Correlations and screening of topological charges in Gaussian random fields

Effective pair-interaction of phase singularities in random waves

In two-dimensional random waves, phase singularities are point-like dislocations with a behavior reminiscent of interacting particles. This-qualitative-consideration stems from the spatial arrangement of these entities, which finds its hallmark in a pair correlation reminiscent of a liquid-like system. Starting from their pair correlation function, we derive an effective pair-interaction for phase singularities in random waves by using a reverse Monte Carlo method. This study initiates a new, to the best of our knowledge, approach for the treatment of singularities in random waves and can be generalized to topological defects in any system.

Screening and fluctuation of the topological charge in random wave fields

Vortices, phase singularities, and topological defects of any kind often reflect information that is crucial for understanding physical systems in which such entities arise. With near-field experiments supported by numerical calculations, we determine the fluctuations of the topological charge for phase singularities in isotropic random waves as a function of the size R of the observation window. We demonstrate that for two-dimensional fields such fluctuations increase with a superlinear scaling law, consistent with a R log R behavior. Additionally, we show that such scaling remains valid in the presence of anisotropy.

Singularity screening in generic optical fields

We revisit the widely studied subject of screening in optical fields by topological charges and show that screening does not depend on charge ordering. Instead, for an array of N charges, screening requires that the variance of the charge fluctuations be small compared to N. We show by means of explicit examples that, when this requirement is met, screening can be complete, even for a spatially random arrangement of charges. We derive a minimal screening constraint on the charge correlation function and show that it is this constraint that is met in practice, rather than the more stringent constraints previously assumed.

Critical and umbilical points of a non-Gaussian random field

Random fields in nature often have, to a good approximation, Gaussian characteristics. For such fields, the number of maxima and minima are the same. Furthermore, the relative densities of umbilical points, topological defects which can be classified into three types, have certain fixed values. Phenomena described by nonlinear laws can, however, give rise to a non-Gaussian contribution, causing a deviation from these universal values. We consider a random surface, whose height is given by a nonlinear function of a Gaussian field. We find that, as a result of the non-Gaussianity, the density of maxima and minima no longer match and we calculate the relative imbalance between the two. We also calculate the change in the relative density of umbilics. This allows us not only to detect a perturbation, but to determine its size as well. This geometric approach offers an independent way of detecting non-Gaussianity, which even works in cases where the field itself can not be probed directly.

Extrema statistics in the dynamics of a non-Gaussian random field

When the equations that govern the dynamics of a random field are nonlinear, the field can develop with time non-Gaussian statistics even if its initial condition is Gaussian. Here, we provide a general framework for calculating the effect of the underlying nonlinear dynamics on the relative densities of maxima and minima of a two-dimensional field. Using this simple geometrical probe, we can identify the size of the non-Gaussian contributions in the random field, or alternatively the magnitude of the nonlinear terms in the underlying equations of motion. We demonstrate our approach by applying it to an initially Gaussian field that evolves according to the deterministic KPZ equation, which models surface growth and shock dynamics.

Stochastic geometry and topology of non-Gaussian fields

Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher-order correlation functions. In the present work, we take a different route. We show how geometric and topological properties of Gaussian fields, such as the statistics of extrema, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. In our treatment, we consider both local and nonlocal mechanisms that generate non-Gaussian fields, both statically and dynamically through nonlinear diffusion.

Density and correlation functions of vortex and saddle points in open billiard systems

We present microwave measurements for the density and spatial correlation of current critical points in an open billiard system and compare them with new and previous predictions of the random-wave model (RWM). In particular, due to an improvement of the experimental setup, we determine experimentally the spatial correlation of saddle points of the current field. An asymptotic expression for the vortex-saddle and saddle-saddle correlation functions based on the RWM is derived, with experiment and theory agreeing well. We also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the RWM and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying Dirichlet conditions.

Singularities in speckled speckle: screening

We study screening of optical singularities in random optical fields with two widely different length scales. We call the speckle patterns generated by such fields speckled speckle, because the major speckle spots in the pattern are themselves highly speckled. We study combinations of fields whose components exhibit short- and long-range correlations and find unusual forms of screening.

Statistics of nodal points of in-plane random waves in elastic media

We consider the nodal points (NPs) u=0 and v=0 of the in-plane vectorial displacements u=(u,v) which obey the Navier-Cauchy equation. Similar to the Berry conjecture of quantum chaos, we present the in-plane eigenstates of chaotic billiards as the real part of the superposition of longitudinal and transverse plane waves with random phases. By an average over random phases we derive the mean density and correlation function of NPs. Consequently we consider the distribution of the nearest distances between NPs.

Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields

There exists a substantial body of theory that predicts mutual screening of signed topological singularities (topological charges) in random optical fields (speckle patterns). Such screening appears to be rather mysterious because there are neither energetic nor entropic reasons for its existence. We present the first experimental confirmation of mutual screening by the stationary points of the intensity, the canonical optical scalar field, and of mutual screening by C points in elliptically polarized light, the generic optical vector field. We also elucidate specific aspects of the geometry and topology of these fields that we argue give rise to screening.

Observation of singularities in multiply scattered microwave fields

Speckle patterns of arbitrary resolution are obtained by applying the sampling theorem to measurements of two orthogonal components of the microwave field transmitted through multiply scattering samples. Core structures of phase singularities, phase critical points, and polarization singularities are explored. We find that equiphase lines connect phase singularities with opposite topological signs except for the bifurcation lines, which run through a phase saddle point, in agreement with predictions by Freund [Phys. Rev. E25, 2348 (1995)]. We observe hyperbolic equiphase lines near phase saddle points and elliptical equiphase lines around phase extrema. Polarization singularities of the vector field with the three morphologies predicted are observed.