A topical review of numerical and experimental studies of supercontinuum generation in photonic crystal fiber is presented over the full range of experimentally reported parameters, from the femtosecond to the continuous-wave regime. Results from numerical simulations are used to discuss the temporal and spectral characteristics of the supercontinuum, and to interpret the physics of the underlying spectral broadening processes. Particular attention is given to the case of supercontinuum generation seeded by femtosecond pulses in the anomalous group velocity dispersion regime of photonic crystal fiber, where the processes of soliton fission, stimulated Raman scattering, and dispersive wave generation are reviewed in detail. The corresponding intensity and phase stability properties of the supercontinuum spectra generated under different conditions are also discussed.

Supercontinuum generation in photonic crystal fiber

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

Understanding the formation of stars in galaxies is central to much of modern astrophysics. However, a quantitative prediction of the star formation rate and the initial distribution of stellar masses remains elusive. For several decades it has been thought that the star formation process is primarily controlled by the interplay between gravity and magnetostatic support, modulated by neutral-ion drift (known as ambipolar diffusion in astrophysics). Recently, however, both observational and numerical work has begun to suggest that supersonic turbulent flows rather than static magnetic fields control star formation. To some extent, this represents a return to ideas popular before the importance of magnetic fields to the interstellar gas was fully appreciated. This review gives a historical overview of the successes and problems of both the classical dynamical theory and the standard theory of magnetostatic support, from both observational and theoretical perspectives. The outline of a new theory relying on control by driven supersonic turbulence is then presented. Numerical models demonstrate that, although supersonic turbulence can provide global support, it nevertheless produces density enhancements that allow local collapse. Inefficient, isolated star formation is a hallmark of turbulent support, while efficient, clustered star formation occurs in its absence. The consequences of this theory are then explored for both local star formation and galactic-scale star formation. It suggests that individual star-forming cores are likely not quasistatic objects, but dynamically collapsing. Accretion onto these objects varies depending on the properties of the surrounding turbulent flow; numerical models agree with observations showing decreasing rates. The initial mass distribution of stars may also be determined by the turbulent flow. Molecular clouds appear to be transient objects forming and dissolving in the larger-scale turbulent flow, or else quickly collapsing into regions of violent star formation. Global star formation in galaxies appears to be controlled by the same balance between gravity and turbulence as small-scale star formation, although modulated by cooling and differential rotation. The dominant driving mechanism in star-forming regions of galaxies appears to be supernovae, while elsewhere coupling of rotation to the gas through magnetic fields or gravity may be important.

Control of star formation by supersonic turbulence

▪ Abstract Turbulence affects the structure and motions of nearly all temperature and density regimes in the interstellar gas. This two-part review summarizes the observations, theory, and simulations of interstellar turbulence and their implications for many fields of astrophysics. The first part begins with diagnostics for turbulence that have been applied to the cool interstellar medium and highlights their main results. The energy sources for interstellar turbulence are then summarized along with numerical estimates for their power input. Supernovae and superbubbles dominate the total power, but many other sources spanning a large range of scales, from swing-amplified gravitational instabilities to cosmic ray streaming, all contribute in some way. Turbulence theory is considered in detail, including the basic fluid equations, solenoidal and compressible modes, global inviscid quadratic invariants, scaling arguments for the power spectrum, phenomenological models for the scaling of higher-order structu...

http://www.ifa.hawaii.edu/%7Ereipurth/reviews/elmegreen1_araa.pdf

Interstellar Turbulence I: Observations and Processes

/pdf/control-of-star-formation-by-supersonic-turbulence-7wf7lt35b2.pdf

Control of Star Formation by Supersonic Turbulence

Aspects of the interaction of the Kerr nonlinearity and polarization mode dispersion (PMD) are reviewed. The basic equation that governs this interaction on the length scale of interest in optical fiber communications systems is the Manakov-PMD equation. This equation is derived using multiple-length-scale techniques. The focus of the derivation is the elucidation of common misunderstandings and pitfalls rather than mathematical rigor. It is shown that the scalar nonlinear Schro/spl uml/dinger equation is valid when PMD is absent and the signal is initially in a single polarization state. Two examples are then presented that illustrate the complexity of the interaction between nonlinearity and PMD. The first example considers the interaction of a nonlinearly induced chirp with PMD. As the power increases, one can obtain an improved eye opening relative to the case when PMD is absent. The second example considers the effect of nonlinear polarization rotation in a wavelength-division-multiplexed system. When nonlinear polarization rotation is important, the principal states of polarization become time dependent and PMD compensation becomes ineffective. This problem can be mitigated through the use of line codes.

/pdf/interaction-of-polarization-mode-dispersion-and-nonlinearity-2mtq6f8esx.pdf

Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems

Evolution of cold shock-bounded slabs

We calculate the pulse compression in a tapered microstructure optical fiber with four layers of holes. We show that the primary limitation on pulse compression is the loss due to mode leakage. As a fiber’s diameter decreases due to the tapering, so does the air-hole diameter, and at a sufficiently small diameter the guided mode loss becomes unacceptably high. For the four-layer geometry we considered, a compression factor of 10 can be achieved by a pulse with an initial FWHM duration of 3 ps in a tapered fiber that is 28 m long. We find that there is little difference in the pulse compression between a linear taper profile and a Gaussian taper profile. More layers of air-holes allows the pitch to decrease considerably before losses become unacceptable, but only a moderate increase in the degree of pulse compression is obtained.

/pdf/pulse-compression-using-a-tapered-microstructure-optical-2jpxpmyuu5.pdf

Pulse compression using a tapered microstructure optical fiber.

Nonlinearity in p-i-n photodetectors leads to power generation at harmonics of the input frequency, limiting the performance of RF-photonic systems. We use one-dimensional and two-dimensional simulations of the drift-diffusion equations to determine the physical origin of the saturation in a simple heterojunction p-i-n photodetector at room temperature. Incomplete ionization, external loading, impact ionization, and the Franz–Keldysh effect are all included in the model. Impact ionization is the main source of nonlinearity at large reverse bias (
 $>$
 10 V in the device that we simulated). The electron and hole current contributions to the second harmonic power were calculated. We find that impact ionization has a greater effect on the electrons than it does on the holes. We also find that the hole velocity saturates slowly with increasing reverse bias, and the hole current makes a large contribution to the harmonic power at 10 V. This result implies that decreasing the hole injection will decrease the harmonic power.

/pdf/modeling-sources-of-nonlinearity-in-a-simple-p-i-n-4vbwpn9uz6.pdf

Modeling Sources of Nonlinearity in a Simple p-i-n Photodetector

We develop robust computational methods, referred to as boundary tracking algorithms, that can rapidly determine the existence and stability of pulses in passively mode-locked laser systems over a broad parameter range. Applying the boundary tracking algorithms to the cubic–quintic mode-locking equation, we find a rich dynamical structure.

/pdf/boundary-tracking-algorithms-for-determining-the-stability-57k8510zmt.pdf

Brian S. Marks

Papers

Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems

Evolution of cold shock-bounded slabs

Pulse compression using a tapered microstructure optical fiber.

Modeling Sources of Nonlinearity in a Simple p-i-n Photodetector

Boundary tracking algorithms for determining the stability of mode-locked pulses