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

Packet-drop function for a time-division multiplexing network using 100-Gb/s, 8-b words is experimentally demonstrated by integrating all-optical header processing and payload demultiplexing with electrooptic packet routing. The header processor consists of two levels of all-optical logic gates based on low birefringent nonlinear optical loop mirrors (NOLMs), and the payload demultiplexer is a two-wavelength NOLM. Both devices are driven by synchronized lasers with timing jitter under 1 ps. The contrast ratios for both header processor and demultiplexer are 10:1 and that of the packet router is 17 dB. The switching energies for header processing and payload reading are 10 and 1 pJ/pulse, respectively.

All-optical packet-drop demonstration using 100-Gb/s words by integrating fiber-based components

Disclosed is an all waveguide fiber wavelength converter which makes use of modulational instability to convert signal wavelength over a conversion bandwidth while maintaining a low pump laser power relative to other wavelength conversion devices such as those which make use of four wave mixing. The device is operated in the anomalous dispersion regions of the waveguides and the zero dispersion of the waveguide in which conversion occurs is less than the pump wavelength so that conversion may occur for signal wavelengths above and below the zero dispersion wavelength. Conversion efficiency is in the range of 25 dB to 30 dB.

Modulation instability wavelength converter

In this letter, we demonstrate a photonic analog-to-digital converter with time stretch (TS) preprocessor that has a sampling rate of 130 GSa/s. The system has a signal-to-noise ratio (SNR) exceeding seven effective number of bits over a 1-GHz bandwidth at 18 GHz. We present an analytical model of the SNR in the TS preprocessor which shows that over the specified bandwidth, the SNR is limited by the amplified spontaneous emission beat noise.

130-GSa/s photonic analog-to-digital converter with time stretch preprocessor

We demonstrate a high-power, multi-wavelength, short pulse source at 10 Gb/s based on spectral slicing of supercontinuum (SC) generated in short fibers. We show that short fiber SC can be used for dense wavelength division multiplexing applications because of its >7.9 dBm/nm power spectral density, 140 nm spectral bandwidth, and /spl plusmn/0.5 dB spectral uniformity over 40 mn. Pulse carving up to 60 nm away from the pump wavelength and CW generation by longitudinal mode carving indicates that the coherence of the SC is maintained. By using high nonlinearity fibers, the spectral bandwidth is increased to 250 nm, which can accommodate >600 wavelength channels with 50 GHz channel spacing and >6 Tb/s aggregate data rate. We also calculate the coherence degradation due to amplification of incoherent energy during the SC generation. Theoretical results show that the SC generation in short fibers has 13 dB higher signal-to-noise ratio (SNR) compared to the SC generated in long fiber.

10 Gb/s multiple wavelength, coherent short pulse source based on spectral carving of supercontinuum generated in fibers

Combinatorial (Boolean) logic functions are provided by ultrafast optical logic devices which utilize soliton trapping between two optical signals propagating in a birefringent fiber. The logic devices are three terminal devices having orthogonally polarized soliton input signals and a single output signal. Optically filtering the output from the fiber permits the desired combinatorial logic operation to be performed on the input optical signals. Logic operations include AND, exclusive-OR, NOT, and NOR functions. In operation, the devices exhibit phase insensitivity, low switching energy, high contrast ratio between output logic levels, and cascadability. In one embodiment of the invention, a first optical signal and a second optical signal are optically coupled into the principal axes of a birefringent fiber. A Fabry Perot etalon centered at the center frequency of both the first and second signals is utilized to realize a exclusive-OR operation whereas centering the etalon on the frequency related to the spectral shift caused by soliton trapping realizes an AND operation.

Mohammed N. Islam

Papers

All-optical packet-drop demonstration using 100-Gb/s words by integrating fiber-based components

Modulation instability wavelength converter

130-GSa/s photonic analog-to-digital converter with time stretch preprocessor

10 Gb/s multiple wavelength, coherent short pulse source based on spectral carving of supercontinuum generated in fibers

Ultra-fast optical logic devices