A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

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

Recent observations show that the probability of encountering an extremely large rogue wave in the open ocean is much larger than expected from ordinary wave-amplitude statistics. Although considerable effort has been directed towards understanding the physics behind these mysterious and potentially destructive events, the complete picture remains uncertain. Furthermore, rogue waves have not yet been observed in other physical systems. Here, we introduce the concept of optical rogue waves, a counterpart of the infamous rare water waves. Using a new real-time detection technique, we study a system that exposes extremely steep, large waves as rare outcomes from an almost identically prepared initial population of waves. Specifically, we report the observation of rogue waves in an optical system, based on a microstructured optical fibre, near the threshold of soliton-fission supercontinuum generation--a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input. We model the generation of these rogue waves using the generalized nonlinear Schrodinger equation and demonstrate that they arise infrequently from initially smooth pulses owing to power transfer seeded by a small noise perturbation.

Optical rogue waves

Metastasis, the dissemination and growth of neoplastic cells in an organ distinct from that in which they originated, is the most common cause of death in cancer patients. This is particularly true for pancreatic cancers, where most patients are diagnosed with metastatic disease and few show a sustained response to chemotherapy or radiation therapy. Whether the dismal prognosis of patients with pancreatic cancer compared to patients with other types of cancer is a result of late diagnosis or early dissemination of disease to distant organs is not known. Here we rely on data generated by sequencing the genomes of seven pancreatic cancer metastases to evaluate the clonal relationships among primary and metastatic cancers. We find that clonal populations that give rise to distant metastases are represented within the primary carcinoma, but these clones are genetically evolved from the original parental, non-metastatic clone. Thus, genetic heterogeneity of metastases reflects that within the primary carcinoma. A quantitative analysis of the timing of the genetic evolution of pancreatic cancer was performed, indicating at least a decade between the occurrence of the initiating mutation and the birth of the parental, non-metastatic founder cell. At least five more years are required for the acquisition of metastatic ability and patients die an average of two years thereafter. These data provide novel insights into the genetic features underlying pancreatic cancer progression and define a broad time window of opportunity for early detection to prevent deaths from metastatic disease.

SF-010-4 Distant metastasis occurs late during the genetic evolution of pancreatic cancer

Ultrafast lasers, which generate optical pulses in the picosecond and femtosecond range, have progressed over the past decade from complicated and specialized laboratory systems to compact, reliable instruments. Semiconductor lasers for optical pumping and fast optical saturable absorbers, based on either semiconductor devices or the optical nonlinear Kerr effect, have dramatically improved these lasers and opened up new frontiers for applications with extremely short temporal resolution (much smaller than 10 fs), extremely high peak optical intensities (greater than 10 TW/cm2) and extremely fast pulse repetition rates (greater than 100 GHz).

Recent developments in compact ultrafast lasers

We theoretically and experimentally study the chirping mechanism responsible for the time shifts in soliton-dragging logic gates. Cross-phase modulation during the first one or two walk-off lengths in a birefringent optical fiber causes most of the frequency shift that translates into a time shift after propagation in a soliton dispersive delay line. Introducing gain or loss during the pulse interaction asymmetrizes the pulse walk-off and, consequently, increases the temporal window over which soliton dragging can result in a time shift. Analytic formulas for the time shift provide the scaling laws as a function of various fiber parameters.

Chirp mechanisms in soliton-dragging logic gates.

We demonstrate a novel self-synchronization scheme for high-speed packet time-division-multiplexed (TDM) networks using a fast saturated, but slowly recovered gain (or loss) combined with intensity discrimination. The marker pulse used in this self-synchronization scheme is the same as other pulses in the packets. In particular, we use a semiconductor optical amplifier (SOA) as the fast saturated but slowly-recovered gain element, and an unbalanced nonlinear optical loop mirror or a dispersion-shifted fiber/optical filter combination as an intensity discriminator. The contrast ratio of the first pulse to the remaining pulses of a 100 Gb/s packet, "10111000" is >3 dB at the output of the SOA and >20 dB at the outputs of both intensity discriminators.

Novel self-synchronization scheme for high-speed packet TDM networks

Third order cascaded Raman shifting is used to generate light to 1867 nm in sulfide fibers, and the nonlinearity is measured to be ~5.7 times 10-12 (m/W). Damage at ~1 GW/cm2 limits the wavelength shift range.

Third order cascaded Raman wavelength shifting in chalcogenide fibers

A mid-infrared system for optical probing is disclosed that comprises a mid-infrared fiber laser based on cascaded Raman wavelength shifting, a sample volume, and a detector or detection system. The cascaded Raman wavelength shifting process in optical fibers involves the emission of a plurality of optical phonons for at least some of the pump photonics involved in the process. As one example, using the cascaded Raman wavelength shifting process a pump laser wavelength between 1 and 2 μm can be shifted down to between 2.5 to 10 μm. In one embodiment, the mid-infrared fiber laser comprises a pump laser with a wavelength between 1 and 2 μm, one or more stages of cascaded Raman oscillators implemented in fused silica fiber, and one or more stages of cascaded Raman oscillators implemented in mid-infrared fiber that transmits beyond 2 μm. Examples of mid-infrared fibers include chalcogenides, fluorides and tellurite fibers. The output wavelength from the mid-infrared fiber laser is at an exemplary wavelength longer than 2.5 μm. The mid-infrared optical probing system can be used in applications such as semiconductor process control, combustion monitoring for engines, and defense or homeland security applications such as chemical sensing and infrared counter-measures.

Mid-infrared fiber laser using cascaded Raman wavelength shifting

Remember when we optimized our computer programs to minimize the memory usage, even at the expense of readability of the code? Now, with an abundance of memory, we no longer worry about the memory requirements. Remember when we minimized the gate count in a circuit by using techniques such as Karnaugh maps? Now, with VLSI and ULSI chips, gate count is no longer an issue. Yet in telecommunications many applications still need to be optimized to fit within bandwidth limitations. Researchers in optics and photonics dream of making bandwidth an inexpensive, virtually limitless commodity, so that we will be able to tailor applications to optimize their functionality without regard to bandwidth.

Mohammed N. Islam

Papers

Chirp mechanisms in soliton-dragging logic gates.

Novel self-synchronization scheme for high-speed packet TDM networks

Third order cascaded Raman wavelength shifting in chalcogenide fibers

Mid-infrared fiber laser using cascaded Raman wavelength shifting

Ultrafast Switching with Nonlinear Optics