Angular Intensity Correlation Research Articles

Bayesian structure determination from fluctuation X-ray scattering experiments

Fluctuation X-ray scattering is an emerging method for biomolecular structure determination, where scattering data of an ensemble of molecules in a dilute solution is collected using ultra-short X-ray pulses below rotational tumbling times. This allows to capture structural information in angular intensity correlations that are absent in traditional solution X-ray scattering methods, while also keeping the more biological conditions in solution compared to single-molecule X-ray scattering. However, the reconstruction of the molecular structure of the sample using the scattering images poses a significant challenge, since the orientations of the individual molecules in the solution are unknown and the low signal-to-noise ratio has to be overcome. We present a rigorous Bayesian framework that finds the molecular structure that has the largest probability given all recorded scattering images. In this approach, the orientation of each molecule explicitly appears in the likelihood function. We formally integrate out the unknown individual orientations, and to reduce the computational cost, this integration is approximated by a finite sum over randomly chosen orientations of each molecule. We show that our method can recover the molecular structure of a fictitious 12-atom molecule up to 4-Å resolution, using 2000 synthetic noise-free images of 50 randomly oriented copies each.

Model-independent particle species disentanglement by X-ray cross-correlation scattering

Mixtures of different particle species are often investigated using the angular averages of the scattered X-ray intensity. The number of species is deduced by singular value decomposition methods. The full disentanglement of the data into per-species contributions requires additional knowledge about the system under investigation. We propose to exploit higher-order angular X-ray intensity correlations with a new computational protocol, which we apply to synchrotron data from two-species mixtures of two-dimensional static test nanoparticles. Without any other information besides the correlations, we demonstrate the assessment of particle species concentrations in the measured data sets, as well as the full ab initio reconstruction of both particle structures. The concept extends straightforwardly to more species and to the three-dimensional case, whereby the practical application will require the measurements to be performed at an X-ray free electron laser.

Angular correlations of photons from solution diffraction at a free-electron laser encode molecular structure.

During X-ray exposure of a molecular solution, photons scattered from the same molecule are correlated. If molecular motion is insignificant during exposure, then differences in momentum transfer between correlated photons are direct measurements of the molecular structure. In conventional small- and wide-angle solution scattering, photon correlations are ignored. This report presents advances in a new biomolecular structural analysis technique, correlated X-ray scattering (CXS), which uses angular intensity correlations to recover hidden structural details from molecules in solution. Due to its intense rapid pulses, an X-ray free electron laser (XFEL) is an excellent tool for CXS experiments. A protocol is outlined for analysis of a CXS data set comprising a total of half a million X-ray exposures of solutions of small gold nanoparticles recorded at the Spring-8 Ångström Compact XFEL facility (SACLA). From the scattered intensities and their correlations, two populations of nanoparticle domains within the solution are distinguished: small twinned, and large probably non-twinned domains. It is shown analytically how, in a solution measurement, twinning information is only accessible via intensity correlations, demonstrating how CXS reveals atomic-level information from a disordered solution of like molecules.

X-ray cross-correlation analysis and local symmetries of disordered systems: General theory

In a recent article (P.Wochner et al., PNAS (2009)) x-ray scattering intensity correlations around a ring, in the speckle diffraction pattern of a colloidal glass, were shown to display a remarkable ~ cos(n $\phi$) dependence on the angular coordinate $\phi$ around the ring, with integer index n depending on the magnitude of the scattering wavevector. With an analytical derivation that preserves full generality in the Fraunhofer diffraction limit, we clarify the relationship of this result to previous x-ray studies of bond-orientation order, and provide a sound basis to the statement that the angular intensity correlations deliver information on local bond arrangements in a disordered (or partially ordered) system. We present a detailed analysis of the angular cross-correlation function and show its applicability for studies of wide range of structural properties of disordered systems, from local structure to spatial correlations between distant structural elements.

Optics of surface disordered systems

No surface is perfectly flat at every length scale. However, all natural and man-made surfaces show some degree of roughness. Therefore, it is imperative to know how surface disorder may affect physical process at, or in the immediate vicinity, of the surface. In this review, we focus on the long-standing problem of electromagnetic wave scattering from randomly rough surfaces. This topic has implications and practical applications in fields of science and engineering ranging from observational astronomy to the electronic and medical industry. How randomly rough surfaces can be described statistically is outlined, and we introduce the theoretical and computational methods most frequently used in the study of light scattering from randomly rough metal or dielectric surfaces. A large part of the review is devoted to the description and the discussion of the physical origin behind various multiple scattering phenomena that can exist when light interacts with a random surface. Some of the addressed phenomena are; the enhanced backscattering and satellite peak phenomena; forward scattering (specular peak) enhancement; coherent effects in the angular intensity correlation functions and the second harmonic generated light (nonlinear effect).

High-order correlations from rough-surface scattering.

