We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Rayleigh, Brillouin and Raman scatterings in fibers result from the interaction of photons with local material characteristic features like density, temperature and strain. For example an acoustic/mechanical wave generates a dynamic density variation; such a variation may be affected by local temperature, strain, vibration and birefringence. By detecting changes in the amplitude, frequency and phase of light scattered along a fiber, one can realize a distributed fiber sensor for measuring localized temperature, strain, vibration and birefringence over lengths ranging from meters to one hundred kilometers. Such a measurement can be made in the time domain or frequency domain to resolve location information. With coherent detection of the scattered light one can observe changes in birefringence and beat length for fibers and devices. The progress on state of the art technology for sensing performance, in terms of spatial resolution and limitations on sensing length is reviewed. These distributed sensors can be used for disaster prevention in the civil structural monitoring of pipelines, bridges, dams and railroads. A sensor with centimeter spatial resolution and high precision measurement of temperature, strain, vibration and birefringence can find applications in aerospace smart structures, material processing, and the characterization of optical materials and devices.

Recent progress in distributed fiber optic sensors.

We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized da-ta points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approxi-mation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data pointswith respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can bequite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimension-al Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.

Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment

In the 2015 review paper 'Petawatt Class Lasers Worldwide' a comprehensive overview of the current status of highpower facilities of >200 TW was presented. This was largely based on facility specifications, with some description of their uses, for instance in fundamental ultra-high-intensity interactions, secondary source generation, and inertial confinement fusion (ICF). With the 2018 Nobel Prize in Physics being awarded to Professors Donna Strickland and Gerard Mourou for the development of the technique of chirped pulse amplification (CPA), which made these lasers possible, we celebrate by providing a comprehensive update of the current status of ultra-high-power lasers and demonstrate how the technology has developed. We are now in the era of multi-petawatt facilities coming online, with 100 PW lasers being proposed and even under construction. In addition to this there is a pull towards development of industrial and multidisciplinary applications, which demands much higher repetition rates, delivering high-average powers with higher efficiencies and the use of alternative wavelengths: mid-IR facilities. So apart from a comprehensive update of the current global status, we want to look at what technologies are to be deployed to get to these new regimes, and some of the critical issues facing their development.

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Petawatt and exawatt class lasers worldwide

Random distributed feedback fibre lasers

Revealing the Buildup Dynamics of Harmonic Mode-Locking States in Ultrafast Lasers

Self-assembly of fundamental elements through weak, long-range interactions plays a central role in both supramolecular DNA assembly and bottom-up synthesis of nanostructures. Optical solitons, analogous in many ways to particles, arise from the balance between nonlinearity and dispersion and have been studied in numerous optical systems. Although both short- and long-range interactions between optical solitons have attracted extensive interest for decades, stable soliton supramolecules, with multiple aspects of complexity and flexibility, have thus far escaped experimental observation due to the absence of techniques for enhancing and controlling the long-range inter-soliton forces. Here we report that long-range soliton interactions originating from optoacoustic effects and dispersive-wave radiations can be precisely tailored in a fibre laser cavity, enabling self-assembly of large numbers of optical solitons into highly-ordered supramolecular structures. We demonstrate several features of such optical structures, highlighting their potential applications in optical information storage and ultrafast laser-field manipulation. Optical solitons have been studied in a variety of systems for their unique properties. Here, the authors report on optically observed solitonic supramolecules, made up of large-scale structures of many solitons interacting within a fiber cavity, and study their fundamental characteristics.

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Formation of optical supramolecular structures in a fibre laser by tailoring long-range soliton interactions

Optomechanically coupled modes in a fibre laser cavity make it possible to generate and robustly store gigahertz-rate soliton sequences.

All-optical bit storage in a fibre laser by optomechanically bound states of solitons

We observed narrow-linewidth lasing spikes in a coherent Brillouin random fiber laser, which uses a continuous-wave fiber laser as the pump source. In this random laser configuration, stimulated Brillouin scattering in a section of conventional single-mode fiber acts as the gain mechanism, while Rayleigh scattering in another section of the nonuniform fiber provides randomly distributed feedback. Above the lasing threshold, coherent random lasing spikes, with a linewidth of ~10 Hz, are measured at the top of the Brillouin gain spectrum. It results from high quality coherent resonance of the Brillouin Stokes light in this open laser cavity.

Observation of narrow linewidth spikes in the coherent Brillouin random fiber laser

We report a single frequency lasing phenomenon with a narrow linewidth of ∼3  kHz in cascaded fiber that is composed of three types of low-loss communication fibers. The Rayleigh scattering of the Brillouin Stokes light created in the middle fiber section along both directions is enhanced by the other two fiber sections. When the Brillouin gain of the middle fiber exceeds the effective loss of the Brillouin stokes light in a roundtrip, a narrow linewidth lasing is observed on the top of the Brillouin spectrum line of the middle fiber. To the best of our knowledge, it is the first report on Rayleigh scattering-assisted Brillouin lasing with single frequency and narrow linewidth in cascaded low-loss communication fibers.

Meng Pang

Papers

Revealing the Buildup Dynamics of Harmonic Mode-Locking States in Ultrafast Lasers

Formation of optical supramolecular structures in a fibre laser by tailoring long-range soliton interactions

All-optical bit storage in a fibre laser by optomechanically bound states of solitons

Observation of narrow linewidth spikes in the coherent Brillouin random fiber laser

Rayleigh scattering-assisted narrow linewidth Brillouin lasing in cascaded fiber.