Attractor-driven matter.

The state of a classical point-particle system may often be specified by giving the position and momentum for each constituent particle. For non-pointlike particles, the center-of-mass position may be augmented by an additional coordinate that specifies the internal state of each particle. The internal state space is typically topologically simple, in the sense that the particle's internal coordinate belongs to a suitable symmetry group. In this paper, we explore the idea of giving internal complexity to the particles, by attributing to each particle an internal state space that is represented by a point on a strange (or otherwise) attracting set. It is, of course, very well known that strange attractors arise in a variety of nonlinear dynamical systems. However, rather than considering strange attractors as emerging from complex dynamics, we may employ strange attractors to drive such dynamics. In particular, by using an attractor (strange or otherwise) to model each particle's internal state space, we present a class of matter coined "attractor-driven matter." We outline the general formalism for attractor-driven matter and explore several specific examples, some of which are reminiscent of active matter. Beyond the examples studied in this paper, our formalism for attractor-driven dynamics may be applicable more broadly, to model complex dynamical and emergent behaviors in a variety of contexts.

Imaging Breast Microcalcifications Using Dark-Field Signal in Propagation-Based Phase-Contrast Tomography.

Breast microcalcifications are an important primary radiological indicator of breast cancer. However, microcalcification classification and diagnosis may be still challenging for radiologists due to limitations of the standard 2D mammography technique, including spatial and contrast resolution. In this study, we propose an approach to improve the detection of microcalcifications in propagation-based phase-contrast X-ray computed tomography of breast tissues. Five fresh mastectomies containing microcalcifications were scanned at different X-ray energies and radiation doses using synchrotron radiation. Both bright-field (i.e. conventional phase-retrieved images) and dark-field images were extracted from the same data sets using different image processing methods. A quantitative analysis was performed in terms of visibility and contrast-to-noise ratio of microcalcifications. The results show that while the signal-to-noise and the contrast-to-noise ratios are lower, the visibility of the microcalcifications is more than two times higher in the dark-field images compared to the bright-field images. Dark-field images have also provided more accurate information about the size and shape of the microcalcifications.

Tomographic phase and attenuation extraction for a sample composed of unknown materials using x-ray propagation-based phase-contrast imaging.

Propagation-based phase-contrast x-ray imaging (PB-PCXI) generates image contrast by utilizing sample-imposed phase-shifts. This has proven useful when imaging weakly attenuating samples, as conventional attenuation-based imaging does not always provide adequate contrast. We present a PB-PCXI algorithm capable of extracting the x-ray attenuation  ß and refraction  δ, components of the complex refractive index of distinct materials within an unknown sample. The method involves curve fitting an error-function-based model to a phase-retrieved interface in a PB-PCXI tomographic reconstruction, which is obtained when Paganin-type phase retrieval is applied with incorrect values of δ and ß. The fit parameters can then be used to calculate true δ and ß values for composite materials. This approach requires no a priori sample information, making it broadly applicable. Our PB-PCXI reconstruction is single-distance, requiring only one exposure per tomographic angle, which is important for radiosensitive samples. We apply this approach to a breast-tissue sample, recovering the refraction component  Î´, with 0.6-2.4% accuracy compared with theoretical values.

Relative roles of multiple scattering and Fresnel diffraction in the imaging of small molecules using electrons, Part II: Differential Holographic Tomography.

It has been argued that in atomic-resolution transmission electron microscopy (TEM) of sparse weakly scattering structures, such as small biological molecules, multiple electron scattering usually has only a small effect, while the in-molecule Fresnel diffraction can be significant due to the intrinsically shallow depth of focus. These facts suggest that the three-dimensional reconstruction of such structures from defocus image series collected at multiple rotational orientations of a molecule can be effectively performed for each atom separately, using the incoherent first Born approximation. The corresponding reconstruction method, termed here Differential Holographic Tomography, is developed theoretically and demonstrated computationally on several numerical models of biological molecules. It is shown that the method is capable of accurate reconstruction of the locations of atoms in a molecule from TEM data collected at a small number of random orientations of the molecule, with one or more defocus images per orientation. Possible applications to cryogenic electron microscopy and other areas are briefly discussed.