It is proposed to apply an optical setup of a randomly weak rough dielectric film on a reflecting metal substrate for the measurement of high-order correlations from rough-surface scattering. The angular amplitude and intensity correlations are measured. Because of multiple scattering, when the input laser beam size is comparatively small or close to the travel pass length inside the film, C(2) and C(3) are measured by subtraction of the amplitude correlation from the intensity correlation.

The angular intensity correlation functions C (1) and C (10) for the scattering of light from randomly rough dielectric and metal surfaces

We study the statistical properties of the scattering matrix S(q|k) for the problem of the scattering of light of frequency ω from a randomly rough one-dimensional surface, defined by the equation x 3=ζ(x 1), where the surface profile function ζ(x 1) constitutes a zero-mean, stationary, Gaussian random process. This is done by studying the effects of S(q|k) on the angular intensity correlation function C(q,k|q′,k′)=⟨I(q|k)I(q′|k′)⟩−⟨I(q|k)⟩⟨I(q′|k′)⟩, where the intensity I(q|k) is defined in terms of S(q|k) by I(q|k)=L −1 1(ω/c)|S(q|k)|2, with L 1 the length of the x 1 axis covered by the random surface. We focus our attention on the C (1) and C (10) correlation functions, which are the contributions to C(q,k|q′,k′) proportional to δ(q−k−q′+k′) and δ(q−k+q′−k′), respectively. The existence of both of these correlation functions is consistent with the amplitude of the scattered field obeying complex Gaussian statistics in the limit of a long surface and in the presence of weak surface roughness. We show that the deviation of the statistics of the scattering matrix from complex circular Gaussian statistics and the C (10) correlation function are determined by exactly the same statistical moment of S(q|k). As the random surface becomes rougher, the amplitude of the scattered field no longer obeys complex Gaussian statistics but obeys complex circular Gaussian statistics instead. In this case the C (10) correlation function should therefore vanish. This result is confirmed by numerical simulation calculations.

The angular intensity correlation function C 0 for the scattering of light from a one-dimensional randomly rough surface

The angular intensity correlation function C 0 for the scattering of light from a one-dimensional randomly rough surface

The angular intensity correlation function C 0 for the scattering of light from a one-dimensional randomly rough surface

Abstract We calculate the contribution C 0 to the normalized angular intensity correlation function C(q, k/q′, k′) for the scattering of light from a one-dimensional random surface. We show that the C0 contribution comes from the correlations of the elastic initial or final single-scattering processes and can be the leading contribution to the long-range correlation function.

Multiple-Scattering Phenomena in the Scattering of Light from Randomly Rough Surfaces

The incorporation of multiple scattering into theories of rough surface scattering has led to the prediction of a number of new effects, many of which have now been verified experimentally. Several of these new multiple-scattering effects will be described in this paper, in particular the presence of satellite peaks in the angular dependence of light scattered from, or transmitted through, films whose illuminated surface is randomly rough; new features in the angular intensity correlation functions of light scattered from randomly rough metal surfaces, and surfaces that scatter light uniformly within a specified range of scattering angles, and produce no scattering outside this range. In addition, some suggestions for directions that research in the area of rough surface scattering could take in the next few years will be presented.

Multiple-Scattering Phenomena in the Scattering of Light from Randomly Rough Surfaces

The incorporation of multiple scattering into theories of rough surface scattering has led to the prediction of a number of new effects, many of which have now been verified experimentally. Several of these new multiple-scattering effects will be described in this paper, in particular the presence of satellite peaks in the angular dependence of light scattered from, or transmitted through, films whose illuminated surface is randomly rough; new features in the angular intensity correlation functions of light scattered from randomly rough metal surfaces, and surfaces that scatter light uniformly within a specified range of scattering angles, and produce no scattering outside this range. In addition, some suggestions for directions that research in the area of rough surface scattering could take in the next few years will be presented.

Speckle correlations in the light reflected and transmitted by metal films with rough surfaces: surface wave effects

Diagrammatic perturbation theory is used to compute the angular intensity correlation function for p-polarized light reflected from, and transmitted through, a thin, free-standing, metallic film. The illuminated surface of the film is taken to be a weakly rough, one-dimensional, random surface and the back surface is planar. Contributions to the angular intensity correlation function in both reflection and transmission include: (a) short-range memory effect and time-reversed memory effect terms, C (1); (b) an additional short-range term of comparable magnitude C (10); (c) a long-range term C (2); (d) an infinite-range term C (3); and (e) a term C (1.5) that along with C (2) displays peaks associated with the excitation of the two surface plasmon polaritons supported by the film. The new features in these correlation functions for the reflection of light from, and its transmission through, the rough film, in comparison with those for the reflection of light from a semi-infinite metal with a random surface, are the presence of additional peaks in C (1) and C (2) occurring when conditions involving the difference of the wave numbers of the two surface plasmon polaritons at the frequency of the incident light are satisfied. New, additional, resonant peaks are also present in C (1.5). These, however, are a consequence of the existence of two surface plasmon polaritons instead of one, and have the same origin as the peaks in C (1.5) in reflection from a semi-infinite metal with a random surface.

