Threshold For Pattern Formation Research Articles

Theory of nanoscale ripple topographies produced by ion bombardment near the threshold for pattern formation.

Nanoscale pattern formation on the surface of a solid that is bombarded with a broad ion beam is studied for angles of ion incidence, θ, just above the threshold angle for ripple formation, θ_{c}. We carry out a systematic expansion in powers of the small parameter ε≡(θ-θ_{c})^{1/2} and retain all terms up to a given order in ε. In the case of two diametrically opposed, obliquely incident beams, the equation of motion close to threshold and at sufficiently long times is rigorously shown to be a particular version of the anisotropic Kuramoto-Sivashinsky equation. We also determine the long-time, near-threshold scaling behavior of the rippled surface's wavelength, amplitude, and transverse correlation length for this case. When the surface is bombarded with a single obliquely incident beam, linear dispersion plays a crucial role close to threshold and dramatically alters the behavior: highly ordered ripples can emerge at sufficiently long times and solitons can propagate over the solid surface. A generalized crater function formalism that rests on a firm mathematical footing is developed and is used in our derivations of the equations of motion for the single and dual beam cases.

Influence of particle dynamics on the instability for pattern formation in shallow pulsed beds

A granular layer can form standing-wave patterns, such as squares, stripes, and hexagons, when it is fluidized with a pulsed gas flow. These patterns resemble the well-known patterns formed in vertically vibrated granular layers, but are governed by different dimensionless numbers. Recent research [de Martin et al., Phys. Rev. Fluids 3, 034303 (2018)] reveals that the onset to pattern formation in shallow pulsed beds can be understood in terms of the dimensionless number Gamma(p) = u(a )/ u(t)(phi) over bar, where u(a), is the amplitude of the gas velocity, u(t) is the terminal velocity of the particles, and (phi) over bar is the average solids volume fraction. In contrast, pattern formation in vertically vibrated granular layers in vacuo is governed by the dimensionless number Gamma(v) = 4 pi(2) f(2) d/g, where f and d are the frequency and displacement of the vibrated plate, respectively, and g is the gravitational acceleration. In addition, the threshold for pattern formation in pulsed beds exhibits a strong dependence with the frequency of the excitation that is not observed in the threshold for pattern formation in vibrated systems. This work explores the origin of these differences by simulating the dynamics of a one-dimensional pulsed array of particles. Simulations reproduce well the experimental stability curves, and reveal that the criterion for instability in shallow pulsed and vibrated systems is actually the same; the layer flight time must be equal to 1/f. In pulsed beds, this criterion is determined by the traveling time of the kinematic wave that forms in each flow pulse. These results provide a theoretical basis to the recent experimental observations and highlights commonalities between the mechanisms behind pattern formation in thin vibrated granular layers and shallow pulsed fluidized beds.

Modeling population patterns of chemotactic bacteria in homogeneous porous media

The spatio-temporal distribution of subsurface microorganisms determines their efficiency in providing essential ecosystem services such as the degradation of organic matter, the remineralization of carbon and nitrogen, or the remediation of anthropogenic contaminants. Populations of motile, chemotactic bacteria have been shown to be capable of pattern formation even in the absence of environmental heterogeneities. Focusing on the water saturated domain of the subsurface ( e.g., aquatic sediments, porous aquifers), we analyze this innate capability of bacterial populations in an idealized model of a homogeneous, saturated porous medium. Considering a linear array of connected, identical microhabitats populated by motile, chemotactic bacterial cells, we identify prerequisites for pattern formation, analyze types of patterns, and assess their impact on substrate utilization. In our model, substrate supplied to the microhabitats facilitates bacterial growth, and microbial cells can migrate between neighboring microhabitats due to (i) random motility, (ii) chemotaxis towards substrate, and (iii) self-attraction. A precondition for inhomogeneous population patterns is analytically derived, stating that patterns are possible if the self-attraction exceeds a threshold defined by the random motility and the steady state population density in the microhabitats. An individual-based implementation of the model shows that static and dynamic population patterns can unfold. Degradation efficiency is highest for homogeneous bacterial distributions and decreases as pattern formation commences. If during biostimulation efforts the carrying capacity of the microhabitats is successively increased, simulation results show that degradation efficiency can unexpectedly decrease when the pattern formation threshold is crossed.

Effects of polarization on wavefront shape in a LC-SLM diffractive feedback system

The effect of polarization on the phase distribution across a wavefront is discussed for a liquid crystal spatial light modulator based nonlinear diffractive single-feedback system. Theoretical analysis and numerical simulations show that when the polarization of the input light beam and interference effects are considered the form of the spectral phase ratio is altered considerably. It is found that this ratio can become zero or even negative for phase distributions with a wide range of spatial frequencies. This may have implications for high-resolution adaptive optics applications where, by careful choice of the polarization angle, very good suppression of phase distortions can be obtained. This can be achieved at relatively low feedback intensities and before the pattern formation threshold is reached.

Order Parameter Fragmentation after a Symmetry-Breaking Transition

As a nonlinear optical system consisting of a Kerr medium inserted in a feedback loop is exposedto a light intensity growing linearly from below to above the threshold for pattern formation, thecritical slowing down around threshold freezes the defect population. The measured number of defectsimmediately after the transition scales with the quench time as predicted by Zurek for a two-dimensionalGinzburg-Landau model. The further temporal evolution of the defect number is in agreement with asimple annihilation model, once the drift of defects specific for our system is taken into account.

Selection of Unstable Patterns and Control of Optical Turbulence by Fourier Plane Filtering

We report on selection and stabilization of transverse optical patterns in a feedback mirror experiment. Amplitude filtering in the Fourier plane is used to select otherwise unstable spatial patterns. Optical turbulence observed for nonlinearities far above the pattern formation threshold is stabilized by low-pass filtering of the optical power spectrum. Experimental observations obtained with a photorefractive nonlinearity are consistent with calculations based on a generic, cubic nonlinearity.

Threshold for pattern formation in a medium with a local photorefractive response

Threshold equations for the instability of counterpropagating waves against spontaneous initiation of small-angle light patterns are analyzed for a medium with a local photorefractive response and various boundary conditions. These equations take into account all actual contributions to the reflection phase gratings. The form of the threshold equation is directly related to the spatial symmetry of the problem. We study the dependence of the threshold thickness of the crystal and the angle of light emission on experimentally controllable parameters and discuss the prospects for experimental studies of pattern formation in LiNbO3 crystals.

Optical pattern formation in a Kerr-like medium with feedback

Abstract We present a series of experiments on pattern formation in an optical system including a Liquid Crystal Light Valve (LCLV), that with good approximation behaves like a Kerr medium, inserted in a feedback loop. We discuss the physical origin of the length scales and of the various symmetries that are observed close to the threshold for pattern formation.