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

This paper describes the design and implementation of wide-band Raman amplifiers for fiber-optic telecommunications systems. All-Raman amplifiers permit 100nm wide systems over spans of over 1500km due to the low noise figure and reduced nonlinear system penalties. First, the enabling technologies for realizing 50-100nm transmission systems are reviewed. Then, the focus is on the key building block for wideband systems, namely the all-Raman optical amplifier. The advantages and challenges of all-Raman wideband amplifiers (WBA) are first reviewed. Then, Section 3 describes the physical principles and engineering design rules for construction of all-Raman WBAs that satisfy gain and noise figure performance requirements of typical long-haul and ultra-long-haul fiber-optic transmission systems. Exemplary wideband system experiments are described in Section 4, and then the paper is summarize and concluded in Section 5.

Raman amplifiers for telecommunications: physical principles to systems

We describe a novel architecture for an all-optical time-domain chirp switch in which digital logic is based on timeshift keying. This architecture is a generalization of fiber soliton-dragging logic gates that have a switching energy approaching 1 pJ. By using solitons we separate the nonlinear chirping from the time shifting and, consequently, reduce the required phase shift during the nonlinear interaction. We discuss the scaling laws for energy and latency versus pulse width and show that the chirp switches have low switching energies for high-bit-rate applications.

All-optical time-domain chirp switches.

In one aspect of the invention, a polarization controller includes a first polarization beam splitter operable to receive an input optical signal having an input state of polarization and to separate the signal into a first and a second principal mode of polarization. The polarization controller further includes at least three stages of phase shifters each operable to introduce a phase shift between the first and second principal modes, at least one phase shifter comprising a beam splitter that is shared with at least one other of the phase shifters. The at least three stages of phase shifters include a first stage coupled to the first polarization beam splitter and a last stage coupled to a second polarization beam splitter. The second polarization beam splitter is operable to receive phase shifted copies of the first and second principal modes, and to align the phase shifted copies of the principal modes to an output state of polarization.

Apparatus and method for controlling polarization of an optical signal

Recent experiments in birefringent optical fibers in which a signal pulse in one polarization is used to switch a control pulse in another polarization are affected by the Raman self-frequency shift. These experiments are modeled numerically, with the experimentally measured Raman profiles used as input to the simulations. Both parallel and perpendicular Raman gain are kept. The effect of keeping the full Raman response rather than just an often-used linear approximation is discussed. The experimental results are in good agreement with theory, although some discrepancies exist. The possibility that these discrepancies could be due to errors in the measurements of the low-frequency portion of the perpendicular Raman gain is examined and ruled out. Other possible sources of this discrepancy are then discussed.

Numerical study of the Raman effect and its impact on soliton-dragging logic gates.

We synchronized two passively mode-locked erbium-doped fiber lasers using a phase lock loop with a large dynamic range and bandwidth, which is realized by using a novel acoustooptic-modulator-grating scheme. Cross-correlation of the two lasers shows the interlaser jitter is under 2 ps (same as the laser pulse width) for period as long as hours. To prove the quality of phase locking, we apply synchronized lasers in two all-optical network applications, one of which requires the lasers to have the same wavelength and the second requires the lasers to be at different wavelengths. In the single wavelength application, the synchronized lasers drive a cascade of two low-birefringence, polarization maintaining, optical logic gates with switching timing window of 4 and 5 ps, respectively. We obtain nonlinear transmission of /spl sim/50% at a switching energy of 8 pJ and contrast ratio of 16 dB, which are comparable performance as that obtained using a single laser. In the different wavelength application, we use 0.8 ps pulses to switch 2 ps pulses in a two-wavelength nonlinear optical loop mirror demultiplexer with timing window of 5.5 ps. Stable switching is reached at a efficiency as high as 90% at switching energy of 0.8 pJ, and a contrast ratio of 20 dB. Excellent agreement is found between the experimental data and the simulated results, which exclude the timing jitter.

Mohammed N. Islam

Papers

Raman amplifiers for telecommunications: physical principles to systems

All-optical time-domain chirp switches.

Apparatus and method for controlling polarization of an optical signal

Numerical study of the Raman effect and its impact on soliton-dragging logic gates.

Synchronization of passively mode-locked erbium-doped fiber lasers and its application to optical communication networks