Angular intensity correlations in reflection of light from a random medium

The correlation of the angular reflection coefficients and their whole distribution function are studied. The intensity fluctuations for different directions are shown to be negatively correlated. A deviation of the distribution function from the Rayleigh law indicates that large fluctuations of the reflected intensity are suppressed. All these peculiarities result from the constraints imposed by conservation of the energy flux.

Speckle correlations in the light scattered from weakly rough random metal surfaces

Diagrammatic perturbation theory and computer simulation methods are used to compute the angular intensity correlation function C(q, k|q′,k′)=([I(q|k) - (I(q|k))] × [I(q′|k′) - (I(q′|k′))]) for p-polarized light scattered from a weakly rough, one-dimensional random metal surface. I(q|k) is the squared modulus of the scattering matrix for the system, and q, q′ and k, k′ are the projections on the mean scattering surface of the wavevectors of the scattered and incident light, respectively. Contributions to C include: (a) short-range memory effect and time-reversed memory effect terms, C (1); (b) an additional short-range term of comparable magnitude C (10); (c) a long-range term C (2); (d) an infinite-range term C (3); and (e) a term C (1.5) that along with C (2) displays peaks associated with the excitation of surface plasmon polaritons. The diagrammatic methods are also extended to treat the angular intensity correlation function for the scattering of p to p, p to s, s to p, and s to s polarizations of light from a two-dimensional randomly rough surface. These correlations are again described in terms of C (1), C (10), C (1.5), C (2), and C (3) contributions to C for the two-dimensional surfaces. Short-range memory and time-reversed memory effects are observed in the two-dimensional C (1) correlations, and peaks associated with the excitation of surface polaritons are observed in the two-dimensional C (1.5) and C (2) correlations. Most of the results for the one- and two-dimensional systems are presented for incident electromagnetic plane waves. In addition, results for one-dimensional systems are presented for incident electromagnetic beams of finite width. Some of the results for one-dimensional surfaces are corroborated by means of computer simulation techniques.

Speckle correlations in the light scattered from a weakly rough one-dimensional random metal surface

Perturbation theory is used to compute the angular-intensity correlation function C(q, k|q(?), k(?)) = ?[I(q|k) - ?I(q|k)?][I(q(?)|k(?)) - ?I(q(?)|k(?))]? for p-polarized light scattered from a weakly rough, one-dimensional random metal surface. I(q|k) is the squared modulus of the scattering matrix for the system, and q , q(?) and k , k(?) are the projections on the mean scattering surface of the wave vectors of the scattered and the incident light, respectively. Contributions to C include (a) short-range memory effect and time-reversed memory effect terms, C((1)) ; (b) an additional short-range term of comparable magnitude C((10)) ; (c) a long-range term C((2)) ; (d) an infinite-range term C((3)) ; and (e) a new term C((1.5)) that along with C((2)) displays peaks associated with the excitation of surface polaritons. These new features arise when the factorization approximation is not made in calculating the correlation function C .

Degenerate optical memory effect in dielectric films with randomly rough surfaces

By means of perturbation theory we calculate the angular intensity correlations of linearly polarized electromagnetic waves scattered from a thin dielectric film deposited on a planar perfectly conducting substrate. The illuminated surface is a one-dimensional, randomly rough surface with a power spectrum given either by a Gaussian, or by a rectangle defined to enhance the coupling into guided waves. The dielectric film supports, at least, two guided waves in the absence of the roughness that have distinct wave numbers at the frequency \ensuremath{\omega} of the incident wave. We analyze the angular dependence of the optical memory effect: In addition to its two well known peaks, we demonstrate that, due to degeneracy of the excitation of guided waves, there appear two pairs of peaks symmetrically located about the former absolute maxima at positions related to the difference between the guided wave numbers. This degenerate memory effect is shown to be a multiple scattering effect.

Angular intensity correlations in the double passage of waves through a random phase screen

<p>The problem of light scattering in folded-path or double-passage configurations is studied theoretically. Assuming as the random medium a deep phase screen that introduces Gaussian-distributed phase fluctuations, we study the motion of the speckle as the source is moved. Some attention is also given to the phenomenon of backscattering enhancement. Our analysis is based on a novel expression for the complex amplitude that has a simple physical interpretation. For simplicity, only the one-dimensional case is considered, but an extension of the analysis to two-dimensional screens is not difficult.</p><p>Using the factorization properties of the moments of a complex Gaussian process, we are able to derive analytical expressions for the mean intensity and the intensity correlation of the backscattered radiation. We find that, in most cases, the speckle field decorrelates rapidly as one moves the angle of incidence and shifts toward the direction of specular reflection with a rate of motion that is different from that of the angle of incidence. We also find conditions under which, when the angle of incidence is modified, the speckle pattern produced in the region of observation tracks the backscattering direction.</p>

Long-range angular-intensity correlations: The influence of time-reversal symmetry.

La fonction de correlation de la reflexion est calculee a l'aide d'une approche diagrammatique de Feynman en regime faiblement localise. On prend en compte une nouvelle classe de diagrammes representant une symetrie reversible dans le temps