Factors limiting quantitative phase retrieval in atomic-resolution differential phase contrast scanning transmission electron microscopy using a segmented detector.

Quantitative differential phase contrast imaging of materials in atomic-resolution scanning transmission electron microscopy using segmented detectors is limited by various factors, including coherent and incoherent aberrations, detector positioning and uniformity, and scan-distortion. By comparing experimental case studies of monolayer and few-layer graphene with image simulations, we explore which parameters require the most precise characterisation for reliable and quantitative interpretation of the reconstructed phases. Coherent and incoherent lens aberrations are found to have the most significant impact. For images over a large field of view, the impact of noise and non-periodic boundary conditions are appreciable, but in this case study have less of an impact than artefacts introduced by beam deflections coupling to beam scanning (imperfect tilt-shift purity).

Suppressing dynamical diffraction artefacts in differential phase contrast scanning transmission electron microscopy of long-range electromagnetic fields via precession.

It is well known that dynamical diffraction varies with changes in sample thickness and local crystal orientation (due to sample bending). In differential phase contrast scanning transmission electron microscopy (DPC-STEM), this can produce contrast comparable to that arising from the long-range electromagnetic fields probed by this technique. Through simulation we explore the scale of these dynamical diffraction artefacts and introduce a metric for the magnitude of their contribution to the contrast. We show that precession over an angular range of a few milliradian can suppress this contribution to the contrast by one-to-two orders of magnitude. Our exploration centres around a case study of GaAs near the [011] zone-axis orientation using a probe-forming aperture semiangle on the order of 0.1 mrad at 300 keV, but the trends found and methodology used are expected to apply more generally.

Dark-field signal extraction in propagation-based phase-contrast imaging.

A method for extracting the dark-field signal in propagation-based phase-contrast imaging is proposed. In the case of objects consisting predominantly of a single material, or several different materials with similar ratios of the real decrement to the imaginary part of the complex refractive index, the proposed method requires a single image for extraction of the dark-field signal in two-dimensional projection imaging. In the case of three-dimensional tomographic imaging, the method needs only one image to be collected at each projection angle. Initial examples using simulated and experimental data indicate that this method can improve visualization of small sharp features inside a larger object, e.g. the visualization of microcalcifications in propagation-based x-ray breast cancer imaging. It is suggested that the proposed approach may be useful in other forms of biomedical imaging, where it can help one to obtain additional small-angle scattering information without increasing the radiation dose to the sample.

High contrast at low dose using a single, defocussed transmission electron micrograph.

For many soft-matter specimens, transmission electron microscopists face the double-bind of low contrast images, due to weakly-scattering specimens, alongside severe limits on the electron dose that can be used before the specimen is damaged by the electron beam. The combination of these effects causes the resultant micrographs to have very low signal-to-noise. It is well known that varying the defocus aberration can enhance image contrast in electron microscopy. For single-material objects where the variation of absorption and phase contrast are functions of one another, since both are governed by the variation in thickness profile, we show that the thickness profile can be reconstructed at very low dose. The algorithm, first established in X-ray imaging, requires some a priori information but only a single defocussed image of the region of interest, making it more dose efficient than either a conventional transport-of-intensity phase reconstruction (which would require two images and tends to amplify noise), or an absorption-contrast analysis of a single in-focus image recorded at the same electron dose (which does not benefit from the significant signal-to-noise enhancement of the present algorithm). These findings are presented through both simulations and experimental data.

Simple Wave-Optical Superpositions as Prime Number Sieves.

We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically generated wave field for integer valued momenta. Similarly, primes are encoded in the diffraction pattern from a simple single aperture and in the harmonics of a single vibrating resonator. Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin primes, and how to construct wave fields with amplitudes equal to the divisor function at integer spatial frequencies. Remarkably, all of these basic diffraction phenomena reveal that the naturally irregular sequence of primes can arise from trivially ordered wave superpositions.

