We investigate the spatio-temporal dynamics of coupled chaotic systems with nonlocal interactions, where each element is coupled to its nearest neighbors within a finite range. Depending upon the coupling strength and coupling radius, we find characteristic spatial patterns such as wavelike profiles and study the transition from coherence to incoherence leading to spatial chaos. We analyze the origin of this transition based on numerical simulations and support the results by theoretical derivations, identifying a critical coupling strength and a scaling relation of the coherent profiles. To demonstrate the universality of our findings, we consider time-discrete as well as time-continuous chaotic models realized as a logistic map and a R\"ossler or Lorenz system, respectively. Thereby, we establish the coherence-incoherence transition in networks of coupled identical oscillators.

Transition from spatial coherence to incoherence in coupled chaotic systems.

Single-molecule biophysics has transformed our understanding of biology, but also of the physics of life. More exotic than simple soft matter, biomatter lives far from thermal equilibrium, covering multiple lengths from the nanoscale of single molecules to up to several orders of magnitude higher in cells, tissues and organisms. Biomolecules are often characterized by underlying instability: multiple metastable free energy states exist, separated by levels of just a few multiples of the thermal energy scale k B T, where k B is the Boltzmann constant and T absolute temperature, implying complex inter-conversion kinetics in the relatively hot, wet environment of active biological matter. A key benefit of single-molecule biophysics techniques is their ability to probe heterogeneity of free energy states across a molecular population, too challenging in general for conventional ensemble average approaches. Parallel developments in experimental and computational techniques have catalysed the birth of multiplexed, correlative techniques to tackle previously intractable biological questions. Experimentally, progress has been driven by improvements in sensitivity and speed of detectors, and the stability and efficiency of light sources, probes and microfluidics. We discuss the motivation and requirements for these recent experiments, including the underpinning mathematics. These methods are broadly divided into tools which detect molecules and those which manipulate them. For the former we discuss the progress of super-resolution microscopy, transformative for addressing many longstanding questions in the life sciences, and for the latter we include progress in 'force spectroscopy' techniques that mechanically perturb molecules. We also consider in silico progress of single-molecule computational physics, and how simulation and experimentation may be drawn together to give a more complete understanding. Increasingly, combinatorial techniques are now used, including correlative atomic force microscopy and fluorescence imaging, to probe questions closer to native physiological behaviour. We identify the trade-offs, limitations and applications of these techniques, and discuss exciting new directions.

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Single-molecule techniques in biophysics: a review of the progress in methods and applications.

The first article by Turunen and Friberg deals with a class of fields with propagation-invariant properties such as the optical intensity distribution. Coherent and partially coherent stationary and pulsed solutions are treated in view of scalar and electromagnetic approaches. Approximations of ideal propagation-invariant fields and methods for their generation are discussed. Finally, some application areas are covered.

Propagation-Invariant Optical Fields

We describe the programmable spatial beam shaping of 100-kHz, 4-?J amplified femtosecond pulses in a focal plane by wave-front modulation. Phase distributions are determined by a numerical iterative procedure. A nonpixelated optically addressed liquid-crystal light valve is used as a programmable wave-front tailoring device. Top-hat, doughnut, square, and triangle shapes of 20-?m size are obtained in a focal plane. Their suitability for femtosecond laser machining is demonstrated.

Programmable focal spot shaping of amplified femtosecond laser pulses.

We consider a variant of the Kuramoto model in which all the oscillators are now assumed to have the same natural frequency, but some of them are negatively coupled to the mean field. These contrarian oscillators tend to align in antiphase with the mean field, whereas, the positively coupled conformist oscillators favor an in-phase relationship. The interplay between these effects can lead to rich dynamics. In addition to a splitting of the population into two diametrically opposed factions, the system can also display traveling waves, complete incoherence, and a blurred version of the two-faction state. Exact solutions for these states and their bifurcations are obtained by means of the Watanabe-Strogatz transformation and the Ott-Antonsen ansatz. Curiously, this system of oscillators with identical frequencies turns out to exhibit more complicated dynamics than its counterpart with heterogeneous natural frequencies.

Conformists and contrarians in a Kuramoto model with identical natural frequencies.

A laser beam with phase singularities is an interesting object to study in optics and may have important applications in guiding atoms and molecules. We explore the characteristics of a singularity in a nondiffracting Bessel beam experimentally by use of a programmable spatial light modulator with 64-level phase holograms. The diffraction efficiency with 64-level phase holograms is greatly improved in comparison with that obtained with a binary grating. The experiments show that the size and deflection angle of the beam can be controlled in real time. The observations are in agreement with scalar diffraction theory.

Generation of nondiffracting Bessel beams by use of a spatial light modulator

We demonstrate generalized synchronization in a spatiotemporal chaotic system, a liquid crystal spatial light modulator with optoelectronic feedback.

Generalized Synchronization of Spatiotemporal Chaos in a Liquid Crystal Spatial Light Modulator

A pair of coupled erbium doped fiber ring lasers is used to explore the dynamics of coupled spatiotemporal systems. The lasers are mutually coupled with a coupling delay less than the cavity round-trip time. We study synchronization between the two lasers in the experiment and in a delay differential equation model of the system. Because the lasers are internally perturbed by spontaneous emission, we include a noise source in the model to obtain stochastic realizations of the deterministic equations. Both amplitude synchronization and phase synchronization are considered. We use the Hilbert transform to define the phase variable and compute phase synchronization. We find that synchronization increases with coupling strength in the experiment and the model. When the time series from two lasers are time shifted in either direction by the delay time, approximately equal synchronization is frequently observed, so that a clear leader and follower cannot be identified. We define an algorithm to determine which laser leads the other when the synchronization is sufficiently different with one direction of time shift, and statistics of switches in leader and follower are studied. The frequency of switching between leader and follower increases with coupling strength, as might be expected since the lasers mutually influence each other more effectively with stronger coupling.

Synchronization and time shifts of dynamical patterns for mutually delay-coupled fiber ring lasers.

Two-dimensional spatial light modulators have been employed to create static and dynamic phase masks for embedding multiple vortices and exotic intensity-void structures in laser beams. A variety of patterns of singularities, producing dark longitudinal and transverse intensity channels, have been created. The uniformity, quality, and suitability of these patterns as elements for atom optics (e.g., atom-tunnel beam splitters) have been studied as a function of the phase quantization level and spatial resolution of the phase mask. Specifically, we show that (1) high-quality modes, those that propagate long distances and can be focused, can be generated when the number of phase steps between 0 and 2pi on the phase mask exceed four and (2) atom confinement increases with the charge of the vortex.

Elizabeth A. Rogers

Papers

Generation of nondiffracting Bessel beams by use of a spatial light modulator

Generalized Synchronization of Spatiotemporal Chaos in a Liquid Crystal Spatial Light Modulator

Synchronization and time shifts of dynamical patterns for mutually delay-coupled fiber ring lasers.

Laser beams with embedded vortices: tools for atom optics