone of the internet. At large signal-to-noise ratio we present the mutual information through the path-integral, which is convenient for the perturbative expansion in nonlinearity. In the limit of small noise and small nonlinearity we derive analytically the first nonzero nonlinear correction to the mutual information for the channel.

QPSK 3R regenerator using a phase sensitive amplifier.

A black box phase sensitive amplifier based 3R regeneration scheme is proposed for non-return to zero quadrature phase shift keyed formatted signals. Performance improvements of more than 2 dB are achieved at the presence of input phase distortion.

Unveiling Temporal Correlations Characteristic of a Phase Transition in the Output Intensity of a Fiber Laser.

We use advanced statistical tools of time-series analysis to characterize the dynamical complexity of the transition to optical wave turbulence in a fiber laser. Ordinal analysis and the horizontal visibility graph applied to the experimentally measured laser output intensity reveal the presence of temporal correlations during the transition from the laminar to the turbulent lasing regimes. Both methods unveil coherent structures with well-defined time scales and strong correlations both, in the timing of the laser pulses and in their peak intensities. Our approach is generic and may be used in other complex systems that undergo similar transitions involving the generation of extreme fluctuations.

Pattern Generation by Dissipative Parametric Instability.

Nonlinear instabilities are responsible for spontaneous pattern formation in a vast number of natural and engineered systems, ranging from biology to galaxy buildup. We propose a new instability mechanism leading to pattern formation in spatially extended nonlinear systems, which is based on a periodic antiphase modulation of spectrally dependent losses arranged in a zigzag way: an effective filtering is imposed at symmetrically located wave numbers k and -k in alternating order. The properties of the dissipative parametric instability differ from the features of both key classical concepts of modulation instabilities, i.e., the Benjamin-Feir instability and the Faraday instabiltyity. We demonstrate how the dissipative parametric instability can lead to the formation of stable patterns in one- and two-dimensional systems. The proposed instability mechanism is generic and can naturally occur or can be implemented in various physical systems.

Method for computing the optimal signal distribution and channel capacity.

An iterative method for computing the channel capacity of both discrete and continuous input, continuous output channels is proposed. The efficiency of new method is demonstrated in comparison with the classical Blahut - Arimoto algorithm for several known channels. Moreover, we also present a hybrid method combining advantages of both the Blahut - Arimoto algorithm and our iterative approach. The new method is especially efficient for the channels with a priory unknown discrete input alphabet.

Stochasticity, periodicity and localized light structures in partially mode-locked fibre lasers.

Physical systems with co-existence and interplay of processes featuring distinct spatio-temporal scales are found in various research areas ranging from studies of brain activity to astrophysics. The complexity of such systems makes their theoretical and experimental analysis technically and conceptually challenging. Here, we discovered that while radiation of partially mode-locked fibre lasers is stochastic and intermittent on a short time scale, it exhibits non-trivial periodicity and long-scale correlations over slow evolution from one round-trip to another. A new technique for evolution mapping of intensity autocorrelation function has enabled us to reveal a variety of localized spatio-temporal structures and to experimentally study their symbiotic co-existence with stochastic radiation. Real-time characterization of dynamical spatio-temporal regimes of laser operation is set to bring new insights into rich underlying nonlinear physics of practical active- and passive-cavity photonic systems.

Wave kinetics of random fibre lasers.

Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak non-linear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with non-linear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by non-uniform double-scale dynamics with strong periodic changes of the energy spectrum and slow evolution from cycle to cycle to a statistically steady state. Taking a practically important example-random fibre laser-we show that a model describing such a system is close to integrable non-linear Schrödinger equation and needs a new formalism of wave kinetics, developed here. We derive a non-linear kinetic theory of the laser spectrum, generalizing the seminal linear model of Schawlow and Townes. Experimental results agree with our theory. The work has implications for describing kinetics of cyclical systems beyond photonics.

Nonlinear pulse combining and pulse compression in multi-core fibers.

We demonstrate light pulse combining and pulse compression using a continuous-discrete nonlinear system implemented in a multi-core fiber (MCF). It is shown that the pulses initially injected into all of the cores of a ring MCF are combined by nonlinearity into a small number of cores with simultaneous pulse compression. We demonstrate the combining of 77% of the energy into one core with pulse compression over 14× in a 20-core MCF. We also demonstrate that a suggested scheme is insensitive to the phase perturbations. Nonlinear spatio-temporal pulse manipulation in multi-core fibers can be exploited for various applications, including pulse compression, switching, and combining.

Conditional probability calculations for the nonlinear Schrödinger equation with additive noise.

The method for the computation of the conditional probability density function for the nonlinear Schrödinger equation with additive noise is developed. We present in a constructive form the conditional probability density function in the limit of small noise and analytically derive it in a weakly nonlinear case. The general theory results are illustrated using fiber-optic communications as a particular, albeit practically very important, example.