Measuring nanometre-scale electric fields in scanning transmission electron microscopy using segmented detectors.

Electric field mapping using segmented detectors in the scanning transmission electron microscope has recently been achieved at the nanometre scale. However, converting these results to quantitative field measurements involves assumptions whose validity is unclear for thick specimens. We consider three approaches to quantitative reconstruction of the projected electric potential using segmented detectors: a segmented detector approximation to differential phase contrast and two variants on ptychographical reconstruction. Limitations to these approaches are also studied, particularly errors arising from detector segment size, inelastic scattering, and non-periodic boundary conditions. A simple calibration experiment is described which corrects the differential phase contrast reconstruction to give reliable quantitative results despite the finite detector segment size and the effects of plasmon scattering in thick specimens. A plasmon scattering correction to the segmented detector ptychography approaches is also given. Avoiding the imposition of periodic boundary conditions on the reconstructed projected electric potential leads to more realistic reconstructions.

Effect of specimen orientation on the accuracy of vector field electron tomography.

Vector field electron tomography (VFET) reconstructs vector fields based on phase maps recorded from two or more orthogonal tilt series. The tomographic reconstruction of vector fields involves considerations beyond those involved in the reconstruction of scalar fields. Here we examine the effect of initial magnetization orientation on reconstruction errors. The orientation of a magnetic particle affects the contrast in the phase maps. This, in turn, affects the accuracy of the reconstructed vector fields. We derive expressions that model the dependence of reconstruction errors on initial specimen orientation when using a filtered backprojection algorithm to reconstruct a vector potential from two tilt series. We compare these analytical results with those from numerical simulations. Our results can inform experimental procedures, such as sample preparation techniques and the choice of tilt series orientations. Specimen orientation can be a significant source of error in VFET, and our results can provide the means to minimize these errors.

Phase contrast x-ray velocimetry of small animal lungs: optimising imaging rates.

Chronic lung diseases affect a vast portion of the world's population. One of the key difficulties in accurately diagnosing and treating chronic lung disease is our inability to measure dynamic motion of the lungs in vivo. Phase contrast x-ray imaging (PCXI) allows us to image the lungs in high resolution by exploiting the difference in refractive indices between tissue and air. Combining PCXI with x-ray velocimetry (XV) allows us to track the local motion of the lungs, improving our ability to locate small regions of disease under natural ventilation conditions. Via simulation, we investigate the optimal imaging speed and sequence to capture lung motion in vivo in small animals using XV on both synchrotron and laboratory x-ray sources, balancing the noise inherent in a short exposure with motion blur that results from a long exposure.

Analytical description of partially coherent propagation and absorption losses in x-ray planar waveguides.

We present an analytical approach to describe field propagation along a planar x-ray waveguide (WG) in the presence of absorption losses. The method utilizes the complete expression for the complex index of refraction in solving the Helmholtz equation describing the guided modes. In this way, the propagation modes for the WG are no longer imposed to be standing waves and the energy flow from the core to the cladding, a consequence of the absorption in the cladding, can be calculated. In addition, the method accurately describes the field coupling between a plane wave and the WG, reproducing the self-imaging phenomenon. The case of partially coherent illumination has also been calculated for a realistic laboratory x-ray source.

Electron vortex production and control using aberration induced diffraction catastrophes.

An aberration corrected electron microscope is used to create electron diffraction catastrophes, containing arrays of intensity zeros threading vortex cores. Vortices are ascribed to these arrays using catastrophe theory, scalar diffraction integrals, and experimentally retrieved phase maps. From measured wave function phases, obtained using focal-series phase retrieval, the orbital angular momentum density is mapped for highly astigmatic electron probes. We observe vortex rings and topological reconnections of nodal lines by tracking the vortex cores using the retrieved phases.

Iterative retrieval of one-dimensional x ray wave field using a single intensity measurement.

