Fast nonlinear Fourier transform algorithms for optical data processing.

The nonlinear Fourier transform (NFT) is an approach that is similar to a conventional Fourier transform. In particular, NFT allows to analyze the structure of a signal governed by the nonlinear Schrödinger equation (NLSE). Recently, NFT applied to NLSE has attracted special attention in applications of fiber-optic communication. Improving the speed and accuracy of the NFT algorithms remains an urgent problem in optics. We present an approach that allows to find all variants of symmetric exponential splitting schemes suitable for the fast NFT (FNFT) algorithms with low complexity. One of the obtained schemes showed good numerical results in computing the continuous spectrum compared with other fast fourth-order NFT schemes.

Mechanism of brightness enhancement in multimode LD-pumped graded-index fiber Raman lasers: numerical modeling.

We develop a comprehensive theory for describing the experimental beam profiles from multimode fiber Raman lasers. We take into account the presence of random linear mode coupling, Kerr beam self-cleaning and intra-cavity spatial filtering. All of these factors play a decisive role in shaping the Stokes beam, which has a predominant fundamental mode content. Although the highly multimode pump beam is strongly depleted, it remains almost insensitive to the different physical effects. As a result, the intensity of the output Stokes beam is an order of magnitude higher than the pump intensity at its maximum, in quantitative agreement with the experimental results and in contrast with the simplified balance model.

Spatio-spectral beam control in multimode diode-pumped Raman fibre lasers via intracavity filtering and Kerr cleaning.

Multimode fibres provide a promising platform for boosting the capacity of fibre links and the output power of fibre lasers. The complex spatiotemporal dynamics of multimode beams may be controlled in spatial and temporal domains via the interplay of nonlinear, dispersive and dissipative effects. Raman nonlinearity induces beam cleanup in long graded-index fibres within a laser cavity, even for CW Stokes beams pumped by highly-multimode laser diodes (LDs). This leads to a breakthrough approach for wavelength-agile high-power lasers. However, current understanding of Raman beam cleanup is restricted to a small-signal gain regime, being not applicable to describing realistic laser operation. We solved this challenge by experimentally and theoretically studying pump-to-Stokes beam conversion in a graded-index fibre cavity. We show that random mode coupling, intracavity filtering and Kerr self-cleaning all play a decisive role for the spatio-spectral control of CW Stokes beams. Whereas the depleted LD pump radiation remains insensitive to them.

Nonlinear Fourier transform for characterization of the coherent structures in optical microresonators.

We propose and demonstrate, in the framework of the generic mean-field model, the application of the nonlinear Fourier transform (NFT) signal processing based on the Zakharov-Shabat spectral problem to the characterization of the round trip scale dynamics of radiation in optical fiber- and microresonators.

Conservative multi-exponential scheme for solving the direct Zakharov-Shabat scattering problem.

The direct Zakharov-Shabat scattering problem has recently gained significant attention in various applications of fiber optics. The development of accurate and fast algorithms with low computational complexity to solve the Zakharov-Shabat problem (ZSP) remains an urgent problem in optics. In this Letter, a fourth-order multi-exponential scheme is proposed for the Zakharov-Shabat system. The construction of the scheme is based on a fourth-order three-exponential scheme and Suzuki factorization. This allows one to apply the fast algorithms with low complexity to calculate the ZSP for a large number of spectral parameters. The scheme conserves the quadratic invariant for real spectral parameters, which is important for various telecommunication problems related to information coding.

Exponential fourth order schemes for direct Zakharov-Shabat problem.

Nowadays, improving the accuracy of computational methods to solve the initial value problem of the Zakharov-Shabat system remains an urgent problem in optics. In particular, increasing the approximation order of the methods is important, especially in problems where it is necessary to analyze the structure of complex waveforms. In this work, we propose two finite-difference algorithms of fourth order of approximation in the time variable. Both schemes have the exponential form and conserve the quadratic invariant of Zakharov-Shabat system. The second scheme allows applying fast algorithms with low computational complexity (fast nonlinear Fourier transform).

Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem.

We propose a finite-difference algorithm for solving the initial problem for the Zakharov-Shabat system. This method has the fourth order of accuracy and represents a generalization of the second-order Boffetta-Osborne scheme. Our method permits the Zakharov-Shabat spectral problem to be solved more effectively for continuous and discrete spectra.

Theoretical analysis of solutions of cubic-quintic Ginzburg-Landau equation with gain saturation.

We study highly chirped analytical solutions of the cubic-quintic Ginzburg-Landau equation with the gain saturation. Based on the analysis, we propose the analytical method of estimating the stable generation area in a long fiber laser. The results allow us to predict the stable generation in long-cavity fiber lasers without performing full mathematical modeling.

Soliton content in the standard optical OFDM signal.

The nonlinear Schrödinger equation (NLSE) is often used as a master path-average model for fiber-optic transmission lines. In general, the NLSE describes the co-existence of dispersive waves and soliton pulses. The propagation of a signal in such a nonlinear channel is conceptually different from linear systems. We demonstrate here that the conventional orthogonal frequency-division multiplexing (OFDM) input optical signal at powers typical for modern communication systems might have soliton components statistically created by the random process corresponding to the information content. Applying the Zakharov-Shabat spectral problem to a single OFDM symbol with multiple subcarriers, we quantify the effect of the statistical soliton occurrence in such an information-bearing optical signal. Moreover, we observe that at signal powers optimal for transmission, an OFDM symbol incorporates multiple solitons with high probability. The considered optical communication example is relevant to a more general physical problem of the generation of coherent structures from noise.

All-fiber highly chirped dissipative soliton generation in the telecom range.

A high-energy (0.93 nJ) all-fiber erbium femtosecond oscillator operating in the telecom spectral range is proposed and realized. The laser cavity, built of commercially available fibers and components, combines polarization maintaining (PM) and non-PM parts providing stable generation of highly chirped (chirp parameter 40) pulses compressed in an output piece of standard PM fiber to 165 fs. The results of the numerical simulation agree well with the experiment. The analyzed intracavity pulse dynamics enables the classification of the generated pulses as dissipative solitons.

Spectral comb of highly chirped pulses generated via cascaded FWM of two frequency-shifted dissipative solitons.

Dissipative solitons generated in normal-dispersion mode-locked lasers are stable localized coherent structures with a mostly linear frequency modulation (chirp). The soliton energy in fiber lasers is limited by the Raman effect, but implementation of the intracavity feedback at the Stokes-shifted wavelength enables synchronous generation of a coherent Raman dissipative soliton. Here we demonstrate a new approach for generating chirped pulses at new wavelengths by mixing in a highly-nonlinear fiber of these two frequency-shifted dissipative solitons, as well as cascaded generation of their clones forming in the spectral domain a comb of highly chirped pulses. We observed up to eight equidistant components in the interval of more than 300 nm, which demonstrate compressibility from ~10 ps to ~300 fs. This approach, being different from traditional frequency combs, can inspire new developments in fundamental science and applications such as few-cycle/arbitrary-waveform pulse synthesis, comb spectroscopy, coherent communications and bio-imaging.

Light self-focusing in the atmosphere: thin window model.

Ultra-high power (exceeding the self-focusing threshold by more than three orders of magnitude) light beams from ground-based laser systems may find applications in space-debris cleaning. The propagation of such powerful laser beams through the atmosphere reveals many novel interesting features compared to traditional light self-focusing. It is demonstrated here that for the relevant laser parameters, when the thickness of the atmosphere is much shorter than the focusing length (that is, of the orbit scale), the beam transit through the atmosphere in lowest order produces phase distortion only. This means that by using adaptive optics it may be possible to eliminate the impact of self-focusing in the atmosphere on the laser beam. The area of applicability of the proposed "thin window" model is broader than the specific physical problem considered here. For instance, it might find applications in femtosecond laser material processing.