The problem of retrieving a complex function from the modulus of its Fourier transform has non-unique solutions in one dimension. Therefore iterative phase retrieval methods cannot in general be confidently applied to one-dimensional problems, due to the presence of ambiguities. We present a method for a posteriori reduction of the ambiguities based on the correlation analysis of the solution of a large number of runs of an iterative phase retrieval algorithm with different random starting phases. The method is applied to experimentally measured diffraction patterns from an x ray waveguide illuminated by hard x rays. We demonstrate the possibility of retrieving the complex wave field at the exit face of the waveguide and compare the result with theoretical prediction.

Characterizing the geometry of InAs nanowires using mirror electron microscopy.

Mirror electron microscopy (MEM) imaging of InAs nanowires is a non-destructive electron microscopy technique where the electrons are reflected via an applied electric field before they reach the specimen surface. However strong caustic features are observed that can be non-intuitive and difficult to relate to nanowire geometry and composition. Utilizing caustic imaging theory we can understand and interpret MEM image contrast, relating caustic image features to the properties and parameters of the nanowire. This is applied to obtain quantitative information, including the nanowire width via a through-focus series of MEM images.

Interface-specific x-ray phase retrieval tomography of complex biological organs.

We demonstrate interface-specific propagation-based x-ray phase retrieval tomography of the thorax and brain of small animals. Our method utilizes a single propagation-based x-ray phase-contrast image per projection, under the assumptions of (i) partially coherent paraxial radiation, (ii) a static object whose refractive indices take on one of a series of distinct values at each point in space and (iii) the projection approximation. For the biological samples used here, there was a 9-200 fold improvement in the signal-to-noise ratio of the phase-retrieved tomograms over the conventional attenuation-contrast signal. The ability to 'digitally dissect' a biological specimen, using only a single phase-contrast image per projection, will be useful for low-dose high-spatial-resolution biomedical imaging of form and biological function in both healthy and diseased tissue.

Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object.

Single defocused transmission electron microscope phase contrast images are used to reconstruct the projected thickness map of a single-material object. The algorithm is non-iterative and stable, and we extend it to account for the presence of spherical aberration in the objective optics. The technique can reconstruct the projected thickness map of general single-material objects in the strong phase/weak amplitude regime. It is sensitive to any excursions in the projected thickness from the average, and ideal for examining voids and free volume accumulation in amorphous/glassy materials at the nanometer scale. The resolution of the technique depends on the choice of defocus and the thickness of the specimen. In a certain regime, we demonstrate that variations in the transverse projected thickness with a lateral diameter of ∼ 0.25 nm may be detected. We use our algorithm to quantitatively reconstruct the projected thickness of latex sphere test specimens from single defocused electron micrographs. We demonstrate that the reconstruction has a large tolerance for error in the input parameters. Simulations confirm that the technique is quantitative, and demonstrate that the origin of low-frequency artifacts is an instability due to noise. We show that the autocorrelation of the projected thickness map may be used to measure the size of open structures in the object using both simulation and latex sphere data.

The effect of vacancy ordering on quadrupole shifts in Mössbauer spectra of maghemite.

Electric field gradient (EFG) lattice sums for the vacancy-disordered and vacancy-ordered forms of maghemite were evaluated using a point charge model. The calculated EFGs produced a wide range of magnitudes and principal directions for the quadrupole interaction for both forms of maghemite. Small perturbations of the crystallographic parameters were shown to have a significant effect on the quadrupole shift. The effects observed were significant enough to show that quadrupole shifts should be considered when fitting the Mössbauer spectra of maghemite.

Caustic imaging of gallium droplets using mirror electron microscopy.

We discuss a new interpretation of mirror electron microscopy (MEM) images, whereby electric field distortions caused by surface topography and/or potential variations are sufficiently large to create caustics in the image contrast. Using a ray-based trajectory method, we consider how a family of rays overlaps to create caustics in the vicinity of the imaging plane of the magnetic objective lens. Such image caustics contain useful information on the surface topography and/or potential, and can be directly related to surface features. Specifically we show how a through-focus series of MEM images can be used to extract the contact angle of a Ga droplet on a GaAs (001) surface.