Mitigation of nonlinear transmission effects for OFDM 16-QAM optical signal using adaptive modulation.

The impact of the fiber Kerr effect on error statistics in the nonlinear (high power) transmission of the OFDM 16-QAM signal over a 2000 km EDFA-based link is examined. We observed and quantified the difference in the error statistics for constellation points located at three power-defined rings. Theoretical analysis of a trade-off between redundancy and error rate reduction using probabilistic coding of three constellation power rings decreasing the symbol-error rate of OFDM 16-QAM signal is presented. Based on this analysis, we propose to mitigate the nonlinear impairments using the adaptive modulation technique applied to the OFDM 16-QAM signal. We demonstrate through numerical modelling the system performance improvement by the adaptive modulation for the large number of OFDM subcarriers (more than 100). We also show that a similar technique can be applied to single carrier transmission.

Cascaded generation of coherent Raman dissipative solitons.

The cascaded generation of a conventional dissipative soliton (at 1020 nm) together with Raman dissipative solitons of the first (1065 nm) and second (1115 nm) orders inside a common fiber laser cavity is demonstrated experimentally and numerically. With sinusoidal (soft) spectral filtering, the generated solitons are mutually coherent at a high degree and compressible down to 300 fs. Numerical simulation shows that an even higher degree of coherence and shorter pulses could be achieved with step-like (hard) spectral filtering. The approach can be extended toward a high-order coherent Raman dissipative soliton source offering numerous applications such as frequency comb generation, pulse synthesis, biomedical imaging, and the generation of a coherent mid-infrared supercontinuum.

Inverse four-wave-mixing and self-parametric amplification effect in optical fibre.

An important group of nonlinear processes in optical fibre involves the mixing of four waves due to the intensity dependence of the refractive index. It is customary to distinguish between nonlinear effects that require external/pumping waves (cross-phase modulation and parametric processes such as four-wave mixing) and self-action of the propagating optical field (self-phase modulation and modulation instability). Here, we present a new nonlinear self-action effect, self-parametric amplification (SPA), which manifests itself as optical spectrum narrowing in normal dispersion fibre, leading to very stable propagation with a distinctive spectral distribution. The narrowing results from an inverse four-wave mixing, resembling an effective parametric amplification of the central part of the spectrum by energy transfer from the spectral tails. SPA and the observed stable nonlinear spectral propagation with random temporal waveform can find applications in optical communications and high power fibre lasers with nonlinear intra-cavity dynamics.

Feedback-controlled Raman dissipative solitons in a fiber laser.

Energy of chirped dissipative solitons (DS) generated in fiber lasers may exceed a threshold of stimulated Raman scattering (SRS) leading to formation of a noisy Raman pulse (RP). As we demonstrated recently, a feedback loop providing re-injection of the Raman pulse into the laser cavity can form a Raman dissipative soliton (RDS) with similar characteristics to those of the main dissipative soliton. Here, we present the results of feedback optimization of the generated RDS spectra. First experimental results of coherent combining of DS and RDS are also shown.

Multicolour nonlinearly bound chirped dissipative solitons.

The dissipative soliton regime is one of the most advanced ways to generate high-energy femtosecond pulses in mode-locked lasers. On the other hand, the stimulated Raman scattering in a fibre laser may convert the excess energy out of the coherent dissipative soliton to a noisy Raman pulse, thus limiting its energy. Here we demonstrate that intracavity feedback provided by re-injection of a Raman pulse into the laser cavity leads to formation of a coherent Raman dissipative soliton. Together, a dissipative soliton and a Raman dissipative soliton (of the first and second orders) form a two (three)-colour stable complex with higher total energy and broader spectrum than those of the dissipative soliton alone. Numerous applications can benefit from this approach, including frequency comb spectroscopy, transmission lines, seeding femtosecond parametric amplifiers, enhancement cavities and multiphoton fluorescence microscopy.